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DEFECT STRUCTURE AND TRANSPORT

PROPERTIES OF NARROW GAP

SEMICONDUCTOR PbTe AND RELATED

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SALAMEH M. AHMAD

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DEFECT STRUCTURE AND TRANSPORT PROPERTIES OF NARROW
GAP SEMICONDUCTOR PbTe AND RELATED SYSTEMS

By

Salameh M. Ahmad

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the Degree of

DOCTOR OF PHILOSOPHY
Department of Physics

2007

ABSTRACT

DEFECT STRUCTURE AND TRANSPORT PROPERTIES OF NARROW
GAP SEMICONDUCTOR PbTe AND RELATED SYSTEMS

By

Salameh M. Ahmad

Understanding the detailed electronic structure of deep defect states in narrow
band-gap semiconductors has been a challenging problem. Self-consistent ab initio
calculations within density functional theory (DFT) using supercell models have been
quite successful in tackling this problem. In this thesis, we carry out such calculations in
PbTe and SnTe, two well-known narrow band-gap semiconductors, for a large class of
defects: cationic and anionic substitutional impurities of different valence, and cationic
and anionic vacancies. For the cationic defects, we study the chemical trends in the
position of defect levels by looking at a series of compounds RQz..-1Te2n, where Q is Pb
or Sn, R is vacancy or monovalent, divalent, or trivalent atom. Similarly, for anionic
defects, we study compounds MsznTeZM, where M is vacancy, S, Se or I. We find that
the density of states (DOS) near the top of the valence band and the bottom of the
conduction band get significantly modified for most of these defects. This suggests that
the transport properties of PbTe and SnTe in the presence of impurities may not always
be interpreted by simple carrier doping (from bound impurity states in the gap) concepts,

confirming such ideas developed from qualitative and semi-quantitative arguments.

Transport coefficient calculations using the Boltzmann equation within energy-
dependent relaxation time approximations have been carried out for n-type PbTe over a

wide temperature (T) and concentration (72) range, 300 K S T S. 900 K and 1 S n/no S

10, no = 5 x 1019cm‘3. The nonparabolic Kane model for the energy dispersion was
used in these calculations following the earlier work of Ravich et. al. Although the T
dependence of the electrical conductivity 0 comes from several sources (band structure
parameters, chemical potential u, scattering relaxation time T), we find that the
T dependence of r dominates. We represent the temperature dependence and the energy

of the electrons (8) dependence of the total relaxation time rm by a scaling function

 

“I“
(1' ~ hwy), where a, b, c, p, r are T and a independent parameters but depend on the

carrier concentration. Using this simple scaling function in the calculation of a, we find
that for these concentrations changing the parameter r which governs the energy
dependence of scattering does not appreciably affect the T dependence of a. In addition
to the study of the T dependence of a, a careful electronic thermal conductivity
calculation (both at constant current J and constant electric field E) was done to
reexamine the validity of the Wiedemann-Franz (WF) law in PbTe, extending the earlier
work of Bhandari and Rowe. We point out that using the standard WF law to estimate the
electronic contribution of the thermal conductivity (K81) usually overestimates this
contribution by more than 0.5 W K'lm", and hence underestimates the lattice thermal
conductivity. This has important implications in the question of how low the lattice

thermal conductivity can go in the presence of nanostructures in PbTe.

To My Parents,

My Wife Haya My Son Omar
My Brothers My sisters
And
My Best Friend Mustafa

iv

ACKNOWLEDGMENT

This work would not have been possible without the support and encouragement
of my advisor Prof. S. D. Mahanti, under whose supervision I chose this topic and began
the thesis. I would like to acknowledge Prof. M. G. Kanatzidis in Chemistry and Prof.
Tim Hogan in Engineering for collaboration on the thermoelectricity project. Deepest
appreciation goes to Dr. D. Bilc who has helped me a great deal in almost all aspects
when I just started my research. Special thanks for K. Hoang and Z. Rak for the valuable
assistance. I would also like to acknowledge Prof. Tessmer, Prof. Piermarocchi, Prof.
Pratt and Prof. Schmidt for the support as members of my Guidance Committee. I would
like to thank D. Simmons and C. Cords for taking care of all aspects of my graduate
study and research. I would like to greatly acknowledge funding from the Office of Naval
Research (MURI program) for financial support.

This joyful moment is especially for my family. My father, my mother, my wife,
my brothers, and my sisters. I should have spent all the time with you all. This moment

belongs to you all.

Table of Content

List of Tables ............................................................................................................................. viii
List of Figures ............................................................................................................................. ix
Chapter 1
Introduction ................................................................................................................................. 1
1.1Why Study IV-VI Narrow Gap Semiconductors ................................................................ 1
1.2 Defects in Semiconductors: Shallow and Deep ................................................................ 3
1.3 Transport Properties of Thermoelectrics ............................................................................. 6
Chapter 2
Defects in Semiconductors ...................................................................................................... 11
2.1 Point Defects ......................................................................................................................... 11
2.1.1 Intrinsic Point Defects ................................................................................................. 11
2.1.2 Extrinsic Point Defects ............................................................................................... 14
2.2 Evolution of Theory of Defects in Semiconductors ........................................................ 17
2.3 Density Functional Theory for Defect Calculations ........................................................ 20
2.3.1 Density Functional Theory ......................................................................................... 21
2.3.2 Full Relativistic Single Particle Dirac Equation ...................................................... 23
2.3.3 Scalar Relativistic Approximation ............................................................................ 24
2.3.4 Method of calculations ................................................................................................ 27
2.4 Supercell Method ................................................................................................................. 31
2.5 Issues with the Supercell Method in Defects Studies ...................................................... 33
Chapter 3
Electronic Structure of Defects in PbTe .............................................................................. 36
3.1 General Introduction ............................................................................................................ 36
3.2 Electronic Structure of PbTe ............................................................................................... 38
3.3 Supercell Model to Calculate Defect Structure ................................................................ 39
3.4 Pb-site Defects ...................................................................................................................... 43
3.4.1 Pb Vacancy in PbTe .................................................................................................... 45
3.4.2 Monovalent impurities (Na, K, Rb, Cs, Cu, Ag) in PbTe ...................................... 48
3.4.3 Divalent impurity (Zn, Cd, Hg, Sn, Ge) in PbTe .................................................... 50
3.4.4 Trivalent impurity (Ga, In, T], As, Sb, Bi) in PbTe ................................................ 54
3.5 Te-site defects ....................................................................................................................... 63
3.5.1 Te vacancy in PbTe ..................................................................................................... 63
3.5.2 S and Se impurities in PbTe ....................................................................................... 66
3.5.3 I impurity in PbTe ....................................................................................................... 66
3.6 Impurity-impurity interaction and the dilute defect limit ......................................... 70
3.7 Comparison with Experiments ........................................................................................... 74
3.8 Summary ............................................................................................................................... 76
Chapter 4
Electronic Structure of Defects in SnTe .............................................................................. 78
4.1 General Introduction ............................................................................................................ 78

vi

4.2 Electronic Structure of SnTe ............................................................................................... 79

4.3 Defects in SnTe .................................................................................................................... 81
4.3.1 Sn vacancy in SnTe ..................................................................................................... 82
4.3.2 Te vacancy in SnTe ..................................................................................................... 84
4.3.3 Ag impurity in SnTe .................................................................................................... 86
4.3.4 Cd impurity in SnTe .................................................................................................... 87
4.3.5 Sb impurity in SnTe .................................................................................................... 89
4.3.6 In impurity in SnTe ..................................................................................................... 90

4.4 Summary ............................................................................................................................... 91

Chapter 5

Energy and Temperature dependence of relaxation time and Wiedemann-Franz law

in PbTe ........................................................................................................................................ 93

5.1 General Introduction ............................................................................................................ 93

5.2 Transport Coefficients using the Boltzmann Transport Equation ................................. 95

5.3 Nonparabolic Kane Model for Energy Dispersion ........................................................ 100

5.4 Transport Coefficients in the Kane Model ...................................................................... 101

5.5 Relaxation Time in the Kane Model ................................................................................ 106

5.6 The Electronic Thermal Conductivity ............................................................................. 115

5.7 Results and Discussions .................................................................................................... 116
5.7.1 Energy and Temperature Dependence of the Relaxation time ............................ 118
5.7.2 Electronic Thermal Conductivity and the Validity of Wiedemann-Franz law.. 124

5.8 Summary ............................................................................................................................. 128

Chapter 6

General Summary .................................................................................................................. 130

References .......................................................................................... 1 36

vii

List of Tables

Table 3.1 Characteristics of the hyper-deep and deep defect states in PbTe with In
impurity. Results obtained in PAW calculations using two different supercell
sizes, (2x2x2) supercell (64 atoms/cell) and (3x3x3) supercell (216
atoms/cell). E0v and E0c are respectively at the bottom of the valence band
(used as the reference for the hyper-deep states) and at the bottom of the
conduction band (used as the reference for the deep states). E1 and E2 are
respectively at the top and the bottom of the lower (upper) localized band
formed by the hyper-deep (deep) defect state. W and E“, are respectively the
width and the average position of the lower and upper localized bands. All
these quantities are measured in eV ................................................. 74

Table 5.1 Parameters used to calculate the relaxation times for PbTe. [52, 106] .....1 11

Table 5.2 Inverse of the relaxation time of different scattering mechanisms in PbTe for
different temperatures T and different carrier energy 8 ........................ 113

viii

List of Figures

Figure 2.1 Schematic representation of electronic states associated with the Vacancy. The
shaded regions represent the conduction and valance states of the bulk. Left:
the vacancy configuration before any relaxation of the neighbors: each
broken bond contains one electron denoted by an up or a down arrow for the
spin states, and all levels are degenerate, coincident with the Fermi level
which is indicated by a dashed line. Right: the reconstructed vacancy, with
pairs of broken bonds formatting bonding and antibonding states; the former
are fully occupied, the latter empty, and the two sets of states are separated
by a small gap. [54] ....................................................................................... 15

Figure 2.2 Schematic illustration of shallow donor and acceptor impurity states in
semiconductor with direct gap. The light and heavy electron and hole states
are also indicated; the bands corresponding to the light and heavy masses are
split in energy for clarity. [54] ....................................................................... 16

Figure 2.3 Division of a unit cell in muflin tin regions and the interstitial region, for a
case with two atoms. The black dot is the origin of the axis system. ............ 29

Figure 2.4 Schematic presentation of the supercell construction of lattice vacancy in a
solid ................................................................................................................ 31

Figure 3.1 (a) fcc Brillouin zone of PbTe (b) Band structure of PbTe in fee Brillouin zone
at V0,”. [23] ..................................................................................................... 40

Figure 3.2 Supercell model of RPb31Te32 where R is either a vacancy or an impurity atom.
Small balls are for Pb and Te, large balls are for R. ...................................... 42

Figure 3.3 (a) The total DOS of PbTe with and without Pb vacancy, (b) the difference in
the partial DOS between PbTe with and without Pb vacancy where the six
nearest neighbor Te atoms of the Pb vacancy are denoted as Te2 and there are
three other inequivalent Te atoms (denoted as Tel, Te3, and Te4), and (c)
Atomic spheres (Apat), and interstitial components Api of Apmt. The energy
origin in (b) and (c) is chosen to be the top of the valence band of PbTe;
whereas the energy origin in (a) is the highest occupied Kohn-Sham single-
particle state (which will be denoted as the Fermi energy) of the system with
defect .............................................................................................................. 46

Figure 3.4 The difference in the total DOS (Aptot) between PbTe with and without
monovalent impurities: Na, K, Rb, Cs, Cu, and Ag. The energy origin (0 eV)
is chosen to be at the top of the valence band of PbTe. ................................. 49

Figure 3.5 The difference in the total DOS (Aptot) of PbTe with and without s-type
divalent defects: Zn, Cd and Hg. The energy origin (0 eV) is chosen to be at
the top of the valence band of PbTe. ............................................................. 51

ix

Figure 3.6 (a) The total DOS of PbTe with and without Cd impurity, (b) their difference
Aptot, (c) partial DOS per Te associated with nearest neighbor Te atoms, and
((1) Cd partial DOS of s-character. The Fermi energy of PbTe with Cd
impurity is at the energy origin (0 eV). ......................................................... 52

Figure 3.7 The difference in the total DOS (Apmt) of PbTe with and without p-type
divalent defects: Sn and Ge. The energy origin (0 eV) is chosen to be at the
top of the valence band of PbTe. ................................................................... 53

Figure 3.8 (a) The total DOS of PbTe with and without In impurity, (b) their difference
Aptot, (0) partial DOS per Te associated with nearest neighbor Te atoms, and
(d) In partial DOS of s-character. The Fermi energy of PbTe with In impurity
is at the energy origin (0 eV). ........................................................................ 56

Figure 3.9 The difference in the total DOS (Aptot) between PbTe with and without
trivalent group III impurities: Ga, In, and T1. The energy origin (0 eV) is
chosen to be at the top of the valence band of PbTe. .................................... 58

Figure 3.10 (a) The total DOS of PbTe with and without Ga impurity, (b) their difference
Aptot, (c) partial DOS per Te associated with nearest neighbor Te atoms, and
((1) Ga partial DOS of s character. The Fermi energy of PbTe with Ga
impurity is at the energy origin (0 eV). ......................................................... 59

Figure 3.11 The difference in the total DOS (Apmt) between PbTe with and without
trivalent group V impurities: As, Sb, and Bi. The energy origin (0 eV) is
chosen to be at the top of the valence band of PbTe. .................................... 61

Figure 3.12 (a) The total DOS of PbTe with and without As impurity, (b) their difference
Aptot, (c) partial DOS per Te associated with nearest neighbor Te atoms, and
((1) As partial DOS of p character. The Fermi energy of PbTe with As
impurity is at the energy origin (0 eV). ......................................................... 62

Figure 3.13 (a) The total DOS of PbTe with and without Te vacancy, (b) their difference
Aptot, and (c) partial DOS per Pb associated with nearest neighbor Pb atoms.
The Fermi energy of PbTe with Te vacancy is at the energy origin (0 eV). . 65

Figure 3.14 (a) The total DOS of PbTe with and without S impurity, (b) their difference
Aptot, (c) partial DOS per Pb associated with nearest neighbor Pb atoms, and
(d) S partial DOS of p character. The Fermi energy of PbTe with S impurity
is at the energy origin (0 eV). ........................................................................ 67

Figure 3.15 (a) The total DOS of PbTe with and without Se impurity, (b) their difference
Apmt, (c) partial DOS per Pb associated with nearest neighbor Pb atoms , and
(d) Se partial DOS of p character. The Fermi energy of PbTe with Se
impurity is at the energy origin (0 eV). ......................................................... 68

Figure 3.16 (a) The total DOS of PbTe with and without I impurity, (b) their difference
Apmt, (c) partial DOS per Pb associated with nearest neighbor Pb atoms, ((1) I

X

partial DOS of p character. The Fermi energy of PbTe with I impurity is at
the energy origin (0 eV). ................................................................................ 69

Figure 3.17 The total DOS of PbTe with In impurity obtained in PAW calculations using
two different models: InPb31Te32 with one In impurity at the comer, and
Insz3oTe32 with one In at the corner and the other one at the center of a
(2x2x2) supercell; the distance between the two In impurities being 13.1 A
and 11.345 A, respectively. The two lines are almost identical except at ~-O.5
eV and ~4.5 eV where the integrated DOS of the deep and hyper-deep defect
states of Insz3oTe32 are larger than those of InPb31Te32 by a factor of 2. The
Fermi energy is at the top of the upper localized band formed by the deep
defect state. [50] ............................................................................................. 72

Figure 3.18 The total DOS of PbTe with In impurity obtained in PAW calculations using
two different supercell sizes, (2x2x2) supercell [InPb31Te32] and (3x3x3)
supercell [InPb107Temg]; the distance between the two In impurities being
13.1 A and 19.65 A, respectively. The Fermi energy is at the top of the upper

localized band formed by the deep defect state. [50] .................................... 73
Figure 4.1 Electronic energy band structure of SnTe calculated using GGA including
SOI.[80] ......................................................................................................... 80
Figure 4.2 Total Density of state of SnTe calculated using GGA including 801 ............. 81

Figure 4.3 (a) The total DOS of SnTe with and without Sn vacancy, (b) their difference
Apmt, and (c) partial DOS per Sn associated with nearest neighbor Te atoms.
The Fermi energy of SnTe with Sn vacancy is at the energy origin (0 eV). . 83

Figure 4.4 (a) The total DOS of SnTe with and without Te vacancy, (b) their difference
Aptot, and (c) partial DOS per Te associated with nearest neighbor Sn atoms.
The Fermi energy of SnTe with Te vacancy is at the energy origin (0 eV). . 85

Figure 4.5 (a) The total DOS of SnTe with and without Ag impurity, (b) their difference
Apmb. The F errni energy of SnTe with Ag impurity is at the energy origin (0
eV). ................................................................................................................ 86

Figure 4.6 (a) The total DOS of SnTe with and without Cd impurity, (b) their difference
Aptot, (0) partial DOS per Te associated with nearest neighbor Te atoms, and
(d) Cd partial DOS of s-character. The Fermi energy of SnTe with Cd
impurity is at the energy origin (0 eV). ......................................................... 88

Figure 4.7 (a) The total DOS of SnTe with and without Sb impurity, (b) their difference
Apmt, (c) partial DOS per Te associated with nearest neighbor Te atoms, and
((1) Sb partial DOS of p-character. The Fermi energy of SnTe with Sb
impurity is at the energy origin (0 eV). ......................................................... 89

Figure 4.8 (a) The total DOS of SnTe with and without In impurity, (b) their difference
Apmt, (c) partial DOS per Te associated with nearest neighbor Te atoms, and

xi

(d) In partial DOS of s-character. The Fermi energy of SnTe with In impurity
is at the energy origin (0 eV). ........................................................................ 91

Figure 5.1 Energy dependence of the inverse of the relaxation time of the three dominant
scattering mechanisms (Ta, 1:0 and IPO ) in PbTe for electron carrier density n
=5><1o19 cm’3 at different temperatures (a) 300K (b) 600K (0) 900 K. The
smooth curves gives the (- afo/de) indicating the energy region contributing
to the transport at different temperatures. .................................................... 114

Figure 5.2 Temperature dependence of the chemical potential ,1: for n-type PbTe at
different concentrations n =5><1019 cm'3 and n =5><1020 cm'3. ..................... 117

Figure 5.3 Temperature dependence of the electrical conductivity 0' and thennopower S
for n-type PbTe at concentration It =5><1019 cm'3. The experimental values are
shown as solid points from Ref [128]. ......................................................... 118

Figure 5.4 Energy dependence of the total relaxation time 1,0,, calculated from Eq. 5.80,
for n-type PbTe at concentration n =5><1019 cm'3. for different temperatures.

..................................................................................................................... 119
Figure 5.5 Energy dependence of the scaled (to 300K) total relaxation time 1,0, at
different temperatures. ................................................................................. 1 19
Figure 5.6 Energy dependence of the total relaxation time 1:,“ for T=3 00 K. ................ 120
Figure 5.7 Temperature dependence of the total relaxation time rm, at £= 0.1 eV. ....... 120

Figure 5.8 A comparison between the electrical conductivity 0' using the scattering
mechanisms of the relaxation time with the scaled formula ........................ 122

Figure 5.9 A comparison between the therrnopower S using the scattering mechanisms of
the relaxation time with the scaled formula. ................................................ 122

Figure 5.10 (a) Temperature dependence of 0', using the scaling formula of the total
relaxation time for different r parameter. (b) Temperature dependence of the
scaled 0' (to r =1.74). ................................................................................. 123

Figure 5.11 Temperature dependence of the electronic thermal conductivity (Kel) at
constant electric field E (191,5) and at constant current density J (Keu) using
Boltzmann transport equations. We also give their values using WF law. . 118

Figure 5.12 Temperature dependence of ZT for n-type PbTe at concentration n =5><1019
cm'3 assuming 1c, = 0. ................................................................................... 125

Figure 5.13 Temperature dependence of the scaled effective Lorenz number (L/Lo) for n-
type PbTe at different concentration n. (Lo= 2.45 X 10-8 WQK'Z) ............. 128

xii

Chapter 1

 

Introduction

 

1.1 Why Study IV-VI Narrow Gap Semiconductors

Narrow-gap IV-VI semiconductors are materials unique in their physical
properties, the diversity of effects observed in them, and the possibilities for practical use.
In particular, these compounds are the basis for creating infrared semiconductor lasers
that can operate at near-room temperatures, highly efficient thermoelectric converters,
and radiation detectors in the far-infrared and submillimeter ranges [88]. All the above
possibilities are the consequence of a nontrivial combination of properties characteristic
of the materials under consideration. These properties include the direct gap whose width
can be smoothly varied by creating solid solutions, the high efficiency of radiative
recombination, very high values of the static permittivity 6 up to 10", and small effective
masses of carriers ~10’2m0, where m0 is the mass of free electron.[88] As a result, the
Coulomb potential of charged impurities in IV—VI semiconductors is almost completely
screened. Therefore, it has no substantial effect on the mobility of carriers.

One of the important disadvantages of type IV—VI semiconductors is the high
concentration of defects in their structure. For standard synthesis methods, the
concentration of vacancies and interstitial atoms reaches 1018 — 1019 cm‘3.[88] All the
growth defects are electroactive, because of which the low carrier concentration

necessary for a number of applications cannot be reached in undoped compounds.

Doping is one of the main methods for controlling the concentration of free
carriers in semiconductors. The use of this method as applied to IV—VI semiconductors
and, in particular, to lead telluride—based solid solutions not only allows the concentration
of electrons and holes to be changed but also leads to the appearance of radically new
properties uncharacteristic of the starting material. For example, when PbTe and a
number of PbTe-based alloys are doped with group III impurities (inditun, gallium, or
thallium) the phenomena of Fermi-level pinning is observed. In this case, the Fermi-level
position is determined by the alloy composition alone and is independent of the
concentration of doping impurities and lattice defects [45]. In particular, a situation can
arise when the Fermi level is stabilized within the band gap and a semi-insulating state is
accomplished at low temperatures. Such a state is not observed in alloys that have not

been specially doped.

PbTe was one of the first materials studied by Ioffe and his colleagues in the
middle of the last century when there was a revival of interest in thermoelectricity.[4]
This compound, its alloys with SnTe and PbSe, and related compounds called TAGS

(alloys of AngTez and GeTe) were for many years the best thermoelectric materials at

temperatures ~700 K.[5] In recent years, quantum wells of PbTe/Pb1-xEuxTe,

PbSeo,9gTeo,02/PbTe superlattices.[6] and quaternary compounds AngPmeTe2m+2

(m=9,10) [7] have attracted considerable attention because of their large thermoelectric
figure of merit (FOM). Similar quaternary systems with Ag replaced by Na are also
promising high temperature therrnoelectrics.[8] Most of the above systems have

stoichiometry closer to the parent compound PbTe.

1.2 Defects in Semiconductors: Shallow and Deep

The band gap, the energy region that is free of electronic states and is bounded by
the valence and conduction bands, is the hallmark that differentiates metals,
semiconductors, and insulators. Dopants and defects can dramatically modify the
electrical conductivity of semiconductors as they add states in this band gap, which can
provide electrical charge carriers (electrons or holes) to the respective bands at energies
very much lower than the gap energy. Electrically active defects in semiconductor
crystals have different characteristics, depending on the location of their energy states
relative to the conduction- or valence-band edge. Shallow defects have energy levels
within a few tens of millielectron volts from the respective band edges, whereas deep
defects typically reside within the middle third of the semiconductor energy band gap. In
fact the above simple definition and classification are no longer generally applicable.
Instead one uses the degree of localization of the defect state to characterize this state.
Deep levels have highly localized wave functions whereas shallow-level wave functions
are as extended as the far reaching Coulomb potential. According to this definition,
oxygen in GaN is a deep defect even though it is located inside the conduction band.[89]
A shallow defect is easily explained. For example, a shallow donor resembles a hydrogen
atom with a positive nucleus binding an electron. Two modifications to the hydrogenic
model arise inside a solid. First, the mass of the electron is renorrnalized because of its
interactions with the lattice. This “effective mass” is usually smaller than that of the free
electron. Second, the crystal reduces the Coulomb potential caused by the dielectric
screening. Combined, these two effects reduce the ground-state binding energy

(ionization energy) of a donor, from the well-known 13.6 eV of the isolated hydrogen

atom, down to, for example, merely 45 meV in Si. Other hydrogenic features remain: a
series of bound excited states, which are observable with far-infrared spectroscopy, and
the ionization into the energy continuum, that is, the conduction band in the case of a
donor. This strikingly simple and effective hydrogenic model of shallow-level impurities
or dopants has been amply confirmed with a broad range of electrical and optical studies
of a vast number of dopants in numerous semiconductor materials.[90] The deep defects
are much more difficult to understand.[91] Badly fitting impurity atoms distort the host
lattice so extensively that the binding energy greatly exceeds that of shallow, hydrogenic
defects. In wide band-gap semiconductors, the energy level lies “deep” within the
forbidden energy band gap. Despite the availability of highly sophisticated computational
tools, deep defects still represent one of the most challenging of problems of current
semiconductor theory. Many atoms around the distorting defect must be considered when
a first-principles calculation is attempted; the defect extends over many lattice constants.
The carriers can interact strongly with the lattice. The effects of deep defects are to
drastically reduce minority carrier lifetimes and to act as traps for charge carriers. Some
deep defects can be used to pin the Fermi level near the middle of the energy gap and
thus lead to highly resistive materials.

In Recent years a lot of progress has been made in tackling the deep defect states
in wide band-gap semiconductors, using ab initio methods. For example, Lany et. al.
[134] investigated theoretically using density functional theory the In DX center and the
DX-like configuration of the Cd host atom in CdTe. Another well known example is P
and As impurities in GaN. Mettila et. al. [135] found that P and As substitutional

impurities in GaN induce deep levels in the band gap in contrast with all other

conventional III-V systems. The calculated exciton binding energy for (P)GaN has an
excellent agreement with the available experimental data. In contrast to the wide band-
gap semiconductors, the problem of defects in narrow-gap semiconductors, particularly
the physics underlying the deep defect states, is far from understood. In these systems
both shallow defects caused by the long—range Coulombic potential and deep defects
formed by short range interactions and mixtures of these may coexist.

The theory of deep defects in narrow band-gap semiconductors has a long history.
For substitutional defects in PbTe, Lent et al[14] presented a simple chemical theory of s-
and p-bonded substitutional impurities almost 20 years ago. This theory naturally gave
deep defect states near the fundamental band gap and also predicted resonant levels
further away from the gap. In this thesis we review the basic ideas underlying this theory
and discuss the results of our recent attempts to understand the physics behind these deep
defect states using first principle ab initio electronic structure calculations. We also
discuss how these defect states get modified when different types of impurity
microstructures are present. It is well-known that transport and optical properties of
semiconductors are dominated by the states in the neighborhood of the band gap. Thus it
is important to have a fundamental understanding of the changes in the electronic states
near the band-gap region in PbTe when it is mixed with other binary and ternary
compounds. Before being able to understand the nature of the electronic states near the
band gap and/or the Fermi energy in the complex systems mentioned above, one must
understand how the electronic states of PbTe change when Pb or Te is replaced by

impurity atom or by vacancy.

Before studying the defect states in narrow band-gap semiconductors, one must
understand their band structure without any defect. In recent years ab initio electronic
structure calculation using density functional theory (within local density or generalized
gradient approximations, LDA [61] or GGA [32]) have been extremely successful in
unraveling the electronic structure (including the band gap) of many narrow band gap
semiconductors with complex crystal structures. Examples are: skutterudites [117], and
ternary bismuth chalcogenides systems (BaBiTe3 [119] and CsBi4Te6 [120]). Although
the fundamental reason for this success is not known, the subtle nature of the origin of
gap formation (i.e. hybridization, spin-orbit interaction etc.) in these compounds may be
responsible for the above mentioned quantitative agreement. [119, 120] Even in systems
where the quantitative predictions of LDA/GGA are not that good, one has been able to
understand the physics of gap formation, the nature of bonding, and the parentage of
states contributing to the transport properties of these narrow band gap
semiconductors.[121] It is clear that ab intio electronic structure calculations are able to

provide great insight about the physical properties of complex systems.

1.3 Transport Properties of Thermoelectrics

The efficiency 77 of thermoelectric (TE) energy conversion depends on the
transport coefficients of the constituent materials through the figure of merit (FOM)
Z T = USzT/(Kl + Ice”), where a is the electrical conductivity and S is the therrnopower
(Seebeck coefficient) [9]. The quantity in the denominator is the thermal conductivity; it
is given by the sum of contributions from the electronic carriers at constant electrical
current i (K31, J ) and the lattice contribution K]. (n —> ncamot when ZT -9 00) Z has units

of inverse temperature, so it is generally quoted as a dimensionless number Z T, with Tthe

absolute operating temperature of the TE device [9]. TE devices are usually made from
semiconductors. From the definition of ZT, it is clear that, to increase ZT we have to
decrease the thermal conductivity of the material and/or increase the therrnopower and
the electrical conductivity. Among the four quantities involved in ZT, three of them (a, S
and K3,, 1) are mainly related to the electronic structure of the material and the fourth one
(Kl) is mainly related to the lattice. One possible way to improve the FOM is to reduce it]
without significantly altering the electronic properties of the materials, referred to as
electron crystal phonon glass concept [92]. This approach has been explored extensively
in the past [93, 94] through the enhancement of phonon scattering. The other way to
improve the thermoelectric properties of a material is to increase the power factor 052 by
varying the doping concentration and manipulating the electronic structure in the
neighborhood of the chemical potential [95, 96]. In this thesis we focus on the electronic
contributions to the thermoelectric properties of PbTe.

PbTe is a well known for its excellent thermoelectric properties. It is used for
power generation in the temperature range 400 to 800 K [97]. In general, iodine and PM;
are the ordinary dopants for n-type PbTe used to optimize the carrier concentration [98].
But the lattice thermal conductivities of PbTe with these dopants are too high (~ 2.2
WK'lm" at 300 K and ~ 1.1 WK'lm" at 650 K) [132] to restrain application (ZT has only
been ~ 0.1 at 300 K and ~ 0.9 at 650 K). [98, 132] Therefore many methods have been
used to reduce K, of PbTe, such as hot pressing and spark plasma sintering (SPS)
techniques [99]. The lattice thermal conductivity of PbTe samples prepared by hot
pressing is about 2WK"m_l which is much higher than the ideal value of 1 WK’lm'l

needed for good thermoelectric materials as pointed out by Mahan [100]. Although there

are many successful examples of reducing the thermal conductivity of PbTe by alloying
such as in Pb1_xSnxTe, a product of PbTe alloyed with SnTe, and (Bi1_bex)2Te3 solutions
[100], recently, novel bulk quaternary systems AngmeTem+2 (LAST-m) with special m
values [7, 19], and MBE (molecular beam epitaxy) grown PbSeTe/PbTe quantum dot
(QD) superlattices [101] have been found to give a large Z T ~ 2 in the temperature range
(400-800 K). LAST-m gives large ZT values at the higher temperature end.

A different approach to increase ZT by increasing the power factor has been
suggested recently by Heremans and collaborators [102] and by Kanatzidis and
collaborators[103]. The proposal is to alter the energy dependence of the carrier
scattering rate (and consequents the T dependence of a) by inserting metallic nano-
particles such as Pb and/or Sb inside bulk PbTe. The precise effect of these nano-particles
(~ 30 — 50 nm size) on electronic and phononic transport properties are not understood.
One of the motivation of the present work is to alter the value of the parameter that
controls the energy dependence of the effective scattering time (rather than individual
scattering processes) and see how the T-dependence of electronic conductivity changes

From the theoretical side, several calculations of the transport coefficients of
PbTe and related systems have been reported over the years [52, 95, 102, 104, 105].
These calculations have been performed using the Boltzmann transport equation within

the energy-dependent relaxation time approximation and the nonparabolic Kane model
[133] for the energy dispersion 8; vs If The energy and temperature dependence of
different scattering mechanisms were incorporated in these calculations. Depending on

the temperature, different scattering mechanisms (from impurities, acoustic and optical

phonons) contributed to the relaxation time ‘1’. At high temperatures when optical phonons

dominate the scattering mechanism, it was found that both polar and deformation
couplings with optical phonons played equally important role. [106, 107] Earlier work by
Bhandari and Rowe [104] focused on the electronic thermal conductivity Ice”; they
indicated that the inclusion of non-parabolicity in energy dispersion had a pronounced
effect on Ice” and neglecting it would overestimate Ice], 1° Also they pointed out that
acoustic phonon scattering was the dominant carrier scattering mechanism in PbTe at
room temperature (RT). In a later work, Bilc et al [52] work focused on the power factor
at high temperatures, and showed that the strength of the deformation coupling constant
had to be reduced by about 30% from its earlier value (used in fitting to the room
temperature data) to get a better quantitative fit to the high T transport data. All these
earlier work did not explore the relationship between the energy and temperature
dependence of the effective relaxation rate and the T-dependence of o, Keu and Ice”,- .
Also a careful analysis of the difference between these two different electronic
conductivities and its impact on the Wiedemann Franz (WF) law in doped PbTe was not
made. In this thesis we address these issues by focusing on (1) a careful analysis of the
energy and T dependence of r to develop a simple scaling form, (2) understanding the
relation between the energy dependence of T and the T dependence of a, S, and the

power factor (052), and (3) finding out the difference between electronic thermal

conductivity at constant E and at constant T and reexamine the validity of WF law
ice, E Ice“,- = LOUT, where the Lorentz number L0 = 2.45 x 10"8 WQ/K. The last point
is quite important because in many papers in the current literature the WP law is used to
obtain ice] and subtracting it from Km, one gets Kl. This method gives unusually small

values of 1c, (<< 1 WK‘lm—l). One of the motivations behind this reexamination of WF

using realistic scattering models for PbTe is to check whether these ultra low values of K1

are reasonable.

10

Chapter 2

 

Defects in Semiconductors

 

2.1 Point Defects

Point defects can be either intrinsic, that is, atoms of the same type as the host
crystal but in a wrong position, or extrinsic, that is, atoms of a different type than those of
the ideal crystal. The most common intrinsic point defects are the vacancy (a missing
atom from a crystalline site) and interstitial (an atom in a non-crystalline site). The most
common extrinsic defects are substitutional impurities, that is, atoms which are foreign to
the crystal but are situated at crystalline sites, substituting for a regular crystal atom. The
presence of defects ofien distorts significantly the atomic structure of the crystal in the

immediate neighborhood of the defects.

2.1.1 Intrinsic Point Defects

Vacancies are quite common in many crystals, both in close-packed metallic
structure, ionic solids and open covalent structure. The creation of a vacancy costs some
energy, due to the breaking of bonds, which is called the formation energy 6;“. This is
defined as the ground state energy of the crystal containing a vacancy, that is, one atom
fewer than the ideal crystal which has N atoms, denoted as EX“ (N — 1), plus the energy
per atom for an ideal crystal, minus the energy of an equal number of atoms in a perfect

crystal (with N —> 00 for an infinite crystal) [54]:

11

 

6;“ = limNro, [(153an — 1) + 5°13”) — 50m] , (2.1)

where Eo (N) is the ground state energy of a crystal consisting of N atoms at ideal
positions. Sometimes instead of using (E0 (N) / N) in Eq. 2.1, one uses 6‘”, the energy of
an isolated atom, the difference between these two definitions is the binding energy per
atom of the perfect crystal. Vacancies can move in the crystal, jumping from one
crystalline site to the next — what actually happens is that an atom jumps from one of the
sites surrounding the vacancy, into the vacant site. The concentration of vacancies per
site in the crystal at temperature T under equilibrium condition is determined by the

vacancy formation energy, and is given by [54]
c”“C(T) = C exp(—e}’“‘/kBT) , (2.2)

where k3 is the Boltzmann constant and C takes account of the appropriate unit. The
concentration of defects in a crystal is usually determined by the balance between the
entropy gain due to the various ways of arranging the defects in the crystal and the energy
cost to introduce each defect.

Interstitial atoms are extra atoms that exist inside the crystal at positions that do
not coincide with regular crystalline sites. These defects are common in crystals with
open structures since in close-packed crystals there simply is not enough room to

accommodate these extra atoms: it costs a large energy to squeeze the extra atom into a
close-packed crystal. The energy to create an interstitial, its formation energy 6"", by

analogy to the formation energy of the vacancy, is given by [54]

Eo(N)
N

 

a?“ = [1,1590 [(Egntm + 1) - )— E0(N)] , (2.3)

12

where E3” (N + 1) is the ground state energy of the solid containing a total of N + 1
atoms, N of them at regular crystalline positions (allowing for small relaxation) plus one
at the interstitial position. Interstitials in crystals with open structures can also move,
undergoing a thermally activated process by which they are displaced from one stable
position to another, either through a direct jump, or by exchanging positions with an atom
of the host lattice. By analogy to the vacancy, there is a thermal activation energy
int

associated with motion of interstitials, called the interstitial migration energy 6 , and

the concentration of interstitials per site in the crystal at temperature T under equilibrium

conditions is given by [54]
cintCl') = C exp(—e}"t/kp T). (2.4)

The presence of intrinsic defects in a crystal has an important consequence: it
introduces electronic states in the solid beyond those of the ideal crystal. If the energy of
such defect-related states happens to be in the same range of energy as that of states of
the perfect crystal, the presence of the defect does not make a big difference to the
electronic properties of the solid. However, when the defect states happen to have an
energy different than that of the crystal states, for example, in the band gap of
semiconductors, or resonant states near the valence band maximum or conduction band
minimum, their effect can be significant. To illustrate this point, consider a vacancy in Si:
the presence of the vacancy introduces four broken bonds (called "dangling bonds"), the
energy of which is in the middle of the band gap, that is, outside the range of energy that
corresponds to crystal states. By counting the available electrons, it becomes clear that

each dangling bond state is occupied by one electron, since two electrons are required to

13

form a covalent bond. If the neighbors of the vacant site are not allowed to distort from
their crystalline positions, symmetry requires that the four dangling bond states are
degenerate in energy, and are each half filled (every electronic level can accommodate
two electrons with opposite spin). The actual physical situation is more complex but
interesting [54]: the four dangling bonds combine in pairs, forming two bonding. and two
antibonding combinations. The four electrons end up occupying the two bonding states,
leaving the antibonding ones empty. This is achieved by slight distortion in the position
of the immediate neighbors of the vacancy, which move in pairs closer to each other as
shown in Fig. 2.1. The net effect is to produce a more stable structure, with lower energy
and a new band gap between the occupied bonding and unoccupied antibonding states, as
illustrated schematically in F ig.2.l. This is called a Jahan-Teller distortion; it is common
in defect configurations when the undistorted defect structure corresponds to partially

occupied, degenerate electronic states.

2.1.2 Extrinsic Point Defects

The most common type of extrinsic point defect is a substitutional impurity, that
is, a foreign atom that takes the place of a regular crystal atom at a crystalline site. A
common application of extrinsic point defects is the doping of semiconductor crystals.
This is a physical process of crucial importance to the operation of modern electronic
devices, so we discuss its basic aspects. The energy of states introduced by the impurities,
which is relevant to doping, lies within the band gap of the semiconductor. If the energy
of the impurity related states is near the middle of the band gap, these are called "deep"
states; if it lies near the band extrema (Valance Band Maximum-VBM- or Conduction

Band Minimum-CBM) they are called "shallow" states [54]. If the impurity atom has

14

C“ “any \Riied V319

Conduction Bands

......+.......+ ------------- +--~--~-- ....... 8.17 I . . .
‘, ’1':- . " - H
. , .
t i .
\ I V I
.

’ "-17.35,” Valence Bands _.i=." . 3"" "‘1‘ "'

 

     

Figure 2.1 Schematic representation of electronic states associated with the Vacancy. The
shaded regions represent the conduction and valance states of the bulk. Left: the vacancy
configuration before any relaxation of the neighbors: each broken bond contains one
electron denoted by an up or a down arrow for the spin states, and all levels are
degenerate, coincident with the Fermi level which is indicated by a dashed line. Right:
the reconstructed vacancy, with pairs of broken bonds formatting bonding and
antibonding states; the former are fully occupied, the latter empty, and the two sets of
states are separated by a small gap. [54]

more valence electrons than the host atom, as P impurity in Si , in which case it is called
a donor, these extra electrons occupy states near the CBM. At finite T, these electrons get
excited to the conduction band and contribute to transport (n-type). In the opposite case,
the impurity creates states near the VBM, which is empty at T = 0. At finite T it is
occupied, which leaves empty states in the VBM of the crystal called "holes"; this type of

impurity is called acceptor and the charge carriers are holes (p-type). One important

15

light demons

 
 
  

 

Conduction Bands

 

 

 

CBM Donor state

8801’

 

VBM Acceptor state

     
 
 

 

 

D

Valence Bands

 

light holes

 

 

k

Figure 2.2 Schematic illustration of shallow donor and acceptor impurity states in
semiconductor with direct gap. The light and heavy electron and hole states are also
indicated; the bands corresponding to the light and heavy masses are split in energy for
clarity. [54]
aspect of the impurity related states is the effective mass of charge carriers (extra
electrons or holes): due to band-structure effects, these can have an effective mass which
is smaller or larger than the bare electron mass, described as "light" or "heavy" electrons
or holes, respectively. These features are illustrated in Fig. 2.2. [54]

Electronic transitions between impurity levels through absorption or emission of
photons, reveal a signature that is characteristic of, and therefore can uniquely identify,

the impurity. Such transitions are often responsible for the color of crystals that otherwise

would be transparent. This type of impurity is called a color center.

16

2.2 Evolution of Theory of Defects in Semiconductors

The first device-related problem that required understanding was the creation of
electrons or holes by dopants. These (mostly substitutional) impurities are a small
perturbation to the perfect crystal and are well described by Effective Mass Theory
(EMT) [129]. The Schrbdinger equation for the nearly-free charge carrier, trapped very
close to a parabolic band edge, is written in hydrogenic form with an effective mass (m‘)
determined by the curvature of the band. The calculated binding energy of the charge
carrier is that of a hydrogen atom but reduced by the square of the static dielectric
constant of 60 of the host material. The binding energy is given by[(m‘/moeg)Ryd].
Typically (50 z 10) and (m‘~ 0.1 m0) where mo is the free electron mass. As a result,
the binding energy ~ meV instead of eV and the associated wavefunction is substantially
delocalized, with an effective Bohr radius some 100 times larger than that of the free
hydrogen atom. EMT has been refined in a variety of ways [130] and provides a basic
understanding of doping in many broad band-gap semiconductors. However, it cannot be
applied to defects that have energy eigenvalues far from band edges. These so-called
deep state defects are not weak perturbations to the crystal and often involve substantial
relaxations and distortions. [69]

The first theoretical tool used to describe the localized defects in semiconductors
involved Green’s functions.[109-111] These calculations begin with the Hamiltonian Ho
of the perfect crystal. Its eigenvalues give the crystals band structure and the
eigenfunctions are Bloch functions. One constructs a set of localized functions (Wannier
function's) formed by linear combination of Bloch functions and uses these localized

basis to study the defect problem. In principle, the defect-free host crystal is perfectly

17

described. The localized defect is represented by a Hamiltonian H ' which includes the
defect potential U. The Green’s function of the perturbed crystal is G(E) = 1 / (E - H ').
The perturbed energies E, in the presence of defect, coincide with its poles. The new
eigenvalues include the levels of the defect and the corresponding eigenfunctions are the
defect wavefunctions. In principle, Green’s functions provide a correct description of a
defect in its crystalline environment. In practice, there are many difficulties associated
with the construction of perfect-crystal eigenfunctions that can be used as a basis set for
the defect calculation, [108] and the construction of the defect potential itself. This is
specially true for deep defect states and those defects which induce large lattice
relaxations and/or distortions.

The first successful Green’s functions calculations for semiconductors date back
to the late 19703. They were used to study charged defects and to calculate forces and
total energies. These calculations also provided important clues about the role of native
defects in impurity diffusion. However, while Green’s functions do provide a near ideal
description of a single defect in a perfect crystal, their implementation is difficult and not
very intuitive. Alternative methods have been introduced to treat the defect problem;
these are cluster and supercell methods. Clusters or supercells are much easier to use and
provide a physically and chemically appealing description of the defect and its immediate
surroundings. In the cluster approach one takes finite size cluster of the host system and
introduces a defect. [131] These types of calculations were performed in real space with
basis sets consisting of localized functions such as Gaussians or LCAOs. The dangling
bonds on the surface atoms of the cluster must be tied up in some way, most often with H

atoms. However, without the underlying crystal and its periodicity, the band structure is

18

missing and the defect’s energy eigenvalues cannot be placed precisely within a gap.
Further, the finite size of the cluster artificially confines the wavefirnctions. This affects
the charged defects the most, as the charge tends to distribute itself on the surface of the
cluster. However, the local covalent interactions are well described by the cluster
approach. [69]

The Schrodinger equation for a cluster containing a defect can be solved using
almost any electronic structure methods. The early work was empirical or semiempirical,
with heavily approximated quantum chemical methods. At first, semiempirical extended
Hiickel theory [113] then self-consistent Hartree-Fock method. Geometries could be
optimized, albeit often with symmetry assumptions. The methods suffered from a variety
of problems such as cluster size and surface effects, basis set limitations, lack of electron
correlation, and the use of adjustable parameters. Their values are normally fitted to
atomic or molecular data, and transferability is a big issue.

Substantial progress in the theory of defects in semiconductors occurred in the
mid 1980’s with the combination of periodic supercells to represent the host crystal, ab-
initio type pseudopotentials [114] for the core regions, density functional theory (DFT)
for the valence regions, and ab-initio molecular dynamics (MD) simulations [115] for
nuclear motion. This combination is now referred to as first-principles in contrast to
semiempirical. There are a few parameters in the theory. They include the size of the
supercell, k-point sampling, type and size of the basis set, chosen by the user, as well as
the parameters associated with the basis sets and pseudopotentials. However, these
parameters and user inputs are not fitted to an experimental database. Instead, some are

determined self-consistently, other are calculated from first principles or obtained from

19

high-level atomic calculations. For more accurate calculations one uses full potential, all
electrons, rather than the pesudupotential. Note that although the first supercell
calculations were done in the 1970’s in conjunction with approximate electronic structure
methods, [116] density firnctional theory and ab initio electronic structure methods using

large supercells are just becoming possible.

2.3 Density Functional Theory for Defect Calculations

Density-functional theory (DF T) [55, 56] has become the most successful theory
for studying atomic-scale computational condensed matter physics and is also widely
used to study defects in semiconductors, predict their structures and energetics,
vibrational and diffusional dynamics, and elucidate their electronic and optical properties,
as observed by various experimental probes. The central quantity in DFT is the electron
density. The ground-state total energy is a functional of the electron density and can be
obtained via variational minimization of the functional. The wave-mechanical kinetic
energy part of the total-energy functional is obtained not from the density directly but
through a mapping to a non-interacting Kohn-Sham system.

The basis of electronic structure calculations lies in the nature of electrons in a
solid. Electrons interact through long range Coulomb forces which makes it very difficult
to describe them as distinct entities. However, the electrons in solids do not act as "bare"
particles, rather they are screened out by positively charged polarization clouds. The
electron and its screening cloud form a "quasiparticle" which weakly interacts with other
quasiparticles in the solid. The effective weak interaction between these quasiparticles

allows for an effective single particle approximation, particularly for low energy

20

properties where they can be treated as acting in the mean field of the other
quasiparticles. Such an approximation is valid in a large class of solids (metals,
semiconductors, and insulators) but not in solids with strong electron-electron

interactions and weak screening (such as magnetic insulators).

2.3.1 Density Functional Theory

Because of its great successful in making computational advances in electronic
structure of solids, I will briefly discuss the basics of DFT. This theory states that the
ground state electronic properties of a solid are determined solely through its electronic

density p(r) in terms of which the total energy is:

Etot(p(r)) = T5(p(r)) + Eee(p(r)) + ENe(p(r)) + Exc(p(r)) + ENN (25)

where T, is the kinetic energy of the noninteracting particles having the same density
p(r), Eee is the electron-electron repulsive Coulomb energy, E Ne is the nuclear-electron
attractive Coulomb energy, Exc is the exchange-correlation energy, and 5va is the

repulsive Coulomb energy of the fixed nuclei.

In the local density approximation (LDA) [61], E,“ is expressed as a function of

the local exchange-correlation energy density Mn:

5:914 = [ uxe(p(r))p(r)d3r. (2.6)

This approximation is generalized for both the spin densities pT(r) and pl(r) in local

spin density approximation (LSDA):

5:5“ = f u... (p'(r). pm) [p'(r) + p*(r)]d3r. (2.7)

21

Recent progress has been made by going beyond LSDA where one adds the gradient
terms of the electron density to E xc. This has led to generalized gradient approximation

(GGA)[32, 63]

555‘ = I lure (p'(r).Vp‘(r).p‘(r).\7p‘(r)) [p’(r) + p‘(r)]d3r . (2.8)

To solve the energy minimization problem Kohn and Sham proposed a method in

which Ts is given by:

N
’12
Ts = Z: I 11130“) (-§;V2) ¢t(r)d3l‘r (2.9)

where till-(1') are the one-electron orbitals and the sum goes over the all occupied orbitals
of an N-electron system. Minimization of the total energy Etot with respect to charge

density p(r) leads to the Kohn-Sham single particle equations:

2

h
-fiV2 + VcCI‘) + ch il’iU‘) = EiII’iCI‘) . (2°10)

where Va is the classical Coulomb potential and ch is the exchange-correlation potential.
VC + ch = eff is an effective single particle potential. The single particle Kohn-Sham

Eqs. 2.10 has to be solved self consistently with the condition:

p(r) = ZiVII/Jt(r)|2 - (2-11)

It should be emphasized that the total energy Em:

Etot it 2 figs 1 (2.12)
i

22

where 6,!“ are Khon-Sham single particle eigenvalues; certain corrections have to be
made. [67] 6,!“ are not the true quasiparticle energy states seen in experiment. However,
for broad band systems they provide a very good description of the physics except for

band gaps and effective masses. [68] Band gaps are usually underestimated.

2.3.2 Full Relativistic Single Particle Dirac Equation

For systems containing heavy atoms, such as Pb and Bi, relativistic effects are
very important and one should in principle use Dirac theory. In the single particle
description, one has to solve the Dirac equation for an electron because the full
relativistic effects applied to electronic structure are included in this equation. The Dirac

Hamiltonian is given by [57]:
HD = ca.p + ([5’ — 1)mc2 + V(r) , (2.13)

where a and 3 are 4 X 4 matrices expressed as:

a = (g g) (2.14)
p = (‘1) 3) (2.15)

with ax, 0y, and oz being the Pauli spin matrices. p is the four component momentum, m 4
is the bare electron mass, c is the speed of light and V(r) is the effective single particle
potential. The eigen vectors ‘1’ of the Dirac Hamiltonian can be written as four
component spinors consisting of the two two-component spinors: large component <1) and

small component x.

23

111 = (‘1’) (2.16)

This gives a set of coupled equations for (I) and x:
c(a.p)x = (e — V)<I> (2.17)
C(O'. p)<l> = (e — V + chz);( (2.18)

Solving for CD the above two equations give the equation for the large component <1):

-1

) (a.p)<l> + V<I> = 6(1) (2.19)

 

1( )(1+6
2m a.p ch2

The above equation is exact.

2.3.3 Scalar Relativistic Approximation

Eq. 2.19 is the exact Dirac equation. This equation serves as the basis for
relativistic electronic structure calculations. [5 8, 59, 60] However, for valence electrons,
where the relativistic effects are not usually strong, one can make the following

approximation:

 

 

(1 + E — V)—1 :-:- (1 - E _ V) (2.20)
ch2 ch2
and using the identities:
pv = Vp — ihVV (2.21)
(a.VV)(o-. p) = (W. p) + 1'0. [17, p] , (2.22)

24

the Dirac Eq. 2.19 can be written as a linear second-order differential equation for (I).

This is given by:

 

(1 €_V)p2+v¢ hz (VVV‘DH hz ([VV ]<1>)- (b 223
chz 2m 4m2c2 4m2c2 0' ’p —6 ' ( ' )

For a spherically symmetric potential V(r), Eq. 2.23 can be simplified further:

p2+V p4 1'12 dVd+ 1 dVl
2m 8m3c2 4m2c2 dr 0r Zmzc2 dr 7'

 

(1.5) 42 = ed) , (2.24)

where the first two terms represent the nonrelativistic Schrodinger Hamiltonian, the third
is the mass velocity term, the fourth is the Darwin term, and the last term is the spin-orbit
interaction ($01). In the presence of the SOI term, j= 1 + s is the good quantum number.

The four-component wave fimction is now written as:

_ <1> _ 9(r)Y,-{’
w ' (X) - (titan-{2) (2'25)

where g and f are the radial functions, Y,{’ is the r-independent eigen function of j 2, jz,

(2, and 52 formed by the combination of the Pauli spinor with the spherical harmonics.
In practice one first solves Eq. 2.24 without the last term [57], which reduces to:

p2 + V p4 h2 dV 0
2m 8m3c2 4m2c2 dr 6r

 

<1> = ab. (2.26)

Eq. 2.26 is called the scalar relativistic Schrodinger equation. Since there is no spin-orbit

coupling in Eq. 2.26, l and s are individually good quantum numbers. The four-

25

component wave function ‘17 can be expressed in terms of radial functions (f'(r) and

,‘g' (r) ) and spherical harmonics, Ylm as:

~ (TS 9'0.) Ylm
w=(,)=( ,. ) , 2.27
x -lf(r)th *5 ( )
. . 1 0 . . . .
where x, are the two component Pauli sprnors (0), (1). The scalar relatrvrsuc Eq. 2.26 15

solved with this choice of 513. The functions ‘P are not eigenfunctions of the Dirac
Hamiltonian H0 and their deviation from eigenfunctions is used to define the spin-orbit

Hamiltonian H 50:

~

H0117 = 5LT! + Hmtp. (2.28)

In the basis set of functions 513, H30 has the form:

 

 

h dV 1 l. S 0
5° — 2Mc2 dr;( 0 0) (2'29)
where M is the relativistic mass given by:
M — E _ V 2 3o
_ m 2C2 o ( . )

Spin-orbit interaction (SOI) effects are important, especially for heavier elements. To
include 801, one method is to double the entire basis set obtained while solving Eq. 2.26
to accommodate H50. This approach is usually computationally expensive. The second
variational method is an alternative approach, which does not require the doubling of the
entire basis set. The spin up and down states are considered separately using the

eigenvectors of the scalar relativistic calculation for only a finite number of states near

26

the Fermi energy. This reduces the number of basis sets involved in the calculation in the

presence of 801. The spin up and down states can be expressed as:

~t

‘T’t=(—gifircii'3fm)” ' “(3) (2'3“
~l

‘T’l=(-girirci~§lfm)i‘ ' “(3) (2'32)

The states given in Eq. 2.31, and 2.32 are used as new basis set for the expansion of the
wave function <19 in Eq. 2.24. The total Hamiltonian including 801 can be solved with
less computational effort since the ntunber of (PT and i1“ states are usually much less

than in the original eigenvalue problem. [20, 57]

2.3.4 Method of calculations

Electronic structure calculations were performed using the self-consistent full-
potential linearized augmented plane wave methods (F LAPW) [20] within DFT [31, 55],
using GGA of Perdew, Burke and Emzerhof [32] for the exchange and correlation
potential. Since the method and the approximations are very well discussed in the
literature, I will not discuss them here. As most of the systems of our interest contain the
heavy atom (Pb) relativistic effects are very important. Relativistic effects are marginally
important for Te, and not important for Se and Sn. The way these effects are included in
the electronic structure code used to obtain the electronic structure (WIENZK program)
[3 7] is discussed very briefly below.

The core states are treated in a fully relativistic way (i.e solving the Dirac
Eq.2.19).[76] The valence and local orbitals, which are defined within the atomic spheres

are first obtained within the scalar relativistic approximation (see section 2.2.4). This

27

approximation does not include 801 but takes into account the effects of mass-velocity
and Darwin corrections.[76] In the absence of SOI spin up and spin down eigen functions
separate. In the presence of 801, the spin up and spin down eigen functions mix and this
is incorporated in a rather efficient way using a second variational method.[77, 78]

The linearized augmented plane wave (LAPW) method [37] which was used in
most of my studies, is among the most accurate methods for performing electronic
structure calculations for crystals. Before we describe the LAPW method it is helpful to
discuss the augmented plane wave (APW) method first. In the region far away from the
nuclei, the electrons are more or less ‘free’. Free electrons are described by plane waves.
Close to the nuclei, the electrons behave as if they were in a free atom, and they could be
described more efficiently by atomic like functions. Space is therefore divided now in
two regions: around each atom a sphere with radius Rd is drawn (call it 5“). Such a
sphere is often called a muflin tin sphere and the part of space occupied by the spheres is
called the muflin tin region. The remaining space outside the spheres is called the
interstitial region (call it I), see Fig. 2.3. One augmented plane wave (APW) used in the

expansion of lp—E is defined as [3 7]:

MG) = W - r (2.33)
( iii” uflr’. 5)) 141.0“) r e s,
l,m

-r

where K are the reciprocal lattice vectors of the plane wave expansion, k are wave

vectors within the first Brillouin zone, V is the volume of the unit cell, Y,§,(f’) are

spherical harmonic functions referred to the center of the sphere located at Ra, u,“ (r’, E )

are the regular solutions of the radial Schrodinger equation for energy E, and Afif‘LR are

28

the expansion coefficients that are determined by the matching and normalizing

conditions for the wavefunction.

The problem with the APW method is that u? (r’, E ) has to be constructed at the
yet unknown eigen energy E = 8;? of the searched eigenstate. It would be helpful if we
were able to recover u? (r’, 8-,?) on the fly from known quantities. That is exactly what

the Linearized Augmented Plane Wave method enables us to do. If we calculate u? at

some energy E0 then we can make a Taylor expansion for energies not far away from it

 

 

 

[37]:
Bu“ r',E
uf’(r’,££ = uf'(r’,Eo) + (E0 — £7?) lag? ) + 0(E0 — E£)Z(2.34)
a 5:52
aware)

 

 

 

 

 

Figure 2.3 Division of a unit cell in muflin tin regions and the interstitial region, for a
case with two atoms. The black dot is the origin of the axis system.

29

Substituting the first two terms of the expansion in the APW for a fixed Eo, gives the

definition of an LAPW. This has a price: the energy difference (50 — 87?) is unknown,

and hence a yet undetermined Biff“? has to be introduced [3 7]:
1 . —. .
TBIU‘HO'T i" E I
41526) = V 2+2: 2+2 (2'35)
m (.433, uf‘(r’,Eo) + 33;, 11,“(r’,Eo)) 12,50“) r e 5,,

'—o+l—(o '-O —0
(1k and Ba k+K

In order to determine both A (m (m

, one requires that the firnction in the sphere
matches the plane wave both in value and in slope at the sphere boundary.

An important parameter that goes into the electronic structure calculation in the
LAPW method is the value of the atomic radius of each component. For multi-component
systems the values of the atomic radii are chosen differently. In some cases the values of

atomic radii are taken to be the same for all the atoms, and this value is chosen to fill the

space between the atoms. In other cases different values are chosen for different atoms.
The number of R points used in the calculations of the self-consistent charge density is

usually taken to be in the range 18 to 30, which corresponds to 400-600 It; points in the
Brillioun zone for cubic systems. The number of plane waves used in the interstitial
region is characterized by a parameter RKmax = leKmax, where leis the smallest
muffin tin radius and Kmax is the maximum plane wave vector. Typically RKmax is
chosen between 7 and 9, and for our calculations it was taken to be 8. Convergence of the
self-consistent iterations are usually performed within 0.0001 Ry with cutoff -6.0 Ry

between the valance and core states.

30

2.4 Supercell Method

A common approach to calculate defect energetics from first principles (i.e.,
starting from electronic degrees of freedom) is based on the supercell idea. In this method
one confines the atoms defining the defect area of interest into an otherwise arbitrary box,
which is then repeated infinitely in one or more spatial directions (see Fig.2.4). In other
words, this box (the supercell) now becomes the new unit cell of the system, and periodic
boundary conditions are applied at one or more of its boundaries. For a point-defect
assembly in a three-dimensional solid, the system now becomes a three-dimensional
periodic defect array. For a line defect (e.g., a dislocation) the result is a regular line-
defect network. For a surface, the system becomes a sandwich of material slabs interlaced
by vacuum regions. Typical sizes of the supercell in all electron calculations are
nowadays 64 atoms or larger in a cubic system, and the increasing computational power

has enabled supercells with hundreds of atoms.[69]

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.4 Schematic presentation of the supercell construction of lattice vacancy in a
solid.

31

The great advantage of supercell calculations is that the periodic boundary
conditions allow the utilization of many efficient techniques derived for the quantum
physics of periodic systems. The wavevectors of the Brillouin zone (BZ) in the reciprocal
space of the supercell are good quantum numbers, and the standard "band-structure
methods" of periodic solids can be applied in full force. Particularly fast computation
methods can be performed by using Fourier analysis (plane-wave basis sets), as they
adopt naturally to periodic boundary conditions and offer spatially uniform resolution.

The supercell approach enables the full relaxation of the structure to minimize
total energy, and the calculation of the formation and migration energies of defects in
different charge states as a function of the Fermi-level position of the host semiconductor,
and as a function of the chemical potentials of the atoms building up the material. Thus,
the method can be conveniently adapted to describe such effects as host materials
stoichiometry on the energetics of isolated defects in compound semiconductors [75].
From supercell calculations one can extract several useful physical properties, such as:
(1) the probabilities of certain types of defect to form under given chemical and
thermodynamical conditions during the growth, (2) the basic nature (acceptor, donor,
deep level) of the defect electronic states, (3) the migration and recombination barriers,
(4) vibrational modes, hyperfine fields, positron-annihilation parameters and other
ground state properties, and (5) within certain limitations, excited-state (for example

optical) properties as well. The ground state properties are those most reliably accessible.

32

2.5 Issues with the Supercell Method in Defects Studies

The obvious drawback of the supercell methods for defect calculations is that the
periodicity is artificial and can lead to spurious interactions between the defects. They
have a finite density, and do not necessarily mimic the true physical situation with an
aperiodic, very low density defect distribution. The supercell method, widely used and
with demonstrated success in defect studies, requires a critical examination of the finite-
size and periodicity effects.

The first consequence of the finite-size supercell approximation is the broadening
of the defect-induced electronic levels to "defect bands", with a bandwidth of the order of
0.1 eV for 64-atom supercells. This translates into a difficulty in accurately placing the
ionization levels. The one-electron Kohn-sham states associated with the defect are often
assigned a position by averaging over the supercell Brillouin zone, or just values at the
Brillouin zone origin ( the I‘ point) as they exhibit the full symmetry of the defect. The
defect-state dispersion usually has a smaller effect on the total energies that include
summations and integrations of the Kohn-Sham states over the entire Brillouin zone.
Thus, ionization levels determined from the total-energy differences is expected to be less
sensitive to the defect-band dispersion. However, this is hard to prove systematically.

Another source of difficulty associated with supercell calculations is the accurate
determination of the host crystal band edges (valence-band maximum, conduction-band
minimum), which are the natural reference energies for the defect-induced gap states. In
the defect-containing supercell, the band-edge states themselves are affected by the
defect. The band-edge positions are often determined by aligning a chosen reference level

between the defect-containing and perfect solid. For example, the effective potential in a

33

localized region far from the defect can be aligned with the potential in the same region
in a perfect crystal. The calculated band-edge distances from the reference energy in a
perfect crystal can then be used to align the band edges in the system containing the
defect. Alternatively, one can consider a deeper, core-level energy of an atom as a local
reference.

Another, more difficult problem arises with charged defects. In order to avoid
divergences in electrostatic energies, a popular solution is to introduce a homogeneous
neutralizing background charge to the supercell array, which enables the evaluation of
electrostatic (Coulomb) energies. This, however, introduces an electrostatic interaction
between the periodic charge distribution in the supercells and the background, which
vanishes only in the limit of infinitely large supercells. The influence of their fictitious
charge has to be subtracted in the end, and this is a highly nontrivial task.

Point defects induce lattice relaxation in the host lattice. The lattice-relaxation
pattern is restrained by the supercell geometry. The argument, often used in supercell
calculations, is that the ion displacement vanish near the borders of the supercell.
However, this does not necessarily guarantee that the long-range ionic relaxations are
correctly described, as the supercell symmetry itself may fix the positions of the
boundary atoms. According to elastic continuum theory, the strain field at large distances
from the point defect fall of as lrl’3 and the ionic displacements falls off as Irl’z. The
finite size of the supercell thus restricts also the ionic relaxations around the defect. The
relaxation pattern is truncated midway between a defect and its nearest periodic replica.

In the case of long-range relaxations this cutoff may be reflected dramatically close to the

34

defect, as was demonstrated by Puska et al. [79] by detailed calculations for vacancies in
silicon.

Further comparison of the supercell to the finite-cluster and Green's function
methods reveals the following. The supercell and Green's function methods have a well-
defined electron chemical potential (they are coupled to a reservoir of electrons) where as
the cluster method does not. This has consequences for the treatment of ionization levels,
which is somewhat problematic for the cluster method. The finite-size effect is there in
the cluster method as well, as the clusters can contain spurious surface-related effects.
The Green's function method is mathematically elegant and in principle the best for
isolated defects, but its computational implementation is difficult in view of the accuracy
required for total electronic energies and derived quantities.

Finally, it is important to note that defect and impurity calculations should, as a
rule, be carried out using the theoretical lattice constant, optimized for the bulk unit cell.
This is crucial in order to avoid spurious elastic interactions with defects or impurities in
the neighboring supercells. The purpose is to investigate properties of isolated defects or
impurities in a dilute limit. If the volume of the defect-containing supercell is relaxed (in
addition to relaxing the positions of the atoms near the defect), the calculation would in
fact correspond to finding the lattice parameter of the system containing an ordered array
of defects at high concentration. In spite of these limitations, the supercell method has
been, extremely successful in predicting physical properties of defects in solids [74, 79],

so we are going to use it in this thesis.

35

Chapter 3

 

Electronic Structure of Defects in
PbTe

 

3.1 General Introduction

A detailed microscopic understanding of the nature of the defect states in PbTe is
an old but a challenging problem. Naively, one expects to see bound shallow defect states
in the band gap (pinned to the valence band maximum or conduction band minimum)
when a Pb or a Te atom is replaced by either a donor- or acceptor-like impurity or by a
vacancy. However, it is known that, in narrow band-gap semiconductors like PbTe, one
does not see these shallow defect states but deep defect states (DDS) which can be

outside the band gap.[11-15]

One of the earlier attempts to understand the origin of deep defect states in PbTe
was made by Lent et al.[13] For substitutional defects in PbTe, they presented a simple
chemical theory for a large class of substitutional defects and vacancies in terms of the
atomic energy levels and tight binding concepts. Unfortunately, these calculations did not
take into account the potentials generated by the impurities in a self-consistent manner
and, therefore, were only qualitative in their prediction. In fact, an even earlier attempt to
understand the origin of vacancy-induced deep defect states in PbTe was made by Parada
and Pratt.[14] They used a Slater-Koster model [110] and Wannier function basis

constructed out of a finite number of PbTe bands. For Pb vacancies, they predicted defect

36

states outside the band gap and were able to explain the low temperature n- and p-type
transport properties in nominally pure PbTe. The defect potential used in the Slater-
Koster model was assumed to be a constant and was not calculated self-consistently.
Therefore the precise positions of the defect states predicted by this calculation are not
expected to be quantitatively reliable. The other major efforts in understanding the
physics of deep defect states are for In-doped PbTe. Inhomogeneous mixed-valence
models for In and other group III impurities (like Ga and T1) have been proposed to
explain several experimental data.[17] In these models, In impurities are assumed to exist
in two valence states, one trivalent and the other monovalent, stabilized by strong
electronic and ionic relaxations. My recent calculations have cast some doubts on the
validity of this model.[18]

In this Chapter, I discuss the results of my calculations to understand the physics
behind the formation of deep defect states in PbTe using self-consistent ab initio
electronic structure calculations within density functional theory (DFT) and supercell
models. Only a few ab initio studies in narrow band gap systems (such as PbTe, PbSe,
etc.) have been made recently. One is by Bilc et al. [19] who have recently reported their
results on Ag and Sb substitutional defects in PbTe using all-electron full-potential
linearized augmented planewave (PP-LAPW) method.[20] Another is by Ahmad et
al.[18] who looked at the In impurity problem. Similar calculations have been carried out
by Edwards et al. [21] who looked at the Ge and Te vacancy and anti-site defects in GeTe.
The purpose of my thesis project is to extend the previous studies of the defects in PbTe
to different types of defects. The defects studied are: Pb and Te vacancies, Pb substituted

by monovalent atoms (Na, K, Rb, Cs, Cu), divalent atoms (s-type Zn, Cd, Hg and p-type

37

Ge, Sn), or trivalent atoms (group III Ga, In, T1, and group V As, Sb, Bi), and Te
substituted by S, Se, or I. My main focus is to develop a systematic understanding of
these defect states as one goes across or down the periodic table to select different
impurity atoms. The results of this study have been recently reported.[18, 24, 50] The
arrangement of this chapter is as follows. In Sec. 2, I discuss the electronic structure of
PbTe. In Sec. 3, I describe the structural model and briefly review the method used to
calculate the electronic structure (The details are already given in chapter 2). In Sec. 4, I
present my results of calculations for Pb—site defects. In Sec. 5 results on Te-site defects
are presented. In Sec. 6, I discuss the impurity-impurity interaction issue and its
implications on the dilute defect limit. In Sec. 7, I discuss my theoretical results in the
light of recent measurements of therrnopower in several PbTe based compounds. Finally,

a brief summary is given in Sec. 8.

3.2 Electronic Structure of PbTe

The rock-salt structure semiconductors PbS, PbSe and PbTe are quite interesting,
they show a series of electronic anomalies relative to the usual II-VI semiconductors
(such as CdTe). For example, the direct gap occurs at the L point and the order of the
band gap and valence band maximum energies in going from S to Te are anomalous
(1'3“g [PbS] = 0.29 eV, Eg[PbSe] = 0.17 eV, Eg[PbTe] = 0.19 eV, at T = OK) [30]. The
band gap pressure coefficient is also anomalous, it is negative. Because of these
peculiarities and their practical applications, there have been many electronic structure
calculations in these compounds [25, 26, 27, 28, 29, 30]. Wei and Zunger [28] have

carried out extensive electronic structure calculations using local density approximation

38

(LDA) [31] as implemented through the linearized augmented plane-wave (LAPW)
method [20]. They have argued that the above anomalous features can be ascribed to the
presence of the Pb s-band below the top of the valance band, setting up coupling and
level repulsion at the L point of the face centered cubic (fcc) Brillouin zone (Fig. 3.1).
Albanesi et al [29] have calculated the frequency dependent dielectric constant of PbSe
and PbTe using the electronic structure (obtained with LAPW) within both LDA and
generalized gradient approximation (GGA) [32]. They found a large dielectric constant
for PbTe in agreement with experiment. [33, 34]. Bilc et al have carried out extensive all
electron LAPW calculations within both LDA and GGA. The band structure results show
that PbTe has a direct band gap at the L point, but with a small gap value of ~ 0.09 eV at
the experimental volume Vexp [23]. The volume optimization increases the gap to ~ 0.19
eV in much better agreement with experiment. The GGA (LDA) optimal
volume Vopt correspond to 5% increase (decrease). The Orbital analysis shows that Te
and Pb p states strongly hybridize, suggesting covalent interaction between these two
atoms. They have also computed the ionicity of Pb and Te atoms according to Bader's
"Atoms in Molecules" theory [35]. The results show an ionic state of PbJ'O'MTe'O'64 which
is much smaller than that of the pure ionic model Pb+2Te_2. Therefore in PbTe covalent
interactions play an important role. The reduction in formal charges from 2(-2) to 0.64

(—0.64) is consistent with the large dielectric constant of PbTe.

3.3 Supercell Model to Calculate Defect Structure

Before we discuss the structural model used to study the impurity states, we

would like to briefly describe the quaternary systems one constructs in developing

39

(a) if

 

 

 

(b)

 

3.0

nanljnrn

uln

 

EF

/ \/ \7
E 0.0:
a “4% \

-10.? Z
W L A F A XZWK

 

 

 

 

 

 

 

Figure 3.1 (a) fcc Brillouin zone of PbTe (b) Band structure of PbTe in fee Brillouin zone
at V0,”. [23]

therrnoelectrics starting from the binary compound PbTe and replacing Pb atoms by other
atoms. The new compounds are described by the formula RR'szn.2Te2n, where R and R’
are chosen to maintain charge neutrality. Here two divalent Pb atoms are replaced by one
monovalent (R) and one trivalent (R') atom. If R is divalent then R' is also divalent (for
example R=R'=Cd). The material is therefore charge compensated and most likely will

maintain its semiconducting character. One then adjusts the concentrations of R and R' to

40

control the sign of the charge carriers. In a recent work, Bilc et al. [19] have discussed the
electronic structures of a series of charge compensated AngPb2n.2Te2n compounds (n=6,
9, 16) and also charge uncompensated compounds AngzmTeZn, Sbe2n.1Te2n (n=16)
where only one of the elements R or R' is present. They found resonant states overlapping
conduction and valence bands caused by Sb and Ag impurities, respectively. In this
chapter, we extend their results to a large class of defects, but focus on defects of one
type (either R or R'), investigating the electronic structure of RPbZMTeZn. We also carry
out similar calculations in compounds of the type MsznTeZn-1, where M is a defect at the
Te site. Our aim is to explore systematically the impurity-induced changes in the
electronic DOS as one goes down or across the periodic table.

We use a supercell model where the defects are periodically arranged in a PbTe
lattice (see Sec. 2.4). This corresponds to the formula RPb2n.1Te2n or MsznTezn-1 as
mentioned above. To increase the distance between the defects as much as possible (in
order to reduce the interaction between them) without increasing the size of the supercell
enormously we have chosen n=16. This corresponds to a unit cell containing 64 atoms
with the minimum distance between two defects being ~13A (see Fig. 3.2). Due to the
large dielectric constant of the host PbTe, we expect long-range Coulomb effects to be
strongly screened, thereby reducing the impurity-impurity interaction. However, short-
range interactions are still present. Before describing the details of our calculations we
would like to point out that there are some fundamental problems regarding the
inadequacy of using the supercell model to calculate the energy of isolated defect states
in semiconductors; [36] as discussed in Sec. 2.4. However, they are usually for extended

“shallow” impurity states and for charged defects. Since most of the defect states in

41

 

Figure 3.2 Supercell model of RPb31Te32 where R is either a vacancy or an impurity atom.
Small balls are for Pb and Te, large balls are for R.

narrow band-gap semiconductors of our interest are of localized “deep” type, we expect
the supercell model to be reasonable. We discuss in details the supercell size dependence
of the defect states in Sec. 6 of this chapter.

Electronic structure calculations were performed within DFT using all-electron
FP-LAPW plus local orbital method [20] incorporated in WIEN2k. [37] We used the
generalized-gradient approximation (GGA) [32] for the exchange and correlation
potential. Scalar relativistic corrections were included and spin-orbit interaction (SOI)
was implemented using a second variational procedure. [38] Convergence of the self-
consistent iterations was performed using 20 k-points inside the reduced Brillouin zone to
within 0.0001 Ry (1.36 meV) with a cutoff energy of -6.0 Ry separating the valence and

the core states. For this initial survey, calculations were performed using the experimental

42

lattice constant of PbTe, which is 6.454A, [39] and no relaxation (volume or internal)
studies were carried out. For a few systems, we also carried out calculations using the
projector augmented wave (PAW) [40] method incorporated in Vienna Ab-initio
Simulation Package (V ASP) [41] to compare with the FP-LAPW results and also to
increase the impurities separation by going to larger supercells. The optimal lattice
constant of PbTe obtained from volume optimization (which is 6.55 A) was used in these
PAW calculations, and ionic relaxations were performed. I should mention here that
choosing between the experimental or optimal lattice constant should not change the
basic physics of the impurities states, since the difference between these two values is

rather small (~0.096A).

3.4 Pb-site Defects

Before presenting the results of the ab initio electronic structure calculations, we
discuss some basic physics of the PbTe system. In this compound Pb and Te 65 and 55
states are quite deep and the conduction and valence bands are formed primarily out of 6p
states of Pb and 5p states of Te respictively. The Pb-Te bonds have both covalent and
ionic character. The removal of one Pb atom or replacing it by a monovalent cation leads
to a local charge disturbance and alters the covalent bonding with the neighboring Te
(denoted as Te2) p states. As a result, the valence band states get strongly perturbed. One
expects acceptor-like states to appear near the top of the valence band. However, their
precise positions can only be obtained after a detailed self-consistent calculation. Also,
because of the local nature of the perturbation we expect the entire valence band region to

be affected. In the case of the monovalent Ag impurity, one has, in addition, the filled d

43

states which interact with the Te p states and change the valence band. The Cd impurity,
although divalent like Pb, has a SS state that may give rise to new states near the bottom
of the conduction band. Indium (In) and other trivalent sp atoms may also give rise to a
bound 3 state below the valence band and therefore act more like a monovalent impurity
than a trivalent one. It is important to emphasize that simple shallow impurity-state
models [16] and Slater-Koster type models [110] that do not calculate the impurity
potential self-consistently [13-15,17], are not capable of describing the short-range
physics of these impurity states quantitatively. Only a carefiil self-consistent ab initio
calculation is able to do so. In this section (3.4) I focus on the cationic site defects. This is
because in several experimental systems of our interest, e. g., AngPb2n.2Te2n [7] and
Nao_95PbmeTem+2,[8] the Ag, Na and Sb atoms are located on the Pb-cation sites. The
removal of one Pb atom (i.e., creation of one vacancy at the Pb site) or replacing it by
another atom with different valence structure charge introduces a local charge
disturbance and alters the bonding with the neighboring Te p states. Due to the long-
range nature of the former and the short-range of the latter, we expect the entire valence
and conduction band states to get perturbed in the presence of the impurity. Although
anionic vacancies (discussed in Sec. 3.5) have been considered to play a significant role
in the transport properties of PbTe, substitutional defects at the anionic sites have been
less well-investigated in the past and there are plans for future investigation of these

defects.

44

3.4.1 Pb Vacancy in PbTe
The total DOS of PbTe with and without Pb vacancy and the difference between the two
are shown in Fig. 3.3(a)-(c). These were calculated using 20 inequivalent k-points in the
cubic Brillouin zone. For the sake of comparison, we shifted (upward) the DOS of PbTe
by a small amount (~ 0.2 eV) so that the lower edge of the valence-band DOS of the
systems with and without Pb vacancy matched. We found that the first peak below the
valence band maximum (V BM) also matched after this shift. This small shift appears to
be reasonable since with this shift the Te core 5s bands (not shown in Fig. 3.3) also
matched perfectly. As we can see from Fig. 3.3(a)-(c), the entire valence-band and
conduction-band DOS get modified by the presence of the Pb vacancy. It is important to
note that the states near the bottom of both the valence and conduction band do not
change appreciably in the presence of the Pb vacancy. Since transport properties are
controlled by the states near the VBM and conduction band minimum (CBM), let us
focus on the band gap region within an energy range of i 0.5 eV about the center of the
band gap. We see that there is very little change near the bottom of the conduction—band
DOS whereas there is an increase in the DOS near the top of the valence band. We
analyze this increase in more detail below. From Fig. 3.3(a), we can clearly see that the
Fermi energy (at 0 eV) cuts through the valence band top and therfore Pb vacancies give
rise to holes as expected.

To see which atoms contribute to the change in the DOS near the VBM, we show
in Fig. 3.3(b) the difference in the partial DOS between PbTe with and without Pb

vacancy. There are six nearest-neighbor Te atoms of the Pb vacancy (denoted as Te2) and

45

 

80«
60‘

4o-

DOS (states/eV cell)

. --—— PbTe

+ PbTe with Pb vacancy

 

 

 

- + Te2
j + Te1+Te3+Te4

 

Ap (states/eV cell)

 

 

 

 

it
o

A)
o

_ _ Total AptOt

1 + Atomic Apat

 

Ap (states/eV cell)

 

—O— Interstitial Api

    

 

-5 4 -3 -2 -1

Energy [eV]

46

Figure 3.3 (a) The total DOS of PbTe with and without Pb vacancy, (b) the difference in
the partial DOS between PbTe with and without Pb vacancy where the six nearest
neighbor Te atoms of the Pb vacancy are denoted as Te2 and there are three other
inequivalent Te atoms (denoted as Tel, Te3, and Te4), and (c) Atomic spheres (Apat),

and interstitial components Api of Aptot. The energy origin in (b) and (c) is chosen to be
the top of the valence band of PbTe; whereas the energy origin in (a) is the highest
occupied Kohn-Sham single-particle state (which will be denoted as the Fermi energy) of
the system with defect.

 

 

three other inequivalent Te atoms (denoted as Tel, Te3, and Te4). Among the Pb
neighbors of the Pb vacancy, Pb2 is the Pb atom that bridges two Te2 atoms each being
the nearest-neighbor of one of the defect sites. In one supercell there are three Pb2 atoms
and twenty-eight other Pb atoms. Of the latter, there are four inequivalent (Pb3, Pb4, Pb5,
Pb6) lead atoms. Clearly the major contribution to the DOS increase near the top of the
valence band comes from the six nearest-neighbor Te2 atoms and we see that Te2 states
in the lower half of the valence band get pushed towards the upper half. This transfer of
the Te2 DOS is partially compensated by the downward shift of the DOS associated with
the other atoms. Also other Te atoms do contribute a small amount to the increased DOS
near the VBM.

It should be noted that, in FP-LAPW calculations, there is an appreciable DOS
associated with charges outside the atomic spheres, i.e., in the interstitial space (as
discussed in chapter 2). [37] In Fig. 3.3(c), we show the difference in the total DOS
(Apia), DOS associated with the charges inside the atomic spheres (Apat), and that
associated with the interstitial space (ApfiApmt-Apat). Apt and Apm have almost the same
energy dependence and are comparable in magnitude. We can interpret the peak in Aptot
near the top of the valence band [Fig.3.3(c)] as a vacancy-induced resonant state. This
should be significant in the hole-doped samples where these states will be involved in
charge and energy transport. The fact that Pb2, Tel, Te3, and Te4 contribute very little to
Aptot suggests that the vacancy-induced resonant states are well localized and hence
weakly interacting with each other.

It is interesting to compare our results with those of Parada and Pratt.[14] They

found that the perturbation due to a Pb vacancy was not strong enough to drive any levels

47

out of the Te p bands. The Pb 5 state associated with the vacancy site was driven about 3
eV above the bottom of the conduction band, hence of no significance for transport. A
doubly degenerate state was driven out of the Te 3 band but only by about 1 eV, hence it
was occupied by two Te s electrons. As a result, a Pb vacancy creates two holes in the Te
p valence band and leads to p-type conduction. These findings are supported by our ab
initio calculations. However, the subtle effects such as the DOS enhancement near the top
of the Te p valence band which can affect the thermopower of holes was not found in this

earlier study.

3.4.2 Monovalent impurities (Na, K, Rb, Cs, Cu, Ag) in PbTe

Now let us discuss the monovalent impurities. They should provide a weaker
perturbation compared to the Pb vacancy because the nominal charge difference is 1. In
Fig. 3.4, instead of giving the total DOS, we only show the difference (Aptot). We find
that K, Rb, Cs, Cu and Ag all show peaks near the top of the valence band, just like a
vacancy. Na is anomalous, it does not seem to change the DOS near the valence band top,
but shows an increase (as compared to other three alkalis) near -3.2 eV. The major
difference between Cu, Ag and the alkalis is seen near the bottom of the valence band in
the energy range (-2, -1) eV for Cu and (-4, -3) eV for Ag. This is due to the Cu and Ag d
states hybridizing with the Te p bands. We will further discuss these results later in Sec.
IV when we compare our theoretical results with experiment. We find that the valence 5
state of the alkalis are all above the valence band, hence all of them donate one electron
to the system. In contrast to the suggestions made in Ref.13, we find that Cs behaves just

like Na in this respect. We should note that even if the alkali atoms donate one electron,

48

 

 

 

A AA AAA
V v VV
1c
-u—~IWN\'J\ AAAA
\r V ‘
d A A
ww—

 

AAA A .'/\A/\

 

N
O o
1 1

4

A

AAA/\AA

‘Vvvvvvvv

A

 

v

vvv V v

V

Aptot (states/eV cell)
<' '
> > <
t
t

AA ;

 

 

1b
| o
01

T

-3

'-2

be at the top of the valence band of PbTe.

when they replace Pb (which is divalent), they give rise to one hole per impurity and

should behave like accepter under ideal conditions.

49

I 1

-1 0
Energy [eV]

Figure 3.4 The difference in the total DOS (Aptot) between PbTe with and without
monovalent impurities: Na, K, Rb, Cs, Cu, and Ag. The energy origin (0 eV) is chosen to

 

 

—

3.4.3 Divalent impurity (Zn, Cd, Hg, Sn, Ge) in PbTe

There are two types of divalent impurities: s-type (Zn, Cd, and Hg) and p-type (Sn
and Ge). The valence configurations of the s-type impurities are 452 (Zn), 5.92 (Cd) and

652 (Hg); whereas those for the p-types are 4p2 (Sn) and 5p2 (Ge). The valence state of

divalent Pb is of course 6p2. Our result for Zn, Cd and Hg are given in Fig. 3.5 and 3.6.
We find that the s-type impurities introduce resonant states near the bottom of the PbTe
conduction band and there are three major changes in the DOS: (i) an enhancement of the
DOS near the bottom of the conduction band caused by a resonant state, (ii) a reduction
in the DOS near the top of the valence band within an energy range ~0.25 eV, and (iii) an
enhancement of the DOS near the bottom of the valence band [see Fig. 3.5 and Fig.
3.6(a)-(d)]. These three changes can be seen clearly if, for example, one looks at the Cd 3
character in Fig. 3.6(d). The Cd atom introduces an s-like state into the system. This state
strongly hybridizes with the p orbitals of the six neighboring Te2 atoms and the bonding
state appears at about -4 eV [see Fig. 3.6(c) and 3.6(d)]. The corresponding anti-bonding
state appears as a resonant state near the conduction band bottom, reflecting the deep
defect nature of these impurity states. The states near the top of the valence band (anti-
bonding between Pb and Te2 p’s) are depressed when Pb is replaced by Cd. Because of
the resonant state near the bottom of the conduction band, we expect electron-doped Cd
systems to exhibit large thermopower. Its experimental implication will be discussed
later. In contrast to the s-type divalent impurities, the p-type impurities, Sn and Ge, do
not alter the DOS near the band gap (see Fig. 3.7). They behave very much like Pb near

the band gap region, which is understandable since Pb is also a p-type divalent atom.

50

 

 

 

 

 

 

A ‘ Hg
'5 _ AW

0
> '1
£2 _

8 Cd

% A'AVAVAVAVAVMVAVAVAVAVA w

:9: 1 5 .

Q_ Zn

4 0 _
'1 5 ' 1 1 1 r I ' 1 v . T -
-5 -4 -3 -2 -1 0 1 2

Energy (eV)

Figure 3.5 The difference in the total DOS (Aptot) of PbTe with and without s-type
divalent defects: Zn, Cd and Hg. The energy origin (0 eV) is chosen to be at the top of the
valence band of PbTe.

51

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

=-’-‘ 100 ,1 15
8 (a) -+- (Cd)PbTe = 10* (b) Difference in total DOS
> 80- ———PbTe 8 ‘
9 . > 5
‘0 60‘ 3 WA
.93 (I) 0 - . AAUVAAA W‘AUVAA
g 1 9, ‘ vvlkvv VAAAA‘fvva
a) 40‘ E _5
Z . 82. T
20 - .
8 ‘ § 10‘
0-.-..4 _ q-15 .............
-5 -4 -3 -2 -1 0 1 2 < -5 -4 -3 -2 -1 0 1 2
.3 Energy [eV] Energy [eV]
E 14 g 2.0
9 12(0) Te partiaI DOS > .(d) ca partial cos
8 . +(Cd)PbTe % 1,5. of s-character
45 10: ———PbTe p 4
373', g? 31.0.
(D m .
8 4j 8 0.5
.. 2‘ 3 JM
.5 01M Eo.o-.....-.--..
15 -5-4-3-2-1o123-5-4-3-2-1012
0..

Energy [eV] Energy [9V1

Figure 3.6 (a) The total DOS of PbTe with and without Cd impurity, (b) their difference
Apia, (c) partial DOS per Te associated with nearest neighbor Te atoms, and ((1) Cd
partial DOS of s-character. The Fermi energy of PbTe with Cd impurity is at the energy
origin (0 eV).

52

 

 

 

 

 

 

 

:3 Sn A

8 ,M, All/l mil
> \IVVVVVVVV

Q 1

$1310—

co

1;; iGe

V 0 .. A,/\r\/\xv.r\ Avm
:9: W V \lV \.
CL

<-1o.,-,.,.,.

-5 —4 -3 -2 -1 0 1 2

Energy (eV)

Figure 3.7 The difference in the total DOS (Aptot) of PbTe with and without p-type
divalent defects: Sn and Ge. The energy origin (0 eV) is chosen to be at the top of the
valence band of PbTe.

53

3.4.4 Trivalent impurity (Ga, In, T1, As, Sb, Bi) in PbTe

Like the divalent case, we also have two types of trivalent impurities: group III
(Ga, In, T1) and group V (As, Sb, Bi). The valence configurations of the group III
trivalent atoms are 452 4p', 55“2 5p1 and 6s2 6p1 for Ga, In and T1, respectively; whereas
the valence configurations of the group V trivalent atoms are 4p3, 5p3 and 6p3 for As, Sb
and Bi, respectively.

Indium and other group III impurities (gallium and thallium) in IV-VI
semiconductors have been known to exhibit anomalous behavior. [42, 43] For example
they are n-type in PbTe and p-type in SnTe. When PbTe is doped with group III
impurities, one observes the pinning of the Fermi energy and mixed-valence behavior.
[17] The same impurity can be either donor or acceptor depending on the specific
composition of the semiconductor. The F ermi-level stabilization within the band gap
appears to be essential for understanding the unusual behavior of these systems and also
for their practical application. Indium metal is nominally trivalent, and hence when In
replaces a divalent Pb, one expects an n-type donor. This assumes that the number of
states in the Te p band does not change when an In impurity is present. If the In s-state is
strongly bound and lies below the valence p-band, then In will behave as an acceptor
because it gives only one electron to the Te-p band instead of needed two. According to
the current theoretical picture, these group III impurities exist in two valence states in the
solid, one trivalent and the other monovalent (referred to as s°p3 and 32pl states
respectively where 3 refers to an impurity s-state and p refers to a band like p-state the
precise relative of which is not known), an inhomogeneous mixed-valence picture. [44,

45] The divalent impurity state (s'pz) is an excited state. The trivalent and monovalent

54

states (two holes or two electrons in the impurity s-state) are degenerate and stabilized by
the negative correlation energy [44, 45] as in the well-known negative-U Anderson
model. [46] Thus half of the In impurities are trivalent On“) and the other half are
monovalent On“). The Fermi level gets pinned between the empty and doubly occupied
single particle defect states.

As I will show, my ab initio calculations give a different picture of the deep
defect states associated with In and similar trivalent impurities in PbTe. In Fig. 3.8(a) and
(b), we plot, respectively, the total density of states (DOS) for PbTe and InPb31Te32 and
the difference (after making a small relative rigid shift by 0.1 eV so that the Pb 6s core
state and the highest peak in the valence band match). Clearly, we see a band of states
split off below the valence-band minimum and a band of states split off above the
valence-band maximum (V BM) and lying in the band gap region. The integrated DOS in
both these bands of states is 2 (including spin degeneracy). The lower band is doubly
occupied whereas the upper band is half full. The width of these two bands is ~ 0.25 eV.
We will refer to these bands as “localized” bands. What is most interesting is that the
Fermi energy (at 0 eV) is at the middle of the upper “localized” band (the one in the gap).
This can be easily understood if the Te-p valence band looses one state per/In impurity.
Naively, one can say that the In 5s state is strongly bound and appears below the valence
band minimum and is occupied by two In valence electrons. If the number of states in the
valence band remained same as in PbTe, then the third In valence electron will go to this
band, since Pb gives two electrons to fill the valence band in PbTe. The Fermi energy
will be in the valence band giving 1 hole per In. The only way the Fermi energy can

move to the upper “localized” band in the gap is if the valence band looses 1 state (2

55

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(.3100 A 20
8 (a) —+—(ln)PbTe § (b) Difference in DOS
% 80: -—-PbTe > .
g 60: §
5 40- E
3 ‘ e
8 20: g
o.

a 0-6 ' .11 1 -'2 ' o ' 2 4

Energy [eV] Energy [eV]
ii 12 . "is 4

. 0 Te PartialDOS d .

10:" 1...... 3 3.‘ ’ term?
8 3. —--PbTe g p 1
1‘3, 6‘ *3; 2-
m 4. (D
o . O 1.
9 2‘ 9 ll l
N N
"- 0 ' 1 ' r ' v . ' . ‘1' - 7 “r“ —:‘
fie -4 -2 o 23 0.6 .4 -2 0 2

Energy [eV] Energy [eV]

Figure 3.8 (a) The total DOS of PbTe with and without In impurity, (b) their difference
Aptot, (c) partial DOS per Te associated with nearest neighbor Te atoms, and (d) In partial
DOS of s-character. The Fermi energy of PbTe with In impurity is at the energy origin (0
eV).

56

including spin). This is precisely what happens, and the reason for the observed Fermi
surface pinning in the band gap in In doped PbTe is a strong interaction between In and
neighboring Te orbitals which removes one state from the valence band and gives rise to

a new hybridized state in the gap, a classic example of a deep defect state (DDS) [11].

In order to understand the parentage of the two “localized” bands, we plot in Fig.
3.8(c) the partial DOS associated with Te atoms which are neighbors of In. Clearly, the
Te p valence band of PbTe has changed dramatically in the presence of In. The two new
“localized bands” have nearly similar Te p character. In Fig. 3.8(d), we show the In 3
character which contribute primarin to the “localized bands”, the lower one has more In
s-character. In addition, in FPLAPW calculations there is a substantial contribution to the
charge density from the interstitial regions. The atomic radii have been chosen to be 1.53
A for both In and Te and the distance between In and its nearest neighbor Te is 3.23 A,
leaving a substantial interstitial charge distribution. The upper “localized” band has more
interstitial contribution, hence it is more delocalized. Thus the “localized” bands consist
of strongly hybridized In s and Te p states. The lower one can be thought of bonding state
between In s and neighboring Te p states and the upper one is the corresponding
antibonding state.

Results for Ga and T1 atoms show that their deep defect states are resonant states
near the top of the PbTe valence band (Fig. 3.9). The difference between In and the other
two may explain the experimental anomalies seen in the case of In impurities in PbTe
such as [17]. The resonant states at the top of the valence band for the Ga and T1 is
caused by the overlap of the upper localized band of states with the states near the top of

the valance band. We show, in Fig. 3.10 (a)-(d), the total and partial DOS of PbTe with

57

 

 

 

 

 

 

A .Ga
g _ . __lqu a?
>
m --
TD .. "1
~93 . I/\ A -4
.59 ' v *Ww
L”,
*5 204
6." .TI A
< 0 'W'MVA‘A'AAVA AV AV - VILVJ - -A
'20 I l r l

 

-6‘-5'-21'-331:2'-1'0 1'2
Energy[eV]

Figure 3.9 The difference in the total DOS (Aptot) between PbTe with and without
trivalent group III impurities: Ga, In, and TI. The energy origin (0 eV) is chosen to be at
the top of the valence band of PbTe.

58

 

 

 

 

    

 

 

 

 

100
S 4(a) -—«-—(Ga)PbTe
8 804 —-—-—PbTe
> t
g 60:
£3. .
5; 4°:
0) .
8 204
eel . - . .
-6 .4 -2 o 2
9 Energy [eV]
78' 12
% 10M0) Te Partial DOS
2; . —~—(Ga)PbTe
+0 8- _ '
g ‘ PbTe
v 6«
m 4
o 4.
a .
.‘é 2‘,
‘l'.’ 0.
£ -6 2

Energy [eV]

  

 

Aptot (states/9V cell)

Partial DOS (states/av cell)

 

‘ (b) Difference in total DOS

 

 

 

 

a ' .21 ' .32 6 ' 2
Energy[eV]

 

. (d) Ga Partial DOS
of s-character

 

 

 

 

 

Energy [eV]

Figure 3.10 (a) The total DOS of PbTe with and without Ga impurity, (b) their difference
Aptot, (c) partial DOS per Te associated with nearest neighbor Te atoms, and ((1) Ga
partial DOS of 3 character. The Fermi energy of PbTe with Ga impurity is at the energy

origin (0 eV).

59

and without Ga impurity, the difference Aptot, and 5 character of the partial DOS of Ga
impurity. Results for T1 impurity are similar except that the lower localized band (hyper
deep state HDS) is further below the valence band minimum. The “hyper-deep” level at
about -5.0 eV is predominantly impurity (Ga)-like whereas the resonant state centered
around 0 eV is predominantly host (Te2)—like, which is consistent with the picture
proposed by Hjalmarson et al. [48] The peak of the resonant state is about 0.25 eV below
the VBM. This is consistent with the current understanding of the Tl-doped PbTe systems
(see Ref. 17 and references therein). According to our calculations, Ga and T1 impurities
should show rather similar behavior.

Results for group V impurities (As, Sb and Bi) show that they introduce resonant
states near the bottom of the conduction band (see Fig. 3.11). In contrast to the group III
impurities which have s and p valence electrons, the resonant states for these p valence
electron impurities are rather broad, the sharpest structure being for As. In Fig. 3.12(a)-
(d), we show the total and partial DOS of PbTe with the As impurity. The resonant state
is a mixture of the p states of As and the p states of the neighboring Te2 atoms in almost

equal proportions.

6O

 

 

'M‘J‘VAA 1..., A A4 VA. A A
v v V WV VVV

 

. BI
AVAAAML ' 4'.
VVV V V

j I 1 ' I

-5'-4'-3'-'2'-101'2
Energy [eV]

0
|

 

: 1r
F5 4
O
> d
g 4 Sb
.. A

.93 «1" V
(U
'55
v 20..

H

.9
Q
<

 

 

 

r'o
o

 

Figure 3.11 The difference in the total DOS (Aptot) between PbTe with and without
trivalent group V impurities: As, Sb, and Bi. The energy origin (0 eV) is chosen to be at
the top of the valence band of PbTe.

61

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A100 3

= . —+—(As)PbTe 8 15 . .

8 80‘(a)———-—PbTe > 10j(b) Difference In total DOS
%, ‘ Q 5:

g :3? g oiMWMA IVAMVH
g 204. E -5.‘ VVV
8 1 33-10-

0 M2,-.. C3--15‘ .............
0-5-4-3-2-10124 -5-4-3-2-1012
A Energy [eV] A Energy [eV]
s12 . a 3

. C Te partial DOS d
E 10“ ) —*—(As)PbTe i, ( )

g 8~ —PbTe g3 2* As Partial DOS

:6 5. E l of p—chatacter

e . a.

8 4. 8 1~

. :1 . . l

.‘E' -5r-ll'-a'-r2'-'1'o'i‘2 TE“ -5 43211 0 i 2
3 Energy [eV] 3 Energy [eV]

Figure 3.12 (a) The total DOS of PbTe with and without As impurity, (b) their difference
Apmt, (c) partial DOS per Te associated with nearest neighbor Te atoms, and (d) As
partial DOS of p character. The Fermi energy of PbTe with As impurity is at the energy

origin (0 eV).

62

3.5 Te—site defects

As in the case of Pb-site defects, the removal of one Te atom (i.e., creation of one
vacancy) or replacing it by another atom introduces a local charge disturbance and alters
the covalent bonding with the neighboring Pb (denoted as Pb2) p states. As a result, we

expect both the valence and conduction bands states to get strongly perturbed.

3.5.1 Te vacancy in PbTe

The total and partial DOS of PbTe with and without vacancy and the difference
Aptot are shown in Fig. 3.13(a)-(c). For the sake of comparison, we shifted (downward)
the DOS of PbTe with defect by ~ 0.455 eV such that the core bands match perfectly. As
we can see, the entire valence and conduction band DOS get perturbed by the vacancy.
Since transport properties are controlled by the states near the Fermi energy (at 0 eV), let
us focus on this region. As seen in Fig. 3.13(a)-(c), new states appear in the band gap
region and near the bottom of the conduction band in the energy range -O.5 to 0 eV.
Partial DOS analysis indicates that these states are predominantly of Pb p character [see
Fig. 3.13(c)]. The valence band looses 6 states (including spin) per Te vacancy. These 6
states are occupied by 4 Te electrons and 2 Pb electrons. Removal of Te atom takes 4
valence electrons away leaving two Pb valence electrons. As a result, two Pb electrons
occupy the new states mentioned above. The system then behaves like an n-doped
system. Our picture is qualitatively consistent with that of Parada [15] who argued that
each Te vacancy causes 4 states (8 states including spin) to be removed from the valence
band and to move near the conduction band bottom. This picture includes the Te 53 core

state. Since Te atom has 6 electrons, the extra two electrons (coming from Pb) occupying

63

the 8 states in the valence band (in PbTe) have to move into the conduction band when a
Te vacancy is present resulting in an n-type conduction. However, the detailed nature of
the DOS near the bottom of the conduction band could not be obtained in earlier non-self

consistent calculations.[15]

64

.A
O
O

 

£5 . (a) + PbTe with Te vacancy
0 30. —— PbTe
>
E .
*8, 60 -
m .
1;; 4o-
8 20-
o .
o T I -

 

 

 

 

-'5'J4r-l'3 -'2"-'1'0'1' 2
20- (b) Difference in total DOS

101
0: A AMA Av.
. w W V

-5'41'-':s'-'2'—'1'o'i'2
, (C) Pb partial DOS

—~— PbTe with Te vacancy
3‘ —— PbTe

 

 

 

 

 

 

 

 

 

 

    

 

-5‘J4r-aT-‘2'-1'o'i 2
Energy [eV]

Figure 3.13 (a) The total DOS of PbTe with and without Te vacancy, (b) their difference
Aptot, and (c) partial DOS per Pb associated with nearest neighbor Pb atoms. The Fermi
energy of PbTe with Te vacancy is at the energy origin (0 eV).

Partial Dos (states/eV cell) A Ptot (states/9V cell)

65

3.5.2 S and Se impurities in PbTe

S and Se, like Te, need two electrons to fill their valence p shell. Thus one expects
mainly the valence band to be perturbed when one replaces Te by S or Se and there
should be a small perturbation of the conduction band (indirectly through changes in the
hybridization effects). This is clearly seen in Fig. 3.14(a)-(c) and Fig. 3.15(a)-(c). The
change in the valence-band DOS takes place over the entire valence-band energy range
with a reduction in the DOS in the energy range of 0.5 eV from the top of the valence
band. S and Se p states appear in the entire valence-band region peaking around 1.2-1.5
eV below the VBM [Fig. 3.14(d) and Fig. 3.15(d)]. The conduction band is hardly

affected within about 1.0 eV from its minimum.

3.5.3 I impurity in PbTe

Iodine (1) appears to act as an ideal donor in PbTe, in agreement with what is
known about the halogen impurities in PbTe.[17, 49] It does not perturb the states either
near the top of the valence band nor near the bottom of the conduction band as can be
seen in Fig. 3.16(a)-(d). The major change in the DOS occurs near the bottom of the
valence band around -5.0 eV (predominantly Ip-character), near -1.1 eV and +1.0 eV due
to changes in the Pb p hybridization effect. The Fermi energy is shifted to the conduction

band making it an ideal n-type impurity (donor), consistent with experiments.

66

 

(a)
O

 

‘(a)----- (S)PbTe

100‘, — PbTe 20‘

    

 

 

 

DOS (states/eV cell)
0)
O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

i
o
3
co
. 9
. (u .
404 ‘23 -10‘
‘ ‘1, _ Difference in
20‘. 2 20.Total 003
0W 3-30 .............
-5 .4 -3 -2 -1 o 1 -5 -4 -3 -2 -1 0 1 2
___»_~ Energy [eV] 3 Energy [eV]
8 8
> 0.8 _ > 4
Q . (C) PD partlal DOS 9 (d)
g 0.6« AHWHAEDSgib-re E; 3‘Spartial DOS
3 . e 9, ofpcharacter
£3 04. , l J!” 2
CD . . In (D l
8 0.2- ’ 8 1<
'3 ‘ '5 ‘
-- ......... . . -- 0 vvvvvvvv 4"?“
goo-5.4-3-2401 5-5-4-3-2-101
O. 0..
Energy [eV] Energy [9V]

Figure 3.14 (a) The total DOS of PbTe with and without S impurity, (b) their difference
prt, (c) partial DOS per Pb associated with nearest neighbor Pb atoms, and (d) S partial
DOS of p character. The Fermi energy of PbTe with S impurity is at the energy origin (0

eV).

67

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

S 120 2 30 Difference in
8 1ooq‘(a‘)‘“‘“‘*‘”~ fibeerTe E 2000) Total 003
E 801 e % 1°:
g 60; .93 0-
m . E ‘
E 40. 3 -10)
8 20. § -20~‘
‘ vvvvvv .- -, Q-30-,-,_,.,_,-.-
00.54-34-101 <1 -5-4—3-2-1o12
Energy [eV] 2 Energy [eV]
E 0.8 :3 4
> l(c) Pb partial DOS 9 .(d) Se partial 003
2 0.64 ~—+—~(Se)PbTe an) 3. ofpcharacter
g . —-—PbTe E
(U
E 04 f 2
U) 0
o 0.2" o 1‘
D ‘ To ‘
a 0_0 - f - . - , _ r - - i ._ 0 ......... 141“”
E-54-3-2-1o1 15-5-4-3-2-101
a, a.
0. Energy [eV] Energy [eV]

Figure 3.15 (a) The total DOS 'of PbTe with and without Se impurity, (b) their difference
Aptot, (c) partial DOS per Pb associated with nearest neighbor Pb atoms , and (d) Se
partial DOS of p character. The Fermi energy of PbTe with Se impurity is at the energy

origin (0 eV).

68

 

 

 

 

 

 

 

 

 

 

Q 120 (a)

a, .

° 100j—-—(|)PbTe

E 80-———Pbre

,9 60«

.39. 4°:

a) 20-

8 ow
~6-5-4-3-2-1012

#2; Energy [eV]

> (c) Pb partial DOS

2 —--—(l)PbTe

3 —PbTe

g 0.51

w

0

o

5 0.0 ..............

t

N

n.

8
E
U)
93
.9
3
E
o.
<
9 Energy [eV]
8
>
E
(D
(D
‘5
3L
a)
O
o
E
t
(B
n.

 

‘(b)

‘ Difference in total DOS
-6-5J413-‘2-‘10i 2

 

 

 

 

(d) I partial 008
of p—character

Nee

 

 

 

 

 

o
-6-5-4-3-2-1012
Energy[eV]

Figure 3.16 (a) The total DOS of PbTe with and without I impurity, (b) their difference
prt, (c) partial DOS per Pb associated with nearest neighbor Pb atoms, (d) I partial DOS
of p character. The Fermi energy of PbTe with I impurity is at the energy origin (0 eV).

69

3.6 Impurity-impurity interaction and the dilute defect limit

In the calculation of energy of the defect states using the supercell model
containing 64 atoms, the defect-defect interaction appears to be appreciable, because the
width of the so called localized bands in the In case, for example, is ~ 0.25eV. It is
therefore important to understand how much of the defect physics remains as one
increases the distance between the impurities (when one goes to the dilute limit). Clearly
the spectral weight of the impurity-induced changes in the DOS will scale with impurity
concentration. In addition, we expect the impurity bandwidth to decrease. But the most
significant question is how the position of the impurity band center changes relative to
the conduction or valence band extrema. In order to see how the impurity-impurity
interaction affects the DOS when two impurities are closer or farther than 13.16 A, we
have studied two different systems. One is Insz31Te32, i.e., two impurities in a (2x2x2)
supercell, where one In is located at the corner and the other at the center of the supercell,
the distance between two In impurities being 11.345 A. The second is InPmeelog with
In located at the comer of a (3x3x3) cubic supercell, the distance between the two In
impurities being 19.65 A. For these two systems, we used the PAW [40] method since it
is’more computationally efficient than FP-LAPW [20] in handling large supercells like
the (3x3x3) one containing 216 atoms. For the sake of clarity, we present here the results

of calculations without the spin-orbit interaction (801).

In the case of Insznge32, the bandwidth of the hyper-deep defect state is the same as
that of InPbnge32 (see Fig. 3.17) [50]. This can be understood using a symmetry
argument. Since the impurity states have s-symmetry and the dominant contribution to

the overlap between two impurity states comes through Te-p and Pb-p orbitals, one

70

expects the hopping amplitude between the two impurity states (in an effective tight-
binding model) to be small due to the 90° relative arrangement of the overlap paths
connecting the two impurities. In the case of InPb107T6103, our main results are
summarized in Fig. 3.18 and Table 3.1 [50]. As we had expected, both the hyper-deep
and deep defect states get narrower since the overlap between the In atoms becomes
smaller. The bandwidths (as measured by Ez-El) change from 0.323 to 0.049 eV for the
hyper-deep (a factor of 6.5 reduction) and from 0.654 to 0.120 eV for the deep defect
state (a factor of 5.45 reduction). In order to see how the changes in the impurity
separation affects their positions of the impurity band centers, we have used the bottom
of the valence band (conduction band) as the reference point EOV (Ecc) for the hyper-deep
(deep) defect states, since the valence and conduction bands are expected to be less
sensitive to the size of the supercell than the impurity states. As seen in Table 3.1, E0v
changes by only 0.001 eV and E0c by 0.036 eV in going from (2x2x2) to (3x3x3)
supercell; the energy separation between the hyper-deep level and Em, (as measured by
Bay-Boy) increases from 0.425 to 0.449 eV, a rather small change of 0.024 eV (5.6%
change); and the corresponding separation between the deep state and EOC (as measured
by Eav-EOC) increases from 0.511 to 0.525 eV, a yet smaller change of 0.014 eV (2.7%).
We then speculate that if we increase the distance between the impurities further, the
main effect will be to reduce the impurity bandwidths without further changing their
positions (peaks) with respect to the band minima. Therefore we can use the results of

smaller supercell calculations to discuss the qualitative physics.

71

100

 

 

 

. _. 1" ln1pb:s1Te32
=3 80 _‘ InszwTe32
8 [relaxed, wlo SOI] p
S 60 i -.*~
g «a; V iii, ‘ .
R ‘ _
.93 Hyper-deep 1
4‘! 40 q state ‘11,
m 2 .
8 q ‘ .
D 20 - .
o .. l
-1 O 1 2 3
Energy (eV)

Figure 3.17 The total DOS of PbTe with In impurity obtained in PAW calculations using
two different models: InPb31Te32 with one In impurity at the comer, and Insz30Te32 with
one In at the corner and the other one at the center of a (2x2x2) supercell; the distance
between the two In impurities being 13.1 A and 11.345 A, respectively. The two lines are
almost identical except at ~-0.5 eV and ~4.5 eV where the integrated DOS of the deep
and hyper-deep defect states of In2Pb30TC32 are larger than those of InPb31Te32 by a factor
of 2. The Fermi energy is at the top of the upper localized band formed by the deep defect

state. [50]

72

”Tan-'4 '1.

Deep state

i 1

 

 

 

250 . - , , » -
‘ (ln)PbTe
[relaxed, Mo 80!] k h

     

 

 

  
   

 

  

E 200 .. __ , __ 2x2x2 supercell (64 atoms/cell)
; - 3x3x3 supercell (216 atoms/cell)
0 l

.93

£3 .

g Hyper-deep '

a) 100 ‘ State

0 .

D

 

 

 

Energy (eV)

Figure 3.18 The total DOS of PbTe with In impurity obtained in PAW calculations using
two different supercell sizes, (2x2x2) supercell [InPb31Te32] and (3x3x3) supercell
[InPb107Temg]; the distance between the two In impurities being 13.1 A and 19.65 A,
respectively. The Fermi energy is at the top of the upper localized band formed by the
deep defect state. [50]

73

 

(2x2x2) supercell (3x3x3) supercell

 

Hyper-deep state Deep state Hyper-deep state Deep state

 

 

 

 

 

E0" E0c -0077 4.903 -0.076 4.939

El -0340 4.065 -0549 4.354

E2 -0.663 4.719 -0.500 4.474
W=E2-E1 0.323 0.654 0.049 0.120
E“: 0.32.451)” .0502 4.392 -0525 4.414

 

Table 3.1 Characteristics of the hyper-deep and deep defect states in PbTe with In
impurity. Results obtained in PAW calculations using two different supercell sizes,
(2x2x2) supercell (64 atoms/cell) and (3x3x3) supercell (216 atoms/cell). Em, and Eco are
respectively at the bottom of the valence band (used as the reference for the hyper-deep
states) and at the bottom of the conduction band (used as the reference for the deep
states). E1 and E2 are respectively at the top and the bottom of the lower (upper) localized

band formed by the hyper-deep (deep) defect state. W and EN are respectively the width
and the average position of the lower and upper localized bands. All these quantities are
measured in eV.

3.7 Comparison with Experiments

One of the remarkable properties of the lead chalcogenide group of narrow band-
gap semiconductors is that they have a range of non-stoichiometry accompanied by either
Pb or Te vacancies. Our theoretical results predict that Pb vacancies produce p-type PbTe
whereas Te vacancies produce n-type PbTe, which are consistent with the earlier model
calculations.[14, 15] However, in contrast to these earlier semi-empirical or model
studies, present ab initio calculations predict significant changes in the DOS near the

band gap region, which should affect the transport properties quantitatively.

74

As regards monovalent impurities, Poudeu et al. [8] found that Nao_95PbmeTem+2
(m=20) was a p-type semiconductor with high thermoelectric FOM, ZTmax of ~ 1.7 at
650K. The observed p-type behavior is consistent with our ab initio calculations and what
is known about Na and Li impurities in PbTe. To approach a Na concentration closer to
that for m=20 compound (NaSbemTem+2), we add one more Na atom at the middle of
the 64 atom super cell. The calculation shows that the DOS in the region of the band gap
does not change by this addition. Thus one can treat the NaogstzoSbTezz as basically
PbTe doped with holes. Its therrnopower should therefore be positive.

The behavior of trivalent impurities such as In, Ga, Tl has been both interesting
and puzzling. Gelbstein et al. [42] have recently argued that the thermoelectric property
of In-doped PbTe can be understood in terms of the existence of deep-lying states
generated by In-doping. We find that In not only introduces deep defect states in the band
gap (perhaps overlap with the bottom of the conduction band) but also strongly reduces
the DOS near the top of the valence band. This results in the reduction in the minority
(hole) contribution to therrnopower both due to DOS reduction and annihilation of the
holes by electrons occupying the localized state in the gap. The net effect can be an

increased n-type therrnopower, as seen in [42].

As, Sb, and Bi give rise to resonant states near the bottom of the conduction band.
Although this resonant state may not take part directly in transport properties because
they are at least 0.5 above the band bottom, they indirectly affect the transport properties
by increasing the DOS near the bottom of the conduction band. This was pointed out in
Fig. 3.11 and Fig. 3.12. The increased DOS can in principle give rise to an increased

therrnopower in n-type systems. A simple calculation [51] of therrnopower by assuming

75

energy-independent relaxation time showed that room temperature therrnopower of
AngPb2,..2Te2n could be large, as seen experimetally.[7] However, detailed calculations
of transport coefficients taking into account the changes in band structure (caused by the
Sb induced resonant states) and energy dependent scattering indicates that this
enhancement of therrnopower is not true in general but depends on the carrier

concentration and temperature range studied.[52]

3.8 Summary

In summary, my ab initio electronic structure calculations in Rszn-1TeZn (n=16),
where R is a vacancy, or monovalent, divalent, or trivalent atom, show that when a Pb
atom is substituted by R, the DOS gets perturbed over the entire valence and conduction
bands. In addition, there are major changes in the DOS near the band gap region for most
R. This should have significant impact on the transport properties of LAST-m compounds
for m z 18. We find that Na does not change the DOS within 0.5 eV of the band
maxima; thus it is an ideal acceptor. In contrast, other alkali atoms and Ag and Cu give
rise to an increase in the DOS near the top of the valence band and negligible change in
the DOS near the bottom of the conduction band. Hg, Cd, Zn give rise to strong resonant
state near the bottom of the conduction band and suppress the DOS near the top of the
valence band (which is good for n-type therrnoelectrics). Group V (As, Sb and Bi)
impurities introduce resonant states near the bottom of the conduction band and should be
good for n-type thermoelectrics. In contrast, the trivalent impurities (Ga, In, and T1)
either introduce bound states in the gap or resonant states near the top of the valence

band.

76

Te vacancy also has a strong effect on the DOS of PbTe near the band gap region.
New states appear in the band gap below the CBM, these states comprise primarily of the
p states of Pb neighboring the vacancy. In contrast, the Pb vacancy increases the DOS
near the top of the valence band but only by a small amount. Te vacancy should be n-type
whereas Pb vacancy should be p-type. Iodine appears to be an ideal donor, it does not
change the DOS of PbTe near the band gap, just shifts the Fermi energy to the conduction
band. The divalent atoms S and Se also do not change the DOS near the CBM. There is
however some depletion of the DOS near the VBM. These results should have important
implications in the thermoelectric properties of, e.g., the n-type ternary compound
PbTeHSx, [53] where one can synthesize mixed Te/S systems without appreciably

affecting the conduction-band states and hence the n-type charge and energy transport.

77

Chapter 4

 

Electronic Structure of Defects in
SnTe

 

4.1 General Introduction

Although the electronic structures of IV—VI semiconductors such as PbS, PbSe
and PbTe [18, 19, 23, 28, 29] have been extensively studied in past using a large variety
of methods (both empirical and ab initio), there is very little information about the
electronic structure of SnTe using ab initio first principles electronic structure techniques
excepting for only one calculations by Okoye [80]. Earlier non ab initio band structure
calculations for SnTe have been carried out using many different techniques which
include the empirical pseudopotential method [81, 82], the augmented plane wave
method [83], the orthogonalized plane wave method [82] and the improved
pseudopoential method [84]. All these calculations give results that show several
common features but they differ in detail in comparison with experimental data.[80]
Okoye has recently carried out an ab initio band structure calculation in SnTe using the
full potential linearized augmented plane wave (PP-LAPW) method within the
framework of the density functional theory (DF T). [80] His calculations have been
performed with and without spin-orbit interaction (SOI); he found that the inclusion of
SO coupling effects generally improved the agreement between the parameters studied

(band gap, lattice constant, bulk modulus) and the experiment data. No defect structure

78

calculations in SnTe using ab initio methods have been done to our knowledge. In this
Chapter, we discuss the results of our attempts to understand the physics behind the deep
defect states formed by different types of defects in SnTe using self-consistent ab initio
electronic structure calculations within DFT. The purpose of this work is to develop a
systematic understanding of the defect states for different types of impurities (Sn and Te
vacancies, substituting for Sn: Ag, Cd, In and Sb) and compare with those obtained in
PbTe (Chapter 3). The arrangement of this chapter is as follows. In Sec. 2 I discuss the
electronic structure of SnTe. In Sec. 3, the structural model and the method used to
calculate the electronic structure are discussed briefly. In Sec. 4, I present the results for

different type of impurities in SnTe. Finally, a brief summary is given in Sec. 5.

4.2 Electronic Structure of SnTe

SnTe crystallizes in the rock salt (NaCl) structure. The experimental lattice
constant for SnTe is 6.327 A. [85] SnTe has a direct minimum energy band gap of 0.23
eV [86], and as in PbTe the gap occurs at the L point in the face centered cubic (fcc)
Brillouin Zone (Fig. 4.1). The theoretical band gap (with SOI and using GGA) is
E9 = 0.212 eV [80]. Orbital analysis shows that Te and Sn p states strongly hybridize,
suggesting covalent interaction between these two atoms. The valance band has more Te
p character whereas the conduction band has more Sn p character. The DOS of SnTe
shows that the Sn s-band overlaps with the valance band, as shown in Fig. 4.2, unlike the
Pb s-band in PbTe, which lies below the valence band minimum by ~ 0.8 eV. This is due

to the larger relativistic effects (Darwin and mass velocity correction) in Pb.

79

 

8.0 ., .9

Irlrlrlrlrlrlr
H‘-
X
‘K

.o
o

 

...............

Energy (eV)

3.}
\

n.-
-----

.lrlrlrlrlrlrlrlrlr

 

rIrlrlrlrIrIrl

 

 

 

 

—17.0

 

 

 

LA F A XZWK

é

Figure 4.1 Electronic energy band structure of SnTe calculated using GGA including
SOI.[80]

80

 

 

 

 

 

 

 

total DOS
A 80 d —- 0 — Te partial DOS
= _ 0— Sn partial DOS
8 Te-p
.H
.E 60 — Sn-s
:
E
8 40 -
to N
m In , f
v *Ariifi l
8 20 .. u ‘ ? z
D :52... ., :3
o I ' A," , r v I
-8 -6 .4 -2

Energy (eV)

Figure 4.2 Total Density of state of SnTe calculated using GGA including 801.

4.3 Defects in SnTe

To model the impurities and the vacancy we have used the 2x2x2 supercell model
consisting of 64 atoms (see Fig. 3.2). We choose the defects to be at the origin of the
supercell with separation of two lattice constants (12.654 A) between the defects.
Convergence of the self-consistent iterations was performed using 10 k-points inside the
reduced cubic Brillouin zone (for cubic supercell) to within 0.0001 Ry (1.36 meV) with a

cutoff -6.0 Ry between the valance and the core states. For all calculations we use the

experimental lattice constant of SnTe.

The removal of a Sn atom (vacancy) or replacing it by a monovalent or trivalent
cation introduces a local charge disturbance and alters the covalent bonding with the

neighboring Te (denoted as Te2) p states. As a result the valence band and conduction

81

Sn-p

 

band states get strongly perturbed. One expects acceptor-like states to appear near the top
of the valence band or donor like states near the bottom of the conduction band, whose
precise positions can only be obtained after a detailed self consistent calculation. Also
because of the local nature of the perturbation we expect the entire valence band and
large part of the conduction band region to be affected. It is important to emphasize that
simple shallow impurity state models may not be able to describe the short range physics
of these impurity systems, as justified in PbTe case. Only a careful self consistent ab

initio calculation will be able to do so.

4.3.1 Sn vacancy in SnTe

The total DOS with one Sn vacancy, without Sn vacancy and the difference
between the two are shown in Fig. 4.3 (a,b). For the sake of comparison we shifted the
DOS of the SnTe upward by ~ 0.25 eV so the core bands Te-s band matched perfectly.
As we can see the entire valence band and conduction band density of states (up to 2 eV)
get modified in the presence of the vacancy. Since transport properties are controlled by
the states near the top of the valence band and the bottom of the conduction band we will
focus on the changes near band gap region within an energy range of :1: 0.5 eV. We see
that there is little change near the bottom of the conduction band DOS whereas there is an
increase in the DOS near the top of the valence band. In order to see which atoms
contribute near the top of the valence band DOS we show the difference in the projected
DOS Ap = p (with defect) — p (SnTe) associated with six Te nearest neighbor neighbors
of the vacancy in Fig 4.3 (c). The major contribution to the increase comes from the six

nearest neighbor Te atoms and we see that Te states near the bottom part of the valence

82

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

% 100
0 . (a) + SnTe with Sn vacancy
a 80« ——SnTe
g .
g 60': ifi‘lug A i ‘ *
V 40. ,4 K '1'! ‘li‘
8 . ) 1' v '4 (1:9
20- .3. i 7
D . I -
-6 -5 -4 -3 -2 -101 2
9 40
g , (b) Difference in total D08
20«
3’ M /\ NM
w .
g 0- . ll W to 11AM
§-20-
Q.
<-40.........,.E..
-6 -5 -4 -3 -2 -1 0 2
S 10
8 . (C C)Te partial DOS
> 8- + SnTe with Sn vacancy
in: . —SnTe
£3 5"
s - i
it” 4. )]3f
8 1
o 2- i
To * 1") i .’ >
‘E’ 0* . . . , .
a“: -6-5—4-3-2-‘1012
Energy [eV]

Figure 4.3 (a) The total DOS of SnTe with and without Sn vacancy, (b) their difference
Apm, and (c) partial DOS per Sn associated with nearest neighbor Te atoms. The Fermi
energy of SnTe with Sn vacancy is at the energy origin (0 eV).

83

band get pushed towards the top. This shift is partially compensated by the DOS
associated with the other atoms. Also other Sn and Te atoms do contribute to the
increased DOS near the valence band maximum. From Fig. 4.3(a), we can clearly see that
the Sn vacancies give rise to holes as expected. A comparison between the Sn vanacny in
SnTe with Pb vanacny in PbTe, we can see that Sn vacancy perturbs the conduction
bottom (reduction in the DOS in the energy range 0.5-1.0 eV), whereas Pb vacancy in

PbTe does not perturb the conduction band (see Fig. 3.3).

4.3.2 Te vacancy in SnTe

The total and partial DOS of SnTe with and without Te vacancy and the
difference Apm. are shown in Fig. 4.4(a)-(c). For the sake of comparison, we shifted
downward the DOS of SnTe with defect by ~ 0.455 eV such that the core bands match
perfectly. As we can see, the entire valence and conduction band DOS get perturbed by
the vacancy. New states appear in the band gap region and near the bottom of the
conduction band in the energy range -0.5 to 0 eV but clser to the valence band maximum
compared to PbTe case (see Fig. 3.13). Partial DOS analysis indicates that these states are
predominantly of Sn p character [see Fig. 4.4(c)]. For each Te vacancy, the valence band
looses 6 states (including spin). These 6 states are occupied by 4 Te electrons and 2 Sn
electrons. Removal of Te atom takes 4 valence electrons away leaving two Sn valence
electrons. As a result, two Sn electrons occupy the new states appearing above the

valence band. The system then behaves like an n-doped system.

84

100

 

 

 

CE: , (a) + SnTe with Te vacancy

O 80 . SnTe

> .

Q 60

.33 . l i“: '1»

m l“ 1' I i! I '3'

E 40 - i \ ‘ *
co j l '

o 20 4 f I ‘

D 0 r . - f ,‘L. -

 

 

 

 

 

 

 

 

 

 

 

 

 

cf; 20- (b) Difference in total DOS
0
E 10- /
9 * l 1
.9, 0 A AA _
s WVVVV
w
v _104
H
O .
H
o. -20-
< . 1 - .
-6 -5 -4 -3 -2 -1 0 1 2
1.0

 

. ((3) Sn partial DOS .
0,81 —*— SnTe with Te vacancy 3
. — SnTe I! j
l

1
. j l. ‘ ' ‘ '
it: I} M J” i.
’0’ ~

.0
a)

 

.0
m

 

J

 

.0
o

I V I

'-5 4'53'32'310
Energy [eV]

Figure 4.4 (a) The total DOS of SnTe with and without Te vacancy, (b) their difference
Aptot, and (c) partial DOS per Te associated with nearest neighbor Sn atoms. The Fermi
energy of SnTe with Te vacancy is at the energy origin (0 eV).

Ad
N

Partial Dos (states/9V cell)
0
. I:
0,-

85

4.3.3 Ag impurity in SnTe

Ag is a monovalent impurity, it provides a weaker perturbation (changes the ionic
charge by 1) compared to the Sn vacancy (change the ionic charge by 2). In Fig. 4.5 (a,b),
we plot the total density of states (DOS) for Sn32Te32 and AgSnnge32 and their difference
(after making a small relative rigid shift by -0.326 eV so that the Te 3 core band match
perfectly). We find that Ag impurity perturbs the valence band states, within energy
range (-4 to -3 eV), due its core d states hybridizing with the Te p bands. The difference
in the total DOS with and without Ag impurity shows a small peak near the top of the
valance band (similar to PbTe- see Fig. 3.4). Since the s state of the Ag is above the
valance band, Ag donates one electron when it replaces Sn (which is divalent), it gives

rise to one hole per impurity and hence acts as an acceptor.

 

 

 
 
 

 

 

 

 

 

 

 

 

e 100 g

8 80(a)—*-(Ag)SnTe g 20,(b)

% i, -—SnTe %

\ 60“ \

"’ a) 0-

a: .. ”Ar-fr

1'6 401 S

15. ‘ 3%.

8 20‘, 05-3 '20‘ Difference in total 008

o °.e 34-513314 1 2 < 433431331444
Energy [eV] Energy [eV]

Figure 4.5 (a) The total DOS of SnTe with and without Ag impurity, (b) their difference
Apmb. The Fermi energy of SnTe with Ag impurity is at the energy origin (0 eV).

86

4.3.4 Cd impurity in SnTe

Cd is a divalent impurity, the valence configurations of Cd is 532 whereas the
valence state of divalent Sn is 5p2. Naively, one can say when Cd replaces Sn, one
expects the Fermi energy position to be unchanged (as in the case of SnTe), if the number
of states in the valence band remains the same as in SnTe. Our ab initio calculations give
a completely different picture when Cd substitutes for Sn in SnTe. In Fig. 4.6(a)-(d), we
show the total and partial DOS of SnTe with Cd impurity. The F errni energy lies in the
conduction band. The only way the Fermi energy can move to the conduction band is if
the valence and the Sn 3 bands loses 2 state (4 including spin) to the conduction band.
This is precisely what happens, and the reason for the observed Fermi energy in the
conduction band in Cd doped SnTe is a strong interaction between Cd and neighboring
Te orbitals which removes one state from the valence band and gives rise to a new
hybridized state near the bottom of the conduction band. The integrated DOS of the
valence and Sn 3 bands show a loss of 2 states (4 including spin). One of these two states
is from missing Sn 5s2 state (two electrons) and the other state moves to the conduction
band giving rise to a new hybridized state near the bottom of the conduction band.
Because of the resonant state near the bottom of the conduction band, we expect electron-
doped Cd systems to exhibit large therrnopower. Cd in PbTe, as discussed in chapter 3,
also shows resonant states near the bottom of the conduction but in contrast to SnTe the
Fermi energy in PbTe remains in the gap, below the resonant state. This difference is
intriguing and more careful calculations are needed to understand the physics behind the

major difference of Cd defect in PbTe and SnTe.

87

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3100 g
8 . (a) —+—(Cd)SnTe 8 20. (13) Difference in total DOS
> 80- —SnTe >
m ‘ g: 10
. . 1 .1 11111
9 . .9 o. -
.4 .0. g . 1 111 V1
33. . v -10-
a) 20« § .
8 0‘ Q '20‘
-23*-'7'-e'-s'.21'-'3'-'2' '1'0 1' 2 < -5111515‘43‘32 1 o 1 2
A Energy [eV] A Energy [eV]
210 . a 2.0
> . (C) Te partial DOS > . (d) Cd partial DOS
2 a. ~—+—(Cd)SnTe 9 _
m B 1.51 of 5 character
93, ———SnTe 93
8 6* 8 10‘
a 4* a ' +
o 2: o 0.15"
'3 0. ........ I- 193’ OOMLN
a“; 057-6-541—321012 g 076-54-3-2-1012
Energy [eV] Energy [eV]

Figure 4.6 (a) The total DOS of SnTe with and without Cd impurity, (b) their difference
Apt“, (c) partial DOS per Te associated with nearest neighbor Te atoms, and ((1) Cd
partial DOS of s-character. The Fermi energy of SnTe with Cd impurity is at the energy
origin (0 eV).

88

4.3.5 Sb impurity in SnTe

Sb is a trivalent impurity, the valence configurations of the Sb is 5p3, The valence
state of divalent Sn is 5p2. Results for Sb impurity show broad resonant states (width ~1
eV) near the bottom of the SnTe conduction band. In Fig. 4.7(a)—(d), we show the total
and partial DOS of SnTe with the Sb impurity. The resonant state is a mixture of the p-
states of Sb and the p states of the neighboring Te2 atoms in almost equal proportions.

The Sb impurity act as donor.

 

100

 

, (b) Difference in

1 a _._. Sb SnTe
( ) ( ) 10. total DOS

801 — SnTe
601

4o-

 

 

20-

 

 

DOS (states/eV cell)

 

 

 

 

 

 

w 1 v F r I v ' 1 ‘— T

0-6-5-4-3--2-1o'1"2 0-6-5~4-3-2-1o'i'2
Energy [eV] Energy [eV]
2.0

Aptot (states/ev supercell)

 

 

.(c) Te partial DOS
8« + (Sb)SnTe
—— SnTe

. (d) Sb partial DOS
1.5+ of p-character

1.0-

0.5-

 

 

 

 

 

 

 

 

'''''''''''

-6 -5 .4 -3 -2 -1oii'2
Energy [eV]

0.0 . , .
-6-5-4-3-2-1012
Energy [eV]

Figure 4.7 (a) The total DOS of SnTe with and without Sb impurity, (b) their difference
Apmt, (c) partial DOS per Te associated with nearest neighbor Te atoms, and ((1) Sb
partial DOS of p-character. The Fermi energy of SnTe with Sb impurity is at the energy
origin (0 eV).

Partial DOS (states/eV cell)
0 . 43 - . -
Partral DOS (states/eV cell)

89

4.3.6 In impurity in SnTe
Indium impurity in IV-VI semiconductors have been known to exhibit anomalous
behavior.[42, 43] For example it act as n-type in PbTe and p-type in SnTe. We discussed
in details in the previous chapter, the anomalous behavior of In impurity in PbTe and we
presented a new picture in order to understand these anomalous behavior. Indium metal is
nominally trivalent, and hence when In replaces a divalent Sn, one expects an n-type
donor. This assumes that the number of states in the Te-p band does not change when an
In impurity is present. If the In s-state is strongly bound and lies below the valence p-
band, then In will behave as an acceptor because it gives only one electron to the Te-p
band instead of needed two. Results for In impurity in SnTe shows that the deep defect
states are resonant states near the top of the SnTe valence band (act as p-type). In
contrast, in PbTe these states lie in the band-gap region and can act as n-type by
thermally exciting an electron from the DDS to the conduction band. This difference in
the position of the deep defect states between SnTe and the PbTe is extremely subtle but
explains the experimental anomalies seen in the case of In impurities (act as n-type in
PbTe and p-type in SnTe). These subtle differences can only be seen in a fully self
consistent ab initio study. We find that the orbital nature of the resonant state near the top
of the valence band is a hybridization of the 3 states of the In with the p state of the
neighboring six Te atoms. We show, in Fig. 4.8 (a)-(d), the total and partial DOS of SnTe
with and without In impurity, the difference Apmt, and 3 character of the partial DOS of In

impurity. These general features are very similar to those for In in PbTe.

9O

 

 

  
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

g 100 '33 . .

8 . (a) +(ln)SnTe 0 (b) Difference In

> 301 —- SnTe % 10‘ total DOS

(D l \

B 60- 8

Q a 0_ 4A!

.59. 40‘ fa; W V

m , 1

v H

a) 20' 3-10.

0 Q.

a ............. q ........... . , .
'6 '5 '4 '3 '2 '1 0 1 2 -6 -5 -4 -3 -2 -1 0 1 2

A Energy [eV] A Energy [9V]

'5 1o . = 1.5

g ‘(C) Te partial DOS E Id) In partial DOS

Q 8‘ ‘ (ln)SnTe Q of s-character

a) 1 -—--—SnTe a) 10,

.fl.’ 61 .9 '

53 - .59

£9. 4- g;

(D 1 a) 0.5

O 2. O

D ‘ o l

E o . . . . - . . ..... 3 0.0 ........... 1:?"7‘

E -6 -5 -4 -3 -2 -1 0 1 2 E -6 -5 -4 -3 -2 -1 0 1 2

n. 0.

Energy [eV] Energy [eV]

Figure 4.8 (a) The total DOS of SnTe with and without In impurity, (b) their difference
Aptm, (c) partial DOS per Te associated with nearest neighbor Te atoms, and (d) In partial
DOS of s-character. The Fermi energy of SnTe with In impurity is at the energy origin (0
eV).

4.4 Summary

In summary my ab initio electronic structure calculations in RSnZMTeZn, n=16, R
= a vacancy, Ag, Cd, Sb, and In show that when a Sn atom is substituted by R, the DOS
of the valence and conduction bands get strongly perturbed. There are significant changes
near the band gap region. Sn vacancy causes very little change near the bottom of the
conduction band DOS whereas there is an increase in the DOS near the top of the valence

band. Te vacancy on the other hand creates new states in the band gap region and near the

91

bottom of the conduction band. Ag impurities perturb the valence band states due to the
Ag d-states hybridizing with the Te p bands. The difference in the total DOS with and
without Ag impurity shows a small peak near the top of the valance band. Sb impurity
shows resonant states near the bottom of the SnTe conduction band. Results for In
impurity shows that, unlike PbTe, the deep defect states in SnTe are resonant states near
the top of the valence band. In PbTe these deep defect states lie in the band-gap region
(act as n-type). This fundamental difference in the position of the deep defect states in
SnTe and PbTe explains the experimental anomalies seen in the case of In impurities (act

as n-type in PbTe and p-type in SnTe).[13, 87]

92

Chapter 5

 

Energy and Temperature
dependence of relaxation time and
Wiedemann-Franz law in PbTe

 

5.1 General Introduction

As discussed in Chapter 1, the efficiency of thermoelectric (TE) energy
conversion depends on the transport coefficients of the constituent materials through the
figure of merit (FOM) Z T = USzT/(Kl + Ice”) where a is the electrical conductivity and
S is the therrnopower (Seebeck coefficient) [9]. The quantity in the denominator is the
thermal conductivity; it is given by the sum of contributions from the electronic carriers
at constant electrical current i (Keu ) and the lattice contribution Kl. Z has units of inverse
temperature, so it is generally quoted as ZT, with T the absolute operating temperature of
the TB device [9]. TE devices are usually made from narrow band gap semiconductors.

Several calculations of the transport coefficients of PbTe and related systems have
been reported over the years [52, 95, 102, 104, 105]. These calculations have been
performed using Boltzmann transport equation [123] within the energy-dependent

relaxation time (t) approximation and the nonparabolic Kane model for energy dispersion

8;,- vs It. [133] The energy and temperature dependence of different scattering mechanisms
were incorporated in these calculations. Depending on the temperature, different

scattering mechanisms (from impurities, acoustic and optical phonons) contribute to the

93

relaxation time. These earlier works did not explore the relationship between the energy
and temperature dependence of the effective relaxation rate and the T -dependence of
0, Ice“ and Kel‘E. Also the validity of using WF law [123] to estimate electronic thermal
conductivity at high temperatures has been questioned in the literature [104], but it is still
widely used to estimate the lattice thermal conductivity from the total experimental
thermal conductivity. The lattice thermal conductivities estimated this way are turning
out to be very small, well below the so called alloy limit. [136] This reduction has been
attributed to the enhanced phonon scattering from the nanostructures present in the

samples. Thus to have a more definite picture one must critically examine the use of WF

law. Furthermore the difference between constant T and constant I? electronic thermal
conductivities in PbTe has not been addressed properly although the fact that the
experimentally measured electronic thermal conductivity ice, appearing in the expression
for ZT is Ice“ 5 LOT, and the modified Lorentz number L is different from L0 is well
known. [104, 122]

In this chapter we address these issues by focusing on (1) a careful analysis of the
energy and T dependence of r to develop a simple scaling form, (2) understanding the
relation between the energy dependence of r and the T dependence of a, S, and the
power factor (052), and (3) finding out the difference between K9), 1 and Ice”,- and to
reexamine the validity of WF law Ice, 5 Ice“; = LoaT, where the Lorentz number L0 =
2.45 x 10"8 WQ/K.

The arrangement of this chapter is as follows. In Sec. 2, I describe the transport
coefficients using the Boltzmann transport equation and define the transport function. In

Sec. 3, I briefly review the non-parabolic Kane model for the energy band dispersion. In

94

Sec. 4, I describe the transport coefficients calculated using in the Kane model. In Sec. 5,
I describe the total relaxation time and different scattering mechanisms contributing to
this quantity in PbTe. In Sec. 6, I discuss the total thermal conductivity K and the
different types of electronic contributions. In Sec. 7, I present and discuss our results and

finally, I give a brief summary in Sec. 8.

5.2 Transport Coefficients using the Boltzmann Transport Equation

The charge carriers in a metal or a semiconductor move under an applied external
field and temperature gradient. We consider the situation where the carriers are
accelerated by the fields, but lose their extra energy and momentum by scattering from
lattice waves (phonons) and impurities. A standard method used to deal with this problem

is to use Boltzmann transport equation. [123] This equation states that in a steady state

situation, at any point in space 1" and for any value of the electronic wave vector E, the

net rate of change of f; (i’, t) is zero, where f; (i’, t) is the probability that an electron

(hole) of wave vector I: is at point 1" in the crystal. Then:

aft 0f; of
[6— :]diff+ {—6—- tklfieza+ +[— :Lwtt'o — ’ (5-1)

where the different contributions to the change are due to diffusion, external fields and
scattering. Eq. 5.1 is valid for the steady-state, not for equilibrium (ffif, t) = $01))
which holds in the absence of external fields, 1:200.) is the equilibrium single particle

Fermi distribution function, where i" dependence comes through the local temperature

TU"). It is given by:

95

1
3"” (killer—(Li) + 1

where a; is the energy, [.1 is the chemical potential, k3 is the Boltzmann constant. For

 

f;°(7)— - (5.2)

 

simplicity we will ignore the spin index and assume that each state If is two-fold spin
degenerate since we do not consider magneto transport.

Assuming Liouville's theorem of the invariance of volume occupied in phase
space, one finds that the number of carriers in the neighborhood of 1’ at time t is equal to
those in neighborhood of i’ — ti}; at time zero:

‘ ma 1) = f;(? — 115;, o). (5.3)
The rate of change of the distribution fimction due to diffusion can then be written as:

[Bf—E _ a? 0f; 13171“

diff at am) k ar (5'4)

using the definition of Fermi distribution function the derivative with respect to

temperature in Eq. 5.4 can be rewritten as:

 

 

 

 

5" - #1
aft ”CM—5,?) e;—u_ _af; ewe (55)
OT _ 8" — 2 kBTz — 087(- T ' '
[exp Err”) + 1l
and then the rate of change of the distribution function due to diffusion is given by:
a__f-°] _( 0f2)875—l1.
— v~ (—VT). 5.6
By analogy to Eq. 5.3 we can apply Liouville's theorem in E—space:
fig. (5’, t) = fE-Et(F’ 0) I (5'7)

considering only electric field E as the external field, the time rate of change of Bloch

_...
wave vector k rs given by:

96

hi2 = eE' . (5.8)

The velocity of the electron in the state If is given by:
hi3}; = —_. . (5.9)

In Eq. 5.8 and 5.9, e is electronic charge, m is free electron mass, and h = h/Zn with h
being the Planck constant. The rate of change of the distribution function due to the field

can be written as:

6 -° aka - _46 68~_ Ea * ..
[aft—k] — f"— f"— —8 4‘15,th =e(- -f—a_’f)13;13. (5.10)
field— —at 6k [(68); 6k_ —h 6—875 ask

The effect of scattering on the change in the distribution function is more
complicated. In this thesis the relaxation time approximation was considered. In the
relaxation time approximation the rate of change of the distribution function due

scattering is given by:

(5.11)

h

[sz1 = _fr- __f__;o ____gg
scatt. Ti"

Tie
where g; is the difference between the steady-state and equilibrium distribution
functions, and I; is the relaxation time.

Using the Eq. 5.6, 5.10, and 5.11, the Boltzmann transport equation (Eq. 5.1) can

be written as:
a " —) a a E” —
git. = e (_ _fl‘.) fikTEE + (— fk) RT ufiETE(—VT). (5.12)

In order to find the expressions for electrical conductivity tensor 3 and

 

therrnopower tensor 3", the microscopic and macroscopic Ohm's laws have to be used,

97

which relate the current density to the applied electric field and temperature gradient. The

microscopic Ohm's law is given by:

j=VZe Raft/299m , (5.13)

since
2 £21336; = o , (5.14)
78

where V is the volume of the crystal in real space. From Eq. 5.12 and Eq. 5.13, the

current density can be expressed as:

«v 1 6f” —c 1 6f" 8" _ u
]= M; e(- 1972;)” kv~E + V E e( 687;) T rkv~k E(— VT), (5.15)

 

 

(—vr) (5.16)

ar~ . ..
1 5f; _, 1 ZE(‘BE_;)(£E—”)TEVEVE
j 721% )rfiaxafi 13+ 0
72

58;;

The macroscopic Ohm's law in the presence of an external electric field E" and
temperature gradient VT is given by:

j’ = 3'” + 5’§(—VT) = 8(12?’ + §(-VT)). (5.17)
Direct comparison of the two Ohm's laws in Eq. 5.16 and 5.17 gives the expressions for

the electrical conductivity and thermopower tensor:

H 32 6f]; —o -o
0': V _. (—-aE-l_E-)T§v-évk: , (5.18)
§= (3)-1A” , (5.19)

where X is given by:

98

H e 6f” -c -v
A = —Z ——5 (872 — 10131275177). (5.20)
VT -
k

as};

Analogous to the current density equations, the heat current density equations for

L; can be written as:
. 1 . 1 .
IQ = 7211182 - #) V2 = 72972092 — #9127; : (5-21)
75 1%

Q s a’E‘Tfi + Rafi—VT) , (5.22)
where 138”; is the electronic thermal conductivity tensor at constant If field
(evaluated at E" = 0). Direct comparison of the two heat current density equations gives

the expression for the electronic thermal conductivity tensor at constant E" field (E = 0):
H 1 af-IE 2 a -+
Keir: = WZ (— 5.9—7".) (£7; '— [1) TEUEUR‘ . (5.23)
It
At constant current density f ( f = 0) (Eq. 5.17), the heat density To is given by:
.. H H 2
10 5 [KM — 0(5) T] (—vr). (5.24)

The electronic thermal conductivity at constant current density i (evaluated at f = 0)
12'3”, is given by:

H H H 2

Ice” = x,” — 0(5) T . (5.25)
Normally in metals and highly degenerate semiconductors the second term on the right

hand side of Eq. 5.25 is quite small and 133),, z Eelig. But as we will see later, for

semiconductors of current interest they can differ dramatically.

99

5.3 Nonparabolic Kane Model for Energy Dispersion

In this section I will briefly describe the Kane model for the band structure of
PbTe valid near the band extrema which occur at the L point of the fcc Brillouin Zone
(BZ), see Fig. 3.1. In narrow band-gap semiconductors (i.e., lead chalcogenides) the
energy region of interest of an electron as measured from the band edge is comparable to
the band gap Eg. Because of this, the dependence of the energy on crystal momentum is
non-parabolic and the effective masses are functions of energy. The nonparabolic Kane
model was introduced [133] to describe the deviations from the quadratic dependence of
energy on the crystal momentum. In this model, the longitudinal effective mass m; and
the transverse effective mass m, depend only on the interaction between the lowest
conduction band and the highest valence band and the contributions of the other bands
are assumed to be small. Here longitudinal and transverse are defined parallel and
perpendicular to the vector joining the center of the El to the L point (see Fig. 3.1). From
now on we will confine ourselves to the conduction band and electron type transport. For

a simple parabolic model the energy dispersion can be expressed as:

—h2 2ktz+k‘2 526
Ek—Z mt m, ’ (' )

where 872 is the energy, and k, and kt are the magnitudes of the longitudinal and

transverse components of k, measured from the band extrema. In the Kane model the

energy dispersion is given by:

8;; r12 Zkt2 klz
_. _ =-- —— — . 7
g,(1+Eg) 2(m.+m. , (52)

100

In this model the effective masses and the mass anisotropy coefficient of electrons and
holes are equal. The constant energy surfaces are ellipsoids, and m; and m, have the same

energy dependence.

5.4 Transport Coefficients in the Kane Model

In order to calculate the transport coefficients we have to compute different
quantities appearing in Eqs. 5.18, 5.19, and 5.23, which include the carrier velocity 13;;
and the relaxation time I}. For the sake of completeness we will derive the expression for
the transport coefficient following the methods used by Ravich et. al. [125] and work of
Bilc et. al.[52] For this, we change the discrete sums over E into integrals, the carrier
velocity 137‘- is calculated using the Kane model for energy dispersion, and the relaxation

time expressions for different scattering mechanisms are obtained within Kane model.
These expressions are given in the section 5 .5. [106, 107, 124, 125]
PbTe systems have cubic symmetry. Therefore, the components 0w and SW of
the electrical conductivity and therrnopower tensors can be expressed as:
0w = W0, (5.28)

s“, = 6),,5, (5.29)

where a and S are related to the trace of 3 and S tensors and they are given by:

1

a = 3115*, (5.30)
1 ..

s = §Tr5, (5.31)

For cubic system, Tr? and TrS are related to {37,4375 = 9;: terms.

101

The Kane model energy (Eq. 5.27) contains two types of effective masses and in

order to simplify the calculations the fallowing substitutions can be made

2 = mt ,2

kt —m;, k, , (5.32)
2 ._

k1 _ ma k1 , (5.33)

where m'd is a scaling mass parameter (see Eq. 5.57 for its definition)

Using the new variables k,’ and k;, Eq. 5.27 can be written as:

 

      

8k, _ hZ ,2 ,2 _ hZ '2
8k, (1 + E9) — 2mg! t ) — 2m; k . (5.34)

 

In terms of the new variable k” the expressions for 3' and 5 are:

82 afkl -O -D
‘VZ( Te.,)’k'”k'”k“ (5'35)
kl
’5” = (3)-12’, (5.36)
where X is given by:

e 6f?! -9 -9

= 1771:; (— 3%) (£72, - ”175,152,123“ (5.37)

kl

Changing the sums into integrals, Eq. 5.36 for therrnopower becomes:

a _.r -v -. "’r
1 I: (.08 fk 37.5) (81;! _ #)T7¢’:v7¢'rvird3k

3 I
—a£k —-::')rk,v-,;,v,~c,dq k

5’: (5.38)

Usrng Eq. 5.34, the infinitesimal unit volume in reciprocal space 6131fI can be expressed
as:

3

1 1
d3k’ = 41rk’2dk' = 4n 1 +— 1 -l- — d8 , (5.39)
r1359 59 -

102

 

where we have dropped the suffix for the carrier energy 83,. The carrier velocity in Eq.

5.9 is given by:
1115;, = 13512 = i—kT. (5.40)
h 6k, md 1 + .5
Ba

2

Using the above equation, the 173, = 133,133, term in the transport equations can be

expressed as:

E
- 2 £(1+E;)

I_ml 2'
11+?)
9

(5.41)

Substituting Eq. 5.39 and 5.41 into Eq. 5.38 and using Eqs. 5.28 and 5.31, the
therrnopower along any one of the cubic axis Sxx (Sxx = S) becomes:
a f 3 2 ‘1
00 E 2 8
1L) (_EE)(£-H)[E(1+E—g)] (1+E—g) T5618

3” = — 3 _1 , (5.42)

[00° (— g) [a (1 + 15.39)]2 (1 + £5) 18 615

where the relaxation time Is depends on energy. Since 5' depends on the ratio of the two

 

integrals (Eq. 5.38) the spin and band degeneracy (y) factors (see the following
paragraph for a definition) cancel out in Eq. 5.38 and 5.42.
Taking into account the spin degeneracy and the band degeneracy y, the

expression for the electrical conductivity 2? can be rewritten as:

H Zye2 6f3, _, 4 Zye2 V °° 0f3, _, .. 3..
= —— —o -o =———— —— -O -0 I ’ .4
a V E ( 6875;)TRIVTUVIU V (2703i) 683, rk,v3,vk,d k (S 3)
kl

and using Eq. 5.39 and 5.41, Tr 5' can be expressed as:

 

103

-1

1 3

(2mg)? °° 6f 8 5 28

H = 2 _ _ _ _—
Tr a 2ye ”2,13 [0 ( as) e 1 + By 1 + E9 r8 d8. (5.44)

 

Again for cubic system the conductivity along any one of the cubic axis axx (on = a) is

given by:

1 3 '1
2 (2mg)? °° 6f 8 5 28
= — 2 — -— —- '—
a 3 ye ”2,13 [0 ( as) [a (1 + 59)] (1 + E) r8 d8 . (5.45)

The expression for the electronic thermal conductivity at constant [3" field (E = 0)

 

We”; can be rewritten as:

H 2}, afil 2 -9 a
Ice“. = VT .. (- 5g?) (53, - 11) 13,123,173, , (5.46)
kl
H 2y V 00 6f}: 2 -v -9 3"!
Kel,E - flan—)3 Io (— 673,) (51;, - #) r3,v3,v3,d k » (5-47)

and using Eq. 5.39 and 5.41, Tr 12'8“,- can be expressed as:

1 3 "1
H _ 2y (2mg)? °° 0f 2 e 5 28
TT Keug —?-;2-f-l-3—L (-3g) (E—fl) 8 14-39- 1+2; T8618 ,(5.48)

the electronic thermal conductivity at constant I? field (E = 0) 133),); along the x-axis

(12235)“ (03835)“ = K835) becomes:

1 3 ‘1
_ 2y (2m;)7 °° 6f 2 s 7 28
Ice“,- — 3—7, ”2,13 IO (— E) (e - u) a 1 + By 1 + 59 re d8 . (5.49)

The three transport coefficients for the cubic system can be expressed in terms of a single

transport function 2(5):

0 = e2 I: (— 13%) 2(8) (18 , (5.50)

104

s = T31: (— g—i) 2(a) (a — p)d8 , (5.51)

x...- = 1107?) 2600 — 102 as (5.52)

where 2(8) = D(8)v§r£, and D(8) is the density of states, defined as D(8) =

zkp 6(8 — 815,0)

The concentration of carriers 11 can be written as:

2y 2y d3k’
n = 723,, =71 8, _ ,, (5.53)

3, exp (4381—) + 1

and using Eq. 5.53 the concentration n can be expressed as:

 

 

o.e(1:[ 73%)]2-(1 + %—8) d8
=n—2—2—h3y m'2 :fo ,, 9 . (5.54)
ET)+1
The expression ym'f/2 is defined as:
m3,” = ym'j/2 , (5.55)

where md is called density of state effective mass because it takes into account the
degeneracy of the conduction band. The relation between scaling mass parameter (m'd),
the density of state effective mass ma and longitudinal and transverse effective masses
(ml, mt) can now be deduced using the Eqs. 5.32, 5.33 and the following equation:

d3k = .132", (5.56)
which sets the equality between the infinitesimal volumes in reciprocal space. Using this
we find the scaling mass parameter m' d = (mlmt)1/3. From Eq. 5.55, we therefore
obtain:

ma = Y2/3m'a = y2/3(m)m§)1/3

105

5.5 Relaxation Time in the Kane Model

Different scattering mechanisms of charge carriers in lead chalcogenides have
been extensively discussed in many papers [106, 107, 124, 125]. It is found that the
dominant scattering mechanism contributing to T are from point defects and thermal
phonons. Their relative contributions to the scattering processes are summarized below.

At low temperatures (liquid helium), charge carriers are scattered mostly by
charged vacancies. At low densities n S S X 1018 cm'3, scattering by Coulomb potential
of the vacancies dominates, whereas for high carrier densities n 2 1019 cm'3, the
Coulomb potential gets screened out and scattering by the short range potential of
vacancies dominates. As the temperature increases, the relative importance of the charged
vacancies decreases and scattering by thermal phonons increases. For temperatures above
300 K, scattering by acoustic phonons and optical phonons (both polar and deformation
potential coupling) contribute mostly to the relaxation time. The expressions for the
dominant contributions to the relaxation time within the Kane model were first worked
out by Ravich and collaborators and later extended by Zayachuck and Freik et al. [106,
107, 124, 125]. They are available in the literature. Therefore we give here only the final
expressions for different scattering mechanisms, discuss the fundamental parameters
involved, and analyze their relative significance for PbTe systems.

(i) Deformation potential of acoustic phonons (Ta)

1

£2 _.2_
mam (e + 3')

Ta: .
(1+2E3)[(1—A)2 —B]

 

(5.58)

106

E
E—g(1 _ Ka)

 

 

 

A = e , (5.59)

(1 + 2 FE)

8 e

8 E_g (1 + F9) Ka
B = 2 , (5.60)
3 (1 + 25—)
£9
21th4C
TO,a(T) = l 3 I (5.61)
EchBT(2md0)§

where Eac is the acoustic deformation potential coupling constant for the conduction
band, C, is a combination of elastic constants, Ka is the ratio of the acoustic deformation
potential coupling constants for the valence and conduction bands K, = Edy/Ede with
the values Ka = 1 for n-type PbTe and Ka = 1.5 for p-type PbTe, and mao is the density
of states effective mass for a single ellipsoid (y = 1), i.e. mdo = m'd.

(ii) Deformation potential of optical phonons (To)

1

£2 "2'
1.0.0 (T) (E + E_g)

 

 

 

r, = 8 , (5.62)
(1 + 2-E—) [(1 —A)2 — B]
9
T?" (1 _ K0)
A = —9——€—, (5.63)
1 + 2..)
( a.
E E
8 E (1 + F9) K0
3 = 2 , (5.64)
3 (1 + 2 3)
E9
2h2a2p(hw )2
16,6(T) = ° 3 . (5.65)
”EchBT(2mdo)§

107

where a is the PbTe lattice constant, p is the PbTe density, wois the frequency of the
optical phonons, K0 is the ratio of the optical deformation potential coupling constants for
valance and conduction bands, KO = Eov/EOC, which are taken to be the same as for the
acoustic phonons, Ka.

(iii) Polar scattering by optical phonons (Tpo)
1

6:2)5 _
£+-— F 1
( Eg

 

 

1”" = 1 E . (5.66)
92(2mao)‘2’K3T(e;1 — 651) (1 + 2F")
9
251(1 + Bi)

F=1—6ln(1+61)— 2 [1—26+261n(1+6 )1, (5.67)

8

(1 + 2E—g
6 = (21%)”, (5.68)

where eoand 6,0 are the static and high frequency dielectric constants, k is the carrier
wave vector and To is the screening length of the optical phonons. The parameters k and

r0 are given by:

 

 

£2

Zmdo (a + F)
k2 = 9 (5 69)

h2 ' °
5 2
Ziezm2 l

_ _ d 0

r0 2 _ 1th3€66 ( Li) , (5.70)

where "L? is the generalized Fermi integral:

m z
n m _ °° _fi .. i E
Ll (1,1,8‘9) —fo ( aJr: [8(1+Eg)] (1+Eg) d8. (5.71)

(iv) Short range deformation potential of vacancies (Iv)

108

TV = To,v(T) (E + g)—§ ’ (5.72)
(1 + 2,—3.3) [(1 —A)2 — B]

 

Fin — K.)
A = 9—— (5.73)

E I
1+2—)
( Eg

 

. (5.74)

 

To” (T) = , (5.75)

UgcmdoamdofiNv
where N, is the vacancy density, K, is the ratio of the short range deformation potential
coupling constants of vacancies for valence and conduction bands, RV = UW/ch which
are taken to be the same as for acoustic phonons, Ka.

(v) Coulomb potential of vacancies (re)

2

 

£3 0me); (a + g—ZY
I, = 9 8 , (5.76)
n(2e2)21v,[zn(1 + 5) — {/(1 + 6)] (1 + 275;)
f = (Zkrv)'2 (5.77)

where Z e is the vacancy charge, and 7”,, is the screening radius of the vacancy potential

 

given by:
41w2
Tv'z = 001), (5-78)
50
2%( f 2 3
mac 2 I1 H
= _ _ _ .7
D01) ”2,1, (u 59) (1 + 2 Eg). (5 9)

109

D (11) being the density of states at the chemical potential.

The total scattering relaxation time ( TE E TM) is expressed as:

1 1 1 1 1 1
=-—+-—-+—+—+-—. (5.80)
Ttot Ta To Tpo Tv TC

 

The parameters used to calculate the relaxation time were taken from Ref. [106] where
the transport data were fitted in the temperature range 300K < T < 900K. They are

given in Table 5.1. Experimentally, one finds that the energy gap EQ and the density of
states effective mass md are temperature dependent. [126] This comes from strong
electron-phonon coupling. Following the work of Bilc et. al. [52], we have incorporated
the T-dependent E9 and ma using the experimental data in all our calculations. It turns
out that the temperature dependence of ma is very important to get a good agreement
between theoretical and measured values of the transport coefficients a and S [52].
Experimentally, it is found that Eg increases linearly with temperature for T 5 400K and
above 400K, it remains constant. The temperature dependence of the band gap Eg[126]

can be approximately given as

E9 = 0.19 + (0.42 x 10‘3)T for T S 400 K, (5.81)
g = 0.358 for T > 400 K. (5.82)

The temperature dependence of the density of states effective mass md comes primarily
through the temperature dependence of the transverse effective mass mt and this is taken

from experiment [127],
m
Ht = 0.024 59 + (8.659 341 x 10-5)T, (5.83)

where m is the bare electron mass.

110

 

 

Parameter Unit of measurement Value
m, /m 0.241
60 400
600 32.6
C, N/m2 0.71 x10“
hwo eV 0.0136
a A 6.461
p g/cm3 8.24
Z 0.14
Eac eV 15
N, cm’3 2.5x1019
Eoc eV 15
ch Erg cm3 3><10'34
Ka ,Ko ,Kv for n-type 1.0
K0 ,K0 ,Kv for p-type 1.5

 

Table 5.1 Parameters used to calculate the relaxation times for PbTe. [52, 106]

Before discussing the detail results of our present study we make a few brief
comments on the work of Bhandari and Rowe [104] and Bilc et al.[52] Bhandari et al.
considered the deformation potential of acoustic phonon as the dominant scattering
mechanism (Ta) at room temperature although they claimed that scattering by polar
optical modes was not insignificant. On the otherhand, Bilc et al.[52] focused on the high
temperature transport coefficients (but only a and S) and included all the scattering
contributions. The latter authors did not explicitly explore the energy and T-dependence
of the relaxation time and how they contributed to the T-dependence of a and S. In this
thesis we do so.

In Table 5.2, we give the contribution of each scattering mechanism to the
scattering rate for different temperatures (T 2 300K) and different carrier energy E.

Scattering by acoustic phonons and optical phonons (both polar and deformation

111

potential coupling) make significant contributions at temperatures above 300 K, whereas
the contribution of the short range deformation potential of vacancies (Iv)and Coulomb
potential of vacancies (TC) at (T 2 300K) are weak and insignificant. At low energy
(8 << 0.10 eV), the polar scattering by optical phonons (Tpo) dominates the scattering
mechanism, whereas in the energy range (0.05 2 8 2 0.30) scattering by acoustic
phonons and optical phonons (both polar and deformation potential coupling) are
comparable. At high energy (8 > 0.30 eV), the acoustic phonon scattering is the
dominant scattering mechanism. At constant temperature, the contribution of the
scattering by the deformation potential of acoustic (Ta) and optical (To) phonons increase
with increasing energy 8; in contrast the contribution of polar scattering by optical
phonons (rpo) decreases with increasing energy a.

In Fig. 5.1, we plot the inverse scattering relaxation time associated with the three
dominant scattering mechanisms (Ta, To and Tpo) as a function of carrier energy 8, at
different T (300K, 600K, and 900K) for electron carrier density n = 5 x 1019 cm‘3.
The chemical potential it was calculated for each T using Eq. 5.54 (T -dependence of y
will be discussed latter in section 5.7). Also, we have drawn (— afo/BE) about the [1-
values, in order to see the range of energy E that contribute to the transport coefficients
(as discussed before is Sec. 5.4) and hence check the scattering mechanisms that
contribute in this range. We can see from Fig 5.1 that, in energy range (0 < E < 0.3), it
is not appropriate to ignore the deformation potential (1 / To) and polar (1 / rpo)
scattering by optical phonons to get a proper understanding of the transport coefficients
and subsequent comparison with experiment (see Ref. 52). At room temperature T =

300K where [.1 = 0.146 eV, Ta and 7120 are comparable and To is not insignificant.

112

 

 

 

 

 

1/‘ra 1/1'0 l/tpo l/Tv l/Tc e

(10'1 sec") (10" sec") (1011 sec") (1011 sec") (1011 sec") (EV)
300 K 14.90 7.12 75.70 0.996 0.25
600 K 45.10 21.50 198.00 1.51 0.22 0.02
900 K 129.00 61.70 315.00 2.88 0.09
300 K 36.60 17.50 56.60 2.44 0.02
600 K 109.00 51.80 136.00 3.62 0.02 0.10
900 K 246.00 118.00 221.00 5.48 0.02
300 K 63.60 30.30 48.60 4.24 0.01
600 K 183.00 87.20 112.00 6.10 0.01 0.20
900 K 400.00 191.00 186.00 8.91 0.01
300 K 97.10 46.30 45.70 6.48 0.004
600 K 272.00 130.00 103.00 9.08 0.004 0.30
900 K 587.00 280.00 172.00 13.10 0.004

 

 

 

 

 

 

 

 

 

Table 5.2 Inverse of the relaxation time of different scattering mechanisms in PbTe for
different temperatures T and different carrier energies 8.

Increasing temperature reduces [.1 and hence shifts the (— afo/ae) peak towards lower
energy, as shown from Fig. 5.1 (b) and (c). This increases the contribution of the polar

scattering by optical phonons to the net scattering rate.

113

 

=o.14 v
“ . 66 1o

14- T=300 K 2 “""Ta 1
. 5 /
‘*—1’To

 

 

 

 

 

 

 

 

 

 

11:0. 091 eV
4 5
-611]
68
11:0. 016 eV
(C) 807 —*_1/Ta T=900 :[(/1
A // ~ 3
'o
Q)
m
51‘ _%J
2 ('96
3.3
ti

 

 

 

 

. - . - . . 0
0.1 0.2 0.3 0.4
8 (eV)

Figure 5.1 Energy dependence of the inverse of the relaxation time of the three dominant

. scattering mechanisms (Ta, To and 1.‘ p0 ) 1n PbTe for electron carrier density n =5><10'9 cm

3at different temperatures (a) 300K (b) 600K (c) 900 K. The smooth curves gives the
(- arc/as) indicating the energy region contributing to the transport at different

temperatures.

114

5.6 The Electronic Thermal Conductivity

We will now evaluate the electronic thermal conductivity using the T and a
dependence of r, as discussed in Sec. 5.5. But before this we would like to make some
general observation. The total thermal conductivity Km is usually defined and measured
at zero electric current and is a sum of the lattice contribution K, and the electronic part at
zero current Ice” and is given by (using Eqs. 5.25, 5.28 and 5.29) and Km, = K5,“, for
different types of electronic thermal conductivity

1cm = K, + Ice” (5.84)

= K, + Ice”; — T052 . (5.85)

This Km appears in the definition of ZT. [9] Since the thermal conductivity is a positive

quantity for a thermally stable system (K3,, 2 0), it is therefore necessary that Ice”,- 2
T052. The dimensionless figure of merit which is then given by

27 - T052 5 86
_ Kl + Kel,E - T052 ( . )

 

can be optimized by manipulating Kel'E. Sofo and Mahan [96] found that for a given
lattice thermal conductivity K1, the best thermoelectric is when Ice”; = T052, which is
equivalent to KeU = 0. If we look at Eqs. (5.50)-(5.52) this equality occurs when
2(8, T) = 116(8 — so), i.e. the energy and charge transport take place in a single energy
channel, 8 = so, i.e. perfect energy filtering. Even in this case one has to adjust the
chemical potential such that In — 801 z 2.5kBT to maximize ZT. In this case ZT -> 00
when the lattice thermal conductivity is ignored (K, = 0). In a recent work Humphrey
and Linke [137] have explored a similar energy filtering idea, but for an inhomogeneous

system. They have shown that in the presence of a temperature gradient if one can

115

 

s(i")-u .
remarns constant
RBTO’)

construct a graded thermoelectric such that the quantity (
throughout the sample, then the energy and charge transport is adiabatic, which gives rise

to perfect Carnot efficiency and this correspond to ZT = 00, when the lattice thermal

conductivity is ignored. The thermoelectric efficiency is given by [9]

_ Tho, — Tc,” \/1 + ZT - 1

Th“ \/1 + ZT + If—O‘i '
hot

 

 

(5.87)

To achieve high efficiencies, high ZT materials and large temperature differences

AT = That _ colds are needed ( 77 '9 7Icarnot = AT/Thot When ZT ‘7 0°)

5.7 Results and Discussions

In order to calculate the electrical conductivity 0, S and Ice“,- for a given
concentration 11 at different temperatures, Eqs. 5.42, 5.45, 5.49, and 5.53 have to be
solved simultaneously, the common link being the chemical potential (.1. We used
Mathematica program to solve these set of coupled equations.

As mentioned in the section 5.5 of this chapter, we have taken into account the

temperature dependence of the energy gap E9 and the density of states effective mass

mdwhich are given in Eqs. 5.81, 5.82, and 5.83. For PbTe, as mentioned before, the gap
occurs at the L point in the fcc B2 and y = 4. At high concentrations, 71 ~ 1019cm'3,
which are required for good thermoelectric, the n- or p-type PbTe can be described using

a single (either conduction or valence) band model since for these concentrations,

contributions to transport come primarily from one type of carrier. (Note: that for p-type

116

transport all transport equations have to be interpreted in terms of hole occupation
function) Here we give the results for the n-type PbTe. Fig. 5.2 shows the temperature

3and

dependence of the chemical potential [.1 for concentrations 71 = 5 x 1019cm’
n = 5 X 102°cm‘3. The change in 11 over the temperature range of 600 K is quite
significant. This plays an important role in the observed T-dependence of o, S and Ice“,-
over this temperature range. Results of a and S with a comparison with experimental
values, taken from Ref. [128], are shown in Fig. 5.3. The parameters used to calculate

Tm were taken from Bilc et a1 's. work. [52]. Our theoretical results for Ice”,- and Ice”

will be discussed in Sec. 5.9.

0.6

 

 
 

o y n=5.0X10190m'3
0.5~ —Fittedy n=5.0x1019 cm"3
. 20 -3
E x 916 y n=5.0x10 cm
0.4.. x X 4. x
A MD=A+BTC 9‘ 9K x
> 0.3- x x >6 x
G.) A 0.17i0.001
v ..
3‘ 0 2‘ B -2.75x1o-6—12.6x1o-7
' C 1.61:1. 0.01
0.1-
0.0

 

 

 

300 5460 ' 560 ' 600 ' 760 ' 800 ' 960
Temperature (K)

Figure 5.2 Temperature dependence of the chemical potential ,u for n-type PbTe at
different concentrations n =5 x10l9 cm'3 and n =5><1020 cm'3.

117

8000

 

\

7000 l
6000 ~
5000 ~
4000 «

o(S/cm)

3000 -

1
2000 -

1000-

 

 

 

 

0 ' I T r ' l v 1 ' 1 ' t V I '
200 300 400 500 600 700 800 900
Temperature (K)

Figure 5.3 Temperature dependence of the electrical conductivity 0' and therrnopower S
for n-type PbTe at concentration 11 =5>< 1019 cm'3. The experimental values are shown as
solid points from Ref [128].

5.7.1 Energy and Temperature Dependence of the Relaxation time

In order to understand the temperature and energy dependence of the total
relaxation time in PbTe we plot the total relaxation time not (see Eq. 5.80) as a function
of carriers energy 8 for different temperatures. Fig. 5.4 shows the energy dependence of
not for deferent temperatures. As we can see, the relaxation time decreases with
increasing a, and decreases with increasing temperature for a given a. We scaled “rm for
different temperatures in order to see how the scaled rm depends on 8. Fig. 5.5 shows
that for different T, the energy dependence of rm is almost the same; therefore we fit

Ttot(T=300, 8) to find the energy dependence of rm (see Fig. 5.6). After that it was
straight forward to find the T dependent of the not (see Fig. 5.7). The total relaxation

time of PbTe can be approximated extremely well by a scaling function

118

 

    

 

 

1 .0 _ 000 o T=300 K
d O _
’5‘ 06 0 L450 K
0.8 - O
3 00 a T=600 K
m J O _-
‘T W . 1-900 K
O I... 00
‘— . .- O
i: 0.4 -
*1: fi ' .I
:§ “'4'. II...-
“ 0.2 —l H+++++ *fit‘,* I‘ll..-
+
++++ i t t a t y g t =:::.=‘:h:h.h-slu.~.".
“titties
0.0 ' ' T ' fl

 

0.0 0.2 0.4 ' 018 ‘ 0:8 ' 1:0
8(eV)

Figure 5.4 Energy dependence of the total relaxation time 1,0,, calculated from Eq. 5.80,
for n-type PbTe at concentration n =5 ><1019 cm'3 for different temperatures.

 

-13

Ttot ( x10 sec)

 

 

 

 

. . fl . .
0.0 0.2 0.4 0.6 0.8 1.0
6' (eV)

Figure 5.5 Energy dependence of the scaled (to 300K) total relaxation time 1,0, at
different temperatures.

119

 

  
   
 
  
 

 

 

 

 

 

 

10‘ 561(k) =F(T)/ (b + c 8’)
13 11 b (9.9 1 0.01) x1012
49”) . c (7.8 1 0.03) x1013
1..- 6_ 1.74 1 0.004
O
‘— 1
25
:5..- 49
L. 1 T“
2_ a 1,0, (T=300 K) -
. Harris fit of 7,0, (T=3OO K)
O I l ' l r l ‘ I ‘
0.0 0.2 0.4 0.6 0.8 1.0
8 (eV)
Figure 5.6 Energy dependence of the total relaxation time 1,0, for T=300 K.
1.00 -
F (T) = a T p
a (2.95 1 0.20)x103
075" p 1.40 1 0.01
A
L: 0.50-
1.1.
0.25-
0.00

 

 

 

300 400 500 600 700 800 900
Temperature (K)

Figure 5.7 Temperature dependence of the total relaxation time 1,0, at 8 = 0.1 eV.

120

_ aT‘l”
7“” ‘ (b + cs") ’

(5.88)
where a, b, c, p and r are the fitted parameters, which depend on the carrier
concentration. We used this expression to check the calculated electrical conductivity 0
and the therrnopower S, (see Fig. 5 .8 and F ig. 5.9). This expression of rm gives
excellent agreement (within 5%) with the values of a and. S calculated directly from the
scattering mechanisms.

To check how sensitive the electrical conductivity 0 is to the energy dependence
of rm, we plot a for different values of the r parameter (see Eq. 5.88) as shown in Fig.
5.10. The absolute values of a change with 1'. As 1‘ increases from 0 scattering in the
energy range of interest (0.0-0.3 eV as discussed in Sec. 5.5) gets suppressed leading to
an increase of a for all T. However, after scaling 0', we notice that it is not sensitive to the
energy dependence of rm. In order to understand why the temperature dependence of a
is so weekly dependent on the parameter r of Eq. 5.88, we look in detail the different
physical quantities that give rise to the observed T-dependence. The scaling form for the

total scattering time not gives p = 1.4. This power law dependence comes fi'om two

sources, T‘1 dependence from the phonon occupation number at high T (kBT >> hm),
[T‘1~ 1%] and a T""4 from the temperature dependence of the density of state through

the parameter mao (see Sec. 5.5 for different phonon induced relaxation rates). The
question is where the additional T'o'8 dependence of 0 comes from: In order to find this,
we made a simple calculation. We assumed rm to be constant and then calculated the T-
dependence of a coming from other sources. We find that 0 ~ T‘O's, the T—dependence

of 11 playing a crucial role in this T -dependence. In this respect, PbTe at the doping

121

 

 
 
  
 

 

 

 

  

6000- 0 o 906* /
>\ 0.04 //
S b 0.02. /
A 0' 3' 0.00 //
E a; 002.\ /
0 40001 3004 /’
E?) .008 . - g ,
v 400 600 800
b Temperature (K)
2000-

 

 

 

300 ' 400 ' 500 ' 600 - 700 V 800 ' 900 '
Temperature (K)

Figure 5.8 A comparison between the electrical conductivity 0' using the scattering
mechanisms of the relaxation time with the scaled formula.

410
-609
-1003
-1205
-140l
4603
-1803
-200l
300 ' 400 ' 500 T 600 ' 700 ' 800 ' 900
Temperature (K)

Figure 5.9 A comparison between the therrnopower S using the scattering mechanisms of
the relaxation time with the scaled formula.

 

 

 

   

 

400 ' 800 800
Temperature (K)

  

s (uV/K)

 

 

 

122

 

 

 

 

 

 

+ (a) 1- (b) O r=0.00
a _
12000_ A 8000- ii * r-1.74
916+ E U
0
. + \ 9K
8:916 a) 4000
+ v +
A 73*
E 8000- 494* b ,.
2 + $00 ' 400 ' 600 ' 800 1
U) “D * +
V DD 1 +++ Temperature (K)
b 4000-1 DD **..+
+
DD *,,_+
EDD *‘ir * 4
[3 DE] *’*‘§-*‘
00 BUD ‘ * 1h... 41""
OO 000 DDDDUDU ‘1' * ¥A2s¥11};};~A-Au~A...
O 0000000000000000000g855'Qflflqqaé

 

200 300 400 500 600 700 800 900
Temperature (K)

Figure 5.10 (3) Temperature dependence of 0', using the scaling formula of the total

relaxation time for different 1' parameter. (b) Temperature dependence of the scaled 0'
(to r =1.74).

levels of thermoelectric interest, differs significantly from both highly degenerate limit

and the non-degenerate limit [9], and is somewhere in between.

123

5.7.2 Electronic Thermal Conductivity and the Validity of Wiedemann-

Franz law

We will now use the full T and 8 dependence of the effective relaxation time (using our
scaled form Eq. 5.88), T dependence of chemical potential and other parameters to look
at the electronic thermal conductivity. Our work is a generalization of the earlier work by
Bhandari and Rowe [104] who looked at this problem but only with acoustic phonon
scattering. They had claimed that the acoustic phonon scattering was the dominant
scattering mechanism in PbTe at high temperature. Our calculations for 7,0,, as we
discussed earlier in Sec. 5.5, show that it is not appropriate to ignore the deformation
potential (To) and polar (1pc) scattering by optical phonons to get a proper

understanding of the transport coefficients. Our results for the electronic thermal
conductivity match perfectly with their results if we only consider the acoustic phonon
scattering. In Fig. 5.11 we give the T dependence of Ice”,- and K9,, 1° Both decrease with

T but the difference increases with T. One question is how close PbTe is to a perfect

thermoelectric(ZT = 00 when K, = 0, see Eq. 5.87). In this limit, ZT = aSzT/Keu =

(Kel.E/Kel.1) - 1. We plot this quantity in Fig. 5.12, it goes from ~O.l at 300K (metallic

limit) to ~ 3 at 900K. PbTe does approach a perfect thermoelectric at high temperatures
but is far from it.
Next we address the validity of Wiedemann-Franz (WF) law. The first question is

whether Ice”; = LOO'T or K3,, = LOO'T where L0 = 2.45 x 10‘8WQ/K2? For a metal or

a highly degenerate semiconductor it is the first equation that is correct. [9, 123] We can

124

 

* K >K L O'T
¥* cl,E 0
_ X
4 .4) My” I K61 J A (LooT-SZO'T)
E a 9K ****
3.1 A Xx *******
3- *x *******
H A... XXXX
t3 2" AA II. XXX
A I 9K>K
a A I.. XXXX
\/ A I.- *Xx
1 A .I. Xié
Q” AAA I
A
0 A“

 

 

 

 

'300 ' 400 ' 500 ' 600 ' 700 ' 800 ' 900T
Temperature (K)

Figure 5.11 Temperature dependence of the electronic thermal conductivity (K91) at

constant electric field E (new) and at constant current density J (Keu) using Boltzmann
transport equations. We also give their values using WF law.

125

31)

 

d

215-

2.0 ~

1.5~

q

1.0~

(27 )0

0.5-

.(

 

0.0 . .
300 400

 

‘1

 

500 v 600 v 700 ' 800 U 900
Temperature (K)

Figure 5.12 Temperature dependence of ZT for n-type PbTe at concentration n =5><10l9
cm'3 assmning K1: 0.

use the calculated a (Fig. 5.8) and this equation to estimate Kew. However there is a
fundamental problem! If we use this as Kc”; then estimate K9,, using the calculated
values of a and S (the plot LOO’T — 520T in Fig. 5.11), we find that for T > 700 K,
K8” < 0. But since K6,] is a response function it must be positive definite. So we cannot
use the equation Kc“; = LoaT to estimate K3”; from experimental a and then calculate
K8,, using the experimental value of a and S.

The next thing we can do is to use the second equation, K8,, = LoaT. This

appears to be more reasonable and is closer to the values of properly calculated KB”. This

126

is what is usually done in the estimation of electronic contribution to the total thermal
conductivity Kw, using experimental values of a, S. As we can see in Fig. 5.11, the use
of WF law overestimates the electronic contribution Ke, -:- K8,, and therefore
underestimates the lattice contribution. The proper lattice contribution should be actually
higher (by more than 0.5 W K'lm").

In order to find the temperature dependence of electronic thermal conductivity at
constant current (Keu), one defines an effective Lorenz number as [122]

“cu

L= .
0T

(5.89)

Fig. 5.13 shows the effective Lorenz number vs. temperature for different carrier
concentrations. As we can see, the effective Lorenz number L approaches Lo (metallic

like electric thermal conductivity) with increasing concentration and decreasing

temperature. As pointed out in Sec. 5.6, for a perfect thermoelectric KelJ = 0, i.e. L = 0.

127

 

1.00

‘ _ 18 J
0.95-%§§§g§§§§ X n-5x10l9 cm3
‘fi‘.fe§** § 1!; 0 n=5x10 cm-
090- a 4 e 911 0+ _!
' a 1 OQxxA0513: 19 -3
‘ *9 ‘4 e axAAO++:: 1* n=10x10 cm
0 fl ‘ e x AOO+ :-
0.85“ . fl { 9 *‘(AAQT‘FfiI 19 '3
. O. fir» {‘QeéxxA205:5:. ‘ n=15xl0 cm
_ o 4 e )6 A 0 ti _‘g
0'80 . 0. *fifi ‘4‘$e*xiagsgtgg.l 99 11=20x10I9 cm‘
0 tr 4 ea 3" A 0 +*t' l9 -
L/L00.75- o. a, 4‘ 99*1ixgog1fi: x n=25x10 cm3
-. *1 a 1, ,9 -1
0.709 XX 0. *1. ‘< 9631,2320 A n=30x10 cm”
' Xxx .0 fife: “< 696*an l9 -3
055- ‘x 0. 9 < ‘96 3“ O n=35x10 cm
. XX 0 *a d. 9’9
. XXX ... fififi “ _ 19 '3
060— XXX 0.. a,” 4 + n—40xlO cm
' XXXX . fit? _ l9 --
- xxXX Do... * n-45x10 cm
0.55— X _ 19 -3
. I n—50x10 cm
0.50

 

 

 

300 I 400 ' 500 ' 600 ' 700 ' 800 ' 900 '
Temperature (K)

Figure 5.13 Temperature dependence of the scaled effective Lorenz number (L/Lo) for n-
type PbTe at different concentration n. (Lo= 2.45 X 10-8 WQK'Z)

5.8 Summary

Transport calculations using the nonparabolic Kane model for energy dispersion
in PbTe show that the deformation potential of acoustic phonons (ta), the deformation
potential of optical phonons (To), and the polar scattering by optical phonons (rpo), are
comparable and make significant contributions at temperatures above 300 K.
Temperature and energy dependence of the total relaxation time 7,0, calculations show
that the T—dependence comes from two sources, one from the high-T limit of phonon
occupation number and the other from the density of states associated with the carriers. In
the temperature range 300K < T < 900K, 0(T) for n-type PbTe ~ T—2.2_ This T-

dependence of 0 comes from several sources (band structure parameters i.e. effective

128

mass, chemical potential 11 and relaxation time T ), the T-dependence of I (~ T‘”)

however dominates, the other important source being a strong T -dependence of 1.1. It
turns out that energy dependence of 1,0,(T, s) does not seem to control the T-dependence
of a in this temperature range and for carrier concentration (11 ~ 1019cm'3). This should
have implications in current attempts to change the energy dependence of electron
scattering to control the T-dependence of a.

Careful electronic thermal conductivity calculations were done using the same

model and parameters which were used to fit a and S. Our calculations show that using
WF law (K8,, = LOUT) to estimate this contribution of the thermal conductivity will

overestimate the electronic contribution and hence underestimate the lattice contribution
to the total thermal conductivity. This suggests that the very low values of lattice thermal

conductivity found from using WF law may not be quite right.

129

Chapter 6

 

General Summary

 

PbTe was one of the first materials studied in the middle of the last century when
there was a revival of interest in thermoelectricity.[4] This compound, its alloys with
SnTe and PbSe, and related compounds called TAGS (alloys of AngTez and GeTe)

were for many years the best thermoelectric materials at temperatures ~700 K.[5] In

recent years, quantum wells of PbTe/Pb,-xEuxTe, PbSGoggTerz/PDTC superlattices.[6]

and quaternary compounds AngszmTe2m12 (m=9,10) [7] have attracted considerable

attention because of their large thermoelectric figure of merit (FOM). Similar quaternary
systems with Ag replaced by Na are also promising high temperature thermoelectric [8].
Most of the above systems have stoichiometry closer to the parent compound PbTe. This
has lead to a great deal of experimental and theoretical activity in PbTe and similar IV-VI
narrow band-gap semiconductors in last few years.

A detailed microscopic understanding of the nature of the defect states in PbTe
and SnTe has been an old but a challenging problem, because it is known that, in narrow
band-gap semiconductors like PbTe and SnTe, one does not see shallow defect states but
deep defect states (DDS) which need not be inside the band gap.[11-15] Inspite of several
theoretical attempts the problem of defects in PbTe, particularly the physics underlying
deep defect states, has not been well understood until my ab initio density functional

studies presented in this thesis.

130

Several calculations of the transport coefficients of PbTe and related systems have
been reported over the years. [52, 95, 102, 104, 105] These earlier works did not explore
the relationship between the energy and temperature dependence of the effective
relaxation rate and the T-dependence of 0, K8” and Kew. Although the validity of using
Wiedemann-Franz (WF) law [123] to obtain electronic thermal conductivity at high
temperatures has been questioned in the literature [104], it is still widely used to estimate
the lattice thermal conductivity from the total experimental thermal conductivity. The
lattice thermal conductivities estimated this way are turning out to be very small, well
below the so called alloy limit [136]. This reduction has been attributed to the enhanced
phonon scattering from the nanostructures present in the samples. Thus to have a more
definite picture one must critically examine the use of WF law.

In my thesis, I have presented two major contributions in PbTe and related
systems. (1) Results of my calculations to understand the physics behind the formation of
deep defect states in PbTe and SnTe using self-consistent ab initio electronic structure
calculations within density functional theory (DFT) and supercell models, and (2) A
careful investigation of the energy and temperature dependent transport processes in n-
doped PbTe, particularly at high temperatures.

Electronic structure calculations on defects in PbTe show that when a Pb atom is
substituted by a defect, the single particle density of states (DOS) gets perturbed over the
entire valence and conduction bands. In addition, there are major changes in the DOS
near the band gap region for most of the defects. This should have significant impact on
the transport properties of these new compounds. We find that Na does not change the

DOS within 0.5 eV of the valence and conduction band extrema; thus it is an ideal

131

acceptor. In contrast, other alkali atoms and Ag and Cu give rise to an increase in the
DOS near the top of the valence band and negligible change in the DOS near the bottom
of the conduction band. The divalent defects (sz-type) Hg, Cd, Zn give rise to resonant
states near the bottom of the conduction band and suppress the DOS near the top of the
valence band (which is good for n-type therrnoelectrics). Group V (As, Sb and Bi)
impurities (p3-type) introduce resonant states in the conduction band and should be good
for n-type thermoelectrics. In contrast, the trivalent impurities (Ga, In, and T1; szpl-type)
either introduce bound states in the gap or resonant states near the bottom of the
conduction band or the top of the valence band. My calculations strongly suggest that the
currently used mixed valence model [43-45] for these defects is not correct. Te vacancy
also has a strong effect on the DOS of PbTe near the band gap region. New states appear
in the band gap below the CBM, these states comprise primarily of the p states of Pb
neighboring the vacancy. In contrast, the Pb vacancy increases the DOS near the top of
the valence band but by a small amount. Te vacancy should be n-type whereas Pb
vacancy should be p-type. Iodine appears to be an ideal donor, it does not change the
DOS of PbTe near the band gap, just shifts the Fermi energy to the conduction band. The
divalent atoms S and Se also do not change the DOS near the CBM. There is however
some depletion of the DOS near the VBM. These results should have important
implications in the thermoelectric properties of, e.g., the n-type ternary compound
PbTeMSx, [53] where one can synthesize mixed Te/S systems without appreciably
affecting the conduction-band states and hence the n-type charge and energy transport.
Electronic structure calculations of defects in SnTe show that when a Sn atom is

substituted by a defect, the DOS of the valence and conduction bands get strongly

132

perturbed, similar ti PbTe. Sn vacancy causes very little change near the bottom of the
conduction band DOS whereas there is an increase in the DOS near the top of the valence
band. Te vacancy on the other hand creates new states in the band gap region and near the
bottom of the conduction band. Ag impurity perturbs the valence band states due to the
Ag d-states hybridizing with the Te p bands. The difference in the total DOS with and
without Ag impurity shows a small peak near the top of the valance band. Sb impurity
shows resonant states near the bottom of the SnTe conduction band. Results for In
impurity shows that, unlike PbTe, the deep defect states in SnTe are resonant states near
the top of the valence band. In PbTe these deep defect states lie in the band-gap region
(act as n-type). This fundamental difference in the position of the deep defect states in
SnTe and PbTe explains the experimental anomalies seen in the case of In impurities (act
as n-type in PbTe and p-type in SnTe).[13, 87] In addition, we also find a major
difference in Cd-induced defect states in PbTe and in SnTe. This has to do with the
valence s-states of Sn which are strongly perturbed by the defects. In contrast, Pb valence
s do not get perturbed by the defect. However, more careful calculations are needed to
understand the physics behind this difference between Cd defects states in PbTe and
SnTe.

Transport calculations in PbTe show that scattering rates associated with
deformation potential of acoustic phonons (Ta), the deformation potential of optical
phonons (To), and the polar scattering by optical phonons (Tpo), are all comparable. All
these make significant contributions at temperatures above 300 K. Temperature and
energy dependence of the total relaxation time rm, calculations show that the T-

dependence comes primarily from two sources, one from the high-T limit of phonon

133

occupation number and the other from the density of states associated with the carriers. In
the temperature range 300K < T < 900K, for n-type PbTe we find 0(T)~ T’Z'Z, in
agreement with experiment. This T-dependence of 0 comes from several sources (band
structure parameters i.e. effective mass, chemical potential 11 and relaxation time 1,0,).
The T-dependence of 1,0, (~ T'“) however dominates, the other important source
being a strong T-dependence of the chemical potential 11. It turns out that energy
dependence of 1,0,(T,£) does not seem to control the T-dependence of o in this
temperature range and for carrier concentration (11 ~ 1019cm'3). This should have
implications in current attempts to change the energy dependence of electron scattering to
control the T-dependence of a. [102, 103]

Careful electronic thermal conductivity calculations were done using the same

model parameters which were used to fit a and S. Our calculations show that using WF
law (K8,, = LOoT) overestimates the electronic contribution. Subtracting this from

measured total thermal conductivity therefore underestimates the lattice contribution.
This suggests that the very low values of lattice thermal conductivity found from using
WF law may not be quite right.

Finally, I would like to make few general remarks. Electronic structure
calculations within density functional theory (DFT) offer varieties of tools to determine
the stability of compounds, their atomic geometries, their band structure and transport
properties. Within DFT the stability of compounds (their total energies) are well
predicted. Therefore, the theory can indeed help in predicting new compounds with new
electronic properties. One can use these calculations not only to understand why already

synthesized good thermoelectric materials show these properties, but also to search for

134

new potential thermoelectric materials. However, the issue of how accurate the DFT
studies of defect states using supercell models, one needs further calculations with larger
supercell. More quantitative studies (experimental) and comparison with theoretical
calculations are needed to check that.

In order to characterize the thermoelectric properties of a material, transport
calculations for a, S, and K8, have to be performed carefully. Unfortunately, to perform
such calculations one has to make approximations whose validity is not always fully
under control. By this I mean that we do not have an estimate of the magnitude of the
errors made. Approximation such as the rigid band model, where the electronic band
structure of doped semiconductors is assumed to be the same as for the undoped
semiconductors, needs to be critically examined. We saw that the host band structures do
get modified strongly for defect concentrations 2 1%. The other approximation is the
relaxation time approximation made in describing the electronic scattering mechanisms.

The other problem we would like to mention is the problem of LDA/GGA in
underestimating the band-gap. [32, 61] For narrow band-gap semiconductors, which most
good thermoelectric are, this could be a problem in calculating the transport coefficients.
Although methods like GW approximation [68] have developed in recent years, their
applications to study the electronic structure and transport properties of complex
materials are still at their its infancy. One expects considerable theoretical effort to be put
in this direction in the future. I believe strongly that for a proper understanding of
transport properties of complex systems a close interaction between theory and

experiment is necessary.

135

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