www.unamur.be

Temporal incoherence of solar radiation: First-principle theory & application to solar

cell optical simulations

Olivier Deparis, Michaël Sarrazin, Aline Herman

Project review meeting, 23-24 April 2014

www.unamur.be

Outline

• Problem statement• Existing computational methods• A method derived from first principles

• A simplified case study• The full treatment

• Application to solar cell optical simulations

Apologies to those who do not like (too much) mathematics after lunch

www.unamur.be

Sunlight has both spatial and temporal incoherence

• Spatial incoherence• coherence length estimated to 60 m• not critical in thin films with lateral/vertical dimensions of the order

of 1-10 m

• Temporal incoherence• coherence time estimated to 3 fs• implicitly taken into account in solar cell efficiency measurements

(solar simulators)• usually not taken into account in solar-cell optical simulations

• Impact of temporal incoherence on efficiency still unclear• Rarity of theoretical investigations due to large computational

demands with existing methods

www.unamur.be

Existing methods perform statistics on coherent calculations

• Multiple runs of coherent calculations + statistical averaging• Each optical carrier frequency treated independently (assumption!)• Incident carrier phase selected randomly at each run• Statistical averaging of many independent runs (large computational demand)

• Each coherent run relies on solutions of Maxwell’s equations in laterally periodic stratified media using standard methods (RCWA, FDTD)

• Hundreds of runs required to simulate temporal (phase) incoherence

W. Lee, S.-Y. Lee, J. Kim, S. C. Kim, B. Lee Optics Express 20 (2012) A941A953

•Practical limitations (CPU time) for complex solar cell structures• Time required to compute one run increases dramatically with solar cell

complexity

Because of both of solar cell structure complexity and statistical treatment, accurate modeling of state-of-the-art solar cells under incoherent light is a formidable task!

www.unamur.be

A question of methodology

• Quantity of interest (accessible to measurements): photocurrent

(100% internal conversion efficiency and perfect carrier collection assumed)

• Illumination: solar power spectral density spectrum S()• Intermediate quantity : absorption spectrum A() in active layer

•Coherent illumination: A() represents the “coherent absorption”•Incoherent illumination: A() represents the “incoherent absorption”

ehcJ A S d

Would it be possible to compute Aincoh without multiple runs and therefore to deduce directly the photocurrent under incoherent illumination?

eincoh incohhcJ A S d

www.unamur.be

A direct method

• The method requires only one single run of coherent calculation!• Two independent steps:

1. calculation of the coherent absorption Acoh at each carrier wavelength c

2. incoherent absorption spectrum Aincoh(c) deduced directly from Acoh(c)

• convolution product in frequency domain with the power spectral density (PSD) of the random process spectrum

• PSD is assumed to be Gaussian, the same for each carrier frequency and depends solely on the sunlight coherence time c

1. Step #1 is time-consuming but performed once for all (no multiple runs)2. Step #2 is straightforward and fast

incoh c coh c cA A I

2 2ln 2

2

3ln 2 c

cI e

: convolution product

M. Sarrazin, A. Herman, O. Deparis, Optics Express 20 (2012) A941A953

c

1/c

www.unamur.be

Preliminary remarks

• Principle of the method can be catch by establishing the response of a linear system with 1 input/1 output channels in the frame of random signal theory

• Generalization to a linear system in scattering configuration (1 input/2 outputs) is tedious (only the great lines will be highlighted hereafter)

• Each individual frequency component of the solar spectrum can be regarded as a quasi-monochromatic signal whose spectral width is defined by random process

• All random processes related to each carrier frequency are independent (each carrier frequency can be treated individually)

• This is the basic assumption made in multiple run statistical methods

c

www.unamur.be

Incoherent response of a linear system with 1 input/1 output

G(ω): transfer functiong(t): impulse response G(ω): transfer functiong(t): impulse response

Linear system

Input signal(excitation)

Output signal(response)

out in out inx t g t x t X G X

Calculation of the response in time or frequency domain

www.unamur.be

Linking the incoherent response of a linear system to the coherent response: theoretical framework

• Random signal theory basic concepts• Real stationary random signal: x(t) (electric field of the electromagnetic radiation)• Autocorrelation function of the random signal

• E[]: expectation value (ensemble average)• Mean square value (average power transported by the optical carrier wave)

• Power spectral density (PSD) = Fourier transform of autocorrelation (Wiener-Khinchine)

• In order to define the PSD, the signal must be truncated within a span of time T, i.e. the sampling interval: xT(t) is one realization of the random signal

• Stochastic quantity corresponding to the Fourier transform of the truncated signal

• For T large enough, it can be shown that

• Normalized average power (deduced by integrating PSD)

XR E x t x t

20XR E x t

iX XS R e d

i tTX x t e dt

21X TS E X

21 1 12 20X X X TP R S d E X d

www.unamur.be

Remark about the sampling interval

• The sampling interval T is used to define a realization of the random signal

• In the context of solar cells, T is effectively the photo-detector response time which is very long at the time scale of the random process assuming T is large is fully satisfied

[0, ]

0 elsewhereT

x t t Tx t

t

T

www.unamur.be

Trivial case of coherent excitation

• Linear response

• Coherent input signal

• Coherent output power

G(ω): transfer functiong(t): impulse response G(ω): transfer functiong(t): impulse response

Linear system

Input signal(excitation)

Output signal(response)

2

0 cos2X c

ER 0 cos

cohin cx t E t

2

2 2 0, , 2coh cohx out c x in c

EP G P G

c: carrier frequency

out in out inx t g t x t X G X

2

0, 0

2cohx in X

EP R

www.unamur.be

Non-trivial case of incoherent excitation

• Incoherent input signal: carrier with randomly modulated amplitude

• Incoherent input power

0 0ci tincoh incoh

in in cx t E m t e X E M

2

02 221 1 1 1

, 02 2 2

Eincoh incohx in in c M cT TP E X d E E M d S d

21M TS E M

(random process PSD)

MS d

(normalization condition*)

*Both coherent and incoherent input signals must have the same power

www.unamur.be

Linking incoherent and coherent output powers: the convolution formula

• Incoherent output power

• Linking this to coherent output power

2 20 0

2 20 0

2 221 1 1 1

, 2 2

2 2 2 21 1

2 2

2 2 21

2 2

incoh incoh incohx out out inT T

E Ec c cT T

E Ec c c M cT

P E X d E G X d

E G M d E G M

G E M G S

1, , ,incoh coh cohx out c x out c M c x out c cP P S P I

with

M

M

SI

S d

(normalized random process PSD)

www.unamur.be

Generalization to the scattering problem

• We have considered the transfer function of a linear system with 1 input/1output• This formalism obviously does not allow us to calculate reflectance (R),

transmittance (T), hence absorption (A =1−R−T)• To do this, we must consider the scattering matrix of the linear system• Though the derivation is complicated, it also ends up with a convolution formula!

|Fin> |Fsca,R>

|Fsca,T>

S() S: scattering matrix|F: field super-vector

sca inF S F

www.unamur.be

Scattering matrix formalism (e.g. RCWA method)

Applicable to laterally periodic , arbitrarily stratified mediumField expanded in spatial Fourier series according to lateral periodicity of dielectric constantFourier components of the field expansion gathered in a super-vectorMaxwell’s eqs recast in matrix form relating incident and scattered super-vectorsQuantity of interest in “photonic” solar cells: photocurrentIntermediate quantities: Poynting vector fluxes, reflectance, transmittance, absorption

z0z1

zL

zj-1zjk=1

k=L

k=j

z

xy

I: incidence medium (z<z0)II: laterally periodic stratified medium (z0<z<zL)III: emergence medium (z>zL)

Unit cell pattern periodically repeated in x,y directions

www.unamur.be

Coherent response (deterministic process)

General response (coherent or incoherent) of the linear system in the frequency domain:

Incident flux

Scattered fluxes (X=R: reflected, X=T: transmitted)

Coherent incident field

Coherent scattered fields

102 cosin I in in in inJ c F F F F

†X sca X X sca X XJ F C C F F F

CX: connection matrix between stratified medium and incidence (emergence) mediumS: scattering matrix of stratified medium

X X inF S F X XS C S

sca inF S F

: angular frequency: unit cell area: incidence angle

0 0c ci t i tX X in X c in XF t S t F t e S F F e

0 0X X c inF S Fwhere

0 ci tin inF t F e

Carrier

www.unamur.be

Coherent reflectance/transmittance

Power fluxes

Reflectance (X=R) & transmittance (X=T)

Remark: the time-averaged incident flux is constant in the coherent case

0 0†0 0

in X c X c inX XX

coh cin in in

F S S FF FJX

J J J

0 01 10 02 2cos cos ctein I in in I in inJ t c F t F t c F F

0 ci tin inF t F e

2 0 01 1022

cosc

cc

T

in in I in inT TJ J t dt c F F

0 0 0 0c ci t i t

in X in X in X in X in X in X in XJ F t F t F F e e F F

www.unamur.be

Incoherent response (random process)

General response (coherent or incoherent) of the linear system in the time domain:

Incoherent incident field (=randomly amplitude modulated carrier)

Incoherent scattered fields

X X inF t S t F t

tiin

tiXin

tiXX

ccc eFetStmFetmtStF 00

tiX

tiinXX

cc etFeFtUtF 00

tiXX

cetStmtU

0 ci tin inF t F m t e

Power spectral density: 2D m Fourier transform: i tm m t e dt

Random process !

Carrier

www.unamur.be

Temporal averaging

The device response under incoherent excitation (finite coherence time c) is the time averaged value of its response recorded during a sampling time Tc>> c

In a solar cell device:• excitation=sunlight, response=photo-generated current

• Tc is fixed by recombination/generation time of charged carriers (0.1 ns to 1 ms in Silicon)

• since c 3fs, Tc>>c is fully satisfied

Temporal averaging of scattered fluxes

2 2 sin 21 1 1 1, 2 2 22 2

c c c

c c cc c

T T Ti tX incoh X X XT T TT TJ J t dt J e d dt J d

011,

XTXTincohX JdJJ

cc

Since JX has spectral width 1/c (around =0) and since Tc >>c, we can take the limit Tc

www.unamur.be

Incoherent time-averaged scattered fluxes

Instantaneous scattered fluxes

Time-averaged scattered fluxes

0 0 0 0†X X X in X X inJ t F t F t F U t U t F

0 0†1, 2 cX incoh c in X X inTJ D F S S F

0 01, 0

cX incoh in X inTJ F I F

0 0 0 0† †

0 0 0 0†12

i t i tX in X X in in X X in

in X X in in X in

J F U t U t F e dt F U t U t e dt F

F U U F F I F

Fourier transform:

From Fourier transform of UX(t): †12X X c X cI m S m S

developing explicitly the convolution product and then setting =0:

†120 ' ' ' ' 'X X c X cI m S m S d

and since m(t) and SX(t) are real functions:

2 †120 ' ' ' 'X c X XI m S S d

we find finally

†

' '

' 'tX c X c

m m

S S

2D m

PSD of random process

www.unamur.be

Incoherent time-averaged incident flux

Instantaneous incident flux

Time-averaged incident flux

2 20 01, 02 cosin incoh I in in inJ t c m t F F J m t

1 1 1 1, , 2 20

0c c cin incoh in incoh in inT T TJ J J m m J D d

Coherent incident flux

Integration of the PSD of the random process leads to time-averaged incoherent flux

Fourier transform:

1 1, 2 2 ' ' 'i t

in incoh in in inJ J m t m t e dt J m m J m m d

2

' '

' '

m m

D m

www.unamur.be

Putting all together …

Ratio of time-averaged incident/scattered leads to X=R or T

Link with coherent R or T

At each carrier frequency, the incoherent R/T is given by the convolution product (in frequency domain) between the coherent R/T and the normalized PSD of the random process

0 0†

,

,

' '' '

in X X in

cX incoh in

incoh cin incoh

F S S FD d

J JX

J D d

'cohX

'' ' ' ' 'c

incoh c coh c coh

DX X d I X d

D d

incoh c c coh cX I X

dD

DI

www.unamur.be

Photocurrent

Incoherent absorption spectrum

Photocurrent directly deduced from coherent absorption spectrum and PSD of random process (convolution product in frequency domain)

1 1

1

incoh c incoh c incoh c incoh c c incoh c c

coh c coh c c coh c c

A R T R I T I

R T I A I

incoh c c coh cA I A

1I d

eincohhcJ A S d

www.unamur.be

Case study: flat/1D grating structures

FDTD+statistical analysis RCWA+direct method

Sarrazin M, Herman A and Deparis O 2013 Optics Express 21 A616

Lee W, Lee S.-Y, Kim J, Kim S. C and Lee B 2012 Optics Express 20 A941A953

www.unamur.be

Solar cell thin corrugated active layer

Sarrazin M, Herman A and Deparis O 2013 Optics Express 21 A616

www.unamur.be

Further reading