tight binding parameterization from ab-initio calculations
TRANSCRIPT
TIGHT BINDING PARAMETERIZATION FROM AB-INITIO CALCULATIONS
AND ITS APPLICATIONS
A Dissertation
Submitted to the Faculty
of
Purdue University
by
Yaohua Tan
In Partial Fulfillment of the
Requirements for the Degree
of
Doctor of Philosophy
May 2016
Purdue University
West Lafayette, Indiana
ii
ACKNOWLEDGMENTS
I would like to express my gratitude to my advisor, Professor G. Klimeck, for
the valuable insights he provided for this research and the continuous support of my
PhD. I would like to thank him for providing all the necessary computing resources.
He introduced me to the world of high performance computing and its applications
to problem solving in Physics and Engineering which is invaluable experience for my
future career pursuits.
I would also like to thank Professor M. Povolotskyi, Professor Timothy.B.Boykin,
Professor M. Lundstrom and Professor A. Strachan for serving on my advisory com-
mittee.
I am grateful to Prof. Michael Povolotskyi and Prof. Tillmann Kubis who pro-
vided a lot of support during the past a few years of my PhD. They provided great
help to improve the quality of my work in many aspects. Prof. Timothy. B. Boykin
is the one who is continuously providing great intellectual support to my project. Dr.
Chris Bowen usually suggested constructive physical insights and provided encourag-
ing suggestions to my project. I would also like to thank our research faculties team
of Prof. Rajib Rahmann, Prof. Jim Fonseca, Dr. Bozidar Novakovic and Dr. Jun
Huang for their great support.
I am grateful to my girl friend Yuling Hsueh. With her my life in West Lafayette
became colorful. I would also like to thank group members and good friends Zheng-
ping Jiang, Junzhe Geng, Yu He, Kai Miao, Xufeng Wang and Dr. Lang Zeng, We
have spent many happy hours together for interesting discussions and events.
I would like to thank former group members Dr. Parijat Sengupta who discussed
and introduced interesting physics topics with me. I would like to thank Daniel Mejia
for the technical support he provided. I would like to thank Mehdi Salmani, Hesamed-
din Ilatikhameneh, Evan Wilson, Daniel Valencia, Santiago Perez, Dr. Ganesh Hegde,
iii
Dr. Saumitra Mehrotra, Dr. Abhijeet Paul, Dr.Seung Hyun Park, Prof. Mathieu
Luisier, Dr. Hoon Ryu, Dr. Sunhee Lee, Dr. Hong-Hyun Park, Sicong Chen, Pengyu
Long, Fan Chen, Yu Wang, Tarek Ameen, Kuangchung Wang, Samik Mukherjee, Har-
shad Sahasrabudhe and Prasad Sarangapani, for interesting and helpful discussions
and technical support.
I would like to thank all my colleagues from the NCN and the Klimeck group
for providing a stimulating and fun environment in which to learn and grow. Lastly,
and most importantly, I wish to thank my parents and my brother for their love and
supports.
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TABLE OF CONTENTS
Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background: Device scaling and the role of empirical tight bindingmodeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Challenges for empirical tight binding modeling: parameterization . 2
1.3 Contribution of the present work . . . . . . . . . . . . . . . . . . . 5
1.4 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 TIGHT BINDING PARAMETERIZATION FROM AB-INITIO CALCU-LATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Algorithm: tight binding parameterization from ab-initio calculations 8
2.3 Transformation of basis functions . . . . . . . . . . . . . . . . . . . 14
2.4 Simplified fitting process . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Parameterization algorithm of multiple materials . . . . . . . . . . 18
2.6 Implementation of the parameterization algorithm . . . . . . . . . . 19
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 PARAMETERIZATION OF UNSTRAINED SEMICONDUCTORS ANDUTBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Parameterization of unstrained materials . . . . . . . . . . . . . . . 24
3.2.1 Band structure of bulk materials . . . . . . . . . . . . . . . 24
3.2.2 Explicit tight binding basis functions . . . . . . . . . . . . . 27
3.3 Tight binding analysis of UTBs . . . . . . . . . . . . . . . . . . . . 32
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3.3.1 Explicit passivation model . . . . . . . . . . . . . . . . . . . 32
3.3.2 Band structures and wave functions of UTBs . . . . . . . . . 35
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 PARAMETERIZATION OF STRAINED MATERIALS (ROOM TEM-PERATURE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Strain model description . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.1 Multipole expansion of atomic potentials . . . . . . . . . . . 43
4.2.2 Strain dependent tight binding Hamiltonian . . . . . . . . . 44
4.2.3 Onsite elements . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.4 Interatomic couplings . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Parameterization of strained group III-V and IV materials . . . . . 48
4.4 Tight binding analysis of strained materials . . . . . . . . . . . . . 51
4.4.1 Unstrained band structures and hydrostatic strain behavior 51
4.4.2 Arbitrary strain behavior . . . . . . . . . . . . . . . . . . . 53
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.6 Tight binding parameters . . . . . . . . . . . . . . . . . . . . . . . 65
4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 TIGHT BINDING ANALYSIS TO HETEROSTRUCTURES . . . . . . . 75
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Tight binding modeling of superlattices . . . . . . . . . . . . . . . . 76
5.3 Band structures of selected superlattices . . . . . . . . . . . . . . . 78
5.4 Transferability of the tight binding model . . . . . . . . . . . . . . . 82
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6 PARAMETERIZATION OF 2D MATERIALS . . . . . . . . . . . . . . 84
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.2 Geometry of transition metal dichalcogenide and black Phosphorus 85
6.3 Tight binding parameterization . . . . . . . . . . . . . . . . . . . . 86
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6.3.1 Transition metal dichalcogenides . . . . . . . . . . . . . . . 86
6.3.2 Black Phosphorus . . . . . . . . . . . . . . . . . . . . . . . . 87
6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7 SUMMARY AND OUTLOOK . . . . . . . . . . . . . . . . . . . . . . . . 94
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
A Introduction to tight binding model . . . . . . . . . . . . . . . . . . . . . 103
B Fast Fourier transformation of radial functions . . . . . . . . . . . . . . . 108
C Wave function projections . . . . . . . . . . . . . . . . . . . . . . . . . . 109
D Solution of quadratic form . . . . . . . . . . . . . . . . . . . . . . . . . . 111
E Analytical expression of radial basis functions . . . . . . . . . . . . . . . 113
F Generation of high resolution band structure by hybrid functional calculation 115
G Expression of M(l)α,γ
(d)
in chapter 2 . . . . . . . . . . . . . . . . . . . . 118
H Interatomic coupling due to dipole potentials . . . . . . . . . . . . . . . . 119
I Generation of ab-initio results matching Room temperature experiments 121
J Definition of deformation potentials . . . . . . . . . . . . . . . . . . . . . 124
K External resources for parameterization of more materials . . . . . . . . . 126
L Agreements for reuse of published papers . . . . . . . . . . . . . . . . . . 127
VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
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LIST OF TABLES
Table Page
3.1 Slater-Koster type ETB parameters of bulk Si and GaAs, and passiva-tion parameters of UTBs. All presented parameters except for the latticeconstants are in the unit of eV. The lattice constants are in Angstrom.The Hydrogen atoms which passivate surface As and Ga are denoted byHa and Hc respectively. Onsite energies of surface As and Ga atoms areshifted by δa and δc respectively. . . . . . . . . . . . . . . . . . . . . . 26
3.2 Targets comparison of bulk Si and GaAs. Critical band edges and effec-tive masses at Γ, X and L points by ETB and HSE06 calculations arecompared. The Eg and ∆SO are in the units of eV; effective masses arescaled by free electron mass m0. The data in error column summarizesthe discrepancies between HSE06 and TB results. . . . . . . . . . . . . 28
4.1 Targets comparison of bulk Si and Ge. Critical band edges and effectivemasses at Γ, X and L from ETB and HSE06 calculations are compared.The Eg and ∆SO are in the unit of eV; effective masses are scaled by freeelectron mass m0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Targets comparison of bulk Phosphides. Critical band edges and effectivemasses at Γ, X and L from ETB and HSE06 calculations are compared.The Eg and ∆SO are in the unit of eV; effective masses are scaled by freeelectron mass m0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Targets comparison of bulk Arsenides. Critical band edges and effectivemasses at Γ, X and L from ETB and HSE06 calculations are compared.The Eg and ∆SO are in the unit of eV; effective masses are scaled by freeelectron mass m0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Targets comparison of bulk Antimonides. Critical band edges and effectivemasses at Γ, X and L from ETB and HSE06 calculations are compared.The Eg and ∆SO are in the unit of eV; effective masses are scaled by freeelectron mass m0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.5 Comparison of deformation potentials of group IV materials (ETB vsHSE06). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6 Comparison of deformation potentials of group III-V materials (ETB vsHSE06). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
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4.7 Atom type dependent onsite and spin orbit coupling parameters for groupIV and III-V elements. All parameters in this table have the unit of eV. 65
4.8 Environment dependent onsite parameters for group IV and part of groupIII-V materials. In Si and Ge, both ’a’ and ’c’ denote the same atom.For Si-Ge bond, ’a’ corresponds to Si and ’c’ corresponds to Ge. Theparameters I’s and O’s are in the unit of eV. parameters λ’s are in theunit of A−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.9 Environment dependent onsite parameters for part of group III-V materi-als. The parameters I’s and O’s are in the unit of eV. parameters λ’s arein the unit of A−1. The nonzero δdij is introduced to match ETB resultswith experimental targets under room temperature. . . . . . . . . . . 67
4.10 Bond length dependent interatomic coupling parameters for group IV ma-terials. In Si and Ge, both ’a’ and ’c’ denote the same atom. For Si-Gebond, ’a’ correspond to Si and ’c’ correspond to Ge. The parameters V ’sare in the unit of eV. parameters η’s are in the unit of A−1. The nonzeroδdij is introduced to match ETB results with experimental targets underroom temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.11 Bond length dependent interactomic coupling parameters V ’s for groupIII-V materials. The parameters V ’s are in the unit of eV. . . . . . . . 69
4.12 Bond length dependent interactomic coupling parameters η’s for groupIII-V materials. parameters η’s are in the unit of A−1 . . . . . . . . . 70
4.13 Off-diagonal onsite parameters due to multipole potentials. In Si and Ge,both ’a’ and ’c’ denote the same atom, parameters Cαaβcσ are left emptydue to relation Cαaβcσ = Cαcβaσ. For Si-Ge bond, ’a’ correspond to Si and’c’ correspond to Ge. All parameters are in the unit of eV. . . . . . . . 71
4.14 Off-diagonal onsite parameters due to dipole and quadrupole potentials.All parameters are in the unit of eV. . . . . . . . . . . . . . . . . . . . 71
4.15 Interatomic coupling parameters due to multipole potentials for group IVand part of group III-V materials. In Si and Ge, both ’a’ and ’c’ denotethe same atom. For Si-Ge bond, ’a’ corresponds to Si and ’c’ correspondsto Ge. All parameters are in the unit of eV. . . . . . . . . . . . . . . . 72
4.16 Interatomic coupling parameters due to multipole potentials for part ofgroup III-V materials. All parameters are in the unit of eV. . . . . . . 73
6.1 ETB parameters of MoS2 and WSe2. The d orbitals are defined as d0 :d2
z, d1 : dxz, dyz and d2 : dxy, dx2−y2; while the p orbitals are definedas p0 : pz,p1 : px, py. . . . . . . . . . . . . . . . . . . . . . . . . . . 88
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6.2 ETB parameters of MoTe2 and WTe2. The d orbitals are defined as d0 :d2
z, d1 : dxz, dyz and d2 : dxy, dx2−y2; while the p orbitals are definedas p0 : pz,p1 : px, py. . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3 Tight binding parameters of Black Phosphorus. The ETB model usessp3d5s* basis set. Scaling parameters η is used to model the bond lengthdependency of the ETB parameters. . . . . . . . . . . . . . . . . . . . 90
6.4 ETB and ab-initio (HSE06) band gaps Eg, valence Ev and conduction Ecband edges in eV, and effective masses for electrons me and holes mh atthe point along X and Y direction for the monolayer, bilayer. . . . . . . 93
A.1 Slater Koster interatomic interactions of s-p, s-d, p-p and p-d orbitals . 106
A.2 Slater Koster interatomic interactions of d orbitals . . . . . . . . . . . 107
I.1 Experimental lattice constants and band gaps of group IV and III-V ma-terials under room temperature; required changes of lattice constants δain order to match HSE06 band gap with experiments. . . . . . . . . . . 123
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LIST OF FIGURES
Figure Page
1.1 Different definitions of materials at a GaAs/AlAs interface which lead todifferent results. Regions of AlAs and GaAs materials are separated bythe dashed line. In definitions (a) and (b), the left part is AlAs and theright part is GaAs. In definition (a), the interface As atoms are definedas atoms in AlAs; in definition (b), the interface As atoms are defined asatoms in GaAs. In the case (c), the interface As atoms are defined as Asatoms in an average material of AlAs and GaAs. . . . . . . . . . . . . 3
1.2 ab-initio and tight binding wave functions in a As terminated GaAs UTB.ETB calculation shows unphysical top valence states. ETB wave functionsby existing parameters show different envelope with the ab-initio wavefunctions. In a As terminated GaAs UTB, ab-initio wave functions (a) ofthe top valence bands are sine-like confined states, while the ETB valencestates (b) turn out to be exponential-like states which is unphysical. (a0
is the lattice constant) . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 The process of mapping from ab-initio calculations to tight binding model.This process extract tight binding parameters and basis functions from ab-initio results iteratively. In this process, tight binding band structure andwave functions are fitted to the corresponding ab-initio results. . . . . 9
2.2 The Simplified process of mapping from ab-initio calculations to tightbinding model. In the simplified process, prototype tight binding basisfunctions are optimized in the first step. In the second step, the tightbinding Hamiltonian is optimized after the basis functions are obtained.The basis functions are kept unchanged in the second step. And exactbasis functions are extracted in the last step. . . . . . . . . . . . . . . 17
2.3 Different stages of code implementation. ab-initio calculations in this workare performed by VASP which was implemented in Fortran by the VASPteam. Formatted VASP output is parsed only once by Python procedure.The following ab-initio mapping process is also implemented in Python.Multiscale atomistic simulation tool NEMO5 [31] provides the capabilityfor tight binding calculation. . . . . . . . . . . . . . . . . . . . . . . . 20
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Figure Page
2.4 The mapping algorithm is separated into initialization, execution and out-put levels. The initialization and final output are needed to be done once.Execution level can perform different tasks according to user’s require-ment. The tasks of band structure calculation and wave function projec-tion are one shot tasks, while the parameters optimization is a iterativeprocess. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 Band structure and density of states of bulk Si. ETB band structureagree with the HSE06 band structure (a), especially for bottom conductionbands (b) and top valence bands (c). . . . . . . . . . . . . . . . . . . . 24
3.2 Band structure and density of states of bulk GaAs. ETB band structureagree with the HSE06 band structure (a), especially for bottom conductionbands (b) and top valence bands (c). . . . . . . . . . . . . . . . . . . . 25
3.3 Contours of selected highly localized ETB basis functions of Si((a),(d)),Ga ((b),(e)) and As ((c),(f)) atoms. (a),(b) and (c) correspond to thecontours of s orbitals of Si, Ga and As atoms in x-y plane. (d),(e) and(f) correspond to the contours of py orbitals of Si, Ga, and As atomsrespectively. (g),(h) and (i) show the contribution of different angularmomentums in basis functions of Si ,Ga, and As atoms respectively. TheETB basis functions of Si and GaAs are highly localized basis functionswith one dominant angular momentum. . . . . . . . . . . . . . . . . . . 29
3.4 Probability amplitude of bulk Si and GaAs by ETB and HSE06 calcula-tions are in good agreement. The Probability amplitude is plotted along[111] direction along which neighboring atoms are located. For bulk Si,
silicon atoms locate at 0,√
34a and
√3a ; while for bulk GaAs, Ga locates
at 0 and√
3a; As locates at√
34a. For bulk Si, probability amplitudes of
conduction states at X points (a) and valence states at Γ (c) are shown.For bulk GaAs, probability amplitudes of conduction states at Γ points(b) and valence states at Γ (d) are shown. . . . . . . . . . . . . . . . . 30
3.5 Planar averaged local potentials of Hydrogen terminated (a) Si and (b)GaAs UTBs. The dashed lines correspond to the envelopes of the localpotentials, the dots on dashed lines correspond to centers of atoms. Theenvelopes of the potentials are flat inside the UTBs. Significant deviationcan be seen at the surface atoms. . . . . . . . . . . . . . . . . . . . . . 31
3.6 Band structures of [001] (a) Si and (b) As terminated GaAs UTBs byETB agree with HSE06 band structures, demonstrating the bulk Si andGaAs ETB parameters are transferable to UTB cases. All UTBs contain17 Si/GaAs atomic layers(with thickness 4a0). . . . . . . . . . . . . . 33
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3.7 Planar averaged real space probability amplitudes of lowest conductionand topmost valance states of [001] Si (a) and As terminated GaAs UTBs(b) by HSE06 and ETB calculations. With the real space TB basis func-tions, the realspace probability amplitudes of TB calculations show rea-sonable agreement with the HSE06 probability amplitudes. UTBs contain17 Si/GaAs atomic layers (with thickness 4a0). . . . . . . . . . . . . . 34
3.8 ETB atom site resolved probability amplitudes of Si ((a),(c)), and Asterminated GaAs ((b),(d)) UTBs using ETB parameters in this chapterand previous work [14,36]. The ETB atom site probability using differentparameters are qualitatively similar in Si UTB, while the ETB atom siteprobability in As terminated GaAs are more sensitive to the parametersets and passivation models, i.e. the valence states with parameters andpassivation model by previous work are not confined. UTBs contain 17atomic layers(thickness is 4a0). . . . . . . . . . . . . . . . . . . . . . . 35
3.9 Comparison of ETB wave functions using different ETB parameters andpassivation model. (a) and (b) use ETB parameters by Boykin et al.[14]. (a) and (c) correspond to implicit passivation model [16]; (b) and(d) correspond to explicit passivation model. The ETB parameters withthe explicit passivation model shows the most confined states, while theprevious parameters and implicit passivation model lead to less confinedstates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.10 Comparison of ETB wave functions using ETB parameters by Boykin etal. [14] (a) and modified parameters (b). In (b), the modified parametersare V ′padcσ = Vpadcσ + 0.3eV , V ′padcπ = Vpadcπ − 0.3eV . With modifiedparameters, the unconfined valence states become confined ones. . . . 38
3.11 Band gaps of Si UTBs (a) and As terminated UTBs (b) by HSE06 andETB calculations. For the presented UTBs with thickness ranging from1nm to 4.5nm, the ETB band gaps have discrepancies of less than 10meVcompared with band gaps by HSE06. The band gap changes by effectivemass calculation show agreement with HSE06 for Si UTBs thicker than3nm. While the effective mass calculations has significant discrepanciesfor all GaAs UTBs. The HSE06 and ETB calculations using parametersin this chapter consider Hydrogen atoms explicitly, while the ETB calcu-lations using parameters by previous work is based on implicit passivationmodel [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1 Strained systems considered in the parameterization process of strainedmaterials. (a) hydrostatic strain, (b) with two bond length changes, (c)diagonal strain with εxx = εyy = −0.5εzz, (d) off-diagonal strain withεxy 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
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4.2 Band structure of group IV materials with ETB and HSE06 calculations.Presented band structures of IV materials include Si (a) and Ge (b). ETBband structures of group IV materials are in good agreement with HSE06results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Band structure of III-V materials with ETB and HSE06 calculation. Pre-sented band structures of III-V materials include (a) AlP , (b) GaP , (c)InP ,(d) AlAs , (e) GaAs,(f) InAs ,(g) AlSb ,(h) GaSb ,(i) InSb. ETB bandstructures of group III-V materials are in good agreement with HSE06 re-sults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4 Band structure of Si with different lattice constants( with hydrostaticstrain ). (a) Si with a lattice constant of 5.4 A, (b)Si with a lattice con-stant of 5.8 A, (c) direct and indirect band gaps of Si with different latticeconstants. When lattice constant is 5.4 A, Si is a indirect gap semicon-ductor, the X conduction valley is the lowest conduction valley. As latticeconstant increase, the band gap at of X valley (Eg(X)) increase slightly,while the bandgap of L valleys (Eg(L)) and direct band gap (Eg(G) sband) decrease significantly. The lowest conduction bands transit from Xvalleys to L valleys at about 5.8 A. When lattice constant reaches 6.0 A,lowest conduction band become Γ valley. . . . . . . . . . . . . . . . . 58
4.5 HSE06 band structure of unstrained (a) and strained (b) GaAs top valencebands. The valence band splitting (c) at Γ point v.s. strain componentεxx. The strain imposed on GaAs in (b) satisfies εxx = εyy = −0.5εzz.The top valence bands of unstrained GaAs are degenerate heavy and lighthole bands. The degeneracy of top valence bands is broken in the strainedGaAs, As a result, both band edges and effective masses of top valencebands change due to strain. The ETB calculations of the band edge split-ting of top valence bands agree well with the HSE06 results. . . . . . . 59
4.6 X conduction valleys in unstrained (a) and strained (b) GaAs. The con-duction band edges (c) at X points vs strain component εxx. The strainimposed on GaAs in (b) satisfies εxx = εyy = −0.5εzz. The six X conduc-tion valleys of unstrained GaAs are degenerate due to crystal symmetry.The degeneracy of conduction valleys is broken in the strained GaAs. TheETB calculations of the band edge for conduction valleys (X points) agreewell with the HSE06 results. . . . . . . . . . . . . . . . . . . . . . . . . 60
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4.7 Conduction band edge splitting (a) and effective mass changes (b) in Siunder [110] shear strain. The unstrained Si conduction bands are 4 folddegenerate at X points. While under shear strain with εxy, the degeneracyat X point(with k = [0, 0, 1]2π
a) is broken as shown in (a). As a result, the
transverse effective masses of the conduction valley at k = [0, 0, 0.84]2πa
arechanged. The ETB calculations of the band edge splitting and effectivemass of conduction bands agree well with the HSE06 results. . . . . . 61
4.8 Band edge splitting of selected conduction bands and valence bands atΓ ((a),(b)), X((c),(d)) and L((e),(f)) points of InAs. At Γ point, 6 topmost valence bands and 2 lowest conduction bands are shown. 4 lowestconduction bands at X points are shown. The lowest conduction bandat L points are included in the figures. The valence bands at X andL points are not shown as those points are of low energy. Two differentstrain is applied on the InAs lattice, including strain caused by stress along[123] and strain caused by biaxial strain along [111]; The ETB band edgesplitting are in good agreement with the corresponding HSE06 results. 62
5.1 Atom structure of Si/Ge and XAs/YAs superlattices. X,Y can be differentcations. (a) Si/Ge superlattice with 4 layers in the primitive unit cell; (b)Si/Ge superlattice with 8 layers in the primitive unit cell. (c) XAs/YAssuperlattice with 4 layers in the unit cell; (d) XAs/YAs superlattice with8 layers in the unit cell. The primitive unit cells are marked by the dashedlines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 Three types of heterojunctions formed by two semiconductors( denotedby A and B ). (a) type-I heterojunctions with straddling gap, (b) type-IIheterojunctions with staggered gap and (c) type-III heterojunctions withbroken gap. The shaded area stands for the states with electrons occu-pied. For type-I and type-II, there is no obvious charge transfer betweentwo different materials. However there is significant charge transfer fromvalence bands of material A to conduction bands of material B, since thetop valence band in material A is higher than the conduction band ofmaterial B in a type-III heterojunction. . . . . . . . . . . . . . . . . . . 77
5.3 Local potential of cations in a GaAs/AlAs 001 superlattice. (a) Superlat-tice considered contains contains As (black dots), Ga (white dots) and Al(grey dots) atoms. The local potential of cations along the dashed straightline shown in (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
xv
Figure Page
5.4 Band structures of Si/Ge superlattices by ETB and HSE06 calculations.ETB and HSE06 bandstructures are in good agreement. Figures corre-spond to band structures of superlattices which contain (a) 4 atoms and(b) 8 atoms, and band gaps of Si/Ge superlattices verse the number ofatoms in the supercell (c). . . . . . . . . . . . . . . . . . . . . . . . . 79
5.5 Band structures of Arsenides superlattices by ETB and HSE06 calcula-tions. ETB and HSE06 bandstructures are in good agreement. Pre-sented band structures include band structures of superlattices of 001AlAs/GaAs((a),(d)), InAs/GaAs((b),(e)) and InAs/AlAs((c),(f)). Upperfigures correspond to supercells which contain 4 atoms(Fig 5.1 (a)), whilelower figures corresponds to supercells with 8 atoms (Fig 5.1 (b)). . . 80
5.6 Band structures InAs/GaSb superlattices by ETB and HSE06 calcula-tions. Present superlattices include 4 layer (a) and 8 layer (b) InAs/GaSbsuperlattices. Corresponding band structures are shown in (c) 4 layerInAs/GaSb and (d) 8 layer InAs/GaSb. . . . . . . . . . . . . . . . . . 81
5.7 Band gaps of III-V superlattices by ETB and HSE06 calculations. TheETB band gaps of different superlattices show good agreement with HSE06results, demonstrating the ETB parameters have good transferability. Thepresented band gaps include superlattices of (a) XP/YP , (b) XAs/YAsand (c) XSb/YSb with ( X and Y stand for different cations, X,Y = Al,Ga or In) and (e) AlX/AlY , (f) GaX/GaY and (g) InX/InY with ( X andY stand for different anions, X,Y = P, As or Sb). . . . . . . . . . . . . 82
6.1 Top view and side view of atom structures of monolayer (a) TMD and (b)black Phosphorus. The first Brillouin zone of TMDs is hexagonal (c), theblack Phosphorus Brillouin zone is rectangular (d). High symmetry pointsconsidered in this work are defined in figures (c) and (d). . . . . . . . 85
6.2 ETB and GGA band structure of monolayer MoS2(a) and WSe2(b). ETBband structures agree well with the GGA result. . . . . . . . . . . . . 91
6.3 ETB and GGA band structure of bilayer MoS2(a) and WSe2(b). ETBband structures agree well with the GGA result. . . . . . . . . . . . . 91
6.4 ETB and HSE06 band structure of (a) monolayer and (b) bilayer blackPhosphorus. The ETB results agree well with the corresponding HSE06results of both monolayer and bilayer black Phosphorus. . . . . . . . . 93
A.1 Tight binding orbitals used in spd tight binding model. (a) s orbitals, (b)p orbitals, (c) d orbitals. The green color correspond to positive sign whilethe blue color correspond to negative sign. . . . . . . . . . . . . . . . 103
xvi
Figure Page
A.2 Two center integrals among p orbitals for chemical bond along z direction.(a) pz−pz coupling gives Vppσ , (b) px−px coupling gives Vppπ , (c) pz−pxis 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
C.1 Orbital weighted band structures of single layer MoS2 . (a) d0 = dz2 (Moatom) weighted band structure, the width of blue region represent theweight of d0 ; (b) d2 = dxy, dx2−y2 (Mo atom) weighted band structure, thewidth of red region represent the weight of d2. . . . . . . . . . . . . . 110
I.1 Contribution of p orbitals to the top valence states (a) and s orbitals tothe lowest conduction states of InAs. The p orbitals of In and As atomscontribute to the top valence bands. The modification of lattice constantwithin 2% only change wave functions sightly, thus it is reasonable toadjust band structure by changing lattice constant to match experiments. When lattice constant change one percent, p orbitals contribution arechanged by less than 0.0002. The s orbitals of In and As atoms contributeto the lowest conduction bands. When lattice constant is changed by onepercent, s orbitals contribution are changed by less than 0.02. . . . . . 122
xvii
ABSTRACT
Tan, Yaohua PhD, Purdue University, May 2016. Tight binding parameterizationfrom ab-initio calculations and its applications. Major Professor: Gerhard Klimeck.
Modern semiconductor nanodevices have reached critical device dimensions in
the range of several nanometers. Innovative nano-electronic devices make use strain
techniques and low dimensional geometries such as nanowire and superlattices to
strengthen device performances. Except for typical semiconductors, esoteric materi-
als such as transition metal dichalcogenide and black Phosphorus enter close device
considerations. To predict device performances, it is critical to have a model that is
capable to handle strains, interfaces and new materials. Ab-initio methods are com-
putationally too expensive for device level simulations; instead, the empirical tight
binding(ETB) method is computationally much cheaper and thus is preferred in de-
vice level simulations. The reliability of ETB simulations depends strongly on the
choice of basis sets and the transferability of ETB parameters. The traditional way
of ETB parameterization is by fitting to experimental data rather than a founda-
tional mapping. The parameters parameterized by traditional ways have potential
transferability issues when applied to nano-structures such as ultrathin bodies and
heterostructures. A further shortcoming of traditional ETB is the lack of explicit
basis functions.
In the present work, an algorithm that constructs ETB parameters and explicit
basis functions from ab-initio calculations is developed. This method takes account of
wave functions and band structures in the parameterization process. Parameters ob-
tained by this process are more transferable than those obtained by traditional fitting.
The algorithm is applied to group IV and III-V semiconductors. Unstained materials
and corresponding ultra-thin bodies are studied by ETB and ab-initio calculations.
xviii
The ab-initio band structures of bulk materials matches the experimental results un-
der room temperature. Tight binding band structures and wave functions agree well
with ab-initio calculations in both bulk and ultra-thin bodies cases, while unphysical
states are observed tight binding analysis of ultra-thin bodies using existing param-
eters and passivation models. For strained materials, a tight binding model that
models the effect of arbitrary strain is developed. The tight binding parameters are
obtained by considering ab-initio results of multiple strained systems in the presented
parameterization algorithm. The tight binding parameters show good transferability,
as the tight binding calculations of strained superlattices show agreement with corre-
sponding ab-initio calculations well. Further more, the parameterization algorithm is
applied to newly appeared 2D materials transition metal dichalcogenides and black
phosphorus. Tight binding model and parameters that are generic for multilayer 2D
materials are obtained.
1
1. INTRODUCTION
1.1 Background: Device scaling and the role of empirical tight binding
modeling
Scaling of semiconductor transistors has continued in the past few decades due
to the immense progress made in fabrication technologies. Modern semiconductor
nanodevices have reached critical device dimensions in the sub-10 nanometer range.
To surpass the limits of downscaling of field effect transistor, innovative devices de-
signs make use of complicated two or three dimensional geometries and are composed
of multiple materials. Geometries such as Ultra Thin Body(UTB) [1], FinFETs [2],
nanowires [3] and superlattices [4, 5] structures are adopted in nanometer scale de-
vice design to maintain the device performance. Except for typical semiconductors
such as Silicon and Germanium, additional materials families such as III-V Sb, P,N
and C families as well as more esoteric materials including Topological Insulator(TI),
Transition Metal Dichalcogenides(TMD) [6] and Black Phosphorus [7] enter close de-
vice considerations. Typically, there are about 10 thousands to 10 million atoms
in the active device region with contacts controlling the current injection. Most of
the electrically conducting devices are of finite extent with contacts controlling the
current injections and potential modulation. These finite sized structures suggest an
atomistic, local and orbital-based electronic structure representation for device level
simulation. Quantitative device design requires the reliable prediction of the mate-
rials’ band gaps and band offsets within a few meV and important effective masses
within a few percent in the active device regions.
Ab-initio methods that have a few adjustable parameters offer atomistic represen-
tations with subatomic resolution for variety of materials. However, ab-initio methods
are in general too expensive to be applied to systems containing millions of atoms.
2
Accurate models such as Hybrid functionals [8], GW [9] and BSE approximations [10]
are computationally far too expensive and can only be applied to systems containing
a few tens of atoms. Furthermore ab-initio methods based on the typical density func-
tional theory approach are fundamentally an equilibrium theory which cannot truly
model out-of-equilibrium device conditions where a large voltage might have been
applied to drive carriers from the source to the drain. At best an ab-initio based
calculation can deliver completely coherent transport that does not mix energies and
momentum. Empirical methods such as the empirical tight binding methods are nu-
merically much more efficient. ETB models usually make use of a few localized basis
functions per atom, and consider short range interactions among atoms. ETB has
been successfully applied to electronic structures of millions of atoms [11] as well as
on non-equilibrium transport problems that even involve inelastic scattering [12].The
accuracy of the ETB methods depend critically on the careful calibration of the em-
pirical parameters. The traditional way to determine the ETB parameters is to fit
ETB results to experimental data of bulk materials [13] [14].
1.2 Challenges for empirical tight binding modeling: parameterization
Most of the existing ETB models are fitted to unstrained or strained bulk systems
with pure material. However, when those models are applied to nano-structures with
confinement and interfaces, the transferability of the ETB models and parameters is
questionable.
First of all, there are ambiguities when applying traditional ETB model to het-
erostructures. Traditional ETB parameters depend on materials, while material type
adjacent to interfaces can not be clearly defined. Fig.1.1 shows three possible defini-
tions of materials near a GaAs/AlAs interface. Interface As atoms can be interpreted
as (a) As in AlAs or (b) As in GaAs. A third assumption, shown by definition (c), is
to take the interface As atoms to have an average of the onsite potentials. All those
definitions are customarily used but with no hard data to justify.
3
(a)
(b)
Al: Ga: As:
AlAs GaAs
GaAsAlAs
Interface atoms are
considered as AlAs
Interface atoms are
considered as GaAs
AlAs GaAs
Interface atoms are considered
as averaged As
(c)
Fig. 1.1. Different definitions of materials at a GaAs/AlAs interface whichlead to different results. Regions of AlAs and GaAs materials are sepa-rated by the dashed line. In definitions (a) and (b), the left part is AlAsand the right part is GaAs. In definition (a), the interface As atoms aredefined as atoms in AlAs; in definition (b), the interface As atoms aredefined as atoms in GaAs. In the case (c), the interface As atoms aredefined as As atoms in an average material of AlAs and GaAs.
Secondly, the ETB parameters obtained by direct fitting possibly lead to unphys-
ical results [15] when applyed to nano-structures such as UTBs, nanowires and more
complicated geometries. The application of existing ETB parameters [14] with an im-
plicit passivation‘ model [16] to As terminated GaAs UTB shows unphysical valence
states as it is shown in Fig.1.2. The real space eigen functions of the topmost valence
4
bands are sine like confined states, while the corresponding ETB results with existing
parameters and passivation model turn out to be exponential-like unconfined states.
The unphysical ETB valence states are more dependent on the quality of bulk param-
eters rather than the passivation parameters, as these states are changed only sightly
when the passivation parameters are tuned. Furthermore, the ETB basis functions
position (a0 )0 2 4
0.05
0.1
0
0.05
0 2 4
position (a0 )
HSE
06
Re
al s
pac
e |ψ
|2(r
)
ETB
Ato
m s
ite
re
solv
ed
|ψ|2
(b) (a)
Fig. 1.2. ab-initio and tight binding wave functions in a As terminatedGaAs UTB. ETB calculation shows unphysical top valence states. ETBwave functions by existing parameters show different envelope with theab-initio wave functions. In a As terminated GaAs UTB, ab-initio wavefunctions (a) of the top valence bands are sine-like confined states, whilethe ETB valence states (b) turn out to be exponential-like states which isunphysical. (a0 is the lattice constant)
.
remain implicitly defined in the traditional fitting process. The lack of explicit basis
functions makes it difficult to predict wave function dependent quantities like optical
matrix elements with high precision.
Therefore, a more fundamental fitting process that relates both the band struc-
ture and the wave functions of ETB models with ab-initio calculations is desirable
to generate tranferable ETB models. Existing approaches to construct localized ba-
sis functions and tightbinding-like Hamiltonians from ab-initio results include maxi-
mally localized Wannier functions(MLWF) [17–19], quasi-atomic orbitals [20, 21], or
DFT-TB analysis [22]. The MLWFs are constructed using Bloch states of either iso-
5
lated bands [17] or entangled bands [18]. However, these methods do not eliminate
the transferability problem of the commonly used orthogonal sp3d5s* ETB models
with first nearest neighbour interactions, since these methods typically include in-
teratomic interactions beyond first nearest neighbors. Moreover, these approaches
usually disregard excited orbitals (i.e. s* and d orbitals for diamond and zincblende
semiconductors) which are often needed to correctly parameterize conduction bands
of semiconductors.
1.3 Contribution of the present work
In this thesis, a parameterization algorithm is presented that ”maps” ab-initio
results (i.e. eigenenergies and eigenfunctions) to tight binding models. In present
mapping algorithm, wavefunction-derived ETB parameters for the Hamiltonian and
highly localized basis functions are obtained. It is important to mention that the
ETB Hamiltonian of this method can be limited to first nearest neighbor interac-
tions. The mapping process is applied to unstrained and strained group IV and III-V
semiconductors. Tight binding model for arbitrarily strained material is developed
and parameterized. The application of ETB model to nano-structures such as UTB
and superlattices show that the ETB parameters by this work have good transfer-
ability. This method was also applied to additional materials such as MgO [23] ,
SmSe [24], transition metal dichacolgenides [25] and Black Phosphorus. Good agree-
ment between bulk ETB and ab-initio band structures is achieved for those materials.
1.4 Thesis organization
This Thesis is organized as follows.
• Chapter 2: This chapter focus on the algorithm of generating tight binding pa-
rameters and basis functions from ab-initio calculations. The method of trans-
formation between ab-initio and tight binding basis functions is described. The
6
process of tight binding parameterization from ab-initio calculation is described.
The implementation of the method is described in this chapter.
• Chapter 3: This chapter focuses on tight binding parameterization of unstrained
bulk and application of tight binding to ultra thin body (UTB) systems. The
parameterization algorithm introduced in chapter 2 is applied to unstrained bulk
materials. Both ETB parameters and explicit basis functions are generated.
The ETB parameters are applied to Hydrogen passivated UTBs. An explicit
passivation model is introduced and applied to UTBs. The ETB band structure
and wave functions of the UTB systems are compared with ab-initio results.
• Chapter 4: This chapter focuses on the ETB model for arbitrarily strained
materials. In this chapter, a generic tight binding model for arbitrarily strain
materials is described. The ETB parameters for strained group IV and III-V
materials are parameterized with respect to ab-initio calculations of different
strained systems. ETB parameters are also adjusted to match experimental
results.
• Chapter 5: This chapter investigates the transferability of the ETB parameters
obtained in chapter 4 using different superlattices. The ETB band structures
of superlattices formed by group IV and IIIV materials are compared with
corresponding Hybrid functional results.
• Chapter 6: This chapter focuses on tight binding parameterization of 2D mate-
rials. 2D materials including transition metal dichacolgenides(TMDs) and black
Phosphorus are introduced. Tight binding models for TMDs and black Phos-
phorus are developed. Corresponding tight binding parameters are obtained
with respect to ab-initio calculations.
• Chapter 7: Summary of the contribution in this thesis. Outlook of the oppor-
tunities and challenges for tight binding parameterization.
7
2. TIGHT BINDING PARAMETERIZATION FROM
AB-INITIO CALCULATIONS
2.1 Introduction
There are intensive needs to model nano-electronic devices atomistically. Accurate
ab-initio methods are in general too expensive to be applied to device level systems
which contain ten thousands to millions of atoms in the active domains. Numerically
efficient empirical models such as empirical tight binding(ETB) model requires trans-
ferable parameters in order to perform reliable device simulations. However tight
binding parameters generated by direct fitting do not guarantee good transferability
and have no explicit basis functions. Furthermore, reliable experimental data for
newly discovered exotic materials is not adequate for tight binding parameterization.
Thus a method that extract transferable ETB parameters from ab-initio calculations
is required. For this purpose, a prototype algorithm was developed to generate ETB
parameters from ab-initio results by Ref. [23]. This prototype method was already
applied to several materials such as GaAs, MgO [23] and SmSe [24] and yielded a
good agreement between bulk ETB and ab-initio band structures. The ETB models
parameterized considers first or second nearest neighbor interactions. However, the
resulting wave functions did not satisfactorily agree with the ab-initio wave functions.
This chapter outlines an improved process of generating tight binding parameters
from ab-initio calculations. The main issues covered by this chapter include (1)
details of the ab-initio mapping algorithm that generates tight binding parameters
and basis functions from ab-initio calculations; (2) the method of transforming ETB
basis functions to ab-initio basis functions since the ab-initio calculations considered
in this work are mainly based on plane wave formalism; (3) the extended method to
parameterize multiple materials; (4) implementation of the method.
8
2.2 Algorithm: tight binding parameterization from ab-initio calcula-
tions
The general process of the parameter mapping from ab-initio results to ETB
models is shown in Fig. 2.4. As will be shown in the following, the ETB parameters
and basis functions are obtained in an iterative fitting procedure that spans over 5
steps (with steps 3 through 4 being iterated). The resulting 1st nearest neighbor
Hamiltonian HTB (k) is of Slater-Koster table type. The Slater-Koster type tight
binding hamiltonian is introduced in appendix A. . The resulting basis Bfinal is
composed of orthonormal real space functions Bfinal =
Ψfinaln,l,m (r)
which have the
shape (vectors are given in bold type)
Ψa,n,l,m (r) = Yl,m (θ, φ) Ra,n,l (r) +∑l′,m′
(l′,m′)6=(l,m)
Yl′,m′ (θ, φ) Ra,n,l,l′,m′ (r) . (2.1)
Here, a labels the atom type, whereas the n, l and m are principle, angular and mag-
netic quantum numbers, respectively. All materials considered in this work contain
no magnetic polarization. Therefore, the basis functions are spin independent. The
tesseral spherical harmonics Yl,m (θ, φ) describe the dependence of the basis functions
on the angular coordinates θ and φ. The functions Ra,n,l (r) and Ra,n,l,l′,m′ (r) define
the radial r dependence of the basis functions. The contribution of Ra,n,l,l′,m′ to the
basis functions is much smaller than the contribution of Ra,n,l. The detailed shapes
of the radial functions Ra,n,l (r) and Ra,n,l,l′,m′ (r) are subject to the fitting algorithm.
Step 1: First, electronic band structures εAbj (k) and wave functions ψAbj,k are
solved which serve as fitting targets to the overall mapping algorithm
HAb (k)∣∣ψAbj,k⟩ = εAbj (k)
∣∣ψAbj,k⟩ . (2.2)
The index j corresponds to the band index and k represents a momentum vector
in the first Brillouin zone. In principle, any method that is capable of solving band
9
Step 1: Ab-initio calculations,
ab-initio eigen states and energies are obtained
Step 3: Project ab-initio wave functions on TB
basis functions
Step 5: Get the exact TB basis functions
Step 2: Initial guess of TB basis functions
and Slater Koster type TB parameters
Step 4: calculate TB band structures and wave
functions; compares targets a) TB band structures,
b) TB effective masses and c) TB wave functions
with ab-initio results.
updated basis
functions and
TB parameters
.
Fig. 2.1. The process of mapping from ab-initio calculations to tightbinding model. This process extract tight binding parameters and basisfunctions from ab-initio results iteratively. In this process, tight bindingband structure and wave functions are fitted to the corresponding ab-initioresults.
diagrams and explicit basis functions can provide these fitting targets. Throughout
this work, however, hybrid functional calculations are performed for step 1 [26].
Step 2: In the second step, initial guesses for the ETB basis functions and ETB
parameters are defined. During the fitting process, the ETB basis Binitial is spanned
by non-orthogonal functions Φa,n,l,m (r) given by
Φa,n,l,m (r) = Yl,m (θ, φ)Ra,n,l (r) . (2.3)
The Ra,n,l (r) in Eq. (2.3) differ from the Ra,n,l (r) of the final basis functions in
Eq. (2.1). The Ra,n,l (r) can be represented by linear combination of functions gn (r)
by Ra,n,l (r) =∑
n sigi (r), where si are adjustable parameters and gi (r) are analytical
functions. The general requirement of Ra,n,l (r) is that it has to be a localized function,
meaning it should decays as r increase. The function with a tail that decay rapidly
is preferred in this work. Some of the analytical formulas are suggested in appendix
E, including Gaussian, Slater and numerical type basis functions. Those different
definitions lead to very similar results following the process in this chapter. However,
analytical formulas that are not listed can possibly be used as well. The details of the
10
initial guesses for the diagonal and off-diagonal elements of the Hamiltonian HTB (k)
are not essential for the overall algorithm. Nevertheless, initial guesses that follow the
framework of existing ETB parameter sets improve the overall fitting convergence.
Urban et al. and Lu et al. discuss that interactions up to third nearest neighbors
might be needed to exactly reproduce ab-initio results [21, 22]. In contrast, we find
that the interatomic interaction elements of HTB (k) can be limited to first nearest
neighbor interactions throughout this work while still reproducing ab-initio results
very well.
Step 3: The nonorthogonal basis functions Φa,n,l,m (r) in position space are trans-
formed into the Bloch representation [27]Φa,n,l,m,k (r)
|Φα,k〉 ≡ Φa,n,l,m,k (r)
=∑R
exp [ik· (R + τ a)] Φa,n,l,m (r−R− τ a) , (2.4)
where τa is the position of atom type a in the unit cell and the sum runs over all unit
cells of the system with R, the position of the respective cell. To improve readability
of all formulas in the Dirac notation, the indices of atom type and quantum numbers
are merged into Greek indices α = (a, n, l,m). For the further steps, an orthogonal
basis Bortho = |Ψα,k〉 is created out of the basis Binitial with Lowdin’s symmetrical
orthogonalization algorithm [28]. Since steps 4 and 5 are formulated in the basis
Bortho, the wave functions∣∣ψAbj,k⟩ of step 1 must be transformed into this basis∣∣ψAbj,k⟩ ≈ P (k)
∣∣ψAbj,k⟩ =∑α
cj,α (k) |Ψα,k〉 , (2.5)
where
cj,α (k) =⟨Ψα,k
∣∣ψAbj,k ⟩ , (2.6)
cj,α (k) =⟨Ψα,k
∣∣ψAbj,k ⟩ ,with the projection operator
P (k) =∑α
|Ψα,k〉 〈Ψα,k| . (2.7)
Equation (2.5) contains an approximation of the ab-initio wave functions in so far
that the sum over α extends only over those orbitals that are included in the tight
11
binding basis Bortho. This basis and Bortho of similar ETB models have much fewer
basis vectors than the input ab-initio calculation. This rank reduction is a typical
outcome of rectangular transformations such as P and is well known in the field of
low rank approximations [29].
In order to calculate product⟨Ψα,k
∣∣ψAbj,k⟩, basis transformation between tight
binding basis functions and ab-initio basis functions is needed. Since the ab-initio
calculations considered in this work use plane wave basis functions. The basis trans-
formation is the Fourier transform of the tight binding basis functions. This is dis-
cussed in detail in the following section 2.3.
Step 4: Here, the quality of the ETB fitting is assessed. In this step, the band
structures of the current ETB model εTBj (k) and the ab-initio input εAbj (k) are com-
pared. If these sufficiently agree, the phases of the ETB wave functions are modulated
to agree with the ab-initio ones and both wave functions are compared after that.
The ETB Hamiltonian of step 2 is diagonalized in the basis Bortho of step 3 to obtain
ETB band structures εTBj (k) and eigen vectors∣∣ψTBj,k ⟩
HTB (k)∣∣ψTBj,k ⟩ = εTBj (k)
∣∣ψTBj,k ⟩ , (2.8)
with ∣∣ψTBj,k ⟩ =∑α
dj,α (k) |Ψα,k〉 . (2.9)
To assess the quality of the ETB results is assessed, different fitness functions
Fε,Fm and Fψ are defined for energies, masses and wave functions respectively. The
Fε and Fm are given by
Fε =∑j,k
wεj (k)∣∣εTBj (k)− εAbj (k)
∣∣2 . (2.10)
Fm =∑m
wm
∣∣∣∣mAb −mTB
mAb
∣∣∣∣2 . (2.11)
where wεj (k) and wm are weights defined for each target.
12
As a convention for wave functions phases, another set of ETB wave functions∣∣∣ψTBj,k ⟩ is introduced ∣∣∣ψTBj,k ⟩ =∑i
Vj,i (k)∣∣ψTBi,k ⟩ . (2.12)
The unitary transformation V (k) is defined by
Vj,i (k) =
⟨ψTBj,k
∣∣ψAbi,k ⟩λ (k)
, (2.13)
with
λ (k) =
√1
N
∑q,p
∣∣⟨ψAbq,k ∣∣ψTBp,k ⟩∣∣2. (2.14)
Here, the sum over p and q runs over all N ETB states∣∣ψTBp,k ⟩ and N ab-initio states⟨
ψAbq,k∣∣ with equivalent energies εTBp (k) ≈ εAbq (k). With this transformation, the
equation holds ⟨ψAbi,k
∣∣∣ψTBj,k ⟩ = λ (k) , (2.15)
for equivalent states. This phase adaption can only work if the ETB band structure
is close enough to the ab-initio result. The ETB wave function fittness is given by
Fψ =∑j,k
wψj (k)∥∥∥∣∣ψAbj,k⟩− ∣∣∣ψTBj,k ⟩∥∥∥2
. (2.16)
The weights wψj (k) are varying depending on respective fitting focusses. Deviations
of∣∣∣ψTBν,k⟩ from
∣∣ψAbν,k⟩ have in general two reasons: inadequate basis functions and/or
eigenfunctions of a poorly approximated ETB Hamiltonian. Therefore, Fψ can be
estimated as ∥∥∥∣∣ψAbj,k⟩− ∣∣∣ψTBj,k ⟩∥∥∥2
≤ 2∥∥∥[I − P (k)
] ∣∣ψAbj,k⟩∥∥∥2
+ 2∥∥∥P (k)
∣∣ψAbj,k⟩− ∣∣∣ψTBj,k ⟩∥∥∥2
. (2.17)
The first right hand side term of the last equation describes the deviation of the low-
rank approximated ab-initio wave functions. This becomes obvious with the projector
property P 2 (k) = P (k)∥∥∥[I − P (k)] ∣∣ψAbj,k⟩∥∥∥2
=⟨ψAbj,k
∣∣∣[I − P (k)]∣∣∣ψAbj,k⟩ . (2.18)
13
It should be noted that equation (2.18) is a quadratic form of tight binding basis
functions. The second term on the right hand side of Eq. (2.17) contains information
about the quality of the eigenfunctions of the approximate ETB Hamiltonian HTB (k).
This is understandable when Eqs. (2.5) and (2.12) are inserted into this term∥∥∥P (k)∣∣ψAbj,k⟩− ∣∣∣ψTBj,k ⟩∥∥∥2
=
2− 2 Re[∑
α,i c†j,α (k)Vj,i (k) di,α (k)
]. (2.19)
The fitness function Fψ represents the major improvement over the traditional ETB
eigenvalue fitting (e.g. typically limited to energies and effective masses). All fitness
functions are minimized by iterating over the steps 3 and 4: the Slater-Koster type
parameters for the ETB Hamiltonian HTB (k) and the parameters of the radial ETB
basis functions Ra,n,l (r) are adjusted for every iteration of step3.
In general, more localized basis functions lead to more transferable tight binding
parameters. However, the minimization of Fψ does not guarantee the most localized
basis functions. To get more localized basis functions, simple parameter constraint
can be added to ensure basis functions have rapidly vanishing tails for large r. It is
also helpful to put more emphasis in the region near each atom when comparing the
tight binding wave functions with the ab-initio ones.
Step 5: Once the fitness functions are small enough to cease the iterations, it
is assumed that those eigenfunctions of the ETB Hamiltonian HTB (k) that were
subject to the fitting are identical to the eigenfunctions of the ab-initio Hamiltonian
HAb (k) after a transformation A (k)∣∣ψTBj,k ⟩ ≈∑i
Aj,i (k)∣∣ψAbi,k⟩ . (2.20)
This transformation A is determined by a singular value decomposition of the rect-
angular overlap matrix of ab-initio eigenstates with ETB eigenstates⟨ψAbi,k
∣∣ψTBj,k ⟩ =∑p
Ui,p (k) Σp,p (k)Wp,j (k) . (2.21)
The row index i runs over all ab-initio eigenstates - exceeding those that served as
fitting targets, whereas the column index j covers all the ETB eigenfunctions. The
14
Σ and W are square and U is a rectangular matrix. The transformation A is then
defined as
Aj,i (k) =∑p
Wj,p (k)U †p,i (k) . (2.22)
A is constructed from relevant columns of a unitary transformation. Combining
Eqs. (2.20) and (2.9) allows to determine the Bloch periodic final basis functions∣∣Ψfinalα,k
⟩=∑i,j
d†α,j (k)Aj,i (k)∣∣ψAbi,k⟩ . (2.23)
The real space counterpart of∣∣Ψfinal
α,k
⟩is given by
Ψfinalα (r−R− τ) =
V
(2π)3
∫BZ
dke−ik·(R+τ)Ψfinalα,k (r) . (2.24)
2.3 Transformation of basis functions
In order to relate ab-initio wave functions which are represented by plane waves
with tight binding wave functions which are represented by localized basis functions.
Basis transformation between ab-initio wave functions and tight binding wave func-
tions is needed. In this work, the considered ab-initio wave functions are presented
in plane wave basis functions, thus the basis transformation can be done through
Fourier transform between localized basis functions and plane wave basis functions.
By Fourier transformation of tight binding Bloch basis functions, tight binding Bloch
basis functions and ab-initio are both presented by plane wave basis functions.
For a given localized basis function Ψlm (r) = Ylm (Ωr)ψ (r), its Fourier transform
and inverse Fourier transform are given by
Ψlm (k) =1
(2π)3/2
∫Ψlm (r) e−ikrdr (2.25)
Ψlm (r) =1
(2π)3/2
∫Ψlm (k) eikrdk. (2.26)
These transformations involved integrals in 3 dimensional real space or k-space. It
is numerically expensive to store the basis function and to perform 3 dimensional
integral with high precision.
15
However, given the form of the localized basis function Ylm (Ωr)ψ (r), the general-
ized 3 dimensional integrals can be written in numerically convenient expression. To
achieve this, following expansion which relate a plane wave function with wave vector
k to spherical harmonics
eikr =∞∑l=0
l∑m=−l
4πiljl (kr)Y∗lm (Ωk)Ylm (Ωr) (2.27)
is useful. Here the jl (x) is the spherical Bessel function.
By inserting equation (2.27) in to Fourier transform (2.25), the Ψlm (k) can be
written as
Ψlm (k) = Ylm (Ωk)ψlm (k) . (2.28)
where
ψlm (k) =
√2
π(−i)l
∫ ∞0
r2drjl (kr)ψlm (r) (2.29)
It should be noted that the Fourier transform of Ψlm (r) do not change the angular
quantum number l and m. It is numerically much more convenient to store the
function given by equation (2.28) since only the one dimensional radial part needs
to be stored. As the spherical bessel functions jl (x)’s can be written as jl (x) =
P (x) sin (x) +Q (x) cos (x), fast Fourier transform exist for one dimentional integral
given by equation (2.29) [30]. The details of fast Fourier transform is describe in
appendix B. Since equation (2.29) involves integrals in one dimension, much higher
precision can be achieved compared with 3 dimensional integrals in equations (2.26)
and (2.25).
In periodic systems, the basis function are Bloch basis function rather than a
stand alone localized basis function. The Bloch basis function is given by
|Ψk〉 =∑R
exp [−ik · (R + τ)]Ψ (r−R− τ) , (2.30)
where R + τ is the center of each atoms. By using the equation (2.26), the Bloch
basis function can be represented in plane wave basis functions
|Ψk〉 =1
(2π)3/2
∑G
exp (−iG · τ)Ψ (k + G) |k + G〉, (2.31)
16
where the plane wave basis function |k + G〉 = 1√V
exp i (k + G) · r. This equation
relate the tight binding like basis functions (Bloch basis functions) to ab-initio basis
functions (plane wave functions).
Alternative way to relate ab-initio wave functions and tight binding wave functions
is to transform both of them into 3 dimensional realspace. While the difficulty of
transforming Bloch tight binding basis functions into realspace is: a lot of neighbors
are possibly needed to be considered in order to represent the realspace Bloch tight
binding basis functions exactly unless the basis functions is highly localized. To make
the problem easier, assumption that the tight binding basis functions of each atoms
are orthogonal to all its neighbors in the non-overlapping regions can be made. By
this assumption, the contributions from neighbors to each region is neglected. The
details of the approach that relate ab-initio wave functions and tight binding wave
functions in realspace is discussed in appendix C.
2.4 Simplified fitting process
In the process flow described in section 2.2, the tight binding basis functions
and tight binding Hamiltonian in principle need to be optimized at the same time.
However, this process could involve a large number of fitting parameters. To make
the fitting easier, it is convenient to fit basis functions and tight binding parameters
separately.
However, the difficulty of fitting basis and tight binding parameters independently
comes from the comparison of wave functions. To compare tight binding wave func-
tions with corresponding ab-initio wave functions, both tight binding basis functions
and tight binding Hamiltonian are needed. The inequality (2.17) decouples the contri-
bution of tight binding basis functions and tight binding Hamiltonian in the compar-
ison of wave functions. The first term in the inequality (2.17) describes the quality
17
of tight binding basis functions. Here we define the sum of the first term over all
considered bands as F(1)ψ
F(1)ψ =
∑j,k
⟨ψAbj,k
∣∣∣[I − P (k)]∣∣∣ψAbj,k⟩ . (2.32)
To estimate F(1)ψ , the knowledge of tight binding Hamiltonian is not required since
only tight binding basis functions appear in the expression. In the second term of
inequality (2.17), the tight binding basis functions do not appear explicitly. Here we
define the sum of the second term over all considered bands as F(2)ψ
F(2)ψ =
∑j,k
[2− 2 Re
(∑α,i
c†j,α (k)Vj,i (k) di,α (k)
)]. (2.33)
The origin of F(2)ψ is the error introduced by the tight binding model itself. The F
(2)ψ
are non-zero as the shortest range interactions are considered in the tight binding
model in this work. In fact, if the tight binding models do not neglect long range
interactions, the term F(2)ψ can vanish. In this case, the tight binding Hamiltonian
can be obtained through basis transformation of the the ab-initio Hamiltonian, and
no fitting of Hamiltonian is needed.
.
Fig. 2.2. The Simplified process of mapping from ab-initio calculationsto tight binding model. In the simplified process, prototype tight bindingbasis functions are optimized in the first step. In the second step, the tightbinding Hamiltonian is optimized after the basis functions are obtained.The basis functions are kept unchanged in the second step. And exactbasis functions are extracted in the last step.
18
The minimization of F(1)ψ corresponds to basis functions optimization. While
for tight binding Hamiltonian optimization, F(2)ψ and other band structure dependent
fitness are considered. Thus instead of minimize the left hand side of inequality (2.17),
it is more convenient to minimized F(1)ψ and F
(2)ψ . The fitting process can be simplified
as process shown by Fig. 2.2. In the simplified process, three steps are involved. 1)
tight binding basis functions are optimized by minimizing F(1)ψ ; 2) Tight binding
Hamiltonian are optimized by minimizing F(2)ψ and other band structure dependent
fitness after tight binding basis functions are obtained; 3) Exact basis functions are
extracted (This step is the same as the step 5 in the process described in Fig. 2.4).
Since F(1)ψ is a quadratic form of tight binding basis functions, it can be minimized
relatively easily. To understand this quadratic form, the F(1)ψ can rewritten as
F(1)ψ =
∑j,k
1−∑α,k
〈Ψα,k|Qk|Ψα,k〉 (2.34)
where the Qk’s are positive definite matrices, given by
Qk =∑j
|ψAbj,k〉〈ψAbj,k| (2.35)
If the considered regions are limited to non-overlapping spherical regions, F(1)ψ can
be minimized in straightforward way which is described in Appendix D. It turns out
that the simplified process can generate tight binding parameters of similar quality
compared with the full fledged process. This fact also suggests that it is important
for tight binding basis functions to present the ab-initio wave functions near each
atoms. This is also coincident with the observation mentioned in section 2.2 that
more localized basis function lead to better transferability. In fact, the non-overlap
region is equivalent to put weight 0 to the interstitial region while put weight 1 inside
the spherical region.
2.5 Parameterization algorithm of multiple materials
Good transferablity of the tight binding parameters implies the tight binding cal-
culations of different systems with the same parameter set are accurate. To ensure
19
the tight binding parameters are transferable, multiple systems have to be considered
in the parameterization process. The process flow described in section 2.2 can be
applied to either single system or multiple systems. The non-orthogonal basis func-
tions set given by equation 2.3 are required to be atom type dependent. For multiple
systems, ab-initio calculations has to be carried out for all considered systems in step
1. The fitness function to be minimized which was defined in step 4 become a sum
of the fitness of all systems. The total fitness calculated of section 2.2 is defined as a
summation of fitness of all systems considered (labeled by index s).
Ftot =∑s
Fs (2.36)
The fitness Fs is defined to capture important targets of each system considered in
the parameterization process. For simplified procedure described in section 2.4, the
fitness Fs can also be defined for step 1 and step 2.
For example, in order to obtain parameters for a generic tight binding model
which work for arbitrarily strained materials, multiple strained systems have to be
considered. The strained systems used in this work include a) systems with hydro
static change(1 equivalent patterns), b) systems with pure bond length changes, c)
systems under diagonal strains and d) non-diagonal strains. More details about tight
binding parameterization of strained systems are discussed in chapter 4.
2.6 Implementation of the parameterization algorithm
According to the description of the parameterization algorithm in section 2.2, the
algorithm have two stagess: ab-initio calculations and fitting process.
ab-initio calculations The first stage of the mapping process is the ab-initio calcu-
lations. In principle, any ab-initio tools that are able to deliver band structure
and wave functions can be used. In this work, the ab-initio calculations are
performed by the Vienna Ab initio Simulation Package(VASP) [32]. VASP is
a Fortran based computer program for atomic scale materials modelling from
20
.
Fig. 2.3. Different stages of code implementation. ab-initio calculations inthis work are performed by VASP which was implemented in Fortran bythe VASP team. Formatted VASP output is parsed only once by Pythonprocedure. The following ab-initio mapping process is also implementedin Python. Multiscale atomistic simulation tool NEMO5 [31] provides thecapability for tight binding calculation.
21
.
Fig. 2.4. The mapping algorithm is separated into initialization, executionand output levels. The initialization and final output are needed to bedone once. Execution level can perform different tasks according to user’srequirement. The tasks of band structure calculation and wave functionprojection are one shot tasks, while the parameters optimization is a iter-ative process.
first principles which is implemented by the VASP team. The wave functions
of VASP are presented in projector augmented plane wave basis functions. The
complex plane wave coefficients are outputted as a binary data file WAVECAR.
However, the plane wave basis functions is not outputted by default. A function
was added to VASP source code in order to output the plane wave basis vectors
whenever the wave function is outputted.
Implementation of the parameterization algorithm Post processing of VASP
output in order to parse the formatted vasp outputs is done only once by through
Python or Matlab scripts. Python compatible binary output is generated for
later steps.
The main process of ab-initio mapping is implemented in Python. The opti-
mization algorithm that can be used include gradient dependent optimization
algorithm and non-gradient dependent optimization algorithm, for example,
simplex method. Multiscale atomistic simulation tool NEMO5 [31] is used since
it provides the capability of performing tight binding calculations for varieties
of different systems.
22
The algorithm implemented has initialization, execution and output levels.
Initialization: Operations in initialization level mainly involve parsing input
options and read in required ab-initio data. All initialization operations
are only done once.
Execution: Operations in execution level varies according to the requirement
of user. It can be single shot band structure calculation or wave function
projections. Or it can be iterative process of parameters optimization
which is described in section 2.2.
Output: Output level operations include outputting readable results and fig-
ures. The tight binding parameters, basis functions, tight binding and
ab-initio band structures are output as readable data, files or figures.
2.7 Summary
The process of generating tight binding parameters from ab-initio calculations and
the details of basis transformation are outlined. This process iteratively optimizing
tight binding parameters and basis functions. During the optimization process, tight
binding band structures and wave functions are compared with corresponding ab-
initio results. This process can be applied to single system and multiple systems. The
prototype algorithm was published in Ref. [23]. The improved algorithm presented
in this chapter was published in Ref. [15].
23
3. PARAMETERIZATION OF UNSTRAINED
SEMICONDUCTORS AND UTBS
3.1 Introduction
In order to model transistors atomistically, a computationally efficient and trans-
ferable model is needed. For this purpose, empirical tight binding (ETB) models are
good candidates due to its numerical efficiency. Most of the existing empirical tight
binding models are parameterized by fitting directly to experimental or theoretical
data. While the ETB parameters obtained by direct fitting may lead to unphysical
results [15]. For example, the application of existing TB parameters [14] with the im-
plicit passivation model [16] to As terminated GaAs UTBs shows unconfined valence
states(as it is shown in Fig.1.2). This observation implies the transferability of tight
binding parameters obtained by direct fitting is questionable.
To generate tight binding parameters with good transferability, the ab-initio map-
ping algorithm is introduced by Chapter 2. This algorithm extracts ETB parameters
and explicit basis functions from ab-initio band structures and wave functions. This
chapter focuses on the application of the ab-initio mapping method to unstrained
materials and UTBs. The unstrained bulk Si and GaAs are parameterized in this
chapter. ETB parameters and explicit basis functions are obtained. To investigate
the transferability of the ETB model and parameters, the ETB parameters are applied
to Hydrogen passivated Si and GaAs UTBs. For UTB systems, ETB calculations in-
clude Hydrogen passivation explicitly. ETB band structures and wave functions are
compared with hybrid functional calculations.
24
G X W L G K X
−10
−5
0
5
10
Energy (eV)
HSE06ETB refETB
b) Si
1
2
3
Energy (eV)
L G X
−0.8
−0.4
0
Si band structure near band gap
c) Si VB
b) Si CB
DOS (eV−1nm
−3)
0 20 40
−10
−5
0
5
10
DOS (eV
−1nm
−3)
20 40
Fig. 3.1. Band structure and density of states of bulk Si. ETB bandstructure agree with the HSE06 band structure (a), especially for bottomconduction bands (b) and top valence bands (c).
3.2 Parameterization of unstrained materials
3.2.1 Band structure of bulk materials
In this work, ab-initio level calculations of Si and GaAs systems were performed
with VASP [32]. The screened hybrid functional of Heyd, Scuseria, and Ernzerhof
(HSE06) [33] is used to produce band gaps [26] comparable with experiments in both
the bulk and the UTB cases. In the HSE06 hybrid functional method scheme, the
total exchange energy incorporates 25% short-range Hartree-Fock (HF) exchange and
75% Perdew-Burke-Ernzerhof(PBE) exchange [34]. The screening parameter µ which
defines the range separation is empirically set to 0.2 A for both the HF and PBE
parts. The correlation energy is described by the PBE functional. In all presented
HSE06 calculations, a cutoff energy of 350eV is used. Γ-point centered Monkhorst
Pack kspace grids are used for both bulk and UTB systems. The size of the kspace
grid for bulk calculations is a 6× 6× 6, while the one for UTB is 6× 6× 1. k-points
25
with integration weights equal to zero are added to the original 6× 6× 6 or 6× 6× 1
grids in order to generate energy bands with higher k-space resolution. PAW [35]
pseudopotentials are used in all HSE06 calculations. The pseudopotentials for Si, Ga
and As atoms include the outermost occupied s and p atomic states as valence states.
The low lying 3d states of Ga are treated as core states since the incorporation of
3d states as valence states leads to less than 1% changes to fitting targets shown in
table 3.2 for bulk materials. The spin orbit coupling is included in band structure
calculations. Small hydrostatic strains up to 0.3% are introduced to adjust the bulk
band gaps in order to match experimental results. The lattice constant used in this
chapter is given by table 3.1.
G X W L G K X
−10
−5
0
5
10
Energy (eV)
HSE06ETB refETB
GaAs
1.4
1.8
2.2
L G X
−0.8
−0.4
0
Energy (eV) b) GaAs CB
c) GaAs VB
a)
GaAs band structure near band gap
DOS (eV−1nm
−3)
DOS
(eV−1nm
−3)
0 20 40
−10
−5
0
5
10
0 20
Fig. 3.2. Band structure and density of states of bulk GaAs. ETB bandstructure agree with the HSE06 band structure (a), especially for bottomconduction bands (b) and top valence bands (c).
For bulk Si and GaAs, fitting targets include the band structures of the lowest
16 bands (with spin degeneracy) along high symmetry directions, important effective
masses and wave functions at high symmetry points such as Γ, L and X points. ETB
26
Table 3.1.Slater-Koster type ETB parameters of bulk Si and GaAs, and passiva-tion parameters of UTBs. All presented parameters except for the latticeconstants are in the unit of eV. The lattice constants are in Angstrom.The Hydrogen atoms which passivate surface As and Ga are denoted byHa and Hc respectively. Onsite energies of surface As and Ga atoms areshifted by δa and δc respectively.
Si GaAs
a0 5.43A
Es −2.803316
Ep 4.096984
Es∗ 25.163115
Ed 12.568228
∆ 0.021926
Vssσ −2.066560
Vs∗s∗σ −4.733506
Vss∗σ −1.703630
Vspσ 3.144266
Vs∗pσ 2.928749
Vsdσ −2.131451
Vs∗dσ −0.176671
Vppσ 4.122363
Vppπ −1.522175
Vpdσ −1.127068
Vpdπ 2.383978
Vddσ −1.408578
Vddπ 2.284472
Vddδ −1.541821
a0 5.6307A
Esa −8.063758 Esc −1.603222
Epa 3.126841 Epc 4.745896
Es∗a 21.930865 Es∗c 23.630466
Eda 13.140998 Edc 14.807586
∆a 0.194174 ∆c 0.036594
Vsascσ −1.798514
Vs∗as∗cσ −4.112848
Vsas∗cσ −1.258382 Vscs∗aσ −1.688128
Vsapcσ 3.116745 Vscpaσ 2.776805
Vs∗apcσ 1.635158 Vs∗cpaσ 3.381868
Vsadcσ −0.396407 Vscdaσ −2.151852
Vs∗adcσ −0.145161 Vs∗cdaσ −0.810997
Vpapcσ 4.034685
Vpapcπ −1.275446
Vpadcσ −1.478036 Vpcdaσ −0.064809
Vpadcπ 1.830852 Vpcdaπ 2.829426
Vdadcσ −1.216390
Vdadcπ 2.042009
Vdadcδ −1.829113
HSi Ha(passivate As) and Hc(passivate Ga)
EsH −3.056510
VsHsSiσ −4.859509
VsHpSiσ 3.776178
VsHs∗Siσ 0.0
VsHdSiσ −0.007703
δSi −0.276789
EsHa 2.758428 EsHc −0.308397
VsHasaσ −2.960420 VsHcscσ −3.151427
VsHapaσ 5.490764 VsHcpcσ 3.539284
VsHas∗aσ 0.0 VsHcs∗cσ −0.129904
VsHadaσ −1.727690 VsHcdcσ −0.252733
δa −0.266815 δc −0.586952
27
basis functions in real space is reconstructed on 6× 6× 6 Γ center k space grid using
Eq (2.24).
The band structures and DOS of bulk Si and GaAs (HSE06 vs ETB) are shown
in Fig. 3.1 and 3.2 respectively. The band structures using existing Si and GaAs
ETB parameters [14, 36] are also shown in corresponding figures. The ETB band
structures and DOS using parameters generated in this chapter show better agree-
ment with the corresponding hybrid functional results compared with the existing
parameterizations. For bulk Si, the existing parameterization shows a unexpected
low s∗ band around 5 eV above topmost valence bands. In the traditional fitting pro-
cess, the s∗ band shows a strong preference for moving downward [36]. Due to large
number of parameters to be determined, traditional (energy-gap and effective-mass)
based fitting procedures can find local minima in their fitness functions corresponding
to wave functions significantly different from those predicted by ab-initio methods.
The present method has the important advantage that optimization involves not only
masses and gaps but also wavefunctions. Thus the ETB wavefunctions can be kept
close to their ab-initio counterparts. For GaAs, the existing parameterization shows 2
eV higher s-type low lying valence bands. The ETB parameters of bulk Si and GaAs
are listed in table 3.1. It can be seen from table 3.2, the anisotropic hole masses by
ETB show a remarkable agreement with HSE06 results. The principal authors of the
previous works [14, 36] explicitly pointed out that fitting hole masses had been very
difficult with the previous methods.
3.2.2 Explicit tight binding basis functions
The orthogonal ETB basis functions Bfinal of Si, Ga and As atoms are shown
in Fig. 3.3. The ETB basis functions are slightly environment dependent because
they are orthogonal. Thus the ETB basis functions are not invariant under arbitrary
rotations but invariant under symmetry operations within Td group, as pointed out
by Slater and Koster [37]. It can be seen from Fig. 3.3.(a) to (f) that the s and p
28
Table 3.2.Targets comparison of bulk Si and GaAs. Critical band edges and effec-tive masses at Γ, X and L points by ETB and HSE06 calculations arecompared. The Eg and ∆SO are in the units of eV; effective masses arescaled by free electron mass m0. The data in error column summarizesthe discrepancies between HSE06 and TB results.
Si GaAs
targets
Eg(Γ)
Eg(X)
Eg(L)
∆SO
mhh100
mhh110
mhh111
mlh100
mlh110
mlh111
mso100
mso110
mso111
mcXl
mcXt
TB Ref HSE06 TB error (%)
3.399 3.302 3.244 1.8
1.131 1.142 1.139 0.2
2.383 2.247 2.188 2.6
0.047 0.051 0.052 0.8
0.299 0.281 0.282 0.097
0.633 0.566 0.572 0.977
0.796 0.704 0.714 1.433
0.232 0.206 0.204 1.001
0.165 0.151 0.149 0.937
0.156 0.143 0.142 0.927
0.266 0.244 0.242 0.809
0.266 0.244 0.242 0.795
0.267 0.244 0.242 0.770
0.887 0.928 0.857 7.615
0.225 0.207 0.215 3.544
targets
Eg(Γ)
Eg(X)
Eg(L)
∆SO
mhh100
mhh110
mhh111
mlh100
mlh110
mlh111
mso100
mso110
mso111
mc100
mc110
mc111
mcXl
mcXt
mcLl
mcLt
TB Ref HSE06 TB error(%)
1.424 1.418 1.416 0.2
1.900 1.919 1.910 0.5
1.707 1.702 1.708 0.3
0.326 0.368 0.367 0.1
0.383 0.310 0.337 8.510
0.667 0.573 0.619 7.879
0.853 0.750 0.813 8.507
0.085 0.082 0.083 0.744
0.078 0.073 0.074 1.614
0.076 0.071 0.072 1.715
0.166 0.164 0.160 1.998
0.166 0.164 0.160 2.037
0.166 0.164 0.160 2.041
0.068 0.065 0.067 2.787
0.068 0.066 0.067 2.790
0.068 0.065 0.067 2.781
1.526 1.577 1.480 6.142
0.177 0.215 0.204 5.083
1.743 1.626 1.446 11.055
0.099 0.111 0.136 22.614
29
−1 0 1−1
0
1
−1 0 1−1
0
1
−1 0 1−1
0
1
−1 0 1−1
0
1
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
−1 0 1
−1
−1 0 1
−1
0
1
0
1Si - s
Si - py
Ga - s
Ga - py As- py
As - s
Angular quantum number l
We
igh
t o
f l i
n o
rbit
als
Si - orbitals As - orbitals Ga - orbitals
(a) (b) (c)
(i)
(f)
(h)
(e)
(g)
(d)
0 2 40
0.2
0.4
0.6
0.8
1
ss*pd
ss*pd
ss*pd
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0 2 4 0 2 4
(h) (i) (j)
Fig. 3.3. Contours of selected highly localized ETB basis functions ofSi((a),(d)), Ga ((b),(e)) and As ((c),(f)) atoms. (a),(b) and (c) correspondto the contours of s orbitals of Si, Ga and As atoms in x-y plane. (d),(e)and (f) correspond to the contours of py orbitals of Si, Ga, and As atomsrespectively. (g),(h) and (i) show the contribution of different angularmomentums in basis functions of Si ,Ga, and As atoms respectively. TheETB basis functions of Si and GaAs are highly localized basis functionswith one dominant angular momentum.
orbitals show s and p features near the atom. More complicated patterns in the area
further away from the atom can be observed. These complicated patterns correspond
to components with high angular momentums. The feature of orthogonal ETB basis
function resembles the augmented basis functions used in ab-initio level calculations
such as Augmented Plane Waves(APWs) and Muffin Tin Orbitals(MTOs). The or-
30
thogonal ETB basis functions have multiple angular parts in each orbital as shown
by Fig. 3.3.(g),(h) and (i). The s, p and d type ETB basis functions are dominated
by components with l = 0, 1 and 2 respectively. More than 90% for the s,p and d
orbitals are comprised of their l = 0, 1 and 2 components respectively. The excited s∗
type ETB basis functions have higher angular momentum and the l = 0 components
have contributions of 60% to 70%. The second largest contribution in s∗ orbitals is
the f component with l = 3. The f component attached to the s∗ orbitals have
angular part equivalent to real space function xyz. This is a result of the existence
of xyz-like crystal field near each atom in zincblende and diamond structures.
0 0.5 1 1.50
0.1
0.2
0.3
0.4
position (a0
)
|ψ|2
0 0.5 1 1.50
0.02
0.04
0.06
0.08
|ψ|2
0 0.5 1 1.50
0.1
0.2
0.3
0.4
position (a0
)
HSE06
ETBHSE06
ETB
HSE06
ETB
0 0.5 1 1.50
0.1
0.2
0.3
HSE06
ETB
Si conduction
band (X)
Si
GaAs conduction
band (G)
(a)
(c)
(b)
(d)
Si valence
band (G)GaAs valence
band (G)
Fig. 3.4. Probability amplitude of bulk Si and GaAs by ETB and HSE06calculations are in good agreement. The Probability amplitude is plottedalong [111] direction along which neighboring atoms are located. For
bulk Si, silicon atoms locate at 0,√
34a and
√3a ; while for bulk GaAs,
Ga locates at 0 and√
3a; As locates at√
34a. For bulk Si, probability
amplitudes of conduction states at X points (a) and valence states at Γ(c) are shown. For bulk GaAs, probability amplitudes of conduction statesat Γ points (b) and valence states at Γ (d) are shown.
31
Fig.3.4 compares the probability amplitudes of valence and conduction states of
bulk Si and GaAs. It can be seen that the ETB result agree with the HSE06 re-
sults. For bulk Si, the highest probability shown in Fig.3.4 (a) corresponds to the
Si-Si bond. While the conduction states in Si appear differently, the probability is
distributed in the place with no chemical bond. In bulk GaAs, most of the probability
of valence states distributes near the anion, while the probability of conduction states
has significant distribution near both cations and anions.
0 1 2 3 4
−10
−5
0
position (a0)
0 1 2 3 4
−15
−10
−5
0
a) Si UTB
Local potential (eV)
b) GaAs UTB
Local potential (eV)
.
Fig. 3.5. Planar averaged local potentials of Hydrogen terminated (a) Siand (b) GaAs UTBs. The dashed lines correspond to the envelopes of thelocal potentials, the dots on dashed lines correspond to centers of atoms.The envelopes of the potentials are flat inside the UTBs. Significant de-viation can be seen at the surface atoms.
32
3.3 Tight binding analysis of UTBs
3.3.1 Explicit passivation model
The ETB interaction range is an important concern when applying the ETB model
to nanostructures. In order to assess whether the ETB model with first nearest neigh-
bours interactions can be transferable to nanostructures like UTBs, ab-initio local
potentials are studied.The ab-initio local potentials of Si and GaAs UTBs averaged
over the transverse plane are shown in Fig.3.5. It turns out that the envelopes of ab-
initio local potential are flat inside the UTBs. Obvious variation of the local potential
can be observed at the surface atoms only. Local potential profiles similar to those
in Fig.3.5 have been also obtained by ab-initio studies of transition metal oxides [38]
and graphene-metal interfaces [39]. This character in local potential suggests that
Hydrogen atoms mainly affect their first nearest neighbours, while the atoms inside
the UTBs are weakly affected by Hydrogen atoms. Therefore, longer range interac-
tions beyond first nearest neighbours are negligible and the ETB models including
only the first nearest neighbours are capable of modeling the UTB systems correctly.
To validate the transferability of the ETB model, band structures and eigen func-
tions of [001] UTBs passivated by Hydrogen atoms are calculated by both HSE06 and
ETB models. The current calculations assume no strain in the UTBs. In the HSE06
calculations, charged hydrogen atoms are used to passivate the dangling bonds of
the surface atoms in GaAs UTBs. The surface As and Ga atoms are passivated by
charged hydrogen atoms with 3/4 ( denoted by HAs ) and 5/4 ( denoted by HGa
) electron respectively. The charged hydrogen atoms neutralize most of the surface
induced electric field in the UTBs. As a result, the charge distribution and local
potential shows almost flat envelopes inside the UTBs. Small deviation of potential
can only be observed at the surface Si/Ga/As atoms. The nearly flat potential en-
velope suggests geometry dependent build-in potentials are needed only for surface
atoms. Thus the comparisons between self-consistent hybrid functional calculations
and single shot ETB calculations are fair.
33
X G K
En
erg
y(e
V)
−2
−1
0
1
2
3
Bulk Ev
Bulk Ec
X G K
Bulk Ec
Bulk Ev
−2
−1
0
1.5
2.5
3.5
(a) Si UTB (b) As terminated GaAs
HSE06ETB
HSE06ETB
Fig. 3.6. Band structures of [001] (a) Si and (b) As terminated GaAsUTBs by ETB agree with HSE06 band structures, demonstrating thebulk Si and GaAs ETB parameters are transferable to UTB cases. AllUTBs contain 17 Si/GaAs atomic layers(with thickness 4a0).
The HSE06 calculations show that the Hydrogen orbitals contribute to the deep
valence bands, thus Hydrogen atoms are considered explicitly into the ETB Hamil-
tonian of UTBs in this work. The 1s orbital is used as the ETB basis function for
Hydrogen atoms. The explicit passivation model includes extra Slater-Koster type
ETB parameters for Hydrogen and Hydrogen bonds EsHs, VsHsσ, VsHpσ,VsHs∗σ and
VsHdσ. For the rest of the UTBs, the bulk Si/GaAs parameters listed in table 3.1
are used. Furthermore, a geometry and element dependent potential δ is included
for surface atoms. The onsite energies of the surface atoms are shifted by δ. The
onsite energies of the surface Ga and As atoms thus become Eαc + δc and Eαa + δa
respectively. Here the α stands for s,p,d and s∗ orbitals. ETB parameters of Si/GaAs
in Si/GaAs UTBs are identical with the parameters of unstrained bulk materials
provided in section 3.2.1. To determine the passivation parameters, an extra fitting
process is needed: the band structure and wave functions of UTBs with 17 Si/GaAs
atomic layers are considered. Targets considered in the fitting process include the
direct and indirect band gaps of the UTBs, top valence and lowest conduction states
34
and band structures from 0.5eV below the top valence bands to 0.5eV above the
lowest conduction band.
To determine the ETB parameters of H-passivation, band structures and real space
wave functions of selected bands near the Fermi level of the UTBs are considered
as fitting targets. The inclusion of wave functions as targets serves the purpose of
correcting possible problematic states. The target Si/GaAs UTBs contain 17 Si/GaAs
atomic layers. Parameters for Hydrogen atoms are also shown in table 3.1. In GaAs
UTBs, As and Ga are passivated by Hydrogen atoms with different charge, thus
the Hydrogen atoms have different onsite energies when different types of atoms are
passivated. The Hydrogen atoms bonding with As atoms are charged positively while
the ones bonding with Ga atoms are charged negatively. Consequently, the Hc which
forms bond with As have a higher onsite energy than the Ha which forms bond with
Ga.
0 2 4
0.05
0.1
0.05
0.1
0.05
0.1
0 2 4
0.05
0.1
Re
al s
pac
e |ψ
|2(r
)
position (a0) position (a0)
(a) Si UTB (b) As terminated GaAs UTB
CB CB
VBVB
HSE06ETB
HSE06ETB
Fig. 3.7. Planar averaged real space probability amplitudes of lowestconduction and topmost valance states of [001] Si (a) and As terminatedGaAs UTBs (b) by HSE06 and ETB calculations. With the real space TBbasis functions, the realspace probability amplitudes of TB calculationsshow reasonable agreement with the HSE06 probability amplitudes. UTBscontain 17 Si/GaAs atomic layers (with thickness 4a0).
35
3.3.2 Band structures and wave functions of UTBs
0.05
0.05Ato
m s
ite
re
solv
ed
|ψ|2
0 2 4position (a0)
0.1
0.1
0.05
0.05
0 2 4position (a0)
Si UTB As terminated GaAs
CB CB
VB VB
0.1
0.1
Si, ETB ref
Si, ETB this work
(a) (b)
(c) (d)
Ga, ETB this workAs, ETB this work
Ga, ETB refAs, ETB ref
Fig. 3.8. ETB atom site resolved probability amplitudes of Si ((a),(c)), andAs terminated GaAs ((b),(d)) UTBs using ETB parameters in this chapterand previous work [14,36]. The ETB atom site probability using differentparameters are qualitatively similar in Si UTB, while the ETB atom siteprobability in As terminated GaAs are more sensitive to the parametersets and passivation models, i.e. the valence states with parameters andpassivation model by previous work are not confined. UTBs contain 17atomic layers(thickness is 4a0).
Band structures of Si/GaAs UTBs are shown in Fig. 3.6. The ETB band struc-
tures match the HSE06 band structures well for energies ranging from 1eV below
the topmost valence bands to 1eV above the lowest conduction bands. Using the
explicit ETB basis functions, ETB wave functions of UTBs with subatomic resolu-
tion are obtained and can be compared with corresponding HSE06 wave functions.
Planar averaged probability amplitudes of wave functions of the lowest conduction
band and top most valence bands in Si/GaAs UTBs are shown in Fig. 3.7. It can
be seen that not only the envelope but also details in subatomic resolution of the
36
A
tom
sit
e r
eso
lve
d |ψ
|2
(a) (b)
(c) (d)
0
0.05
0.1
0 1 2 3 4
position (a0 )
0
0.05
0.1
0 1 2 3 4
ETB ref + implicit passivation ETB ref + explicit passivation
ETB this work + explicit passivation
Envelopes of ETB valence states in GaAs UTB (As-t)
Ga, ETB this workAs, ETB this work
Ga, ETB refAs, ETB ref
ETB this work + implicit passivation
Fig. 3.9. Comparison of ETB wave functions using different ETB parame-ters and passivation model. (a) and (b) use ETB parameters by Boykin etal. [14]. (a) and (c) correspond to implicit passivation model [16]; (b) and(d) correspond to explicit passivation model. The ETB parameters withthe explicit passivation model shows the most confined states, while theprevious parameters and implicit passivation model lead to less confinedstates.
ETB planar averaged |ψ|2 show agreement with corresponding HSE06 results.On the
other hand, Fig. 3.8 compares the ETB atom site resolved probability amplitudes
among ETB models in present and previous works [14, 36]. The cations and anions
in GaAs UTBs form different envelopes for all of the presented states. The lowest
conduction and highest valence states turn out to be well confined states in Si UTBs
in all of the calculations. While, in GaAs UTBs, the lowest conduction states has
significant contribution from the surface atoms. In Si ETB probability amplitudes by
parametrizations by Boykin et al [36] show similar envelopes compared to the ETB
and HSE06 probability amplitudes in this work. Fig. 3.8 (d) shows the problematic
valence states in As terminated GaAs UTB by parameters by Boykin et al [36] The
37
corresponding valence states shown by this work turn out to be a well confined ones.
To investigate this issue in more detail, in Fig. 3.9, ETB atom site resolved probability
amplitudes for the topmost valence states of the four possible As-terminated GaAs
UTBs are plotted: (a)parameters by Boykin et al [14] and implicit passivation [16];
(b) parameters by Boykin et al [14] and explicit passivation; (c) new parameters and
implicit passivation; (d) new parameters and explicit passivation. It is clear that, for
a given set of bulk parameters, the implicit passivation model leads to wavefunctions
that are less-confined than those of the explicit passivation model. On the other hand,
with the same passivation model, the ETB parameters in this chapter shows more
confined top valence states than the existing ETB parameters. Thus the un-confined
ETB state using the existing parameter set and implicit passivation model appears
to be due to both the bulk GaAs parameters and the passivation model. The implicit
model [16] replaces the s- and p-orbitals of the surface atoms by sp3 hybrids and
raises the energy of the dangling hybrids by δsp3 = 30eV . The d- and s∗-orbitals
are left completely un-passivated, and the unconfined states of Fig. 3.9 (a) are only
slightly affected by changing the value of δsp3. The impact of alternate implicit pas-
sivation model to explicit passivation model is obvious by comparing sub-figures (a)
to (b), as well as (c) to (d). To better understand the impact of bulk parameters on
this behavior, the contribution of orbitals to the bulk bands by different parameter
sets is compared. Obvious differences are found at the d-orbital contributions of the
topmost bulk valence states at Γ point. The bulk valence states using parameters
by Boykin et al. [14] have about 16% contribution from the d orbital of Ga, while
the ones by parameters of this chapter have only 8.5%. This discrepancy suggests
either d-orbital onsites energies are excessively low or coupling of pa-dc are excessively
strong in the parameter set by Boykin et al [14]. It turns out that the couplings of
pa-dc are the major problematic parameters in the previous parameter set: as it is
shown by Fig.3.10, by reducing the magnitude of the nearest-neighbor pa-dc coupling
parameters in both sets as Vpadcσ → Vpadcσ + 0.3eV , Vpadcπ → Vpadcπ− 0.3eV , remark-
ably, in both cases the topmost valence-band state became much more confined. Bulk
38
0 2 40
0.02
0.04
0.06
0.08
0.1
position (a0
)
Ato
m s
ite
re
solv
ed
|ψ
|2
Ga envelope
As envelope
0 2 40
0.02
0.04
0.06
0.08
0.1
position (a0
)
Ga envelope
As envelope
with modified
parameters
with original
parameters
Fig. 3.10. Comparison of ETB wave functions using ETB parameters byBoykin et al. [14] (a) and modified parameters (b). In (b), the modifiedparameters are V ′padcσ = Vpadcσ + 0.3eV , V ′padcπ = Vpadcπ − 0.3eV . Withmodified parameters, the unconfined valence states become confined ones.
valence band wave functions in modified and original parameter sets tell the story:
The general trend is that bulk sets which generate more p-like top of VB states give
better confinement under passivation (and especially implicit passivation) than do
those with higher d-content. The reduction of |Vpadcσ| and |Vpadcπ| lead to more p-like
top VB states. The Ga terminated case has less passivation problems because its
top-of-VB bulk states have more contribution from the As atoms than from the Ga
atoms.
Fig. 3.11 shows the band gaps of the Si and GaAs [001] UTBs as functions of
UTB thickness. With the ETB parameters by this work, the ETB bandgaps of Si
and GaAs UTBs with thickness from 0.5nm to 4nm agree well with the gaps by
HSE06 calculations. The ETB bandgaps of Si UTBs using parameters from previous
work also show good agreement with the HSE06 results. However the ETB bandgaps
of GaAs UTBs using parameters from previous work and implicit passivation model
are of around 20% lower than the Hybrid functional results. The gaps of GaAs UTBs
terminated with Ga and As atoms are very close in value for both Hybrid functional
and ETB results in this work, however the gaps of GaAs UTBs terminated with Ga
39
Si UTBs
2
4
6
8
Erro
r (me
V)
8
16
24
8
1 2 3 4 50
0.5
1
1.5
2
UTB Thickness (nm)
EG
(UT
B)
− E
G(b
ulk
) (e
V)
effective massETBETB refHSE06Error ETB vs HSE06
Si [001] UTBs
1 2 3 4 50
0.5
1
1.5
2
UTB Thickness (nm)
EG
(UT
B)
− E
G(b
ulk
) (e
V)
As terminated [001] GaAs UTBs
effective massETBETB refHSE06Error ETB vs HSE06
(a) (b)
Erro
r (me
V)
Fig. 3.11. Band gaps of Si UTBs (a) and As terminated UTBs (b) byHSE06 and ETB calculations. For the presented UTBs with thicknessranging from 1nm to 4.5nm, the ETB band gaps have discrepancies of lessthan 10meV compared with band gaps by HSE06. The band gap changesby effective mass calculation show agreement with HSE06 for Si UTBsthicker than 3nm. While the effective mass calculations has significantdiscrepancies for all GaAs UTBs. The HSE06 and ETB calculations usingparameters in this chapter consider Hydrogen atoms explicitly, while theETB calculations using parameters by previous work is based on implicitpassivation model [16].
and As atoms by previous parameterizations and implicit passivation model show 0.1
to 0.2eV discrepancies. The band gap change in Si UTBs thicker than 3nm can be
modeled by effective mass model( assuming parabolic E-k relation ). While in the
GaAs UTBs, the discrepancies between effective mass calculations and HSE06 or TB
calculations are obvious for all GaAs UTBs presented, suggesting the non-parabolic
feature of the GaAs valleys have a significant impact on GaAs nanostructures. The
gaps obtained by the previous parameterization with implicit passivation model of
As terminated GaAs UTBs has lower confined energies due to the unconfined valence
states.
40
3.4 Summary
The mapping process of generating ETB parameters from ab-initio calculations is
applied to unstrained Si and GaAs. The ETB band structure show good agreement
with Hybrid functional results. The ETB basis functions turn out to be highly local-
ized orbitals with s, p and d features. The Si/GaAs ETB parameters are applied to
Hydrogen-passivated Si/GaAs ultra thin bodies with an explicit passivation model.
The ETB band structure of ultra thin bodies agree well with the corresponding Hy-
brid functional calculations. The explicit passivation model turns out to be important
to model passivated systems. Furthermore, the ETB probabilities of important band
edges agree with the corresponding Hybrid functional results, while the ETB proba-
bilities by ETB parameters of existing work [14,36] show a qualitative disagreement.
This comparison suggests that the ETB parameters obtained by the ab-initio map-
ping process can in principle achieve good transferability. The corresponding results
are published in Ref. [15].
41
4. PARAMETERIZATION OF STRAINED MATERIALS
(ROOM TEMPERATURE)
4.1 Introduction
Strain is in general inevitable in realistic nano-electronic devices due to compli-
cated geometries. The introducing of proper strain to the channel of transistors is
a key approach to maintain device performances for transistors of nanometer scale.
To have reliable prediction of performances in nano-electronic devices, it is critical to
have an accurate atomistic model that can handle strained materials. For this pur-
pose, efficient semiempirical approaches, such as the k · p [40,41], the empirical pseu-
dopotential [42] and the empirical tight binding (ETB) methods [13, 14] are actively
developed. Among these empirical approaches, ETB has established itself as the stan-
dard state-of-the-art for realistic device simulations [31]. For strained systems, ETB
models take into account the altered environment in terms of both bond angle and
length. In existing ETB models, generalized Harrison’s law [13,43] is usually adopted
to describe bond-length dependency of the nearest-neighbor interatomic parameters.
Changes of bond angles in interatomic interactions are automatically incorporated
through the Slater-Koster formulas [37, 44]. With generalized Harrison’s law, ETB
model can reproduce some hydrostatic and uniaxial deformation potentials [45], while
much higher accuracy can be achieved with the introduction of strain-dependent on-
site parameters. Boykin et al. [14] introduced position-dependent diagonal orbital
energies in the sp3d5s* Hamiltonian to reproduce correct deformations under 001
strains. Off-diagonal onsite corrections are suggested by Boykin et al [46] and Niquet
et al. [47] in order to handle 110 strains.
Those existing ETB models are fitted to strained bulk systems by traditional
fitting. Previous chapters has shown that parameters by traditional fitting has trans-
42
ferability problem when applied to nanostructures. While the parameterization al-
gorithm presented by Chapter 2 leads to better transferability. In this chapter, an
ETB model for arbitrarily strained materials is introduced. In this model, the onsite
elements and interatomic interactions depend on the environment of each atom. The
model is applied to group IV and III-V materials. ETB parameters are parameterized
with respect to hybrid functional calculations of different strained materials based on
the ab-initio mapping algorithm introduced in Chapter 2. Deformation potentials
and band edge splitting are compared with corresponding ab-initio results. To have
ETB parameters for materials under room temperature, ab-initio results are adjusted
to match experimental targets under room temperature.
4.2 Strain model description
The ETB model of strained materials in this work is based on the multipole ex-
pansion [48] of the local potential near each atom. This ETB model has environment
dependency, and it does not rely on the selection of coordinates. It can be applied
to arbitrarily strained and rotated systems. In this work, the tight binding model is
applied to group IV and III-V semiconductors with diamond or zincblende structures.
However, this model is in principle not limited to group IV and III-V semiconduc-
tors. For the presented materials, the interaction range considered in the tight binding
model is limited to the first nearest neighbors. In the following sections, letters in
bold such as r and d are used for three dimensional vectors; correspondingly, r and d
are used to denote the lengths of r and d. Ω stands for polar angle and θ and azimuth
angle φ of a three dimensional vector. α, β,γ correspond to angular and magnetic
quantum numbers l1m1,l2m2 and l3m3 of ETB orbitals respectively. Dirac notation
is used for ETB basis functions, e.g. |ψαi〉 stands for α orbital of atom i.
43
4.2.1 Multipole expansion of atomic potentials
The local potential near atom i is approximated by a summation of the potential
of atom i and potential of its nearest neighbors j
U toti (r) = Ui (|r|) +
∑j∈ NNs
Uj (|r− dij|) , (4.1)
where the relative position between atoms i and j is dij. The potential at r due to
atom at dij is approximated by general spherical potential. This general spherical
potential Uj(|r− dij|) centered at dij has multipole expansion given by
Uj(|r− dij|) =∑l
U(l)j (r, dij)
l∑m=−l
Y ∗lm(Ωr)Ylm(Ωdij), (4.2)
where Ωr and Ωdij stands for angles θ and φ of vectors r and dij. The U (l)(r, dij)
is the radial part of multipole potential with angular momentum l. By substituting
Uj (|r− dij|) in eq (4.1) by equation (4.2), the total potential near atom i given by
equation (4.1) can be written as sum of multipole potentials
U toti (r) =
∑l
U(l)i (r), (4.3)
where the multipole potentials U(l)j (r)’s are given by
U(0)i (r) = Ui (|r|) +
∑j U
(0)j (r, dij)
U(l)i (r) =
∑m Y
∗lm(Ωr)
(∑j U
(l)j (r, dij)Ylm(Ωdij)
)(4.4)
The U(l)i ’s are summations of multipoles over nearest neighbors. The strain induced
multipole potentials up to quadrupole (with l = 2) are considered in this work.
The U(0)i describes the crystal potential under hydrostatic strain. U
(0)i depends only
bond lengths. For unstrained or hydrostatically strained zincblende and diamond
structures, both dipole potential U(1)i (r) and quadrupole potential U
(2)i (r) are zero
due to the crystal symmetry of zincblende and diamond structures. For strained
systems with traceless diagonal strain component like εxx, U(2)i (r) is induced due to
44
angle change; while for strained systems with off-diagonal strain component like εxy,
both U(1)i (r) and U
(2)i (r) are induced.
4.2.2 Strain dependent tight binding Hamiltonian
The strain dependent ETB Hamiltonian is constructed according to the multipole
expansion of U toti . Similar to the multipole expansion of the total potential given by
eq (4.3), the strain dependent ETB Hamiltonian is written as
H = H(0) +H(1) +H(2), (4.5)
where the H(l) depends on multipole potential U (l)(r). Matrix element Hαi,βj is thus
written as Hαi,βj = H(0)αi,βj
+H(1)αi,βj
+H(2)αi,βj
.
4.2.3 Onsite elements
The U(0)i has contribution from atom i and its neighbors. Similar to U
(0)i , the
diagonal onsite energiesH(0)αi,αi also has contribution Eαi from atom i and contributions
from its neighbors. The contribution of neighbors to diagonal onsites energies is
separated to orbital dependent part Iαi,j(dij) and orbital independent part Oi,j(dij).
The onsite elements due to U(0)i is given by
H(0)αi,αi
= Eαi +∑j∈NNs
Iαi,j(dij) +∑j∈NNs
Oi,j(dij), (4.6)
with
Iαi,j(dij) = Iαi,je−λαi,j(dij+δdij−d0) (4.7)
Oi,j(dij) = Oi,je−λij(dij+δdij−d0) (4.8)
Here the d0 is the reference bond length. The parameter δdij is introduced to modulate
discrepancy between ab-initio results and experimental results. Non-zero δdij’s are
introduced to match the ETB results in this work with experimental data under room
temperature; while with zero δdij, ETB results match the zero temperature ab-initio
45
results. The term Eαi depends on orbital and atom type instead of material type.
The summation over Iαi,j(dij) and Oi,j(dij) are the environment dependent part of
diagonal onsite energies H(0)αi,αi . Oi,j(dij) is used to modulate the band offset and it
satisfies Oi,j(dij) = Oji(dji). Similar expression is also applied to spin-orbit coupling
terms ∆SOCi = ∆i+
∑j∈NNs ∆i,j. In this work, only spin-orbit interaction of p orbitals
is considered. The bond length dependency of ∆i,j is neglected as well.
Due to dipole and quadrupole potentials, non-zero off-diagonal onsite elements
appear. Off-diagonal onsite elements due to multipole potentials is given by
Eαiβi = 〈ψαi(r)|U (l)(r)|ψβi(r)〉. l > 1 (4.9)
Since the U (l)(r) given by eq (4.4) is non-spherical, to estimate these terms, following
relation is used
Yα(Ω)Yβ(Ω) =∑γ
Gγα,βYγ(Ω), (4.10)
where the Gγα,β is the Gaunt coefficient [49] defined by
Gγα,β =
∫Yα(Ω)Yβ(Ω)Y ∗γ (Ω)dΩ (4.11)
where dΩ = sin θdθdφ.
With eq (4.4), off-diagonal onsite elements of atom i can be written as summation
of terms from neighbors j of atom i
Eαiβi =∑j
Mα,β(dij)C(l)αiβi,j
(dij), (4.12)
where the C(l)αiβi,j
is the integral of radial parts of |ψαi〉, U (l) and |ψβi〉, given by
C(l)αiβi,j
= 〈Rαi(r)|U (l)(r, dij)|Rβi(r)〉 (4.13)
The Mα,γ is given by
M(l)α,γ(dik) =
∑m′
Gγα,α′Ylm′(Ωdik), α′ = l,m′ (4.14)
The explicit form of M(l)α,γ(dik)’s due to multipole potentials are given by appendix
G.
46
Since this work limits orbitals α and β to s,p,d and s*, the dipole potentials lead
to non-zero off-diagonal onsite among s-p, and p-d orbitals. While the quadrupole
potential lead to non-zero off-diagonal onsite among p-p, and d-d orbitals. Therefore,
there is no confusion to use Cαiβi,j instead of C(l)αi,βi,j
. In this work, it turns out the
bond length dependency of Cαiβi,j can be neglected, since the strain considered in this
work has amplitudes up to 4%. The off-diagonal onsite elements depend completely
on the bond angle changes. Fitting parameters for onsite elements introduced in this
work include Eαi , Iαi,j, λαi,j, Cαiβi,j. For atoms in alloys or material interfaces, where
an atom might has different type of neighbors, an averaged Cαiβi,j over neighbors j
is used in this work.
4.2.4 Interatomic couplings
Interatomic couplings H(0)αi,βj
due to U (0) which couple orbital α of atom i and
orbital β of atom j follows the Slater Koster formulas [37,44]. Bond length dependent
two center integrals in this work are approximated by exponential law
Vαiβj |m|(dij) = Vαiβj |m|e−ηαiβj |m|(dij+δdij−d0)
. (4.15)
The δdij is the parameter introduced in order to match the ETB band structure with
experimental results.
The interatomic coupling due to multipole potential U (l) are written as
V(l)αi,βj
= 〈ψα(r)|U (l)(r) + U (l)(r− dij)|ψβ(r− dij)〉. (4.16)
By substituting U (l) with equation (4.4), this integral can be written as
V(l)αi,βj
=∑
γ,kM(l)α,γ(dik)Q
(l)γi,βj
(dik) + (4.17)∑γ′,k′ Q
(l)
αi,γ′j(djk′)M(l)
γ′,β(djk′)
47
where k’s denotes the nearest neighbors of atom i, k′ denotes the nearest neighbors
of atom j. The Q(l)γ,β(dik) and Q
(l)α,γ′(djk′) are given by
Q(l)γi,βj
(dik) = 〈ψγ(r)|U (l)(r, dik)|ψβ(r− dij)〉 (4.18)
Q(l)
αi,γ′j(djk′) = 〈ψα(r)|U (l)(|r− dij|, djk′)|ψγ′(r− dij)〉
The |ψγ(r)〉 has the same radial part as |ψα(r)〉 , although γ and α are different.
Q(l)γi,βj
(dik) and Q(l)
αi,γ′j(djk′) are three center integrals involving orbitals of atom i,j
and potential U (l) from atom k or k′. However, since the quadrupole potential U (l)
are centered either at atom i or j, the Q(l)γi,βj
(dik) and Q(l)
αi,γ′j(djk′) has the expression
of two center integrals describing by Slater Koster formulas. To simplify the formula,
we approximate the effect of U (l)(r, dik)’s by using averaged potential over k and k′ to
remove the dependency of atom k and k′, U (l)(r) = 1nk
∑k U
(l)(r, dik) , U (l)(|r−dij|) =
1nk′
∑k′ U
(l)(|r− dij|, djk′).
For dipole potentials, the complete explicit expressions of equation(4.17) are quite
lengthy. In this work, we find it is sufficient to approximated them with Slater Koster
formula. The impact of U (1) appears in term of strain correction δV(1)αiβj |m| to two center
integral Vαiβj |m|(dij) given by equation (4.15). The δV(1)αiβj |m| are have the expression
δV(1)αiβj |m| =
4π
3Pαi,βj ,|m| (pij + pji) +
4π
3Sαi,βj ,|m| (qij + qji) , (4.19)
where the pij and qij estimate the dipole potential along bond dij. Pαi,βj ,|m| and
Sαi,βj ,|m| are fitting parameters. pij and qij are given as
pij =∑k,m
Y1,m
(Ωdi,k
)Y1,m
(Ωdi,j
)(4.20)
qij =∑k,m
Y1,m
(Ωdi,k
)Y1,m
(Ωdi,j
) δdikd0
. (4.21)
pji =∑k′,m
Y1,m
(Ωdj,k
)Y1,m
(Ωdj,i
)(4.22)
qji =∑k′,m
Y1,m
(Ωdj,k′
)Y1,m
(Ωdj,i
) δdjk′d0
.
48
The d0 is the reference bond length. More discussion of his approximation is given
in appendix H. pij and qij estimate the impact of dipole moment to neighbors. The
non-zero pij correspond to non-zero off-diagonal strain components, while the nonzero
term qij corresponds to bond length changes which break crystal symmetry.
For quadrupole potentials, we find it is sufficient to drop the bond length depen-
dency of U (2)(r) and U (l)(|r − dij|) from equation (4.18) since we consider strain up
to 4% in this work. Thus Qγi,βj(dik) and Qαi,γ′j(dk′j) can be simplified by
Qγi,βj = 〈ψγ(r)|U (2)(r)|ψβ(r− dij)〉 (4.23)
Qαi,γ′j= 〈ψα(r)|U (2)(|r− dij|)|ψγ′(r− dij)〉 (4.24)
Here fitting parameters in Slater Koster form Qαi,βj ,|m| are introduced.
4.3 Parameterization of strained group III-V and IV materials
In this work, ab-initio level calculations of group IV and III-V systems were per-
formed with VASP [32]. The screened hybrid functional of Heyd, Scuseria, and Ernz-
erhof (HSE06) [33] is used to produce the bulk and the superlattices band structures
with band gaps comparable with experiments [26]. In the HSE06 hybrid functional
method scheme, the total exchange energy incorporates 25% short-range Hartree-Fock
(HF) exchange and 75% Perdew-Burke-Ernzerhof(PBE) exchange [34]. The screening
parameter µ which defines the range separation is empirically set to 0.2 A for both the
HF and PBE parts. The correlation energy is described by the PBE functional. In
all presented HSE06 calculations, a cutoff energy of 350eV is used. Γ-point centered
Monkhorst Pack kspace grids are used for both bulk and superlattice systems. The
size of the kspace grid for strained bulk calculations is 6 × 6 × 6, while one for 001
superlattices is 6 × 6 × 3. k-points with integration weights equal to zero are added
to the original uniform grids in order to generate energy bands with higher k-space
resolution. PAW [35] pseudopotentials are used in all HSE06 calculations. The pseu-
dopotentials for all atoms include the outermost occupied s and p atomic states as
valence states. Ab-inito band structures of strained and unstrained bulk materials
49
are aligned based on model solid theory [50,51]. With the model solid theory, relative
band offsets are determined by using different superlattices.
Ab-initio calculations based on LDA or GGA approximation severely underesti-
mate semiconductor band gap. Compared with LDA or GGA approximations, Hybrid
functional greatly improves the band gap, however these methods still have discrep-
ancies about 0.1 to 0.2 eV compared with experiments. Furthermore, ab-initio cal-
culations usually model systems under zero temperature. Finite temperature targets
can not be truly obtained by most ab-initio calculations. In order to get ab-initio
band structure matching room temperature experiments, artificial thermal expansion
is applied to to mimic the effect of finite temperature and to correct the error of
HSE06 calculations. More detail of this correction is discussed in Appendix I.
(a) (b)
(c) (d)
z
x
y
Fig. 4.1. Strained systems considered in the parameterization processof strained materials. (a) hydrostatic strain, (b) with two bond lengthchanges, (c) diagonal strain with εxx = εyy = −0.5εzz, (d) off-diagonalstrain with εxy 6= 0
50
Since the sp3d5s* ETB model with nearest neighbor interactions has proven to be
a sufficient model for bulk zincblende and diamond structures [14, 15, 36], the ETB
model in this work makes use of sp3d5s* basis functions with nearest neighbor inter-
actions. To parameterize the ETB model from ab-initio results, both ab-initio band
structure and wave functions are considered as fitting targets. The process of parame-
terization from ab-initio results was described by Ref [15]. Previous work applied the
ab-initio mapping algorithm to single material. In this work the ab-initio mapping
algorithm is applied to multiple strained systems in order to obtain ETB parameters
for strained materials. To consider multiple systems in the fitting process, a total
fitness to be minimized is defined as a summation of fitness of all systems considered
(labeled by index s) Ftotal =∑
s Fs. The fitness Fs is defined to capture important
targets of each stained system considered in the fitting process. The strained systems
considered in this work are shown by Fig. 4.1, including zincblende or diamond struc-
tures with a) hydro static strain, b) pure bond length changes, c) diagonal strains
and d) off-diagonal strain. For the sHydrostatic strain cases, materials with different
lattice constant ranging from 5.2 to 6.6 A are considered. While for other kind of
strains, strains with amplitudes from −4% to 4% are considered.
For hydrostatically strained materials, fitting targets includes band structures,
important band edges, effective masses and wave functions at high symmetry points.
Those targets were considered in previous work (Ref. [15]) in order to get ETB pa-
rameters for unstrained bulk materials. To extract tight binding parameters for arbi-
trarily strained materials, wave functions and energies at high symmetry points are
also considered as fitting targets. For strained systems, it is sufficient to use the strain
induced band edge splitting at high symmetry points as targets. Effective masses at
those points are not considered as fitting targets since effective masses in strained
materials are related to the splitting of band edges and effective masses of unstrained
systems. For example, the effective masses of valence bands in a strained group III-V
or IV material can be well described by a Luttinger model [40]. The well known
conduction band effective mass change under shear strain( with strain component εxy
51
) can also described by a two band k.p model for X valleys [41, 52]. Those models
include the strain effect as k-independent perturbation terms. The strain induced
terms correspond to the band edge splitting at high symmetry points.
In order to ease reading, ETB parameters are presented in the last section of
this chapter( section 4.6). The atom type dependent onsite parameters are listed on
table 4.7. Table 4.8, 4.9 list the environment dependent diagonal site parameters
I’s, O’s and λ’s. Tables 4.10, 4.11 and 4.12 summarizes the bond length dependent
interatomic coupling parameters V ’s and η’s. From tables 4.10, 4.11 and 4.12, it
can be seen that interatomic parameters for different III-V materials have similar
values. Multipole dependent off-diagonal onsite parameters and interatomic coupling
parameters are listed in table 4.14, 4.15 and 4.16 respectively. The parameters P ’s
, Q’s and S’s in principle contain the same number of parameters as interatomic
interaction parameter V . However, it turns out that it is sufficient to consider only
s − p, s − d, p − p and d − d interactions for parameters P ’s, Q’s and S’s. Others,
such as s∗ − p, s∗ − d and p− d interactions, are constrained to zero.
4.4 Tight binding analysis of strained materials
4.4.1 Unstrained band structures and hydrostatic strain behavior
Fig.4.3 shows the band structure of unstrained bulk band structure for group IV
and III-V materials. The presented materials include XP,XAs and XSb with X =
Al,Ga,In. It can be seen that the ETB results of unstrained bulk group IV and
III-V materials match corresponding HSE06 results well. Tables 4.1,4.2, 4.3 and
4.4 compare the effective masses and critical band edges between ETB and HSE06
calculations. Effective masses of important valence and conduction valleys are within
15% error. Critical band edges at high symmetric points are within 10meV.
Fig.4.4 shows Si band structures under hydrostatic strain. The hydrostatic strain
does not change crystal symmetry, thus the degeneracies at high symmetry points
are conserved under hydro static strain. However, it can be observed by comparing
52
L G X
−2
0
2
4
6
L G X
−2
0
2
4
6
Energy (eV)
(a) Si (b) Ge
HSE06
ETBHSE06
ETB
Fig. 4.2. Band structure of group IV materials with ETB and HSE06calculations. Presented band structures of IV materials include Si (a) andGe (b). ETB band structures of group IV materials are in good agreementwith HSE06 results.
Fig.4.4 (a) and (b) that the hydrostatic strains change the band edges significantly.
With a lattice constant of 5.4A, the lowest conduction bands of Si are X valleys, the
L and s-type Γ valleys are of more than 1eV above the X valleys. However, with a
larger lattice constant of 5.8A, the L and Γ gap descends dramatically, and the X gap
even increase slightly. The change of band gaps are shown clearly by Fig.4.4 (c). The
band gaps change almost linearly. From Fig.4.4 (c), it can be seen that at around
5.8A, the L and s-type Γ valley become lower than the X valleys. As the lattice
constant is increased, Si becomes a direct gap material (lowest conduction band is Γ
valley). In fact if the lattice constant is sufficiently large, Si becomes a metal as the
s-type Γ valley conduction band becomes even lower than the valence bands. The
trend shown by Fig.4.4 is valid for other group IV and III-V materials which have
diamond or zincblende structures.
53
L G X
−2
0
2
4
L G X
−2
0
2
4
L G X
−2
0
2
4
L G X
−2
0
2
4
6
L G X
−2
0
2
4
6
L G X
−2
0
2
4
6
L G X
−2
0
2
4
L G X
−2
0
2
4
L G X
−2
0
2
4
Energy (eV)
(h) AlSb
(i) GaSb (j) InSb
HSE06
ETBHSE06
ETB
HSE06
ETB
Energy (eV)
(a) AlP (b) GaP (c) InP
(d) AlAs (e) GaAs (g) InAs
HSE06
ETB
HSE06
ETBHSE06
ETB
HSE06
ETB
HSE06
ETB
HSE06
ETB
Energy (eV)
Fig. 4.3. Band structure of III-V materials with ETB and HSE06 calcu-lation. Presented band structures of III-V materials include (a) AlP , (b)GaP , (c) InP ,(d) AlAs , (e) GaAs,(f) InAs ,(g) AlSb ,(h) GaSb ,(i) InSb.ETB band structures of group III-V materials are in good agreement withHSE06 results.
4.4.2 Arbitrary strain behavior
Fig.4.5 shows the impact of diagonal strain components to valence bands of GaAs.
The strain imposed satisfies εxx = εyy = −0.5εzz. For unstrained GaAs, the valences
bands have 4 fold degenerate heavy holes and light holes bands at Γ point. Under
the diagonal strain, z direction is no longer equivalent to the x and y direction. As a
result, this 4 fold valence bands degeneracy is lifted( Fig.4.5 (b)). The effective masses
54
Table 4.1.Targets comparison of bulk Si and Ge. Critical band edges and effectivemasses at Γ, X and L from ETB and HSE06 calculations are compared.The Eg and ∆SO are in the unit of eV; effective masses are scaled by freeelectron mass m0.
Si Ge
targets
Eg(Γ)
Eg(X)
Eg(L)
∆SO
mhh100
mhh110
mhh111
mlh100
mlh110
mlh111
mso
mcΓ
mcXl
mcXt
mcLl
mcLt
HSE06 ETB error(%)
3.302 3.289 0.4
1.142 1.158 1.5
2.247 2.266 0.8
0.051 0.051 0.3
0.260 0.266 2.562
0.522 0.535 2.401
0.649 0.672 3.527
0.190 0.179 5.950
0.139 0.134 3.784
0.132 0.127 3.680
0.225 0.218 2.855
- -
0.856 0.754 11.915
0.191 0.194 1.227
1.641 1.774 8.103
0.130 0.147 13.227
HSE06 ETB error(%)
0.755 0.746 1.2
0.974 0.968 0.7
0.710 0.711 0.2
0.313 0.313 0.0
0.203 0.197 2.700
0.378 0.381 0.637
0.506 0.523 3.290
0.040 0.040 1.015
0.037 0.037 0.313
0.035 0.035 0.199
0.093 0.091 2.211
0.032 0.033 3.777
0.840 0.768 8.524
0.189 0.203 7.513
1.577 1.738 10.236
0.081 0.101 23.871
55
Table 4.2.Targets comparison of bulk Phosphides. Critical band edges and effectivemasses at Γ, X and L from ETB and HSE06 calculations are compared.The Eg and ∆SO are in the unit of eV; effective masses are scaled by freeelectron mass m0.
AlP GaP InP
targets
Eg(Γ)
Eg(X)
Eg(L)
∆SO
mhh100
mhh110
mhh111
mlh100
mlh110
mlh111
mso
mcΓ
mcXl
mcXt
mcLl
mcLt
HSE06 ETB error(%)
3.576 3.572 0.1
2.391 2.379 0.5
3.353 3.269 2.5
0.063 0.065 1.8
0.529 0.530 0.142
1.020 1.005 1.460
1.303 1.301 0.178
0.230 0.217 5.720
0.190 0.181 4.482
0.183 0.174 4.495
0.329 0.313 4.711
0.172 0.166 3.466
0.851 0.855 0.425
0.249 0.237 4.725
1.540 1.605 4.193
0.174 0.186 6.597
HSE06 ETB error(%)
2.746 2.739 0.3
2.266 2.250 0.7
2.484 2.453 1.2
0.099 0.101 2.8
0.358 0.353 1.365
0.670 0.657 1.862
0.846 0.839 0.809
0.158 0.151 4.056
0.131 0.126 3.510
0.126 0.121 3.598
0.228 0.221 3.388
0.129 0.130 0.376
1.510 1.289 14.663
0.225 0.231 2.923
1.672 1.722 2.985
0.138 0.163 18.152
HSE06 ETB error(%)
1.387 1.382 0.4
2.400 2.370 1.2
2.207 2.128 3.5
0.124 0.124 0.1
0.412 0.411 0.248
0.738 0.739 0.135
0.935 0.956 2.309
0.108 0.105 3.172
0.097 0.094 2.951
0.094 0.091 3.040
0.184 0.180 1.862
0.082 0.079 3.156
1.479 1.367 7.574
0.246 0.252 2.761
1.959 1.949 0.505
0.143 0.165 15.418
56
Table 4.3.Targets comparison of bulk Arsenides. Critical band edges and effectivemasses at Γ, X and L from ETB and HSE06 calculations are compared.The Eg and ∆SO are in the unit of eV; effective masses are scaled by freeelectron mass m0.
AlAs GaAs InAs
targets
Eg(Γ)
Eg(X)
Eg(L)
∆SO
mhh100
mhh110
mhh111
mlh100
mlh110
mlh111
mso
mcΓ
mcXl
mcXt
mcLl
mcLt
HSE06 ETB error(%)
2.612 2.608 0.2
2.042 2.035 0.3
2.720 2.707 0.5
0.315 0.319 1.3
0.418 0.449 7.373
0.829 0.848 2.357
1.083 1.113 2.757
0.156 0.153 2.133
0.134 0.132 1.859
0.129 0.127 1.462
0.256 0.249 2.618
0.120 0.116 2.930
0.891 0.894 0.337
0.233 0.225 3.496
1.544 1.606 3.982
0.143 0.157 10.175
HSE06 ETB error(%)
1.418 1.410 0.6
1.919 1.915 0.2
1.702 1.693 0.5
0.368 0.366 0.4
0.308 0.317 2.963
0.569 0.581 2.177
0.744 0.762 2.455
0.081 0.081 0.766
0.073 0.072 0.267
0.070 0.070 0.242
0.162 0.156 3.755
0.065 0.066 1.301
1.564 1.331 14.895
0.213 0.216 1.435
1.613 1.669 3.474
0.110 0.129 17.941
HSE06 ETB error(%)
0.351 0.347 1.2
2.052 2.024 1.4
1.515 1.490 1.6
0.391 0.393 0.4
0.344 0.352 2.216
0.625 0.639 2.317
0.835 0.865 3.562
0.026 0.026 1.017
0.026 0.026 1.078
0.025 0.025 0.880
0.102 0.095 6.708
0.022 0.021 1.586
1.458 1.275 12.520
0.232 0.238 2.363
1.904 1.820 4.430
0.114 0.131 15.016
57
Table 4.4.Targets comparison of bulk Antimonides. Critical band edges and effectivemasses at Γ, X and L from ETB and HSE06 calculations are compared.The Eg and ∆SO are in the unit of eV; effective masses are scaled by freeelectron mass m0.
AlSb GaSb InSb
targets
Eg(Γ)
Eg(X)
Eg(L)
∆SO
mhh100
mhh110
mhh111
mlh100
mlh110
mlh111
mso
mcΓ
mcXl
mcXt
mcLl
mcLt
HSE06 ETB error(%)
2.303 2.296 0.3
1.584 1.586 0.1
1.860 1.859 0.1
0.657 0.658 0.1
0.312 0.320 2.709
0.590 0.614 4.074
0.756 0.803 6.206
0.128 0.125 2.782
0.107 0.105 2.123
0.103 0.101 2.166
0.240 0.221 8.121
0.111 0.113 1.421
1.485 1.242 16.319
0.216 0.208 3.575
1.491 1.548 3.808
0.121 0.132 9.050
HSE06 ETB error(%)
0.726 0.721 0.6
1.203 1.184 1.5
0.872 0.877 0.6
0.715 0.717 0.2
0.231 0.250 8.323
0.425 0.455 7.006
0.563 0.604 7.282
0.042 0.041 2.146
0.039 0.039 1.337
0.038 0.038 1.166
0.138 0.127 8.259
0.038 0.037 1.209
2.342 1.799 23.186
0.194 0.219 12.570
1.588 1.566 1.385
0.091 0.108 19.284
HSE06 ETB error(%)
0.173 0.170 1.6
1.567 1.550 1.1
0.891 0.867 2.8
0.754 0.760 0.7
0.245 0.277 12.859
0.452 0.507 12.156
0.609 0.694 13.923
0.012 0.013 6.063
0.013 0.014 4.723
0.012 0.012 6.615
0.117 0.108 7.506
0.011 0.012 8.707
0.877 0.790 10.004
0.219 0.230 4.992
1.685 1.575 6.500
0.096 0.111 15.670
58
L G X
−10
−5
0
5
10
Energy (eV)
Lattice constant = 5.4 Angstrom
HSE06
ETB
L G X
−10
−5
0
5
10 Lattice constant = 5.8 Angstrom
HSE06
ETB
(a) (b)
5.4 5.6 5.8 60
1
2
3
4
5
Lattice constant (Angstrom)
Energy (eV)
HSE06
ETB
Eg(X)
Ecp(G) - Ev
Eg(L)
Ecs(G) - Ev
(c) gaps vs lattice constant
Ec(L)
Ecs(G)
Ecp(G)
Fig. 4.4. Band structure of Si with different lattice constants( with hy-drostatic strain ). (a) Si with a lattice constant of 5.4 A, (b)Si with alattice constant of 5.8 A, (c) direct and indirect band gaps of Si withdifferent lattice constants. When lattice constant is 5.4 A, Si is a indi-rect gap semiconductor, the X conduction valley is the lowest conductionvalley. As lattice constant increase, the band gap at of X valley (Eg(X))increase slightly, while the bandgap of L valleys (Eg(L)) and direct bandgap (Eg(G) s band) decrease significantly. The lowest conduction bandstransit from X valleys to L valleys at about 5.8 A. When lattice constantreaches 6.0 A, lowest conduction band become Γ valley.
of valence bands are changed dramatically. From Fig .4.5 (a), it can be seen clearly
that the topmost unstrained valence bands are heavy hole bands with a large effective
mass and lower valence bands are light hole bands with smaller effective mass. All
bands in Fig .4.5 (a) are doubly degenerate. While for a strained valence bands shown
by Fig .4.5 (b), the top valence bands (2 fold) has a much lighter effective mass along
[100] direction than the lower valence bands. The degeneracy of the bands along [111]
splits sightly. Fig .4.5 (c) shows that the ETB in this work actually capture the strain
induced valence bandedge splitting well for strains with εxx up to 3%.
Fig.4.6 show the impact of strain conduction bands ( X valleys ) of GaAs due to
diagonal strains. The strain imposed satisfies εxx = εyy = −0.5εzz. The conduction
valleys along 6 X directions are equivalent due to crystal symmetry shown by Fig .4.6
(a). The x, y and z directions in a unstrained zincblende structure are equivalence.
This lead to a equivalence [100],[010] and [001] directions in the first Brillouin zone.
59
−0.4
−0.2
0
0.2
En
erg
y (
eV
)
−0.4
−0.2
0
0.2
(a) (b)
G [100][111] G [100][111]
VB (unstrained) VB (strained)
−4 −2 0 2 4−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
εxx (% )
En
erg
y (
eV
)
(c)
HSE06
TB
Fig. 4.5. HSE06 band structure of unstrained (a) and strained (b) GaAstop valence bands. The valence band splitting (c) at Γ point v.s. straincomponent εxx. The strain imposed on GaAs in (b) satisfies εxx = εyy =−0.5εzz. The top valence bands of unstrained GaAs are degenerate heavyand light hole bands. The degeneracy of top valence bands is broken inthe strained GaAs, As a result, both band edges and effective masses oftop valence bands change due to strain. The ETB calculations of the bandedge splitting of top valence bands agree well with the HSE06 results.
Under the diagonal strain, z direction is no longer equivalent to the x and y directions.
As a result, the 6 fold degeneracy of conduction valleys is also lifted ( Fig.4.6 (b)).
Resulting valleys along [100] and [010] direction form 4 fold degeneracy, while the
valleys along [001] direction form 2 fold degeneracy. The [100] and [010] valleys are
still degenerate since the strain imposed does not break the equivalency of x and y
directions. Fig .4.6 (c) shows that the ETB in this work actually capture the strain
induced valence bandedge splitting well for εxx up to 3%.
Fig.4.7 show the impact of [110] shear strain to Si conduction bands ( X valleys
). The strain imposed with a non-trivial εxy. The Si conduction bands at X points
are degenerate if there is no shear strain components εij. With εxy, the degeneracy
of the X point(with k = [0, 0, 1]2πa
) is lifted as it is shown in Fig.4.7 (a). As a result,
the transverse effective masses of the X conduction valleys at k = [0, 0, 0.84]2πa
are
changed. This change of effective masses is shown in Fig.4.7 (b). This effect can be
described by a two band k · p model which is shown by Appendix . From Fig.4.7,
it can be seen that the current ETB model also captures the effect of shear strains
60
(b)
−2 0 2−0.4
−0.2
0
0.2
0.4
εxx (% )
En
erg
y (
eV
)
(a)
(c)
HSE06
TB
Equivalent X valleys in
unstrained GaAs
Inequivalent X valleys in
strained GaAs
x,y
z
Fig. 4.6. X conduction valleys in unstrained (a) and strained (b) GaAs.The conduction band edges (c) at X points vs strain component εxx. Thestrain imposed on GaAs in (b) satisfies εxx = εyy = −0.5εzz. The sixX conduction valleys of unstrained GaAs are degenerate due to crystalsymmetry. The degeneracy of conduction valleys is broken in the strainedGaAs. The ETB calculations of the band edge for conduction valleys (Xpoints) agree well with the HSE06 results.
to conduction bands accurately. Both the band edge splitting and effective masses
changes are modeled by the ETB model.
Fig. 4.8 shows the band edge splitting at Γ, X and L points of InAs under dif-
ferent strains. In this figure, two kinds of strains which are not considered during
the fitting are presented. The strain produce by [001] stress contains only diagonal
strain components while the strain produce [123] and [111] biaxial strain have both
diagonal and off-diagonal strain components. It can be seen that the ETB band edge
splittings are in good agreement with the corresponding HSE06 results. The tight
binding model not only captures the strain induced splitting for special strains, it
also captures the strain behavior by arbitrary strains. To quantitatively estimate
the discrepancies between ETB and HSE06 calculations for strained materials, the
deformation potentials are extracted from both ETB and HSE06 results. The defor-
mation potentials of group IV and III-V materials are compared in tables 4.5 and 4.6.
It can be seen that the important deformation potentials by ETB agree well with
61
−1 0 1−40
−20
0
20
40
60
80
exy
(% )
∆ m
t/mt (
%)
−1 0 1
−0.2
−0.1
0
0.1
0.2
0.3
εxy
(%)
En
erg
y (
eV
)
(a) (b)
Splitting of Si conduction band
at X point
Transverse effective mass
of Si conduction band (X valley)
HSE06
TB
HSE06
TB
Fig. 4.7. Conduction band edge splitting (a) and effective mass changes(b) in Si under [110] shear strain. The unstrained Si conduction bandsare 4 fold degenerate at X points. While under shear strain with εxy,the degeneracy at X point(with k = [0, 0, 1]2π
a) is broken as shown in
(a). As a result, the transverse effective masses of the conduction valleyat k = [0, 0, 0.84]2π
aare changed. The ETB calculations of the band edge
splitting and effective mass of conduction bands agree well with the HSE06results. .
the HSE06 results. The discrepancies are within 1%. The deformation potentials
bv and dv describe the band edge splitting of valence bands under diagonal and off-
diagonal strain components respectively [40]. Ξ001 and Ξ110 describe the conduction
band edge splitting at X points due to diagonal and off-diagonal strain components
respectively [41, 52, 53]. The definitions of those deformation potentials are specified
in Appendix J.
4.5 Summary
An environment dependent ETB model for arbitrarily strained materials is pre-
sented. The onsite and interatomic interactions of the presented ETB model depend
on the environment of each atom. The model is applied to group IV and III-V strained
62
−2 0 2−2 0 2−0.5
0
0.5
1−0.5
0
0.5−0.5
0
0.5
Strained InAs
(e) 123 stress, CB (X) (f) 111 biaxial, CB (X)
(c) 123 stress, CB (L)
(a) 123 stress, VB and CB (G) (b) 111 biaxial, VB and CB (G)
(d) 111 biaxial CB (L)
HSE06
ETB
HSE06
ETB
Ba
nd
ed
ge
s sp
litt
ing
(e
V)
123 stress (GPa) 111 biaxial strain (%)
Fig. 4.8. Band edge splitting of selected conduction bands and valencebands at Γ ((a),(b)), X((c),(d)) and L((e),(f)) points of InAs. At Γ point,6 top most valence bands and 2 lowest conduction bands are shown. 4lowest conduction bands at X points are shown. The lowest conductionband at L points are included in the figures. The valence bands at X andL points are not shown as those points are of low energy. Two differentstrain is applied on the InAs lattice, including strain caused by stress along[123] and strain caused by biaxial strain along [111]; The ETB band edgesplitting are in good agreement with the corresponding HSE06 results.
materials with arbitrary strains. The ETB parameters are fitted with respect to hy-
brid functional calculations of different strained materials. Band edge splitting under
63
Table 4.5.Comparison of deformation potentials of group IV materials (ETB vsHSE06).
Si Ge
bv
dv
Ξ001
Ξ110
HSE06 ETB error (%)
3.10 3.12 0.8%
3.32 3.29 1.0%
1.60 1.59 0.8%
6.02 6.04 0.4%
HSE06 ETB error (%)
3.37 3.37 0.1%
3.34 3.34 0.0%
0.87 0.87 0.0%
6.13 6.13 0.0%
strain and corresponding deformation potentials show good agreement with Hybrid
functional results. Ab-initio targets that match finite temperature experimental data
are obtained by introducing lattice constant adjustment. Corresponding ETB param-
eters match experiments are generated. Initial results in this chapter were published
in Ref. [55]. Detailed discussion on strain model will be submitted to Physical Review
B [56].
64
Table 4.6.Comparison of deformation potentials of group III-V materials (ETB vsHSE06).
AlP GaP InP
targets
bv
dv
Ξ001
Ξ110
HSE06 ETB error
2.09 2.10 0.3%
2.18 2.15 1.3%
2.97 2.97 0.2%
6.14 6.18 0.7%
HSE06 ETB error
2.47 2.47 0.0%
2.86 2.84 0.8%
2.35 2.35 0.1%
6.61 6.62 0.2%
HSE06 ETB error
2.07 2.08 0.8%
2.26 2.27 0.5%
2.04 2.03 0.5%
6.18 6.12 0.9%
AlAs GaAs InAs
targets
bv
dv
Ξ001
Ξ110
HSE06 ETB error
2.14 2.14 0.1%
2.04 2.03 0.6%
2.87 2.87 0.1%
5.98 5.97 0.3%
HSE06 ETB error
2.53 2.48 2.0%
2.62 2.61 0.4%
2.28 2.29 0.3%
6.36 6.37 0.1%
HSE06 ETB error
2.10 2.07 1.2%
1.97 1.97 0.2%
1.86 1.87 0.5%
5.72 5.75 0.5%
AlSb GaSb InSb
targets
bv
dv
Ξ001
Ξ110
HSE06 ETB error
2.19 2.17 0.8%
2.12 2.15 1.5%
2.40 2.40 0.4%
5.05 4.98 1.4%
HSE06 ETB error
2.57 2.56 0.6%
3.26 3.22 1.0%
1.85 1.85 0.3%
5.34 5.34 0.1%
HSE06 ETB error
2.16 2.14 1.0%
3.04 3.04 0.0%
1.57 1.57 0.5%
4.78 4.72 1.2%
65
4.6 Tight binding parameters
The ETB parameters for strained group IV and III-V materials are presented in
this section.
Table 4.7.Atom type dependent onsite and spin orbit coupling parameters for groupIV and III-V elements. All parameters in this table have the unit of eV.
Atom
Es
Ep
Es∗
Ed
∆
Si
1.1727
10.1115
12.4094
13.8987
0.0215
Ge
-0.1105
9.8495
12.9983
13.3211
0.1234
Al
2.5246
8.8642
12.701
13.540
0.0015
Ga
1.4880
8.6528
12.7318
13.5576
0.02434
Atom
Es
Ep
Es∗
Ed
∆
In
1.6787
8.9987
12.7742
13.5664
0.1301
P
-2.3788
7.6742
12.5016
13.0781
0.0252
As
-3.5206
7.6037
12.5733
13.1056
0.1293
Sb
-2.3695
6.8994
12.6421
13.1316
0.2871
66
Table 4.8.Environment dependent onsite parameters for group IV and part of groupIII-V materials. In Si and Ge, both ’a’ and ’c’ denote the same atom.For Si-Ge bond, ’a’ corresponds to Si and ’c’ corresponds to Ge. Theparameters I’s and O’s are in the unit of eV. parameters λ’s are in theunit of A−1.
bond
Isc,a
Ipc,a
Is∗c ,a
Idc,a
∆ca
λsc,a
λpc,a
λs∗c ,a
λdc,a
Isa,c
Ipa,c
Is∗a,c
Ida,c
∆ac
λsa,c
λpa,c
λs∗a,c
λda,c
Oac
λac
δdij(A)
Si-Si
3.1457
2.5307
6.7086
3.5979
0.0
1.3389
1.4197
0.9522
1.1200
-
-
-
-
-
-
-
-
-
-2.1211
1.3004
-0.0118
Ge-Ge
2.4312
2.0823
6.3232
3.4105
0.0
1.4676
1.5634
1.0074
1.1411
-
-
-
-
-
-
-
-
-
-1.5926
1.5457
-0.0043
Ge-Si
2.8305
2.3208
6.5890
3.6100
0.0
1.4230
1.5026
0.9195
1.2181
2.6494
2.2079
6.8296
3.2390
0.0
1.4527
1.5253
0.6896
1.0880
-1.5267
1.4566
-0.08
Al-P
3.4070
2.8113
5.8451
3.1639
0.0023
1.3373
1.2648
0.9476
1.1719
2.1735
1.8851
5.6415
3.2039
0.0003
1.5196
1.4219
1.0031
1.0961
-2.0783
1.1878
0.0537
Al-As
3.3106
2.7621
5.8293
3.0491
0.0
1.4021
1.3069
0.9414
1.1437
2.2018
1.8730
5.7855
3.0758
0.0017
1.4805
1.4592
1.0470
1.2393
-1.9981
1.2167
0.0217
Al-Sb
3.9775
3.5125
6.1929
3.4694
0.0019
1.4438
1.3917
0.8225
1.3307
3.1709
2.9550
6.5076
3.6241
0.0045
1.5553
1.4960
1.0632
1.4126
-2.7374
1.2930
-0.0081
67
Table 4.9.Environment dependent onsite parameters for part of group III-V materi-als. The parameters I’s and O’s are in the unit of eV. parameters λ’s arein the unit of A−1. The nonzero δdij is introduced to match ETB resultswith experimental targets under room temperature.
bond
Isc,a
Ipc,a
Is∗c ,a
Idc,a
∆ca
λsc,a
λpc,a
λs∗c ,a
λdc,a
Isa,c
Ipa,c
Is∗a,c
Ida,c
∆ac
λsa,c
λpa,c
λs∗a,c
λda,c
Oac
λac
δdij(A)
Ga-P
2.5762
2.3797
5.6393
2.8883
0.0142
1.4641
1.1584
0.9348
1.1312
1.7691
1.5278
5.5291
2.9033
0.0015
1.5613
1.6773
0.9848
1.1407
-1.6875
1.2657
0.0043
Ga-As
2.4389
2.3491
5.6115
2.8751
0.0097
1.4410
1.2383
0.9153
1.2365
1.8772
1.5867
5.5735
2.8658
0.0053
1.5505
1.6838
1.0037
1.0993
-1.6467
1.2697
-0.0098
Ga-Sb
2.6906
2.7446
5.5030
3.0666
0.0003
1.5640
1.5635
1.0551
1.4301
2.5078
2.2820
5.7372
2.8997
0.0085
1.6001
1.6629
0.8877
1.3835
-1.9763
1.2931
-0.0019
In-P
3.7423
2.9385
5.7125
3.4549
0.0191
1.3365
1.2082
0.9125
1.2202
2.2406
2.0409
5.5215
3.4065
0.0008
1.4194
1.6325
0.9084
1.2028
-2.2511
1.2338
0.0182
In-As
3.5655
2.9008
5.9270
3.4482
0.0147
1.3568
1.2211
0.8811
1.2880
2.2916
2.1313
5.7498
3.3732
0.0023
1.3955
1.6602
0.9131
1.2172
-2.2073
1.2546
0.0096
In-Sb
4.1432
3.7455
6.1345
4.0470
0.0017
1.3823
1.4496
0.8716
1.3863
3.3104
3.1702
6.2579
3.8005
0.0038
1.6355
1.6168
0.9284
1.3715
-2.9363
1.2860
0.0176
68
Table 4.10.Bond length dependent interatomic coupling parameters for group IV ma-terials. In Si and Ge, both ’a’ and ’c’ denote the same atom. For Si-Gebond, ’a’ correspond to Si and ’c’ correspond to Ge. The parameters V ’sare in the unit of eV. parameters η’s are in the unit of A−1. The nonzeroδdij is introduced to match ETB results with experimental targets underroom temperature.
bond
Vscsaσ
Vs∗cs∗aσ
Vscs∗aσ
Vscpaσ
Vs∗cpaσ
Vscdaσ
Vs∗cdaσ
Vpcpaσ
Vpcpaπ
Vpcdaσ
Vpcdaπ
Vdcdaσ
Vdcdaπ
Vdcdaδ
Vsas∗cσ
Vsapcσ
Vs∗apcσ
Vsadcσ
Vs∗adcσ
Vpadcσ
Vpadcπ
Si-Si
-1.7377
-4.2881
-1.7587
2.9260
2.5379
-2.0901
-0.1627
3.7002
-1.2896
-0.9729
2.1919
-0.9507
1.8412
-1.3776
Ge-Ge
-1.7530
-4.4947
-1.4865
2.9146
2.3919
-1.9432
-0.1556
3.8013
-1.3517
-0.7001
2.1684
-0.4385
1.5738
-1.6745
Ge-Si
-1.7411
-4.6183
-1.6734
2.8349
2.5087
-2.2045
-0.2007
3.6856
-1.2686
-1.0464
1.9985
-0.3279
1.6931
-1.6394
-1.5824
2.8553
2.0593
-2.2859
-0.3354
-0.9837
2.0199
bond
ηscsaσ
ηs∗cs∗aσ
ηscs∗aσ
ηscpaσ
ηs∗cpaσ
ηscdaσ
ηs∗cdaσ
ηpcpaσ
ηpcpaπ
ηpcdaσ
ηpcdaπ
ηdcdaσ
ηdcdaπ
ηdcdaδ
ηsas∗cσ
ηsapcσ
ηs∗apcσ
ηsadcσ
ηs∗adcσ
ηpadcσ
ηpadcπ
Si-Si
1.5188
0.7884
0.9121
1.0267
0.6723
1.2901
0.7353
0.9903
1.3057
0.7324
0.8449
0.8837
1.4832
1.4183
Ge-Ge
1.5938
0.7628
0.9936
1.1150
0.6652
1.2611
0.7792
1.0020
1.3256
0.4988
0.7391
0.6221
1.4947
1.5345
Ge-Si
1.5187
0.5629
1.1773
1.0444
0.7828
1.2553
0.7795
0.9412
1.2571
0.7486
0.8194
0.6172
1.4207
1.5080
0.8371
1.1317
0.9643
0.9601
0.7171
0.7872
0.8921
69
Table 4.11.Bond length dependent interactomic coupling parameters V ’s for groupIII-V materials. The parameters V ’s are in the unit of eV.
bond
Vscsaσ
Vs∗cs∗aσ
Vscs∗aσ
Vscpaσ
Vs∗cpaσ
Vscdaσ
Vs∗cdaσ
Vpcpaσ
Vpcpaπ
Vpcdaσ
Vpcdaπ
Vdcdaσ
Vdcdaπ
Vdcdaδ
Vsas∗cσ
Vsapcσ
Vs∗apcσ
Vsadcσ
Vs∗adcσ
Vpadcσ
Vpadcπ
Al-P
-1.7682
-4.0139
-2.0131
2.9402
2.1206
-2.2681
-0.3042
3.5838
-1.2121
-0.7139
2.2351
-0.9666
1.9252
-1.5266
-1.2241
2.5861
2.6252
-2.1557
-0.5445
-1.2443
1.8639
Al-As
-1.8219
-4.3097
-2.0242
3.1045
2.1783
-2.2634
-0.3051
3.7366
-1.3318
-0.6818
2.2795
-0.7343
1.8295
-1.6782
-1.2520
2.5919
2.6105
-2.1862
-0.4197
-1.1628
1.9673
Al-Sb
-2.1063
-4.2962
-1.8153
3.3534
2.2283
-2.4048
-0.3387
4.1011
-1.6433
-0.9318
2.4007
-0.7374
1.7864
-1.8053
-1.5371
2.9884
2.5435
-2.0941
-0.2418
-0.9421
2.0986
Ga-P
-1.7010
-4.1464
-1.8778
2.8997
2.0854
-2.2303
-0.2808
3.5451
-1.1631
-0.8561
2.1997
-0.4721
1.5643
-1.4702
-1.1986
2.6045
2.6205
-1.7346
-0.4906
-0.7510
1.8737
Ga-As
-1.7842
-4.3164
-1.8820
2.9935
2.1256
-2.1456
-0.2812
3.7312
-1.2992
-0.7416
2.2874
-0.4906
1.4887
-1.6107
-1.1588
2.7008
2.5674
-1.9422
-0.3828
-0.6656
2.0486
Ga-Sb
-2.0232
-4.2066
-1.7410
3.2439
2.4986
-2.2758
-0.1848
4.1685
-1.5846
-1.1356
2.3716
-0.5153
1.6402
-1.8241
-1.6281
3.0092
2.2691
-2.1687
-0.3829
-0.3859
2.1917
In-P
-1.9110
-3.7944
-2.2047
3.0736
2.2361
-2.2543
-0.3446
3.6073
-1.2755
-0.5488
2.2517
-0.4615
1.6186
-1.6310
-1.1401
2.5465
2.6249
-1.6800
-0.7584
-0.5816
1.8626
In-As
-1.9667
-4.2049
-2.1482
3.2715
2.2493
-2.2986
-0.2867
3.9261
-1.4074
-0.6025
2.2879
-0.4708
1.6103
-1.8837
-1.1581
2.6184
2.6070
-1.7252
-0.4789
-0.5791
1.9421
In-Sb
-2.2797
-4.1696
-1.8748
3.5395
2.2701
-2.4392
-0.1813
4.2661
-1.7708
-0.9446
2.4045
-0.6675
1.7524
-2.0733
-1.3964
3.0903
2.3266
-2.0149
-0.3659
-0.3351
2.0716
70
Table 4.12.Bond length dependent interactomic coupling parameters η’s for groupIII-V materials. parameters η’s are in the unit of A−1
bond
ηscsaσ
ηs∗cs∗aσ
ηscs∗aσ
ηscpaσ
ηs∗cpaσ
ηscdaσ
ηs∗cdaσ
ηpcpaσ
ηpcpaπ
ηpcdaσ
ηpcdaπ
ηdcdaσ
ηdcdaπ
ηdcdaδ
ηsas∗cσ
ηsapcσ
ηs∗apcσ
ηsadcσ
ηs∗adcσ
ηpadcσ
ηpadcπ
Al-P
1.5395
0.7239
0.9612
1.1504
0.8908
1.0099
0.6760
0.9720
1.4131
0.7045
0.9310
0.7986
1.3402
1.3826
1.0682
1.0207
0.9204
1.1400
0.6734
0.7138
0.9125
Al-As
1.5402
0.7385
0.9635
1.1291
0.9000
0.9765
0.6901
0.9481
1.4223
0.6716
0.9336
0.8016
1.2909
1.4205
1.0682
1.0266
0.9233
1.1880
0.6640
0.7090
0.8956
Al-Sb
1.5484
0.6720
1.0249
0.9883
0.9711
0.8921
0.6394
0.9539
1.3508
0.5149
0.9104
0.8906
1.2642
1.5074
1.0043
1.0507
0.8024
1.2410
0.6954
0.7175
0.7612
Ga-P
1.5399
0.7270
0.9639
1.0862
0.8632
1.1882
0.6625
0.9887
1.4554
0.6995
0.9056
0.7629
1.4121
1.4383
0.9752
1.0821
0.9074
1.1570
0.6609
0.7059
0.9149
Ga-As
1.5565
0.7447
0.9515
1.1004
0.7836
1.1300
0.6818
0.9646
1.3846
0.6976
0.8730
0.6990
1.2959
1.4491
0.9898
1.1126
0.8269
1.0945
0.6838
0.6976
0.8941
Ga-Sb
1.5076
0.6439
1.0117
1.0413
0.9136
1.1453
0.6042
1.0211
1.4392
0.5096
0.9348
0.6763
1.4977
1.4208
0.9824
1.0806
0.8240
0.9333
0.7762
0.7726
0.8046
In-P
1.5274
0.7325
0.9559
1.0960
0.8578
1.1067
0.6949
1.0454
1.4932
0.7044
0.8241
0.8025
1.3955
1.3471
0.9630
1.0298
0.8790
1.0923
0.6906
0.7041
0.9100
In-As
1.5436
0.7794
0.9384
1.0707
0.8618
1.0693
0.6982
1.0434
1.4411
0.6964
0.7977
0.8020
1.4221
1.3581
0.9941
1.0809
0.8193
1.1253
0.6837
0.6993
0.9198
In-Sb
1.5461
0.6794
0.9793
1.0835
0.9525
0.9973
0.7439
0.9518
1.4457
0.5439
0.8398
0.7115
1.3794
1.2748
0.9732
1.1634
0.7068
0.9660
0.7474
0.7927
0.8251
71
Table 4.13.Off-diagonal onsite parameters due to multipole potentials. In Si and Ge,both ’a’ and ’c’ denote the same atom, parameters Cαaβcσ are left emptydue to relation Cαaβcσ = Cαcβaσ. For Si-Ge bond, ’a’ correspond to Si and’c’ correspond to Ge. All parameters are in the unit of eV.
bond
Csapc
Cpadc
Cdadc
Cscpa
Cpcda
Cdcda
Si-Si
1.2234
3.4303
9.9099
Ge-Ge
1.1939
3.3684
9.8628
Si-Ge
1.2030
3.3930
9.8856
1.2030
3.3930
9.8856
Table 4.14.Off-diagonal onsite parameters due to dipole and quadrupole potentials.All parameters are in the unit of eV.
bond
Csapc
Cpadc
Cdadc
Cscpa
Cpcda
Cdcda
Al-P
1.5306
3.5101
8.4800
1.2755
3.7066
10.1674
Ga-P
1.2321
3.3655
8.9391
1.2266
3.3529
9.1512
In-P
1.5843
3.2494
8.2225
1.0321
3.8671
8.8370
Al-As
1.9559
3.6671
6.9304
1.4219
3.8677
9.0338
Ga-As
1.2601
3.4064
9.2562
1.2327
3.5647
9.9997
In-As
1.1396
3.3227
9.1776
1.1388
3.3128
9.7860
Al-Sb
1.5751
3.5628
8.4919
1.2914
3.5603
8.5971
Ga-Sb
1.9561
3.8564
6.9425
1.9606
3.8573
6.7043
In-Sb
1.0291
3.3380
8.7305
1.1456
3.3593
8.6512
72
Table 4.15.Interatomic coupling parameters due to multipole potentials for group IVand part of group III-V materials. In Si and Ge, both ’a’ and ’c’ denotethe same atom. For Si-Ge bond, ’a’ corresponds to Si and ’c’ correspondsto Ge. All parameters are in the unit of eV.
bond
Psapcσ
Psadcσ
Ppapcσ
Ppapcπ
Pscpaσ
Pscdaσ
Ssapcσ
Ssadcσ
Spapcσ
Spapcπ
Sdadcσ
Sdadcπ
Sdadcδ
Sscpaσ
Sscdaσ
Qsapcσ
Qsadcσ
Qpapcσ
Qpapcπ
Qdadcσ
Qdadcπ
Qdadcδ
Qscpaσ
Qscdaσ
Si-Si
-1.5396
0.7752
-0.9283
1.6156
0.7491
1.4609
1.6103
-3.8712
0.7450
4.0875
3.9344
6.5771
-1.3985
-2.5641
-0.9290
1.9700
6.9775
-0.4367
Ge-Ge
-1.5663
0.7925
-0.6865
1.2451
0.8861
1.5098
1.6759
-2.6283
0.6304
3.2465
3.2883
5.1614
-1.4161
-1.9725
-0.7786
2.0320
6.8269
-0.4345
Si-Ge
-1.5006
0.8145
-0.7794
1.5188
-1.5006
0.8145
0.8200
1.4848
1.4812
-3.4877
0.7508
3.8909
3.7768
0.8200
1.4848
6.2119
-1.3773
-2.2944
-0.9155
2.0051
6.9180
-0.2475
6.2119
-1.3773
Al-P
-1.6592
0.3091
-1.0469
1.8003
-2.3325
0.3045
1.9743
1.5210
3.1829
-4.5544
0.9623
3.6546
3.7809
1.1003
2.3270
6.5773
-2.2243
-2.6508
-0.4430
2.0628
6.3774
-0.8822
7.3014
-1.6439
Ga-P
-1.5167
0.7372
-0.3635
1.6262
-1.5468
0.6986
0.7668
1.4103
2.0255
-4.4913
0.7014
3.7256
4.0881
0.7325
1.4101
5.6126
-1.0040
-2.3040
-0.5811
2.0977
6.1846
-0.2823
6.3718
-1.3167
In-P
-1.3417
1.1406
-0.2978
1.0269
-2.7879
0.4468
1.7468
1.3737
2.6098
-4.4096
0.2811
3.7468
3.5400
0.8607
2.0861
4.3389
-2.1996
-2.3389
-0.0174
2.3898
7.1001
-0.7120
6.0554
-1.3153
73
Table 4.16.Interatomic coupling parameters due to multipole potentials for part ofgroup III-V materials. All parameters are in the unit of eV.
bond
Psapcσ
Psadcσ
Ppapcσ
Ppapcπ
Pscpaσ
Pscdaσ
Ssapcσ
Ssadcσ
Spapcσ
Spapcπ
Sdadcσ
Sdadcπ
Sdadcδ
Sscpaσ
Sscdaσ
Qsapcσ
Qsadcσ
Qpapcσ
Qpapcπ
Qdadcσ
Qdadcπ
Qdadcδ
Qscpaσ
Qscdaσ
Al-As
-0.9448
1.0546
-1.5842
1.3440
-2.6736
0.3458
1.8554
1.8819
2.8324
-4.3925
0.7210
4.2782
3.7232
0.4794
2.3560
4.6049
-2.0707
-2.6020
-0.0880
2.4063
5.9092
-1.3089
7.0724
-0.8685
Ga-As
-1.3555
0.5127
-0.4684
1.0823
-2.2816
0.5314
1.4927
1.8221
1.8866
-4.2555
0.7340
3.1996
3.6569
0.5577
2.2435
5.2229
-1.5659
-1.2315
-1.1158
2.4369
7.0035
-0.7043
6.1072
-1.0584
In-As
-1.5821
1.0312
-0.4048
0.7503
-1.7354
0.7030
1.2099
1.5578
2.4993
-4.2825
0.6007
3.4492
3.9674
0.7424
1.4634
4.7711
-1.4199
-1.1147
-0.7130
1.7380
6.9446
-0.4713
5.4079
-1.1532
Al-Sb
-1.1122
0.8430
-1.1008
0.4365
-2.4051
0.4151
1.5753
1.4029
2.7592
-3.6569
0.4945
2.9925
2.8669
0.7568
2.2419
4.7149
-1.5381
-2.4559
0.1086
1.9471
6.3621
-1.2464
5.3544
-0.7379
Ga-Sb
-1.2324
0.6116
-0.9431
0.4087
-2.6815
0.4081
1.1479
1.8530
2.2525
-3.4164
0.1092
4.0625
2.6015
0.1379
2.3834
3.6729
-2.0985
-1.3465
-0.6194
2.3476
5.9208
-1.1962
6.9797
-0.7086
In-Sb
-1.0398
1.0719
-0.6557
0.5719
-1.8459
0.8668
1.9147
0.7074
2.7781
-2.8257
0.3453
3.3261
3.6276
-0.1957
2.0629
3.5491
-1.4815
-1.8447
-0.0713
2.2434
6.6177
-0.6252
5.2978
-1.1249
74
4.7 Summary
An environment dependent ETB model for arbitrarily strained materials is pre-
sented. The onsite and interatomic interactions of the presented ETB model depend
on the environment of each atom. The model is applied to group IV and III-V strained
materials with arbitrary strains. The ETB parameters are fitted with respect to hy-
brid functional calculations of different strained materials. Band edge splitting under
strain and corresponding deformation potentials show good agreement with Hybrid
functional results. Ab-initio targets that match finite temperature experimental data
are obtained by introducing lattice constant adjustment. Corresponding ETB param-
eters match experiments are generated. Initial results in this chapter were published
in Ref. [55]. Detailed discussion on strain model will be submitted to Physical Review
B [56].
75
5. TIGHT BINDING ANALYSIS TO
HETEROSTRUCTURES
5.1 Introduction
To surpass the oncoming limit of downscaling for field effect transistors, recent
innovative device designs such as tunneling field-effect transistors (TFET) [57–59]
and superlattice field-effect transistors [4, 5] are being actively investigated. Those
devices rely strongly on the usage of hetero-structures and strain techniques. To have
reliable prediction of device performance, it is critical to have an accurate atomistic
model that can handle strained heterostructures.
Efficient empirical tight binding(ETB) methods [13, 14] are preferred for device
level simulations for its numerical efficiency. Existing ETB strain models are fitted to
strained bulk systems with single material. However, when those models are applied
to structures with interfaces, the transferability of the ETB models and parameters is
questionable. On one hand, traditional ETB parameters depend on materials, while
material type around interfaces can not be clearly defined for compounds heterostruc-
tures. On the other hand, it was shown that ETB parameters obtained by traditional
direct fitting may lead to unphysical results [15]. To improve the transferability of
ETB parameters, ab-initio mapping algorithm are developed [15] in Chapter 2. An
new strain model was proposed in chapter 4 for arbitrarily strained materials. This
model is parameterized using the ab-initio mapping algorithm.
This chapter focuses on investigating the transferability of the ETB models and
parameters obtained in Chapter 4. For this purpose, the ETB model and parameters
shown in Chapter 4 are applied to strained superlattices. Superlattices of group IV
and III-V materials are studied by both ETB and hybrid functional calculations.
Band structure of superlattices obtained by ETB model are compared with HSE06
76
results. Excellent transferability of the ETB model is demonstrated as ETB and
hybrid functional band structures of strained superlattices get good agreement.
(c) (d)( )
4 layer XAs/YAs
(a) (b)
(f)(e)
8 layer XAs/YAs
4 layer Si/Ge 8 layer Si/Ge
4 layer AX/BY 8 layer AX/BY
Fig. 5.1. Atom structure of Si/Ge and XAs/YAs superlattices. X,Y canbe different cations. (a) Si/Ge superlattice with 4 layers in the primitiveunit cell; (b) Si/Ge superlattice with 8 layers in the primitive unit cell.(c) XAs/YAs superlattice with 4 layers in the unit cell; (d) XAs/YAssuperlattice with 8 layers in the unit cell. The primitive unit cells aremarked by the dashed lines.
5.2 Tight binding modeling of superlattices
To investigate the transferability of our ETB parameters, group IV and group
III-V superlattices are studied by both ETB and HSE06 models. The structures
of the superlattices considered in this work are shown in Fig.5.1. The superlattices
77
considered in this work grow along 001 direction. These superlattices contain only a
few atomic layers (with thickness from about 0.5 nm to 1.5 nm).
Group IV and III-V materials can form different types of hetero-junctions which
are defined by Fig.5.2. To model those superlattices by ETB method, in principle,
self-consistent ETB calculations coupled with Possion equation should be applied if
there is significant charge redistribution in the hetero-structures. Charge redistri-
bution must happen if there is a type-III heterojunction as it is shown in Fig.5.2
(c).However the presented superlattices turn out to be either type-I or type-II hetero-
junctions as the ab-initio band structures shows band gap of at least 0.5eV for all the
presented superlattices. The charge redistribution in type-I or type-II heterostruc-
tures under zero temperature is negligible because the valence bands of both materials
are perfectly occupied. The negligible build-in field can also be realized by looking
at the envelope of ab-initio local potentials. The local potential of a GaAs/AlAs
(a) (c)(b)
Type I
heterojunction
Type II
heterojunction
Type III
heterojunctionConduction
band (CB)
Valence
band (VB)
A B BA
BA
Fig. 5.2. Three types of heterojunctions formed by two semiconductors(denoted by A and B ). (a) type-I heterojunctions with straddling gap, (b)type-II heterojunctions with staggered gap and (c) type-III heterojunc-tions with broken gap. The shaded area stands for the states with elec-trons occupied. For type-I and type-II, there is no obvious charge transferbetween two different materials. However there is significant charge trans-fer from valence bands of material A to conduction bands of material B,since the top valence band in material A is higher than the conductionband of material B in a type-III heterojunction.
001 superlattice from HSE06 calculation is shown in Fig .5.3. It can be seen that
78
the potential envelope of Ga atoms and Al atoms are flat, suggesting small build-in
electric field. Similar potential profiles were observed in other group III-V and IV
superlattices with different size. This feature in local potential profile is also observed
in earlier work done by Van de Walle, et al [50,51].
0 0.5 1 1.5 2
−100
−50
0
z (nm)
U(r) (eV)
(a)
(b)
Fig. 5.3. Local potential of cations in a GaAs/AlAs 001 superlattice. (a)Superlattice considered contains contains As (black dots), Ga (white dots)and Al (grey dots) atoms. The local potential of cations along the dashedstraight line shown in (b).
5.3 Band structures of selected superlattices
Fig. 5.4 and Fig. 5.5 show the comparison of band structures of Si/Ge and Ar-
senides superlattices by ETB and Hybrid functional calculations respectively. In these
figures, band structures of Si/Ge, GaAs/AlAs, GaAs/InAs and InAs/AlAs superlat-
tices are presented. In both ETB and hybrid functional calculations, zero temperature
is assumed and spin orbital coupling is also included. For each type of superlattices,
band structure of two different unit cells are shown. It can be seen that the ETB
band structures are in good agreement for energy from -2eV to 1eV above lowest
conduction bands. ETB band structures are obtained with the parameters given by
previous sections without introducing extra fitting parameters. From Fig. 5.4 and
79
X G K
−2
0
2
4
Energy (eV)
X G K
−2
0
2
4
Energy (eV)
Si/Ge
HSE06
ETB
HSE06
ETB
4 layers 8 layers
(a) (b) (c)
HSE06
ETB
4 8 12# of layers
0.5
1
1.5
Band Gap (eV)
Fig. 5.4. Band structures of Si/Ge superlattices by ETB and HSE06calculations. ETB and HSE06 bandstructures are in good agreement.Figures correspond to band structures of superlattices which contain (a)4 atoms and (b) 8 atoms, and band gaps of Si/Ge superlattices verse thenumber of atoms in the supercell (c).
Fig. 5.5, it can be seen that ETB calculations without solving Poisson equation (zero
build-in potential is added ) match the HSE06 results well.
More complicated cases include InAs/GaSb superlattices which contain no com-
mon cations or anions at material interface. The InAs/GaSb superlattices with 4
atomic layers can also be interpreted as InSb/GaAs superlattice. From Fig. 5.6, it
can be seen that ETB calculations match the HSE06 results well even for interfaces
with no common cations or anions.
In 001 superlattices, the primitive unit cells are defined by vectors u1 = [0.5, 0.5, 0]a,u2 =
[−0.5, 0.5, 0]a and u3 = [0, 0, N ]a, where N can be any integer number. According to
the theory of Brillouin zone folding [60–62], the X points along [001] direction in a fcc
Brillouin zone is folded to the k = [0, 0, 0] point in the Brillouin zone of superlattices.
As a result, the lowest few conduction states at k = [0, 0, 0] of 001 superlattices can
have the feature of Γ and X conduction valleys of pure materials. The Γ and X
conduction states can be easily distinguished by corresponding ETB wave functions.
Considering the valleys in the Brillouin zone of a face center cubic lattice, the lowest
conduction states at Γ point are dominated by s and s* orbitals; while the conduction
80
X G K
−2
0
2
4
Energy (eV)
X G K
−2
0
2
4
Energy (eV)
X G K
−2
0
2
4
Energy (eV)
X G K
−2
0
2
4
Energy (eV)
X G K
−2
0
2
4
Energy (eV)
X G K
−2
0
2
4
Energy (eV)
GaAs/AlAs GaAs/InAs InAs/AlAs
HSE06
ETB
HSE06
ETB
HSE06
ETB
HSE06
ETB
HSE06
ETB
HSE06
ETB
4 layers 4 layers 4 layers
8 layers 8 layers 8 layers
(a) (b) (c)
(d) (e) (f)
Fig. 5.5. Band structures of Arsenides superlattices by ETB and HSE06calculations. ETB and HSE06 bandstructures are in good agreement.Presented band structures include band structures of superlattices of 001AlAs/GaAs((a),(d)), InAs/GaAs((b),(e)) and InAs/AlAs((c),(f)). Upperfigures correspond to supercells which contain 4 atoms(Fig 5.1 (a)), whilelower figures corresponds to supercells with 8 atoms (Fig 5.1 (b)).
states at X points have significant contribution from both s and p orbitals. This
can also be realized by the effective masses of the valleys. The folded X conduction
valleys have anisotropic effective masses as it is shown in Fig.5.5 (a) and (d); while
the Γ valley have isotropic effective masses as in Fig.5.5 (b) and (e). It can be seen
from Fig.5.5 that the lowest conduction state in AlAs/GaAs superlattices have the
feature of X conduction valley; while in InAs/GaAs and InAs/AlAs superlattices, the
lowest conduction state has the feature of Γ valley.
81
X G K
−2
0
2
4
Energy (eV)
X G K
−2
0
2
4
Energy (eV)
HSE06
ETB
HSE06
ETB
8 layers4 layers
(a) (b)(c)
InAs/GaSb
InAs/AlSb
InP/AlAs
InP/GaAs
InAs/GaSb
4 8 12# of layers
0.5
1
1.5
2
Band Gap (eV)
HSE06
ETB
Fig. 5.6. Band structures InAs/GaSb superlattices by ETB and HSE06calculations. Present superlattices include 4 layer (a) and 8 layer (b)InAs/GaSb superlattices. Corresponding band structures are shown in(c) 4 layer InAs/GaSb and (d) 8 layer InAs/GaSb.
82
4 8 120
0.5
1
1.5
2
# of layers4 8 12
1
1.5
2
# of layers4 8 12
1
1.5
2
2.5
Ba
nd
Ga
p (
eV
)
# of layers
4 8 120.5
1
1.5
2
# of layers4 8 12
0.5
1
1.5
2
2.5
# of layers4 8 12
1
1.5
2
2.5
Ba
nd
Ga
p (
eV
)
# of layers
GaP/InP
InP/AlP
GaP/AlP
GaSb/AlSb
GaSb/InSb
InSb/AlSb
(a)
(b) (c)
HSE06
ETBHSE06
ETB HSE06
ETB
HSE06
ETB
HSE06
ETB
HSE06
ETB
(d) (e) (f)AlAs/AlP
AlAs/AlSb
AlP/AlSb
GaAs/GaP
GaP/GaSb
GaAs/GaSbInAs/InSb
InP/InSb
InAs/InP
GaAs/AlAs
InAs/AlAs
GaAs/InAs
Fig. 5.7. Band gaps of III-V superlattices by ETB and HSE06 calcula-tions. The ETB band gaps of different superlattices show good agreementwith HSE06 results, demonstrating the ETB parameters have good trans-ferability. The presented band gaps include superlattices of (a) XP/YP ,(b) XAs/YAs and (c) XSb/YSb with ( X and Y stand for different cations,X,Y = Al, Ga or In) and (e) AlX/AlY , (f) GaX/GaY and (g) InX/InYwith ( X and Y stand for different anions, X,Y = P, As or Sb).
5.4 Transferability of the tight binding model
Fig. 5.4 (c), Fig.5.7 and Fig.5.6 (c) compare the ETB band gap of for different
superlattices with corresponding HSE06 results. Fig. 5.4 (c) shows the band gaps in
Si/Ge superlattices. The compared superlattices in Fig.5.7 include superlattices with
common anions (XP/YP, XAs/YAs and XSb/YSb) and superlattices with common
cations (AlX/AlY, GaX/GaY and InX/InY). Fig.5.6 (c) shows the band gaps of
selected AX/BY type superlattices, including InAs/GaSb, InAs/AlSb, InP/GaAs and
InP/AlAs. For the superlattices shown in the figure, averaged lattice constant is used
to create the unit cell of the superlattices since lattice mismatch always exists in
superlattices. It can be seen that ETB methods in this work delivered accurate band
83
gaps for ultra small superlattices. For ultra small superlattices, the band gaps are
not always monotonic functions of thickness. This non-monotonic dependency of
band gaps can be seen in many of the presented superlattices which have common
cations (Fig.5.7 (d), (e) and (f)). The ETB band gap of superlattices agree well with
corresponding HSE06. For superlattices which contain common cations or anions
(shown in Fig.5.7), the largest discrepancy of about 0.03eV appears in GaP/GaSb
superlattices. While the discrepancy of superlattices which contain no common cation
or anions, the largest discrepancy reaches a slightly higher of about 0.05eV. These
comparisons suggest that the ETB model and parameters by this work has good
transferability.
5.5 Summary
Transferability of ETB model and parameters to material interfaces is studied.
Strained superlattices of group IV and III-V materials are investigated by ETB and
HSE06 calculations. The ETB calculation of superlattice are in good agreement with
corresponding HSE06 results. It turns out that ETB calculation without including
build-in electric field is sufficient for type-I and type-II heterostructures. ETB calcu-
lations of all group IV and III-V superlattices which have common cations or anions
are performed. All of those ETB calculations generate band gaps that agree with
corresponding HSR06 results well. These comparisons suggest that the ETB model
and parameters in this work have good transferability. Initial results in this chapter
were published in Ref. [55]. Detailed discussion on Strain model will be submitted to
PRB [56].
84
6. PARAMETERIZATION OF 2D MATERIALS
6.1 Introduction
A great interest in 2D layered materials was triggered by the discovery of graphene.
In the past decade, intense research activities were conducted to 2D materials like
graphene not only because of rich in novel physics but also because of the technological
promise 2D materials hold in a broad range of applications such as optical, chemical
sensors and nanoelectronic transistors. Transition metal dichalcogenides (TMD) and
black Phosphorus are newly proposed 2D materials in transistor design. In contrast
to graphene, the most attractive feature of those newly proposed 2D materials for
device engineering is the exist of intrinsic band gap. TMD family materials such
as single layer and multilayer MoS2 and black Phosphorus are demonstrated to be
a promising candidate channel material for next generation transistors. The single
layer MoS2 based device shows promising I-V character, high on-off ratio and mobility
comparable to graphene based transistors [6]. Multilayer TMDs are also involved in
recent experiments. Even better device performance can be achieved by choosing
proper metal contacts [63]. In addition to TMD materials, the black Phosphorus,
which has intrinsic direct band gap and high intrinsic mobility, also entered the device
considerations [7,64]. As the newly appeared 2D materials in general have complicated
atomistic structure and electronic band structure, simple effective mass is insufficient
for device prediction. Generic transferable ETB models that can model TMDs and
black Phosphorus with different numbers of layers and under strain are thus desired.
This Chapter focuses on the empirical tight binding(ETB) parameterization of
newly proposed 2D materials such as TMDs and black Phosphorus. Geometries
and fundamental properties of band structures of TMDs and black Phosphorus are
85
introduced. ETB models are created and parameterized for the TMDs and black
Phosphorus systems. The ETB results are compared with ab-initio results.
6.2 Geometry of transition metal dichalcogenide and black Phosphorus
monolayer TMD monolayer Black P
top
view
side
view
1st
BZ
GG X
Y S
M
K
(a) (b)
(c) (d)
Fig. 6.1. Top view and side view of atom structures of monolayer (a)TMD and (b) black Phosphorus. The first Brillouin zone of TMDs ishexagonal (c), the black Phosphorus Brillouin zone is rectangular (d).High symmetry points considered in this work are defined in figures (c)and (d).
The TMDs and black Phosphorus are stacked 2D materials. The geometry of
monolayer TMDs and black Phosphorus are shown in Fig.6.1 (a) and (b) respectively.
It can be seen that the TMDs form hexagonal lattices while the black Phosphorus
forms rectangular lattice. For TMD, it is shown by the side view that each TMD
86
monolayer contains one transition metal layer and two anion layers. Each transition
metal atom forms bonds with neighboring six anions. The top view of TMD shows
that the cations and anions form a hexagonal lattice. From the side view, the black
Phosphorus has a rippled-like structure; the top view shows that the black Phosphorus
has a squeezed honeycomb lattice. The Brillouin zone of TMDs and black Phosphorus
are shown by Fig.6.1 (c) and (d) respectively. TMD’s first Brillouin zone are hexagonal
while the black Phosphorus has a rectangular Brillouin zone.
It is known that TMDs and black Phosphorus [64] have intrinsic band gap, which
make them good candidates of channel materials in novel transistors. The band gap
of TMDs and black Phosphorus can be modulated by the number of stacked layers.
In monolayer TMDs, the direct band gap is at K point. As the number of stacked
layers increase, direct gap will transit to indirect gap. While the black Phosphorus
always has direct gap at Γ point. In black Phosphorus the x and y directions are not
equivalent. As a result, black Phosphorus has different effective masses along x and
y direction.
6.3 Tight binding parameterization
TMDs and black Phosphorus with different layers are done by Ab-initio calcu-
lations. For TMDs we find GGA model with PBE [34] functional gives reasonable
agreement with experiment data. While for the black Phosphorus, Hybrid functional
calculations with HSE06 model is used to produce reasonable band structures.
6.3.1 Transition metal dichalcogenides
The ETB model of TMDs in this work uses an sp3d5 basis set with nearest-
neighbor interactions. The interatomic couplings considered in this model include
couplings between nearest neighbors of Mo-S, intra-layer Mo-Mo, intra- and inter-
layer S-S. Due to strong crystal field, the p orbitals and d orbitals are splitted. The
d-orbitals are splitted to d0 : d2z, d1 : dxz, dyz and d2 : dxy, dx2−y2; while the
87
p-orbitals split to p0 : pz,p1 : px, py. Here the TMD is assumed to be laid in
the x-y plane. The corresponding on-site energies and the interatomic couplings of p
and d orbitals are dependent on quantum number |m|. The inter layer S-S coupling
correspond to Van de Waals bond, while the in-plane S-S bonds are considered as
covalence bond. Thus the inter and inplane S-S couplings correspond to different
parameters. The TB model in this work is generic for single-layer and multi-layer
TMDs. The TB parameters are obtained by fitting to ab-initio calculations based
on PBE functional [15]. For TMDs, ETB model with s, p and d orbitals gives good
agreement with GGA results for different TMDs and for TMDs with different layers.
In this work, four kinds of TMDs are parameterized including MoS2, WSe2, MoTe2
and WTe2. Fig. 6.3.1 shows the band structure of monolayer MoS2 and WSe2 by
ETB and GGA calculations. It can be seen that the ETB band structures agree well
with corresponding GGA results. Fig. 6.3.1 shows the band structure of bilayer MoS2
and WSe2 by ETB and GGA calculations. The ETB calculations for bilayer TMDs
use the same parameter set as the one for monolayers.
Tables 6.1 and 6.2 show the parameters of TMDs. It turns out that the d orbitals of
anions have trends to get high energy and weakly couple to other low energy orbitals
during the fitting process. We thus decouple the d orbitals for anions with other
orbitals to reduce the number of free parameters. It is important to note that the d
orbitals in transition metal atoms have lower energies than the p orbitals of both the
cation and anions since the d orbitals of transition metal are orbitals of inner shell.
With the TMD parameters, transistors based on TMD materials is studied. This
work is published in Ref. [25].
6.3.2 Black Phosphorus
Phosphorus is a group V element, corresponding to five valence electrons in the
outermost shell. Each Phosphorus atom has three Phosphorus as its nearest neigh-
bors; however the bond lengths for these three neighbors are different due to the
88
Table 6.1.ETB parameters of MoS2 and WSe2. The d orbitals are defined as d0 :d2
z, d1 : dxz, dyz and d2 : dxy, dx2−y2; while the p orbitals are definedas p0 : pz,p1 : px, py.
MoS2 WSe2
Mo-Mo S-S S-Mo
Es 5.9449
Ep0 4.5135
Ep1 23.6168
Ed0 −0.1596
Ed1 0.7014
Ed2 −0.1573
Vssσ 0.0090
Vsp1σ 0.0018
Vsd0σ 0.0035
Vsd2σ 0.4118
Vp0p0π 1.0035
Vp0d1π −0.4646
Vp1p1σ −0.4759
Vp1p1π −4.9403
Vp1d0σ −2.5564
Vp1d2σ −1.6673
Vp1d2π 1.4229
Vd0d0δ 0.4541
Vd0d2σ −0.8784
Vd1d1π −0.0539
Vd1d1δ −0.1604
Vd2d2σ −0.5719
Vd2d2π 0.2650
Es 6.5601
Ep0 0.0214
Ep1 −1.5659
Ed0 −
Ed1 −
Ed2 −
Vssσ −0.0328
Vsp1σ 0.0425
Vp0p0σ −0.8864
Vp0p0π −0.3473
Vp1p1σ 0.5815
Vp1p1π −0.0169
Vssσ −0.7241
Vsp0σ 0.0101
Vsp1σ 0.0000
Vp0p0σ 0.2864
Vp0p1σ 0.0014
Vp1p1σ 0.0646
Vp1p1π −0.2731
Vssσ −0.0373
Vsp0σ −2.4738
Vsp1σ 0.9145
Vp0sσ 2.0484
Vp0p0σ −0.9355
Vp0p1σ −1.6004
Vp1sσ −3.0765
Vp1p0σ 3.5430
Vp1p1σ −3.7604
Vp1p1π 1.9706
Vd0sσ −8.0062
Vd0p0π −0.5256
Vd0p1π 1.8761
Vd1sσ 0.0109
Vd1p0σ 2.2803
Vd1p1σ 2.3695
Vd1p1π −0.9671
Vd2sσ 1.1799
Vd2p0σ 2.5001
Vd2p1σ −2.4039
Vd2p1π 0.9454
W-W Se-Se Se-W
Es 5.3987
Ep0 5.3475
Ep1 25.3555
Ed0 0.5662
Ed1 1.8202
Ed2 0.6786
Vssσ 0.0190
Vsp1σ −0.0008
Vsd0σ −0.0011
Vsd2σ 0.5763
Vp0p0π 1.0149
Vp0d1π −0.4353
Vp1p1σ −0.2721
Vp1p1π −5.8515
Vp1d0σ −2.2546
Vp1d2σ −1.4717
Vp1d2π 1.5458
Vd0d0δ 0.4561
Vd0d2σ −0.7826
Vd1d1π −0.0665
Vd1d1δ −0.1368
Vd2d2σ −0.5344
Vd2d2π 0.3831
Es 8.6987
Ep0 1.7485
Ep1 −0.4481
Ed0 −
Ed1 −
Ed2 −
Vssσ −0.0249
Vsp1σ 0.0741
Vp0p0σ −1.6061
Vp0p0π −0.4556
Vp1p1σ 0.5461
Vp1p1π −0.0178
Vssσ −0.7911
Vsp0σ 0.0114
Vsp1σ 0.0041
Vp0p0σ 0.3274
Vp0p1σ −0.0007
Vp1p1σ 0.0699
Vp1p1π −0.1798
Vssσ 0.0009
Vsp0σ −2.6403
Vsp1σ 1.0174
Vp0sσ 2.5358
Vp0p0σ −1.2675
Vp0p1σ −1.7282
Vp1sσ −2.8927
Vp1p0σ 4.7681
Vp1p1σ −4.1337
Vp1p1π 2.1779
Vd0sσ −7.4908
Vd0p0π −0.2153
Vd0p1π 1.9285
Vd1sσ 0.0050
Vd1p0σ 2.5711
Vd1p1σ 2.4009
Vd1p1π −1.0447
Vd2sσ 1.3341
Vd2p0σ 2.1966
Vd2p1σ −2.3308
Vd2p1π 0.9312
89
Table 6.2.ETB parameters of MoTe2 and WTe2. The d orbitals are defined asd0 : d2
z, d1 : dxz, dyz and d2 : dxy, dx2−y2; while the p orbitals aredefined as p0 : pz,p1 : px, py.
MoTe2 WTe2
Mo-Mo Te-Te Te-Mo
Es 6.1453
Ep0 6.5224
Ep1 26.1956
Ed0 1.1687
Ed1 2.0857
Ed2 0.9919
Vssσ 0.0451
Vsp1σ −0.0008
Vsd0σ −0.0008
Vsd2σ 0.1680
Vp0p0π 1.2574
Vp0d1π −0.5057
Vp1p1σ −0.3028
Vp1p1π −5.9098
Vp1d0σ −2.3617
Vp1d2σ −0.7565
Vp1d2π 1.3810
Vd0d0δ 0.4727
Vd0d2σ −0.5041
Vd1d1π −0.0642
Vd1d1δ −0.1509
Vd2d2σ −0.3797
Vd2d2π 0.3337
Es 9.0810
Ep0 1.3172
Ep1 0.2874
Ed0 −
Ed1 −
Ed2 −
Vssσ −0.0484
Vsp1σ 0.1282
Vp0p0σ −1.1333
Vp0p0π −0.2621
Vp1p1σ 0.6970
Vp1p1π −0.0094
Vssσ −0.7323
Vsp0σ 0.0113
Vsp1σ 0.0068
Vp0p0σ 0.3598
Vp0p1σ −0.0002
Vp1p1σ 0.0620
Vp1p1π −0.3463
Vssσ 0.0010
Vsp0σ −2.2437
Vsp1σ 1.0102
Vp0sσ 2.5760
Vp0p0σ −1.2928
Vp0p1σ −1.4803
Vp1sσ −3.1747
Vp1p0σ 3.3528
Vp1p1σ −3.9999
Vp1p1π 1.9569
Vd0sσ −7.4627
Vd0p0π −0.2492
Vd0p1π 1.6191
Vd1sσ 0.0066
Vd1p0σ 1.7550
Vd1p1σ 2.0784
Vd1p1π −0.7646
Vd2sσ 1.2626
Vd2p0σ 2.1165
Vd2p1σ −2.1892
Vd2p1π 0.9587
W-W Te-Te Te-W
Es 6.4285
Ep0 6.9305
Ep1 25.9185
Ed0 1.7556
Ed1 2.9570
Ed2 1.7099
Vssσ 0.0553
Vsp1σ −0.0008
Vsd0σ −0.0007
Vsd2σ 0.2002
Vp0p0π 1.2725
Vp0d1π −0.5713
Vp1p1σ −0.2582
Vp1p1π −5.6565
Vp1d0σ −2.1467
Vp1d2σ −0.7703
Vp1d2π 1.3489
Vd0d0δ 0.4040
Vd0d2σ −0.7968
Vd1d1π −0.0961
Vd1d1δ −0.1233
Vd2d2σ −0.4588
Vd2d2π 0.3347
Es 10.4628
Ep0 2.1664
Ep1 0.7733
Ed0 −
Ed1 −
Ed2 −
Vssσ −0.0407
Vsp1σ 0.1261
Vp0p0σ −1.5545
Vp0p0π −0.3122
Vp1p1σ 0.7117
Vp1p1π −0.0034
Vssσ −0.8142
Vsp0σ 0.0074
Vsp1σ 0.0080
Vp0p0σ 0.3520
Vp0p1σ −0.0002
Vp1p1σ 0.0677
Vp1p1π −0.4373
Vssσ 0.0010
Vsp0σ −2.2370
Vsp1σ 1.1236
Vp0sσ 2.7973
Vp0p0σ −1.4060
Vp0p1σ −1.4727
Vp1sσ −2.8831
Vp1p0σ 4.1974
Vp1p1σ −3.9190
Vp1p1π 2.0160
Vd0sσ −7.1226
Vd0p0π −0.2665
Vd0p1π 1.7039
Vd1sσ 0.0046
Vd1p0σ 1.8288
Vd1p1σ 2.2935
Vd1p1π −0.8663
Vd2sσ 1.2617
Vd2p0σ 2.2071
Vd2p1σ −2.2175
Vd2p1π 0.9362
90
Table 6.3.Tight binding parameters of Black Phosphorus. The ETB model usessp3d5s* basis set. Scaling parameters η is used to model the bond lengthdependency of the ETB parameters.
Intralayer Interlayer Intralayer Interlayer
Es
Epx
Epy
Epz
Es∗
Ed
Vssσ
Vspσ
Vsdσ
Vss∗σ
Vps∗σ
Vppσ
Vppπ
Vpdσ
Vpdπ
Vddσ
Vddπ
Vddδ
Vs∗s∗σ
Vs∗dσ
−7.089
−2.841
−2.125
−3.218
19.967
6.799
−2.268
3.851
−2.462
−1.314
0.417
4.945
−1.489
−2.082
1.431
−1.355
0.423
−0.675
−3.833
1.169
−5.34
15.153
−0.065
−2.506
0.478
3.902
−1.656
−2.566
0.055
0.786
0.399
−0.522
0.116
0.017
ηssσ
ηspσ
ηsdσ
ηss∗σ
ηps∗σ
ηppσ
ηppπ
ηpdσ
ηpdπ
ηddσ
ηddπ
ηddδ
ηs∗s∗σ
ηs∗dσ
4.469
6.399
5.754
4.679
0.266
3.972
8.066
2.402
3.892
1.271
0.769
10.536
2.963
28.313
7.188
6.688
1.972
3.924
0.391
4.585
4.524
2.102
0.969
0.299
0.146
7.584
0.437
2.139
91
GGAETB
−6
−4
−2
0
2
4
En
erg
y (
eV
)
K G M K K G M K
−6
−4
−2
0
2
4
En
erg
y (
eV
)
GGAETB
Band structure of monolayer MoS Band structure of monolayer WSe
(a) (b)
2 2
Fig. 6.2. ETB and GGA band structure of monolayer MoS2(a) andWSe2(b). ETB band structures agree well with the GGA result.
GGAETB
−6
−4
−2
0
2
4
En
erg
y (
eV
)
K G M K K G M K
−6
−4
−2
0
2
4
En
erg
y (
eV
)
GGAETB
Band structure of bilayer MoS Band structure of bilayer WSe
(a) (b)
2 2
Fig. 6.3. ETB and GGA band structure of bilayer MoS2(a) and WSe2(b).ETB band structures agree well with the GGA result.
92
crystal field. These differences in bond length suggest a bond length dependency is
needed in the ETB model. In this work, scaling law (d0/d)η is used to model the
bond length dependency. Moreover, it turns out that second nearest neighbors has
to be included to model the band structure through the whole Brillouin zone. In this
work, the interlayer and intralayer P-P coupling correspond to different parameters,
since interlayer P-P bonds are Van de Waals bonds while intralayer P-P bonds are
covalent bond. The x y z directions are not equivalent directions in black Phosphorus
structure. Consequently, the ETB model uses different onsite energies for px, py and
pz orbitals. The d orbitals use the same onsite energies for simplicity.
Table 6.3 list the ETB parameters for Black Phosphorus. The ETB model and
parameters in this work is generic for Black Phosphorus with different layers. Fig.6.4
show the comparison of ETB and HSE06 band structure of monolayer and bilayer
Black Phosphorus respectively. It can be seen that ETB results agree well with the
corresponding HSE06 results. Table 6.4 compares the effective masses and critical
band edge of black Phosphorus by ETB and HSE06 calculations. It can be seen that
the ETB model reach good agreement with the HSE06 for different layers. The black
Phosphorus parameters trigger application of transport studies in few layer black
Phosphorus, this work is in preparation [65].
6.4 Summary
Recently proposed 2D materials such as transition metal dichalcogenides and black
Phosphorus are parameterized in this chapter. Generic ETB models are developed for
transition metal dichalcogenides and black Phosphorus. For transition metal dichalco-
genides, the ETB model uses spd orbitals and the effect of crystal field is considered.
ETB band structures of transition metal dichalcogenides with different layers shows
agreement with ab-initio results with GGA approximation. An spds* tight binding
model with Harrison’s scaling law is applied for black Phosphorus. ETB band struc-
tures of black Phosphorus with different layers shows agreement with HSE06 results.
93
S G X S Y G
−4
−2
0
2
Energy (eV)
S G X S Y G
−4
−2
0
2
(a) (b)
monolayer black P bilayer black P
HSE06
ETB
HSE06
ETB
Fig. 6.4. ETB and HSE06 band structure of (a) monolayer and (b) bilayerblack Phosphorus. The ETB results agree well with the correspondingHSE06 results of both monolayer and bilayer black Phosphorus.
Table 6.4.ETB and ab-initio (HSE06) band gaps Eg, valence Ev and conduction Ecband edges in eV, and effective masses for electrons me and holes mh atthe point along X and Y direction for the monolayer, bilayer.
Monolayer Bilayer
HSE06 TB HSE06 TB
Ev -1.988 -1.989 -1.568 -1.571
Ec -0.640 -0.616 -0.704 -0.728
Eg 1.348 1.373 0.864 0.843
meX 0.187 0.183 0.186 0.185
mhX 0.171 0.157 0.166 0.160
meY 1.088 1.092 1.146 1.138
mhY 10.08 8.87 1.709 1.424
The TMD parameters and related application was published in Ref. [25]. The black
Phosphorus parameters and related application is in preparation [65].
94
7. SUMMARY AND OUTLOOK
7.1 Summary
The present thesis mainly focuses on the development and application of empirical
tight binding (ETB) parameterization algorithm. The summaries of related areas are
provided below.
The algorithm of generating ETB from ab-initio calculations are developed. This
algorithm match ETB wave functions and band structures with respect to correspond-
ing ab-initio targets. This algorithm can be applied to single material parameteriza-
tion and multi material parameterization, like the strained materials. The ab-initio
tool used in this work is VASP, the algorithm has been implemented in the Python
language.
This algorithm was applied to unstrained bulk systems and corresponding ultra
thin bodies. ETB parameters and highly localized basis functions are generated.
The ETB band structures of unstrained bulk materials show good agreement with
the Hybrid functional calculations. For ultra thin bodies which are passivated with
Hydrogen atoms, an explicit passivation tight binding model is introduced. This
model considers Hydrogen explicitly in the ETB Hamiltonian. ETB band structures
and wave functions of the important bands from this work agree well with HSE06 ones,
while the existing parameters and passivation model gave unphysical valence states.
It turns out both the explicit passivation model and the quality of bulk material
parameters are key factors for physical results in ultra thin bodies. The inclusion of
wave functions in the ETB parameterization helps improving the transferability of
parameters.
We developed an environment dependent ETB model for arbitrarily strained ma-
terials. This model has atom dependent onsite parameters rather than material de-
95
pendent onsites. Therefore ambiguity for material interfaces is eliminated. The strain
effects are included by considering the impact of dipole and quadrupole components
to the onsites and interatomic couplings. This model is applied to group IV and III-V
materials which have diamond or zincblende structures. Strain behavior is correctly
captured by this model. Parameters that match the finite temperature experimental
results can be easily obtained by including an extra artificial lattice constant.
The transferability of the ETB strain model is investigated by the application to
ultra scaled superlattices. It is shown that the ETB calculation without the consid-
eration of extra electric field is sufficient to model ultra scaled superlattices. The
ETB band structure for ultra small superlattices are in good agreement with HSE06
band structures. The band gaps of group IV and III-V superlattices are studied. It
is shown that for superlattices with common cations and superlattices with common
anions, the tight binding band gaps agree well with HSE06 band gaps, the maximum
discrepancy is within 0.1eV.
The ab-initio mapping algorithm is also applied to recently proposed 2D materi-
als such as transition metal dichalcogenides and black Phosphorus. Generic spd ETB
model considering crystal field is developed for transition metal dichalcogenides. ETB
band structures of transition metal dichalcogenides with different layers shows agree-
ment with ab-initio results with GGA approximation. An spds* ETB model with
Harrison’s scaling law is applied for black Phosphorus. ETB band structures of black
Phosphorus with different layers shows agreement with HSE06 results. With these
ETB models, transport calculations to transistors are performed.
7.2 Outlook
This thesis develops an algorithm of generating transferable parameters from ab-
initio calculations. Some of the pioneering work has been done on strained materials
and corresponding nano-structures. However, there are still a lot of opportunities
based on the presented efforts.
96
Study of semiconductor alloy This is extended topic of the work shown in chap-
ter 4 and 5. The proposed ETB model and transferable parameters for strained
materials provide good starting point for the investigation of semi- conductor
alloys. Open issues include: comparison of ETB calculations and ab-initio
calculations for random alloys; investigating the relation between random alloy
and virtue crystal alloy using the transferable ETB pa- rameters.
Strained Nitrides The strain model proposed by this thesis has been applied to
zincblende and diamond structures successfully. The wurtzite structures are
similar to zincblende and diamond structures since each atom in wurtzite struc-
tures also has four neighbor. The proposed strain model can be apply to both
zincblende and wurtzite Nitrides. Transferable parameters for strained Nitrides
can be expected.
Mapping of Extended Huckel Extended Huckel model is similar to tight bind-
ing model, but with explicit non-orthogonal basis function. Extended Huckel
model provides the possibility of handling amorphous systems semi-empirically.
However the existing Extended Huckel parameters have deficiencies when ap-
plied to strained systems and alloys. The algorithm presented in this thesis pro-
vides a straight forward way to improve the transferability of Extended Huckel
model.
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97
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APPENDICES
103
A. INTRODUCTION TO TIGHT BINDING MODEL
Tight binding models make use of localized orthogonal basis functions with form
Ψlm (r) = Ylm (Ωr)ψ (r). For example a sp3d5 model uses 9 orbitals shown by Fig
.A.1. Each orbital is characterized by its angular quantum numbers l and m. Excited
orbitals like s* orbitals are used in the a sp3d5s* model. The s and s* orbitals have
the same angular quantum number but different radial parts.
(a) s orbital
(b) p orbitals
(c) d orbitals
Fig. A.1. Tight binding orbitals used in spd tight binding model. (a)s orbitals, (b) p orbitals, (c) d orbitals. The green color correspond topositive sign while the blue color correspond to negative sign.
The interatomic interactions among orbitals of different atoms are approximated
by two center integrals which can be generalized by Slater Koster formulas [37, 44].
The two center integrals has a simple form if the bond is along z direction, as
〈l,m|H|l′,m′〉 = 〈l,m|H|l′,m〉δm,m′ (A.1)
This can be understood by Fig.A.2
104
(a) (c) (b)
Fig. A.2. Two center integrals among p orbitals for chemical bond along zdirection. (a) pz − pz coupling gives Vppσ , (b) px − px coupling gives Vppπ ,(c) pz − px is 0 .
Two center integrals for generalized bond direction can be related to two center
integrals along z direction through finite rotation matrix. For bond along ~d = [l,m, n]
Yl,m (θ, φ) =∑m′
Dlm,m′
(~d)Yl,m′ (θ
′, φ′) (A.2)
Where, β and γ correspond to the Euler angles between two coordinate systems. θ, φ
and θ′, φ′ corresponds to the angles defined in the old and new coordinates. The finite
rotation matrix is given by
Dlm,m′ (β, γ) = exp (im′γ)dlm,m′ (β) (A.3)
dlm,m′ (β) =
(1 +N
2
)l(1−N1 +N
)m/2−m′/2(A.4)√
(l +m′)!(l −m′)!(l +m)!(1−m)! (A.5)2l+1∑t=0
(−1)t
(l +m′ − t)!(l −m− t)!t!(t+m−m′)!(1−N1 +N
)t
The interatomic interaction Hamiltonian for bond along ~d, H(~d) is thus written
as
H(~d) = D†(~d)H (z)D
(~d)
(A.6)
105
Where D(~d)
is the finite rotation matrix for tight binding orbitals.
For spd orbitals, the D(~d)
has the form
D(~d)
=
1 0 0
0 Dpp
(~d)
0
0 0 Ddd
(~d) . (A.7)
The H (z) is Slater Koster type interatomic interaction matrix for a bond along z
direction
H (z) =
Vssσ Hsp (z) Hsd (z)
H†sp (z) Hpp (z) Hpd (z)
H†sd (z) H†pd (z) Hdd (z)
(A.8)
The corresponding blocks have simple form
Hsp (z) =(
0 Vspσ 0)
(A.9)
Hsd (z) =(
0 0 Vsdσ 0 0)
(A.10)
Hpp (z) =
Vppπ 0 0
0 Vppσ 0
0 0 Vppπ
(A.11)
Hpd (z) =
0 Vpdπ 0 0 0
0 0 Vpdσ 0 0
0 0 0 Vpdπ 0
(A.12)
Hdd (z) =
Vddδ 0 0 0 0
0 Vddπ 0 0 0
0 0 Vddσ 0 0
0 0 0 Vddπ 0
0 0 0 0 Vddδ
(A.13)
Explicit formular of Slater Koster interatomic interactions are given by table A.1
and A.2
106
Table A.1.Slater Koster interatomic interactions of s-p, s-d, p-p and p-d orbitals
Vspy mVspσ Vsdxy lmVsdσ
Vspz nVspσ Vspyz mnVsdσ
Vspx lVspσ Vsdz2
(n2 − l2+m2
2
)Vsdσ
Vsdxz lnVsdσ
Vsdx2−y2
√3
2
(l2 −m2
)Vsdσ
Vpypy m2Vppσ + (1−m2)Vppπ Vpydxy l(√
3m2Vpdσ + (1− 2m2)Vpdπ)
Vpypz mn(Vppσ − Vppπ) Vpydyz n(√
3m2Vpdσ + (1− 2m2)Vpdπ)
Vpypx lm(Vppσ − Vppπ) Vpydz2 m((n2 − 0.5(l2 +m2))Vpdσ −
√3n2Vpdπ
)Vpzpz n2Vppσ + (1− n2)Vppπ Vpydxz lmn
(√3Vpdσ − 2Vpdπ
)Vpzpx ln(Vppσ − Vppπ) Vpydx2−z2 m
(√3
2 (l2 −m2)Vpdσ − (1−m2 + l2)Vpdπ
)Vpxpx l2Vppσ + (1− l2)Vppπ Vpzdxy lmn
(√3Vpdσ − 2Vpdπ
)Vpzdyz m
(√3n2Vpdσ + (1− 2n2)Vpdπ
)Vpzdz2 n
((n2 − 0.5(l2 +m2))Vpdσ −
√3n2Vpdπ
)Vpzdxz l
(√3n2Vpdσ + (1− 2n2)Vpdπ
)Vpzdx2−z2 n
(√3
2 (l2 −m2)Vpdσ − (1−m2 + l2)Vpdπ
)Vpxdxy m
(√3l2Vpdσ + (1− 2l2)Vpdπ
)Vpxdyz lmn
(√3Vpdσ − 2Vpdπ
)Vpxdz2 l
((n2 − 0.5(l2 +m2))Vpdσ −
√3n2Vpdπ
)Vpxdxz n
(√3l2Vpdσ + (1− 2l2)Vpdπ
)Vpxdx2−z2 l
(√3
2 (l2 −m2)Vpdσ − (1−m2 + l2)Vpdπ
)
107
Table A.2.Slater Koster interatomic interactions of d orbitals
Vdxydxy 3l2m2Vddσ + (l2 +m2 − 4l2m2)Vddπ + (n2 + l2m2)Vddδ
Vdxydyz ln(3m2Vddσ + (1− 4m2)Vddπ + (m2 − 1)Vddδ
)Vdxydz2
√3lm((n2 − 0.5(l2 +m2)Vddσ − 2n2Vddπ + 0.5(1 + n2)Vddδ
Vdxydxz mn(3l2Vddσ + (1− 4l2)Vddπ + (l2 − 1)Vddδ
)Vdxydx2−z2 lm(l2 −m2) (1.5Vddσ − 2Vddπ + 0.5Vddδ)
Vdyzdyz 3n2m2Vddσ + (n2 +m2 − 4n2m2)Vddπ + (l2 + n2m2)Vddδ
Vdyzdz2√
3mn((n2 − 0.5(l2 +m2))Vddσ + (l2 +m2 − n2)Vddπ − 0.5(l2 +m2)Vddδ
Vdyzdxz ml(3n2Vddσ + (1− 4n2)Vddπ + (n2 − 1)Vddδ
)Vdyzdx2−z2 mn
(1.5(l2 −m2)Vddσ − (1 + 2(l2 −m2))Vddπ + (1 + 0.5(l2 −m2))Vddδ
)Vdz2dz2 (n2 − 0.5(l2 +m2))2Vddσ + 3n2(l2 +m2)Vddπ + 0.75(l2 +m2)2Vddδ
Vdz2dxz√
3ln((n2 − 0.5(l2 +m2))Vddσ + (l2 +m2 − n2)Vddπ − 0.5(l2 +m2)Vddδ
)Vdz2dx2−z2
√3(l2 −m2)
(0.5(n2 − 0.5(l2 +m2))Vddσ − n2Vddπ + 0.25(1 + n2)Vddδ
)Vdxzdxz 3l2n2Vddσ + (l2 + n2 − 4l2n2)Vddπ + (m2 + l2n2)Vddδ
Vdxzdx2−z2 nl(1.5(l2 −m2)Vddσ + (1− 2(l2 −m2))Vddπ − (1− 0.5(l2 −m2))Vddδ
)Vdx2−y2dx2−z2 0.75(l2 −m2)2Vddσ + (l2 +m2 − (l2 −m2)2)Vddπ + (n2 + 0.25(l2 −m2)2)Vddδ
108
B. FAST FOURIER TRANSFORMATION OF RADIAL
FUNCTIONS
To integrate
ψl (k) =
∫ ∞0
r2drjl (kr)ψl (r) (B.1)
in high accuracy, a fine grid in real space is needed. In fact, fast transformation exist
for this integral [30]. The spherical bessel functions jl (x)’s can be written as
jl (x) = (P (x) sin (x) +Q (x) cos (x)) /xl+1, (B.2)
where P (x) and Q(x) are polynomials. Let P (x) =∑l
n=0 pnxn and Q(x) =
∑ln=0 qnx
n
. The equation (2.29) can be written as
ψlm (k) =l∑
n=0
pn2kl+1−n
∫ ∞−∞
ψl(r)
rl−1−n sin (kr)dr +l∑
n=0
qn2kl+1−n
∫ ∞−∞
ψl(r)
rl−1−n cos (kr)dr.
(B.3)
In this equation, every integral in the right hand are Fourier transformation of function
ψl(r)rl−1−n . The terms with sin correspond to the imaginary part of the Fourier transform
while the terms with cos correspond to the imaginary part With the technique of fast
Fourier transform the computational load reduce from O(N2) to O(N logN)
To get the coefficient pn and qn. A complex polynomial Pl (x) = P cl (x) + iP s
l (x)
can be used. This polynomial satisfy
P0 (x) = i (B.4)
P1 (x) = i− x (B.5)
Pl+1 (x) = (2l + 1)Pl (x)− x2Pl−1 (x) . (B.6)
109
C. WAVE FUNCTION PROJECTIONS
For real space ab-initio wave functions ψn,k (r), realspace analysis can be performed.
For this purpose, non-overlaping regions around atoms located at Ra which satisfies
|r−Ra| < rc are defined. The realspace ab-initio wave functions ψn,k (r) inside the
non-overlaping regions are written as
ψn,k (r) =∑l,m
Rlm(|r−Ra| , n,k)Ylm(Ωr−Ra), for |r−Ra| < rc (C.1)
Many ab-initio tools generate wave functions which are represented by plan wave
basis functions, given by
ψn,k (r) =∑G
CG,kei(G+k)·r. (C.2)
To do realspace analysis of these plane wave represented ab-initio wave functions,
following expansion can be used
eik·r =∞∑l=0
l∑m=−l
4πiljl (kr)Y∗lm (Ωk)Ylm (Ωr) (C.3)
This equation relate a single plane wave to spherical harmonics (as angular parts) and
spherical Bessel functions (as radial parts). With this equation, the Rlm(|r−Ra|) in
equation (C.1) can be written as
Rlm(|r−Ra| , n,k) = 4πil∑G
CG,kei(G+k)·RaY ∗lm (ΩG+k) jl (|G + k| |r−Ra|) (C.4)
Equation (C.4) is essential for realspace analysis of plane wave represented wave
functions. For example, the weight of orbital characterized by l,m of atom a can
be measured as wl,m,a(n,k) =∫ rc
0|Rlm(|r−Ra| , n,k)|2r2dr. Figure C.1 shows a way
to represent this wl,m,a(n,k) using orbital weighted band structure. The weights of
selected orbital for each band are presented by the widths of lines. From Figure
110
C.1 (a) and (b), it can be seen that the orbitals d0 and d2 of Mo atoms contributes
differently to band structures. This figure provides a ”tight binding” view to the
ab-initio band structures.
8
6
4
2
0
2E
ne
rgy
(e
V)
2
0
2
Mo(d2)
(a) (b)
Mo(d0)
Fig. C.1. Orbital weighted band structures of single layer MoS2 . (a)d0 = dz2 (Mo atom) weighted band structure, the width of blue regionrepresent the weight of d0 ; (b) d2 = dxy, dx2−y2 (Mo atom) weighted bandstructure, the width of red region represent the weight of d2.
111
D. SOLUTION OF QUADRATIC FORM
The quadratic form define in equation (2.35)
F(1)ψ =
∑j,k
1−∑α,k
〈Ψα,k|Qk|Ψα,k〉 (D.1)
with Qk given by
Qk =∑j
|ψAbj,k〉〈ψAbj,k| (D.2)
can be solved straightforwardly in non-overlapping spherical regions. For non-overlapping
regions, we have equation (C.1) for ab-initio wave functions.
|ψAbn,k (r)〉 =∑l,m
RAblm(|r−Ra| , n,k)Ylm(Ωr−Ra), for |r−Ra| < rc (D.3)
On the other hand, tight binding bloch basis functions for non-overlaping regions are
given by
|Φα,k〉 =∑R
exp (ik ·Ra)Φa,n,l,m (r−Ra) , for |r−Ra| < rc, (D.4)
Here we limit the tight binding atomic basis function Φa,n,l,m (r−Ra) with the region
for |r−Ra| < rc. With equation (D.4), the F(1)ψ can be written as
F(1)ψ =
∑j,k
1−∑l,m,a
〈RTBlm (|r−Ra| , j,k)|Ql,m,a|RTB
l,m(|r−Ra| , j,k) (D.5)
and
Ql,m,a =∑j,k
|RAblm(|r−Ra| , j,k)〉〈RAb
lm(|r−Ra| , j,k)| (D.6)
To minimize the F(1)ψ using required orbitals ( for example spd orbitals ), it is
convenient to present |RAblm(|r−Ra| , j,k)〉 on a realspace grid. The minima of F
(1)ψ
corresponds to sum of max possible eigen values of Ql,m,a, given by
minF(1)ψ =
∑j,k
1−∑l,m,a
N∑n=1
λ(n)l,m,a. (D.7)
112
Here the λ(n)l,m,a is the eigen value of Ql,m,a, satisfying
Ql,m,a|R(n)l,m,a〉 = λ
(n)l,m,a|R
(n)l,m,a〉, (D.8)
with λ(n′)l,m,a ≤ λ
(n)l,m,a if n′ > n, the N is the number of orbitals required. As the
Ql,m,a with different l,m and at different atom do not couple, each Ql,m,a can be
diagonalized independently. For example, to get s,p and d orbitals, Ql,m,a with l =
0, 1 and 2 needs to be solved respectively. Maximum eigen values of each Ql,m,a
and corresponding eigen vectors are selected for s,p and d orbitals. The prototype
tight binding basis functions which minimizing F(1)ψ are eigen functions of Ql,m,a. If s
and s∗ orbitals are required, the radial part of s and s∗ correspond to the eigen vectors
of the largest and second largest eigen values. The eigen vectors are also presented
on a realspace grid as the ab-initio wave functions. Analytical expressions can be
used to parameterized the radial basis functions after the numerical basis functions
are obtained by solving eigen value of Ql,m,a. Some of the analytical expressions are
suggested in Appendix E.
The approach described in this section minimize the F(1)ψ and get corresponding
tight binding basis functions within non-overlapping spherical regions. It assumes the
tight binding basis functions are only non-trivial within one of the spherical regions.
This assumption hold if the tail of each orbitals are orthogonal with the orbitals in
other spherical regions.
113
E. ANALYTICAL EXPRESSION OF RADIAL BASIS
FUNCTIONS
Radial basis functions can be expressed as analytical functions. For example, Slater
type, Gaussian type functions and piecewise Gaussian functions can be used. The
Slater type functions are given by
Rl(r) = Pl(r) exp (−αr) (E.1)
with Pl(r) = rl (∑
i siri). And the Gaussian type functions are given by
Rl(r) = Pl(r) exp(−αr2
)(E.2)
with Pl(r) = rl (∑
i siri). Here the si and α are adjustable parameters. The Slater
type and Gaussian type functions can reach similar accuracy, while the Gaussian type
functions can be in principle more localized.
To give degree of freedom to regions near the core of the atoms and regions far away
from the cores, piecewise functions can be used. Piecewise functions are also used as
basis functions in LAPW or LMTO calculations. In this work, piecewise Gaussian
functions are preferred as analytical expression of basis functions. Piecewise Gaussian
functions are given by
Rl(r) =
Pl(r) exp (−αr2) r ≤ rc
(Rl(rc) +R′l(rc)r) exp (−β(r − rc)2) r > rc
(E.3)
with Pl(r) = rl (∑
i siri). The piece-wise Gaussian function Rl(r) and its first order
derivative R′l(r) are continuous at rc. Here the si, α and β are adjustable parameters.
For Slater and Gaussian basis functions, fitting parameters affect the shape of basis
functions globally. Compared with pure Gaussian or Slater type functions, the piece-
wise Gaussian functions provide more degree of freedom for r ≤ rc and at the same
114
time fast decaying tails for r > rc. This is because the parameters of piece-wise basis
functions either affect the region of r > rc or r ≤ rc. The piece-wise functions are
good candidates to present the basis functions obtained by Appendix D.
115
F. GENERATION OF HIGH RESOLUTION BAND
STRUCTURE BY HYBRID FUNCTIONAL
CALCULATION
This chapter provides details of generating high resolution band structure by Hy-
brid functional calculation with VASP. Typical LDA and GGA calculations calculate
density self consistently. Once density is obtained, one shot non-selfconsistent calcu-
lation is needed to generate band structure with any k-space resolution. While the
difficulty of generating high resolution band structure by hybrid functional calcula-
tion is that hybrid functional calculation requires self-consistent calculation for any
k-point. VASP can be used to calculate high resolution band structures using HSE06
hybrid functional approximation. This requires an explicit KPOINTS file as input. In
the KPOINTS file, the k-points and corresponding weights are listed explicitly. The
k-points include k-points in a uniform k-space mesh and additional k-points. Those
additional k-points can be any k-point to be calculated, and they have 0-weights.
Those additional k-points do not affect the self-consistent calculation because of the
0-weights.
Example of KPOINTS file used in HSE06 calculations:
Automatically generated mesh
53
Reciprocal lattice
0.0000 0.0000 0.0000 1
0.1667 0.0000 0.0000 2
0.3333 0.0000 0.0000 2
........
4
0.3333 0.5000 0.3333 4
116
0.5000 0.5000 0.3333 2
0.5000 0.5000 0.0000 0.0
0.4917 0.4917 0.0000 0.0
0.4833 0.4833 0.0000 0.0
There are four digits in each line specifying the k-point and corresponding weight.
The Fourth digit in each line specifies weight of each k-point used in the HSE06
calculations. The non-zero fourth digits will affect the self-consistent HSE06 results.
Additional k points with zero weights can be added according to the need of users.
Those points are calculated and outputted but does not affect the self consistent
calculations. Note the additional k-points have to have 0-weight, otherwise the results
will dependent on the additional k-points added.
With the explicit KPOINTS file, two steps are suggested to calculate band struc-
tures.
1. Self consistent GGA/LDA calculation (no spin orbit coupling). the final wave
functions are used as initial wave function for next step.
2. Self consistent HSE06 calculation (no spin orbit coupling). to get band struc-
tures. The wave functions from self consistent GGA/LDA calculation is used as
initial guess.
3. Self consistent HSE06 calculation to get band structures ( with spin orbit cou-
pling). The wave functions from self consistent HSE06 calculation obtained from
previous step is used as initial guess.
Note that All the steps use the same KPOINTS file. In principle, one can start
from step 2 or 3 directly. However that requires much longer time and converge
much slowly. Usually, the step 1 is much faster than the step 2 and 3. Following
the suggested steps above, the step 2 and 3 usually requires less than 10 iteration to
finish. Directly start from step 2 or 3 requires much more iterations.
Hybrid function calculations are in general much more expensive than normal
density functional calculations. In order to get high resolution band structure with
117
Hybrid functional approach, the explicit KPOINTS file possibly contains a large
number of zero weighted k-points. It will dramatically increase the computational
time. However, since the zero weighted k-points do not affect the self-consistent
calculations, it is convenient to make use of multiple different KPOINTS files. Each
KPOINT file contains different set of zero weighted k-points and with the same set of
non-zero weighted k-points. Those KPOINTS files can be calculated independently
or in parallel.
118
G. EXPRESSION OF M(L)α,γ
(D)
IN CHAPTER 2
For a unit vector d = [x, y, z], the explicit form ofM(l)α,γ
(d)
are given as follows. For
p and d orbitals, the order of orbitals are arranged according to quantum number m,
with py, pz, px and dxy, dyz, d2z2−x2−y2 , dxz, dx2−y2. Here theM(l)α,γ
(d)
are written
as matrices with α and γ as row and column indices respectively.
The matrix [M(1)00,1m′
(d)
] is given by
√3
4π
[y z x
]. (G.1)
The matrix [M(1)1m,2m′
(d)
] is given by
√3
4√
5π
√
3x√
3z −y 0 −√
3y
0√
3y 2z√
3x 0√
3y 0 −x√
3z√
3x
. (G.2)
The matrix [M(2)1m,1m′
(d)
] is given by
3
4π
2y2−x2−z2
3yz yx
yz 2z2−x2−y23
xz
yx xz 2x2−y2−z23
. (G.3)
The matrix [M(2)2m,2m′
(d)
] is given by
M(2)m,m′
(d)
=15
28π
−2z2−x2−y23
xz − 2√3xy yz 0
xz −2x2−y2−z23
1√3yz xy −yz
− 2√3xy 1√
3yz 2z2−x2−y2
31√3yz −x2−y2√
3
yz xy 1√3xz −2y2−x2−z2
3xz
0 −yz −x2−y2√3
xz −2z2−x2−y23
.
(G.4)
119
H. INTERATOMIC COUPLING DUE TO DIPOLE
POTENTIALS
The interatomic coupling due to multipole was given by equation (4.17). For dipole
moment, the term M(1)α,γ(d) are given by equations (G.1) and (G.2). The explicit
form of V(1)α,β are given in this appendix. For example, the px − py couplings V
(1)x,y is
given by
V (1)x,y =
∑kM
(1)x,s(dik)Q
(1)s,y(dik) (H.1)
+∑
k′ Q(1)x,s(djk′)M(1)
s,y(djk′)
+∑
d,k′M(1)x,d(dik)Q
(1)d,y(dik)
+∑
d,kQ(1)x,d(djk′)M
(1)d,y(djk′),
The Q(1)α,β’s are two center integrals given by equation (4.18). Using the explicit
expression of M and Slater Koster formula of Q(1), the terms in equation (H.1) are
written as ∑k
M(1)x,s(dik)Q
(1)s,y(dik) =
∑k xijyikQ
(1)spσ(dik) (H.2)∑
k′
Q(1)x,s(djk′)M(1)
s,y(djk′) =∑
k′ xijyjk′Q(1)spσ(djk′),
∑d,k′
M(1)x,d(dik)Q
(1)d,y(dik) =
1√15xijyijpij,k
(3Q
(1)pdσ(dik)− 2
√3Q
(1)pdπ(dik)
)(H.3)
+xikyij
(−Q(1)
pdσ(dik) + 3√
3Q(1)pdπ(dik)
)∑d,k
Q(1)x,d(djk′)M
(1)d,y(djk′) =
1√15xijyijpij,k′
(3Q
(1)pdσ(djk′)− 2
√3Q
(1)pdπ(djk′)
)(H.4)
−xjk′yij(−Q(1)
pdσ(djk′) + 3√
3Q(1)pdπ(djk′)
),
The pij,k =∑
m Y1,m(Ωdij)Y1,m(Ωdik
) and pji,k′ =∑
m Y1,m(Ωdji)Y1,m(Ωdjk′
), satisfying∑k pij,k = pij and
∑k pji,k′ = pji with pij and pji given by equations (4.20) and
120
(4.22). It can be seen that the terms with pij or pji has resemblance with Slater
Koster formula of Vxy = xy (Vppσ − Vppπ). To make the expression simpler, in this
work, only the terms with pij,k and pji,k′ are preserved. Let
3Q(1)pdσ√15
(dik) =4π
3
(Pppσ +
δdikd0
Sppσ
)(H.5)
2√
3Q(1)pdπ√
15(dik) =
4π
3
(Pppπ +
δdikd0
Sppπ
)(H.6)
The V(1)x,y can be approximated by
V (1)x,y = xijyij(δV
(1)ppσ − δV (1)
ppπ), (H.7)
here the δV(1)ppσ and δV
(1)ppπ are defined by
δV (1)ppσ = 4π
3(pij + pji)Pppσ + 4π
3(qij + qji)Sppσ (H.8)
δV (1)ppπ = 4π
3(pij + pji)Pppπ + 4π
3(qij + qji)Sppπ (H.9)
The pij, pji, qij and qji are given by equations (4.20) and (4.22). Similar process
can be applied to other V 1α,β’s. The generalized approximation was summarized by
equation (4.19).
121
I. GENERATION OF AB-INITIO RESULTS MATCHING
ROOM TEMPERATURE EXPERIMENTS
Ab-initio calculations based on LDA or GGA approximation severely underestimate
semiconductor band gap. More accurate methods such as Hybrid functional and
GW approximation greatly improve the band gap, however these methods still have
discrepancies about 0.1 to 0.2 eV compared with experiments. Furthermore, ab-initio
calculations usually assume zero temperature. Finite temperature targets can not be
truly obtained by most ab-initio calculations.
In order to get ab-initio band structure matching room temperature experiments,
artificial hydrostatic strain is applied to individual material to mimic the effect of
finite temperature and to compensate for the error of ab-initio calculations. With
hydrostatic strain, lattice constants change from a0 to a0 + δa. This artificial lattice
constant adjustment can be used to modulate the ab-initio band gap of semiconduc-
tors to match corresponding room temperature experimental results. Table I.1 shows
the required δa in order to match HSE06 band gaps with room temperature experi-
mental data. It can be seen that most of the required δa are in general less than 1%
hydrostatic strain. The AlP requires δa up to 2%a0. By this adjustment, band gaps
of most of the presented semiconductors reach less than 0.05eV mismatch compared
with experiments. The largest mismatch appears in AlAs which has the mismatch of
about 0.1eV.
Since the parameterization algorithm in this work relies on the ab-initio wave func-
tions, the concern of this artificial adjustment is that whether it will change ab-initio
wave functions significantly. Figure I.1 shows the contribution of different orbitals
in ab-initio wave functions as a function of lattice constant. Here the ab-initio wave
functions of InAs with different lattice constants are represented by the same basis
functions. It can be seen that every percent of hydrostatic strain introduced changes
122
the contribution of orbitals up to 0.02. Thus the artificial adjustment introduces
negligible changes to wave functions.
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
hydrostatic strain (%)
pro
ba
bili
ty o
f o
rbit
als
As, s orbital
In, s orbital
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
hydrostatic strain (%)
pro
ba
bili
ty o
f o
rbit
als
As, p orbitals
In, p orbitals
(a) (b)
Contribution of p orbitals to
top valence bands of InAs
Contribution of s orbitals to
lowest conductin band of InAs
slope: -0.015/(1%)
slope: 0.016/(1%)
slope: -0.00006/(1%)
slope: 0.00015/(1%)
Fig. I.1. Contribution of p orbitals to the top valence states (a) and sorbitals to the lowest conduction states of InAs. The p orbitals of In andAs atoms contribute to the top valence bands. The modification of latticeconstant within 2% only change wave functions sightly, thus it is reasonableto adjust band structure by changing lattice constant to match experiments. When lattice constant change one percent, p orbitals contribution arechanged by less than 0.0002. The s orbitals of In and As atoms contributeto the lowest conduction bands. When lattice constant is changed by onepercent, s orbitals contribution are changed by less than 0.02.
123
material a0 (A) gap (eV) δa (A) δa/a0(%) gap (eV)
exp,300K exp,300K HSE06 HSE06 HSE06
Si 5.43 1.12 -0.0273 0.5 1.141
Ge 5.658 0.66 -0.010 -0.2 0.755
AlP 5.4672 2.488 0.124 2.3 2.391
GaP 5.4505 2.273 0.01 0.2 2.256
InP 5.8697 1.353 0.042 0.7 1.397
AlAs 5.6611 2.164 0.05 0.9 2.05
GaAs 5.6533 1.422 -0.0226 -0.4 1.418
InAs 6.0583 0.354 0.0221 0.4 0.350
AlSb 6.1355 1.616 -0.0186 0.3 1.597
GaSb 6.0959 0.727 -0.0045 -0.1 0.707
InSb 6.4794 0.174 0.0406 0.6 0.172
Table I.1.Experimental lattice constants and band gaps of group IV and III-V ma-terials under room temperature; required changes of lattice constants δa inorder to match HSE06 band gap with experiments.
124
J. DEFINITION OF DEFORMATION POTENTIALS
For zincblende and diamond semiconductors , strain induced band edges splitting at
high symmetry points can be modeled by k.p models.
• Strain induced band edges splitting of top valence bands (at Γ point ) can be
described by a 4 band Luttinger k.p Hamiltonian [66].
Hε = −
Pε +Qε −Sε Rε 0
−S†ε Pε −Qε 0 Rε
R† 0 Pε −Qε Sε
0 R†ε S†ε Pε +Qε
(J.1)
with
Pε = −av (εxx + εyy + εzz) Qε = − bv2
(εxx + εyy − 2εzz) (J.2)
Rε =
√3bv2
(εxx − εyy)− idvεxy Sε = −dv(εxz − εyz) (J.3)
This Hamiltonian describes the interactions among heavy and light holes. The
basis function of this Hamiltonian is bloch functions characterized by total an-
gular momentum j = 3/2. The order of the basis functions is |j = 3/2,m =
3/2〉, |j = 3/2,m = 1/2〉, |j = 3/2,m = −1/2〉, |j = 3/2,m = −3/2〉 The pa-
rameters bv and dv are so-called deformation potentials. The bv describes the
the valence bandedge splitting under strains satisfying εxx = εyy = −0.5εzz or
εxx = −εyy. The dv describes the valence bandedge splitting under strain with
shear components (εxy, εyz, εxz).
• For conduction bands at near X points, the strain induced band edge change is
modeled by (X valleys) [53] ,
Ec = Ξu(k · ε · k) (J.4)
125
where ε is the 3 × 3 strain tensor, k is a unit vector along the direction of one
of the conduction band minima (X valley).
While the deformation potential of conduction X valleys due to off-diagonal
strain component εxy is described by 2 by 2 Hamiltonian [52] Eu Ξεxy
Ξεxy El
(J.5)
This Hamiltonian describes the upper and lower conduction bands at X point of
zincblende and diamond structures. The energy difference ∆E between the up-
per and lower conduction bands has the relation ∆E =√
(Eu − El)2 + 4Ξ2110ε
2xy.
It is well known that εxy modulate the effective mass of X valleys ([001] valley)
conduction bands. On the other hand, the degeneracy at X point is splitting
due to this strain component.
126
K. EXTERNAL RESOURCES FOR
PARAMETERIZATION OF MORE MATERIALS
Additional material parameters were parameterized during the PhD period. With
the parameterization algorithm by this work, materials such as metals, oxides, Ni-
trides and exotic materials such as SmSe are parameterized. Parameters of exotic
material SmSe and corresponding applications is published in Ref. [24, 67]. Pa-
rameters of Oxide material MgO are published in Ref. [23]. More details about
these additional parameterization can be found through nanohub resource [68] :
https://nanohub.org/resources/15173.
127
L. AGREEMENTS FOR REUSE OF PUBLISHED PAPERS
The licenses for the reuse of contents of published papers in this dissertations are
provided in this appendix
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VITA
Yaohua Tan was born in Guilin, Guangxi province, China. He received the Bachelor
of Engineering degree in Electrical Engineering from Tsinghua University, Beijing,
China in 2006. He continued his study and received the Master of Engineering degree
in Nanoelectronics from Tsinghua University, Beijing, China, in 2008. He joined
Purdue University in 2008 to pursue a Ph.D. degree in Electrical and Computer
Engineering. Yaohua’s PhD research focuses on algorithm and application of tight
binding parameterization from ab-initio calculations. After receiving his Ph.D. degree
in December, 2015, Yaohua Tan will work in University of Virginia as postdoc fellow.