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ELECTRONIC STRUCTURE OF DEFECTS IN III-VI AND II-VI
SEMICONDUCTORS AND NOVEL Yb—BASED INTERMETALLICS

By
Zsolt Rak

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Physics and Astronomy

2009

ABSTRACT

ELECTRONIC STRUCTURE OF DEFECTS IN III-VI AND II-VI
SEMICONDUCTORS AND NOVEL Yb-BASED INTERMETALLICS

By
Zsolt Rék

In recent years there has been a revival of interest in the III-VI family of semicon-
ductors (GaS, GaSe, GaTe and InSe) due to their exciting nonlinear optical properties
and their possible application in detector devices. These materials crystallize in lay-
ered crystal structure and their physical properties display a quasi two-dimensional
character. An important characteristic of these systems is the existence of Ga—Ga
(or In-In) dimers. It is well known that defects control the physical properties of
semiconductors. In this thesis, we have carried out electronic structure calculations
to study the nature of defect states in these materials. The defects we have studied
include substitutional impurities at the cation and the anion sites as well as cationic
and anionic vacancies. The failure of the hydrogenic effective mass approximation
(EMA) to reproduce the experimental binding energies for the substitutional Cd and
Sn defect states in GaSe, indicates the presence of large central cell corrections and
the necessity of incorporating short range interactions in the calculation of defect
binding energy. This has been done using a supercell model and self-consistent ab
initio electronic structure method within density functional theory (DFT), which is
known to be quite successful in tackling the problem of defects in semiconductors.
Analyzing the defects from first-principles, we have been able to explain the detailed
microscopic mechanism of the formation of Ga—site defects in GaSe and GaTe. When
Ga is replaced by an impurity or when it is removed from the system to create a
vacancy, the Ga dimer states can be strongly perturbed and this perturbation can
give rise to defect states in the band gap.

Defect formation energy calculations, based on total energy differences between

the pure and defect containing systems, can give valuable insight into the solubility
of different impurities in a host compound. The formation energies of Ge and Sn
impurities reveal that under Ga—rich growth conditions it is easier to incorporate
Sn in GaTe, whereas in the Te-rich limit Ge becomes more soluble than Sn. This
information can be used to reduce the large leakage current due to the presence of
native acceptors (Ga vacancies) in GaSe and GaTe by Ge or Sn doping. Furthermore,
the formation energy calculations provide information about the preferred location
of an impurity inside the host lattice. Using this idea, we developed a model which
explains the experimentally observed improvement in the mechanical properties of
In doped GaSe. In p-type GaSe, In becomes positively charged and can occupy an
interstitial site, improving drastically the shear rigidity of the layered material.

Using the same theoretical methods we have investigated the nature of H defects
in CdTe. The formation energy calculations indicate that the ground state position of
H inside the CdTe lattice depends on the charge state: the lowest energy positions for
H0 and H+ is at the bond center site, while H— prefers the low electron density site
surrounded by Cd cations. H in CdTe acts as an amphoteric impurity as expected.
In the case of H on Cd site, the system undergoes Jahn-Teller distortion, due to
the presence of a partially occupied degenerate t2 state at the top of the VB. The
symmetry of the system is lowered (the H atom moves closer to one of the four nearest
neighbor Te atoms) and the t2 level is split by ~74 meV at the F-point.

In order to study the properties of strongly correlated systems, one has to go
beyond the local density approximation (LDA)to the DFT and take into consider-
ation the strong Coulomb interaction within the localized electronic shell. In this
thesis we have used the LDA+U formalism to investigate the electronic, magnetic
and structural properties of several Yb-base systems, which involve highly localized
and strongly correlated f electrons. We find that the configuration of the f shell

plays a crucial role in the physical properties of many Yb containing intermetallics.

DEDICATION

To my lovely wife, Enik6, who kept my spirits up.

iv

ACKNOWLEDGMENT

Foremost, I would like to express my sincere gratitude to my advisor Prof. S. D.
Mahanti for the continuous support of my Ph.D study and research, for his patience,
motivation, enthusiasm, and immense knowledge. His guidance helped me in all the
time of research and writing of this thesis. I could not have imagined having a better
advisor and mentor for my Ph.D study.

My sincere thanks go to Dr. K. C. Mandal form EIC Laboratories, with whom
we collaborated and worked on diverse exciting projects, most of which are included
in my PhD thesis.

Special thanks also go to Prof. Stuart Tessmer, Dr. Alecsandra Tomic and Josh
Veazey for the interesting discussions and the collaboration on CeTe3 system.

I would like to thank the rest of my thesis committee: Prof. Ruby Ghosh, Prof.
Carlo Piermarocchi and Prof. Pavel Danielewicz for their encouragement and insight-
ful comments and questions during the Guidance Committee Meetings.

I thank my former and present group mates: Dr. Daniel Bilc, Dr. Salameh
Ahmad, Dr. Khang Hoang, Dr. Mal-Soon Lee and Dat Than Do for the stimulating
discussions.

Last but not the least, I want to thank my parents Istvan and Julianna and my

brother Istvan who encouraged me during all these years.

1

2

TABLE OF CONTENTS

List of Tables .................................

List of Figures ................................

Introduction ...........................
1.1 The III—VI layered semiconductors: GaS,

GaSe, GaTe, InSe .............................
1.2 The II-VI Wide band gap semiconductors:

CdTe and CdZnTe ............................

1.3 Ytterbium containing intermetallics ...................

Defects in Semiconductors ....................
2.1 Crystal defects: classification and basic concepts ............
2.1.1 Point defects ...........................
2.1.2 Donors and acceptors .......................
2.1.3 Shallow versus deep defects ...................
2.1.4 Shallow defects in semiconductors ................
2.1.5 Deep defects in semiconductors .................
2.1.6 The role of defects in semiconductors ..............
2.2 Theoretical methods to analyze defects .................
2.2.1 Basic quantum mechanics ....................
2.2.2 Hydrogenic—model (effective mass approximation) .......
2.2.3 Green’s function methods ....................
2.2.4 Scattered wave method ......................
2.2.5 Partial retention of diatomic differential overlap
method ..............................
2.2.6 Atomic cluster models ......................
2.2.7 The supercell model .......................
2.3 Density functional theory (DFT) ....................
2.3.1 The theorems of Hohenberg and Kohn .............
2.3.2 The Kohn-Sham equations ....................
2.3.3 The exchange-correlation functional ...............
2.4 The supercell technique for defect calculations .............
2.4.1 Introduction ............................
2.4.2 Defect formation energies and ionization levels .........
2.4.3 Equilibrium growth conditions .................
2.4.4 Energy-level reference and valence band alignment .......
2.4.5 Summary and conclusions ....................

ix

1

1

6

10
10
11
12
13
14
17
2O
22

36
38
41
41
42
45
46
47

3 Electronic structure of non-isovalent impurities in GaSe ..... 49

3.1 Introduction ................................ 49
3.2 Method of calculation ........................... 51
3.3 Pure GaSe ................................. 53
3.3.1 Crystal structure and geometric relaxation ........... 53
3.3.2 The electronic structure of GaSe ................. 55
3.4 Defects in GaSe .............................. 60
3.4.1 Atomic relaxations in the presence of defects .......... 61
3.4.2 Hydrogenic effective mass approximation ............ 62
3.4.3 Ab initio calculations of defects in GaSe ............ 65
Substitutional Cd on Ca site (CdGa) ........... 66
Substitutional Sn on Ca site (SnGa). .......... 71
Ga vacancy in GaSe (VGa)- ............... 74
Se vacancy in GaSe (VSe)- ................ 77
3.5 Summary ................................. 80
4 Electronic structure of isovalent impurities in GaSe ....... 82
4.1 Introduction ................................ 82
4.2 Computational method .......................... 83
4.3 Theoretical crystal structure ....................... 85
4.4 Elastic properties of GaSe1_xTeg; and
Ga1_$InxSe ................................ 87
4.5 Formation energies and electronic structure of defects ......... 91
4.6 Te induced defects ............................ 93
4.7 In induced defects ............................. 95
4.8 Rigidity enhancement of GaSe by In doping .............. 100
4.9 Impurity clustering ............................ 102
4.10 Summary ................................. 102
5 Electronic structure of defects in GaTe ............. 104
5.1 Introduction ................................ 104
5.2 Computational details .......................... 107
5.3 Crystal and electronic structures of pure GaTe ............. 109
5.4 Defects in GaTe .............................. 114
5.4.1 Substitutional impurities: SnGa and GeGa ........... 114
5.4.2 Vacancies VGa and VTe ..................... 115
5.4.3 Defect formation energies .................... 118
5.5 Summary ................................. 122
6 Hydrogen in CdTe ........................ 124
6.1 Introduction ................................ 124
6.2 Method of calculation ........................... 126
6.3 Location of hydrogen in CdTe ...................... 127
6.3.1 Substitutional hydrogen in CdTe ................ 128

6.3.2 Interstitial hydrogen at the tetrahedral sites: TCd and TTe . 134

6.3.3 Interstitial hydrogen in CdTe: the bond center (BC) site . . . 136

6.4 J ahn-Teller effect ............................. 139

6.5 Defect formation energies ......................... 140

6.6 Comparison between the LDA and GGA results ............ 142

6.7 Summary ................................. 144

7 Ytterbium containing systems .................. 145

7.1 Introduction: the LDA+U method ................... 145

7.2 Electronic structure of YbNiFeAlg ................... 150
7.3 The valence state of Yb in Yb3Au31n3

and Yb3AuGe2In3 ............................ 155

7.4 Summary ................................. 160

8 Summary and future directions ................. 162

Bibliography .......................... 169

viii

3.1

3.2

4.1

4.2

4.3

4.4

5.1

5.2

LIST OF TABLES

The theoretical lattice constants of GaSe and the Ga—Ga, Se-Se and
Ga-Se bond lengths calculated by FPLAPW and PAW methods. For
comparison the experimental values are also given as in Ref.[153]. All
distances are given in angstroms. .................... 54

The changes in the Ga—Ga, Ga-Se and SeSe bond lengths when Ga
is replaced by an impurity (Cd, In or Sn). All distances are given in
angstroms.(alimited relaxation where only the defects’ nearest neigh-
bor atoms were allowed to relax ............. . ........ 62

Strains and elastic moduli for crystals with hexagonal symmetry. AB is
the change in energy due to the specific strain and V0 is the equilibrium
unit cell volume. ............................. 85

Optimized theoretical lattice parameters of GaSe1_xTex and Ga1_xlnx Se.

b

(“theory; experiment) .......................... 86

Elastic contants of GaSe1_$Te$ and Ga1_xIna;Se. All values are given
in GPa. (“theory; bexperiment) ..................... 88

Formation energies of neutral defects and the charge transition levels.
All values are given in eV. ........................ 93

The experimental and theoretical lattice parameters of bulk GaTe. The
distances are given in angstroms and the monoclinic angle 7 in degrees.
(1%; a_ Ga and dHGa- Ga refer to the Ga-Ga dimer lengths that are per-

pendicular and parallel to the layer planes, respectively ......... 110

Optimized impurity-host bond lengths. The last row contains the per-
cent increase in the bond lengths relative to the ones obtained for the
bulk GaTe.The notations Tel and Te2 refer to the N N and next NN
Te atoms of the defect .......................... 115

ix

5.3 The distances between the vacancies and their nearest neighbors. The

6.1

7.1

notations of Gal and Ga2 refer to the Ga atoms from the dimers ori-
ented perpendicular and parallel to the atomic layers respectively. The
bond lengths of the pure GaTe are given in parenthesis. The distances
are given in angstroms ...........................

The CdTe band gap, formation enthalpy and the formation energies of

117

the lowest energy defects and charge states. All values are given in eV. 142

The calculated f occupation for different values of the Coulomb corre-
lation U ...................................

159

2.1

2.2

3.1

3.2

3.3

3.4

LIST OF FIGURES

Images in this dissertation are presented in color

Rectangular, one-dimensional well of depth V0 and width d ......

Electron eigenstates (a) in a one-dimensional square well potential and
(b) in a Coulomb potential ........................

Schematic view of the crystal structure of fl-GaSe. The unit cell ex-
tends over two layers of GaSe and contains four Ga and four Se atoms.
In each layer two monoatomic sheets of Ga are sandwiched between
two monoatomic sheets of Se. In this figure 9 unit cells are shown . .

Qualitative scheme showing the bands responsible for bonding in GaSe.
The bottom of the valence band has a strong Ga 3 character, followed
by a strong mixture of Ga 3 and Se p—states. Two Se p bands split
off the valence band as a result of this hybridization, and form the
bottom of the conduction band. The lowest conduction bands also
display Ga—pz character. .........................

(a) Band structure of fi-GaSe. In addition to the scheme presented
in Fig. 3.2, the Se s-states are also shown here (subband IV). The
zero energy is chosen at the top of the valence band. (b) The Ga 3
orbital contribution to the band structure. The Ga .9 antibonding state
hybridizes with the Se p states, pushing up some Se p—states into the
conduction band. The lowest conduction bands, therefore also display
Ga 3 character. (c) The Se 10;; + 1);, orbital contribution to the band
structure. (d) The Brillouin zone corresponding to the hexagonal unit
cell. The F—A direction is perpendicular to the GaSe layers .......

The dependence of the energy surface on the variational parameters a
and 6. The minimum is located at -77.5 meV. .............

18

20

54

56

59

65

3.5

3.6

3.7

3.8

3.9

3.10

3.11

3.12

The total DOS of (a) pure GaSe (b) Cd doped and (c) Sn doped GaSe.
(b) CdGa introduces a defect state (indicated by the arrows) near the
valence band maximum and (c) SnGa introduces a state near the con-
duction band minimum. ......................... 66

Partial density of states associated with the impurity (a) Cd 3, p and (e)
Sn 3, p orbitals and those associated with atoms located at different
distances from the defects as indicated in the figure (b-d, f-h). The
atoms Cal and Se1 are the nearest neighbors (NN) of the defects,
while the atoms Ga2 and Se2 are the next NN of the same defects.
The contribution of the Sn to the DOS is larger compared to that of
Cd, suggesting a more localized nature for the defect state associated
with the Sn impurity ............................ 67

The electronic band structure of GaSe(Cd) along the I‘ — A direction,
calculated with the non-relaxed (a) 3x3x1 and (b) 5x5x2 supercells.
The width of the topmost VB decreases from 0.55 to 0.11 eV in going
from (a) to (b), indicating that this is a defect level. As a guide to the
eye, the topmost VB calculated using the two supercells is enclosed in
red rectangles. The host bands show folding (e.g. see the lowest CB)
and their dispersion does not change considerably ........... 69

Electronic charge density associated with the Cd impurity induced de-
fect band. It extends to the impurity’s next NN Se atomn (Se pz
orbitas). The Cd impurity is represented by the red sphere in the figure. 71

Electronic charge density associated with the Sn impurity induced de-
fect band. The contribution to this defect state comes from Sn 3 and
its NN Ga 3 and Se pa; + py orbitals ................... 72

Electronic band structure of GaSe with Sn impurity on the Ga site,
showing the Sn 3 orbital character. The impurity induced band lies in
the CB; therefore it is strongly hybridized with the CB states. . . . . 73

(a) Electronic band structure of GaSe with Ga vacancy. There is a deep
defect state located in the band gap (D1) as well as a defect induced
state at the top of the VB (D2, D3,4). (b) The defect levels D1, D2
and D3,4 are shown along the band structure of the pure GaSe . . . 75

Electronic charge density associated with the Ga vacancy-induced de-
fect band D1. The contribution to the charge density comes from the
Ga 3 and Se pz orbitals from the vicinity of the vacancy. ....... 76

xii

3.13

3.14

4.1

4.2

4.3

4.4

4.5

4.6

4.7

(a) Electronic band structure of GaSe with Se vacancy. There are two
defect levels (D1 and D2) in the band gap (hybridized with the CB) as
well as a defect induced state (D3) at the top of the VB. (b) The three
defect levels D1, D2 and D3 are shown along with the band structure
of Pure GaSe, after matching the Ga 3d semicore levels of the pure
and vacancy containing GaSe. We have also shifted the CB of GaSe

upwards until the band gap reached the experimental value (~2.05 eV) 78

(a) Electronic charge density associated with the defect bands D1 and
D2 showing the 3 character of the Ga atoms closest to the vacancy. (b)
The charge density associated with the band D3. This state extends
to the vacancy’s next NN atoms and shows Ga 3 and Se pz orbital
character. .................................

The elastic constants of Ga1_mIna;Se depend linearly on the composi-
tion. The increase in C13, C33 and C44 suggests the strengthening of
GaSe in the direction perpendicular to the atomic layers as the con-
centration of In increases .........................

The calculated band structures of GaSe and InSe. A-I‘ and F—M

79

89

directions are perpendicular and parallel to the atomic layers respectively. 90

(a) Formation energies of Te impurity in GaSe. For all values of E F
within the band gap, Te prefers to occupy the Ga site. (b) Formation
energies of In impurity in GaSe. When E F is very close to VBM, the In
impurity becomes positively charged (+3) and moves to the interstitial
site. ....................................

The total DOS of GaSe with In,- and the projected DOS of the In 3-
orbital, showing the positions of the HDDS (-5.5 eV) and DDS (just
above E F) introduced by the charged Ini defect .............

Charge density distribution associated with the localized band intro-
duced by Ini in the band gap of GaSe. The dominant contribution
comes from the In 3 orbital, hybridized with the NN Se 19; orbitals.

The total density of states associated with InGa — VGa defect complex,
for different charge states. When the defect complex is triply negatively
charged, the defect state is located in the gap (lower panel). .....

Charge density distribution associated with the localized band intro-
duced by InGa — VGa in the band gap of GaSe. The dominant con—
tribution comes from the In 3 orbital, hybridized with the NN Se p2
orbitals. ..................................

xiii

96

97

98

4.8

4.9

5.1

5.2

5.3

5.4

5.5

5.6

5.7

The DOS projected on the In s orbital for the neutral and -3 charge
state of the Inca — VGa defect complex. The splitting between the
bonding and antibonding states decreases mainly because the distance
between the In and Se atoms increases. .................

The energy barrier which must be overcome in order to cleave the
GaSe crystal increases dramatically when In occupies the interstitial
site compared to the case when In occupies substitutional site. For
comparison the case of pure GaSe is also shown. The asymmetry of
the energy barrier associated with InGa is due to the geometry of the
2x2x1 supercells considered in the calculations: while in the case of
the pure GaSe and Ini the atomic configurations are symmetrical with
respect to the relative displacement of the atomic layers by d / a = 0.5,
in the case of InGa the atomic configuration is only symmetrical with
respect to the displacement of d/ a = 1.0 .................

The crystal structure of GaTe. Dark spheres are the Ga atoms and
the light spheres are Te atoms. In each layer one can see the Ga—Ga

100

101

dimers two oriented normal to the layer and one oriented along the layer. 105

(a) Total density of states (DOS) of the Gate crystal calculated using
PAW method; (b) and (c) Projected DOS of s and p orbitals respec-
tively of the Ga and Te. .................... . .....

Band structure of GaTe crystal along high symmetry directions, cal-
culated using FP-LAPW method. The Brillouin zone used in this
calculation is identical to the one described in Ref. [27] .........

Band structure of GaTe (a) without and (b) with spin-orbit interaction.
The bands with pa; — py character, which lie below the topmost VB,
split under spin-orbit interaction and those with j = 3/ 2 shift up in
energy by ~0.1 eV. ............................

(a)The band structure of GaTe doped with Ge (or Sn) on the Ga site.

111

112

113

(b) The charge density associated with the impurity induced defect band. 115

(a) The band structure of GaTe with Ga vacancy. (b) The charge
density associated with the vacancy induced deep defect band (lying
in the gap near energy 0.8 eV). .....................

(a) The band structure of GaTe with Te vacancy. (b) The charge

density associated with the two, vacancy induced deep defect bands
which lie below the CB. .........................

118

5.8

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

Calculated defect formation energies in GaTe as a function of Fermi
energy, under (a) Ga—rich and (b) Te-rich conditions. The slope of the
energy lines indicates the charge state of the defect and the value of
E F where the slope changes, represents the charge transition level.

Locations of H impurity inside the CdTe lattice ............

Simple bonding scheme for substtutional hydrogen on the (a) Te site
and (b) Cd site ..............................

The density of states (DOS) of CdTe with substitutional H located at
(a) Te site and (b) Cd site. In both cases The H 3 partial DOS is

shown along with the nearest neighbor (a) Cd 3 and (b) Te p partial
DOS ....................................

The charge density distributions associated with the (a) bonding and
(b) antibonding combinations between H s orbital and a1 singlet state.
The main contribution comes from the H s and the nearest-neighbor
Te p orbitals. (c) The charge density of the three-fold degenerate t2
state showing the Te p contribution (and no H 3 contribution) . . . .

Simple scheme for level splitting due to H impurity at tetrahedral in-
terstitial site. ...............................

The density of states of CdTe with interstitial hydrogen on (a) tetra-
hedral site with Cd NNs and (b) tetrahedral site with Te N Ns. In both
cases the H 3 partial DOS is show along with the nearest-neighbor (a)
Cd 3 and (b) Te p partial DOS. .....................

Bonding scheme for H in CdTe located at the bond center (BC) site.

(a) The partial DOS associated with the s and p orbitals of the Cd
and Te atoms located in the vicinity of the H impurity. The lower
panel shows the H 3 partial DOS. (b) The charge density distribution
associated with the defect state located near the bottom of CB. The
dominant contribution comes from the Te p and H s orbitals. The H
impurity is located at the bond center site ................

The band structures of CdTe with (a) H at the ideal Cd site and (b) H
removed from the quasi-equilibrium site. When H occupies the ideal
Cd vacancy position (panel (a)), the t2 level is 3-fold degenerate at I‘-
point. The degeneracy is removed by the J ahn-Teller distortion (panel
(b)). The Jahn-Teller splitting is ~74 meV at F-point. ........

121

127

129

130

132

134

135

137

138

140

6.10 The formation energies of the H defect located at different sites in

6.11

7.1

7.2

7.3

7.4

7.5

CdTe, as a function of Fermi energy. The lowest energy locations are
at the BC site for H+ and H0 and at the TCd site for H'. ......

Comparison between the LDA and GGA results for the formation ener-
gies of the lowest energy hydrogen defects in CdTe. Despite the small

quantitative variations, there is good qualitative agreement between
the LDA and GGA results .........................

The crystal structure of YbNiFeAlg viewed down on the z-axis. In this
figure the M1 and M2 sites are occupied by Ni and Fe respectively . .

Results from the electronic band calculations for YbNiFeA18.(a) and
(b): Total spin-up and spin-down DOS of YbNiFeAlg. (c) Yb 4f spin-
up DOS, with 6 occupied states and one hole located at 2 eV. ((1) Yb
4f spin-down DOS, with all 7 states occupied. (e) Ni(1) 3d and Ni(2)
3d DOS. (f) Fe(l) 3d and Fe(2) 3d DOS. (g) and (h) p states of Al
atoms located around M(1) (A110 and A19) and located around M(2)
( A111 and A15) ..............................

(a) the top view and the (b) side view of Yb3Au3In3 hexagonal crys-
tal structure. Yb3AuGe21n3 is obtained by replacing the two Au
atoms / cell indicated by the arrows ...................

(a) Band structure of Yb3Au3In3 with the In 3p orbital character
emphasized. The unoccupied 4f level lies at 1 eV above E F- (b) Band
structure of Yb3AuGe21n3 showing the Ge 3p orbital character. The
band structures were calculated for U = 6.75 eV. ...........

Spin polarized density of states associated with the spin-up Yb f
electrons for (a) Yb3Au3In3 and (b) Yb3AuGe21n3. In the case of
Yb3Au3In3 the unoccupied f level is shifted up in energy as the U
value increases. For Yb3AuGe21n3 the Fermi level (zero on the energy
scale) is pinned to the upper f level. ..................

141

143

150

152

155

156

Chapter 1

Introduction

1.1 The III-VI layered semiconductors: GaS,

GaSe, GaTe, InSe

The layered III-VI semiconductors GaS, GaSe, GaTe and InSe form a quite remark-
able class of materials. They have long been studied not only because of fundamental
interest in their electronic structure, but also due to their possible application in non-
linear optics [1—3], detector devices [1—4], solar cells [5—8] and solid state batteries [9].
They crystallize in quasi two-dimensional layered structure and exhibit interesting
optical, electronic and mechanical properties. Each atomic layer is composed of four
hexagonally arranged monoatomic sheets of anion-cation-cation—anion. Because of
the strong, covalent intralayer interaction and weak, van der Waals type interlayer
coupling, most of the physical properties display anisotropic, two-dimensional char-
acters. In this context, the early theoretical investigations, in the 1960’s, using the
tight-binding approach considered the III-VI materials as being two-dimensional in
nature, and totally neglected the interlayer interaction [10,11]. However, despite the
fact that the interlayer coupling in the layered III-VI compounds is small, it is much

greater than in typical layered materials such as graphite, and it cannot be neglected.

For instance, in the case of s—GaSe, the ratio between the elastic constants C33 / 011
in the directions perpendicular and parallel to the layers is ~15 times greater com-
pared to graphite [12], and only ~3 times smaller compared to three-dimensional
(3D) materials. Moreover, the interlayer interaction can also influence the optical
properties of the crystal. Thus, the theoretical investigation of this class of materi-
als requires models which are capable of describing electronic interactions of rather

different nature (covalent, ionic) within the same system.

The first electronic structure calculations, which took into account the inter-
layer interaction, emerged in the 1970’s and used the empirical pseudopotential ap—
proach [13]. Later, 3D calculations of the electronic structure of InSe [14, 15] and
GaSe [15,16] were performed, based on parametrized tight-binding method using 3
and p orbtials as basis (sp3 model). A significant improvement of the tight-binding
formalism was achieved in the 1980’s, when an additional 3* orbital was introduced
in order to mimic the effect of the d orbitals lost in the truncation of the tight bind-
ing Hamiltonian matrix. The 3* orbital allowed the sp3s* model to reproduce the
electronic structure and band gaps of semiconductors [17]. The first band structure
calculations of GaSe and InSe in the framework of the tight-binding approach with
sp3s* model, again considered the interlayer interaction as being of purely van der
Waals type, i.e. the III-VI materials were regarded as 2D [18—21]. Only in a recent
theoretical study the sp3s* tight-binding model was adjusted such that the interlayer
interaction was taken into account [22]; consequently the band dispersion (6n): vs. 1:,
where I: is the wave vector and an; is the energy of the nth band for 1:) determined
by empirical pseudopotential methods for the valence band (VB) and the lower con-
duction band (CB) of the GaSe and InSe (to foit the experimental data) was well

reproduced by this extended tight binding method.

In the late 1990’s, with the increase in the computational power and the develop-

ment of the modern electronic structure packages based on ab initio density functional

theory (DFT), the understanding of the III-VI semiconductors reached a whole new
level. Very good agreement has been found between the ab initio results and angle-
resolved photoemission measurements performed on GaSe [23]. The evolution of the
band structures of the GaSe and InSe under pressure has also been investigated, us-
ing pseudopotential methods within DFT, to explain the behavior of hole transport
parameters under compression [24]. First-principles investigations of the lattice dy-
namics and elastic constants of GaSe has revealed that the dynamical properties of
the GaSe(OOOl) surface are very similar to those of the bulk, supporting the fact that

only weak interaction exists between the atomic layers [25,26].

Despite the development of the modern electronic structure methods, one had to
wait until the early 2000’s to find theoretical calculations involving the least studied
member of the III-VI family: GaTe [27,28]. Among the III-VI layered compounds
GaTe occupies a special place because it has a more complex crystal structure with
higher anisotropy. Unlike GaS, GaSe or InSe in which all the cation-cation bonds
are perpendicular to the atomic layers, in GaTe one-third of the Ga—Ga bonds are
parallel to the layers. Partly because of its more complex crystal structure, GaTe
has not been studied experimentally and theoretically as extensively as the other
members. Consequently the optical properties and even the band structure of GaTe

are less well known.

Modern electronics and optoelectronics rely heavily on semiconductors. The prop—
erty of semiconductors that makes them so unique and most useful for constructing
electronic devices is that their conductivity can be easily modified by introducing im-
purities into their crystal lattice. Defects in semiconductors can significantly enhance
the performance of the host; they can introduce electronic states in the vicinity of
the band edges and therefore can control the electronic and optical properties of the
material. As a result, understanding and controlling the origin and nature of the

electronic states introduced by various impurities and defects is crucial in improving

the efficiency of the semiconductors in high quality device applications. In the case of
the III-VI layered family most of the attention has been concentrated on the experi-
mental investigation of the defect levels in GaSe. Numerous authors have investigated
the electrical and optical properties of GaSe and InSe doped with different elements
(such as Cu, Cd, Zn, Sn, Mn, Cl, In, As, Bi Sb, Te, Si, Ge) and many impurity
levels have been detected [8, 29—41]. The number of experimental studies related to
defects in GaTe, however, is much less. Cui et al. [42] measured the levels associated
native defects, several defect complexes and indium impurities in GaTe using deep-
level transient spectroscopy (DLTS). Further defect levels in unintentionally doped
p—GaTe have been measures by Shigetomi et al. [43,44] and Zubiaga et al. [45].

Despite the large amount of experimental effort devoted to investigate the elec-
tronic and optical properties of defects in III-VI semiconductors, there is significant
lack of theoretical studies concerning the impurities in this class of materials. The-
oretical investigations are very important because they not only help interpreting
experimental data about the origin and nature of the defect levels, but also predict
the solubility, ionization energies, electrical activity and the lattice relaxation associ-
ated with various impurities and defects in semiconductors and insulators.

To our knowledge at this point, there are no theoretical studies in the scientific
literature on the electronic structure of defects and impurities in III-VI layered semi-
conductors. This thesis is a serious attempt to address this issue. In Chapters 3,
4 and 5 of the this thesis we present theoretical electronic structure results and the
energetics obtained for several defects and defect complexes in GaSe and GaTe. In ad-
dition, we also investigate the effect of doping on the structural and elastic properties
of GaSe. This is an important problem, because certain type of defects dramatically
increase the shear rigidity of the layered solids, thereby improving its applicability in

device fabrication.

1.2 The II-VI wide band gap semiconductors:
CdTe and CdZnTe

Research in cadmium telluride (CdTe) dates back to the 1950’s when it was identified
as having a band gap (~1.5 eV) almost perfectly matched to the distribution of
photons in the solar spectrum in terms of optimal conversion to electricity. A simple
heterojunction design was developed in which p—type CdTe was matched with n-type
cadmium sulfide (CdS) and the cell was completed by adding top and bottom contacts.
During the last several decades CdTe, cadmium zinc telluride (CZT) and related II-
VI compounds have been subjected to a tremendous amount of experimental and
theoretical research. These semiconductors possess many attractive properties such
as sufficiently wide band gap, good carrier mobility, large atomic numbers of Cd and
Te, which makes them promising materials for room temperature x-ray and 'y-ray

detector applications [46,47].

The performance of CdTe and CZT in radiation detector devices and solar cells
are often limited by the presence of localized defect states in the band gap, which
act as electron or hole trapping centers and thus reduce the efficiency of free charge
carrier collection. Furthermore, the required high resistivity (> 109 9 cm) for a
detector-grade semiconductor is difficult to reach due to the excess holes and elec-
trons originating from the native defects. The principal intrinsic defect in CdTe and
CZT is the cation vacancy (VCd/Zn), which introduces acceptor states in the band
gap, close to the valence band maximum (VBM). It has been proposed that the excess
holes originating from VCd/Zn can be compensated by introducing donor impurities
in the crystal; however an exact compensation of the shallow donors and acceptors is
impossible. Numerous models involving point defects and defect complexes have been
suggested and studied using experimental [48—52] and theoretical [53—56] methods in

order to understand the carrier compensation and the semi-insulating behavior of

CdTe and CZT. Clearly understanding the carrier compensation in CdTe is an im-
portant step towards the further improvement of detector and solar cell performance.

The passivation of the electrically active defects can also be achieved in princi-
ple by hydrogenation [57]. Hydrogen is a very reactive element, it can easily form
hydrogen-defect complexes in bulk materials. Theoretical studies performed on a
large number of semiconductors such as Si, Ge, GaAs, GaN, ZnSe have shown that
hydrogen typically takes a charge state counteracting the prevailing type of conduc-
tivity in the system: in p-type materials acts as a donor and in n-type materials acts
as acceptor [57—61]. Detailed ab initio theoretical studies of H in CdTe have not been
made. In this thesis we have attempted to address this problem. In Chapter 6 we
discuss the ab initio modeling of hydrogen in CdTe, we describe the electronic struc—
ture associated with the H impurity located at different lattice site, and from total
energy calculations we identify the most stable position of the H defect in different

charge states.

1.3 Ytterbium containing intermetallics

Ytterbium (Yb) is one of the most fascinating element of the periodic table. Ytter-
bium based systems display a great variety of unusual physical properties, related to
the presence of the localized Yb 4f states, which can hybridize with the conduction
band states. The Yb ion usually exists in two valence states: trivalent magnetic
(Yb3+) and divalent nonmagnetic (Yb2+). Depending on the electronic and crys-
tallographic environment of the Yb—ion, the two valence states can be energetically
almost degenerate, giving rise to a rich variety of unusual magnetic and transport
properties such as mixed-valency, valence fluctuation, heavy fermion behavior, Kondo
effect, charge density wave (CDW) instability and even superconductivity [62—69]. A

large amount of experimental and theoretical studies have been carried out in recent

years, which show that equiatomic or even isostructural Yb-systems can display very
different ground state and low energy excitations. One such isostructural series is the
extensively investigated YbMCu4, with M = Ag, Au, Pd, In, Cu, Cd, Mg, T1 and
Zn [63,70—77]. The compounds with M = Ag, Au, Pd are heavy-fermion systems with
approximate electron effective masses of 60 me [74]. The ground state of YbAgCu4
is nonmagnetic, while YbAuCu4, YdeCu4 display long-range magnetic order below
the temperatures 0.6 K and 0.8 K respectively [74,77]. YbInCu4 shows a first-order
isostructural valence transition at 40 K from trivalent Yb state to a mixed valence
state [71, 73,75]. The compounds with M = Cd, Tl, Mg and Zn display a variety
of magnetic properties depending on the degree of hybridization between the 4f and
the band states [72,76]. Experimental investigations of the equiatomic ternary series
YbTX (T = transition metal, X = Sn and Bi) reveal a wealth of interesting physical
properties [78—80]. The compounds YbTBi (T = Cu, Ag, Au) and YbTSn (T = Ag,
Au, Zn) have non-magnetic ground state due to divalent Yb ions, whereas in YthSn
and YthSn the valence state of Yb it is trivalent. In YdeBi the Yb display mixed

valent character.

A fundamental theoretical description of such systems is extremely difficult due
to the mixing between the two valence configurations of the RE ions (interconfigu-
ration fluctuation) [81—83] at certain atomic sites. However, before any attempt to
understand these complex mixing processes, a deeper understanding of the ground
state electronic structure these systems, within the effective mean field approximation
(LDA or LSDA) is extremely important, not only from material science standpoint,

but also from fundamental point of view.

A proper ab initio theoretical description of strongly correlated electrons still
represents a serious challenge. It is well known that the local (spin) density ap-
proximation L(S)DA fails to describe the correct ground state of systems containing

transition metal or rare-earth metal atoms. L(S)DA always puts the partially filled d

or f bands right at the Fermi level E F, predicting metallic character with itinerant
d or f electrons, which is obviously not always correct. Strong Coulomb repulsion
(U) between localized d (or f) electrons, when two electrons are present at the same
state, suppresses the charge fluctuations inherent in an itinerant system. In order to
describe correctly the ground state of such systems, one has to go beyond the stan-
dard L(S)DA, and add the effects of strong electron-electron repulsion. One of the
most successful approaches is the L(S)DA+U method [84—87], in which the localized
d or f electrons and the delocalized s and p electrons are treated differently. The
orbital-dependent Coulomb potential (U) is only taken into account for the local-
ized states, while the delocalized states are treated by orbital-independent L(S)DA
type potential. The L(S)DA+U method has been successfully applied to many bi-
nary and ternary rare earth (RE) systems with a diversity of physical properties like
heavy fermion behavior, Kondo eflect, mixed valency and valence fluctuation, long
range magnetic order and superconductivity [62—69]. However, not many quaternary

systems have been studied using this approach.

In recent years a large number of RE-containing multinary phases have been
discovered, partly due to a continuous effort to enhance magnetic, thermoeletric,
transport and structural characteristics of the materials. New synthetic methods
and crystal growth techniques have been developed to explore novel intermetallic
phases. The use of molten Al, Ga or In as solvents has contributed to the dis-
covery of many ternary and quaternary systems. Examples of such materials are
REAu3Al7 [88], RENiGe2 [89,90], ngGe3Sb5 [91], REC04Si14 [92], RENiAl4Ge2
[93], Yb5Pt6In16Bi2 [94], just to name a few. As discussed in the previous paragraph,
electronic structure calculations on numerous RE-containing binaries and ternaries
have been reported in the literature, however there are only a few examples where
the electronic structure of quaternary systems has been theoretically investigated.

The quaternary compounds do possess a more complex crystal structure and the RE

ions can occupy several inequivalent sites. Depending on the local coordination of
each RE ion, the position of the f level with respect to the Fermi energy can change,
and one can have different valence states: 2+, 3+ or mixed valency. The system can
simultaneously exhibit homogeneous mixed-valent and inhomogeneous mixed-valent
behavior [83].

In the last chapter of the present thesis we address some of the issues associated
with the valence state of the Yb in several intermetallic compounds along with the
electronic structure calculation of the ternary YbAuIn, Yb3AuGe21n3 and quaternary
Yb7C04InGe12 and YbNiFeAlg.

Chapter 2

Defects in Semiconductors

2.1 Crystal defects: classification and basic con-
cepts

A perfect crystalline solid is a material whose constituent atoms, molecules, or ions
are arranged in an orderly repeating pattern extending in all three spatial dimensions.
Such a crystallin solid is described in principle by a many-body Hamiltonian, which
describes the correlated motion of all electrons and nuclei in the material. A conve-
nient way to reduce the complex electron-ion problem to a tractable form is to use the
Born-Oppenheimer approximation, that is to separate the motion of the nuclei from
the electronic motion and neglect the nuclear kinetic energy. After this approxima-
tion we obtain a simpler (but still too difficult to solve) many-body Hamiltonian, in
which the nuclear positions enter as parameters, so that the only players left are the
electrons. The electron-nucleus interactions are not removed, therefore the electrons
feel the Coulomb potential of the nuclei fixed at certain positions in space. From this
Hamiltonian one can deduce an effective one-electron Hamiltonian, which describes in
approximate way the behavior of each electron in the field created by the nuclei and

all other electrons. Several methods exist to solve the one-electron (single-particle)

10

Schrodinger equation, and they will be presented later in this chapter.

In a perfect crystal, due to the periodicity, the single-particle wave functions of
all the states extend over the entire crystal, localized states does not exist. However,
when the periodicity of the crystal lattice is disrupted by a defect or impurity, localized
states are allowed, precisely states whose wave functions decay with distance outside
the unit cell containing the defect (or a finite set of unit cells) can appear. In the
case of semiconductors, two types of defect-induced localized states may exist: (i)
those with energy within the allowed energy bands of the perfect crystal (resonant
states) and (ii) those with energy within the fundamental band gap (bound states).
The defect-induced states can have a dramatic effect on the electronic, optical and
even on the mechanical properties of the semiconductors.

Since there is a large variety of imperfections that can be present in an otherwise
periodic crystal and these imperfections can impact the properties of the material
in many different ways, we begin by classifying the typical defects. They can be
classified, on the one hand, in terms of the modifications they produce in the lat-
tice geometry and on the other hand, in terms of the changes they generate in the
electronic properties of the host semiconductor. For instance, some of them produce
donor or acceptor states, others introduce carrier trapping centers, still others are

responsible for carrier scattering and recombination.

2.1.1 Point defects

In terms of the geometrical changes induced by the defects in the host lattice we
distinguish point defects, such as vacancies or misplaced atoms, line defects, which
correspond to misplaced lined of atoms, also known as dislocations, surface defects
such as stacking faults and volume defects, which are related to small inclusions and
precipitations inside the crystal. In this thesis we only study the electronic structure

of point defects and simple defect complexes, which are formed by a combination of

11

at most two point defects.

There are two main classes of point defects: (i) intrinsic or native lattice defects
and (ii) foreign atoms or impurities present inside the lattice. Within the class of in-
trinsic point defects we distinguish vacancies (missing atoms), self-interstitials (extra
host atom occupying an interstitial site) and antisite defects in compound semicon-
ductors (in an AB compound atom A occupies site B). The defects associated with
foreign atoms can be classified in terms of their physical location inside the host lat-
tice. The case when a single foreign atom occupies the site of a host atom, is called a
substitutional impurity, whereas the case when the impurity is located at an intersti-
tial site is knows as a simple interstitial. Further, the special case when a host atom
is replaced by two symmetrically displaced atoms is called split-interstitial. Finally,
in terms of the geometrical arrangement with respect to each other, the point defects
can form simple defect complexes such as pairs (two impurity atoms are located at
neighboring lattice sites), impurity-vacancy complexes (vacancy and impurity at the
neighboring sites) and divacancies. Evidently, the formation of more extended defect

complexes is also possible but their characterization becomes more diflicult.

2.1.2 Donors and acceptors

Substitutional impurities are often classified in terms of their position in the periodic
table relative to the host elements. Impurities from the same column as the host atom
are identified as isovalent impurities, because they have the same number of valence
electrons as the host atoms. Non-isovalent impurities are frequently referred to as
acceptors or donors, depending if the impurity is located to the left or to the right of
the host atoms in the periodic table of elements. Acceptors / donors have less / more
valence electrons compared to the host atoms, therefore they must accept / donate
electrons from/ to the host, in order to fulfill the local bonding requirements.

The definition of donor and acceptor impurities given above, is somewhat confus-

12

ing, because in principle an impurity can introduce, both donor and acceptor states in
the electronic structure of a semiconductor, in which case the impurity is called am-
photeric. For instance, hydrogen in most semiconductors acts as amphoteric impurity,
as it counteracts the prevailing type of conductivity in the system [57]. Furthermore,
it is known that isovalent impurities, which have the same number of valence elec-
trons as the host atoms, can introduce defect states in the band gap [95], thus the
above definition of donors and acceptors cannot be applied. A more general and un—
ambiguous definition for donors and acceptors can be given by the charged state of
the defect: we define a positively charged defect states as donors and the negatively
charged states as acceptors. This definition allows for a defect to have one or more
acceptor or donor states, and also allows for the possibility that a particular impurity
can introduce both donor and acceptor states, and therefore acts as an amphoteric
impurity. According to this definition one can describe the donor and acceptor states

associated with the isbvalent impurities as well.

2.1.3 Shallow versus deep defects

Another criterion which can be used for the classification of defects is related to
the degree of localization of the eigenfunction of the defect-induced electronic state.
Generally, if the energy of the defect state is close to the VBM or CBM, the defect
state is referred to as shallow acceptor or donor, whereas if the state is located close
to the mid-gap, it is called deep defect state. This definition however, implies that
the binding energy (energy of the defect state measured from the band extrema) of
a deep defect level necessarily has to be large, and that the energy position of the
deep level in the gap has to be far from the band edges. However, it is known that
deep levels also appear in narrow-gap semiconductors [96,97] or close to the band
edges [98]. Therefore a better and more general way of classification is needed: one

which is related to the fundamental nature of the shallow and deep defect states. The

13

shallow defects states are dominated by long-range Coulomb potentials, therefore
their wavefunctions extend over several unit cells in the crystal. Deep defect states,
on the other hand, are dominated by short-range potential and the corresponding

waveftmctions are highly localized, and usually do not extend beyond one or two unit

cells.

In the next subsections we will present some of the basic characteristics of the
shallow and deep defects states. The detailed theoretical description of the models

and methods used for analyzing defects will be addressed in later sections.

2.1.4 Shallow defects in semiconductors

From theoretical point of view the description of shallow defect states is quite straight-
forward. For instance a shallow donor can be regarded as a hydrogen atom with a
positive nucleus binding an electron. In order to understand the elementary aspect
of the hydrogenic model, we can start from the electronic band structure of a perfect
solid. In the case of semiconductors and insulators there is a valence band followed
by the band gap and the conduction band. At OK, the valence band is completely
occupied while the conduction band is completely empty. If an extra electron is in- -
troduced in the system, at UK it will occupy the lowest energy state available, which
is the bottom of the conduction band. In the case of simple, parabolic bands the
energy of the states in the vicinity of the conduction band minimum (CBM) is given

by:

h2k2

*
2me

 

E (k) = ECBM + (2-1)

where EC B M is the energy of the CBM and mg is called effective mass. The compar-
ison between Eq. (2.1) and the corresponding relation for a free electron characterized

by the same wave vector If:

14

h2k2

13(k)._-§;z;

(2.2)
where m0 is the free electron mass, justifies the terminology for the constant mg. The
effect of the crystal potential is included in the effective mass in an average way, such
that the dynamical behavior of an extra electron in the conduction bands resembles

to that of a free electron with a mass mg.

To illustrate the basic concept behind the hydrogenic model, we will describe, as
an example, the case of a single substitutional donor. This means that a host atom in
a perfect crystal is replaced by an impurity from a column in the periodic table located
to the right of the host atom. This impurity contains one extra electron compared to
the host and becomes positively charged after it has given its electron to the system.
If we imagine that somehow we keep the extra electron out of the crystal, then the
crystal has the same number of valence electrons as it had before, so that the valence
band is completely filled and the conduction band is still completely empty. Although
at this point it appears that there is no change in the occupation of the bands, there
is a significant modification in the crystal potential. Since the impurity is necessarily
positively charged, it introduces a Coulomb field in addition to all crystal fields that
were present before the introduction of the impurity. Due to the dielectric medium the
impurity-induced Coulomb potential is given by U = +e/sr, where e is the dielectric
constant of the material. Now if we introduce the extra electron into the system,
according to the above discussion, it can be regarded as a free electron with mass m2,
which moves in a Coulomb potential e/sr. It is now clear that the whole picture is
analogous to the hydrogen atom, except that the nucleus has a charge equal to e/e and
the mass of the electron is m2. Therefore, by analogy with the ionization energies of
the hydrogen atom (E51 = —e4m0/2h2n2, n = 1,2,3, ...oo), we conclude that the

shallow donor introduces bound state below the conduction band, whose energies are

15

given by:

 

4 *
_ 6 me
En — ECBM ‘ m (23)
Of
E — E EH ( m: ) (2 4)
n _ CBM n m0€2 -

In eqs. (2.3) and (2.4) m2 is given in units of mo and the principal quantum number
is n = 1, 2, 3, ...00. Equation (2.4) can be used to obtain an estimate of the ionization
energies of shallow donor levels: if we consider that the typical value of 5 is 10 and
the value of mg ranges from 0.1 mg to 0.5 m0, than the binding energy (E1) of a
shallow donor ranges from about ~10 meV to ~70 meV. These binding energy values
represent a small fraction of the typical bandgap (~1 eV), therefore such a shallow
donor is intimately related to the conduction band. Furthermore, we can also estimate

the quasi-Bohr radius associated with the shallow defect in the hydrogenic model:

 

— — 2.5

where do is the Bohr radius. Using equation (2.5) with the above mentioned typical

values of e and m2, we obtain a range for the quasi-Bohr radius of ~50 A to ~10 A.

These values are larger compared to the typical lattice constant of a semiconductor.

The qualitative picture for a shallow acceptor is similar to that for donors. First
we remove an electron from the perfect crystal, which corresponds to the creation of
a hole at the top of the valence band. The dynamics of the hole at the near the top

of the valence band can be described by an equation similar to (2.1):

h2k2
2m];

 

E (k) = EVBM - (2-6)

16

where EV B M is the energ of the VBM, and m]; is the effective mass of the hole. If
an impurity, with fewer valence electrons than the host, is introduced in the otherwise
perfect crystal, it will capture an electron and will create a negatively charged center.
This impurity-induced Coulomb potential will act upon the positively charged hole,
again creating a picture similar to the hydrogen atom. The energy levels introduced

in the band gap, now are located close to the valence band and they are given by:

 

m*
En = EVBM + E#( ’12) (2.7)
mos

The numerical estimates for the shallow acceptor states are similar to those given in
the case of shallow donors.

The use of hydrogenic model is justified when the impurities introduce approxi-
mately Coulombic potentials in the crystal. Its quantitative success depends on the
complexity of the band structure and on the values of the dielectric constant (e) and
the effective mass (mg/h). It gives good results for single donors and acceptors, when
the band extrema have simple form and the quasi-Bohr radius is large compared to
the lattice constant. The hydrogenic model can be extended to double donors and
acceptors, by simply using the screened Coulomb potential of two charges, so that the
energy levels are four times deeper. However, it has been found that the quantitative
agreement between the binding energies calculated this way and the experimental
results is not very good [99]. Despite its striking simplicity, the hydrogenic model
of shallow-level impurities has been confirmed by a large number of electrical and

optical studies in a vast number of semiconductors and insulators [100].

2.1.5 Deep defects in semiconductors

In contrast to shallow-level impurities, which are quite well understood, it is much

more difficult to describe theoretically the electronic behavior of deep-level centers.

17

V(X)
-d/2 d/2

 

 

 

-Vo

 

 

Figure 2.1: Rectangular, one-dimensional well of depth V0 and width d

They are dominated by short—range potentials and the wavefunctions remain localized
close to the core of the defect. Deep levels are usually connected to more than one
band edge (may be connected to both VB and CB), and in the case of some external
perturbation, they do not follow a specific band edge. States from both VB and CB
may be necessary to construct the ground state eigenfunction of such a deep center
and in addition the deformed lattice environment also has to be considered. The
resulting deep defect levels communicate with both bands and act as charge carrier
trap or recombination center. Several good reviews about deep defects can be found

in references [99,101—103].

To illustrate the substantial difference between the eigenvalue spectrum of the
shallow and deep defect centers, we can consider the very simple case of a one-
dimensional, deep potential well. The chemical identity of the center is introduced,
by assuming a square-well potential of depth —V0 and width d, as represented in

Fig. 2.1 (referred to as central cell potential).

In general, the stationary electronic states in such a quantum well are described

by Schrddinger equation:

 

£1210 2 .
E—k’wr—O Wlth k=\/

2m [E - V (17)]
52

 

(2.8)

18

After simple calculations, for —V0 < E < 0, (V0 > 0) it can be shown that the
solutions of the Eq. (2.8) exist for k values that satisfy the following transcendental

equations:

k d
the? + k = ktan (7d) or \/Is2 + k2 = —k cot (k?) (2.9)

where

2_2mV0
_—h2_

K.

(2.10)

Equations (2.9) can be solved graphically, and for the discrete values obtained for k,

the values of E inside the well are also discrete:

_ 712192, _

E _
n 2m0

 

(2.11)

where kn c: mr/d for the lower values of n(;£ 0). In Eq. (2.11), the electron rest mass
was used, because in a deep-level center, the electron remains close to the core of the

defect. Therefore the eigenvalues can be written as:

2 2
En = —hI-—2ng — V0 where nq = 1,2, (2.12)
' 2mg ((1/2)

From Eq. (2.12) we see that the eigenstates increase quadratically with nq. In con-
trast, according to Eq. (2.3), the eigenstates of the simple hydrogen-like defect de-
crease as l/nz. The electron eigenstates for the two simple potentials are represented
schematically in Fig. 2.2.

We have used the simple example of the square well potential as a tool to illustrate
the difference between the spectrum of shallow and deep defect centers. However, for
a realistic description of the deep defect states, a better central cell potential, V (r),

is required. Furthermore, as we have mentioned earlier in this section, a deep defect

19

 

“—17

 

 

 

 

(a) (b)

Figure 2.2: Electron eigenstates (a) in a one-dimensional square well potential and
(b) in a Coulomb potential

state is usually connected to more than one band, therefore the inclusion of additional
bands in the model of the deep center is very important. In addition, along with the
central cell potential, we also have to consider the long-range Coulomb potential of
the charged deep center as it describes the higher excited states of such a center [103].

Deep defects also present a challenge from experimental point of view. In semi-
conductors and insulators a great variety of deep centers exist, and their identification
and chemical origin is not always an easy task. However, by intensive experimental
efforts it is possible to identify unambiguously some of the deep centers [101,104]. The
experimental methods include optical absorption, luminescence emission and excita-
tion spectroscopy, electron-spin resonance, optical detection of magnetic resonances,
extended X-Ray absorption fine structure (EXAFS), deep-level transient spectroscopy
(DLTS). Despite all the efforts put into understanding deep-centers, some of them

seem to escape the generally accepted identification methods.

2.1.6 The role of defects in semiconductors

The role played by defects and impurities in semiconductors depends on the nature of

the localized levels (shallow or deep) that are introduced in the band-gap and on the

20

concentration in which they can be incorporated into the crystal. The most important
role of shallow donors and acceptors is to control conductivity. This is because at
room temperature the ionization energies of shallow defects are comparable to kBT
(k B is the Boltzmann’s constant and T is the temperature) and therefore almost
all such shallow levels are ionized and contribute to conductivity. A very important
aspect of the semiconductor doping is that the impurities can be incorporated in
arbitrary concentrations, up to 10200m-3. By controlling the impurity concentra-
tion, the conductivities can be varied over a range of twelve orders of magnitude:
from approximately 10—9 (0cm)_1to about 103 (Qcm)_1. To appreciate the bene-
fits of semiconductor doping, these values should be compared with the conductivity
in metals (106 (Qcm)’1) and insulators (10‘22 (Qcm)_1). Furthermore, in most
semiconductors, the conductivity can be dominated either by electrons (n—type), or

holes (p-type).

The most important feature of semiconductor doping is that, by carefully con-
trolled ways, the concentration of shallow donors and acceptors can be made non-
uniform within a sample. The inhomogeneities of electron and hole densities can be
exploited to produce a variety of effects useful for device applications. A simple ex-
ample of such an inhomogeneous semiconductor is the well known p-n junction, which
consists of a region doped with acceptors (p-type) adjacent to a region doped with
donors (n-type). Such a p—n junction permits the flow of electrical current in only
one direction and it can also act as radiation emitter in a light emitting diode (LED).
Other devices such as transistors, modulators, detectors, photocells, etc. consist of
more complicated structures of p- and n-type regions of different acceptor and donor

concentrations.

Deep defect states play an entirely different role. They are incorporated in semi-
conductors in smaller concentrations (1012— 10130m'"3 in Si) and their contribution

to conductivity is usually negligible. In most cases they act as traps and recombina-

21

tion centers for electrons and holes. For an electron and hole to recombine, that is,
an electron from the CB to drop into the VB, an amount of energy equal to the band
gap must be dissipated. The recombination is going to be more likely if this energy
can be dissipated in smaller fractions, and this can be precisely accomplished by a
deep—level present somewhere in the middle of the band-gap. The electron from the
CB can first drop to the deep-level and then to the VB, by losing energy (emitting
photons) in smaller quantities at a time.

The most important role of the recombination centers introduced by deep defects
is to control the lifetime of carriers. When a device (such as a photovoltaic) needs
long carrier life-times, then deep defects have to be avoided. On the other hand, when
a device calls for short carrier lifetime, deep defects must be incorporated carefully in
the semiconductor. An example of such a case is a photocell which is used as a fast
switch, where the current generated by the electron-hole pair must last only a short

period of time.

2.2 Theoretical methods to analyze defects

we will approach the defect problem within an effective single-particle model.Even
within ths approximation the theoretical methods are based on different approxi-
mations for solving Schr6dinger equation associated with the defect. Schrédinger
equation can be solved using any electronic structure methods. The defect in the
crystal environment can be modeled by a considering only the atoms surrounding the
defect, thus defining a cluster or by using the supercell technique, in which periodic
boundary conditions are applied to the surface of a cluster. The potential of the
defect Hamiltonian can be constructed using empirical (or semiempirical) parameters
or by self-consistent methods.

In this section we will address some of the issues associated with the approxima-

22

' "a . 2.. 1 A .u. . I . i z u
. . .- I . ‘9 n u a r
m «ct-«nu 34‘ 0M 5 ‘n n. “It“ I 3‘ "1331:”

 
   

_ .
|I
9' a».

"I I 'Ikl
1‘4?

 

1:: ,m‘.

. . . .o. . , . ~
' ~ 1 1 A 9 . . . - . . .. . . . ,
.mmmmraszmwmw'.» ‘ '.-. . . :.'

tions used for solving Schrédinger equation. We will describe the hydrogenic model,
which is used to treat the shallow defects and then, in subsequent sections, we will
talk abut theoretical methods applicable to deep defects as well. First we discuss
general, elementary formulation of the quantum mechanical problem associated with

crystal defects.

2.2.1 Basic quantum mechanics

After mapping the many-body ground-state problem onto single-particle problems,

we end up with a set of eigenvalue equations of the following type:

Hip =[—fiv2+v 1p =E1/2 (213)
V 2m 11 V V -

 

If we start from the premise that the electronic structure of the perfect host solid
is known then the total crystal potential and the Hamiltonian of the system can be

written as:

V=V0+U with H=H°+U (2.14)

where H 0 is the Hamiltonian of the unperturbed lattice and U is the perturbation
introduced by the defect. By using the term “perturbation” we do not imply that we
use the well known perturbation theory (based on a power series in the perturbation
U), but rather that we want to calculate the changes produced by the perturbation.

The corresponding eigenvalue problem which has to be solved is given by:

Hwy = (H0 + U) a. = Em (2.15)

Equations identical to Eq. (2.15) can be written down whether the above quantities

are regarded as true, pseudo, or model. The eigenvalues E, and the eigenfunctions

23

5,75
1‘”
«“1
.1-
25?
$2.:
a
. ‘7

 

#:134; ‘Lo

-

6

. saw «was. - -

e... I l; .v I .1 up rum ..
iwv-‘rh l‘Jm" V" ‘ ‘ '9 -5 '-

..
...
,cp-
. .
. ..'. 7‘1!

a v . . . 1 . . 4 .
' ‘ ' tr . o- - .on I u I .u o . -. .!
w. - - .~ — m . t . . .
I b.0051 I ..- at}. on— I!_:tdi!m‘::i,: V; * . ilfntmiw.fim:ml€.'

 

 

11),, can be determined by using the secular matrix method, that is expanding 1111/ in

terms of a complete set of functions (15),:

1,111; = ZFA¢A (2.16)
A

where we have omitted the index V in the FA coefficients. By substituting (2.16) in
(2.15), multiplying on the left by d): and integrating over all space, we get a set of

coupled linear equations:

2: [<¢A’IHOI¢A) + <¢AIIUI¢Al — EU<¢AII 9%)] FA = 0 (2.17)
z\

If the basis set is orthonormal, that is (gb/V] (by) = 52V» than the energies EV are

just the eigenvalues of the secular matrix:

Haul—UNA:<¢XIHOI¢A>+<¢NlUI¢A> (2.18)

The set of functions (by can be chosen to be Bloch functions, Wannier functions or

other orthonormalized functions, such as simple exponentials of Gaussian orbitals. In

many cases one of the most natural choices for ¢,\ is the set of Bloch functions 7,120,?
Tl

which are eigenfunctions of H 0 and are orthonormal. In this case (2.16) becomes:

V(=’F) 22F” 1:31;; .. (2.19)

and the set of coupled equations (2.17) is:

E0517” ~+ Z (201.5]U]¢0, E'>Fn’ 1‘." = Burn]; (2.20)
n’k’

The secular matrix which needs to be diagonalized is:

24

 

Egt‘lnn'ial + (filial/It” > (2.21)

n’ lit"
The size of this matrix is equal to the product between the number of k-point in the
BZ and the number of bands included in the expansion (2.19). Since in a practical
implementation the sampling of the BZ is necessarily finite, this method is not very

useful for states for which Fnl-c' varies rapidly with Is.

Different methods have been developed, in an effort to reduce the size of the secular
matrix which has to be diagonalized. Simple models, specially designed for cases when
the range of perturbing potential is short, have been constructed by Koster and Slater,

using Wannier basis functions and other localized representations [105,106].

2.2.2 Hydrogenic-model (effective mass approximation)

Shallow defect states are described by the hydrogenic model or effective mass approx-
imation (see also section 2.1.4). As an example we will discuss the electronic states
associated with a shallow, hydrogen-like donor (such as a substitutional phosphorus
atom in the silicon lattice). In this case the Hamiltonian of the defect containing

crystal can be written as:

82

H = H0 — 57 (2.22)

where H0 is the Hamiltonian of the host lattice, to which the attractive Coulomb
potential of the defect, which is added as a perturbation to H0. The static dielectric
constant e is introduced in the expression of the Coulomb potential, because of the
screening action of the host. The eigenfunctions of H near the defect can be con-
structed, using the Bloch functions $0 (1.5,?) = no (13,?) ell—5"? of the nearest band

(which is the conduction band in the case of a shallow donor):

25

[FA Mi: ‘

 

a 2 2 cc ()2) we (13,?) (2.23)
75

As we have mentioned in Section 2.1.4, the eigenfunctions of a shallow-state extend
over several lattice constants, therefore the values of if are restricted close to the
center of the Brillouin zone (BZ). Since uc (15) changes slowly with If, we can extract
uc (if = 1:0,?) as a constant from the summation in Eq. (2.23). E0 is the value of

the wavevector at which E (it) reaches its minimum. We have:

212 = no (130, F) E Cc (I?) e275"? (2.24)
E

For 120 = 0 (F-point) we have: 22,; (E0 = 0,1") = uc (0,7'") eiO'F = uc (130 = 0,77).

Therefore Eq. (2.24) can be written as:

-o

a = we (130,?) Zoo (1?) eik'f’} = F(r~‘)¢c (120,7?) (2.25)
k

where we (130,17) is the Bloch function in the minimum of E (If) and F (7") = 21: cc (1;) eiki’

is an envelope function, which satisfies the modified Schrddinger equation for the

hydrogenic-model:

 

”Zr/23 Fin-(E—Ewm (226)
2mg er — C '

Equation (2.26) is identical to the Schrdinger equation for the hydrogen atom, but

with an electron with effective mass m; in a medium with dielectric constant 5. In

particular the envelope eigenfunction of the Is ground state is given by:

 

F (r) = 1 exp (w—L) (2.27)

where the effective Bohr radius aq H is given by:

 

525 m0 m0
aqH = *62 = 50.0 (E?) = 0.529 X E (T—n—IT) (2.28)

me e e
In Eq. (2.28) the electron effective mass m; is given in units of free electron mass mg.
The eigenstates of the envelope function are bound states located below conduction
band minimum (CBM) and the energy eigenvalues are given by the quasi-hydrogen

energy spectrum:

4 =1:
_ 6 rm, 1
EqH — EC ‘ m (7) (229)

Equations (2.28) and (2.29) are identical to (2.5) and (2.3) respectively, which were
derived from the qualitative picture of shallow defects in Section 2.1.4.

In our description of the hydrogenic model, so far, we have considered that the
defect potential is spherically symmetric. In this case the entire energy spectrum of
the hydrogen-like defect (Eq. (2.29)) is described by the principal quantum number n,
and all the states characterized by the same quantum number are degenerate. How-
ever, when the defect potential deviates from the spherical symmetry, the degeneracy
of the hydrogenic levels is lifted and the energies of the s, p, d,... states are shifted.
We will discuss this case in more detail in Chapter 3 (Section 3.4.2), when we apply

the hydrogenic-approximation to the highly anisotropic compound GaSe.

2.2.3 Green’s function methods

The first theoretical method which addressed the problem of deep defect states in
semiconductors was the Green’s functions method [103,105, 107,108]. In this method
the defect free host crystal with Hamiltonian H O is perfectly described. The localized
defect is represented by the Hamiltonian H, which includes the defectpotential V.

The Green’s function calculations determine the differences between the ideal crys-

27

tal and the changes introduced by the defect. The Green’s function is defined by
G (E) = 1/ (E — H), so that the perturbed energies E are given by its poles. The
new eigenvalues contain the gap levels of the defect, while the corresponding eigen-
functions are the defect wave functions. If the new eigenvalues overlap the eigenvalues
of the unperturbed Hamiltonian H0, then one has resonant states. Despite the fact
that Green’s functions, in principle provide a near ideal description of the defect
in a crystal, their implementation is difficult, especially when it comes to defects
that induce large relaxations and distortions in the otherwise perfect crystal. There
are difficulties associated with the construction of the defect potential and with the
eigenfuctions of the perfect crystal that can be used as basis set for the defect calcula-
tion [107,109]. In recent years, due to their computational complexity, there has been
a decrease in the use of the Green’s function calculations, particularly for complex
systems. However, their revival within the GW formalism [110] is probable, if one

wants to go beyond an effective single particle model for describing the defect states.

2.2.4 Scattered wave method

Another method to solve the general eigenvalue problem (2.13), is the Xa-scattered
wave method, which is a particular technique for small collection of atoms. The po-
tential is calculated self—consistently for the entire system, including the core electrons,
using Slater’s Xa exchange [111]. The approximation used in this method is that the
potential is spherically averaged within atom centered, muffin-tin spheres and vol-
ume averaged within the interstitial region. The eigenvalue problem is then solved by
scattering methods [111,112]. The Xa-scattered wave method has been successful in
describing the one-electron spectra of molecules, and it has also been applied to study
interstitial transition metal impurities and hydrogen-alkali metal complexes [113,114].
Despite its relative success, it was diflicult to bring the method to self-consistency

and the rather arbitrarily chosen muffin-tin spheres made it unsuitable for calculation

28

of atomic positions by minimizing total energies.

2.2.5 Partial retention of diatomic differential overlap

method

A successful self-consistent Hartree-Fock method in the mid-19803 was the partial re-
tention of diatomic difl'erential overlap (PRDDO) [115,116]. It contained no empirical
parameters, allowed for geometry optimizations to be performed without symmetry
constraints, the convergence was efficient. This method was implemented using a
cluster model (see below) and relatively large clusters could be used (44 atoms). It
has been used to study defects in diamond and silicon, and it was the first theoretical
tool which demonstrated the stability of hydrogen at the bond-centered site in Si. It
was unexpected and surprising that H could occupy the site between two Si atoms
and force the Si—Si bond by more than 1 A [117, 118]. On the negative side, PRDDO

is a minimal basis-set technique and ignores electron-electron correlation.

2.2.6 Atomic cluster models

The eigenstates of a defect center can be described by considering only a few host
atoms in its neighborhood, thus defining an atomic cluster [119]. The first cluster cal-
culations were performed using basis sets consisting of linear combination of atomic
orbitals (LCAO) [119,120] or other localized functions such as Gaussians. The early
work was empirical or semiempirical, that is the potential V was not calculated explic-
itly, but instead a basis set was chosen and the matrix elements of H were determined
from a direct fit to experiment or evaluated using a well-tested prescription.

A very simple LCAO method for the determination of energies of molecular or-
bitals of 1r-electrons in conjugated hydrocarbon systems was the Hiickel method, pro—

posed by Erich Hiickel in 1930. The extended Hiickel theory (EHT) is a semiempirical

29

quantum chemistry method, develedoped by Hoffmann since 1963 [121]. It is based on
the Hiickel method but, while the original Hiickel method only considers 7r-orbitals,
the extended method also includes the (Jr-orbitals. The extended Hiickel method can
be used for determining the molecular orbitals, but it is not very successful in deter-
mining the structural geometry of an organic molecule. It can however determine the
relative energy of different geometrical configurations. It involves calculations of the
electronic interactions in a rather simple way where the electron-electron repulsions
are not explicitly included and the total energy is just a sum of terms for each electron
in the cluster.

The usefulness of the cluster models is reflected by the numerous theoretical stud-
ies within Hartree-Fock or density functional theory, which use the cluster model.

Although, the implementation of the cluster calculations is straightforward, they
converge slowly with the size of the cluster and they suffer from a variety of problems
such as cluster size, surface effects, basis-set limitations, lack of electron correlation.
Furthermore the adjustable parameters are very sensitive to the conditions at the
cluster surfaces. The finite size of the cluster confines the wavefunction, which affects
charged defects the most, as the charge tends to distribute at the surface of the

cluster.

2.2.7 The supercell model

In order to overcome the surface problem in the cluster calculations, the supercell
technique was designed, which replaces the problematic boundary conditions for clus-
ters with less problematic Born-von-Karman periodic boundary conditions. We will
describe the supercell technique in more detail in subsequent sections, as it is the
principal approach which has been used throughout this thesis.

The great progress in the theory of defects in semiconductors occurredin the mid-

19803 with the introduction of the theoretical approach known as “first-principles”. In

30

general the first principles methods use a combination of ab-initio pseudOpotentials
for the electronic core region and density functional theory (DFT) for the valence
region. These methods use periodic supercells to represent both the host crystal
and the crystal with defect. The parameters in this approach include the size of the
supercell, k-point sampling of the Brillouin zone (BZ) in carrying out summation
over wave vectors If, size of the basis set and other parameters associated with the
pseudopotentials. However, these parameters are not fitted to an experimental data,
but rather determined self-consistently from first-principles.

Since the results presented in this thesis have been obtained using DFT in combi-
nation supercell technique in the next two sections we will provide brief descriptions

of the DFT and supercell approach to solve the defect problem.

2.3 Density functional theory (DFT)

For ab initio electronic structure calculations in solids, the overwhelming method of
choice is Density Functional Theory (DFT) due to Kohn, Hohenberg and Sham [122,
123]. The greatest simplification introduced by this theory is that it replaces the
ground state problem of a complicated many-body system and the associated many-
particle Schréidinger equation, with the single-particle density, which is a function of
position. We review for completeness the description of DFT with the presentation of
two important theorems proved by Hohenberg and Kohn [122]. In this presentation

we follow the same line of development as in Ref. [124].

2.3.1 The theorems of Hohenberg and Kohn

The first theorem of Hohenberg and Kohn states that the external potential, v (F)
acting on a fully interacting many-particle system in its ground state is determined

within an additive constant by the electron density n (F).

31

To prove this theorem let us consider the Hamiltonian of an N-electron system

moving in some external potential 12 (r)1:

. N 1 2 N N 1
H=Z(—§Vi) +Zv(r,)+:; (2.30)
i=1 i=1 Z<J ~7

The first term in Eq. (2.30) is the kinetic energy operator and the last term describes

 

the Coulomb interaction between the electrons, with rij = r]: — SI. The second term
can be considered, for instance, to be the potential v (r‘,) = — Ea Za/rz-a acting on
an electron located at 1", due to the presence of the nuclear charges Z0, at position

Ra, With Tia =

 

77,; — Ha I. After applying the Born-Oppenheimer approximation, this
potential created by the fixed nuclei becomes external to the electron cloud. Let us
suppose that there are two different external potentials v (f) and v’ (7‘) corresponding
to the same ground state density n (7"). Also let H and H ’ be the corresponding
many-body Hamiltonians, with 1,0 and 112’ the associated ground state wavefunctions.
If E0 and E6 are the ground state energies of the N-electron system in the presence

of v (f) and v’ (r) we can write:

E0= <¢IHIw> < (w’lfllw’) = (w’IH’lib’HWlH-H’lw’)

(2.31)
= E6+fn(1’) [v(7") —v’(7")]d3r

The integral in Eq. (2.31) comes from the second term in Eq. (2.30) and describes
the interaction of the charge distribution with the external potential. Similarly, if we

take 1/2’ as a trial wavefunction for the system described by H ' , we have:

 

1In order to simplify the notations throughout this section, we use atomic units in which the unit
of length is the first Bohr radius of the hydrogen atom a0 = 0.529 A, the unit of mass is the mass
of the electron, and the unit of charge is the electronic charge. We also set h = 1.

32

 

 

E6=<w|H|w)<<w|F1’lw> = <41H14>+<21H’-H14>

(2.32)
= E0 — Inc) [M —v’0=)] .13.
Adding Eq. (2.31) and Eq. (2.32) we get:
E0 + E6 < E0 + E6 (2.33)

which is clearly a contradiction. Therefore there cannot be two different external
potentials that give the same ground state density for the system. This implies that
there is a one-to-one correspondence between the ground state density and the ex-
ternal potential of a many-electron system. On the one hand, a given many-electron
system has a unique external potential which gives, by the Hamiltonian (2.30) and
corresponding Schriidinger equation, a unique ground state many-particle wave func-
tion, from which the ground state density is easily calculated. An external potential,
therefore leads in a well defined way to a unique ground state density. On the other
hand the first theorem of Hohenberg and Kohn states that the density n (7"), of a
many-particle system contains exactly as much information as the wave function;
that it is possible to find a unique external potential if only the ground state density
is given. The ground state observables can be obtained uniquely way from the density
alone, since they can be written as functionals of the density.

Among the ground state properties of the many-particle system, the total energy
and the different contributions to the total energy are of particular interest. Thus
the kinetic energy, T [n], the potential energy, V [n], and the total energy, E [n] are

determined as functional of the density n (F):

33

E [”1 = T [”1 + Heat in] ‘l' U86 lnl
= fn(7")v(7‘”)d3T+FHKlnl

Here the integral describes the interaction between the electron cloud and the external

(2.34)

potential and:

FHK [n] = T [n] + Uee [n] (2.35)

is the exact functional representing the kinetic part and the mutual interaction be-
tween the electrons. All the ground state many-body effects are incorporated in this
functional. The second term in Eq. (2.35) contains the classical repulsive interaction
of a charge distribution with itself (Hartree term) as well as the contributions arising
from the fact that the electrons are indistinguishable particles which obey Fermi-Dirac

statistics. We can write:

Uee [n] = J [n] + (non — classical terms) (2.36)
where
J[n] = l / Md3r1d3r2 (2.37)
2 |T1- 2|

The non-classical term in Eq. (2.36) is of extreme importance and although its
exact form is not known, it contains a major contribution to the exchange-correlation
energy. In principle, both exchange and correlation is included in a self-interaction

of the form:

Uee [n2] /d3r1/d3 WW?) (2.33)

7"2—"----'
In - 7‘2|

where n (171,153) is the exact two-particle density, giving the number of particles in

34

unit volume simultaneously at F1 and F2. Formally, the non-classical term in (2.36)

includes the difference between J [n] and U86 [n2].

The second theorem of Hohenberg and Kohn provides a variational principle for
the energy. It can be formulated as follows: For a trial density a (F) such that a. (F) 2 0
and fem d3r = N,

%5EW am)

where E [n] is the energy functional in Eq. (2.34). This theorem can be proved based
on the first theorem of Hohenberg and Kohn according to which each ii (7") determines
its own 0 (7") and wave function ii. For a system described by the Hamiltonian H and

with external potential v (r), we can write:

Mimi) = / a (n v (0 .13. + PM [41

=EWZEM=%

aw)

In Eq. (2.40) n is the correct ground state density, therefore by definition E [n] = E0,
which proves the theorem. According to this second theorem, out of the infinite
number of possible densities, the one which minimizes E [n] is the ground-state density

corresponding to the external potential v (7").

We note that the functional FH K [n] defined in Eq. (2.34) is a universal functional
of n. It does not contain any information about the external potential (e.g. nuclei
and their positions) therefore it is universal to any many-electron system. This means
that in principle an expression for FH K [n] exists, which can be used for every atom,
molecule or solid. However, an explicit expression for the ftmctional FH K [n] is not
known. One approximate approach has been to use the form of PH K [n] appropriate
to a uniform electron gas, but with the constant density replaced by the local, varying

density of the interacting electron system at position 7". This local approximation,

35

referred to as Local Density Approximation (LDA), has proven to be a very successful
approach for describing the properties of numerous systems.2 At the same time, the
existence of an exact functional has given a momentum to ongoing work for the

development of more accurate forms of PH K [n].

2.3.2 The Kohn-Sham equations

In the previous section we have seen that the theorems of Hohenberg and Kohn guar-
antee that the ground state energy of an interacting many particle system is the
minimum of the energy functional defined by Eq. (2.34). Although the theorems
provided an exact treatment for the interacting electron system, the numerical imple-
mentation of the theory was not obvious until Kohn and Sham published in 1965 a
set of equations, which turned DFT (LDA) into a practical tool to obtain the ground
state density and energy [123].

Kohn and Sham proposed a method in which the kinetic energy component of the
total energy is replaced by the kinetic energy functional T0 [n] of a non-interacting
electron gas. Any difference between the exact kinetic energy T [n] and T0 [n], along
with any difference in the energy due to exchange and correlation effects (see Eqs.
(2.37) and (2.38) for J [n] and U66 [n2]) are treated as a separate contributions to the

energy. Therefore we have:

FHK [n] = To in] + J [n] + Eric [71] (2.41)

where J [n] defined in Eq. (2.37), is the classical interaction of the electron-cloud with
itself (commonly known as Hartree term). The last term Exc [n], usually referred to

as the exchange-correlation energy, contains the difference T [n] — T0 [n], and any

 

2For a large class of materials, commonly referred to as strongly correlated systems, LDA fails.
One method to overcome this problem is the LDA+U approach, discussed in Section 2.3.3 and
Chapter 7.

36

diflerence between the exact self-interaction U86 [n2], defined by Eq. (2.38) and its
classical approximation J [n]. Using (2.41), we have a new expression for the energy

functional:

E [n] = TO [n] + J [n] + Uext [n] + E330 [n] (2.42)

We can now interpret the expression (2.42) as the energy functional of a non-
interacting electron-gas, subject to two external potentials: one external potential
such as due to the nuclear charges and another due to exchange and correlation effects.
The corresponding single-particle Hamiltonian is called Kohn-Sham Hamiltonian and

has the form:

fiKS = 73+ 3+0...” + 1.73“
= -1 2V? + 6‘] [77’] + 5Uext lnl + 5Exc [Tl]
2 ' 6n 6" 5” (2 43)
z I

7
1 n 7'
-_-——§ V22+/——(——_).—d3r’+v(7f)+vxc(f)
2 i lF-T’l

Now we can state that the ground state density n (r) of an interacting many-electron

system is.

N
n (o = Z 4’; (o 4.- (n (2.44)

i=1
where the single particle wave functions 35,- (r) are the N lowest-energy solutions of

the Kohn-Sham equation

HKSCPz' = 544% (2-45)

As a result, to find the ground state density, all we have to do is to solve the

Schriidinger-like noninteracting single-particle equations. We note that the single-

37

particle wave functions, 45,- (F), are not the single-particle electronic wave functions;
they describe mathematical “single-particles states”, without any direct physical
meaning. The only thing we know is that the density associated with these “single-
particles” is guaranteed to be equal to the exact electron density. Also the single-

particle energies, e,- are not the singe—electron energies.

In the expression of Kohn-Sham equations, both J [n] and Exc [n] operators de-
pend on the density n (7‘), which in turn depends of (l’i (F) which are being searched.
This means that we are dealing with a self-consistent problem, which can be solved
using an iterative process. Some starting density n0 is guessed, and a Hamiltonian
H K 51 is constructed with it. The eigenvalue problem is solved and using the result-
ing set of (b1, a new density n1 can be derived. If no differs from n1, then n1 is used
to construct H K 32 which will give n2 etc., and the is repeated until the convergence

of the density is reached. This final density is consistent with the Hamiltonian.

Kohn-Sham equations accomplish a remarkable thing: in principle, they allow an
exact treatment of the ground state of the many-body problem. However, in reality
the exact treatment is not possible, because the exchange-correlation functional is

not known exactly.

2.3.3 The exchange-correlation functional

The solution of the Kohn-Sham equations requires an explicit form of the exchange
correlation term. A widely used approximation, called Local Density Approximation
(LDA, briefly mentioned at the end of the Section 2.3.1), is to postulate that the

exchange-correlation functional has the following form:

E39414 = f n (n in 021 (131: (2.46)

38

The function Eq.-c (n) for the homogeneous electron gas is numerically known. This
postulate implies that the exchange-correlation energy associated with a particular
density n (F) can be found by dividing the material in infinitesimally small volumes
with uniform densities, and then the exchange-correlation energy is calculated by
summing the contributions of all the small volume over all space. By construction,
LDA is expected to perform well for systems "with slowly varying densities, but sur-

prisingly, it gives very accurate results in many realistic cases too.

In order to improve on LDA, one can consider the dependence of the exchange-
correlation energy on the density of the neighboring infinitesimal volumes as well. In
other words, the gradient of the density can also be taken into account. This ap-
proximation is therefore called Generalized Gradient Approximation (GGA). Within
GGA, the exchange-correlation is a functional not of the density alone, but also of its

local spatial variations:

E3.“ in] = f n (n e€§A1n<0.Vn(0id3r. (247)

Despite the fact that in general GGA performs slightly better than LDA, there are
situations when LDA is preferred over GGA. As we will see in Chapter 3, GGA has
the tendency to underestimate the binding between atoms and as a result it usually
overestimates the lattice constants. This effect of GGA is especially problematic in
the case of layered crystals, such as the III-VI family of semiconductors, where due
to the weak interlayer interaction the lattice constant becomes highly overestimated

in the direction perpendicular to the atomic layers.

One of the classic and well known failures of the LDA is the underestimation
of band gaps in semiconductor . There are two origins of this failure: (i) the self-
interaction error inherent in LDA potential, and (ii) the vanishing discontinuity of the

LDA exchange-correlation potential as a function of occupation at the Fermi level.

39

They lead to wrong absolute energy positions and too small or entirely absent bang
gap for many materials. This failure is not corrected by GGA, where the discontinuity
also vanishes. In fact it has been found that in many cases, due to the above mentioned

under-binding tendency of GGA, it produces even smaller band gaps than LDA.

Another deficiency of LDA and GGA is related to the description strongly cor-
related materials. Such systems usually contain transition metal (TM) or rare earth
(RE) elements, with partially filled d or f electronic shells. When we apply a one-
electron method with orbital-independent potential such as LDA, to TM or RE sys-
tems, the result is partially filled d or f band with metallic type electronic structure
and itinerant d or f electrons. This is a definitely wrong description of transition-
metal-oxidesand RE metal compounds, where the d and f electrons are well localized,
with a large energy separation between the occupied and unoccupied bands. One way
to overcome this deficiency of LDA (and GGA) is to take into account the strong
Coulomb repulsion within the localized shells. This has been done in the LDA+U
method [84—87], in which the orbital dependent Coulomb interaction is considered
for localized electrons by the addition of the Hubbard-like term 1 / 2U 22-75 j ninj (n,-
are orbital occupancies), while the delocalized electrons are treated by the usual,
orbital-independent, one-electron LDA functional. In order to avoid double counting
one has to subtract from the LDA functional the corresponding Coulomb d—d (or f—
f) interaction energy. As a result (neglecting the exchange), we have the following

functional:

1
E=ELDA—UN(N—1)/2+§U2ninj (2.48)
irj
where N is the total number of electrons in the localized shell. The above energy

functional allows us to calculate orbital energies and orbital dependent one-electron

potentials, which reproduce qualitatively the correct physics for many TM and RE

40

systems.

2.4 The supercell technique for defect calculations

2.4. 1 Introduction

The supercell method is a widely used approach to calculate properties associated
with the presence of defects in solids. In this method the defect and the surrounding
atoms are placed in a suitably chosen box, which is repeated infinitely in one or
more spatial directions. Therefore this box becomes the new unit cell for the system
and periodic boundary conditions are applied. The supercell method transforms an
isolated point defect or defect complex in a three dimensional periodic array of defects.
For surface calculation the system becomes a repeating array of slabs separated by
vacuum regions. In each case, one has to be careful about how to choose of the
size of the supercell (separation between the slabs), so that the effect of impurity-
impurity (or surface-surface) interaction is reduced. Typical sizes of the supercells
are 64-atoms, however, the increasing computational power has allowed for supercells
with several hundred atoms.

Using supercells in combination with periodic boundary conditions allow the ex—
ploitation of many techniques derived for the quantum physics of periodic systems.
The wave vectors of the BZ in the reciprocal space of the periodic supercell are good
quantum numbers, thus the standard band structure methods can be applied. Fur-
thermore the supercell method enables the full relaxation of the crystal structure by
minimizing the total energy. Defect formation energies and migration barriers can be
calculated as a function of Fermi level position of the host material and as a function
of chemical potentials of the atoms which form the material. Using supercell calcula-
tions, useful physical properties can be extracted such as probabilities of certain type

of defect to form under particular thermodynamical growth conditions, basic nature

41

of the defect state (acceptor, donor, deep, shallow), local vibrational modes (LVM),

hyperfine fields, and other ground state properties and optical properties.

2.4.2 Defect formation energies and ionization levels

In first-principle total energy calculations, the defect formation energies are usually
described as a function of atomic chemical potentials of the constituent elements [125,
126] and Fermi energy, which is the chemical potential of the electrons [127]. To
understand the dependence of the formation energy on the chemical potential and
Fermi level, we give some simple and illustrative examples of defects. First, let us
consider a charge-neutral cation vacancy (denoted as V8) in a binary compound. In
order to form such a defect one cation must be removed from the compound and place
in an atomic “reservoir” of energy equal to the chemical potential of the cation in its

bulk form, ,uC. Therefore the formation energy of the cation vacancy is:

AH) (V8) = Etot (V8) — Etot (0) + #0 (2.49)

where Etot (V8) is the total energy of the crystal with the vacancy present and
Etot (0) is the total energy of the pure host material. The chemical potential pa in
Eq. (2.49) is kept as a parameter, and it can vary over a certain range, depending
on the grth conditions. If the crystal is grown under cation-rich conditions (pc is
high), the formation energy of cation vacancy according to Eq. (2.49) will increase,
so that it will be less probable for the cation vacancy to form. Therefore, in order to
avoid the formation of an undesired defect it is useful to prepare the growth conditions
such that the formation energy of that defect is maximized.

The dependence of the formation energy on the Fermi level can be understood
as follows. In order to create a positively charged defect (D+) we have to remove

an electron from the corresponding neutral defect (D0) and place it in an electronic

42

reservoir of energy equal to the Fermi energy (E F). Thus the formation energy of
(D‘l') is equal to the formation energy of (DO), minus the energy 6 (0/+) required

to ionize (DO), plus the energy of the electron reservoir (E F):

AHf (0+) = AHf (DO) — 5 (0/+) + EF (2.50)

Equation (2.50) shows that the higher the E F, the more energy is required to form
a positively charged defect. This makes sense because in electron-rich materials (n-
type or with high E F) it is more difficult to create donors (which produce even more
electrons by: D0 —> D+ + e‘). In the case of acceptors, on the other hand, the

formation energy decreases with the increase of E F3

AHf (A‘) = AHf (A0) + e (—/0) -— EF (2.51)

Equation (2.51) shows that it is more diflicult to create acceptors in hole-rich (p—type)

materials.

In general, the formation energy of a defect X in charge state q is given by:

AH): (X4) = Eta, (X9) — Etc, (0) + 2 71,41; + (13],.

Z

= Etot (Xq) - Etot (0) + Z niEi + qEVBM + Z "M + qEF ( )
i i 2.52

x J
v

313(th

= AE(X‘1)+ 272,11,- + qEF

Z

In Eq. (2.52) Etot (X 9) and Etot (0) are the total energies of the defect containing and
defect free supercells, n; indicates the number of atoms added (77., < 0) or removed

(nz- > 0) from the supercell and )1; denote the corresponding chemical potentials.

43

They represent the energies of the reservoirs with which atoms are exchanged when
assembling the crystal. Usually the chemical potential (pg) can vary over a certain
range, with the upper limit being the energy (E,) of the elemental solid or gas. This
happens because if a; > E,- then the elemental solid will form, which will prevent the
further increase of the chemical potential. For convenience, in Eq. (2.52), we have
set )1; = E,- + u,- so that the upper limit to the newly defined chemical potentials #71
is zero (a,- S 0). Similarly we have set E]; = By B M + EF, where EV B M is the
energy of the VBM of the bulk material as calculated using the defect free supercell.
The VBM of the defect containing supercell must be aligned with that in the pure

supercell. The importance of this alignment will be discussed in Section 2.4.4.

Following Eq. (2.50) and (2.51) we can define the defect transition level or ion-
ization level a (q / q’ ), as the Fermi energy at which two charge states q and q’, of the

same defect X have the same formation energy, AH (X ‘1) = AH X q’ . Thus
f f

E (q/q’) : [AE (Xq) _ AE (xq )] (2.53)

q’—q

 

If the total energy of the final state q’ is calculated after allowing the atomic
positions in the supercell to relax, then the transition level given by Eq. (2.53), is
the one observed in DLTS experiments or as thermal ionization energies derived from
temperature-dependent conductivity or Hall effect data. If the energy of the final
state with charge q’ is calculated using the geometry of the initial state q, then
the level given by Eq. (2.53) corresponds to the optical level observed in “vertical”
absorption experiments, where the final charge state cannot relax to equilibrium.
In emission experiments, however the initial excited state has evolved towards its
equilibrium configuration thus the emitted photon has less energy than the absorbed
one. Therefore the peak positions of the optical absorption and emission spectra of

the same electronic transition are separated. This separation, the so—called Stokes

44

shift, poses challenges to both the experimental assignment of the peaks and the

theoretical analysis.

2.4.3 Equilibrium growth conditions

We have mentioned in the previous section that the atomic chemical potentials of
the host elements and impurities can vary over certain range. The achievable values
of the chemical potential are determined by the equilibrium growth conditions [55,
128]. First, as explained in Section 2.4.2, in order to avoid precipitations, the atomic
chemical potentials of the constituent and dopants are bound by the total energies of
the constituents in their bulk metallic (or gaseous) form. For example if we consider
a ternary compound AleC'n, and for convenience we set the total energies of the

constituents as reference, we have:

#A S 0. MB S 0, MC S 0 (2-54)

For instance, if the chemical potential of component A is greater than zero, than the

component A will form rather the compound AleCn.

Second, the chemical potentials are limited to those values that maintain a ther-

modynamically stable host AleC’n compound:

luA + mp3 + 72/10 = AH (AleCn) (2.55)

where AH (AleCn) is the heat of formation (enthalpy of formation) of host solid
AleC'n. In first-principles calculations the heat of formation can be calculated

from:

E (AleCn) = lE (A) + mE (B) + nE (C) + AH (AleCn) (2.56)

45

where the E’s are the total energies of the host and the constituent in their bulk
(solid or gas) form.
Finally, to avoid the formation of secondary phases between a dopant X and a

host element H, the chemical potentials are limited by:

max + W}; S AH (XmHn) (2-57)

In Eq. (2.57), AH (XmHn) is the heat of formation of the secondary phase XmHn.

The above discussion demonstrates that by changing the chemical potentials of
the host and impurity elements or the Fermi energy, one can control the dopant
solubility. It is well known that there are intrinsic thermodynamic limitations to the
ability to dope semiconductors, such as p-type doping in wide gap semiconductors.
To overcome these fundamental limitations, first-principles simulations can be used
to design new materials with desired doping properties. One such example is the
recent success in fabricating p-type transparent conductive oxides (TCOs), where the

mechanism that led to p-type conductivity was explained by Nie et al [129].

2.4.4 Energy-level reference and valence band alignment

It is obvious that from Eq. (2.52), that an important parameter in supercell calcula-
tions is the position of VBM. Usually in almost all theoretical calculations the VBM
is considered to be the reference for the electron chemical potential (cf. Eq. (2.52)).
However an important issue is how to find the energy of the VBM in a finite size su-
percell, where the eigenvalues of the bulk could be strongly perturbed by the presence
of a periodic array of defects. One method of valence band alignment was proposed
by Zhang and Northrup [126]: a region of the supercell was chosen away from the
defect to represent the bulk and the average self-consistent potential calculated for

this region was aligned with the average potential of the same region in the defect free

46

supercell. Consequently the magnitude of the correction to the VBM can be written

as:

AEVBM=E$BM—E{}BM=(VD>—(VH> (2.58)

where E? B M and E5 B M are the energies of the VBM calculated in the defect con-
taining and defect free supercells, respectively. A widely applied practice in defect
calculations is to assign Ell/2131;! to the single-particle VBM eigenvalue, which is often
located at the F-point. However, it has been suggested that within the framework
of supercell approach, it is often better to calculate the energies EV B M and EC B M
as averages of the band edges over the special k-point used in total energy calcula-
tion [130].

Other energy references are also possible: in all electron calculations, one can
align the electronic core or semicore level energy, or define all energy references with

respect to the so—called crystal zero, the potential energy at the surface of a neutral

Wigner-Seitz cell.

2.4.5 Summary and conclusions

First-principles calculations within DFT and supercell method enable us to quantita-
tively estimate the defect formation energies, diffusion barriers, structural parameters,
detect ionization levels in semiconductors. However, the desired accuracy in total-
energy calculations is not always easy to achieve. In particular, the calculation of the
energy position of the defect induced states in the semiconducting band gap can be
quite challenging and requires careful considerations of possible sources of error. One
possible source of error can be related to the underlying theory, because of the treat-
ment of electronic exchange and correlation. As mentioned in Section 2.3.3, local and

semilocal approximations contribute to the underestimation of the fundamental band

47

gap. The second source of error is associated with the finite-size effects inherent in
the supercell approach. These errors include spurious defect-defect interaction which
leads to dispersion of the defect related electronic states, incomplete sampling of the
reciprocal cell, and the constrained relaxations of the atoms surrounding the defects.
The third source of errors is related to the approximations needed for specific im-
plementations such as generation of pseudopotentials, construction of basis sets and
numerical accuracy of the algorithms.

The superiority of the DFT methods consists in revealing systematic trends in
the electronic and structural properties of defects in semiconductors. These include
the nature of the defect levels (donor, acceptor, deep, shallow), their spin structure,
point symmetry, energetics.

Obtaining quantitatively accurate results using supercell calculations is an ex-
tremely difficult task and requires significant computational resources. As a result,
many confusing and seemingly contradictory results have been published in the sci-
entiflc literature. However, over the time the computational methods have matured,
and with the significant increase in the computational power available for such cal-

culations it is possible to perform calculation with critical predictive power.

48

Chapter 3

Electronic structure of

non-isovalent impurities in GaSe

3. 1 Introduction

Gallium Selenide (GaSe) is a wide band-gap semiconductor that has been studied for
a long time due to its unique properties associated with different layered structures.
Strong covalent bonding within the layers and weak bonding between them (van der
Waals and small ionic-covalent component) make GaSe highly anisotropic. Because
of its large nonlinear optical coefficient (d22 = 75 pm/ V), GaSe has been investigated
as a non-linear optical material [131]. One of the important applications of nonlinear
optical effects is the generation of broadband mid-infrared electromagnetic waves and

detection of coherent broadband THz radiation.

Another important application of GaSe is in the area of radiation detectors. Since
the first report by Manfredotti et al., in 1974, GaSe has been studied by several groups
due to easy fabrication, high resistance for radiation damage, and high temperature
operation [29,132—136]. It has room temperature resistivity of 103-109 0cm [132].

The large leakage current for pure GaSe made it unsuitable for efficient radiation

49

detectors. However doping GaSe with Ge or Sn gives rise to high resistivity by
reducing the leakage current [137]. In a different experiment, Yamazaki et al. reported
3—7 orders of magnitude decrease in leakage current and dramatic improvement in

energy resolution of GaSe detectors by doping with 0005-01 at.% Ge [29,136].

Since defects play a dominant role in the ultimate performance of a semiconductor,
it is extremely important to understand the nature of the impurity-induced states. As
we described in the previous chapter, defect states in semiconductors can be broadly
classified as shallow defect states (SDS) and deep defect states (DDS). These are of
course two extremes and one can have defect states somewhere in between. SDS in
wide band gap semiconductors associated with donors or acceptors are well described
by effective mass approximation (EMA) with small central cell corrections associated
with deviations from the hydrogenic potential in the impurity cell and its immediate
neighborhood. On the other hand DDS cannot be describ ed adequately by the EMA.
Also the EMA has difficulties when the central cell corrections are large or comparable

to the EMA binding energy(BE).

In this chapter we discuss the nature of the defect states in GaSe when Ga is sub-
stituted by impurities of different nominal valence. The impurities we have considered
are Cd (divalent), and Sn (tetravalent). Cd gives rise to acceptor state(s) whereas
Sn gives rise to a donor state. The experimental values of the Cd acceptor binding
energies (BEs) are 0.13 eV and 0.18 eV [41] while the Sn donor BE is 0.36 eV [34].
From these BE values it appears that the donor state introduced by Sn is more likely
a deep defect state whereas the acceptor states due to Cd are somewhere between
a shallow and deep defect state [99]. In addition to the above substitutional defects
we have also looked at the electronic structure of Ga and Se vacancies. Micocci et
al. [138], have found a localized level at ~0.2 eV above the valence band from their
hole trapping measurement which they have assigned to Ca vacancy. Also they find

another defect level at ~0.8 eV above the valence band maximum. The latter was

50

assigned to some extended defect region. Micocci et a1. [34] have also argued that Se
vacancies produce electron trapping centers at ~0.52 eV below the conduction band
minimum. All these defect levels assigned to vacancies in GaSe appear to be of deep

defect type.

To understand the nature of the defect states mentioned above we first used EMA
in the case of substitutional defects to see how large the central cell and nonparabolic
corrections are [99]. We then used ab initio methods and supercell model to calculate
the defect state energies (defect bands) and also looked at the degree of localization
of the defect states. A comparison of the results obtained with these two extreme
approaches can help us to get a better understanding of the nature of defect states in
GaSe. For the vacancies we only use the supercell model. It is well known that due
to the underestimation of the band gap by GGA and due to the limitations of the
supercell model, accurate calculations of defect BE represents a serious challenge [139].
However, in the case of acceptor states, derived mostly from the valence band, their
position relative to the valence band maximum (VBM) can be obtained more reliably

and with reasonable accuracy.

3.2 Method of calculation

For EMA we have used the experimental values of the dielectric constant and effec-
tive mass tensors [140]. In addition, we have also used the effective mass values ob-
tained from our theoretical calculations. Electronic structure calculations have been
performed within density functional theory (DFT) using all electron full-potential
Linearized Augmented Plane Wave (FPLAPW) plus local orbital [141] and Projec-
tor Augmented Wave (PAW) [142] methods implemented through WIEN 2k [143] and
VASP [144—147] packages respectively. In both cases we have used the generalized

gradient approximation (GGA) [148] for the exchange and correlation potential.

51

In the FPLAPW calculations the 3d, 4s, 4p electrons of Ga and Se atoms and
the 4p, 4d, 5s, 5p electrons of the impurity atoms (Cd, In, Sn) were treated as
valence electrons. Convergence was assumed when the energy difference between
the self-consistent cycles was less then 0.0001 Ry (1.36 meV). The product between
the smallest muffin tin radius and the largest reciprocal lattice vector was chose as
RMTKmax = 7.0. In the case of bulk GaSe, where the atomic radii were chosen as
2.17 a.u. and 2.18 a.u. for Ga and Se respectively, this corresponds to a cutoff energy
of 141.7 eV. The accuracy (see next paragraph) in the bulk energy calculation was
achieved by using a relatively dense k-mesh of 19x19x4, which gives 120 k—points in
the irreducible Brillouin zone (BZ). The defect calculations using FPLAPW methods
were performed on 3x3x1 supercells (72 atoms) with one impurity atom (Cd, In, Sn)
replacing a Ga atom. The smallest muffin tin radius was chosen to be 2.06 a.u., which
gives an energy cutoff of 157.2 eV, and the BZ was sampled by a 6x6x3 mesh (17
k-points in the irreducible BZ).The k-point meshes used are suitable for the modified
tetrahedron integration scheme [149].

Since the PAW method is computationally less demanding than the FPLAPW,
we use it for calculations involving large supercells (up to 400 atoms). The Ga and
Se vacancy calculations were also performed using PAW methods. However, in order
to compare the two methods, we have performed calculations on bulk GaSe and on
3x3x1 supercells with Cd and Sn impurity. In the case of PAW method the outermost
s and p electrons of all the atoms were treated as valence electrons but in addition,
the 3d electrons of Ga and Sn were also included in the valence region. The electronic
relaxations were performed until the total energy difference relative to the previous
cycle was less than 10‘4 eV. In the PAW calculations the energy cutofl was set to
300 eV. For the smaller supercells (3x3x1) we have carried out geometric relaxation
but for the large ones (containing 400 atoms/supercell), due to limited computing

time, we have not been able to do full atomic relaxation and complete band structure

52

calculations.
Since the spin-orbit effects are expected to be small for III-VI compounds [22], we

did not include spin-orbit interaction in our calculations.

3.3 Pure GaSe

3.3.1 Crystal structure and geometric relaxation

The crystal structure of GaSe is quite interesting. It consists of Ga—Se layers that can
be stacked in different ways leading to different poly-types (B, 'y, 6 or c). All these
poly-types are wide band gap semiconductors with room temperature band gaps of
1.996 eV (for e and 7), 2.026 eV (for 6) and 2.046 eV (for H) [150]. Since all these
structures contain Ga—Ga dimers as a common motif, the origin of the ~2.00 eV band
gap is most likely related to this geometry.

Several energy band calculations have been done for e-GaSe [16,23,24,26, 151, 152]
and a few for ,B-GaSe [16,22]. These prior works show that the differences in the
electronic structure between different polytypes are negligible, which means that we
have the choice of the polytype to perform our calculations. We only consider the
fl-type lattice, which has the highest symmetry and contains the smallest number of
molecules (eight) in the unit cell and it will make easier the comparison between our
results and prior works. ,B-GaSe crystallizes in layered, hexagonal system, having the
space group P63 / mmc. The GaSe layers consist of two planes of Ga atoms sandwiched
between two planes of Se atoms as illustrated in Fig. 3.1. In each plane the Ga or
Se atoms are arranged in a two-dimensional hexagonal lattice. The nearest neighbor
coordination of the Ga atoms differ from that of Se atoms: each Ga has three Se
nearest neighbors (NN) and one Ga NN, while each Se is only coordinated by three
Ga atoms and no other Se atoms.

For the structure optimization we first relaxed the volume of the unit cell, keeping

53

 

Figure 3.1: Schematic view of the crystal structure of [i-GaSe. The unit cell extends
over two layers of GaSe and contains four Ga and four Se atoms. In each layer two
monoatomic sheets of Ga are sandwiched between two monoatomic sheets of Se. In
this figure 9 unit cells are shown

Table 3.1: The theoretical lattice constants of GaSe and the Ga—Ga, Se—Se and Ga-
Se bond lengths calculated by FPLAPW and PAW methods. For comparison the
experimental values are also given as in Ref. [153]. All distances are given in angstroms.

 

3: b C dGa—Ga, dSe—Se dGa—Se
FPLAPW 3.83 16.29 2.45 3.83 2.50
PAW 3.80 16.17 2.45 3.80 2.49
Exp. 3.74 15.92 2.32 3.74 2.48

 

the ratio of the lattice constants a:b:c fixed and then we optimized the internal struc-
tural parameters by minimizmg the total energy as a function of atomic positions,
keeping the volume of the unit cell constant. As a second step, after obtaining the
optimized theoretical crystal structure, we performed an additional iteration in the
volume relaxation (keeping a:b:c constant) and analyzed the changes in the lattice
parameters. Since we have found that these changes are quite small (less than 0.3%)
we assumed that the unit cell volume and the internal structural parameters were

at their optimal values. We note that we have also tried to optimize the c/a ratio

54

keeping the volume of the unit cell fixed. We observed a large elongation of the unit
cell along the z-axis (perpendicular to the GaSe layers): for example, the lattice con-
stants obtained after the c/ a optimization using the PAW method, were a = 3.77 A
and c = 16.39 A. These values are ~0.7% smaller and ~1.4% greater compared to the
values obtained after the volume relaxation, respectively. As pointed out in Ref. [23],
this result is a consequence of the fact that DFT cannot describe van der Waals
interactions adequately. Therefore, in this chapter, we present the results obtained
using the theoretical crystal structure resulted after the first volume optimization and
relaxation of the atomic positions at the experimental c/ a ratio.

The difference between the FPLAPW and PAW lattice parameters and bond
lengths are less than 0.5% (listed in Table 3.1).The theoretical lattice constants of
bulk GaSe are larger than the experimental ones reported in Ref. [153]. This is due to
the well known overestimation of the lattice parameters by GGA. The shortest Se—Se
distance is between Se atoms located in the same monoatomic sheet and it is more
than 50% longer than the Ga-Ga dimer. On the other hand, the length of the Ga—Se
bond is comparable with the Ga-Ga bond length, leading to a strong hybridization

between the orbitals of Ga and Se.

3.3.2 The electronic structure of GaSe

Before describing the detailed band structure of the GaSe, we will discuss the bonding
scheme in this system qualitatively as shown in Fig. 3.2. .This will help us in develop—
ing an intuitive picture of the band gap formation in this valence mismatched system.
The states mainly responsible for bonding are Ga 43 and Se 4p valence orbitals. If
the mixing between Ga 3 and Se p states is turned off, then for each Ga—Ga dimer we
have a bonding 8 state and an antibonding 3 state. In the unit cell there are 4 Ga
atoms (2 Ga dimers) and 4 Se atoms, so there are 2 bonding s states, 2 antibonding

3 states and additional 12 p states coming from the 4 Se. In the presence of Ga 3 and

55

_ Lowest CBs (Se p

 

 

 

 

 

 

 

 

,. ' with Ga 3 and pz
4 Ga pz states =— - character)
41 {5. ....... - EF
12 Se p states
' Valence band
...... “. (mixture of Ga 8
2 Ga 8 antibonding ............................. , and 39 P states)
states — ------------ . - m
2 Ga 3 bonding __ """""""" T
E".rgy states — ---------- ' -!
Mixing is Mixing is
turned off turned on

Figure 3.2: Qualitative scheme showing the bands responsible for bonding in GaSe.
The bottom of the valence band has a strong Ga 3 character, followed by a strong
mixture of Ga 3 and Se p-states. Two Se p bands split off the valence band as a
result of this hybridization, and form the bottom of the conduction band. The lowest
conduction bands also display Ga—pz character.

Se p hybridization the Ga s bands pick up Se p character and vice-versa. The main
effect of this mixing is the splitting off of 2 Se p derived bands near the top, giving
rise to an energy gap in the band structure of GaSe. The electron counting is such
that 4 Ga—Ga s bands and 10 Se p bands (per unit cell) can accommodate 28 electrons
(12 from 4 Ga and 16 from 4 Se). Thus the gap in the DOS discussed above is the
true semiconducting gap. Both the valence bands and the lowest conduction band
are derived from the Se p orbitals mixed with Ga 3 states. There is however some
additional complexity due to the presence and subsequent mixing of Ga pz bands,
which contribute primarily to the lowest conduction bands with some admixture with

states near the top of the valence band.

As pointed out before, the electronic structure of GaSe has been extensively inves-

tigated during the last two decades, using empirical tight-binding approach [16,22] as

56

well as ab initio methods [23,24,26, 151,152]. We briefly review some of the general
features of the band structure for fi-GaSe, obtained with the full potential method
(Fig. 3.3(a), (b), (c) and (d)), and explore the nature of bonding and atomic characters
of the valence and lowest conduction band states to confirm our qualitative picture
described above. We further note that the electronic structure of pure ,B-GaSe was
also calculated with the PAW method and it is very slightly different than the one
obtained by the FPLAPW scheme as regards the occupied states. The differences in
the energy eigenvalues at the I‘ point are less than 50 meV (when the energy of the

valence band maximum was set to zero in both calculations).

Let us look at Fig. 3.3(a) first. The semiconducting band gap (0.65 eV) is direct,
located at the F—point and it is significantly smaller than the experimental value of
2.046 eV reported for fi-GaSe [150]. This is the well-known limitation of the LDA
and GGA, which underestimate the band gap in semiconductors [154]. The valence
band is split into three sub-bands: one between -7 .4 eV and -6.4 eV (denoted III),
originating from the s states of Ga slightly hybridized with pa,- and py states of Se;
sub-band II, between -6.5 eV and -3.5 eV, which is also derived from the Ga 3 orbitals,
but in this case the mixing with Se p is much stronger, resulting in a larger dispersion
of this sub-band; finally the much broader band between -4.6 eV and 0 eV (denoted
as I) which comes from the p states of Se hybridized with the s and pz states of Ga.
Whereas the states near the middle and the bottom of this band come primarily from
the Se pa; and py orbitals, the states near the top of this sub-band (which is also the
top of the valence band) correspond to the Se pz orbitals. There are four low lying
conduction bands between 0.6 and 3.0 eV which consist of mixtures of Se p, Ga 3 and
Ga pz orbitals. The lowest bands (IV) in the Fig. 3.3(a) correspond to the Se 33 core

states.

To get a better understanding of the chemical bonding in GaSe, we have analyzed

the orbital character of different energy bands (in which each eigenvalue is associated

57

Energy (eV)

Energy (eV)

:5
o

 

 

l"
o

 

P
c>

 

 

 

 

1'0
o

 

.5
o

 

-6.0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 
  

 
   

.- -- - .....&.., ----------
2'0 gGa s antibonding+ S .
"m", .....

0.0 W BF

 

 

 

 

 

 

 

 

 

 

 

 

,Ga s bonding]
I
r KH Ar ML A
(b)

 

 

58

4.0

 

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

2.0
0.0
A
>
o
v
a -2.0 -:
E E
m -4.0 -_
I
-6.0 -_
-3.0 '.
l

 

 

 

 

.. 35‘
' : o
. r. l .
T l 3 o
i
M ° K
A—

 

Figure 3.3: (a) Band structure of ,B-GaSe. In addition to the scheme presented in
Fig. 3.2, the Se s-states are also shown here (subband IV). The zero energy is chosen at
the top of the valence band. (b) The Ga 3 orbital contribution to the band structure.
The Ga 3 antibonding state hybridizes with the Se p states, pushing up some Se p-
states into the conduction band. The lowest conduction bands, therefore also display
Ga 8 character. (0) The Se pa; + py orbital contribution to the band structure. ((1)
The Brillouin zone corresponding to the hexagonal unit cell. The I‘-A direction is
perpendicular to the GaSe layers.

59

with a circle, whose radius is proportional to the specific orbital character of that
state - the fat band representation). Fig. 3.3(b) shows the contribution of the Ga
3 orbitals to the band structure. The “fat” bands III and II, with strong Ga 3
character, correspond to Ga—Ga bonding (III) and anti-bonding (II) states. The anti-
bonding state (II) is strongly hybridized with a linear combination of Se p orbitals
of appropriate symmetry (F2— at the I‘ point) and is broad compared to the bonding
band (III). As a result of this strong hybridization two of the Se p bands split off from
the rest of the Se p bands leading to the formation of a gap in the DOS of Se derived
p-bands. Therefore the lowest conduction bands originate from Se p orbitals, but
these bands also display Ga s and Ga pz character as a result of hybridization. This
can be observed from Fig. 3.3(b) and (c), which show the contributions of Ga 3 and
Se pa; + py orbitals to the band structure respectively. The relative admixture of Se
p and Ga pz states near the conduction band minimum and valence band maximum
determines the effective mass anisotropy (see discussions below). Note that because
of small LDA band gap the hybridization between Se p and Ca p states may be
overestimated. The Brillouin zone corresponding to the hexagonal unit cell is shown

in Fig. 3.3(d). The I‘-A direction is perpendicular to the GaSe layers.

3.4 Defects in GaSe

When Ga is substituted by another atom or there is a vacancy at the Ga site, the local
electronic structure of the Ga-Ga dimer can change dramatically depending on the
energy mismatch between the energies of Ga 4s and the valence s-state of the defect.
This can perturb the valence band and if the perturbation is sufficiently strong it
can give rise to different types of defect states. One also expects to see strong local

perturbation if there is a Se vacancy in the neighborhood of Ga dimer.

60

3.4.1 Atomic relaxations in the presence of defects

The calculations in the presence of defects have been performed with two different
supercells: 3x3x1 (72 atoms) and 5x5x2 (400 atoms) with one Ga being replaced
by an impurity. This corresponds to 1 impurity/ 36 cations (~3 at.% concentration)
and 1 impurity/ 200 cations (~0.5 at.% concentration) respectively. Supercells of
the same size were also used for the electronic structure calculations associated with
the vacancies. As suggested in Ref. [155], to reduce the spurious strain induced by
the impurity-impurity interactions within neighboring supercells, we performed the
calculations using the theoretical lattice constants and allowed for the relaxation of

the atomic positions.

In the FPLAPW calculations we only relax the positions of the Ga and Se closest
to the impurity (limited relaxation). We find that when a Ca atom is replaced by
Cd (or Sn) there is significant outward relaxation (away from the impurity, along
the impurity-host bond direction) of the Ga and Se atoms surrounding the defect.
The minimum energy corresponds to the situation when the Ga—Cd / Sn bond length
expands by 3 / 7% relative to the Ga—Ga dimer length, while the Se-Cd/ Sn bond length
expands by approximately 5 / 7% compared to the original Se-Ga distance.

Since for large systems we use PAW method (because such calculations are not
feasible within FPLAPW method), we first want to compare these two methods in
terms of atomic relaxation. In the PAW calculations we have performed two types of
relaxation of the internal parameters. First we carried out limited relaxation studies
as in the case of FPLAPW method. Then we relaxed the positions of all the atoms
(full relaxation) in the defect containing 3x3x1 supercell. The results are summarized
in Table 3.2. In the first case, the changes in the bond lengths are nearly identical
in the two methods indicating that PAW method is as good as FPLAPW. When

all the atoms are allowed to relax the outward relaxation of Cd and Sn N Ns are

61

Table 3.2: The changes in the Ga-Ga, Ga—Se and Se-Se bond lengths when Ga is
replaced by an impurity (Cd, In or Sn). All distances are given in angstroms.(alimited
relaxation where only the defects’ nearest neighbor atoms were allowed to relax

 

Cd impurity In impurity Sn impurity
Ga—Cd Se-Cd Se-Se Ga—In Se-In Se-Se Ga—Sn Ga—Sn Se—Se
LAPW“ 0.07 0.14 0.26 0.10 0.15 0.23 0.18 0.18 0.26

PAW“ 0.07 0.15 0.26 0.17 0.18 0.25
PAW 0.12 0.16 0.25 0.31 0.19 0.32

 

 

more pronounced: comparing the Ga—Cd / Sn distance to the Ga—Ga dimer length we
observe an increase by ~5/11% while the Se—Cd/ Sn bond length increases by ~6 / 7%
compared to the original Se—Ga distance. The larger outward relaxation in PAW
calclation is easy to understand, because all the atomic positions in the supercell

were optimized.

In the case of Sn impurity we observe that the increase in the Ga-Sn distance
is much larger than the increase of the Se—Sn distance (~11% compared to ~7%),
suggesting a stronger Coulomb attraction and therefore a more ionic character of - the
Se-Sn bond, compared to Ga—Sn bond. In the case of Ga vacancy, the relaxation
of the atomic positions leads to a contraction of the three N N Se atoms (by ~14%)
toward the vacancy and to an expansion of the NN Ga (by ~4%) away from the
vacancy. The GaSe system with Se vacancy, on the other hand, minimizes its energy

by contracting of the NN Ga and Se atoms toward the vacancy (Ga ~12% and Se
~3%).

3.4.2 Hydrogenic effective mass approximation

In the effective mass approximation, the energy levels of shallow donors or acceptors
in a semiconductor with anisotropic effective mass and dielectric constant, are given

by the solutions of a Schréidinger equation with the Hamiltonian [102,156]:

62

h? 82 32 1 62 e2 1

1
H ——— — —+— +—— +
0 2 l7"4 (332 3v?) mnazzl \/€i€||\/a:2+y2+§-Lz2

5ll

 

(3.1)

 

In the above equation m .L and m” are the effective masses, and 51 and e” are
the static dielectric constants,while the parallel and perpendicular directions are con-

sidered with respect to the c-axis (which is perpendicular to the GaSe layers).

Let us consider the acceptor case first. We have used the experimental values of
these parameters (top most valence band effective mass) given in Ref. [140]. These are:
51 = 10.6 and 5“ = 6.18; mi = 0.8m0 and m“ = 0.2 m0 (where m0 is the free elec-
tron mass). Taking the units of length and energy as a = ’12 \/;?L—EH / m 1.5 132 = 1.64

A and e = m e e4 2h252 e = 166 meV, equation 3.1 becomes dimensionless:
0 J. .L _]_ ||

_ (535+5?+75Z§+§) W) =Ewtn, (3.2)

where the anisotropy factor is: 7 = m 15 .1. / mus“ = 6.86. The calculated theoretical
values of effective mass parameters obtained from ab initio calculations in GaSe are
m J. = 1.4 mg, m” = 0.1 m0 and give a larger value for this anisotropy factor, 7 =
24.01. It is interesting to point out that even if GaSe appears to be layered the
acceptor defect states are more one-dimensional because the anisotropy factor 7 >>
1. This is due to the strong hybridization between the Se p and Ga pz states near
the top of the valence band. The anisotropic Schréidinger equation (Eq. (3.2)) can be
solved using a variational perturbation theory where one expands the wave function

\11 (F) in a basis of anisotropic hydrogen-like orbitals [102,156]:

1/4
WU?) = 2 (g) t/Jnlm (1733/, £270) ' (33)

nlm

63

In Eq. (3.3) wnlm(a:, y, z, a) = Rn)(a, r)Ylm(6, cp) are normalized hydrogenic wave
functions and a and ,8 are variational parameters, which can be varied to minimize the
energy. This approach is known to give accurate energies for the lowest hydrogenic
s and p like states: earlier calculations for the effective mass binding-energies were
done for donor levels in Si (7 = 0.2079) and Ge (7 = 0.0513) [156]. We are not
aware of any earlier calculations for which the anisotropy factor is much larger than
1. Equation (3.2) was diagonalized using this method and the ground state energy
converged surprisingly well for a 6 by 6 Hamiltonian matrix. The dependence of the
energy surfaceon the variational parameters for the 6 x 6 case is shown in Fig. 3.4.
The lowest energy, calculated using the experimental dielectric constants and effective
masses (8] = 10.6, e“ = 6.18; mi = 0.8 m0 and m” = 0.2 m0), is E0 = -77.46 meV.
The corresponding values for the variational parameters are a: 0.6 and 6 =1.7. This
gives a binding energr (BE) of 77.46 meV. If we use the theoretical value of the 7

parameter, the acceptor BB is 41.54 meV (see below)

For the donor case, we only used the experimental values of the effective mass
parameters of the lowest conduction band m _L = 0.5 mg, m” = 1.6 m0 [140]. The
values of the length and energy scale parameters are a = 2.63 A, 80 = 104 meV, and
the anisotropy factor is 7 = 0.54. Note that in contrast to the acceptor state, which is
elongated along the z-axis (cigar shaped), the donor state is more pancake like. The
donor BE from the variational perturbation calculation, turns out to be 126 meV and

the corresponding values for the variational parameters are a = 1.1 and )6 = 0.9.

Comparing the theoretical acceptor BE of 77.46 meV to the energy of the acceptor
levels obtained from photoluminescence measurement by Sighetomi et a1. [41] (130
meV and 180 meV), we conclude that the hydrogenic EMA is not adequate to describe
the acceptor levels in GaSe, even if we assume that Cd related defect BE is the smaller
of the two, namely 130 meV. The disagreement is worse for the anisotropy factor,

7 = 24.01, obtained from the theoretically calculated effective masses. For this case

64

     
 

      

0 A
E
-10
E
-20 v
GD
-30 3
.40 g
-50 5
U)
-60 a,
-70 3
h
-80 0:)
-90 LU
-100
0-0 0.2 1 dial

2
0'4 0-6 0.3 1,0 1.2 3 acct“
0“parameter ’

Figure 3.4: The dependence of the energy surface on the variational parameters a
and 6. The minimum is located at -77.5 meV.

we obtained the BE = 41.54 meV (a = 0.4 and fl = 2.4). Similarly for the donor
state for Sn impurity the experimental BE is 360 meV [34] whereas our theoretical
calculation gives 126 meV. Here again the central cell corrections (234 meV) are quite
large and the EMA is not adequate. Thus we need to carry out calculations that take
proper account of the short distance corrections to the EMA. This has been done

using ab initio DFT calculations and the results are discussed below.

3.4.3 Ab initio calculations of defects in GaSe

To characterize the substitutional impurity states we first compare the total DOS
with and without the impurity (Fig. 3.5), calculated using the 3x3x1 supercells, to
see if there are any noticeable changes in the neighborhood of the band gap. For Cd

(Fig. 3.5(b)) there is a peak near the valence band maximum and for Sn (Fig. 3.5(c))

65

 

 

80- (a) pure GaSe
4o-

 

 

aoq (b)
40-

 

DOS (states/eV)

80 .- (C)
40 -

 

 

 

 

Energy (eV)

Figure 3.5: The total DOS of (a) pure GaSe (b) Cd doped and (0) Sn doped GaSe.
(b) CdGa introduces a defect state (indicated by the arrows) near the valence band
maximum and (c) SnGa introduces a state near the conduction band minimum.

there is a peak near the conduction band minimum. We argue that these peaks are the
impurity-induced states and they are broad (full width at half maximum ~O.15 eV)
due to strong impurity-impurity interaction. For the 3x3x1 supercells the minimum
impurity-impurity distance is 11.4 A. To probe these defect states further we have
analyzed their Cd (or Sn) orbital character and orbital characters associated with

atoms located at different distances from these defects (see Fig. 3.6).

Substitutional Cd on Ga site (CdGa)- As we can see in Fig. 3.6 (a-d), the main
contribution to the impurity-induced peaks are coming from Cd 3 and p states as well
as from the states associated with the NN Ga (denoted Gal), NN and next NN Se
atoms of the defect (Sel and Se2). The next NN Ga (denoted Ga2) has very little

66

 

.o .0
sate

0.21,

P
.L

 

:3 .0
91*:

A L A

 

PPS-7°
.54

 

. . . ‘09
—Ga1p (Q) —-—Ga1p :06
---Ga2p "-9329 W
as
' f i M
—Se1 p (h)— 891 p 10'9
---SeZp —--SeZp j0-3

0.33

DOS (states/eV)

 

.° 99
‘3‘}?

9
9°..

1 L

14

 

 

 

9.09
oA-sn

 

0.0

 

 

' 014 F 0T8 -o'.e -o'.4 olo ' 014 ' 028

Energy (eV)

Figure 3.6: Partial density of states associated with the impurity (a) Cd 3, p and
(e) Sn 3, p orbitals and those associated with atoms located at different distances
from the defects as indicated in the figure (b—d, f-h). The atoms Gal and Sel are
the nearest neighbors (N N) of the defects, while the atoms Ga2 and Se2 are the next
N N of the same defects. The contribution of the Sn to the DOS is larger compared
to that of Cd, suggesting a more localized nature for the defect state associated with
the Sn impurity.

contribution to the states near the peak. However, the contribution of the NN Se p
is comparable to that of the next NN Se p. We find that the greatest contribution to
the DOS near the peak comes from the p orbitals of the next nearest Se atom (~0.38
states/ eV/ atom) followed by the nearest Se p orbitals (~0.30 states/eV/ atom) as
shown in Fig. 3.6 (d). So the defect state associated with Cd impurity is not strictly
localized near the defect and its nearest neighbors, but rather extends to the next

NN Se atoms.

The origin (parentage) of this defect state can be understood in terms of the Ga

67

s bonding-antibonding picture described in the case of pure GaSe. Since there is a
considerable mismatch between the energies of the atomic Ga 3 and Cd 3 orbitals
(—11.15 eV and -7 .21 eV respectively), they will not form strong bonding and an—
tibonding states anymore, but the Ga 3 orbital will rather mix with the pz states
of the neighboring Se atoms. As the result of this hybridization, the Se pz state
is perturbed, pushed up in energy and splits off from the top of the valence band.
This defect-induced band becomes partially occupied since the Cd atom substituting
for a Ga atom, provides one less electron to the Se network. The dispersion of this
band looks qualitatively similar to the highest GaSe valence band. One may therefore
confuse this band with the perturbed host band. To make sure that this is indeed a
defect state band, one has to increase the size of the supercell and take a look at its
dispersion. The increase in the supercell size should decrease the width of the defect
band. On the other hand if this band is a perturbed host band its band width will
not decrease much and it should show the effect of band folding. These points are

discussed below.

As pointed out in the literature (see for example references [95,157]), in supercell
calculations the defects change the host band extreme and one has to correct for this
defect-induced change in order to properly estimate the energy of the defect state. We
note that in the single defect limit (dilute impurity limit) one expects the host bands
to approach the bands of pure GaSe. In order to put the band structures of the defect
containing and the defect free supercells in a common energy scale we matched the
energies of the Se 3d semicore states calculated in both, the defect containing and pure
GaSe. This alignment is based on the assumption that unlike the valence states the
energies of core and semicore states do not change in the presence of defects. Other
energy scale alignments are also possible [26, 55], but in our all-electron calculation
matching some core (or semicore) states seems to be natural choice. We then compare

the energy of the defect-induced band to the energy of the valence band of the pure

68

GaSe(Cd) 3x3x1 supercell GaSe(Cd) 5x5x2 supercell
bi

2.0 (a)
1.6- §+

 

2.0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.6
1.2-\’ a1.2
1 - - -
i 0.8- - -0.8
> . . .
9 0.4- - « L04
2 . .
Lu ~0.55 eV
0.0 / k
-0.4 -
-0.8 - -0.8
A I‘ A r

Figure 3.7: The electronic band structure of GaSe(Cd) along the I‘ — A direction,
calculated with the non-relaxed (a) 3x3x1 and (b) 5x5x2 supercells. The width of the
topmost VB decreases from 0.55 to 0.11 eV in going from (a) to (b), indicating that
this is a defect level. As a guide to the eye, the topmost VB calculated using the two
supercells is enclosed in red rectangles. The host bands show folding (e. g. see the
lowest CB) and their dispersion does not change considerably

(3386 at the P-point to estimate the energy level of the defect. This method was
successfully used in literature; e.g. in Ref. [158] the authors have determined the
single electron energy levels at P—point and aligned them using some core levels, to

study the defects in CdTe.

Our calculation for Cd impurity at the Ga site gives the defect state at 120 meV
above the VBM. This value is in fairly good agreement with experimental BE of 130
meV obtained by Shigetomi et a1. [41]. However this agreement should be critically
examined. There are still several important issues that need to be resolved vis-a-vis
ab initio calculations of defect BEs using supercell model for GaSe. Whether our
identification of the “top most valence band” state with the defect band is justified?
How does this band change when the distance between the defects is increased? If

the impurity band flattens without appreciably affecting the energy at the P-point

69

then our current estimates will be unchanged. If, on the other hand, the impurity
band flattens with its average remaining the same then the impurity levels will merge
into the valence band, making it a resonant state rather than a bound state, in dis-
agreement with experiment. To check these issues we have performed the calculations
using PAW methods in combination with a 5x5x2 supercell (containing 400 atoms)
containing one Cd atom on Ga—site. For computational reasons, we did not perform

atomic relaxations in this large supercell.

We find that the dispersion of the “top most valence band” decreases dramatically
as the distance between the impurities increases from 11.4 A to 19.3 A in the xy plane
and from 16.17 A to 32.34 A along the z-direction. Because of large computation time
we only looked at the band structure along the F—A direction in the B2 (corresponding
to the crystal c-axis), where the dispersion was the largest. As shown in Fig. 3.7, the
width of the band decreases from ~0.55 eV to ~0.11 eV as we double the size of the
cell in the z—direction and there is no band folding. The band just below this defect
band does not narrow, in fact it broadens, and shows zone-folding that is characteristic
of a perturbed host band. We therefore confirm that the top most band as indeed
the defect band and it is half filled because divalent Cd replaces a trivalent Ga. To
confirm this further we have calculated the charge density associated with this band
using the 3x3x1 supercell results and show it in Fig. 3.8. As seen in the figure the
defect state extends to the next N N of the Cd impurity and is formed predominantly
by the nearest Ga 3 orbital and the next nearest Se pz orbital. There is also somewhat
smaller contribution from the nearest Se pz states. This finding is consistent with the

partial DOS shown in Fig. 3.6(d).

The energy of the defect level (in the dilute limit) was also estimated from the
5x5x2 supercell calculations by measuring its value at the F-point from the Ga 3d
semicore state and comparing with the VBM of the pure GaSe after matching the
Ga 3d energies in the two cases. We find it to be 143 meV above the VBM. The Ga

70

 

 

 

 

 

Figure 3.8: Electronic charge density associated with the Cd impurity induced defect
band. It extends to the impurity’s next NN Se atomn (Se pz orbitas). The Cd
impurity is represented by the red sphere in the figure.

3d states are nearly flat (width ~5meV), therefore even in the limit when we match
the extrema of the Ga 3d bands, the defect energy level will not be affected by more
than 10 meV. The difference between 3x3x1 and 5x5x2 (120 meV vs. 143 meV) can
come from many sources, the dominant one being the neglect of atomic relaxation in

the case of the latter.

Substitutional Sn on Ga site (SnGa). To analyze the nature of the defect level
introduced by the Sn impurity, first we look in Fig. 3.6 (e—h). The contribution to
the impurity-induced peak (at 0 eV) of the next nearest Ga 3 orbital (Ga2 s) is
almost 90% less then that of the nearest Ga 3 (Gal 3). The former gives ~0.058
states/eV/ atom compared to ~0.55 state/ eV/ atom for the latter. This indicates the
strongly localized nature of the defect state and is perhaps the reason why EMA
fails badly in this case. The main difference between Cd and Sn is that whereas Cd

contributes ~0.275 states / eV near the maximum of the impurity state, Sn contributes

71

 

 

 

 

 

Figure 3.9: Electronic charge density associated with the Sn impurity induced defect
band. The contribution to this defect state comes from Sn 3 and its NN Ga 3 and Se
1):; + py orbitals

~0.8 states / eV. This suggests that the Sn donor state is much more localized than the
Cd acceptor state, which is consistent with the experimental BEs (0.36 eV for Sn [34]
and 0.13, 0.18 eV for Cd [41]). In Fig. 3.9 we show the charge density associated with
the impurity-induced band. The defect state is indeed much more localized compared
to the Cd acceptor case and is predominantly formed by the Sn 5 orbital mixed with
the nearest neighbor Ga. 3. In contrast to the case of Cd defect however, the Sn .9 is
mixed with the nearest neighbor Se pa: + py orbitals rather than 1); orbital.

In the case of the donor BE, the natural choice of the energy reference for the defect
induced gap state would be the CBM. However, the band gap is considerably smaller
in the LDA/GGA calculations and in the case of Sn impurity in GaSe, the defect
energy level is not sufficiently deep and hybridizes with the bottom of the conduction
band. As a result it contains a strong admixture of the conduction band states. This

is shown in Fig. 3.10 where we plot the band structure of the Sn doped GaSe showing

72

 

 

 

Energy (eV)

 

 

 

 

 

 

 

 

 

Figure 3.10: Electronic band structure of GaSe with Sn impurity on the Ga site,
showing the Sn 3 orbital character. The impurity induced band lies in the CB;
therefore it is strongly hybridized with the CB states.

the Sn .9 orbital character. Due to the underestimation of the band gap, the lowest
conduction band dips below the defect level along the A-F—M direction and makes
the impurity band a resonant state rather than a bound state in the gap. It is therefore
not possible to extract the binding energy of the defect state associated with the Sn
impurity, from the LDA/GGA supercell calculations. Any improved calculation that
gives a better estimation of the band gap (i.e. nonlocal exchange models [159]) will

also affect the donor energy and hence can give an accurate estimation of the donor

BE.

73

Ga vacancy in GaSe (VGa)° The nature of the defect states introduced by the
Ga vacancy in GaSe is similar to that of Cd substitutional impurity and can be
related to the Ga 5 bonding-antibonding scheme. While in the Cd case, there is a
large mismatch in the energies of Ga 3 and Cd 3 orbitals, in the case of the vacancy,
the Ga dimer bonding and antibonding .9 states cannot appear because of the missing
Ga atom. As a result the s orbital of the Ga atom nearest to the vacancy will be
more like that of a Ga atom rather than that of a Ga dimer. Consequently, the
existing hybridization between the dimer bonding and antibonding 3 states with the
nearest Se p orbitals will be drastically affected. This can not only perturb the Se p
states in the valence band but also give rise to new states in the gap arising from the

conduction band which is also derived from the Se p states.

Figure 3.11 (a) shows the band structure of GaSe with Ga vacancy as calculated
for a 3x3x1 supercell (Ga35Se36). There are two major changes in the band structure
compared to the pure GaSe. First, the states near the top of the valence band are
strongly perturbed (bands denoted as D2 and D3,4) and second, a new state appears
in the gap ~O.66 eV above the perturbed valence band maximum, measured at the
F-point (denoted as D1). The Fermi energy cuts the valence band (hole doping).
Let us first discuss the parentage of the defect band D1, lying in the middle of the
gap. This band does not split off from the valence band (occupied states in the pure
GaSe). We confirmed this by counting the number of occupied states at the F—point.
Ignoring the Ga 3d and Se 43 as core states, in the pure GaSe there are 36 Ga 43
(18 bonding and 18 antibonding dimer states) and 90 Se 4p states. The rest 18 Se
4p states are split off from the valence band forming the bottom of the conduction
band. The occupied 126 states contain 252 electrons (108 from Ga and 144 from
Se). In the presence of a Ga vacancy (Ga358e36) we find that there are 35 Ga 43
(nominally 17 bonding, 17 antibonding and 1 non bonding) and 90 Se p states. These

125 states accommodate 249 electrons (105 from Ga and 144 from Se) resulting in

74

(a) (b)

 

\
x
\Sé
>2;

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A
>
a)
3 1./ \
h
8
_D
LU 2-
§D3,4'f~ v EF
-10 i D /$ / N
.l' KH AFMLAF KH AF Ml- A

Figure 3.11: (a) Electronic band structure of GaSe with Ga vacancy. There is a deep
defect state located in the band gap (D1) as well as a defect induced state at the top
of the VB (D2, D3’4). (b) The defect levels D1, D2 and D3,4 are shown along the
band structure of the pure GaSe
a partially filled valence band. The unoccupied defect state in the gap (D1) does
not come from the valence band but from the conduction band formed out of Se p
orbitals. To further analyze the nature of this gap state we plot the charge density
associated with it, in Fig. 3.12. We see that the defect state is quite localized and
just like in the case of Cd impurity most of the contribution to the charge density
comes from the nearest Ga .9 and neighboring Se pz orbitals with some admixture of
Se pm and py orbitals. It is actually an antibonding combination of Ga 43 and 4p and
10 states of neighboring three Se atoms.

We next discuss the vacancy induced changes in the states near the top of the

valence band. We have seen a large inward relaxation of Ga and Se atoms surround-

75

 

 

 

 

 

 

Figure 3.12: Electronic charge density associated with the Ga vacancy—induced defect
band D1. The contribution to the charge density comes from the Ga 5 and Se pz
orbitals from the vicinity of the vacancy.

ing the vacancy (see Section 3.4.1) and this should change the local bonding. In
Fig. 3.11(a) we see that there are three bands which appear to split off from the rest
of the valence band with a small overlap near the A-point. Two of these three bands
(D3,4) are relatively flat where as the third one (D2) has a large dispersion along l"—A
(z—direction). Band D2 is above the other two at the F-point. In fact the two flat
bands are dispersionless along F—A. Whether these three bands are impurity-induced
bands or simply perturbed host bands is an important question. To address this issue
we have done a 5x5x2 supercell calculation. Again due to large computation time we
have calculated energies at three symmetry points I‘, A, and M. We find that band
D2 flattens out considerably (width ~0.08 eV). The other two also flatten but appear
to overlap with folded perturbed host bands and it is not easy to see whether they
are split off from the rest of the valence band states or not. We will therefore focus

on the defect state D2, which splits off from the rest of the valence bands.

Following the method described for the Cd impurity and using the energies at the
F—point obtained in 3x3x1 supercell, we find two defect-induced energy levels in the

gap region: one acceptor level at ~0.26 eV (D2, which splits off from the valence

76

band and is partially occupied) and one at ~O.92 eV (D1, which drops down from the
conduction band and is unoccupied) above the VBM of the pure GaSe. This is visible
in Fig. 3.11 (b) where we plot the bands D1, D2, D3,4 along with the band structure of
pure GaSe. To plot these two band structures in a common energy scale, as discussed
before, we matched the energies of the Ga 3d semicore levels, calculated in the pure
and defect containing supercells. The acceptor level at 0.26 eV seems to compare
well with the hole trapping level measured by Micocci et al. [138] in nonintentionally
doped p—GaSe at 0.2 eV. Since the parentage of this level is the occupied valence
band state DFT calculation may be somewhat more reliable regarding the energy
position of the defect level. In addition to this level Micocci et al. [138] also found a
trapping level at ~0.8 eV above the valence band maximum. These defect levels were
tentatively assigned to Ga vacancy and to some extended defect region (stacking fault
or dislocation) respectively. Whether the defect level we find at 0.92 eV has something

to do with that found at 0.8 eV needs further investigation.

Se vacancy in GaSe (VSe)° In contrast to the Ga vacancy, the Se vacancy does
not remove the Ga 3 dimer states. So the physics of defect formation in this case is
quite different. In the presence of a Se vacancy three p states associated with the
Se atom which was occupying the vacancy site are removed. The net effect is (i) a
reduction in the number of Se p states along with the loss of 4 Se valence electrons
and (ii) the change in the vacancy’s nearest neighbor Ga dimer states and Ga p states.
Fig. 3.13(a) shows the electronic structure obtained for the 3x3x1 supercell of GaSe
containing one Se vacancy (Ga36Se35). There are two major changes in the band
structure compared to the pure GaSe. First, two defect levels (D1, D2) drop down
from the conduction band, one of them (D2) has a very large dispersion along the
F—M direction compared to the other. These two defect bands overlap with the

bottom of the conduction band at the F—point. Second, the top valence band splits

77

3.0 (a) (b)

 

            

\
\w
)l

  
 

ill»;
// 121'"

lit

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9 — __
3 017 E 7
a " ‘i’i
'- E
a: ~~o 4 F
3.
LE >§ ;\
> s5
K H A\1‘ M L A

Figure 3.13: (a) Electronic band structure of GaSe with Se vacancy. There are two
defect levels (D1 and D2) in the band gap (hybridized with the CB) as well as a
defect induced state (D3) at the top of the VB. (b) The three defect levels D1, D2
and D3 are shown along with the band structure of Pure GaSe, after matching the
Ga 3d semicore levels of the pure and vacancy containing GaSe. We have also shifted
the CB of GaSe upwards until the band gap reached the experimental value (~2.05
eV)

off (but with a small overlap) from the rest of the valence bands. We will denote it as
D3. Fermi energy is above D3 but below D1 and D2. To understand the parentage
of these vacancy-induced states we again do the band counting as we did for the Ga
vacancy. We find that removal of one Se removes 3 Se 1) states, two of which come
from the valence band and one from the conduction band. Since the removal of Se
takes away 4 electrons the valence band is filled and the Fermi energy lies above the
valence band. In order to understand the nature of D1, D2, and D3 further we replot
these bands along with the band structure of pure GaSe in Fig. 3.13(b) by matching

78

 

 

 

 

 

 

 

(a) (b)

Figure 3.14: (a) Electronic charge density associated with the defect bands D1 and
D2 showing the 3 character of the Ga atoms closest to the vacancy. (b) The charge
density associated with the band D3. This state extends to the vacancy’s next NN
atoms and shows Ga 3 and Se pz orbital character.

the Ga 3d semicore energies of pure GaSe and the one with Se vacancy. Also to
show the two levels D1 and D2 clearly we have shifted the conduction band of GaSe
upwards to match the experimental value of the gap (~2.05 eV). In Fig. 3.14(a) we
plot the charge densities associated with the bands D1 and D2; they are hybridized
Ga 3 states associated with the three nearest-neighbor Ga atoms of the Se vacancy.
In Fig. 3.14(b) we show the charge density associated with D3; this is not strictly
localized, but rather extends to the vacancy’s next nearest neighbor Se and Ga atoms.
The contribution to the charge density comes from p z and s orbitals of the next nearest
neighbor Se and Ga atoms of the Se vacancy, respectively.

According to the experimental result of Micocci et al. [34], Se vacancies produce
electron trapping centers with energy 520 meV below the conduction band minimum.
In Fig. 3.13, the deep defect states D1 and D2 at P-point are degenerate and lie
~1.1 eV above the valence band maximum. If we compare this energy with that
of the shifted conduction band minimum we find that these two defect states are
~0.95 eV below the conduction band minimum. It is tempting to identify these

two states with the electron trapping centers. But for a quantitative understanding

79

one needs to do a theory (going beyond local DFT) which not only treats the band
gap problem correctly but can be extended to the case of defect states. In general,
nonlocal exchange and correlation which appear to improve the band gap will also
affect the nature and position of the defect state, particularly for those defects which

form out of the conduction band states.

3.5 Summary

In summary we have carried out both hydrogenic effective mass (EMA) and ab ini-
tio electronic structure calculations within density functional theory using supercell
method for Cd and Sn impurities in GaSe. We have also studied the nature of defect
states introduced by Ga and Se vacancies For both donor (Sn) and acceptor (Cd) we
find that compared to experiment, the EMA results give lower values of BE (ranging
from a factor of 4to 2) suggesting large corrections and the necessity of incorporat-
ing short-range interactions to the EMA. This has been done using ab initio DFT
calculations using GGA. The general features of the Ga—site defect states (impurities
or vacancy) can be understood using the Ga dimer bonding and antibonding 3 states
and how they interact with the Se states. In the presence of substitutional atoms or
vacancies at the Ga sites these dimer states are drastically changed and give rise to
localized defect states. In the case of the acceptor impurities (Cd on Ga site and Ga
vacancy) we have calculated the energy levels of the defect states with respect to the
VBM (at the I‘ point) of pure GaSe after matching the energies of the electronic core
states. These results should be more reliable since LDA/GGA gives a correct descrip-
tion of the VBM. In fact, the acceptor BE values thus obtained are found to be in
reasonable agreement with experiment. In contrast, the position of the defect states
arising from donors (as in the case of Sn) or those split off from the conduction band

in the presence of vacancies (both Ga and Se) cannot be given reliably because of the

80

problems associated with the band gap and excited states calculated with LDA/GGA.
However by going to larger supercells one can distinguish between the defect states
from the perturbed host states and analyze the nature of these defect states such as
their charge densities, bonding, and localization etc. This has been successfully done

in the case of a variety of defects in GaSe.

81

Chapter 4

Electronic structure of isovalent

impurities in GaSe

4. 1 Introduction

As we discussed in the previous chapter, the layered GaSe crystal is a highly effi-
cient nonlinear optical material, with applications in second harmonic generation,
frequency mixing, generation and detection of terahertz radiation [3,131,160,161].
However, the mechanical properties (hardness and cleavability) of GaSe are unsatis-
factory: nearly zero hardness by Mohs scale1 and enhanced cleavability along planes
parallel to the atomic layers [162]. Furhermore the nonlinear properties are difficult
to reproduce from sample to sample [160]. All these properties hamper the use of
large GaSe crystals in practical applications. Improved crystal quality and second-
harmonic generation has been reported in GaSe doped with In [162—165]. Further

attempts were made to improve the optical, thermal and mechanical properties GaSe,

 

1The Mohs scale of mineral hardness characterizes the scratch resistance of various minerals
through the ability of a harder material to scratch a softer material. The scale is based on ten
minerals that are all readily available. As the hardest known naturally occurring substance when
the scale was designed (1812), diamonds are at top of the scale. For comparison, on the Mohs scale,
a pencil ”lead” (graphite) has a hardness of 1; a fingernail, 2.5; a copper penny, about 3.5; a knife
blade, 5.5; window glass, 5.5; and a steel file, 6.5.

82

by doping it with Ag and mixing it with AgGaSe2 [165].

Despite the considerable amount of experimental work, there is a lack of theoret-
ical approach to the subject which deals with the electronic, optical and mechanical
properties of GaSe doped with isovalent impurities (Te and In). In this chapter we
investigate the elastic properties of GaSe1_$Tez and Ga1_xlng;Se as a function of
the composition at, and examined the electronic structure of several point defects and

defect complexes associated with Te and In doping in GaSe.

4.2 Computational method

The results presented in this chapter have been obtained using the projector aug-
mented wave (PAW) [142,166] method, within density functional theory (DFT) as
implemented in the Vienna Ab-z’m’tz’o Simulation Package (VASP) [144—147]. The
exchange-correlation potential was approximated by the Ceperley—Adler local density
approximation (LDA) [167]. This exchange-correlation potential was chosen over the
gradient corrected version (GGA) because it is known that GGA underestimates the
binding energies, which results in an overestimation of the lattice parameters. As
we have seen in Chapter 3, since the interaction between the atomic layers of GaSe
is week, van der Waals type, the “GGA effect” becomes much more significant in
the direction perpendicular to the atomic layers, resulting in at theoretical structure
which is overly elongated in the direction of the crystallographic c—axis. This elon-
gated structure would be inappropriate for the theoretical investigation of the elastic
properties of GaSe.

In all calculations the outer s, p, d orbitals of the Ga and In atoms as well as the
s and p orbitals of the Se and Te were included in the valence states, while the rest
were treated as core states. The cut-off energy for the plane wave basis was set to

300 eV and the convergence of self-consistent cycles was assumed when the energy

83

difference between them was less than 10"4 eV.

To explore the effect of the isovalent impurities on the physical properties of
GaSe the dopant atoms were placed at several substitutional (TeSe, TeGa, Inga) and
interstitial (Tei, Ini) siteslinside the host matrix. In addition, we have also examined
the electronic structure and defect formation energies associated with substitutional
indium - gallium vacancy complex (InGa'VGal-

Before discussing the details of electronic structure and elastic stiffness calcula-
tions, we briefly review the elastic constants of GaSe, which has hexagonal symmetry.
A crystal with hexagonal symmetry is characterized by 5 elastic constants: C11, C12,
013, C33 and C44. We have determined linear combinations of these quantities from
total energy calculations for five different strain configurations [168]. When the lattice

is distorted by a small strain, the lattice vectors change according to:

a' = (I + e) a (4.1)

where a and a’ are matrices that contain the components of the old and new lattice
vectors, I is the identity matrix and e is the strain matrix, which has the form:

1 1
81 286 285
286 e2 %84 (4.2)

 

 

_ 285 %e4 e3 _
The specific strain configurations along with the corresponding energy densities used
to determine the elastic moduli of the hexagonal GaSe1_$Te$ and Ga1_xInxSe are

listed in Table 4.1.

In order to calculate of the elastic constants of GaSe1_$Tex (:L' = 0, 0.0625,
0.25) and Ga1__xlnxSe (a: = 0, 0.0625, 0.25, 1) we have computed the theoretical

crystal structures by minimizing the total energies with respect to the lattice constants

84

Table 4.1: Strains and elastic moduli for crystals with hexagonal symmetry. AB is
the change in energy due to the specific strain and V0 is the equilibrium unit cell

mime-

 

 

Strain configuration (unlisted e; = 0) Energy density (AE/VO)

e1 = 82 = 5 (011+Clzl52

el = 62 = —2e3 = 5 (011+C12—4C13 + 2C33)52
e, = 5 1/2x(033> .52

‘36 = 5 1/4><(011-012) 52

e4 = e5 = 6 (C44) 62

 

(at each concentration 3:): first with respect to the volume of the unit cell keeping
the c/ a ratio fixed and then with respect to c/ a keeping the previously obtained
equilibrium volume constant. The elastic constants of GaSe1_$Tex (a: = 0, 0.25) and
Ga1_$lnxSe (as = 0, 0.25, 1) were obtained using small unit cells (8 atoms/ cell) and
the Brillouin zone (BZ) was sampled by a F-centerd 12x12x3 k—mesh. In the case of
x = 0.0625 the calculations were performed on 2x2x1 supercells with the BZ sampled
by a 6x6x3 grid of k-points.

The electronic structure and defect formation energies were calculated using 3x3x1
supercells with the theoretical lattice constants of GaSe and the integration of the
BZ was carried out on a F-centered, 4x4x3 set of k-points. The 3x3x1 supercells
containing one impurity correspond to a composition of cc 2 0.028. For all the
calculations described in this chapter the internal structural parameters were fully

relaxed until the Hellmann-Feynman forces were less than 0.02 eV / A.

4.3 Theoretical crystal Structure

The calculated equilibrium lattice parameters of GaSe1_$Tex and Ga1_xlna;Se are
listed in Table 4.2, along with other available theoretical [25,26] and experimen-
tal [140] data. We observe a monotonic increase in the lattice constants as the con-

centration of the impurities (rt-value) increases. This is not surprising because the

85

Table 4.2: Optimized theoretical lattice parameters of GaSe1_xTex and Ga1_mln$Se.

(“theory; bexperiment)

 

 

 

 

 

 

 

Compound 0: a(A) C(A) c/a dGa—Ga/C dSe—Se/C
Pres. calc. 3.715 15.77 4.244 0.153 0.300
0 Ref. [25]a 3.724 15.68 4.21 0.150 0.350
Ref. [26]a 3.720 15.62 4.199 0.154 0.302
GaSe1_zTee Ref. [140]b 3.755 15.94 4.245
90625 Te: 3:223 13:32 2:33
0.25 Tese 3.301 15.99 4.207
3:33 1:2; :92
Gal-xlnxse 0.25 111G, 3.773 15.93 4.233
Pres. Calc. 3.972 16.49 4.151
Ref. [1401b 4.005 16.64 4.155

 

sizes of the dopant atoms (In and Te) are larger compared to the host atoms. In
the case of the end compounds GaSe and InSe the theoretical lattice constants are
less than 3% smaller compared to experiment, while the c / a ratios are within 0.7%
of the experimental values. The underestimation of the lattice parameters is due to
the well-known overbinding effect of LDA. It has been shown that in GaSel_$Tex a
phase transition takes place from hexagonal (GaSe) to monoclinic (GaTe) structure
in the composition range 0.26 < r < 0.60 [169]. Thus in the case of GaSe1_$Tex
we have limited our calculations to the maximum value of a: = 0.25, because the
comparison between the elastic constants of the monoclinic and hexagonal lattices

not quite meaningful.

86

4.4 Elastic properties of GaSe1_,,Tex and
Ga1_xInxSe

To determine the elastic constants, we have calculated the total energies of the
strained GaSel_xTe$ (:1: = 0, 0.0625, 0.25) and Ga1_$In$Se (a; = 0, 0.0625, 0.25, 1)
crystals for (5 between —0.03 and 0.03 and fit the results to a second order polyno-
mial. The calculated elastic constants are given in Table 4.3, along with previously
calculated theoretical [25,26] results and experimental [170,171] data. The earlier
theoretical calculations were performed using norm conserving pseudopotential and
plane wave (PW) basis set [25] and full-potential augmented plane-wave method with
local orbitals (APW+lo) [26]. Our calculated values of the elastic moduli of GaSe
and InSe using the PAW method are in overall good agreement with the available

experimental values and earlier theoretical calculations.

In the case of the Te doping we do not observe a systematic change in the elastic
constants with the dopant concentration. For example C11 increases by ~0.3% as
concentration goes from a: = 0 to :6 = 0.0625 but when the dopant concentration is
further increased (:1: = 0.25), the elastic modulus C11 decreases by more than 6%
compared to the original value. Similar changes are observed for all other elastic
moduli, except for C44, which behaves in an opposite way: first it decreases and then
increases as the dopant concentration grows. The unsystematic changes in the elastic
constants of GaSe1_mTeg; might be related to the structural phase transition from
hexagonal to monoclinic structure, which takes place in the compositions between
0.26 < a: < 0.60 [169]. At the concentration of at a: = 0.25, which is close to the lower
end of the transition range, the hexagonal phase might be unstable with respect to
the monoclinic one, thus the elastic constants obtained at this composition might not

be realistic.

In the case of substitutional In impurity on Ga site, we observe a systematic

87

Table 4.3: Elastic contants of GaSe1_$Tex and Ga1_$ln$Se. All values are given in

GPa. (“theory; bexperiment)

 

 

 

 

 

 

 

X C11 C12 C13 C33 C44 C11+Clg C1143;
Pres. calc. 100.9 27.0 9.7 33.9 8.3 127.9 73.8
5% Ref. [25](1 12.4 35.4 130.2
9;, 0 Ref. [26]a 13.4 34.4 127.9
5" Ref. [1701b 12.2 35.7 132.5
:34 Ref. [1711b 105.0 32.4 12.6 35.1 10.4 137.4 72.6
0.0625 TeSe 101.2 27.3 10.5 34.3 8.2 128.5 73.9
0.25 Tese 94.5 25.3 10.5 32.2 11.5 119.8 69.2
(a 0.0625 InGa 99.2 27.1 10.5 34.7 8.8 126.3 72.1
58 0.25 InGa 91.3 25.6 11.4 36.1 9.7 116.9 65.7
5'. 1 Pres. calc. 70.3 23.5 14.2 38.5 11.5 93.9 46.9
0 Ref. [1711b 73.0 27.0 36.0 100.0 46.0

 

change in the elastic constants with defect concentration as shown in Fig. 4.1. The
changes in the elastic moduli of Ga1_$lnxSe depend linearly on the concentration of
In. On the one hand, we observe a monotonic decrease in C11 and C12 as the z-value
increases, but on the other hand, the elastic constants C13, C33 and C44 reveal steady
enhancements with the increase in dopant concentration. This indicates that when
In impurity occupies Ga sites, the crystal becomes softer in the a— and b-directions
(parallel to the atomic layers) and stiffer along the c-axis (perpendicular to the atomic

layers) .

The “Vegard—law type” behaviour [172] of substitutional In impurity on the elastic
properties of GaSe can be understood if we examine the connection between structural
and electronic properties of the end-compounds: GaSe and InSe. The elasticity of the
layered Ga1_xln$Se in the c—direction is determined predominantly by the interaction
between the atomic layers. The electronic structure of the III-VI layered compounds
is discussed in detail in several earlier reports (e.g. Ref. [173]); however for an easier

comparison between GaSe and InSe we also show the calculated band structures in

88

 

 

A—l
38888

 

 

 

A '~I\-\ Ga1_xlnxSe
E C11
9 \-
§ 35; “P’s/033M?
g 303
=8 25‘. Wme-i
8 2°" '.
m 15-: C13__'_fl____._.——A j
1°: W C444 '.
5 l I '

 

 

 

' l l

0.0 ' 0.2 ' 0:4 ' 0:6 0.8 1.0
xvalue

Figure 4.1: The elastic constants of Ga1_$lna,-Se depend linearly on the composition.
The increase in C13, C33 and C44 suggests the strengthening of GaSe in the direction
perpendicular to the atomic layers as the concentration of In increases

Fig. 4.2. First let us analyze the top of the valence band (VB), which is mainly of
Se pz character. Since the Se atoms are facing the interlayer region, we expect that
the dispersion of the top VB along the F—A direction to be sensitive to the interlayer
interaction. Indeed, according to our calculated band structure, the F—A dispersion
of the top VB increases by ~40 meV (see Fig. 4.2), in going from GaSe to InSe,
indicating that the interlayer interaction is stronger in InSe compared to GaSe. This
is consistent with the smaller interlayer separation between Se-monoatomic sheets in
InSe (2.96 A) than in GaSe (3.15 A). Thus, when the concentration of In substitutional
impurities increases, the interlayer Se—Se distance becomes smaller, the interaction
becomes stronger, and as a consequence, the crystal becomes stiffer in the direction

perpendicular to the atomic layers.

The softening of Ga1_xlnxSe in the a- and b—direction with the increase in the

89

 

 

2'5: Gase : A InSe

Energy (eV)

 

 

 

 

 

 

 

 

/
M

 

 

Figure 4.2: The calculated band structures of GaSe and InSe. A—I‘ and l"—M direc-
tions are perpendicular and parallel to the atomic layers respectively.

composition as, can also be easily understood, because the intralayer distances are
longer and therefore the intralayer covalent bonds are weaker in InSe than in GaSe
[22,24]. Inspecting Fig. 4.2 we observe that indeed, the lower lying valence bands of
InSe, with predominantly Se pa; and py character, disperse less in the P—M direction
compared to the corresponding bands of the GaSe. Given that in the e and b-
directions there are no “interlayer regions” which could counteract the weakening
of the atomic bonds, the crystal becomes softer as the In concentration increases.
Although the substitutional In impurity seems to enhance the elastic properties of
GaSe along the c-axis, the effect is rather small (e.g. C33 increases by 7.8% from GaSe
to InSe). So, we we have to look for different mechanisms for In induced interlayer

rigidity enhancement as seen experimentally [162—165]. This aspect will be discussed

90

later, in Section 4.8. But before this we discuss the formation energies of different

defects to find out which defect is more likely to be present in GaSe.

4.5 Formation energies and electronic structure of

defects

In order to identify the preferred location of the impurities inside the GaSe matrix,
we calculate the formation energies of the Te and In defects located at different
substitutional and interstitial sites. The formation energies and charge transition
levels are calculated using the formalism described Section 2.4.2.

As pointed out in Section 2.4.4, an important parameter in supercell calculations
is the position of VBM, which is usually considered to be the reference for the electron
chemical potential. In the present calculations EVBM was determined as the average
of the one-electron energy level of the VBM over the k-points where the total energy
was calculated. As pointed out S. B. Zhang [128] this approach has the advantage that
transition levels calculated this way (from total energy differenes) are consistent with
the single-particle energy levels. Furthermore, the band gap of GaSe calculated with
the “average band-edge” approach (Egg/gage = 1.68 eV), is closer to the experimental
value (Egg = 2.13 eV) than the direct gap located at P—point (Eggp = 0.85 eV).
The transition levels associated with the various defects in GaSe are calculated using
Eq. (2.53).

Equation (2.52) shows that the formation energies of the defects depend on the
chemical potential (14) of the constituents as well as on the charge state (q) of the
defect. The values achievable by the chemical potentials are limited by several
conditions (see Section 2.4.3): (a) to avoid precipitations, pi’s must be negative
and (b) to maintain a stable host compound, the chemical potential must satisfy

”Ga + #Se = AH(GaSe), where AH(GaSe) is the formation enthalpy of GaSe. Our

91

VBM CBM CBM (exp)

 

 

 

 

A 3.61 Se rich I“

a, 3.0- (2)/ Tai;

V ‘ +

> 2.4- \

g 131/ ('2)

5 ' 1/—\ \
5 1'2? (+1) T663

:1 O'Gi/l’__(o)—T°Se \

(U

8 0-0‘3 \§\(-2)

‘L -0.6§ §8(+2/+1)=0.08 eV \‘

 

0.0 ' 015 Y 1T0 1.5 2.0

 

 

 

 

 

 

 

EF(9V)
(a)
2 g/BM CBM CBM (exp)
' . Ga rich]
E 2'()‘-(0)—\ “if—(0)“
E 1.5. m/
.‘ +1
g 1.0. /( ) InGa-VGa .
e 0.5 (+3) \\
:8 ‘ ' InGa—(O) \
E 0-0‘7: : (-2)
53-05- 5 Ee(+1/0)=0.28ev \
-1.0 1.! . . , . , .
0.0! 0.5 1.0 1.5 2.0
0.135

EF (8V)
0))

Figure 4.3: (a) Formation energies of Te impurity in GaSe. For all values of E F within
the band gap, Te prefers to occupy the Ga site. (b) Formation energies of In impurity
in GaSe. When E F is very close to VBM, the In impurity becomes positively charged
(+3) and moves to the interstitial site.

theoretical calculation gives AH(GaSe) = —1.12 eV. In order to avoid secondary
phase formation between the host elements and impurities, we can impose some fur-
ther conditions on the chemical potentials of the defects as described in Section 2.4.3.

However, the effect of these conditions would be a constant shift in the formation ener-

92

Table 4.4: Formation energies of neutral defects and the charge transition levels. All
values are given in eV.

 

Defect A E (X0) Trans1tlon levels (measured from the VBM)

5(+3/+2) €(+2/+1) 5(+1/0) s(0/-1) 5(-1/-2) e(-2/-3)

 

Tese 0.65 0.08 0.34 1.45 1.62

TeGa 2.07 0.13 0.45 0.72 1.11

Tei 3.58 1.04 0.70 1.01 1.28

Inca 0.28 —0.10 —0.02 0.28

Ini 1.89 0.34 0.41 1.41

Inca-VG, 1.76 0.39 1.71 2.10

 

gies and since we are only interested in the relative formation energies associated with
the same dopant located at different lattice sites (i.e. we do not compare In induced
defects to Te induced ones), we can consider for all cases “def = ”Te = 1411, = 0.
The calculated formation energies are represented as a function of E F in Fig. 4.3 (a)
and (b) for Te and In impurities, respectively In these figures both the theoretical
(E;heor = 1.68 eV) and experimental (E;Xp = 2.13 eV) band gaps are indicated. The
calculated defect transition levels 5 (q / q’ ) and formation energies AE (X q, q = 0) are
listed in Table 4.4.

4.6 Te induced defects

For the case of Te doping we have considered the configurations with Te located at Se
and Ga sites as well as at the interstitial site. The interstitial configuration with the
lowest energy was found when the impurity atom was located midway between the
Ga-Se-Se-Ga layers at equal distances from the 6 nearest neighbor (NN) Se atoms.
The formation energies of the Te induced defects, under Se-rich growth conditions
(“Se = 0, ”Ga = —1.12 eV) are represented in Fig. 4.3(a). The lowest formation
energy occurs when the Te atom is located at Se site (TeSe). In neutral charge

state the calculated formation energy of TeSe is AHf(Tege) = 0.65 eV. When the

93

Fermi energy (E F) is closer to the valence band (VB)(p—type sample), TeSe becomes
positively charged, and gives rise to two charge transition levels located at €(+2/ +
1) = 0.08 eV and €(+1/0) = 0.34 eV above the VBM. Photoluminescence and Hall
effect measurements have shown that the carrier transport in p—GaSe doped with Te,
is dominated by two acceptor levels at 0.08 and 0.02 eV above the VB [32,39]. Our
calculated €(+2/ + 1) transition level is in excellent agreement one of the experimental
values. When the Fermi energy (E F) is close to the conduction band (CB)(n—type
conductivity), TeSe becomes negatively charged, the transition levels being located

at 5(0/—1) = 1.45 eV and €(-1/-2) = 1.62 eV above the VBM.

The formation energies of the substitutional Te atom at the Ga site (TeGa) is
slightly higher compared to Tese. However, when the defect is —2 charge state
(Teai), the difference between the formation energies AHf (Tegez) and AHf (T963)
becomes small (~ 0.18 eV) suggesting that under Se—rich growth conditions, Te atoms
can fill up the Ga vacancies. Under Ga-rich conditions, according to Eq. (2.52), the
formation energy of TeGa defect is shifted up by 1.12 eV, making it less likely for Te
to occupy Ga site. This is not surprising because under Ga—rich conditions, there are

less Ga vacancies available.

The interstitial Te defect (Tei) has the highest formation energy therefore it is
less likely to occur. As shown in Fig. 4.3(a), the formation energy of the defect in
neutral charge state AHf(Te?) is almost 3 eV higher compared to AHf (Tege), and
the difference becomes somewhat smaller when Tei is in doubly charged negative
(Tei—z) or positive (Tei'l'2) state. However, according to recent experimental results
of Evtodiev et al. [32] at high Te doping concentration, part of the Te atoms localize
in the interstitial sites within the interlayer region. Therefore we cannot exclude the
possibility that the Te atoms can occupy interstitial sites, which clearly would affect
the cleavability of the GaSe crystal. It is also interesting to note that the Te related

defects can behave as either donor or acceptor, depending on the position of E F

94

relative to the band edges.

4.7 In induced defects

We have studied three types of In induced defects, In substituting for Ga (Inga),
interstitial In (Ini), and substitutional In - Ga vacancy complex (InGa — VGa). In
Fig. 4.3(b) we show the calculated formation energies as a function of E F for In
induced defects in GaSe, in different charge states. We find that the formation energy
of lug;a is 0.28 eV and there is one charge transition level associated with this defect:
e(+1/0) = 0.28 eV above the VBM. This value is in fairly good agreement with
the acceptor level at 0.21 eV, measured by Cui et al. using deep level transient
spectroscopy (DLTS) [31]. This, along with the result for TeSe discussed earlier,
gives us confidence in our total energy calculations using DFT and the supercell
model to understand the defect physics.

The defect states with lowest formation energies are: In?+ for EVBM(= 06V) 3
EF 3 0.135 eV, Ina; for 0.135 eV 3 EF 3 0.28 eV and Inaa for 0.28 eV 3 EF- Ing}3L
is the most stable defect for a wide range of E F- However when the Fermi energy is
tuned towards the VBM energy, In;3+ defect becomes most stable. We will discuss the
underlying physics of this change by examining the single particle density of states, in
the next paragraph. As regards the effect of Inga,a defect on the electronic structure,
we find that the band structures near VBM and CBM are affected very little. One
therefore does not expect much change in the transport properties in In doped GaSe
if the impurity goes to a Ga site in the neutral charge state. One can understand
this lack of significant change by looking at the In 53-Ga4s dimer antibonding state
(which hybridizes with the Se p bands to give rise to states in the neighborhood of
the band gap) and observe that it is not significantly different from the Ga 4s-Ga 43

dimer antibonding state (see Section 3.3.2).

95

+3

DOS (states/eV)

 

-10 -8 -6 -4 -2
Energy (ev)

Figure 4.4: The total DOS of GaSe with In; and the projected DOS of the In s—orbital,
showing the positions of the HDDS (-5.5 eV) and DDS (just above EF) introduced
by the charged Ini defect.

To understand why In]3+ has the lowest formation energy when E F g 0.135 eV,
we look at the electronic structure, the single particle density of states (DOS) and
the nature of defect state introduced by Ini. Figure 4.4 gives the total and partial
(associated with In 3) DOS for this case. We see that In; introduces a hyper deep
defect state (HDDS) near the bottom of the Se p bands (at ~ —5.5eV). It is a
bonding state formed out of In 53 and neighboring Se p states. The corresponding
antibonding state splits off from the Se p valence band states and is denoted as the
deep defect state (DDS). The charge density distribution associated with the DDS is
represented in Fig. 4.5 showing this state is indeed an antibonding combination of the
In s and the surrounding Se p orbitals. This picture is very close to what happens
when In is a substitutional defect in PbTe [97]. The strong mixing between In 58
and the neighboring Se p states leads to the removal of one state (per spin) from the

Se p band which becomes the DDS. In terms of electron counting, two of the three

96

 

Figure 4.5: Charge density distribution associated with the localized band introduced
by Ini in the band gap of GaSe. The dominant contribution comes from the In 5
orbital, hybridized with the NN Se pz orbitals.

electrons from In occupy the HDDS and the three electrons (two from the electrons
occupying the valence band in pure GaSe and one from In) fill the DDS and partially
occupy the conduction band. Thus Ini acts like a donor. Since the three electrons
occupy states with energies larger than EVBM, clearly the formation energr of Ini in
charge state q = 0, 1, and 2 are higher than InGa for which neither the band structure
nor the electron count change. By removing three electrons from Ini to obtain q = 3
charge state we can lower its formation energy2.

Figure 4.3(b) also shows the calculated defect formation energy associated with
the substitutional In—Ga vacancy complex (InGa — VGa)- In the neutral charge state,
the formation energy of this defect complex is relatively high. However, as the position
of the E F moves up in energy across the band gap, InGa — VGa becomes negatively
charged and its formation energy decreases considerably. Within the theoretical band
gap we find one transition level associated with this defect complex, located at 5(0/ —

1) = 0.40 eV. This is in fact in good agreement with the acceptor level at 0.44 eV,

 

2The formation energy of neutral In interstitial defect is underestimated in our calculation since
on electron / defect occupies the bottom of the conduction band and the LDA gap is 0.85 eV compared
to the average band gap of 1.68 eV and the experimental value of 2.13 eV.

97

'“Ga'VGa

 

9 -1

% 30 ('"Ga'VGa)

9 0

cu

*-' 60 -2

3 30 ('"Ga'VGa)

O

D (In -v )‘3
30 Ga Ga
0
-10-8-6-4-20246

Energy (eV)

Figure 4.6: The total density of states associated with InGa — VGa defect complex,
for diflerent charge states. When the defect complex is triply negatively charged, the
defect state is located in the gap (lower panel).

measured using DLTS and assigned to InGa — VGa’ by Y. Cui et al. [31]. As shown
in Fig. 4.3(b) the other two calculated transition levels of InGa — VGa. are located
above the theoretical CBM, but within the experimental gap. The values are also
listed in Table 4.4.

To obtain a better understanding of the nature of the defect states introduced
by InGa — VGa) we calculated the total DOS associated with this defect complex in
different charge states. As shown in Fig. 4.6, the DOS reveals an interesting feature
which is only present for the —3 charge state: a localized level appears in the gap
(indicated by the arrow). The origin of this defect state can be understood from a
careful analysis of the relationship between the ionic relaxation and the bonding of In
and its NN Se atoms. From the charge density distribution shown in Fig. 4.7 we see
that the defect state in fact corresponds to the antibonding combination of In 3 and its

NN Se p2 orbital, the dominant contribution coming from the In .9 orbital. To locate

98

 

Figure 4.7: Charge density distribution associated with the localized band introduced
by InGa — VGa in the band gap of GaSe. The dominant contribution comes from the
In 3 orbital, hybridized with the NN Se p2 orbitals.

the corresponding bonding combination, we have calculated at the partial density
of states associated with the In 5 orbital. This is represented in Fig. 4.8 where we
have plotted the In 3 partial DOS corresponding to the neutral and —3 charge states
of InGa — VGa- The origin of the energy scale was chosen at the highest occupied
energy state. We notice that the splitting between the bonding and antibonding levels
decrease by ~1.5 eV as the defect complex becomes triply negatively charged. To pin
down the cause of this energy shift, we have examined the differences between the
ionic relaxations of neutral and triply charged systems. We found that independently
of the charge state, the In atom prefers to occupy the position located at the center
of the Ga-Se-Se-Ga atomic layer, at equal distances from the 6 NN Se atoms (see
Fig. 4.7). However, the distances between In and its NN Se atoms increase as more
negative charge is localized at the defect center. This is due to the increasingly
stronger Coulomb repulsion between the In ion and the neighboring Se anions. We
find that the average In—Se distance increases by ~0.34 A as the state of the defect
center changes from neutral to the triply negatively charged. The splitting between

the In 3 — Se p2 bonding and antibonding states becomes smaller as the separation

99

 

 

g — In s
: neutral state

 

Ins-Sepz Ins-Sepz

‘bonding‘ ] antibonding
' l r I ' 1‘" I Afi‘I/fifr l *

 

O-‘ngm
1 l 14.

 

 

 

DOS (states/eV)
N

 

 

 

 

—lns I
. (~3chargestate)
1..
i
0....4..A:-.A..~,4...
-10-8-6-4-20246

Energy (eV)

Figure 4.8: The DOS projected on the In s orbital for the neutral and -3 charge state of
the Inc,a — VGa defect complex. The splitting between the bonding and antibonding
states decreases mainly because the distance between the In and Se atoms increases.

between the atoms becomes larger. Consequently, at —3 charge state (when the In—Se
distance is the largest), the energy of the antibonding state becomes small enough
such that it appears to be located in the band gap of GaSe. For all the other charge
states the In 3 — Se pz antibonding combination is resonant in the CB. One of the
important predictions of our calculation is the presence of a deep defect state just
above the valence band maximum when In is present as an interstitial defect (see

Fig 4.4). Deep level spectroscopy should be able to see this defect.

4.8 Rigidity enhancement of GaSe by In doping

As we have seen in Section 4.4 the elastic constants do not change appreciably in
In doped GaSe when In goes as a substitutional impurity. This is also seen in the

calculations of energy barrier associated with relative shearing of two atomic blocks

100

 

I V I T I V r 1

'_.'_e;.se -

_A

1

I l _A

 

l

\

 

E-Eo(eV)
O—KNOO 150105 \I

A l

A

A .4I:.-'=I=l-- A ‘

a J n

 

 

 

-0.2' 010 ' 0T2 ' of 016' oisf 171‘ 1.2
d/a

Figure 4.9: The energy barrier which must be overcome in order to cleave the GaSe
crystal increases dramatically when In occupies the interstitial site compared to the
case when In occupies substitutional site. For comparison the case of pure GaSe is
also shown. The asymmetry of the energy barrier associated with InGa is due to the
geometry of the 2x2x1 supercells considered in the calculations: while in the case of
the pure GaSe and Ini the atomic configurations are symmetrical with respect to the
relative displacement of the atomic layers by d / a = 0.5, in the case of Inc;a the atomic
configuration is only symmetrical with respect to the displacement of d/ a = 1.0.

(each block being made up from 4-atomic planes) in a unit cell. Fig. 4.9 compares
the energy barriers involved in this relative shearing. The energy barriers for both
pure GaSe and Ga1_mIn$Se are very small and comparable. Thus substitutional In
does not enhance the shear rigidity of GaSe. In the same figure we show the energy
barrier associated with similar shearing in the presence of an interstitial charged In
defect (In?) The energy barrier and the initial slope increase dramatically (by
factors of ~10 and ~7 respectively) in the presence of hi”. Clearly GaSe is very
soft and inserting interlayer charged In defects can make the crystal rigid against
shear distortion. We note that for certain shear configurations the interstitial In

could not be accommodated by atomic relaxation within the 2x2x1 supercell. Thus

101

for consistency, we have given and compared the results obtained for rigid shear only
(without relaxation) for all three cases: Ini, InGa and pure GaSe. In order to perform
the atomic relaxations, one should either increase the size of the supercell or allow
for volume optimization.

The mechanism responsible for rigidity enhancement discussed here is quite gen-

eral and applicable to a large class of layered materials with weak interlayer bonding.

4.9 Impurity clustering

We have also investigated the possibility of In and Te cluster formation inside the
GaSe host. This was done by performing supercell (3x3x1, 72 atoms) calculations
with impurities located close and far away from each other and comparing the cor-
responding total energies. These calculations were performed using the theoretical
lattice constants of GaSe and relaxing all the internal atomic positions. In the case of
Te doping we find that the total energy is 40 meV/supercell lower when the impurities
are located far from each other, suggesting that Te clustering does not take place. In
the case of In doping, the situation is similar, but the energy difference is smaller: 9
meV/supercell. Considering that the accuracy of our total energy calculations is less

than 10 meV, we cannot exclude the possibility of In clustering in GaSe.

4.10 Summary

We have investigated the dependence of elastic properties of GaSe1_xTeg; and Ga1_$lnxSe
as a function of defect concentration, using first-principles methods, within DFT. In
the case of Te doping we did not observe a systematic change in the elastic con-
stants with the dopant concentration. This is most likely related to the structural

phase transition which occurs in the range of 0.26 < :1: < 0.60. In the case of sub-

102

stitutional In doping (Inga) we find a monotonic increase in the C13, C33 and C44
elastic constants which indicate a strengthening of the crystal in along the direction
perpendicular to the atomic layers.

The defect formation energy calculations show that Te and In prefer the substi-
tutional Se and Ga sites, respectively. Nevertheless, in the case p—type GaSe (when
E F is close to VBM) indium impurity can acquire +3 charge state and can occupy
interstitial sites between the GaSe layers. This strongly influences the cleavability of
the crystal along planes parallel to the atomic layers and it is the main source of the
observed improvement of the structural properties of In doped GaSe [162—165].

We find that in the case of InGa —- VGa defect complex the atomic relaxation plays
a major role in the stabilization of the charge states. The variations in the distance
between the host Se atoms and the In impurity as a function of the charge states,
are responsible for the position of the defect level relative to the band edges of GaSe.

One should be able to probe this defect using experimental methods.

103

Chapter 5

Electronic structure of defects in

GaTe

5.1 Introduction

One of the main characteristics of the III-VI systems is the presence of cation-cation
dimers oriented perpendicular to the layers. GaTe occupies a special place among the
members the III-VI family, since its crystal structure is more complex compared to
the other members [174]. In this compound there are Te—Ga—Ga—Te layers and Ga—
Ga dimers, similar to the rest of the family, but only two-thirds of these dimers are
oriented perpendicular to the atomic layers while the rest one-third the Ga—Ga bonds
lie almost in the layer plane (see Fig. 5.1). GaTe belongs to the B2 / In space group and
crystallizes in the monoclinic system [175]. Although the overall crystal structure of
GaTe appears to be more complicated, the local coordination of the atoms is similar
to that of the other III-VI compounds: each Ga atom is fourfold coordinated by three
Te and one Ga atom, while each Te atom is threefold coordinated by three Ga atoms.
The bonds within the layers have a strong covalent character while the layers are

bound mainly by weak van der Waals type interaction. Unlike GaS, GaSe and InSe,

104

 

Figure 5.1: The crystal structure of GaTe. Dark spheres are the Ga atoms and the
light spheres are Te atoms. In each layer one can see the Ga—Ga dimers two oriented
normal to the layer and one oriented along the layer.

no polytypism has been observed in GaTe [28,176]. Whether this has to do with the
presence and orientation of Ga—Ga (In-In) dimers in the layer is not known at the

present time.

Experimental investigations in GaTe have focused on the structural, optical, and
photo emission properties associated with its layered structure [27,177—181]. Schwartz
et al. [182] have observed that GaTe undergoes a structural phase transition at 10
GPa into a metallic NaCl—type structure. Pellicer-Porres et al. [183] used x-ray ab-
sorption spectroscopy (XAS) to study the evolution of the bond lengths in GaTe
under pressure. The same authors [184] observed a nonlinear pressure dependence of
the direct band gap in their optical absorption measurements. Extensive transport
measurements have been carried out in p—type GaTe by Efeoglu et al. [185]. Exper-
iments dealing specifically with defect energy levels will be discussed along with our

theoretical calculations.

Because of its more complex structure, there is a lack of detailed theoretical in-

105

vestigation of the electronic properties of GaTe. Yamamoto et al. [28] calculated the
electronic structure of GaTe using ab initio tight binding linear muffin-tin orbitals
(TB-LMTO) method within the atomic sphere approximation (ASA). For the ex-
change and correlation potential they have used the von Barth-Hedin local density
approximation (LDA). From a comparison of their theoretical band structure to opti-
cal absorption spectra they concluded that the dominant excitations were associated

with j-j coupled, optically allowed exciton states.

A more detailed analysis of the GaTe band structure was done by Sénchez-Royo et
al. [27], using numerical atomic orbitals and density functional theory (NAO-DFT), in
the local density approximation. The dispersion of the valence bands along different
directions in the Brillouin zone (BZ) was compared with angle-resolved photoemission
spectroscopy (APRES) measurement. Although there were qualitative agreements
between theory and experiment as regards band dispersion, there were also major
differences. To explain this difference Sénchez—Royo et al. [27] suggested that spin-
orbit interaction (SOI) might increase the mixing between the Te pz states from
the top of the valence band (VB) and the deeper Te px, py states, leading to an
upward shift and flattening of the topmost VB near the F—point. We have carried
out electronic structure calculations incorporating spin-orbit interaction to test their

suggestion and the results are discussed later in this chapter.

In general, electronic transport and dominant optical properties of a semiconduc-
tor are determined mainly by the electronic states near band gap region. These states
are easily influenced by the defects present in the system. The defects therefore play
an important role in the performance of a semiconductor. In this chapter we investi—
gate the nature of the defect states in GaTe associated with native point defects (Ga
and Te vacancies) and substitutional impurities (Ge and Sn) on Ga—site. As in other
systems, all calculations have been done within ab initio density functional theory

(DFT) using generalized gradient approximation (GGA). The defects are placed in

106

large super cells to minimize defect-defect interactions. It is well known that due to
the underestimation of the band gap by LDA or GGA and due to the limitations of
the supercell model [139], accurate calculation of the defect energy levels represents
a serious theoretical challenge, particularly when they are formed out of conduction
band states (donor states) or they overlap with conduction band (deep defect states
overlapping the conduction band). However, in the case of acceptor states, derived
mostly from the valence band, their position relative to the valence band maximum
(VBM) can be obtained with reasonable accuracy using supercell models provided
proper corrections are made to take care of the effect of the defects on the host va-
lence band structure. In this work we have estimated the energy of the defect levels
when these are derived mainly from the valence band or when they are deep defect
states lying in the band gap. We have also calculated the formation energies of defects
in different charge states and discuss the dependence of the formation energies on the

chemical potentials of the constituent elements under equilibrium growth conditions.

5.2 Computational details

We have performed structural optimizations and electronic structure calculations us-
ing Projector Augmented Wave (PAW) [149] methods implemented through VASP [144—
147] package. The exchange and correlation potential was approximated by gener-
alized gradient approximation (GGA) [148]. The 3d, 43 and 4p states of Ga and 53
and 5p states of Te were treated as valence states and the rest as cores. The energy
cutoff was set to 300 eV and convergence was assumed, when the energy difference
between consecutive cycles was less than 10‘4 eV. The internal structural parame-
ters of pure and defect containing GaTe were optimized using the conjugate gradient
algorithm and the convergence criterion for atomic relaxation was set to 10‘3 eV

energy difference between two consecutive ionic relaxation steps.

107

For structure optimization and total energy calculation of pure GaTe, we have
used the conventional unit cell containing 12 Ga and 12 Te atoms. The Brillouin
zone (BZ) was sampled by a F-centered Monkhorst-Pack [186] grid of 2x4x10 k-
points (36 k points in the irreducible (IBZ)). To model Sn and Ge substitutional
defects in GaTe, we have constructed 1x2x4 supercells (192 atoms) starting from the
conventional unit cell and replaced one Ga atom by either Sn or Ge. Supercells of the
same size were used to simulate the intrinsic defects (such as Ga and Te vacancies)
in GaTe. These calculations were performed using 3x3x3 F—centered Monkhorst-
Pack k—meshes. In order to reduce the spurious defect-defect elastic interactions
within neighboring supercells, the calculations were performed using the theoretical

(relaxed) lattice constants of the bulk GaTe, as suggested in reference [155].

To check the accuracy of our calculations obtained with the PAW method we have
compared the band structures of pure GaTe with that obtained using full potential
linearized augmented plane wave (FP-LAPW) [141] method, implemented through
Wien2k [143] package. For the full potential calculation we used the following setup:
the 3d, 4s, 4p electrons of Ga and 4d, 53, 5p electrons of Te were treated as valence
electrons, the product between the smallest muflin tin radius (RMT) and the largest
reciprocal lattice vector (Kmax) were chosen such that RMTKmax = 7.0. The atomic
radii were 2.17 a.u. for Ca and 2.35 a.u. for Te and convergence was assumed
when the energy difference between the self-consistent cycles was less then 0.0001
Ry (1.36 meV). The FP-LAPW calculations were performed using the experimental
crystal structure [175]. The small differences in the results obtained by the two
methods are related primarily to the volume relaxation and structure optimization.
The electronic band structures and the density of states, however, are nearly identical,
as expected. Thus, all the results presented in this chapter are obtained with PAW
method except for the band structure of pure GaTe (which was calculated using both
PAW and FPLAPW methods). To verify whether the large effect of 801 on the band

108

structure suggested in reference [27] is reasonable, we have calculated and compared
the electronic structures of bulk GaTe with and without SOI using FPLAPW. We find
SOI effects on the band structure to be quite small and have not therefore included

it in our defect containing supercell calculations.

5.3 Crystal and electronic structures of pure GaTe

To relax the structure of bulk GaTe, using PAW method, first we performed volume
optimization keeping the ratio of the lattice parameters a:b:c constant and then min-
imized the internal structural parameters at constant volume allowing for the shape
of the cell to change. The initial a:b:c ratio was chosen based on the values of the
experimental lattice constans [140] and it was allowed to change when the internal
parameters were varied. The obtained lattice parameters a, b, and c are larger than
the experimental ones by 6.4%, 4.7% and 1.5% respectively. The large overestimation
of the lattice constants a and b are attributed to the underbinding effect of GGA,
which is enhanced by the weak Van der Waals interlayer interaction along the a-
and b—axis. Since the crystal c-axis lies in the plane of the GaTe layers (and the
intralayer interaction is strong, covalent type), the effect of overestimation of the lat-
tice constants by GGA is less evident on the c-axis. After this two-step relaxation
process we have performed one more volume optimization to check if the calculated
lattice constants are close to their optimal values. Since the lattice parameters after
the additional volume optimization increased by only ~0.01 A, we considered that
the crystal structure obtained after the initial two-step process is close enough to
the fully relaxed structure. The theoretical lattice constants and the relaxed Ga-Ga

dimer lengths are given in Table 5.1.

In Fig. 5.2 we show the total density of states (DOS) of GaTe calculated with
the PAW method along with the partial DOS of Ga and Te, with SOI included. The

109

Table 5.1: The experimental and theoretical lattice parameters of bulk GaTe. The
distances are given in angstroms and the monoclinic angle 7 in degrees. déa_ Ga and

dlcly a_ Ga refer to the Ga—Ga dimer lengths that are perpendicular and parallel to the
layer planes, respectively.

 

 

1 II
a b C 7 dGa—Ga dGa—Ga
Exp. [140] 17.44 10.46 4.08 104.4 2.43 2.44
Theor. 18.56 10.96 4.14 107.6 2.46 2.49

 

origin of the energy was chosen at the highest occupied level. The valence band of
GaTe can be divided into three main groups. The first group of peaks located between
—12.5 eV and —10.5 eV originates mainly from Te 5.9 states with a very small Ga 43
and 4p contribution. The second group of bands, from —7.2 eV to —4.0 eV displays a
pronounced Ga 43 character with significant contribution from the Te 5p states. The
highest group of bands from -4.0 eV to 0 eV (energy of the highest occupied state) is
formed primarily by a strong hybridization of Te 5p and Ga 4p states. The bottom
of the conduction band is a mixture of almost equal contributions from Te 5p and Ga

43, with a smaller Ga p character.

For a better understanding of the electronic structure of GaTe, we analyze the
band structure of bulk GaTe, obtained by the FPLAPW method in the primitive
unit cell (6 Ga and 6 Te atoms), with SOI included (Fig. 5.3). We can identify
the three main group of valence bands mentioned in the previous paragraph: (i) the
lowest group of 6 bands (Te 53) is split off from the upper part of the valence band by
approximately 3.0 eV; (ii) the next 6 bands, with strong Ga 43 character, correspond
to GaeGa bonding and antibonding states associated with the Ga—Ga dimers; (iii)
the upper 15 bands originate from Te 5p states (15 out of 18 Te p states associated
with 6 Te atoms/ unit cell) hybridized with Ga 4p states. The electron counting is
such that the 6 Ga 8 and the 6 Te 3 bands together with the 15 hybridized Te-Ga p

bands can accommodate the total number of 54 valence electrons (18 from Ga and

110

 

.5
O

 

 

 

 

 

 

 

 

 

 

  

 

30: (a) —total DOS
20{
A 10‘. [
> 0,
3 1.6- —Gas
93 . -———Tes
.53 0.83
a . [ [m
(D 0.0 I I . EFLR—
8 1.0“ Ga ap :
05;“: —Tep
0.0‘—'—T_¢fi

 

-14-12-10 -8 -6 -4 -2 0 2 4
Energy (eV)

Figure 5.2: (a) Total density of states (DOS) of the Gate crystal calculated using
PAW method; (b) and (c) Projected DOS of s and p orbitals respectively of the Ga
and Te.

36 from Te). Therefore 3 Te 5p states (out of 18) remain empty and are pushed up in
energy giving rise to a semiconducting gap in the band structure of GaTe. If we look
at the dispersion of the Ga 3 bands we see one bonding band is quite flat compared
to the other two. It is also present to some extent for the antibonding 3 bands. This
asymmetry is a result of the structure where two of the three Ga dimers are oriented

differently compared to the third one.

The overall structure of the electronic DOS and band structure calculated by PAW
and FPLAPW methods are similar, and also agree to those obtained previously by
tight-binding (TB-LMTO) [28] and pseudo—potential (NAO-DFT) [27] approaches.

Small differences in the orbital contributions to the DOS, however, exist throughout

111

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

E / E5-
2 0‘; / 7N
0.0: E.
A -2.05 5 , Tep
5 5% Q \5
; 40-2: / _j
gs evseas
s “14% in 3~
-8.0-§
40.0%
‘52 N Te 3
420-; \ x
r

Z BA Pl' A NPZ

Figure 5.3: Band structure of GaTe crystal along high symmetry directions, calculated
using FP-LAPW method. The Brillouin zone used in this calculation is identical to
the one described in Ref. [27]

the valence band. According to our PAW calculations the Ga 3 contribution to
the low-energy valence bands (—12.5 eV to -10.5 eV) is stronger than the Ga p
contribution, in contrast with the results reported by Sanchez-Rcyo et al. [27], where
the Ga p character was found to be stronger than the Ga 3. Also the contribution
of Te 3 and Te p to the DOS in the region —7.2 eV to —6.0 eV is about the same
order in our calculations, whereas in reference [28] the Te p character is much more
pronounced. The quasi-gap in the valence band at about ~4 eV was also present in

the TB-LMTO calculation of Yamamoto et al. [28] and it is in agreement with the

112

without spin-orbit with spin-orbit
I

 

 

15.x..- . -1.5
1.0- - .1 ~1.0
. . . f'
0.5- f - [0.5

Energy (eV)

 

 

 

 

 

 

 

Figure 5.4: Band structure of GaTe (a) without and (b) with spin-orbit interaction.
The bands with pa; — py character, which lie below the topmost VB, split under
spin-orbit interaction and those with j = 3/ 2 shift up in energy by ~0.1 eV.
ultraviolet photoelectron spectroscopy (UPS) measurement [187].

In order to understand the effect of SOI in GaTe in more detail, we plot in Fig. 5.4
the calculated band structures along F—Z direction (a) without S01 and (b) with
SOI. As seen in the figure, the most significant effect is the energy shift—up of the top
valence bands at the F—point when SOI is included in the calculation. This effect has
been explained by Sénchez-Royo et al. [27]; the bands with pz-py character, which lie
below the topmost VB, split under spin—orbit interaction and those with j = 3/ 2 shift
up in energy. However, the magnitude of the energy shift calculated with FPLAPW
method (~0.1 eV) is much less than the energy shift estimated in reference [27] (0.7-
0.9.eV) inferred from the atomic data. Thus in solids where the Te orbitals are much
more diffused, SOI effects are quite small. The effect of SOI on the energy position of
the Te bands has been calculated in PbTe [188], and a similar value of ~0.1 eV was

obtained. The above analysis clearly suggests that the SOI in GaTe is rather small,

113

just like in the other members of the III-VI family [22].

From our band structure calculations we see that GaTe is a direct-gap semicon-
ductor with the gap located at the Z-point, at the edge of the BZ, in agreement
with the previous theoretical results [27,28]. Our band gap of 0.98 eV (and 1.098 eV
obtained in reference [27]) is underestimated by a factor of almost two compared to
the experimental value of 1.799 eV [140], due to the well-known limitation of LDA

(GGA), which underestimates the band gap in semiconductors [154].

5.4 Defects in GaTe

5.4.1 Substitutional impurities: SnGa and GeGa

In the case of simple extrinsic impurities, one should be able to predict whether the
impurity will give rise to a donor state or acceptor state by simply counting the
number of valence electrons of the dopant and the host atoms. The addition of group
IV elements, such as Si, Ge, Sn or Pb, to the Ga site, introduces more electrons in
to the system, giving rise to donor states. Figure 5.5(a) shows the calculated band
structure for the Ge impurity. Since Sn behaves very similar to Ge, we only present
the results for the latter and point out the noticeable differences between the two.

Let’s first discuss the local atomic relaxation in the neighborhood of the impurity.
Since one of the major differences between Ge and Sn atoms is in their atomic sizes,
we observe noticeable differences in the relaxation of the atomic positions around the
Ge and Sn defects. In both cases the atoms surrounding the Ge or Sn relax outward,
but in the case of the Sn, the relaxation is much more pronounced. The optimized
impurity-host bond lengths are listed in Table 5.2.

Next we look at the position of the defect states. The defect level lies deep in
the gap region, closer to the CBM and it is partially filled. Due to the well known
problem of the underestimation of the band gap by LDA/GGA, the position of this

114

Energy (eV)

 

Figure 5.5: (a)The band structure of GaTe doped with Ge (or Sn) on the Ga site.
(b) The charge density associated with the impurity induced defect band.

Table 5.2: Optimized impurity-host bond lengths. The last row contains the percent
increase in the bond lengths relative to the ones obtained for the bulk GaTe.The
notations Tel and Te2 refer to the NN and next NN Te atoms of the defect

dGe—Ga dGe—Tel dGe—Te2 dSn—Ga dSn—Tel dSn—Te2
2.56 2.75 2.74 2.73 2.92 2.90
Increased by: 4.5% 2.6% 1.1% 9.8% 8.9% 7.4%

 

level with respect to the CBM can not be obtained reliably. However, if we measure
their position from the respective VBM, we find that they are about ~1 eV above
the VBM and compared to Sn, the Ge level is closer to the VBM by ~50 meV.
Figure 5.5(b) shows that the charge density associated with the impurity state is
localized around the impurity and it originates primarily from Ge and Ga 3 and the

surrounding Te p orbitals.

5.4.2 Vacancies VG,a and VTe

For the vacancy calculations we minimize the energy of the 1x2x4 supercell containing

one Ga vacancy (VGa) or one Te vacancy (VTe)~ For VGa, the supercell contains

115

96 Te, 95 Ga, and a vacant Ga site. The shortest distance between two vacancies
is 16.56 A. After the optimization of the atomic positions we observe a large inward
relaxation of the atoms that are nearest neighbors (NN) of the vacancy. There are
significant changes in the bond lengths and bond angles. To quantify this inward
atomic relaxation we give, in Table 5.3, the distances between the vacancy and the Ga
and Te atoms surrounding the vacancy. For comparison we also give the corresponding
bond lengths for pure GaTe inside parenthesis. The vacancies are considered to be

at the ideal atomic positions.

To understand the nature of the defect states introduced by the vacancies in GaTe,
we have calculated the electronic band structure of the defect containing supercells
and the charge densities associated with the defect induced bands. Introducing a Ga
vacancy means that there is one less Ga 3 state in the VB and three less electrons in
the system. Thus, for one Ga atom removed, we expect partially filled bands near the
top of the VB. From the band structure plot in Fig. 5.6(a) we see that there is indeed
one partially filled band crossing the E F along F—P and A—I‘ directions. Besides
this, a Ga vacancy introduces a narrow defect band lying deep in the gap region. The
energy of this deep defect state with respect to the VBM of the vacancy containing
supercell (which occurs at the P—point) is ~0.78 eV. This value is consistent with the
hole trapping level measured by deep-level transient spectroscopy at 0.8 eV above the
VB by Sighetomi et al. [43]. Manfredotti et al. [189] also report an acceptor level at
0.74 eV above VB in melt grown samples of GaTe. It is tempting to presume that
this defect level originates from the Ga vacancy. One should however note that in
experiments the defect concentrations are much smaller than what we have in our.
supercell calculations. Consequently one must correct for the changes in the VBM
energy. To estimate these corrections we match the energies of the Ga 3d semicore
states of the pure and vacancy containing GaTe, at F—point. We find that the energy
of the defect level with respect to the VBM of the pure GaTe shifts to ~0.82 eV. This

116

Table 5.3: The distances between the vacancies and their nearest neighbors. The no-
tations of Cal and Ga2 refer to the Ga atoms from the dimers oriented perpendicular
and parallel to the atomic layers respectively. The bond lengths of the pure GaTe are
given in parenthesis. The distances are given in angstroms.

 

Gallium vacancy Tellurium vacancy
Vacancy — Gal 1.87 (2.46) 2.27 (2.73)
Vacancy — Ga2 2.59 (2.71)
Vacancy — Te 2.23 (2.71) 4.07 (4.19)
Te — Te 3.64 (4.19)

 

value is still in good agreement with the experiment, suggesting that the defect level
might indeed come from Ga vacancy. Figure 5.6(b) shows the electronic charge density
associated with this deep defect band. The charge density originates mainly from Ga

3 states and Te p states from the vicinity of the vacancy and it is quite localized.

For VTe we find that the band structure also changes drastically (see Fig. 5.7(a))
compared to pure GaTe. The changes in the band structure introduced by 3 Te
vacancy can be qualitatively understood in terms of the bonding model described
in the case of GaSe in Chapter 3, since the local bonding between Ga and Te is
similar to that between Ga and Se. As discussed earlier, the Ga-Ga dimers form
bonding and antibonding states (the six Ga 3 bands indicated in Fig. 5.3). Due to
the hybridization between the Ga 3 states and Te p states, some of the Te bands are
pushed up in energy, giving rise to the semiconducting gap. In the case of pure GaTe
there are 6 Ga and 6 Te atoms in the unit cell. Three Te p states (out of 18) are
pushed up to the CB. Removing one Te atom from the unit cell (Ga6Te5) gives rise
to one less Te 3 state and two less Te p states in the VB and one less Te p state
in the CB. Since the number of electrons is reduced by 6 due to the Te vacancy we
expect that all the valence bands will be fully occupied (no partial occupation). To
check this point we counted the number of valence bands obtained in the pure and in
the vacancy containing supercells. The number of valence bands indeed decreases by

three (disregarding spin degeneracy) when one Te atom is removed from the 192-atom

117

 

 

 

Energy (eV)

   

(a)

Figure 5.6: (a) The band structure of GaTe with Ga vacancy. (b) The charge density
associated with the vacancy induced deep defect band (lying in the gap near energy
0.8 eV).

system. In Fig. 5.7(a) we see that Te vacancy introduces two gap states right below
the CB and one nearly non-dispersive resonant state near the top of the VB. From the
charge density plots associated with the two gap states below the conduction band
shown in Fig. 5.7(b) we see that these bands originate mainly from the 3 states of
Ga atoms surrounding the Te vacancy. A Te vacancy acts like an attractive potential
and lowers the energies of the neighboring Ga states. An interesting prediction is

that doping these two states can give rise to magnetism.

5.4.3 Defect formation energies

In this Section we discuss the formation energies of different types of defects in GaTe
and see how they depend on the atomic chemical potentials. We adopt the formalism
presented earlier in Sections 2.4.2 and 2.4.3. As an example, we will describe the
details of calculations for the case of substitutional Ge impurity on Ga site (GeGa).

The formation energies and the charge transition levels for the other defects under

118

2.0

 

 

 

 

 

1.5
’>‘
9, 1.0 W‘\—’—
> 0.5
8
0.0
5 E
-0.5
-1.0
I‘ P 2 A F

 

(a)

Figure 5.7: (a) The band structure of GaTe with Te vacancy. (b) The charge density
associated with the two, vacancy induced deep defect bands which lie below the CB.

consideration can be obtained in a similar way, using the values given in the footnotel.

The calculated enthalpy of formation for GaTe compound is AH (GaTe) = —0.72 eV,
therefore the 11’s must satisfy: ”Ga + I‘Te = —0.72 eV and p,- S 0 (i = Ga,Te , Ge).
Since Ge can form secondary phase with Te, with calculated AH(GeTe) = —0.14
eV, the maximum achievable value of the chemical potentials are further limited by
the condition: I‘Ge + #Te = —0.14 eV. Under Garrich growth conditions (I‘Ga = 0,
”Te = —0.72 eV) this gives ”Ge = —0.14 + 0.72 = 0.58 eV. However, in order
to avoid precipitations of Ge, we have to impose I‘Ge = 0. Under Te-rich conditions
(”Ga = —0.72 eV, ”Te = 0) on the other hand, “Ge is reduced to —0.14 eV. Using Eq.
(2.52), the formation energy of GeGa in neutral charge state is AHf(Ge%a) = 1.18
eV + ”Ca — #Ge' Under Ga-rich conditions (I‘Ga = 0, #Te = —0.72 eV and “Ge =
0) this gives AHf(Gega) = 1.26 eV, whereas under Te-rich conditions AHf(Ge%a)

 

1The calculated formation energies AE(Xq) using Eq. (2.52) are: AE(V%e) =
2.17 eV, naval) = 2.15 eV, AE(V$§) = 2.12 eV, AE(Gega) = 1.18
eV, AE(Geg;) = 0.43 eV, AE(Ge5§) = 0.42 eV, AE(Sn0Ga) = 1.02 eV,
AE(sng;)=0.24 eV, AE(sng§)=0.18 eV. The calculated enthalpies of for-
mation are AH(GeTe) = —0.14 eV and AH(SnTe) = —0.61 eV.

119

decreases to 0.60 eV. This is because under Te-rich (Garpoor) conditions, Ga vacancies
are more likely to appear making it easy for Ge atoms to occupy these vacancies. In
the case of the charged defects, the formation energy depends also on the Fermi
level, because in order to ionize the defect, electrons must be taken from or added
to the electron reservoir with energy E F- The donor transition levels for GeGa are

calculated using Eq. (2.53): 5(+/0) = 0.01 eV and €(+2/+) = 0.75 eV.

Figure 5.8 shows the calculated formation energies as a function of E F for different
defects. For VGa we find two acceptor levels 5(0/ — 1) and €(—1/ — 2) in the band
gap at 0.13 eV and 0.83 eV above the VBM, respectively. The latter is in very good
agreement with the experimental results reported in Refs. [43,189]. VTe appears to be
in neutral charge state for almost all values of E F across the band gap. Nevertheless,
it introduces a deep donor level at 0.03 eV above VBM. Furthermore, our calculations
(not shown in the figure) show that the donor level e(+2/ + 1) is slightly higher in
energy than E(+l/0), meaning that VTe is a negative-U center [190], with unstable
+1 charge state. A defect has negative-U properties if it can trap two electrons (or
holes) with the second bound more strongly than the first. The subsitutional defect
SnGa behaves similarly to GeGa. The deep donor levels €(+2/ + 1) and €(+1/0)
associated with SnGa are located at 0.06 eV and 0.78 eV above the VBM.

Ftom Fig. 5.8 we observe that the defect formation energies are quite sensitive to
the growth conditions. First let us look at the intrinsic defects VGa and VTe- At the
Te-rich limit (Fig. 5.8(b)), the formation energr of VGa is always less compared to
VTeI indicating that Ga vacancies are more likely to appear during the Te-rich growth
process. However, at the Ga-rich limit (Fig. 5.8(a)), we find that the formation energy
of VGa relative to VTe depends on the position of the Fermi level. If E F is closer to
VBM (p-type material) the Te-vacancy will be the dominant intrinsic defect, whereas
if E F is above the midgap (n—type material), the acceptor VGa will dominate over

the donor VTe-

120

 

 

 

 

 

 

 

 

 

 

VBM CBM
2.0-
\VG Ga-rich
a conditions
1.5./_ Te \ *
3 1.0- k
“" . Ge S"Ga
1: Ga
< 0.5-/
0.0 . . . . . . , .
0.0 0.2 0.4 0.6 0.8
EF(°V)
(a)
VBM CBM
3'0] Te-rich
2.51 VT conditions
2.0- '
91.5-
9, ‘_\ V
0.0—10‘ .
a s a”)?
- ‘ n
0.0-
-0.5] . . . . . . . . -
0.0 0.2 0.4 0.6 0.8

Figure 5.8: Calculated defect formation energies in GaTe as a function of Fermi
energy, under (a) Ga—rich and (b) Te—rich conditions. The slope of the energy lines
indicates the charge state of the defect and the value of E F where the slope changes,

EF (eV)
(b)

represents the charge transition level.

We observe that the growth conditions also influence the solubility of Sn relative to
the solubility of Ge. Under Ga—rich conditions it is easier to incorporate Sn in GaTe,
whereas at the Te-rich limit Ge becomes more soluble than Sn.

the Sn-Te bond being stronger than the Ge-Te bond (AH(SnTe) = —0.61 eV and

121

This is because

AH(GeTe) = —0.14 eV), it will be more likely for Sn to form secondary phases with
the Te atoms, during the Te-rich growth process. In order to avoid this, one has to
decrease the atomic chemical potential of the Sn, which in turn, increases the defect

formation energy. This prediction has to be checked by experiment.

5.5 Summary

In summary, we have performed electronic structure calculations in pure and defect
containing GaTe to understand the nature of local bondings and how they effect
the overall band structure of pure GaTe and the defect states. The results obtained
for the pure system indicate that it is a direct gap semiconductor. As in other III-
VI semiconductors containing Ga-Ga dimers, the presence of these dimers and the
interaction of the dimer states with Te p states lead to the formation of the band
gap. Our results obtained without SOI agree with previous works [27,28]. In contrast
to the suggestion made by Sanchez-Royo et al. [27] we do not find the SOI to be
important in GaTe; the shifts in the energy levels near VBM and CBM are ~0.1
eV rather than ~0.8 eV. To investigate the nature of various defects in GaTe, we
constructed large supercell models (192 atoms). We find that Ga vacancy introduces
a deep defect state in band gap region at ~0.78 (~0.82 eV) above the VBM of the
defect containing (defect free) system, at the F—point. The charge density associated
with this defect level is strongly localized around the vacancy’s NN Ga atom. The
energy of the Ga vacancy induced defect band is in quite good agreement with the
experimental observation [43,189]. Te vacancy introduces two localized states near the
CBM and a resonant state just below E F: Charge densities associated with the gap
states reveal the nearly localized character for these bands. Ge and Sn substitutional
impurities behave almost identically in the GaTe host, introducing a deep donor state

in the band gap region.

122

Our calculations show that the defect formation energies depend not only on
the charge state of the defect but also on the growth conditions (Ge-rich or Te-
rich). In the case of n-type samples, however, VGa is always the dominant intrinsic
acceptor, which can compensate the intentional donors. The calculated acceptor level
e(—1/ — 2) = 0.83 eV above VBM is in good agreement with the experiment [43, 189].
We find that VTe is a negative-U center [190] with a deep e(+2/0) donor transition
level located at 0.03 eV above VBM. Our results also indicate that as the growth
conditions change from Ga-rich to Te-rich, the solubilities of Sn and Ge interchange
with respect to each other. Both, GeGa and SnGa, are deep donors with the €(+2 / + 1)
level located at 0.01 eV and 0.06 eV above VBM and €(+1/0) level located at 0.75
eV and 0.78 eV above VBM.

123

Chapter 6

Hydrogen in CdTe

6.1 Introduction

Cadmium telluride (CdTe) is a semiconducting material which, under normal, condi-
tions crystallizes in the zincblende structure. It is a direct gap semiconductor, with
the smallest gap at F—point, in the center of the B2. The experimentally determined
gap at room temperature is 1.475 eV [140]. CdTe has been extensively studied ex-
perimentally and theoretically during the last several decades due to the possible
applications in room-temperature X-ray and 'y-ray detector [46,47]. As we have men-
tioned in the introduction (Section 1.2), the performance of the CdTe based radiation
detectors and solar cells is often limited by the presence of the defect states in the
band gap of the CdTe, which act as recombination centers for the charge carriers.
Furthermore, the excess holes and electrons originating from the native defects re—
duce the required high resistivity for a detector grade semiconductor (>108 9cm).
One way to improve the efficiency of CdTe in detector devices and solar cells is to
achieve the passivation of electrically active defects by hydrogen. The best known
property of atomic hydrogen in many semiconductors is the increase of resistivity by

passivation of donors and acceptors [191].

124

In CdTe, hydrogen is usually introduced in order to purify the material from
oxygen impurities (mainly cadmium and tellurium oxide compounds incorporated
in CdTe). During the hydrogenation process, hydrogen can be trapped inside the
crystalline lattice and can significantly influence the electrical and optical properties
of the semiconductor. Extensive studies have been carried out to investigate the
infrared (IR) vibrational spectra of CdTe using samples exposed to different levels
of hydrogenation in an effort to understand the distribution of H in CdTe [192—195].
Using ab initio modeling for hydrogen-group V complexes in CdTe and calculating
the local vibrational mode (LVM) frequencies, it has been found that the ground state

location of H is near the bond center, closer to the acceptor (group V) atom [196].

From theoretical point of view it is attractive to study H, the most elementary
atom, as an impurity in semiconductors. It is known that H does not behave as a
“hydrogenic” effective-mass—type impurity, but rather introduces deep levels in the
semiconducting band gaps. In most semiconductors, atomic hydrogen acts as am-
photeric impurity, being either in positively charged state (H+, donor) or negatively
charged state (H‘, acceptor), depending on the value of the electron chemical po-
tential (position of the fermi level within the band gap). The neutral charge state
is usually unstable, indicating that H introduces a negative-U center in most materi-

als [58,59,59—61,191].

The interaction of H with defects in semiconductors is still an area of active re-
search. State—of-the—art calculations have helped reveal many of the fascinating prop-
erties of H in semiconductors and made H a model system for defect studies [103, 197].
Recently, Du et al. [54] reported that complex of OTeH may play an important role
in the carrier compensation in CdTe. Hydrogen easily forms complexes impurities
and native defects in semiconductors. Takenaka et al. [198] briefly reported the exis-
tence of hydrogen-cation vacancy complex in CdTe. The stability and the electronic

properties H defects in CdTe and the mechanism of hydrogenation have not been

125

reported systematically as yet, to our knowledge. To this end, we performed the total
energy calculations based on density functional theory to investigate the stability and

electronic properties of the hydrogen related defects in CdTe.

6.2 Method of calculation

We have performed electronic structure calculations based on density functional the-
ory (DFT), using both the generalized gradient approximation (GGA) [148] and the
local density approximation (LDA) [167] with projector augmented wave (PAW) [167]
pseudopotentials as implemented in VASP [144—147] code. The 4d and 53 states of
Cd and the 5s and 5p states of Te were treated as valence electrons. The cutoff energy
for the plane wave basis was set to 300 eV and convergence was assumed when the
total energy difference between consecutive cycles was less than 10—4 eV. The defect
calculations were performed on 2x2x2 supercells using the theoretical lattice constants
' with the defect located near the center of the cell. To see if the 2x2x2 supercell (64
atoms) is adequate for the calculations of defect formation energies and transition lev-
els, we have also performed several calculations using 3x3x3 supercells (216 atoms).
The atomic positions in the defect containing supercells were relaxed until the quan-
tum mechanical forces were smaller than 0.02 eV/ A The BZs of the 64-atom and the
216-atom supercells were sampled by 4x4x4 and 2x2x2 F-centered Monkhorst—Pack
grids, respectively. In the case of charged defect calculations, a uniform background
charge was added to preserve the charge neutrality of the supercell.

The formation energies and charge transition levels were calculated using the for-
malism from Sections 2.4.2 and 2.4.3. The chemical potential of the hydrogen #H (=
—3.38 eV) was taken as half of the calculated ground state energy of the H2 molecule.
The enthalpy of formation of pure CdTe calculated using GGA (AH(CdTe) = —0.93

eV) is in good agreement with the experimental value of —0.96 eV [199].

126

 

Figure 6.1: Locations of H impurity inside the CdTe lattice
6.3 Location of hydrogen in CdTe

In order to study the behavior of H in CdTe, we have performed total energy and
electronic structure calculations for several different positions and configurations of
the impurity, as shown in Fig. 6.1. CdTe crystallizes in zinc—blend structure. Inside
the zinc—blend structure there are two tetrahedrally coordinated interstitial sites: one
surrounded by anions and the other surrounded by cations. In Fig. 6.1, if we consider
that the red spheres represent the Cd atoms, then the hydrogen (small blue sphere)
located at the center of the cell is tetrahedrally coordinated by anions (Te atoms),
whereas the one located at the lower right area of the cell is tetrahedrally coordinated
by cations (Cd atoms). We denote these two sites as TTe and TCdv respectively. We
have also investigated the configurations in which H is placed at the Cd—Te bond
center (BC) and when it substitutes one Cd atom (HCd) or one Te atom (HTe)- For
each location of the H atom, the host lattice was fully relaxed. This is essential in
order to obtain the low energy position of the H impurity in the CdTe host. For
example, the BC site in Si turned out to be the lowest energy position for H only

after the relaxation of Si atoms were allowed [59].

127

Hydrogen in semiconductors can occur in different charge states: H+, HO and
H". In order to produce a certain charge state, electrons have to be exchanged with
a reservoir of energy equal to the Fermi level (E F), therefore the preferred charge
state of the impurity depends on the position of the E F- Van de Walle et al. [58]
found that the location of the H impurity in a Si lattice depends strongly on the
charge state of H. For H+ and H0 the high-electron-density region located at the BC
site is more stable, while H“ prefers the low-electron—density interstitial regions in
the Si lattice located at the tetrahedral site. We anticipate that H behaves similarly
in CdTe. However, in the case of the BC site since the two neighbor atoms are not
identical, H will prefer to move closer to the anion or cation depending on its charge
state.

In the following sections we will discuss the results of electronic structure calcu-
lations (obtained with the GGA) for a H impurity occupying the above mentioned

lattice sites in CdTe.

6.3.1 Substitutional hydrogen in CdTe

The electronic structure of substitutional H in CdTe can be understood in term of
simple molecular orbital theory. When a vacancy is created in the zinc blend lattice,
four anion-cation bonds are broken, thus four dangling bonds are created on the
atoms surrounding the vacancy. As shown in Fig. 6.2, these four dangling bonds
will give rise to a singlet state a1 (s—like) and a threefold-degenerate state t2 (p-like)
(not including spin degeneracy). In the case of a Te-vacancy, the al-combination
is occupied by two electrons and lies below the VBM, while the empty t2 states are
located above the conduction band minimum (CBM) [55]. The s orbital of the H atom
placed at the vacancy site interacts with the singlet state a1 and creates a bonding
state deep in the VB and an antibonding state located in the band gap, closer to

CBM (see Fig. 6.2(a)). There are three electrons associated with a HTe defect center:

128

t2
.............................................. CBM
Cd 3 - H s
antibonding
state

     
     
  
  

------- VBM

.- Te p - H s ....................
antibonding
state

CBM

   
 

------ VBM

dangling H s dangling H 5
31
Cd 3 - H 3 Te 91H 3
bonding bondIng

(a)

(b)

Figure 6.2: Simple bonding scheme for substtutional hydrogen on the (a) Te site and
(b) Cd site

two electrons coming from the Cd 3 dangling bonds and one from the H s orbital.
Two of these three electrons will stabilize the defect center by occupying the bonding
state and one electron will be promoted to the high-lying antibonding state, located
below the CBM.

The electronic structure of HCd is slightly different (Fig. 6.2(b)). The 01 and t2
combinations are now created from the Te dangling bonds surrounding the vacancy.
They have significant p character. The singlet state a1 lies deeper in the VB while
the threefold-degenerate t2 state is located in the band gap, close to VBM [55]. Two
out of the six electrons coming from the four Te dangling bonds are accommodated
by the al state while the remaining four electrons are accommodated by the t2 state.
When a hydrogen atom is placed at the Cd-vacancy site, the H s orbital and the al
state will interact and give rise to a bonding combination located at the bottom of

the VB and occupied by two electrons and an antibonding state located just below

129

the CBM. The electron, which would have occupied the antibonding state, lowers the
energy of the system by transferring to the t2 level, located at the top of VB (see
Fig. 6.2(b)).

 

 

 

 

 

 

 

 

 

  
  
   
 
  

 

 

160 .
‘ ' H total DOS
120'. ll antibondlng t Te
80: bonding ]2
A 40'. U W
> 0 . . . , . g - m 4
° 4 Hslocal DOS
2 0.24
E 013
3 ' -
(0 0-0 ' I 1 I I
O 1.2: CdslocalDOS
D 03-:
0.4-
0.0‘ MAW“ . + f
-6 .4 .2 4
Energy (eV)
(8)
160
- ; HCdtotalDOS
120‘ : t2
80: bonding :/ antlbondlng
401 \ : MM
0. ' I f I I r 1 I '
0.75: Hslocel DOS

9
5?

DOS (states/eV)
o s: .o
3: $.93.

 

 

 

 

0 2 74 6
Energy (eV)
(b)
Figure 6.3: The density of states (DOS) of CdTe with substitutional H located at (a)

Te site and (b) Cd site. In both cases The H 3 partial DOS is shown along with the
nearest neighbor (3) Cd 3 and (b) Te p partial DOS

:5
.L
,5.

130

We will now analyze the electronic structure of substitutional H in CdTe based on
the results of ab initio DFT calculations, and see whether the arguments presented
above are indeed true. Figure 6.3 shows the total density of states (DOS) of CdTe
with H substitutional impurity on the Cd and Te sites as well as the partial DOS
associated with the H s orbital and the nearest-neighbor (NN) Cd 3 and Te p or-
bitals. In Fig. 6.3(a) and (b), we can identify the states discussed in the previous
paragraphs. Common to both HTe and HCd is the peak located at the bottom of
the VB corresponding to the H s — a1 bonding combination. From the partial DOS
plots it is evident that this bonding state is a superposition of H s and the nearest

neighbor Cd 3 and Te p orbitals (see Fig. 6.3(a) and 6.3(b), respectively).

In the case of HTe (Fig. 6.3(a)), the H s - a1 antibonding state is located in the
band gap, close to the CBM, and it is half-filled. Thus, HTe acts like a donor. The
partial DOS plots in Fig. 6.3(a) show that both H s and Cd .3 orbitals contribute
to this antibonding state. The three-fold degenerate t2 state, with a strong Cd 3
character (and no H contribution), is resonant in the CB, in agreement with the

electronic structure of Te vacancy described by Wei et al. [55].

For HCd’ the peak located at the bottom of the CB corresponds to the hydrogen
3 -— a1 antibonding state. The electron which would have occupied this state is
transferred to the threefold-degenerate t2 level located at the top of the VBM (see
Fig. 6.3(b)). As a result of this electron transfer, two t2 states will be doubly occupied
and one state will remain half-filled. Consequently, HCd is an acceptor. We also have
to mention that due to partial occupation of the t2 level, HCd should undergo Jahn-
Teller distortion. This would result in the splitting of the t2 level into an occupied
6 level (pxy-like) and a half occupied upper a1 level (pg-like) [200], along with a
symmetry lowering atomic relaxation. We will address the issue associated with the

Jahn-Teller effect in Section 6.4.

To further analyze the nature of the defect states introduced by substitutional H

131

 

Figure 6.4: The charge density distributions associated with the (a) bonding and
(b) antibonding combinations between H s orbital and a1 singlet state. The main
contribution comes from the H s and the nearest-neighbor Te p orbitals. (c) The
charge density of the three-fold degenerate t2 state showing the Te p contribution
(and no H .3 contribution)

in CdTe, we illustrate in Fig. 6.4 the three-dimensional visualization of the charge
densities associated with the bonding, antibonding and t2 states for the case of HCd-
Figure 6.4 (a) and (b) show the charge density distributions of the low-lying bonding
state located at the bottom of the VB and the high-lying antibonding state at the
bottom of the CB, respectively. The dominant contribution to these states comes from
the H s orbital and the NN Te p orbitals, as described in the previous paragraphs.

The charge density associated with the state t2, located at the top of the VB, is

132

shown in Fig. 6.4(c). We observe that the only contribution to this state comes from
the Te p orbitals (no H 3 contribution), showing that the H s orbital interacts mostly
with the al combination of the Te dangling bonds, as required by symmetry.

From the charge density distributions shown in Fig. 6.4 (a) and (b), we see that
H is equally bonded to all nearest neighbors. This suggests the possibility that sub—
stitutional H in CdTe can develop multi-coordinated bonds, similar to the case of
substitutional H in ZnO and MgO [201]. This is specially true for the case of HTe,a
where no Jahn—Teller distortion is expected, since this configurations has a partially
occupied s-like state (01 level) in the band gap (and no partially unoccupied, de-
generate p-like states). In contrast to ZnO and MgO, in CdTe, however, the Cd-Te
bondlength (~2.87 A) is much larger than the Cd—H or Te-H bondlengths (~1.7 A)
and so we cannot exclude the possibility that two or more H atoms will occupy the
vacancy site to saturate the dangling bonds.

For the substitutional H impurity at the Cd site we have also carried out calcu-
lations using 3x3x3 supercell (216 atoms) to check the energy convergence and the
atomic relaxation with respect to the size of the supercell. We have calculated the
formation energy of H defect in neutral charge state and we find that the values pro-
duced by the 64- and 216-atom supercells differ by less than 4% (3.47 and 3.33 eV
respectively). Further, we have found that the atomic relaxations around the defect
are not very sensitive to the size of the supercell. In the case of the 216-atoms sys-
tem the defect’s NN and next NN atoms move 0.7% and 0.5% closer to the H atom
compared to the 64—atoms system. Since the formation energy calculations are based
on total energies differences, we expect the errors associated with the supercell size

and k—point sampling to mostly cancel out [155].

133

 

NN H orbitals
Te or Cd

hybrid obitals

 

31

Figure 6.5: Simple scheme for level splitting due to H impurity at tetrahedral inter-
stitial site.

6.3.2 Interstitial hydrogen at the tetrahedral sites: TCd and
TTe

In the case of a H interstitial impurity in CdTe, the defect levels appear to be mainly
due to the interaction between the H orbitals and the hybrid orbitals of the NN host
atoms. The splitting of the energy levels is shown schematically in Fig. 6.5. Two
pairs of al-tg levels are created. The first pair, originating mainly from the hybrid
orbitals of the impurity’s NN host atoms, is situated deep inside the VB. The second
pair, derived mostly from the H orbitals, lies above the VBM [200]. The electron of

the H atom is accommodated by the al state in the band gap. The position of this

134

state depends on the potential of the defect center. For example, it has been found
that in the case of Cd and Te interstitial impurities at the tetrahedral site in CdTe,

the (11 state is located closer to the CBM and below the VBM respectively [55].

 

 

 

0.2-

 

DOS (states/eV)
_O
‘L

0.3: CdslocalDOS

 

 

 

b
.b-I
0:

20Energy (eV)2
(3)

80 ‘ Hat TTO
' total \DOS
40 -I
4
0

3.? Hslocal \DOS

 

 

 

8‘ *- JP . E . , ‘2 , .
0 ‘ TepxpypzlocalDOS

DOS (states/eV)

 

 

 

 

Energy (eV)
(b)

Figure 6.6: The density of states of CdTe with interstitial hydrogen on (a) tetrahedral
site with Cd NNs and (b) tetrahedral site with Te NNs. In both cases the H 3 partial
DOS is show along with the nearest-neighbor (3) Cd 3 and (b) Te p partial DOS.

135

Figure 6.6 shows the calculated DOS associated with the H impurity located at the
two tetrahedral interstitial sites (TCd and TTe in Fig. 6.6 (a) and (b) respectively).
In both cases we observe that the sharp peaks located at E F originate primarily
from the contribution of H s orbital to the DOS. The contributions of Cd 3 and Te
p orbitals to this state (al) are insignificant, suggesting that the interaction between
H at the tetrahedral site and the surrounding host atoms is rather weak. This is also
indicated by the very little relaxation of the Cd and Te atoms in the vicinity of H:
when H is at the T Te site, the Te atoms relax towards the H by ~0.07 A, whereas
for H at TCd site, the Cd atoms move towards the H by ~0.18 A.

Hydrogen at TCd (Fig. 6.6(a)) introduces a defect state (0.1) near the top of VB.
As discussed earlier in this section, this state is half filled and we therefore expect H
at TCd to act as an acceptor impurity. On the other hand, H at TTe introduces a
deep defect state near the middle of the band gap closer to CBM, which could act as

a donor state.

6.3.3 Interstitial hydrogen in CdTe: the bond center (BC)

site

The electronic states introduced by H located at the center of the Cd-Te bond (BC-
site) can be understood (to a first approximation) by considering only the states of
the H atom and the nearest Te and Cd atoms. If the impurity is not present, the Cd
3 and Te p orbitals form bonding and antibonding states located in the VB and CB
respectively, as illustrated schematically in Fig. 6.7. If H is placed at the BC site, the
H s orbital couples to the bonding state, lowering its energy and giving rise to a new

(antibonding) state near the CBM.

To check the validity of this idea, we plot in Fig. 6.8(a) the partial DOS associated
with Cd .9 and Te p orbitals located in the vicinity of the H impurity. We also

136

Cds-Tep

 

antibonding state

 
  
   
   

................................................ CBM

................................... VBM
Te p states

:33: H ,5

Cds-Tep

bonding state

Figure 6.7: Bonding scheme for H in CdTe located at the bond center (BC) site.

show the partial DOS of the H s orbital. The peak at the bottom of the VB in
Fig. 6.8(a) corresponds to the Cd-Te bonding state, which is lowered in energy due
to the interaction with the H s orbital. We can also identify the peak near the
bottom of the CB as the new (antibonding) state created by the antibonding coupling
of the H s and the original Cd-Te bonding states. This can be further verified by
looking at Fig. 6.8(b) where we show the three-dimensional charge density distribution
associated with the defect state located near the bottom of the CB. The dominant

contribution comes from the Te p and H s orbitals.

We observe a significant relaxation of the atoms around the H impurity placed at

the BC site. Initially, the H atom was placed midway between the Cd and Te atoms.

137

D

 

 

 

 

 

 

0.4: Cd 3 local DOS

0.2: J
9 0.9. . . . r . . .
d) .
E 0. 4_ Te px, py, pz local DOS
iii _'
8 02 H 3 local DOS

0.1 l

M JMM
0.0 - 4 . .
I 2

 

 

 

Figure 6.8: (a) The partial DOS associated with the s and p orbitals of the Cd and
Te atoms located in the vicinity of the H impurity. The lower panel shows the H
3 partial DOS. (b) The charge density distribution associated with the defect state
located near the bottom of CB. The dominant contribution comes from the Te p and
H s orbitals. The H impurity is located at the bond center site.

After relaxation, Cd-Te distance increases by 1.07 A, while the H atom (in neutral
charge state) moves closer to the Te atom (H-Te distance is 1.72 A and the H-Cd

distance is 2.21 A).

138

6.4 J ahn-Teller effect

We have mentioned in Section 6.3.1 that in the case of the substitutional H on Cd
site (HCd), due to the partial occupation of the t2 level located at the top of the
VB (Fig. 6.2(b)) the system is susceptible to Jahn—Teller distortion. The electronic
structure and atomic relaxation calculations, performed with H at the ideal Cd site
as starting configuration, showed neither the splitting of the t2 nor the expected
symmetry lowering atomic distortion [155]. This is similar to the case of Cd—vacancy
(VCd) in CdTe, described by Lany et al. [202], where the experimentally observed
J ahn-Teller effect could not be confirmed theoretically by DFT calculations.

In order to investigate the J ahn-Teller effect theoretically we have performed cal-
culations starting from a configuration with lower symmetry: we have displaced the
H atom from the ideal Cd vacancy position and then let the system relax. During the
relaxation process we observed that the H atom moved closer to one of the Te atoms,
lowering the tetragonal symmetry of he system. As a result of the relaxation, the
shortest H-Te distance becomes 1.69 A and the total energy of the system is lowered
by 1.22 eV/(64 atom-supercell) compared to the case when H is located at the high

symmetry site (Cd-site).

To further ananlyse the J ahn-Teller effect, we compare the band structures of the
distorted and undistorted HCd configurations. In Fig. 6.9 we observe several diffrences
between the two band structures, but what is relevant to the present discussion is the
splitting of the top VB, when H is moved out of the high symmetry configuration.
As a result of the relaxation, the t2 level located at the top of the VB splits into
a doubly-degenerate level and a single band (see Fig. 6.9. The removal of the 3-
fold degeneracy of the t2 level, associated with the lowering of the spatial symmetry
of the defect center, followed by the lowering of the total energy can be attributed
to the Jahn¥Teller effect. At the I‘ - point, the Jahn—Teller splitting (see Fig. 6.9

139

(a) (b)

2.5
R40

 

 

I
_A_

N go
o a:
%

  

   
 
 

 

 

 

 

 

 

A 1.5- +1.5
> . .
3
> 1.0 -+ . - 1.0
9 ‘ the t2 Ievel '8 the t2 level
2 0.5- 340“ degenerate splits at the r-point 40.5
Lu , at the F-point _
0.01
-0.5-
M F

 

Figure 6.9: The band structures of CdTe with (a) H at the ideal Cd site and (b)
H removed from the quasi-equilibrium site. When H occupies the ideal Cd vacancy
position (panel (a)), the t2 level is 3-fold degenerate at F—point. The degeneracy
is removed by the J ahn-Teller distortion (panel (b)). The J ahn-Teller splitting is
~74 meV at F—point.

(b)) is approximately 74 meV. We have also analyzed the charge density distribution
associated with t2 hands. We found that in the Jahn-Teller distorted system, the t2
levels are localized on only three Te atoms from around the defect center. The Te
atom which captures the H, is not involved in the re—hybridization of the dangling

bonds around the Cd vacancy.

6.5 Defect formation energies

In the previous sections we have discussed the properties of a neutral H impurity
located at several different sites in CdTe. However, since the formation energy of a
defect configuration depends on its charge state (see Eq. (2.52)), in order to establish
the preferred location of the H, one has to calculate the formation energies of the

defect in different charge states. Depending on the position of the Fermi level (E F),

140

4.0 -

305- l“‘
f “
.
30 ~
\
o “
‘
‘\ _-
H ‘ a’ 6“
. ‘ r s
- --. - In 3 -~ ~
A ' s I " ~ \
V \ I - ’ ““
Q l“ " ~ ‘
f s ’ § \
f s ’ ‘ \
I § ‘ ‘ ‘
' ~ 'i \
e V ’ s \ ~
, s \‘V
' ‘ q
l \
1 , 1
I - -
a
I ” ( r ‘s
a v \
< 15‘ ' ’ I
0 fl

. /
0.5 4/("'1 )
0.0 . . . . . . . .
0.0 0.4 0.8 1.2 1.6

EF(eV)

 

 

 

Figure 6.10: The formation energies of the H defect located at different sites in CdTe,
as a function of Fermi energy. The lowest energy locations are at the BC site for H+
and H0 and at the TCd site for H‘.

a hydrogen impurity can generally occur in three different charge states: H+, H0 or
H“. The dependence on the Fermi energy is due to the fact that, in order to produce
a certain charge state, electrons must be exchanged with the electron reservoir with

energy equal to E F-

Figure 6.10 shows the calculated formation energies of the H defect located at
different sites in CdTe as a function of Fermi energy, using Eq. (2.52). We observe
that for H0 and H+ the lowest formation energies occur when H is located at the
bond center (BC) site. This indicates that the energy gained due to the formation
of the strong H-Te bond compensates the energy lost due to the significant strain
induced in the lattice due to the presence of H at the BC site. H" on the other hand,
prefers to occupy the tetrahedral interstitial site with Cd nearest neighbors (TCd).
This is because the defect level induced by H at TCd is located near the VBM (see
Fig 6.6(a)) and it therefore requires less energy to create H“ by placing a second

electron on this level, compared to the case when H is at TTe or BC sites, where the

141

Table 6.1: The CdTe band gap, formation enthalpy and the formation energies of the
lowest energy defects and charge states. All values are given in eV.

 

CdTe Formation energies
Egap (at I‘) Egap (average)1 AH(CdTe) H+ at BC site H‘ at TCd site
LDA 0.63 1.92 —0.74 0.29+EF 2.42—EF
GGA 0.56 1.74 —0.93 0.22+EF 2.63—BF

 

defect levels are closer to CBM (Fig. 6.6(b) and Fig. 6.8(a)).

Our calculations suggest that H0 is not stable in CdTe. As shown in Fig. 6.10,
the formation energies of H+ and H— are less compared to H0 for all values of E F
throughout the band gap. When E F is closer to VBM (p—type samples), H gives up
its electron and therefore acts as a donor, compensating the holes in the VB. If E F
is raised above the midgap (n—type samples), H acquires an electron, acting as an
acceptor. This suggests that H acts as an amphoteric impurity, always compensating

the intentional donors or acceptors in CdTe.

6.6 Comparison between the LDA and GGA re-

sults

It is known that the results of the DFT electronic structure calculations on a given sys-
tem can be slightly different, depending on the approximation used for the exchange-
correlation functional. The LDA and the GGA usually produce different values for
the lattice parameters, band gaps, cohesive energies, formation enthalpies. Due to
the overbinding character of the LDA, it provides smaller lattice constants, compared
to the experimental values. The GGA on the other hand, tends to overestimate the
structural parameters. In the case of CdTe we find that the theoretical lattice con-
stant calculated with the LDA and the GGA are ~0.6% smaller and ~3% larger than

the experimental value of 6.46 A [140], respectively.

142

AHf (eV)

 

 

 

 

0.0 . . e . . . 5 .
0.0 0.4 0.8 1.2 1.6

EF(eV)

Figure 6.11: Comparison between the LDA and GGA results for the formation ener-
gies of the lowest energy hydrogen defects in CdTe. Despite the small quantitative
variations, there is good qualitative agreement between the LDA and GGA results.

In order to see how the defect formation energies depend on the choice of the
exchange-correlation functional, we performed both LDA and GGA calculations on
the systems with the lowest defect formation energies (i.e. hydrogen at BC and TCd
sites). The comparison between the LDA and GGA results is shown in Fig. 6.11 and
the calculated numerical values are listed in Table 6.6. We observe that the LDA
provides higher formation energy for H+ at BC site compared to the GGA. On the
other hand, the formation energy of H“ at TCd site as calculated with the LDA, is
lower than the GGA result. As pointed out in Ref. [203], these variations in the defect
formation energies can be ascribed to the LDA/GGA differences in cohesive energies,
binding energies, formation enthalpies and band gaps. For example we find that in
the case of pure CdTe, the heat of formation is --0.74 eV in the LDA and —0.93 eV in

the GGA, the latter being in better agreement with the experimental value of —0.96

 

1the VBM and CBM are taken as average over all special k-points where the total energy was
calculated (see Ref. [128])

143

eV [199]. Also the chemical potential of hydrogen is 0.04 eV smaller in the GGA [204].
Since the energy position of the defect levels relative to the band edges is influenced
by the size of the gap, the differences between the LDA/GGA band gaps could also
influence the formation energies. However, despite the quantitative differences, there

is good qualitative agreement between the LDA and GGA results.

6.7 Summary

The formation energy calculations indicate that the ground state position of the H
inside the CdTe lattice depends on its charge state. The lowest energy position for
H0 and H+ is at the BC site while H- prefers the TCd site. We find that H in CdTe
acts as an amphoteric impurity: in p—type materials, H is in a positive charge state
acting as a donor to neutralize the free holes in the VB and in n-type materials, H
acquires an electron, compensating the donors in the sample. We find that despite
some small quantitative variations, the LDA and the GGA provides qualitatively
similar results. The configuration in which H occupies the Cd vacancy site undergoes
J ahn—Teller distortion due to the presence of the partially filled t2 (p-like) level at the
top of the VB. As a result, the H atom moves closer to one Te atom, lowering the
symmetry of the defect center. The degeneracy of the t2 level is removed and the
total energy of the supercell is lowered by 1.22 eV, which is an extremely large effect.
The Jahn-Teller splitting of the t2 level at the F—point is 74 meV.

144

Chapter 7

Ytterbium containing systems

7 .1 Introduction: the LDA+U method

We have mentioned in Sections 1.3 and 2.3.3 that the LDA, as an approximation,
cannot be successful for all systems, although in principle, the exact DFT should
be capable of obtaining the ground-state properties. The deficiency of LDA is most
clearly seen in the case of strongly correlated materials, such transition metal (TM)
or rare-earth (RE) metal containing systems, with partially filled d (or f) shells.
When a one-electron method with an orbital-independent potential (like the LDA)
is applied to TM (or RE) compounds, the result is a partially filled d (or f) band
with metallic-type electronic structure and itinerant d (or f) electrons. This result is
incorrect for the late-transition—metal oxides and rare-earth metal compounds where
d (and f) electrons are well localized and usually there is a large energy separation

between occupied and unoccupied subbands.

There were several attempts to improve the LDA in order to take into account
strong electronelectron correlations. One of the most popular approaches is the self-
interaction correction (SIC) method [205]. It reproduces quite well the localized

nature of the d (or f) electrons in transition metal (rare earth metal) compounds,

145

but the SIC one-electron energies are usually in strong disagreement with spectroscopy

data.

The Hartree-Fock (HF) method [206] is appropriate for describing Mott insulators
because it explicitly contains a term which cancels the self-interaction. The fact that
the self-interaction is treated in an averaged way in the LDA is the main reason for
which the LDA gives the qualitative disagreement with experimental data. However,
a serious problem of the Hartree-Fock approximation is the unscreened nature of the
Coulomb interaction used in this method. The “bare” value of Coulomb interaction
parameter U is rather large (15-20 eV) while screening in a solid leads to much smaller
values: 7-8 eV or less [207, 208]. Due to the neglect of screening, the HF energy gap

values are a factor of 2-3 larger than the experimental values [206].

The problem of screening is addressed in the GW approximation [110,209], which
may be regarded as a HartreeFock theory with an orbital-dependent screened Coulomb
interaction. The CW has been applied with success to real systems ranging from sim-
ple metals to transition metals but applications to more complex systems have not

been feasible up to now due to the large computational task.

In Section 2.3.3 we indicated that a successful way to overcome the LDA deficiency
in describing strongly correlated systems is the so-called LDA+U method [84—86]. In
this method the orbital dependent Coulomb interaction is taken into account for
the localized states, while the delocalised states are treated by the standard LDA
potential. This idea is reflected in Eq. (2.48), in which the on—site Coulomb interaction
is considered by the addition of the Hubbard like term to the LDA energy functional.
Naturally, in order to avoid double counting, the corresponding average d—d (or f - f )
Coulomb interaction energy must be subtracted. This Coulomb interaction energy
given by LDA is a good approximation, however, the orbital energies (eigenvalues)

are wrong.

If we rewrite Eq. (2.48), in which (for simplicity) the exchange interaction and the

146

anisotropy of the Coulomb interaction within the locaized shell are neglected:

I
E=ELDA—UN(N—I)/2+§UE ninj, (7.1)
2.75.7
where N = Z,- n;, we can calculate the orbital energies e,- as derivatives of (7.1) with

respect to the orbital occupancies 72;:

3E 3E I I
€i=%;=—3%?A+U(§_ni)=€LDA,i+U(_2'—ni) (7.2)

This above equation shows that the energy of the occupied orbitals (712 = 1) is shifted
by —U / 2 and the energy of the unoccupied orbitals (n, = 0) is shifted by +U / 2. The
orbital dependent potential can be calculated in a similar way starting from (7 .1),
but the variation is taken not on the total charge (occupation) associated with the

orbital i, but the charge density of the ith orbital:

V.- (a = 33% = VLDA,z’ + U (é -— n) (7.3)

The additional contribution to the LDA potential is independent of F. The one-
electron potential given by the above expression has a discontinuous behavior as a
flmction of the occupation number. From Eq. (7.3) we see that the one-electron,
orbital-dependent potential is attractive for orbitals occupied by more than 1 / 2 elec-

trons and it is repulsive for orbitals occupied by less than 1 / 2 electrons.

If the exchange interaction is taken into account, then the interaction energy for
electrons with the same spin projection a will be (U — J). Furthermore, the non-
sphericity of the Coulomb and exchange interactions (dependence on which particular
orbital m and m' are occupied or empty) can also be considered by introducing the

matrices Umm/ and Jmml. The energy functional in this case takes the form:

147

E = ELDA — [UN(N —1)/2 — JN(N — 2)/4] +

1
+5 2:, Umm/nmanm/,_U+

m,m ,0

Z (Umm’ - Jmm’)nm0nm’a

maém’ ,m’ ,0

(7.4)

+

[\DIH

The above scheme was developed by Anisimov et al. [85]. The doublecounting
term (the second term in Eq. (7 .1) and (7.4)) has been chosen to ensure an atomiclike
limit of the LDA, which approximately removes the electron self-interaction, hence
it is usually referred to as the “self-interaction corrected” (SIC) LDA+U method.
A slightly different version of the LDA+U method was formulated by CzyZyk and
Sawatzky [86], in the literature, referred to as an “around mean field” (AMF) cor-
rection. The former scheme (SIC) is usually more appropriate for highly localized
electrons while the latter (AMF) might be better for weakly correlated systems. The
detailed description of the two methods can be found in the literature [85,86], but we
can easily illustrate the difference between AMF and SIC models by considering only
the diagonal elements of the density matrix and neglecting the exchange interaction.

The orbital-dependent corrections to the LDA potential have then the following form:

VAMF = Manhunt). -4...) (7.5)

1
VSIC = VL(S)DA + U (5 — nm) (7.6)

where 723 = 2T1¥I Em 737710 is the average occupation and nma' is the occupation
number of orbital m with spin a. The difference between the two methods is clear:
assuming positive U, the AMF correction to the potential is attractive for orbitals
occupied with more than the average and repulsive for states occupied with less than

the average occupancy. The SIC potential (as we have already mentioned) does the

148

same for states occupied by more or less than one-half.

Despite its considerable success, the L(S)DA+U approach has several shortcom-
ings. First, strictly speaking, it is not a fully ab initio method, since it contains two
parameters which determine the orbital potential felt by the localized d or f states:
the Hubbard and exchange parameters U and J. The values of these parameters can
be determined by comparing the positions of the d (or f) levels obtained from x-
ray photoemission spectroscopy (XPS) measurements to the theoretical value. It has
been found that the physically reasonable values of the Coulomb term U for the f
states are in the range of 5-9 eV and the values of the exchange term J are ~10% of
U [210]. There are also theoretical methods to find the value of U and J based on
electronic structure calculations using supercell methods [208]. The second problem
with L(S)DA+U comes from the fact that in the case of partially filled d or f states,
the electrons can be arranged in many different ways over the different orbitals, and
it is possible that in each case a converged solution is obtained. In other words, the
self-consistent cycle can easily be trapped in a local energy minimum, which may not
necessarily be the true ground state. In order to overcome this problem of L(S)DA+U,
one can perform “constrained density matrix” (CDM) calculations [66,211]. The main
idea in the CDM method is that one can do a series of calculations keeping the density
matrices unchanged (the diagonal elements of the density matrix represent the orbital
occupation number) and then, based on the fact that L(S)DA+U (for fixed U) is a
variational method [212], one can use the total energy criterion to obtain the ground
state configuration. In Ref. [66] the method was applied successfully to CeSb and it
was found that the lowest-energy Ce f1 configuration corresponds to the case when
m = —3 spin-up state was occupied (in agreement with Hund’s rules). This result is
relevant to our case, since Yb is often considered as “one f-hole” analogue the Ce f1
configuration. A more recent example of CDM calculation can be found in Ref. [211],

where the properties of RE—ion impurities in Fe host lattice have been investigated.

149

 

Figure 7.1: The crystal structure of YbNiFeAlg viewed down on the z-axis. In this
figure the M1 and M2 sites are occupied by Ni and Fe respectively

In the next sections we apply the LDA+U method combined with the CDM ap-
proach to several Yb—containing intermetallics to investigate the role played by the

Yb ion in the electronic and magnetic properties of these materials.

7.2 Electronic structure of YbNiFeAlg

YbNi2_xFexA18 crystallizes in the Ca002A18 structure type in the space group Pbam.
The existence of a large number of isotypical ternary phases [213—216] indicates that
this is a rather stable structural arrangement. YbNi2_xFexA18 seems to be the first
quaternary analog of this structure type. In the structure of YbNi2_xFexA18, a
total of 12 atomic sites including one Yb, two transition metal sites (M(1) and M(2)
occupied by Ni / Fe) and nine Al sites were identified. Figure 7.1 depicts the structure
of YbN12_xFexA18, viewed down the z—axis. The M(1)— and M(2)-centered polyhedra,
composed of Al atoms, form a three-dimensional framework with Yb ions sitting in

small channels.

150

In order to understand the nature of bonding between different atoms and the
orbital distributions of Yb f electrons and Ni and Fe d electrons, we have performed
electronic structure calculations in a series of compounds. According to the single-
crystal structure refinement, the transition metal sites M(1) and M(2) are occupied
by a mixture of Ni/Fe: 42%/58% on M(1) and 62%/38% on M(2) [217]. To model
this mixed occupancy we constructed a 1x1x2 supercell, which corresponds to placing
two unit cells next to each other along the [001] direction. In one unit cell the M(1)
and M(2) sites were assigned as Ni and Fe respectively, while in the second unit cell
the assignments were switched, i.e. Fe on M(1) and Ni on M(2). This supercell model
corresponds to 50% / 50% occupancy of Ni / Fe on both M(1) and M(2) sites in a quasi-
periodic structure, which is close to the experimental observation. The total number

of atoms in the supercell is 88 (8 Yb, 8 Ni, 8 Fe and 64 Al atoms).

Before discussing the results of our calculations we make a few general remarks
about the ability of LSDA+U calculations to describe the physics of Yb (f 14 and f 13)
systems, particularly the mixed valence ground state. There are many calculations
which give the Yb3+ (f 13) state properly (e.g. Refs. [218,219]). In these calculations
13 f states lie below the Fermi level (EF) and 1 f state lies above E F, the splitting
between the occupied and unoccupied levels being ~U. The calculations by Antonov
et al. [219] on YbInCu4 system, using the LMTO method indicate that the above
single f level is pinned to the Fermi energy thereby giving a mixed-valent ground
state for this compound, which is in agreement with experiment. This pinning was
found to be not very sensitive to the exact value of U as long as U was greater than 5
eV, but it depended sensitively on the density of states associated with non- f states.
We have redone the calculation in this compound using all-electron FPLAPW method
and found the same result. Thus we conclude that LSDA+U calculation is capable

of giving a mixed-valent ground state.

The total density of spin—up and spin-down states for YbNiFeAlg are shown in

151

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

120 r . + 120 1
90 (a) —— total spIn-up DOS 80‘ (b) — total spin-dn DOS
60
A 40
.s 30
Q 0 . 0 + i
Q 18 (c) — _spln-Up: Yb1 f —.spin-dn: Yb1 f
> 12 — Spln-dn: Yb2 f - spin-dn: Yb2 f
g 6 ][
9 l I JL ,
(U 3.3 ; spin-up: Ni1 d spin-dn: Fe1 d
*0; (e) 5'. —— spin-up: Ni2 d - spln—dn: F92 d
v 2.2 1 7:
U) 1.1 , i.
0.0 I 0.0 . - . . -
D ——spin-up: Al10 p a spin-dn: Al9 p
3'3: (9) ; :2 spin-”fig? p 8.2): (h) ' ’- . spin-dn: Al? p .3
00910-8 -6l-4 -2 0 2 4 6 80'0910-8 35 4 -2 0 2 4 6 3
Energy (eV) Energy (eV)

Figure 7.2: Results from the electronic band calculations for YbNiFeA18.(a) and (b):
Total spin-up and spin-down DOS of YbNiFeAlg. (c) Yb 4f spin-up DOS, with 6
occupied states and one hole located at 2 eV. (d) Yb 4f spin-down DOS, with all 7
states occupied. (e) Ni(1) 3d and Ni(2) 3d DOS. (f) Fe(l) 3d and Fe(2) 3d DOS. (g)
and (h) p states of Al atoms located around M(1) (A110 and A19) and located around
M(2) ( A111 and A15)

Fig. 7.2 (a) and (b) respectively. The main features of these two DOS figures are the
sharp peaks located between —5 eV and —7 eV below E F) which is set to be at 0 eV.
They are split because of the spin-orbit (SO) interaction and the anisotropy of the
Coulomb interaction within the 4f shell of the Yb ions. The peak (width ~0.2eV)
located nearly 2 eV above the E F, which is only present in Fig. 7.2 (a), represents the
main difference between the spin-up and spin-down DOS. This band originates from

the hole in the 4f spin-up state of Yb with ml = +2 character. Convergent solutions

152

were also obtained for unoccupied [3, T), [1, T) and [0, T) states, but it was found that

the lowest energy configuration corresponded to the unoccupied [2, T) state.

The splitting within the f shell can be seen more clearly in Fig. 7.2 (c) and (d),
which show the projected DOS of Yb 4f states. The left panel (Fig. 7.2 (c)) displays
the spin-up DOS, with six out of seven states fully occupied, while the seventh spin-
up state, located at 2 eV above E F, is empty. In Fig. 7.2 (d), all seven 4f spin-down
states are situated below E F From this picture we can conclude that the valence
state of Yb is 3+. Another character shown in both total spin-up and spin-down DOS
Fig. 7.2 (a) and (b) is the uniform background due to Al, Ni, and Fe states, as well as
a finite DOS at the Fermi level, indicating the metallic nature of the system. This is
consistent with very small value of thermopower observed in this system (~-1 [JV/ K
at room temperature) [217]. The broad hump in the DOS located between E F and -4
eV comes from the d states of Ni and Fe. The Ni 3d bands (Fig. 7 .2 (c)) are located

well below E F, suggesting a filled 3d10

configuration. Therefore, the Ni is essentially
reduced to a diamagnetic state (d8s2 —> dlo). This is also evident from the spin-up
and spin-down DOS, which are almost identical for both Ni atoms (only the spin-up
DOS is shown). This type of behavior in late transition metals has been observed in
several intermetallic systems containing more electropositive elements [220]. The Fe
3d spin-up states, shown in Fig. 7.2 (f), have a significant contribution to the DOS
near the Fermi level, indicating that these states are not fully occupied. However,
the spin-up and spin-down DOS of the Fe d levels are identical, just as they were in

the case of Ni. Thus Ni and Fe do not show a local magnetic moment.They, however,

contribute to Pauli paramagnetic susceptibility.

An interesting feature, shared by both Ni and Fe d states, can be observed from
Fig. 7.2 (e) and (f): they become energetically more stable when the atom occupies the
M(2) site. The difference in the DOS of the d levels of the Fe occupying different sites

in the supercell is in agreement with the result of the Méissbauer spectroscopy [217]

153

which shows different local charge densities at two different Fe sites in the structure.
The coordination environments of M(1) and M(2) sites are similar in that both are
surrounded by 9 Al atoms, but there are fundamental differences. The transition
metal atoms located at the M(1) sites form dimers. It seems that the dimer geometry
reduces the degree of mixing between some Al p and N i/ Fe d orbitals. This can be
verified by analyzing the contribution to the DOS of the p orbitals of several different
Al atoms. Figure 7 .2 (g) and (b) show the p states of Al(lO) and Al(9) from the tri-
capped trigonal prism surrounding M(2) as well as the p states of Al(ll) and Al(5)
surrounding M(1). It is evident that the Al p states hybridize more with the d states
of Ni(2) and Fe(2), than with the d states of Ni(1) and Fe(l). Because of the higher
degree of p — d hybridization between A1 and N i / Fe located on M(2), these states are

energetically better stabilized.

Next we discuss the magnetic properties of the system coming primarily from the
unfilled Yb f shell. The calculated spin and orbital magnetic moments for the two
Yb ions are: ,uspm = 1.0059143; 1401.), = 1.98111IB for Yb1 and uspm = 1.0044113;
no”, = 1.981014 B for Yb2. This adds up to approximately 2.99 p B per Yb ion
suggesting a strong atomic character and trivalent configuration. The fact that the
13 occupied and the 1 empty f states are far removed from the Fermi energy argues
against a mixed-valence picture. A similar result showing a hole in the 4f shell of Yb
was recently obtained on Yth2812 [218]. This kind of electronic structure, where
the empty and occupied f states are well separated and located far from E F, together
with the absence of long range magnetic order, suggest the possibility of Kondo—lattice
scenario. We have to emphasize, however, that DFT with LSDA+U approximation is
an effective single particle theory and therefore cannot describe the complex correlated
many-body ground state of a Kondo lattice system. Finally, the results presented here
point to the need to investigate in more detail the parent compound YbC02A18 with

respect to magnetism, the role of Co atoms and the electronic state of Yb.

154

   

(8)

Figure 7.3: (a) the top view and the (b) side view of Yb3Au3In3 hexagonal crystal
structure. Yb3AuGe21n3 is obtained by replacing the two Au atoms/ cell indicated
by the arrows

7.3 The valence state of Yb in Yb3Au3In3
and Yb3AuGegIn3

Yb3Au3In3 crystallizes in a hexagonal system in the space group P62m [221]. The
crystal structure, shown in Fig. 7.3, can be regarded as three Au-In dimensional
framework, in which one dimensional channels run along the crystallographic c—axis.
These channels are occupied by 1D chains of Yb atoms. The figure also indicates that
Yb3AuGe21n3 is obtained by replacing two Au atoms in a unit cell by two Ge atoms.
The crystallographic analysis of Yb3Au3In3 and Yb3AuGe21n3 reveals a peculiar
feature: on the one hand the a- and b-axis of the Yb3Au3In3 are larger than those of
Yb3AuGe21n3. This, of course, can be related to the fact that Au atoms are larger
than Ge atoms. On the other hand, the c-axis of Yb3Au3In3 is 10% shorter than
that of Yb3AuGe21n3, which cannot be understod by the size difference between Au
and Ge. We will try to explain this feature, using electronic structure calculations on
Yb3Au3In3 and Yb3AuGe21n3 focusing on the role played by the Yb f electrons in

the electronic and structural properties of the two systems.

155

 

2.0
1.0
0.0 E:

—1.0

-2.0

—3.0

40; \

-5.0§

30;

4.05

-8.0 =\

 

 

 

 

 

Energy (eV)
Energy (eV)

 

/(

 

 

 

 

 

 

 

 

 

(a) (b)

Figure 7.4: (a) Band structure of Yb3Au3In3 with the In 3p orbital character em-
phasized. The unoccupied 4f level lies at 1 eV above EF. (b) Band structure of
Yb3AuGe21n3 showing the Ge 3p orbital character. The band structures were calcu-
lated for U = 6.75 eV.

The band structure obtained for Yb3Au3In3 is shown in Fig. 7.4(a). The states
near the Fermi level are associated with In p, Au p and Yb d orbitals. Yb 5d bands
are mostly unoccupied, but they hybridize with In p states therefore we find Yb
5d character even below E F- The flat band located at 1 eV above E F represents an
unoccupied level in the 4f shell of the Yb atom. The occupied Yb 4f bands lie between
—5 eV and —7 eV below E F: they are split into 4 f5 /2 and 4 f7 /2 manifolds under
spin orbit (SO) interaction of approximately 1.3 eV. The two spin-orbit complexes
are further split due to the anisotropy of the Coulomb interaction within the 4f
shell [212] and possibly due to crystal field eflect. All the bands originating from
Yb f orbitals are quite flat, and hybridize weakly with the other electronic states.
Unlike in YbInCu4 and YbNiFeAlg we find that the splitting between the empty and
occupied 4f levels is very sensitive to the value of the Coulomb repulsion U. This is

shown in Fig. 7.5(a) where we plot the spin-up DOS associated with the Yb f states

156

in Yb3Au3In3 for different values of U. We observe that the empty f level is the one,
which shifts up or down on the energy scale as the value of U increases or decreases,
while the occupied bands remain mainly at the same position. However, at band-
theory level, for physically acceptable values of U [210], the occupied and empty 4f
states are well separated (the separation between them being approximately equal to
the U value) and located sufficiently far from E F, indicating a trivalent configuration
for the Yb ion. In order to calculate the f occupation number we have integrated
the DOS associated with the Yb 4f bands from —15 eV to E F: The results obtained
for different U’s are listed in Table 7.1 indicating that in Yb3Au31n3, 13 electrons
occupy the Yb 4f bands.

In contrast, the electronic structure calculations performed on Yb3AuGe21n3 sug-
gest quite a different picture for the Yb valency. The band structure of Yb3AuGe21n3,
shown in Fig. 7.4(b), displays an interesting feature: one flat band, originating from
Yb f levels is located right at E F- An important aspect of this observation is that the
position of the Yb f level, which is pinned at E F, does not depend on the U value.
This can be observed in Fig. 7.5(b), which shows the DOS associated with the Yb
f levels in Yb3AuGe21n3, calculated for different U’s. The upper f state is pinned
to E F, while the other f levels are slightly shifted up or down in energy, depend-
ing on the value of the Coulomb repulsion U. Since Yb3Au3In3 and Yb3AuGe21n3
are isostructural systems and the latter is obtained by replacing two Au atoms in
Yb3Au3In3 by two Ge atoms, the Fermi level pinning must be related to the pres-
ence of Ge. When Au (5d10631) is replaced by Ge (3d104s24p2) the number of valence
electrons in the system increases, therefore the Fermi level is shifted up. The Ge p
band is not completely filled, so that, the bands near E F have mostly Ge p character.
Since the upper Yb f level, which was initially located above E F, now lies below
the top of Ge p band, becomes partially occupied and pinned to E F- Therefore the

position of the upper Yb f level with respect to the Ge p states becomes essential in

157

 

 

 

 

 

 

 

A14; U=5.44 eV ]

.E l

3 71

S 0, ............... L I

§ 18« U =6.75 eV

... 9-

:3, 0; ............... 4L, ,,,,,,

a) 16- U =9.52 eV

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0‘ . JL 1
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Energy(eV)

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01163 U= 8.16eV
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(b)

Figure 7 .5: Spin polarized density of states associated with the spin-up Yb f electrons
for (a) Yb3Au31n3 and (b) Yb3AuGe21n3. In the case of Yb3Au31n3 the unoccupied
f level is shifted up in energy as the U value increases. For Yb3AuGe21n3 the Fermi
level (zero on the energy scale) is pinned to the upper f level.

understanding the Fermi level pinning in Yb3AuGe21n3.

To find the f occupation number and the degree of Yb valency, we have inte-
grated the DOS of associated with Yb 4f. The results for difierent U’s are listed

in Table 7 .1. We observe that the 4f shell is partially occupied, with an occupe

158

 

Table 7.1: The calculated f occupation for different values of the Coulomb correlation
U

 

 

U value (eV/ Ry) f occupation number
Yb3Au3In3 Yb3AuGe21n3

5.44/04 13.01 13.50

6.75/0.5 12.92 13.31

816/06 13.42

9.52/07 12.92

 

tion number of between 133-13.5. Therefore, according to our electronic structure
calculation, which includes magnetic and relativistic effects as well as intra-atomic
correlation (in L(S)DA+U approximation), there is an essential difference between
the two systems: in the Yb3Au3In3 the Yb ion is in trivalent configuration (Yb3+),
with one hole in the 4 f shell, while in Yb3AuGe21n3, Yb is calculated to be interme-
diate valent (Yb2+/3+), with a valency of approximately ~2.6. As a restult, a series
of physical properties, closely related to the Yb valency, will be different in the two

systems.

The interesting behavior of the lattice parameters, mentioned at the beginning of
this section, can also be associated with the valence configuration of the Yb atoms.
Since Ge is smaller than Au, one would expect an overall decrease in the lattice
parameters of Yb3AuGe21n3 compared to those of Yb3Au3In3. However, in the
presence of Ge, the valency of Yb becomes Yb2+/3+. Since Yb2'l'/3+ is larger than
Yb3+, the overall decrease in the lattice constants will be balanced by the increase in
the size of Yb. If we consider the crystal structure of Yb3Au3In3 as being formed by
alternating monoatomic sheets of In-Au and Yb—Au stacked along the c-axis we have
the following situation: on the one hand the In-Au sheet shrinks when Au is replaced
by Ge, but on the other hand, the Yb-Au sheet expands due to the increase in size
of the Yb. The net effect is a decrease in the a- and b- and an increase in the c-

parameter. To verify this point we have performed structural relaxation calculations

159

why

 

on two hypothetical systems La3Au31n3 and Lu3Au3In3. The two systems were
chosen to simulate the change in the size of the Yb atom and to avoid complications
arising from the presence of the partially filled f levels, at the same time. The
obtained theoretical lattice parameters for La3Au3In3 were a = 7.89 A; c = 4.36 A
and for Lu3Au31n3 a = 7.85 A; c = 3.76 A. We observe that when La is replaced
by Lu the decrease in the lattice constant a is ~0.5%, while the decrease in the c
parameter is ~14%. From this theoretical study we conclude that the size of the Yb

atom primarily influences the length of the crystallographic c-axis.

7 .4 Summary

In summary, YbNi2_xFexA18, is the first quaternary analog of the YbC02A18 struc-
ture type and can crystallize from excess liquid A1. A mixed Ni / Fe occupancy appears
to be present on the two transition metal sites (M1 and M2) in the structure. The
studies of the magnetization behavior shows that the ab plane is the easy plane on
which the magnetic moments are confined. Both resistivity and thermopower mea-
surements on YbNi2_xFexA18 point to a strongly metallic character. Temperature
dependent magnetic susceptibility measurements and electronic structure calculations
(at T=OK) using the LSDA+U formalism suggest that in these systems the Yb atoms
are most likely in the f13 configuration. All magnetism in the compound derives from
Yb3+ species and not from the Ni or Fe atoms. 57Fe Mossbauer spectroscopy con-
firms that the Fe atoms do not have a local magnetic moment, consistent with the
theoretical calculations.

Using ab initio electronic structure calculations, we found that the differences be-
tween the physical properties of two isostructural, Yb-containing systems are strongly
related to the valence configuration of the Yb atom. Yb3Au3In3 is an integer valent

system, with one hole in the Yb 4f shell. We can also mention that the calcu-

160

 

lated electronic structure of Yb3Au3In3, with the occupied and unoccupied f levels
well separated, and located far from E F suggests Kondo behavior for the system.
Yb3AuGe21n3, on the other hand, was calculated to be an intermediate-valent com-
pound, with approximately Yb2'6+ configuration. Fermi level pinning was observed

in this system, indicating that Yb3AuGe21n3 is possibly a heavy fermion system [69].

161

Chapter 8

Summary and future directions

The III-VI semiconductors (GaS, GaSe, GaTe, InSe) have been subject of extensive
research because their potential for use in optical applications. GaSe crystals show
great promise in nonlinear optics (frequency mixing, second harmonic generation),
and has been proposed as a far-infared conversion material. There is research interest
in the structure and electronic properties of GaTe because of the possibility that it,
or related compounds, may have applications in the electronics industry. InSe has
the potential for use in photovoltaic devices and it has been the subject of extensive
investigations. The crystal structure of these semiconductors are similar. They are all
layered systems: the strong covalent interactions within the layers and the weak, van
der Waals-type coupling between them provide an interesting, quasi—two—dimensional

character to the physical properties of these materials.

Throughout the first part of this thesis, the importance of doping and its impact
on the properties of semiconductors have been emphasized and explored. In the
case of the III-VI family, a vast body of experimental knowledge is available on
the subject of defects and impurities, especially in GaSe and InSe. The electrical
and optical properties in the presence of numerous impurities (such as Cu, Cd, Zn,

Sn, Mn, Cl, In, As, Bi, Sb, Te, Si, Ge) have been investigated and many impurity

162

 

induced defect levels and binding energies have been detected and measured [8,29—41].
Comparatively little theoretical work has been devoted to the this class of materials
and until now, there was a complete lack of theoretical approach to the problem of

defects and impurities in III-VI family of semiconductors.

The detailed microscopic understanding of the nature of defect states in semicon-
ductors, in general, is a difficult problem and it is specially challenging in the case of
the III-VI family of valence mismatched systems. In this thesis the defect problem has
been addressed by carrying out electronic structure calculations (hydrogenic effective
mass approximation and self-consistent DFT) in pure and defect containing GaSe
and GaTe. The failure of the effective mass approximation to predict the binding
energies of Cd and Sn substitutional impurities in GaSe, indicated the necessity of
incorporating the short range interaction into the theory [173]. The formation mech-
anism of the semiconducting band gap in these systems has been explained based
on the interaction between the Ga 3 dimers states (bonding-antibonding states) and
the surrounding Se (or Te in case of GaTe) p orbitals. Using this simple molecular
orbital model, the general features of the Ga—site defects can be understood: when
Ga is replaced by an impurity or it is removed from the system to form a vacancy, the
dimer states are perturbed giving rise to localized states in the band gap. The posi-
tion of the defect state in the band gap, of course, depends on the energy mismatch
between the impurity orbitals and the Ga .9 orbitals. In the case of acceptors impu-
rities, such as substitutional Cd on Ga site (CdGa), the position in the bandgap of
the single particle eigenstates associated with the defects are in reasonable agreement
with the experimental binding energies [41]. In contrast, the binding energies of the
donor states (SnGa) and of those split off from the conduction band in the presence
of vacancies, can not be estimated reliably, because of the problems associated with
the band gap and excited states calculated with LDA/GGA [154]. It is also possible

that the energy mismatch between a Ga—site defect orbitals and the Ga 3 orbitals is

163

 

'v- . ur wary

 

not large enough to produce defect levels in the gap. This is the case of Inga where
the difference between the Ga 3 (-11.55 eV) and In 3 (-10.14 eV) orbitals is too small

to perturb the dimer states and to give rise to defect states in GaSe.

Formation energy calculations associated with defects in GaTe show that the dom—
inant intrinsic defect is the gallium vacancy (VGa)- It introduces a deep (acceptor)
defect state in the gap, located at 0.78 eV above the VBM, as measured at the I‘-
point. This value is in good agreement with the calculated charge transition level
€(-1/ — 2) = 0.83 eV and experimental results [43,189]. The formation energies of
Ge and Sn impurities reveal the under Ga-rich growth conditions it is easier to incor-
porate Sn in GaTe, whereas at the Te-rich limit Ge becomes more soluble than Sn.
This information about the solubility is important because Ge and and Sn can be used

to reduce the large leakage current in GaSe and GaTe by carrier compensation [136].

The electronic structure and mechanical properties of GaSe doped with isova-
lent impurities (In and Te) has also been investigated in this thesis. According to
the formation energy calculations, Te and In prefer to occupy the substitutional Se
and Ga sites in GaSe, respectively. The calculated charge transition levels associ-
ated with different Te and In defect configurations have been compared to available
experimental data [31, 32,39]. The good agreement between the theoretical and ex-
perimental findings indicates that the ab initio electronic structure method is quite
robust and can be used as a predictive tool for certain defect propertiae in semicon-
ductors. In order to understand the mechanism of rigidity enhancement observed
experimentally in doped GaSe [163,164], the elastic properties of GaSe1_xTex and
Ga1_xInxSe have been studied as a function of composition 2:. The calculated elastic
constants of Ga1_xInXSe show a very small increase in the elastic stiffness in the
direction perpendicular to the atomic layers. The detailed microscopic description
of mechanism responsible for the rigidity enhancement can be explained by the In

interstitial defect configuration._ In p-type GaSe, In impurity can acquire +3 charge

164

 

 

state and can occupy an interstitial site between the atomic layers. The calculations
show that the energy barrier which must be overcome in order to shear the GaSe
crystal increases dramatically when In occupies the interstitial site. Therefore thi is
most likely the main source of the observed improvement of the structural properties
of In doped GaSe [163,164]. Further careful measurements of GaSe doped with con-
trolled amount of In under different carrier doping concentrations (n-type od p-type)

are suggested.

Understanding the nature of H defects in semiconductors has been a long stand-
ing physics problem. There have been extensive studies of H defects in group IV
and III-V semiconductors. Similar studies in II-VI semiconductors are limited. In
this thesis we look at the problem of H defect in CdTe. The location and the elec-
tronic structure of H in CdTe has been investigated using first-principles theoretical
methods. The mechanism of defect formation of H in CdTe is explained using simple
molecular orbital models. The electronic structure of the different defect configura-
tions were analyzed using ab initio DFT calculations. The location of the H inside
the CdTe lattice was obtained by calculating total energy valuae with H located at
many different sites and with the defect in different charge state configurations (-1,
0, +1) . It has been found that H0 and H+ prefers to occupy the bond-center site
(BC) located between the Cd and Te atoms, while H— prefers the low charge density
site tetrahedrally coordinated by Cd atoms (TCd). The configuration in which H is
located at the substitutional Cd site is susceptible to Jahn-Teller distortion, due to
the presence of the partially occupied t2-level (p-like state) at the top of the VB.
Electronic structure calculations show that by lowering the symmetry of the system,
the degeneracy of the t2-level is removed and the total energy of the supercell is de-
creased by 1.22 eV. The calculated J ahn-Teller splitting of the tg-level at the F—point
is 74 meV. H in CdTe acts as amphoteric impurity; it always takes the charge state

which counteracts the prevailing type of conductivity in the system.

165

 

The electronic structure and the valence state of Yb in novel ternary and qua-
ternary Yb—based systems have been investigated within the LDA+U formalism. It
has been found that electronic, magnetic and structural properties of these systems
strongly depend on the valence configuration of Yb. The LDA+U approach sug-
gests that the Yb atoms in YbNIFeA18 are in the trivalent state and all magnetism
derivrs from the Yb3+ species. The Ni and Fe atoms do not have local magnetic
moments. The differences in the crystal structures of Yb3Au3In3 and Yb3AuGe21n3
have been explained based on the size difference between Yb3+ and Yb2+/3+ ions.
Even though LDA+U is an effective single particle method, it can be useful in pre-
dicting complicated many-body effects in strongly correlated systems. For example,
the integer valent system Yb3Au3In3, with well separated occupied and unoccupied
f levels, located far from the Fermi level is susceptible to Kondo effect. On the other
hand, the intermediate valent system Yb3AuGe21n3 with the Fermi level pinned to

the partially occupied f level is likely to be a heavy fermion compound.

Finally I would like to make some comments on the use of DFT/LDA and super-
cell methods to understand the defect physics. Density-functional theory [122,123]
has proven to be a very successful tool for the study of a wide variety of problems in
solid state physics. However, in many cases, achieving the required accuracy in total
energy calculations (especially when one has to deal with charged defects) is not an
easy task and requires careful analysis of several sources of errors. In the Kohn-Sham
formalism of DFT (which is most commonly used) the electron-electron interactions
are described with an exchange correlation potential, which is a functional of the
charge density. Since the exact form of this functional is not known, for practical
purposes it has to be approximated, which implicitly introduces sources of errors in
the theory. In the local density approximation (LDA) the exchange and correlation
potential at a particular point is only a function of the charge density at the same

point. A slightly improved version is the generalized gradient approximation (GGA)

166

 

which takes into account not only the density alone, but also its local spatial varia-
tions. Such an approximation fails when the charge density varies rapidly in space.
The LDA and GGA have been extensively tested and found to provide a reliable de-
scription of a wide variety of solid-state properties, for wide band systems. However,
even for such systems, the LDA and GGA have several shortcomings. The classic fail-
ure of LDA and GGA is the underestimation of the band gap in semiconductor and
insulators, which makes it very difficult to calculate the binding energies of various
defects (especially when the defect states are derived from the CB). Another source
of error in DFT calculations is related to the choice of numerical cutofi parameters,
intergration schemes, sampling of the Brillouin zone, construction of basis sets and
pseudopotentials. However, since all these parameters are tractable by explicit con-
vergence tests, the DFT calculations can provide quite accurate estimates of total

energies (~0.01 eV / atom).

In the case of defect calculations using supercell models, one has to deal with other
technical problems such as the spurious interaction between defects in neighboring
supercells, which leads to the dispersion of the defect related electronic states. This
finite size effect can be reduced by a proper scaling analysis as a function of the cell
size. However, for computational reasons, one can not consider very large supercells,
therefore the host bandstructure, which is the natural reference for the defect states,
will always be affected by the presence of defects. Consequently, one has to choose
a reference energy-level and align the bandstructures of the defect containing and
defect free systems in order to get a proper estimate of the defect ionization levels
defects. All these issues make it challenging to obtain quantitatively accurate re-
sults for defects in semiconductors, using supercell calculations. The power of DFT
however, relies in bringing out systematic trends in the properties of defects in semi—
conductors, such as the nature of the deep levels, their spin structure point symmetry,

energetics, local vibrational modes, potential energy surfaces for defects moving in

167

 

the lattice. The results of DF T calculation provide a good starting point for more
advanced calculations where many-electron effects are treated beyond the mean-field
approximation, such as the GW method [110].

Many recent theoretical studies, such as those presented in this thesis, establish
the combination of DFT with the supercell technique as a powerful tool for investi-
gating the properties of defects in semiconductors. One area in which theory is not
yet able to make accurate predictions is the position of defect levels in the band struc-
ture. Methods that go beyond the one-particle description are available (for example
the GW method [110]), but presently too computationally demanding. Increasing
computer power and / or the development of simplified schemes will allow the first-
principles prediction of the defect spectra in the future. The dynamics of impurities
in semiconductors (especially that of H) also requires more scrutiny: investigations
of quantum diffusion, incorporating tunneling effects, and also finite temperature
studies will provide a better understanding of how H defects affect the physical prop—
erties of semiconductors. While many qualitative results presented in this thesis are
doubtlessly of general validity, it is necessary to obtain quantitative answers and
broaden our insights, especially where technological applications are very promising,
such as the study of other defects in III-VI semiconductors, or the behavior of H in

III-V and other II-VI compound semiconductors.

168

 

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