Simulation of Organic Light-Emitting Diodes

and Organic Photovoltaic Devices

by

Hui Wang

Submitted in Partial Fulfillment

of the

Requirements for the Degree

Doctor of Philosophy

Supervised by

Professor Ching W. Tang

and

Professor Lewis J. Rothberg

Department of Physics and Astronomy

Arts, Sciences and Engineering

School of Arts and Sciences

University of Rochester

Rochester, New York

2012

ii

Curriculum Vitae

Hui Wang was born in Shucheng, Anhui, China on October 5th, 1987. He

graduated with a Bachelor of Science degree in Physics from the University of

Science and Technology of China in 2006, where he attended the Special Class for

Gifted Youths. In fall 2006, he came to the University of Rochester to pursue the

Doctor of Philosophy degree in Physics under the supervision of Professor Ching W.

Tang. He received a Master of Arts degree in Physics in 2008 and was awarded the

Susumu Okubo Prize by the Department of Physics and Astronomy for excellent

performance in graduate course work and on the Preliminary Exam. His field of

research was in simulation of organic light-emitting diodes and organic photovoltaic

devices.

Publications

Hui Wang, Kevin P. Klubek, C. W. Tang, “Current Efficiency in Organic Light-

emitting Diodes with a Hole-injection Layer”, Applied Physics Letter, 93,

093306 (2008)

Minlu Zhang, Hui Wang, C. W. Tang, “Effect of the highest occupied molecular

orbital energy level offset on organic heterojunction photovoltaic cells”, Applied

Physics Letter, 97, 143503 (2010)

Minlu Zhang, Hui Wang, Hongkun Tian, Yanhou Geng, C. W. Tang, “Bulk

heterojunction photovoltaic cells with low donor concentration”, Advanced

Materials, 23, 4960 (2011)

iii

Minlu Zhang, Hui Wang, C. W. Tang, “Hole transport limited S-shape I-V curves

in organic photovoltaic cells”, Applied Physics Letter, 99, 213506 (2011)

Minlu Zhang, Hui Wang, C. W. Tang, “Enhanced efficiency in multi-junction

TAPC doped C60 photovoltaic cells”, Organic Electronics, 13, 249 (2011)

iv

Acknowledgments

First and foremost, I would like to express my gratitude to my advisors,

Professor Ching W. Tang and Professor Lewis J. Rothberg, for their invaluable

guidance and support in my PhD study and research. What I have learned from them

is not only the comprehensive knowledge-set they have in the subject, but also their

critical way of thinking, initiative of innovation, and keen insight into research. Their

generous and patient guidance to me has always made my research more efficient and

productive. It is a great honor for me to finish my PhD study under the supervision of

such wonderful mentors.

I would like to thank all my professors in the Department of Physics and

Astronomy for the knowledge and tools they taught me during the graduate

coursework. They have provided a solid foundation for my research work.

I am also very thankful to collaborators Dr. Minlu Zhang and Kevin P.

Klubek for their cooperation in providing experimental results for my simulation

analysis. I would like to acknowledge Joseph K. Madathil and Dr. Jason U.

Wallace for their help when I first joined the group. I greatly appreciate the insightful

discussions with Dr. David S. Weiss, Dr. Ralph H. Young, and Dr. Alfred P.

Marchetti, as well as their willingness to answer all my questions. I would also like

to thank all other members in our group, including Hao Lin, Dr. Sang Min Lee, Wei

Xia, Hsiang Ning Sunny Wu, Felipe Angel, Mohan Ahluwalia, Dr. Lichang Zeng,

Eric Glowacki, Matthew Smith, Jonathan A. Welt, Qing Du, Chieh Chang Mark

v

Hsu, William Finnie, Dr. Meng-Huan Kinneas Ho, Prashant Kumar Singh, Guy

Mongelli, Charles Chan, Laura Ciammaruchi, Yung-Hsin Thomas Lee, Chris

Favaro, who made my study and research a enjoyable experience. I learned a lot

from the regular group meeting and everyday discussions with them. It was a great

pleasure to work in such a great team.

Finally, I want to thank my girlfriend, Changxin Zhao, whom I met here at

the University of Rochester, for her love, understanding, and support. I also owe the

most special gratitude to my parents, Shengfeng Pan and Honggui Wang, for their

continuous and unconditional love and support.

vi

Abstract

This thesis focuses on the simulation of organic light-emitting diodes (OLEDs)

and organic photovoltaic devices (OPV). By building the model and choosing

appropriate parameters, I reproduced the experimental data collected by my

colleagues and interpreted the results qualitative and quantitatively.

We begin by simulating single layer devices to establish a good understanding of

the charge carrier injection, transport and recombination. Efficiency of single layer

OLEDs is sensitive to the mobilities of electrons and holes. Charge carrier traps can

be introduced to balance the transport. We then systematically investigate the effect

of the layer structure on the current efficiency in bilayer and trilayer OLEDs, and

conclude that inserting a hole injection layer can effectively reduce the quenching by

charge carriers near the recombination zone and hence improve the current efficiency.

Mixed host OLEDs with different device structures have been simulated and

compared.

We next investigate the effect of the highest occupied molecular orbital (HOMO)

energy level offset on planar heterojunction OPV devices, where dissociation at the

donor/acceptor (DA) interface controls the device performance. Bound charge-

transfer (CT) states are produced when excitons arrive at the DA interface. The

following dissociation of CT states is simulated using the Braun-Onsager model. Two

fitting parameters, the initial separation distance r0 and the CT state decay rate kf, are

used to explain the effect of the HOMO offset. The S-shape current-voltage

vii

characteristics and the donor layer thickness dependence of the device performance

are explained by the hole transport limitation in the donor layer.

For bulk heterojunction (BHJ) OPV devices that mix the donor and acceptor

materials in the BHJ layer, the device performance is sensitive to the donor

concentration. We explained this finding by considering the donor concentration

dependence of parameters in the BHJ layer, including the absorption coefficient,

dielectric constant, and hole/electron mobilies. A good match between the simulation

and experimental results has been achieved when all the parameters are set properly.

Finally, we simulate the two-stack tandem OPV devices and predict the optimal

combination of the BHJ layer thickness for both subcells, which is confirmed by

experiments.

viii

Table of Contents

Curriculum Vitae ii

Acknowledgements iv

Abstract vi

List of Tables xii

List of Figures xiii

List of Symbols and Abbreviations xxi

Foreword 1

Chapter 1 Introduction 3

1.1 Introduction to OLEDs and OPV devices 3

1.2 Introduction to the simulation of OLEDs and OPV devices 6

Chapter 2 Simulation of Single Layer Devices 8

2.1 Charge carrier injection, transport and recombination 8

2.1.1 Charge carrier injection at the metal/organic interface 8

2.1.2 Charge carrier transport and recombination 11

2.2 Single layer devices with unipolar transport materials 15

2.2.1 Space charge limited current 15

2.2.2 Injection barrier effect 20

2.2.3 Trap limited current 22

ix

2.3 Single layer OLED devices with bipolar transport materials 26

2.3.1 Recombination zone in single layer OLED devices 26

2.3.2 Recombination efficiency in single layer OLED devices 29

2.3.3 Single layer OLED devices with charge carrier traps 31

Chapter 3 Simulation of Multilayer Organic Light-Emitting Diodes 37

3.1 Interfaces between organic layers 37

3.1.1 Charge carrier transport across organic interfaces 37

3.1.2 Cross interface recombination 40

3.2 Exciton diffusion, decay and quenching 41

3.3 Quenching by charge carriers in OLEDs 42

3.3.1 Effect of the hole injection layer on the drive voltage of

OLEDs

42

3.3.2 Effect of the hole injection layer on the device efficiency

of OLEDs

48

3.4 Mixed host OLEDs 52

3.4.1 Parameter assumptions for the mixed host layer 52

3.4.2 Recombination zone in mixed host OLEDs 53

3.4.3 Quenching by charge carriers in mixed host OLEDs 60

3.5 Photon extraction 62

3.5.1 Cavity model for OLEDs 63

3.5.2 Mode decomposition of the dipole emission in OLEDs 72

3.5.3 Weak cavity versus strong cavity OLEDs 76

x

Chapter 4 Simulation of Organic Photovoltaic Devices 84

4.1 Photon absorption in OPV devices 84

4.2 Exciton diffusion and dissociation in OPV devices 87

4.3 Charge carriers transport in OPV devices 89

4.4 Effect of the HOMO offset on planar heterojunction OPV

devices

90

4.5 Effect of the donor layer thickness on planar heterojunction

OPV devices

97

4.6 Effect of the donor concentration on bulk heterojunction OPV

devices

102

4.7 Simulation of tandem OPV devices 110

Chapter 5 Summary and Future Work 114

References 119

Appendix A Program for the Simulation of OLEDs

(Electrical Part and Exciton Part)

142

A.1 Flow diagram 143

A.2 Codes 144

A.3 Sample input (input.txt) 170

Appendix B Program for the Simulation of OLEDs

(Optical Part)

173

B.1 Codes 174

xi

B.2 Sample input (input.txt) 269

B.3 Sample input (Al.txt) 270

B.4 Sample input (PL_red.txt) 272

Appendix C Program for the Power Consmption Model of OLED

Screens

274

C.1 Codes 275

Appendix D Program for the Simulation of OPV devices

(Optical Part, Exciton Part and Electrical Part)

280

D.1 Flow diagram 281

D.2 Codes 282

D.3 Sample input (input.txt) 310

xii

List of Tables

Table Title

Table 3.1 Input parameters used in the simulation of MTDATA/NPB/Alq

OLEDs.

44

Table 3.2 Input parameters used in the simulation of mixed host OLEDs. 54

Table 4.1 List of donors, HOMO levels and CAS names used in donor (3

nm)/C60 (40 nm) OPV devices.

92

Table 4.2 Experimental photovoltaic parameters (Jsc, Voc and FF) for

donor (3 nm) /C60 (40 nm) OPV devices. Vbi, r0 and kf are the

built-in potential, the initial separation distance and the decay

rate of the CT state used in the simulation.

93

Table 4.3 Experimental photovoltaic parameters (Jsc, Voc and FF) for

ITO/MoOx(2nm)/NPB(x nm)/C60(40 nm)/BPhen(8 nm)/LiF/Al

cells with NPB layer thickness varied from 3 nm to 100 nm.

Vbi, r0 and kf are the built-in potential, the initial separation

distance and the decay rate of the charge transfer state used in

the simulation.

98

Table 4.4 Experimental Photovoltaic parameters of TAPC:C60 cells

(AM1.5 @ 100mW/cm2) and calculated hole and electron

mobilities.

105

xiii

List of Figures

Figure Title

Figure 2.1 Simulation and analytical results of (a) the current density versus

voltage (J-V) characteristics and (b) the electric field distribution

under current density of 1000 mA/cm2. The layer thickness is

100 nm and the carrier mobility is 2.010-4

cm2V

-1s

-1.

17

Figure 2.2 (a) Simulation and analytical results (calculated from equation

2.18) of the J-V characteristics for the device with a field

dependent mobility (mobility parameters:

, ) and a layer

thickness of 100 nm. (b) Electric field distribution across the

devices with a field independent mobility (mobility parameters:

, ) and a field

dependent mobility (mobility parameters:

, ) under current

density of 1000 mA/cm2 and a layer thickness of 100 nm.

19

Figure 2.3 Simulated J-V curves for devices with injection barrier varied

from 0 eV to 0.4 eV and mobility parameters: (a)

and (b)

. The layer thickness is 100

nm for all devices.

21

xiv

Figure 2.4 (a) Simulated J-V curves for devices with hole trap

concentration varied from 0% to 2%. The hole trap energy is set

as the hole mobility parameters are

, . The layer

thickness is 100 nm for all devices. (b) The density of trapped

holes and free holes at 1000 mA/cm2 in the device with a hole

trap concentration of 1%.

25

Figure 2.5 Distribution of the charge carrier density, the electric field and

the recombination zone in single layer OLED devices. The layer

thickness for all devices is 100 nm and the results are simulated

under current density of 100 mA/cm2. Left figures are simulated

with . Right figure are

simulated with ,

. All mobilities are set as field independent and

the injection contacts for electrons and holes are set as Ohmic.

28

Figure 2.6 Recombination efficiency (J_Recombination/J_Injection) of

single layer OLED devices with balanced and unbalanced

mobilities. (1) , hole mobility

, electron mobility

and (2) , hole

mobility , electron mobility

. All mobilities are set as field

independent and injection contacts for holes and electrons are

set as Ohmic.

30

xv

Figure 2.7 Distribution of the hole trap concentration, the charge carrier

density, the electric field, the recombination between free holes

and free electrons, and the recombination between trapped holes

and free electrons in single layer OLED devices under current

density of 100 mA/cm2. Left panel has hole trap concentration

across the whole layer. Right panel has hole trap

concentration for the 20 nm in the middle of the

layer, and for the rest.

33

Figure 2.8 (a) Recombination efficiency and (b) normalized recombination

zone at100 mA/cm2 for single layer OLED devices with and

without hole traps.

35

Figure 2.9 Simulated J-V curves for single layer OLED devices with and

without hole traps.

36

Figure 3.1 Diagram of the hole transport across the HTL/ETL interface.

40

Figure 3.2 Structure of MTDATA/NPB/Alq OLEDs and materials used.

HOMO energy level: MTDATA (-5.1eV), NPB (-5.5eV), Alq (-

5.7eV). d is the variable thickness of the NPB layer.

44

Figure 3.3 Experimental (symbols) and simulated (solid curves) J-V curves

for MTDATA/NPB/Alq devices with variable NPB layer

thickness.

45

xvi

Figure 3.4 The energy level diagram and the distribution of the charge

carrier density, the electric field, and the recombination at 80

mA/cm2 in the bilayer (left side) and trilayer (right side)

devices.

47

Figure 3.5 Experimental (symbols) and simulated (solid curves) external

quantum efficiency versus current density for the series of

MTDATA/NPB/Alq devices with variable NPB layer thickness.

48

Figure 3.6 Simulated J-V curves for bilayer, trilayer, uniformly mixed host

and graded mixed host OLED devices.

56

Figure 3.7 The charge carrier density and recombination zone distribution

in various OLED devices: (a) Bilayer; (b) Trilayer; (c) UM-

H20; (d) UM-H50; (e) UM-H80 and (f) Graded.

59

Figure 3.8 Simulated quantum efficiency of bilayer, trilayer, uniformly

mixed host and graded mixed host OLED devices.

61

Figure 3.9 The schematic multilayer structure for the cavity model, where

dipole located in the ne layer with thickness de. The distance

between the dipole and the nearest interfaces on both sides are

z+ and z- respectively. The half-infinite media surrounding the

OLED structure has optical refractive indexes n+ and n-,

respectively. One or more intermediate layers stand between the

emissive layer and the outside media. Any light with specific

direction can be identified with the wave vector ke, which can

be decomposed into the in-plane projection κ and the z-axis

projection kz,e.

64

xvii

Figure 3.10 The illustration of the light out-coupling from the substrate into

the air. and are the power

transmission and reflection coefficient for light injection from

the substrate into the air. is the power

reflection coefficient for light injection from the substrate into

the OLED stack, which effectively considers the multi-layer

structure of the OLED already.

71

Figure 3.11 PL spectrum of the emitter to be used in the simulation. The PL

spectrum is generated from a Gaussian distribution with a peak

wavelength at 620nm and FWHM of 80nm. Inset: The device

structure of the OLED devices in the simulation, with glass

substrate/ITO (100 nm)/Organic Layer (20 nm to 600 nm)/Al

(100 nm), and the dipole emitter locates in the middle of the

organic layer.

74

Figure 3.12 (a) Optical refractive indexes of the glass, ITO, organic material

and Al used in the simulation. The optical refractive index of the

organic material is assumed to be 1.7 at all wavelengths. The

indexes of other materials are retrieved from literature. (b) The

contribution of each mode from the dipole emission in OLED

devices with organic layer thickness varying from 20 nm to 600

nm. Inset: the device structure of the OLED devices. Emitters

locate in the middle of the organic layer, with a PL spectrum

shown in Figure 3.11.

75

xviii

Figure 3.13 The device structure of the weak cavity and strong cavity

OLEDs used in the simulation. The HTL and ETL thickness of

the devices is tuned to improve the luminance at the normal

direction. Red emitters locate in the middle of the EML in both

devices with a PL spectrum shown in Figure 3.11. All organic

layers including the HTL, EML, ETL and capping layer are

assumed to have an optical refractive index of 1.7 at all

wavelengths.

78

Figure 3.14 The EL spectrum of the weak cavity and strong cavity OLEDs

(a) at normal direction and (b) over all angles.

80

Figure 3.15 EL spectrum at normal direction (0 degrees) and at high

viewing angles (60 degrees) for the (a) weak cavity and (b)

strong cavity OLEDs.

81

Figure 3.16 The luminance at normal direction from the (a) weak cavity and

(b) strong cavity OLEDs with HTL thickness and ETL thickness

varied from 0 nm to 400 nm, respectively.

83

Figure 4.1 Diagram of the photon absorption in OPV devices. The device

structure is glass substrate/ITO anode/Donor layer/Acceptor

layer/LiF/Al cathode. and

are the components of the

optical electric field propagating in the positive and negative

directions in the jth monolayer, and

are the

corresponding parameters for the adjacent kth monolayer.

85

xix

Figure 4.2 Diagram of the exciton dissociation process at the DA interface.

kd(r0,E) is the dissociation rate depending on the initial

separation distance r0 and the electric field E. kf is the decay rate

of the charge transfer state.

88

Figure 4.3 Experimental (symbols) and simulation results (solid curves) of

the current-voltage characteristics of ITO/MoOx(2 nm)/donor(3

nm)/C60(40 nm)/ BPhen(8 nm)/LiF(1 nm)/Al OPV cells.

92

Figure 4.4 Experimental (solid squares) and simulated (open squares)

results of Jsc, FF and Voc vs ∆EHOMO for donor (3 nm)/C60 (40

nm) OPV devices.

94

Figure 4.5 The initial separation distance r0 and the charge transfer state

decay rate kf used in the simulation of donor (3 nm)/C60 (40 nm)

OPV devices.

96

Figure 4.6 (a) Experimental (symbols) and simulated (solid curves) J-V

curves of ITO/MoOx(2 nm)/NPB(x nm)/C60(40 nm)/BPhen(8

nm)/LiF/Al OPV devices with the NPB layer thickness varied

from 3 nm to 100 nm. (b) Experimental (solid square) and

simulated (open circle) values of Jsc, FF and Voc for NPB/C60

OPV devices.

101

Figure 4.7 (a) Experimental and (b) simulated current-voltage

characteristics of ITO/MoOx/TAPC:C60/BPhen/Al OPV devices

with various TAPC concentration from 1.2% to 50%.

104

xx

Figure 4.8 Photovoltaic characteristics of TAPC:C60 cells under AM1.5 at

100 mW/cm2 illumination: (a) Experimental data of Voc versus

TAPC concentration; (b) Experimental and simulated data of

Jsc versus TAPC concentration; (c) Experimental and simulated

data of FF versus TAPC concentration.

106

Figure 4.9 (a) Schematic structure of the 2-stack tandem OPV devices. (b)

Contour plot of simulated current density of 2-stack tandem

cells with various BHJ layer thickness in top and bottom

subcells. Star points are the experimental results with the

current density indicated accordingly.

111

xxi

List of Symbols and Abbreviations

A Richardson constant 9

ADN 9,10-di(2-naphthyl) anthracene 92

Al Aluminum 3

Alq tris(8-quinolinolato)aluminum 42

( )TAPCc absorption coefficient of the mixed material 107

60( )C absorption coefficient of C60 107

BHJ bulk heterojunction 5

BPhen 4,7-diphenyl-1,10-phenanthroline 90

c speed of light in the vacuum 87

CBP 4,4'-bis(carbazol-9-yl)-biphenyl 92

const constant given by meeting the condition G(0)=1 39

hole trap concentration 23

CT charge transfer 88

TAPCc TAPC concentration in volume percentage 107

non-radiative decay rate in infinite media 69

Γp (Γn) electronic wave function overlap factors 108

xxii

radiative decay rate in infinite media 69

Poole-Frenkel field dependence factor 11

Poole-Frenkel factor for electrons 13

Poole-Frenkel factor for holes 44

d NPB layer thickness 44

d distance between the monolayer and the exciton 50

DA donor-acceptor 5

de thickness of the layer where dipole locates 63

jd

thickness of the jth

monolayer 86

dm molecular diameter 23

diffusion constant of electrons 13

diffusion constant of excitons 42

E effective energy barrier blocking holes 38

∆EHOMO HOMO offset 90

∆ELUMO LUMO offset 90

E electric field 9

e elementary charge 9

xxiii

optical electric fields of the positive and negative

propagation light in of the 0th

monolayer

86

work function of the anode 12

EBL exciton blocking layer 4

work functions of the cathode 12

,HOMO HTLE HOMO level of the HTL 38

,HOMO ETLE HOMO level of the ETL 38

EIL electron injection layer 4

intE electric field at the HTL/ETL interface 38

,

optical electric fields of the positive and negative

propagation light in the jth

monolayer

85

,

optical electric fields of the positive and negative

propagation light in the kth

monolayer

85

EL electroluminescence 79

EML emissive layer 3

optical electric fields of the positive and negative

propagation light in the N+1th

monolayer

86

hole trap energy. 23

ETL electron transport layer 3

xxiv

ETM electron transport material 53

electric field distribution 15

vacuum dielectric constant 9

relative dielectric constant 9

r TAPCc

dielectric constant of the mixed material 108

εr(C60) dielectric constant of C60 108

εr(TAPC)

dielectric constants of TAPC 108

FF fill factor 93

spinF factor for spin statistics 42

FWHM full width at half maximum 73

( )G E the pre-factor considering the energy barrier for holes 38

h Plank constant 9

HAT-CN

1,4,5,8,9,11-hexaazatriphenylene hexacarbonitrile 110

HIL hole injection layer 4

HIM hole injection material 54

HOMO highest occupied molecular orbital 37

HTL hole transport layer 3

xxv

HTM hole transport material 53

θ angle between the light direction and the z-axis in the n+

media

68

, angle between the light direction and the z-axis in the air

and in the substrate

70

ICL inter-connecting layer 110

ITO indium tin oxide 3

electron current 13

,p crossJ crossing interface drift current of holes 38

Jsc short circuit current 93

space charge limited current density 15

thermionic injection current 8

J-V current density versus voltage 16

k Boltzmann constant 9

kd dissociation rate of the charge transfer state 88

ke wave vector 64

decay rate of the excitons 49

kf decay rate of the charge transfer state 88

xxvi

quenchingk quenching rate of excitons 42

Forster energy transfer rate 49

kze z-axis projection of the wave vector 64

z,ik, z,i+1k z-axis projection of wave vectors in the i

th layer and the

i+1th

layer

65

TAPCc

imaginary part of the optical refractive index of the

mixed material

107

imaginary part of the optical refractive index of C60 107

absolute value of the wave vector in the n+ media 68

TM

RNDK ,TE

RNDK power density of the TM and TE waves emitted from

random oriented dipoles

68

TM

+,T,RNDK ,TE

+,T,RNDK power density of the out-coupling TM and TE waves in

the n+ media emitted from random oriented dipoles

68

TMK ,TEK power densities of the TM and TE waves emitted from

the dipole with a dipole moment perpendicular to the

OLED surface

66

TMK ,TEK power densities of the TM and TE waves emitted from

the dipole with a dipole moment parallel to the OLED

surface

67

xxvii

TM

, ,K T ,TE

,T,K power density of the out-coupling TM and TE waves in

the n+ media emitted from the dipole with a dipole

moment perpendicular to the OLED plane

67

TM

,T,K ,TE

,T,K power density of the out-coupling TM and TE waves in

the n+ media emitted from the dipole with a dipole

moment parallel to the OLED plane

68

κ in-plane projection of the wave vector 64

L total organic layer thickness 12

LiF lithium fluoride 3

LUMO lowest unoccupied molecular orbital 37

wavelength 72

m effective mass of charge carriers 9

MTDATA 4,4',4''-tris[N-(3-methylphenyl)-N-

phenylamino]triphenylamine

42

mobility of organic material 10

mobility under electric field E 11

mobility under zero electric field 11

hole (electron) mobility of the electron transport material 53

hole (electron) mobility of the hole transport material 53

xxviii

hole (electron) mobility of the mixed host 53

electron mobility 13

electron mobility under electric field E 12

electron mobility under zero electric field 12

,n ETL electron mobility of the ETL 41

hole mobility 13

hole mobility under zero electric field 44

µp0 (µn0) hole (electron) mobility at zero electric field in pure

TAPC and C60

108

,p HTL hole mobility on the HTL side 38

,p ETL hole mobility on the ETL side 38

electron density 12

, optical refractive indexes of the air and the substrate 70

ETLn electron density on the ETL side of the interface 41

density of states in the HOMO level 23

jn , kn complex refractive index in the jth

monolayer and the kth

monolayer

86

optical refractive index of organic materials 72

xxix

density of chargeable states 10

n+ , n- optical refractive index of the infinite media outside

OLEDs

63

NDP 6,13-dihydro-6,13-di-2-naphthalenyl-

Dibenzo[b,i]phenazine

92

ne optical refractive index of the layer where dipole locates 63

NPB 4,4'-bis[N-(1-naphthyl)-N-phenylamino]biphenyl 42

OLEDs organic light-emitting diodes 3

OPV organic photovoltaic devices 3

hole density 12

angular distribution of light power density in the air 70

angular distribution of light power density in the

substrate

70

PHJ planar heterojunction 5

HTLp hole density on the HTL side 38

PL photoluminescence 73

density of trapped holes 22

P θ angular distribution of the light power density in the n+

media

68

xxx

charge carrier density 50

jQ absorption rate in the jth

monolayer 87

R recombination rate 13

r distance between the exciton and the quenching center 49

crossR cross interface recombination rate 40

R_free holes recombination between free holes and free electrons 32

TM

i i 1r ,TE

i i 1r the reflection coefficient at the interface between the ith

layer and the i+1th

layer for TM wave and TE wave

65

recombination rate between trapped holes and free

electrons

22

power reflection coefficient for light injection from the

substrate into the OLED stack

70

power reflection coefficient for light injection from the

substrate into the air

70

R_trapped holes recombination between trapped holes and free electrons 32

R0 Forster distance 49

initial separation distance 88

SCLC space charge limited current 15

ETL energy width of the ETL 38

xxxi

HTL energy width of the HTL 38

cross section of charge carrier trapping 23

T temperature 9

TM

i i 1t ,TE

i i 1t transmission coefficient at the interface between the ith

layer and the i+1th

layer for TM wave and TE wave

65

TAPC 1,1-bis-4-bis4-methyl-phenyl-amino-phenyl-cyclohexane 92

TCTA 4,4,4-trisN-carbazolyl-triphenyl amine 92

TE Transverse Electric 63

TM Transverse Magnetic 63

power transmission coefficient for light injection from

the substrate into the air

70

exciton lifetime 42

applied voltage across the device 12

Vbi built-in potential 93

Voc open circuit voltage 93

injection barrier 9

X exciton density 42

z+ , z- distance between the dipole and the nearest interfaces 64

1

Foreword

The following chapters of this thesis were jointly produced. My participation

and contribution to the research is as follows:

I am the primary author of section 3.3 in Chapter 3. I collaborated with

Professor Ching W. Tang and fellow graduate student Kevin P. Klubek. The devices

in section 3.3 were made and measured by Kevin P. Klubek in Eastman Kodak

Company. My contribution to this section was the analysis of the results, building the

model and providing simulation results. The section has been published in Applied

Physics Letter, 2008, 93, 090036.

Chapter 4 of my dissertation was in collaboration with Professor Ching W.

Tang, and fellow graduate student Dr. Minlu Zhang at the University of Rochester. Dr.

Minlu Zhang carried out all the experiments and measurements reported in this

chapter. Dr. Hongkun Tian, and Professor Yanhou Geng from the Changchun

Institute of Applied Chemistry provided NAT5 material for section 4.6. I built the

model and provided simulation results to quantitatively explain the device

performance.

Section 4.4 has been published in Applied Physics Letter, 2010, 97, 143503,

and was co-authored with Professor Ching W. Tang and Dr. Minlu Zhang.

Section 4.5 has been published in Applied Physics Letter, 2011, 99, 213506

and was co-authored with Professor Ching W. Tang and Dr. Minlu Zhang.

2

Section 4.6 has been published in Advanced Materials, 2011, 23, 4960, and

was co-authored with Professor Ching W. Tang, Dr. Minlu Zhang, Dr. Hongkun Tian,

and Professor Yanhou Geng.

Section 4.7 has been published in Organic Electronics, 2012, 13, 249, and was

co-authored with Professor Ching W. Tang and Dr. Minlu Zhang.

3

Chapter 1

Introduction

Energy has become a more and more important issue these years due to the fast

development of our society and the limited fossil-fuel resources on earth. There are

two approaches to solving the energy problem. One is to reduce the energy

consumption by improving the efficiency of energy utilization and the other is to

generate more energy, especially from long-term renewable sources. Organic light-

emitting diodes (OLEDs) and organic photovoltaic (OPV) devices can help ease the

energy scarcity issue by utilizing those two approaches, respectively.

1.1 Introduction to OLEDs and OPV devices

Ever since Tang et al. invented them in 1987 [1], organic light-emitting diodes

have received a lot of attention because of their potential use as full-color display

panels and lighting devices. Thanks to the tremendous development during the last

two decades, OLED technology has been widely used in cell phones and large screen

TV displays. Typical OLED device structure is: glass substrate/transparent

anode/hole transport layer (HTL)/emissive layer (EML)/electron transport layer (ETL)

/reflective cathode, where organic layers are sandwiched between the anode and

cathode. While ITO is the most popular transparent anode used to inject holes into the

HTL, LiF/Al is usually used as the cathode to inject electrons into the ETL [2]. After

4

the transport through the HTL and ETL, respectively, holes and electrons recombine

with each other in the emissive layer (EML) and emit light.

Over the last two decades, much research has been devoted to improving the

efficiency, reliability, and color quality of OLEDs, as well as to making them

commercially viable. By doping the EML with high quantum yield molecules, high

efficiency and color varieties can be achieved [3]. Other functional layers such as the

hole injection layer (HIL) [4-7], the electron injection layer (EIL) [2, 8] and the

exciton blocking layer (EBL) [9] were introduced to increase the device efficiency

and lifetime. The device lifetime can be further improved by constructing a uniformly

mixed [10] or graded mixed host layer [11]. When several EMLs were incorporated

together, a white OLED can be fabricated and become attractive for lighting

applications [12-14]. While the significant progress in materials and device structures

led to a higher efficiency and longer lifetime for OLEDs, the detailed mechanisms

underlying these improvements are not fully understood.

Organic photovoltaic devices [15], also known as organic solar cells, are devices

that use organic active layers to absorb sunlight and generate electricity. Due to the

low dielectric constant (2-4) of organic materials, strongly bound electron hole pairs,

also called excitons, are generated instead of free charge carriers by the absorption of

photons. Because the exciton binding energy is significantly higher than the thermal

energy (kT), another driving force (usually a strong electric field) is required to

dissociate the excitons into free charge carriers. In 1986, Tang et al. invented the

5

planar heterojunction (PHJ) OPV device with a structure of glass substrate/anode/

donor layer (HTL)/acceptor layer (ETL)/cathode [15]. A donor-acceptor (DA)

interface was introduced to dissociate the excitons. In this case, the device efficiency

is further limited by the exciton diffusion length of the organic material since only

those excitons arriving at the DA interface can dissociate efficiently. In the 1990s, a

bulk heterojunction (BHJ) structure was introduced, which mixed the donor and

acceptor material together in the active layer. Hence most excitons generated can

reach the nearby DA interface and dissociate [16]. However, the low mobility of the

organic materials will enhance the recombination loss during the extraction of the

charge carriers. Meanwhile, a high electric field is still favorable for the dissociation

of excitons at the DA interface which limits the device thickness and photon

absorption as well.

OLEDs and OPV devices have the similar device structure and completely

reversed working process--OLEDs generate photons from charge carriers, while OPV

devices do the reverse job. Such a correlation between the two types of devices

provides a perspective to study them together, especially through simulation.

6

1.2 Introduction to the simulation of OLEDs and OPV devices

With systematic study of the device performance of OLEDs and OPV devices,

people can explain some effects qualitatively. For example, the improved current

efficiency in OLEDs with a HIL can be explained by the reduction of the radical

cations quenching at the HTL/EML interface [17]. However, in order to explain the

effect quantitatively, we should simulate the device performance and obtain some

numerical results.

Since the working process of OLEDs and OPV devices is complicated and several

interfaces (singularities) are presented, it is difficult to get any analytical results

directly. This empirical drawback makes numerical simulation with computer

programming more viable. Simulation is a powerful method to investigate the

underlying mechanism of devices. More importantly, it can tell us to what amplitude

the mechanism can affect the device performance. Meanwhile, simulation can retrieve

detailed information, such as the charge carrier distribution, which is difficult to

directly measure through experiments. Furthermore, a good simulation program can

predict the performance of new devices without if being necessary to do the

experiment, which can save material, energy and time.

In the simulation of OLEDs and OPV devices, the first step is to divide the

organic layers into a stack of discrete monolayers, which are sandwiched between an

anode and a cathode. Because the cross section of the devices (around 1 mm) is

several orders of magnitude larger than the device thickness (around 1 um), it is

7

reasonable to assume that each monolayer has an infinite area. With this assumption,

we can treat the thin film devices as one-dimensional cells along the direction normal

to the electrode surfaces. The same treatment has been accepted for almost all

publications modeling OLEDs and OPV devices [18-38].

Simulation of OLEDs includes three parts: the electrical part, the exciton part and

the optical part. The electrical part describes the charge carrier injection, transport,

recombination, and the electric field distribution. The exciton part simulates the

exciton generation, diffusion, quenching, radiative decay and non-radiative decay.

The optical part models the cavity effect and the out-coupling process.

In contrast, simulation of OPV devices also includes the same three parts in a

reverse order: the optical part, the exciton part and the electrical part. The optical part

calculates the photon absorption in the active layer and the interference effect due to

the reflective cathode. The exciton part describes the exciton diffusion, decay, and

dissociation at the DA interface. After the exciton dissociation, free charge carriers

are generated. The electrical part models the charge carrier transport, recombination

and extraction.

8

Chapter 2

Simulation of Single Layer Devices

Single layer devices have only one organic layer sandwiched between electrodes.

They are basic devices we can simulate as a starting point. Despite their simple

structure, they still provide us meaningful information on how to optimize the

performance of devices with more complicated structures. In addition, the

mechanisms of the charge carrier injection, transport and recombination in single

layer devices are identical to those in multilayer devices. A solid understanding of

these mechanisms establishes a good foundation for research on multilayer devices.

2.1 Charge carrier injection, transport and recombination

2.1.1 Charge carrier injection at the metal/organic interface

Charge carrier injection from a metal electrode into an inorganic semiconductor

can be described by the thermionic emission model [39], which assumes that the

current flow across the barrier is solely dependent on the barrier height. The injection

current based on this model can be calculated by the following equations:

2 bth

B EJ AT exp

kT

(2.1)

9

2

3

4 emkA

h

(2.2)

3

0/ 4 rB e (2.3)

where

: thermionic injection current

A: Richardson constant

T: temperature

: injection barrier

E: electric field

k: Boltzmann constant, 8.61710−5

eV/K

e: elementary charge, 1.610−19

C

m: effective mass of charge carriers

h: Plank constant, 4.13610−15

eV*s, hc=1240 eV*nm

: vacuum dielectric constant, 8.85410−15

C/(V*cm)

: relative dielectric constant of the material

Equation (2.1) includes the term

that accounts for the lowering of

the barrier height by an image potential. Given mc2=0.511 MeV, T=300 K, =3,

=0.2eV, E=1 MV/cm, we can calculate the injection current density as

, which is obviously much higher than the working current range

10

(1 mA/cm2 to 1000 mA/cm

2) in organic semiconductor devices. Thus, the thermionic

emission model needs to be modified for organic semiconductor devices.

The problem of applying the original thermionic emission model in organic

semiconductor devices lies in the highly amorphous structure and low mobility of

organic materials. The injected holes (electrons) cannot transport away from the

anode (cathode) quickly enough. Therefore, part of them will diffuse back or

recombine with the charges at the electrode. Smith et al. [21, 40, 41] first proposed

the back diffusion current that will cancel most injection current and leave the net

contribution current as the true injection current. Later, Scott and Malliaras [42]

proposed a more unified model to calculate the charge carrier injection from a metal

electrode into an organic layer. The model has considered the balance between the

thermionic injection current and the surface recombination current. The electric field

and image potential at the interface lower the injection barrier and increase the

thermionic injection current. For surface recombination, the Coulomb capture

distance at the interface is defined at which the energy is kT below the maximum

energy barrier. The difference between the thermionic injection current and surface

recombination current--the net injection current--is calculated as:

2

04 * exp( )bthJ N e E exp f

kT

(2.4)

1 1 1 2 f

ff f

(2.5)

11

3

0/ 4 rf e E kT (2.6)

wherethJ is the injection current, is the elementary charge, is the mobility of the

organic material, is the density of chargeable states, is the injection barrier,

is the electric field at the metal/organic interface, and are the relative and

vacuum dielectric constant, is the Boltzmann constant, and is the temperature.

Equation 2.5 denotes an intermediate parameter when solving for the Coulomb

capture distance, and equation 2.6 represents the lowering of energy barrier due to the

electric field and image potential.

2.1.2 Charge carrier transport and recombination

Due to the amorphous structure of organic materials, charge carriers mainly

move by hopping between molecules. The energy and position disorder impedes the

transport and cause a low mobility for organic materials. Bässler [43] used the Monte-

Carlo method to analyze the transport behavior in disordered materials. As a result of

the disordered structure, most organic materials have electric field dependent mobility.

The electric field dependence of mobility can be described by the Poole-Frenkel

equation [43-45] as follows

0( ) *exp( * )E E (2.7)

where E is the electric field, is the charge carrier mobility under electric field

E, is the mobility under zero electric field, and is the Poole-Frenkel field

12

dependence factor. Since and are both constant, the mobility of the organic

material under any given electric field can be calculated through the above equation.

To correctly model the charge carrier transport in organic semiconductor

devices, we need to calculate the electric field and charge carrier density distributions

across the whole device. As we mentioned in the introduction, we treat the thin film

devices as one-dimensional cells along the direction normal to the electrode surfaces.

Therefore, the one-dimensional Poisson equation describes the electric field

distribution across the organic layer, and the applied voltage across the device

provides the necessary boundary condition [18, 19, 21, 22, 25, 26]. The electric field

in the device is calculated by the following equations

0r

e p nE

x

(2.8)

L

applied cathode anode

0

V E E E x *dx (2.9)

where is the electric field, and are the hole and electron densities, is the

applied voltage across the device, and are the work functions of the

cathode and the anode, which may not be the bulk work functions due to the possible

dipolar layers at the metal/organic interfaces [18], and L is the total organic layer

thickness. Equation 2.8 shows that the electric field distribution in the device is

determined by the charge carrier density at each position. Equation 2.9 presents a

boundary condition for the device.

13

Charge carriers (holes and electrons) transport through drift (electric field

driven) and diffusion (density gradient driven). The mobilities of organic materials

are calculated by the Poole-Frenkel equation as mentioned above, and the diffusion

constant is assumed to follow the Einstein relation [46, 47]. For instance, the electron

transport in devices is calculated by the equations below

0( ) *exp( * )n n nE E (2.10)

* /n nD u kT e (2.11)

* * * *n n n

nJ e u E n D

x

(2.12)

where is the electron mobility under electric field E, is the electron

mobility at zero electric field, is the Poole-Frenkel factor describing the field

dependence, is the diffusion constant of electrons, and denotes the electron

current considering both drift and diffusion. The hole transport can be described by

similar equations.

Electrons and holes recombine with each other if they are both present in the

device and are close to each other. The bulk recombination rate of free electrons and

holes is calculated in accordance with Langevin’s theory [48]

0

* * *n p

r

eR n p

(2.13)

14

where R is the recombination rate, and are electron and hole mobilities, and

are electron and hole densites. The recombination rate in organic material depends

on how fast the electrons and holes can find each other. Thus the mobilities of the

charge carriers play a significant role in the recombination process.

The drift and diffusion currents move the charge carriers in the device, while

the recombination reduces the charge carrier density. Combining all of them together,

we can calculate the change of charge carrier density over time through the

continuity equation [18] below

1

* nJnR

t e x

(2.14)

Equations 2.8 to 2.14 above are sufficient to simulate the current density versus

voltage characteristics and the recombination distribution in single layer organic

semiconductor devices. Charge carriers in the device determine the electric field

distribution, and the electric field will modify the charge carrier distribution. The

interplay between the charge carriers and the electric field is the key issue in organic

semiconductor devices, while the field dependent mobility further complicates the

problem. In the model, we will calculate the change of the electric field and charge

carrier density at each position over time. The model stops at the equilibrium state

where the electric field and charge carrier density at each position remain constant.

15

2.2 Single layer devices with unipolar transport materials

We first simulate single layer devices with unipolar transport materials. Unipolar

means that there is only one type of charge carriers (either holes or electrons, but not

both) in the device. Therefore, we can ignore the recombination for now and focus on

the injection and transport of charge carriers.

2.2.1 Space charge limited current

Space charge limited current (SCLC) takes place in single layer devices with a

unipolar transport material. With an Ohmic injection contact, the bulk transport of the

charge carriers determines the current density of the device. Since the injected charge

carriers (space charges) from the Ohmic contact also modify the electric field across

the layer, the charge carrier density, and therefore the current density, in the device is

limited. Mott-Gurney Law [49] [50] is an analytical theory describing the SCLC

current.

The SCLC current density and the electric field distribution in the devices can be

calculated through the Mott-Gurney Law as follows

2

0 0

3

9

8

rSCLC

VJ

L

(2.15)

0 0

2* *( ) SCLC

r

J xE x

(2.16)

16

where denotes the SCLC current density, is the charge carrier mobility, V is

the applied voltage across the layer, L is the layer thickness, and is the electric

field distribution across the device. Figure 2.1(a) compares the simulation and

analytical results of the current density versus voltage (J-V) characteristics. The

simulation results are obtained through the model discussed in the previous section,

while the analytical J-V curve are calculated by the SCLC theory (equation 2.15). The

thickness of the layer is 100 nm and the carrier mobility is 2.010-4

cm2V

-1s

-1. The

injection barrier is set to be 0 eV in the simulation to form an Ohmic contact. Figure

2.1(b) shows the simulation and analytical results of the electric field distribution in

the same device under the current density of 1000 mA/cm2. The good agreement

between the simulation and the analytical results indicates that the model works well

for single layer devices when SCLC conditions are met.

17

Equation 2.15 and 2.16 are derived with the assumption that the mobility is

constant across the whole layer. Since the electric field inside the layer varies at each

position, the mobility is not constant across the layer if it is field dependent. Thus

there is no analytical solution for the SCLC current with field dependent mobility.

However, the current can be estimated through the empirical equations [51] as

follows

Figure 2.1. Simulation and analytical results of (a) the current density versus

voltage (J-V) characteristics and (b) the electric field distribution under current

density of 1000 mA/cm2. The layer thickness is 100 nm and the carrier mobility

is 2.010-4

cm2V

-1s

-1.

0 20 40 60 80 100

0

1x105

2x105

3x105

4x105

5x105

6x105

7x105

(b)

2.0*10

-4 cm

2V

-1s

-1

=0 (V/cm)-1/2

SCLC theory

SimulationEle

ctr

ic f

ield

(V

/cm

)

Position (nm)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

10

100

1000

2.0*10

-4 cm

2V

-1s

-1

=0 (V/cm)-1/2

J (

mA

/cm

2)

Voltage (V)

SCLC theory

Simulation

(a)

18

0( ) *exp( * )E E (2.17)

2

0 0

3

9exp(0.89* * )

8

rSCLC

V VJ

L L

(2.18)

Figure 2.2 (a) shows the simulation and analytical results (calculated from

equation 2.18) of the SCLC current for a device with a 100 nm layer. The mobility

parameters are: , . The

good agreement between the simulation and the analytical results confirms that the

simulation model works well for a single-layer and unipolar device where the carrier

mobility is field dependent.

Figure 2.2(b) compares the simulated electric field distribution in devices with

a field independent mobility (mobility parameters

) and a field dependent mobility (mobility parameters

). The results are simulated under the

current density of 1000 mA/cm2 and the layer thickness for both devices is 100 nm.

Under non-zero electric field, the mobility in the device with a field dependent

mobility is higher than that in the device with a field independent mobility. Therefore

the electric field required to maintain the same current density is lower. The field

dependence of mobility plays a significant role in determining the device

performance, especially the J-V curves, of organic semiconductor devices.

19

Figure 2.2. (a) Simulation and analytical results (calculated from equation 2.18) of

the J-V characteristics for the device with a field dependent mobility (mobility

parameters: , ) and a layer

thickness of 100 nm. (b) Electric field distribution across the devices with a field

independent mobility (mobility parameters: ,

) and a field dependent mobility (mobility parameters:

, ) under current density of 1000

mA/cm2 and a layer thickness of 100 nm.

0 20 40 60 80 100

0

1x105

2x105

3x105

4x105

5x105

6x105

7x105

2.0*10

-4 cm

2V

-1s

-1

(b)

Ele

ctr

ic f

ield

(V

/cm

)

Position (nm)

0 (V/cm)-1/2

2.0*10-3 (V/cm)

-1/2

0.0 0.5 1.0 1.5 2.0 2.5 3.0

10

100

1000

2.0*10

-4 cm

2V

-1s

-1

=2.0*10-3 (V/cm)

-1/2

(a)

J (

mA

/cm

2)

Voltage (V)

SCLC theory

Simulation

20

2.2.2 Injection barrier effect

The previous section investigates the SCLC that requires an Ohmic injection

contact. When injection barrier is high enough, the current in organic semiconductor

devices will show very different voltage dependence. We have simulated devices with

various injection barriers to investigate the injection barrier effect. Figure 2.3(a)

compares the J-V curves for devices with injection barrier varied from 0 eV to 0.4 eV.

The mobility is , and the layer

thickness is 100 nm for all devices. The simulation results demonstrate that when

injection barrier is lower than 0.2 eV, the barrier is low enough and the contact can be

practically treated as Ohmic contact. The effect has also been reported by other

groups [19, 21]. However, when injection barrier is high enough, the current density

is limited by the injection barrier and higher drive voltage is required, as evident from

Figure 2.3(a).

Figure 2.3(b) shows similar simulation results for devices with lower charge

carrier mobility: , For devices with

low charge carrier mobility, the electric field that is required to drive the current

density is high. As a result, the high electric field near the injection interface

effectively lowers the injection barrier and hence helps the injection of charge carriers.

21

Figure 2.3. Simulated J-V curves for devices with injection barrier varied from 0 eV

to 0.4 eV and mobility parameters: (a)

and (b) . The layer

thickness is 100 nm for all devices.

0 20 40 60 80 100 120 140 1601

10

100

1000

2.0*10

-7 cm

2V

-1s

-1

=0 (V/cm)-1/2

J (

mA

/cm

2)

Voltage (V)

SCLC theory

Simulation, B=0 eV

Simulation, B=0.2 eV

Simulation, B=0.3 eV

Simulation, B=0.4 eV

(b)

0 5 10 151

10

100

1000

(a)

2.0*10

-4 cm

2V

-1s

-1

=0 (V/cm)-1/2

J (

mA

/cm

2)

Voltage (V)

SCLC theory

Simulation, B=0 eV

Simulation, B=0.2 eV

Simulation, B=0.3 eV

Simulation, B=0.4 eV

22

2.2.3 Trap limited current

After observing the effect of the injection barrier on J-V curves, we expand the

model to investigate the effect of charge carrier traps on single layer devices. The

contact of the device is set to be Ohmic with zero injection barriers in this section, so

that we can focus on the effect of the charge carrier traps and ignore the injection

barrier effect. Doping with luminophores [3] is widely used in OLEDs to obtain a

high quantum yield and good color purity. Some dopant molecules also act as hole

(electron) traps in the host material. These trap centers will severely alter the transport

behavior of organic semiconductor devices even with a small fraction. Staudigel et.al

[18] proposed a model to calculate the trapped charge carrier density in organic

materials. For devices with hole traps, the recombination between trapped holes and

free electrons is calculated through the modified Langevin recombination rate as

follows

0

* * *pt n t

r

eR n p

(2.19)

where is the recombination rate between trapped holes and free electrons, is

the density of the trapped holes, and n is the electron density. Only electron mobility

n is used in the equation, because the trapped holes are assumed to be still.

The transformation between free holes and trap holes are simulated by the

equations below

23

 

**

[ * * *( )*exp ]

free trap

t p

m

pt

HOMO pt t t HOMO

dp kTE

dt e d

Ep N C p p N p

kT

(2.20)

25 mt d (2.21)

where is the cross section of charge carrier trapping. Trapping is assumed to occur

on the trap center itself and its four neighbors in the same monolayer, thus is

estimated as 5 times the square of molecular diameter dm [18]. is the density

of states in the HOMO levels, which can be approximated as 1/3

md . denotes the

hole trap concentration and denotes the hole trap energy. The trapped hole has a

probability of

to pop out from the trapped state. Combining the equations

above, we can calculate the change of trapped holes density over time by the

following equation

free trapt

pt

dpdpR

dt dt

(2.22)

Figure 2.4(a) shows the J-V curves for devices with hole trap concentration

varied from 0% to 2%. The devices are set to only transport holes. The hole trap

energy is set as ; the hole mobility is ,

and the layer thickness is 100 nm. When trap concentration is 0%,

the current density in device is the same as the SCLC because we set the contact to be

Ohmic. Increasing the trap concentration will dramatically increase the drive voltage

24

of the device. The high drive voltage is caused by the space charge effect of the

trapped charge carriers.

Figure 2.4(b) shows the density of trapped holes and free holes in the device

with a hole trap concentration of 1%. The results are simulated under current density

of 1000 mA/cm2. The simulation indicates that the trapped holes become the majority

part of the space charges. Consequently, the accumulated trapped holes dramatically

increase the electric field and hence the drive voltage of the device.

25

Figure 2.4. (a) Simulated J-V curves for devices with hole trap concentration varied

from 0% to 2%. The hole trap energy is set as the hole mobility

parameters are , . The layer

thickness is 100 nm for all devices. (b) The density of trapped holes and free holes at

1000 mA/cm2 in the device with a hole trap concentration of 1%.

0 20 40 60 80 100

0.0

5.0x1017

1.0x1018

1.5x1018

2.0x1018

2.5x1018

3.0x1018

(b)

p2.0*10

-4 cm

2V

-1s

-1

p=0 (V/cm)

-1/2

Ept=0.2 eV

Cpt=1%

trapped holes

free holes

ch

arg

e c

arr

ier

de

nsity (

1/c

m3)

Position (nm)

0 5 10 15 20 25 301

10

100

1000

p2.0*10

-4 cm

2V

-1s

-1

p=0 (V/cm)

-1/2

Ept=0.2 eV

J (

mA

/cm

2)

Voltage (V)

Cpt=0%

Cpt=0.1%

Cpt=1%

Cpt=2%

(a)

26

2.3 Single layer OLED devices with bipolar transport materials

2.3.1 Recombination zone in single layer OLED devices

Bipolar materials can support the transport of both electrons and holes, and

recombination takes place in the device when electrons and holes meet with each

other. The recombination rate can be calculated by the Langevin recombination rate

[48] (equation 2.13).

Single layer bipolar transport devices are the original organic electroluminescent

(EL) devices [52]. Without a heterojunction structure, the charge carrier mobilities

control the location of the recombination zone in single layer OLED devices.

Figure 2.5 shows the distribution of the charge carrier density, the electric field

and the bulk recombination zone in single layer OLED devices with balanced (left

side of the figure,

) and unbalanced (right side of

the figure,

,

) electron and

hole mobilities. All mobilities are set as field independent and the injection contacts

for holes and electrons are set as Ohmic. The layer thickness for both devices is 100

nm and the results are simulated under current density of 100 mA/cm2. If the material

has perfect balanced bipolar transport properties (

), electrons and holes

distribute in the device symmetrically, and the recombination takes place across the

whole layer. Expanding the recombination zone away from the electrodes is favorable

for a high efficiency and long lifetime.

27

However, if the hole mobility is much higher than the electron mobility in the

device (right side of Figure 2.5), holes that moves faster than electrons will arrive at

the cathode/organic interface. As consequency, most of the recombination takes place

near the cathode/organic interface before electrons can move even further. It is well

known that recombination that takes place close to the cathode can be quenched by

the metal electrode [53-57]. Therefore, the efficiency of this type of EL devices is

very low. The electrode quenching was the limiting factor for organic EL devices [52]

before Tang et.al introduced the heterojunction structure [1] that effectively move the

recombination zone away from the electrodes.

28

µp: µn=100:1 µp: µn=1:1

Figure 2.5. Distribution of the charge carrier density, the electric field and the

recombination zone in single layer OLED devices. The layer thickness for all devices

is 100 nm and the results are simulated under current density of 100 mA/cm2. Left

figures are simulated with . Right figures are

simulated with , . All

mobilities are set as field independent and the injection contacts for electrons and

holes are set as Ohmic.

29

2.3.2 Recombination efficiency in single layer OLED devices

Recombination efficiency, defined by the ratio of recombined charge carriers

(J_Recombination) over total injected charge carriers (J_Injection), represents the

balance factor for OLED devices. The injected charge carrier can either recombine in

the device or pass through the device without recombination; the latter part of the

charge carriers forms the leakage current and is not contributing to the light emission.

The mobility mismatch can play a significant role in determining the balance factor of

single layer OLED devices.

Figure 2.6 compares the recombination efficiency (J_Recombination/J_Injection))

in single layer OLED devices with balanced and unbalanced mobilities. The mobilies

for the two devices are (1) µp: µn=1:1, ,

and (2) µp: µn=100:1, ,

, respectively. All mobilities are set as field independent and

injection contacts for holes and electrons are set as Ohmic.

For devices with the hole mobility equal to the electron mobility (µp: µn=1:1), at

high current density, the recombination efficiency is high since a majority of the

charge carriers will recombine in the bulk before reaching the other side of the device.

However, the recombination efficiency drops as current density decreases. This drop

is attributed to the reduced charge carrier density in the device. When current density

decreases, electrons see fewer holes on their way passing through the device.

Therefore, the probability of recombination reduces.

30

While for devices with unbalanced mobilities (µp: µn=100:1), the recombination

efficiency is always low even at high current density. Because of the mobility

mismatch, most recombination takes place near the cathode. The holes have a high

probability of passing through the narrow recombination zone without recombining

with electrons. As a result, the leakage current is high, and the recombination

efficiency is low. The efficiency of the device with unbalanced mobilities would be

even lower if we considered the electrode quenching mechanism [53].

Figure 2.6. Recombination efficiency (J_Recombination/J_Injection) of single

layer OLED devices with balanced and unbalanced mobilities. (1) ,

hole mobility , electron mobility

and (2) , hole mobility

, electron mobility . All mobilities

are set as field independent and injection contacts for holes and electrons are set

as Ohmic.

31

From the simulation results presented in section 2.3.1 and 2.3.2, we can tell that

for single layer OLED devices, a balanced mobility is necessary to achieve a

favorable recombination zone away from the electrodes and high recombination

efficiency as well. By setting the injection contacts as Ohmic, we can ignore injection

barrier effects. However, the injection contacts for holes and electrons can alter the

device performance as well. A high injection barrier for electrons would shift the

recombination zone towards the cathode and decrease the recombination efficiency.

2.3.3 Single layer OLED devices with charge carrier traps

In the previous section, we have shown that when the hole mobility is much

higher than the electron mobility in single layer OLED devices, recombination takes

place near the organic/cathode interface. In order to achieve high quantum efficiency,

we need to move the recombination zone away from the electrodes to reduce the

electrode quenching and leakage current. To balance the electron and hole transport

without introducing the heterojunction structure, we can add some hole traps [53, 58]

into the device to slow down the transport of holes.

Figure 2.7 shows the hole trap concentration, the charge carrier density, the

electric field and recombination distribution for devices with hole traps. Parameters

for the simulation include: the layer thickness is 100 nm, µp: µn=100:1, the hole

mobility is , the electron mobility is

, and the hole trap energy is . All mobilities are set as

electric field independent and the contacts for injections are set as Ohmic. The

32

recombination between free holes and free electrons (R_free holes) as well as the

recombination between trapped holes and free electrons (R_trapped holes) are plotted

separately.

The left side of Figure 2.7 are simulation results for the device with hole traps

that are uniformly distributed across the layer with a hole trap concentration of

. The recombination zone of free holes and electrons are wider than the case

without traps (right side of Figure 2.5), and the hole traps effectively slow down the

hole transport. Because the whole layer doped with hole traps, the recombination

between trapped holes and free electrons still take place near the cathode.

The right side of Figure 2.7 shows the device performance in a situation where

the 20 nm in the middle of the layer is heavily doped with hole traps ( ) and

the rest of the layer is doped with a hole trap concentration . Doping hole

traps heavily in the middle can effectively move the recombination zone between

trapped holes and free electrons away from the cathode. If the recombination between

the trapped holes and free electrons can emit photons efficiently, we can achieve a

high efficiency by reducing the electrode quenching. We can increase the amplitude

of recombination between trapped holes and free electrons by increasing the hole trap

concentration and/or hole trap energy. Furthermore, we can choose to heavily dope

the region where cavity effect can be magnified.

33

Figure 2.7. Distribution of the hole trap concentration, the charge carrier density, the

electric field, the recombination between free holes and free electrons, and the

recombination between trapped holes and free electrons in single layer OLED

devices under current density of 100 mA/cm2. Left panel has hole trap concentration

across the whole layer. Right panel has hole trap concentration

for the 20 nm in the middle of the layer, and for the rest.

34

Figure 2.8 (a) compares the recombination efficiency

(J_Recombination/J_Injection) of the three devices with unbalanced mobilities (µp:

µn=100:1) we have simulated so far:

Device No-Trap: single layer OLED device without hole traps.

Device Uniform-Trap: hole trap concentration across the whole layer

Device Middle-Heavy-Trap: hole trap concentration for the 20 nm in

the middle of the layer, and for the rest.

The recombination in devices with hole traps combining the recombination of free

holes and trapped hole together. Clearly, both devices with hole traps improve the

recombination efficiency by slowing down the hole transport and achieve a better

balance between the hole transport and electron transport [53, 58].

Figure 2.8 (b) compares the normalized recombination zone of the three devices.

Wider recombination zone is achieved through more balanced charge carrier transport

in devices with hole traps. By heavily doping the middle layer, we have observed a

slightly more recombination in the middle of the layer.

35

Figure 2.8.(a) Recombination efficiency and (b) normalized recombination zone

at100 mA/cm2 for single layer OLED devices with and without hole traps.

(b)

(a)

36

In conclusion, doping the bipolar transport material with hole (or electron) traps

provide an approach to control the recombination zone and improve the device

performance. However, the traps will severely increase the drive voltage of the device

which should be considered before applying the method. Figure 2.9 compares the J-V

curves of the three single layer OLED devices with and without hole traps. As shown

in the figure, doping the layer with hole traps dramatically increases the drive voltage.

Figure 2.9. Simulated J-V curves for single layer OLED devices with and without

hole traps.

0 1 2 3 4 5

1

10

100

J(m

A/c

m2)

Voltage (V)

No-Trap

Uniform-Trap

Middle-Heavty-Trap

37

Chapter 3

Simulation of Multilayer Organic Light-

Emitting Diodes

As we mentioned in the previous chapter, single layer OLED devices are usually

not efficient due to unbalanced charge carrier transport and electrode quenching of

the recombination. In 1987, Tang et al. [1] introduced the bilayer heterojunction

structure in OLEDs, which effectively increase the efficiency of organic EL devices.

Since then, the heterojunction structure has been widely used in OLED devices and

layers with more complicated function have been introduced [2, 4-9]. To study

OLEDs with such multilayer structures, we would need to extend the model that we

used in single layer devices

3.1 Interfaces between organic layers

3.1.1 Charge carrier transport across organic interfaces

The most important characteristic of OLED devices is the multilayer structure

where each layer has different HOMO (highest occupied molecular orbital)/LUMO

(lowest unoccupied molecular orbital) levels and hole/electron mobilities. When holes

(electrons) arrive at the interface between two adjacent organic layers, they may face

the energy barrier generated by the HOMO (LUMO) level difference between the

38

layers. In this case, a pre-factor of

is usually multiplied to the drift current

and diffusion current to represent the energy barrier effect. Some other factors have

been introduced for the crossing interface transport; the geometric average [25] of the

mobilities on both sides of the interface is used as the crossing interface mobility; the

energy disorder of organic materials should be considered which can effectively

lower the real energy barrier for charge carriers to jump over [18].

Figure 3.1 shows the diagram of the hole transport across the HTL/ETL

interface. The crossing interface drift current can be calculated through the following

equations

, int , ,* * * ( )p cross HTL p HTL p ETLJ e p E G E (3.1)

, , int *HOMO HTL HOMO ETL mE E E E d (3.2)

2 2

( ) * *exp( )*exp( )2 2

1, ( )

( )exp( ), ( )

HTL ETLHTL ETL

HTL ETL

ETL HTL

ETL HTLETL HTL

x xG E const dx dx

if x x E

E x xif x x E

kT

(3.3)

where

,p crossJ : crossing interface drift current of holes

e : elementary charge

HTLp : hole density on the HTL side

39

intE : electric field at the HTL/ETL interface

,p HTL : hole mobility on the HTL side

,p ETL : hole mobility on the ETL side

( )G E : the pre-factor considering the energy barrier for holes

E : effective energy barrier blocking holes

md : molecule diameter, which is also the monolayer thickness

,HOMO HTLE : HOMO level of the HTL

,HOMO ETLE : HOMO level of the ETL

const : constant given by meeting the condition G(0)=1

HTL : energy width of the HTL

ETL : energy width of the ETL

k : Boltzmann constant

T : temperature

The electron transport across the interface is calculated in a similar way except that

the LUMO level difference should be used to calculate the energy barrier.

40

3.1.2 Cross interface recombination

For OLEDs with multilayer structure, holes will accumulate at one side of the

interface while electrons will accumulate on the other side. The cross interface

recombination in this case needs to be treated correctly. The Langevin recombination

rate [48] is used for bulk recombination of holes and electrons. The intuitive way to

calculate the cross interface recombination is to use the sum of the hole mobility on

one side of the interface (e.g. HTL) and the electron mobility on the other side of the

interface (e.g. ETL). Cross interface recombination of holes on the HTL side and

electrons on the ETL side can be calculated as

, ,

0

*)( *n ETL p HTLcross E TL

r

HTL

en pR

(3.4)

Figure 3.1. Diagram of the hole transport across the HTL/ETL interface.

41

where crossR is the cross interface recombination rate, e is the elementary charge, r

and 0 are the relative and vacuum dielectric constant, ,n ETL is the electron mobility

of the ETL, ETLn is the electron density on the ETL side of the interface, ,p HTL is the

hole mobility of the HTL, and HTLp is the hole density on the HTL side of the

interface. However, Blom et al. proposed that the smaller value of the two

mobilities( ,n ETL and ,p HTL ) should be used to calculate the cross interface

recombination rate [59], the same treatment has been adopted in this thesis

, ,

0

*m , )in( * *n ETL p HTLcross H

r

TE LTL

en pR

(3.5)

Only part of the excitons generated through the cross interface recombination can

transform to bulk excitons and emit photons [18]. Since the interface excitons have

lower energy than bulk excitons, the probability of forming bulk excitons after cross

interface recombination is assumed to be

or

,

depending on which side the bulk excitons locate. However, exciplex formed at the

strong donor/acceptor interface can emit by themselves, which has been observed in

OLED devices [17, 60, 61].

3.2 Exciton diffusion, decay and quenching

Excitons are formed when electrons and holes recombine with each other. The

following equation describes the exciton diffusion, decay and quenching

42

2

2( )* * ( )*spin X quenching

X X XR x F D k x X

t x

(3.6)

where X is the exciton density, is the recombination rate, spinF is the factor for

spin statistics, which is assumed to be 25% for singlet and 75% for triplet, is the

exciton lifetime due to radiative decay and non-radiative decay, the ratio of radiative

decay rate over total decay rate is the quantum yield of the material, is the

diffusion constant for excitons, and ( )quenchingk x is the quenching rate of excitons.

3.3 Quenching by charge carriers in OLEDs

3.3.1 Effect of the hole injection layer on the drive voltage of OLEDs

The basic OLED device has a bi-layer organic thin-film structure such as

ITO/NPB/Alq/LiF/Al, where indium-tin-oxide (ITO) is the anode and LiF/Al is the

cathode, and NPB(4,4'-bis[N-(1-naphthyl)-N-phenylamino]biphenyl) and Alq (tris(8-

quinolinolato)aluminum) are the hole-transport layer (HTL) and electron-transport

layer (ETL), respectively. Meanwhile, Alq also acts as an EML in the device. It has

been shown that much improved OLED performance can be realized using a

HIL/HTL structure where HIL is the “hole-injection” layer inserted between the

anode and the HTL. For example, with CuPc [62] as the HIL as in CuPc/NPB/Alq

where Alq also functions as the emissive layer, long-lived OLEDs have been obtained.

Another common HIL material is MTDATA(4,4',4''-tris[N-(3-methylphenyl)-N-

phenylamino]triphenylamine) [63], with which enhanced current efficiency and

43

operational stability have been demonstrated. High-efficiency OLEDs have also been

reported in various HIL/HTL configurations [64-68].

To systematically study the effect of the HIL on OLED devices, we have made a

series of OLED devices. Figure 3.2 shows the multilayer OLED structure and the

molecular structures for the HIL (MTDATA), HTL (NPB) and ETL (Alq). The

OLED devices share a common structure of ITO/MTDATA(70-d nm)/NPB(d

nm)/Alq(70 nm)/LiF/Al where the NPB layer thickness (d) varies from 0 nm to 70 nm.

All devices were fabricated and measured in Eastman Kodak Company by my

colleague Kevin Klubek [17].

Figure 3.3 shows the experimental (symbols) and simulated (solid curves) current

density versus voltage (J-V) characteristics for the series of OLED devices. Bilayer

devices with NPB (d=70 nm) and MTDATA (d=0 nm) as the single HTL,

respectively, have a lower drive voltage compared to trilayer devices (d=5 nm, 25 nm,

45 nm and 65 nm). Furthermore, the drive voltage for trilayer devices increases as the

thickness of the NPB layer increases. The trend of the voltage shift was reproduced

very well in the simulation with the input parameters shown in Table 3.1 [18, 69-72].

However, at low current density, the simulated drive voltages are lower than the

experimental ones. The deviation may arise from the overestimation of the hole

mobility of the NPB layer, especially at the low electric field region.

44

Table 3.1. Input parameters used in the simulation of MTDATA/NPB/Alq OLEDs.

MTDATA NPB Alq

HOMO (eV) -5.1 -5.5 -5.7

LUMO (eV) -1.9 -2.4 -3.1

p0 (cm2/(Vs)) 7.0*10

-6 2.0*10

-4 1.5*10

-9

p (cm1/2

/V1/2

) 3.0*10-3

2.0*10-3

3.0*10-3

n0 (cm2/(Vs)) 7.0*10

-8 2.0*10

-6 1.5*10

-7

n (cm1/2

/V1/2

) 3.0*10-3

2.0*10-3

3.0*10-3

energy width (eV) 0.06 0.03 0.06

Figure 3.2. Structure of MTDATA/NPB/Alq OLEDs and materials used. HOMO

energy level: MTDATA (-5.1eV), NPB (-5.5eV), Alq (-5.7eV). d is the variable

thickness of the NPB layer.

45

Figure 3.4 shows the detailed information of the bilayer (left) and trilayer (right)

devices from the simulation including the charge carrier density, electric field and

recombination distribution. In bilayer devices, since the hole mobility in NPB is much

higher than the electron mobility in Alq, the voltage drop in NPB is negligible

comparing to the one in Alq. In trilayer devices, the HOMO level offset between

MTDATA and NPB presents an energy barrier for the hole transport. As a result, the

holes accumulated at the MTDATA/NPB interface will generate a strong electric

field in the NPB layer and increase the device drive voltage. Furthermore, increasing

the NPB layer thickness will increase the voltage drop across the NPB layer and the

Figure 3.3. Experimental (symbols) and simulated (solid curves) J-V curves for

MTDATA/NPB/Alq devices with variable NPB layer thickness.

NPB 70nm

MTDATA 70nm

MTDATA 65nm/ NPB 5nm

MTDATA 45nm/ NPB 25nm

MTDATA 25nm/ NPB 45nm

MTDATA 5nm / NPB 65nm

2 4 6 8 10 12 14 16 18

0.1

1

10

100

Voltage (V)

J (

mA

/cm

2)

46

device drive voltage as well. This explains the voltage shift that we have observed for

trilayer devices in Figure 3.3.

Figure 3.4 also shows the hole density at the NPB/Alq interface in bilayer and

trilayer devices. In trilayer devices, the HOMO offset at MTDATA/NPB interface

effectively block the holes. As a consequence, the hole density at the NPB/Alq

interface is less than that in bilayer devices. The reduced hole density, and hence the

reduced charge carrier quenching, increases the device efficiency as we will see in the

next section.

47

Figure 3.4. The energy level diagram and the distribution of the charge carrier

density, the electric field, and the recombination at 80 mA/cm2 in the bilayer (left

side) and trilayer (right side) devices.

48

3.3.2 Effect of the hole injection layer on the device efficiency of OLEDs

Figure 3.5 shows the experimental (symbols) and simulated (solid curves) device

efficiency of the series of MTDATA/NPB/Alq OLEDs. The low efficiency for

device MTDATA (70 nm)/ Alq(70 nm) has been attributed to the formation of

exciplexes [73, 74], at the MTDATA/Alq interface, where the EL emission quantum

yield is controlled by the exciplex emission with a lower quantum yield than Alq.

The NPB/Alq device has a higher efficiency than the MTDATA/Alq device. This can

be attributed to the lack of exciplex formation, which is consistent with NPB being a

weaker donor compared to MTDATA.

Figure 3.5. Experimental (symbols) and simulated (solid curves) external

quantum efficiency versus current density for the series of MTDATA/NPB/Alq

devices with variable NPB layer thickness.

0 20 40 60 80

J (mA/cm2)

NPB 70nm

MTDATA 70nm

MTDATA 65nm/ NPB 5nm

MTDATA 45nm/ NPB 25nm

MTDATA 25nm/ NPB 45nm

MTDATA 5nm / NPB 65nm

0.0

0.5

1.0

1.5

2.0

2.5

Exte

rnal Q

uantu

m E

ffic

iency (

%)

49

The device efficiency in the tri-layer devices is substantially improved over

NPB/Alq bi-layer device despite the fact that they have in common the NPB/Alq

interface. It has been reported that charge quenching can play a significant role in

determining the current efficiency of EL devices under various device configurations

and drive conditions [75-77]. It has been suggested that exciton quenching at the

NPB/Alq interface due to the accumulation of NPB+ radical cations [78, 79] at the

interface largely determines the current efficiency.

We used the NPB+ radical cation quenching to explain that all trilayer devices

have the same efficiency that is higher than that of bilayer devices. Recombination

taken place near the NPB/Alq interface can be quenched by the accumulated NPB+

radical cations. The simulation results in Figure 3.4 shows that the trilayer structure

can dramatically reduce the hole density at the NPB/Alq interface and hence improve

the efficiency of the device.

To quantitatively simulate the device efficiency, we need to come up with a

correct model to calculate the charge carrier quenching. Forster energy transfer [80-

82] is widely used to describe the charge carrier quenching in OLEDs, where the

exciton energy is transferred to the quenching centers (NPB+ radical cations in this

case). Forster energy transfer rate can be calculated by the following equation

6

0transfer f 6

Rk k

r (3.7)

50

where is the Forster energy transfer rate, is the decay rate of the exciton,

R0 is the Forster distance and r is the distance between the exciton and the quenching

center. Equation 3.7 only calculates the quenching rate due to a single quenching

center, whereas in OLED devices, lots of charge carriers (quenching centers) are

present. In the model, we have divided the organic layer into many mono-layers, each

of which has an infinite area. With the charge carrier density we have simulated for

each monolayer, we can calculate the effective quenching rate of the charge carriers

in the monolayer through surface integration. For example, given a monolayer with

thickness dm, the distance between the monolayer and exciton is d. The charge carrier

density in the monolayer is , and charge carriers are assumed to be uniformly

distributed in the monolayer. Then the effective quenching rate due to the charge

carriers in the monolayer is

6 6

0 f 0transfer f 42 2 60

R πρ k Rk k (ρ )2π

2d( d +r )( ) m

m

dd rdrd

(3.8)

Equation 3.8 demonstrates that the quenching effect of the charge carriers decreases

quickly as the distance between the charge carrier (mono-layer) and the exciton

increases. The total quenching rate is the sum of the quenching rates of all

monolayers.

In addition to the charge carrier quenching, we notice that the device efficiency

in our cells increases as the current density (drive voltage) increases. At first we

proposed that under high current density, the recombination zone shift further into the

51

ETL (Alq) layer and away from the NPB radical cations, which give a higher

efficiency to the device. However, the simulation shows that the recombination zone

doesn’t change much under different current densities. To capture this device

efficiency increasing phenomenon, we refer to the cross interface recombination.

When holes in the NPB layer recombine with electrons in the Alq layer across the

interface, interface excitons are formed. They transform to bulk excitons and emit

photons. The energy mismatch between interface excitons and bulk excitons presents

an energy barrier for the transformation, and hence reduces the efficiency. Under high

current density, the electric field across the NPB/Alq interface is high, which make

the cross interface transport of holes easier. Consequently, a big portion of the holes

transport across the interface and recombine with the electrons on the Alq side, and

form bulk excitons directly. Our simulation shows that, the ratio of bulk

recombination over total recombination increases as the current density increases,

which would result in a higher efficiency in devices at higher current density.

Meanwhile, the charge carrier densities ( e.g. NPB+ radical cations) increases as the

current density increases, which causes a more severe quenching effect and reduces

the efficiency at high current density. Figure 3.5 shows the simulated (solid curves)

quantum efficiency that include both the charge carrier quenching effect and the cross

interface recombination effect. The Forster distance R0=3 nm is used as input

parameter for all devices to control the quenching effect.

To scale the efficiency and fit the external quantum efficiency of the device, we

have assumed that the Alq quantum yield is 0.4. Only singlet excitons (25% of total

52

excitons generated) contribute to the radiative emission, and the light out coupling

efficiency is 25%. For MTDATA/Alq bilayer device, the large HOMO level

difference between MTDATA (-5.1 eV) and Alq (-5.7 eV) will effectively block the

holes at the interface. Cross interface recombination and exciplex emission dominate

even at higher current density. The exciplex emission is assumed to have a lower

quantum yield. The deviation between the simulated and experimental results is

probably due to the overestimation of the cross interface recombination rate,

especially at low current density region. A deeper understanding of the cross interface

recombination is an area of interest for future study.

3.4 Mixed host OLEDs

After investigating the bilayer and trilayer OLEDs, we are interested to expand

our model to investigate the mixed host OLEDs. Mixed host OLEDs are devices that

have hole transport material and electron transport material mixed together in the

EML. They provide a new approach to improving device lifetime [10, 83-86].

3.4.1 Parameter assumptions for the mixed host layer

To model the mixed host OLEDs, the first problem that arises is how to set the

parameters for the mixed host layer. The parameters that need to be defined for the

materials are energy levels and mobilities. Usually mixed host contains two type of

materials, hole transport and electron transport materials. The design is very similar to

the bulk heterojunction structure in organic photovoltaic devices (OPVs) where donor

and acceptor materials are mixed together. Therefore, we apply the energy level

53

treatment for bulk heterojunction OPV [87] to the mixed host OLEDs. The HOMO of

the mixed host is assumed to be the HOMO of the hole transport material, while the

LUMO of the mixed host is assumed to be the LUMO of the electron transport

material. For the bipolar mobilities in the mixed host, there is no unified model to

describe. However, it is reasonable to assume that the mobilities should lie between

the mobilities of each material. Some experimental results [88] of the mixed host

mobilities show a power dependence on the concentration of the components.

Therefore, we use the following equations to calculate the mobilities for the mixed

host layer

HTM HTMC 1 C

mix,p HTM,p ETM,pμ μ μ (3.9)

HTM HTMC 1 C

mix,n HTM,n ETM,nμ μ μ (3.10)

where is the hole (electron) mobility of the mixed host layer, is

the hole (electron) mobility of the hole transport material and is the hole

(electron) mobility of the electron transport material. After defining the energy level

and mobilities of the mixed host layer, we can simulate the device performance and

analyze the numerical results.

3.4.2 Recombination zone in mixed host OLEDs

To investigate the performance of the mixed host OLEDs, we use two

hypothetical materials in the simulation. Table 3.2 lists the energy level and

mobilities of the hole transport material (HTM) and the electron transport material

54

(ETM) that we used for simulation. We set the hole transport property of the HTM at

a same level as the electron transport property of the ETM. The HOMO offset is the

same as the LUMO offset between the two materials.

Table 3.2. Input parameters used in the simulation of mixed host OLEDs.

By changing the percentage of the HTM in the mixed host from 20% to 80%,

we can generate mixed host material with different transport property as shown in

Table 3.2. All mixed OLEDs in the simulation shares the same structure of

ITO/HTM(20 nm)/Mixed host(100 nm)/ ETM(20 nm)/LiF/Al. Three uniformly

mixed host OLEDs are simulated with various hole concentration in the mixed layer

(20%, 50% and 80%). In additional to the uniformly mixed host OLEDs, we also

simulated a mixed host device with graded concentration [11, 84, 89, 90]. The HTM

concentration decrease from 80% (near the HTM/Mixed host interface) to 20% (near

HIM HTM ETM Mixed host

CHTM=20%

Mixed host

CHTM=50%

Mixed host

CHTM=80%

HOMO (eV) -5.2 -5.5 -5.7 -5.5 -5.5 -5.5

LUMO (eV) -1.9 -2.9 -3.1 -3.1 -3.1 -3.1

p0 (cm2/(Vs) 2.0*10

-4 2.0*10

-4 2.0*10

-6 5.023*10

-6 2.0*10

-5 7.962 *10

-5

p0 (cm1/2

/V1/2

) 2.0*10-3

2.0*10-3

2.0*10-3

2.0*10-3

2.0*10-3 2.0*10

-3

n0 (cm2/(Vs) 2.0*10

-6 2.0*10

-6 2.0*10

-4 7.962*10

-5 2.0*10

-5 5.023*10

-6

n0 (cm1/2

/V1/2

) 2.0*10-3

2.0*10-3

2.0*10-3

2.0*10-3

2.0*10-3

2.0*10-3

energy width (eV) 0 0 0 0 0 0

55

the Mixed host/ETM interface). For comparison, we also simulate the bilayer and

trilayer devices with the same material. A hole injection material (HIM) that has

higher HOMO level than the HTM is used. The mobility of the HIM is set to be the

same as the mobility of the HTM as shown in Table 3.2. Detailed device structures

are listed below,

Device Bilayer: ITO/ HTM (70 nm)/ ETM (70 nm)/LiF/Al

Device Trilayer: ITO/HIM (20 nm)/HTM (50 nm)/ ETM (70 nm)/LiF/Al

Device UM-H20: ITO/HTM (20 nm)/Mixed host (20% HTM, 100nm)/ETM (20 nm)/

LiF/Al

Device UM-H50: ITO/HTM (20 nm)/Mixed host (50% HTM, 100nm)/ETM (20 nm)/

LiF/Al

Device UM-H80: ITO/HTM (20 nm)/Mixed host (80% HTM, 100nm)/ETM (20 nm)/

LiF/Al

Device Graded: ITO/HTM (20 nm)/Mixed host (80% HTM, 30 nm)/Mixed host (50%

HTM, 40 nm)/Mixed host (20% HTM, 30 nm)/ETM (20 nm)/ LiF/Al

56

Figure 3.6 shows the simulated J-V curve of the OLED devices, and the drive

voltage can be summarized as, Device Bilayer < Device Graded < Device UM-20 =

Device UM-H80 < Device UM-50 < Device Trilayer.

Device Bilayer has the lowest drive voltage because the HTM and the ETM have

good transport property for holes and electrons, respectively. However, at low current

density, the drive voltage of Device Bilayer is higher than the mixed host OLEDs due

to the sharp energy barrier at the HTM/ETM interface. All mixed host OLEDs blur

the interface barriers with the mixed material.

By breaking the EML into three different zones with different HTM

concentration, Device Graded can utilize the transport advantage of holes on the

Figure 3.6. Simulated J-V curves for bilayer, trilayer, uniformly mixed host

and graded mixed host OLED devices.

2 3 4 5 6 7 8

0.1

1

10

100

J(m

A/c

m2)

Voltage (V)

Bilayer

Trilayer

UM-H20

UM-H50

UM-H80

Graded

57

HTM side and electron transport on the ETM side. Therefore, the drive voltage is still

low.

Device UM-H20 has higher drive voltage, since the mixed host with 20% HTM

(which means 80% ETM) favors the electron transport. However, the electron

mobility in the mixed host is lower than the pure ETM, which causes a higher voltage

drop in the EML. Device UM-H80 and Device UM-H20 have exactly the same J-V

curve since they are symmetric devices. Device UM-H80 has good hole transport

property, while Device UM-20 has good electron transport property.

Device UM-H50 has balanced hole and electron mobilities, but both of them are

relatively low. Thus, Device UM-H50 requires a higher drive voltage than Device

UM-H20 and Device UM-H80. Device Trilayer has the highest drive voltage due to

the same HIL effect that we discussed in previous sections. The holes accumulate at

the HIM/HTM interface cause a strong electric field in the HTM and increase the

overall drive voltage.

Figure 3.7 shows the charge carrier density and recombination distribution of all

devices. Bilayer and trilayer devices have similar characteristics as we saw in the

previous section. Device UM-20 has better electron transport property, which pushes

the electrons to the HTM/Mixed host interface, where most recombination takes place.

On contrast, Device UM-H80, with a good hole transport property, pushes the holes

and recombination zone to the Mixed host/ETM interface. The recombination zone in

Device UM-20 and Device UM-H80 is wider than in Device Bilayer, but most of the

58

recombination still takes place near the interface, which might be quenched by the

accumulated charge carriers. Device UM-50 has a balanced hole and electron

mobility in the mixed layer, which effectively balances the hole and electron transport.

Both type of charge carriers are widely distributed in the mixed layer. A well

expanded recombination zone is also obtained.

Device Graded also generate a relative balanced charge carrier distribution and

recombination zone with the help of graded structure. Device Graded has multiple

interfaces. Each of them can block certain amount of charge carriers and effectively

distribute the charge carriers across the whole graded mixed layer. We can introduce

more steps/interfaces to magnify the effect and further expand the recombination

zone.

59

Figure 3.7. The charge carrier density and recombination zone distribution in

various OLED devices: (a) Bilayer; (b) Trilayer; (c) UM-H20; (d) UM-H50; (e)

UM-H80 and (f) Graded.

(a)

(b)

(c)

(d)

(e)

(f)

60

3.4.3 Quenching by charge carriers in mixed host OLEDs

It is worthwhile to apply the charge carrier quenching effect in our previous

discussion to the mixed host OLEDs to see which device structure provides the best

quantum efficiency. Here I ignore the cross interface recombination effect and focus

only on the charge carrier quenching.

Figure 3.8 shows the efficiency curve for all devices assuming only the radical

cations can quench the emission. Device Bilayer has the lowest quantum efficiency

since the narrow recombination zone is close to the hole accumulation region, as

evident from Figure 3.7. Devices UM-20, UM-50 and UM-80 improve the efficiency

by expanding the recombination zone. Device UM-80 has the highest efficiency

among the three uniformly mixed host devices, because the recombination takes place

near the electron accumulation zone and we have assumed that only radical cations

can quench the emission, not anions. The quantum efficiency of Device Graded is

very close the Device UM-50 due to the similar distribution of charge carriers and

recombination zone. It is interesting to see that Device Trilayer still has the highest

quantum efficiency due to the effectively separation of recombination zone and the

hole accumulation region. All efficiency curves decreases as the current density

increases because of the increased charge carrier density under higher current

densities, and hence the increased charge carrier quenching.

After comparing the performance of all different devices, we can conclude that

high efficiency can only be achieved when quenching effect is deminished. If we can

61

identify the quenching mechanism or quenching species, we should design the device

structure to effectively move the recombination zone away from the quenching

centers. Consequently, we can obtain a high device efficiency. The ratio between the

hole transport material and the electron transport material in the mixed host layer

plays a significant role in determining the device performance.

It has been said that expanded recombination zone can improve the lifetime [10,

83, 84] of OLED devices. A uniform mixed host with balanced hole and electron

mobilities can do the best job based on our simulation. However, in the real world,

the mobilities of the hole transport material and the electron transport material vary a

lot. Thus, finding the best mixed ratio that provides the balanced hole and electron

mobilities could be difficult, the simulation also proves that a graded junction

Figure 3.8. Simulated quantum efficiency of bilayer, trilayer, uniformly mixed host

and graded mixed host OLED devices.

62

structure can effectively expand the recombination zone, which could be a good

solution to apply in the device especially when mobilities of the materials are

unknown.

3.5 Photon extraction

In OLEDs, photons are generated through the radiative decay of excitons.

However, the photon extraction from the devices is complicated due to the cavity

effect. Most OLED devices have multilayer structures. The thickness of organic

layers and electrodes (10 nm-200 nm) is less than the wavelength of visible light (380

nm-780 nm). Meanwhile, one electrode is usually designed to be reflective to

improve the light collection on the other side. Thus, multiple reflections and

transmissions that take place in the device will modulate the light emission of OLED

devices. Meanwhile, the cavity structure also alters the radiative decay rate of the

excitons [91-94].

The cavity effect in OLED causes the angular dependence of the emission

characteristics. The luminance drop and color shift at high viewing angles are

important especially when applying the OLED technology in display panels. By

tuning the device structure, people can improve the color quality and increase the out-

coupling efficiency. Neyts et al. [92, 93] have proposed the cavity model for OLED

devices, where they quantitatively calculated the dipole emission in cavity structure.

We have rebuilt the model and improved the simulation efficiency with the help of

Gaussian quadrature technique [95]. The model will be first briefly introduced.

63

Several common cavity effects in OLEDs are tested including the light mode

decomposition in OLEDs, comparison between weak cavity and strong cavity OLEDs,

and the layer thickness dependence in OLEDs.

3.5.1 Cavity model for OLEDs

OLED emission comes from the random oriented dipoles in a multilayer cavity

structure. Since any specific oriented dipole can be decomposed into the vertically

oriented dipole (dipole moment perpendicular to the OLED surface) and horizontally

oriented dipole (dipole moment parallel to the OLED surface), we first investigate the

emission characteristics from these two special oriented dipoles. The emission light

with different polarizations, Transverse Electric (TE) wave and Transverse Magnetic

(TM) wave, need to be calculated separately since the transmission and reflection

coefficient depends on the light polarization.

The typical multilayer cavity structure of OLED devices is shown in Figure 3.9,

assuming the direction perpendicular to the OLED surface is the z-axis. The dipole

(exciton) locates in the emissive layer, with optical refractive index ne and thickness

de. The half-infinite media surrounding the OLED structure has optical refractive

indexes of n+ and n-, respectively. Several intermediate layers stand between the

emissive layer and outside media and introduce multiple interfaces. The reflection

and transmission of light at the internal interfaces are the main reason of the cavity

effect in OLED. As we will see later, the cavity effect of OLED depends on the exact

64

location of the dipole in the emissive layer, which is defined by the distance between

the dipole and the nearest interfaces z+ and z-, respectively, where z++z-=de.

Dipoles can emit light into all directions, and the direction of the light can be

identified with the wave vector ke, as shown in Figure 3.9, or more conveniently, with

the in-plane projection value κ. The advantages of using κ to represent the light

Figure 3.9. The schematic multilayer structure for the cavity model, where dipole

located in the ne layer with thickness de. The distance between the dipole and the

nearest interfaces on both sides are z+ and z- respectively. The half-infinite media

surrounding the OLED structure has optical refractive indexes n+ and n-, respectively.

One or more intermediate layers stand between the emissive layer and the outside

media. Any light with specific direction can be identified with the wave vector ke,

which can be decomposed into the in-plane projection κ and the z-axis projection kz,e.

65

direction includes (1) κ remains constant at all layers, and (2) κ can be used to

identify the modes of the light as we will see in the next section. The z-axis projection

of the wave vector kze is calculated as

2 2

z,e ek k κ (3.11)

With the optical refractive index of each layer, we can calculate the reflection and

transmission coefficient at each interface for the TM wave and TE wave, assuming

light inject from the ith

layer with the refractive index ni to the i+1th

layer with the

refractive index ni+1, the reflection and transmission coefficient is calculated as

z,i z,i 1

2 2TM i i 1i i 1

z,i z,i 1

2 2

i i 1

k k

n nr

k k

n n

(3.12)

z,i

2TM ii i 1

z,i z,i 1

2 2

i i 1

k2

nt

k k

n n

(3.13)

z,i z,i 1TE

i i 1

z,i z,i 1

k kr

k k

(3.14)

z,iTE

i i 1

z,i z,i 1

2 kt

k k

(3.15)

66

where TM

i i 1r and TE

i i 1r are the reflection coefficient for the TM wave and TE wave,

respectively, TM

i i 1t and TE

i i 1t are the transmission coefficient at the interface, z,ik and

z,i+1k are the z-axis projection of wave vectors in the ith

layer and the i+1th

layer,

respectively. Having the thickness of each layer, we can calculate the effective

reflection and transmission coefficient, from the ith

layer to the i+2th

layer, assuming

that the i+1th

layer stands between them with the thickness di+1,

i i 1 i 1 i 2 z,i 1 i 1

i i 2

i i 1 i 1 i 2 z,i 1 i 1

r r exp(2*j*k *d )r

1 r *r exp(2*j*k *d )

(3.16)

i i 1 i 1 i 2 z,i 1 i 1

i i 2

i i 1 i 1 i 2 z,i 1 i 1

t *t *exp( j*k *d )t

1 r *r *exp(2*j*k *d )

(3.17)

where j is the imaginary unit. By applying the equation above iteratively, we can

calculate the effective reflection and transmission coefficient from the emissive layer

to the outside media: , . Then, we can calculate the light

emission (both TM wave and TE wave) of the two basic dipoles, vertically oriented

and horizontally oriented. Neyes et al. [92, 93] has calculated the power density K of

the light emission from the basic dipoles as below

TM TM2

TM

3 TM

e z,e

1 α (1 α )3 κK *Re[ * ]

4 k *k 1 α

(3.18)

TEK 0 (3.19)

67

TM TM

z,eTM

3 TM

e

1 α (1 α )k3K *Re[ * ]

8 k 1 α

(3.20)

TE TE

TE

TE

e z,e

1 α (1 α )3 1K *Re[ * ]

8 k *k 1 α

(3.21)

where

TM,TE TM,TE

e z,eα r *exp(2*j*k *z ) (3.22)

TM,TE TM,TE

e z,eα r *exp(2*j*k *z ) (3.23)

TM,TE TM,TE TM,TEα α *α (3.24)

TMK and TEK are the power densities of the TM wave and TE wave emitted from the

dipole with a dipole moment perpendicular to the OLED plane, TMK and

TEK are the

power densities of the TM wave and the TE wave emitted from the dipole with a

dipole moment parallel to the OLED plane. The power density of out-coupling light

into the n+ media is calculated as,

2TM 22

2 z,TM TM e, , e3 2TM2

e z,e z,e

1 α kn3 κ 1K * * * * t * *

4 k *k 2 n k1 αT

(3.25)

TE

,T,K 0 (3.26)

68

2TM 2

2z,e z,TM TM e,T, e3 2TM2

e z,e

1 αk kn3 1K * * * * t * *

8 k 2 n k1 α

(3.27)

2TE

2 z,TE TE

,T, eTE2e z,e z,e

1 α k3 1K * * * t *

8 k *k k1 α

(3.28)

The angular distribution of the light power density in the n+ media can be

calculated as

2

,T

k *cos(θ)P θ *K (κ k *sin(θ))

π

(3.29)

where θ is the angle between the light direction and the z-axis in the n+ media, and k+

is the absolute value of the wave vector in the n+ media.

All the above equations are used to calculate the light for the two basic

oriented dipoles, considering the contribution of a set of random oriented incoherent

dipoles, the power density of light emitted is calculated as

TM TM TM

RND

1 2K K K

3 3 (3.30)

TE TE TE

RND

1 2K K K

3 3 (3.31)

The out-coupling light into the n+ media from random oriented dipoles follows

similar equations

69

TM TM TM

+,T,RND , , , ,

1 2K K K

3 3T T (3.32)

TE TE TE

+,T,RND , , , ,

1 2K K K

3 3T T (3.33)

Exciton can decay radiatively or non-radiatively in the device. Assuming that

the radiative decay rate and non-radiative decay rate in infinite media are and ,

respectively, the radiative efficiency in infinite media is . When dipole is

located in a micro cavity structure, the cavity can affect the radiative decay rate of the

exciton but not the non-radiative one [91-94]. The new radiative efficiency in the

cavity is calculated as

rrad

r nr

F*Γη

F*Γ Γ

(3.34)

2

0

F K κ dκ

(3.35)

where K(κ) is the power density we calculated from equation 3.25 -3.31. The out-

coupling efficiency into the n+ media is calculated as

k2

,T0

2

0

K κ dκ

K κ dκoutcoupling efficiency

(3.36)

where k+ is the absolute value of the wave vector in the n+ media. Notice that we only

integrate the out-coupling light from κ=0 to κ=k+, since any light with κ> k+ is not

able to escape into the outside media due to the total internal reflection. Using

70

Gaussian quadrature technique and consider the integration in the complex plane can

effectively reduce the simulation time [95].

For most bottom emission OLED devices, light emitted from the organic layer

enters into the glass substrate first, then into the air. The thickness of substrate is

around 1 mm, which is several orders larger than the wavelength of visible light (380

nm-780 nm). Thus, the substrate should not be considered as part of the micro cavity.

Therefore, in a cavity model, we treat the substrate as an infinite medium and

calculate the power density of light into the substrate first. To calculate the light out-

coupling from the substrate into the air, we consider the multiple reflections of light

at the substrate/air interface and at the substrate/OLED interface as shown in Figure

3.10. Meanwhile, the direction of light in substrate and air is different due to the

different optical refractive indexes. Thus, the angular distribution of the light output

in air is calculated as

2

air air substrate airair substrate 2

substrate substrate substrate air substrate OLED

n cos(θ ) TP θ P θ * * *

n cos(θ ) 1 R *R

(3.37)

air air substrate substraten *sin θ n *sin θ (3.38)

where and is the angle between the light and the z-axis (normal to

OLED plane) in the air and substrate, respectively, and are the

angular distribution of light power density in the air and substrate, and

are the optical refractive indexes of the air and substrate, and

are the power reflection and transmission coefficient for light injection

71

from the substrate into the air, is the power reflection coefficient for

light injection from the substrate into the OLED stack, which effectively considers

the multi-layer structure of the OLED already.

Both the reflection and transmission coefficient are wavelength dependent.

Thus to calculate the OLED electroluminescence spectrum, we need to calculate the

cavity effect for each wavelength, and multiply the results with the

photoluminescence spectrum of the emitters. Furthermore, the light emission from the

excitons located in the cavity depends on the exciton location. Excitons at different

Figure 3.10. The illustration of the light out-coupling from the substrate into the

air. and are the power transmission and reflection

coefficient for light injection from the substrate into the air. is

the power reflection coefficient for light injection from the substrate into the OLED

stack, which effectively considers the multi-layer structure of the OLED already.

72

locations are assumed to be incoherently correlated with each other. Thus in the

model, we calculate the light output from all emitters/excitons separately and sum the

light output from all emitters together.

3.5.2 Mode decomposition of the dipole emission in OLEDs

The out-coupling efficiency of typical OLED is still low (around 20% to 30%

without any out-coupling technique) due to the high optical refractive index of

organic material (around 1.7 to 1.9). Only part of the light emitted from OLEDs can

come into the air, and most light is trapped in the device due to the total internal

reflection at the organic/glass interface or glass/air interface. As we mentioned earlier,

it is convenient to use the in-plane projection κ of the wave vector to identify the

modes. Light emitted from the dipole in OLED devices can be decomposed into

different modes as below [91, 96, 97], where is the wavelength, and ,

and are the optical refractive indexes of the air, substrate and

organic materials.

(1)

, air mode, the light can be extracted into air if not absorbed by

any layer.

(2)

, substrate mode, the light can reach the substrate

but not the air.

(3)

, waveguide mode, the light is trapped in the

organic layers.

73

(4)

, surface plasmon mode, the light is coupled into the

surface plasmon on electrodes.

To calculate the contribution of each mode in the cavity model, we integrate the

power density of light emission over the specific range for the mode, and divide it by

the total contribution of all modes. For example, the weight of the air mode can be

calculated as

air 2

2π*n

0

2

λ

0

( )mode

( )

K dweight of air

K d

(3.39)

To investigate the mode decomposition, we have simulated a series of OLED

devices and calculated the mode contributions. The inset in Figure 3.11 shows the

structure of a typical bottom emission OLED device: glass substrate/ITO (100

nm)/organic layer (x nm)/ Al (100 nm), with organic layer thickness varying from 20

nm to 600 nm. One emitter is located at the center of the organic layer with a

photoluminescence (PL) spectrum shown in Figure 3.11. The PL spectrum is

generated from a Gaussian distribution with a peak wavelength at 620 nm and full

width at half maximum (FWHM) of 80 nm. The optical refractive index of the

organic material is assumed to be 1.7 at all wavelengths, while the refractive indexes

of the glass, ITO and Al are taken from literature [98, 99]. We have monitored the

contribution of each mode in devices where the organic layer thickness ranges from

74

20 nm to 600 nm. Non-radiatve rate of the devices is assumed to be 0 for all devices

discussed in this thesis.

Figure 3.11. PL spectrum of the emitter to be used in the simulation. The PL

spectrum is generated from a Gaussian distribution with a peak wavelength at

620nm and FWHM of 80nm. Inset: The device structure of the OLED devices in

the simulation, with glass substrate/ITO (100 nm)/Organic Layer (20 nm to 600

nm)/Al (100 nm), and the dipole emitter locates in the middle of the organic layer.

75

Figure 3.12. (a) Optical refractive indexes of the glass, ITO, organic material and Al

used in the simulation. The optical refractive index of the organic material is assumed

to be 1.7 at all wavelengths. The indexes of other materials are retrieved from

literature. (b) The contribution of each mode from the dipole emission in OLED

devices with organic layer thickness varying from 20 nm to 600 nm. Inset: the device

structure of the OLED devices. Emitters locate in the middle of the organic layer,

with a PL spectrum shown in Figure 3.11.

(a)

(b)

76

Clearly, the contribution of each mode changes as the organic layer thickness

increases. The plasmon mode drops as the organic layer thickness increases, the main

reason is that the dipole is located in the middle of organic layer, increasing the

organic layer thickness can effectively move the dipole away from the electrodes,

which suppressing the plasmon mode. However, the reduction of plasmon mode is

compensated by the increase of waveguide mode instead of the air mode [91].

The air mode contribution, which determines the out-coupling efficiency,

increases and decreases as the organic layer thickness increases. Therefore, the cavity

structure of OLED devices should be tuned carefully to improve the out-coupling

efficiency. However, the maximum air mode contribution obtained by varying the

organic layer thickness is only about 30%. The majority of the light is still trapped in

the device, about 20% in substrate mode and 50% in the waveguide and surface

plasmon mode. Tuning the cavity structure cannot extract the trapped modes in the

substrate or organic layers. Techniques that can effectively change the light injection

angle at the substrate/air interface or at the OLED/substrate interface have been

introduced to extract the trapped modes [100-107].

3.5.3 Weak cavity versus strong cavity OLEDs

OLED devices can be categorized by the cavity structure they use. Weak cavity

(also called non-cavity) OLED devices have one reflective electrode and one

transparent electrode. While strong cavity (also called micro-cavity) OLED devices

has one reflective electrode and one semi-transparent electrode. The semi-transparent

77

electrode can be a thin film metal or a set of quarter wavelength stacks (Distributed

Bragg Reflectors) in contact with a transparent electrode [108-111]. The semi-

transparent electrode increases the reflection of light back into the OLEDs and hence

increases the interference effect. The strong cavity structure is typically used in top

emission OLEDs. Because of the high luminance and good color purity at normal

direction, strong cavity OLEDs are widely used in hand-held display panels such as

cell phones. However, the strong cavity OLEDs usually have a strong luminance drop

and large color shift at high viewing angles, which impedes their application in large

screen displays, such as TVs.

78

OLED emission has angular dependence due to the cavity effect, thus to compare

the emission characteristics between weak cavity OLEDs and strong cavity OLEDs,

we either compare the light from the same viewing angle or compare the total light

output over all angles. Figure 3.13 shows the device structure of weak cavity and

strong cavity OLEDs we have simulated. Thin film Ag is used as the semi-transparent

electrode in the strong cavity device. A capping layer is included to reduce the index

Figure 3.13. The device structure of the weak cavity and strong cavity OLEDs

used in the simulation. The HTL and ETL thickness of the devices is tuned to

improve the luminance at the normal direction. Red emitters locate in the

middle of the EML in both devices with a PL spectrum shown in Figure 3.11.

All organic layers including the HTL, EML, ETL and capping layer are

assumed to have an optical refractive index of 1.7 at all wavelengths.

79

mismatch between the Ag and the air and hence improve the out-coupling efficiency

[112-115].

Figure 3.14(a) compares the electroluminescence (EL) spectrum at normal

direction between the weak cavity and strong cavity OLEDs. At normal direction, the

strong cavity device has higher luminance and narrower spectrum (which means

higher color purity) than the weak cavity device. An improved light output at the

normal direction make strong cavity OLEDs attractive for hand-held displays.

Figure 3.14(b) compares the total light output from the two devices. The

simulation shows that the total light output from the strong cavity device is close to

that from the weak cavity device, and the total light output spectrum is blue shifted

relative to the weak cavity device. The behavior can be explained by the strong

luminance drop and color shift at high viewing angles, as evident from Figure 3.15.

80

Figure 3.14. The EL spectrum of the weak cavity and strong cavity OLEDs

(a) at normal direction and (b) over all angles.

(b)

(a)

81

Figure 3.15 shows the simulated EL spectrum at normal direction (0 degrees) and

at high viewing angles (60 degrees) for the weak cavity and strong cavity OLEDs. A

strong luminance drop and spectrum shift (color shift) has been observed in the strong

cavity device.

Figure 3.15. EL spectrum at normal direction (0 degrees) and at high viewing

angles (60 degrees) for the (a) weak cavity and (b) strong cavity OLEDs.

(a)

(b)

82

So far we have investigated the emission characteristics in the weak cavity and

strong cavity OLEDs with fixed layer thickness. The thickness of organic layers is the

key parameter for the cavity structure, and the OLED emission characteristics heavily

depend on the layer thickness. To investigate the thickness dependence effect, we use

the same device structure shown in Figure 3.13 and vary the thickness of the HTL and

ETL from 0 nm to 400 nm respectively. Figure 3.16 shows the thickness dependence

of the normal direction luminance for (a) the weak cavity and (b) the strong cavity

devices. The luminance varies a lot as the layer thickness changes. In weak cavity

devices, the ETL thickness, which controls the distance between emitter and

reflective electrode, has more effect on the light output. On contrast, for strong cavity

devices, both HTL and ETL thickness have strong effect on the light output. The

diagonal shape of the plot means that the total thickness of organic layers controls the

emission of strong cavity OLEDs.

83

Figure 3.16. The luminance at normal direction from the (a) weak cavity and (b)

strong cavity OLEDs with HTL thickness and ETL thickness varied from 0 nm to

400 nm, respectively.

(a)

(b)

84

Chapter 4

Simulation of Organic Photovoltaic Devices

While OLED devices transform electricity into light, organic photovoltaic

devices (also called organic solar cells) convert the power of sunlight into electricity.

As we mentioned earlier, the device structure and the mechanisms of these two types

of devices are closely related to each other. The experience and knowledge we have

learned from the simulation of OLED devices establish a solid foundation for

building a model for organic photovoltaic devices.

4.1 Photon absorption in OPV devices

The optical part in OPV devices is easier to model than the cavity effect in OLED

devices because we are only interested in the absorption of the device. Most of the

time, we only focus on the light injection at normal angle. The injection light is

assumed to be plane waves which is much easier to handle than dipole emission.

Because of the low mobility of the organic materials and the strong electric field

requirement for the exciton dissociation, OPV devices are usually very thin (50nm to

1um). This nature brings an interference effect in the device and hence affects the

absorption.

Since excitons generated in OPV devices can diffuse to the donor/acceptor (DA)

interface and dissociate, the location of the exciton generation has impact on the

85

device performance. Therefore, to correctly model the OPV devices, we need to

calculate the photon absorption at each position. In other words, we need to calculate

the photon absorption at each monolayer after dividing the device into discrete

monolayers.

The absorption in OPV devices is modeled by the transfer matrix theory [36, 37].

Figure 4.1 shows the diagram of the photon absorption in OPV devices. Because of

the reflectance of the back cathode, the optical electric field at any position can be

decomposed into two components; one component propagating in the positive

direction and the other in the negative position. Given that and

are the

components of the optical electric field propagating in the positive and negative

directions in the jth

monolayer, and

are the corresponding parameters for the

Figure 4.1. Diagram of the photon absorption in OPV devices. The device structure is

glass substrate/ITO anode/Donor layer/Acceptor layer/LiF/Al cathode. and

are

the components of the optical electric field propagating in the positive and negative

directions in the jth

monolayer, and

are the corresponding parameters for the

adjacent kth

monolayer.

86

adjacent kth

monolayer. The optical electric fields in these two adjacent layers are

connected by the following matrix equation

2exp( * ) 0

/ 2 / 2

/ 2 / 2 20 exp( * )

j j

j k j j k jj k

j k j j k jj j j k

n di

n n n n n nE E

n n n n n nE n d Ei

(4.1)

where jn and kn are the complex refractive index in the j

th monolayer and the k

th

monolayer, respectively, the imaginary part of jn controls the absorption strength,

is the wavelength, and jd is the thickness of the jth

monolayer . After calculating all

the transfer matrixes in the devices and multiplying them together, we can achieve a

final matrix. The final matrix connects the optical electric fields of the 0th

monolayer

(layer in front of the OPV device) and the N+1th

monolayer (layer at the back of the

OPV device).

0 1

0 1

N

N

E ES

E E

(4.2)

Given that is related to the incident light, and that

because no light

come back after it get out of the device, we can figure out the other two variables ( 0E

and 1NE

) by solving the matrix equation. With the optical electric fields in the 0th

layer, we can calculate the optical electric fields for each monolayer by applying the

transfer matrix iteratively. The average absorbed light power for each monolayer is

calculated as

87

204

 2

j j

j j j

c k nQ E E

(4.3)

where jQ is the absorption rate in jth

monolayer, c is the speed of light in the vacuum,

0 is the vacuum dielectric constant,

jn and jk re the real part and imaginary part of

the optical refractive index in the jth

monolayer, and

are the optical electric

fields of the positive and negative propagation light in the jth

monolayer. Similar to

the cavity model for OLEDs, the absorption in OPV devices is also wavelength

dependent. To calculate the total absorbed photons in OPV devices, we need the

injection light power density (usually AM 1.5G with integrated intensity 100

mW/cm2 [116]) and the refractive indexes of materials at all wavelengths.

4.2 Exciton diffusion and dissociation in OPV devices

Excitons, rather than free charge carriers, are formed after the photon absorption

in OPV devices because of the low dielectric constant and strong exciton binding

energy. Excitons can diffuse to the donor acceptor (DA) interface and dissociate

efficiently. The exciton diffusion in OPV devices is the same as the exciton diffusion

in the OLED simulation. For planar hetero-junction OPV devices, the dissociation of

excitons in bulk is negligible compared to the exciton dissociation at the DA interface.

We can solve the diffusion equation first and calculate the exciton flux arriving at the

DA interface. This flux is the upper limit of the current density we can achieve for

planar heterojunction OPV devices, since only excitons that could arrive at the DA

88

interface can dissociate efficiently. Thus for planar heterojunction OPV devices, the

exciton diffusion and dissociation can be treated as two consecutive steps.

Figure 4.2 demonstrates the exciton dissociation process at the DA interface. To

calculate the exciton dissociation, we assume that the charge transfer (CT) process

takes place immediately after excitons arrive at the DA interface [87, 117, 118]. The

new CT state can either dissociate into free charge carriers with a field dependent rate

kd or decay to the ground state with a constant rate kf. The dissociation rate kd of CT

state is calculated through the Braun-Onsager model [ 87, 119-121]

2 3

p n

d 3

0 r 0 r 0 0

e*(μ μ )3 e b bk * *exp (1 b )

4πr ε ε 4πε ε r kT 3 18

(4.4)

Figure 4.2. Diagram of the exciton dissociation process at the DA interface. kd(r0,E)

is the dissociation rate depending on the initial separation distance r0 and the electric

field E. kf is the decay rate of the charge transfer state.

89

where , E is the electric field,

and

are the hole and

electron mobilities, respectively, is the charge carrier initial separation distance in

the CT state, and are the relative and vacuum dielectric constant, e is the

elementary charge, is the Boltzmann constant, and T is the temperature. The model

was originated by Onsager for analysis of geminate recombination of an ion pair in

an isotropic medium [118, 119]. Braun later refined the model by introducinga charge

transfer state with a finite lifetime [121]. The model has been used by Blom et al. [87]

for calculating the photogeneration of charges in bulk hetorojunction OPV cells. The

two key parameters that control the dissociation efficiency are and kf: a large

can help to dissociate the CT state while a large kf can make the CT state more likely

to decay rather than dissociate.

4.3 Charge carriers transport in OPV devices

The charge carrier transport in OPV devices is similar to its counterpart in

OLEDs. The only difference is that, OPV devices run at an applied voltage lower than

the built-in potential. The injection of charge carriers in the OLEDs part cannot be

used here. One way to treat the Metal/Organic interface is to assume an Ohmic

boundary condition at the interface [87]. Thus the contacts are in thermodynamic

equilibrium and we can set the hole (electron) density at the hole (electron) contact

constant.

90

4.4 Effect of the HOMO offset on planar heterojunction OPV

devices

After integrating all the parts discussed above, we are able to simulate OPV

devices and compare simulation results with experimental results. The energy offset

[122-124] (∆EHOMO or ∆ELUMO) between donor and acceptor material at the DA

interface has always been considered as the driving force that dissociates the excitons

in OPV devices. To systematically investigate the effect of energy offset on OPV

devices, my colleague Dr. Minlu Zhang at the University of Rochester has made a

series of devices [125]. The device structure is ITO(90 nm)/MoOx(2 nm)/donor(3

nm)/C60(40 nm)/BPhen (4,7-diphenyl-1,10-phenanthroline )(8 nm)/LiF/Al where a set

of donor materials with variable HOMO levels [60, 126-130] are used as shown in

Table 4.1. The HOMO level of C60 is -6.2eV according to the literature [131].

In the donor/C60 OPV devices, photons are absorbed in the C60 layer where

excitons are generated. Generated excitons can diffuse to the donor/C60 interface and

dissociate. C60 is the acceptor [131] where electrons prefer to stay, while the donor

material provides a lower energy level to attract holes at the DA interface. Therefore,

excitons that arrive at the interface can transform to charge transfer complex with

bound holes in the donor side and bound electrons in the C60 side. Another way to

look at the charge transfer process is that, the bound holes step up from the HOMO of

C60 to the HOMO of the donor. Therefore, the HOMO offset between the donor and

91

C60, rather than the LUMO offset, is the driving force for the charge transfer process

in these devices.

Since the C60 layer thickness is constant, the photon absorption in C60 and the

exciton flux reaching the DA interface is the same for all devices. The good electron

transport property of C60 [132] and the extremely small thickness of the donor layer

guarantee that the transport of charge carriers is not a limiting factor in these devices.

Thus the device performance is solely controlled by the dissociation process at the

DA interface, which ultimately depends on the HOMO offset.

Figure 4.3 shows the experimental and simulated current density versus voltage

(J-V) characteristics of the series of OPV devices with different donor material. The

HOMO level of the donor material varies from -5.9 eV (CBP) to -5.1 eV (MTDATA).

Clearly, varying the HOMO level of the donor will change the ∆EHOMO and the device

performance. A good fit between experimental and simulated J-V curves has been

achieved. The input parameters we choose for different devices in the simulation will

be discussed in details below.

92

Table 4.1. List of donors, HOMO levels and CAS names used in donor (3 nm)/C60

(40 nm) OPV devices.

Donor HOMO

(eV)

CAS Names

MTDATA -5.1 4,4',4" -Tris(N-3-methylphenyl-N-phenyl-amino)triphenylamine

NDP -5.3 6,13-dihydro-6,13-di-2-naphthalenyl-Dibenzo[b,i]phenazine

NPB -5.4 N’ ,N’-Di-[(1-naphthyl)-N’,N’-diphenyl]-1,1’-biphenyl)-4,4’-diamine

TAPC -5.5 1,1-bis-4-bis4-methyl-phenyl-amino-phenyl-cyclohexane

TCTA -5.7 4,4,4-trisN-carbazolyl-triphenyl amine

ADN -5.8 9,10-di(2-naphthyl) anthracene

CBP -5.9 4,4'-bis(carbazol-9-yl)-biphenyl

Figure 4.3. Experimental (symbols) and simulation results (solid curves) of the

current-voltage characteristics of ITO/MoOx(2 nm)/donor(3 nm)/C60(40 nm)/

BPhen(8 nm)/LiF(1 nm)/Al OPV cells.

CBP

ADN

TCTA

TAPC

NPB

NDP

MTDATA

-1.0 -0.5 0.0 0.5 1.0

-5

-4

-3

-2

-1

0

1

2

3

4

5

J (

mA

/cm

2)

Voltage(V)

93

Table 4.2 Experimental photovoltaic parameters (Jsc, Voc and FF) for donor (3 nm)

/C60 (40 nm) OPV devices. Vbi, r0 and kf are the built-in potential, the initial

separation distance and the decay rate of the CT state used in the simulation.

Donor Jsc

(mA/cm2)

Voc

(V)

FF

(%)

Vbi

(V)

r0

(nm)

kf

(1/s)

MTDATA 2.39 0.26 51.59 0.6 1.34 5.50×107

NDP 3.42 0.44 59.74 0.8 1.22 1.60×107

NPB 3.53 0.60 62.00 0.9 1.16 1.00×107

TAPC 3.85 0.65 61.25 1.0 1.10 5.00×106

TCTA 3.05 0.89 49.39 1.2 0.98 5.00×106

ADN 2.53 0.97 38.21 1.3 0.92 5.00×106

CBP 1.88 0.99 36.08 1.4 0.86 5.00×106

94

Table 4.2 lists the experimental photovoltaic parameters. Figure 4.4 shows both

experimental and simulated photovoltaic parameters for all devices. Both Jsc and FF

increases and then decreases as the ∆EHOMO increases from 0.3eV (CBP/C60 devices)

to 1.1eV (MTDATA/C60 devices). The Voc drops almost linearly as the ∆EHOMO

increases, because Voc and Vbi are determined by the difference between the HOMO

Figure 4.4. Experimental (solid squares) and simulated (open squares) results of Jsc,

FF and Voc vs ∆EHOMO for donor (3 nm)/C60 (40 nm) OPV devices.

95

of the donor and the LUMO of C60 [122]. Vbi is taken as the energy gap between the

HOMO of the donor and the LUMO of the C60 in the simulation

60bi C donorV LUMO HOMO (4.5)

The Braun-Onsager model [35, 120, 121] is used to model the CT state dissociation at

the DA interface. The initial separation distance r0 and the CT decay rate kf are the

key parameters controlling the dissociation process. Table 4.2 and Figure 4.5 show

the values of two fitting parameters, r0 and kf, for each device. We assume that the

initial separation distance r0 in the Braun-Onsager model increases linearly as the

∆EHOMO increases. We consider that the energy loss (∆EHOMO) during the CT process

at the DA interface can help separate the bounded charge carriers [124]. For devices

with ∆EHOMO > 0.7eV, we use a large kf to offset the large r0 to get reduced Jsc and FF

to fit the experimental data. It is possible that because of the strong DA interaction at

the interface with large ∆EHOMO, the charge-transfer states are more effectively

bounded, which results in an increased decay rate to the ground states.

Ohmic boundary conditions are assumed at the contacts, where the hole density

at the anode/donor contact and the electron density at the cathode/acceptor contact are

both set as . At 100 mW/cm2 illumination, an exciton flux of

at the DA interface (equivalent to a current density of 4.5

mA/cm2) is used for all the devices with a 40 nm layer of C60. For electron mobility

of C60, we use , . We

have found that the variation of the hole mobility in the donor layer does not

96

significantly change the simulation results as long as it is within the range of

to and the donor layer is kept sufficiently

thin (~ 3 nm). This finding further supports the argument that the dissociation process

at different donor/C60 interfaces explains the device performance difference, rather

than the transport property of different donor material. However, when the donor

layer thickness increases, the hole transport may become a factor that limits the

device performance, as we will see in the next section.

Figure 4.5. The initial separation distance r0 and the charge transfer state decay

rate kf used in the simulation of donor (3 nm)/C60 (40 nm) OPV devices.

97

4.5 Effect of the donor layer thickness on planar heterojunction

OPV devices

In the previous section, we kept the donor layer thickness constant and varied the

donor material to see the device performance, where the hole transport is not a

limiting factor because the donor layer is thin (3 nm). However, when donor layer

become thick, the charge carrier transport may limit the device performance due to

the relatively low mobility of organic materials.

Figure 4.6 (a) shows the experimental and simulated J-V curve of a set of OPV

devices with a device structure of ITO/MoOx(2 nm)/NPB(x nm)/C60(40 nm)/BPhen(8

nm)/LiF/Al where the NPB layer thickness varies from 3 nm to 100 nm. The devices

are made by my colleague Dr. Minlu Zhang at the University of Rochester [133]. The

current density drops quickly as the NPB layer thickness increases. Table 4.3 lists the

photovoltaic parameters observed from the devices. Increasing the NPB layer

thickness can quickly reduce the level of Jsc and FF. This effect is also attributed to

hole transport limitations in the NPB layer.

98

Table 4.3. Experimental photovoltaic parameters (Jsc, Voc and FF) for ITO/MoOx(2

nm)/NPB(x nm)/C60(40 nm)/BPhen(8 nm)/LiF/Al cells with NPB layer thickness

varied from 3 nm to 100 nm. Vbi, r0 and kf are the built-in potential, the initial

separation distance and the decay rate of the charge transfer state used in the

simulation.

Notice that the Voc for devices with thin NPB layer (3 nm and 5 nm) is about 0.6

V, which suddenly increases to about 0.9 V for devices with thick NPB layer (10 nm

to 100 nm) and remain unchanged afterwards. The dramatic change of Voc cannot be

explained by the hole transport behavior in the donor layer, since the hole transport

properties between 5 nm thick NPB and 10 nm thick NPB is not very different. One

possible explanation is that when NPB layer is thin (3 nm or 5 nm), the MoOx that

diffuses into the NPB layer effectively dope the NPB layer. The MoOx doped NPB

NPB Jsc Voc FF Vbi r0 kf

(nm) (mA/cm2) (V) (%) (V) (nm) (1/s)

3 3.56 0.61 62 1 1.16 1.0 × 107

5 3.21 0.62 55 1 1.16 1.0 × 107

10 2.93 0.89 37 1.3 1.04 1.0 × 107

30 1.95 0.91 27 1.3 1.04 1.0 × 107

50 1.33 0.90 20 1.3 1.04 1.0 × 107

100 0.141 0.90 17 1.3 1.04 1.0 × 107

99

layer may provide a HOMO level above the pure NPB layer at the DA interface,

which reduces the Vbi and Voc of the device [133]. Meanwhile, the effective HOMO

level of the donor at the DA interface provides a larger HOMO offset and a good

driving force for exciton dissociation as we discussed in the previous section, which

has been confirmed by the higher FF in devices with thin NPB layer. When the NPB

layer is thick enough, the diffusing MoOx cannot reach the NPB/C60 interface and

will not affect the Vbi for the device. Built-in potential (Vbi) and r0 are two fitting

parameters adjusted to explain the two-stage Voc values. For cells with a thin NPB

layer, the Vbi and r0 are set as 1.0 V and 1.16 nm, respectively. For cells with a thick

NPB layer, the Vbi and r0 are set as 1.3 V and 1.04 nm, respectively. The CT state

decay rate kf is set at 1.0 107s

-1 for all devices.

Ohmic boundary conditions are still assumed at the contacts, where the hole

density at the anode/donor contact and the electron density at the cathode/acceptor

contact are both set as . At 100 mW/cm2 illumination, an exciton

flux of at the DA interface (equivalent to a current density of

4.5 mA/cm2) is used for all the devices. For the electron mobility of C60, we use

, . In order to fit the

experimental curve, I have to set the NPB layer hole mobility as low as

, . The hole mobility of NPB used in

the simulation is much lower than the values generally reported from the time of

flight experiments [70]; however, people have reported much lower hole mobility

100

values for ultra thin NPB film [134]. The low hole mobility and large thickness of the

NPB layer will impede the extraction of holes generated at the NPB/C60 interface.

The holes accumulated in the NPB layer will dramatically reduce the electric field at

the NPB/C60 interface and consequently reduce the charge carrier generation

efficiency. The S-shape J-V curve for devices with large NPB layer thickness, and the

photovoltaic parameters, are reproduced well in the simulation as shown in Figure

4.6(a) and Figure 4.6(b).

101

Figure 4.6. (a) Experimental (symbols) and simulated (solid curves) J-V curves of

ITO/MoOx(2 nm)/NPB(x nm)/C60(40 nm)/BPhen(8 nm)/LiF/Al OPV devices with

the NPB layer thickness varied from 3 nm to 100 nm. (b) Experimental (solid

square) and simulated (open circle) values of Jsc, FF and Voc for NPB/C60 OPV

devices.

(a)

(b)

0.0

0.2

0.4

0.6

0.8

0 20 40 60 80 100

0.6

0.8

1.0

0

1

2

3

4

FF

Vo

c(V

)

NPB (nm)

Experiment

Simulation

Jsc(m

A/c

m2)

-1.0 -0.5 0.0 0.5 1.0 1.5-5

-4

-3

-2

-1

0

1

2

3

4

5

NPB 3nm

NPB 5nm

NPB 10nm

NPB 30nm

NPB 50nm

NPB 100nmJ (

mA

/cm

2)

Voltage(V)

102

4.6 Effect of the donor concentration on bulk heterojunction

OPV devices

In planar heterojunction OPV devices, only excitons arriving at the DA interface

can dissociate efficiently and those that cannot reach the DA interface are wasted.

Thus, the exciton diffusion length limits the photo current. Bulk hetorojunction (BHJ)

[135] OPV devices mix the donor and acceptor materials together in the active layer

and effectively solve the problem. DA interfaces that are uniformly distributed across

the active layer act as dissociation centers for excitons.

For BHJ OPV devices, photon absorption, exciton dissociation, charge carrier

transport, and recombination occur everywhere in the BHJ layer. Thus, all of them

need to be incorporated together to achieve a valid simulation. Exciton diffusion is

ignored in the model assuming that all excitons are transformed to the charge transfer

states immediately after generation [87, 118].

In the active layer, excitons (CT states) act as the intermediate states between

photons and charge carriers. Photon absorption generates excitons. The following

exciton dissociation produces free charge carriers, and the recombination of charge

carriers can form excitons again. These dynamic transformations can be expressed in

the following equations [87]

f d

dXG k k *X R

dt (4.6)

103

d

1* k *XnJn

Rt e x

(4.7)

where X is the exciton density, G is the photon absorption rate, is the decay

and dissociation rate of the CT states, R is the recombination rate of charge carriers, n

is the electron density, is the electron current, and e is the elementary charge. The

change of hole density in the layer is calculated through a similar equation.

To study the bulk heterojunction OPV devices, my colleague Dr. Minlu Zhang at

the University of Rochester has made a series of devices with the structure: ITO (90

nm)/MoOx(2 nm)/BHJ (TAPC:C60) (40 nm)/BPhen(8 nm)/Al(100 nm) [136]. The

BHJ layer is a mixed film of TAPC [137] (donor) and C60 (acceptor) with TAPC

concentration varying from 1.2% to 50%. The concentration of TAPC is indicated in

volume fraction. Figure 4.7 shows the (a) experimental and (b) simulated J-V curves

of bulk heterojunction devices with different TAPC concentration. Table 4.4 and

Figure 4.8 present experimental and simulated photovoltaic parameters for the series

of devices. The device performance changes a lot at different TAPC concentration.

104

-1.0 -0.5 0.0 0.5 1.0

-6

-4

-2

0

2

J (

mA

/cm

2)

Voltage (V)

1.2%

2.5%

5%

12.5%

25%

33%

50%

(a) Experiment

-1.0 -0.5 0.0 0.5 1.0

-6

-4

-2

0

2

J (

mA

/cm

2)

Voltage (V)

(b) Simulation

Figure 4.7. (a) Experimental and (b) simulated current-voltage characteristics of

ITO/MoOx/TAPC:C60/BPhen/Al OPV devices with various TAPC concentration

from 1.2% to 50%.

105

Table 4.4. Experimental Photovoltaic parameters of TAPC:C60 cells (AM1.5 @

100mW/cm2) and calculated hole and electron mobilities.

TAPC

(%)

Jsc

(mA/cm2)

Voc

(V)

FF

ηPCE

(%)

µha

(cm2/Vs)

µea

(cm2/Vs)

1.2 2.88 1.02 0.36 1.06 9.09×10-8

4.91×10-3

2.5 3.90 0.96 0.40 1.50 1.54×10-7

4.66×10-3

5.0 5.94 0.91 0.52 2.81 1.32×10-6

4.19×10-3

12.5 5.25 0.88 0.54 2.50 1.01×10-5

2.98×10-3

25.0 4.12 0.80 0.47 1.55 2.94×10-5

1.53×10-3

33.0 3.27 0.78 0.43 1.10 4.11×10-5

9.11×10-4

50.0 1.45 0.74 0.29 0.31 6.24×10-5

2.15×10-4

a. Calculated hole and electron mobilities under an electric field of 1.0105 V/cm by using

Bässler’s model according to Equation 3 and 4. Parameters used in the calculation: µp0 =

1.010-4

cm2V

-1s

-1, Γp = 3.7, p = 1.010

-4 cm

1/2V

-1/2 and µn0 = 5.010

-3 cm

2V

-1s

-1, Γn = 14,

n = 1.010-4

cm1/2

V-1/2

.

106

Voc drops from 1.03 V to 0.74 V as TAPC concentration increases from 1.2% to

50%. For conventional BHJ cells, the built-in potential and Voc are generally

governed by the HOMO-LUMO gap [138, 139] between the donor and acceptor

material in the active layer. The observed Voc dependence on TAPC concentration in

the TAPC:C60 cells suggests that the effective HOMO level of TAPC is not fixed, but

Figure 4.8. Photovoltaic characteristics of TAPC:C60 cells under AM1.5 at 100

mW/cm2 illumination: (a) Experimental data of Voc versus TAPC concentration;

(b) Experimental and simulated data of Jsc versus TAPC concentration; (c)

Experimental and simulated data of FF versus TAPC concentration.

107

shifted to a higher energy with increasing TAPC concentration in C60. The built-in

potential Vbi used in the simulation is based on linear extrapolation of the

experimental Voc values. Both Jsc and FF increase with TAPC concentration at first

and drop their levels afterwards.

To correctly model the effect of the TAPC concentration on device performance,

we need to identity what parameters in the devices will change at different TAPC

concentration, and more importantly, how to model the change.

Note that TAPC is a practically non-absorbing material in the visible spectrum.

Thus, the photon absorption is primarily contributed by the C60 component. The

absorption coefficients of the mixed layer are approximated as

60( ) ( ) (1 )TAPC TAPCc C c (4.8)

where ( )TAPCc and α(C60) are absorption coefficient of the mixed material and C60

respectively at wavelength , and TAPCc is the TAPC concentration in volume

percentage. Since the photon absorption is calculated according to the transfer matrix

theory where the complex refractive index (n,k) of the mixed material is required.

Intuitively, I assume that

60( ) ( ) (1 )TAPC TAPCk c k C c (4.9)

where TAPCc is the imaginary part of the optical refractive index for the mixed

material, and is the value for C60 [140, 141]. The real part of the refractive

108

index of the mixed material is set the same as the C60 which is relatively constant at

all wavelengths. AM 1.5 spectrum with integrated intensity 100mW/cm2 is assumed

for the light source [116].

Furthermore, one important parameter that affects the dissociation efficiency is

the dielectric constant of the material. The concentration-dependent dielectric

constant of the TAPC:C60 mixed film r TAPCc is calculated as [142]

1 1

33 3

60( (1 ) )r TAPC TAPC r TAPC rc c TAPC c C (4.10)

where, εr(TAPC) = 3.5 [143] and εr(C60) = 4.5 [144] are the dielectric constants of

TAPC and C60, respectively. Increase in the TAPC concentration will lower the

dielectric constant of the material and increases the Coulomb force between the

charge pairs in CT states. Consequently, the dissociation of CT states becomes more

difficult [145].

Further, we assume that holes and electrons in the TAPC:C60 layer are transported

through TAPC and C60, respectively. The concentration dependence of the hole and

electron mobility can be calculated according to Bässler’s model[43]:

2

0 p

/3 1/3 1/6  Γ 1 ( )p p TAPC TAPC p TAPCE c exp c exp c E (4.11)

2 1

3 3

1

60 n(1 ) (1 ) 1  ( (1 ) )TAPC TAPCn APCn Tn E exp exc cpc E

(4.12)

109

where µp0 (µn0) are the hole (electron) mobility at zero electric field in pure TAPC and

C60, respectively, Γp (Γn) are the electronic wave function overlap factors, γp (γn) are

the electric field dependence of hole (electron) mobility, and E is the electric field.

Table 4.4 shows the mobility parameters we used for the simulation as well as the

calculated electron and hole mobility of the bulk layer under an electric field of 1×105

V/cm.

For the TAPC:C60 cells, we assume that, a donor concentration of ~ 1% by the

volume of TAPC in C60 provides sufficient dissociation sites for complete capture

excitons generated by the light absorption in C60. Exciton dissociation is via a charge

transfer state with an initial separation distance of r0 = 1.08 nm and decay rate of kf =

5.0106 s

-1 for all devices.

In Figure 4.8 (b) and Figure 4.8(c) the simulated results for Jsc and FF are plotted

against TAPC concentration for all TAPC:C60 cells along with the experimental

results. The agreement between the simulated and experimental results appears to be

fairly good. At very low TAPC concentration (≤ 2.5%), the hole mobility in

TAPC:C60 is so low (< 10-7

cm2V

-1s

-1) that the photocurrent is more limited by the

field-dependent hole transport than by exciton dissociation in this layer. The hole

mobility in TAPC:C60 BHJ layer rapidly increases with TAPC concentration. At

about 5 - 10% TAPC level, the hole mobility is sufficiently high so that the hole-

transport is no longer the limiting factor for the photogeneration of charge carriers in

the BHJ. Above this concentration range, the reduction in the dielectric constant and

110

the electron mobility of the TAPC:C60 layer becomes significant, resulting in reduced

exciton dissociation efficiency and hence adversely affecting the Jsc and FF. The

electron mobility of the TAPC:C60 decreases by about an order of magnitude as the

TAPC concentration is increased from 5% to 50%. This similar trend has also been

found in polymeric BHJs such as PPV:PCBM [118].

4.7 Simulation of tandem OPV devices

The thickness of OPV devices is usually small, because (1) thick layer reduces

the electric field and hence reduces the dissociation efficiency, and (2) thick layer

impedes the extraction of charge carriers due to the low mobility of organic materials.

However, a thin layer will limit the absorption in the OPV devices and reduce the

photo current. One way to solve the dilemma is to apply a tandem cell structure [146-

150].

My colleague Dr. Minlu Zhang at the University of Rochester has made 2-stack

tandem cells as shown in Figure 4.9 (a) [151]. The tandem cell composed of two

subcells connected serially by an inter-connecting layer (ICL) with structure: ITO(90

nm)/MoOx (2 nm)/ BHJ (x nm)/ Bphen(6 nm)/Ag(0.3 nm)[152]/1,4,5,8,9,11-

hexaazatriphenylene hexacarbonitrile (HAT-CN) (1.5 nm)[153]/ MoOx (2 nm)/ BHJ

(y nm)/ Bphen(6 nm)/Al (100nm). The BHJ layer is a mixed film TAPC (5% by

volume) doped in C60 layer as we used in the previous section. The thickness of the

BHJ in the bottom and top subcells are varied to achieve optimal results.

111

Figure 4.9 (a) Schematic structure of the 2-stack tandem OPV devices. (b) Contour

plot of simulated current density of 2-stack tandem cells with various BHJ layer

thickness in top and bottom subcells. Star points are the experimental results with

the current density indicated accordingly.

(a)

(b)

112

Modeling tandem OPV devices is straightforward especially when we have a

model for single stack BHJ OPV devices. Using the transfer matrix theory [36, 154]

and including the ICL layer in the structure, we can calculate the photon absorption in

both subcells. The excition dissociation and charge carrier transport in each subcell is

calculated respectively as in single stack devices. The built-in voltage of the tandem

OPV cells is doubled due to the tandem structure. Subcells are connected with the

ICL, which we treated as an intermediate electrode and a recombination center as

well. Electrons generated in the bottom subcell and holes generated in the top subcell

can meet at the ICL and cancel each other, leaving only the net charges in the ICL.

The net charges in ICL can adjust the electric field in subcells and balance the photo

current generated from them [155]. This self-adjusting mechanism is necessary for

the tandem OPV devices to achieve a stationary state – same level of current from

both subcells. For example, if the bottom subcell generates more photo current than

the top subcell, there would be more electrons arriving at the interface from the

bottom subcell than holes coming from the top subcell. After the recombination, the

excess of electrons accumulate at the ICL, which increase the electric field as well as

the dissociation efficiency in top cell. Simultaneously, the electrons in the ICL will

reduce the electric field in bottom cell and hence reduce the dissociation efficiency.

With this self-adjusting mechanism, the photo current from both subcells will

eventually converge to the same value.

113

The BHJ layer thickness of both subcells can affect the photon absorption,

exciton dissociation and charge carriers transport. Therefore the tandem OPV device

performance is sensitive to the thickness of both layers.

Figure 4.9(b) shows the simulated Jsc for 2-stack tandem OPV devices with

various BHJ layer thickness in bottom and top subcells. Jsc is maximized when the

BHJ layer thickness in both cells is around 70 nm. The experimental data of tandem

cells are shown as the start points with the values indicated accordingly, which quite

match the simulation results.

114

Chapter 5

Summary and Future Work

In this thesis, we have successfully built a model for simulating the conversion

process from charge carriers to photons in OLED devices.

Charge carrier injection, transport, and recombination are investigated first in

single layer devices. With Ohmic injection contacts, the mobility mismatch between

electrons and holes determines the recombination zone in the device. A balanced

electron and hole transport is necessary to achieve high recombination efficiency as

well as expanded recombination zone in single layer OLEDs.

Trilayer devices with MTDATA/NPB/Alq structure where MTDATA is the hole

injection layer (HIL) have shown improved current efficiency compared to bilayer

NPB/Alq devices. Charge carrier quenching is introduced to explain the effect of the

hole injection layer. Forster energy transfer rate is used to calculate the quenching

rate in our model. The simulation has quantitatively reproduced the voltage shift and

the efficiency improvement in devices with such a hole injection layer. The increase

of current efficiency at high current density is attributed to the reduced cross interface

recombination. For trilayer devices, the drive voltage is reduced as the NPB layer

thickness is decreased, while the current efficiency is almost independent of the NPB

115

layer thickness. The power efficiency is therefore optimized by mimimizing the NPB

layer thickness.

Improved current efficiency in OLEDs has been observed almost universally

with various fluorescent and phosphorescent emitters with the insertion of a suitable

HIL in the device structure. We also observed charge carrier quenching plays a

critical role in limiting the efficiency of most OLED devices and should be avoided or

controlled in the future device design.

In mixed host OLEDs, the ratio between the hole transport material and the

electron transport material controls the mobilities of the mixed host. Balanced

mobilities between holes and electrons are preferred to achieve an expanded

recombination zone. Graded mixed host structures provide a means to achieve an

expanded recombination zone without introducing sharp boundaries between the

transport materials. However, high efficiency in OLEDs with either uniformly mixed

or graded mixed is not necessarily guaranteed if these device structures fail to exclude

quenching centers from the recombination zone.

The cavity effect in OLEDs is important for enhancing device performance. The

out-coupling efficiency of OLEDs is particularly sensitive to the layer configuration

and thickness. In weak cavity OLEDs, we found that the distance between the emitter

and the reflective cathode dominates the cavity effect, while in strong cavity OLEDs

we have to take into consideration the total thickness of all organic layers in

optimizaing the device performance.

116

We simulated the performance of various OPV devices. The Braun-Onsager

model was used to simulate the dissociation of excitons at donor-acceptor interfaces.

The initial separation distance r0 and the decay rate of charge transfer states kf are the

key parameters in determining the exciton dissociation efficiency according to the

model.

We showed that improved exciton dissociation efficiency can be achieved at a

donor-acceptor interface with a high HOMO offset, which effectively separates the

charge-transfer states formed at the interface. However, we also found that exciton

dissocation can be impeded in a donor-acceptor interface with excessive HOMO

offset because of the increasing decay rate of the CT states.

We showed that charge carrier transport in the hole transport layer such as NPB

can severely limit the efficiency in planar heterojunction OPV devices due to the fact

that extraction of holes from the low-mobility hole-transport layer is impeded.

Consequently hole accumulation in the hole-transport layer causes the reduction of

electric field and exciton dissociation efficiency at the donor-acceptor interface.

We also observed that hole transport can limit the efficiency of bulk

heterojucntion (BHJ) TAPC:C60 OPV devices when the donor (TAPC) concentration

is lower than 5%. Increasing the TAPC concentration above 5% would reduce the

photon absorption, which reduces the photo current, and lower the dielectric constant

and electron mobility of the BHJ layer, both of which reduce the exciton dissociation

efficiency.

117

We also simulated two-stack tandem OPV devices and optimized the power

conversion efficiency by varying the layer thickness of both subcells.

In summary, we have successfully modelled a series of OLED and OPV devices

and found quantitative agreement between experimental results and theoretical

simulation. Baed on our simulation work, we have established a better understanding

of the underlying mechanism and identified the limiting factors of the device

performance. Furthermore, we are able to optimize the device performance through

sensitivity analysis of the device layer structures. Our device simulation model can be

used to provide directions for future OLED and OPV device research.

The performance of OLEDs and OPVs heavily depends on the device structure

and material used. New quenching centers as well as new quenching mechanism

should be included in the model once they are identified. In our current simulation,

we did not consider the exciton-exciton annihilation processes in OLED devices.

These processes could play a critical role in high current density and may severely

lower and quantum efficiency and the ultimate brightness achievable in OLED

devices. The energy transfer between host and dopant molecules should be

investigated and simulated in future studies, including emissive layers with multiple

dopants, such as those used in white OLEDs.

It should be noted that the success of simulation is dependent on the validity of

input parameters. Accurate measurements of material parameters, especially the

carrier mobility and its dependence on the layer thickness and the electric field, are

118

extremely important in achieving meaningful simulation results. Charge transport and

recombination in mixed materials as well as the transport and recombination across

the heterojunction interfaces should be investigated in more detail to achieve a better

understanding of these processes and their effect on the device performance.

For the BHJ OPV model, we assume that the BHJ layer is consisted of uniformly

mixed donor and acceptor materials and ignore the possiblility of phase separation. It

is possible to expand the current model to include the effect of phase separation on

the device performance. This expanded model should be applicable to polymer BHJ

OPV devices. Currently, the efficiency of OPV devices is still considerably lower

than that of the inorganic solar cells largely because of the low dielectric constant and

mobility of organic materials. Materials research should be directed at producing

organic semiconductors with a higher dielectric constant and carrier mobility.

Another topic of interest is to improve the absorption of OPV through optical or

geometrical designs.

119

References

1. C. W. Tang, S. A. Vanslyke, "Organic Electroluminescent Diodes", Applied

Physics Letters, 51, 913 (1987).

2. L. S. Hung, C. W. Tang, M. G. Mason, "Enhanced electron injection in organic

electroluminescence devices using an Al/LiF electrode", Applied Physics

Letters, 70, 152 (1997).

3. C. W. Tang, S. A. Vanslyke, C. H. Chen, "Electroluminescence of Doped

Organic Thin-Films", Journal of Applied Physics, 65, 3610 (1989).

4. H. Jiang, Y. Zhou, B. S. Ooi, Y. Chen, T. Wee, Y. L. Lam, J. Huang, S. Liu,

"Improvement of organic light-emitting diodes performance by the insertion

of a Si3N4 layer", Thin Solid Films, 363, 25 (2000).

5. D. Liu, C. G. Zhen, X. S. Wang, D. C. Zou, B. W. Zhang, Y. Cao, "Enhancement

in brightness and efficiency of organic electroluminescent device using novel

N,N-di(9-ethylcarbaz-3-yl)-3-methylaniline as hole injecting and transporting

material", Synthetic Metals, 146, 85 (2004).

6. Y. Shirota, Y. Kuwabara, H. Inada, T. Wakimoto, H. Nakada, Y. Yonemoto, S.

Kawami, K. Imai, "Multilayered organic electroluminescent device using a

novel starburst molecule, 4, 4 ', 4 ''-tris (3-methylphenylphenylamino)

triphenylamine, as a hole transport material", Applied Physics Letters, 65, 807

(1994).

120

7. S. A. VanSlyke, C. H. Chen, C. W. Tang, "Organic electroluminescent devices

with improved stability", Applied Physics Letters, 69, 2160 (1996).

8. L. S. Hung, C. H. Chen, "Recent progress of molecular organic

electroluminescent materials and devices", Materials Science & Engineering

R-Reports, 39, 143 (2002).

9. B. W. D'Andrade, M. E. Thompson, S. R. Forrest, "Controlling exciton diffusion

in multilayer white phosphorescent organic light emitting devices", Advanced

Materials, 14, 147 (2002).

10. V. E. Choong, S. Shi, J. Curless, C. L. Shieh, H. C. Lee, F. So, J. Shen, J. Yang,

"Organic light-emitting diodes with a bipolar transport layer", Applied Physics

Letters, 75, 172 (1999).

11. D. G. Ma, C. S. Lee, S. T. Lee, L. S. Hung, "Improved efficiency by a graded

emissive region in organic light-emitting diodes", Applied Physics Letters, 80,

3641 (2002).

12. F. W. Guo, D. G. Ma, "White organic light-emitting diodes based on tandem

structures", Applied Physics Letters, 87, 173510 (2005).

13. G. Schwartz, M. Pfeiffer, S. Reineke, K. Walzer, K. Leo, "Harvesting triplet

excitons from fluorescent blue emitters in white organic light-emitting

diodes", Advanced Materials, 19, 3672 (2007).

14. M. Strukelj, R. H. Jordan, A. Dodabalapur, "Organic multilayer white light

emitting diodes", Journal of the American Chemical Society, 118, 1213 (1996).

121

15. C. W. Tang, "Two-layer organic photovoltaic cell", Applied Physics Letters, 48,

183 (1986).

16. G. Yu, J. Gao, J. C. Hummelen, F. Wudl, A. J. Heeger, "Polymer Photovoltaic

Cells - Enhanced Efficiencies Via a Network of Internal Donor-Acceptor

Heterojunctions", Science, 270, 1789 (1995).

17. H. Wang, K. P. Klubek, C. W. Tang, "Current efficiency in organic light-

emitting diodes with a hole-injection layer", Applied Physics Letters, 93,

093306 (2008).

18. J. Staudigel, M. Stossel, F. Steuber, J. Simmerer, "A quantitative numerical

model of multilayer vapor-deposited organic light emitting diodes", Journal

of Applied Physics, 86, 3895 (1999).

19. G. G. Malliaras, J. C. Scott, "Numerical simulations of the electrical

characteristics and the efficiencies of single-layer organic light emitting

diodes", Journal of Applied Physics, 85, 7426 (1999).

20. B. K. Crone, I. H. Campbell, P. S. Davids, D. L. Smith, C. J. Neef, J. P. Ferraris,

"Device physics of single layer organic light-emitting diodes", Journal of

Applied Physics, 86, 5767 (1999).

21. P. S. Davids, I. H. Campbell, D. L. Smith, "Device model for single carrier

organic diodes", Journal of Applied Physics, 82, 6319 (1997).

122

22. S. J. Konezny, D. L. Smith, M. E. Galvin, L. J. Rothberg, "Modeling the

influence of charge traps on single-layer organic light-emitting diode

efficiency", Journal of Applied Physics, 99, 064509 (2006).

23. H. Houili, E. Tutis, H. Lutjens, M. N. Bussac, L. Zuppiroli, "MOLED: Simulation

of multilayer organic light emitting diodes", Computer Physics

Communications, 156, 108 (2003).

24. B. Masenelli, E. Tutis, M. N. Bussac, L. Zuppiroli, "Numerical model for

injection and transport in multilayers OLEDs", Synthetic Metals, 122, 141

(2001).

25. E. Tutis, M. N. Bussac, B. Masenelli, M. Carrard, L. Zuppiroli, "Numerical

model for organic light-emitting diodes", Journal of Applied Physics, 89, 430

(2001).

26. B. Ruhstaller, T. Beierlein, H. Riel, S. Karg, J. C. Scott, W. Riess, "Simulating

electronic and optical processes in multilayer organic light-emitting devices",

IEEE Journal of Selected Topics in Quantum Electronics, 9, 723 (2003).

27. B. Ruhstaller, S. A. Carter, S. Barth, H. Riel, W. Riess, J. C. Scott, "Transient

and steady-state behavior of space charges in multilayer organic light-

emitting diodes", Journal of Applied Physics, 89, 4575 (2001).

28. C. H. Hsiao, Y. H. Chen, T. C. Lin, C. C. Hsiao, J. H. Lee, "Recombination zone in

mixed-host organic light-emitting devices", Applied Physics Letters, 89,

163511 (2006).

123

29. C. C. Lee, M. Y. Chang, P. T. Huang, Y. C. Chen, Y. Chang, S. W. Liu, "Electrical

and optical simulation of organic light-emitting devices with fluorescent

dopant in the emitting layer", Journal of Applied Physics, 101, 114501 (2007).

30. C. C. Lee, M. Y. Chang, Y. D. Jong, T. W. Huang, C. S. Chu, Y. Chang,

"Numerical simulation of electrical and optical characteristics of multilayer

organic tight-emitting devices", Japanese Journal of Applied Physics Part 1-

Regular Papers Short Notes & Review Papers, 43, 7560 (2004).

31. C. C. Lee, Y. D. Jong, P. T. Huang, Y. C. Chen, P. J. Hu, Y. Chang, "Numerical

simulation of electrical model for organic light-emitting devices with

fluorescent dopant in the emitting layer", Japanese Journal of Applied Physics

Part 1-Regular Papers Brief Communications & Review Papers, 44, 8147

(2005).

32. C. C. Wu, C. L. Lin, P. Y. Hsieh, H. H. Chiang, "Methodology for optimizing

viewing characteristics of top-emitting organic light-emitting devices",

Applied Physics Letters, 84, 3966 (2004).

33. P. W. M. Blom, M. J. M. de Jong, C. Liedenbaum, "Device physics of polymer

light-emitting diodes", Polymers for Advanced Technologies, 9, 390 (1998).

34. L. J. A. Koster, V. D. Mihailetchi, P. W. M. Blom, "Ultimate efficiency of

polymer/fullerene bulk heterojunction solar cells", Applied Physics Letters, 88,

093511 (2006).

124

35. L. J. A. Koster, E. C. P. Smits, V. D. Mihailetchi, P. W. M. Blom, "Device model

for the operation of polymer/fullerene bulk heterojunction solar cells",

Physical Review B, 72, 085205 (2005).

36. L. A. A. Pettersson, L. S. Roman, O. Inganas, "Modeling photocurrent action

spectra of photovoltaic devices based on organic thin films", Journal of

Applied Physics, 86, 487 (1999).

37. P. Peumans, A. Yakimov, S. R. Forrest, "Small molecular weight organic thin-

film photodetectors and solar cells", Journal of Applied Physics, 93, 3693

(2003).

38. J. A. Barker, C. M. Ramsdale, N. C. Greenham, "Modeling the current-voltage

characteristics of bilayer polymer photovoltaic devices", Physical Review B,

67, 075205 (2003).

39. W. Schottky, "The influence of the structural effects, especially the Thomson

graphic quality, on the electron emission of metals.", Physikalische Zeitschrift,

15, 872 (1914).

40. I. H. Campbell, P. S. Davids, D. L. Smith, N. N. Barashkov, J. P. Ferraris, "The

Schottky energy barrier dependence of charge injection in organic light-

emitting diodes", Applied Physics Letters, 72, 1863 (1998).

41. B. K. Crone, I. H. Campbell, P. S. Davids, D. L. Smith, "Charge injection and

transport in single-layer organic light-emitting diodes", Applied Physics

Letters, 73, 3162 (1998).

125

42. J. C. Scott, G. G. Malliaras, "Charge injection and recombination at the metal-

organic interface", Chemical Physics Letters, 299, 115 (1999).

43. H. Bassler, "Charge Transport in Disordered Organic Photoconductors - a

Monte-Carlo Simulation Study", Physica Status Solidi B-Basic Research, 175,

15 (1993).

44. D. H. Dunlap, P. E. Parris, V. M. Kenkre, "Charge-dipole model for the

universal field dependence of mobilities in molecularly doped polymers",

Physical Review Letters, 77, 542 (1996).

45. H. C. F. Martens, P. W. M. Blom, H. F. M. Schoo, "Comparative study of hole

transport in poly(p-phenylene vinylene) derivatives", Physical Review B, 61,

7489 (2000).

46. S. Selberherr, Analysis and simulation of semiconductor devices. (1984).

47. M. A. Lampert, P. Mark, Current Injection in Solids (Academic Press, New

York), (1970).

48. P. Langevin, "The recombination and mobilities of ions in gases", Annales De

Chimie Et De Physique, 28, 433 (1903).

49. C. D. Child, "Discharge from hot CaO.", Physical Review, 32, 0492 (1911).

50. M. A. Lampert, P. Mark, Current injection in solids (Academic Press, New

York), 1970).

51. Murgatro.Pn, "Theory of Space-Charge-Limited Current Enhanced by Frenkel

Effect", Journal of Physics D-Applied Physics, 3, 151 (1970).

126

52. W. Helfrich, Schneide.Wg, "Recombination Radiation in Anthracene Crystals",

Physical Review Letters, 14, 229 (1965).

53. S. J. Konezny, D. L. Smith, M. E. Galvin, L. J. Rothberg, "Modeling the

influence of charge traps on single-layer organic light-emitting diode

efficiency", Journal of Applied Physics, 99, 064509 (2006).

54. Y. Wu, Y. C. Zhou, H. R. Wu, Y. Q. Zhan, J. Zhou, S. T. Zhang, J. M. Zhao, Z. J.

Wang, X. M. Ding, X. Y. Hou, "Metal-induced photoluminescence quenching

of tri-(8-hydroxyquinoline) aluminum", Applied Physics Letters, 87, 044104

(2005).

55. Y. Wu, H. R. Wu, M. L. Wang, M. Lu, Q. L. Song, X. M. Ding, X. Y. Hou, "Metal-

induced photoluminescence quenching in thin organic films originating from

noncontact energy transfer between single molecule and atom", Applied

Physics Letters, 90, 154105 (2007).

56. T. Dienel, H. Proehl, R. Forker, K. Leo, T. Fritz, "Metal-induced

photoluminescence quenching of organic molecular crystals", Journal of

Physical Chemistry C, 112, 9056 (2008).

57. A. L. Burin, M. A. Ratner, "Exciton migration and cathode quenching in

organic light emitting diodes", Journal of Physical Chemistry A, 104, 4704

(2000).

58. S. J. Konezny, L. J. Rothberg, M. E. Galvin, D. L. Smith, "The effects of

energetic disorder and polydispersity in conjugation length on the efficiency

127

of polymer-based light-emitting diodes", Applied Physics Letters, 97, 143305

(2010).

59. L. J. A. Koster, V. D. Mihailetchi, P. W. M. Blom, "Bimolecular recombination

in polymer/fullerene bulk heterojunction solar cells", Applied Physics Letters,

88, 052104 (2006).

60. Y. Shirota, Y. Kuwabara, H. Inada, T. Wakimoto, H. Nakada, Y. Yonemoto, S.

Kawami, K. Imai, "Multilayered Organic Electroluminescent Device Using a

Novel Starburst Molecule, 4,4',4''-Tris(3-

Methylphenylphenylamino)Triphenylamine, as a Hole Transport Material",

Applied Physics Letters, 65, 807 (1994).

61. G. Li, C. H. Kim, Z. Zhou, J. Shinar, K. Okumoto, Y. Shirota, "Combinatorial

study of exciplex formation at the interface between two wide band gap

organic semiconductors", Applied Physics Letters, 88, 253505 (2006).

62. S. A. VanSlyke, C. H. Chen, C. W. Tang, "Organic electroluminescent devices

with improved stability", Applied Physics Letters, 69, 2160 (1996).

63. Y. Shirota, Y. Kuwabara, H. Inada, T. Wakimoto, H. Nakada, Y. Yonemoto, S.

Kawami, K. Imai, "Multilayered organic electroluminescent device using a

novel starburst molecule, 4, 4 ', 4 ''-tris (3-methylphenylphenylamino)

triphenylamine, as a hole transport material", Applied Physics Letters, 65, 807

(1994).

128

64. Z. B. Deng, X. M. Ding, S. T. Lee, W. A. Gambling, "Enhanced brightness and

efficiency in organic electroluminescent devices using SiO buffer layers",

Applied Physics Letters, 74, 2227 (1999).

65. H. Jiang, Y. Zhou, B. S. Ooi, Y. Chen, T. Wee, Y. L. Lam, J. Huang, S. Liu,

"Improvement of organic light-emitting diodes performance by the insertion

of a Si3N4 layer", Thin Solid Films, 363, 25 (2000).

66. D. Liu, C. G. Zhen, X. S. Wang, D. C. Zou, B. W. Zhang, Y. Cao, "Enhancement

in brightness and efficiency of organic electroluminescent device using novel

N,N-di(9-ethylcarbaz-3-yl)-3-methylaniline as hole injecting and transporting

material", Synthetic Metals, 146, 85 (2004).

67. S. F. Chen, C. W. Wang, "Influence of the hole injection layer on the

luminescent performance of organic light-emitting diodes", Applied Physics

Letters, 85, 765 (2004).

68. J. Li, C. Ma, J. Tang, C. S. Lee, S. Lee, "Novel Starburst Molecule as a Hole

Injecting and Transporting Material for Organic Light-Emitting Devices",

Chemistry of Mateials, 17, 615 (2005).

69. C. Giebeler, H. Antoniadis, D. D. C. Bradley, Y. Shirota, "Influence of the hole

transport layer on the performance of organic light-emitting diodes", Journal

of Applied Physics, 85, 608 (1999).

129

70. S. Naka, H. Okada, H. Onnagawa, Y. Yamaguchi, T. Tsutsui, "Carrier transport

properties of organic materials for EL device operation", Synthetic Metals,

111, 331 (2000).

71. J. U. Wallace, R. H. Young, C. W. Tang, S. H. Chen, "Charge-retraction time-of-

flight measurement for organic charge transport materials", Applied Physics

Letters, 91, 152104 (2007).

72. S. C. Tse, K. C. Kwok, S. K. So, "Electron transport in naphthylamine-based

organic compounds", Applied Physics Letters, 89, 262102 (2006).

73. C. Giebeler, H. Antoniadis, D. D. C. Bradley, Y. Shirota, "Influence of the hole

transport layer on the performance of organic light-emitting diodes", Journal

of Applied Physics, 85, 608 (1999).

74. K. Itano, H. Ogawa, Y. Shirota, "Exciplex formation at the organic solid-state

interface: Yellow emission in organic light-emitting diodes using green-

fluorescent tris (8-quinolinolato) aluminum and hole-transporting molecular

materials with low ionization potentials", Applied Physics Letters, 72, 636

(1998).

75. D. Y. Kondakov, J. R. Sandifer, C. W. Tang, R. H. Young, "Nonradiative

recombination centers and electrical aging of organic light-emitting diodes:

Direct connection between accumulation of trapped charge and luminance

loss", Journal of Applied Physics, 93, 1108 (2002).

130

76. R. H. Young, C. W. Tang, A. P. Marchetti, "Current-induced fluorescence

quenching in organic light-emitting diodes", Applied Physics Letters, 80, 874

(2002).

77. T. Haskins, A. Chowdhury, R. H. Young, J. R. Lenhard, A. P. Marchetti, L. J.

Rothberg, "Charge-Induced Luminescence Quenching in Organic Light-

Emitting Diodes", Chemistry of Materials, 16, 4675 (2004).

78. M. Sims, S. W. Venter, I. D. Parker, "A Novel Techique to Study OLED

Function", SID Symposium Digest of Technical Papers, 39, 223 (2008)

79. R. H. Young, J. R. Lenhard, D. Y. Kondakov, T. K. Hatwar, "Luminescence

Quenching in Blue Fluorescent OLEDs", SID Symposium Digest of Technical

Papers, 39, 705 (2008).

80. P. K. Wolber, B. S. Hudson, "Analytic Solution to the Forster Energy-Transfer

Problem in 2 Dimensions", Biophysical Journal, 28, 197 (1979).

81. T. Haskins, A. Chowdhury, R. H. Young, J. R. Lenhard, A. P. Marchetti, L. J.

Rothberg, "Charge-induced luminescence quenching in organic light-emitting

diodes", Chemistry of Materials, 16, 4675 (2004).

82. T. Forster, "Experimentelle Und Theoretische Untersuchung Des

Zwischenmolekularen Ubergangs Von Elektronenanregungsenergie",

Zeitschrift Fur Naturforschung Section a-a Journal of Physical Sciences, 4, 321

(1949).

131

83. H. Aziz, Z. D. Popovic, N. X. Hu, A. M. Hor, G. Xu, "Degradation mechanism of

small molecule-based organic light-emitting devices", Science, 283, 1900

(1999).

84. A. B. Chwang, R. C. Kwong, J. J. Brown, "Graded mixed-layer organic light-

emitting devices", Applied Physics Letters, 80, 725 (2002).

85. W. B. Gao, J. X. Sun, K. X. Yang, H. Y. Liu, J. H. Zhao, S. Y. Liu, "Improved

performances of the organic light-emitting devices by doping in the mixed

layer", Optical and Quantum Electronics, 35, 1149 (2003).

86. J. H. Lee, C. I. Wu, S. W. Liu, C. A. Huang, Y. Chang, "Mixed host organic light-

emitting devices with low driving voltage and long lifetime", Applied Physics

Letters, 86, 103506 (2005).

87. L. J. A. Koster, E. C. P. Smits, V. D. Mihailetchi, P. W. M. Blom, "Device model

for the operation of polymer/fullerene bulk heterojunction solar cells",

Physical Review B, 72, 085205 (2005).

88. S. W. Liu, J. H. Lee, C. C. Lee, C. T. Chen, J. K. Wanga, "Charge carrier mobility

of mixed-layer organic light-emitting diodes", Applied Physics Letters, 91,

142106 (2007).

89. C. W. Chen, T. Y. Cho, C. C. Wu, H. L. Yu, T. Y. Luh, "Fuzzy-junction organic

light-emitting devices", Applied Physics Letters, 81, 1570 (2002).

132

90. A. Gusso, D. G. Ma, I. A. Hummelgen, M. G. E. da Luz, "Modeling of organic

light-emitting diodes with graded concentration in the emissive multilayer",

Journal of Applied Physics, 95, 2056 (2004).

91. R. Meerheim, M. Furno, S. Hofmann, B. Lussem, K. Leo, "Quantification of

energy loss mechanisms in organic light-emitting diodes", Applied Physics

Letters, 97, 253305 (2010).

92. K. A. Neyts, "Simulation of light emission from thin-film microcavities",

Journal of the Optical Society of America a-Optics Image Science and Vision,

15, 962 (1998).

93. S. Mladenovski, S. Hofmann, S. Reineke, L. Penninck, T. Verschueren, K. Neyts,

"Integrated optical model for organic light-emitting devices", Journal of

Applied Physics, 109, 083114 (2011).

94. S. Mladenovski, S. Reineke, L. Penninck, K. Neyts, "Detailed analysis of

exciton decay time change in organic light-emitting devices caused by optical

effects", Journal of the Society for Information Display, 19, 80 (2011).

95. X. W. Chen, W. C. H. Choy, S. L. He, "Efficient and rigorous modeling of light

emission in planar multilayer organic light-emitting diodes", Journal of

Display Technology, 3, 110 (2007).

96. B. C. Krummacher, S. Nowy, J. Frischeisen, M. Klein, W. Brutting, "Efficiency

analysis of organic light-emitting diodes based on optical simulation",

Organic Electronics, 10, 478 (2009).

133

97. M. Furno, R. Meerheim, S. Hofmann, B. Lussem, K. Leo, "Efficiency and rate

of spontaneous emission in organic electroluminescent devices", Physical

Review B, 85, 115205 (2012).

98. J. D. Kotlarski, P. W. M. Blom, L. J. A. Koster, M. Lenes, L. H. Slooff,

"Combined optical and electrical modeling of polymer : fullerene bulk

heterojunction solar cells", Journal of Applied Physics, 103, 084502 (2008).

99. http://refractiveindex.info/.

100. K. Saxena, V. K. Jain, D. S. Mehta, "A review on the light extraction techniques

in organic electroluminescent devices", Optical Materials, 32, 221 (2009).

101. N. K. Patel, S. Cina, J. H. Burroughes, "High-efficiency organic light-emitting

diodes", Ieee Journal of Selected Topics in Quantum Electronics, 8, 346 (2002).

102. C. F. Madigan, M. H. Lu, J. C. Sturm, "Improvement of output coupling

efficiency of organic light-emitting diodes by backside substrate

modification", Applied Physics Letters, 76, 1650 (2000).

103. S. Moller, S. R. Forrest, "Improved light out-coupling in organic light emitting

diodes employing ordered microlens arrays", Journal of Applied Physics, 91,

3324 (2002).

104. M. Fujita, T. Ueno, T. Asano, S. Noda, H. Ohhata, T. Tsuji, H. Nakada, N.

Shimoji, "Organic light-emitting diode with ITO/organic photonic crystal",

Electronics Letters, 39, 1750 (2003).

134

105. D. K. Gifford, D. G. Hall, "Emission through one of two metal electrodes of an

organic light-emitting diode via surface-plasmon cross coupling", Applied

Physics Letters, 81, 4315 (2002).

106. D. K. Gifford, D. G. Hall, "Extraordinary transmission of organic

photoluminescence through an otherwise opaque metal layer via surface

plasmon cross coupling", Applied Physics Letters, 80, 3679 (2002).

107. Y. J. Lee, S. H. Kim, J. Huh, G. H. Kim, Y. H. Lee, S. H. Cho, Y. C. Kim, Y. R. Do,

"A high-extraction-efficiency nanopatterned organic light-emitting diode",

Applied Physics Letters, 82, 3779 (2003).

108. E. F. Schubert, N. E. J. Hunt, M. Micovic, R. J. Malik, D. L. Sivco, A. Y. Cho, G. J.

Zydzik, "Highly Efficient Light-Emitting-Diodes with Microcavities", Science,

265, 943 (1994).

109. K. Neyts, P. De Visschere, D. K. Fork, G. B. Anderson, "Semitransparent metal

or distributed Bragg reflector for wide-viewing-angle organic light-emitting-

diode microcavities", Journal of the Optical Society of America B-Optical

Physics, 17, 114 (2000).

110. A. Dodabalapur, L. J. Rothberg, T. M. Miller, E. W. Kwock, "Microcavity Effects

in Organic Semiconductors", Applied Physics Letters, 64, 2486 (1994).

111. A. Dodabalapur, L. J. Rothberg, R. H. Jordan, T. M. Miller, R. E. Slusher, J. M.

Phillips, "Physics and applications of organic microcavity light emitting

diodes", Journal of Applied Physics, 80, 6954 (1996).

135

112. H. Riel, S. Karg, T. Beierlein, W. Riess, K. Neyts, "Tuning the emission

characteristics of top-emitting organic light-emitting devices by means of a

dielectric capping layer: An experimental and theoretical study", Journal of

Applied Physics, 94, 5290 (2003).

113. L. S. Hung, C. W. Tang, M. G. Mason, P. Raychaudhuri, J. Madathil,

"Application of an ultrathin LiF/Al bilayer in organic surface-emitting diodes",

Applied Physics Letters, 78, 544 (2001).

114. H. Riel, S. Karg, T. Beierlein, B. Ruhstaller, W. Riess, "Phosphorescent top-

emitting organic light-emitting devices with improved light outcoupling",

Applied Physics Letters, 82, 466 (2003).

115. R. S. Cok, J. D. Shore, "Microcavity white-emitting OLED devices", Journal of

the Society for Information Display, 17, 617 (2009).

116. http://rredc.nrel.gov/solar/spectra/am1.5/.

117. V. D. Mihailetchi, L. J. A. Koster, J. C. Hummelen, P. W. M. Blom,

"Photocurrent generation in polymer-fullerene bulk heterojunctions",

Physical Review Letters, 93, 216601 (2004).

118. P. W. M. Blom, V. D. Mihailetchi, L. J. A. Koster, D. E. Markov, "Device physics

of polymer : fullerene bulk heterojunction solar cells", Advanced Materials,

19, 1551 (2007).

119. L. Onsager, "Deviations from Ohm's Law in Weak Electrolytes", Journal of

Chemical Physics, 2, 599 (1934).

136

120. L. Onsager, "Initial recombination of ions", Physics Review, 54, 554 (1938).

121. C. L. Braun, "Electric-Field Assisted Dissociation of Charge-Transfer States as a

Mechanism of Photocarrier Production", Journal of Chemical Physics, 80,

4157 (1984).

122. B. P. Rand, D. P. Burk, S. R. Forrest, "Offset energies at organic semiconductor

heterojunctions and their influence on the open-circuit voltage of thin-film

solar cells", Physical Review B, 75, 115327 (2007).

123. M. C. Scharber, D. Wuhlbacher, M. Koppe, P. Denk, C. Waldauf, A. J. Heeger,

C. L. Brabec, "Design rules for donors in bulk-heterojunction solar cells -

Towards 10 % energy-conversion efficiency", Advanced Materials, 18, 789

(2006).

124. P. Peumans, S. R. Forrest, "Separation of geminate charge-pairs at donor-

acceptor interfaces in disordered solids", Chemical Physics Letters, 398, 27

(2004).

125. M. L. Zhang, H. Wang, C. W. Tang, "Effect of the highest occupied molecular

orbital energy level offset on organic heterojunction photovoltaic cells",

Applied Physics Letters, 97, 143503 (2010).

126. S. Barth, P. Muller, H. Riel, P. F. Seidler, W. Riess, H. Vestweber, H. Bassler,

"Electron mobility in tris(8-hydroxy-quinoline)aluminum thin films

determined via transient electroluminescence from single- and multilayer

organic light-emitting diodes", Journal of Applied Physics, 89, 3711 (2001).

137

127. S. W. Culligan, A. C. A. Chen, J. U. Wallace, K. P. Klubek, C. W. Tang, S. H. Chen,

"Effect of hole mobility through emissive layer on temporal stability of blue

organic light-emitting diodes", Advanced Functional Materials, 16, 1481

(2006).

128. S. H. Kim, J. Jang, J. Y. Lee, "Relationship between host energy levels and

device performances of phosphorescent organic light-emitting diodes with

triplet mixed host emitting structure", Applied Physics Letters, 91, 083511

(2007).

129. J. Meyer, S. Hamwi, T. Bulow, H. H. Johannes, T. Riedl, W. Kowalsky, "Highly

efficient simplified organic light emitting diodes", Applied Physics Letters, 91,

113506 (2007).

130. Y. H. Niu, M. S. Liu, J. W. Ka, J. Bardeker, M. T. Zin, R. Schofield, Y. Chi, A. K. Y.

Jen, "Crosslinkable hole-transport layer on conducting polymer for high-

efficiency white polymer light-emitting diodes", Advanced Materials, 19, 300

(2007).

131. R. Mitsumoto, T. Araki, E. Ito, Y. Ouchi, K. Seki, K. Kikuchi, Y. Achiba, H.

Kurosaki, T. Sonoda, H. Kobayashi, O. V. Boltalina, V. K. Pavlovich, L. N.

Sidorov, Y. Hattori, N. Liu, S. Yajima, S. Kawasaki, F. Okino, H. Touhara,

"Electronic structures and chemical bonding of fluorinated fullerenes studied

by NEXAFS, UPS, and vacuum-UV absorption spectroscopies", Journal of

Physical Chemistry A, 102, 552 (1998).

138

132. R. Konenkamp, G. Priebe, B. Pietzak, "Carrier mobilities and influence of

oxygen in C-60 films", Physical Review B, 60, 11804 (1999).

133. M. L. Zhang, H. Wang, C. W. Tang, "Hole-transport limited S-shaped I-V

curves in planar heterojunction organic photovoltaic cells", Applied Physics

Letters, 99, 213506 (2011).

134. T. Y. Chu, O. K. Song, "Hole mobility of N,N '-bis(naphthalen-1-yl)-N,N '-

bis(phenyl) benzidine investigated by using space-charge-limited currents",

Applied Physics Letters, 90, 203512 (2007).

135. N. S. Sariciftci, L. Smilowitz, A. J. Heeger, F. Wudl, "Photoinduced Electron-

Transfer from a Conducting Polymer to Buckminsterfullerene", Science, 258,

1474 (1992).

136. M. Zhang, H. Wang, H. Tian, Y. Geng, C. W. Tang, "Bulk Heterojunction

Photovoltaic Cells with Low Donoe Concentration", Advanced Materials, 23,

4960 (2011).

137. M. L. Zhang, H. Wang, C. W. Tang, "Effect of the highest occupied molecular

orbital energy level offset on organic heterojunction photovoltaic cells",

Applied Physics Letters, 97, 143503 (2010).

138. C. J. Brabec, A. Cravino, D. Meissner, N. S. Sariciftci, T. Fromherz, M. T.

Rispens, L. Sanchez, J. C. Hummelen, "Origin of the open circuit voltage of

plastic solar cells", Advanced Functional Materials, 11, 374 (2001).

139

139. C. J. Brabec, A. Cravino, D. Meissner, N. S. Sariciftci, M. T. Rispens, L. Sanchez,

J. C. Hummelen, T. Fromherz, "The influence of materials work function on

the open circuit voltage of plastic solar cells", Thin Solid Films, 403, 368

(2002).

140. T. Stubinger, W. Brutting, "Exciton diffusion and optical interference in

organic donor-acceptor photovoltaic cells", Journal of Applied Physics, 90,

3632 (2001).

141. F. Monestier, J. J. Simon, P. Torchio, L. Escoubas, B. Ratier, W. Hojeij, B. Lucas,

A. Moliton, M. Cathelinaud, C. Defranoux, F. Flory, "Optical modeling of

organic solar cells based on CuPc and C-60", Applied Optics, 47, C251 (2008).

142. H. Looyenga, "Dielectric constants of heterogenous mixtures", Physica, 31,

401 (1965).

143. A. P. Marchetti, K. E. Sassin, R. H. Young, L. J. Rothberg, D. Y. Kondakov,

"Integer charge transfer states in organic light-emitting diodes: Optical

detection of hole carriers at the anode vertical bar organic interface", Journal

of Applied Physics, 109, 013709 (2011).

144. R. R. Zope, T. Baruah, M. R. Pederson, B. I. Dunlap, "Static dielectric response

of icosahedral fullerenes from C-60 to C-2160 characterized by an all-electron

density functional theory", Physical Review B, 77, 115452 (2008).

140

145. M. M. Mandoc, W. Veurman, L. J. A. Koster, B. de Boer, P. W. M. Blom,

"Origin of the reduced fill factor and photocurrent in MDMO-PPV : PCNEPV

all-polymer solar cells", Advanced Functional Materials, 17, 2167 (2007).

146. B. de Boer, A. Hadipour, P. W. M. Blom, "Organic tandem and multi-junction

solar cells", Advanced Functional Materials, 18, 169 (2008).

147. D. Cheyns, B. P. Rand, P. Heremans, "Organic tandem solar cells with

complementary absorbing layers and a high open-circuit voltage", Applied

Physics Letters, 97, 033301 (2010).

148. M. Riede, C. Uhrich, J. Widmer, R. Timmreck, D. Wynands, G. Schwartz, W. M.

Gnehr, D. Hildebrandt, A. Weiss, J. Hwang, S. Sundarraj, P. Erk, M. Pfeiffer, K.

Leo, "Efficient Organic Tandem Solar Cells based on Small Molecules",

Advanced Functional Materials, 21, 3019 (2011).

149. S. Sista, Z. R. Hong, L. M. Chen, Y. Yang, "Tandem polymer photovoltaic cells-

current status, challenges and future outlook", Energy & Environmental

Science, 4, 1606 (2011).

150. J. Yang, R. Zhu, Z. R. Hong, Y. J. He, A. Kumar, Y. F. Li, Y. Yang, "A Robust Inter-

Connecting Layer for Achieving High Performance Tandem Polymer Solar

Cells", Advanced Materials, 23, 3465 (2011).

151. M. L. Zhang, H. Wang, C. W. Tang, "Tandem photovoltaic cells based on low-

concentration donor doped C-60", Organic Electronics, 13, 249 (2012).

141

152. J. G. Xue, S. Uchida, B. P. Rand, S. R. Forrest, "Asymmetric tandem organic

photovoltaic cells with hybrid planar-mixed molecular heterojunctions",

Applied Physics Letters, 85, 5757 (2004).

153. L. S. Liao, W. K. Slusarek, T. K. Hatwar, M. L. Ricks, D. L. Comfort, "Tandem

organic light-emitting mode using hexaazatriphenylene hexacarbonitrile in

the intermediate connector", Advanced Materials, 20, 324 (2008).

154. N. K. Persson, H. Arwin, O. Inganas, "Optical optimization of polyfluorene-

fullerene blend photodiodes", Journal of Applied Physics, 97, 034503 (2005).

155. A. Hadipour, B. de Boer, P. W. M. Blom, "Device operation of organic tandem

solar cells", Organic Electronics, 9, 617 (2008).

142

Appendix A

Program for the Simulation of OLEDs

(Electrical Part and Exciton Part)

143

A.1 Flow diagram

144

A.2 Codes

//------------------------------------------------------------------------------------------ // OLED model: electrical part and excition part // C++ codes to calculate the charge carrier injection, transport and // recombination, and excition generation, quenching, diffusion and decay in // OLED devices //------------------------------------------------------------------------------------------ #include <math.h> #include <cmath> #include <iostream> #include <fstream> #include <string> #include <sstream> using namespace std; //========================================================= void Initiate_Array(double* layerorder, double* position,double* Eh0, double* El0,double* up0,double* un0,double* Fp0,double* Fn0,double* qyield,double* hole_quench_distance, double* electron_quench_distance,double* sigma0, int N,int NL,int* NLayer,double Eanode,double Ecathode,double* Eh00, double* El00, double* up00,double* un00,double* Fp00,double* Fn00,double* qyield00,double* hole_quench_distance00, double* electron_quench_distance00,double* sigma00,double dmA) // divide the OLED device into multiple mono-layers { int i; int j; layerorder[0]=0; position[0]=0; Eh0[0]=Eanode; El0[0]=Eanode; up0[0]=0; un0[0]=0; Fp0[0]=0; Fn0[0]=0; qyield[0]=0; sigma0[0]=0; for (j=1;j<=NL;j++) { for (i=NLayer[j-1]+1;i<=NLayer[j];i++) {

145

layerorder[i]=j; Eh0[i]=Eh00[j]; El0[i]=El00[j]; up0[i]=up00[j]; un0[i]=un00[j]; Fp0[i]=Fp00[j]; Fn0[i]=Fn00[j]; qyield[i]=qyield00[j]; hole_quench_distance[i]=hole_quench_distance00[j]; electron_quench_distance[i]=electron_quench_distance00[j]; sigma0[i]=sigma00[j]; position[i]=position[i-1]+dmA; } } position[N+1]=position[N]+dmA; layerorder[N+1]=NL+1; Eh0[N+1]=Ecathode; El0[N+1]=Ecathode; up0[N+1]=0; un0[N+1]=0; Fp0[N+1]=0; Fn0[N+1]=0; qyield[N+1]=0; sigma0[N+1]=0; } //========================================================== //========================================================== void Write_Files (double* position,double* p,double* n, double* F,double V, double* jt, double* R, double* Rif, double* Rnew,double* s,double* Eh,double* El,double* up,double* un, double* jpR, double* jnR,double* jp, double* jn, double* jp_drift, double* jp_diff, double* jn_drift, double* jn_diff, int N,int NJ) { ofstream myfile; ostringstream ss; ss << "Jstep" << NJ << ".dat"; myfile.open( ss.str().c_str() ); myfile<< "Number" << " " << "Position(A)" << " " << "Holes(1/cm3)" << " " << "Electrons(1/cm3)" << " " << "ElectricField(V/cm)" << " " << "Volage" << " " << "J_Total(A/cm2)" << " " << "RecombinationBulk(1/cm3s)" << " " << "RecombinatinInterface(1/cm3s)" << " " << "RecombinationTotal(1/cm3s)" << "

146

" << "ExcitionDecay(1/cm3s)" << " "<< "HOMO(ev)" << " " << "LUMO(ev)" << " " << "HoleMobility(cm2/Vs)" << " " << "ElectronMobility(cm2/Vs)" << " "<< "J_Hole_Corrected(A/cm2)" << " " << "J_Electron_Corrected(A/cm2)" << " " << "J_Hole(A/cm2)" << " " << "J_Electron(A/cm2)" << " " << "Jp_drift(A/cm2)" << " " << "Jp_diff(A/cm2)" << " " << "Jn_drift(A/cm2)" << " " << "Jn_diff(A/cm2)" << " "<< endl; int i; for(i=0;i<=N+1;i++) {

myfile << i << " " << position[i] << " " << p[i] << " " << n[i] << " " << F[i] << " " << V << " "<<jt[i]<< " "<<R[i]<< " "<<Rif[i]<< " "<<Rnew[i]<< " "<<s[i]<< " " << Eh[i] << " " << El[i] << " " << up[i]<< " " << un[i] << " " << jpR[i] << " " << jnR[i] << " "<< jp[i] << " " << jn[i] << " " << jp_drift[i] << " "<< jp_diff[i] << " "<< jn_drift[i] << " "<< jn_diff[i] << " "<< endl;

} myfile.close(); } //========================================================== //========================================================== double Estimate_dt (double* p, double* n,double* F, double* Fint,double* up, double* un, int N,double dm, double diel_0, double diel_r,double e,double ts_factor,double Hole_Injection_Factor, double Electron_Injection_Factor, double Hole_Injection_B,double Electron_Injection_B) // estimate the discrete time scale for the simulation { double uFmax,x,dt; int i; double kT=0.026; double dE; double f=0,phi=0; uFmax=0; for (i=1;i<=N;i++) { x=abs(up[i]*F[i]); if (uFmax<=x) uFmax=x; x=abs(un[i]*F[i]); if (uFmax<=x) uFmax=x; x=abs(up[i]*kT/dm); if (uFmax<=x) uFmax=x; x=abs(un[i]*kT/dm);

147

if (uFmax<=x) uFmax=x; } if (Fint[0]>0) { f=(1.438e-7)*Fint[0]/(diel_r*kT*kT); phi=1/f+1/sqrt(f)-sqrt(1+2*sqrt(f))/f; dE=-Hole_Injection_B+sqrt(1.438e-7*Fint[0]/diel_r); if (dE>=0) dE=0;

if(Hole_Injection_Factor*4*phi*phi*up[0]*Fint[0]*exp(dE/kT)/p[0]>uFmax) uFmax=Hole_Injection_Factor*4*phi*phi*up[0]*Fint[0]*exp(dE/kT)/p[0];

} if (Fint[N]>0) { f=(1.438e-7)*Fint[N]/(diel_r*kT*kT); phi=1/f+1/sqrt(f)-sqrt(1+2*sqrt(f))/f; dE=-Electron_Injection_B+sqrt(1.438e-7*Fint[N]/diel_r); if (dE>=0) dE=0;

if(Electron_Injection_Factor*4*phi*phi*un[N+1]*Fint[N]*exp(dE/kT) /n[N+1]>uFmax) uFmax=Electron_Injection_Factor*4*phi*phi*un[N+1]*Fint[N]*exp(dE/kT) /n[N+1]; } dt=dm/(uFmax*ts_factor); return dt; } //========================================================= double Calculate_Voltage( double* p,double* n,double* F,double* Fint,double* Eh0,double* El0,double* Eh,double* El, double Eanode, double Ecathode, int N, double L, double* up, double* un, double* up0, double* un0, double* Fp0,double* Fn0, double dm, double diel_0, double diel_r,double e) // calculate the voltage and electric field inside the device // Fint is the electric field at the interface between monolayers // F is the electric field inside the monolayer { int i; int j; for (i=0;i<=N;i++)

148

{ Fint[i]=0; for(j=0;j<=N+1;j++) { if (j<=i) Fint[i]=Fint[i]+dm*e*(p[j]-n[j])/(2*diel_0*diel_r); if (j>i) Fint[i]=Fint[i]-dm*e*(p[j]-n[j])/(2*diel_0*diel_r); } } for (i=1;i<=N;i++) { F[i]=(Fint[i-1]+Fint[i])/2; } double V=0; for (i=1;i<=N;i++) { V=V+F[i]*dm; Eh[i]=Eh0[i]+V; El[i]=El0[i]+V; // so far V is only consider the electric potential drop up[i]=up0[i]*exp(abs(Fp0[i]*sqrt(abs(F[i])))); un[i]=un0[i]*exp(abs(Fn0[i]*sqrt(abs(F[i])))); } up[0]=up0[1]*exp(abs(Fp0[1]*sqrt(abs(Fint[0])))); un[N+1]=un0[N]*exp(abs(Fn0[N]*sqrt(abs(Fint[N])))); Eh[0]=Eh0[0]; El[0]=El0[0]; Eh[N+1]=V+Eh0[N+1]; El[N+1]=V+El0[N+1]; V=V+(Ecathode-Eanode);

// consider the built-in potential+ potential drop return V; } //========================================================== double Kapa(double E, double sigma) { double pi=3.14159265359; return exp((E*E)/(2*sigma*sigma))/(sigma*sqrt(2*pi));

149

} double G_Absolute(double deltaE, double sigma1,double sigma2){ double E1,E2,dE=0.01; double kT=0.026; double g=0; double integration1=0; if(sigma1==0 && sigma2==0) g=exp(-deltaE/kT); else { if(sigma1==0 && sigma2!=0) { E1=0; integration1=0; for(E2=-1;E2<=1;E2=E2+dE) { if (E2+deltaE>E1) integration1=integration1+Kapa(E2,sigma2)*exp(-(E2+deltaE-E1)/kT)*dE; else integration1=integration1+Kapa(E2,sigma2)*dE; } g=integration1; } else { if (sigma1!=0 && sigma2==0) { E2=0; integration1=0; for (E1=-1;E1<=1;E1=E1+dE) { if (E2+deltaE>E1) integration1=integration1+exp(-(E2+deltaE-E1)/kT)*Kapa(E1,sigma1)*dE; else integration1=integration1+Kapa(E1,sigma1)*dE; } g=integration1; } else { for (E1=-1;E1<=1;E1=E1+dE) {

150

integration1=0; for(E2=-1;E2<=1;E2=E2+dE) { if (E2+deltaE>E1) integration1=integration1+Kapa(E2,sigma2)*exp(-(E2+deltaE-E1)/kT)*dE; else integration1=integration1+Kapa(E2,sigma2)*dE; } g=g+integration1*Kapa(E1,sigma1)*dE; } } } } return g; } double G(double deltaE,double sigma1,double sigma2) { return G_Absolute(deltaE,sigma1,sigma2)/G_Absolute(0,sigma1,sigma2); // if ((deltaE-sigma1-sigma2)>=0) return exp(-(deltaE-sigma1-sigma2)/0.026);

// else return 1; } void Calculate_ddR(double* p, double* n, double* F, double* Fint, double* up, double* un, double* up0,double* un0,double* Eh0, double* El0, double* sigma00, double* R, double* Rif, double* Rnew,double* s, double* dp_plus_drift, double* dp_minus_drift, double* dn_plus_drift, double* dn_minus_drift, double* dp_plus_diff, double* dp_minus_diff, double* dn_plus_diff, double* dn_minus_diff, int* NLayer, int N, int NL, double dm, double dt,double diel_0, double diel_r,double e, double Hole_Injection_Factor, double Electron_Injection_Factor,double Recombination_Interface_Factor,double Hole_Injection_B,double Electron_Injection_B ) // calculate the charge carrier transport (drift and diffusion) and recombination { double dE; int i; double ssdE_p_plus,ssdE_p_minus,ssdE_n_plus,ssdE_n_minus; int order; double kT=0.026;

151

double f=0,phi=0; for (i=0;i<=N+1;i++)

{ dp_plus_drift[i]=0; dp_minus_drift[i]=0; dn_plus_drift[i]=0; dn_minus_drift[i]=0; }// clear the current array

for(i=0;i<=N+1;i++)

{ dp_plus_diff[i]=0; dp_minus_diff[i]=0; dn_plus_diff[i]=0; dn_minus_diff[i]=0; }

//************************************************* Holes if (Fint[0]>0) { f=(1.438e-7)*Fint[0]/(diel_r*kT*kT); phi=1/f+1/sqrt(f)-sqrt(1+2*sqrt(f))/f; dE=-Hole_Injection_B+sqrt(1.438e-7*Fint[0]/diel_r); if (dE>=0) dE=0; dp_plus_drift[0]=(dt/dm)*Hole_Injection_Factor*4*phi*phi*up[0]*Fint[0]*exp(dE/kT); // time e and devide e, so get rid of the e } else { dp_plus_drift[0]=0; // remember that at this time, the equation have F[1] so negative F no injection } if (F[1]>0) dp_plus_drift[1]=abs(up[1]*F[1]*dt/dm)*p[1]; //p[1] else dp_minus_drift[1]=abs(up[1]*F[1]*dt/dm)*p[1]; dp_plus_diff[1]=abs(up[1]*kT*dt/(dm*dm))*p[1]; int j; NLayer[0]=0; for(j=1;j<=NL;j++)//p[2]----p[N-1] {

152

for(i=NLayer[j-1]+2;i<=NLayer[j]-1;i++)// inside each layer { if (F[i]>0) dp_plus_drift[i]=abs(up[i]*F[i]*dt/dm)*p[i]; else dp_minus_drift[i]=abs(up[i]*F[i]*dt/dm)*p[i]; dp_plus_diff[i]=abs(up[i]*kT*dt/(dm*dm))*p[i]; dp_minus_diff[i]=dp_plus_diff[i]; } if(j!=NL)// if j!=NL, inter layer drift { i=NLayer[j]; dE=Eh0[i]-Eh0[i+1]-(F[i]+F[i+1])*dm/2; if (dE>0) { ssdE_p_plus=G(abs(dE),sigma00[j],sigma00[j+1]);ssdE_p_minus=1; } else { ssdE_p_minus=G(abs(dE),sigma00[j+1],sigma00[j]);ssdE_p_plus=1; } // calculate the interface cross probablity

if (F[i]>0) dp_plus_drift[i]=abs(sqrt(abs(up[i]*F[i]*up[i+1]*F[i+1]))*dt/dm) *p[i]*ssdE_p_plus;

else dp_minus_drift[i]=abs(up[i]*F[i]*dt/dm)*p[i];

dp_plus_diff[i]=abs(sqrt(up[i]*up[i+1])*kT*dt/(dm*dm))*p[i]*ssdE_p_plus; dp_minus_diff[i]=abs(up[i]*kT*dt/(dm*dm))*p[i]; // finish the ith layer i=NLayer[j]+1; if (F[i]>0)

dp_plus_drift[i]=abs(up[i]*F[i]*dt/dm)*p[i]; else

dp_minus_drift[i]=abs(sqrt(abs(up[i]*F[i]*up[i-1]*F[i-1]))*dt/dm) *p[i]*ssdE_p_minus;

dp_plus_diff[i]=abs(up[i]*kT*dt/(dm*dm))*p[i];

dp_minus_diff[i]=abs(sqrt(up[i]*up[i-1])*kT*dt/(dm*dm)) *p[i]*ssdE_p_minus;

153

// finish the i+1th layer } } //p[N],there is no holes on N+1 if (F[N]>0) dp_plus_drift[N]=abs(up[N]*F[N]*dt/dm)*p[N]; else dp_minus_drift[N]=abs(up[N]*F[N]*dt/dm)*p[N]; dp_plus_diff[N]=abs(up[N]*kT*dt/(dm*dm))*p[N]; dp_minus_diff[N]=dp_plus_diff[N]; //*************************************************************** //********************************** Electrons if (Fint[N]>0) { f=(1.438e-7)*Fint[N]/(diel_r*kT*kT); phi=1/f+1/sqrt(f)-sqrt(1+2*sqrt(f))/f; dE=-Electron_Injection_B+sqrt(1.438e-7*Fint[N]/diel_r); if (dE>=0) dE=0; dn_minus_drift[N+1]=(dt/dm)*Electron_Injection_Factor*4*phi*phi*un[N+1]*Fint[N]*exp(dE/kT); // time e and devide e, so get rid of the e } else { dn_minus_drift[N+1]=0; } //n[N] if (F[N]>0) dn_minus_drift[N]=abs(un[N]*F[N]*dt/dm)*n[N]; else dn_plus_drift[N]=abs(un[N]*F[N]*dt/dm)*n[N]; dn_minus_diff[N]=abs(un[N]*kT*dt/(dm*dm))*n[N]; //dn_plus_diff[N]=dn_minus_diff[N]; for(j=NL;j>=1;j--) //n[N-1]--n[2] { for(i=NLayer[j]-1;i>=NLayer[j-1]+2;i--)//inside layer {

154

if (F[i]>0) dn_minus_drift[i]=abs(un[i]*F[i]*dt/dm)*n[i]; else dn_plus_drift[i]=abs(un[i]*F[i]*dt/dm)*n[i]; dn_minus_diff[i]=abs(un[i]*kT*dt/(dm*dm))*n[i]; dn_plus_diff[i]=dn_minus_diff[i]; } if(j!=1)// between layers { i=NLayer[j-1]+1; dE=El0[i-1]-El0[i]-(F[i-1]+F[i])*dm/2; if (dE>0) { ssdE_n_minus=G(abs(dE),sigma00[j],sigma00[j-1]);ssdE_n_plus=1; } else { ssdE_n_minus=1; ssdE_n_plus=G(abs(dE),sigma00[j-1],sigma00[j]); } // caculate crossing factor if (F[i]>0)

dn_minus_drift[i]=abs(sqrt(abs(un[i]*F[i]*un[i-1]*F[i-1]))/dm*dt) *n[i]*ssdE_n_minus;

else dn_plus_drift[i]=abs(un[i]*F[i]*dt/dm)*n[i];

dn_minus_diff[i]=abs(sqrt(un[i]*un[i-1])*kT*dt/(dm*dm)) *n[i]*ssdE_n_minus; dn_plus_diff[i]=abs(un[i]*kT*dt/(dm*dm))*n[i]; // finish +1 layer i=NLayer[j-1]; if (F[i]>0)

dn_minus_drift[i]=abs(un[i]*F[i]*dt/dm)*n[i]; else

dn_plus_drift[i]=abs(sqrt(abs(un[i]*F[i]*un[i+1]*F[i+1]))*dt/dm) *n[i]*ssdE_n_plus;

dn_minus_diff[i]=abs(un[i]*kT*dt/(dm*dm))*n[i]; dn_plus_diff[i]=abs(sqrt(un[i]*un[i+1])*kT*dt/(dm*dm))*n[i]*ssdE_n_plus; // finish -1 layer }

155

} //n[1] if (F[1]>0) dn_minus_drift[1]=abs(un[1]*F[1]*dt/dm)*n[1]; else dn_plus_drift[1]=abs(un[1]*F[1]*dt/dm)*n[1]; dn_minus_diff[1]=abs(un[1]*kT*dt/(dm*dm))*n[1]; dn_plus_diff[1]=dn_minus_diff[1]; for(i=1;i<=N;i++) { R[i]=p[i]*n[i]*e*(un[i]+up[i])/(diel_0*diel_r); } if(Recombination_Interface_Factor>0.01) { for(j=1;j<=NL-1;j++) { i=NLayer[j]; if(Eh0[i]>Eh0[i+1] && El0[i]>El0[i+1]) // cross interface recombination only takes place when energy barrier // block both types of charge carriers { if (up[i]>un[i+1]) Rif[i]=p[i]*n[i+1]*e*(un[i+1])/(diel_0*diel_r);

// use umin(up,un) to calculate interface recombination else Rif[i]=p[i]*n[i+1]*e*(up[i])/(diel_0*diel_r); } } } for(i=1;i<=N;i++) { p[i]=p[i]-R[i]*dt; n[i]=n[i]-R[i]*dt; } for(j=1;j<=NL-1;j++) { i=NLayer[j]; p[i]=p[i]-Rif[i]*dt; n[i+1]=n[i+1]-Rif[i]*dt; }

156

p[0]=p[0]-dp_plus_drift[0]+dp_minus_diff[1]-dn_minus_diff[1]+dp_minus_drift[1]-dn_minus_drift[1]; //*** begin to calculate the change of p and n for each monolayer n[N+1]=n[N+1]-dn_minus_drift[N+1]+dn_plus_diff[N]-dp_plus_diff[N]+dn_plus_drift[N]-dp_plus_drift[N]; for (i=1;i<=N;i++) { p[i]=p[i]-dp_plus_diff[i]-dp_minus_diff[i]+dp_plus_diff[i-1]+ dp_minus_diff[i+1]-dp_plus_drift[i]-dp_minus_drift[i]+dp_plus_drift[i-1]+ dp_minus_drift[i+1]; n[i]=n[i]-dn_plus_diff[i]-dn_minus_diff[i]+dn_plus_diff[i-1]+ dn_minus_diff[i+1]-dn_plus_drift[i]-dn_minus_drift[i]+dn_plus_drift[i-1]+ dn_minus_drift[i+1]; } } //============================================= void Calculate_Current( double* jt,double* jpR, double* jnR,double* jp, double* jn, double* jp_drift, double* jp_diff, double* jn_drift, double* jn_diff, double* R, double* Rif, double* Rnew,double* s,double* IntR,double* IntRif, double* dp_plus_drift, double* dp_minus_drift, double* dn_plus_drift, double* dn_minus_drift, double* dp_plus_diff, double* dp_minus_diff, double* dn_plus_diff, double* dn_minus_diff, int* NLayer, int N, int NL,double dm, double dt,double diel_0, double diel_r,double e, double converge, int* check,int NJ,double* Fjtave, double* Fdjt, double* FjpRave, double* FdjpR,double* FjnRave,double* FdjnR, double* FIntR,double* FIntRif,int* FCheck) // calculate the current density in the device { int i; IntR[0]=0; IntRif[0]=0; for(i=1;i<=N;i++) { IntR[i]=IntR[i-1]+R[i]*e*dm; IntRif[i]=IntRif[i-1]+Rif[i]*e*dm; }

157

for (i=0;i<=N;i++) // remember to calculate the interface current flow, not the layer, this is the key // point, i is the ith interface, between i and i+1 { jp_drift[i]=e*dm*(dp_plus_drift[i]-dp_minus_drift[i+1])/dt; jn_drift[i]=-e*dm*(dn_plus_drift[i]-dn_minus_drift[i+1])/dt; jp_diff[i]=e*dm*(dp_plus_diff[i]-dp_minus_diff[i+1])/dt; jn_diff[i]=-e*dm*(dn_plus_diff[i]-dn_minus_diff[i+1])/dt; jp[i]=jp_drift[i]+jp_diff[i]; jn[i]=jn_drift[i]+jn_diff[i]; jt[i]=jp[i]+jn[i]+Rif[i]*e*dm; jpR[i]=jp[i]+IntR[i]+IntRif[i]; if (i!=0) jnR[i]=jn[i]+IntR[N]-IntR[i]+IntRif[N]-IntRif[i-1]; else jnR[i]=jn[i]+IntR[N]-IntR[i]+IntRif[N]; } double jpRmax,jpRmin,jpRave,jnRmax,jnRmin,jnRave,jtmax,jtmin,jtave; jpRmax=jpRmin=jpRave=jpR[0]; jnRmax=jnRmin=jnRave=jnR[0]; jtmax=jtmin=jtave=jt[0]; for(i=1;i<=N;i++) { if (jpRmax<=jpR[i]) jpRmax=jpR[i]; if (jpRmin>=jpR[i]) jpRmin=jpR[i]; if (jnRmax<=jnR[i]) jnRmax=jnR[i]; if (jnRmin>=jnR[i]) jnRmin=jnR[i]; if (jtmax<=jt[i]) jtmax=jt[i]; if (jtmin>=jt[i]) jtmin=jt[i]; jpRave=jpRave+jpR[i]; jnRave=jnRave+jnR[i]; jtave=jtave+jt[i]; } jpRave=jpRave/(N+1); jnRave=jnRave/(N+1); jtave=jtave/(N+1);

158

double djpR,djnR,djt; if (abs(jpRave)<1.0e-8) djpR=0; else djpR=(jpRmax-jpRmin)/jpRave; if (abs(jnRave)<1.0e-8) djnR=0; else djnR=(jnRmax-jnRmin)/jnRave ; if (abs(jtave)<1.0e-8) djt=0; else djt=(jtmax-jtmin)/jtave;

if (abs(djt)<=converge && abs(djpR)<=converge && abs(djnR)<=converge) check[0]=1;

FIntR[NJ]=IntR[N];

// turns the recombination to a unit of current, recombination current FIntRif[NJ]=IntRif[N]; Fjtave[NJ]=jtave; Fdjt[NJ]=djt; FjpRave[NJ]=jpRave; FdjpR[NJ]=djpR; FjnRave[NJ]=jnRave; FdjnR[NJ]=djnR; FCheck[NJ]=check[0]; } //======================================================= void Calculate_Exciton( double* p, double* n, double* F, double* Fint, double* up, double* un, double* up0,double* un0,double* Eh0, double* El0, double* sigma00, double* R, double* Rif, double* Rnew,double* s,int* NLayer, int N, int NL, double dm, double diel_0, double diel_r,double e, double converge, int Nmax,double Ldiffusion, double tlife,double* qyield, double* hole_quench_distance, double* electron_quench_distance, int NJ,double* FIntS,int* FCheckExciton) // calculate the exction diffusion, quenching and decay { int i,j,k; double quench[10000]={0}; double snew[10000]={0};

159

int check=0; double dE=0; for (i=0;i<=N+1;i++){s[i]=0;quench[i]=0;snew[i]=0;Rnew[i]=R[i];} for(j=1;j<=NL-1;j++) { i=NLayer[j]; dE=Eh0[i]-Eh0[i+1]; if( abs(Eh0[i]-Eh0[i+1]) < abs(El0[i]-El0[i+1]))

Rnew[i+1]=Rnew[i+1]+Rif[i]*G(abs(Eh0[i]-Eh0[i+1]), sigma00[j],sigma00[j+1]);

else Rnew[i]=Rnew[i]+Rif[i]*G(abs(El0[i]-El0[i+1]), sigma00[j+1],sigma00[j]);

} for(i=1;i<=N;i++) { for(j=1;j<=N;j++) { if (j!=i) quench[i]=quench[i]+(p[j]/tlife)*(3.1415926/2.0)*dm*pow(hole_quench_distance[j]*1.0e-8,6.0)/pow(abs(j-i)*dm,4.0)+ (n[j]/tlife)*(3.1415926/2.0)*dm*pow(electron_quench_distance[j]*1.0e-8,6.0)/pow(abs(j-i)*dm,4.0); // not same layer quenching else quench[i]=quench[i]+(p[j]/tlife)*(3.1415926/2.0)*dm*pow(hole_quench_distance[j]*1.0e-8,6.0)/pow(dm,4.0)+(n[j]/tlife)*(3.1415926/2.0)*dm*pow(electron_quench_distance[j]*1.0e-8,6.0)/pow(dm,4.0); // in the same layer, quenching } } double dmax=1/tlife; if (Ldiffusion*Ldiffusion/(dm*dm*tlife)>dmax) dmax=Ldiffusion*Ldiffusion/(dm*dm*tlife); for(i=1;i<=N;i++) { if(quench[i]>dmax) dmax=quench[i];

160

} double dt=1/(10*dmax); NLayer[0]=0; for (k=0;k<=Nmax && check==0; k++) { for(j=1;j<=NL;j++) { i=NLayer[j-1]+1; if(j!=1) { dE=(El0[i-1]-Eh0[i-1])-(El0[i]-Eh0[i]); snew[i]=s[i]+dt*((s[i+1]+s[i-1]*G(-dE,sigma00[j-1], sigma00[j])-s[i]-s[i]*G(dE,sigma00[j],sigma00[j-1])) *Ldiffusion*Ldiffusion/(dm*dm*tlife)-s[i]/tlife-quench[i]*s[i]+Rnew[i]); // consider for the cross interface excitons } else snew[i]=s[i]+dt*((s[i+1]-s[i]) *Ldiffusion*Ldiffusion/(dm*dm*tlife)-s[i]/tlife-quench[i]*s[i]+Rnew[i]); for(i=NLayer[j-1]+2;i<=NLayer[j]-1;i++) { snew[i]=s[i]+dt*((s[i+1]+s[i-1]-2*s[i]) *Ldiffusion*Ldiffusion/(dm*dm*tlife)-s[i]/tlife-quench[i]*s[i]+Rnew[i]); } i=NLayer[j]; if(j!=NL) { dE=(El0[i+1]-Eh0[i+1])-(El0[i]-Eh0[i]); snew[i]=s[i]+dt*((s[i+1]*G(-dE, sigma00[j+1],sigma00[j])+s[i-1]-s[i]-s[i]*G(dE,sigma00[j],sigma00[j+1])) *Ldiffusion*Ldiffusion/(dm*dm*tlife)-s[i]/tlife-quench[i]*s[i]+Rnew[i]); // consider for the cross interface excitons } else snew[i]=s[i]+dt*((s[i-1]-s[i]) *Ldiffusion*Ldiffusion/(dm*dm*tlife)-s[i]/tlife-quench[i]*s[i]+Rnew[i]); }

161

check=1; for (i=1;i<=N && check==1; i++) { if (snew[i]!=0 && abs((snew[i]-s[i])/snew[i])>converge*dt) check=0; } for(i=1;i<=N;i++) { s[i]=snew[i]; } } FCheckExciton[NJ]=check; FIntS[NJ]=0; for(i=1;i<=N;i++) { s[i]=s[i]/tlife; FIntS[NJ]=FIntS[NJ]+s[i]*e*dm*qyield[i]; } } //========================================== int main() { //*************** Part I: set up input parameters ******** double dm,dmA,e,diel_0,diel_r,sigma,kT; double dt,converge,ts_factor,NTMAX,Hole_Injection_Factor,Electron_Injection_Factor,Hole_Injection_B,Electron_Injection_B,Recombination_Interface_Factor; double Eanode,Ecathode; double JStart,JEnd,J,V; int NJStep; int NL; double LayerWidth[20]={0},Eh00[20]={0},El00[20]={0},up00[20]={0},un00[20]={0},Fp00[20]={0},Fn00[20]={0},hole_quench_distance00[20]={0},electron_quench_distance00[20]={0},qyield00[20]={0},sigma00[20]={0}; string LayerName[20]; int i=1; string AnodeName,CathodeName; int Nmax; double Ldiffusion,tlife,hole_quench_factor,electron_quench_factor;

162

int hole_quench_range,electron_quench_range; //*************** Part II: read input parameters ************* string line; ifstream myfile ("input.txt"); if (myfile.is_open()) { getline (myfile,line); cout << line << endl; getline (myfile,line); cout << line << endl; myfile >> JStart; getline (myfile,line); cout << JStart << endl;

getline (myfile,line); cout << line << endl; myfile >> NJStep; getline (myfile,line); cout << NJStep << endl; getline (myfile,line); cout << line << endl; getline (myfile,line); cout << line << endl; //JStart, JEnd, NJStep getline (myfile,line); cout << line << endl; getline (myfile,line); cout << line << endl; myfile >> ts_factor; getline (myfile,line); cout << ts_factor << endl; getline (myfile,line); cout << line << endl; myfile >> NTMAX; getline (myfile,line); cout << NTMAX << endl; getline (myfile,line); cout << line << endl;

163

myfile >> converge; getline (myfile,line);

cout << converge << endl;

getline (myfile,line); cout << line << endl; myfile >> Hole_Injection_Factor; getline (myfile,line); cout << Hole_Injection_Factor << endl; getline (myfile,line); cout << line << endl; myfile >> Hole_Injection_B; getline (myfile,line); cout << Hole_Injection_B << endl;

getline (myfile,line); cout << line << endl; myfile >> Electron_Injection_Factor; getline (myfile,line); cout << Electron_Injection_Factor << endl; getline (myfile,line); cout << line << endl; myfile >> Electron_Injection_B; getline (myfile,line); cout << Electron_Injection_B << endl; getline (myfile,line); cout << line << endl; myfile >> Recombination_Interface_Factor; getline (myfile,line); cout << Recombination_Interface_Factor << endl; getline (myfile,line); cout << line << endl; myfile >> Nmax; getline (myfile,line); cout << Nmax << endl;

getline (myfile,line); cout << line << endl;

164

myfile >> Ldiffusion; getline (myfile,line); cout << Ldiffusion << endl;

getline (myfile,line); cout << line << endl;

myfile >> tlife; getline (myfile,line);

cout << tlife << endl;

getline (myfile,line); cout << line << endl;

myfile >> diel_r; getline (myfile,line);

cout << diel_r << endl;

getline (myfile,line); cout << line << endl;

getline (myfile,line); cout << line << endl; // control parameters ts_factor,NTMAX,converge,sigma getline (myfile,line); cout << line << endl; getline (myfile,line); cout << line << endl; myfile >> dmA; getline (myfile,line); cout << dmA << endl;

getline (myfile,line); cout << line << endl; myfile >> NL; getline (myfile,line); cout << NL << endl;

getline (myfile,line); cout << line << endl; getline (myfile,AnodeName); cout << AnodeName << endl;

165

getline (myfile,line);

cout << line << endl; myfile >> Eanode; getline (myfile,line); cout << Eanode << endl; getline (myfile,line); cout << line << endl; getline (myfile,CathodeName ); cout << CathodeName << endl; getline (myfile,line); cout << line << endl; myfile >> Ecathode; getline (myfile,line); cout << Ecathode << endl; getline (myfile,line); cout << line << endl;// dmA,NL,Anode,Cathode for (i=1;i<=NL;i++){ getline (myfile,line); cout << line << endl; getline (myfile,LayerName[i]); cout << LayerName[i] << endl; getline (myfile,line); cout << line << endl; myfile >> LayerWidth[i]; getline (myfile,line); cout << LayerWidth[i] << endl; getline (myfile,line); cout << line << endl; myfile >> Eh00[i]; getline (myfile,line); cout << Eh00[i] << endl; getline (myfile,line); cout << line << endl; myfile >> El00[i]; getline (myfile,line); cout << El00[i] << endl; getline (myfile,line); cout << line << endl; myfile >> up00[i];

166

getline (myfile,line); cout << up00[i] << endl; getline (myfile,line); cout << line << endl; myfile >> Fp00[i]; getline (myfile,line); cout << Fp00[i] << endl; getline (myfile,line); cout << line << endl; myfile >> un00[i]; getline (myfile,line); cout << un00[i] << endl; getline (myfile,line); cout << line << endl; myfile >> Fn00[i]; getline (myfile,line); cout << Fn00[i] << endl; getline (myfile,line); cout << line << endl; myfile >> hole_quench_distance00[i]; getline (myfile,line); cout << hole_quench_distance00[i] << endl; getline (myfile,line); cout << line << endl; myfile >> electron_quench_distance00[i]; getline (myfile,line); cout << electron_quench_distance00[i] << endl; getline (myfile,line); cout << line << endl; myfile >> qyield00[i]; getline (myfile,line); cout << qyield00[i] << endl; getline (myfile,line); cout << line << endl; myfile >> sigma00[i]; getline (myfile,line); cout << sigma00[i] << endl; getline (myfile,line); cout << line << endl;

167

} myfile.close(); } else cout << "Unable to open file"; Ldiffusion=Ldiffusion*1.0e-7; //unit cm dm=dmA*1.0e-8; // unit cm e=1.6e-19; // unit C diel_0=8.854e-14; // vacuum dielectric constant 8.854e-14F/cm or 8.854e-14 C/V/cm kT=0.026; // unit eV //********************************************* //************************ Part III: define parameters to simulate *** double layerorder[2010]={0},position[2010]={0}; double p[2010]={0},n[2010]={0},Fint[2010]={0},F[2010]={0}, Eh[2010]={0},El[2010]={0},Eh0[2010]={0},El0[2010]={0},up[2010]={0}, un[2010]={0},up0[2010]={0},un0[2010]={0},Fp0[2010]={0},Fn0[2010]={0}, qyield[2010]={0},sigma0[2010]={0},hole_quench_distance[2010]={0}, electron_quench_distance[2010]={0}; double dp_plus_drift[2010]={0},dp_minus_drift[2010]={0}, dp_plus_diff[2010]={0},dp_minus_diff[2010]={0}; double dn_plus_drift[2010]={0},dn_minus_drift[2010]={0}, dn_plus_diff[2010]={0},dn_minus_diff[2010]={0}; double vp_drift_plus[2010]={0},vp_drift_minus[2010]={0}, vp_diff_plus[2010]={0},vp_diff_minus[2010]={0},vn_drift_plus[2010]={0}, vn_drift_minus[2010]={0},vn_diff_plus[2010]={0},vn_diff_minus[2010]={0}; double jp[2010]={0},jn[2010]={0},jpR[2010]={0},jnR[2010]={0}, jt[2010]={0},jn_drift[2010]={0},jn_diff[2010]={0},jp_drift[2010]={0}, jp_diff[2010]={0},s[2010]={0},R[2010]={0},Rif[2010]={0},Rnew[2010]={0}, IntR[2010]={0},IntRif[2010]={0}; int check[2]={0}; double Fjtave[50]={0},Fdjt[50]={0},FjpRave[50]={0},FdjpR[50]={0}, FjnRave[50]={0}, FdjnR[50]={0},FIntR[50]={0},FIntRif[50]={0},FIntS[50]={0}; int FCheck[50]={0},FCheckExciton[50]={0}; int N=0; double L=0; double t; int NLayer[10]={0};

168

//*************************** Part IV: build the device ************ int Nx; for (i=1;i<=NL;i++) { Nx=(int) (LayerWidth[i]/dmA); NLayer[i]=NLayer[i-1]+Nx; L=L+LayerWidth[i]*1.0e-8; N=N+Nx; } Initiate_Array(layerorder,position,Eh0,El0,up0,un0,Fp0,Fn0,qyield, hole_quench_distance,electron_quench_distance,sigma0,N,NL,NLayer,Eanode, Ecathode,Eh00,El00,up00,un00,Fp00,Fn00,qyield00,hole_quench_distance00,electron_quench_distance00,sigma00,dmA); //****************************** Part V: start J steps******************** int NJ=1; J=JStart; ofstream finalfile; finalfile.open("FinalFile.dat"); finalfile << "NJstep" << " " << "V" << " " << "J_Total(A/cm2)" << " " << "Bulk_Recombination(A/cm2)" << " " <<"Interface_Recombination(A/cm2)"<< " " << "Decay(A/cm2)" << " " << "J_Hole_Corrected(A/cm2)" << " " << "J_Electron_Corrected(A/cm2)" << " " << "dJ_total" << " " << "dJ_Hole_Corrected" << " " << "dJ_Electron_Corrected" << " " << "Check" << " "<<"checkExciton" <<" " << "t(s)" <<" " <<"dt(s)" <<endl; int j; p[0]=J*1.0e-14/(e*dm); // give the elctrode some carriers to begin the step, dt=1.0e-14 // here, maybe not accurate, we can change later n[N+1]=p[0]; for (NJ=1;NJ<=NJStep;NJ++) { check[0]=0; t=0; for(i=0;i<=N+1;i++){R[i]=0;Rif[i]=0;Rnew[i]=0;s[i]=0;} for (i=1;i<=NTMAX && check[0]==0;i++)

169

{ for (j=1;j<=100;j++) {

V=Calculate_Voltage(p,n,F,Fint,Eh0,El0,Eh,El,Eanode,Ecathode,N,L,up, un, up0, un0, Fp0,Fn0,dm,diel_0,diel_r,e); dt=Estimate_dt

(p,n,F,Fint,up,un,N,dm,diel_0,diel_r,e,ts_factor,Hole_Injection_Factor,Electron_Injection_Factor,Hole_Injection_B,Electron_Injection_B);

t=t+dt; Calculate_ddR(p, n, F, Fint,up, un, up0,un0,Eh0, El0,sigma00,R, Rif, Rnew,s,dp_plus_drift, dp_minus_drift, dn_plus_drift, dn_minus_drift,dp_plus_diff, dp_minus_diff, dn_plus_diff,dn_minus_diff, NLayer,N, NL, dm, dt,diel_0,diel_r,e,Hole_Injection_Factor,Electron_Injection_Factor,Recombination_Interface_Factor,Hole_Injection_B,Electron_Injection_B); p[0]=p[0]+J*dt/(e*dm); n[N+1]=n[N+1]+J*dt/(e*dm); //***** the importance of current control } Calculate_Current(jt,jpR, jnR,jp, jn, jp_drift, jp_diff, jn_drift, jn_diff,R, Rif, Rnew,s,IntR,IntRif, dp_plus_drift, dp_minus_drift, dn_plus_drift, dn_minus_drift,dp_plus_diff, dp_minus_diff, dn_plus_diff, dn_minus_diff, NLayer, N, NL,dm,dt,diel_0, diel_r,e,converge,check,NJ,Fjtave, Fdjt, FjpRave,FdjpR,FjnRave,FdjnR, FIntR,FIntRif,FCheck); } Calculate_Exciton(p, n, F, Fint, up, un, up0,un0,Eh0, El0, sigma00,R, Rif, Rnew,s,NLayer,N, NL, dm, diel_0, diel_r, e, converge, Nmax, Ldiffusion, tlife, qyield, hole_quench_distance, electron_quench_distance,NJ,FIntS,FCheckExciton); finalfile << NJ << " " << V << " " << Fjtave[NJ] << " " << FIntR[NJ] << " "<<FIntRif[NJ] <<" " << FIntS[NJ] << " " << FjpRave[NJ] << " " << FjnRave[NJ] << " " << Fdjt[NJ] << " " << FdjpR[NJ] << " " << FdjnR[NJ] << " " << FCheck[NJ] << " " <<FCheckExciton[NJ] <<" " << t <<" " << dt <<endl; Write_Files ( position,p,n,F,V,jt,R,Rif,Rnew,s,Eh,El,up,un,jpR,jnR,jp,jn,jp_drift,jp_diff,jn_drift,jn_diff,N,NJ); J=J/2; // set the current density to be half for the next step } //********************Part V: End ***** finalfile.close(); return 0; }

170

A.3 Sample input (input.txt)

************ Voltage Parameters ************************ Start Current density (unit A/cm2) JStart= 0.08 Number of steps NJStep= 10 ************************************************************* *************** Control Parameters ********************** Control time distance ts_factor= 5 Numbers of calculations NTMAX= 10000 Test of convergence converge= 0.01 Hole_Injection_Factor= 1.0e21 Hole_Injection_B= 0.2 Electron_Injection_Factor= 1.0e21 Electron_Injection_B= 0.2 Recombination_Interface_Factor= 1 Exciton calculate Nmax= 100000 Exciton diffusion length(nm)= 20 Exciton life time (s)= 1.0e-8 Relative dielectric constant= 3 ************************************************************* *************** Device Parameters************************ Monolayer thichness (unit A) dmA= 10 number of organic layers NL= 2 Anode Name: ITO

171

Anode energy level (unit ev) Eanode= -5.0 Cahode Name: Al/LiF Cathode energy level ( unit ev) Ecathode= -3.55 First layer: HTL Thickness (unit A) L1= 700 HOMO (unit ev) Eh1= -5.5 LUMO (unit ev) El1= -2.9 Hole moblity ( unit cm2/vs) up0= 2.0e-4 Hole moblity (unit (cm/V)^1/2) Fp0= 1.0e-3 Electron moblity ( unit cm2/vs) un0= 2.0e-6 Electron moblity (unit (cm/V)^1/2) Fn0= 1.0e-3 Hole Quench distance (unit A)= 30 Electron Quench distance (unit A)= 0 Quantum Yield= 0 Energy width (unit eV) sigma= 0.03 Second layer: ETL Thickness (unit A) L1= 700 HOMO (unit ev) Eh1= -5.7 LUMO (unit ev) El1= -3.1 Hole moblity ( unit cm2/vs) up0= 2.0e-8 Hole moblity (unit (cm/V)^1/2) Fp0=

172

1.0e-3 Electron moblity ( unit cm2/vs) un0= 2.0e-6 Electron moblity (unit (cm/V)^1/2) Fn0= 1.0e-3 Hole Quench distance (unit A)= 0 Electron Quench distance (unit A)= 0 Quantum Yield= 0.025 Energy width (unit eV) sigma= 0.03 ***************************************************

173

Appendix B

Program for the Simulation of OLEDs

(Optical Part)

174

B.1 Codes

// ----------------------------------------------------------------------------------- // Cavity model for OLEDs, optical part of the OLED simulation: // C++ codes to calculate the photon extraction, // angular dependence of the light output from OLED devices // ----------------------------------------------------------------------------------- #include <math.h> #include <float.h> #include <cmath> #include <iostream> #include <fstream> #include <string> #include <sstream> #include <stdlib.h> #include <stdio.h> #include <complex> using namespace std; double findpeak( double* wv, double x[][50]); double findpeak2( double* wv, double x[][200]); double findpeak3( double* wv, double* x); double findwidth( double* wv, double x[][50]); double findwidth2( double* wv, double x[][200]); double findwidth3( double* wv, double* x); double deltauv(double x1, double y1, double x2, double y2); void LumColor(double CIExyz[][3],double *PL, double& Lum, double& CIEx, double& CIEy); double MyGauss1(int order, double kbegin, double kend, double lambda, int position_Layer,double position_x,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE); // integrate K[] use complex curve double MyGauss2(int order, double kbegin, double kend, double lambda, int position_Layer,double position_x,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE); // integrate K[] along the real axix

175

double MyGauss3(int order, double kbegin, double kend, double lambda, int position_Layer,double position_x,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE); // integrate KPT[] along the complex curve, better not use it double MyGauss4(int order, double kbegin, double kend, double lambda, int position_Layer,double position_x,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE); // integrate KPT[] along the real axix complex <double> R(string ss,complex <double> n1,complex <double> n2,complex <double> kz1,complex <double> kz2); complex <double> T(string ss,complex <double> n1,complex <double> n2,complex <double> kz1,complex <double> kz2); void Calculate_Matrix(double lambda, complex <double> kx, int position_Layer,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE); void Calculate_P(double lambda, double theta, int position_Layer,double position_x,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE,double* result); double Etotal2(double kmin, double lambda, int position_Layer,double position_x,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE); void Read(double CIExyz[][3], double CF[][3],complex<double> n[][50], double* wv, double PL[][50], string* PLname, int NEmitter, string* layername, int NLayer); void Read(double CIExyz[][3], double CF[][3],complex<double> n[][50], double* wv, double PL[][50], string* PLname, int NEmitter, string* layername, int NLayer) // read the OLED layer structure and emitter information { double lambda; double tempD; string word, line; int i,layer,j; double nr,kr; stringstream ss;

176

for(i=0;i<=80;i++) { wv[i]=380+i*5; } //--------------------------------------------------------- ifstream myfile ("CIExyz.txt"); if (myfile.is_open()) { myfile >>word >> word >> word >> word; for(i=0;i<=80;i++) { myfile >> lambda >> CIExyz[i][0] >> CIExyz[i][1] >> CIExyz[i][2]; } } else { cout << "Unable to open CIExyz.txt"<< endl; return; } myfile.close(); //--------------------------------------------------------- myfile.open("CF.txt"); if (myfile.is_open()) { myfile >>word >> word >> word >> word; for(i=0;i<=80;i++) { myfile >> lambda >> CF[i][0] >> CF[i][1] >> CF[i][2]; } } else { cout << "Unable to open CF.txt"<< endl; return; } myfile.close(); //--------------------------------------------------------- // PL1 => PL NEmitter for( j=1;j<=NEmitter;j++) { ss.str("");

177

ss <<PLname[j] <<".txt"; myfile.open(ss.str().c_str()); if (myfile.is_open()) { myfile >>word >> word; for(i=0;i<=80;i++) { myfile >> lambda >> PL[i][j]; } } else { cout << "Unable to open " << ss.str() << endl; return; } myfile.close(); } //--------------------------------------------------------- // Layer 0 => Layer NLayer+1 for( layer=0;layer<=(NLayer+1);layer++) { if(layername[layer][0]=='n' && layername[layer][1]=='=')

// if the name is "n=x.xx" { tempD=atof(layername[layer].substr(2,layername[layer].length()-1).c_str()); for(i=0;i<=80;i++) { n[i][layer]=complex<double>(tempD,0); } } else{ ss.str(""); ss <<layername[layer] <<".txt"; myfile.open(ss.str().c_str()); if (myfile.is_open()) {

178

myfile >>word >> word >> word; for(i=0;i<=80;i++) { myfile >> lambda >> nr >> kr; n[i][layer]=complex<double>(nr,kr); } } else { cout << "Unable to open "<< ss.str() << endl; return; } myfile.close(); } } } double findpeak( double* wv, double x[][50]) // find the peak of a spectrum { double temp=x[0][0]; int index=0; for(int i=1;i<=80;i++) { if (x[i][0]>temp) {temp=x[i][0];index=i;}; } return wv[index]; }; double findpeak2( double* wv, double x[][200]) { double temp=x[0][0]; int index=0; for(int i=1;i<=80;i++) { if (x[i][0]>temp) {temp=x[i][0];index=i;}; } return wv[index]; }; double findpeak3( double* wv, double* x) {

179

double temp=x[0]; int index=0; for(int i=1;i<=80;i++) { if (x[i]>temp) {temp=x[i];index=i;}; } return wv[index]; }; double findwidth( double* wv, double x[][50]) // find the FWHM of a spectrum { double temp=x[0][0]; int index=0; for(int i=1;i<=80;i++) { if (x[i][0]>temp) {temp=x[i][0];index=i;}; } int a=0,b=0; for(int i=1;i<=80;i++) { if (x[i][0]>=0.5*temp && x[i-1][0]<0.5*temp ) {a=i;}; if (x[i-1][0]>=0.5*temp && x[i][0]<0.5*temp ) {b=i;}; } return wv[b]-wv[a]; }; double findwidth2( double* wv, double x[][200]) { double temp=x[0][0]; int index=0; for(int i=1;i<=80;i++) { if (x[i][0]>temp) {temp=x[i][0];index=i;}; } int a=0,b=0;

180

for(int i=1;i<=80;i++) { if (x[i][0]>=0.5*temp && x[i-1][0]<0.5*temp ) {a=i;}; if (x[i-1][0]>=0.5*temp && x[i][0]<0.5*temp ) {b=i;}; } return wv[b]-wv[a]; }; double findwidth3( double* wv, double* x) { double temp=x[0]; int index=0; for(int i=1;i<=80;i++) { if (x[i]>temp) {temp=x[i];index=i;}; } int a=0,b=0; for(int i=1;i<=80;i++) { if (x[i]>=0.5*temp && x[i-1]<0.5*temp ) {a=i;}; if (x[i-1]>=0.5*temp && x[i]<0.5*temp ) {b=i;}; } return wv[b]-wv[a]; }; double deltauv(double x1, double y1, double x2, double y2) // from the CIEx,y values of two color points, calculate their distance in the CIEu'v'frame { double u1=4*x1/(-2*x1+12*y1+3); double v1=9*y1/(-2*x1+12*y1+3); double u2=4*x2/(-2*x2+12*y2+3); double v2=9*y2/(-2*x2+12*y2+3); return sqrt((u1-u2)*(u1-u2)+(v1-v2)*(v1-v2));

181

} void LumColor(double CIExyz[][3],double *PL, double& Lum, double& CIEx, double& CIEy) // calculate the luminance and CIExy color points from any spectrum { double x=0,y=0,z=0,total=0; int i; for(i=0;i<=80;i++) { x+=CIExyz[i][0]*PL[i]; y+=CIExyz[i][1]*PL[i]; z+=CIExyz[i][2]*PL[i]; } total=x+y+z; Lum=y; CIEx=x/total; CIEy=y/total; } complex <double> R(string ss,complex <double> n1,complex <double> n2,complex <double> kz1,complex <double> kz2) // the complex reflection coefficient at the interface for TE and TM waves { complex <double> reflection; if (ss=="TE") { reflection=(kz1-kz2)/(kz1+kz2); } if(ss=="TM") { reflection=(n2*n2*kz1-n1*n1*kz2)/(n2*n2*kz1+n1*n1*kz2); } return reflection; } complex <double> T(string ss,complex <double> n1,complex <double> n2,complex <double> kz1,complex <double> kz2)

182

// the complex transmission coefficient at the interface for TE and TM waves { complex <double> transmission; if (ss=="TE") { transmission=(kz1+kz1)/(kz1+kz2); } if(ss=="TM") { transmission=(n2*n2*kz1+n2*n2*kz1)/(n2*n2*kz1+n1*n1*kz2);

// this is the transition from Neyts paper // they use different transmision definition

// therefore the power transmission efficient is also changed a little bit // the index mismatch between two layers need to considered carefully // to get the correct power transmission

} return transmission; } void Calculate_Matrix(double lambda, complex <double> kx, int position_Layer,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE) // calculate the total reflection and transmission coefficient // for the whole OLED stack // the value is used to calculate the dipole emission from the OLED cavity { complex <double> kz[50],cd[50]; double pi =3.1415926; complex <double> kk (2*pi/lambda,0); int i; for( i=0;i<=NLayer+1;i++) { kz[i]=sqrt(kk*n[i]*kk*n[i]-kx*kx); } for( i=1;i<=NLayer;i++)

183

{ cd[i]=complex<double> (d[i],0);

// cd is the complex version of thickness } complex <double> r,t; complex <double> a,b; complex <double> c1 (1,0); complex <double> c2 (2,0); complex <double> cj (0,1);

// define the complex version of 1 2 and j //****** r+, t+, r-, t-, R, T, for TM ************************ // r+,t+ r=R("TM",n[1],n[0],kz[1],kz[0]);

// The R and T at the interface closest to air t=T("TM",n[1],n[0],kz[1],kz[0]); for ( i=2;i<=position_Layer;i++) { a=R("TM",n[i],n[i-1],kz[i],kz[i-1]); b=T("TM",n[i],n[i-1],kz[i],kz[i-1]); t=(b*t*exp(cj*kz[i-1]*cd[i-1]))/(c1+a*r*exp(c2*cj*kz[i-1]*cd[i-1])); r=(a+r*exp(c2*cj*kz[i-1]*cd[i-1]))/(c1+a*r*exp(c2*cj*kz[i-1]*cd[i-1])); } TM[1]=r; // r+ for TM from emission to substrate air TM[2]=t; // t+ // r-,t- r=R("TM",n[NLayer],n[NLayer+1],kz[NLayer],kz[NLayer+1]); t=T("TM",n[NLayer],n[NLayer+1],kz[NLayer],kz[NLayer+1]); for ( i=(NLayer-1);i>=position_Layer;i--) { a=R("TM",n[i],n[i+1],kz[i],kz[i+1]); b=T("TM",n[i],n[i+1],kz[i],kz[i+1]); t=(b*t*exp(cj*kz[i+1]*cd[i+1]))/(c1+a*r*exp(c2*cj*kz[i+1]*cd[i+1])); r=(a+r*exp(c2*cj*kz[i+1]*cd[i+1]))/(c1+a*r*exp(c2*cj*kz[i+1]*cd[i+1])); } TM[3]=r; // r- for TM from emission to Al air TM[4]=t; // t- // R,T

184

r=R("TM",n[NLayer],n[NLayer+1],kz[NLayer],kz[NLayer+1]); t=T("TM",n[NLayer],n[NLayer+1],kz[NLayer],kz[NLayer+1]); for ( i=(NLayer-1);i>=0;i--) { a=R("TM",n[i],n[i+1],kz[i],kz[i+1]); b=T("TM",n[i],n[i+1],kz[i],kz[i+1]); t=(b*t*exp(cj*kz[i+1]*cd[i+1]))/(c1+a*r*exp(c2*cj*kz[i+1]*cd[i+1])); r=(a+r*exp(c2*cj*kz[i+1]*cd[i+1]))/(c1+a*r*exp(c2*cj*kz[i+1]*cd[i+1])); } TM[5]=r; // R for the whole matrix for TM from substrate air to Al air TM[6]=t; // T for the whole matrix //*******r+, t+, r-, t-, R, T, for TE ************************************ // r+,t+ r=R("TE",n[1],n[0],kz[1],kz[0]);

// The R and T at the interface closest to air t=T("TE",n[1],n[0],kz[1],kz[0]); for ( i=2;i<=position_Layer;i++) { a=R("TE",n[i],n[i-1],kz[i],kz[i-1]); b=T("TE",n[i],n[i-1],kz[i],kz[i-1]); t=(b*t*exp(cj*kz[i-1]*cd[i-1]))/(c1+a*r*exp(c2*cj*kz[i-1]*cd[i-1])); r=(a+r*exp(c2*cj*kz[i-1]*cd[i-1]))/(c1+a*r*exp(c2*cj*kz[i-1]*cd[i-1])); } TE[1]=r; // r+ for TE from emission to substrate air TE[2]=t; // t+ // r-,t- r=R("TE",n[NLayer],n[NLayer+1],kz[NLayer],kz[NLayer+1]); t=T("TE",n[NLayer],n[NLayer+1],kz[NLayer],kz[NLayer+1]); for ( i=(NLayer-1);i>=position_Layer;i--) { a=R("TE",n[i],n[i+1],kz[i],kz[i+1]); b=T("TE",n[i],n[i+1],kz[i],kz[i+1]); t=(b*t*exp(cj*kz[i+1]*cd[i+1]))/(c1+a*r*exp(c2*cj*kz[i+1]*cd[i+1]));

r=(a+r*exp(c2*cj*kz[i+1]*cd[i+1]))/(c1+a*r*exp(c2*cj*kz[i+1]*cd[i+1])); } TE[3]=r; // r- for TE from emission to Al air TE[4]=t; // t-

185

// R,T r=R("TE",n[NLayer],n[NLayer+1],kz[NLayer],kz[NLayer+1]); t=T("TE",n[NLayer],n[NLayer+1],kz[NLayer],kz[NLayer+1]); for ( i=(NLayer-1);i>=0;i--) { a=R("TE",n[i],n[i+1],kz[i],kz[i+1]); b=T("TE",n[i],n[i+1],kz[i],kz[i+1]); t=(b*t*exp(cj*kz[i+1]*cd[i+1]))/(c1+a*r*exp(c2*cj*kz[i+1]*cd[i+1])); r=(a+r*exp(c2*cj*kz[i+1]*cd[i+1]))/(c1+a*r*exp(c2*cj*kz[i+1]*cd[i+1])); } TE[5]=r; // R for the whole matrix for TE from substrate air to Al air TE[6]=t; // T for the whole matrix }

void Calculate_P(double lambda, double theta, int position_Layer,double position_x,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE,double* result) // calculate the angular light power density // in the substrate and in the air, respectively { double pi=3.1415926; double realde=d[position_Layer]; double realzp=position_x; double realzm=realde-realzp; complex <double> kx,ke,kp,km,kze,kzp,kzm; complex <double> de,zp,zm; complex <double> c2j(0,2); complex <double> c1(1,0); complex <double> aTM,aTMp,aTMm,aTE,aTEp,aTEm; double KPT[10]={0},P[10]={0}; de=complex <double> (realde,0); zp=complex <double> (realzp,0); zm=complex <double> (realzm,0); complex <double> kk (2*pi/lambda,0); ke=kk*n[position_Layer]; kp=kk*n[0]; km=kk*n[NLayer+1];

186

// ******************************* calculate P[angle=theta] // assume the theta is in the substrate kx=complex <double> (2*pi*real(n[0])*sin(theta)/lambda,0); kze=sqrt(ke*ke-kx*kx); kzp=sqrt(kp*kp-kx*kx); kzm=sqrt(km*km-kx*kx); Calculate_Matrix(lambda,kx,position_Layer, NLayer,d, n,TM,TE); aTMp=TM[1]*exp(c2j*kze*zp); aTMm=TM[3]*exp(c2j*kze*zm); aTM=aTMp*aTMm; aTEp=TE[1]*exp(c2j*kze*zp); aTEm=TE[3]*exp(c2j*kze*zm); aTE=aTEp*aTEm; KPT[1]=0.75*real(kx*kx/(ke*ke*ke*kze))*0.5*norm((c1+aTMm)/(c1-aTM))*norm(TM[2])*abs(kzp/kze)*norm(n[position_Layer]/n[0]); KPT[2]=0; KPT[3]=0.375*real(kze/(ke*ke*ke))*0.5*norm((c1-aTMm)/(c1-aTM))*norm(TM[2])*abs(kzp/kze)*norm(n[position_Layer]/n[0]); KPT[4]=0.375*real(c1/(ke*kze))*0.5*norm((c1+aTEm)/(c1-aTE))*norm(TE[2])*abs(kzp/kze); KPT[5]=KPT[1]/3.0+2.0*KPT[3]/3.0; KPT[6]=KPT[2]/3.0+2.0*KPT[4]/3.0; KPT[7]=KPT[5]+KPT[6]; P[1]=norm(kp)*cos(theta)*KPT[1]/pi; P[2]=norm(kp)*cos(theta)*KPT[2]/pi; P[3]=norm(kp)*cos(theta)*KPT[3]/pi; P[4]=norm(kp)*cos(theta)*KPT[4]/pi; P[5]=P[1]/3.0+2.0*P[3]/3.0; P[6]=P[2]/3.0+2.0*P[4]/3.0; P[7]=P[5]+P[6]; result[1]=P[7]; // ok, this is the P of angle theta in the substrate // now calculate the theta in the air double airtheta=theta;

187

complex<double> nair (1,0); complex<double> kzair,crair; double Rair,Roled; theta=asin(1.0*sin(airtheta)/real(n[0])); kx=complex <double> (2*pi*real(n[0])*sin(theta)/lambda,0); kze=sqrt(ke*ke-kx*kx); kzp=sqrt(kp*kp-kx*kx); kzm=sqrt(km*km-kx*kx); kzair=sqrt(kk*nair*kk*nair-kx*kx); Calculate_Matrix(lambda,kx,position_Layer, NLayer,d, n,TM,TE); aTMp=TM[1]*exp(c2j*kze*zp); aTMm=TM[3]*exp(c2j*kze*zm); aTM=aTMp*aTMm; aTEp=TE[1]*exp(c2j*kze*zp); aTEm=TE[3]*exp(c2j*kze*zm); aTE=aTEp*aTEm; KPT[1]=0.75*real(kx*kx/(ke*ke*ke*kze))*0.5*norm((c1+aTMm)/(c1-aTM))*norm(TM[2])*abs(kzp/kze)*norm(n[position_Layer]/n[0]); KPT[2]=0; KPT[3]=0.375*real(kze/(ke*ke*ke))*0.5*norm((c1-aTMm)/(c1-aTM))*norm(TM[2])*abs(kzp/kze)*norm(n[position_Layer]/n[0]); KPT[4]=0.375*real(c1/(ke*kze))*0.5*norm((c1+aTEm)/(c1-aTE))*norm(TE[2])*abs(kzp/kze); KPT[5]=KPT[1]/3.0+2.0*KPT[3]/3.0; KPT[6]=KPT[2]/3.0+2.0*KPT[4]/3.0; KPT[7]=KPT[5]+KPT[6]; P[1]=norm(kp)*cos(theta)*KPT[1]/pi; P[2]=0; P[3]=norm(kp)*cos(theta)*KPT[3]/pi; P[4]=norm(kp)*cos(theta)*KPT[4]/pi; crair=R("TM",n[0],nair,kzp,kzair); Rair=norm(crair); Roled=norm(TM[5]);

188

P[1]=P[1]*norm(nair/n[0])*cos(airtheta)/cos(theta)*(1-Rair)/(1-Rair*Roled); crair=R("TE",n[0],nair,kzp,kzair); Rair=norm(crair); Roled=norm(TE[5]); P[3]=P[3]*norm(nair/n[0])*cos(airtheta)/cos(theta)*(1-Rair)/(1-Rair*Roled); P[4]=P[4]*norm(nair/n[0])*cos(airtheta)/cos(theta)*(1-Rair)/(1-Rair*Roled); P[5]=P[1]/3.0+2.0*P[3]/3.0; P[6]=P[2]/3.0+2.0*P[4]/3.0; P[7]=P[5]+P[6]; result[4]=P[7]; } double Etotal2(double kmin, double lambda, int position_Layer,double position_x,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE) // function to do part of the integration of the light power density { double pi=3.1415926; double realde=d[position_Layer]; double realzp=position_x; double realzm=realde-realzp; complex <double> kx,ke,kp,km,kze,kzp,kzm; complex <double> de,zp,zm; complex <double> c2j(0,2); complex <double> c1(1,0); complex <double> aTM,aTMp,aTMm,aTE,aTEp,aTEm; double K[10]={0},KPT[10]={0},P[10]={0}; de=complex <double> (realde,0); zp=complex <double> (realzp,0); zm=complex <double> (realzm,0); complex <double> kk (2*pi/lambda,0);

189

ke=kk*n[position_Layer]; kp=kk*n[0]; km=kk*n[NLayer+1]; //********************************************************** calculate Etot double Etot=0; kx=complex <double> (kmin,0); kze=sqrt(ke*ke-kx*kx); kzp=sqrt(kp*kp-kx*kx); kzm=sqrt(km*km-kx*kx); Calculate_Matrix(lambda,kx,position_Layer, NLayer,d, n,TM,TE); aTMp=TM[1]*exp(c2j*kze*zp); aTMm=TM[3]*exp(c2j*kze*zm); aTM=aTMp*aTMm; aTEp=TE[1]*exp(c2j*kze*zp); aTEm=TE[3]*exp(c2j*kze*zm); aTE=aTEp*aTEm; K[1]=0.375/real(ke)/real(ke)/real(ke)*(imag(TM[1])*exp(-2*kmin*realzp)/realzp/realzp/realzp*(1+2*kmin*realzp+2*kmin*kmin*realzp*realzp)+imag(TM[3])*exp(-2*kmin*realzm)/realzm/realzm/realzm*(1+2*kmin*realzm+2*kmin*kmin*realzm*realzm)); K[2]=0; K[3]=0.5*K[1]; K[4]=0; K[5]=K[1]/3.0+2.0*K[3]/3.0; K[6]=K[2]/3.0+2.0*K[4]/3.0; K[7]=K[5]+K[6]; Etot=K[7]; return Etot; } // the Gaussion quadrature method code // is modified from Pavel Holoborodko's code

190

/* Numerical Integration by Gauss-Legendre Quadrature Formulas of high orders. High-precision abscissas and weights are used. Project homepage: http://www.holoborodko.com/pavel/?page_id=679 Contact e-mail: pavel@holoborodko.com Copyright (c)2007-2010 Pavel Holoborodko All rights reserved. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. Redistributions of any form whatsoever must retain the following acknowledgment: " This product includes software developed by Pavel Holoborodko Web: http://www.holoborodko.com/pavel/ e-mail: pavel@holoborodko.com " 4. This software cannot be, by any means, used for any commercial purpose without the prior permission of the copyright holder. Any of the above conditions can be waived if you get permission from the copyright holder. THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT

191

SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. Contributors Konstantin Holoborodko - Optimization of Legendre polynomial computing. */ #ifndef __GAUSS_LEGENDRE_H__ #define __GAUSS_LEGENDRE_H__ #ifdef __cplusplus extern "C" { #endif /* Numerical computation of int(f(x),x=a..b) by Gauss-Legendre n-th order high precision quadrature [in]n - quadrature order [in]f - integrand [in]data - pointer on user-defined data which will be passed to f every time it called (as second parameter). [in][a,b] - interval of integration return: -computed integral value or -1.0 if n order quadrature is not supported */ double gauss_legendre(int n, double (*f)(double,void*), void* data, double a, double b); /* 2D Numerical computation of int(f(x,y),x=a..b,y=c..d) by Gauss-Legendre n-th order high precision quadrature [in]n - quadrature order [in]f - integrand [in]data - pointer on user-defined data which will

192

be passed to f every time it called (as third parameter). [in][a,b]x[c,d] - interval of integration return: -computed integral value or -1.0 if n order quadrature is not supported */ double gauss_legendre_2D_cube(int n, double (*f)(double,double,void*), void* data, double a, double b, double c, double d); /* Computing of abscissas and weights for Gauss-Legendre quadrature for any(reasonable) order n [in] n - order of quadrature [in] eps - required precision (must be eps>=macheps(double), usually eps = 1e-10 is ok) [out]x - abscisass, size = (n+1)>>1 [out]w - weights, size = (n+1)>>1 */ void gauss_legendre_tbl(int n, double* x, double* w, double eps); #ifdef __cplusplus } #endif #endif /* __GAUSS_LEGENDRE_H__ */ #ifndef PI #define PI 3.1415926535897932384626433832795028841971693993751 #endif #ifndef FABS #define FABS(a) ((a)>=0?(a):-(a)) #endif /* n = 2 */ static double x2[1] = {0.5773502691896257645091488}; static double w2[1] = {1.0000000000000000000000000};

193

/* n = 4 */ static double x4[2] = {0.3399810435848562648026658,0.8611363115940525752239465}; static double w4[2] = {0.6521451548625461426269361,0.3478548451374538573730639}; /* n = 6 */ static double x6[3] = {0.2386191860831969086305017,0.6612093864662645136613996,0.9324695142031520278123016}; static double w6[3] = {0.4679139345726910473898703,0.3607615730481386075698335,0.1713244923791703450402961}; /* n = 8 */ static double x8[4] = {0.1834346424956498049394761,0.5255324099163289858177390,0.7966664774136267395915539,0.9602898564975362316835609}; static double w8[4] = {0.3626837833783619829651504,0.3137066458778872873379622,0.2223810344533744705443560,0.1012285362903762591525314}; /* n = 10 */ static double x10[5] = {0.1488743389816312108848260,0.4333953941292471907992659,0.6794095682990244062343274,0.8650633666889845107320967,0.9739065285171717200779640}; static double w10[5] = {0.2955242247147528701738930,0.2692667193099963550912269,0.2190863625159820439955349,0.1494513491505805931457763,0.0666713443086881375935688}; /* n = 12 */ static double x12[6] = {0.1252334085114689154724414,0.3678314989981801937526915,0.5873179542866174472967024,0.7699026741943046870368938,0.9041172563704748566784659,0.9815606342467192506905491}; static double w12[6] = {0.2491470458134027850005624,0.2334925365383548087608499,0.2031674267230659217490645,0.1600783285433462263346525,0.1069393259953184309602547,0.0471753363865118271946160};

194

/* n = 14 */ static double x14[7] = {0.1080549487073436620662447,0.3191123689278897604356718,0.5152486363581540919652907,0.6872929048116854701480198,0.8272013150697649931897947,0.9284348836635735173363911,0.9862838086968123388415973}; static double w14[7] = {0.2152638534631577901958764,0.2051984637212956039659241,0.1855383974779378137417166,0.1572031671581935345696019,0.1215185706879031846894148,0.0801580871597602098056333,0.0351194603317518630318329}; /* n = 16 */ static double x16[8] = {0.0950125098376374401853193,0.2816035507792589132304605,0.4580167776572273863424194,0.6178762444026437484466718,0.7554044083550030338951012,0.8656312023878317438804679,0.9445750230732325760779884,0.9894009349916499325961542}; static double w16[8] = {0.1894506104550684962853967,0.1826034150449235888667637,0.1691565193950025381893121,0.1495959888165767320815017,0.1246289712555338720524763,0.0951585116824927848099251,0.0622535239386478928628438,0.0271524594117540948517806}; /* n = 18 */ static double x18[9] = {0.0847750130417353012422619,0.2518862256915055095889729,0.4117511614628426460359318,0.5597708310739475346078715,0.6916870430603532078748911,0.8037049589725231156824175,0.8926024664975557392060606,0.9558239495713977551811959,0.9915651684209309467300160}; static double w18[9] = {0.1691423829631435918406565,0.1642764837458327229860538,0.1546846751262652449254180,0.1406429146706506512047313,0.1225552067114784601845191,0.1009420441062871655628140,0.0764257302548890565291297,0.0497145488949697964533349,0.0216160135264833103133427}; /* n = 20 */ static double x20[10] = {0.0765265211334973337546404,0.2277858511416450780804962,0.3737060887154195606725482,0.5108670019508270980043641,0.6360536807265150254528367,0.7463319064601507926143051,0.8391169718222188233945291,0.9122344282513259058677524,0.9639719272779137912676661,0.9931285991850949247861224};

195

static double w20[10] = {0.1527533871307258506980843,0.1491729864726037467878287,0.1420961093183820513292983,0.1316886384491766268984945,0.1181945319615184173123774,0.1019301198172404350367501,0.0832767415767047487247581,0.0626720483341090635695065,0.0406014298003869413310400,0.0176140071391521183118620}; /* n = 32 */ static double x32[16] = {0.0483076656877383162348126,0.1444719615827964934851864,0.2392873622521370745446032,0.3318686022821276497799168,0.4213512761306353453641194,0.5068999089322293900237475,0.5877157572407623290407455,0.6630442669302152009751152,0.7321821187402896803874267,0.7944837959679424069630973,0.8493676137325699701336930,0.8963211557660521239653072,0.9349060759377396891709191,0.9647622555875064307738119,0.9856115115452683354001750,0.9972638618494815635449811}; static double w32[16] = {0.0965400885147278005667648,0.0956387200792748594190820,0.0938443990808045656391802,0.0911738786957638847128686,0.0876520930044038111427715,0.0833119242269467552221991,0.0781938957870703064717409,0.0723457941088485062253994,0.0658222227763618468376501,0.0586840934785355471452836,0.0509980592623761761961632,0.0428358980222266806568786,0.0342738629130214331026877,0.0253920653092620594557526,0.0162743947309056706051706,0.0070186100094700966004071}; /* n = 64 */ static double x64[32] = {0.0243502926634244325089558,0.0729931217877990394495429,0.1214628192961205544703765,0.1696444204239928180373136,0.2174236437400070841496487,0.2646871622087674163739642,0.3113228719902109561575127,0.3572201583376681159504426,0.4022701579639916036957668,0.4463660172534640879849477,0.4894031457070529574785263,0.5312794640198945456580139,0.5718956462026340342838781,0.6111553551723932502488530,0.6489654712546573398577612,0.6852363130542332425635584,0.7198818501716108268489402,0.7528199072605318966118638,0.7839723589433414076102205,0.8132653151227975597419233,0.8406292962525803627516915,0.8659993981540928197607834,0.8893154459951141058534040,0.9105221370785028057563807,0.9295691721319395758214902,0.9464113748584028160624815,0.9610087996520537189186141,0.9733268277899109637418535,0.9833362538846259569312993,0.9910133714767443207393824,0.9963401167719552793469245,0.9993050417357721394569056}; static double w64[32] = {0.0486909570091397203833654,0.0485754674415034269347991,0.048344

196

7622348029571697695,0.0479993885964583077281262,0.0475401657148303086622822,0.0469681828162100173253263,0.0462847965813144172959532,0.0454916279274181444797710,0.0445905581637565630601347,0.0435837245293234533768279,0.0424735151236535890073398,0.0412625632426235286101563,0.0399537411327203413866569,0.0385501531786156291289625,0.0370551285402400460404151,0.0354722132568823838106931,0.0338051618371416093915655,0.0320579283548515535854675,0.0302346570724024788679741,0.0283396726142594832275113,0.0263774697150546586716918,0.0243527025687108733381776,0.0222701738083832541592983,0.0201348231535302093723403,0.0179517157756973430850453,0.0157260304760247193219660,0.0134630478967186425980608,0.0111681394601311288185905,0.0088467598263639477230309,0.0065044579689783628561174,0.0041470332605624676352875,0.0017832807216964329472961}; /* n = 96 */ static double x96[48] = {0.0162767448496029695791346,0.0488129851360497311119582,0.0812974954644255589944713,0.1136958501106659209112081,0.1459737146548969419891073,0.1780968823676186027594026,0.2100313104605672036028472,0.2417431561638400123279319,0.2731988125910491414872722,0.3043649443544963530239298,0.3352085228926254226163256,0.3656968614723136350308956,0.3957976498289086032850002,0.4254789884073005453648192,0.4547094221677430086356761,0.4834579739205963597684056,0.5116941771546676735855097,0.5393881083243574362268026,0.5665104185613971684042502,0.5930323647775720806835558,0.6189258401254685703863693,0.6441634037849671067984124,0.6687183100439161539525572,0.6925645366421715613442458,0.7156768123489676262251441,0.7380306437444001328511657,0.7596023411766474987029704,0.7803690438674332176036045,0.8003087441391408172287961,0.8194003107379316755389996,0.8376235112281871214943028,0.8549590334346014554627870,0.8713885059092965028737748,0.8868945174024204160568774,0.9014606353158523413192327,0.9150714231208980742058845,0.9277124567223086909646905,0.9393703397527552169318574,0.9500327177844376357560989,0.9596882914487425393000680,0.9683268284632642121736594,0.9759391745851364664526010,0.9825172635630146774470458,0.9880541263296237994807628,0.9925439003237626245718923,0.9959818429872092906503991,0.9983643758631816777241494,0.9996895038832307668276901}; static double w96[48] = {0.0325506144923631662419614,0.0325161187138688359872055,0.0324471637140642693640128,0.0323438225685759284287748,0.0322062047940302506686671,0.0320344562319926632181390,0.0318287588944110065347537,0.0315893307707271685580207,0.0313164255968613558127843,0.0310103325863138374232498,0.0306713761236691490142288,0.03029991542

197

08275937940888,0.0298963441363283859843881,0.0294610899581679059704363,0.0289946141505552365426788,0.0284974110650853856455995,0.0279700076168483344398186,0.0274129627260292428234211,0.0268268667255917621980567,0.0262123407356724139134580,0.0255700360053493614987972,0.0249006332224836102883822,0.0242048417923646912822673,0.0234833990859262198422359,0.0227370696583293740013478,0.0219666444387443491947564,0.0211729398921912989876739,0.0203567971543333245952452,0.0195190811401450224100852,0.0186606796274114673851568,0.0177825023160452608376142,0.0168854798642451724504775,0.0159705629025622913806165,0.0150387210269949380058763,0.0140909417723148609158616,0.0131282295669615726370637,0.0121516046710883196351814,0.0111621020998384985912133,0.0101607705350084157575876,0.0091486712307833866325846,0.0081268769256987592173824,0.0070964707911538652691442,0.0060585455042359616833167,0.0050142027429275176924702,0.0039645543384446866737334,0.0029107318179349464084106,0.0018539607889469217323359,0.0007967920655520124294381}; /* n = 100 */ static double x100[50] = {0.0156289844215430828722167,0.0468716824215916316149239,0.0780685828134366366948174,0.1091892035800611150034260,0.1402031372361139732075146,0.1710800805386032748875324,0.2017898640957359972360489,0.2323024818449739696495100,0.2625881203715034791689293,0.2926171880384719647375559,0.3223603439005291517224766,0.3517885263724217209723438,0.3808729816246299567633625,0.4095852916783015425288684,0.4378974021720315131089780,0.4657816497733580422492166,0.4932107892081909335693088,0.5201580198817630566468157,0.5465970120650941674679943,0.5725019326213811913168704,0.5978474702471787212648065,0.6226088602037077716041908,0.6467619085141292798326303,0.6702830156031410158025870,0.6931491993558019659486479,0.7153381175730564464599671,0.7368280898020207055124277,0.7575981185197071760356680,0.7776279096494954756275514,0.7968978923903144763895729,0.8153892383391762543939888,0.8330838798884008235429158,0.8499645278795912842933626,0.8660146884971646234107400,0.8812186793850184155733168,0.8955616449707269866985210,0.9090295709825296904671263,0.9216092981453339526669513,0.9332885350430795459243337,0.9440558701362559779627747,0.9539007829254917428493369,0.9628136542558155272936593,0.9707857757637063319308979,0.9778093584869182885537811,0.9838775407060570154961002,0.9889843952429917480044187,0.9931249370374434596520099,0.9962951347331251491861317,0.9984919506395958184001634,0.9997137267734412336782285}; static double w100[50] = {0.0312554234538633569476425,0.0312248842548493577323765,0.031163

198

8356962099067838183,0.0310723374275665165878102,0.0309504788504909882340635,0.0307983790311525904277139,0.0306161865839804484964594,0.0304040795264548200165079,0.0301622651051691449190687,0.0298909795933328309168368,0.0295904880599126425117545,0.0292610841106382766201190,0.0289030896011252031348762,0.0285168543223950979909368,0.0281027556591011733176483,0.0276611982207923882942042,0.0271926134465768801364916,0.0266974591835709626603847,0.0261762192395456763423087,0.0256294029102081160756420,0.0250575444815795897037642,0.0244612027079570527199750,0.0238409602659682059625604,0.0231974231852541216224889,0.0225312202563362727017970,0.0218430024162473863139537,0.0211334421125276415426723,0.0204032326462094327668389,0.0196530874944353058653815,0.0188837396133749045529412,0.0180959407221281166643908,0.0172904605683235824393442,0.0164680861761452126431050,0.0156296210775460027239369,0.0147758845274413017688800,0.0139077107037187726879541,0.0130259478929715422855586,0.0121314576629794974077448,0.0112251140231859771172216,0.0103078025748689695857821,0.0093804196536944579514182,0.0084438714696689714026208,0.0074990732554647115788287,0.0065469484508453227641521,0.0055884280038655151572119,0.0046244500634221193510958,0.0036559612013263751823425,0.0026839253715534824194396,0.0017093926535181052395294,0.0007346344905056717304063}; /* n = 128 */ static double x128[64] = {0.0122236989606157641980521,0.0366637909687334933302153,0.0610819696041395681037870,0.0854636405045154986364980,0.1097942311276437466729747,0.1340591994611877851175753,0.1582440427142249339974755,0.1823343059853371824103826,0.2063155909020792171540580,0.2301735642266599864109866,0.2538939664226943208556180,0.2774626201779044028062316,0.3008654388776772026671541,0.3240884350244133751832523,0.3471177285976355084261628,0.3699395553498590266165917,0.3925402750332674427356482,0.4149063795522750154922739,0.4370245010371041629370429,0.4588814198335521954490891,0.4804640724041720258582757,0.5017595591361444642896063,0.5227551520511754784539479,0.5434383024128103634441936,0.5637966482266180839144308,0.5838180216287630895500389,0.6034904561585486242035732,0.6228021939105849107615396,0.6417416925623075571535249,0.6602976322726460521059468,0.6784589224477192593677557,0.6962147083695143323850866,0.7135543776835874133438599,0.7304675667419088064717369,0.7469441667970619811698824,0.7629743300440947227797691,0.7785484755064119668504941,0.7936572947621932902433329,0.8082917575079136601196422,0.8224431169556438424645942,0.8361029150609068471168753,0.8492629875779689691636001,0.8619154689395484605906323,0.8740527969580317

199

986954180,0.8856677173453972174082924,0.8967532880491581843864474,0.9073028834017568139214859,0.9173101980809605370364836,0.9267692508789478433346245,0.9356743882779163757831268,0.9440202878302201821211114,0.9518019613412643862177963,0.9590147578536999280989185,0.9656543664319652686458290,0.9717168187471365809043384,0.9771984914639073871653744,0.9820961084357185360247656,0.9864067427245862088712355,0.9901278184917343833379303,0.9932571129002129353034372,0.9957927585349811868641612,0.9977332486255140198821574,0.9990774599773758950119878,0.9998248879471319144736081}; static double w128[64] = {0.0244461801962625182113259,0.0244315690978500450548486,0.0244023556338495820932980,0.0243585572646906258532685,0.0243002001679718653234426,0.0242273192228152481200933,0.0241399579890192849977167,0.0240381686810240526375873,0.0239220121367034556724504,0.0237915577810034006387807,0.0236468835844476151436514,0.0234880760165359131530253,0.0233152299940627601224157,0.0231284488243870278792979,0.0229278441436868469204110,0.0227135358502364613097126,0.0224856520327449668718246,0.0222443288937997651046291,0.0219897106684604914341221,0.0217219495380520753752610,0.0214412055392084601371119,0.0211476464682213485370195,0.0208414477807511491135839,0.0205227924869600694322850,0.0201918710421300411806732,0.0198488812328308622199444,0.0194940280587066028230219,0.0191275236099509454865185,0.0187495869405447086509195,0.0183604439373313432212893,0.0179603271850086859401969,0.0175494758271177046487069,0.0171281354231113768306810,0.0166965578015892045890915,0.0162550009097851870516575,0.0158037286593993468589656,0.0153430107688651440859909,0.0148731226021473142523855,0.0143943450041668461768239,0.0139069641329519852442880,0.0134112712886163323144890,0.0129075627392673472204428,0.0123961395439509229688217,0.0118773073727402795758911,0.0113513763240804166932817,0.0108186607395030762476596,0.0102794790158321571332153,0.0097341534150068058635483,0.0091830098716608743344787,0.0086263777986167497049788,0.0080645898904860579729286,0.0074979819256347286876720,0.0069268925668988135634267,0.0063516631617071887872143,0.0057726375428656985893346,0.0051901618326763302050708,0.0046045842567029551182905,0.0040162549837386423131943,0.0034255260409102157743378,0.0028327514714579910952857,0.0022382884309626187436221,0.0016425030186690295387909,0.0010458126793403487793129,0.0004493809602920903763943}; /* n = 256 */ static double x256[128] = {0.0061239123751895295011702,0.0183708184788136651179263,0.0306149687799790293662786,0.0428545265363790983812423,0.05508765569463

200

39841045614,0.0673125211657164002422903,0.0795272891002329659032271,0.0917301271635195520311456,0.1039192048105094036391969,0.1160926935603328049407349,0.1282487672706070947420496,0.1403856024113758859130249,0.1525013783386563953746068,0.1645942775675538498292845,0.1766624860449019974037218,0.1887041934213888264615036,0.2007175933231266700680007,0.2127008836226259579370402,0.2246522667091319671478783,0.2365699497582840184775084,0.2484521450010566668332427,0.2602970699919425419785609,0.2721029478763366095052447,0.2838680076570817417997658,0.2955904844601356145637868,0.3072686197993190762586103,0.3189006618401062756316834,0.3304848656624169762291870,0.3420194935223716364807297,0.3535028151129699895377902,0.3649331078236540185334649,0.3763086569987163902830557,0.3876277561945155836379846,0.3988887074354591277134632,0.4100898214687165500064336,0.4212294180176238249768124,0.4323058260337413099534411,0.4433173839475273572169258,0.4542624399175899987744552,0.4651393520784793136455705,0.4759464887869833063907375,0.4866822288668903501036214,0.4973449618521814771195124,0.5079330882286160362319249,0.5184450196736744762216617,0.5288791792948222619514764,0.5392340018660591811279362,0.5495079340627185570424269,0.5596994346944811451369074,0.5698069749365687590576675,0.5798290385590829449218317,0.5897641221544543007857861,0.5996107353629683217303882,0.6093674010963339395223108,0.6190326557592612194309676,0.6286050494690149754322099,0.6380831462729113686686886,0.6474655243637248626170162,0.6567507762929732218875002,0.6659375091820485599064084,0.6750243449311627638559187,0.6840099204260759531248771,0.6928928877425769601053416,0.7016719143486851594060835,0.7103456833045433133945663,0.7189128934599714483726399,0.7273722596496521265868944,0.7357225128859178346203729,0.7439624005491115684556831,0.7520906865754920595875297,0.7601061516426554549419068,0.7680075933524456359758906,0.7757938264113257391320526,0.7834636828081838207506702,0.7910160119895459945467075,0.7984496810321707587825429,0.8057635748129986232573891,0.8129565961764315431364104,0.8200276660989170674034781,0.8269757238508125142890929,0.8337997271555048943484439,0.8404986523457627138950680,0.8470714945172962071870724,0.8535172676795029650730355,0.8598350049033763506961731,0.8660237584665545192975154,0.8720825999954882891300459,0.8780106206047065439864349,0.8838069310331582848598262,0.8894706617776108888286766,0.8950009632230845774412228,0.9003970057703035447716200,0.9056579799601446470826819,0.9107830965950650118909072,0.9157715868574903845266696,0.9206227024251464955050471,0.9253357155833162028727303,0.9299099193340056411802456,0.9343446275020030942924765,0.9386391748378148049819261,0.9427929171174624431830761,0.9468052312391274813720517,0.9506755153166282763638521,0.9544031887

201

697162417644479,0.9579876924111781293657904,0.9614284885307321440064075,0.9647250609757064309326123,0.9678769152284894549090038,0.9708835784807430293209233,0.9737445997043704052660786,0.9764595497192341556210107,0.9790280212576220388242380,0.9814496290254644057693031,0.9837240097603154961666861,0.9858508222861259564792451,0.9878297475648606089164877,0.9896604887450652183192437,0.9913427712075830869221885,0.9928763426088221171435338,0.9942609729224096649628775,0.9954964544810963565926471,0.9965826020233815404305044,0.9975192527567208275634088,0.9983062664730064440555005,0.9989435258434088565550263,0.9994309374662614082408542,0.9997684374092631861048786,0.9999560500189922307348012}; static double w256[128] = {0.0122476716402897559040703,0.0122458343697479201424639,0.0122421601042728007697281,0.0122366493950401581092426,0.0122293030687102789041463,0.0122201222273039691917087,0.0122091082480372404075141,0.0121962627831147135181810,0.0121815877594817721740476,0.0121650853785355020613073,0.0121467581157944598155598,0.0121266087205273210347185,0.0121046402153404630977578,0.0120808558957245446559752,0.0120552593295601498143471,0.0120278543565825711612675,0.0119986450878058119345367,0.0119676359049058937290073,0.0119348314595635622558732,0.0119002366727664897542872,0.0118638567340710787319046,0.0118256971008239777711607,0.0117857634973434261816901,0.0117440619140605503053767,0.0117005986066207402881898,0.0116553800949452421212989,0.0116084131622531057220847,0.0115597048540436357726687,0.0115092624770394979585864,0.0114570935980906391523344,0.0114032060430391859648471,0.0113476078955454919416257,0.0112903074958755095083676,0.0112313134396496685726568,0.0111706345765534494627109,0.0111082800090098436304608,0.0110442590908139012635176,0.0109785814257295706379882,0.0109112568660490397007968,0.0108422955111147959952935,0.0107717077058046266366536,0.0106995040389797856030482,0.0106256953418965611339617,0.0105502926865814815175336,0.0104733073841704030035696,0.0103947509832117289971017,0.0103146352679340150682607,0.0102329722564782196569549,0.0101497741990948656546341,0.0100650535763063833094610,0.0099788230970349101247339,0.0098910956966958286026307,0.0098018845352573278254988,0.0097112029952662799642497,0.0096190646798407278571622,0.0095254834106292848118297,0.0094304732257377527473528,0.0093340483776232697124660,0.0092362233309563026873787,0.0091370127604508064020005,0.0090364315486628736802278,0.0089344947837582075484084,0.0088312177572487500253183,0.0087266159616988071403366,0.0086207050884010143053688,0.0085135010250224906938384,0.0084050198532215357561803,0.0082952778462352254251714,0.0081842914664382699356198,0.0080720773628734995009470,0.0079586523687543483536132,0.00784

202

40334989397118668103,0.0077282379473815556311102,0.0076112830845456594616187,0.0074931864548058833585998,0.0073739657738123464375724,0.0072536389258339137838291,0.0071322239610753900716724,0.0070097390929698226212344,0.0068862026954463203467133,0.0067616333001737987809279,0.0066360495937810650445900,0.0065094704150536602678099,0.0063819147521078805703752,0.0062534017395424012720636,0.0061239506555679325423891,0.0059935809191153382211277,0.0058623120869226530606616,0.0057301638506014371773844,0.0055971560336829100775514,0.0054633085886443102775705,0.0053286415939159303170811,0.0051931752508692809303288,0.0050569298807868423875578,0.0049199259218138656695588,0.0047821839258926913729317,0.0046437245556800603139791,0.0045045685814478970686418,0.0043647368779680566815684,0.0042242504213815362723565,0.0040831302860526684085998,0.0039413976414088336277290,0.0037990737487662579981170,0.0036561799581425021693892,0.0035127377050563073309711,0.0033687685073155510120191,0.0032242939617941981570107,0.0030793357411993375832054,0.0029339155908297166460123,0.0027880553253277068805748,0.0026417768254274905641208,0.0024951020347037068508395,0.0023480529563273120170065,0.0022006516498399104996849,0.0020529202279661431745488,0.0019048808534997184044191,0.0017565557363307299936069,0.0016079671307493272424499,0.0014591373333107332010884,0.0013100886819025044578317,0.0011608435575677247239706,0.0010114243932084404526058,0.0008618537014200890378141,0.0007121541634733206669090,0.0005623489540314098028152,0.0004124632544261763284322,0.0002625349442964459062875,0.0001127890178222721755125}; /* n = 512 */ static double x512[256] = {0.0030649621851593961529232,0.0091947713864329108047442,0.0153242350848981855249677,0.0214531229597748745137841,0.0275812047119197840615246,0.0337082500724805951232271,0.0398340288115484476830396,0.0459583107468090617788760,0.0520808657521920701127271,0.0582014637665182372392330,0.0643198748021442404045319,0.0704358689536046871990309,0.0765492164062510452915674,0.0826596874448871596284651,0.0887670524624010326092165,0.0948710819683925428909483,0.1009715465977967786264323,0.1070682171195026611052004,0.1131608644449665349442888,0.1192492596368204011642726,0.1253331739174744696875513,0.1314123786777137080093018,0.1374866454852880630171099,0.1435557460934960331730353,0.1496194524497612685217272,0.1556775367042018762501969,0.1617297712181921097989489,0.1677759285729161198103670,0.1738157815779134454985394,0.1798491032796159253350647,0.1858756669698757062678115,0.1918952461944840310240859,0.1979076147616804833961808,0.2039125467506523717658375,0.2099098165

203

200239314947094,0.2158991987163350271904893,0.2218804682825090362529109,0.2278534004663095955103621,0.2338177708287858931763260,0.2397733552527061887852891,0.2457199299509792442100997,0.2516572714750633493170137,0.2575851567233626262808095,0.2635033629496102970603704,0.2694116677712385990250046,0.2753098491777350342234845,0.2811976855389846383013106,0.2870749556135979555970354,0.2929414385572244074855835,0.2987969139308507415853707,0.3046411617090842500066247,0.3104739622884204453906292,0.3162950964954948840736281,0.3221043455953188263048133,0.3279014912994984240551598,0.3336863157744371275728377,0.3394586016495210024715049,0.3452181320252866497799379,0.3509646904815714220351686,0.3566980610856456291665404,0.3624180284003264285948478,0.3681243774920730946589582,0.3738168939390633631820054,0.3794953638392505477003659,0.3851595738184011246011504,0.3908093110381124851478484,0.3964443632038105531190080,0.4020645185727269675414064,0.4076695659618555307670286,0.4132592947558876229222955,0.4188334949151262845483445,0.4243919569833786700527309,0.4299344720958265754056529,0.4354608319868747443376920,0.4409708289979766581310498,0.4464642560854375149423431,0.4519409068281941054521446,0.4574005754355712925046003,0.4628430567550148032795831,0.4682681462798000434299255,0.4736756401567166435172692,0.4790653351937284489919577,0.4844370288676086658851277,0.4897905193315498753147078,0.4951256054227486308513615,0.5004420866699643537454866,0.5057397633010522419821678,0.5110184362504699101074361,0.5162779071667574777562819,0.5215179784199908258105606,0.5267384531092077401231844,0.5319391350698066637637706,0.5371198288809177797701793,0.5422803398727461474300859,0.5474204741338866161668468,0.5525400385186102421644070,0.5576388406541219339368088,0.5627166889477890541289656,0.5677733925943407059267120,0.5728087615830374335557009,0.5778226067048110674604360,0.5828147395593744458765762,0.5877849725623007456415722,0.5927331189520721562306608,0.5976589927970976321572046,0.6025624090026994600382737,0.6074431833180683777981926,0.6123011323431869846644595,0.6171360735357211818019505,0.6219478252178793846326095,0.6267362065832392490988318,0.6315010377035416553494506,0.6362421395354516935575740,0.6409593339272863978194482,0.6456524436257089753330001,0.6503212922823892793136899,0.6549657044606302753737317,0.6595855056419602523685720,0.6641805222326905300017078,0.6687505815704384167754210,0.6732955119306151731807642,0.6778151425328787363350998,0.6823093035475509635996236,0.6867778261019991540425409,0.6912205422869816079558685,0.6956372851629569859851427,0.7000278887663572307915895,0.7043921881158238155354902,0.7087300192184070848475163,0.7130412190757284553416507,0.7173256256901052441189100,0.7215830780706378951153816,0.7258134162392593745610389,0.73001648

204

12367465082373380,0.7341921151286930346516885,0.7383401610114441496854630,0.7424604630179923197192207,0.7465528663238341416942072,0.7506172171527880300329109,0.7546533627827725118134392,0.7586611515515449130726824,0.7626404328624002206015913,0.7665910571898299050923647,0.7705128760851404930018538,0.7744057421820316760079998,0.7782695092021337484565606,0.7821040319605041647237048,0.7859091663710830099561901,0.7896847694521071791947507,0.7934306993314830614379285,0.7971468152521175267628422,0.8008329775772070161862372,0.8044890477954845355235412,0.8081148885264243560855026,0.8117103635254042266412553,0.8152753376888249026732770,0.8188096770591868005536242,0.8223132488301235858819787,0.8257859213513925068443721,0.8292275641338212850768968,0.8326380478542113781512150,0.8360172443601974294381733,0.8393650266750627227522641,0.8426812690025104608329811,0.8459658467313906883792422,0.8492186364403826820199251,0.8524395159026326312771384,0.8556283640903464362590494,0.8587850611793374495058711,0.8619094885535289911058997,0.8650015288094114678982387,0.8680610657604539292849800,0.8710879844414698938880857,0.8740821711129372830049576,0.8770435132652722985416439,0.8799718996230570848337538,0.8828672201492210155023745,0.8857293660491754482355527,0.8885582297749017921351663,0.8913537050289927340242104,0.8941156867686464718706125,0.8968440712096138052506156,0.8995387558300979345474886,0.9021996393746068223597927,0.9048266218577579723776075,0.9074196045680354827749729,0.9099784900714992329623006,0.9125031822154460643436214,0.9149935861320228175302595,0.9174496082417910902748409,0.9198711562572435822074657,0.9222581391862718942794141,0.9246104673355856526489486,0.9269280523140828285786768,0.9292108070361711277546193,0.9314586457250403242837002,0.9336714839158854164789745,0.9358492384590804834007204,0.9379918275233031229867813,0.9400991705986093544775539,0.9421711884994588697201555,0.9442078033676905198230562,0.9462089386754479255274304,0.9481745192280551015654245,0.9501044711668419871894147,0.9519987219719197769813274,0.9538572004649059479887372,0.9556798368115988811866200,0.9574665625246019772327448,0.9592173104658971684737507,0.9609320148493677311718534,0.9626106112432703039637754,0.9642530365726560206402068,0.9658592291217406674540047,0.9674291285362237773389233,0.9689626758255565756615864,0.9704598133651586944555050,0.9719204848985835745206522,0.9733446355396324773464471,0.9747322117744170315712560,0.9760831614633702416830300,0.9773974338432058899681861,0.9786749795288262664309572,0.9799157505151781656726285,0.9811197001790570947322311,0.9822867832808596419166429,0.9834169559662839640681455,0.9845101757679783590716126,0.9855664016071379024692622,0.9865855937950491429603684,0.9875677140345828729848910,0.9885127254216350200148487,0.98942

205

05924465157453777048,0.9902912809952868962106899,0.9911247583510480415528399,0.9919209931951714500244370,0.9926799556084865573546763,0.9934016170724147657859271,0.9940859504700558793702825,0.9947329300872282225027936,0.9953425316134657151476031,0.9959147321429772566997088,0.9964495101755774022837600,0.9969468456176038804367370,0.9974067197828498321611183,0.9978291153935628466036470,0.9982140165816127953876923,0.9985614088900397275573677,0.9988712792754494246541769,0.9991436161123782382453400,0.9993784092025992514480161,0.9995756497983108555936109,0.9997353306710426625827368,0.9998574463699794385446275,0.9999419946068456536361287,0.9999889909843818679872841}; static double w512[256] = {0.0061299051754057857591564,0.0061296748380364986664278,0.0061292141719530834395471,0.0061285231944655327693402,0.0061276019315380226384508,0.0061264504177879366912426,0.0061250686964845654506976,0.0061234568195474804311878,0.0061216148475445832082156,0.0061195428496898295184288,0.0061172409038406284754329,0.0061147090964949169991245,0.0061119475227879095684795,0.0061089562864885234199252,0.0061057354999954793256260,0.0061022852843330780981965,0.0060986057691466529805468,0.0060946970926976980917399,0.0060905594018586731119147,0.0060861928521074844014940,0.0060815976075216427620556,0.0060767738407720980583934,0.0060717217331167509334394,0.0060664414743936418598512,0.0060609332630138177841916,0.0060551973059538766317450,0.0060492338187481899521175,0.0060430430254808039978627,0.0060366251587770195404584,0.0060299804597946507400317,0.0060231091782149633972884,0.0060160115722332929281516,0.0060086879085493424136484,0.0060011384623571610896056,0.0059933635173348036527221,0.0059853633656336707715812,0.0059771383078675312031423,0.0059686886531012259272183,0.0059600147188390547233923,0.0059511168310128456267588,0.0059419953239697077107922,0.0059326505404594676575446,0.0059230828316217905872556,0.0059132925569729856313229,0.0059032800843924967444267,0.0058930457901090792634301,0.0058825900586866627324847,0.0058719132830099005255609,0.0058610158642694068093892,0.0058498982119466814015496,0.0058385607437987230901727,0.0058270038858423319934219,0.0058152280723381015486124,0.0058032337457741007324836,0.0057910213568492471257818,0.0057785913644563714469284,0.0057659442356649741911390,0.0057530804457036750229319,0.0057400004779423555815070,0.0057267048238739963699973,0.0057131939830962084110906,0.0056994684632924603629882,0.0056855287802130018011102,0.0056713754576554833823756,0.0056570090274452746202723,0.0056424300294154800102991,0.0056276390113866542566918,0.0056126365291462173626557,0.0055974231464275703576030,0.0055819994348889124461425,0.0055663659740917603747899,0.00555

206

05233514791708235538,0.0055344721623536666407146,0.0055182130098548677502395,0.0055017465049368275723757,0.0054850732663450758090285,0.0054681939205933684565648,0.0054511091019401459196852,0.0054338194523647001109732,0.0054163256215430514316688,0.0053986282668235365401123,0.0053807280532021078251738,0.0053626256532973455128155,0.0053443217473251833447318,0.0053258170230733487787774,0.0053071121758755186716175,0.0052882079085851914147269,0.0052691049315492765055207,0.0052498039625814025460136,0.0052303057269349446719890,0.0052106109572757724261988,0.0051907203936547190996206,0.0051706347834797735752665,0.0051503548814879957194620,0.0051298814497171563759039,0.0051092152574771030281542,0.0050883570813208522065339,0.0050673077050154097256505,0.0050460679195123198490183,0.0050246385229179444874178,0.0050030203204634735477834,0.0049812141244746675595135,0.0049592207543413337151533,0.0049370410364865364724225,0.0049146758043355438745290,0.0048921258982845107556462,0.0048693921656689000083132,0.0048464754607316430993636,0.0048233766445910410307843,0.0048000965852084069516609,0.0047766361573554516370718,0.0047529962425814130594576,0.0047291777291799312876071,0.0047051815121556699579709,0.0046810084931906855725376,0.0046566595806105458869828,0.0046321356893501986622283,0.0046074377409195920619320,0.0045825666633690479877601,0.0045575233912543896535753,0.0045323088656018247089130,0.0045069240338725852313010,0.0044813698499273259161146,0.0044556472739902818017469,0.0044297572726131868769073,0.0044037008186389549258496,0.0043774788911651239762643,0.0043510924755070657234522,0.0043245425631609613132305,0.0042978301517665448748000,0.0042709562450696162035304,0.0042439218528843240022977,0.0042167279910552210986262,0.0041893756814190930634598,0.0041618659517665616659011,0.0041341998358034646067195,0.0041063783731120129818357,0.0040784026091117279353449,0.0040502735950201579699371,0.0040219923878133783908191,0.0039935600501862743674273,0.0039649776505126091053562,0.0039362462628048786290012,0.0039073669666739546834366,0.0038783408472885172720108,0.0038491689953342783540510,0.0038198525069729982349166,0.0037903924838012961884344,0.0037607900328092568594835,0.0037310462663388340021755,0.0037011623020420531166926,0.0036711392628390145554094,0.0036409782768756986764252,0.0036106804774815746300758,0.0035802470031270143713799,0.0035496789973805134987000,0.0035189776088657205261605,0.0034881439912182762045767,0.0034571793030424645127888,0.0034260847078676769483860,0.0033948613741046917538288,0.0033635104750017697209450,0.0033320331886005682236783,0.0033004306976918751358177,0.0032687041897711642972145,0.0032368548569939741987234,0.0032048838961311115627642,0.0031727925085236815030060,0.0031405819000379459532169,0.0031082532810200120618074,0.00

207

30758078662503522550163,0.0030432468748981576780527,0.0030105715304755267298129,0.0029777830607914904130339,0.0029448826979058762279357,0.0029118716780830123435331,0.0028787512417452737868732,0.0028455226334264723964728,0.0028121871017250922921949,0.0027787458992573726197173,0.0027452002826102393336092,0.0027115515122940877888456,0.0026778008526954179163600,0.0026439495720293237639656,0.0026099989422918391896635,0.0025759502392121415000167,0.0025418047422046148318992,0.0025075637343207750815413,0.0024732285022010581903898,0.0024388003360264736029032,0.0024042805294701247170072,0.0023696703796485981535706,0.0023349711870732236769383,0.0023001842556012066042973,0.0022653108923866345474810,0.0022303524078313603367724,0.0021953101155357629823745,0.0021601853322493885355395,0.0021249793778214727179358,0.0020896935751513471947536,0.0020543292501387313744068,0.0020188877316339116255770,0.0019833703513878098109153,0.0019477784440019430461334,0.0019121133468782766036998,0.0018763764001689718921795,0.0018405689467260314557679,0.0018046923320508429542037,0.0017687479042436241015783,0.0017327370139527705642995,0.0016966610143241088445575,0.0016605212609500562072903,0.0016243191118186897474239,0.0015880559272627267421479,0.0015517330699084184928942,0.0015153519046243599371387,0.0014789137984702174059640,0.0014424201206453770259886,0.0014058722424375164225552,0.0013692715371711025869345,0.0013326193801558190401403,0.0012959171486349257824991,0.0012591662217335559930561,0.0012223679804069540808915,0.0011855238073886605549070,0.0011486350871386503607080,0.0011117032057914329649653,0.0010747295511041247428251,0.0010377155124045074300544,0.0010006624805390909706032,0.0009635718478212056798501,0.0009264450079791582697455,0.0008892833561045005372012,0.0008520882886004809402792,0.0008148612031307819965602,0.0007776034985686972438014,0.0007403165749469818962867,0.0007030018334087411433900,0.0006656606761599343409382,0.0006282945064244358390880,0.0005909047284032230162400,0.0005534927472403894647847,0.0005160599690007674370993,0.0004786078006679509066920,0.0004411376501795405636493,0.0004036509265333198797447,0.0003661490400356268530141,0.0003286334028523334162522,0.0002911054302514885125319,0.0002535665435705865135866,0.0002160181779769908583388,0.0001784618055459532946077,0.0001408990173881984930124,0.0001033319034969132362968,0.0000657657316592401958310,0.0000282526373739346920387}; /* n = 1024 */ static double x1024[512] = {0.0015332313560626384065387,0.0045996796509132604743248,0.0076660846940754867627839,0.0107324176515422803327458,0.01379864968998

208

44401539048,0.0168647519770217265449962,0.0199306956814939776907024,0.0229964519737322146859283,0.0260619920258297325581921,0.0291272870119131747190088,0.0321923081084135882953009,0.0352570264943374577920498,0.0383214133515377145376052,0.0413854398649847193632977,0.0444490772230372159692514,0.0475122966177132524285687,0.0505750692449610682823599,0.0536373663049299446784129,0.0566991590022410150066456,0.0597604185462580334848567,0.0628211161513580991486838,0.0658812230372023327000985,0.0689407104290065036692117,0.0719995495578116053446277,0.0750577116607543749280791,0.0781151679813377563695878,0.0811718897697013033399379,0.0842278482828915197978074,0.0872830147851321356094940,0.0903373605480943146797811,0.0933908568511667930531222,0.0964434749817259444449839,0.0994951862354057706638682,0.1025459619163678143852404,0.1055957733375709917393206,0.1086445918210413421754502,0.1116923886981416930665228,0.1147391353098412365177689,0.1177848030069850158450139,0.1208293631505633191883714,0.1238727871119809777282145,0.1269150462733265659711591,0.1299561120276415015747167,0.1329959557791890421802183,0.1360345489437231767245806,0.1390718629487574087024745,0.1421078692338334288514767,0.1451425392507896747338214,0.1481758444640297746894331,0.1512077563507908736360111,0.1542382464014118381930443,0.1572672861196013386077717,0.1602948470227058049622614,0.1633209006419772551419632,0.1663454185228409920472972,0.1693683722251631675310675,0.1723897333235182105457458,0.1754094734074561169859457,0.1784275640817695987127083,0.1814439769667610892475458,0.1844586836985096036255346,0.1874716559291374498981239,0.1904828653270767897777182,0.1934922835773360459175133,0.1964998823817661533215037,0.1995056334593266523810493,0.2025095085463516210358758,0.2055114793968154435588961,0.2085115177825984134657778,0.2115095954937521680517391,0.2145056843387649520596422,0.2174997561448267079850562,0.2204917827580939905255947,0.2234817360439547026834844,0.2264695878872926510320010,0.2294553101927519176581055,0.2324388748850010462953415,0.2354202539089970401627982,0.2383994192302491690277166,0.2413763428350825830111093,0.2443509967309017306575811,0.2473233529464535787923793,0.2502933835320906316905658,0.2532610605600337470850902,0.2562263561246347465424530,0.2591892423426388177365829,0.2621496913534467061535080,0.2651076753193766937613805,0.2680631664259263621824189,0.2710161368820341379053566,0.2739665589203406170790369,0.2769144047974496674298651,0.2798596467941893048479266,0.2828022572158723421886958,0.2857422083925568078394062,0.2886794726793061316013119,0.2916140224564490954412652,0.2945458301298395466682397,0.2974748681311158710926665,0.3004011089179602237287060,0.3033245249743575146018584,0.3062450888108541472266190,0.3091627729648165073212094,0.3120775500

209

006891993287636,0.3149893925102530283167230,0.3178982731128827248285835,0.3208041644558044102645582,0.3237070392143528003701590,0.3266068700922281444141618,0.3295036298217528976399056,0.3323972911641281245763845,0.3352878269096896307981228,0.3381752098781638207253743,0.3410594129189232790587667,0.3439404089112420734451077,0.3468181707645507759736923,0.3496926714186912011050938,0.3525638838441708576370887,0.3554317810424171123150528,0.3582963360460310626968790,0.3611575219190411168852009,0.3640153117571562777424605,0.3668696786880191292071420,0.3697205958714585223322883,0.3725680364997419586702471,0.3754119737978276686304337,0.3782523810236163824397703,0.3810892314682027913383487,0.3839224984561266966457784,0.3867521553456238443366159,0.3895781755288764427662286,0.3924005324322633611914264,0.3952191995166100067331951,0.3980341502774378774318886,0.4008453582452137890482864,0.4036527969855987732669841,0.4064564400996966449616823,0.4092562612243022361850445,0.4120522340321492945489319,0.4148443322321580436639788,0.4176325295696824033106488,0.4204167998267568670171117,0.4231971168223430347225035,0.4259734544125757982073747,0.4287457864910091769763965,0.4315140869888618022816824,0.4342783298752620469783905,0.4370384891574927989076034,0.4397945388812358755048319,0.4425464531308160773358662,0.4452942060294448782650898,0.4480377717394637499647905,0.4507771244625871184774399,0.4535122384401449505463744,0.4562430879533249674337895,0.4589696473234144839484647,0.4616918909120418704091584,0.4644097931214176352731591,0.4671233283945751261630457,0.4698324712156108470282980,0.4725371961099243891820077,0.4752374776444579739565725,0.4779332904279356047259052,0.4806246091111018260453658,0.4833114083869600876643171,0.4859936629910107111699206,0.4886713477014884570245255,0.4913444373395996897627612,0.4940129067697591391182235,0.4966767308998262548534419,0.4993358846813411530706387,0.5019903431097601517846292,0.5046400812246908935430768,0.5072850741101270528831987,0.5099252968946826264179220,0.5125607247518258033484145,0.5151913329001124142038603,0.5178170966034189556133159,0.5204379911711751889184691,0.5230539919585963104401304,0.5256650743669146912153147,0.5282712138436111840258187,0.5308723858826459955432696,0.5334685660246891214197081,0.5360597298573503421568799,0.5386458530154087775915395,0.5412269111810419978382210,0.5438028800840546885350993,0.5463737355021068682427603,0.5489394532609416558499039,0.5515000092346125858442412,0.5540553793457104693110943,0.5566055395655897985264809,0.5591504659145946930157566,0.5616901344622843849532002,0.5642245213276582417822586,0.5667536026793803239405196,0.5692773547360034755778519,0.5717957537661929461605442,0.5743087760889495408586850,0.5768163980738322976184566,0.5793185961411806888254667,0.58181534

210

67623363454697137,0.5843066264598643017272666,0.5867924118077737578782574,0.5892726794317383594853053,0.5917474060093159907610475,0.5942165682701680800580147,0.5966801429962784154186793,0.5991381070221714681281111,0.6015904372351302222163013,0.6040371105754135078618616,0.6064781040364728366534687,0.6089133946651687366701116,0.6113429595619865853458987,0.6137667758812519380899084,0.6161848208313453506363029,0.6185970716749166931046915,0.6210035057290989537555048,0.6234041003657215304299416,0.6257988330115230076688675,0.6281876811483634175098794,0.6305706223134359819666081,0.6329476340994783351992008,0.6353186941549832233898213,0.6376837801844086803419153,0.6400428699483876768269192,0.6423959412639372417070377,0.6447429720046670528676835,0.6470839401009874959981582,0.6494188235403171892641570,0.6517476003672899719207013,0.6540702486839613549191454,0.6563867466500144315669620,0.6586970724829652463040876,0.6610012044583676196647058,0.6632991209100174274984589,0.6655908002301563325302097,0.6678762208696749663426270,0.6701553613383155598710345,0.6724282002048740205051479,0.6746947160974014538975312,0.6769548877034051285838219,0.6792086937700488815250166,0.6814561131043529626873631,0.6836971245733933167806834,0.6859317071045003002812397,0.6881598396854568318705713,0.6903815013646959744270519,0.6925966712514979467122689,0.6948053285161865628996815,0.6970074523903250980984011,0.6992030221669115780303307,0.7013920172005734910243170,0.7035744169077619204963997,0.7057502007669450960906928,0.7079193483188013616608982,0.7100818391664115582779368,0.7122376529754508204546805,0.7143867694743797837842896,0.7165291684546352021941915,0.7186648297708199730232898,0.7207937333408925681355609,0.7229158591463558692887801,0.7250311872324454059827217,0.7271396977083169940167956,0.7292413707472337729927181,0.7313361865867526410034676,0.7334241255289100847554419,0.7355051679404074033764222,0.7375792942527953241676460,0.7396464849626580085640129,0.7417067206317964465721772,0.7437599818874112379620360,0.7458062494222847584928838,0.7478455039949627094612890,0.7498777264299350488635483,0.7519028976178163024713854,0.7539209985155252531253957,0.7559320101464640065565832,0.7579359136006964320521972,0.7599326900351259762879594,0.7619223206736728486546595,0.7639047868074505764130149,0.7658800697949419280166093,0.7678481510621742029486694,0.7698090121028938864243967,0.7717626344787406673165402,0.7737089998194208176678866,0.7756480898228799321603470,0.7775798862554750259163361,0.7795043709521459890141759,0.7814215258165863961053031,0.7833313328214136695271245,0.7852337740083385943114429,0.7871288314883341834944720,0.7890164874418038921405657,0.7908967241187491784979139,0.7927695238389364107105941,0.7946348689920631175175217,0.7964927420379235813750136,0.79834

211

31255065737724458586,0.8001860019984956219039900,0.8020213541847606330100649,0.8038491648071928284194859,0.8056694166785310321906380,0.8074820926825904849673728,0.8092871757744237908160400,0.8110846489804811942036542,0.8128744953987701856100790,0.8146566981990144342734272,0.8164312406228120465742028,0.8181981059837931485700490,0.8199572776677767911993239,0.8217087391329271766780945,0.8234524739099092046215225,0.8251884656020433364270094,0.8269166978854597764628854,0.8286371545092519686128428,0.8303498192956294067327593,0.8320546761400697575830038,0.8337517090114702948057846,0.8354409019522986425235764,0.8371222390787428271411563,0.8387957045808606359402829,0.8404612827227282810625704,0.8421189578425883674826439,0.8437687143529971635802028,0.8454105367409711729261812,0.8470444095681330059047621,0.8486703174708565497995875,0.8502882451604114359791023,0.8518981774231068028225812,0.8535000991204343530350070,0.8550939951892107040056078,0.8566798506417190298715048,0.8582576505658499939545848,0.8598273801252419702463831,0.8613890245594205526224495,0.8629425691839373504743648,0.8644879993905080694542896,0.8660253006471498760336444,0.8675544584983180445842596,0.8690754585650418856970762,0.8705882865450599544602407,0.8720929282129545374252050,0.8735893694202854169962281,0.8750775960957229119854680,0.8765575942451801930826613,0.8780293499519448719952049,0.8794928493768098630212838,0.8809480787582035158255322,0.8823950244123190181935674,0.8838336727332430675485994,0.8852640101930838100201983,0.8866860233420980458621863,0.8880996988088177000235219,0.8895050233001755566829532,0.8909019836016302565651375,0.8922905665772905558628607,0.8936707591700388455969280,0.8950425484016539302522575,0.8964059213729330645356690,0.8977608652638132471078410,0.8991073673334917701488930,0.9004454149205460236240486,0.9017749954430525531228459,0.9030960963987053701523781,0.9044087053649335137720782,0.9057128099990178624646022,0.9070083980382071951444166,0.9082954572998335002127549,0.9095739756814265315746820,0.9108439411608276105410847,0.9121053417963026725455006,0.9133581657266545576127977,0.9146024011713345435238301,0.9158380364305531206273175,0.9170650598853900072573273,0.9182834599979034047218800,0.9194932253112384908353520,0.9206943444497351509745089,0.9218868061190349456451742,0.9230705991061873135537215,0.9242457122797550091847637,0.9254121345899187738936182,0.9265698550685812395293315,0.9277188628294700636112689,0.9288591470682402950895005,0.9299906970625759697264543,0.9311135021722909341445515,0.9322275518394288975917975,0.9333328355883627104845635,0.9344293430258928687940732,0.9355170638413452433503852,0.9365959878066680331449597,0.9376661047765279417201973,0.9387274046884055757416456,0.9397798775626900648558921,0.9408235135027729019444869,0.94

212

18583026951410028915762,0.9428842354094689849902736,0.9439013019987106631201510,0.9449094928991897628355911,0.9459087986306898495121205,0.9468992097965434727052183,0.9478807170837205248834878,0.9488533112629158137054760,0.9498169831886358470168335,0.9507717237992848297519245,0.9517175241172498719314184,0.9526543752489854069548347,0.9535822683850968193944507,0.9545011948004232815044368,0.9554111458541197976665483,0.9563121129897384560011695,0.9572040877353088863799924,0.9580870617034179240840996,0.9589610265912884783587268,0.9598259741808576051234879,0.9606818963388537831043733,0.9615287850168733926613630,0.9623666322514563965930439,0.9631954301641612222071790,0.9640151709616388439537466,0.9648258469357060659245549,0.9656274504634180035311332,0.9664199740071397636802195,0.9672034101146173227737943,0.9679777514190476018682591,0.9687429906391477383350273,0.9694991205792235533724866,0.9702461341292372147270016,0.9709840242648740939883669,0.9717127840476088178328839,0.9724324066247705125950353,0.9731428852296072415565604,0.9738442131813496343496072,0.9745363838852737078785517,0.9752193908327628781730396,0.9758932276013691625928266,0.9765578878548735718130775,0.9772133653433456910269459,0.9778596539032024498104955,0.9784967474572660801033674,0.9791246400148212617670490,0.9797433256716714551911835,0.9803527986101944204270933,0.9809530530993969223366037,0.9815440834949686212533729,0.9821258842393351486632952,0.9826984498617103674201996,0.9832617749781478160230522,0.9838158542915913364912672,0.9843606825919248853856025,0.9848962547560215275335618,0.9854225657477916120303537,0.9859396106182301300994116,0.9864473845054632544104222,0.9869458826347940594679517,0.9874351003187474227003598,0.9879150329571141058970610,0.9883856760369940166627304,0.9888470251328386495802522,0.9892990759064927068006818,0.9897418241072348978090276,0.9901752655718179181502248,0.9905993962245076069415402,0.9910142120771212830473891,0.9914197092290652598522332,0.9918158838673715386394944,0.9922027322667336806727008,0.9925802507895418581838653,0.9929484358859170846092543,0.9933072840937446245820355,0.9936567920387065844051246,0.9939969564343136839997662,0.9943277740819362116746914,0.9946492418708341635125525,0.9949613567781865697596566,0.9952641158691200113800912,0.9955575162967363309635588,0.9958415553021395435525955,0.9961162302144619548145649,0.9963815384508894965215124,0.9966374775166862927999356,0.9968840450052184754903082,0.9971212385979772738362093,0.9973490560646014135491635,0.9975674952628988745188845,0.9977765541388680773265018,0.9979762307267185998745420,0.9981665231488915727109186,0.9983474296160799746514418,0.9985189484272491654281575,0.9986810779696581776171579,0.9988338167188825964389443,0.9989771632388403756649803,0.9991111161818228462260355,0

213

.9992356742885348165163858,0.9993508363881507486653971,0.9994566013984000492749057,0.9995529683257070064969677,0.9996399362654382464576482,0.9997175044023747284307007,0.9997856720116889628341744,0.9998444384611711916084367,0.9998938032169419878731474,0.9999337658606177711221103,0.9999643261538894550943330,0.9999854843850284447675914,0.9999972450545584403516182}; static double w1024[512] = {0.0030664603092439082115513,0.0030664314747171934849726,0.0030663738059349007324470,0.0030662873034393008056861,0.0030661719680437936084028,0.0030660278008329004477528,0.0030658548031622538363679,0.0030656529766585847450783,0.0030654223232197073064431,0.0030651628450145009692318,0.0030648745444828901040266,0.0030645574243358210601357,0.0030642114875552366740338,0.0030638367373940482295700,0.0030634331773761048702058,0.0030630008112961604635720,0.0030625396432198379186545,0.0030620496774835909559465,0.0030615309186946633309249,0.0030609833717310455112352,0.0030604070417414288079918,0.0030598019341451569616257,0.0030591680546321751827342,0.0030585054091629766484119,0.0030578140039685464545661,0.0030570938455503030247440,0.0030563449406800369760227,0.0030555672963998474425352,0.0030547609200220758572342,0.0030539258191292371925135,0.0030530620015739486603347,0.0030521694754788558725307,0.0030512482492365564619779,0.0030502983315095211653578,0.0030493197312300123682482,0.0030483124576000001133114,0.0030472765200910755723677,0.0030462119284443619831693,0.0030451186926704230517109,0.0030439968230491688209395,0.0030428463301297590067471,0.0030416672247305038021562,0.0030404595179387621506312,0.0030392232211108374894710,0.0030379583458718709642643,0.0030366649041157321154111,0.0030353429080049070377385,0.0030339923699703840142628,0.0030326133027115366251721,0.0030312057191960043331307,0.0030297696326595705460252,0.0030283050566060381583022,0.0030268120048071025720655,0.0030252904913022221991274,0.0030237405303984864452325,0.0030221621366704811776946,0.0030205553249601516777118,0.0030189201103766630786495,0.0030172565082962582916016,0.0030155645343621134195681,0.0030138442044841906616068,0.0030120955348390887083441,0.0030103185418698906302495,0.0030085132422860092601062,0.0030066796530630300711306,0.0030048177914425515522176,0.0030029276749320230818149,0.0030010093213045803019478,0.0029990627485988779939449,0.0029970879751189204574353,0.0029950850194338893942123,0.0029930539003779692985814,0.0029909946370501703558363,0.0029889072488141488505262,0.0029867917552980250862041,0.0029846481763941988183689,0.0029824765322591622023349,0.0029802768433133102577897,0.0029780491302407488518214,0.0029757934139891002022209,0.0029735097157693059028890,0.0029711980570554274731990,0.002968858459584444

214

4331918,0.0029664909453560499065010,0.0029640955366324437529314,0.0029616722559381232326340,0.0029592211260596712038487,0.0029567421700455418562030,0.0029542354112058439815854,0.0029517008731121217846274,0.0029491385795971332348581,0.0029465485547546259626151,0.0029439308229391107008170,0.0029412854087656322747309,0.0029386123371095381418860,0.0029359116331062444843108,0.0029331833221509998552933,0.0029304274298986463828860,0.0029276439822633785324025,0.0029248330054184994301727,0.0029219945257961747508486,0.0029191285700871841705750,0.0029162351652406703883623,0.0029133143384638857180205,0.0029103661172219362530391,0.0029073905292375236068160,0.0029043876024906842306667,0.0029013573652185263120627,0.0028982998459149642555740,0.0028952150733304507490135,0.0028921030764717064173001,0.0028889638846014470665859,0.0028857975272381085212091,0.0028826040341555690560623,0.0028793834353828694269858,0.0028761357612039305018167,0.0028728610421572684947521,0.0028695593090357078067012,0.0028662305928860914743281,0.0028628749250089892305081,0.0028594923369584031789413,0.0028560828605414710856927,0.0028526465278181672904478,0.0028491833711010012402964,0.0028456934229547136488796,0.0028421767161959702837564,0.0028386332838930533848701,0.0028350631593655507170153,0.0028314663761840422592303,0.0028278429681697845340603,0.0028241929693943925796601,0.0028205164141795195677262,0.0028168133370965340702726,0.0028130837729661949782821,0.0028093277568583240752928,0.0028055453240914762689974,0.0028017365102326074839556,0.0027979013510967402185435,0.0027940398827466267692845,0.0027901521414924101257281,0.0027862381638912825390663,0.0027822979867471417676962,0.0027783316471102450029635,0.0027743391822768604783394,0.0027703206297889167653083,0.0027662760274336497592617,0.0027622054132432473587211,0.0027581088254944918412282,0.0027539863027083999392661,0.0027498378836498606195970,0.0027456636073272705694208,0.0027414635129921673927833,0.0027372376401388605206822,0.0027329860285040598383428,0.0027287087180665020331547,0.0027244057490465746667821,0.0027200771619059379749851,0.0027157229973471443987056,0.0027113432963132558499974,0.0027069380999874587163979,0.0027025074497926766073634,0.0026980513873911808464073,0.0026935699546841987126055,0.0026890631938115194351518,0.0026845311471510979446691,0.0026799738573186563850015,0.0026753913671672833892344,0.0026707837197870311237119,0.0026661509585045101038391,0.0026614931268824817854798,0.0026568102687194489357814,0.0026521024280492437872770,0.0026473696491406139791397,0.0026426119764968062894804,0.0026378294548551481626046,0.0026330221291866270351630,0.0026281900446954674651512,0.0026233332468187060677353,0.0026184517812257642618999,0.0026135456938180188319369,0.0026086150307283703078113,0.0026036598383208

215

091684657,0.0025986801631899798721388,0.0025936760521607427178014,0.0025886475522877335418257,0.0025835947108549212540321,0.0025785175753751632172710,0.0025734161935897584747222,0.0025682906134679988291122,0.0025631408832067177780710,0.0025579670512298373098703,0.0025527691661879125638030,0.0025475472769576743594882,0.0025423014326415695994010,0.0025370316825672995489502,0.0025317380762873559984451,0.0025264206635785553113127,0.0025210794944415703629476,0.0025157146191004603745948,0.0025103260880021986466869,0.0025049139518161981960773,0.0024994782614338353016280,0.0024940190679679709626349,0.0024885364227524702745874,0.0024830303773417197267843,0.0024775009835101424263432,0.0024719482932517112531633,0.0024663723587794599504176,0.0024607732325249921551741,0.0024551509671379883737605,0.0024495056154857109065099,0.0024438372306525067265426,0.0024381458659393083172574,0.0024324315748631324732279,0.0024266944111565770692147,0.0024209344287673158020275,0.0024151516818575909099866,0.0024093462248037038747545,0.0024035181121955041103265,0.0023976673988358756439882,0.0023917941397402217940673,0.0023858983901359478493246,0.0023799802054619417548485,0.0023740396413680528093376,0.0023680767537145683786720,0.0023620915985716886306938,0.0023560842322189992961374,0.0023500547111449424606655,0.0023440030920462853929883,0.0023379294318275874140606,0.0023318337876006648123684,0.0023257162166840538103394,0.0023195767766024715869239,0.0023134155250862753614165,0.0023072325200709195436049,0.0023010278196964109553481,0.0022948014823067621287099,0.0022885535664494426857857,0.0022822841308748288053830,0.0022759932345356507817318,0.0022696809365864386804193,0.0022633472963829660967620,0.0022569923734816920218464,0.0022506162276392008214839,0.0022442189188116403333494,0.0022378005071541580875846,0.0022313610530203356561684,0.0022249006169616211363732,0.0022184192597267597736437,0.0022119170422612227292520,0.0022053940257066339981005,0.0021988502714001954820607,0.0021922858408741102242558,0.0021857007958550038097087,0.0021790951982633439377969,0.0021724691102128581719720,0.0021658225940099498722195,0.0021591557121531123157498,0.0021524685273323410114303,0.0021457611024285442134846,0.0021390335005129516400021,0.0021322857848465214018174,0.0021255180188793451473363,0.0021187302662500514289029,0.0021119225907852072963166,0.0021050950564987181231273,0.0020982477275912256713511,0.0020913806684495044002679,0.0020844939436458560249764,0.0020775876179375023304007,0.0020706617562659762464561,0.0020637164237565111901030,0.0020567516857174286800274,0.0020497676076395242297101,0.0020427642551954515246552,0.0020357416942391048895728,0.0020286999908050000513193,0.0020216392111076532034194,0.0020145594215409583780096,0.0020074606886775631310555,0.0020003430792

216

682425467160,0.0019932066602412715667394,0.0019860514987017956507927,0.0019788776619311997736447,0.0019716852173864757651327,0.0019644742326995879988655,0.0019572447756768374356240,0.0019499969142982240274419,0.0019427307167168074883601,0.0019354462512580664378677,0.0019281435864192559230531,0.0019208227908687633255086,0.0019134839334454626590447,0.0019061270831580672642844,0.0018987523091844809062265,0.0018913596808711472808775,0.0018839492677323979370705,0.0018765211394497986196010,0.0018690753658714940398285,0.0018616120170115510799024,0.0018541311630493004367905,0.0018466328743286767122991,0.0018391172213575569552912,0.0018315842748070976623218,0.0018240341055110702429247,0.0018164667844651949558009,0.0018088823828264733221690,0.0018012809719125190225581,0.0017936626232008872833327,0.0017860274083284027592567,0.0017783753990904859184165,0.0017707066674404779358362,0.0017630212854889641021349,0.0017553193255030957535871,0.0017476008599059107299616,0.0017398659612756523665312,0.0017321147023450870266539,0.0017243471560008201813452,0.0017165633952826110422716,0.0017087634933826857546100,0.0017009475236450491562317,0.0016931155595647951096823,0.0016852676747874154134422,0.0016774039431081072989678,0.0016695244384710795200224,0.0016616292349688570408253,0.0016537184068415843295541,0.0016457920284763272637533,0.0016378501744063736542136,0.0016298929193105323938983,0.0016219203380124312385075,0.0016139325054798132252838,0.0016059294968238317366751,0.0015979113872983442154825,0.0015898782522992045381361,0.0015818301673635540527516,0.0015737672081691112886347,0.0015656894505334603439125,0.0015575969704133379579831,0.0015494898439039192754876,0.0015413681472381023085203,0.0015332319567857911038062,0.0015250813490531776215856,0.0015169164006820223329593,0.0015087371884489335424584,0.0015005437892646454426166,0.0014923362801732949073323,0.0014841147383516970308228,0.0014758792411086194189814,0.0014676298658840552399621,0.0014593666902484950408286,0.0014510897919021973371136,0.0014427992486744579821480,0.0014344951385228783230315,0.0014261775395326321501237,0.0014178465299157314469528,0.0014095021880102909474427,0.0014011445922797915073771,0.0013927738213123422970256,0.0013843899538199418218713,0.0013759930686377377783877,0.0013675832447232857518263,0.0013591605611558067629844,0.0013507250971354436709363,0.0013422769319825164387192,0.0013338161451367762689788,0.0013253428161566586165863,0.0013168570247185350852537,0.0013083588506159642151809,0.0012998483737589411687807,0.0012913256741731463215379,0.0012827908319991927650686,0.0012742439274918727294554,0.0012656850410194029319476,0.0012571142530626688591208,0.0012485316442144679896043,0.0012399372951787519644928,0.0012313312867698677125706,0.0012227136999117975374834,0.0012140846

217

156373981740056,0.0012054441150876388205601,0.0011967922795108381551550,0.0011881291902619003419159,0.0011794549288015500353964,0.0011707695766955663898644,0.0011620732156140160807669,0.0011533659273304853455891,0.0011446477937213110513287,0.0011359188967648107958214,0.0011271793185405120501566,0.0011184291412283803494364,0.0011096684471080465391373,0.0011008973185580330843445,0.0010921158380549794491381,0.0010833240881728665534171,0.0010745221515822403144596,0.0010657101110494342805238,0.0010568880494357913638046,0.0010480560496968846800697,0.0010392141948817375023057,0.0010303625681320423357186,0.0010215012526813791214350,0.0010126303318544325762649,0.0010037498890662086758941,0.0009948600078212502888805,0.0009859607717128519688418,0.0009770522644222739122264,0.0009681345697179550890732,0.0009592077714547255541688,0.0009502719535730179460261,0.0009413272000980781811114,0.0009323735951391753507612,0.0009234112228888108282347,0.0009144401676219265933610,0.0009054605136951127822476,0.0008964723455458144695262,0.0008874757476915376906225,0.0008784708047290547115472,0.0008694576013336085537138,0.0008604362222581167813022,0.0008514067523323745586954,0.0008423692764622569855308,0.0008333238796289207169173,0.0008242706468880048763834,0.0008152096633688312691343,0.0008061410142736039032099,0.0007970647848766078261514,0.0007879810605234072847989,0.0007788899266300432158601,0.0007697914686822300749096,0.0007606857722345520114971,0.0007515729229096583980656,0.0007424530063974587204051,0.0007333261084543168373926,0.0007241923149022446178008,0.0007150517116280949619884,0.0007059043845827542163241,0.0006967504197803339882351,0.0006875899032973623698204,0.0006784229212719745780188,0.0006692495599031030193850,0.0006600699054496667875923,0.0006508840442297606018626,0.0006416920626198431946113,0.0006324940470539251567018,0.0006232900840227562488244,0.0006140802600730121876541,0.0006048646618064809156059,0.0005956433758792483631993,0.0005864164890008837132649,0.0005771840879336241764943,0.0005679462594915592881427,0.0005587030905398147360662,0.0005494546679937357307118,0.0005402010788180699282026,0.0005309424100261499182844,0.0005216787486790752896494,0.0005124101818848942860548,0.0005031367967977850677401,0.0004938586806172365939677,0.0004845759205872291441124,0.0004752886039954144966810,0.0004659968181722957880391,0.0004567006504904070755681,0.0004474001883634926336095,0.0004380955192456860150653,0.0004287867306306889171352,0.0004194739100509498966958,0.0004101571450768429896514,0.0004008365233158462997325,0.0003915121324117206363681,0.0003821840600436882993131,0.0003728523939256121308821,0.0003635172218051749865499,0.0003541786314630598135175,0.0003448367107121305776064,0.0003354915473966143456333,0.0003261432293912849189248,0.00031679

218

18446006485317858,0.0003074374809581322877037,0.0002980802264252762217455,0.0002887201689909301727620,0.0002793573966704570567274,0.0002699919975049447012834,0.0002606240595604292032823,0.0002512536709271339139118,0.0002418809197187298044384,0.0002325058940716253739001,0.0002231286821442978268308,0.0002137493721166826096154,0.0002043680521896465790359,0.0001949848105845827899210,0.0001855997355431850062940,0.0001762129153274925249194,0.0001668244382203495280013,0.0001574343925265138930609,0.0001480428665748079976500,0.0001386499487219861751244,0.0001292557273595155266326,0.0001198602909254695827354,0.0001104637279257437565603,0.0001010661269730276014588,0.0000916675768613669107254,0.0000822681667164572752810,0.0000728679863190274661367,0.0000634671268598044229933,0.0000540656828939400071988,0.0000446637581285753393838,0.0000352614859871986975067,0.0000258591246764618586716,0.0000164577275798968681068,0.0000070700764101825898713}; /* n = 3 */ static double x3[2] = {0.0000000000000000000000000,0.7745966692414833770358531}; static double w3[2] = {0.8888888888888888888888889,0.5555555555555555555555556}; /* n = 5 */ static double x5[3] = {0.0000000000000000000000000,0.5384693101056830910363144,0.9061798459386639927976269}; static double w5[3] = {0.5688888888888888888888889,0.4786286704993664680412915,0.2369268850561890875142640}; /* n = 7 */ static double x7[4] = {0.0000000000000000000000000,0.4058451513773971669066064,0.7415311855993944398638648,0.9491079123427585245261897}; static double w7[4] = {0.4179591836734693877551020,0.3818300505051189449503698,0.2797053914892766679014678,0.1294849661688696932706114}; /* n = 9 */ static double x9[5] = {0.0000000000000000000000000,0.3242534234038089290385380,0.6133714327005903973087020,0.8360311073266357942994298,0.9681602395076260898355762};

219

static double w9[5] = {0.3302393550012597631645251,0.3123470770400028400686304,0.2606106964029354623187429,0.1806481606948574040584720,0.0812743883615744119718922}; /* n = 11 */ static double x11[6] = {0.0000000000000000000000000,0.2695431559523449723315320,0.5190961292068118159257257,0.7301520055740493240934163,0.8870625997680952990751578,0.9782286581460569928039380}; static double w11[6] = {0.2729250867779006307144835,0.2628045445102466621806889,0.2331937645919904799185237,0.1862902109277342514260976,0.1255803694649046246346943,0.0556685671161736664827537}; /* n = 13 */ static double x13[7] = {0.0000000000000000000000000,0.2304583159551347940655281,0.4484927510364468528779129,0.6423493394403402206439846,0.8015780907333099127942065,0.9175983992229779652065478,0.9841830547185881494728294}; static double w13[7] = {0.2325515532308739101945895,0.2262831802628972384120902,0.2078160475368885023125232,0.1781459807619457382800467,0.1388735102197872384636018,0.0921214998377284479144218,0.0404840047653158795200216}; /* n = 15 */ static double x15[8] = {0.0000000000000000000000000,0.2011940939974345223006283,0.3941513470775633698972074,0.5709721726085388475372267,0.7244177313601700474161861,0.8482065834104272162006483,0.9372733924007059043077589,0.9879925180204854284895657}; static double w15[8] = {0.2025782419255612728806202,0.1984314853271115764561183,0.1861610000155622110268006,0.1662692058169939335532009,0.1395706779261543144478048,0.1071592204671719350118695,0.0703660474881081247092674,0.0307532419961172683546284}; /* n = 17 */ static double x17[9] = {0.0000000000000000000000000,0.1784841814958478558506775,0.3512317634538763152971855,0.5126905370864769678862466,0.65767115921669

220

07658503022,0.7815140038968014069252301,0.8802391537269859021229557,0.9506755217687677612227170,0.9905754753144173356754340}; static double w17[9] = {0.1794464703562065254582656,0.1765627053669926463252710,0.1680041021564500445099707,0.1540457610768102880814316,0.1351363684685254732863200,0.1118838471934039710947884,0.0850361483171791808835354,0.0554595293739872011294402,0.0241483028685479319601100}; /* n = 19 */ static double x19[10] = {0.0000000000000000000000000,0.1603586456402253758680961,0.3165640999636298319901173,0.4645707413759609457172671,0.6005453046616810234696382,0.7209661773352293786170959,0.8227146565371428249789225,0.9031559036148179016426609,0.9602081521348300308527788,0.9924068438435844031890177}; static double w19[10] = {0.1610544498487836959791636,0.1589688433939543476499564,0.1527660420658596667788554,0.1426067021736066117757461,0.1287539625393362276755158,0.1115666455473339947160239,0.0914900216224499994644621,0.0690445427376412265807083,0.0448142267656996003328382,0.0194617882297264770363120}; /* Merge all together */ typedef struct tagGLAW{ int n; double* x; double* w; }GLAW; static GLAW glaw[] = { {2,x2,w2}, {3,x3,w3}, {4,x4,w4}, {5,x5,w5}, {6,x6,w6}, {7,x7,w7}, {8,x8,w8}, {9,x9,w9}, {10,x10,w10}, {11,x11,w11}, {12,x12,w12}, {13,x13,w13}, {14,x14,w14},

221

{15,x15,w15}, {16,x16,w16}, {17,x17,w17}, {18,x18,w18}, {19,x19,w19}, {20,x20,w20}, {32,x32,w32}, {64,x64,w64}, {96,x96,w96}, {100,x100,w100}, {128,x128,w128}, {256,x256,w256}, {512,x512,w512}, {1024,x1024,w1024} }; static const int GLAWSIZE = sizeof(glaw)/sizeof(glaw[0]); /* Gauss-Legendre n-points quadrature, exact for polynomial of degree <=2n-1 1. n - even: int(f(t),t=a..b) = A*sum(w[i]*f(A*x[i]+B),i=0..n-1) = A*sum(w[k]*[f(B+A*x[k])+f(B-A*x[k])],k=0..n/2-1) A = (b-a)/2, B = (a+b)/2 2. n - odd: int(f(t),t=a..b) = A*sum(w[i]*f(A*x[i]+B),i=0..n-1) = A*w[0]*f(B)+A*sum(w[k]*[f(B+A*x[k])+f(B-A*x[k])],k=1..(n-1)/2) A = (b-a)/2, B = (a+b)/2 */ double gauss_legendre(int n, double (*f)(double,void*), void* data, double a, double b) { double* x = NULL; double* w = NULL; double A,B,Ax,s;

222

int i, dtbl, m; m = (n+1)>>1; /* Load appropriate predefined table */ dtbl = 0; for (i = 0; i<GLAWSIZE;i++) { if(n==glaw[i].n) { x = glaw[i].x; w = glaw[i].w; break; } } /* Generate new if non-predefined table is required */ /* with precision of 1e-10 */ if(NULL==x) { dtbl = 1; x = (double*)malloc(m*sizeof(double)); w = (double*)malloc(m*sizeof(double)); gauss_legendre_tbl(n,x,w,1e-10); } A = 0.5*(b-a); B = 0.5*(b+a); if(n&1) /* n - odd */ { s = w[0]*((*f)(B,data)); for (i=1;i<m;i++) { Ax = A*x[i]; s += w[i]*((*f)(B+Ax,data)+(*f)(B-Ax,data)); } }else{ /* n - even */

223

s = 0.0; for (i=0;i<m;i++) { Ax = A*x[i]; s += w[i]*((*f)(B+Ax,data)+(*f)(B-Ax,data)); } } if (dtbl) { free(x); free(w); } return A*s; } /* 2D Numerical computation of int(f(x,y),x=a..b,y=c..d) by Gauss-Legendre n-th order high precision quadrature [in]n - quadrature order [in]f - integrand [in]data - pointer on user-defined data which will be passed to f every time it called (as third parameter). [in][a,b] - interval of integration return: -computed integral value or -1.0 if n order quadrature is not supported 1. n - even: int(f(t,p),t=a..b,p=c..d) = C*A*sum(w[i]*w[j]*f(A*x[i]+B,C*y[j]+D),i=0..n-1,j=0..n-1) = C*A*sum(w[k]*w[l]*[f(B+A*x[k],C*y[l]+D)+f(B-A*x[k],C*y[l]+D)],k=0..n/2-1,l=0..n-1) = C*A*sum(w[k]*w[l]*[f(B+A*x[k],C*y[l]+D)+f(B+A*x[k],D-C*y[l])+ +f(B-A*x[k],C*y[l]+D)+f(B-A*x[k],D-C*y[l])],k=0..n/2-1,l=0..n/2-1) A = (b-a)/2 B = (a+b)/2 C = (d-c)/2 D = (d+c)/2

224

2. n - odd: int(f(t,p),t=a..b,p=c..d) = C*A*sum(w[i]*w[j]*f(A*x[i]+B,C*y[j]+D),i=0..n-1,j=0..n-1) = C*A*[w[0]*w[0]*f(B,D)+sum(w[0]*w[j]*(f(B,C*y[j]+D)+f(B,D-C*y[j])),j=1..(n-1)/2)+ +sum(w[i]*w[0]*(f(B+A*x[i],D)+f(B-A*x[i],D)),i=1..(n-1)/2)+ +sum(w[k]*w[l]*[f(B+A*x[k],C*y[l]+D)+f(B+A*x[k],D-C*y[l])+ +f(B-A*x[k],C*y[l]+D)+f(B-A*x[k],D-C*y[l])],k=1..(n-1)/2,l=1..(n-1)/2)] A = (b-a)/2, B = (a+b)/2 C = (d-c)/2 D = (d+c)/2 */ double gauss_legendre_2D_cube(int n, double (*f)(double,double,void*), void* data, double a, double b, double c, double d) { double* x = NULL; double* w = NULL; double A,B,C,D,Ax,Cy,s,t; int i,j, dtbl, m; m = (n+1)>>1; /* Load appropriate predefined table */ dtbl = 0; for (i = 0; i<GLAWSIZE;i++) { if(n==glaw[i].n) { x = glaw[i].x; w = glaw[i].w; break; } } /* Generate new if non-predefined table is required */ /* with precision of 1e-10 */ if(NULL==x) {

225

dtbl = 1; x = (double*)malloc(m*sizeof(double)); w = (double*)malloc(m*sizeof(double)); gauss_legendre_tbl(n,x,w,1e-10); } A = 0.5*(b-a); B = 0.5*(b+a); C = 0.5*(d-c); D = 0.5*(d+c); if(n&1) /* n - odd */ { s = w[0]*w[0]*(*f)(B,D,data); for (j=1,t=0.0;j<m;j++) { Cy = C*x[j]; t += w[j]*((*f)(B,D+Cy,data)+(*f)(B,D-Cy,data)); } s += w[0]*t; for (i=1,t=0.0;i<m;i++) { Ax = A*x[i]; t += w[i]*((*f)(B+Ax,D,data)+(*f)(B-Ax,D,data)); } s += w[0]*t; for (i=1;i<m;i++) { Ax = A*x[i]; for (j=1;j<m;j++) { Cy = C*x[j]; s += w[i]*w[j]*( (*f)(B+Ax,D+Cy,data)+(*f)(Ax+B,D-Cy,data)+(*f)(B-Ax,D+Cy,data)+(*f)(B-Ax,D-Cy,data)); } }

226

}else{ /* n - even */ s = 0.0; for (i=0;i<m;i++) { Ax = A*x[i]; for (j=0;j<m;j++) { Cy = C*x[j]; s += w[i]*w[j]*( (*f)(B+Ax,D+Cy,data)+(*f)(Ax+B,D-Cy,data)+(*f)(B-Ax,D+Cy,data)+(*f)(B-Ax,D-Cy,data)); } } } if (dtbl) { free(x); free(w); } return C*A*s; } /* Computing of abscissas and weights for Gauss-Legendre quadrature for any(reasonable) order n [in] n - order of quadrature [in] eps - required precision (must be eps>=macheps(double), usually eps = 1e-10 is ok) [out]x - abscisass, size = (n+1)>>1 [out]w - weights, size = (n+1)>>1 */ /* Look up table for fast calculation of Legendre polynomial for n<1024 */ /* ltbl[k] = 1.0 - 1.0/(double)k; */ static double ltbl[1024] = {0.00000000000000000000,0.00000000000000000000,0.50000000000000000000,0.66666666666666674000,0.75000000000000000000,0.80000000000000004000,0.83333333333333337000,0.85714285714285721000,0.87500000000000000000,0.88888888888888884000,0.90000000000000002000,0.90909090909090906000,0.91666666666666663000,0.92307692307692313000,0.92857142857142860000,0.93333333333333335000,0.937500000000000000

227

00,0.94117647058823528000,0.94444444444444442000,0.94736842105263164000,0.94999999999999996000,0.95238095238095233000,0.95454545454545459000,0.95652173913043481000,0.95833333333333337000,0.95999999999999996000,0.96153846153846156000,0.96296296296296302000,0.96428571428571430000,0.96551724137931039000,0.96666666666666667000,0.96774193548387100000,0.96875000000000000000,0.96969696969696972000,0.97058823529411764000,0.97142857142857142000,0.97222222222222221000,0.97297297297297303000,0.97368421052631582000,0.97435897435897434000,0.97499999999999998000,0.97560975609756095000,0.97619047619047616000,0.97674418604651159000,0.97727272727272729000,0.97777777777777775000,0.97826086956521741000,0.97872340425531912000,0.97916666666666663000,0.97959183673469385000,0.97999999999999998000,0.98039215686274506000,0.98076923076923073000,0.98113207547169812000,0.98148148148148151000,0.98181818181818181000,0.98214285714285710000,0.98245614035087714000,0.98275862068965514000,0.98305084745762716000,0.98333333333333328000,0.98360655737704916000,0.98387096774193550000,0.98412698412698418000,0.98437500000000000000,0.98461538461538467000,0.98484848484848486000,0.98507462686567160000,0.98529411764705888000,0.98550724637681164000,0.98571428571428577000,0.98591549295774650000,0.98611111111111116000,0.98630136986301364000,0.98648648648648651000,0.98666666666666669000,0.98684210526315785000,0.98701298701298701000,0.98717948717948723000,0.98734177215189878000,0.98750000000000004000,0.98765432098765427000,0.98780487804878048000,0.98795180722891562000,0.98809523809523814000,0.98823529411764710000,0.98837209302325579000,0.98850574712643680000,0.98863636363636365000,0.98876404494382020000,0.98888888888888893000,0.98901098901098905000,0.98913043478260865000,0.98924731182795700000,0.98936170212765961000,0.98947368421052628000,0.98958333333333337000,0.98969072164948457000,0.98979591836734693000,0.98989898989898994000,0.98999999999999999000,0.99009900990099009000,0.99019607843137258000,0.99029126213592233000,0.99038461538461542000,0.99047619047619051000,0.99056603773584906000,0.99065420560747663000,0.99074074074074070000,0.99082568807339455000,0.99090909090909096000,0.99099099099099097000,0.99107142857142860000,0.99115044247787609000,0.99122807017543857000,0.99130434782608701000,0.99137931034482762000,0.99145299145299148000,0.99152542372881358000,0.99159663865546221000,0.99166666666666670000,0.99173553719008267000,0.99180327868852458000,0.99186991869918695000,0.99193548387096775000,0.99199999999999999000,0.99206349206349209000,0.99212598425196852000,0.99218750000000000000,0.99224806201550386000,0.99230769230769234000,0.99236641221374045000,0.99242424242424243000,0.99248120300751874000,0.99253731343283580000,0.99259259259259258000,0.99264705882352944000,0.99270072992700731000,0.99275

228

362318840576000,0.99280575539568350000,0.99285714285714288000,0.99290780141843971000,0.99295774647887325000,0.99300699300699302000,0.99305555555555558000,0.99310344827586206000,0.99315068493150682000,0.99319727891156462000,0.99324324324324320000,0.99328859060402686000,0.99333333333333329000,0.99337748344370858000,0.99342105263157898000,0.99346405228758172000,0.99350649350649356000,0.99354838709677418000,0.99358974358974361000,0.99363057324840764000,0.99367088607594933000,0.99371069182389937000,0.99375000000000002000,0.99378881987577639000,0.99382716049382713000,0.99386503067484666000,0.99390243902439024000,0.99393939393939390000,0.99397590361445787000,0.99401197604790414000,0.99404761904761907000,0.99408284023668636000,0.99411764705882355000,0.99415204678362579000,0.99418604651162790000,0.99421965317919070000,0.99425287356321834000,0.99428571428571433000,0.99431818181818177000,0.99435028248587576000,0.99438202247191010000,0.99441340782122900000,0.99444444444444446000,0.99447513812154698000,0.99450549450549453000,0.99453551912568305000,0.99456521739130432000,0.99459459459459465000,0.99462365591397850000,0.99465240641711228000,0.99468085106382975000,0.99470899470899465000,0.99473684210526314000,0.99476439790575921000,0.99479166666666663000,0.99481865284974091000,0.99484536082474229000,0.99487179487179489000,0.99489795918367352000,0.99492385786802029000,0.99494949494949492000,0.99497487437185927000,0.99500000000000000000,0.99502487562189057000,0.99504950495049505000,0.99507389162561577000,0.99509803921568629000,0.99512195121951219000,0.99514563106796117000,0.99516908212560384000,0.99519230769230771000,0.99521531100478466000,0.99523809523809526000,0.99526066350710896000,0.99528301886792447000,0.99530516431924887000,0.99532710280373837000,0.99534883720930234000,0.99537037037037035000,0.99539170506912444000,0.99541284403669728000,0.99543378995433796000,0.99545454545454548000,0.99547511312217196000,0.99549549549549554000,0.99551569506726456000,0.99553571428571430000,0.99555555555555553000,0.99557522123893805000,0.99559471365638763000,0.99561403508771928000,0.99563318777292575000,0.99565217391304350000,0.99567099567099571000,0.99568965517241381000,0.99570815450643779000,0.99572649572649574000,0.99574468085106382000,0.99576271186440679000,0.99578059071729963000,0.99579831932773111000,0.99581589958159000000,0.99583333333333335000,0.99585062240663902000,0.99586776859504134000,0.99588477366255146000,0.99590163934426235000,0.99591836734693873000,0.99593495934959353000,0.99595141700404854000,0.99596774193548387000,0.99598393574297184000,0.99600000000000000000,0.99601593625498008000,0.99603174603174605000,0.99604743083003955000,0.99606299212598426000,0.99607843137254903000,0.99609375000000000000,0.99610894941634243000,0.99612403100775193000,0.99613899613899

229

615000,0.99615384615384617000,0.99616858237547889000,0.99618320610687028000,0.99619771863117867000,0.99621212121212122000,0.99622641509433962000,0.99624060150375937000,0.99625468164794007000,0.99626865671641796000,0.99628252788104088000,0.99629629629629635000,0.99630996309963105000,0.99632352941176472000,0.99633699633699635000,0.99635036496350360000,0.99636363636363634000,0.99637681159420288000,0.99638989169675085000,0.99640287769784175000,0.99641577060931896000,0.99642857142857144000,0.99644128113879005000,0.99645390070921991000,0.99646643109540634000,0.99647887323943662000,0.99649122807017543000,0.99650349650349646000,0.99651567944250874000,0.99652777777777779000,0.99653979238754320000,0.99655172413793103000,0.99656357388316152000,0.99657534246575341000,0.99658703071672350000,0.99659863945578231000,0.99661016949152548000,0.99662162162162160000,0.99663299663299665000,0.99664429530201337000,0.99665551839464883000,0.99666666666666670000,0.99667774086378735000,0.99668874172185429000,0.99669966996699666000,0.99671052631578949000,0.99672131147540988000,0.99673202614379086000,0.99674267100977199000,0.99675324675324672000,0.99676375404530748000,0.99677419354838714000,0.99678456591639875000,0.99679487179487181000,0.99680511182108622000,0.99681528662420382000,0.99682539682539684000,0.99683544303797467000,0.99684542586750791000,0.99685534591194969000,0.99686520376175547000,0.99687499999999996000,0.99688473520249221000,0.99689440993788825000,0.99690402476780182000,0.99691358024691357000,0.99692307692307691000,0.99693251533742333000,0.99694189602446481000,0.99695121951219512000,0.99696048632218848000,0.99696969696969695000,0.99697885196374625000,0.99698795180722888000,0.99699699699699695000,0.99700598802395213000,0.99701492537313430000,0.99702380952380953000,0.99703264094955490000,0.99704142011834318000,0.99705014749262533000,0.99705882352941178000,0.99706744868035191000,0.99707602339181289000,0.99708454810495628000,0.99709302325581395000,0.99710144927536237000,0.99710982658959535000,0.99711815561959649000,0.99712643678160917000,0.99713467048710602000,0.99714285714285711000,0.99715099715099720000,0.99715909090909094000,0.99716713881019825000,0.99717514124293782000,0.99718309859154930000,0.99719101123595510000,0.99719887955182074000,0.99720670391061450000,0.99721448467966578000,0.99722222222222223000,0.99722991689750695000,0.99723756906077343000,0.99724517906336085000,0.99725274725274726000,0.99726027397260275000,0.99726775956284153000,0.99727520435967298000,0.99728260869565222000,0.99728997289972898000,0.99729729729729732000,0.99730458221024254000,0.99731182795698925000,0.99731903485254692000,0.99732620320855614000,0.99733333333333329000,0.99734042553191493000,0.99734748010610075000,0.99735449735449733000,0.99736147757255933000,0.99736842105263157000,0.9

230

9737532808398954000,0.99738219895287961000,0.99738903394255873000,0.99739583333333337000,0.99740259740259740000,0.99740932642487046000,0.99741602067183466000,0.99742268041237114000,0.99742930591259638000,0.99743589743589745000,0.99744245524296671000,0.99744897959183676000,0.99745547073791352000,0.99746192893401020000,0.99746835443037973000,0.99747474747474751000,0.99748110831234260000,0.99748743718592969000,0.99749373433583965000,0.99750000000000005000,0.99750623441396513000,0.99751243781094523000,0.99751861042183620000,0.99752475247524752000,0.99753086419753090000,0.99753694581280783000,0.99754299754299758000,0.99754901960784315000,0.99755501222493892000,0.99756097560975610000,0.99756690997566910000,0.99757281553398058000,0.99757869249394671000,0.99758454106280192000,0.99759036144578317000,0.99759615384615385000,0.99760191846522783000,0.99760765550239239000,0.99761336515513122000,0.99761904761904763000,0.99762470308788598000,0.99763033175355453000,0.99763593380614657000,0.99764150943396224000,0.99764705882352944000,0.99765258215962438000,0.99765807962529274000,0.99766355140186913000,0.99766899766899764000,0.99767441860465111000,0.99767981438515085000,0.99768518518518523000,0.99769053117782913000,0.99769585253456217000,0.99770114942528731000,0.99770642201834858000,0.99771167048054921000,0.99771689497716898000,0.99772209567198178000,0.99772727272727268000,0.99773242630385484000,0.99773755656108598000,0.99774266365688491000,0.99774774774774777000,0.99775280898876406000,0.99775784753363228000,0.99776286353467558000,0.99776785714285710000,0.99777282850779514000,0.99777777777777776000,0.99778270509977829000,0.99778761061946908000,0.99779249448123619000,0.99779735682819382000,0.99780219780219781000,0.99780701754385970000,0.99781181619256021000,0.99781659388646293000,0.99782135076252720000,0.99782608695652175000,0.99783080260303691000,0.99783549783549785000,0.99784017278617709000,0.99784482758620685000,0.99784946236559136000,0.99785407725321884000,0.99785867237687365000,0.99786324786324787000,0.99786780383795304000,0.99787234042553197000,0.99787685774946921000,0.99788135593220340000,0.99788583509513740000,0.99789029535864981000,0.99789473684210528000,0.99789915966386555000,0.99790356394129975000,0.99790794979079500000,0.99791231732776620000,0.99791666666666667000,0.99792099792099798000,0.99792531120331951000,0.99792960662525876000,0.99793388429752061000,0.99793814432989691000,0.99794238683127567000,0.99794661190965095000,0.99795081967213117000,0.99795501022494892000,0.99795918367346936000,0.99796334012219956000,0.99796747967479671000,0.99797160243407712000,0.99797570850202433000,0.99797979797979797000,0.99798387096774188000,0.99798792756539234000,0.99799196787148592000,0.99799599198396793000,0.99800000000000000000,0.99800399201596801000,0.9980079681

231

2749004000,0.99801192842942343000,0.99801587301587302000,0.99801980198019802000,0.99802371541501977000,0.99802761341222879000,0.99803149606299213000,0.99803536345776034000,0.99803921568627452000,0.99804305283757344000,0.99804687500000000000,0.99805068226120852000,0.99805447470817121000,0.99805825242718449000,0.99806201550387597000,0.99806576402321079000,0.99806949806949807000,0.99807321772639690000,0.99807692307692308000,0.99808061420345484000,0.99808429118773945000,0.99808795411089868000,0.99809160305343514000,0.99809523809523815000,0.99809885931558939000,0.99810246679316883000,0.99810606060606055000,0.99810964083175802000,0.99811320754716981000,0.99811676082862522000,0.99812030075187974000,0.99812382739212002000,0.99812734082397003000,0.99813084112149530000,0.99813432835820892000,0.99813780260707630000,0.99814126394052050000,0.99814471243042668000,0.99814814814814812000,0.99815157116451014000,0.99815498154981552000,0.99815837937384899000,0.99816176470588236000,0.99816513761467895000,0.99816849816849818000,0.99817184643510060000,0.99817518248175185000,0.99817850637522765000,0.99818181818181817000,0.99818511796733211000,0.99818840579710144000,0.99819168173598549000,0.99819494584837543000,0.99819819819819822000,0.99820143884892087000,0.99820466786355477000,0.99820788530465954000,0.99821109123434704000,0.99821428571428572000,0.99821746880570406000,0.99822064056939497000,0.99822380106571940000,0.99822695035460995000,0.99823008849557526000,0.99823321554770317000,0.99823633156966496000,0.99823943661971826000,0.99824253075571179000,0.99824561403508771000,0.99824868651488619000,0.99825174825174823000,0.99825479930191974000,0.99825783972125437000,0.99826086956521742000,0.99826388888888884000,0.99826689774696709000,0.99826989619377160000,0.99827288428324701000,0.99827586206896557000,0.99827882960413084000,0.99828178694158076000,0.99828473413379071000,0.99828767123287676000,0.99829059829059830000,0.99829351535836175000,0.99829642248722317000,0.99829931972789121000,0.99830220713073003000,0.99830508474576274000,0.99830795262267347000,0.99831081081081086000,0.99831365935919059000,0.99831649831649827000,0.99831932773109244000,0.99832214765100669000,0.99832495812395305000,0.99832775919732442000,0.99833055091819700000,0.99833333333333329000,0.99833610648918469000,0.99833887043189373000,0.99834162520729686000,0.99834437086092720000,0.99834710743801658000,0.99834983498349839000,0.99835255354200991000,0.99835526315789469000,0.99835796387520526000,0.99836065573770494000,0.99836333878887074000,0.99836601307189543000,0.99836867862969003000,0.99837133550488599000,0.99837398373983743000,0.99837662337662336000,0.99837925445705022000,0.99838187702265369000,0.99838449111470118000,0.99838709677419357000,0.99838969404186795000,0.99839228295819937000,0.9983948635634029000

232

0,0.99839743589743590000,0.99839999999999995000,0.99840255591054317000,0.99840510366826152000,0.99840764331210186000,0.99841017488076311000,0.99841269841269842000,0.99841521394611732000,0.99841772151898733000,0.99842022116903628000,0.99842271293375395000,0.99842519685039366000,0.99842767295597479000,0.99843014128728413000,0.99843260188087779000,0.99843505477308292000,0.99843749999999998000,0.99843993759750393000,0.99844236760124616000,0.99844479004665632000,0.99844720496894412000,0.99844961240310082000,0.99845201238390091000,0.99845440494590421000,0.99845679012345678000,0.99845916795069334000,0.99846153846153851000,0.99846390168970811000,0.99846625766871167000,0.99846860643185298000,0.99847094801223246000,0.99847328244274813000,0.99847560975609762000,0.99847792998477924000,0.99848024316109418000,0.99848254931714719000,0.99848484848484853000,0.99848714069591527000,0.99848942598187307000,0.99849170437405732000,0.99849397590361444000,0.99849624060150377000,0.99849849849849848000,0.99850074962518742000,0.99850299401197606000,0.99850523168908822000,0.99850746268656720000,0.99850968703427723000,0.99851190476190477000,0.99851411589895989000,0.99851632047477745000,0.99851851851851847000,0.99852071005917165000,0.99852289512555392000,0.99852507374631272000,0.99852724594992637000,0.99852941176470589000,0.99853157121879588000,0.99853372434017595000,0.99853587115666176000,0.99853801169590639000,0.99854014598540142000,0.99854227405247808000,0.99854439592430855000,0.99854651162790697000,0.99854862119013066000,0.99855072463768113000,0.99855282199710560000,0.99855491329479773000,0.99855699855699853000,0.99855907780979825000,0.99856115107913668000,0.99856321839080464000,0.99856527977044474000,0.99856733524355301000,0.99856938483547930000,0.99857142857142855000,0.99857346647646217000,0.99857549857549854000,0.99857752489331442000,0.99857954545454541000,0.99858156028368794000,0.99858356940509918000,0.99858557284299854000,0.99858757062146897000,0.99858956276445698000,0.99859154929577465000,0.99859353023909991000,0.99859550561797750000,0.99859747545582045000,0.99859943977591037000,0.99860139860139863000,0.99860335195530725000,0.99860529986053004000,0.99860724233983289000,0.99860917941585536000,0.99861111111111112000,0.99861303744798890000,0.99861495844875348000,0.99861687413554634000,0.99861878453038677000,0.99862068965517237000,0.99862258953168048000,0.99862448418156813000,0.99862637362637363000,0.99862825788751719000,0.99863013698630132000,0.99863201094391241000,0.99863387978142082000,0.99863574351978168000,0.99863760217983655000,0.99863945578231295000,0.99864130434782605000,0.99864314789687925000,0.99864498644986455000,0.99864682002706362000,0.99864864864864866000,0.99865047233468285000,0.99865229110512133000,0.99865410497981155000,0.99865591397849462000,0.998657

233

71812080539000,0.99865951742627346000,0.99866131191432395000,0.99866310160427807000,0.99866488651535379000,0.99866666666666670000,0.99866844207723038000,0.99867021276595747000,0.99867197875166003000,0.99867374005305043000,0.99867549668874167000,0.99867724867724872000,0.99867899603698806000,0.99868073878627972000,0.99868247694334655000,0.99868421052631584000,0.99868593955321949000,0.99868766404199472000,0.99868938401048490000,0.99869109947643975000,0.99869281045751634000,0.99869451697127942000,0.99869621903520212000,0.99869791666666663000,0.99869960988296491000,0.99870129870129876000,0.99870298313878081000,0.99870466321243523000,0.99870633893919791000,0.99870801033591727000,0.99870967741935479000,0.99871134020618557000,0.99871299871299868000,0.99871465295629824000,0.99871630295250324000,0.99871794871794872000,0.99871959026888601000,0.99872122762148341000,0.99872286079182626000,0.99872448979591832000,0.99872611464968153000,0.99872773536895676000,0.99872935196950441000,0.99873096446700504000,0.99873257287705952000,0.99873417721518987000,0.99873577749683939000,0.99873737373737370000,0.99873896595208067000,0.99874055415617125000,0.99874213836477987000,0.99874371859296485000,0.99874529485570895000,0.99874686716791983000,0.99874843554443049000,0.99875000000000003000,0.99875156054931336000,0.99875311720698257000,0.99875466998754669000,0.99875621890547261000,0.99875776397515525000,0.99875930521091816000,0.99876084262701359000,0.99876237623762376000,0.99876390605686027000,0.99876543209876545000,0.99876695437731200000,0.99876847290640391000,0.99876998769987702000,0.99877149877149873000,0.99877300613496933000,0.99877450980392157000,0.99877600979192172000,0.99877750611246940000,0.99877899877899878000,0.99878048780487805000,0.99878197320341044000,0.99878345498783450000,0.99878493317132444000,0.99878640776699024000,0.99878787878787878000,0.99878934624697335000,0.99879081015719473000,0.99879227053140096000,0.99879372738238847000,0.99879518072289153000,0.99879663056558365000,0.99879807692307687000,0.99879951980792314000,0.99880095923261392000,0.99880239520958081000,0.99880382775119614000,0.99880525686977295000,0.99880668257756566000,0.99880810488676997000,0.99880952380952381000,0.99881093935790721000,0.99881235154394299000,0.99881376037959668000,0.99881516587677721000,0.99881656804733732000,0.99881796690307334000,0.99881936245572611000,0.99882075471698117000,0.99882214369846878000,0.99882352941176467000,0.99882491186839018000,0.99882629107981225000,0.99882766705744430000,0.99882903981264637000,0.99883040935672518000,0.99883177570093462000,0.99883313885647607000,0.99883449883449882000,0.99883585564610011000,0.99883720930232556000,0.99883855981416958000,0.99883990719257543000,0.99884125144843572000,0.99884259259259256000,0.99884393063583810000,0.998845265588914

234

51000,0.99884659746251436000,0.99884792626728114000,0.99884925201380903000,0.99885057471264371000,0.99885189437428246000,0.99885321100917435000,0.99885452462772051000,0.99885583524027455000,0.99885714285714289000,0.99885844748858443000,0.99885974914481190000,0.99886104783599083000,0.99886234357224113000,0.99886363636363640000,0.99886492622020429000,0.99886621315192747000,0.99886749716874290000,0.99886877828054299000,0.99887005649717520000,0.99887133182844245000,0.99887260428410374000,0.99887387387387383000,0.99887514060742411000,0.99887640449438198000,0.99887766554433222000,0.99887892376681620000,0.99888017917133254000,0.99888143176733779000,0.99888268156424576000,0.99888392857142860000,0.99888517279821631000,0.99888641425389757000,0.99888765294771964000,0.99888888888888894000,0.99889012208657046000,0.99889135254988914000,0.99889258028792915000,0.99889380530973448000,0.99889502762430937000,0.99889624724061810000,0.99889746416758540000,0.99889867841409696000,0.99889988998899892000,0.99890109890109891000,0.99890230515916578000,0.99890350877192979000,0.99890470974808321000,0.99890590809628010000,0.99890710382513659000,0.99890829694323147000,0.99890948745910579000,0.99891067538126366000,0.99891186071817195000,0.99891304347826082000,0.99891422366992400000,0.99891540130151846000,0.99891657638136511000,0.99891774891774887000,0.99891891891891893000,0.99892008639308860000,0.99892125134843579000,0.99892241379310343000,0.99892357373519913000,0.99892473118279568000,0.99892588614393130000,0.99892703862660948000,0.99892818863879962000,0.99892933618843682000,0.99893048128342243000,0.99893162393162394000,0.99893276414087517000,0.99893390191897657000,0.99893503727369537000,0.99893617021276593000,0.99893730074388953000,0.99893842887473461000,0.99893955461293749000,0.99894067796610164000,0.99894179894179891000,0.99894291754756870000,0.99894403379091867000,0.99894514767932485000,0.99894625922023184000,0.99894736842105258000,0.99894847528916930000,0.99894957983193278000,0.99895068205666315000,0.99895178197064993000,0.99895287958115186000,0.99895397489539750000,0.99895506792058519000,0.99895615866388310000,0.99895724713242962000,0.99895833333333328000,0.99895941727367321000,0.99896049896049899000,0.99896157840083077000,0.99896265560165975000,0.99896373056994814000,0.99896480331262938000,0.99896587383660806000,0.99896694214876036000,0.99896800825593390000,0.99896907216494846000,0.99897013388259526000,0.99897119341563789000,0.99897225077081198000,0.99897330595482547000,0.99897435897435893000,0.99897540983606559000,0.99897645854657113000,0.99897750511247441000,0.99897854954034726000,0.99897959183673468000,0.99898063200815490000,0.99898167006109984000,0.99898270600203454000,0.99898373983739841000,0.99898477157360410000,0.99898580121703850000,0.99898682877406286000,0.99

235

898785425101211000,0.99898887765419619000,0.99898989898989898000,0.99899091826437947000,0.99899193548387100000,0.99899295065458205000,0.99899396378269623000,0.99899497487437183000,0.99899598393574296000,0.99899699097291872000,0.99899799599198402000,0.99899899899899902000,0.99900000000000000000,0.99900099900099903000,0.99900199600798401000,0.99900299102691925000,0.99900398406374502000,0.99900497512437814000,0.99900596421471177000,0.99900695134061568000,0.99900793650793651000,0.99900891972249750000,0.99900990099009901000,0.99901088031651830000,0.99901185770750989000,0.99901283316880551000,0.99901380670611439000,0.99901477832512320000,0.99901574803149606000,0.99901671583087515000,0.99901768172888017000,0.99901864573110888000,0.99901960784313726000,0.99902056807051909000,0.99902152641878672000,0.99902248289345064000}; void gauss_legendre_tbl(int n, double* x, double* w, double eps) { double x0, x1, dx; /* Abscissas */ double w0, w1, dw; /* Weights */ double P0, P_1, P_2; /* Legendre polynomial values */ double dpdx; /* Legendre polynomial derivative */ int i, j, k, m; /* Iterators */ double t0, t1, t2, t3; m = (n+1)>>1; t0 = (1.0-(1.0-1.0/(double)n)/(8.0*(double)n*(double)n)); t1 = 1.0/(4.0*(double)n+2.0); for (i = 1; i <= m; i++) { /* Find i-th root of Legendre polynomial */ /* Initial guess */ x0 = cos(PI*(double)((i<<2)-1)*t1)*t0; /* Newton iterations, at least one */ j = 0; dx = dw = DBL_MAX; do { /* Compute Legendre polynomial value at x0 */ P_1 = 1.0; P0 = x0;

236

#if 0 /* Simple, not optimized version */ for (k = 2; k <= n; k++) { P_2 = P_1; P_1 = P0; t2 = x0*P_1; t3 = (double)(k-1)/(double)k; P0 = t2 + t3*(t2 - P_2); } #else /* Optimized version using lookup tables */ if (n<1024) { /* Use fast algorithm for small n*/ for (k = 2; k <= n; k++) { P_2 = P_1; P_1 = P0; t2 = x0*P_1; P0 = t2 + ltbl[k]*(t2 - P_2); } }else{ /* Use general algorithm for other n */ for (k = 2; k < 1024; k++) { P_2 = P_1; P_1 = P0; t2 = x0*P_1; P0 = t2 + ltbl[k]*(t2 - P_2); } for (k = 1024; k <= n; k++) { P_2 = P_1; P_1 = P0; t2 = x0*P_1;

237

t3 = (double)(k-1)/(double)k; P0 = t2 + t3*(t2 - P_2); } } #endif /* Compute Legendre polynomial derivative at x0 */ dpdx = ((x0*P0-P_1)*(double)n)/(x0*x0-1.0); /* Newton step */ x1 = x0-P0/dpdx; /* Weight computing */ w1 = 2.0/((1.0-x1*x1)*dpdx*dpdx); /* Compute weight w0 on first iteration, needed for dw */ if (j==0) w0 = 2.0/((1.0-x0*x0)*dpdx*dpdx); dx = x0-x1; dw = w0-w1; x0 = x1; w0 = w1; j++; } while((FABS(dx)>eps || FABS(dw)>eps) && j<100); x[(m-1)-(i-1)] = x1; w[(m-1)-(i-1)] = w1; } return; } double fk1(double x, double origin, double radius, double lambda, int position_Layer,double position_x,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE) { double realde=d[position_Layer]; double realzp=position_x; double realzm=realde-realzp;

238

complex <double> kx,ke,kp,km,kze,kzp,kzm; complex <double> de,zp,zm; complex <double> c2j(0,2); complex <double> cj(0,1); complex <double> c1(1,0); complex <double> cx(x,0); // x is the coutour integal angle from pi to 2pi complex <double> aTM,aTMp,aTMm,aTE,aTEp,aTEm; double K[10]={0}; de=complex <double> (realde,0); zp=complex <double> (realzp,0); zm=complex <double> (realzm,0); complex <double> kk (2*PI/lambda,0); complex <double> korigin (origin,0); complex <double> kradius (radius,0); ke=kk*n[position_Layer]; kp=kk*n[0]; km=kk*n[NLayer+1]; kx=korigin+kradius*exp(cj*cx);// kx is the complex value of the point kze=sqrt(ke*ke-kx*kx); kzp=sqrt(kp*kp-kx*kx); kzm=sqrt(km*km-kx*kx); Calculate_Matrix(lambda,kx,position_Layer, NLayer,d, n,TM,TE); aTMp=TM[1]*exp(c2j*kze*zp); aTMm=TM[3]*exp(c2j*kze*zm); aTM=aTMp*aTMm; aTEp=TE[1]*exp(c2j*kze*zp); aTEm=TE[3]*exp(c2j*kze*zm); aTE=aTEp*aTEm; K[1]=0.75*real(kx*kx/(ke*ke*ke*kze)*(c1+aTMp)*(c1+aTMm)/(c1-aTM)*kradius*exp(cj*cx)*cj*(c1+c1)*kx); // addition 2kdk=2k*kr*exp(i*theta)*i d theta K[2]=0; K[3]=0.375*real(kze/(ke*ke*ke)*(c1-aTMp)*(c1-aTMm)/(c1-aTM)*kradius*exp(cj*cx)*cj*(c1+c1)*kx);

239

K[4]=0.375*real(c1/(ke*kze)*(c1+aTEp)*(c1+aTEm)/(c1-aTE)*kradius*exp(cj*cx)*cj*(c1+c1)*kx); K[5]=K[1]/3.0+2.0*K[3]/3.0; K[6]=K[2]/3.0+2.0*K[4]/3.0; K[7]=K[5]+K[6]; return K[7]; } double MyGauss1(int order, double kbegin, double kend, double lambda, int position_Layer,double position_x,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE) { double* x = NULL; double* w = NULL; double A,B,Ax,s; int i, dtbl, m; double origin=0.5*(kbegin+kend); double radius=0.5*(kend-kbegin); double a=PI; double b=2*PI; m = (order+1)>>1; /* Load appropriate predefined table */ dtbl = 0; for (i = 0; i<GLAWSIZE;i++) { if(order==glaw[i].n) { x = glaw[i].x; w = glaw[i].w; break; } } /* Generate new if non-predefined table is required */ /* with precision of 1e-10 */ if(NULL==x) { dtbl = 1; x = (double*)malloc(m*sizeof(double));

240

w = (double*)malloc(m*sizeof(double)); gauss_legendre_tbl(order,x,w,1e-10); } A = 0.5*(b-a); B = 0.5*(b+a); if(order&1) /* n - odd */ { //s = w[0]*((*f)(B,data)); s = w[0]*fk1(B, origin, radius,lambda, position_Layer,position_x, NLayer,d, n, TM,TE); for (i=1;i<m;i++) { Ax = A*x[i]; //s += w[i]*((*f)(B+Ax,data)+(*f)(B-Ax,data)); s += w[i]*(fk1(B+Ax, origin, radius, lambda, position_Layer,position_x, NLayer,d, n, TM,TE)+fk1(B-Ax,origin, radius, lambda, position_Layer,position_x, NLayer,d, n, TM,TE)); } }else{ /* n - even */ s = 0.0; for (i=0;i<m;i++) { Ax = A*x[i]; //s += w[i]*((*f)(B+Ax,data)+(*f)(B-Ax,data)); s += w[i]*(fk1(B+Ax, origin, radius, lambda, position_Layer,position_x, NLayer,d, n, TM,TE)+fk1(B-Ax,origin, radius, lambda, position_Layer,position_x, NLayer,d, n, TM,TE)); } } if (dtbl) { free(x); free(w); } return A*s; }

241

double fk2(double x, double lambda, int position_Layer,double position_x,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE) { double realde=d[position_Layer]; double realzp=position_x; double realzm=realde-realzp; complex <double> kx,ke,kp,km,kze,kzp,kzm; complex <double> de,zp,zm; complex <double> c2j(0,2); complex <double> cj(0,1); complex <double> c1(1,0); complex <double> aTM,aTMp,aTMm,aTE,aTEp,aTEm; double K[10]={0}; de=complex <double> (realde,0); zp=complex <double> (realzp,0); zm=complex <double> (realzm,0); complex <double> kk (2*PI/lambda,0); ke=kk*n[position_Layer]; kp=kk*n[0]; km=kk*n[NLayer+1]; kx=complex <double> (x,0); // kx is the complex value of the point kze=sqrt(ke*ke-kx*kx); kzp=sqrt(kp*kp-kx*kx); kzm=sqrt(km*km-kx*kx); Calculate_Matrix(lambda,kx,position_Layer, NLayer,d, n,TM,TE); aTMp=TM[1]*exp(c2j*kze*zp); aTMm=TM[3]*exp(c2j*kze*zm); aTM=aTMp*aTMm; aTEp=TE[1]*exp(c2j*kze*zp);

242

aTEm=TE[3]*exp(c2j*kze*zm); aTE=aTEp*aTEm; K[1]=0.75*real(kx*kx/(ke*ke*ke*kze)*(c1+aTMp)*(c1+aTMm)/(c1-aTM)*(c1+c1)*kx); //additional 2k*dk // and I put the additional abs() function to make sure it is positive, which is not the same as the original paper // maybe the complex calculation in the paper, it not the defaul calculation in the C++ // for example, sqrt(-1) can be +1j or -1j, but c++ alwasy give +1j, which can be different when consider the real part or imaginary part // and the problem mostly come from dipole lies in a material have non-zero k, or dipole is close to a non-zero k, so i suggest put the organic material k==0 all the times K[2]=0; K[3]=0.375*real(kze/(ke*ke*ke)*(c1-aTMp)*(c1-aTMm)/(c1-aTM)*(c1+c1)*kx); K[4]=0.375*real(c1/(ke*kze)*(c1+aTEp)*(c1+aTEm)/(c1-aTE)*(c1+c1)*kx); K[5]=K[1]/3.0+2.0*K[3]/3.0; K[6]=K[2]/3.0+2.0*K[4]/3.0; K[7]=K[5]+K[6]; return K[7]; } double MyGauss2(int order, double kbegin, double kend, double lambda, int position_Layer,double position_x,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE) { double* x = NULL; double* w = NULL; double A,B,Ax,s; int i, dtbl, m; m = (order+1)>>1; /* Load appropriate predefined table */ dtbl = 0; for (i = 0; i<GLAWSIZE;i++) {

243

if(order==glaw[i].n) { x = glaw[i].x; w = glaw[i].w; break; } } /* Generate new if non-predefined table is required */ /* with precision of 1e-10 */ if(NULL==x) { dtbl = 1; x = (double*)malloc(m*sizeof(double)); w = (double*)malloc(m*sizeof(double)); gauss_legendre_tbl(order,x,w,1e-10); } A = 0.5*(kend-kbegin); B = 0.5*(kend+kbegin); if(order&1) /* n - odd */ { //s = w[0]*((*f)(B,data)); s = w[0]*fk2(B,lambda,position_Layer,position_x, NLayer,d, n, TM,TE); for (i=1;i<m;i++) { Ax = A*x[i]; //s += w[i]*((*f)(B+Ax,data)+(*f)(B-Ax,data)); s += w[i]*( fk2(B+Ax, lambda, position_Layer,position_x, NLayer,d, n, TM,TE)+fk2(B-Ax, lambda, position_Layer,position_x, NLayer,d, n, TM,TE)); } }else{ /* n - even */ s = 0.0; for (i=0;i<m;i++) { Ax = A*x[i]; //s += w[i]*((*f)(B+Ax,data)+(*f)(B-Ax,data));

244

s += w[i]*( fk2(B+Ax, lambda, position_Layer,position_x, NLayer,d, n, TM,TE)+fk2(B-Ax, lambda, position_Layer,position_x, NLayer,d, n, TM,TE)); } } if (dtbl) { free(x); free(w); } return A*s; } double fk3(double x, double origin, double radius, double lambda, int position_Layer,double position_x,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE) { double realde=d[position_Layer]; double realzp=position_x; double realzm=realde-realzp; complex <double> kx,ke,kp,km,kze,kzp,kzm; complex <double> de,zp,zm; complex <double> c2j(0,2); complex <double> cj(0,1); complex <double> c1(1,0); complex <double> cx(x,0); // x is the coutour integal angle from pi to 2pi complex <double> aTM,aTMp,aTMm,aTE,aTEp,aTEm; double KPT[10]={0}; de=complex <double> (realde,0); zp=complex <double> (realzp,0); zm=complex <double> (realzm,0); complex <double> kk (2*PI/lambda,0); complex <double> korigin (origin,0); complex <double> kradius (radius,0);

245

ke=kk*n[position_Layer]; kp=kk*n[0]; km=kk*n[NLayer+1]; kx=korigin+kradius*exp(cj*cx);// kx is the complex value of the point kze=sqrt(ke*ke-kx*kx); kzp=sqrt(kp*kp-kx*kx); kzm=sqrt(km*km-kx*kx); Calculate_Matrix(lambda,kx,position_Layer, NLayer,d, n,TM,TE); aTMp=TM[1]*exp(c2j*kze*zp); aTMm=TM[3]*exp(c2j*kze*zm); aTM=aTMp*aTMm; aTEp=TE[1]*exp(c2j*kze*zp); aTEm=TE[3]*exp(c2j*kze*zm); aTE=aTEp*aTEm; // kpt seems can not calculate with the "Complex" gauss quadrature , maybe because it is not complex analytical KPT[1]=0.75*real(kx*kx/(ke*ke*ke*kze)*kradius*exp(cj*cx)*cj*(c1+c1)*kx*0.5*norm((c1+aTMm)/(c1-aTM))*norm(TM[2])*abs(kzp/kze)*norm(n[position_Layer]/n[0]));//additional 2k*dk KPT[2]=0; KPT[3]=0.375*real(kze/(ke*ke*ke)*kradius*exp(cj*cx)*cj*(c1+c1)*kx*0.5*norm((c1-aTMm)/(c1-aTM))*norm(TM[2])*abs(kzp/kze)*norm(n[position_Layer]/n[0])); KPT[4]=0.375*real(c1/(ke*kze)*kradius*exp(cj*cx)*cj*(c1+c1)*kx*0.5*norm((c1+aTEm)/(c1-aTE))*norm(TE[2])*abs(kzp/kze)); KPT[5]=KPT[1]/3.0+2.0*KPT[3]/3.0; KPT[6]=KPT[2]/3.0+2.0*KPT[4]/3.0; KPT[7]=KPT[5]+KPT[6]; return KPT[7]; }

246

double MyGauss3(int order, double kbegin, double kend, double lambda, int position_Layer,double position_x,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE) { double* x = NULL; double* w = NULL; double A,B,Ax,s; int i, dtbl, m; double origin=0.5*(kbegin+kend); double radius=0.5*(kend-kbegin); double a=PI; double b=2*PI; m = (order+1)>>1; /* Load appropriate predefined table */ dtbl = 0; for (i = 0; i<GLAWSIZE;i++) { if(order==glaw[i].n) { x = glaw[i].x; w = glaw[i].w; break; } } /* Generate new if non-predefined table is required */ /* with precision of 1e-10 */ if(NULL==x) { dtbl = 1; x = (double*)malloc(m*sizeof(double)); w = (double*)malloc(m*sizeof(double)); gauss_legendre_tbl(order,x,w,1e-10); } A = 0.5*(b-a); B = 0.5*(b+a);

247

if(order&1) /* n - odd */ { //s = w[0]*((*f)(B,data)); s = w[0]*fk3(B, origin, radius,lambda, position_Layer,position_x, NLayer,d, n, TM,TE); for (i=1;i<m;i++) { Ax = A*x[i]; //s += w[i]*((*f)(B+Ax,data)+(*f)(B-Ax,data)); s += w[i]*(fk3(B+Ax, origin, radius, lambda, position_Layer,position_x, NLayer,d, n, TM,TE)+fk3(B-Ax,origin, radius, lambda, position_Layer,position_x, NLayer,d, n, TM,TE)); } }else{ /* n - even */ s = 0.0; for (i=0;i<m;i++) { Ax = A*x[i]; //s += w[i]*((*f)(B+Ax,data)+(*f)(B-Ax,data)); s += w[i]*(fk3(B+Ax, origin, radius, lambda, position_Layer,position_x, NLayer,d, n, TM,TE)+fk3(B-Ax,origin, radius, lambda, position_Layer,position_x, NLayer,d, n, TM,TE)); } } if (dtbl) { free(x); free(w); } return A*s; } double fk4(double x, double lambda, int position_Layer,double position_x,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE) { double realde=d[position_Layer]; double realzp=position_x;

248

double realzm=realde-realzp; complex <double> kx,ke,kp,km,kze,kzp,kzm; complex <double> de,zp,zm; complex <double> c2j(0,2); complex <double> cj(0,1); complex <double> c1(1,0); complex <double> aTM,aTMp,aTMm,aTE,aTEp,aTEm; double KPT[10]={0}; de=complex <double> (realde,0); zp=complex <double> (realzp,0); zm=complex <double> (realzm,0); complex <double> kk (2*PI/lambda,0); ke=kk*n[position_Layer]; kp=kk*n[0]; km=kk*n[NLayer+1]; kx=complex <double> (x,0); // kx is the complex value of the point kze=sqrt(ke*ke-kx*kx); kzp=sqrt(kp*kp-kx*kx); kzm=sqrt(km*km-kx*kx); Calculate_Matrix(lambda,kx,position_Layer, NLayer,d, n,TM,TE); aTMp=TM[1]*exp(c2j*kze*zp); aTMm=TM[3]*exp(c2j*kze*zm); aTM=aTMp*aTMm; aTEp=TE[1]*exp(c2j*kze*zp); aTEm=TE[3]*exp(c2j*kze*zm); aTE=aTEp*aTEm; KPT[1]=0.75*real(kx*kx/(ke*ke*ke*kze)*(c1+c1)*kx)*0.5*norm((c1+aTMm)/(c1-aTM))*norm(TM[2])*abs(kzp/kze)*norm(n[position_Layer]/n[0]);//additional 2k*dk KPT[2]=0;

249

KPT[3]=0.375*real(kze/(ke*ke*ke)*(c1+c1)*kx)*0.5*norm((c1-aTMm)/(c1-aTM))*norm(TM[2])*abs(kzp/kze)*norm(n[position_Layer]/n[0]); KPT[4]=0.375*real(c1/(ke*kze)*(c1+c1)*kx)*0.5*norm((c1+aTEm)/(c1-aTE))*norm(TE[2])*abs(kzp/kze); KPT[5]=KPT[1]/3.0+2.0*KPT[3]/3.0; KPT[6]=KPT[2]/3.0+2.0*KPT[4]/3.0; KPT[7]=KPT[5]+KPT[6]; return KPT[7]; } double MyGauss4(int order, double kbegin, double kend, double lambda, int position_Layer,double position_x,int NLayer,double *d, complex <double> *n, complex <double> *TM,complex <double> *TE) { double* x = NULL; double* w = NULL; double A,B,Ax,s; int i, dtbl, m; m = (order+1)>>1; /* Load appropriate predefined table */ dtbl = 0; for (i = 0; i<GLAWSIZE;i++) { if(order==glaw[i].n) { x = glaw[i].x; w = glaw[i].w; break; } } /* Generate new if non-predefined table is required */ /* with precision of 1e-10 */ if(NULL==x) { dtbl = 1; x = (double*)malloc(m*sizeof(double));

250

w = (double*)malloc(m*sizeof(double)); gauss_legendre_tbl(order,x,w,1e-10); } A = 0.5*(kend-kbegin); B = 0.5*(kend+kbegin); if(order&1) /* n - odd */ { //s = w[0]*((*f)(B,data)); s = w[0]*fk4(B,lambda,position_Layer,position_x, NLayer,d, n, TM,TE); for (i=1;i<m;i++) { Ax = A*x[i]; //s += w[i]*((*f)(B+Ax,data)+(*f)(B-Ax,data)); s += w[i]*( fk4(B+Ax, lambda, position_Layer,position_x, NLayer,d, n, TM,TE)+fk4(B-Ax, lambda, position_Layer,position_x, NLayer,d, n, TM,TE)); } }else{ /* n - even */ s = 0.0; for (i=0;i<m;i++) { Ax = A*x[i]; //s += w[i]*((*f)(B+Ax,data)+(*f)(B-Ax,data)); s += w[i]*( fk4(B+Ax, lambda, position_Layer,position_x, NLayer,d, n, TM,TE)+fk4(B-Ax, lambda, position_Layer,position_x, NLayer,d, n, TM,TE)); } } if (dtbl) { free(x); free(w); } return A*s; }

int main()

251

{ double pi = 3.14159265359;; int theta=0,deltatheta=0; double CIExyz[100][3]={0},CF[100][3]={0}, PL[100][50]={0}, F[100][50]={0},Etot[100][50]={0},Esubstrate[100][50]={0}, Effisub[100][50]={0}, factor[100][50]={0},wv[100]={0}; double Psubstrate[100][200]={0},Pair[100][200]={0}; double ModeAir[100][50]={0},ModeSubstrate[100][50]={0},ModeWaveGuide[100][50]={0},ModePlasmon[100][50]={0}; int i=0,j=0; double totalwv[10]={0}; double Eoutsubstrate[100]={0},Eoutair[100]={0}; double result[50]={0}; complex<double> n[100][50]; complex <double> TM[50],TE[50]; double realk0=0; double realkr=2*realk0; double realkmin=40*realkr; double final[100]={0}; //----------------------------------- OLED structure --------------------------- // maximum 50 emitters string PLname[50]; int position_Layer[50]={0}; double position_x[50]={0}; // maximum 50 layers string layername[50]; double d[50]={0}; string line; int NLayer=0; int NEmitter=0; // the relative non_radiative_rate, assuming radiative_rate=1

252

// the cavity modified factor = (1)/(non_rad_rate+F*rad_rate) double nonrad=0; ifstream myfile("input.txt"); if (myfile.is_open()) { getline(myfile,line); getline(myfile,line); myfile >> nonrad; getline(myfile,line); getline(myfile,line); getline(myfile,line); myfile >> NLayer; getline(myfile,line); getline(myfile,line); getline(myfile,line); myfile >> layername[0]; getline(myfile,line); getline(myfile,line); for( i=1;i<=NLayer;i++) { getline(myfile,line); myfile >> layername[i]; getline(myfile,line); myfile >> d[i]; getline(myfile,line); getline(myfile,line); } getline(myfile,line); myfile >> layername[NLayer+1]; getline(myfile,line); getline(myfile,line); getline(myfile,line); myfile >> NEmitter; getline(myfile,line); getline(myfile,line);

253

for( i=1;i<=NEmitter;i++) { getline(myfile,line); myfile >> PLname[i]; getline(myfile,line); myfile >> position_Layer[i]; getline(myfile,line); myfile >> position_x[i]; getline(myfile,line); getline(myfile,line); } cout << "finish reading input.txt..." << endl; } else { cout << "Unable to open input.txt" << endl; } Read(CIExyz,CF,n,wv,PL, PLname, NEmitter, layername, NLayer); cout << "finish reading PL and material..." << endl; //--------------------------------------------------------------------------- //writing R_T.txt, which include the transmission and reflection of the stack // calculate the light injection from the substrate into the OLED stack // the reflection/transmission at 0,30,60 degrees and over all angles ofstream outfile; complex <double> kx,kzbegin,kzend,kbegin,kend,k0; kx=complex<double> (0,0); double temp1,temp2,temp3,temp4; double total1,total2,total3,total4; outfile.open("R_T.txt"); outfile << "lambda(nm)(theta=0)" << " " << "R_TM" << " " << "T_TM" << " " << "R_TE" << " " << "T_TE" << " ";

254

outfile << "lambda(nm)(theta=30)" << " " << "R_TM" << " " << "T_TM" << " " << "R_TE" << " " << "T_TE" << " "; outfile << "lambda(nm)(theta=60)" << " " << "R_TM" << " " << "T_TM" << " " << "R_TE" << " " << "T_TE" << " "; outfile << "lambda(nm)(all_theta)" << " " << "R_TM" << " " << "T_TM" << " " << "R_TE" << " " << "T_TE" << " "; outfile << endl ; for( i=0;i<=80;i++) { k0=complex<double>(2*pi/wv[i],0); kbegin=k0*n[i][0]; kend=k0*n[i][NLayer+1]; theta=0; kx=complex<double>(2*pi/wv[i]*sin(theta*pi/180.0),0); kx=kx*n[i][0]; kzbegin=sqrt(kbegin*kbegin-kx*kx); kzend=sqrt(kend*kend-kx*kx); Calculate_Matrix(wv[i], kx, position_Layer[1], NLayer,d,n[i],TM,TE); if(abs(imag(kzend))<1.0e-9) {

outfile << wv[i] << " " << norm(TM[5]) << " " << norm(TM[6])*norm(n[i][0]/n[i][NLayer+1])*abs(kzend/kzbegin)<< " " << norm(TE[5]) << " " << norm(TE[6])*abs(kzend/kzbegin) << " " ;

} else { outfile << wv[i] << " " << norm(TM[5]) << " " << 0 << " " << norm(TE[5]) << " " << 0 << " " ; } theta=30; kx=complex<double>(2*pi/wv[i]*sin(theta*pi/180.0),0); kx=kx*n[i][0]; kzbegin=sqrt(kbegin*kbegin-kx*kx); kzend=sqrt(kend*kend-kx*kx); Calculate_Matrix(wv[i], kx, position_Layer[1], NLayer,d,n[i],TM,TE); if(abs(imag(kzend))<1.0e-9) {

255

outfile << wv[i] << " " << norm(TM[5]) << " " << norm(TM[6])*norm(n[i][0]/n[i][NLayer+1])*abs(kzend/kzbegin)<< " " << norm(TE[5]) << " " << norm(TE[6])*abs(kzend/kzbegin) << " " ; } else { outfile << wv[i] << " " << norm(TM[5]) << " " << 0 << " " << norm(TE[5]) << " " << 0 << " " ; } theta=60; kx=complex<double>(2*pi/wv[i]*sin(theta*pi/180.0),0); kx=kx*n[i][0]; kzbegin=sqrt(kbegin*kbegin-kx*kx); kzend=sqrt(kend*kend-kx*kx); Calculate_Matrix(wv[i], kx, position_Layer[1], NLayer,d,n[i],TM,TE); if(abs(imag(kzend))<1.0e-9) { outfile << wv[i] << " " << norm(TM[5]) << " " << norm(TM[6])*norm(n[i][0]/n[i][NLayer+1])*abs(kzend/kzbegin)<< " " << norm(TE[5]) << " " << norm(TE[6])*abs(kzend/kzbegin) << " " ; } else { outfile << wv[i] << " " << norm(TM[5]) << " " << 0 << " " << norm(TE[5]) << " " << 0 << " " ; } deltatheta=1; total1=0;total2=0;total3=0;total4=0; for(theta=0;theta<=89;theta=theta+deltatheta) { kx=complex<double>(2*pi/wv[i]*sin(theta*pi/180.0),0); kx=kx*n[i][0]; kzbegin=sqrt(kbegin*kbegin-kx*kx); kzend=sqrt(kend*kend-kx*kx); Calculate_Matrix(wv[i], kx, position_Layer[1], NLayer,d,n[i],TM,TE); if(abs(imag(kzend))<1.0e-9) {

256

temp1= norm(TM[5]);temp2= norm(TM[6])*norm(n[i][0]/n[i][NLayer+1])*abs(kzend/kzbegin); temp3= norm(TE[5]); temp4= norm(TE[6])*abs(kzend/kzbegin); } else { temp1= norm(TM[5]);temp2= 0; temp3= norm(TE[5]); temp4= 0; } total1+=temp1*sin(theta*pi/180.0)*deltatheta*pi/180.0; total2+=temp2*sin(theta*pi/180.0)*deltatheta*pi/180.0; total3+=temp3*sin(theta*pi/180.0)*deltatheta*pi/180.0; total4+=temp4*sin(theta*pi/180.0)*deltatheta*pi/180.0; } outfile << wv[i] << " " << total1 << " " << total2 << " "<< total3 << " "<< total4<< " "; outfile << endl; } outfile.close(); cout << "finish writing R_T.txt..." << endl; //--------------------------------------------------------------------------- // writing Spectrum.txt //calculate the EL spectrum over all angles, and at 0,10,20....80,90 degrees // also calcualte the out-coupling efficiency at each wavelength // all calculations are done for the substrate and the air, respectively deltatheta=10; outfile.open("Spectrum.txt"); outfile << "lambda(nm)" << " " ; outfile << "PL_total" <<" " ; outfile << "Etot_total" <<" " ; outfile << "Esubstrate_total" <<" " ; outfile << "Eair_total" << " "; outfile << "Etot/PL" << " "; outfile << "Esubstrate/Etot" << " "; outfile << "Eair/Etot" << " "; outfile << "lambda(nm)" << " "; for( theta=0;theta<=91;theta+=deltatheta) {

257

outfile << "substrate_theta_" << theta << " "; } outfile << "lambda(nm)" << " "; for( theta=0;theta<=91;theta+=deltatheta) { outfile << "air_theta_" << theta << " "; } for(j=1;j<=NEmitter;j++) { outfile << "F_" << j <<" " ; } for(j=1;j<=NEmitter;j++) { outfile << "PL_" << j <<" " ; } for(j=1;j<=NEmitter;j++) { outfile << "Etot_" << j <<" " ; } for(j=1;j<=NEmitter;j++) { outfile << "Esubstrate_" << j <<" " ; } outfile << endl; for( i=0;i<=80;i++) { theta=0; realk0=2*pi/wv[i]*real(n[i][0]); realkr=2*realk0; realkmin=40*realkr; Etot[i][0]=0; Esubstrate[i][0]=0; PL[i][0]=0; //--------------------------------------------------------------------------- for(j=1;j<=NEmitter;j++)

258

{ result[2]=MyGauss4(1024, 0,realk0, wv[i], position_Layer[j], position_x[j], NLayer,d, n[i], TM,TE); result[3]=MyGauss1(256,0,2*realkr,wv[i],position_Layer[j],position_x[j],NLayer,d,n[i],TM,TE) +MyGauss2(256,2*realkr,realkmin, wv[i],position_Layer[j],position_x[j],NLayer,d,n[i],TM,TE) +Etotal2(realkmin,wv[i],position_Layer[j],position_x[j],NLayer,d,n[i],TM,TE); result[10]=MyGauss1(256,0,2*pi/wv[i]*1.0,wv[i],position_Layer[j],position_x[j],NLayer,d,n[i],TM,TE); // into air result[11]=MyGauss1(256,2*pi/wv[i]*1.0,2*pi/wv[i]*1.5, wv[i], position_Layer[j], position_x[j], NLayer,d, n[i], TM,TE); // into substrate result[12]=MyGauss1(256,2*pi/wv[i]*1.5,2*pi/wv[i]*1.7, wv[i], position_Layer[j], position_x[j], NLayer,d, n[i], TM,TE); // into substrate result[13]=MyGauss2(256,2*pi/wv[i]*1.7,2*pi/wv[i]*30, wv[i], position_Layer[j], position_x[j], NLayer,d, n[i], TM,TE)+Etotal2(2*pi/wv[i]*30,wv[i],position_Layer[j],position_x[j],NLayer,d,n[i],TM,TE);; // into substrate F[i][j]=result[3]; factor[i][j]=1.0/(nonrad+result[3]); Etot[i][j]=result[3]*factor[i][j]*PL[i][j]; Esubstrate[i][j]=result[2]*factor[i][j]*PL[i][j]; Effisub[i][j]=result[2]/result[3]; ModeAir[i][j]=result[10]*factor[i][j]*PL[i][j]; ModeSubstrate[i][j]=result[11]*factor[i][j]*PL[i][j]; ModeWaveGuide[i][j]=result[12]*factor[i][j]*PL[i][j]; ModePlasmon[i][j]=result[13]*factor[i][j]*PL[i][j]; Etot[i][0]+=Etot[i][j]; Esubstrate[i][0]+=Esubstrate[i][j]; PL[i][0]+=PL[i][j]; ModeAir[i][0]+= ModeAir[i][j]; ModeSubstrate[i][0]+= ModeSubstrate[i][j];

259

ModeWaveGuide[i][0]+= ModeWaveGuide[i][j]; ModePlasmon[i][0]+= ModePlasmon[i][j]; } Effisub[i][0]=Esubstrate[i][0]/Etot[i][0]; Eoutsubstrate[i]=0; Eoutair[i]=0; deltatheta=1; for(theta=0;theta<90;theta+=deltatheta) // not calculting theta=90 { Psubstrate[i][theta]=0; for(j=1;j<=NEmitter;j++) { Calculate_P (wv[i], theta*pi/180.0,position_Layer[j],position_x[j], NLayer,d, n[i],TM,TE,result); Psubstrate[i][theta]+=result[1]*factor[i][j]*PL[i][j]; // sum of all j emitters for, each theta, each wavelength Pair[i][theta]+=result[4]*factor[i][j]*PL[i][j]; } Eoutsubstrate[i]+=Psubstrate[i][theta]*2*pi*sin(theta*pi/180.0)*deltatheta*pi/180.0; Eoutair[i]+=Pair[i][theta]*2*pi*sin(theta*pi/180.0)*deltatheta*pi/180.0; } outfile << wv[i] << " " << PL[i][0] << " " << Etot[i][0]<< " " << Eoutsubstrate[i] << " " << Eoutair[i] << " " << Etot[i][0]/PL[i][0] << " " << Eoutsubstrate[i]/Etot[i][0] << " " << Eoutair[i]/Etot[i][0] << " " ; outfile << wv[i] << " "; deltatheta=10; for(theta=0;theta<=91;theta+=deltatheta) { outfile << Psubstrate[i][theta] << " " ;} outfile << wv[i] << " "; for(theta=0;theta<=91;theta+=deltatheta) { outfile << Pair[i][theta] << " " ;} for(j=1;j<=NEmitter;j++) { outfile << F[i][j] <<" " ;}

260

for(j=1;j<=NEmitter;j++) { outfile << PL[i][j] <<" " ;} for(j=1;j<=NEmitter;j++) { outfile << Etot[i][j] <<" " ;} for(j=1;j<=NEmitter;j++) { outfile << Esubstrate[i][j] <<" " ;} outfile << endl; } outfile.close(); cout << "finish writing Spectrum.txt..." << endl; //--------------------------------------------------------------------------- // writing Angular.txt // calculate the angular dependence of the color point and luminance // all calclations are done in the substrate and in the air, respectively outfile.open("Angular.txt"); outfile << "theta" " "<< "PL/(2*pi)" << " " << "P(substrate)" << " " << "P(air)" << " "<< "theta" <<" "; outfile << "Lum(PL_total)/pi" <<" "<< "PL_total_CIEx" << " " << "PL_total_CIEy" <<" " ; outfile<< "Lum(substrate)" << " " "substrate_CIEx" << " "<< "substrate_CIEy" << " " << "Lum(air)" << " " << "air_CIEx" << " "<< "air_CIEy" <<" " << "Lum(red)" << " " << "red_CIEx" << " " << "red_CIEy" << " " << "Lum(green)" << " " << "green_CIEx" << " " << "green_CIEy" << " " << "Lum(blue)" << " " << "blue_CIEx" << " " << "blue_CIEy" << " " <<endl; double PLlum[50]={0},PLx[50]={0},PLy[50]={0}; double substratelum,airlum,redlum,redx,redy,greenlum,greenx,greeny,bluelum,bluex,bluey; double substratex=0,substratey=0,airx=0,airy=0; double R[100]={0},G[100]={0},B[100]={0}; double x=0,y=0,z=0,total=0; for(j=0;j<=NEmitter;j++) { x=0;y=0;z=0;total=0; for(i=0;i<=80;i++) { x+=CIExyz[i][0]*PL[i][j];

261

y+=CIExyz[i][1]*PL[i][j]; z+=CIExyz[i][2]*PL[i][j]; } total=x+y+z; PLlum[j]=y/pi; PLx[j]=x/total; PLy[j]=y/total; } totalwv[0]=0; totalwv[1]=0; totalwv[2]=0; totalwv[6]=0; totalwv[7]=0; for(i=0;i<=80;i++) { totalwv[0]+=Etot[i][0]; totalwv[1]+=Esubstrate[i][0]; totalwv[2]+=PL[i][0]; totalwv[6]+=Eoutsubstrate[i]; totalwv[7]+=Eoutair[i]; } deltatheta=3; double temp; for( theta=0;theta<90;theta+=deltatheta) { totalwv[3]=0; totalwv[4]=0; for(i=0;i<=80;i++) { realk0=2*pi/wv[i]*real(n[i][0]); realkr=2*realk0; realkmin=40*realkr; Psubstrate[i][theta]=0; Pair[i][theta]=0; for(j=1;j<=NEmitter;j++)

262

{ Calculate_P (wv[i], theta*pi/180.0,position_Layer[j],position_x[j], NLayer,d, n[i],TM,TE,result); totalwv[3]+=result[1]*factor[i][j]*PL[i][j]; // P theta in substrate, all wavelength, each theta, all emitters totalwv[4]+=result[4]*factor[i][j]*PL[i][j]; // P theta in air Psubstrate[i][theta]+=result[1]*factor[i][j]*PL[i][j]; // P theta in substrate, each theta, each wavelength, all emitters Pair[i][theta]+=result[4]*factor[i][j]*PL[i][j]; // P theta in air } temp=sin(theta*pi/180.0)*1.0/1.5; // calculate the path in CF temp=sqrt(1.0-temp*temp); temp=1.0/temp; R[i]=Pair[i][theta]*pow(CF[i][0],temp); G[i]=Pair[i][theta]*pow(CF[i][1],temp); B[i]=Pair[i][theta]*pow(CF[i][2],temp); } outfile << theta << " " << totalwv[2]/(2*pi) << " " << totalwv[3] << " " << totalwv[4] << " "; x=0;y=0;z=0;total=0; for(i=0;i<=80;i++) { x+=CIExyz[i][0]*Psubstrate[i][theta]; y+=CIExyz[i][1]*Psubstrate[i][theta]; z+=CIExyz[i][2]*Psubstrate[i][theta]; } total=x+y+z; substratelum=y; substratex=x/total; substratey=y/total; x=0;y=0;z=0;total=0; for(i=0;i<=80;i++) { x+=CIExyz[i][0]*Pair[i][theta]; y+=CIExyz[i][1]*Pair[i][theta];

263

z+=CIExyz[i][2]*Pair[i][theta]; } total=x+y+z; airlum=y; // get the lum per solid angle airx=x/total; airy=y/total; LumColor(CIExyz,R, redlum, redx, redy); LumColor(CIExyz,G, greenlum, greenx, greeny); LumColor(CIExyz,B, bluelum, bluex, bluey); if( theta==0) { final[1]= airlum/cos(theta*pi/180.0); final[2]=airx; final[3]=airy; final[4]=redlum/cos(theta*pi/180.0); final[5]=redx; final[6]=redy; final[7]=greenlum/cos(theta*pi/180.0); final[8]=greenx; final[9]=greeny; final[10]=bluelum/cos(theta*pi/180.0); final[11]=bluex; final[12]=bluey; } if( theta==60) { final[21]= airlum/cos(theta*pi/180.0); final[22]=airx; final[23]=airy; final[24]=redlum/cos(theta*pi/180.0); final[25]=redx; final[26]=redy; final[27]=greenlum/cos(theta*pi/180.0);

264

final[28]=greenx; final[29]=greeny; final[30]=bluelum/cos(theta*pi/180.0); final[31]=bluex; final[32]=bluey; } outfile << theta <<" "; outfile << PLlum[0] << " " << PLx[0] << " " << PLy[0] << " "; outfile << substratelum/cos(theta*pi/180.0) << " " << substratex<< " " << substratey << " "<< airlum/cos(theta*pi/180.0) << " " <<airx<< " " << airy << " " << redlum/cos(theta*pi/180.0) << " " << redx<< " " << redy << " "<< greenlum/cos(theta*pi/180.0) << " " << greenx<< " " << greeny << " "<< bluelum/cos(theta*pi/180.0) << " " << bluex<< " " << bluey << " " <<endl; } outfile.close(); cout << "finish writing Angular.txt..." << endl; //--------------------------------------------------------------------------------------------- // writing Modes.txt // calculate the mode contribution from the dipole emission // assuming that the in-plane wave vector projection kaapa // k0=2pi/lambda // 0<kappa<k0 air mode // k0<kappa<1.5k0 glass/substrate mode // 1.50<kappa<1.7k0 organic mode // 1.7k0<kappa<infinite surface plasmon mode // also calculate the mode contribution at each wavlength final[0]=totalwv[7]/totalwv[0]; // power EQE final[39]=0;final[40]=0;final[41]=0;final[42]=0;final[43]=0; for(i=0;i<=80;i++) { final[39]+=Etot[i][0]; final[40]+=ModeAir[i][0]; final[41]+=ModeSubstrate[i][0]; final[42]+=ModeWaveGuide[i][0];

265

final[43]+=ModePlasmon[i][0]; } outfile.open("Modes.txt"); outfile << "lambda(nm)" << " " <<"PL" << " " << "Etot" << " " << "Esubstrate" << " " << "ModeAir" << " "<< "ModeSubstrate" << " "<< "ModeWaveGuide" << " " << "ModePlasmon" << " " << "Eoutinair/Etot" << " " << "ModeAir/Etot" << " " << "ModeSubstrate/Etot" << " "<<"ModeWaveGuide/Etot" <<" "<< "ModePlasmon/Etot" <<endl; i=0; outfile << wv[i] << " " << PL[i][0] << " "<< Etot[i][0] << " "<< Esubstrate[i][0] << " "<<ModeAir[i][0] << " "<< ModeSubstrate[i][0] << " "<< ModeWaveGuide[i][0] << " "<< ModePlasmon[i][0] << " " << final[0] << " " <<final[40]/final[39]<< " " <<final[41]/final[39]<< " " <<final[42]/final[39] << " " <<final[43]/final[39] << endl; for(i=1;i<=80;i++) { outfile << wv[i] << " " << PL[i][0] << " "<< Etot[i][0] << " "<< Esubstrate[i][0] << " "<<ModeAir[i][0] << " "<< ModeSubstrate[i][0] << " "<< ModeWaveGuide[i][0] << " "<< ModePlasmon[i][0] << " " << endl; } outfile.close(); cout <<"finish writing Modes.txt... " << endl << endl; //--------------------------------------------------------------------------- // display the useful information, and write the information in result.txt // show the wavelength,peak and width // the OLED outcoupling peak and width double PsubstrateEff[100]={0},PairEff[100]={0}; for( i=0;i<=80;i++) { PsubstrateEff[i]=Psubstrate[i][0]/PL[i][0]; PairEff[i]=Pair[i][0]/PL[i][0]; } cout << "PL=" << totalwv[2] << endl; cout << "Etot=" << totalwv[0] << endl;

266

cout << "Eoutsubstrate=" << totalwv[6] << endl; cout << "Eoutair=" << totalwv[7] << endl; cout << endl; cout << "Etot/PL=" << totalwv[0]/totalwv[2] << endl ; cout << "Eoutsubstrate/PL=" << totalwv[6]/totalwv[2] << endl ; cout << "Eoutair/PL=" << totalwv[7]/totalwv[2]<< endl ; cout << endl; cout << "Eoutsubstrate/Etot=" << totalwv[6]/totalwv[0] << endl ; cout << "Eoutair/Etot=" << totalwv[7]/totalwv[0]<< endl ; cout << endl; cout << "air(0)=(" << final[2] <<"," << final[3] <<")" <<endl; cout << "air(60)=(" << final[22] <<"," << final[23] <<")" <<endl; cout << "deltauv=" << deltauv(final[2],final[3],final[22],final[23]) <<endl; cout << endl; cout << "Lum(0)=" << final[1] << endl; cout << "Lum(60)=" << final[21] << endl; cout << "Lum(60)/Lum(0)=" << final[21]/final[1] << endl; cout << endl; cout << "(peak,width)" << endl; cout << "PL=( " << findpeak(wv,PL) <<" nm, " << findwidth(wv,PL) << " nm)" <<endl; cout << "Total_substrate=( " << findpeak3(wv,Eoutsubstrate) <<" nm, " << findwidth3(wv,Eoutsubstrate) << " nm)" <<endl; cout << "Total_air=( " << findpeak3(wv,Eoutair) <<" nm, " << findwidth3(wv,Eoutair) <<" nm)" <<endl << endl; cout << "Normal_substrate=(" << findpeak2(wv,Psubstrate) <<" nm, " <<findwidth2(wv,Psubstrate) <<" nm)" <<endl; cout << "Normal_air=(" << findpeak2(wv,Pair) <<" nm, " <<findwidth2(wv,Pair) << " nm)" <<endl << endl; cout << "Normal output/PL efficiency:" << endl; cout << "Eff_substrate=("<< findpeak3(wv,PsubstrateEff) <<" nm, " <<findwidth3(wv,PsubstrateEff) <<" nm)" <<endl; cout << "Eff_air=(" << findpeak3(wv,PairEff) <<" nm, " <<findwidth3(wv,PairEff) <<" nm)" <<endl; //---------------------------------------------------------------------------

267

outfile.open("result.txt"); outfile << "PL=" << totalwv[2] << endl; outfile << "Etot=" << totalwv[0] << endl; outfile << "Eoutsubstrate=" << totalwv[6] << endl; outfile << "Eoutair=" << totalwv[7] << endl; outfile << endl; outfile << "Etot/PL=" << totalwv[0]/totalwv[2] << endl ; outfile << "Eoutsubstrate/PL=" << totalwv[6]/totalwv[2] << endl ; outfile << "Eoutair/PL=" << totalwv[7]/totalwv[2]<< endl ; outfile << endl; outfile << "Eoutsubstrate/Etot=" << totalwv[6]/totalwv[0] << endl ; outfile << "Eoutair/Etot=" << totalwv[7]/totalwv[0]<< endl ; outfile << endl; outfile << "air(0)=(" << final[2] <<"," << final[3] <<")" <<endl; outfile << "air(60)=(" << final[22] <<"," << final[23] <<")" <<endl; outfile << "deltauv=" << deltauv(final[2],final[3],final[22],final[23]) <<endl; outfile << endl; outfile << "Lum(0)=" << final[1] << endl; outfile << "Lum(60)=" << final[21] << endl; outfile << "Lum(60)/Lum(0)=" << final[21]/final[1] << endl; outfile << endl; outfile << "(peak,width)" << endl; outfile << "PL=( " << findpeak(wv,PL) <<" nm, " << findwidth(wv,PL) << " nm)" <<endl; outfile << "Total_substrate=( " << findpeak3(wv,Eoutsubstrate) <<" nm, " << findwidth3(wv,Eoutsubstrate) << " nm)" <<endl; outfile << "Total_air=( " << findpeak3(wv,Eoutair) <<" nm, " << findwidth3(wv,Eoutair) <<" nm)" <<endl << endl; outfile << "Normal_substrate=(" << findpeak2(wv,Psubstrate) <<" nm, " <<findwidth2(wv,Psubstrate) <<" nm)" <<endl; outfile << "Normal_air=(" << findpeak2(wv,Pair) <<" nm, " <<findwidth2(wv,Pair) << " nm)" <<endl << endl; outfile << "Normal output/PL efficiency:" << endl;

268

outfile << "Eff_substrate=("<< findpeak3(wv,PsubstrateEff) <<" nm, " <<findwidth3(wv,PsubstrateEff) <<" nm)" <<endl; outfile << "Eff_air=(" << findpeak3(wv,PairEff) <<" nm, " <<findwidth3(wv,PairEff) <<" nm)" <<endl; getchar(); return 0; }

269

B.2 Sample input (input.txt)

========================================= nonrad=(the relative non_radiative_rate, assuming radiative_rate=1) 0 ========================================= NLayer= 5 Layer 0: glass Layer 1: ITO 100 Layer 2: n=1.7 69 Layer 3: n=1.7 20 Layer 4: n=1.7 69 Layer 5: Al 100 Layer 6: air ========================================= NEmitter= 1 Emitter 1: PL_red 3 10 ========================================

270

B.3 Sample input (Al.txt)

lambda(nm) n k 380 0.442168 4.61474 385 0.454122 4.67866 390 0.466095 4.74084 395 0.478073 4.80105 400 0.490123 4.8606 405 0.502471 4.92014 410 0.514861 4.98033 415 0.527196 5.04178 420 0.539393 5.10508 425 0.551729 5.16803 430 0.56433 5.22952 435 0.577197 5.28952 440 0.590531 5.34753 445 0.604087 5.4073 450 0.617921 5.46933 455 0.632065 5.52964 460 0.64627 5.59015 465 0.660457 5.65359 470 0.674848 5.71603 475 0.689515 5.77714 480 0.704641 5.8385 485 0.720042 5.89999 490 0.735933 5.95972 495 0.751983 6.01888 500 0.76864 6.07814 505 0.785439 6.13743 510 0.802831 6.1986 515 0.820383 6.26027 520 0.838673 6.32327 525 0.857309 6.38691 530 0.876416 6.4472 535 0.896032 6.50387 540 0.915959 6.56164 545 0.937245 6.62425 550 0.95853 6.68685 555 0.980983 6.74732 560 1.00375 6.80723 565 1.02654 6.86636 570 1.04943 6.92357 575 1.07231 6.98078

271

580 1.09693 7.03627 585 1.12242 7.0909 590 1.14791 7.14552 595 1.17231 7.201 600 1.19662 7.25656 605 1.22101 7.31214 610 1.24746 7.36835 615 1.27392 7.42457 620 1.3004 7.48075 625 1.32872 7.53424 630 1.35704 7.58773 635 1.38536 7.64123 640 1.41499 7.69247 645 1.44487 7.74328 650 1.47476 7.79409 655 1.50528 7.84639 660 1.53646 7.90024 665 1.56764 7.9541 670 1.59882 8.00795 675 1.63617 8.06166 680 1.67376 8.11537 685 1.71135 8.16908 690 1.75028 8.22088 695 1.79347 8.26662 700 1.83666 8.31235 705 1.87985 8.35808 710 1.92667 8.40305 715 1.98188 8.44626 720 2.03709 8.48946 725 2.0923 8.53267 730 2.1483 8.57154 735 2.20936 8.58284 740 2.27042 8.59415 745 2.33148 8.60546 750 2.39254 8.61677 755 2.44345 8.61696 760 2.4903 8.6127 765 2.53715 8.60844 770 2.58399 8.60418 775 2.63061 8.59946 780 2.66463 8.56945

272

B.4 Sample input (PL_red.txt)

lambda(nm) PL_red 380 1.45519E-11 385 4.07157E-11 390 1.1148E-10 395 2.98691E-10 400 7.83146E-10 405 2.00935E-09 410 5.045E-09 415 1.23954E-08 420 2.98023E-08 425 7.01187E-08 430 1.6144E-07 435 3.63731E-07 440 8.01941E-07 445 1.73021E-06 450 3.65297E-06 455 7.54721E-06 460 1.52588E-05 465 3.01888E-05 470 5.84475E-05 475 0.000110733 480 0.000205297 485 0.00037246 490 0.000661258 495 0.001148825 500 0.001953125 505 0.003249368 510 0.005290061 515 0.008427819 520 0.013139006 525 0.020044844 530 0.029925103 535 0.043718115 540 0.0625 545 0.08743623 550 0.11970041 555 0.160358752 560 0.210224104 565 0.269690199 570 0.338563887 575 0.415919145

273

580 0.5 585 0.588198496 590 0.677127773 595 0.762799075 600 0.840896415 605 0.907126088 610 0.957603281 615 0.989228013 620 1 625 0.989228013 630 0.957603281 635 0.907126088 640 0.840896415 645 0.762799075 650 0.677127773 655 0.588198496 660 0.5 665 0.415919145 670 0.338563887 675 0.269690199 680 0.210224104 685 0.160358752 690 0.11970041 695 0.08743623 700 0.0625 705 0.043718115 710 0.029925103 715 0.020044844 720 0.013139006 725 0.008427819 730 0.005290061 735 0.003249368 740 0.001953125 745 0.001148825 750 0.000661258 755 0.00037246 760 0.000205297 765 0.000110733 770 5.84475E-05 775 3.01888E-05 780 1.52588E-05

274

Appendix C

Program for the Power Consumption Model of

OLED Screens

275

C.1 Codes

//------------------------------------------------------------------- // OLED screens power model: // C++ codes to calculate the power consumption of OLED screens //------------------------------------------------------------------- #include <math.h> #include <cmath> #include <iostream> #include <fstream> #include <string> #include <sstream> #include <stdlib.h> #include <stdio.h> using namespace std; void ColorRatio(double* CIEx, double* CIEy, double* Ratio) // function to break the target color into RGB ratios { double rR_x,rR_z,rG_x,rG_z,rB_x,rB_z,rM_x,rM_z; // R G B => TargetW (3-RGB panel) // CIE [0,1,2] red, green, blue of 3-RGB panel // CIE[3] TargetW for 3-RGB panel rR_x=CIEx[0]/CIEy[0]; rR_z=(1-CIEx[0]-CIEy[0])/CIEy[0]; rG_x=CIEx[1]/CIEy[1]; rG_z=(1-CIEx[1]-CIEy[1])/CIEy[1]; rB_x=CIEx[2]/CIEy[2]; rB_z=(1-CIEx[2]-CIEy[2])/CIEy[2]; rM_x=CIEx[3]/CIEy[3]; rM_z=(1-CIEx[3]-CIEy[3])/CIEy[3]; Ratio[0]=((rM_x-rB_x)*(rG_z-rB_z)-(rM_z-rB_z)*(rG_x-rB_x))/((rR_x-rB_x)*(rG_z-

rB_z)-(rR_z-rB_z)*(rG_x-rB_x)); Ratio[1]=((rM_x-rB_x)-Ratio[0]*(rR_x-rB_x))/(rG_x-rB_x); Ratio[2]=1-Ratio[0]-Ratio[1]; // R G B => EmitW (4-RGBW panel)

// Ratio [4,5,6] R,G,B ratio for EmitW in 4-RGBW panel // used when normalize the color to white OLED

// CIE [4,5,6] red, green, blue of 4-RGBW panel

276

// CIE[7] EmitW color for 4-RGBW panel rR_x=CIEx[4]/CIEy[4]; rR_z=(1-CIEx[4]-CIEy[4])/CIEy[4]; rG_x=CIEx[5]/CIEy[5]; rG_z=(1-CIEx[5]-CIEy[5])/CIEy[5]; rB_x=CIEx[6]/CIEy[6]; rB_z=(1-CIEx[6]-CIEy[6])/CIEy[6]; rM_x=CIEx[7]/CIEy[7]; rM_z=(1-CIEx[7]-CIEy[7])/CIEy[7]; Ratio[4]=((rM_x-rB_x)*(rG_z-rB_z)-(rM_z-rB_z)*(rG_x-rB_x))/((rR_x-rB_x)*(rG_z-rB_z)-(rR_z-rB_z)*(rG_x-rB_x)); Ratio[5]=((rM_x-rB_x)-Ratio[4]*(rR_x-rB_x))/(rG_x-rB_x); Ratio[6]=1-Ratio[4]-Ratio[5]; // R G B => TargetW (4-RGBW panel)

// Ratio [14,15,16] R,G,B ratio for TargetW in 4-RGBW panel // ratio info to calculate the max lum for each color channel in display

// CIE [4,5,6] red, green, blue of 4-RGBW panel // CIE[8] TargetW color for 4-RGBW panel rR_x=CIEx[4]/CIEy[4]; rR_z=(1-CIEx[4]-CIEy[4])/CIEy[4]; rG_x=CIEx[5]/CIEy[5]; rG_z=(1-CIEx[5]-CIEy[5])/CIEy[5]; rB_x=CIEx[6]/CIEy[6]; rB_z=(1-CIEx[6]-CIEy[6])/CIEy[6]; rM_x=CIEx[8]/CIEy[8]; rM_z=(1-CIEx[8]-CIEy[8])/CIEy[8]; Ratio[14]=((rM_x-rB_x)*(rG_z-rB_z)-(rM_z-rB_z)*(rG_x-rB_x))/((rR_x-rB_x)*(rG_z-rB_z)-(rR_z-rB_z)*(rG_x-rB_x)); Ratio[15]=((rM_x-rB_x)-Ratio[14]*(rR_x-rB_x))/(rG_x-rB_x); Ratio[16]=1-Ratio[14]-Ratio[15]; }

277

int main() { double DisplaySize=40; // unit inch int height=1920,width=1080; // screen resolution double Polarizer=0.42; // 42% light transmission of the polarizer double PeakNit=400; // screen peak lumiannce, unit nit

double Gamma=2.2; // gamma correction curve, relate the digital level to luminance

int R=250,G=166,B=188; // digital value of the pixel, (0-255) string ColorName[10]={"Red","Green","Blue","TargetWhite","Red","Green","Blue","EmitWhite","TargetWhite"}; double CIEx[10]={0},CIEy[10]={0},Eff[10]={0},V[10]={0},Ratio[30]={0}; CIEx[0]=0.64;CIEy[0]=0.36; // 3-RGB red CIEx[1]=0.31;CIEy[1]=0.63; // 3-RGB green CIEx[2]=0.14;CIEy[2]=0.12; // 3-RGB blue CIEx[3]=0.313;CIEy[3]=0.329; // 3-RGB target white CIEx[4]=0.66;CIEy[4]=0.33; // 4-RGBW red CIEx[5]=0.23;CIEy[5]=0.70; // 4-RGBW green CIEx[6]=0.14;CIEy[6]=0.07; // 4-RGBW blue CIEx[7]=0.33;CIEy[7]=0.33; // 4-RGBW emit white CIEx[8]=0.313;CIEy[8]=0.329; // 4-RGBW target white // efficiency, unit cd/A Eff[0]=30; Eff[1]=85; Eff[2]=10; Eff[4]=10; Eff[5]=20; Eff[6]=3; Eff[7]=57; // toal voltage drop on OLED and TFT, unit V V[0]=10; V[1]=10; V[2]=10; V[4]=14; V[5]=14; V[6]=14; V[7]=14; double PixelArea=DisplaySize*DisplaySize*2.54*2.54

/(height*height+width*width)/10000; // Pixel Area, unit m^2 double PixelLum[10]={0},ScreenLum[10]={0}; // unit cd double PixelCurrent[10]={0},ScreenCurrent[10]={0}; // unit A double PixelPower[10]={0},ScreenPower[10]={0}; // unit W

//---------------------------------------------- end read inputs ------------------

278

// calculate color channel ratio ColorRatio(CIEx, CIEy, Ratio); //3-RGB panel // each color channel PixelLum[0]=PeakNit*PixelArea*Ratio[0]*pow(R*1.0/255.0,Gamma); PixelLum[1]=PeakNit*PixelArea*Ratio[1]*pow(G*1.0/255.0,Gamma); PixelLum[2]=PeakNit*PixelArea*Ratio[2]*pow(B*1.0/255.0,Gamma); for(int i=0;i<=2;i++) { PixelCurrent[i]=PixelLum[i]/Eff[i]/Polarizer; PixelPower[i]=PixelCurrent[i]*V[i]; } // the sum of all color channels PixelLum[3]=PixelLum[0]+PixelLum[1]+PixelLum[2]; PixelCurrent[3]=PixelCurrent[0]+PixelCurrent[1]+PixelCurrent[2]; PixelPower[3]=PixelPower[0]+PixelPower[1]+PixelPower[2]; // 4-RGBW panel

// calcualte the luminance contribution of each color channel // without the contribution from the EmitWhite channel

double Rlm=PeakNit*PixelArea*Ratio[14]*pow(R*1.0/255.0,Gamma); double Glm=PeakNit*PixelArea*Ratio[15]*pow(G*1.0/255.0,Gamma); double Blm=PeakNit*PixelArea*Ratio[16]*pow(B*1.0/255.0,Gamma); // normalized the luminance of R G B back to EmitWhite // determine the maximum EmitWhite lumiannce contribution double min=Rlm/Ratio[4]; if( Glm/Ratio[5] < min) {min=Glm/Ratio[5];} if( Blm/Ratio[6] < min) {min=Blm/Ratio[6];} // each color channel PixelLum[7]=min; // emit white luminance PixelLum[4]=Rlm-PixelLum[7]*Ratio[4]; PixelLum[5]=Glm-PixelLum[7]*Ratio[5]; PixelLum[6]=Blm-PixelLum[7]*Ratio[6]; for(int i=4;i<=7;i++) { PixelCurrent[i]=PixelLum[i]/Eff[i]/Polarizer; PixelPower[i]=PixelCurrent[i]*V[i]; }

279

// the sum of all color channels PixelLum[8]=PixelLum[4]+PixelLum[5]+PixelLum[6]+PixelLum[7]; PixelCurrent[8]=PixelCurrent[4]+PixelCurrent[5]+PixelCurrent[6]+PixelCurrent[7]; PixelPower[8]=PixelPower[4]+PixelPower[5]+PixelPower[6]+PixelPower[7]; // the screen power consumption // assuming that each pixel display the same digital value (R,G,B) for(int i=0;i<=8;i++) { ScreenLum[i]=PixelLum[i]*height*width; ScreenCurrent[i]=PixelCurrent[i]*height*width; ScreenPower[i]=PixelPower[i]*height*width; } //-------------------------------------------------------------------- cout << "3-RGB ScreenPower=" <<ScreenPower[3] <<" W" <<endl; cout << "4-RGBW ScreenPower=" << ScreenPower[8] << " W" <<endl; //-------------------------------------------------------------------- getchar(); return 0; }

280

Appendix D

Program for the Simulation of OPV devices

(Optical Part, Exciton Part and Electrical Part)

281

D.1 Flow diagram

282

D.2 Codes

//---------------------------------------------------------------------------------------------- // OPV model: // C++ codes to calculate the photon absorption, excition dissociation, // charge carrier generation and transport in OPV devices //---------------------------------------------------------------------------------------------- #include <cmath> #include <iostream> #include <fstream> #include <string> #include <sstream> #include <complex> using namespace std; void Write_Files (int NJ, int N, int check, double* p,double* n,double* un, double* up, double* F, double* Fint, double* jt, double* jp, double*jn, double* jpdrift, double* jpdiff, double* jndrift, double* jndiff, double* G, double *R, double* U,double* P,double* X, double* Xnew,double* result) { ofstream myfile; ostringstream ss; int j; ss << "Vstep" << NJ << ".dat"; myfile.open( ss.str().c_str() ); myfile << "Number" << " " <<"check" << " " << "Holes(1/cm3)" << " " << "Electrons(1/cm3)" << " " << "F(V/cm)"<< " " << "Fint(V/cm)" << " " << "jt(mA/cm2)" << " " << "jp(mA/cm2)" << " " << "jn(mA/cm2)" << " " << "jpdrift(mA/cm2)" << " " << "jpdiff(mA/cm2)" << " " << "jndrift(mA/cm2)" << " " << "jndiff(mA/cm2)" << " " << "G(1/cm3)"<< " " << "R(1/cm3)" << " " << "up(cm2/Vs)" << " "<< "un(cm2/Vs)" <<" " << "netGenration(1/cm3)" << " " << "Probability" << " " << "X[](1/cm3)" <<" " << "Xnew[](1/cm3)" <<" " << "result[](mA/cm2)"<<endl; for(j=0;j<=N+1;j++) { myfile << j << " " << check << " " << p[j] << " " << n[j] << " " << F[j]<< " " << Fint[j] << " " << jt[j] << " " << jp[j] << " " << jn[j] << " " << jpdrift[j] << " " << jpdiff[j] << " " << jndrift[j] << " " << jndiff[j] << " " << G[j] <<" " << R[j] << " " <<" "

283

<< up[j] << " " << un[j] << " " << U[j]<<" " << P[j] << " " << X[j] << " " << Xnew[j] << " " << result[j] <<endl; } myfile.close(); } double Calculate_E(double* p,double* n,double* un,double* un0, double* Fn0, double* up,double* up0, double* Fp0, double* F,double *Fint,double* Dn, double* Dp,double Vbi, double V, int N, double dl,double ts_factor,double e, double diel_0, double diel_r) { int i; int j; for (i=0;i<=N;i++) { Fint[i]=0; for(j=0;j<=N+1;j++) { if (j<=i) Fint[i]=Fint[i]+dl*e*(p[j]-n[j])/(2*diel_0*diel_r); if (j>i) Fint[i]=Fint[i]-dl*e*(p[j]-n[j])/(2*diel_0*diel_r); } } for (i=1;i<=N;i++) { F[i]=(Fint[i-1]+Fint[i])/2; } double VF=0; for (i=1;i<=N;i++) { VF=VF+F[i]*dl; } VF=Vbi+VF; n[0]=n[0]+(VF-V)*diel_0*diel_r/(e*dl*N*dl); p[N+1]=p[N+1]+(VF-V)*diel_0*diel_r/(e*dl*N*dl);

// adjust electrons on ITO and holes on Al if(n[0]<p[0]) {p[0]=p[0]-n[0];n[0]=0;}

// balance the holes and electrons on the electrodes

284

else {n[0]=n[0]-p[0];p[0]=0;} if(n[N+1]<p[N+1]) {p[N+1]=p[N+1]-n[N+1];n[N+1]=0;} else {n[N+1]=n[N+1]-p[N+1];p[N+1]=0;} for (i=1;i<=N;i++) { F[i]=F[i]-(VF-V)/(N*dl); } for (i=1;i<=N;i++) { un[i]=un0[i]*exp(Fn0[i]*sqrt(abs(F[i]))); Dn[i]=un[i]*0.026; up[i]=up0[i]*exp(Fp0[i]*sqrt(abs(F[i]))); Dp[i]=up[i]*0.026; } double uFmax,x,dt; uFmax=0; for (i=1;i<=N;i++) { x=abs(un[i]*F[i]); if (uFmax<=x) uFmax=x; x=abs(up[i]*F[i]); if (uFmax<=x) uFmax=x; x=abs(Dn[i]/dl); if (uFmax<=x) uFmax=x; x=abs(Dp[i]/dl); if (uFmax<=x) uFmax=x; } dt=dl/(uFmax*ts_factor); return dt; } double kd(double u, double a, double er, double E) { E=abs(E); double e=1.6e-19; double e0=8.854e-14; double kT=0.026;

285

double R=e*u/(e0*er); double pi=3.1415926; double Eb=e/(4*pi*er*e0*a*1.0e-7); double b=E*e/(8*pi*er*e0*kT*kT); double kdiss; kdiss=3*R*exp(-Eb/kT)*(1+b+(b*b)/3)/(4*pi*a*a*a*1.0e-21); return kdiss; } int move(double* p,double* n, double* pnew, double* nnew, double *X, double* Xnew, double kf, double a,double* un, double* up, double* F, double* Fint, double* Dn, double* Dp, double* G,double* result, double* jp, double*jn, double* jt,double* jpdrift, double* jpdiff, double* jndrift, double* jndiff, double* R,double* interRnp, double* interRpn,double* U,double* P, double dt, int Nmax, double converge,int N, double dl,double e, double diel_0,double diel_r,double* current,double Nc, double Nv, double Eg) { int j; int check=1; double dp=0,dn=0; double nisquare=Nc*Nv*exp(-Eg/0.026); for(j=2;j<=N-1;j++) { pnew[j]=p[j]+dt*((Dp[j+1]*p[j+1]+Dp[j-1]*p[j-1]-2*Dp[j]*p[j])/(dl*dl)-p[j]*up[j]*abs(F[j])/dl); if (F[j+1]<0) pnew[j]=pnew[j]+dt*p[j+1]*up[j+1]*abs(F[j+1])/dl; if (F[j-1]>0) pnew[j]=pnew[j]+dt*p[j-1]*up[j-1]*abs(F[j-1])/dl; nnew[j]=n[j]+dt*((Dn[j+1]*n[j+1]+Dn[j-1]*n[j-1]-2*Dn[j]*n[j])/(dl*dl)-n[j]*un[j]*abs(F[j])/dl); if (F[j+1]>0) nnew[j]=nnew[j]+dt*n[j+1]*un[j+1]*abs(F[j+1])/dl; if (F[j-1]<0) nnew[j]=nnew[j]+dt*n[j-1]*un[j-1]*abs(F[j-1])/dl; } j=1; pnew[j]=p[j]+dt*(Dp[j+1]*p[j+1]-Dp[j]*p[j])/(dl*dl); if (F[j+1]<0) pnew[j]=pnew[j]+dt*p[j+1]*up[j+1]*abs(F[j+1])/dl;

286

if (F[j]>0) pnew[j]=pnew[j]-dt*p[j]*up[j]*abs(F[j])/dl; nnew[j]=n[j]+dt*(Dn[j+1]*n[j+1]-Dn[j]*n[j])/(dl*dl); if (F[j+1]>0) nnew[j]=nnew[j]+dt*n[j+1]*un[j+1]*abs(F[j+1])/dl; if (F[j]<0) nnew[j]=nnew[j]-dt*n[j]*un[j]*abs(F[j])/dl; j=N; pnew[j]=p[j]+dt*(Dp[j-1]*p[j-1]-Dp[j]*p[j])/(dl*dl); if (F[j-1]>0) pnew[j]=pnew[j]+dt*p[j-1]*up[j-1]*abs(F[j-1])/dl; if (F[j]<0) pnew[j]=pnew[j]-dt*p[j]*up[j]*abs(F[j])/dl; nnew[j]=n[j]+dt*(Dn[j-1]*n[j-1]-Dn[j]*n[j])/(dl*dl); if (F[j-1]<0) nnew[j]=nnew[j]+dt*n[j-1]*un[j-1]*abs(F[j-1])/dl; if (F[j]>0) nnew[j]=nnew[j]-dt*n[j]*un[j]*abs(F[j])/dl; for(j=1;j<=N;j++) {

if (n[j]*p[j]>nisquare) R[j]=e*(un[j]+up[j])*(p[j]*n[j]-nisquare)/(diel_r*diel_0);

else R[j]=0; Xnew[j]=X[j]+dt*(G[j]/(dl*e*1000)-kf*X[j]-kd(un[j]+up[j],a,diel_r,F[j])*X[j]+R[j]); pnew[j]=pnew[j]-R[j]*dt+kd(un[j]+up[j],a,diel_r,F[j])*X[j]*dt; nnew[j]=nnew[j]-R[j]*dt+kd(un[j]+up[j],a,diel_r,F[j])*X[j]*dt; P[j]=kd(un[j]+up[j],a,diel_r,F[j])/(kd(un[j]+up[j],a,diel_r,F[j])+kf); U[j]=P[j]*G[j]/(dl*e*1000)-(1-P[j])*R[j]; } p[0]=p[0]+(n[1]-nnew[1]); n[0]=n[0]+(p[1]-pnew[1]); p[N+1]=p[N+1]+(n[N]-nnew[N]); n[N+1]=n[N+1]+(p[N]-pnew[N]); for (j=1;j<=N && check==1; j++) { if (abs(Xnew[j]-X[j])>abs(converge*dt*Xnew[j])) check=0; } for (j=2;j<=N-1 && check==1; j++)

287

{ if (abs(p[j]-pnew[j])>abs(converge*dt*pnew[j])) check=0; if (abs(n[j]-nnew[j])>abs(converge*dt*nnew[j])) check=0; } if (check==1) //calculate current { jp[0]=(p[1]-pnew[1])*dl*e*1000/dt; jn[0]=-(n[1]-nnew[1])*dl*e*1000/dt; jt[0]=jp[0]+jn[0]; jp[N]=-(p[N]-pnew[N])*dl*e*1000/dt; jn[N]=(n[N]-nnew[N])*dl*e*1000/dt; jt[N]=jp[N]+jn[N];

if (abs(jt[0]-jt[N])>abs(converge*(abs(jt[N])+abs(jt[0])))) check=0; // check again if the electron current equal the hole current

for(j=1;j<=N-1;j++) { dp=dt*(Dp[j]*p[j]-Dp[j+1]*p[j+1])/(dl*dl); jpdiff[j]=dp*dl*e*1000/dt; dp=0; if(F[j]>0) dp=dt*p[j]*up[j]*abs(F[j])/dl; if (F[j+1]<0) dp=dp-dt*p[j+1]*up[j+1]*abs(F[j+1])/dl; jpdrift[j]=dp*dl*e*1000/dt; dn=dt*(Dn[j]*n[j]-Dn[j+1]*n[j+1])/(dl*dl); jndiff[j]=-dn*dl*e*1000/dt; dn=0; if(F[j]<0) dn=dt*n[j]*un[j]*abs(F[j])/dl; if (F[j+1]>0) dn=dn-dt*n[j+1]*un[j+1]*abs(F[j+1])/dl; jndrift[j]=-dn*dl*e*1000/dt; jp[j]=jpdiff[j]+jpdrift[j]; jn[j]=jndiff[j]+jndrift[j]; jt[j]=jp[j]+jn[j]; } }

288

for(j=2;j<=N-1;j++) // notice that, for j=1 and N , it is the thermal equilibium, no change is made { p[j]=pnew[j]; n[j]=nnew[j]; X[j]=Xnew[j]; } X[1]=Xnew[1]; X[N]=Xnew[N]; return check; } void data(int* lamda, double* AM, double* ITOnr, double* ITOni, double* C60nr, double* C60ni,double* Alnr,double* Alni) { // wavelenth unit nm lamda[ 1 ]= 300 ; lamda[ 2 ]= 310 ; lamda[ 3 ]= 320 ; lamda[ 4 ]= 330 ; lamda[ 5 ]= 340 ; lamda[ 6 ]= 350 ; lamda[ 7 ]= 360 ; lamda[ 8 ]= 370 ; lamda[ 9 ]= 380 ; lamda[ 10 ]= 390 ; lamda[ 11 ]= 400 ; lamda[ 12 ]= 410 ; lamda[ 13 ]= 420 ; lamda[ 14 ]= 430 ; lamda[ 15 ]= 440 ; lamda[ 16 ]= 450 ; lamda[ 17 ]= 460 ; lamda[ 18 ]= 470 ; lamda[ 19 ]= 480 ; lamda[ 20 ]= 490 ; lamda[ 21 ]= 500 ; lamda[ 22 ]= 510 ;

289

lamda[ 23 ]= 520 ; lamda[ 24 ]= 530 ; lamda[ 25 ]= 540 ; lamda[ 26 ]= 550 ; lamda[ 27 ]= 560 ; lamda[ 28 ]= 570 ; lamda[ 29 ]= 580 ; lamda[ 30 ]= 590 ; lamda[ 31 ]= 600 ; lamda[ 32 ]= 610 ; lamda[ 33 ]= 620 ; lamda[ 34 ]= 630 ; lamda[ 35 ]= 640 ; lamda[ 36 ]= 650 ; lamda[ 37 ]= 660 ; lamda[ 38 ]= 670 ; lamda[ 39 ]= 680 ; lamda[ 40 ]= 690 ; lamda[ 41 ]= 700 ; lamda[ 42 ]= 710 ; lamda[ 43 ]= 720 ; lamda[ 44 ]= 730 ; lamda[ 45 ]= 740 ; lamda[ 46 ]= 750 ; lamda[ 47 ]= 760 ; lamda[ 48 ]= 770 ; lamda[ 49 ]= 780 ; lamda[ 50 ]= 790 ; lamda[ 51 ]= 800 ; lamda[ 52 ]= 810 ; lamda[ 53 ]= 820 ; lamda[ 54 ]= 830 ; lamda[ 55 ]= 840 ; lamda[ 56 ]= 850 ; lamda[ 57 ]= 860 ; lamda[ 58 ]= 870 ; lamda[ 59 ]= 880 ; lamda[ 60 ]= 890 ; lamda[ 61 ]= 900 ; //AM 1.5 flux unit mW/cm^2*eV

290

AM[ 1 ]= 0.000246895 ; AM[ 2 ]= 0.01273475 ; AM[ 3 ]= 0.052972903 ; AM[ 4 ]= 0.125450565 ; AM[ 5 ]= 0.137590323 ; AM[ 6 ]= 0.149026613 ; AM[ 7 ]= 0.173662258 ; AM[ 8 ]= 0.225303145 ; AM[ 9 ]= 0.214752097 ; AM[ 10 ]= 0.25066621 ; AM[ 11 ]= 0.359387097 ; AM[ 12 ]= 0.346681452 ; AM[ 13 ]= 0.38043871 ; AM[ 14 ]= 0.303295645 ; AM[ 15 ]= 0.478996774 ; AM[ 16 ]= 0.565947581 ; AM[ 17 ]= 0.567246774 ; AM[ 18 ]= 0.571466935 ; AM[ 19 ]= 0.62636129 ; AM[ 20 ]= 0.641109677 ; AM[ 21 ]= 0.623024194 ; AM[ 22 ]= 0.636718548 ; AM[ 23 ]= 0.638929032 ; AM[ 24 ]= 0.660191935 ; AM[ 25 ]= 0.645604839 ; AM[ 26 ]= 0.683020161 ; AM[ 27 ]= 0.665677419 ; AM[ 28 ]= 0.681058065 ; AM[ 29 ]= 0.702548387 ; AM[ 30 ]= 0.652283065 ; AM[ 31 ]= 0.713854839 ; AM[ 32 ]= 0.722456452 ; AM[ 33 ]= 0.73695 ; AM[ 34 ]= 0.707429032 ; AM[ 35 ]= 0.740129032 ; AM[ 36 ]= 0.71258871 ; AM[ 37 ]= 0.744735484 ; AM[ 38 ]= 0.767041935 ; AM[ 39 ]= 0.766041935 ; AM[ 40 ]= 0.657781452 ; AM[ 41 ]= 0.723879032 ; AM[ 42 ]= 0.754375 ; AM[ 43 ]= 0.572225806 ;

291

AM[ 44 ]= 0.664358871 ; AM[ 45 ]= 0.727766129 ; AM[ 46 ]= 0.746431452 ; AM[ 47 ]= 0.163056774 ; AM[ 48 ]= 0.720819355 ; AM[ 49 ]= 0.731941935 ; AM[ 50 ]= 0.695072581 ; AM[ 51 ]= 0.691935484 ; AM[ 52 ]= 0.689741129 ; AM[ 53 ]= 0.569952903 ; AM[ 54 ]= 0.613135726 ; AM[ 55 ]= 0.688054839 ; AM[ 56 ]= 0.612630645 ; AM[ 57 ]= 0.685336774 ; AM[ 58 ]= 0.678845565 ; AM[ 59 ]= 0.666791613 ; AM[ 60 ]= 0.663143306 ; AM[ 61 ]= 0.538983871 ; //ITO,C60,Al nr,ni ITOnr[ 1 ]= 2.200 ; ITOnr[ 2 ]= 2.176 ; ITOnr[ 3 ]= 2.152 ; ITOnr[ 4 ]= 2.128 ; ITOnr[ 5 ]= 2.104 ; ITOnr[ 6 ]= 2.080 ; ITOnr[ 7 ]= 2.060 ; ITOnr[ 8 ]= 2.040 ; ITOnr[ 9 ]= 2.020 ; ITOnr[ 10 ]= 2.000 ; ITOnr[ 11 ]= 1.980 ; ITOnr[ 12 ]= 1.966 ; ITOnr[ 13 ]= 1.952 ; ITOnr[ 14 ]= 1.938 ; ITOnr[ 15 ]= 1.924 ; ITOnr[ 16 ]= 1.910 ; ITOnr[ 17 ]= 1.900 ; ITOnr[ 18 ]= 1.890 ; ITOnr[ 19 ]= 1.880 ; ITOnr[ 20 ]= 1.870 ; ITOnr[ 21 ]= 1.860 ; ITOnr[ 22 ]= 1.848 ;

292

ITOnr[ 23 ]= 1.836 ; ITOnr[ 24 ]= 1.824 ; ITOnr[ 25 ]= 1.812 ; ITOnr[ 26 ]= 1.800 ; ITOnr[ 27 ]= 1.792 ; ITOnr[ 28 ]= 1.784 ; ITOnr[ 29 ]= 1.776 ; ITOnr[ 30 ]= 1.768 ; ITOnr[ 31 ]= 1.760 ; ITOnr[ 32 ]= 1.752 ; ITOnr[ 33 ]= 1.744 ; ITOnr[ 34 ]= 1.736 ; ITOnr[ 35 ]= 1.728 ; ITOnr[ 36 ]= 1.720 ; ITOnr[ 37 ]= 1.706 ; ITOnr[ 38 ]= 1.692 ; ITOnr[ 39 ]= 1.678 ; ITOnr[ 40 ]= 1.664 ; ITOnr[ 41 ]= 1.650 ; ITOnr[ 42 ]= 1.640 ; ITOnr[ 43 ]= 1.630 ; ITOnr[ 44 ]= 1.620 ; ITOnr[ 45 ]= 1.610 ; ITOnr[ 46 ]= 1.600 ; ITOnr[ 47 ]= 1.594 ; ITOnr[ 48 ]= 1.588 ; ITOnr[ 49 ]= 1.582 ; ITOnr[ 50 ]= 1.576 ; ITOnr[ 51 ]= 1.570 ; ITOnr[ 52 ]= 1.556 ; ITOnr[ 53 ]= 1.542 ; ITOnr[ 54 ]= 1.528 ; ITOnr[ 55 ]= 1.514 ; ITOnr[ 56 ]= 1.500 ; ITOnr[ 57 ]= 1.488 ; ITOnr[ 58 ]= 1.476 ; ITOnr[ 59 ]= 1.464 ; ITOnr[ 60 ]= 1.452 ; ITOnr[ 61 ]= 1.440 ; ITOni[ 1 ]= 0.050 ; ITOni[ 2 ]= 0.038 ; ITOni[ 3 ]= 0.032 ;

293

ITOni[ 4 ]= 0.028 ; ITOni[ 5 ]= 0.024 ; ITOni[ 6 ]= 0.020 ; ITOni[ 7 ]= 0.018 ; ITOni[ 8 ]= 0.014 ; ITOni[ 9 ]= 0.012 ; ITOni[ 10 ]= 0.010 ; ITOni[ 11 ]= 0.008 ; ITOni[ 12 ]= 0.005 ; ITOni[ 13 ]= 0.003 ; ITOni[ 14 ]= 0.001 ; ITOni[ 15 ]= 0.001 ; ITOni[ 16 ]= 0.001 ; ITOni[ 17 ]= 0.001 ; ITOni[ 18 ]= 0.001 ; ITOni[ 19 ]= 0.001 ; ITOni[ 20 ]= 0.001 ; ITOni[ 21 ]= 0.001 ; ITOni[ 22 ]= 0.001 ; ITOni[ 23 ]= 0.001 ; ITOni[ 24 ]= 0.001 ; ITOni[ 25 ]= 0.001 ; ITOni[ 26 ]= 0.001 ; ITOni[ 27 ]= 0.001 ; ITOni[ 28 ]= 0.001 ; ITOni[ 29 ]= 0.001 ; ITOni[ 30 ]= 0.001 ; ITOni[ 31 ]= 0.001 ; ITOni[ 32 ]= 0.001 ; ITOni[ 33 ]= 0.001 ; ITOni[ 34 ]= 0.001 ; ITOni[ 35 ]= 0.001 ; ITOni[ 36 ]= 0.001 ; ITOni[ 37 ]= 0.001 ; ITOni[ 38 ]= 0.001 ; ITOni[ 39 ]= 0.005 ; ITOni[ 40 ]= 0.008 ; ITOni[ 41 ]= 0.010 ; ITOni[ 42 ]= 0.011 ; ITOni[ 43 ]= 0.012 ; ITOni[ 44 ]= 0.012 ; ITOni[ 45 ]= 0.013 ; ITOni[ 46 ]= 0.014 ;

294

ITOni[ 47 ]= 0.015 ; ITOni[ 48 ]= 0.015 ; ITOni[ 49 ]= 0.016 ; ITOni[ 50 ]= 0.017 ; ITOni[ 51 ]= 0.018 ; ITOni[ 52 ]= 0.018 ; ITOni[ 53 ]= 0.019 ; ITOni[ 54 ]= 0.020 ; ITOni[ 55 ]= 0.021 ; ITOni[ 56 ]= 0.021 ; ITOni[ 57 ]= 0.022 ; ITOni[ 58 ]= 0.023 ; ITOni[ 59 ]= 0.024 ; ITOni[ 60 ]= 0.024 ; ITOni[ 61 ]= 0.025 ; C60nr[ 1 ]= 2.097 ; C60nr[ 2 ]= 1.981 ; C60nr[ 3 ]= 1.902 ; C60nr[ 4 ]= 1.85 ; C60nr[ 5 ]= 1.989 ; C60nr[ 6 ]= 2.203 ; C60nr[ 7 ]= 2.25 ; C60nr[ 8 ]= 2.174 ; C60nr[ 9 ]= 2.086 ; C60nr[ 10 ]= 2.07 ; C60nr[ 11 ]= 2.143 ; C60nr[ 12 ]= 2.203 ; C60nr[ 13 ]= 2.218 ; C60nr[ 14 ]= 2.246 ; C60nr[ 15 ]= 2.272 ; C60nr[ 16 ]= 2.303 ; C60nr[ 17 ]= 2.313 ; C60nr[ 18 ]= 2.315 ; C60nr[ 19 ]= 2.321 ; C60nr[ 20 ]= 2.326 ; C60nr[ 21 ]= 2.32 ; C60nr[ 22 ]= 2.294 ; C60nr[ 23 ]= 2.254 ; C60nr[ 24 ]= 2.213 ; C60nr[ 25 ]= 2.181 ; C60nr[ 26 ]= 2.159 ; C60nr[ 27 ]= 2.147 ;

295

C60nr[ 28 ]= 2.134 ; C60nr[ 29 ]= 2.121 ; C60nr[ 30 ]= 2.111 ; C60nr[ 31 ]= 2.104 ; C60nr[ 32 ]= 2.1 ; C60nr[ 33 ]= 2.097 ; C60nr[ 34 ]= 2.092 ; C60nr[ 35 ]= 2.087 ; C60nr[ 36 ]= 2.081 ; C60nr[ 37 ]= 2.074 ; C60nr[ 38 ]= 2.066 ; C60nr[ 39 ]= 2.058 ; C60nr[ 40 ]= 2.053 ; C60nr[ 41 ]= 2.046 ; C60nr[ 42 ]= 2.039 ; C60nr[ 43 ]= 2.032 ; C60nr[ 44 ]= 2.026 ; C60nr[ 45 ]= 2.021 ; C60nr[ 46 ]= 2.018 ; C60nr[ 47 ]= 2.013 ; C60nr[ 48 ]= 2.009 ; C60nr[ 49 ]= 2.005 ; C60nr[ 50 ]= 2.002 ; C60nr[ 51 ]= 1.999 ; C60nr[ 52 ]= 1.999 ; C60nr[ 53 ]= 1.994 ; C60nr[ 54 ]= 1.991 ; C60nr[ 55 ]= 1.989 ; C60nr[ 56 ]= 1.986 ; C60nr[ 57 ]= 1.984 ; C60nr[ 58 ]= 1.981 ; C60nr[ 59 ]= 1.98 ; C60nr[ 60 ]= 1.978 ; C60nr[ 61 ]= 1.976 ; C60ni[ 1 ]= 0.612 ; C60ni[ 2 ]= 0.55 ; C60ni[ 3 ]= 0.571 ; C60ni[ 4 ]= 0.701 ; C60ni[ 5 ]= 0.839 ; C60ni[ 6 ]= 0.766 ; C60ni[ 7 ]= 0.582 ;

296

C60ni[ 8 ]= 0.475 ; C60ni[ 9 ]= 0.48 ; C60ni[ 10 ]= 0.531 ; C60ni[ 11 ]= 0.577 ; C60ni[ 12 ]= 0.532 ; C60ni[ 13 ]= 0.499 ; C60ni[ 14 ]= 0.483 ; C60ni[ 15 ]= 0.462 ; C60ni[ 16 ]= 0.413 ; C60ni[ 17 ]= 0.362 ; C60ni[ 18 ]= 0.32 ; C60ni[ 19 ]= 0.281 ; C60ni[ 20 ]= 0.231 ; C60ni[ 21 ]= 0.193 ; C60ni[ 22 ]= 0.14 ; C60ni[ 23 ]= 0.103 ; C60ni[ 24 ]= 0.086 ; C60ni[ 25 ]= 0.082 ; C60ni[ 26 ]= 0.082 ; C60ni[ 27 ]= 0.081 ; C60ni[ 28 ]= 0.08 ; C60ni[ 29 ]= 0.079 ; C60ni[ 30 ]= 0.078 ; C60ni[ 31 ]= 0.078 ; C60ni[ 32 ]= 0.076 ; C60ni[ 33 ]= 0.073 ; C60ni[ 34 ]= 0.068 ; C60ni[ 35 ]= 0.062 ; C60ni[ 36 ]= 0.055 ; C60ni[ 37 ]= 0.05 ; C60ni[ 38 ]= 0.045 ; C60ni[ 39 ]= 0.041 ; C60ni[ 40 ]= 0.039 ; C60ni[ 41 ]= 0.037 ; C60ni[ 42 ]= 0.035 ; C60ni[ 43 ]= 0.034 ; C60ni[ 44 ]= 0.033 ; C60ni[ 45 ]= 0.032 ; C60ni[ 46 ]= 0.032 ; C60ni[ 47 ]= 0.031 ; C60ni[ 48 ]= 0.031 ; C60ni[ 49 ]= 0.031 ; C60ni[ 50 ]= 0.03 ;

297

C60ni[ 51 ]= 0.03 ; C60ni[ 52 ]= 0.03 ; C60ni[ 53 ]= 0.029 ; C60ni[ 54 ]= 0.029 ; C60ni[ 55 ]= 0.028 ; C60ni[ 56 ]= 0.028 ; C60ni[ 57 ]= 0.027 ; C60ni[ 58 ]= 0.027 ; C60ni[ 59 ]= 0.027 ; C60ni[ 60 ]= 0.026 ; C60ni[ 61 ]= 0.026 ; Alnr[ 1 ]= 0.276 ; Alnr[ 2 ]= 0.294 ; Alnr[ 3 ]= 0.314 ; Alnr[ 4 ]= 0.334 ; Alnr[ 5 ]= 0.355 ; Alnr[ 6 ]= 0.380 ; Alnr[ 7 ]= 0.400 ; Alnr[ 8 ]= 0.420 ; Alnr[ 9 ]= 0.440 ; Alnr[ 10 ]= 0.440 ; Alnr[ 11 ]= 0.490 ; Alnr[ 12 ]= 0.510 ; Alnr[ 13 ]= 0.540 ; Alnr[ 14 ]= 0.560 ; Alnr[ 15 ]= 0.590 ; Alnr[ 16 ]= 0.620 ; Alnr[ 17 ]= 0.650 ; Alnr[ 18 ]= 0.680 ; Alnr[ 19 ]= 0.700 ; Alnr[ 20 ]= 0.740 ; Alnr[ 21 ]= 0.770 ; Alnr[ 22 ]= 0.800 ; Alnr[ 23 ]= 0.840 ; Alnr[ 24 ]= 0.880 ; Alnr[ 25 ]= 0.920 ; Alnr[ 26 ]= 0.960 ; Alnr[ 27 ]= 1.000 ; Alnr[ 28 ]= 1.050 ; Alnr[ 29 ]= 1.100 ; Alnr[ 30 ]= 1.150 ;

298

Alnr[ 31 ]= 1.200 ; Alnr[ 32 ]= 1.250 ; Alnr[ 33 ]= 1.300 ; Alnr[ 34 ]= 1.360 ; Alnr[ 35 ]= 1.410 ; Alnr[ 36 ]= 1.470 ; Alnr[ 37 ]= 1.540 ; Alnr[ 38 ]= 1.600 ; Alnr[ 39 ]= 1.670 ; Alnr[ 40 ]= 1.750 ; Alnr[ 41 ]= 1.830 ; Alnr[ 42 ]= 1.930 ; Alnr[ 43 ]= 2.040 ; Alnr[ 44 ]= 2.150 ; Alnr[ 45 ]= 2.270 ; Alnr[ 46 ]= 2.400 ; Alnr[ 47 ]= 2.490 ; Alnr[ 48 ]= 2.580 ; Alnr[ 49 ]= 2.660 ; Alnr[ 50 ]= 2.730 ; Alnr[ 51 ]= 2.800 ; Alnr[ 52 ]= 2.780 ; Alnr[ 53 ]= 2.760 ; Alnr[ 54 ]= 2.720 ; Alnr[ 55 ]= 2.670 ; Alnr[ 56 ]= 2.610 ; Alnr[ 57 ]= 2.530 ; Alnr[ 58 ]= 2.430 ; Alnr[ 59 ]= 2.310 ; Alnr[ 60 ]= 2.190 ; Alnr[ 61 ]= 2.060 ; Alni[ 1 ]= 3.610 ; Alni[ 2 ]= 3.740 ; Alni[ 3 ]= 3.867 ; Alni[ 4 ]= 3.996 ; Alni[ 5 ]= 4.118 ; Alni[ 6 ]= 4.240 ; Alni[ 7 ]= 4.370 ; Alni[ 8 ]= 4.490 ; Alni[ 9 ]= 4.410 ; Alni[ 10 ]= 4.610 ;

299

Alni[ 11 ]= 4.860 ; Alni[ 12 ]= 4.980 ; Alni[ 13 ]= 5.100 ; Alni[ 14 ]= 5.230 ; Alni[ 15 ]= 5.350 ; Alni[ 16 ]= 5.470 ; Alni[ 17 ]= 5.590 ; Alni[ 18 ]= 5.710 ; Alni[ 19 ]= 5.840 ; Alni[ 20 ]= 5.960 ; Alni[ 21 ]= 6.080 ; Alni[ 22 ]= 6.200 ; Alni[ 23 ]= 6.320 ; Alni[ 24 ]= 6.450 ; Alni[ 25 ]= 6.560 ; Alni[ 26 ]= 6.690 ; Alni[ 27 ]= 6.810 ; Alni[ 28 ]= 6.920 ; Alni[ 29 ]= 7.040 ; Alni[ 30 ]= 7.150 ; Alni[ 31 ]= 7.260 ; Alni[ 32 ]= 7.370 ; Alni[ 33 ]= 7.480 ; Alni[ 34 ]= 7.590 ; Alni[ 35 ]= 7.690 ; Alni[ 36 ]= 7.790 ; Alni[ 37 ]= 7.900 ; Alni[ 38 ]= 8.010 ; Alni[ 39 ]= 8.120 ; Alni[ 40 ]= 8.220 ; Alni[ 41 ]= 8.310 ; Alni[ 42 ]= 8.400 ; Alni[ 43 ]= 8.490 ; Alni[ 44 ]= 8.570 ; Alni[ 45 ]= 8.600 ; Alni[ 46 ]= 8.620 ; Alni[ 47 ]= 8.610 ; Alni[ 48 ]= 8.600 ; Alni[ 49 ]= 8.570 ; Alni[ 50 ]= 8.510 ; Alni[ 51 ]= 8.450 ; Alni[ 52 ]= 8.390 ; Alni[ 53 ]= 8.340 ;

300

Alni[ 54 ]= 8.300 ; Alni[ 55 ]= 8.260 ; Alni[ 56 ]= 8.220 ; Alni[ 57 ]= 8.200 ; Alni[ 58 ]= 8.190 ; Alni[ 59 ]= 8.190 ; Alni[ 60 ]= 8.240 ; Alni[ 61 ]= 8.300 ; } void TransferMatrix(double lamda,double dl,double nglass, double ITOnr,double ITOni,double C60nr,double C60ni, double Alnr, double Alni, int* d, double* mono) // return the non-unit absorption in each layer for this wavelength { complex <double> Ep[2000],En[2000],n[2000]; complex <double> s11[2000],s12[2000],s21[2000],s22[2000]; complex <double> sp11[2000],sp12[2000],sp21[2000],sp22[2000]; complex <double> I11[2000],I12[2000],I21[2000],I22[2000]; complex <double> L11[2000],L12[2000],L21[2000],L22[2000]; double nr[2000]={0}; double k[2000]={0}; double q[2000]={0}; double qex[2000]={0}; double pi = 3.14159265359; int j; double total=0; double R1=0; // glass reflectivity double R2=0; // multilayer reflectivity //double qx=(4*pi*3*8.854*0.5/lamda)*1.0e5; double qx=0; double x=2*pi*dl/lamda; complex <double> a,b,c,f; complex <double> cx (x,0); complex <double> c0 (0,0); complex <double> c1 (1,0); complex <double> c2 (2,0); complex <double> ci (0,1); // define the complex time of x,0,1,i int N=d[1]+d[2]+d[3]+d[4]+d[5];

301

for(j=1;j<=d[1];j++) {nr[j]=ITOnr;k[j]=ITOni;}// ITO for(j=d[1]+1;j<=d[1]+d[2];j++) {nr[j]=2;k[j]=0;} // MoOx for(j=d[1]+d[2]+1;j<=d[1]+d[2]+d[3];j++) {nr[j]=C60nr;k[j]=C60ni;} // C60 BHJ for(j=d[1]+d[2]+d[3]+1;j<=d[1]+d[2]+d[3]+d[4];j++) {nr[j]=2;k[j]=0;} // BPhen for(j=d[1]+d[2]+d[3]+d[4]+1;j<=d[1]+d[2]+d[3]+d[4]+d[5];j++) {nr[j]=Alnr;k[j]=Alni;} // Al nr[0]=nglass;k[0]=0;// Glass nr[N+1]=1;k[N+1]=0; // air, initiate n,k parameter for(j=0;j<=N+1;j++) { n[j]=complex <double> (nr[j],k[j]); // initiate the complex type of n } for(j=0;j<=N;j++)

// calculate Ijk matrix, from 0,1 interface -> N,N+1 interface { I11[j]=(n[j]+n[j+1])/(c2*n[j]); I22[j]=I11[j]; I12[j]=(n[j]-n[j+1])/(c2*n[j]); I21[j]=I12[j]; } for(j=1;j<=N;j++) // calculate Ljk matrix, from 1 layer -> N layer { L11[j]=c1/exp(ci*cx*n[j]); L12[j]=c0; L21[j]=c0; L22[j]=exp(ci*cx*n[j]); } s11[1]=I11[0];s12[1]=I12[0];s21[1]=I21[0];s22[1]=I22[0];

// calculate s- matrix for(j=2;j<=N+1;j++) { a=s11[j-1]*L11[j-1]+s12[j-1]*L21[j-1]; b=s11[j-1]*L12[j-1]+s12[j-1]*L22[j-1];

302

c=s21[j-1]*L11[j-1]+s22[j-1]*L21[j-1]; f=s21[j-1]*L12[j-1]+s22[j-1]*L22[j-1]; s11[j]=a*I11[j-1]+b*I21[j-1]; s12[j]=a*I12[j-1]+b*I22[j-1]; s21[j]=c*I11[j-1]+f*I21[j-1]; s22[j]=c*I12[j-1]+f*I22[j-1]; } Ep[0]=c1; En[N+1]=c0; En[0]=Ep[0]*s21[N+1]/s11[N+1]; Ep[N+1]=Ep[0]/s11[N+1]; R1=(1-nglass)*(1-nglass)/((1+nglass)*(1+nglass)); R2=norm(s21[N+1]/s11[N+1]); qx=4*pi*(1-R1)/(lamda*nglass*(1-R1*R2)); for(j=0;j<=N+1;j++){mono[j]=0;}; for(j=1;j<=N;j++) { Ep[j]=(s22[j]*Ep[0]-s12[j]*En[0])/(s11[j]*s22[j]-s12[j]*s21[j]); En[j]=(-s21[j]*Ep[0]+s11[j]*En[0])/(s11[j]*s22[j]-s12[j]*s21[j]); mono[j]=qx*nr[j]*k[j]*norm(Ep[j]+En[j])*dl;

//consider the I0, the unit here is non unit } } int main() { double dl,e,diel_0,diel_r,kT; double dt,converge,ts_factor; double V,Vbi,Vbegin,Vmore,dV; double up00,Fp00,un00,Fn00; int check=0,N=0,Nmax=0,NJ=0; double Nc,Nv,Eg; double p[1000]={0},n[1000]={0},un[1000]={0},un0[1000]={0},Fn0[1000]={0}, up[1000]={0},up0[1000]={0},Fp0[1000]={0},F[1000]={0},Fint[1000]={0}, pnew[1000]={0},nnew[1000]={0},Dn[1000]={0},Dp[1000]={0}; double jn[1000]={0},jp[1000]={0},jt[1000]={0},jpdrift[1000]={0},jpdiff[1000]={0}, jndrift[1000]={0},jndiff[1000]={0},R[1000]={0},interRnp[1000]={0}, interRpn[1000]={0},U[1000]={0},P[1000]={0};

303

double current[10]={0}; double X[1000]={0},Xnew[1000]={0}; double G[1000]={0},result[1000]={0}; double a; double kf; double dn=0; double dp=0; int lamda[100]={0}; int d[100]={0}; double mono[2000]={0},range[2000]={0}; double AM[100]={0},ITOnr[100]={0},ITOni[100]={0},C60nr[100]={0},C60ni[100]={0}, Alnr[100]={0},Alni[100]={0}; data(lamda,AM,ITOnr,ITOni,C60nr,C60ni,Alnr,Alni); dl=1.0e-7; // unit cm e=1.6e-19; // unit C diel_0=8.854e-14; // vacuum dielectric constant 8.854e-14 C/V/cm kT=0.026; // unit eV double ch; // concentraion of hole material double diel_r1; double diel_r2; string line; ifstream myfile ("input.txt"); if (myfile.is_open()) {

getline (myfile,line); // time control converge parameters getline (myfile,line); myfile >> ts_factor;

getline (myfile,line);

getline (myfile,line); myfile >> Nmax;

getline (myfile,line);

getline (myfile,line); myfile >> converge;

304

getline (myfile,line);

getline (myfile,line); myfile >> dV;

getline (myfile,line);

getline (myfile,line); myfile >> Vbegin;

getline (myfile,line);

getline (myfile,line); myfile >> Vmore;

getline (myfile,line);

getline (myfile,line); // votlage, dissociation parameters

getline (myfile,line); myfile >> ch;

getline (myfile,line); getline (myfile,line); myfile >> diel_r1;

getline (myfile,line);

getline (myfile,line); myfile >> diel_r2;

getline (myfile,line);

getline (myfile,line); myfile >> Vbi;

getline (myfile,line);

getline (myfile,line); myfile >> a;

getline (myfile,line);

getline (myfile,line); myfile >> kf;

getline (myfile,line);

getline (myfile,line); // active layer parameters getline (myfile,line);

myfile >> N;

305

getline (myfile,line);

getline (myfile,line); myfile >> Nc;

getline (myfile,line);

getline (myfile,line); myfile >> Nv;

getline (myfile,line);

getline (myfile,line); myfile >> Eg;

getline (myfile,line);

getline (myfile,line); myfile >> up00;

getline (myfile,line);

getline (myfile,line); myfile >> Fp00;

getline (myfile,line);

getline (myfile,line); myfile >> un00;

getline (myfile,line);

getline (myfile,line); myfile >> Fn00;

getline (myfile,line);

getline (myfile,line); // end myfile.close();

} else cout << "Unable to open file"; int i,j; double FF=0; double power=0; double pmax=0; double Vpmax=0; double Jpmax=0; double Jsc=0;

306

double Joc=100; double Voc=0; double Vocbot=0; double Voctop=0; double Jocbot=1000; double Joctop=1000; double nglass=1.42; d[1]=100; // ITO thickness d[2]=3; // MoOx thickness d[3]=N; // Active Layer thickness d[4]=8; // BPhen thickness d[5]=100; // Al thickness int totalN=d[1]+d[2]+d[3]+d[4]+d[5]; for(j=1;j<=totalN;j++) {range[j]=0;} for(i=1;i<=41;i++) // for each wavelength { TransferMatrix(lamda[i],dl*1.0e7,nglass,ITOnr[i],ITOni[i],C60nr[i],C60ni[i]*(1-ch),Alnr[i],Alni[i],d,mono); // 2 changes, dl-> dl*1.0e7, and C60ni-> C60ni*(1-ch) for(j=1;j<=totalN;j++) { range[j]=range[j]+AM[i]*mono[j]; } } diel_r=pow(ch*pow(diel_r1,1.0/3.0)+(1.0-ch)*pow(diel_r2,1.0/3.0),3.0); for( j=1;j<=N;j++) { G[j]=range[100+3+j]; // light absorption in the active layer Fp0[j]=Fp00;up0[j]=up00;Fn0[j]=Fp00; un0[j]=un00; } ofstream finalfile; finalfile.open("finalfile.dat"); finalfile << "V" << " " << "holecurrent(mA/cm2)" << " " << "electroncurrent(mA/cm2)" << " " << "power(mW/cm2)" << " " << "check"<<" " << "dt" << endl; for(j=0;j<=N+1;j++)

307

{ p[j]=0;n[j]=0;F[j]=0;Fint[j]=0;pnew[j]=0;nnew[j]=0;un[j]=0;Dn[j]=0;up[j]=0;Dp[j]=0;jn[j]=0;jp[j]=0;jt[j]=0;jndrift[j]=0;jndiff[j]=0;jpdrift[j]=0;jpdiff[j]=0;R[j]=0;interRnp[j]=0;interRpn[j]=0;U[j]=0;P[j]=0;X[j]=0;Xnew[j]=0; } for( V=Vbegin;V<=Vbi+Vmore;V=V+dV,NJ++) { p[1]=Nv; n[1]=Nc*exp(-Eg/kT); p[N]=Nv*exp(-Eg/kT); n[N]=Nc; p[0]=n[1]; n[0]=p[1]; p[N+1]=n[N]; n[N+1]=p[N]; // initial stuff check=0; dt=Calculate_E(p,n,un,un0,Fn0,up,up0,Fp0,F,Fint,Dn,Dp,Vbi,V,N,dl,ts_factor,e,diel_0,diel_r); for (i=0;i<=Nmax && check==0;i++) { check=move(p,n,pnew,nnew,X,Xnew,kf,a,un, up, F, Fint,Dn, Dp ,G,result,jp, jn, jt,jpdrift,jpdiff,jndrift,jndiff,R,interRnp,interRpn,U,P,dt, Nmax,converge, N, dl,e,diel_0,diel_r,current,Nc,Nv,Eg); dt=Calculate_E(p,n,un,un0,Fn0,up,up0,Fp0,F,Fint,Dn,Dp,Vbi,V,N,dl,ts_factor,e,diel_0,diel_r); } if (check==0) //still calculate the current if they didn't calculate in the move subfunction { jp[0]=(p[1]-pnew[1])*dl*e*1000/dt; jn[0]=-(n[1]-nnew[1])*dl*e*1000/dt; jt[0]=jp[0]+jn[0]; jp[N]=-(p[N]-pnew[N])*dl*e*1000/dt; jn[N]=(n[N]-nnew[N])*dl*e*1000/dt; jt[N]=jp[N]+jn[N]; if (abs(jt[0]-jt[N])>abs(converge*(abs(jt[N])+abs(jt[0])))) check=0; // check again if the electron current equal the hole current

308

for(j=1;j<=N-1;j++) { dp=dt*(Dp[j]*p[j]-Dp[j+1]*p[j+1])/(dl*dl); jpdiff[j]=dp*dl*e*1000/dt; dp=0; if(F[j]>0) dp=dt*p[j]*up[j]*abs(F[j])/dl; if (F[j+1]<0) dp=dp-dt*p[j+1]*up[j+1]*abs(F[j+1])/dl; jpdrift[j]=dp*dl*e*1000/dt; dn=dt*(Dn[j]*n[j]-Dn[j+1]*n[j+1])/(dl*dl); jndiff[j]=-dn*dl*e*1000/dt; dn=0; if(F[j]<0) dn=dt*n[j]*un[j]*abs(F[j])/dl; if (F[j+1]>0) dn=dn-dt*n[j+1]*un[j+1]*abs(F[j+1])/dl; jndrift[j]=-dn*dl*e*1000/dt; jp[j]=jpdiff[j]+jpdrift[j]; jn[j]=jndiff[j]+jndrift[j]; jt[j]=jp[j]+jn[j]; } } Write_Files (NJ,N,check,p,n,un,up,F,Fint,jt,jp,jn,jpdrift,jpdiff,jndrift,jndiff,G,R,U,P,X,Xnew,result); power=-V*jt[0]; finalfile << V << " " << jt[0] << " " << jt[N] << " " << power << " " << check <<" " << dt << endl; cout << V << endl; if(V>0 && power>pmax) {pmax=power;Vpmax=V;Jpmax=jt[0];}; if(abs(V)<1.0e-3) Jsc=jt[0]; if(V>=0 && jt[0]<0 && abs(jt[0])<abs(Jocbot)) { Jocbot=jt[0];Vocbot=V;} if(V>=0 && jt[0]>0 && abs(jt[0])<abs(Joctop)) { Joctop=jt[0];Voctop=V;} } finalfile.close(); Voc=(Joctop*Vocbot-Jocbot*Voctop)/(Joctop-Jocbot); FF=pmax/(abs(Jsc)*Voc); finalfile.open("Voc-Jsc-FF.txt");

309

finalfile << "Voc=" << Voc << " V" << endl; finalfile << "Jsc=" << abs(Jsc) << " mA/cm2" << endl; finalfile << "FF=" << FF*100 << " %" << endl; return 0; }

310

D.3 Sample input (input.txt)

*************** Control Parameters **********************

Control time distance ts_factor=

3

Numbers of calculations Nmax=

30000000

Test of convergence converge=

0.2

Voltage increase Step dV=

0.2

Begin Voltage Vbegin=

-1

Additional voltage after built-in Vmore=

0.2

******************************************************************

Donor concentration=

0.125

Donor dielectric constant diel_r1=

3.5

Acceptor dielectric constant diel_r2=

4.5

Build-in Potential Vbi=

1.3258

Initial seperation distance a(nm)=

1.08

CT state decay rate kf (1/s)=

5.0e6

******************************************************************

311

Active Layer thickness N1 (nm)=

40

Conduction band Nc1 (1/cm3)=

1.0e18

Valence band Nv1 (1/cm3)=

1.0e18

Energy Gap Eg1 (eV)=

1.3

Hole mobility up0 (cm2/Vs)=

1.01e-5

Hole field dependent Fp0 ((V/cm)^(-1/2))=

1.41e-4

Electron mobility un0 (cm2/Vs)=

2.99e-3

Electron field dependent Fn0 ((V/cm)^(-1/2))=

1.02e-4

******************************************************************