Electronic and Optical Properties of Semiconductors: a Study Based on the Empirical Tight Binding Model

Electronic and Optical Properties of Semiconductors: A Study Based on the Empirical Tight Binding Model

by Lok C. Lew Yan Voon

ISBN: 0-9658564-4-5

DISSERTATION.COM

1997

ELECTRONIC AND OPTICAL PROPERTIES OF

SEMICONDUCTORS: A STUDY BASED ON

THE EMPIRICAL TIGHT BINDING MODEL

Lok C. Lew Yan Voon, B.A., M.A., M.Sc.

A Thesis

Submitted to the Faculty

of the

WORCESTER POLYTECHNIC INSTITUTE

in partial ful llment of the requirements for the

Degree of Do ctor of Philosophy

Physics

May 1993

APPROVED:

Professor L. R. Ram-Mohan, Thesis Advisor

Professor P.K.Aravind, Committee Member

Professor A. K. McCurdy, Committee Member

ELECTRONIC AND OPTICAL PROPERTIES OF

SEMICONDUCTORS: A STUDY BASED ON

THE EMPIRICAL TIGHT BINDING MODEL

Lok C. Lew Yan Voon, Ph.D.

Worcester Polytechnic Institute, 1993

Sup ervisor: Professor L. R. Ram-Mohan

Abstract

This study is a theoretical investigation of the electronic and optical prop-

erties of intrinsic semiconductors using the orthogonal empirical tight binding

mo del. An analysis of the bulk prop erties of semiconductors with the zincblende,

diamond and ro cksalt structures has b een carried out. Wehave extended the

work of others to higher order in the interaction integrals and derived new pa-

rameter sets for certain semiconductors which b etter t the exp erimental data

over the Brillouin zone. The Hamiltonian of the heterostructures is built up

layer bylayer from the parameters of the bulk constituents.

The second part of this work examines a numb er of applications of the

theory.We present a new microscopic derivation of the intervalley deformation

potentials within the tight binding representation and computes a number of

conduction-band deformation p otentials of bulk semiconductors. Wehave also

studied the electronic states in heterostructures and haveshown theoretically ii

the p ossibilityofhaving barrier lo calization of ab ove-barrier states in a mul-

tivalley heterostructure using a multiband calculation. Another result is the

prop osal for a new \typ e-I I" lasing mechanism in short-p erio d GaAs/AlAs su-

p erlattices. As for our work on the optical prop erties, a new formalism, based

on the generalized Feynman-Hellmann theorem, for computing interband optical

matrix elements has b een obtained and has b een used to compute the linear and

second-order nonlinear optical prop erties of a numb er of bulk semiconductors

and semiconductor heterostructures. In agreement with the one-band e ective-

mass calculations of other groups, our more elab orate calculations show that the

intersubband oscillator strengths of quantum wells can b e greatly enhanced over

the bulk interband values. iii

Acknowledgments

Ithankmy advisor, Professor L. R. Ram-Mohan, for his patient guidance and

constant encouragement during the course of my research.

Many p eople have made it p ossible for me to pursue physics. They stretch

from Mauritius to the United States, via England and Canada. Particular thanks

go to the various facultymemb ers, sta p ersons, and fellow students at all the

institutions that I have attended.

Iwould also liketothankDr.JoelN.Schulman, of Hughes Research

Lab oratories, for numerous helpful conversations.

I wish to thank my family and non-physics friends for their continued

supp ort, and for having faith that I will always reapp ear despite rep eated b outs

of silence and retrusion.

The thesis researchwas supp orted through a grant from the U. S. Naval

Research Lab oratory,Grant No: N00014-87-K-20 31- LRR, and by the Depart-

mentofPhysics at Worcester Polytechnic Institute. iv

Table of Contents

Abstract ii

Acknowledgments iv

List of Figures ix

ListofTables xi

I EMPIRICAL TIGHT BINDING MODEL OF ELEC-

TRONIC STATES xiv

Chapter 1. GENERAL FORMALISM 1

1.1 One-Band Mo del : :: :: :: ::: :: :: :: :: ::: :: :: :: :: 2

1.2 Multiband Mo del : :: :: :: ::: :: :: :: :: ::: :: :: :: :: 6

1.3 Treatment of Symmetry : :: ::: :: :: :: :: ::: :: :: :: :: 9

1.3.1 Hermiticity : :: :: :: ::: :: :: :: :: ::: :: :: :: :: 9

1.3.2 Inversion :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: 9

1.3.3 Point symmetry :: :: ::: :: :: :: :: ::: :: :: :: :: 10

1.4 Spin-Orbit Coupling : :: :: ::: :: :: :: :: ::: :: :: :: :: 10

1.5 Nonorthogonal Mo del : :: :: ::: :: :: :: :: ::: :: :: :: :: 13

Chapter 2. BULK HAMILTONIANS 17

2.1 Zincblende ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: 17

2.1.1 Spin-free case :: :: :: ::: :: :: :: :: ::: :: :: :: :: 17

2.1.2 Spin-orbit interaction : ::: :: :: :: :: ::: :: :: :: :: 25

2.2 Diamond : ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: 30

2.3 Ro cksalt : ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: 31

2.3.1 Spin-free case :: :: :: ::: :: :: :: :: ::: :: :: :: :: 32

2.3.2 Spin-orbit interaction : ::: :: :: :: :: ::: :: :: :: :: 35 v

2.4 Strained Hamiltonian : :: :: ::: :: :: :: :: ::: :: :: :: :: 36

2.5 Random Alloys :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: 40

Chapter 3. SUPERLATTICE HAMILTONIANS 42

3.1 [001] Zincblende :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: 44

3.2 [110] Zincblende :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: 47

3.3 [111] Zincblende :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: 50

3.4 [001] Ro cksalt : :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: 52

3.5 [110] Ro cksalt : :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: 54

3.6 [111] Ro cksalt : :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: 56

3.7 Sp ecial Cases of the Sup erlattice Matrix : :: :: ::: :: :: :: :: 56

3.8 Strained Sup erlattice : :: :: ::: :: :: :: :: ::: :: :: :: :: 58

3.8.1 Biaxial strain tensor :: ::: :: :: :: :: ::: :: :: :: :: 59

3.8.2 Uniaxial strain : :: :: ::: :: :: :: :: ::: :: :: :: :: 62

3.8.3 Internal displacement : ::: :: :: :: :: ::: :: :: :: :: 63

II APPLICATIONS 65

Chapter 4. BAND PARAMETERS 66

4.1 Energies : ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: 66

4.1.1 Zincblende : :: :: :: ::: :: :: :: :: ::: :: :: :: :: 67

4.1.2 Diamond :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: 69

4.1.3 Ro cksalt :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: 72

4.2 Group Velo city :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: 72

4.3 Deformation Potentials :: :: ::: :: :: :: :: ::: :: :: :: :: 75

4.4 Tight Binding Parameters :: ::: :: :: :: :: ::: :: :: :: :: 80

Chapter 5. INTERVALLEY SCATTERING 92

5.1 Tight Binding Deformation Potential Theory : :: ::: :: :: :: :: 94

5.2 Results :: ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: 97

5.2.1 GaAs: D ;D :: :: ::: :: :: :: :: ::: :: :: :: :: 97

X L

5.2.2 E ect of hydrostatic pressure :: :: :: :: ::: :: :: :: :: 98

5.2.3 Si Ge :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::100

x 1x

5.3 Summary : ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::100 vi

Chapter 6. OPTICAL MATRIX ELEMENTS 102

6.1 Previous Approaches : :: :: ::: :: :: :: :: ::: :: :: :: ::103

6.2 Theory and Discussions :: :: ::: :: :: :: :: ::: :: :: :: ::105

6.3 Applications :: :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::113

6.3.1 Bulk semiconductors :: ::: :: :: :: :: ::: :: :: :: ::113

6.3.2 Heterostructures :: :: ::: :: :: :: :: ::: :: :: :: ::124

6.4 Summary : ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::129

Chapter 7. HETEROSTRUCTURE PHYSICS 135

7.1 HgCdTe System :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::136

7.2 WaveFunction Lo calizatio n : ::: :: :: :: :: ::: :: :: :: ::140

7.2.1 Theoretical mo del : :: ::: :: :: :: :: ::: :: :: :: ::141

7.2.2 Results : :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::142

7.2.3 Summary :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::145

7.3 GX Laser ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::151

7.3.1 Design considerations : ::: :: :: :: :: ::: :: :: :: ::152

7.3.2 Calculations :: :: :: ::: :: :: :: :: ::: :: :: :: ::154

7.3.3 Summary :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::155

Chapter 8. SECOND-ORDER NONLINEAR OPTICAL PRO-

CESSES 158

(2)

8.1 Theory of : :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::160

8.1.1 Bulk second-harmonic generation :: :: :: ::: :: :: :: ::166

(2)

8.1.2 QW :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::167

8.2 Results :: ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::170

(2)

8.2.1 Bulk (0) :: :: :: ::: :: :: :: :: ::: :: :: :: ::170

8.2.2 Asymmetric GaAs/AlGaAs QW :::::::::::::::::172

App endices

App endix A. CRYSTAL STRUCTURES 176

A.1 Zincblende ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::176

A.1.1 Pointgroup: :: :: :: ::: :: :: :: :: ::: :: :: :: ::179

A.1.2 Irreducible representations : :: :: :: :: ::: :: :: :: ::179 vii

A.2 Diamond : ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::179

A.3 Ro cksalt : ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::180

A.3.1 Pointgroup: :: :: :: ::: :: :: :: :: ::: :: :: :: ::182

A.3.2 Irreducible representations : :: :: :: :: ::: :: :: :: ::182

App endix B. TWO-CENTER APPROXIMATION 190

B.1 Cubic and Spherical Harmonics :: :: :: :: :: ::: :: :: :: ::191

B.2 Two-Center Approximation : ::: :: :: :: :: ::: :: :: :: ::193

App endix C. HAMILTONIANS 197

C.1 Zincblende ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::197

C.2 Ro cksalt : ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::207

C.3 Hamiltonian in JM Basis : :: ::: :: :: :: :: ::: :: :: :: ::214

C.4 Sup erlattice :: :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::217

C.4.1 Zincblende-based SL :: ::: :: :: :: :: ::: :: :: :: ::217

C.4.2 Ro cksalt-based SL : :: ::: :: :: :: :: ::: :: :: :: ::223

App endix D. CRYSTAL ELASTICITY 229

Bibliography 232 viii

List of Figures

1.1 One-Band, One-Dimensional, Nearest-Neighb or, Empirical Tight

Binding Mo del :: :: :: ::: :: :: :: :: ::: :: :: :: :: 5

3.1 Distribution of Atoms in an [001] Zincblende-Based Sup erlattice : 46

3.2 Distribution of Atoms in a [110] Zincblende-Based Sup erlattice :: 48

3.3 Distribution of Atoms in a [111] Zincblende-Based Sup erlattice :: 51

3.4 Distribution of Atoms in an [001] Ro cksalt-Based Sup erlattice :: 53

3.5 Distribution of Atoms in a [110] Ro cksalt-Based Sup erlattice ::: 55

3.6 Distribution of Atoms in a [111] Ro cksalt-Based Sup erlattice ::: 57

3.7 Sp eci c Form of the Sup erlattice Matrix : :: ::: :: :: :: :: 64

4.1 Typical Band Structure for a Zincblende Semiconductor :: :: :: 70

4.2 Typical Band Structure for a Diamond Semiconductor :: :: :: 73

4.3 Typical Band Structure for a Ro cksalt Semiconductor ::::::: 74

4.4 Lattice ConstantasaFunction of Pressure :: ::: :: :: :: :: 78

4.5 Chemical Trend in the p-Like Energies for Zincblende and Dia-

mond Semiconductors :: ::: :: :: :: :: ::: :: :: :: :: 83

5.1 Intervalley Scattering Pro cess : :: :: :: :: ::: :: :: :: :: 93

6.1 Disp ersion Curves for the Conduction- and Valence-Band Energies

and Momentum Matrix ElementofaTwo-Band k p Mo del :: ::111

6.2 x Comp onent of the Squared Optical Matrix Elements Along the

LandX Directions for Bulk GaAs : :: :: ::: :: :: :: ::116

6.3 Chadi-Cohen Sp ecial Points for a Cubic Crystal :: :: :: :: ::120

6.4 Imaginary Part of the Dielectric Constant asaFunction of Energy

for GaAs :: :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::121

6.5 Joint Density of States as a Function of Energy for GaAs :::::122

6.6 Rationalized Joint Density of States as a Function of Energy for

GaAs : ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::123

6.7 Calculated Fundamental Transition in a 65-A (001) GaAs Quan-

tum Well as a Function of Barrier Width : :: ::: :: :: :: ::127 ix

7.1 Subband Disp ersion for a 50-A{40-A (001) HgTe/CdTe Sup erlat-

tice with = 40 meV :: ::: :: :: :: :: ::: :: :: :: ::138

7.2 Subband Disp ersion for a 50-A{40-A (001) HgTe/CdTe Sup erlat-

tice with = 350 meV : ::: :: :: :: :: ::: :: :: :: ::139

7.3 Conduction-Band Edges for Al Ga As-Al Ga As Sup erlattices 146

x 1x y 1y

7.4 Ab ove-Barrier States for Both the and X Conduction-Band

Edges in a Sup erlattice :: ::: :: :: :: :: ::: :: :: :: ::149

7.5 Intersubband- and Interband-Transition Oscillator Strengths for

a GaAs-Al Ga As Sup erlattice ::::::::::::::::::150

0:3 0:7

7.6 Three-Level Diagram for Lasing :: :: :: :: ::: :: :: :: ::156

7.7 Energy-Level Diagram for a Typ e-I I Lasing Mechanism :: :: ::157

8.1 Feynman Diagrams for Sum-Frequency Generation :: :: :: ::163

A.1 First Brillouin Zone of the Face-Centered Cubic Lattice :: :: ::177

C.1 General Form of a Sup erlattice Matrix for [001], [110] and [111]

Growth Direction : :: :: ::: :: :: :: :: ::: :: :: :: ::218 x

List of Tables

2.1 Tight Binding Parametrizations for Zincblende and Diamond Semi-

conductors : :: :: :: :: ::: :: :: :: :: ::: :: :: :: :: 24

4.1 Sources of Tight Binding Parameters for Orthogonal Nearest-

Neigb or Mo dels :: :: :: ::: :: :: :: :: ::: :: :: :: :: 86

4.2 Sources of Tight Binding Parameters for Orthogonal Second-Nearest-

Neighbor Models : :: :: ::: :: :: :: :: ::: :: :: :: :: 87

4.3 Tight Binding Parameters for GaAs :: :: :: ::: :: :: :: :: 88

4.4 Tight Binding Parameters for AlAs :: :: :: ::: :: :: :: :: 89

4.5 Tight Binding Parameters for HgTe :: :: :: ::: :: :: :: :: 90

4.6 Tight Binding Parameters for CdTe :: :: :: ::: :: :: :: :: 91

5.1 Deformation Potentials for Bulk GaAs :: :: ::: :: :: :: :: 99

5.2 Variation of Deformation Potential with Pressure for GaAs :: :: 99

5.3 Intervalley Deformation Potential for the Si Ge Alloys :::::101

x 1x

6.1 Interband Matrix Elements and Key Band Energies and E ective

Masses for GaAs at the Point Calculated Using Three Di erent

Tight Binding Mo dels :: ::: :: :: :: :: ::: :: :: :: ::131

6.2 Interband Matrix Elements and Key Band Energies and E ective

Masses for AlAs at the Point Calculated Using Three Di erent

Tight Binding Mo dels :: ::: :: :: :: :: ::: :: :: :: ::131

6.3 Band-O set Ratio, Transition Energy, Oscillator Strength, Dip ole

Matrix Element and Interband Momentum for the Fundamental

Intersubband Transition in the Conduction Band of a 65-A GaAs

Quantum Well : :: :: :: ::: :: :: :: :: ::: :: :: :: ::132

6.4 Conduction-Band Well Depth, Transition Energy, Oscillator Strength,

Dip ole Matrix Element, and Interband Momentum for the Fun-

damental Intersubband Transition in the Conduction Band of a

65-A GaAs Quantum Well ::: :: :: :: :: ::: :: :: :: ::132

6.5 Same as for Table 6.3 for a 82-AWell : :: :: ::: :: :: :: ::133

6.6 Same as for Table 6.4 for a 82-AWell : :: :: ::: :: :: :: ::133 xi

6.7 Band-O set Ratio, Transition Energy, Oscillator Strength, Dip ole

Matrix Element, and Interband Momentum for the Fundamental

Intersubband Transition in the Conduction Band of a 70-A{31-A

GaAs-AlAs Sup erlattice : ::: :: :: :: :: ::: :: :: :: ::133

6.8 Conduction-Band Well Depth, Transition Energy, Oscillator Strength,

Dip ole Matrix Element, and Interband Momentum for the Fun-

damental Intersubband Transition in the Conduction Band of a

70-A{31-A GaAs-AlAs Sup erlattice :: :: :: ::: :: :: :: ::134

7.1 Transition Energies for (GaAs) -(AlAs) Sup erlattices :: :: ::156

m m

7.2 Oscillator Strengths for (GaAs) -(AlAs) Sup erlattices :: :: ::156

m m

(2)

8.1 Calculated Bulk (0) for GaAs, InAs, and AlAs :::::::::174

8.2 Computed Zone-Center E ective Masses for Subband Levels in an

Asymmetric Quantum Well :: :: :: :: :: ::: :: :: :: ::174

8.3 Optical Transitions in an Asymmetric Quantum Well :::::::174

A.1 Symmetry Op erations of the Point Group T : ::: :: :: :: ::181

A.2 Character Table of the Double Group T : :: ::: :: :: :: ::183

A.3 Explicit (x; y ; z ) Irreducible Representation of the T Group 184

15 d

2 2 2 2

A.4 Explicit (x y ; 3z r ) Irreducible Representation of

the T Group :::::::::::::::::::::::::::::185

A.5 Explicit (xy; yz; zx) Irreducible Representation of the T

15 d

Group ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::186

A.6 Explicit (R ;R ;R ) Irreducible Representation of the T

25 x y z d

Group ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::187

A.7 Rotational Symmetry Op erations for Ro cksalt and Diamond :::188

: :: ::: :: :: :: ::189 A.8 Character Table of the Double Group O

B.1 s; p and d Spherical and Cubic Harmonics :: ::: :: :: :: ::195

B.2 Energy Integrals in the Two-Center Approximation :: :: :: ::196

C.1 Tight Binding Parameters for the 20-Band, Second-Nearest-Neighb or,

sp s Mo del for Zincblende ::: :: :: :: :: ::: :: :: :: ::204

C.2 Additional Parameters Up to Nearest-Neighbor in d Interactions

for Zincblende : :: :: :: ::: :: :: :: :: ::: :: :: :: ::205

C.3 Third- and Fourth-Neighbor Parameters for sp Interactions in

Zincblende : :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::206 xii

C.4 Tight Binding Parameters for the 36-Band, Nearest-Neighb or,

3 5

sp d Mo del for Ro cksalt : ::: :: :: :: :: ::: :: :: :: ::212

C.5 Additional Parameters for the Next-Nearest-Neighbor, sp Inter-

actions for Ro cksalt :: :: ::: :: :: :: :: ::: :: :: :: ::213

C.6 Clebsch-Gordan Coupling Co ecients for Spherical States Made

Up From L =0; 1; 2andS =1=2 States : :: ::: :: :: :: ::215

C.7 Decomp osition of JM States Into Cubic-Harmonic States for J =

3=2; 5=2 ::: :: :: :: :: ::: :: :: :: :: ::: :: :: :: ::216

C.8 f -Functions for Zincblende-Based Sup erlattice ::: :: :: :: ::220

C.9 h-Functions for Zincblende-Based Sup erlattice ::: :: :: :: ::222

C.10 g -Functions for Zincblende-Based Sup erlattice ::: :: :: :: ::224

C.11 g -Functions for Ro cksalt-Based Sup erlattice : ::: :: :: :: ::227

C.12 g -Functions for Ro cksalt-Based Sup erlattice : ::: :: :: :: ::228 xiii

Part I

EMPIRICAL TIGHT

BINDING MODEL OF

ELECTRONIC STATES xiv

Chapter 1

GENERAL FORMALISM

The tight binding (TB) scheme is an extension of the metho d of linear com-

bination of atomic orbitals of molecular chemistry as applied to solids, with

the explicit inclusion of the Blo ch p erio dicity condition for the crystalline state.

This connection is worth keeping in mind as one can address the question of the

chemistry of the electronic bands.

The empirical formulation of the tight binding mo del was intro duced

by Slater and Koster [1] and amounts to treating the matrix elements of the

Hamiltonian b etween atomic orbitals as free parameters, to b e determined by

tting energy gaps and e ective masses to exp erimental data. Some of the

reasons for the p opularity of the mo del are:

1. it is fairly straightforward to implement (the most dicult stage is, often,

in the initial parameter tting);

2. the real-space atomic picture is convenient and provides insights, for ex-

ample, in surface and cluster calculations [2, 3], and for short-p erio d su-

p erlattices;

3. it incorp orates the correct space-group symmetry of the crystal and, hence,

repro duces band-mixing and inversion-asymmetry e ects [4]; 1

4. its full Brillouin zone treatment can reveal e ects of the folding of the bulk

disp ersion in heterostructures;

5. the full rst Brillouin zone analysis also allows for a uni ed treatmentof

many-valley scattering phenomena;

6. the overall accuracy in the band structure is generally acceptable, and can

b e improved systematically [5];

7. the mo del can b e made to exhibit chemical trends [6].

General features of tight binding bands include their narrowness and non-

parab olicity. Hence, they have often b een used to describ e transition metals and

indirect band-gap semiconductors. The atomic description obviously provides

more detailed information than e ective-mass mo dels. For example, this is par-

ticularly useful in investigating the e ect of di erentinterfacial b ond typ es in

heterostructures with no common ion [7]. Also, items (3) and (4) ab ove are

readily seen to b e of great interest for sup erlattice structures, esp ecially within

the context of band-gap engineering and atomic growth of synthetic materials.

In the following sections, we will brie y describ e some of the general asp ects of

the theory. Sp eci c mo del Hamiltonians are derived in the next chapter.

1.1 One-Band Mo del

Most of the ideas p ertaining to the empirical tight binding mo del (ETBM) can

b e displayed by referring to a one-band, one-dimensional mo del. This also allows easy comparison with other band-structure calculations.

Thus, given an in nite linear chain, the exercise is to obtain a one-electron

state over the whole chain. Following the molecular calculations, one p ossible

pro cedure involves taking a linear combination of atomic orbitals (LCAO) cen-

tered at all the chain sites. This is a generalization of, e.g., the molecular-b enzene

problem, with rotational invariance b eing replaced by translational symmetry

with lattice constant a . Hence, a convenient starting function | one which

diagonalizes the translation group | is

ik R

e jRi; R = la (l 2Z) : (1.1) jk i =

In the ab ove, wehave replaced the b oundary condition at in nityby an equiva-

lent p erio dic one for N unit cells (Born-von Karman b oundary condition). The

discrete translational symmetry is represented mathematically bythewavenum-

ber quantum number k .Theaverage energy of an electron in such a state is then

ik (R R) 0

E (k ) hk jH jk i = e hRjH jR i

R;R

X X

ik R ik R

= e h0jH jRi e H (R) ; (1.2)

R R

where a choice of the origin has b een made (R = 0). The H (R) are material con-

stants (energy integrals) and are tted to actual data in the empirical approach.

Note that wehave assumed the orthonormality of the TB states in Eq. (1.2); in

fact,

X X

ik (R R) 0 0 ik R

hk jk i = e hRjR i =1+ e h0jRi ; (1.3)

R;R

where h0jRi are overlap integrals. In an empirical treatment, one is tempted to

think of the normalization as b eing absorb ed into the parameters. This is only

exact if the latter are recognized to havea k dep endence.

As an illustration of how the parametrization is carried out, consider a

nearest-neighb or mo del with time-reversal symmetry and the inclusion of the

normalization factor. The energy can then b e written as

H (0) + 2 H (R) cos kR

R>0

E (k ) = (1.4)

1+2 h0jRi cos kR

R>0

H (0) + 2H (1) cos

( = a k ): (1.5)

1+2h0j1i cos

Using the fact that the extrema can only b e at the zone edges or the zone

center in a one-band mo del and tting to the zone-center energy (chosen to b e

0), zone-center e ectivemass m , and the bandwidth E , one then has the

following disp ersion relation:

(1 cos )H (0)

E (k )= ; (1.6)

1+2I cos

with

1 1

I = ; (1.7)

2 +1

E (1 2I ) ; and, (1.8) H (0) =

m a E

= : (1.9)

Wehave plotted Eq. (1.6) in Fig. 1.1 for twovalues of the overlap integral I .A

numb er of p oints are worth noting in this simplistic example. Weobserve that

wave function overlap intro duces k -space disp ersion even in an onsite mo del (i.e.,

where H (1) = 0). This re ects the fact that onsite energies are mo di ed from the

free-atom values bytwo separate contributions: one due to the lattice p otential,

and the other due to the distortion of the lo cal orbitals by neighb oring ones. The

latter contribution leads to the so-called Lowdin orbitals used in the multiband

Figure 1.1. One-band, one-dimensional, nearest-neighb or, empirical tight bind-

ing mo del. Wehave plotted the two cases of an orthogonal (I =0)anda

nonorthogonal mo del (I =0:3).

problem. Such a mo del can also have a disp ersion relation whichmimicks an

orthogonalized mo del if one approximates the denominator by a series expan-

sion. The orthogonalized nearest-neighbor model intro duces a mo del-dep endent

constraint among m ;a and E such that, in Eq. (1.9), =1. Indeed, the

0 X

zone-center energy has already imp osed a constraintonthenumerical value of

the nearest-neighb or parameter. Such arti cial constraints will app ear again

when we analyze an orthogonalized nearest-neighbor multiband mo del for three-

dimensional (3D) crystals. Finally, a measure of orthogonali tycanbeintro duced

into Eq. (1.6) by de ning a new k -dep endent energy integral. The latter asp ect

is less evident in implementing for a multiband calculation since the disp ersion-

relation equation then takes the form of a generalized-eigenvalue problem.

1.2 Multiband Mo del

Consider rst the treatment without spin-orbit interaction. Given a one-electron

Hamiltonian op erator, a matrix representation in terms of a complete set of states

is required. One such set is the set of all the atomic states on all the atoms of

the crystal. Such a set, however, has two fundamental problems with resp ect to

applications: 1) the basis states are nonorthogonal, and 2) the set is countably

in nite.

The rst diculty can b e resolved by using orthogonali zed linear com-

binations of the original atomic states [8]. The new states are called Lowdin

orbitals. The second diculty can b e partially resolved by another linear com-

bination, this time over all translational ly-equiva lent atoms in the crystal:

ik(R+ )

e jR;b; i ; (1.10) jb; ; ki =

N R

where

: atomic-symmetry lab els

b : lab els the atoms in a primitive unit cell

R : lattice p oint p ositions

: p osition vector of basis atoms

N : numb er of unit cells :

The ket on the LHS is a Blo ch sum while that on the RHS is a Lowdin orbital.

It is easy to show that the new Blo ch-sum states satisfy the Blo ch condition and

are orthonormalized. This diagonali zes the Hamiltonian in k-space, and hence

the simpli cation.

However, the incompatibility of translational and p oint symmetry leads

to mixing of the Blo ch states in a band-structure calculation (i.e., where the

pseudomomentum k isagoodquantum numb er). As a result, the exact repre-

sentation of the Hamiltonian is still 1-dimensional. In analogy to the cuto

scheme of pseudop otential theory and in view of the near-band-gap interest

in semiconductor physics, the philosophy of TB calculations is to restrict the

treatment to a nite set of states. This approximation can b e justi ed within

the ETBM since, ultimately, a reduced set of energy parameters present in the

Hamiltonian matrix element (the TB parameters) is tted to exp erimental band

gaps and e ective masses. The contributions of distant orbitals are then viewed

as having b een accounted for,alaL owdin [9], in the renormalized parameters.

One can now obtain the exact solutions by applying a variational argu-

ment to the Schrodinger equation:

H jnki = E jnki ; (1.11)

which leads to solving the determinantal equation

jjhb; ; kjH E jb ; ;kijj =0 (1.12)

for the energies, with

hb; ; kjb ; ;ki = ; (1.13)

and the Blo ch states are

X X

jb; ; kihb; ; kjnki C (b; )jb; ; ki : (1.14) jnki =

b; b;

Using the representation of Eq. (1.10), one can actually obtain the explicit k

dep endence of the Hamiltonian matrix element:

) 0 ik(R+ 0

b b

h0;b; jH jR;b ; i e hb; ; kjH jb ; ;ki ( = ) =

ikR

0 bb

b b

e E (R) : (1.15)

The states in which the Hamiltonian is represented are thus the atomiclike or-

bitals of each of the translationall y non-equivalent atoms in a primitive unit cell

of the crystal. Furthermore, in the ETBM, the E (R) are treated as adjustable

parameters. Hence, there is no need for explicit expressions for the Hamiltonian

op erator and for the Lowdin orbitals (except when spin-orbit is included | see

x 1.4).

In order to obtain a more physical picture of the matrix element, one can

rewrite Eq. (1.15) as a shell summation:

X X

E e ( ) ( = ) =

X X

0 0

bb ik ik

bb bb

1 2

= E + e E ( )+ e E ( )+ (1.16)

1 2