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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.
CHAPTER 2. METHODS FOR LOW RESISTANCE CONTACTS 6
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.
CHAPTER 2. METHODS FOR LOW RESISTANCE CONTACTS 7
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
CHAPTER 2. METHODS FOR LOW RESISTANCE CONTACTS 8
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]
CHAPTER 2. METHODS FOR LOW RESISTANCE CONTACTS 9
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].
CHAPTER 2. METHODS FOR LOW RESISTANCE CONTACTS 10
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
CHAPTER 2. METHODS FOR LOW RESISTANCE CONTACTS 11
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
CHAPTER 2. METHODS FOR LOW RESISTANCE CONTACTS 12
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]
CHAPTER 2. METHODS FOR LOW RESISTANCE CONTACTS 13
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].
CHAPTER 2. METHODS FOR LOW RESISTANCE CONTACTS 14
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
CHAPTER 2. METHODS FOR LOW RESISTANCE CONTACTS 15
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]
CHAPTER 3. MEASUREMENT OF CONTACT PARAMETERS 18
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
CHAPTER 3. MEASUREMENT OF CONTACT PARAMETERS 19
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
CHAPTER 3. MEASUREMENT OF CONTACT PARAMETERS 20
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
CHAPTER 3. MEASUREMENT OF CONTACT PARAMETERS 21
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
CHAPTER 3. MEASUREMENT OF CONTACT PARAMETERS 22
Figure 3.4: Linearized CTLM Data
Slope = Rsh/2πL
Y-intercept = 2RC
X-intercept = -2lt
Figure 3.5: Parameter Extraction for Circular TLM
CHAPTER 3. MEASUREMENT OF CONTACT PARAMETERS 23
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
CHAPTER 3. MEASUREMENT OF CONTACT PARAMETERS 24
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]
CHAPTER 3. MEASUREMENT OF CONTACT PARAMETERS 25
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)
CHAPTER 3. MEASUREMENT OF CONTACT PARAMETERS 26
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)
CHAPTER 3. MEASUREMENT OF CONTACT PARAMETERS 27
3.2 Hall Effect Structures
Table 3.3: Symbols used in calculations of Hall measurements
Name Description Units
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
CHAPTER 3. MEASUREMENT OF CONTACT PARAMETERS 28
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.
CHAPTER 3. MEASUREMENT OF CONTACT PARAMETERS 29
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
CHAPTER 3. MEASUREMENT OF CONTACT PARAMETERS 30
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
CHAPTER 3. MEASUREMENT OF CONTACT PARAMETERS 31
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
CHAPTER 3. MEASUREMENT OF CONTACT PARAMETERS 32
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)
CHAPTER 3. MEASUREMENT OF CONTACT PARAMETERS 33
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
Name Description Units
(|) 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
CHAPTER 4. MODELING OF CONTACT TYPES 36
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)
CHAPTER 4. MODELING OF CONTACT TYPES 37
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
CHAPTER 4. MODELING OF CONTACT TYPES 38
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.
CHAPTER 4. MODELING OF CONTACT TYPES 39
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].
CHAPTER 4. MODELING OF CONTACT TYPES 40
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)
CHAPTER 4. MODELING OF CONTACT TYPES 41
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
CHAPTER 4. MODELING OF CONTACT TYPES 42
“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]
CHAPTER 4. MODELING OF CONTACT TYPES 43
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
CHAPTER 4. MODELING OF CONTACT TYPES 44
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.
CHAPTER 4. MODELING OF CONTACT TYPES 45
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
CHAPTER 4. MODELING OF CONTACT TYPES 46
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.
CHAPTER 4. MODELING OF CONTACT TYPES 47
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.
CHAPTER 4. MODELING OF CONTACT TYPES 48
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
CHAPTER 4. MODELING OF CONTACT TYPES 49
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
CHAPTER 4. MODELING OF CONTACT TYPES 50
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.
CHAPTER 4. MODELING OF CONTACT TYPES 51
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]
CHAPTER 4. MODELING OF CONTACT TYPES 52
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)
CHAPTER 4. MODELING OF CONTACT TYPES 53
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]
CHAPTER 4. MODELING OF CONTACT TYPES 54
eff
MSeff
IS
MS
Figure 4.11: Effective MIS structure for thin nonzero oxides.3
CHAPTER 4. MODELING OF CONTACT TYPES 55
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.
CHAPTER 4. MODELING OF CONTACT TYPES 56
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
CHAPTER 4. MODELING OF CONTACT TYPES 57
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.
CHAPTER 4. MODELING OF CONTACT TYPES 58
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
Name Description Units
(|) Without image charge lowering —
(⊕|) With image charge lowering —
ΦB0 Schottky barrier height (|) eV
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
Ex Energy of electron ⊥ to interface eV
Eτ Energy of electron ‖ to interface eV
E Total energy of electron (Ex + Eτ ) eV
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].
CHAPTER 5. EM ≈ SLOPE MODEL 61
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.
CHAPTER 5. EM ≈ SLOPE MODEL 62
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
CHAPTER 5. EM ≈ SLOPE MODEL 63
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.
CHAPTER 5. EM ≈ SLOPE MODEL 64
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.
CHAPTER 5. EM ≈ SLOPE MODEL 65
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)
CHAPTER 5. EM ≈ SLOPE MODEL 66
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.
CHAPTER 5. EM ≈ SLOPE MODEL 67
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.
CHAPTER 5. EM ≈ SLOPE MODEL 68
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
CHAPTER 5. EM ≈ SLOPE MODEL 69
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
CHAPTER 5. EM ≈ SLOPE MODEL 70
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
CHAPTER 5. EM ≈ SLOPE MODEL 71
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.
CHAPTER 5. EM ≈ SLOPE MODEL 72
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.
CHAPTER 5. EM ≈ SLOPE MODEL 73
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
Name Description Units
(|) Without image charge lowering -
(⊕|) With image charge lowering -
ΦB0 Schottky barrier height (|) eV
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
Ex Energy of electron ⊥ to interface eV
Eτ Energy of electron ‖ to interface eV
E Total energy of electron (Ex + Eτ ) eV
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.
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 76
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.
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 77
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.
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 78
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.
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 79
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.
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 80
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.
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 81
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
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 82
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.
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 83
−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).
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 84
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(ξ/kBT )1+exp(δ/kBT )
)(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)
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 85
∫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
(−bm − cm(Em − Ex)− fm(Em − Ex)2
))]dEx (6.22)
∫Eeep2
= −∫ Eβ
Eα
[exp
(δ − ExkBT
)− 1
πexp
(π
2
(δ − ExkBT
))]
×[exp
(−bm − cm(Em − Ex)− fm(Em − Ex)2
)− 1
2exp
(π2
(−bm − cm(Em − Ex)− fm(Em − Ex)2
))]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
(bm + cm(Em − Ex) + fm(Em − Ex)2
))]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.
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 86
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)
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 87
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.
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 88
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)
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 89
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
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 90
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
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 91
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.
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 92
−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.
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 93
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.
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 94
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.
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 95
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
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 96
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)
↑ Specified E00/kTmin
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.
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 97
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)
↑ Specified E00/kTmin
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
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 98
(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
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 99
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
CHAPTER 6. DERIVATION OF ANALYTICAL RICHARDSON’S PLOT 100
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.
CHAPTER 7. RESEARCH SUMMARY AND FUTURE WORK 103
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
CHAPTER 7. RESEARCH SUMMARY AND FUTURE WORK 104
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)
APPENDIX B. J AND ρC DERIVATION 110
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)
APPENDIX B. J AND ρC DERIVATION 111
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)
APPENDIX B. J AND ρC DERIVATION 112
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)
APPENDIX B. J AND ρC DERIVATION 113
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.
APPENDIX B. J AND ρC DERIVATION 114
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)
APPENDIX B. J AND ρC DERIVATION 115
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
APPENDIX B. J AND ρC DERIVATION 116
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)
APPENDIX B. J AND ρC DERIVATION 117
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
)(
1 + exp(Eτs+Exs−Ef
kBT
))2 dθdEτsdExs
−1
=
mτsm0q2
2π2~3kBT
∫ ∞Exs=Exsmin
∫ ∞Eτs=0
Tsm
exp(Eτs+Exs−Ef
kBT
)(
1 + exp(Eτs+Exs−Ef
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.
APPENDIX B. J AND ρC DERIVATION 118
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)
APPENDIX C. Ψ AND 1M∗
∂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
APPENDIX C. Ψ AND 1M∗
∂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)
APPENDIX C. Ψ AND 1M∗
∂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.
APPENDIX C. Ψ AND 1M∗
∂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)
APPENDIX D. RICHARDSON’S PLOT ANALYTICAL MODEL 127
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(ξ/kBT )1+exp(δ/kBT )
)(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)
APPENDIX D. RICHARDSON’S PLOT ANALYTICAL MODEL 128
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
(−bm − cm(Em − Ex)− fm(Em − Ex)2
)− 1
2exp
(π2
(−bm − cm(Em − Ex)− fm(Em − Ex)2
))]dEx (D.19)
∫Eeep2
= −∫ Eβ
Eα
[exp
(δ − ExkBT
)− 1
πexp
(π
2
(δ − ExkBT
))]
×[exp
(−bm − cm(Em − Ex)− fm(Em − Ex)2
)− 1
2exp
(π2
(−bm − cm(Em − Ex)− fm(Em − Ex)2
))]dEx (D.20)
∫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
(bm + cm(Em − Ex) + fm(Em − Ex)2
))]dEx (D.21)
APPENDIX D. RICHARDSON’S PLOT ANALYTICAL MODEL 129
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)
APPENDIX D. RICHARDSON’S PLOT ANALYTICAL MODEL 130
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
APPENDIX G. MASK STRUCTURE DETAILS 138
s L
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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
APPENDIX G. MASK STRUCTURE DETAILS 139
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
APPENDIX G. MASK STRUCTURE DETAILS 140
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
APPENDIX G. MASK STRUCTURE DETAILS 141
1 3
4
2
L
z
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L = 100µm, z = 2L-δ = 195µm
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L = 150µm, z = 2L-δ = 295µm L = 200µm, z = 2L-δ = 395µm
Figure G.6: Greek Cross Structures
APPENDIX G. MASK STRUCTURE DETAILS 142
1 3
4
2
L
sz
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L = 100µm, z = 2L = 200µm,
s = 3L+2δ = 310µm
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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
APPENDIX G. MASK STRUCTURE DETAILS 143
1 5 3
4
2
L
c
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L = 100µm, c = 2L-δ = 195µm
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L = 150µm, c = 2L-δ = 295µm L = 200µm, c = 2L-δ = 395µm
Figure G.8: Kelvin Structures
APPENDIX G. MASK STRUCTURE DETAILS 144
1 2
3
4
5
6
7
8
c
a
b d bd
W
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
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L = 200µm, W = 900µm, a = 210µm, c = 900µm, b = 1395µm, d = 1290µm
Figure G.9: Hall Bar Structure
APPENDIX G. MASK STRUCTURE DETAILS 145
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
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L = 400µm
Figure G.10: SIMS Area
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