exploiting non-linear arrhenius … kenney...exploiting non-linear arrhenius dependence of diode iv...

EXPLOITING NON-LINEAR ARRHENIUS DEPENDENCE OF

DIODE IV CURVES TO DETERMINE SCHOTTKY BARRIER

BAND DIAGRAMS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ELECTRICAL

ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Crystal Kenney

November 2012

Abstract

Accurate extraction of Schottky barrier height is imperative to the development of low

resistance contacts. An analytical model for current density is found that accurately

accounts for conduction in the thermionic emission (TE), thermionic field emission

(TFE), and field emission (FE) regimes. Use of this model in non-linear regression al-

lows more information to be extracted from diode current-voltage-temperature (IVT)

curves than previously possible. The proposed model uses the Arrhenius non-linear

dependence experimental diode IV curves to regress the Schottky barrier height (φB0),

steepness factor (E00), and Fermi level (ξ), enabling band diagrams of the measured

interfaces to be determined. This model is tested against both simulated interfaces

using the transmission matrix method (TMM) and experimental data. This complete

picture of band information allows material interface behavior to be understood more

completely, ultimately facilitating more efficient contact engineering.

iv

Contents

Abstract iv

1 Introduction 1

2 Methods for low resistance contacts 4

2.1 Cleaning and Passivation . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.2 Germanium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.3 III-V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 High Doping Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Doped Growth . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Ion Implantation . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.3 Gas Immersion Laser Doping (GILD) . . . . . . . . . . . . . . 10

2.2.4 Mono-Layer Doping (MLD) . . . . . . . . . . . . . . . . . . . 10

2.3 Silicide, Germanosilicide & Germanocide . . . . . . . . . . . . . . . . 10

2.4 Alloyed Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Barrier Height Lowering Methods . . . . . . . . . . . . . . . . . . . . 13

2.5.1 Bandgap Modulation . . . . . . . . . . . . . . . . . . . . . . . 13

2.5.2 Dielectric Dipole Mitigation (DDM) . . . . . . . . . . . . . . . 13

2.5.3 Dipole Modulation . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Measurement of Contact Parameters 16

3.1 Resistivity and Contact Measurements . . . . . . . . . . . . . . . . . 16

3.1.1 Linear (Rectangular) Transfer Length Method (TLM) . . . . . 18

v

3.1.2 Circular TLM . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.3 Kelvin Structure . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.4 4 point probe . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.5 Specific Contact Resistivity Extraction . . . . . . . . . . . . . 26

3.2 Hall Effect Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.1 Hall Bar Structure . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.2 Greek Cross and Box Cross . . . . . . . . . . . . . . . . . . . 30

3.3 SIMS Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Modeling of Contact Types 34

4.1 Metal Semiconductor Junction . . . . . . . . . . . . . . . . . . . . . . 34

4.1.1 Tunneling Formalism . . . . . . . . . . . . . . . . . . . . . . . 34

4.1.2 Historical Approximations . . . . . . . . . . . . . . . . . . . . 37

4.1.3 Transfer Matrix Method . . . . . . . . . . . . . . . . . . . . . 41

4.1.4 Metal-Semiconductor Specific Definitions . . . . . . . . . . . . 44

4.2 Metal Insulator Semiconductor (MIS) Junctions . . . . . . . . . . . . 45

4.2.1 Analytical Model . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Modified Silicided Junction . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Em ≈ slope model 59

5.1 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2 Current emission theory . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.3 Simulation Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.4 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6 Derivation of Analytical Richardson’s Plot 74

6.1 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.1.1 Transmission Probability T . . . . . . . . . . . . . . . . . . . 76

6.2 Approximations for Fermi-Dirac Statistics Related Functions . . . . . 81

6.3 Analytical Current Density J . . . . . . . . . . . . . . . . . . . . . . . 84

vi

6.4 Arrhenius (Richardson’s) plot and ideality n plot . . . . . . . . . . . 88

6.5 Regression capability of analytical model . . . . . . . . . . . . . . . . 93

6.5.1 Regression capability against simulated data . . . . . . . . . . 93

6.5.2 Regression capability against experimental data . . . . . . . . 98

7 Research Summary and Future Work 101

7.1 Research Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

A Approximation for F 12

105

B J and ρc Derivation 108

B.1 Current Density Derivation . . . . . . . . . . . . . . . . . . . . . . . 112

B.1.1 Current density independent of transverse energy . . . . . . . 114

B.1.2 Thermionic emission limit of Current Density . . . . . . . . . 115

B.2 Full Specific Contact Resistivity Derivation . . . . . . . . . . . . . . . 115

B.2.1 ρc independent of transverse energy . . . . . . . . . . . . . . . 117

B.2.2 Thermionic emission limit of ρc . . . . . . . . . . . . . . . . . 117

C Ψ and 1m∗

∂Ψ∂x

versus 1√m∗

Ψ and 1√m∗

∂Ψ∂x

119

C.1 Ψ and 1m∗

∂Ψ∂x

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

C.2 1√m∗

Ψ and 1√m∗

∂Ψ∂x

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

D Richardson’s plot analytical model 125

D.1 Analytical no image charge lowering . . . . . . . . . . . . . . . . . . . 125

E Techniques for Successful Regression 131

F Helpful Integrals 134

G Mask Structure Details 136

Bibliography 146

vii

List of Tables

1.1 Max Specific Contact Resistivity ρc (Ω cm2) . . . . . . . . . . . . . . 2

2.1 Stable Doping Limits for Several Semiconductors . . . . . . . . . . . 6

2.2 Steps in alloyed contact formation . . . . . . . . . . . . . . . . . . . . 11

3.1 Parameters measured by mask structures . . . . . . . . . . . . . . . . 17

3.2 Symbols used in calculations of resistivity and contact resistance . . . 18

3.3 Symbols used in calculations of Hall measurements . . . . . . . . . . 27

4.1 Symbols and variables for modeling contacts . . . . . . . . . . . . . . 35

4.2 Estimated n-type N values for boundary condition of kBT ≈ E00 using

relative permittivity εs and effective mass m∗ values from literature. . 37

4.3 Semiconductor Electron Affinity and Metallic Workfunctions (eV) . . 38

4.4 Fermi level pinning from experimental literature values and dipole model. 39

4.5 Metal “conduction” band in reference to metal Fermi level for some

“simple” metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.6 Assumptions of MIS analytical model . . . . . . . . . . . . . . . . . . 48

5.1 Symbols and variables used in Arrhenius plot derivation . . . . . . . . 59

5.2 Extracted SBH and error % from Richardson’s plot using different

extraction methods with Nd=1016, 1017, and 1018cm−3. The labels

Odd3, Odd5 and Odd7 represent the highest order term of the odd

order polynomial used in the extraction (Equation 5.9). . . . . . . . . 68

5.3 Summary of non-linear Richardson’s plot literature. . . . . . . . . . . 70

viii

6.1 Symbols and variables used in Arrhenius plot derivation . . . . . . . . 75

7.1 Research Accomplishments . . . . . . . . . . . . . . . . . . . . . . . . 101

B.1 Assumptions of J and ρc derivation . . . . . . . . . . . . . . . . . . . 108

B.2 Symbols and variables for complete J and ρc derivations . . . . . . . . 109

D.1 Substitutions for different conduction regimes . . . . . . . . . . . . . 127

ix

List of Figures

2.1 Interface between semiconductor and metal is of the form of a Schottky

barrier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Higher doping levels near the interface of the semiconductor make the

Schottky barrier thinner and slightly lower due to image charge lowering. 7

2.3 Dielectric dipole mititation (DDM) . . . . . . . . . . . . . . . . . . . 14

2.4 Dipole modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1 Linear TLM Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Parameter Extraction for Linear TLM . . . . . . . . . . . . . . . . . 20

3.3 Circular TLM Structure . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Linearized CTLM Data . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.5 Parameter Extraction for Circular TLM . . . . . . . . . . . . . . . . 22

3.6 Kelvin Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.7 Four Point Probe Structure . . . . . . . . . . . . . . . . . . . . . . . 25

3.8 Sign Conventions for Hall Setup for N-type Semiconductor . . . . . . 28

3.9 1-3-3-1 Hall Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.10 Greek and Box Cross Structures . . . . . . . . . . . . . . . . . . . . . 31

4.1 Energy band diagram of a forward biased Schottky junction . . . . . 36

4.2 Step-wise approximation of potential barrier . . . . . . . . . . . . . . 41

4.3 Notation scheme for TMM formalism . . . . . . . . . . . . . . . . . . 42

4.4 Detailed description of pertinent variables for one layer of structure . 42

4.5 Sample Vx description of Schottky MS Junction . . . . . . . . . . . . 45

4.6 ρc of N-type semiconductors . . . . . . . . . . . . . . . . . . . . . . . 46

x

4.7 ρc of P-type semiconductors . . . . . . . . . . . . . . . . . . . . . . . 47

4.8 Description of Schottky MIS junction with surface states. . . . . . . . 48

4.9 Definition of parameters for determination of pinning factor S . . . . 50

4.10 MIS structure for thin nonzero oxides. . . . . . . . . . . . . . . . . . 53

4.11 Effective MIS structure for thin nonzero oxides. . . . . . . . . . . . . 54

4.12 TMM simulation ρc results for TaN/SiO2/n-Si < 100 > . . . . . . . . 55

4.13 Pinning factor comparison of CNL and Tung models . . . . . . . . . 56

4.14 Dual Silicide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1 Definition of Em in reference to energy distribution of emitted electrons

Eee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2 Simulated Em behavior as a function of applied voltage at and temper-

ature for NiSi/n-Si for (a) Nd=1015cm−3 and (b) Nd=1018cm−3. Plot

(a) at Nd=1015cm−3 shows distinctly TE behavior for most tempera-

tures with Em only going below the SBH at about 50K. Plot (b) at

Nd=1019cm−3 shows TFE is the dominant mechanism as the Em is

below the SBH for most temperatures and only goes above it at about

800K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.3 Map of diode area with area in homogeneous and inhomogeneous models 66

5.4 Simulation of Richardson’s plot for NiSi/n-Si . . . . . . . . . . . . . . 67

5.5 Simulated Tcrit values showing line between TE and TFE regimes for

different metal/semi pairs . . . . . . . . . . . . . . . . . . . . . . . . 71

5.6 Example analysis of Richardson’s plot data . . . . . . . . . . . . . . . 72

6.1 Setup of band diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.2 Definition of Em in reference to energy distribution of emitted electrons

Eee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.3 Potential barrier with classical turning points . . . . . . . . . . . . . 77

6.4 Setup of band diagram with Eee . . . . . . . . . . . . . . . . . . . . . 79

6.5 Comparison of full form and historical approximations for 1exp (x)+1

. . 82

6.6 Comparison of full form and approximation for 1exp (x)+1

. . . . . . . . 83

6.7 Comparison of full form and approximation for log (exp (x) + 1) . . . 83

xi

6.8 Current density for TE, TFE and FE cases . . . . . . . . . . . . . . . 88

6.9 When ln (Jlin) shows non-linearity the substitutional and extrapolated

saturation linearized current density give different results. . . . . . . . 89

6.10 Richardson’s plot and ideality factor for TE, TFE and FE dominated

conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.11 Analytical approximations for Richardson’s plot and ideality factor . 91

6.12 Piecewise and analytical approximation comparison . . . . . . . . . . 92

6.13 Schottky barrier height regression behavior of several models on simu-

lated data (|) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.14 Extracted SBH and E00 for simulated data (|) . . . . . . . . . . . . 94

6.15 Schottky barrier height regression behavior of several models on simu-

lated data (⊕|) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.16 Extracted SBH and E00 for simulated data (⊕|) of NiSi/n-Si . . . . 96

6.17 Extracted SBH and E00 for simulated data (⊕|) of Al/n-Ge . . . . . 97

6.18 Extracted SBH and E00 for experimental data of NiSi/n-Si as function

of As dose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.19 Extracted SBH and E00 for experimental data of NiSi/n-Si as function

of Sb co-implant dose . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

A.1 Numerical and analytical approximation of Fermi-Dirac integral of or-

der 12

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

C.1 Ψ and 1m∗

boundary condition for interface transmission limit . . . . 121

C.2 1√m∗

Ψ and 1√m∗

boundary condition for interface transmission limit . 124

G.1 Legend for Mask Layers . . . . . . . . . . . . . . . . . . . . . . . . . 136

G.2 Completed Mask Design . . . . . . . . . . . . . . . . . . . . . . . . . 137

G.3 Four Point Probe Structures . . . . . . . . . . . . . . . . . . . . . . . 138

G.4 Circular Transmission Line Structures . . . . . . . . . . . . . . . . . . 139

G.5 Linear Transmission Line Structures . . . . . . . . . . . . . . . . . . 140

G.6 Greek Cross Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 141

G.7 Box Cross Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

xii

G.8 Kelvin Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

G.9 Hall Bar Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

G.10 SIMS Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

xiii

Chapter 1

Introduction

As devices scale down, contact resistance plays a large role in device parasitics [1, 2]

and obtaining low specific contact resistivity becomes critical [3]. Total parasitic

resistance (Equation 1.1) of a transistor is the sum of the the channel resistance (RCH)

and the source/drain resistance (RSD). As devices scale down, the channel resistance

stays approximately constant assuming the ratio of channel width to channel length

(W/L) remains constant1, while the contact resistance is dependent on the contact

hole size which scales as the square of the lithographic dimension. This leads to

the interface between the semiconductor and silicide/metal dominating the contact

resistance and increasing dramatically for small feature sizes. [3]

RC = RCH +RSD (1.1)

Specific contact resistivity ρc (Ω cm2) is a contact size independent measure of the

quality of the interface. The ITRS [3] suggested values for future technology nodes

are outlined in Table 1.1. The value of 1e-8 Ω cm2 is significant as the next technology

hurdle.

It is vital to the future of device scaling to understand and address this critical

interface. There are several methods proposed by the ITRS [3] to improve the quality

of this interface including maximizing dopant concentration at the interface, using

1Also assuming the doping of the channel remains constant.

1

CHAPTER 1. INTRODUCTION 2

Table 1.1: Max Specific Contact Resistivity ρc (Ω cm2) [3]

Year of Production 2013 2014 2015 2016 2017

Printed Gate Length (nm) 28 25 22 20 17.7

Bulk 2e-8 1e-8 8e-9

FDSOI & Multi-Gate 4e-8 2e-8 1e-8 8e-9 7e-9

a lower barrier-height junction material such as SiGe used as the contact junction

and/or low-barrier-height dual metal silicides to be used to contact n+ and p+ junc-

tions2. All of these suggestions seek to lower the contact resistance by either lowering

or thinning the Schottky barrier found at the semiconductor/silicide interface. Other

possibilities to lower the barrier height found in literature include dielectric dipole

mitigation (DDM) [4] and dipole modulation through ion implantation [5]. This dis-

sertation presents tools and models to quantify the effects of these different schemes.

This research outlines a method to exploit non-linear Arrhenius dependence of diode

current-voltage (IV) curves to determine Schottky barrier band diagrams. Under-

standing the effects of these contact resistance lowering methods on the Schottky

barrier interface diagrams is conducive to engineering lower resistance contacts.

Chapter 2 discusses methods of lowering contact resistance in more detail includ-

ing benefits and drawbacks of each. Chapter 3 goes over structures that are used

to measure contact parameters and that are found on the contact mask that was

designed. Chapter 4 goes over modeling different contact types and the methods that

were used to model them and the derivations used to derive the models. Chapter 5

discusses the observations that led to the first attempt at extracting a more accurate

Schottky barrier height. Chapter 6 shows the derivation, and implementation of a

analytical modelling of the Arrhenius plot and a better extraction of Schottky barrier

height as well as the method to extract Schottky barrier band diagrams. Chapter 7

summarizes the work that was done as well as highlighting areas for future improve-

ment. The Appendices include other pertinent information such as detailed layout of

2Also proposed but not practically demonstrated is using Schottky barriers that serve as junctionsand contacts simultaneously [3]

CHAPTER 1. INTRODUCTION 3

the contact mask and detailed mathematical derivations for the models used.

Chapter 2

Methods for low resistance

contacts

Low resistance contacts are critical towards the future scaling of devices and the

dominant component of contact resistance is the interface between the semiconductor

and contact metal [3]. This interface is in the form of a metal-semiconductor Schottky

barrier (Figure 2.1).

Figure 2.1: Interface between semiconductor and metal is of the form of a Schottkybarrier.

Reduction of contact resistance consists of either lowering the Schottky barrier

4

CHAPTER 2. METHODS FOR LOW RESISTANCE CONTACTS 5

height (ΦB0) or thinning the Schottky barrier1. This chapter briefly addresses many

topics that are of interest to Schottky contacts and discusses different methods of

lowering contact resistance found in literature. Looking forward, it is also important

to be able to form low resistance contacts on materials other than silicon as III-V

and Ge high mobility channel materials are being explored and therefore includes

information on these alternate channel materials as well.

2.1 Cleaning and Passivation

Since the interface between the semiconductor and slicide/metal dominates the con-

tact resistance, any contaminants located at this interface can have dramatic effects

on the quality, stability and reproducibility of the contact resistance. Therefore, semi-

conductor cleaning and passivation can have a critical impact on contact resistance.

In particular unintentional oxide layers are of concern since any residual oxide will

degrade silicide formation and possibly lead to a discontinuous silicide.

2.1.1 Silicon

Cleaning of silicon is primarily achieved through 2 cleaning methods labeled SC1 and

SC2 cleans. The SC1 clean is a mixture of 5:1:1 Water/38% ammonium hydroxide

(NH4OH)/30% hydrogen peroxide (H2O2) that is used as an organic clean. The SC2

clean is a mixture of 5:1:1 water/38% hydrochloric acid (HCl)/30% hydrogen peroxide

(H2O2) that removes metal contaminants [6]. Passivation of silicon during processing

is achieved by a hydrofluoric acid (HF) based solution.

2.1.2 Germanium

H2O2 aqueous solutions such as SC1 and SC2 are unacceptable for Ge due to high

etch rates and increased surface roughness [7]. Ozone oxidation and thermal treat-

ments are effective as removing organic contamination on Ge. Native oxide and metal

1This is typically achieved by increasing the dopant concentration of the semiconductor.


contaminent removal can be achieved with HCl, HBr or HCl/HBr mixture [7]. Pas-

sivation of germanium is best achieved using HBr or HI [8].

2.1.3 III-V

Ozone oxidation is useful for removing stubborn sub-monolayers of hydrocarbons

on III-V materials. Oxide removal is obtained by use of HF, HCl or NH4OH:H2O,

while metal contamination is best removed by NH4OH:H2O [9]. Without additional

passivation, oxide regrowth will occur quickly (≈ 10 minutes) [9]. Passivation of III-V

materials (e.g. GaAs & InP) has been shown to work with ammonium sulfide wet

chemistry or a H2S plasma [10].

2.2 High Doping Methods

Table 2.1: Stable Doping Limits for Several Semiconductors

Material Ntype Doping Limit (cm−3) Ptype Doping Limit (cm−3)

Si 2e21 [11] 2e20 [11]

Ge 9e19a [12] 2e20 [12]

GaAs 2e18 [13] 1-2e20 [14,15]

GaSb 3e19 [16] 5e19 [16]

In0.53Ga0.47As 5e19 [14] 1e20 [14]

In0.8Ga0.2As 5e19 [14] 1e20 [14]

InP 5e19 [14] 1e20 [14]a Meta-stable concentration of 1e20 cm−3 achieved by [17] by laser annealing.

A very effective method of reducing the contact resistance of the Schottky barrier is

by increasing the dopant concentration at the surface of the semiconductor. As shown

in Figure 2.2, a higher doping concentration thins the Schottky barrier (allowing more

tunneling current) and also slightly lowers the barrier due to image charge lowering.

For many methods of semiconductor doping, the electricallly active concentration of

dopants is limited to those found in Table 2.1.


Figure 2.2: Higher doping levels near the interface of the semiconductor make theSchottky barrier thinner and slightly lower due to image charge lowering.

2.2.1 Doped Growth

Silicon and germanium are grown epitaxily through a variety of different methods

such as chemical vapor deposition (CVD), molecular beam epitaxy (MBE), atomic

layer deposition (ALD) [6, 18]. Growth of III-V substrates is generaally done by one

of two methods; molecular beam epitaxy (MBE) and metal-organic chemical vapor

deposition (MOCVD) [9]. When dopants are introduced during growth, the resulting

semiconductor is doped. However, dopants introduced during growth are subject to

the limits outlined in Table 2.1.

2.2.2 Ion Implantation

Ion implantation (as its name suggests) implants ions into the target semiconductor.

The form of the ions after implantation is roughly gaussian with the distribution

defined by implant energy, dose (number of ions cm−2), and implant angle. A good

estimate for the distribution of the implant can be obtained by SRIM (Stopping and

Range of Ions in Matter) and TRIM (Transport of Ions in Matter) software [19].

Ion implantation is a versatile important part of semiconductor processing. High


energy implants (deep implants) are used for isolation structures2 while low energy

implants (shallow) are used to form shallow junctions for source/drain contacts. Ion

implantation can be used in a self-aligned process by using photoresist as a mask to

limit areas to be implanted [6].

Ion implantation damages the target semiconductor during implantation causing

vacancies and interstitials and possibly amorphizes the target. The target then has

to be recrystalized by using a thermal or laser annealing process. During this anneal-

ing process, the implanted profile will change as diffusion occurs and the implanted

species is activated (substitutional). The amount of diffusion is highly dependent

on the thermal budget of the anneal. Higher temperatures and longer times usually

repair more damage at the cost of larger amounts of diffusion. The semiconductor

industry has moved to higher temperatures and short times (e.g. RTA - Rapid Ther-

mal Annealing and Laser Annealing) as the best way to repair damage and yet allow

a minimum of diffusion [6].

III-V Considerations

III-V semiconductors have additional considerations with ion implantation. Implan-

tation into a III-V semiconductor results in the elements recoiling differently due to

their difference in mass. The lighter element will recoil further into the substrate leav-

ing an excess of the heavier element shallower than the peak of the implant (<Rp).

This makes it harder to recombine the substrate during annealing, especially if the

III-V component mobilities are not very high. An increase in temperature during

implantation will increase the mobility of point defects allowing more to be on site

before the anneal, therefore increasing activation. An increase in implantation tem-

perature isn’t necessary for GaAs implantation [13]. This is due to the fact that the

masses of Ga and As are fairly close together and therefore the recoiling differences

are not as drastic as some of the other common III-V combinations.

2To create SI (semi-insulating) GaAs wafers, EL2 defects (related to As on Ga site) are pur-posefully introduced. This creates high resistivity substrate to isolate electrical structures. It isimportant to keep in mind when choosing annealing conditions for an implantation, the effect thiswill have on the EL2 defect. Annealing at 600-800oC will generate EL2 defects, while a anneal at1200oC followed by rapid cooling will eliminate EL2 defects. [20]


As loss during temperature processing above 400oC (e.g. annealing) is a concern

with III-V arsenides. As overpressure during the annealing process will help prevent

As loss [21].

III-V materials can also be doped by implantation of their own constituents [22].

For example, assuming no defect formation, As would make a good n-type dopant in

GaAs.

Laser annealing

Laser annealing has been shown to improve dopant activation beyond the values in

Table 2.1 [21]. In the case of n-type Ge, laser annealing has been shown to improve

the electrically active dopant concentration to >1020 cm−3 [17]. High power laser

annealing (HPLA) is unacceptable with use of GaAs due to the volatility of As when

the GaAs substrate is melted. However, low-power pulsed laser annealing (LPPLA)

has been shown to recrystallize the GaAs substrate without significant changes to the

sociometry of the substrate. [23] Pulsed electron beam annealing (EPBA) has been

shown to achieve higher activity levels of dopants than thermal annealing as well as

reduced dopant diffusion. [24] The use of an electron beam has also been shown to

reactivate Si dopants by disassociation of SiH complexes. [25]

Solid Phase Epitaxy

Higher activation can be achieved when the target is fully amporphized during im-

plantation and subsequently annealed (i.e. solid phase epitaxy). Lighter dopants

shuch as B often do not amorphize the substrate, therefore non-doping species (e.g

Si implant in Si target) are implanted with the purpose of amorphization. In sili-

con p-type meta-stable dopant activation of 3e20 cm−3 can be achieved. Remaining

defects are an issue near the amorphous/crystalline interface. Care should be taken

to place this interface so that it does not degrade device performance (e.g junction

leakage) [6].


2.2.3 Gas Immersion Laser Doping (GILD)

Gas-immersion laser doping (GILD) uses an excimer laser to melt the semiconductor

substrate while in a dopant gas ambient forming ultra-shallow junctions. This method

directly incorporates the dopants into the semiconductor with a 100% activation and

needs no additional annealing step. A larger number of laser pulses results in a higher

dose of dopant being incorporated, but too many pulses will crack the substrate [26].

2.2.4 Mono-Layer Doping (MLD)

In mono-layer doping (MLD), dopant containing molecules are self-assembled onto the

semiconductor surface and then annealed through rapid thermal annealing (RTA) to

diffuse into the semiconductor. This process form ultra-shallow junctions without

damage to the underlying semiconductor and can be used for any device geometry

(e.g. nanowires) [27]. This process has also been shown to work with III-V materials

such as InAs [28].

2.3 Silicide, Germanosilicide & Germanocide

Silicided contacts are a particularly attractive solution to low resistance contacts in

particular due their ability to be used in a self-aligned process (salicide). Historically,

silicided contacts have been made from several different metals including W, Pt,

Ti, Co and Ni, however, Ni has become the metal of choice for several different

reasons. NiSi is a lower resistance phase that consumes less substrate3. Its one weak

characteristic is its morphological stability at higher temperatures, which has been

shown to be rectified from the use of a small percentage of Pt. Also attractive is

it’s ability to react with all concentrations of Si1−xGex4. NiPt alloy is also shown to

improve temperature stability for Si1−xGex germanosilicides as well5 [6].

3This is important for shallower junction requirements from the ITRS.4Ge outdiffusion is a concern leading to Si1−uGeu instead.5A small amount of C doping (<1%) aslo helps


2.4 Alloyed Contacts

III-V semiconductors have conventionally use alloyed contacts in HEMT fabrication.

As III-V materials are used for MOSFETs the drawbacks (e.g. laterial diffusion,

spiking, reproducibility and aging [29]) prove prohibitive to low contact reistance.

However, these types of contacts form low resistance contacts to III-V semiconductors

and are in wide use.

There are seven common steps that occur during n-type GaAs alloyed contact

formation. Not all types of contacts use all of these steps, however, most have all of

their steps listed in these seven. The seven steps are shown in Table 2.2.

1. An element of the contact reacts with the GaAs native oxide.

2. An element of the contact reacts with GaAs at low temperatures to form a

X-GaAs complex.

3. An element of the contact diffuses into GaAs mediated by defects created in

step 2 and dopes GaAs.

4. An element of the contact diffuses into GaAs and forms a low bandgap semi-

conductor phase.

5. Ga outdiffuses from the contact and reacts, usually with Au.

6. Elements of the contact react and form their thermodynamically preferred com-

pounds.

7. As diffuses from the GaAs to the contact surface where it resides or vaporizes.

Table 2.2: Steps in alloyed contact formation [9]

The most common contact used is GaAs/Ge/Au/Ni and most likely includes six

out of the seven above steps. In the first step the Ni reacts with the GaAs native

oxide. In the second step the Ni forms temporary complexes with GaAs that disturb

the lattice and act as a catalyst for Au and Ge reactions. Step 3 involves Ge diffusing

into the GaAs. Step 4 is currently believed not to occur in these reactions, though

previously it was believed that NiAs were possible low bandgap phases. The Au-Ga

reaction in step 5 with the Au-Ga alloy very thermodynamically stable and favored

above 250C. Step 6 involves reactions between GaAs and other contact materials


including compounds such as NiAs and NiGe. Step 7 involves the transport to the

surface and desorption of As. The voids from As are filled with the compounds such

as AuGa, NiGe and others. [9]

There are some common misconceptions about GaAs contacts and particularly

GaAs/Ge/Au/Ni contacts. The first of which is that the “eutectic is chosen so that

contacts will melt at the chosen temperature becoming the molten alloy that allows

Ge to diffuse better into GaAs”. The reality is that the mechanisms for Ge and other

materials to diffuse into GaAs work efficiently at temperatures below the eutectic

melting of GeAu. The Ni-Ge-GaAs complexes act as a catalyst allowing Ge to diffuse

rapidly at much lower temperatures than if the Ni had not been present. The second

misconception is that “Ge acts as the n-type dopant in the GaAs”, whereas the reality

is that Ge/Ni reactions are also important. The last misconception is that “Ni acts

as a wetting agent so that the melted AuGe will not ball up.” In reality the Ni reacts

with the oxide to improve morphology and reduce variability as well as acting as a

catalyst for Au and Ge reactions, and aiding in the diffusion of Ge by forming lattice

disturbing compounds. [9]

RTA machines are usually used for the annealing step. It is very important not

to have any oxygen in the chamber while annealing as As-oxides are volatile and can

consume large amounts of As from arsenide semiconductors (e.g. GaAs, InGaAs) as

well as reacting with the contact materials. Therefore it is important to purge the

system with an inert gas prior to annealing. [9]

Solid phase regrowth (SPR) is a mechanism in which part of the metal contact

reacts with the GaAs to form low-temperature intermediate phases. These are not

stable at higher temperatures and the metal constituent is removed, reforming the

GaAs lattice. These types of contacts are typically characterized by low contact

resistivity and excellent morphology. Refractory contacts use refractory metals and

typically either act as a cap to increase the In content at the GaAs surface or act as

a barrier to the Au in the contact. [30]


2.5 Barrier Height Lowering Methods

Reduction of contact resistance consists of either lowering the Schottky barrier height

(ΦB0) or thinning the Schottky barrier. This section goes over methods used to lower

the Schottky barrier of the metal-semiconductor junction.

2.5.1 Bandgap Modulation

One method of obtaining a lower barrier is to replace the material near the metal with

a semiconductor that has a lower pinned barrier. One III-V example uses a thin layer

of n+ doped Ge, which gives a higher donor concentration than GaAs (1020 cm−3

rather than 5x1019 cm−3) and a small barrier height to electrons (0.5eV rather than

0.7-0.8eV). [21] Both of these properties lead to a lower resistance contact. Another

option involves grading from x=0 of Ga1−xInxAs by MBE to 0.8<x<1. This will lead

to an “ohmic” structure due to pinning at the conduction band for high values of

x. This is a commonly used alternative to alloyed contacts on III-V [31]. Another

benefit of using a InGaAs layer on top of GaAs layer is the easier doping to >1019

cm−3 when In >50% [32].

2.5.2 Dielectric Dipole Mitigation (DDM)

Dielectric dipole mititation (DDM) inserts a thin dielectric between the metal and

semiconductor with the aim of unpinning the Schottky barrier height (Figure 2.3).

When the Schottky barrier is unpinned, the barrier height can be changed by changing

to a metal with a different work function. The downside of this method is that

there is added resistance because the carriers need to tunnel through the dielectric.

Therefore an optimum dielectric thickness exists for each material system for the

lowest resistance to be obtained. This method has been shown to work with Si [33],

Ge [34], and III-V materials (e.g. GaAs and InGaAs) [35].


Figure 2.3: Dielectric dipole mititation (DDM) inserts a thin dielectric between themetal and semiconductor with the aim of unpinning the Schottky barrier height.Surface states are shown at the insulator-semiconductor interface.6


2.5.3 Dipole Modulation

Dipole modulation creates an additional dipole at the M-S interface to modulate the

Schottky barrier height achieved by implanting a species close to the interface (Fig-

ure 2.4). There are two methods of forming this dipole, one is with the implantation

of a species which is neutral in the semiconductor (e.g. Ge or C into Si) and the other

is with a species that acts as a dopant. With the neutral species the dipole is formed

from the difference in the electronegativity [36]. If the species also acts as a dopant, it

is expected that the buildup of electrically active dopants within a few monolayers of

the interface form image charges in the metal. This forms a dipole across the interface

and modulates the Schottky barrier height as well as possibly thinning the barrier

due to the increased doping effect [37].

Figure 2.4: Dipole modulation changes the pinning at the interface by adding in anadditional dipole which can raise or lower the barrier.

6Although not shown, there is likely a field in the insulator which can be in either directiondepending on the interaction between the doping level in the semiconductor, the surface statesof the insulator-semiconductor interface and induced charge on the metal surface. If the insulatorcompletely unpins the metal-semiconductor interface giving rise to a barrier height that is determinedby the Schottky-Mott relation, then there is no field across the insulator. [38]

Chapter 3

Measurement of Contact

Parameters

Characterization of semiconductor contacts is done though a variety of measurement

structures. These structures are used to determine contact parameters such as spe-

cific contact resistivity ρc, sheet resistance Rsh, and hall mobility µH . A simple two

layer mask was designed to be compatible with III-V processing (i.e. mesa etch for

isolation). A variety of structures which would measure the contact parameters was

desired. Redundancy in parameter measurement is useful to determine reliability of

results. If the results from different structures measuring the same parameter match,

then the result is more reliable than if it was measured by only one type of structure.

Table 3.1 shows a summary of the defining parameters found by each of the structures

included in the mask design.

3.1 Resistivity and Contact Measurements

Table 3.2 lists the symbols used in calculations of resistivity and contact resistance.

Specific contact resistivity (ρc) is the most useful parameter for device metallurgy. It

is dependent on the Schottky barrier height (φB), semiconductor dielectric constant

(εs), doping density (N), and effective mass of the carrier (m∗) (Equation 3.1) and

has units of Ω-cm2. [39, 40] A four point probe method eliminates the resistance of

16

CHAPTER 3. MEASUREMENT OF CONTACT PARAMETERS 17

Table 3.1: Parameters measured by mask structures include specific contact resistivityρc, sheet resistance Rsh, and hall mobility µH .

ρc Rsh µH

Linear TLM X X

Circular TLM X X

Kelvin Structure X X

4 Point Probe X

Hall Bar X X

Greek + Box Cross X X

the probes from the data. It is best to use this method whenever possible. [9]

ρc = ρco exp

(2φB~

√εsm∗

N

)(3.1)

The transfer length is shown in Equation 3.2 and is very useful because it de-

termines the point in the length of the contact at which 63% of the current has

transferred into the metal. [39, 40]

`t =

√ρcRsh

(3.2)

A check to find out how inaccurate the model used is by plotting specific contact

resistance versus contact area. Since specific contact resistance is a fundamental

property of the metal and semiconductor interface it should be independent of contact

area. If the plot shows otherwise, a more accurate model for specific contact resistance

should be used. [39]

It is assumed by most researchers that the sheet resistance of the semiconductor

directly under the contact (Rsk) is equal to the sheet resistance of the intervening

semiconductor(Rsh). In many cases such as alloyed contacts the value of Rsk is actu-

ally lower than that of Rsh [9] and for silicided contacts on a SOI substrate the value

of Rsk is much higher than Rsh due to the reduction in the silicon depth. If it is much

different than a more complicated model is needed for added accuracy. [39, 40]


Table 3.2: Symbols used in calculations of resistivity and contact resistance

Name Description Units

`t Contact transfer length µm

R Measured resistance from V-I graph Ω

Rce Contact end resistance Ω

RC Contact resistance Ω

rc Normalized contact resistance Ω mm

ρc Specific contact resistivity Ω cm2

ρ Semiconductor resistivity Ω cm

Rsh Sheet resistance of semiconductor Ω/

Rsk Sheet resistance of semiconductor under contact Ω/

L Length of contact in dimension of main current flow or radius of

circular contact

µm

W Linear contact array dimension perpendicular to current flow µm

d Distance between contacts µm

s Distance between contacts in four-point probe measurements µm

F Correction factor for four point probe calculation

δ Distance between contacts and edge of mesa isolation µm

The resistance extraction from V-I curves is best done at the lowest current pos-

sible to avoid non-linear high current effects. [41] Care should also be taken when

determining the thickness of samples since the actual layer thickness may differ from

the undepleted layer thickness due to Fermi pinned surfaces and bending at the layer-

substrate interface. [41] [42]

3.1.1 Linear (Rectangular) Transfer Length Method (TLM)

A linear contact array is the most commonly used test structure for contact resistance.

It consists of many contacts in a row with different contact spacings as shown in

Figure 3.1. Using a transmission line derivation, the total resistance measured is


shown by Equation 3.3 where RC is the contact resistance, Rsh is the sheet resistance

of the semiconductor, d is the length between two contacts, and W is the width of the

contacts. The substitution of Equation 3.4 into Equation 3.3 is valid when L≥ 1.5`t,

where L is the length of the contact in the same direction as the current. When

graphed versus the distance spacing d, it is very easy to find the contact resistance

RC , the sheet resistance Rsh and `t parameters as shown in Figure 3.2.

d1

L

d2 d3

W

Figure 3.1: Linear TLM Structure

R = 2RC +Rshd

W

=Rsh

W(d+ 2`t) (3.3)

RC =ρc`tW

(3.4)

3.1.2 Circular TLM

A circular TLM structure operates on the same principle as a linear TLM structure

but does not require a mesa etch for isolation. The first type of circular TLM structure

was analyzed by Reeves [43] and consisted of concentric metal donuts. This structure

is not included in this contact mask but is useful due to keeping Rsh and Rsk separate

in the calculations. The second type of circular TLM structure was analyzed by


Slope = Rsh/W

Y-intercept = 2RC

X-intercept = -2lt

Figure 3.2: Parameter Extraction for Linear TLM

Marlow and Das [44] and is shown in Figure 3.3 where L is the radius of the inner

contact and d is the circular separation.

This structure takes up less space than the concentric circle approach and when

the assumption is made that Rsh and Rsk are the same, the calculations are simpler

[45]. The most accurate equation contains Bessel functions (Equation 3.5), but when

L> 4`t, it can be simplified to Equation 3.61 [41].

R =Rsh

2π

`tL

I0

(L`t

)I1

(L`t

) +`t

L+ d

K0

(L`t

)K1

(L`t

) + ln

(1 +

d

L

) (3.5)

R =Rsh

2π

[`tL

+`t

L+ d+ ln

(1 +

d

L

)](3.6)

When dL<< 1 Equation 3.6 reduces to R = Rsh

2πL(d+ 2`t). When d

L< 1, R can be

described by Equation 3.7 where C is defined by Equation 3.8. [46] [41]

R =Rsh

2πL(d+ 2`t)C (3.7)

1Beware: Baca’s [9] book contains a typo in Equation 6.13


d L+dL

Figure 3.3: Circular TLM Structure

C =L

dln

(1 +

d

L

)(3.8)

Fitting data to Equation 3.6 is mathematically prone to up to 15% error depending

on which extrapolation methods are used in MATLAB. Linearization of the data is

obtained by by dividing the data by the correction factor in Equation 3.8 as shown

in Figure 3.4. Linearizing the data first results in a mathematical underestimation `t

(when data up to L=d is included) of less than 0.2%. Therefore most of the error in

this calculation will be the extraction of R from the I-V graph at each point.

Parameter extraction is simple when viewing the linearized data in Figure 3.5.

The only difference from the linear TLM extraction is that W is replaced by 2πL

when determining Rsh from the slope of the graph.

3.1.3 Kelvin Structure

Figure 3.6 shows the Kelvin structure used in this mask. All of the contacts are

square with side length L.

Contacts forming a 90o angle (Ex pads 1,5,2) form cross bridge Kelvin resistance


Figure 3.4: Linearized CTLM Data

Slope = Rsh/2πL

Y-intercept = 2RC

X-intercept = -2lt

Figure 3.5: Parameter Extraction for Circular TLM


1 5 3

4

2

L

c

Figure 3.6: Kelvin Structure

test structures. By passing a current from pad 5 to pad 1, a voltage can be measured

from pad 5 to pad 2 (V51,52). A one dimensional estimate of the contact resistance

is shown in Equation 3.9, however, this requires that L ≤ `t which is most likely not

the case since L is large in this mask. [41] Therefore 2D or 3D numerical simulations

should be used to get an accurate representation of ρc. [47] [48]

RC =V51,52

I51

=ρcL2

(3.9)

The front contact resistance and the end contact resistance assume that ρc >

0.2Rsht2. This is true in most cases, but should be tested to be sure. These resistances

also assume that Rsh and Rsk are equal, which is not the case for alloyed contacts.

Alloyed contacts have a more complicated structure and therefore require the trilayer

transmission line model (TLTLM) for an accurate representation. [49] [50]

The front contact resistance which is usually referred to as the contact resistance


is measured by measuring the the voltage from pad 5 to pad 1 when the current is

forced from pad 5 to pad 1 (V51,51). When L ≥ 1.5`t, Equation 3.10 is an accurate

representation for non-alloyed contacts.

RC =V51,51

I51

=ρc`tL

(3.10)

The end contact resistance can be measured from this structure by forcing a

current from pad 5 to pad 1 and measuring the voltage from pad 5 to pad 3 (V51,53).

The calculation is shown in Equation 3.11. However, when L >> `t, Rce becomes

very small and the accuracy is limited by the measurement apparatus. [41]

Rce =V51,53

I51

=ρc`tL

1

sinh(L`t

) (3.11)

3.1.4 4 point probe

Four point probe measurements are advantageous because they eliminate the volt-

age drop from the resistance of the probe as well as contact resistance. This allows

accurate measurements of the test device. In the case of this mask, a four point

probe structure of square contacts of equal spacing is included to measure the semi-

conductor sheet resistance as shown in Figure 3.7. For a thin sample where t≤s/2,

Equation 3.12 accurately describes the sheet resistance. For a very thick(t≥10s) ho-

mogeneous sample as in a highly doped substrate, Equation 3.13 describes the sheet

resistance. 2 [41]

Rsh =πF

ln(2)

V

I≈ 4.532F

V

I(3.12)

Rsh =2πsF

t

V

I(3.13)

The correction factor F is actually the product of several correction factors. The

correction for sample thickness is already taken into account in the above equations.

2For samples of thicknesses between these extremes please refer to the correction factors sectionof Chapter 1 of Schroder’s book. [41]


s L

Figure 3.7: Four Point Probe Structure

The Other major correction factor applicable to this mask set is the correction factor

for proximity to the edge of sample boundary. In the case of this mask the sample

boundary is the edge of the mesa isolation. Since the ratio of δ/s << 1, the value of

F = (0.7)(0.7)(0.5)(0.5) = 0.1125. Other correction factors are not applicable in this

case.

Alternatively, a dual configuration can be used. In the first configuration, current

is forced between the first and last pads while the voltage is measured between the

second and third pads. Measurements for forward and reverse current are obtained.

The second set of measurements are obtained with the current forced between the

first and third pads and voltage measured between the second and last pads. The

application of the calculations detailed in Equations 3.14 & 3.15 then result in an

accurate value of sheet resistance without correction factors. [51] [41]

Ra =1

2

(Vf23

If14

+Vr23

Ir14

)Rb =

1

2

(Vf24

If13

+Vr24

Ir13

)(3.14)

Rsh =

[−14.696 + 25.173

(Ra

Rb

)− 7.872

(Ra

Rb

)2]∗Ra (3.15)


3.1.5 Specific Contact Resistivity Extraction

Specific contact resistivity is extracted from front and end contact resistance mea-

surements by use of the tranmission line method (TLM). This model is a pseudo-2D

model which takes into account current crowding effects. Equations 3.16 & 3.17 are

used to obtain both Rsk and ρc.

RC =

√RskρcW

coth

(d

√Rsk

ρc

)(3.16)

RE =

√RskρcW

1

sinh(d√

Rskρc

) (3.17)


3.2 Hall Effect Structures

Table 3.3: Symbols used in calculations of Hall measurements


t Thickness of sample or doped region cm

a Hall bar arm width cm

b Distance from edge of outer Hall bar arms cm

c Hall bar arm length cm

d Distance between interior Hall bar arms cm

B Applied magnetic field Gauss

VH Hall voltage V

Vρ Voltage measured to determine resistivity V

I Current through sample A

RH Hall bar coefficient cm3/C

ρav Average resistivity of semiconductor Ω cm

Rsh Semiconductor sheet resistance Ω/

µH Hall mobility cm2/Vs

µn Majority carrier electron mobility cm2/Vs

r Hall scattering factor

L Length of contact in Greek and Box cross structures µm

z Parameter of Greek and Box cross structure µm

s Parameter of Box cross structure µm

Table 3.3 shows the symbols commonly used in Hall type measurements. The

units given are the ones used in the equations to get the correct units out of the

calculations.3

Hall structures are used to measure mobility, resistivity, and carrier density of a

sample by making use of the Hall effect. The Hall effect is an induced voltage in the

direction perpendicular to both current flow and applied magnetic field as shown in

310,000 Gauss = 1 Tesla


Figure 3.8. [52] A p-type sample will result in a Hall voltage VH of opposite sign.

z

y

x

→

B

I

VH

+ Vρ–

Figure 3.8: Sign Conventions for Hall Setup for N-type Semiconductor

3.2.1 Hall Bar Structure

The 1-3-3-1 Hall bar design is modified from the ASTM standard F76. The structure

is shown in Figure 3.9 with the additional dimension of t being the thickness of

the sample. The numbering notation is the same as the ASTM standard. [53] The

measurement notation of VAB,CD refers to the voltage VC-VD when the current is in

the direction from A to B. The value IAB refers to the current in the direction from

A to B.

Resistivity measurements can be obtained without use of an electric field. Mea-

surements of V12,46, V21,46, V12,57, V21,57 are taken and the resistivity is calculated

according to Equation 3.18.4 The semiconductor sheet resistance is then simply

Rsh = ρav/t.

4Note that the sign of the Hall voltage will switch when the direction of the current switches.Also, I12 + I21 should approximately equal 2I not zero.


1 2

3

4

5

6

7

8

c

a

b d bd

W

Figure 3.9: 1-3-3-1 Hall Bar

ρA =V12,46 − V21,46

I12 + I21

wt

d

ρB =V12,57 − V21,57

I12 + I21

wt

d

ρav =ρA + ρB

2(3.18)

The Hall coefficient RH can be used to determine carrier density and mobility.

These measurements require the use of a magnetic field where a positive magnetic

field is as defined in Figure 3.8. Typical fields used are between 0.05T and 1T. [41]

Equation 3.19 shows the measurements needed and how to calculate RH with resulting

units of cm3

C. By measuring all of the conditions presented in Equation 3.19, misalign-

ment voltages, external Seebeck effect voltage, Nerst effect, and Righi-Leduc voltages

are eliminated. The Ettingshausen effect which is due to a voltage induced from a

temperature gradient between hot and cool electrons is not eliminated but is much

smaller than VH in most cases.5 [52] If a fast measurement is needed, Equation 3.20

could be used, but will suffer from the previously mentioned effects.

5Exception: When there is a change from an n-type to p-type within the sample causing RH tochange sign


RH = 108 t

B

V12,65(+B)− V21,65(+B)− V12,65(−B) + V21,65(−B)

I12(+B) + I21(+B) + I12(−B) + I21(−B)(3.19)

RH = 108 t

B

V12,65(+B)

I12(+B)(3.20)

The Hall scattering factor (r) is a parameter that is dependent on the scattering

mechanisms dominant in the semiconductor. For GaAs this value varies from 1.17 at

B = 0.01T to 1.006 at B = 8.3T. [41]

µH =|RH |ρav

(3.21)

µH = rµn (3.22)

Carrier concentration can be determined from the Hall coefficient but requires

an estimate of the Hall scattering factor, as well as assuming energy-independent

scattering mechanisms. [41]

n = − r

qRH

(3.23)

3.2.2 Greek Cross and Box Cross

The Greek cross and the Box cross are symmetrical van der Pauw structures often used

in integrated test designs. Figure 3.10 shows the layout and applicable dimensions

for both cross structures. For a Greek cross a ratio of z/L = 2 is often cited due to

the small error (<0.1%) such a ratio achieves. [54] However, there is evidence that

the ratio should not be excessively large due to joule heating. [55] With a box cross

of dimensions z/s = 2/3 & L/s = 1/3 the error is also very small (<<0.1%). [54] The

box cross has less current crowding around the interior corners of the structure. [56]

The calculations of resistivity and the hall coefficient are very similar to the cal-

culations with the Hall Bar structure, but the values are labeled differently since the


1 3

4

2

L

z

(a) Greek Cross Structure

1 3

4

2

L

sz

(b) Box Cross Structure

Figure 3.10: Greek and Box Cross Structures


structures are numbered differently. [53] To determine the resistivity the long method

may be used to eliminate the effects mentioned in the Hall Bar structure section.

This requires measurement of V21,34, V12,34, V32,41, V23,41, V43,12, V34,12, V14,23, and

V41,23. The resistivity is then calculated as in Equation 3.24. If a quick determination

is desirable, measurement of V21,34 and calculation with Equation 3.25 should suffice.

Rsh1 =π

ln(2)

V21,34 − V12,34 + V32,41 − V23,41

I12 + I21 + I32 + I23

Rsh2 =π

ln(2)

V43,12 − V34,12 + V14,23 − V41,23

I43 + I34 + I14 + I41

Rsh =Rsh1 +RRsh2

2(3.24)

Rsh =π

ln(2)

V34

I21

≈ 4.532V34

I21

(3.25)

To obtain the hall coefficient (RH), measure V31,42(+B), V13,42(+B), V42,13(+B),

V24,13(+B), V31,42(−B), V13,42(−B), V42,13(−B), V24,13(−B). The hall coefficient is

then determined by the calculations shown in Equation 3.26. If the mobility is re-

quested, then it is determined as in Equation 3.27.

RH1 = 108 t

B

V31,42(+B)− V13,42(+B)− V31,42(−B) + V13,42(−B)

I31(+B) + I13(+B) + I31(−B) + I13(−B)

RH2 = 108 t

B

V42,13(+B)− V24,13(+B)− V42,13(−B) + V24,13(−B)

I42(+B) + I24(+B) + I42(−B) + I24(−B)

RHav =RH1 +RH2

2(3.26)

µH =|RHav| tRsh

(3.27)


3.3 SIMS Area

Secondary ion mass spectrometry (SIMS) is a technique used to measure impurities.

Removal of material is done by sputtering and then ions of dopant materials are

measured. [41] This gives a profile of the different layers of the sample. There is a

special SIMS area on the mask of 400µm x 400µm for substrate, mesa, ohmic metal

and ohmic metal on mesa.

3.4 Diodes

There are no diodes in the mask shown in Appendix G, but they are a common tool

for analysis. Testing diodes can be built laterally or vertically. Vertical diodes are

built on a highly conductive substrate and back of the wafer is deposited with metal to

make a back contact. Despite the highly conductive substrate, this method can often

have high series resistance. Lateral contacts are built to reduce this series resistance

and to be able to be able to characterize higher quality contacts (low specific contact

resistivity). Lateral diodes are often built back to back so that the current flow is

actually through two diodes in series. This will affect the observed measurement

results and require some additional processing. The most commonly used method of

extraction uses current-voltage (IV) measurements of contact diodes. A current is

measured as the voltage is sweeped between the two contacts.

Chapter 4

Modeling of Contact Types

4.1 Metal Semiconductor Junction

A metal semiconductor junction is formed when metal is brought into intimate con-

tact with a semiconductor material with a negligible interfacial layer. If the surface is

unpinned, then a Schottky or Ohmic contact can result depending on the differences

between the electron or hole affinity and metal work functions. In actual practice the

junction is almost always pinned as a Schottky junction as shown in Figure 4.1. Fig-

ure 4.1 shows a forward biased junction where the voltage is applied from the metal

to the semiconductor and the electrons are moving from the semiconductor into the

metal. There are three conduction regimes of thermionic emission(TE), thermionic

field emission(TFE) and field emission(FE). The parameter φB0 represents the barrier

height, V is the applied voltage, and φn represents the Fermi level relative to the con-

duction band with the value being negative for degenerately doped semiconductors.

4.1.1 Tunneling Formalism

Current density of a metal-semiconductor junction for both forward and reversed bi-

ases is described by Equations 4.1& 4.2 respectively. The subscript notation describes

the direction of the particle current. The complete current for a positive applied volt-

age and 1-D elastic tunneling can be described by Equation 4.3 [57–59]. mt and Et

34

CHAPTER 4. MODELING OF CONTACT TYPES 35

Table 4.1: Symbols and variables for modeling contacts


(|) Without image charge lowering -

(⊕|) With image charge lowering -

ΦB0 Schottky barrier height (|) eV

ΦBn, ΦBp Schottky barrier height (⊕|) for n-type and p-type eV

φb0 Schottky barrier height (|) V

φbn, φbp Schottky barrier height (⊕|) for n-type and p-type V

φn, φp Fermi level from conduction band and valance band V

V Applied voltage V

T Temperature K

Ex Energy of electron ⊥ to interface eV

Eτ Energy of electron ‖ to interface eV

E Total energy of electron (Ex + Eτ ) eV

Tsm Transmission from semi to metal —

Tms Transmission from metal to semi —

Fs Fermi probability function for semi —

Fm Fermi probability function for metal —

EFm Fermi level of metal eV

ECm “Conduction” band level of metal eV

EFs Fermi level of semiconductor eV

ECs Conduction band level of semiconductor eV

m0 Electron rest mass kg

m∗ Density of states (DOS) effective mass units of m0

mf Tunneling effective mass units of m0


FE

TFE

TE

qΦBn

-qΦn EC

EFm

V

EF

Figure 4.1: Energy band diagram of a forward biased Schottky junction showing thedirection of electron particle current and relative energy placement for thermionicemission(TE), thermionic field emission(TFE) and field emission(FE).

are the effective mass and energy of electrons with momentum in transverse direction

of the barrier on the side of the originating material. Tsm is the transmission probabil-

ity of an electron from the semiconductor to the metal for a particular energy. Fs and

Fm are the fermi probability functions for the semiconductor and metal respectively

(Equation 4.4). See Appendix B for more details.

Jsm =mτsq

2π2~3

∫ ∞Ex=0

∫ ∞Et=0

FsTsm (1− Fm) dEτdEx (4.1)

Jms =mτmq

2π2~3

∫ ∞Ex=0

∫ ∞Et=0

FmTms (1− Fs) dEτdEx (4.2)

J =qm0

π3~3

∫ +∞

Ex=0

∫ +∞

Eτ=0

∫ π2

θ=0

F1T (1− F2)mτdEτdθdEx (4.3)

F =1

1 + exp(E−EFkBT

) (4.4)


4.1.2 Historical Approximations

Historically different approximations to the value of the transmission probability

(Tsm) as well as non-degenerate approximations in certain cases have led to some

analytical expressions for current through a metal-semiconductor junction.

The parameter E00 shown in Equation 4.5 is useful for delineating the three con-

duction regimes. N is the active doping level in the semiconductor, εs is the relative

permittivity of the semiconductor, ε0 is the permittivity of free space, m∗ is the effec-

tive mass of the semiconductor, and m0 is the electron rest mass . If kBT >> E00 then

thermionic emission is the dominate conduction mechanism, whereas, if kBT << E00

then field emission (or tunneling) is the dominate mechanism. For the case of

kBT ≈ E00, the dominate mechanism is thermionic field emission (or thermally as-

sisted tunneling).

E00 =q~2

√N

m∗m0εsε0(4.5)

Table 4.2: Estimated n-type N values for boundary condition ofkBT ≈ E00 using relative permittivity εs and effective mass m∗ valuesfrom literature.

Material Permittivity(εs) Effective Mass (m∗) N (cm−3)

InP 12.61 [60] 0.079 [61] 1.9e18

GaAs 12.9 [62] 0.063 [62] 1.6e18

In0.53Ga0.47As 13.9 [62] 0.041 [62] 1.1e18

In0.8Ga0.2As 14.55 [62] 0.031 [62] 8.7e17

InAs 15.15 [62] 0.023 [62] 6.8e17

In0.52Al0.48As 12.82 [60]a 0.73 [63] 1.8e19

In0.20Al0.80Sb 12.5 [60]a 0.786 [63] [61]ab 1.9e19a Bowing parameters not found, linear interpolation usedb Indirect bandgap, effective mass of valley X used

The parameter of choice for describing the interfacial quality of the junction is

described by the specific contact resistivity (ρc) and is in units of Ω cm2. This


Table 4.3: Semiconductor Electron Affinityand Metallic Workfunctions (eV)

Material Electron Affinity (eV)

Si 4.05 [64]

Ge 4.0 [64]

InP 4.38 [64]

GaAs 4.07 [64]

In0.53Ga0.47As 4.51 [62]

In0.8Ga0.2As 4.73 [62]

InAs 4.9 [64]

GaSb 4.06 [64]

InSb 4.59 [64]

Material Metallic Workfunction (eV)

Al 4.20 [60]

Au 5.47 [60]

Ag 4.64 [60]

Cu 5.10 [60]

parameter is defined by Equation 4.6 where J is the current density and V is the

applied voltage across the interface.

ρc =

(dJ

dV

)−1∣∣∣∣∣V=0

(4.6)

In the case of pure thermionic emission, a non-degenerate approximation is used

for the Fermi functions and a simple step function is used as the transmission proba-

bility (Tsm) for electrons over the barrier height φBn. The current density is described

by Equation 4.7 and the corresponding value for the specific contact resistivity is de-

scribed by Equation 4.8. Pure thermionic emission can occur for low doping levels or

elevated temperatures (kBT >> E00). A∗ (Equation 4.9) is the standard Richard-

son’s constant where k is Boltzmann’s constant.


Table 4.4: Fermi level pinning from experimental literature values and dipolemodel.

Material ΦBn ΦBp = Eg − ΦBn ΦBp from dipole model (eV) [65]

Si 0.75 [66] 0.37 -0.04

Ge 0.6 [34] 0.06 -0.32

InP 0.5 [67] 0.84 +0.77

GaAs 0.95 [68] 0.47 +0.34

In0.53Ga0.47As 0.2 [67] 0.54 —

In0.8Ga0.2As 0.0 [68] 0.5 —

InAs -0.2 [68] 0.56 +0.47

In0.52Al0.48As 0.7 [67] 0.75 —

GaSb 0.55 [69] 0.18 +0.14

InSb 0.1 [70] 0.07 +0.28

JTE = A∗∗T 2 exp

(−qφBnkBT

)[exp

(qV

kBT

)− 1

](4.7)

ρc,te =kB

A∗∗Tqexp

(qφBnkBT

)(4.8)

A∗ =4πqm∗m0k

2B

h3(4.9)

Modifications can be made to the Richardson’s constant A∗∗ (Equation 4.10) to

take into account additional effects such as diffusion. The relative values of effec-

tive recombination velocity (νR) and effective diffusion velocity (νD) determine the

percentage contributions of thermionic emission and diffusion (νR >> νD is diffusion

limited). Additional terms correct for quantum mechnicanical reflection over the top

of the barrier where fQ takes into account the quantum mechanical reflection and fp

taking into account the probability of emission over the barrier [71].


A∗∗ =fpfQA

∗

1 +(fpfQ

νRνD

) (4.10)

The cases of thermionic field emission and field emission are far more complicated

and are highly dependent on both the barrier height at the interface and the doping

properties of the semiconductor. The forward current densities are described by

Equations 4.11 & 4.14 [71, 72] where E0 is described by Equation 4.13 and c1 is

described by Equation 4.16. The corresponding specific resistivity values are described

by Equations 4.12 & 4.15 [71].

JTFE =A∗∗T

√πE00q (φBn − φn − V )

k cosh(E00

kBT

) exp

[−qφnkBT

− q (φBn − φn)

E0

]exp

(qV

E0

)(4.11)

ρc,tfe =k√E00 cosh

(E00

kBT

)coth

(E00

kBT

)A∗∗Tq

√πq (φBn − φn)

exp

q (φBn − φn)

E00 coth(E00

kBT

) +qφnkBT

(4.12)

E0 = E00 coth

(E00

kBT

)(4.13)

JFE =A∗∗Tπ exp

[−q(φBn−V )

E00

]c1k sin (πc1kT )

[1− exp (−c1qV )] (4.14)

ρc,fe =kB sin (πc1kT )

A∗∗Tqπexp

(qφBnE00

)(4.15)

c1 =1

2E00

log

[4 (φBn − VF )

−φn

](4.16)


4.1.3 Transfer Matrix Method

The transfer matrix method (TMM) is a way to describe tunneling through a barrier

of arbitrary shape and materials for a specific value of energy (E ). The following

description of the TMM is primarily taken from Miller’s book [73] with some notation

changes.

The first step in the TMM is to break up the barrier shape into segments with

an average height for each segment as shown in Figure 4.2. The accuracy of TMM

is based heavily on the width of the segments and the width of the segments need

not be constant as long as the widths are known. The barrier shape together with

the starting and exiting materials describes the full TMM structure. The notation

scheme for layers and interfaces is shown in Figure 4.3 assuming an incident wave

from the left. A closeup of one layer (Figure 4.4) shows the pertinent variables for

the general notation.

X

Vx

Figure 4.2: Step-wise approximation of potential barrier

For each layer m there is a value for electron momentum perpendicular to the

barrier (km) as seen in Equation 4.17. This value is calculated from the energy of the

electron perpendicular to the barrier (Ex) as well as the average barrier height for this


“entering”material

“exiting”material

N layers

Layer 1 2 3 4 N N+1 N+2

Interface 1 2 3 4 N-1 N N+1

TransmittedIncident

Reflected

Position z1 z2 z3 z4 zN-1 zN zN+1

Figure 4.3: Notation scheme for TMM formalism [73]

Interface (m-1) Interface (m)

Layer (m) Layer (m+1)

Am

Bm

AL

BL

Am+1

Bm+1

dm

Figure 4.4: Detailed description of pertinent variables for one layer of structure [73]


Table 4.5: Metal “conduction” band in reference to metal Fermi level for some “sim-ple” metals [60,77]. The metal “conduction” band is not a true conduction band, buta tool to describe the momentum of the electrons in the metal of layer m (km).

Metal EFm − ECm (eV)

Cu 7.00

Ag 5.49

Au 5.53

Al 11.7

layer (Vm) and the effective tunneling mass for this layer (mfm). When the energy

of the electron is less than the barrier height, the value of km is imaginary, which is

usually allowed. However, the value of km in layer 1 (the “entering” material) must

be real. If it is not real, than that particular value of the energy is not allowed for

this scenario. The values of qVm for the “entering” and “exiting” materials are the

conduction band values (EC) of the material.

km =

√2mfm

~2(Ex − qVm) (4.17)

The important material parameter is the km value. To approximate the km

value in a metal, a free electron approximation with parabolic bands is sometimes

used [60, 74, 75] yielding a “conduction” band value for the metal (ECm). This is

appropriate for the alkali metals as well as some transitional metals. The metals that

this approximation is valid for are called “simple” metals. A few of the most com-

monly used “simple” metals in semiconductor manufacturing are listed in Table 4.5.

Other metals (such as Ta, Ti, and W) have much more complicated fermi surfaces

and need a more complex way to determine the appropriate km values as a function

of energy such as the augmented plane-wave method (APW) or the Green’s function

method (KKR) [74,76,77].

Once the value of km is defined in all layers, the tranfer matrix T can be determined

from multiplying the transmission through each layer and interface. Equation 4.19

describes the transmission at an interface between layer m and layer m+1. The


propagation matrix for layer m is determined by Equation 4.20. The total transfer

matrix is assembled by T = D1P2D2P3D3 · · ·PN+1DN+1. The transmission for single

energy Ex is then described by Equation 4.21.

∆m =km+1

km

mfm

mfm+1

(4.18)

Dm =

(1+∆m

21−∆m

21−∆m

21+∆m

2

)(4.19)

Pm =

(exp (−ikmdm) 0

0 exp (ikmdm)

)(4.20)

T = 1− |T21|2

|T11|2(4.21)

4.1.4 Metal-Semiconductor Specific Definitions

The depletion approximation with image charge lowering is used to determine the

band diagram input into the TMM method (Figure 4.5). Vx is the description for the

band diagram in the semiconductor and a flat value for ECm is used from Table 4.5

for the choice of metal.

Vx =−qND

εs

(Wx− x2

2

)− q

16πεsx(4.22)

W =

√2εsqND

(Vbi − V −

kBT

q

)(4.23)

Vbi = φb0 − φn (4.24)

φn = ECs − EFs (4.25)

The full equations simulated can be found in Appendix B where T is determined

from the TMM method described above.


ECNL

Figure 4.5: Sample Vx description of Schottky MS Junction

Figures 4.6 & 4.7 show the results of the TMM simulation of specific contact

resistivity for n-type and p-type IV and III-V semiconductors. Results are provided

for both directions of simulation (integration over metal and semiconductor energy

space). Group IV semiconductors are denoted with dash-dot, binary III-V with solid

line and ternary III-V with dashed line. Best results for n-type are obtained for InAs,

InSb and Ga0.2In0.8As. Best results for p-type are obtained for Ge and GaSb.

4.2 Metal Insulator Semiconductor (MIS) Junc-

tions

4.2.1 Analytical Model

In a metal insulator semiconductor junction where the insulator is very thin (<5nm)

tunneling can occur through the insulator. The assumptions in Table 4.6 are used to

obtain a model that can be represented by modifications to the thermionic emission


1016

1017

1018

1019

1020

1021

10−10

10−8

10−6

10−4

10−2

100

102

104

106

108

1010

DopingpDensitypofpSemiconductorpcm−3

Sp

ecif

icpC

on

tact

pRes

isti

vity

pΩcm

2

ρc

bypTMMpMethodpN−type

p

1e−8 Ω cm2

SipC0N6eVR

Ge

InSb

InP

InAs

GaSb

GaAs

Al0N48In0N52As

Ga0N47In0N53As

Ga0N20In0N80As

Figure 4.6: ρc of N-type semiconductors. Results are provided for both directionsof simulation (integration over metal and semiconductor energy space). Group IVsemiconductors are denoted with dash-dot, binary III-V with solid line and ternaryIII-V with dashed line. The 2011 ITRS critical value of 1e-8 Ω cm2 is represented bya dotted line (See Table 1.1). Best n-type results are obtained for InAs, InSb andGa0.2In0.8As.


1016

1017

1018

1019

1020

1021

10−10

10−8

10−6

10−4

10−2

100

102

104

106

108

DopingpDensitypofpSemiconductorpcm−3

Sp

ecif

icpC

on

tact

pRes

isti

vity

pΩcm

2

ρc

bypTMMpMethodpP−type

p

1e−8 Ω cm2

SipC0.6eVR

Ge

InSb

InP

InAs

GaSb

GaAs

Al0.48In0.52As

Ga0.47In0.53As

Ga0.20In0.80As

Figure 4.7: ρc of P-type semiconductors. Results are provided for both directionsof simulation (integration over metal and semiconductor energy space). Group IVsemiconductors are denoted with dash-dot, binary III-V with solid line and ternaryIII-V with dashed line. The 2011 ITRS critical value of 1e-8 Ω cm2 is represented bya dotted line (See Table 1.1). Best p-type results are obtained for Ge and GaSb.


SemiconductorMetal

EFm

ECm

ECs

EFs

φBn

Insulator

Figure 4.8: Description of Schottky MIS junction with surface states. A thin insulatoris inserted between the metal and semiconductor for the purpose of unpinning thebarrier height. Although not shown, there is likely a field in the insulator whichcan be in either direction depending on the interaction between the doping level inthe semiconductor, the surface states of the insulator-semiconductor interface andinduced charge on the metal surface. If the insulator completely unpins the metal-semiconductor interface giving rise to a barrier height that is determined by theSchottky-Mott relation, then there is no field across the insulator. [38]

Table 4.6: Assumptions of MIS analytical model [71]

1. Thermionic emission for m-s transmission

2. Tunneling through insulator layer

3. Effective mass in insulator approaches free electron mass

4. Tunneling independent of transmission energy


Equation 4.8 [57,58,71,78] shown in Equation 4.26.

JMIS = A∗T 2 exp(−αTd

√qφTd

)exp

(−qφbnkBT

)[exp

(qV

nkBT

)− 1

](4.26)

The transmission probability through the insulator can be approximated by the

tunneling probability for a rectangualr barrier (exp(−αTd

√qφTd

)) with barrier height

qφT and width d . The prefactor of αT of 1.01 eV −12 A−1 is generally omitted, but

requires d to be in A and the barrier height φT to be in V [57].

The ideality factor n is modfied by interface traps in equilibrium with the metal

(Ditm) and semiconductor (Dits) shown in Equation 4.27 where εs and εi are the

permittivity values of the semiconductor and insulator respectively and WD is the

depletion layer thickness.

n = 1 +

(d

εi

) εsWD

+ qDits

1 + dεiqDitm

(4.27)

If thickness of insulator is <3nm, the interface traps tend to be in equilibrium

with the metal [71] and Equation 4.27 reduced to Equation 4.28.

n = 1 +

(d

εi

) εsWD

1 + dεiqDitm

(4.28)

This model is difficult to use because the defect density is broken up into two

portions that are in phase with the semiconductor and the metal. This is hard to

determine where the separation should occur for thicknesses >3nm.

4.2.2 Numerical Model

Numerical simulation of a MIS junction can be accomplished through the TMM

assuming a band diagram and relavant material properties can be determined. Toward

this end the empirical relationship describing pinning factor [79] (Equation 4.29) and

theoretical descriptions of pinning factors [38] (Equations 4.30-4.35) are used.

A empirical model for the pinning factor S is shown in Equation 4.29 where ε∞ is


vacuum

S SΧ

M

ECNL

Figure 4.9: Definition of parameters for determination of pinning factor S

the optical dielectric constant [80].

S =1

1 + 0.1 (ε∞ − 1)2 (4.29)

Determination of the pinned barrier height from the pinning factor S can be

determined from a few different methods. Presented here are two of the models.

The first is a fixed-separation model of metal-induced-gap-states (MIGS) that tends

to pin the barrier near a charge-neutrality-level (CNL) in the semiconductor. This

model is denoted by a subscript of -CNL. The second model is a bond polarization

model suggested by Raymond Tung and is denoted by a subscript of -Tung. Both of

these models are presented fully in [38]. Figure 4.9 shows a band diagram of a MS

interface with the parameters used in both models.


The CNL theoretical model value for the pinning factor is shown in Equation 4.30

where δGS is the fixed separation between the metal and the gap states, DGS is the

density of gap states, and εint is the permittivity of the interface region between the

metal and semiconductor 1.

SCNL =

(1 +

q2δGSDGS

εint

)−1

(4.30)

The barrier heights without image charge lowering for both p-type and n-type

semiconductors are shown in Equations 4.31 & 4.32 respectively where Eg is the

bandgap of the semiconductor.

ΦB0,p−CNL = SCNL (ΦS − ΦM) + (Eg − ΦS + χS) (4.31)

ΦB0,n−CNL = SCNL (Φm − Φs) + (Φs − χs) (4.32)

The Tung theoretical model value for the pinning factor is shown in Equation 4.33

where δMS is the average interface bond length, NB is the density of interface bonds,

εint is the permittivity of the interface region between the metal and semiconductor,

Eg is the semiconductor bandgap and κ is the sum of all the hopping interactions

where κ << Eg is expected [38].

STung = 1− q2δMSNB

εint (Eg + κ)(4.33)

The barrier heights without image charge lowering for both p-type and n-type

semiconductors are shown in Equations 4.34 & 4.35 respectively.

ΦB0,p−Tung = STung

(χS +

Eg2− φM

)+Eg2

(4.34)

ΦB0,n−Tung = STung

(φM − χS −

Eg2

)+Eg2

(4.35)

1Often the permittivity of semiconductor εs is used for εint, but is left generalized in the model.[38]


There are some differences between which model CNL or the Tung model is used

to determine pinning factor. The MIGS model is easier to implement, but the Tung

model is based upon the bonds of the interface and is probably more accurate [38].

The following argument is for n-type contacts using the MIGS CNL model. The

interfaces are denoted by a subscript of ms for metal-semiconductor, mi for metal-

insulator, and is for insulator-semiconductor.

When looking at a very thin (or even zero thickness) insulator layer, the Schottky

behavior of the metal/semiconductor junction should dominate the pinning. This

means:

ΦB0,nms = Sms (ΦM − ΦS) + (ΦS − χS) (4.36)

In the case of a thick insulator, the pinning of the insulator to metal and insulator

to semiconductor should be independent and the following equations hold true. The

Si is chosen in Equation 4.38 because it is assumed that the empirical relationship for

S depends on the ε∞ of the larger bandgap material [79]. Alternatively, a compound

S factor can be used as in Equation 4.39 [81].

ΦB0,nmi = Smi (ΦM − ΦI) + (ΦI − χI) (4.37)

ΦB0,nis = Si (ΦI − ΦS) + (χI − ΦI)− (χS − ΦS) (4.38)

S12 =1

1 + 0.1[

(ε∞1−1)2(ε∞2−1)2

(ε∞1−1)2+(ε∞2−1)2

] (4.39)

In the case of an insulator of in-between thickness, the pinning of the semicon-

ductor should be dependent on both the theoretical ms interface and the is interface.

The pinned barrier height is estimated from the following relationship where the

associated barrier heights are determined from Equations 4.36-4.38.

ΦB0,nmseff=

ΦB0,nms − (ΦB0,nmi − ΦB0,nis)

2(4.40)


SemiconductorMetal

EFm

ECm

ECs

EFs

φBn-ms

Insulator

φBn-is

φBn-mi

Figure 4.10: MIS structure for thin nonzero oxides.2

However, the Sms is modified by the thickness of the oxide. This value can be

extracted as a function of oxide thickness.

The interface that has both an intimate interaction and remote interfaction has

a total of DGS states or NB bonds. Just because there are two interfaces, it does

not mean that the number of bonds/gap states doubles. Instead the number of

states/bonds is used as a weighted average for the determination of the effective

SBH. The value of DGS is modified by distance by the following relation:

2,3Although not shown, there is likely a field in the insulator which can be in either directiondepending on the interaction between the doping level in the semiconductor, the surface statesof the insulator-semiconductor interface and induced charge on the metal surface. If the insulatorcompletely unpins the metal-semiconductor interface giving rise to a barrier height that is determinedby the Schottky-Mott relation, then there is no field across the insulator. [38]


eff

MSeff

IS

MS

Figure 4.11: Effective MIS structure for thin nonzero oxides.3


1016

1018

1020

10−10

10−5

100

105

Doping/Density/of/Semiconductor/cm−3

Sp

ecif

ic/C

on

tact

/Res

isti

vity

/Ωcm

2ρ

cof/TaN/SiO2/n−Si/<100>/contacts

/

1e−8/Ω cm2

tox

=0nm

tox

=0.25nm

tox

=0.69nm

tox

=1.13nm

tox

=1.56nm

tox

=2.0nm

Figure 4.12: TMM simulation ρc results for TaN/SiO2/n-Si < 100 >. The 2011 ITRScritical value of 1e-8 Ω cm2 is represented by a dotted line (See Table 1.1). The bestMIS result is for the thinnest non-zero dielectric thickness of 0.25nm. However, athigher doping concentrations, the MS junction shows lower ρc.

DGS† = DGS exp

(− (x− δGS)

δGS

)(4.41)

where the minimum δGS is equal to the assumed value for the MIGS-CNL model

(2A-2.5A) and x is the insulator thickness. [38,82]

ΦB0,nmseff=DGS†

DGS

ΦB0,nms +

(1− DGS†

DGS

)ΦB0,nis (4.42)

As an example, Figure4.12 shows the TMM simulation ρc results for TaN/SiO2/n-

Si <100>. The best result is for the thinnest non-zero dielectric thickness of 0.25nm.

However, at higher doping concentrations, the MS junction shows lower ρc. This

cross-over point will be different for each material system.


0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

MIGS)CNL)Model

Tung)Bond)Polarization)Model

Distance)between)interfaces)4nm6

Pin

ning

)Fac

tor)

S

Figure 4.13: Pinning factor comparison of CNL and Tung models


2

eff

MSeff

M1S

M2S

1

Figure 4.14: Dual Silicide

Using the Tung model instead of the CNL model will yeild slightly different re-

sults. Figure 4.13 shows the difference bween the pinning factors for a TaN/SiO2/n-Si

<100> material system.

4.3 Modified Silicided Junction

The above numerical model can also be applied to other thin layer interfaces such as

a silicide junction.

In this case, a silicde junction is investigated that has two phases of silicide in

close proximity to the semiconductor interface. It is desirable for the bulk of the

junction to be a low resistance phase. However, the high reistance phase has a lower

barrier height to silicon, therefore it could prove beneficial to have an interlayer of

the high-resistance\low barrier height phase and then the rest of the junction be the

low-resistance higher barrier height phase.


There is a direct correlation between the above section for MIS junctions and a

dual layer silicide.

If the silicide is instead characterized by clustering at the interface so that there

are two different barrier heights for different grains, then the effective barrier height

will be a weighted average, but the pinning factor S (to determine the barrier height)

will not be dependent on distance. Low temperature measurements should be able to

determine if this is what is happening because the apparent SBH versus T plot will

show non-linear characteristics that can be fitted with a double gaussian profile. [83]

Chapter 5

Em ≈ slope model

Table 5.1: Symbols and variables used in Arrhenius plot derivation


(|) Without image charge lowering —

(⊕|) With image charge lowering —


ΦBn Schottky barrier height (⊕|) eV

WD Depletion width m

T Temperature K

Fs Fermi probability function for semiconductor unitless

Fm Fermi probability function for metal unitless

T Transmission function unitless

Eee Energy distribution of emitted electrons unitless




Standard thermionic emission theory predicts a barrier height independent of mea-

surement temperature [71]. However, temperature-dependence of the extracted SBH

59

CHAPTER 5. EM ≈ SLOPE MODEL 60

(or non-linearity of the Richardson’s plot) has been observed for over 55 years [84].

Since inhomogeneity of the interface has been proposed [85,86] as an explanation for

the non-linearity, it has been applied to a wide variety of diodes [84, 86–109] that

show non-linear Richardson’s plots. Many of these do not offer proof of inhomogene-

ity other than the inhomogeneous analysis itself. Presented here is an explanation

of non-linearity without invoking generation-recombination currents, defect-assisted

tunneling or even inhomogeneity. Non-linearity of the Richardson’s plot is a natural

consequence of thermionic-field emission that can be observed in even lightly doped

diodes.

5.1 Simulation details

The simulations presented were done using MATLAB with the transfer matrix method

(See Section 4.1.3). Because this method does not differentiate between Schottky

diode conduction mechanisms, it calculates thermionic emission (TE), thermionic

field emission (TFE) and field emission (FE) simultaneously. Extracted SBH mea-

surements were obtained by simulating the current density as a function of both

applied voltage and temperature and then using the same extraction procedures as

on experimental data. Simulation has the added advantage of being able to compare

the extracted values to the input values that were used to simulate the junction.

5.2 Current emission theory

Schottky junction current can be described by Equation 6.19 [71, 73] where Eee rep-

resents the energy distribution of the emitted electrons by all conduction methods

(Equation 5.2) and A∗ is the effective Richardson’s constant. In Equation 5.2, T is

the emission probability (i.e. transmission probability) of the Schottky barrier at a

particular energy and Fs & Fm are the quasi Fermi functions of the semiconductor

and metal, respectively. Em represents the expected value (mean) energy of the Eee

distribution (Figure 5.1a) [72].


Figure 5.1: (a) Definition of Em in reference to energy distribution of emitted electronsEee. (b) Approximate representations of Eee (arbitrary units) for the three differentconduction methods with negative bias (Thermionic emission (TE), Thermionic FieldEmission (TFE), and Field Emission (FE)).

J =A∗T

k

∫ ∞−∞

EeedE (5.1)

Eee = T (Fs − Fm) (5.2)

Derivations of thermionic emission current equations [71] are based on the as-

sumption that any contribution to Eee of energies below the SBH are negligible. As

a first order approximation, the dominant mechanism is assumed to be determined

by the relationship of the mean energy (Em) to the SBH (ΦBn) and Fermi level of

the metal (EFm). TE is described by Em ≥ ΦBn, FE is described by Em = EFm,

and TFE describes any condition between these two extremes. Figure 5.1b shows a

Schottky barrier under reverse bias with the three conduction regimes represented by

normalized Eee representations. EFm and EFs represent the Fermi levels in the metal

and semiconductor, respectively.


Fig. 5.2 shows simulated Em values for a NiSi/n-Si junction with no inhomogene-

ity, no generation-recombination current, no defect-assisted tunneling and no series

resistance, plotted over functions of both temperature and applied voltage. As the

applied voltage goes from positive to negative, the function of Em with temperature

asmptotically converges.

For the lightly doped case (1015cm−3), Fig. 5.2a shows the voltage converent Em

behavior is definitively above the SBH at room temperature and above, which cor-

responds to TE behavior. However, as the temperature decreases, this convergent

Em behavior crosses below the SBH at about 80K. This means that even though the

Schottky barrier is lightly doped it can have a significant TFE contribution given a low

enough operating temperature. This temperature of intersection (Tcrit) corresponds

to the temperature at which half the current contribution is from energies below the

SBH. For the higher doped case (1018cm−3), Fig. 5.2b shows that the voltage conver-

gent Em behavior is below the SBH at most temperatures and only exceeds the SBH

at about 800K. This junction will be dominated by TFE for all realistic measurement

temperatures.

This behavior should be expected from analysis of the figure of merit used by

Padovani & Stratton [72], which is defined as E00 in Equation 5.3.

E00 =q~2

√N

m∗εs(5.3)

According to [72] the relation of E00 to kBT will help determine whether the

junction is operating in the TE, TFE or FE range. If E00 << kBT then the junction

is TE-dominated, if E00 >> kBT then it is FE-dominated and anywhere in between

is TFE-dominated. This relation is dependent on the doping concentration as well as

temperature. If the temperature is low enough, E00 will appear larger and therefore

TFE will start to influence conduction.

The key point that seems to be neglected by the current literature is that generally

TFE-dominated interfaces are being forced into a TE mold. This affects the extracted

parameters (such as SBH and n) by use of the standard diode equations in current-

voltage (IV) analysis as well as the Richardson’s plot (current-temperature or IT


0 100 200 300 400 500 600 700 8000.5

0.75

1

1.25

1.5

1.75

2

Temperature (K)

Em

(eV

)

SBH V = +1V

V = -1V

(a)

0 100 200 300 400 500 600 700 8000

0.25

0.5

0.75

1

1.25

1.5

Temperature (K)

Em

(eV

)

SBH V = +1V

V = -1V

(b)

Figure 5.2: Simulated Em behavior as a function of applied voltage at and temperaturefor NiSi/n-Si for (a) Nd=1015cm−3 and (b) Nd=1018cm−3. Plot (a) at Nd=1015cm−3

shows distinctly TE behavior for most temperatures with Em only going below theSBH at about 50K. Plot (b) at Nd=1019cm−3 shows TFE is the dominant mechanismas the Em is below the SBH for most temperatures and only goes above it at about800K.


analysis). The SBH extracted from a Richardson’s plot is proposed to be Em rather

than the actual SBH. Em has already been found [72] to have the analytical form

described by Equations 5.4 & 5.5 where φBn is measured from the Fermi level of the

metal and V is the applied voltage for reverse and forward bias respectively.

Em−V =qφBn − qV sinh2

(E00

kBT

)cosh2

(E00

kBT

) (5.4)

Em+V=

qφBn

cosh2(E00

kBT

) (5.5)

The typical TE diode current equation [71] is described by Equation 5.6 where A

is the diode area, A∗ is the Richardson’s constant, V is the applied voltage, ΦBn is

the barrier height, and rs is the series resistance.

I = AA∗T 2 exp

(−qΦBn

kBT

)exp

(q (V − Irs)nkBT

)×(

1− exp

(−q (V − Irs)

kBT

)) (5.6)

Linearization of this curve is obtained by dividing both sides by a factor of(1− exp

(− q(V−Irs)

kBT

)). The Richardson’s plot is obtained by plotting the saturation

current (extrapolated to V=0) versus qkBT

(or sometimes 1000/T). When Equations

5.4 & 5.5 are combined with Equation 5.6, the resulting Islin (linearized current at

V=0) is the same for both forward and reverse bias (Equation 5.7).

Islin = AA∗T 2 exp

− qφBn

kBT cosh2(E00

kBT

) (5.7)

The cosh term as defined above in Equation 5.7 can be series expanded to allow

simpler regression analysis. This expansion is of odd orders of qkBT

only as shown in

Equation 5.8 and more accuracy can be obtained when using more terms.


q

kBT cosh2(E00

kBT

) = −(

q

kBT

)+

(E00

q

)2(q

kBT

)3

−2

3

(E00

q

)4(q

kBT

)5

+17

45

(E00

q

)6(q

kBT

)7

− . . .

(5.8)

Combining (5.7) and (5.8) yields an equation for the Richardson’s plot that can

be used with polynomial regression (5.9).

ln

(IslinT 2

)= ln (AA∗)−

(q

kBT

)φb

+

(E00

q

)2

φb

(q

kBT

)3

− 2

3

(E00

q

)4

φb

(q

kBT

)5

+17

45

(E00

q

)6

φb

(q

kBT

)7

− . . .

(5.9)

In contrast, conventional Richardson’s plot analysis [41] assumes a homogeneous

TE junction and a purely linear relationship of the Richardson’s plot. The slope

of this line is the negative of the barrier height and the intercept can be used to

determine the Richardson’s constant (Equation 5.10).

ln

(IslinT 2

)= ln (AA∗)− qΦB0

kBT(5.10)

A common inhomogeneous model [86] assumes a Gaussian distribution of barrier

heights centered around a mean value, with TE being the only conduction mechanism.

This modifies the standard equation making the SBH the mean SBH and by adding

a quadratic term dependent on the standard deviation of the distribution (Equation

5.11).

ln

(IslinT 2

)= ln (AA∗)−

(q

kBT

)φb +

(q2

2k2T 2

)δ2s (5.11)

Both the inhomogeneous (Equation 5.11) and proposed model (Equation 5.9)


Figure 5.3: Map of diode area with area representation of range of SBH in a homo-geneous model and inhomogeneous (Gaussian) model. The value of µ is the acceptedliterature extracted SBH for the material interface. Many Gaussian extracted valuesyield σ values of about 10% of µ, therefore creating a range of ∼ µ ± 20%µ for thelegend. For a typical SBH of 0.5eV this represents a significant variation (0.4eV to0.6eV).

modifications to the standard Richardson’s plot make the slope (or apparent SBH)

of the Richardson’s plot temperature dependent. As can be seen from Figure 5.3, the

Gaussian inhomogeneous model relies on fluctuations in barrier height both above

and below the nominal value. Therefore, the interface can be perceived to contain

a wide variety of barrier heights. Many Gaussian extracted values yield σ values of

about 10% of µ, therefore creating a range of ∼ µ ± 20%µ for the legend. For a

typical SBH of 0.5eV this represents a significant variation (0.4eV to 0.6eV). The

proposed model results in a non-linear Richardson’s plot without invoking the need

for an inhomogeneous interface.

While inhomogeneity likely exists, it should not be applied without bias to the

large portion of published data with temperature non-linearities. Furthermore, the

magnitude of sigma (Figure 5.3) required to explain the data is often unrealistically

high. Certainly if equivalent behavior is observed in simulation in which no inhomo-

geneity has been introduced, then there is an alternative explanation.


50 100 150 200−160

−140

−120

−100

−80

−60

−40

−20

q/(k*Tempterature) (eV−1)

ln(J

s T−

2 )

ExtractedLinear FitQuadratic FitOdd Order 3Odd Order 5Odd Order 7

Figure 5.4: Simulation of Richardson’s plot for NiSi/n-Si with Nd=1016, 1017, and1018cm−3. The labels Odd3, Odd5 and Odd7 represent the highest order term of theodd order polynomial used in the extraction (Equation 5.9). This simulation includesno generation-recombination current, defect-assisted tunneling, series resistance orinhomogeneity and yet still displays a non-linear Richardson’s plot.

5.3 Simulation Data

An example Schottky barrier junction was simulated with a NiSi/n-Si junction with

dopings of 1016, 1017 and 1018cm−3 in a temperature range of 50K-350K. At the lowest

doping in Figure 5.4, the Richardson’s plot appears linear. The doping of 1017cm−3

yields a theoretical value for E00/kBT of 0.38 which is expected to present some non-

linearity as seen in Figure 5.4. As the doping is increased further, it can be seen in

Figure 5.4 that the resulting Richardson’s plot becomes increasingly non-linear, which

is expected from a theoretical value for E00/kBT of 1.2. All of the non-linear models

seem to fit the data very well at first glance since they appear to perfectly overlay

each other. However, different values for the barrier height of varying accuracy are

extracted for each model.


Table 5.2: Extracted SBH and error % from Richardson’splot using different extraction methods with Nd=1016,1017, and 1018cm−3. The labels Odd3, Odd5 and Odd7represent the highest order term of the odd order polyno-mial used in the extraction (Equation 5.9).

Method (Eq#) SBH (eV)

1016 cm−3 1017 cm−3 1018 cm−3

Linear (5.10) 0.732 0.612 0.330

% error 2.0% 16.0% 52.5%

Gaussian (5.11) 0.761 0.750 0.433

% error 1.9% 3.0% 37.7%

TFE Odd3 (5.9) 0.748 0.685 0.381

% error 0.1% 5.9% 45.2%

TFE Odd5 (5.9) 0.749 0.724 0.456

% error 0.3% 0.5% 34.4%

TFE Odd7 (5.9) 0.751 0.728 0.557

% error 0.5% <0.1% 19.9%

Actual 0.747 0.728 0.695

±0.002 ±0.002 ±0.003


Table 5.2 shows the extracted SBH from Richardson’s plot using different extrac-

tion methods. The linear method is the standard method of extracting SBH (Equation

5.10). The Gaussian method is a popular method for judging inhomogeneous inter-

faces (Equation 5.11). The TFE method is the proposed model based upon the TFE

contribution (Equation 5.9). The labels Odd3, Odd5 and Odd7 represent the highest

order term of the odd order polynomial used in the extraction (Equation 5.9). The

method listed as actual is the actual image charge-lowered SBH from the simulated

voltage profile. The variation comes from the small temperature dependence of this

value.

The linear model underestimates the SBH by 16% at a doping of 1017cm−3 and

53% at a doping of 1018cm−3. It will further degrade as the doping (and therefore

non-linearity) is further increased. The Gaussian and TFE Odd3 model overestimate

by 3% and underestimate by 6% respectively at a doping of 1017cm−3. However,

since no inhomogeneity has been introduced into this simulation, the Gaussian model

has no physical basis in this case. Increasing the number of terms used in the TFE

model by 1 (TFE Odd5) reduces the error to <1%. The addition of a further term

decreases the error to within the small temperature dependence of the SBH itself.

At a doping level of 1018cm−3 the TFE Odd7 model is the best performing model

but still underestimates the SBH by 20%. This shows that the TFE model extends

the range of dopings over which it is possible to extract accurate values of the SBH,

although improvement is still needed at the range of dopings of most interest for

contact research (∼1020cm−3). In contrast to simulation, it is difficult to prove that

experimental data contains no inhomogeneity. However, it is possible to determine if

TFE is a significant cause of Richardson’s plot non-linearity.

5.4 Experimental Data

A survey of inhomogeneous literature is summarized in Table 5.3. Tcrit is defined

as the temperature at which half the current is contributed from energies below

the SBH. At or below this temperature TFE is expected. Figure 5.5 displays the

simulated Tcrit values (solid lines and shapes) showing lines between TE and TFE


Table 5.3: Summary of non-linear Richardson’s plot literature. A vast majority ofthis data can be explained by TFE (see Figure 5.5). E00

kTlowis the theoretical value

with Tlow as the lowest temperature measured.

Interface Composition Nd (cm−3) Tlow (K) E00

kBTlow

PtSi/p-Si [87]a 3.7×1015 85 0.06

Ti/p-Si [88] 1016 208 0.04

Al/SiO2/p-Si [89] 1.8×1015 200 0.02

Al/Si3N4/p-Si [90] 2.2×1016 80 0.16

Au,Ni,Cr/n-Si [91]a 3.6×1016 173 0.07

Ni/n-Si [110] 3×1015 86 0.04

Cu/n-Si [92] 4.9×1014 95 0.01

Ti/SiO2/n-Si [93] 4.9×1014 85 0.02

PtSi/n-Si [86] 2×1015 77 0.03

AuGe/n-InP [95] 1.2×1016 50 0.48

Au/n-InP [96]b 2.6×1015 70 0.16

Al/n-InP [97] 4.8×1015 77 0.19

NiAu/n-InP [98] 4.9×1015 210 0.08

PdPt/n-InP [99] 4.9×1015 230 0.07

Co/p-InP [100]b 6×1017 80 0.74

Ni/n-GaAs [101,111] 7.3×1015 60 0.34

Al/n-GaAs [102] 2×1016 77 0.31

Au/SiO2/n-GaAs [103] 2.5×1018 80 4.72

Au/n-GaAs [104] 2.5×1017 80 1.77

Au/n-GaAs [105] 2.5×1015 77 0.16

Au/n-GaAs [105] 1017 77 0.98

Au/n-GaAs [105] 1018 77 3.10

Au/n-GaAs [84] 1.3×1015 4.2 1.80

Au/n-GaAs [106] 1.3×1017 80 1.08

Ni/n-GaAs [107] 1018 90.5 2.64

Cu/n-GaAs [108] 2.5×1017 80 1.49

a Range of doping given in literature, used highest value for analysis b Dual Gaussian Distribution


1015

1016

1017

1018

0

100

200

300

400

500

600

N−type/Doping/cm−3

Cri

tica

l/Tem

per

atu

re/T

crit

dK)

TaN/n−Si

NiSi/n−Si

TaN/n−Inp

Au/n−GaAs

300K

n−InP/Lit/data

n−GaAs/Lit/data

n−Si/Lit/data

da)

1015

1016

1017

1018

1019

0

100

200

300

400

500

P−typeuDopingucm−3

Cri

tica

luTem

per

atu

reuT

crit

dK)

NiSi/p−Si

Al/p−Si

Au/p−InP

300K

p−SiuLitudata

p−InPuLitudata

db)

Figure 5.5: Simulated Tcrit values (solid lines and shapes) showing line between TEand TFE regimes for different metal/semi pairs for both n-type (a) and p-type (b).The open shapes are associated experimental data from literature of the matchingsimulated curve. TFE should be addressed in the majority of inhomogeneous litera-ture due to the vast number points on the TFE sides of the curves. The non-linearityof the literature points on the TE sides of the curves are not explained by TFE.


Figure 5.6: Analysis of data from [102] includes a Richardson’s plot (a) and thederivative of the Richardson’s plot (b). The best fit lines for Linear, Gaussian andTFE order 3 analysis are included. The label “der Rich” is the numerical derivativeof the Richardson’s plot, “Gaussian” is the common inhomogeneous model (Equation5.11) and “Odd Order 3” is the 3rd order odd polynomial used in the extraction(Equation 5.9). The best fit model of the Richardson’s plot seems inconclusive in (a)but from (b) it can be seen that the TFE result best explains the non-linearity of thisdata set.


regimes for different metal/semi pairs. The open shapes are associated experimental

data from the literature of matching semiconductor simulated curves. Two different

Si curves have been shown to give an idea of the difference in choice of metal. This

analysis defines the regime by placement of Em with reference to the SBH and not

the magnitude of the Eee curve. Therefore, it should also be a good approximation

to estimate the behavior of MIS junctions, although “thermionic emission” in this

case is still tunneling through the oxide. Figure 5.5 shows TFE should be addressed

in the majority of literature on inhomogeneity. The non-linearity observed in the

literature references on the TE side of the curves (upper left) ( [87–89,92,93]) are not

explained by TFE and need another explanation (such as inhomogeneity) to explain

the non-linear Richardson plots reported. Also, the farther away from the simulated

curves, the greater the non-linearity expected. Therefore if the points are very close

to the curve (like the points from [91, 98, 99]) and show severe non-linearity, there

may be an additional contributor to non-linearity other than just TFE.

The data set found in [102] was chosen for detailed analysis due to the compre-

hensive temperature measurements. When looking at the Richardson’s plot found in

Figure 5.6a non-linearity is clearly present but it is uncertain whether the Gaussian

or TFE model fits the data best. However, when the derivative of the Richardson’s

plot is analyzed (Figure 5.6b), the TFE result fits the temperature-dependence of the

data best.

TFE is likely present in the vast majority of surveyed inhomogeneous literature

and is a source of non-linearity in Richardson plots. In the very least this means that a

lot of literature data explained as inhomogeneous needs to be corrected first for a sig-

nificant TFE contribution before an accurate description of inhomogeneity is obtained

(or only use measurement temperatures above Tcrit). At most it fully explains the

non-linearity of the Richardson’s plots without invoking the need for inhomogeneity,

generation-recombination, series resistance or defect-assisted tunneling.

Chapter 6

Derivation of Analytical

Richardson’s Plot

The purpose of regression is to analyze experimental data to extract parameters that

definitivly describe the behavior of a system, thereby enabling prediction of future

behavior. If these parameters are tied to a physical based model they can also be

used to deepen understanding of the underlying physical behavior of the system.

The following is the derivation of a model that usees experimental diode IV curves

to regress Schottky barrier height (ΦB0), steepness factor (E00), and fermi level (ξ),

enabling the determination of band diagrams of the measured interfaces.

6.1 Model Setup

The shape of the Schottky barrier is assumed to follow the depletion approximation

in that it is a parabolic barrier without image charge lowering. It can be derived that

such a barrier follows the form of:

Φ(x) =

(αE00

2x+

√Ψs

)2

(6.1)

where the surface energy Ψs = ΦB0 + qV + ξ, α = 2√

2mDOS~ , ξ = EFs − ECs, and

steepness factor:

74

CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 75

Table 6.1: Symbols and variables used in Arrhenius plot derivation


(|) Without image charge lowering -

(⊕|) With image charge lowering -


ΦBn Schottky barrier height (⊕|) eV

E00 Steepness factor eV

WD Depletion width m

T Temperature K

ξ Fermi level from conduction band (EFs − ECs) eV

δ Fermi level and applied voltage (ξ − qV ) eV

Ψs Surface potential energy (ΦB0 + ξ − qV or ΦB0 + δ) eV

Φ Energy band diagram eV

T Transmission function unitless

P Population based function unitless

Eee Energy distribution of emitted electrons (T(Ex)P(Ex)) unitless




E00 =q~2

√N

mDOSεs=

2q

α

√N

2εs(6.2)

The depletion width in this case is WD = 2√

ΨsαE00

.

The current through a Schottky junction is described [112] by Equation 6.31 where

T(Ex) is the transmission probability of the electron through the Schottky barrier at

energy Ex and A∗ is the Richardson’s constant.

1There is a difference in definition of zero energy point from derivation in appendix to make themath easier for the analytical case. In appendix, EFm = 0 wheras here ECs = Exsmin = 0.


qV

ΦB0

EFm

WD

EFsECs

ξ

Figure 6.1: Setup of band diagram.

Jx =A∗T

k

∫ ∞Ex=0

T (Ex) log

1 + exp(ξ−ExkBT

)1 + exp

(ξ−qV−ExkBT

) dEx (6.3)

This equation can be rewritten as Equation 6.4 where the logrithmic term is a

population based function P (Ex). The multiplication of T (Ex) and P (Ex) yields the

energy distribution of emitted electrons Eee.

Jx =A∗T 2

kBT

∫ ∞Ex=0

T (Ex)P (Ex) dEx =A∗T 2

kBT

∫ ∞Ex=0

Eee dEx (6.4)

The energy at which the peak of Eee occurs(

d(Eee)dEx

= 0)

is defined as Em (Fig-

ure 6.2a). The placement of Em to the band diagram defines the conduction regime

through the Schottky barrier (Figure 5.1b). If Em is at the top of the Schottky barrier

than the conduction is thermionic emission (TE), if Em is at the fermi level EFm than

it is field emission (FE) and anywhere in between is generally considered thermionic

field emission (TFE).

6.1.1 Transmission Probability T

The value of T can be determined in a number of different ways. The transmission

matrix method (TMM) [73] is a highly accurate and versatile method for determining

the transmission through the Schottky barrier, however, its computational complex-

ity limits eliminates its viability for regression. There are two major methods for

determining transmission probability analytically.


Figure 6.2: (a) Definition of Em in reference to energy distribution of emitted electronsEee. (b) Approximate representations of Eee (arbitrary units) for the three differentconduction methods with negative bias (Thermionic emission (TE), Thermionic FieldEmission (TFE), and Field Emission (FE)).

X

qVx E

x

X1 X

2

Figure 6.3: Potential barrier with classical turning points

WKB Method

The Wentzel-Kramers-Brillouin (WKB) Method is a way of determining the trans-

mission probability of of electrons or holes through a “slowly” varying potential. The

value of TWKB is shown in Equation 6.5 [73] where x1 and x2 are the classical turning

points (where qV (x) = Ex) of the barrier as shown in Figure 6.3.


TWKB ≈ exp

(−2

∫ x2

x1

√2mf

~2(Φ(x)− Ex) dx

)

≈ exp

(−α∫ x2

x1

√(Φ(x)− Ex) dx

)≈ exp (−B(Ex)) (6.5)

Since this approach yields value of TWKB = 1 for energies above the maximum

value of qVx, it doesn’t include quantum mechanical reflection in these cases. This

means that it overestimates current conduction at low doping and cannot handle the

case of a zero or negative barrier at the interface. For the case of a high barrier and

moderate to high doping (where thermionic-field emission and field emission are the

dominant mechanisms), it is quite accurate. It is recommended [72] to limit use for

when TWKB <1e.

WKB-type Approximation

An alternate method of derivaiton of the transmission yeilds an expression with iden-

tical exponent but slightly different form [113] shown in Equation 6.7. This alternate

expression has far more accuracy near and above the top of the barrier.

TWKB−TY PE ≈1

1 + exp(α∫ x2x1

√(Φ(x)− Ex) dx

) (6.6)

=1

1 + exp (B(Ex))(6.7)

In general this expression isn’t often used because when multiplied by another function

it is often unintegrible. However, with the series approximations shown in Section 6.2,

it becomes a very useful method.


qV

ΦB0

EFm

WD

EFsECs

ξ

Em

δ

ψs

Figure 6.4: Setup of band diagram with Eee

Taylor Series Expansion of Transmission

Figure 6.4 shows the full form of the band diagram for a reverse biased Schottky

junction. The energies of most interest are around the value of Em (peak of the Eee

profile). A Taylor series expansion2 is used for the shape of the Φ profile around the

value of Em yeilding Equation 6.8. Previously these Taylor series coefficients (bm, cm

and fm) have been derived [72] for the TFE and FE cases separately. The following

equations (Equations 6.8-6.12) are a variation3 on the derivation that provide values

for bm, cm and fm valid in all (TE,TFE and FE) conduction regimes.

B(Ex) =α

∫ x2

x1

√(Φ(x)− Ex) dx

=bm − cm (Ex − Em) + fm (Ex − Em)2

=bm + cm (Em − Ex) + fm (Em − Ex)2 (6.8)

2Definitions of bm, cm and fm are chosen so that all three quantities are positive.3A typo in [72] gives a form of cosh2 (E00cm) for fm instead of the actual

sinh (E00cm) cosh (E00cm) value. The difference is not very significant if application is limited to theTFE regime, but becomes highly significant when extending into the TE regime.


bm = B(Ex)|Ex=Em= α

∫ x2

x1

√(Φ(x)− Em) dx

=1

E00

[Φ (x2) log

( √Φ (x2)√

Φ (x1)− Φ (x2) +√

Φ (x1)

)+√

Φ (x1)√

Φ (x1)− Φ (x2)

]=

Ψs

E00

tanh (cmE00)− Emcm (6.9)

cm =− 1

1!

d (B(Ex))

dEx

∣∣∣∣Ex=Em

=α

2

∫ x2

x1

1√(Φ(x)− Em)

dx

=1

E00

log

(√Φ (x1)− Φ (x2) +

√Φ (x1)√

Φ (x2)

)

=1

E00

log

(√Ψs − Em +

√Ψs√

Em

)(6.10)

Em =Ψs

cosh2 (E00cm)(6.11)

fm =1

2!

d2 (B(Ex))

dE2x

∣∣∣∣Ex=Em

=α

4

[1

x2 − x1

1

Φ′(x1)− 1

Φ′(x2)

∫ x2

x1

dx√(Φ(x)− Em)

−1

2

∫ x2

x1

dx

(Φ(x)− Em)32

1− Φ

′(x)

x2 − x1

(x− x1

Φ′(x2)+x2 − xΦ′(x1)

)]

=1

4E00

√Φ (x1)− Φ (x2)

Φ (x2)√

Φ (x1)

=1

4E00

sinh (E00cm) cosh (E00cm)

Ψs

(6.12)

Equations 6.13 & 6.14 were used along with Φ (x1) = Ψs, Φ (x2) = Em, and

x2 = −2√em+

√Ψs

αE00to simpify bm, cm and fm.


Φ′(x) = αE00

(αE00

2x+

√Ψs

)= αE00

√Φ(x) (6.13)

Φ(x1)− Φ(x2) = Ψs − Em = Ψs −Ψs

cosh2 (E00cm)= Ψs tanh2 (E00cm) (6.14)

In addition, the definition of Eee is also used, where Equation 6.15 is used withdEeedEx

∣∣∣Ex=Em

= 0 to obtain Equation 6.16.

Eee = T (Ex)P (Ex) =log(

1+exp(ξ/kBT )1+exp(δ/kBT )

)(1 + exp(B(E(x))))

(6.15)

cm =

(1 + exp (bm)

kBT exp (bm)

) 1

1+exp(δ−EmkBT

) − 1

1+exp(ξ2−EmkBT

)log

(1+exp

(ξ2−EmkBT

)1+exp

(δ−EmkBT

)) ≈ 1

kBT(6.16)

Previously published results [72] make the approximation of cm = 1kBT

. This limits

the applicability of the bm, cm and fm values to only the TFE regime. Using the full

form in Equation 6.16, along with Equations 6.9 & 6.11 give three equations and

three unknowns which is a solvable linear system of equations that is valid over TFE,

TFE & FE regimes. Since the range of these Equations have been expanded to cover

all regimes, the inclusion of the ξ factor in Ψs becomes mandatory since it is highly

significant at high doping levels.

6.2 Approximations for Fermi-Dirac Statistics Re-

lated Functions

The following analysis presented here proposes analytical approximations for 1exp (x)+1

and log (exp (x) + 1) that could be then substituted into Equation 6.15 to obtain an

analytical expression for current density.

The typical model for approximating 1exp (x)+1

for large x is exp (−x) (equivalent


0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

x

1ex + 1

e− x

e− x − e− 2x

e− x − 12 e− π

2 x

Figure 6.5: Full form of 1exp (x)+1

shown in solid line. Typical single term approxima-

tion of e−x shown as dashed line. First two terms of∑∞

n=1(−1)n+1 exp (−nx) shownas dash-dot line. Proposed two-term model shown as circles provides visual match tofull form.

to Boltzmann statistics). More accurate representations often follow from using more

terms in the 1exp (x)+1

=∑∞

n=1(−1)n+1 exp (−nx) expansion [114]. However, an accu-

rate two term expansion is desirable to limit equation complexity. While trying to

match limits at zero and infinity for a two term expansion, Equation 6.17 for x ≥ 0

was observed to provide a visual match (Fig. 6.5). Further investigations yielded an

expression with similar matching for x ≤ 0. Integration of these expressions yielded

the expressions in Equation 6.18.

1

(exp (x) + 1)=

exp (−x)− 1

2exp

(−π

2x)

x ≥ 0

1− exp (x) + 12

exp(π2x)

x ≤ 0(6.17)

log (exp (x) + 1) =

x+ exp (−x)− 1

πexp

(−π

2x)

x ≥ 0

exp (x)− 1π

exp(π2x)

x ≤ 0(6.18)

Fig 6.6 shows the approximations in Equation 6.17 versus the full form of 1exp (x)+1

,

whereas Fig 6.7 shows the approximations in Equation 6.18 versus the full form of

log (exp (x) + 1). Short expressions that match the curve shape and expression limits

have been found. Analysis of the relative error of these expressions in Equations 6.17

& 6.18 shows that the results that are within 5% for the full range of x.


−5 0 50

0.2

0.4

0.6

0.8

1

x

1/(1

+ex )

−5 0 5−1

012345

x

erro

r %

Figure 6.6: Full form of 1exp (x)+1

shown in solid line. Approximation from Equa-

tion 6.17 shown as circles. Relative error is less than 5% (inset).

−5 0 50

1

2

3

4

5

6

x

log

(1+

ex )

−5 0 5012345

x

erro

r %

Figure 6.7: Full form of log (exp (x) + 1) shown in solid line. Approximation fromEquation 6.18 shown as circles. Relative error is less than 5% (inset).


6.3 Analytical Current Density J

The current density is shown in Equation 6.19 where Eee is defined in Equation 6.15.

J =A∗T

k

∫ ∞−∞

Eee dEx =A∗T

k

∫ ∞−∞

P (Ex)T (Ex) dEx

=A∗T

k

∫ ∞−∞

log(


)(1 + exp(bm + cm (Em − Ex) + fm (Em − Ex)2)

) dEx (6.19)

If applied reverse voltage is at least several kBT away from zero than the function

P ≈ − log(

1 + exp δ−ExkBT

). Left in this form, this equation is non-integrible [115].

However, using the approximations in Equations 6.17 & 6.18 the determination of J

is broken into three parts where Eα = δ and Eβ = Em+cm−√c2m−4bmfm

2fm. Each of these

parts is integrible.

J =A∗T

k

∫ Eα

0

Eee dEx +

∫ Eβ

Eα

Eee dEx +

∫ E∞

Eβ

Eee dEx

J =A∗T

k

∫ Eα

0

P (Ex ≤ Eα)T (Ex ≤ Eβ) dEx +

∫ Eβ

Eα

P (Ex ≥ Eα)T (Ex ≤ Eβ) dEx

+

∫ ∞Eβ

P (Ex ≥ Eα)T (Ex ≥ Eβ) dEx

(6.20)

J =A∗T

k

∫

Eeep1

+

∫Eeep2

+

∫Eeep3

(6.21)


∫Eeep1

= −∫ Eα

0

[δ − ExkBT

+ exp

(−δ − Ex

kBT

)− 1

πexp

(−π

2

(δ − ExkBT

))]

×[exp

(−bm − cm(Em − Ex)− fm(Em − Ex)2

)− 1

2exp

(π2


))]dEx (6.22)

∫Eeep2

= −∫ Eβ

Eα

[exp

(δ − ExkBT

)− 1

πexp

(π

2

(δ − ExkBT

))]

×[exp


)− 1

2exp

(π2


))]dEx (6.23)

∫Eeep3

= −∫ ∞E′β

[exp

(δ − ExkBT

)− 1

πexp

(π

2

(δ − ExkBT

))]

×[1− exp

(bm + cm(Em − Ex) + fm(Em − Ex)2

)+

1

2exp

(π2


))]dEx (6.24)

The integral of∫

Eeep3

is actual done between E′

β and ∞ where E′

β = Em + bmcm

which is the inflection point with the two term taylor series expansion of the T term.

This is because the two term expansion of T behaves much better at ∞ [113] and a

4 terminal expansion is unintegrible. The effect of this difference is minimal because

the values of E′

β and Eβ are almost identical in the TE regime, where∫

Eeep3

is most

significant.


γ1 =1

kBT

2√fm

(6.25)

γ2 =cm

2√fm

(6.26)

γ3 =√fm (Em − δ) (6.27)

γ4 =γ2 −∣∣∣∣√γ2

2 − bm∣∣∣∣ (6.28)

γ5 =δ√fm (6.29)

γ6 =δ − Em − bm

cm

kBT(6.30)

γ7 =− bm − cm (Em − δ)− fm (Em − δ)2 (6.31)

γ1π =1

kBT

2√

π2fm

(6.32)

γ2π =π2cm

2√

π2fm

(6.33)

γ3π =

√π

2fm (Em − δ) (6.34)

γ4π =γ2π −∣∣∣∣√γ2

2π −π

2bm

∣∣∣∣ (6.35)

γ5π =δ

√π

2fm (6.36)

pA = (− erfmin(γ1 − γ2 − γ3, γ3 + γ4) +1

πerfmin(

π

2γ1 − γ2 − γ3, γ3 + γ4)

− erfmin(γ1 + γ2 + γ3, γ5) +1

πerfmin(

π

2γ1 + γ2 + γ3, γ5)

+cm + 2fm(Em − δ)

2fmkBTerfmin(γ2 + γ3, γ5)− exp (−cmδ − 2δfmEm + fmδ

2)− 1√πfmkBT

)

(6.37)


pB = (− erfmin (γ1π − γ2π − γ3π, γ3π + γ4π)− erfmin (γ1π + γ2π + γ3π, γ5π)

+1

πerfmin

(π2γ1π − γ2π − γ3π, γ3π + γ4π

)+

1

πerfmin

(π2γ1π + γ2π + γ3π, γ5π

)+cm + 2fm(Em − δ)

2fmkBTerfmin (γ2π + γ3π, γ5π)

−exp

(−π

2cmδ − δπfmEm+ π

2fmδ

2)− 1

π√

12fmkBT

(6.38)

pC = exp (γ6)

(π

πcmkBT + 2+ π −

1π

+ π

cmkBT + 1

)+ exp

(π2γ6

)( 2

2cmkBT + π− 2

π

)(6.39)

J = −A∗T 2

√

πfm

2kBTexp(γ7)pA−

√2fm

4kBTexp

(π2γ7

)pB +

pC

π

(6.40)

Equation 6.40 shows the full current density equation for reverse biased current with

at least a few kBT from zero. Numerical instability during evaluation is a significant

concern, since there are many instances of substracting two very large numbers to

get a very small number4. The following transformations to make use of MATLAB’s

scaled complementary error function (erfcx) are very useful.

erfcx (x) = x2 erfc(x) = x2 (1− erf(x)) (6.41)

exp(x2)

[erf (x+ a)− erf (x)] = erfcx (x)− exp(−2xa− a2

)erfcx (x+ a) (6.42)

erfmin(x, a) = exp(x2)

(erf(x)− erf(x+ a)) (6.43)

4MATLAB functions of log1p, exp1m and erfcx are very useful in minimizing error.


TE

TFEFE

V V

TE

TFE

FE

| |

(a) (b)

Figure 6.8: (a) shows ln (|J |) and (b) shows ln (Jlin) for TE, TFE and FE cases. TheTE case shows linearity, but the TFE and FE case show non-linearity.

6.4 Arrhenius (Richardson’s) plot and ideality n

plot

Conventional extraction of Schottky barrier height uses an Arrhenius (Richardson’s)

plot. This plot is derived from thermionic emission (TE) equations. According to

thermionic theory [41] Equation 6.44 describes the current density through a Schottky

junction for both forward and reverse biases.

J = A∗T 2 exp

(−qφB0

kBT

)exp

(qV

nkBT

)(1− exp

(−qVkBT

))(6.44)

Figure 6.8 shows the linearized current density (defined as Jlin = J(1−exp

(−qVkBT

)))

and gets it’s name from linearizing the current density in the negative voltage regime.

ln

(JlinT 2

)= ln (A∗)−

(q(φB0 − V

n

)kBT

)(6.45)

The saturation linearized current density is obtained by fitting Jlin linearly and

obtaining the intercept values. However, there is a distinction between extrapolated

and substitutional saturation current density as shown in Figure 6.9.

The substitutional linearized current density is described by:

ln

(JslinT 2

)= ln

(JlinT 2

)∣∣∣∣V=0

= ln (A∗)− qφB0

kBT(6.46)


Extrapolated

Substitutional

V

Figure 6.9: When ln (Jlin) shows non-linearity the substitutional and extrapolatedsaturation linearized current density give different results.

The extrapolated linearized current density is described by:

ln

(JslinT 2

)= ln

(JlinT 2

)− V

d(ln(JlinT 2

))dV

(6.47)

= ln (A∗)−

(q(φB0 − V

n

)kBT

)− V

q 1n

kBT= ln (A∗)− qφB0

kBT(6.48)

In the case of thermionic emission these two values are the same, but as the

doping increases the linearized current density becomes less linear. This will show a

bias dependence to the extrapolated saturation current density.

From the above analysis of the extrapolated saturation current density, the ideality

factor n can be defined by:

n ≡d(ln(JlinT 2

))dV

kBT

q(6.49)

However, the ideality factor n will also have a voltage dependence when the lin-

earized current density becomes non-linear.

These saturation current density points are then plotted as a function of inverse


26 28 30 32 34 36 38 40−25

−20

−15

−10

−5

0

5

log

(Jsl

in/T

2 )

26 28 30 32 34 36 38 400.95

1

1.05

1.1

1.15

Idea

lityq

fact

orq

n

q/kTq(V-1)

q/kTq(V-1)(a)

(b)

TE

TFE

FE

TE

TFE

FE

Figure 6.10: (a) Arrhenius (Richardson’s) plot showing typical curves for TE, TFEand FE dominated conduction for above room temperature measurements. (b) Ide-ality factor for same conditions.

temperature (1000/T or q/kBT) to form the Arrhenius (or Richardson’s) plot. In the

case of TE, the negative slope of this graph is the Schottky barrier height φB0 and

the intercept is ln (A∗), allowing the determination of the Richardson’s constant. In

the case of TFE or FE, the values extracted in this manner are not accurate and may

not even be linear.

Due to assumptions in the derivation5, Equation 6.40 is only valid a few kBT away

from zero bias. Therefore, the extrapolated saturation current density method must

be used. Unlike the TE case, the voltage dependence does not perfectly cancel, how-

ever, this bias dependence can be exploited by using several voltage points for the ex-

trapolation, increasing the number of data points for regression. Equations 6.47&6.49

are used with Equation 6.40 to obtain equations for regression objective functions.

5neglecting log(1 + exp(ξ/kBT )) term of P


0 50 100 150 200 2501

1.05

1.1

1.15

1.2

Idea

lity

fact

or

n

0 50 100 150 200 250−200

−150

−100

−50

0

50

log

(Jsl

in/T

2 )

(a)

(b)q/kT (V-1)

Figure 6.11: (a) Arrhenius (Richardson’s) plot for analytical approximation derivedfrom Equation 6.40 versus numerical (|) and Equation 6.19. (b) Ideality factor forsame conditions.

The result of this is calculation is complex but an analytical representation means no

reliance on an energy mesh for accuracy of the result or slowing down the calculation

time.

Figure 6.11 shows that the matching between the derived analytical model (Equa-

tion 6.40) as the solid lines and the full numberical simulation (|) from Equation

6.19 as the circles show excellent matching over a large temperature range (50K-450K)

and over all conduction regimes (TE,TFE,FE) for both the Arrhenius plots (a) and

ideality plots (b).

Figure 6.12 shows the full numerical simulation (|) from Equation 6.19 as the

circle points. The derived analytical model from Equation 6.40 shows excellent match-

ing as the solid blue line. The TE, TFE & FE analytical piecewise approximations

are derived by using Equations 6.47&6.49 with the analytical models found in liter-

ature [71, 72]. Part (a) shows the Arrhenius plot and (b) shows the ideality (n) plot.


−60

−50

−40

−30

−20

−10

log

VJsl

in/T

2 -

NumericalFV|Θ-AnalyticalFV|Θ-TETFEFE

0 50 100 150 200 2501

1.05

1.1

1.15

Idea

lityF

fact

orF

n

q/kTFVV-1-

NumericalFV|Θ-AnalyticalFV|Θ-TETFEFE

Va-

Vb-

TE TFE FE

Figure 6.12: (a) Arrhenius (Richardson’s) plot for piecewise approximation derivedfrom Equations from [72] verus derived from Equation 6.40 and numerical simulation(|) (Equation 6.19). (b) Ideality factor for same conditions.


The piecewise approximation in the Arrhenius plot (Figure 6.12a) seems to follow

the numerical simulation quite well, however, there are two issues that interfere with

using this type of model for regression. The first issue is because it is a piecewise

model. During the regression, the solver does not smoothly transition from one peice

to another, it tends to get caught around the transition points. The second issue is

deviations between the numerical model and the piecewise model, particularly for the

ideality values around the transition points (TFE/FE) and (TE/TFE). If the regres-

sion is done over a very small q/kBT range (e.g. 25-40 for above room temperature

measurements) there can be very poor matching between the piecewise model and

numerical simulation which would result in erroneous values being regressed.

6.5 Regression capability of analytical model

The derived analytical model is tested by regressing values from simulated data

(|) and (⊕|) . This gives an overview on if the model is successful and where

it starts to break down. Knowledge of breakdown conditions is useful when analyzing

experimental data.

6.5.1 Regression capability against simulated data

Figure 6.13 shows the regression behavior of the conventional (TE) method of ex-

traction, the Em ≈ slope model [116], and the proposed analytical model for a range

of steepness factors (E00) for above room temperature (300K-450K) simulated data

(|) . The proposed analytical model (|) shows improvement6 over both the Em ≈slope model as well as the conventional method. This confirms that the proposed

model accurately describes behavior of (|) Schottky junction.

Simulated data was obtained by using the numerical integration (|) from Equa-

tion 6.19. The SBH was kept constant and several different E00/kBTmin were sim-

ulated. The proposed analytical model was then used to regress values out of the

simulated data. Figure 6.14a shows the extracted E00/kBTmin and SBH values as a

6Non-linear regression is sensitive to initial starting points for regressors but having multiple startpoints helps global convergence.


10−1

100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Specified E00

/kTmin

Ext

ract

ed S

BH

(eV

)Specified SBHNeg Arrhenius slopeEm ≈ slopeProposed analytical

Figure 6.13: Shows the regression behavior of the conventional (TE) method of ex-traction, the Em = slope model [116], and the proposed analytical model for a rangeof steepness factors (E00) for above room temperature (300K-450K) simulated data(|) .

0 10 20 30 40 50 60−0.4

−0.2

0

0.2

0.4

0.6

0.8

MS interface distance (nm)

En

erg

y (e

V)

10−1

100

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Ext

ract

ed S

BH

(eV

)

10−1

100

10−1

100

Specified E00

/kTmin

Ext

ract

ed E

00/k

Tm

in

(a)

(b)

↑ Specified E00/kTmin

Figure 6.14: (a) Extracted SBH and E00 show exact theoretical trend with simulateddata (|) . (b) Regressed band diagram shows SBH constant while thickness of barrierdecreases.


10−2

10−1

100

0

0.5

1

1.5

Specified E00

/kTmin

SB

H (

eV)

ΦB0

ΦBn

Neg Arrhenius slopeEm ≈ slopeProposed analytical

Figure 6.15: Shows the regression behavior of the conventional (TE) method of ex-traction, the Em = slope model [116], and the proposed analytical model for a rangeof steepness factors (E00) for above room temperature (300K-450K) simulated data(⊕|) .

function of specified E00/kBTmin. The extracted SBH and E00 show exact theoretical

trend further proving that the model accurately describes behavior of (|) Schottky

junction. These extracted values are then used to obtain bands diagrams shown in

Figure 6.14b. These diagrams show SBH constant while thickness of barrier decreases

which is consistent with the specified values of the simulation.

Figure 6.15 shows the regression behavior of the conventional method, the Em ≈slope model [116], and the proposed analytical model for a range of steepness factors

(E00) for above room temperature (300K-450K) simulated data (⊕|) for a NiSi/n-Si

Schottky junction. The Em ≈ slope model extracts out the φBn value whereas the

proposed model extracts out the φB0. In the FE regime the proposed model breaks

down due to mismatch between the barrier profile of (⊕|) case and assumed profile

by the analytical (|) model.

From Figure 6.16a it can be seen that the proposed model works well up to an ex-

tracted E00/kBTmin ≈ 1. When the value is higher than this, the SBH value becomes

overestimated. At extracted values of E00/kBTmin > 3, the extracted E00/kBTmin

stops increasing and the extracted SBH sharply reduces. The extracted E00/kBTmin

value is generally higher than the specified value because image charge lowering makes


0 10 20 30 40 50 60 70 80 90 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

M−S Distance (nm)

En

erg

y (e

V)

10−2

10−1

100

101

0

0.5

1

1.5

Ext

ract

ed S

BH

(eV

)

10−2

10−1

100

10110

−2

10−1

100

101

Specified E00

/kTmin

Ext

ract

ed E

00/k

Tm

in

(a)

(b)


Figure 6.16: (a) Regression of simulated (⊕|) data of a NiSi/n-Si Schottky junction.Shows extracted SBH and E00/kBTmin as a function of specified E00/kBTmin values.(b)Band diagram obtained from regressed SBH, E00 and ξ values from a simulated(⊕|) data of a NiSi/n-Si Schottky junction.


10−2

10−1

100

101

0

0.2

0.4

0.6

0.8

Specified E00

/kTmin

Ext

ract

ed S

BH

(eV

)

10−2

10−1

100

10110

−2

10−1

100

101

Ext

ract

ed E

00/k

Tm

in

5 10 15 20 25 30 35

0.1

0.2

0.3

0.4

0.5

0.6

M−S Distance (nm)

En

erg

y (e

V)

(a)

(b)


Figure 6.17: (a) Regression of simulated (⊕|) data of a Al/n-Ge Schottky junction.Shows extracted SBH and E00/kBTmin as a function of specified E00/kBTmin values.(b) Band diagram obtained from regressed SBH, E00 and ξ values from a simulated(⊕|) data of a Al/n-Ge Schottky junction.

the barrier thinner than it would be otherwise. When the barrier is very thin, the

image charge lowering becomes dominant in determining Schottky barrier behavior,

drasticly reducing the percieved SBH. The band diagrams in Figure 6.16b show the

behavior a bit more clearly. Another issue to consider is that at with extremely thin

barriers, the depletion approximation starts to break down.

It is important to note that this proposed model can be used on any semiconductor

system. It can be used with Ge and even III-V semiconductor Schottky barriers. The

only use of material specific parameters is when translating the SBH, E00 and ξ values

into a band diagram, in which case a mDOS value is needed for the α parameter in


(a)

(b)

↑ As dose

Figure 6.18: (a) Extracted SBH and E00/kBTmin values for experimental NiSi/n-Sijunction with co-implant 90keV As and 15keV 5e12 cm−2 Sb. (b) Band diagrams ofexperimental NiSi/n-Si Schottky junction with increased well doping.

Eqation 6.1. Figure 6.17 shows the extracted parameters and band diagrams for

simulated (⊕|) data for an Al/n-Ge Schottky junction. It is particularly important

to notice that the model breaks down at approximately the same point (extracted

E00/kBTmin ≈ 1). This indicates that E00/kBTmin ≈ 1 could be used as a cut-off

guide to determine where to trust extracted values from experimental data.

6.5.2 Regression capability against experimental data

Figure 6.18a shows the extracted SBH andE00/kBTmin values for experimental NiSi/n-

Si junction with co-implant 90keV As and 15keV 5e12 cm−2 Sb. The dotted lines are

the control samples without any As implant. Figure 6.18b shows the band diagrams of

experimental NiSi/n-Si Schottky junction with increased well doping. The extracted


0

0.2

0.4

0.6

0.8

Sb dose (cm−3)

Ext

ract

ed S

BH

(eV

)

1012

1013

1014

1012

1013

101410

−1

100

Ext

ract

ed E

00/k

Tm

in

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

M−S Distance (nm)

En

erg

y (e

V)

no Sb1e125e127e121e135e13

(a)

(b)

↑ Sb dose

Figure 6.19: (a) Extracted SBH and E00/kBTmin values for experimental NiSi/n-Si junction with 15keV Sb. (b) Band diagrams of experimental NiSi/n-Si Schottkyjunction with increased shallow Sb doping.

SBH shows very little change with increased well implant, while E00/kBTmin consis-

tently increases with increased well implant. This is expected with only a change in

well doping. The extracted results stay within E00/kBTmin < 1, so there shouldn’t

be any model breakdown issues. The large error bars in this case are the standard

deviation in data across the wafer.

Figure 6.19a shows the extracted SBH and E00/kBTmin values for experimental

NiSi/n-Si junction with 15keV Sb. The dotted lines are the control samples without

any Sb implant. Figure 6.19b shows the band diagrams of experimental NiSi/n-

Si Schottky junction with increased shallow Sb doping. The SBH shows a general


decreasing trend with increased Sb dose, wheras the E00/kBTmin stays fairly constant

except for the highest dose. All points conform to E00/kBTmin < 1 so the results

shouldn’t be suffering from model breakdown. In the case of shallow Sb implant

conditions it seems that contact improvement is predominately obtained through

reduction in SBH for all doses and an increase in E00/kBTmin at higher doses. [117]

Chapter 7

Research Summary and Future

Work

7.1 Research Summary

Table 7.1: Research Accomplishments

1. Strong physics based simulation from transmission matrix method (TMM) that

can model current density and ρc of Schottky and MIS contacts

2. Results of simulated data led to observation of non-linear Arrhenius plots at

lower than expected doping

3. Analytical approximations to Fermi statisics related functions discovered

4. Development of analytical comprehensive emission (valid in TE, TFE and FE

regimes) numerically stable model for current density through Schottky junc-

tions

5. Regression of Schottky barrier height (ΦB0) and steepness factor (E00) from

diode IVT curves allow band diagrams regressed from data and allow better

understanding of barrier height modulation schemes

Chapter 3 outlines the experimental measurements available to characterize con-

tacts. A mask was developed that includes these structures and that is compatible

101

CHAPTER 7. RESEARCH SUMMARY AND FUTURE WORK 102

with III-V processing including mesa isolation is detailed in Appendix G. This mask

set has been utilized by several Stanford Nanofabrication Facility (SNF) labmembers.

Chapter 4 goes over the variety of methods used to model different contact types.

Appendix B shows the detailed derivation of current density and ρc that preserves

the transverse energy dependence on tranmission probability as well as transverse

effective mass differences. Preservation of these effects allows modelling of contacts

to both type IV and III-V semiconductors. A determination of the transmission ma-

trix method was developed in MATLAB to simulate metal-semiconductor Schottky

junctions as well as metal-insulator-semiconductor junctions for dielectric dipole miti-

gation (DDM). Use of TMM for the transmission probability simultanously takes into

account the contributions from thermionic emission (TE), thermionic field emission

(TFE) and field emission (FE).

Observations of the results from these simulations in a Arrhenius or Richard-

son’s plot, uncovered an unexpected non-linearity in even lightly doped cases. It was

observed that the TFE contributions could be an explanation for non-linearity in

Richardson’s plots that has been observed for decades.

Chapter 5 details a model whereby the extracted barrier height is actually related

to the peak energy of the energy distribution of the emitted electrons (5.4)(5.5) and

can be similarly analytically modeled (5.9). This adds in dependence of the extracted

SBH on temperature. The performance of the proposed model is compared to the

conventional model as well as a popular inhomogeneous model that uses a Gaussian

distribution of barrier heights (Fig. 5.4 & Table 5.2). Both the inhomogeneous and

proposed model modifications to the standard Richardson’s plot make the slope (or

apparent SBH) of the Richardson’s plot temperature dependent. However, the pro-

posed model results in a non-linear Richardson’s plot without invoking the need for an

inhomogeneous interface. Also, it is shown that the proposed TFE model can extend

the range of accurate extraction of the image charge lowered SBH of homogeneous

interfaces (Table 5.2).

Short analytical approximations for Fermi-Dirac statistics related functions have

been obtained that are globally accurate to within 5%. These approximations allow

analytical representations of carrier density and current density to be determined.


The value in this approximation is its versatility in being able to be applied to a wide

range of Fermi-Dirac based phenomena. Application of this method helps bridge the

gap between modeling and experiment by facilitating the development of analytical

models that are compatible with experimental data regression.

Chapter 6 proposes an analytical model that accurately accounts for current con-

duction in the TE, TFE, and FE regimes. It extracts more information from diode

IVT curves than previously possible by exploiting non-linear Arrhenius behavior. This

allows the analytical model to be used as a non-linear regression objective function

to regress Schottky barrier height (ΦB0), steepness factor (E00), and Fermi level (ξ),

enabling band diagrams of the measured interfaces to be determined. This complete

picture of band information allows material interface behavior to be understood more

completely, ultimately facilitating more efficient contact engineering.

7.2 Future Work

It may be possible to alter the analytical model from Chapter 6 to be able to include

image charge lowering. This would change the bm, cm, fm and Em derivations and

there is no guarentee they would become solvable. However, the form of current

density in Equation 6.40 would remain the same. If this derivation is possible, it

would allow a much better extraction of parameters in FE regime, which is ultimately

the regime of most interest.

With the inclusion of the log(1 + exp(ξ/kT )) term of P it would be possible

to adjust Equation 6.40 to be valid near V = 0. This would allow derivation of an

analytical form for ρc

((dJdV

)−1∣∣∣V→0

)to be derived as well as possibly using IV curves

from CBK structures instead of diodes, which would eliminate the contribution from

series resistance.

Finally, it may be possible to use this analytical model with dielectric dipole miti-

gated (DDM) contacts. If the dielectric barrier offset is large enough, than the added

resistance from tunneling through the dielectric could be rolled into the Richardson’s

constant (A∗) [71]. At large thicknesses of dielectric, the dielectric should shield the


contribution to the band diagram from image charge lowering, while with thin di-

electric there may be some image charge lowering. This would mean that dielectric

thickness would have to be a specified parameter in the extraction and would require

a different derivation of bm, cm, fm and Em based on this thickness.

Appendix A

Approximation for F12

Short (2 & 3 term) exponential based mathematical approximation for Fermi-statistics

related functions are presented. These approximations are globally accurate within

5% and allow previously non-integrable functions such as carrier density and current

density to have analytical integrated forms. Applications are widespread (carrier

density, current density, tunneling, etc) and a new analytical approximation for the

Fermi-Dirac integral of order 12

(3D carrier density) is presented as one example.

Equations based on Fermi statistics are commonly found throughout physics.

Fermi-Dirac statistics often result in expressions containing the terms 1exp (x)+1

and/or

log (exp (x) + 1). By themselves these expressions are integrable, but when multi-

plied by another function of x they are often non-integrable. The two most common

uses in solid state device physics are to determine carrier density (1) & (2) (where

ηf = EF−ECkT

and current density (3). In the case of carrier density the only integrable

case is when the electron carriers are confined in a 2D sheet. The cases of 3D and

1D rely on Fermi-Dirac integrals (Fj (ηf )) that have to be numerically obtained [118].

Since these Fermi-Dirac integrals are heavily used, they have received significant at-

tention over the years to obtain highly accurate analytical forms [119–121]. The other

common application of Fermi-Dirac statistics is that of current density (3). Current

density has had analytical forms in the past [71], [72] for specific emission cases, but

no universal emission analytical model has yet been reported.

105

APPENDIX A. APPROXIMATION FOR F 12

106

−10 −5 0 5 1010

−6

10−4

10−2

100

102

ηf

F1/

2−10 0 100

2

4

ηf

erro

r %

Figure A.1: The solid line is the numerically determined Fermi-Dirac integral of order12

using the script found in [118]. The circles represent the approximation defined in(4). The relative error is less than 5% over the whole range of ηf .

n =

N3D

2√πF 1

2(ηf ) , 3D

N2D2√π

ln (1 + exp (ηf )) , 2D

N1D2√πF− 1

2(ηf ) , 1D

(A.1)

Fj (ηf ) =

∫ ∞0

ηj

1 + exp (η − ηf )dη (A.2)

J =

A∗T 2

kT

∫∞0

log

(1+exp(ExkT −ηf)

1+exp(Ex−qVkT−ηf)

)exp (B(Ex))+1

dEx, ηf ≥ 0

A∗T 2

kT

∫∞0

log

(1+exp(ExkT )

1+exp(Ex−qVkT )

)exp (B(Ex))+1

dEx, ηf < 0

(A.3)

The approximations are found to be the following (See 6.2).

1

(exp (x) + 1)=

exp (−x)− 1

2exp

(−π

2x)

x ≥ 0

1− exp (x) + 12

exp(π2x)

x ≤ 0(A.4)

log (exp (x) + 1) =

x+ exp (−x)− 1

πexp

(−π

2x)

x ≥ 0

exp (x)− 1π

exp(π2x)

x ≤ 0(A.5)

3D carrier density is dependent on the Fermi-Dirac integral of order 12

(1). This

non-integrable function has been well-researched and has many highly accurate em-

pirical models for this function [119–121]. However, what those empirical models lack

APPENDIX A. APPROXIMATION FOR F 12

107

is versatility. They cannot be easily extendable applications such as current density.

The following analysis uses F 12(ηf ) as an example of how to apply the proposed ap-

proximations in (4) & (5) to a physically meaningful example. When the value of

ηf ≤ 0, the value of η − ηf ≥ 0 for the whole range of the integration, therefore

the approximation for 1exp (x)+1

when x ≥ 0 is valid for the whole range of the in-

tegration. The result shown in (6) is then a straightforward integration of the two

term integral. If the value of ηf ≥ 0, the Fermi-Dirac integration is split into two

sections. One section is for where η − ηf ≥ 0 and the other is for where η − ηf ≤ 0.

The appropriate approximations from (4) are then used for each integral. The final

results is shown in (7) where the scaled complementary error function is defined as

erfcx(x) = exp (x2) erfc(x) and D+ is the dawson integral related to the imaginary

error function by D+(x) =√π

2exp (−x2) erfi(x).

F 12(ηf ≤ 0) =

∫ ∞0

η

exp (η − ηf ) + 1dη (A.6)

=

√π

2exp (ηf )−

1√2π

exp(π

2ηf

)(A.7)

F 12(ηf ≥ 0) =

∫ ηf

0

η

exp (η − ηf ) + 1dη +

∫ ∞ηf

η

exp (η − ηf ) + 1dη (A.8)

=

√π

2erfcx

(√ηf)− 1

πerfcx

(√π

2ηf

)−√

2

π32

D+

(√π

2ηf

)+D+

(√ηf)

+2

3η

32f (A.9)

Fig A.1 shows the numerically integrated value of F 12(ηf ) [118] as the solid line.

The analytical approximation (6) & (7) is shown as circles. The inset shows that the

relative error is below 5% for the full range of ηf . A similar methodology to that

described above also works to derive analytical expressions for carrier density in 1D

and current density.

Appendix B

J and ρc Derivation

Table B.1: Assumptions of J and ρc derivation

1. Conservation of energy (Exm + Eτm = Exs + Eτs)

2. Conservation of transverse momentum (Specular Transmission) (kτm = kτs)

3. Parabolic bands

4. Zero level energy taken to be metal fermi level

The following is the derivation for current density (J) and specific contact resis-

tivity (ρc) for a M-S semiconductor junction from first principles. The assumptions

and choices for this derivation are listed in Table B.1.

F =1

1 + exp(E−EFkBT

) (B.1)

ρc =

(dJ

dV

)−1∣∣∣∣∣V=0

(B.2)

qV = EFs − EFm (B.3)

%x = exp

(Ex − EfmkBT

)(B.4)

108

APPENDIX B. J AND ρC DERIVATION 109

Table B.2: Symbols and variables for complete J and ρc derivations

Name Description

(|) Without image charge lowering

(⊕|) With image charge lowering

ΦB0 Schottky barrier height (|)

ΦBn Schottky barrier height (⊕|)

φn fermi level from conduction band

E Total energy of electron

Ex Energy ⊥ to barrier

Eτ Energy ‖ or transverse to barrier

EFm Fermi energy for metal

EFs Fermi energy for semiconductor

V Applied Voltage

Tsm Transmission from semi to metal

Tms Transmission from metal to semi

Fs Fermi probability function for semi

Fm Fermi probability function for metal

%τ = exp

(EτkBT

)(B.5)

% (Ex, Eτ ) = exp

(Ex + Eτ − Efm

kBT

)= %x%τ (B.6)

Fs =1

1 + exp(Ex+Eτ−qV−EF

kBT

) =1

1 + % exp(−qVkBT

) (B.7)

Fm =1

1 + exp(Ex+Eτ−EF

kBT

) =1

1 + %(B.8)


Richardson’s Constants:

A∗ =qm∗k2

B

2π2~3(B.9)

A∗τm =qmτmk

2B

2π2~3(B.10)

A∗τs =qmτsk

2B

2π2~3(B.11)

Relations stemmed from conservation of energy and conservation of transverse

momentum:

kz1 = kz2 (B.12)

ky1 = ky2 (B.13)

kz1 =

√2mz1Ez1

~(B.14)

kz2 =

√2mz2Ez2

~(B.15)

ky1 =

√2my1Ey1

~(B.16)

ky2 =

√2my2Ey2

~(B.17)

k2τ = k2

z + k2y (B.18)

kτ1 =√k2z1 + k2

y1 (B.19)

kτ2 =√k2z2 + k2

y2 (B.20)

kτ1 = kτ2 (B.21)

mτ1Eτ1 = mτ2Eτ2 (B.22)

Eτ2 =mτ1

mτ2

Eτ1 (B.23)


θ1 = tan−1

(ky1

kz1

)(B.24)

θ2 = tan−1

(ky2

kz2

)(B.25)

θ1 = θ2 (B.26)

Eτ1 = Ez1 + Ey1 (B.27)

Eτ2 = Ez2 + Ey2 (B.28)

mτ1 =mz1my1

sin2 (θ)mz1 + cos2 (θ)my1

(B.29)

mτ2 =mz2my2

sin2 (θ)mz2 + cos2 (θ)my2

(B.30)

tan θ =kykz

(B.31)

θ1 = θ2 = θ (B.32)

Parabolic bands:

Eτ =~2k2

τ

2mτ

(B.33)

Ex =~2k2

x

2mx

(B.34)

kx1 =

√2mx1 (Vx1 − Ex1)

~(B.35)

kx2 =

√2mx2 (Vx2 − Ex2)

~(B.36)

Eτ1 + Ex1 = Eτ2 + Ex2 (B.37)

Ex2 = Ex1 + Eτ1

(1− mτ1

mτ2

)(B.38)


B.1 Current Density Derivation

Start with basic definition of current density [73] between materials 1 and 2 and

translate integrals from k-space to energy space.

J = 2q

(2π)3

∫ +∞

kx=0

∫ +∞

ky=−∞

∫ +∞

kz=−∞F1T (1− F2) vzdkzdkydkx

= 2q

(2π)3 ~

∫ +∞

Ex=0

∫ +∞

kr=0

∫ 2π

θ=0

F1T (1− F2) krdkrdθdEx

= 2qm0

(2π)3 ~3

∫ +∞

Ex=0

∫ +∞

Eτ=0

∫ 2π

θ=0

F1T (1− F2)mτdEτdθdEx

J =qm0

π3~3

∫ +∞

Ex=0

∫ +∞

Eτ=0

∫ π2

θ=0

F1T (1− F2)mτdEτdθdEx (B.39)

where mτ is a function of θ. Due to the rotational symmetry since both the sine

and cosine components are squared, the integral over θ can be further reduced.

mτ =mymz

sin2 (θ)my + cos2 (θ)mz

(B.40)

Full current density will be subtraction of current density components in both

directions. The range on the integrals are θ = 0→ π2, Eτs = 0→∞, Eτm = 0→∞,

Exm = Exmmin → ∞ and Exs = Exsmin → ∞ where Exsmin is the semiconductor

conduction band level far from the interface.

Exmmin = Exsmin + Eτm

(mτm

mτs

− 1

)(B.41)

For a valid transition from the semiconductor to the metal, the tranmission is the

same in both directions. In other words:

Tsm (Eτs, Exs) = Tms (Eτm, Exm) (B.42)


Jx =m0q

π3~3

∫ ∞Exs=0

∫ ∞Eτs=0

∫ π2

θ=0

mτsFsTsm (1− Fm) dθdEτsdExs

− m0q

π3~3

∫ ∞Exm=0

∫ ∞Eτm=0

∫ π2

θ=0

mτmFmTms (1− Fs) dθdEτmdExm

=m0q

π3~3

∫Exs

∫Eτs

∫θ

mτsTsm

1

1 + % exp(−qVkBT

)(1− 1

1 + %

)dθdEτsdExs

− m0q

π3~3

∫Exm

∫Eτm

∫θ

mτmTms(

1

1 + %

)1− 1

1 + % exp(−qVkBT

) dθdEτmdExm

=m0q

π3~3

∫Exs

∫Eτs

∫θ

mτsTsm

1− 11+%

1 + % exp(−qVkBT

) dθdEτsdExs

− m0q

π3~3

∫Exs

∫Eτs

∫θ

mτmTsm

% exp(−qVkBT

)(1 + %)

(1 + % exp

(−qVkBT

)) dθ

mτs

mτm

dEτsdExs

=m0q

π3~3

∫Exs

∫Eτs

∫θ

mτsTsm

%

(1 + %)(

1 + % exp(−qVkBT

)) dθdEτsdExs

− m0q

π3~3

∫Exs

∫Eτs

∫θ

mτsTsm

% exp(−qVkBT

)(1 + %)

(1 + % exp

(−qVkBT

)) dθdEτsdExs

Jx =m0q

π3~3

∫ ∞Exs=Exsmin

∫ ∞Eτs=0

∫ π2

θ=0

mτsTsm

%(

1− exp(−qVkBT

))(1 + %)

(1 + % exp

(−qVkBT

)) dθdEτsdExs

(B.43)

Jx =m0q

π3~3

∫ ∞Exm=Exmmin

∫ ∞Eτm=0

∫ π2

θ=0mτmTms

%(

1− exp(−qVkBT

))(1 + %)

(1 + % exp

(−qVkBT

)) dθdEτmdExm

(B.44)

Equations B.43 & B.44 offer two methods to determine the current density through

the M-S junction depending on integration in semiconductor or metal energy space.


Conventionally semiconductor energy space is used, but both are possible with de-

termination of Tsm Tms with the TMM method outlined in Section 4.1.3. Solving for

both offers a built-in check without utilizing self-consistency. If the answers agree,

then the assumption for the band diagram holds, if they are different, then the starting

band diagram (e.g. depletion approximation) is not the steady state solution.

B.1.1 Current density independent of transverse energy

Further simplifications can be made if the transmission is assumed to be transverse

energy dependent. The result of this simplification is found in literature [114].

Jx =m0q

π3~3

∫ ∞Exs=Exsmin

∫ ∞Eτs=0

∫ π2

θ=0mτsTsm

%(

1− exp(−qVkBT

))(1 + %)

(1 + % exp

(−qVkBT

)) dθdEτsdExs

=m0qkBT

π3~3

∫ ∞Exs=Exsmin

∫ π2

θ=0mτsTsm

log

1 + %x exp(EτskBT

)%x exp

(EτskBT

)+ exp

(qVkBT

)∣∣∣∣∣∣

∞

Eτs=0

dθdExs

=m0qkBT

π3~3

∫ ∞Exs=Exsmin

∫ π2

θ=0mτsTsm log

%x + exp(qVkBT

)1 + %x

dθdExs

=m0qkBT

π3~3

∫ ∞Exs=Exsmin

∫ π2

θ=0mτsTsm log

1 + exp(qV−ExskBT

)1 + exp

(−ExskBT

) dθdExs

Jx =m0mτsqkBT

2π2~3

∫ ∞Exs=Exsmin

Tsm log

1 + exp(qV−ExskBT

)1 + exp

(−ExskBT

) dExs

=m0mτsqkBT

2π2~3

∫ ∞Ex=0

Tsm log

1 + exp(ξ−ExkBT

)1 + exp

(ξ−qV−ExkBT

) dEx

Jx =A∗T 2

kBT

∫ ∞Ex=0

Tsm log

1 + exp(ξ−ExkBT

)1 + exp

(ξ−qV−ExkBT

) dEx (B.45)


B.1.2 Thermionic emission limit of Current Density

Using a thermionic emission with the transverse energy independent transmission

probability should yeild Equation 4.7. The following makes use of the approximation

of log (1 + x) = x for small x with .

Tsm,x =

1 when Exs ≥ qφBn

0 otherwise(B.46)

Jx =m0mτsqkBT

2π2~3

∫ ∞Exs=Exsmin

Tsm log

1 + exp(qV−ExskBT

)1 + exp

(−ExskBT

) dExs

Jx =A∗T 2

kBT

∫ ∞Exs=Exsmin

Tsm log

1 + exp(qV−ExskBT

)1 + exp

(−ExskBT

) dExs

Jx =A∗T 2

kBT

∫ ∞Ex=qφBn

log

1 + exp(qV−ExskBT

)1 + exp

(−ExskBT

) dExs

Jx =A∗T 2

kBT

∫ ∞Ex=qφBn

exp

(qV − ExskBT

)− exp

(−ExskBT

)dExs

Jx =A∗T 2

kBT

[−kBT exp

(qV − ExkBT

)+ kBT exp

(−ExkBT

)]∣∣∣∣∞Exs=qφBn

Jx = A∗T 2

(exp

(qV − qφBn

kBT

)− exp

(−qφBnkBT

))Jx = A∗T 2 exp

(−qφBnkBT

)[exp

(qV

kBT

)− 1

](B.47)

B.2 Full Specific Contact Resistivity Derivation

Determination of the specific contact resistivity is resolved by substituting Equa-

tion B.43 into Equation B.2. The limits of the integrals are θ = 0→ π2, Eτs = 0→∞,

and Exs = Exsmin → ∞ where Exsmin and the T term is assumed to be Tsm. The

derivative with respect to voltage is brought inside the integral and conviently the


fermi related multiplicative factor of the dTdV

term goes to zero.

ρc =

d

dV

[m0q

π3~3

∫ ∞Exs=Exsmin

∫ ∞Eτs=0

∫ π2

θ=0

mτsTsm

·

%(

1− exp(−qVkBT

))(1 + %)

(1 + % exp

(−qVkBT

)) dθdEτsdExs

V→0

−1

=

m0q

π3~3

∫Exs

∫Eτs

∫θ

d

dV

mτsT

%(

1− exp(−qVkBT

))(1 + %)

(1 + % exp

(−qVkBT

)) dθdEτsdExs

V→0

−1

=

m0q

π3~3

∫Exs

∫Eτs

∫θ

mτsT

− % exp(−qVkBT

)(−qkBT

)(1 + %)

(1 + % exp

(−qVkBT

))

−

%2 exp(−qVkBT

)(1− exp

(−qVkBT

))(−qkBT

)(1 + %)

(1 + % exp

(−qVkBT

))2

+dTdV

%(

1− exp(−qVkBT

))(1 + %)

(1 + % exp

(−qVkBT

))∣∣∣∣∣∣

V→0

dθdEτsdExs

−1

=

m0q2

π3~3kBT

∫Exs

∫Eτs

∫θ

mτsT

% exp(−qVkBT

) [1 + % exp

(−qVkBT

)+ %

(1− exp

(−qVkBT

))](1 + %)

(1 + % exp

(−qVkBT

))2

+dTdV

%(

1− exp(−qVkBT

))(1 + %)

(1 + % exp

(−qVkBT

))∣∣∣∣∣∣

V→0

dθdEτsdExs

−1

=

m0q2

π3~3kBT

∫Exs

∫Eτs

∫θ

mτsT

% exp(−qVkBT

)(

1 + % exp(−qVkBT

))2 dθdEτsdExs

−1

ρc =

m0q

2

π3~3kBT

∫Exs

∫Eτs

∫θ

mτsT

(%

(1 + %)2

)dθdEτsdExs

−1

ρc =

m0q2

π3~3kBT

∫ ∞Exs=Exsmin

∫ ∞Eτs=0

∫ π2

θ=0

mτsT

exp(Eτs+Exs−Ef

kBT

)(

1 + exp(Eτs+Exs−Ef

kBT

))2 dθdEτsdExs

−1

(B.48)


ρc =

m0q2

π3~3kBT

∫ ∞Exm=Exmmin

∫ ∞Eτs=0

∫ π2

θ=0

mτmT

exp(Eτm+Exm−Ef

kBT

)(

1 + exp(Eτm+Exm−Ef

kBT

))2 dθdEτmdExm

−1

(B.49)

B.2.1 ρc independent of transverse energy

ρc =

m0q2

π3~3kBT

∫ ∞Exs=Exsmin

∫ ∞Eτs=0

∫ π2

θ=0

mτsTsm

exp(Eτs+Exs−Ef

kBT

)(


kBT

))2 dθdEτsdExs

−1

=

mτsm0q2

2π2~3kBT

∫ ∞Exs=Exsmin

∫ ∞Eτs=0

Tsm

exp(Eτs+Exs−Ef

kBT

)(


kBT

))2 dEτsdExs

−1

=

mτsm0q2

2π2~3kBT

∫ ∞Exs=Exsmin

Tsm

− kBT(1 + exp

(Eτs+Exs−Ef

kBT

))∣∣∣∣∣∣∞

Eτs=0

dExs

−1

=

qA∗T 2

(kBT )2

∫ ∞Exs=Exsmin

Tsm1(

1 + exp(Exs−EfkBT

))dExs−1

(B.50)

B.2.2 Thermionic emission limit of ρc

Using the same definition above for Tsm for thermionic emission (Equation B.46) and

Boltzmann statistics [58], the value for ρc reduces to the well-known value [71]. The

definition of ρc is at zero bias voltage and EFm = EFs = Ef and is taken as the zero

energy level.


ρc =

qA∗T 2

(kBT )2

∫ ∞Exs=Exsmin

Tsm1(

1 + exp(Exs−EfkBT

))dExs−1

=

qA∗T 2

(kBT )2

∫ ∞Exs=qφBn

1(1 + exp

(ExskBT

))dExs−1

=

qA∗T 2

(kBT )2

∫ ∞Exs=qφBn

exp

(− ExskBT

)dExs

−1

=

qA∗T 2

(kBT )2

[−kBT exp

(−ExskBT

)]∣∣∣∣∞Exs=qφBn

−1

=

qA∗T 2

kBTexp

(−qφBnkBT

)−1

=kBqA∗T

exp

(qφBnkBT

)(B.51)

Appendix C

Ψ and 1m∗

∂Ψ∂x versus 1√

m∗Ψ and 1√

m∗∂Ψ∂x

There are two schools of thought on the boundary conditions needed at a heteroge-

neous interface. Both conserve particle current, but there may be ramifications in

the Transmission properties depending on the choice used. The first is most com-

monly used and is continuity of the Schrodinger equation (Ψ) across the interface

and 1m∗

∂Ψ∂x

[73]. The second is 1√m∗

Ψ and 1√m∗

∂Ψ∂x

[65, 122].

Assuming plane waves of the form:

ψ (z) = Am exp [ikm (z − zm−1)] +Bm exp [−ikm (z − zm−1)] (C.1)

119

APPENDIX C. Ψ AND 1M∗

∂Ψ∂X

VERSUS 1√M∗

Ψ AND 1√M∗

∂Ψ∂X

120

C.1 Ψ and 1m∗

∂Ψ∂x from [73]

Am +Bm = Am+1 +Bm+1

kmmm

(Am −Bm) =km+1

mm+1

(Am+1 −Bm+1)

Am −Bm = ∆m (Am+1 −Bm+1)

Am = Am+1

(1 + ∆m

2

)−Bm+1

(1−∆m

2

)Bm = Am+1

(1−∆m

2

)−Bm+1

(1 + ∆m

2

)

Am

Bm

= Dm

Am+1

Bm+1

(C.2)

Dm =

1+∆m

21−∆m

2

1−∆m

21+∆m

2

(C.3)

∆m =kx2

kx1

meff1

meff2

=

√mx2 (Vx2 − Ex2)

mx1 (Vx1 − Ex1)

meff1

meff2

=

√√√√mx2

(Vx2 −

(Ex1 + Eτ1

(1− mτ1

mτ2

)))mx1 (Vx1 − Ex1)

meff1

meff2

= Emeff1

meff2

(C.4)


∂Ψ∂X

VERSUS 1√M∗

Ψ AND 1√M∗

∂Ψ∂X

121

Limit of transmission (Figure C.1):

T =4M21DV E

(M21 +DV E)2 (C.5)

M21 =

√m2

m1

(C.6)

DV E =

√V2 − E2

V1 − E1

(C.7)

0

0.5

110−2 10−1 100 101 102

0

0.5

1

M21DVE

Inte

rface

Tra

nsm

issio

n

Figure C.1: Ψ and 1m∗

boundary condition for interface transmission limit


∂Ψ∂X

VERSUS 1√M∗

Ψ AND 1√M∗

∂Ψ∂X

122

C.2 1√m∗

Ψ and 1√m∗

∂Ψ∂x from [65,122]

1√mm

(Am +Bm) =1

√mm+1

(Am+1 +Bm+1)

km√mm

(Am −Bm) =km+1√mm+1

(Am+1 −Bm+1)√mm+1

mm

(Am +Bm) = (Am+1 +Bm+1)√mm+1

mm

(Am −Bm) = ∆m (Am+1 −Bm+1)√mm+1

mm

Am = Am+1

(1 + ∆m

2

)−Bm+1

(1−∆m

2

)√mm+1

mm

Bm = Am+1

(1−∆m

2

)−Bm+1

(1 + ∆m

2

)Am

Bm

=

√mm

mm+1

Dm

Am+1

Bm+1

Am

Bm

= Dm′

Am+1

Bm+1

(C.8)

Dm′ =

√meff1

meff2

1+∆m

21−∆m

2

1−∆m

21+∆m

2

(C.9)


∂Ψ∂X

VERSUS 1√M∗

Ψ AND 1√M∗

∂Ψ∂X

123

∆m =kx2

kx1

=

√mx2 (Vx2 − Ex2)

mx1 (Vx1 − Ex1)

=

√√√√mx2

(Vx2 −

(Ex1 + Eτ1

(1− mτ1

mτ2

)))mx1 (Vx1 − Ex1)

= E (C.10)

Limit of transmission (Figure C.2):

T =4M21DV E

(1 +M21DV E)2 (C.11)

Even through the behavior of these two conditions is quite different, the difference

between these two in TMM simulations is small and well within measurement error,

therefore I could not prove experimentally which is the correct boundary condition.


∂Ψ∂X

VERSUS 1√M∗

Ψ AND 1√M∗

∂Ψ∂X

124

0

0.5

1 10−2100

102

0

0.5

1

M21DVE

Inte

rface

Tra

nsm

issio

n

Figure C.2: 1√m∗

Ψ and 1√m∗

boundary condition for interface transmission limit

Appendix D

Non-linear Richardson’s plot

analytical model

D.1 Analytical no image charge lowering

The following equations are valid for the depletion approximation to the voltage profile

while neglecting image charge barrier lowering. It is very close to the derivation found

in [123] but left in a more general form.

cz =1

E00

log

(√φ (x1)− φ (x2) +

√φ (x1)√

φ (x2)− φ (x2) +√φ (x2)

)

=1

E00

log

(√φ (x1)− φ (x2) +

√φ (x1)√

φ (x2)

)(D.1)

c1F =1

E00

log

(√EB − E +

√EB − E + ξ2√ξ2

)(D.2)

c1R =1

E00

log

(√EB +

√EB − E + ξ2√−E + ξ2

)(D.3)

cm =1

E00

log

(√EB − E + ξ2 − Em +

√EB − E + ξ2√

Em

)(D.4)

125

APPENDIX D. RICHARDSON’S PLOT ANALYTICAL MODEL 126

Following from dEeedEx

∣∣∣Ex=Em

= 0 where δ = −E + ξ2. The full form of cm is Equa-

tion D.5, but the approximation in Equation D.6 is valid over most of the thermionic

emission regime and is used by [123]. When using the full form for cm, the values for

bm, cm, fm and Em are valid over TE, TFE and FE regimes (i.e. not limited to TFE).

cm =

(1 + exp (bm)

kBTexp (bm)

) 1

1+exp(δ−EmkBT

) − 1

1+exp(ξ2−EmkBT

)log

(1+exp

(ξ2−EmkBT

)1+exp

(δ−EmkBT

)) (D.5)

cm ≈1

kBT(D.6)

Em =(EB − E + ξ2)

cosh2 (E00cm)(D.7)

bz =1

E00

[φ (x2) log

(√φ (x2)− φ (x2) +

√φ (x2)√

φ (x1)− φ (x2) +√φ (x1)

)+√φ (x1)

√φ (x1)− φ (x2)

−√φ (x2)

√φ (x2)− φ (x2)

]=

1

E00

[φ (x2) log

( √φ (x2)√

φ (x1)− φ (x2) +√φ (x1)

)+√φ (x1)

√φ (x1)− φ (x2)

](D.8)

b1F =1

E00

[ξ2 log

( √ξ2√

−E + EB +√ξ2 − E + EB

)+√EB − E + ξ2

√EB − E

](D.9)

b1R =1

E00

[(ξ2 − E) log

( √ξ2 − E√

EB +√ξ2 − E + EB

)+√EB − E + ξ2

√EB

](D.10)

bm =EB − E + ξ2

E00

tanh (cmE00)− Emcm (D.11)


fz =1

4E00

√φ (x1)− φ (x2)

φ (x2)√φ (x1)

(D.12)

f1F =1

4E00

√EB − E

ξ2

√EB − E + ξ2

(D.13)

f1R =1

4E00

√EB

(ξ2 − E)√EB − E + ξ2

(D.14)

fm =1

4E00

sinh (E00cm) cosh (E00cm)

(EB + ξ2 − E)(D.15)

Table D.1: Substitutions for different conduction regimes

Conduction Regime φ (x1) φ (x2)

Forward and Reverse TFE ξ2 − E + EB = δ + EB Em

Forward FE ξ2 − E + EB = δ + EB ξ2

Reverse FE ξ2 − E + EB = δ + EB ξ2 − E = δ

J =A∗T

kB

∫ ∞−∞

Eee dEx =A∗T

k

∫ ∞−∞

P (Ex)T (Ex) dEx

=A∗T

kB

∫ ∞−∞

log(


)(1 + exp(bm + cm (Em − Ex) + fm (Em − Ex)2)

) dEx (D.16)

J =A∗T

kB

∫ Eα

0

Eee dEx +

∫ Eβ

Eα

Eee dEx +

∫ E∞

Eβ

Eee dEx

J =A∗T

kB

∫ Eα

0

P (Ex ≤ Eα)T (Ex ≤ Eβ) dEx +

∫ Eβ

Eα

P (Ex ≥ Eα)T (Ex ≤ Eβ) dEx

+

∫ ∞Eβ

P (Ex ≥ Eα)T (Ex ≥ Eβ) dEx

(D.17)


J =A∗T

kB

∫

Eeep1

dEx +

∫Eeep2

dEx +

∫Eeep3

dEx

(D.18)

∫Eeep1

= −∫ Eα

0

[δ − ExkBT

+ exp

(−δ − Ex

kBT

)− 1

πexp

(−π

2

(δ − ExkBT

))]

×[exp


)− 1

2exp

(π2


))]dEx (D.19)

∫Eeep2

= −∫ Eβ

Eα

[exp

(δ − ExkBT

)− 1

πexp

(π

2

(δ − ExkBT

))]

×[exp


)− 1

2exp

(π2


))]dEx (D.20)

∫Eeep3

= −∫ ∞E′β

[exp

(δ − ExkBT

)− 1

πexp

(π

2

(δ − ExkBT

))]

×[1− exp


)+

1

2exp

(π2


))]dEx (D.21)


γ1 =1

kBT

2√fm

(D.22)

γ2 =cm

2√fm

(D.23)

γ3 =√fm (Em − δ) (D.24)

γ4 =γ2 −∣∣∣∣√γ2

2 − bm∣∣∣∣ (D.25)

γ5 =δ√fm (D.26)

γ6 =δ − Em − bm

cm

kBT(D.27)

γ7 =− bm − cm (Em − δ)− fm (Em − δ)2 (D.28)

γ1π =1

kBT

2√

π2fm

(D.29)

γ2π =π2cm

2√

π2fm

(D.30)

γ3π =

√π

2fm (Em − δ) (D.31)

γ4π =γ2π −∣∣∣∣√γ2

2π −π

2bm

∣∣∣∣ (D.32)

γ5π =δ

√π

2fm (D.33)

pA = (− erfmin(γ1 − γ2 − γ3, γ3 + γ4) +1

πerfmin(

π

2γ1 − γ2 − γ3, γ3 + γ4)

− erfmin(γ1 + γ2 + γ3, γ5) +1

πerfmin(

π

2γ1 + γ2 + γ3, γ5)

+cm + 2fm(Em − δ)

2fmkBTerfmin(γ2 + γ3, γ5)− exp (−cmδ − 2δfmEm + fmδ

2)− 1√πfmkBT

)

(D.34)


pB = (− erfmin (γ1π − γ2π − γ3π, γ3π + γ4π)− erfmin (γ1π + γ2π + γ3π, γ5π)

+1

πerfmin

(π2γ1π − γ2π − γ3π, γ3π + γ4π

)+

1

πerfmin

(π2γ1π + γ2π + γ3π, γ5π

)+cm + 2fm(Em − δ)

2fmkBTerfmin (γ2π + γ3π, γ5π)

−exp

(−π

2cmδ − δπfmEm+ π

2fmδ

2)− 1

π√

12fmkBT

(D.35)

pC = exp (γ6)

(π

πcmkBT + 2+ π −

1π

+ π

cmkBT + 1

)+ exp

(π2γ6

)( 2

2cmkBT + π− 2

π

)(D.36)

J = −A∗T 2

√

πfm

2kBTexp(γ7)pA−

√2fm

4kBTexp

(π2γ7

)pB +

pC

π

(D.37)

Appendix E

Techniques for Successful

Regression

The first tip to a sucessful regression is to know your problem. What is the goal

of the regression? Is your goal is to estimate more data within the range? If so,

then the best technique will be different than if you are trying to extract physical

meaning out of your regressors. If your goal is to estimate more data points than

your best bet will probably be a polynomial regression or limited order spline based

regression (piecewise polynomial regression). The Splinefit code by Jonas Lundgren

from the mathworks fileexchange is a very useful tool towards this end. If your goal is

to extract physical meaning from your regressor values then the next step is to know

your problem. First of all, you can save yourself a lot of headache by getting your

problem described universally by one curve. If you have a piecewise defined func-

tion, regression techniques tend to get stuck at the transition points. Theoretically

regression tools should be okay with an objective function that is continusous with

with a non-contiuous derivative and jacobian, but that has not been the case in my

experience.

Transformation of variables is a valuable asset in regression. the most commonly

used transformation of variables is taking the logrithim. This changes an exponential

function form into a linear line which is much easier to regress. The common advice

for a non-linear regression is to do a transformation of variables to make it linear.

131

APPENDIX E. TECHNIQUES FOR SUCCESSFUL REGRESSION 132

However, this is not the only use of a transformation of variables. A transformation

of variables can help with centering and scaling of a polynomial regression as well

as a non-linear regression. Extremely importantly, a transformation of variables can

make sure that your regressors do not range into unphysical values without the need

for a regression solver that can use lower and upper limits.

For instance, if you have a regressor x which ranges from 0 to 1, there are several

possible options.

1. Use an upper boundary of 1 and lower boundary of 0 (x=y)

2. Use a exponential transformation on x and then limit the upper boundary on

y to 0 and the lower boundary to -inf (x=exp(y))

3. Use a sin transform (x=(sin(y)+1)/2) - no boundary needed

4. Use a atan transform (x=(atan(y)+π/2)/π) - no boundary needed

5. sech(y) or 1-sech(y) - best for combination of exclusive and inclusive boundary

conditions

Which one is the best one? It depends on your problem. If your objective function

is linear at the beginning or a logrithmic or exponential transform will make it linear,

stick with those to have a robust regression. If your regression can not be transformed

into a linear regression it depends on if your boundary conditions are <> or <=>=

boundaries. Sin transform is good if the boundary values of 0 and 1 are inclusive.

Atan transform is good if the boundary values of 0 and 1 are exclusive.

Also, if your data is very noisy and you want accurate values for confidence in-

tervals you ultimately want to do the regression on a function that has normalized

gaussian error in the data. The above transformations can be used as a very accurate

guess into your normalized error objective function. In most cases your data shouldn’t

be extremely noisy and the second regression is not strictly needed (unless you are

submitting to a statistics journal).

Alternatively, your objective function might present itself with a great transfor-

mation. For instance if you are regressing a funciton such as f(x) = a ∗ asech(b) + c

APPENDIX E. TECHNIQUES FOR SUCCESSFUL REGRESSION 133

where a,b and c are your regressors and b is limite to the range of 0 to 1 (most

likely has one inclusive and one exclusive boundary). You might want to consider

f(x) = a ∗ asech(sech(d)) + c = a ∗ d + c where b = sech(d). This type of inverse

function transformation allows regression without explicit boundary conditions and

simplified the object function at the same time.

Appendix F

Helpful Integrals

Where log is the natural logarithm

∫ (b exp

(xa

))2(1 + b exp

(xa

))3dx = −a(1 + 2b exp

(xa

))2(1 + b exp

(xa

))2 (F.1)

∫ (b exp

(xa

))(1 + b exp

(xa

))3dx = − a

2(1 + b exp

(xa

))2 (F.2)

∫ (b exp

(xa

))(1 + b exp

(xa

))2dx = − a(1 + b exp

(xa

)) (F.3)

∫1(

1 + b exp(xa

))2dx = −a log(b exp

(xa

)+ 1)

+a(

b exp(xa

)+ 1) + x (F.4)

∫ (1 + 2b exp

(xa

))(1 + b exp

(xa

))2 dx = −a log(b exp

(xa

)+ 1)− a(

b exp(xa

)+ 1) + x (F.5)

∫1

2(b exp

(xa

))2dx = −a exp

(−2xa

)4b2

(F.6)

134

APPENDIX F. HELPFUL INTEGRALS 135

∫1 + 2b exp

(xa

)2(b exp

(xa

))2 dx = −12a exp

(−2xa

)+ 2ab exp

(−xa

)2b2

(F.7)

Appendix G

Mask Structure Details

This section lists all the specific dimensions of the structures in the contact mask. In

all cases the space between the edge of contact metal and the edge of the mesa (δ) is

5µm. Figure G.1 shows the contact layers for this mask.

ActiveMesa SDOhmicMetal

Figure G.1: Legend for Mask Layers

136

APPENDIX G. MASK STRUCTURE DETAILS 137

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!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L = 6954µm, W = 8986µm

Figure G.2: Completed Mask Design


s L

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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

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L = 100µm, s = 100µm

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

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L = 150µm, s = 150µm

!!!!!!!!!!!!!!!!!!!!!!!!

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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

L = 200µm, s = 200µm

Figure G.3: Four Point Probe Structures


d L+dL

L = 50µm, d = 2,5,10,15,20,25,30,40,60,80µm

L = 100µm, d = 2,5,10,15,20,25,30,40,60,80µm

Figure G.4: Circular Transmission Line Structures


d1

L

d2 d3

W

!!!!!!!!!!!!

!!!!!!!!!!!!

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L = 60µm, W = 100µm, d = 2,5,10,15,20,25,30,40,60,80µm

!!!!!!!!!!!!

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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

L = 60µm, W = 200µm, d = 2,5,10,15,20,25,30,40,60,80µmµm

Figure G.5: Linear Transmission Line Structures


1 3

4

2

L

z

!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!

!!!!!!!!!!

!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!

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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

L = 100µm, z = 2L-δ = 195µm

!!!!!!

!!!!!!

!!!!!!

!!!!!!

!!!!!!

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!!!!!!

!!!!!!

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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

L = 150µm, z = 2L-δ = 295µm L = 200µm, z = 2L-δ = 395µm

Figure G.6: Greek Cross Structures


1 3

4

2

L

sz

!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!

!!!!!!!!!!

!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

L = 100µm, z = 2L = 200µm,

s = 3L+2δ = 310µm

!!!!!!

!!!!!!

!!!!!!

!!!!!!

!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!

!!!!!!

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!!!!!!

!!!!!!

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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

L = 150µm, z = 2L = 300µm,

s = 3L+2δ = 460µm

L = 200µm, z = 2L = 400µm,

s = 3L+2δ = 610µm

Figure G.7: Box Cross Structures


1 5 3

4

2

L

c

!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!

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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

L = 100µm, c = 2L-δ = 195µm

!!!!!!

!!!!!!

!!!!!!

!!!!!!

!!!!!!

!!!!!!

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!!!!!!

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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

L = 150µm, c = 2L-δ = 295µm L = 200µm, c = 2L-δ = 395µm

Figure G.8: Kelvin Structures


1 2

3

4

5

6

7

8

c

a

b d bd

W

111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

2222222222222222

2222222222222222

22222222222222222222

22222222222222222222

22222222222222222222222222222222222222222222222222222222

22222222222222222222222222222222222222222222222222222222

2222222222222222

222222

222222

2222222222222222

222222

222222

L = 200µm, W = 900µm, a = 210µm, c = 900µm, b = 1395µm, d = 1290µm

Figure G.9: Hall Bar Structure


111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

22222222222222222222222222222222222

22222222222222222222222222222222222

2222222222222222222222222222

22222222222222222222222222222222222

2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222

2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222

L = 400µm

Figure G.10: SIMS Area

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