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ELECTRONIC STRUCTURE AND THERMOELECTRIC
PROPERTIES OF NARROW BAND GAP CHALCOGENIDES

PhD.

presented by
DANIEL BILC

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2/05 czlcficzoam.m—p.1s

ELECTRONIC STRUCTURE AND
THERMOELECTRIC PROPERTIES OF NARROW
BAND GAP CHALCOGENIDES

By

Daniel Bilc

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy

2005

ABSTRACT
ELECTRONIC STRUCTURE AND THERMOELECTRIC
PROPERTIES OF NARROW BAND GAP CHALCOGENIDES
By
Daniel Bilc

In recent years there have been a revival of interest in discovering and understanding the
physical properties of novel thermoelectric (TE) systems with high figure of merit. These
systems are primarily narrow band gap semiconductors. In this thesis, electronic struc-
ture calculations were carried out for several narrow band gap chalcogenide TE materials
in order to understand their electronic and transport properties governing their TE char-
acteristics. These calculations were performed within ab initio density functional theory
(DFT) using full potential linearized augmented plane wave (FLAPW) method. Transport
calculations were carried out using Boltzmann transport equations.

For the binary chalcogenides Bi28e3 and BigTe3 I have studied the effect of quantum
confinement (QC) created by the surfaces on their bulk electronic structure. In the presence
of such confinement, surface states appear, which are a consequence of the strong influence
of the interlayer bonding on the bulk electronic structure of these compounds. I find that
in contrast to standard belief, there is an important covalent contribution to the interlayer
bonding besides the Van der Waals contribution.

(BizTe3)m(Sb2Te3)n superlattices (SL) show very good TE properties at room tem-
perature. To see how the electronic structure of BigTe3 and SbgTe3 are affected by the

formation of SL, I have investigated the electronic properties of (Bi2Te3)m(Sb2Te3)n SL

as compared to those of BigTe3 and Sb2Te3 bulk systems. We find that the formation of SL
does not deteriorate the electronic transport properties along the cross plane direction.

Complex ternary KgBiSSe13 system shows great potential for superior TE performance.
This compound forms in two distinct phases. a and ,3. The ,B-phase, which has two sites
with K/Bi disorder, is a better TE. The calculations show that a-phase is an indirect band
gap semiconductor. For the [3-phase we find that the atoms at the "mixed sites" are very
important in determining the electronic properties. The incorporation of the K/Bi mixed
occupancy at the disordered sites is crucial for the semiconductor behavior. We also find a
strong anisotropy in the hole and electron effective mass.

Complex quaternary AngmeTe2+m (LAST-m) systems are excellent high tempera-
ture TE. These systems form in the rocksalt structure similar to PbTe where Ag and Sb
occupy Pb sites. There are clear experimental evidences that LAST-m systems exhibit mi-
croscopic inhomogeneities rich in Ag-Sb embedded into a PbTe matrix. Our calculations
show that resonant states appear near the PbTe band gap. The common feature of all Ag-Sb
arrangements is that they have a more enhanced density of states (DOS) near the gap as
compared to PbTe. To see how these features in the DOS affect the transport properties I
have carried out transport calculations in PbTe and LAST-m systems. The results for PbTe
show that the temperature dependence of the effective mass md is very important in order
to have good agreement with experiment. The LAST-m systems show an enhancement
of the power factor (520) relative to PbTe. But this enhancement is not large enough to
explain the experimentally observed ZT. This suggests that the reduction in the thermal

conductivity caused by Ag-Sb nanostructures in PbTe matrix may be significant.

Copyright by
DANIEL IOAN BILC
2005

To my family and to Richi

ACKNOWLEDGMENTS

I would like to acknowledge the enormous support from my adviser, Professor S.D. Ma-
hanti, and Professor M.G. Kanatzidis in Chemistry. I would like also to acknowledge
Dr. Tim Hogan in Engineering for collaboration on the therrnoelectricity project, Dr. S.
Tessmer and Dr. S. Urazhdin for discussions and for providing the scanning tunneling mi-
croscopy experimental results. I would like to thank Dr. E. Quarez and Dr. K.F. Hsu in
Chemistry for discussions on modeling the complex crystal structures. I would also like to
acknowledge Professor P. Duxbury and Professor P. Danielewicz in Physics for the support
as members of my committee.

Finally, I would like to greatly acknowledge funding form the Office of Naval Research
(Contract No. N00014-02-1-0867 MURI program) and DARPA (Contract No. N00014-

01-1-0728).

vi

Table of Contents

List of Tables ix
List of Figures x
1 Introduction 1
2 Electronic Structure Calculations Using Density Functional Theory (DFT) 8

2.1 Full Relativistic Single Particle Dirac Equation ............... 9
2.2 Scalar Relativistic Approximation ...................... 10
2.3 Density Functional Theory (DFT) ...................... 14
2.4 Method of Calculation ............................ 16
Surface States in Binary Semiconductors Bi28e3 and Bi2T83 18
3.1 General Introduction for Bi28e3 and BigTe3 ................. 18
3.2 Crystal Structure of BigSeg and BigTe3 ................... 19
3.3 Electronic Structure of Bulk and Slab (to Model the Surface) ........ 20
3.4 Comparison with Scanning Tunneling Microscopy (STM) Results ..... 25
3.5 Summary ................................... 28
Electronic Structure of (BigTe3)m(Sb2Te3)n Superlattice (SL) 30
4.1 General introduction for (Bi2Te3)m(Sb2Te3)n SL .............. 30
4.2 Model Structures and Geometry optimizations for (BigTe3)m (Sb2T63)n SL 31
4.3 Electronic Structure of (BizTe3)m(Sb2Te3)nSL ............... 35
4.4 Effective Masses in Bluk BigTe3, Sb2T63 and (BigTe3)m(Sb2Te3)n SL . . . 38
4.5 Summary ................................... 42
Electronic Structure of Complex Ternary System K2Bi88e13 43
5.] General Introduction for K2Bi88e13 ..................... 43
5.2 Crystal Structure of K2Bi88e13 ........................ 45
5.3 Electronic Structure and Effective Mass ................... 49
5.3.1 a-KgBi85613 ............................ 49
5.3.2 B—KgBigSem ............................ 54
5.4 Summary ................................... 62
Electronic Structure of AngmeTe2+m (LAST-m), Complex Quaternary Sys-
tems 65
6.1 General Introduction ............................. 65
6.2 Electronic Structure of AngTe2 ....................... 68
6.3 Electronic Structure of PbTe ......................... 74
6.4 Effect of Ag-Sb Microstructural Arrangements on Electronic Structure of
LAST-m systems ............................... 78

vii

6.4.1 Defects in PbTe ............................
6.4.2 Models for Ag-Sb Microstructural Arrangement ..........
6.4.3 Electronic Structure of Ag-Sb Microstructural Arrangements . . . .
6.4.4 Summary ...............................

7 Electronic Transport in PbTe and AngmeTe2+m (LAST-m)

7.1
7.2
7.3
7.4
7.5
7.6
7.7

7.8

General Introduction .............................
Boltzmann Transport Equation ........................
Nonparabolic Kane Model for Energy Dispersion ..............
Transport Coefficients in the Kane Model ..................
Relaxation Times in the Kane Model .....................
Transport Coefficients Using ab initio Electronic Structure .........
Results and Discussions ............................
7.7.1 PbTe Using Kane Model .......................
7.7.2 PbTe Using ab initio Electronic Structure ..............
7.7.3 LAST-m Using Modified Kane Model ................
Summary ...................................

8 General Summary and Future Directions for Research

BIBLIOGRAPHY

viii

78
80
82
89

91
91
92
97
99
104
111
112
112
117
120
122

126

130

4.1

4.2

5.1

5.2

5.3

5.4

6.1

7.1

List of Tables

Valence band maximum VBM hole effective masses and mobility anisotropy. . 40

Conduction band minimum CBM electron effective masses and mobility
anisotropy .................................... 41

Energy band gap values in eV corresponding to experimental volume Va], and
optimal volume Vopt for a—KgBi88e13 .................... 52

Effective mass parameters at Vopt for a—KgBiSSe13 .............. 53

Energy band gap values in eV corresponding to experimental volume Vex], and

optimal volume Vopt for fi—KgBigSelg-Supercell. ............. 60
Effective mass parameters at Vopt for 13 —KgBi88e13-Supercell ......... 62
Total energies of AngTe2 ordered model structures. .............. 72

Parameters used to calculate the relaxation times for PbTe and LAST-m sys-
tems at 300K. ................................. 1 10

ix

3.2

3.3

3.4

3.5

4.1

4.2

4.3

5.1

5.2

List of Figures

(a) Bi28e3 (BIQTC3) structure viewed parallel to the layers. 2x1xl hexagonal
unit cell is shown with a solid line. The unit cell used for slab calculations
(for a 3-layer slab) is shown in dash. In a dashed frame one of the ppa
bonded chains is plotted. (b) Hexagonal Brillouin zone of BigSe3 (BigTeg). 20

Calculated band structure: (a) bulk Bi28e3, (b) 3-1ayer slab of BigSe3,(c) bulk
B12TC3, (d) 3-layer slab of BigTe3. The experimental Fermi energies are
shown as dashed lines; zero energy corresponds to the top of the valence

band ....................................... 2]

Partial DOS of Se] in Bi28e3 for various positions of the quintuple layers with
respect to the surface in a 12-slab geometry. ................. 23

(a) 1.3 x 1.3 nm topographic image of Bi28e3 surface acquired at a bias voltage
—0.7 V. Black to white scale is 30 pm. Autocorrelation was used to enhance
the image. (b) Calculated tOpography of 312863 at a bias voltage of —0.7 V. 27

(a) Solid lines: near-gap tunneling differential conductance dI/dVB spec-
trum of 312363, acquired directly with a lock-in detection technique at a
temperature of 4.2 K. Dashed lines: theoretical curves, calculated for a 3
slab-geometry. The curves are normalized by the differential conductance
at V B = 0.3V. (b) Same as (a), for BigTe3 ................... 28

Hole mobility anisotropy in (BigTe3)m(Sb2T€3)n SL. [98] ........... 32

Models for (BigTC3)m(Sb2TC3)n SL structures of (a) (l l), (b) (12), and (c)
(24). Hexagonal Brillowin zone ((1). ..................... 34

Band structure of (a) bulk Bi2Te3, (b) bulk SbgTe3, (c) (1 1) SL, and (d) (12) SL. 36

Projection of the crystal structure of a—K2Bi88e13 viewed down the c-axis
(z-axis). BigTe3-, Cdlg- and Snge3- building blocks in the structure are
highlighted by the shaded areas. ....................... 47

Projection of the crystal structure of ,B—K2B183613 viewed down the c-axis
(z-axis). BigTe3-, Cdlg- and NaCl- building blocks in the structure are
highlighted by the shaded areas. ....................... 48

5.3 Brillouin zone of a—K2Bi8Se13 ................ ' .........
5.4 Brillouin zone of fi—KgBigSem .........................

5.5 Band structure of a—KgBi88e13 at Vopt (a)LDA without SOI, 0))LDA with
SCI, (c)GGA without SCI, and (d)GGA with SCI. .............

5.6 Band structure of B —K2Bi8Se13-Configuration I with 801. Orbital character
of (a) Se9 p, (b) SelO p, (c)Bi8 p, (d) Se4 p. The size of the circles is
directly proportional to the strength of the orbital character. ........

5.7 Band structure of B—KgBigSem-Configuration I with $01 for partially opti-
mized calculations. Orbital character of (a) Se9 p, (b) Se4 p. The size of
the circles is directly proportional to the strength of the orbital character.

5.8 Band structure of B —K2Bi83e13-Configuration II with SCI. Orbital character
of (a) Se4 p, (b) Bi9 p. ............................

5.9 Band structure of fi—KgBigSelg-Supercell at Vopt (a)LDA without SOI,
(b)LDA with SCI, (c)GGA without SCI, and (d)GGA with SCI. ......

5.10 Charge density of 5 —K2Bi88e13-Supercell in planes parallel to the c-axis cor-
responding to (a), (b) CBM states and ( c), (d ) VBM states. The charge
density is represented by closed lines from low charge density regions to
high charge density regions in steps of 5x10‘4 electrons/A3. ........

6.] Electron diffraction pattern showing a lowering of the crystal symmetry from
cubic to orthorhombic. They correspond to (a and c) [111]c, (b) [Il2]c
and (d) [231k zone axis in face center cubic (fcc) symmetry (ac=6.4A).
Superstructure spots can be indexed in orthorhombic symmetry according
to ac ~9.1A, b0 ~15.8A and c0 ~22.3A unit cell. The corresponding zone
axis are (a and c) [001]0, (b) [01210 and (d) [lOI]0. Indexations are given
for both symmetries. .............................

6.2 High resolution electron microscopy image from a LAST-l8 sample (a), and
the fast fourier transforms from the (b) PbTe matrix and (c) the Ag-Sb
microscopic inhomogeneities. The electron beam is parallel to one of the
[100] directions. The unit cells are indicated. ................

6.3 Unit cell lattices of ordered structures: (A) type I, (B) type II, (C) type IIIA,
and (D) type IIIB AF order; (E) type l-AngTeg, (F) type 2-AngTe2, (G)
type 2-AngTe2 in hexagonal symmetry, and (H) type 3-AngTe2. For the
sake of clarity Te atoms are not included. ..................

xi

50

50

56

59

61

6.4 Density of states (DOS) for type I AF model structure. ............. 71

6.5 Total DOS of:(A) type II, (B) type IIIA, and (C) type IIIB AF structure; (D)
type l-AngTeg, (E) type 2-AngTe2, and (F) type 3-AngTe2. ...... 73

6.6 Face centred cubic (fcc) Brillowin zone of PbTe .................. 75

6.7 Band structure of PbTe at Vexp along different directions in the fcc Brillouin
zone. Orbital character of : (a) Pb p and (b)Te p. The size of the circles is
proportional with the orbital character. (c)Band structure of PbTe at Vopt. . 76

6.8 -COHP as a function of energy for PbTe without including spin-orbit interac-
tion (801). .................................. 76

6.9 Unit cell models for (a) single Ag impurity atom in Ang31Te32, (b) Ag-Sb
layer perpendicular to [001] direction in AngPblgTego, (c)A Ag-Sb layer
perpendicular to the fcc [1 l 1] direction in AngPblOTelg, (d) Ag-Sb chain
parallel to the [001] direction in AngPb30T632, and (e) Ag-Sb cluster in
AngPmeelg. For the reason of clarity we show only Pb fcc lattices with
Pb in gray, Ag in white, and Sb in black colors. ............... 81

6.10 Total DOS of (a) Ag and (c) Sb single impurity atom in PbTe. For comparison
the total DOS of bulk PbTe is shown in dashed line; (b) Partial p DOS of
Te2 (Ag nearest neighbor), Te4 (away from Ag impurity), and Te in bulk
PbTe; (d) Partial p DOS of Te2 (Sb nearest neighbor), Te4 (away from Sb
impurity), and Te in bulk PbTe ......................... 83

6.11 VB electronic charge density plots in plane perpendicular to [001] direction
in the (-1,0.25)eV energy range for (a) Ag single impurity, (b) bulk PbTe,
and (c) difference between Ag single impurity and bulk PbTe . CB elec-
tronic charge density plots in plane perpendicular to [001] direction in the
(-0.5,2)eV energy range for (e) Sb single impurity, (b) bulk PbTe, and
(c) difference between Sb single impurity and bulk PbTe . The charge
density is represented by closed lines from low charge density regions (0
electrons/A3) to high charge density regions (0.5 electrons/A3) in steps of
0.01 electrons/A3 ................................ 84

6.12 Total DOS of (a) Ag-Sb pair impurity with the Ag-Sb distance of ~11.19A,
and (b) Ag-Sb pair impurity with Ag-Sb distance of ~4.57A, in PbTe. For
comparison the total DOS of bulk PbTe is shown in dashed line. ...... 86

xii

6.13 Total DOS of (a) Ag-Sb layer model perpendicular to [001] direction, (b) Ag-
Sb layer model perpendicular to [111] direction, (c) Ag-Sb chain model,
and (d) Ag-Sb cluster model. The total DOS of bulk PbTe is shown in
dashed line. .................................. 87

6.14 (a) Simple cubic Brillouin zone, (b) hexagonal Brillouin zone. ......... 88

6.15 (a) Band structure in simple cubic Brillouin zone (see Fig. 6.14a) of (a) bulk
PbTe, (b)-(d) Ag-Sb chain model; orbital character of (b) Sb p, (c) Te5 p
(Sb nearest neighbor), and (d) Te6 p (Ag nearest neighbor), (e) Ag-Sb pair
impurity with the Ag-Sb distance of ~1 1.19A. Band structure in hexagonal
Brillouin zone (see Fig. 6.14b) of (f) Ag-Sb layer model perpendicular to
the [111] direction. .............................. 88

7.1 Band structure of PbTe in face centered cubic Brillouin zone ........... 98

7.2 Temperature dependence of (a)Electrical conductivity, (c)Thermopower, and
(e)Power factor for n-type PbTe at concentration n = 5 x 1019cm’3 ob-
tained using Eac=15eV and Eoc=26eV values of the deformation potential
coupling constants; Temperature dependence of (b)Electrical conductiv-
ity, (d)Thermopower, and (f)Power factor for n-type PbTe at concentration
12 = 5 x lolgcm‘3 using Eac=15eV and Eoc=15eV. Theoretical values
are shown as dashed lines and the experimental values are shown as black
points from Ref. [16] and as gray points from Ref. [76] ............ 114

7.3 Temperature dependence of (a)Electrical conductivity, (c)Thermopower, and
(e)Power factor for n-type PbTe at concentration 72 = 5 x 1019cm—3 ob-
tained using Eac=15eV and Eoc=26eV values of the deformation potential
coupling constants; Temperature dependence of (b)Electrical conductiv-
ity, (d)Thermopower, and (f)Power factor for n-typc PbTe at concentra-
tion n = 5 x 1019cm“3 using coupling constants values Eac=15eV and
Eoc=lSeV. Theoretical values obtained using constant md are shown as
dashed lines. Experimental values are shown as black points from Ref. [16]
and as gray points from Ref. [76]. ...................... 115

xiii

7.4 Temperature dependence of (a)Electrical conductivity, and (c)Thermopower
for n-type PbTe at concentration 11 = 5 x 1019cm‘3 using Eac=15eV and
EOC=26eV values of the deformation potential coupling constants; Temper-
ature dependence of (b)Electrical conductivity, and (d)Thermopower for
n-type PbTe at concentration n = 5 x 1019cm-3 using coupling constant
values EaC=15eV and EOC=15eV. Theoretical values obtained using tem-
perature dependent E9 in the OK-950K range are shown as dashed lines.
Experimental values are shown as black points from Ref. [16] and as gray
points from Ref. [76] .............................. 1 16

7.5 Temperature dependence of (a)Electrical conductivity, (c)Thermopower, and
(e)Power factor for n-type PbTe at concentration 12 = 2.4 x 1019cm-3;
Temperature dependence of (b)Electrical conductivity, (d)Thermopower,
and (f)Power factor for n-type PbTe at concentration 72 = 10 x 1019cm—3.
Theoretical values obtained using temperature dependent E9 and md and
using Eac=15eV and Eoc=15eV values for the deformation potential cou-
pling constants are shown as dashed lines. Experimental values are shown
as black points from Ref. [16] and as gray points from Ref. [76]. ...... l 18

7.6 Temperature dependence of (a)Electrical conductivity and (b)Thermopower for
n-type PbTe at concentration n = 5 x 1019cm—3 and using Eac=ISeV and
Eoc=15eV values of the deformation potential coupling constants. Theo-
retical values obtained using ab initio energy dispersibn and temperature
dependence of E9 and md are shown as dashed lines. Theoretical values
obtained using nonparabolic Kane energy dispersion with constant md in
the expression of n are shown as continuous lines. Experimental values are
shown as black points from Ref. [16] and as gray points from Ref. [76]. . . 120

7.7 Transport coefficients at concentration n = 5 x 1019cm‘3 using temperature
dependent E9 and md and Eac=15eV and Eoc=15eV for the deformation
potential coupling constants. Temperature dependence of (a)E1ectrical con-
ductivity, (c)Thermopower, and (e)Power factor for mfihain=l .Smd; Tem-
perature dependence of (b)Electrical conductivity, (d)TherInOpower, and
(f)Power factor for mflham=2.5md. Theoretical values for bulk PbTe are
shown as dashed lines and for the chain model of LAST-30 as continuous
lines. Experimental values are shown as black points from Ref. [16] and as
gray points from Ref. [76]. .......................... 123

xiv

7.8 Transport coefficients at concentration 72 = 1 x 1020cm-3 using temper-
ature dependent EQ and md and Eac=15eV and EOC=15eV values of
the deformation potential coupling constants. Temperature dependence
of (a)Electrical conductivity, (c)Thermopower, and (e)Power factor for
mghain=2.5md. Theoretical values for bulk PbTe are shown as dashed
lines and for the chain model of LAST-30 as continuous lines. Experimen-

tal values are shown as black points from Ref. [16] and as gray points from
Ref. [76]. ................................... 124

XV

Chapter 1

Introduction

Thennoelectricity is an old field, which was discovered back in 1821 by Seebeck. Un-
fortunately this field was obscured by Seebeck’s own misjudgment trying to relate the ap-
pearance of magnetization as a result of an applied temperature gradient. For more than
one century the field of thermoelectrics did not advance until the 19505 when the first re-
vival took place. [33] During the 19503 the basic science of therrnoelectricty became well
established, the role of doped semiconductors as good thermoelectric materials was ac-
cepted, the thermoelectric material Bi2Te3 was discovered, and the thermoelectric industry
was launched. Over the next three decades 1960-1990 the thermoelectrics industry grew
slowly but continuously. [14] In the 19903, a second revival occurred in the field of ther-
molelectrics, when new theoretical concepts were introduced and new materials using these

concepts were synthesized. [l4]

Thermoelectric materials can be used for both refrigeration or power generation. For
refrigeration one maintains a temperature gradient using an applied electric current or, con-

1

versely for power generation an electric current is generated from an applied temperature
gradient. The devices built with these thermoelectric materials have no moving parts, pro-
duce no waste products to be released into the environment, and are scalable to any size.
The thermoelectric devices have been used in space missions, laboratory equipment, and
medical applications, for which the cost and the efficiency were not as important as the
energy availability, and the reliability. The problem using thermoelectric devices on a wide
scale is that their coefficient of performance for refrigeration (efficiency for power genera-

tion) is much lower than that of current technologies. [13, 56, 93]

The performance of a thermoelectric device can be described by the dimensionless
thermoelectric figure of merit, ZT:

2
ZT: £51", (1.1)

where S is the Seebeck coefficient (thermopower), a is the electrical conductivity, n is the
thermal conductivity, and T is the operating temperature. There are two contributions to
is, “ph from phonons and Kel from electrons. Over the three decades spanning 1960-1990,
the highest values of ZT were found to be ~l in binary chalcogenides, specifically 312Te3,
PbTe, Sb2Te3 and their alloys. These are doped narrow band-gap semiconductors having
a large Seebeck coefficient and electrical conductivity but a low thermal conductivity. [35]
To make thermoelectric devices competitive with the existing technologies for cooling and
power generation, ZT values of 3-4 are required. [13, 56, 93] Since the quantities S, a, and
K, for conventional 3D systems are inter-related, it is very difficult to control these variables
independently so that ZT values could be increased.

2

Several attempts have been made to improve ZT by making novel materials using con-
cepts such as quantum confinement [27] (QC) and phonon glass electron crystal [86]
(PGEC). Within the QC picture, first proposed by Hicks er. al. [27], new physical phe-
nomena are introduced and new opportunities arise to independently vary S, a, and K. as the
dimensionality is decreased from 3D crystalline solids to 2D (quantum wells) , 1D (quan-
tum wires), and 0D (quantum dots). Model calculations predict an increase in ZT for a
two dimensional layer having thickness smaller than about 300A (for example an idealized
BigTe3-multiple quantum well structure) due to enhanced power factor PF (PF=S20). Even

larger power factor enhancement can occur in one-dimensional quantum wires. [14]

Harman er. al. have reported ZT values of 0.9 at 300K and 2.0 at 550K in
PbSe0_98Te0_02/PbTe quantum dot structures . [25] They believe that the increase in the
27' values are due to QC effects, which enhance the power factor PF. However, no theoret-
ical calculations have been performed for these quantum dot structures to support the QC
picture and the enhancement in the ZT may also originate from a strong reduction of the

phonon thermal conductivity mph.

In contrast to the QC idea, within PGEC picture one uses superlattice (SL) structures
consisting of two materials both having favorable S and a (such that electronic properties
are not affected by the SL structure), but reduces mph by engineering the phonon band
structure in a suitably chosen transport direction. ZT values of ~2.4 and ~1.4 have been
achieved in p-type BigTe3/ SbgTe3 and n-type BigTe3/Bi2Te2_838e0,17 SL thin films at
300K. [98] Other classes of systems which support the PGEC picture are skutterudites and
clatharates. These systems have cage-like structures and contain weakly bonded atoms

3

called "rattlers", which have been studied to reduce the phonon thermal conductivity mph
without drastically affecting their thermopower S and electrical conductivity 0. A ZT value
of ~1.35 was reported for the skutterudite CeFe3.5Coo.5Sb12 at ~900K. [19]

A third approach where both the QC and PGEC ideas have been exploited to certain
extent is to chemically synthesize new ternary and quaternary narrow band gap semicon-
ductors containing Bi, Te, Se atoms with different arrangements of Bi-Te-Se blocks which
are referred to as different quantum architectures. [36] These complex chalcogenides with
their large unit cells contain also "rattlers" atoms. [35] Some examples of such systems are
BaBiTe3, CsBi4Te6, AngmeTe2+m (LAST-m) etc. In particular CsBi4Te6 [10] has been
found to be the best bulk thermoelectric at low temperature (ZT~0.8 at 225K) and LAST-m
systems have been found to be the best bulk thermoelectrics at high temperatures (ZT~1.2
for LAST-12 and ~2.2 for LAST-l 8 at 800K). [30]

Electronic structure calculations can serve an important role in determining which ma-
terials will prove useful for thermoelectric applications. The ZT values depend on band
structure parameters such as the band gap, effective mass and its anisotropy, degeneracy of
the band extrema, carrier mobility, and the total thermal conductivity (electron plus lattice)
of the physical system. The importance of anisotropic effective mass was nicely brought

out by Hicks and Dresselhaus [27] through a parameter B given by:

3 1
1 2kBT 2 mg 2 7kBTuzz
B = —— —— —— ————— (1.2)
3W2 ’12 AzxAyyAzz efiph
Aij 2 Tim]: 2 1:, y,z V (1.3)

where A“, AW and A22 are the effective mass parameters along principal symmetry direc-
tions of a lattice (inversely proportional to the effective mass), me is the free electron mass,
7 is the degeneracy of the electron or hole pockets, pzz is the mobility along the current
flow (chosen to be the z-axis) which is proportional with A22, k B is the Boltzmann constant
and rep}, is the lattice thermal conductivity. B is a material parameter and for a fixed value of
B one can maximize ZT by changing the carrier concentration. This optimized value of ZT
increases monotonically with B. As can be seen from Eq. 1.2 the B factor in addition to de-
pending on mobility and mph, is directly proportional to the fraction I‘i=[)\zz/(/\gm¢AW)]l/2
which is a measure of the electronic structure anisotropy. A large value of R is necessary

for large ZT in anisotropic systems.

In recent years ab initio electronic structure calculations using density functional the-
ory (within local density or generalized gradient approximations, LDA [42] or GGA [70])
have been extremely successful in unravelling the electronic structure of many narrow
band gap semiconductors with complex crystal structures. Examples are: skutteru-
dites [65, 87], clatharates [7, 8], and ternary bismuth chalcogenide systems (BaBiTe3 [50]
and CsBi4Te5 [51]). In certain cases they have even been able to predict the band gaps and
effective masses quantitatively in spite of the limitations of LDA and GGA. [2] Although
the fundamental reason for this is not known, the subtle nature of the origin of gap for-
mation (i.e. hybridization, spin-orbit interaction) in these compounds may be responsible
for the above mentioned quantitative agreement. [50, 51] Even in systems where the quan-
titative 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

5

the transport properties of these narrow band gap semiconductors. [52] It is clear that ab
initio electronic structure calculations are able to provide great insight about the physical

properties of complex systems.

In this thesis I have studied the electronic structure and electronic transport proper-
ties of simple (binary) and complex (ternary and quaternary) chalcogenide systems. Many
electronic structure calculations have been performed previously for the binary chalco-
genides BigTe3 [62, 50, 102, 49, 81], Bi28e3 [62, 49], Sb2Te3 [92, 66], Sb2363 [82, 66],
PbTe [67, 60, 97, 99, 1, 47], PbSe [67, 60, 97, 99, l, 47], and PbS. [67, 97, 99, 47] Since ZT
can be enhanced through QC and SL structures, I have investigated the effect of QC cre-
ated by the surfaces on the electronic structure of layered BigSe3 and BigTe3 (the impact of
interlayer bonding on the electronic properties). Although in (BigTe3)m(Sb2Te3)n SL [98]
the PGEC effect is primarily responsible for the increase in ZT, it is important to know
how the electronic properties of BigTe3 and SbgTe3 are affected by the formation of SL. To
answer this problem, I have investigated the electronic properties of (Bi2Te3)m(Sb2Te3)n
SL as compared to those of BigTe3 and SbgTe3 bulk systems. The ternary KgBi85e13 sys-
tem shows potential for superior thermoelectric performance. This compound forms in two
distinct phases, a and B. The B-phase, which has several sites with mixed K/Bi occupancy
is a better thermoelectric. It has been found that 5 phase is a much better thermoelectric.
Therefore, to gain a deeper understanding of the electronic structure of the promising ther-
moelectric B —K2Bi88e13, to see how the difference in the crystal structures of a and B
phases shows up in their electronic structures, and the role of K/Bi mixing, I have carried
out detailed electronic structure calculations in both phases of these compounds. The qua-

6

ternary LAST-m systems are the best known high temperature bulk thermoelectrics. These
systems form in the rocksalt structure similar to PbTe where Ag and Sb occupy Pb sites.
There are clear experimental evidence that LAST-m systems are not solid solutions be-
tween the AngTe2 and PbTe but they exhibit microscopic inhomogeneities rich in Ag-Sb
embedded into a PbTe matrix. To see the effect of Ag-Sb microscopic arrangements on
the electronic structure and transport properties I have performed electronic structure and

transport calculations in these complex systems.

Chapter 2

Electronic Structure Calculations Using

Density Functional Theory (DFT)

The basis of electronic structure calculations lies in the nature of electrons in a solid. Elec-
trons 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
screening cloud form a “quasiparticle” which weakly interact with other quasiparticles in
the solid. The weak interaction between quasiparticles allows for an effective single par-
ticle approximation, particularly for low energy 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 (magnetic insulators). Within a single
particle approximation electronic structure calculations involve solving the single particle

8

Dirac equation. When relativistic effects are not significant as in many solid state systems,

one solves a single particle Schrodinger equation.

2.1 Full Relativistic Single Particle Dirac Equation

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. [43]

The Dirac Hamiltonian is given by:

HD =ca-p+(B—1)mc2+V(r) , (2.1)

where a and B are 4x4 matrices expressed as:

0 0
a = , (2.2)
0 0
I 0
5 = , (2-3)
0 —I

with 0;, cry, and 02 being the Pauli spin matrices. p is the four component momentum,
m 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 compo-
nent spinors consisting of the two two-component spinors: large component (I) and small

component x.

q; : . (2.4)

This gives a set of coupled equations for (I) and x:

(:(a - pix = (6 - V)‘1> , (2.5)

do - p)<I> = (e — V + 2mc2)x . (2.6)

Solving for (I) the above two equations give the equation for the large component (I):

6 — V
2mc2

 

1 ’1 , _.
%(0.p)(1+ ) (a-p)<I>+I<I>—€<I> . (2.7)

2.2 Scalar Relativistic Approximation

Eq. 2.7 is the exact Dirac equation. This equation serves as the basis for relativistic elec-
tronic structure calculations. [78, 32, 18] However, for valence electrons, where the rela-

tivistic effects are not usually strong, one can make the following approximation:

 

 

—1
e-V c—V
1 E’ 1— , 2.8
( + 2m02) ( 2m02) ( )

and using the identities:

10

p1" = Vp — ihVV , (2.9)

(a - VI")((7 - p) 2 (VV - p) + i0 - [V7, p] , (2.10)

the Dirac Eq. 2.7 can be written as a linear second-order differential equation in (I). This is

given by:

 

 

l—é—V)2:+V <I>— I12 (VVV‘P)+—hf—(U[VV ]<I>)--c<I> (211)
2mc2 2m 477L2C2 4rn2c2 ,p — ' '

For a spherically symmetric potential V(r) the Eq. 2.11 can be simplified further:

2 4 2
p_+V p h dVa 1 dVl

‘—_’———— ——-—-—-l- (I): (1) 2,12
2m 8rn3c2 4772.202 dr 8r 2m2c2 dr 7'( S) 6 ’ ( )

where the first two terms are the nonrelativistic Schroedinger Hamiltonian, the third is
the mass velocity term, the fourth is the Darwin term, and the last term is the spin-orbit
interaction (801). Due to the SOI term, j=l+s is the good quantum number. The four-

component function is now written as:

q, z z 3'" , (2.13)

where g and f are the radial functions, Y3; is the r-independent eigen function of 3'2, jz, [2

9

and 32 formed by the combination of the Pauli spinor with the spherical harmonics.

In practice one first solves Eq. 2.12 without the last term [43], which reduces to:

p2 , p4 ’12 dV 0

2m + 8771302 4m2c2 dr 67‘ f ( )

 

Eq. 2.14 is called the scalar relativistic Schroedinger equation. Since there is no spin-orbit
coupling in Eq. 2.14, l and s are still good quantum numbers. However, the spin-orbit
effects are important, especially for heavier elements. The four-component wave function
\I is expressed in terms of radial functions (f('r) and §(r)) and spherical harmonics, Ylm

as:

be"
(Gr
3
S
3

(2.15)

16:
H
Xi
H
J.
:2:
CL
S
3
X
a;

The scalar relativistic Eq. 2.14 is solved with this choice of (I). The functions \I1 are not
eigen functions of the Dirac Hamiltonian H D and their deviation from eigen functions is

used to define the spin-orbit Hamiltonian H30:

HD‘i’ZCqI‘I'HSOql . (2.16)

In the basis set of functions (I), H 30 has the form:

12

h (II/'1 1-3 0

 

 

., = __ 2.17
M) 211102 dr r ’ ( )
0 0
where M is the relativistic mass given by:
6 — V
1.] : + 2.18
I m 262 ( )

To include 801, one method is to double the entire basis set to accommodate H30. This
approach is usually computationally expensive. The second variational method is an alter-
native, which does not require the doubling of basis set. The spin up and down states are
considered separately using the eigenvectors of the scalar relativistic calculation for a finite
number of states near the Fermi energy. This reduces the number of basis sets involved in

the calculation in the presence of $01. The spin up and down states can be expressed as:

T

~ 9 (Tn/1m 1

ct = X, :Xr = , (2.19)
‘sz(T)Ylm 0

j I

~ g (7') lm 0

vi = Xi 2X1 = . (2.20)
—3fl(T)YIm 1

The states given in Eqs. 2.19, and 2.20 are used as new basis set for the expansion of the
wave function (I) in Eq. 2.12. The total Hamiltonian including 801 can be solved with less
computational effort since the number of \IJT and \Ill states are usually much less than in

13

the original eigenvalue problem. [43, 85].

2.3 Density Functional Theory (DFT)

A major theoretical and computational advance in electronic structure of solids was made
by Hohenberg and Kohn [29] and later by Kohn and Sham [42] with the advent of density
functional theory. This theory states that the ground state electronic properties of a solid
are determined solely through its electronic density p(F). Therein the key quantity is p(f')

in terms of which the total energy is:

Etot (9(a) = Ts (PU-'2) + Eee (9(a) + ENe (9(0) + Emc (1001)) + ENN : (2-21)

where T3 is the kinetic energy of the noninteracting particles having the same density p(f'),
Eee is the electron-electron repulsive Coulomb energy, E Ne is the nuclear-electron attrac-
tive Coulomb energy, E“ is the exchange-correlation energy and E N N is the repulsive
Coulomb energy of the fixed nuclei.

In local density approximation (LDA) [34], Exc is expressed as a function of the local

exchange-correlation energy density #:ch

E2?!“ = f 22.. (pm) p(nd3r . (2.22)

This approximation is generalized for both the spin densities p107) and pH?) in local spin
density approximation (LSDA):

14

£2th = / 21.....(pitnwn) [Mm/Jim] «1372' . (2.23)

Recent progress has been made going beyond LSDA by adding gradient terms of the elec-

tron density to ETC. This has led to generalized gradient approximation (GGA) [71, 70]:

EEC“ = f it“ (NWVplvitpimflpifii) [IND +2203] d3? - (2.24)

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

Ts is given by:

N t 52 2 3-2
Tszszn 7;; wxndr . (2.25)

where 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(F) leads to the Kohn-Sham single

particle equations:

’12
—2—n—lv2+vc(n w... IMF) = 6.2/22m . (2.26)

where V0 is the classical Coulomb potential and ch is the exchange-correlation poten-
tial. VC+ch=Veff is an effective single particle potential. The single particle Kohn-Sham
Eqs. 2.26 have to be solved self consistently with the condition:

15

N
M = Z I (2.-(n I‘2 . (2.27)

For a generalization of Kohn-Sham equations to fully relativistic systems see [74, 73, 55].

It should be emphasized that the total energy EM:

E20: ¢ 2 ({‘S , (2.28)
i
where 61K 5 are the Kohn-Sham single particle eigen values. The corrections have to be
made. [24]
K S

e are not the true quasiparticle states seen in experiment. However, for broad band

2'

systems they provide a very good description except for band gaps and effective masses. [2]

2.4 Method of Calculation

Electronic structure calculations were performed using the self-consistent full-potential lin-
earized augmented plane wave method (FLAPW) [85] within DFT [29, 42], using GGA of
Perdew, Burke and Emzerhof [70] for the exchange and correlation potential. Since most of
the systems of our interest contain the heavy atom Bi relativistic effects are very important.
Relativistic effects are marginally important for Sb and Te, and not important for Se. The
way such effects are included in the electronic structure code used to obtain the electronic
structure (WIEN97 and WIENZK programs) [6] is discussed very briefly.

The core states are treated in a fully relativistic way (i.e. solving the Dirac Eq.2.7). [80]
The valence and local orbitals, which are defined within the atomic spheres are first ob-

l6

tained within scalar relativistic approximation (see section 2.2). This approximation does
not include SOI but takes into account the effects of mass-velocity and Darwin correc-
tions. [80] In the absence of SOI spin up and spin down eigen functions separate. In the
presence of 501, the spin up and spin down eigen functions mix and this is incorporated
in a rather efficient way using a second variational method. [38, 54] In this method one
takes advantage of the fact that 801 is usually small and only a smaller subset of basis
functions obtained without 801 are used to obtain the eigen values and eigen functions in
the presence of 801.

An important parameter that goes into the electronic structure calculation in the
FLAPW method is the value of the atomic radius of each component. For different multi-
component systems the values of the atomic radii were chosen differently. In some cases
the values of atomic radii were taken to be the same for all the atoms, and this value is cho-
sen to fill the space between atoms. In other cases different values were chosen for different
atoms. The number of the reciprocal 1: points used in the calculations of the self-consistent
charge density were taken to be in the range 18 to 30. The number of plane waves used
in the interstitial region is characterized by a parameter RKmax=Rm1*Kmax, where Rm)
is the smallest muffin tin radius and KmaJr is the maximum plane wave vector. Typically
RKmax is chosen between 7 to 9, and for our calculations it was taken to be 8. Convergence
of the self-consistent iterations was performed within 0.0001 Ry with a cutoff of -6.0 Ry

between the valence and the core states.

17

Chapter 3

Surface States in Binary Semiconductors

BigSegJ and BigTeg

3.1 General Introduction for BiQSeg and B12T93

Among the best known bulk materials for room and high temperature thermoelectric ap-
plications are binary chalcogenides, specifically BigTeg, SbgTeg, PbTe and their alloys.
Bi2Te3, BigSe3, SbgTe3, and Sb2363 are layered materials consisting of multiple “quintu-
ple layers”, with each “quintuple layer” containing five alternating atomic planes (see sec-
tion 3.2). Inside a “quintuple layer” the interaction between atoms is of covalent type with
covalent intralayer bonds, whereas the interaction between the “quintuple layers” is con-
ventionally termed as Van der Waals-type with Van der Waals interlayer bonds. [91, 102]
Because of such weak interlayer bonding, the effect of the surface on the electronic struc-
ture is expected to be small. Such an assumption is made in most interpretations of the

18

surface probes (e.g. electron tunneling and photoemission) of layered compounds. How-
ever, an increased attention has been recently given to the impact of the weak interlayer
bonding on both bulk and surface electronic properties of layered materials. [102, 37] In a
recent publication, [96] we have shown that surface electronic structure plays an important

role in the observation of near-surface defects in BigSe3.

3.2 Crystal Structure of Bi28e3 and B12T83

The crystal structure of both BigTe3 and Bi2Se3 is rhombohedral with space group
D§d(R3m). [100] It can be represented as a stack of hexagonally arranged atomic planes,
each consisting of only one type of atom. Five atomic planes are stacked in a close-packed
fcc fashion in order (Fig. 3.1) Tel(Se1)-Bi-Te2(Se2)-Bi-Tel(Sel), in a quintuple layer. The
proximity of the surface breaks the equivalence of the Te1(Se1) positions in a layer, so the
Te3(Se3) notation was used for the third Te(Se) atomic plane from the surface. The hexag-
onal unit cell, shown in Fig. 3.1, spans three quintuple layers and contains 15 atoms.

To model the experimental scanning tunneling spectroscopy (STS) data (to be discussed
in section 3.4, calculations in a slab geometry, [90] modeling the surface were performed.
For band structure calculations for bulk Bi28e3 and BigTe3, the hexagonal unit cell consist—
ing of 3 quintuple layers was used, whereas to model the surface a repeating slab geometry
was used, with slabs consisting of 3 to 12 quintuple layers separated by an increased dis-
tance d, as compared to the bulk crystal structure (~0.35nm). The calculated band struc-
tures did not exhibit significant variation with the separation larger than 0.4—0.5nm between
the slabs; therefore the slab separation of d=0.7 nm was found sufficient for modeling the

19

 

 

 

 

 

)quintuple (h) f
layer , A kz: .
F
o L ’ H
M K
3d/2 k1 k2

Figure 3.1: (a) 312883 (BigTe3) structure viewed parallel to the layers. 2x1x1 hexagonal
unit cell is shown with a solid line. The unit cell used for slab calculations (for a 3-layer
slab) is shown in dash. In a dashed frame one of the ppa bonded chains is plotted. (b)
Hexagonal Brillouin zone of Bi28e3 (BigTe3).

surface.

3.3 Electronic Structure of Bulk and Slab (to Model the

Surface)

In Fig. 3.2 the calculated band structures of Bi2Se3 and BigTe3 both in the bulk and slab
geometries are shown. Since the bulk unit cell spans 3 layers, the 3-layer thick slab ge-
ometry used in Fig. 3.2 gives the same number of bands as in the bulk calculations, which
is convenient for a direct comparison. It can be presumed that due to the weak interlayer
bonding the electronic states of the middle layer in a slab are close to the bulk states, and
thus a 3-layer slab might be sufficient for modeling the surface electronic structure of a
bulk sample. However, it is shown below that this conclusion can be accepted with certain
limitations. Calculations of the bulk band structure, shown in Fig. 3.2(a),(c), yield a direct

20

(a) (b)

 

 

‘kl‘

l

.0 -l.0 I
FZIVIKAIAFZMKAIA FZMKAIAFZMKAIA

 

A A
> >
e 3
>5 >5
goo g. _
LL] LLI

 

 

 

 

      

 

f—L} ‘

 

 

 

 

 

 

 

-l

 

 

Figure 3.2: Calculated band structure: (a) bulk Bi28e3, (b) 3-layer slab of Bi28e3,(c) bulk
B12T63, (d) 3-layer slab of BlgTe3. The experimental Fermi energies are shown as dashed
lines; zero energy corresponds to the top of the valence band.

band gap of 0.3eV in Bi25e3 and an indirect gap of 0.14eV in Bi2Te3 with degenerate va-
lence band maximum. These are similar to the earlier calculations. [50, 91, 62, 102] Our
calculations do not reproduce the six—fold degeneracy of the conduction band minimum
seen experimentally in BigTeg. [39, 84, 40] This problem has been encountered in a num-
ber of earlier calculations. [50, 91, 62] However our more recent calculations reproduce
the 6-fold degeneracy of the conduction band minimum when the 121/2 relativistic local
orbitals are incorporated in the LAPW basis. [59] Comparing the calculations of the bulk
band structure (Fig. 3.2(a),(c)) with the slab (Fig. 3.2(b)(d)), one finds that there is a signif-
icant effect of the reduced interlayer bonding between the slabs both in BigSe3 and Bi2Te3.
The most notable differences are lowering of the bulk conduction band states into the gap
and a drastic restructuring of the valence band, with most of the high valence band (HVB)

21

states associated with surface Sel (Tel) atoms lowered in energy as compared to the bulk
band structure. The remaining HVB states are dominated by the Se (Te) orbitals of the
atoms located in the quintuple layers away from the surface, and as a consequence have a

small amplitude at the surface.

The nature of the surface states can be elucidated through the analysis of their or-
bital character in real space and dispersion in the reciprocal space. As follows from
Fig. 3.2(b)(d), the in-gap states are highly dispersive in the plane parallel to the quintu-
ple layers (I‘M and PK directions in the Brillouin zone, Fig. 3.1(b)), indicating that they
originate from the strong in-plane hybridization of the surface atoms. Analysis of the or-
bital character of the in-gap states shows that they are dominated by the two atomic layers
closest to the surface: Se] and Bi in BigSe3, and Tel and Bi in BigTeg. However, there are
significant contributions from all the other atoms in the surface quintuple layer. The in-gap
states do not exhibit dispersion in the direction perpendicular to the quintuple layers (the
FA direction), which is consistent with the negligible interaction between the slabs. The
analysis of the orbital character of the HVB for the bulk BigSe3(Bi2Te3) shows that they
are dominated by Se(Te) states, which are antibonding in the sense of interlayer coupling.
At the surface, the energy separation between the Se(Te) bonding and antibonding states
is reduced as a consequence of the increase in energy of the valence band (VB) bonding
states and decrease in energy of the HVB antibonding states. The lowering of the HVB at
the surface is a robust effect which has been qualitatively understood in terms of a simple

one-dimensional tight-bonding model. [96]

To follow the evolution of the electronic structure with the distance from the surface,

22

0.6
0.5
0.4

80.3

D
0.2
0.1

O

-0.

I

 

 

 

 

 

 

 

 

 

 

l .
Energy [eV]

 

 

I: 1 I I I ' . I
11 E. 3 first layer _
Ii 33;. second layer .....
I * ‘jl third layer mm '
- x
I’ ‘1': fourth layer mu
1"
I1 :I i 0‘1» F fifségye'r— EUR —1
l \ go... mass:
I . é > fourth la er--. .
‘l‘ 2‘0006. ulk_ j .
' .8004: i ,
l E , ,l A r,
| 0.02; "I’m/1;; ; l',’ .
I 90.2 0.1“ '. """"" .. oooo 3 0.4 .
'. Energylevl : .
\ k v»
’ Ate-J
.4 ‘5 p '.
-., h. ’M .4 .. ..8

Figure 3.3: Partial DOS of Se] in Bi28e3 for various positions of the quintuple layers with
respect to the surface in a 12-slab geometry.

in Fig. 3.3 we show the partial "atomic " density of states (DOS) of Sel atoms for various

quintuple layers as a function of their position relative to the surface in a 12 slab-geometry

calculation. The in-gap states associated with the lowering of the bulk conduction band

into the gap (the states between OeV and 0.25eV) are mostly present in the first quintuple

layer. There are also in-gap states associated with the restructuring of the HVB (the states

between -0.07eV and OeV, insert in Fig. 3.3). The bulk HVB states (below -0.07eV) are

almost recovered by the Sel at the second layer from the surface, while the surface Sel

23

contribution to the states near the top of the valence band (between -0.07eV and -0.3eV) is
dramatically suppressed. This is consistent with the antibonding (in the sense of interlayer
bonding) nature of the states near the top of the valence band, as found e. g. in ti ght-binding

calculations. [96]

The sharp peaks, which are present in the DOS calculations in slab geometry, occur
due to the saddle points, where the bands have no dispersion along the PA direction. Ac-
cording to Gofron and Dabrowsky, [23] such saddle points, which they called "extended"
Van Hove singularities, yield a power law divergence in DOS. These are artifacts due to the
approximation of the bulk sample terminated with a surface by a finite thickness slab. The
number of peaks increases with the slab thickness due to the folding of the Brillouin zone
along the PA direction, while their weight decreases. In the limit of an infinite slab, the
DOS should become smooth. The deviation from the bulk DOS in the case of Sel atoms
located at second, third and fourth layer are particularly strong in the HVB between -0.3eV

and -0.1eV and are due to these "extended" Van Hove singularities.

To see the effects of structural relaxation on the electronic states, we performed vol-
ume optimization (i.e. optimization with respect to scaling of the lattice constants), and
force minimization for individual atomic sites up to 0.2mRy/nm for the 3-slab geometry.
Optimization decreases the volume by 1% for Bi28e3 and increases it by 4% for Bi2Te3.
All the optimized bond distances, except for the interlayer bonds, are within 1 to 2.5% of
their bulk values for both 312363 and BigTe3. The interlayer distances within the slabs
increase by 7.9% and 5.5% in 312363 and BigTeg, respectively. The relaxation does not
significantly change the effects of the surface on the electronic band structure. In 312363,

24

the lowering of the surface conduction band into the bulk gap is reduced from ~0.25eV in
unrelaxed calculation to ~0. l 7eV after relaxation is performed, while the other changes in
the band structure are insignificant. In Bi2Te3, the only significant effect of the relaxation is
a slight upward shift of the highest valence band by ~0.03eV. The small effect of structural
relaxation reflects the overall weakness of interlayer bonds, i.e. the relatively small struc—
tural perturbation introduced by the presence of the surface. However, locally the effect of

surface is important, it gives rise to surface states.

3.4 Comparison with Scanning Tunneling Microscopy

(STM) Results

Scanning tunneling microscopy (STM) is an imaging method with diverse possibilities
for real space imaging on a scale which extends to atomic dimensions. [4] The underly-
ing physical basis of the STM is electron tunneling. Electron tunneling occurs between
two conductors (electrodes) separated by a sufficiently thin insulating layer or, in physical
terms, potential barrier. The tunneling current I is a measure of the overlap of the wave
functions of the two electrodes. Therefore, I is a function of the electrode separation and
the nature of the states involved. In practice, the electronic images are obtained by record-
ing dI/dV or d] / ds while scanning and controlling the tunneling distance 3 by keeping the
average current constant, where V is the voltage across tunneling barrier. Because of the
usually close relation of the tip-sample spacing at constant I with the topography, of d] / dV
with the local density of states, and of dI/ds with the local barrier height (work function),

25

associated images are often referred to as topographical or STM images, spectroscopic or
scanning tunneling spectroscopy (STS) images, and work-function profiles, respectively.

The experimental results presented here were acquired by Sergei Urazhdin. [95]

An experimental STM topographic image of B12363 (0,0,1) surface is shown in
Fig. 3.4(a). Fig. 3.4(b) shows a simulated topographic image of B12363 surface. It was
obtained from theoretical calculations in a slab geometry by integrating the energy pro-
jected charge density in a range of 0.7 eV from the top of the valence band, at a plane 2.4 A
above the surface Se atoms. In both the experimental and simulated topographic images,
the highest corrugations are identified with the surface Se atoms. Both images Fig. 3.4(a,b)
reflect the 6-fold rotational symmetry of the surface Se plane. However, this 6-fold sym-
metry is broken by the Bi atoms in the underlying atomic plane. Bi atoms are positioned
in the center of every other triangle formed by the neighboring surface Se atoms, lowering
the rotational symmetry to 3-fold. In the simulated image (Fig. 3.4(b)), the darkest spots
between Se atoms correspond to positions which are not occupied by Bi in the second
atomic plane, while positions occupied by Bi yield less contrast. The experimental image
Fig. 3.4(a) are interpreted similarly. The slight discrepancy between the shapes of the Se
atomic corrugations is, in terms of the Fourier components of the topographic images, due

to the higher harmonics, which are not manifested in the experiment.

The STS conductance spectrum gives information about local DOS. To simulate the
experimental differential conductance spectra, the electronic charge density was calculated
as a function of energy, p(F',E), at a point located at 2.35 A above the surface Se atom. This
was done by calculating p(F,E) for different values of E within an energy mesh of 20meV

26

 

Figure 3.4: (a) 1.3 x 1.3 nm topographic image of Bi28e3 surface acquired at a bias voltage
—0.7 V. Black to white scale is 30 pm. Autocorrelation was used to enhance the image. (b)
Calculated topography of 312563 at a bias voltage of ~0.7 V.

and then averaging over five energy intervals. The theoretical conductance spectra was then
obtained by multiplying p(F,E) by a scale factor to give an overall fit to the experiment. To
simulate the conductance spectra a very dense mesh of 961 irreducible I0 points in the [02:0
plane of the Brillouin zone was required and in order to reduce the computational time we
used the 3 slab—geometry (instead of 12 slab-geometry) to model the surface. The calculated
and experimental conductance spectra of Bi28e3 and Bi2Te3 are given in Fig. 3.5. The
theoretical values were shifted in order to match the Fermi energies. The experimental
conductance spectra show a finite DOS in the gap and a suppressed conduction in the HVB
energy range. The calculated conductance spectra clearly show finite in-gap density of
states in both B12863 and Bi2Te3. The overall agreement in BigSe3 is quite remarkable,
while for Bi2Te3 the agreement is not as good.

27

 

 

1.2-

. 31.0:
J 2: 0.8-
"O

goe-

1 ._ .

E 0.4-
0.2 .

l a l g .l A I . l A 0.0 n - l - l

-0.6 -0.4 -0.2 0.0 0.2 0.4 -0.2 0.0 0.2 0.4

3880/) BIBS (V)
Figure 3.5: (a) Solid lines: near-gap tunneling differential conductance dI/dVB spectrum
of 312363, acquired directly with a lock-in detection technique at a temperature of 4.2 K.

Dashed lines: theoretical curves, calculated for a 3 slab-geometry. The curves are normal-
ized by the differential conductance at V B = 0.3V. (b) Same as (a), for Bi2Te3.

 

 

 

 

 

A

 

 

3.5 Summary

Surface states appear in our theoretical calculations, in agreement with STS observations.
Our calculations show that the surface states extend into the first quintuple layer with a de-
cay length of about lnm. These surface effects are complimentary to the strong influence
of the inter(quintuple) layer bonding on the bulk electronic structure of these compounds,
rendering the intuitive picture of layers weakly bonded with Van der Waals-type interaction
inadequate. There is an important covalent contribution to the inter(quintuple) layer bond-
ing. This can also be seen from the incomplete two-dimensional nature of Fermi surface in
312TC3, which reveals that the interlayer bonding is not only of Van der Waals-type (with a
two-dimensional nature of Fermi surface), but is also covalent. [102] Our electronic struc-
ture calculations suggest that applications of low—dimensional (quantum confinement) nar-
row gap semiconductor structures [27] for thermoelectric devices where interface effects

28

play an important role cannot rely on the assumption that bulk-like electronic properties

persist in low-dimensional structures.

29

Chapter 4

Electronic Structure of

(BigTe3)m(Sb2Te3)n Superlattice (SL)

4.1 General introduction for (BigTe3)m(Sb2Te3)n SL

Following the phonon glass electron crystal (PGEC) concept to improve the efficiency of
a thermoelectric material, great efforts have been made to synthesize SL systems with
low temperature growth techniques such as molecular beam epitaxy (MBE) and chemical
vapor deposition (CVD). As discussed in Chapter 1 according to PGEC concept, a high ZT
value could be obtained in SL systems by introducing phonon scattering at the interface
of SL systems and reducing mph, provided one does not reduce the power factor PF (S20)

significantly.

(Bi2Te3),n(Sb2Te3)n SL structures grown by chemical vapor deposition were found
to be very good thermoelectric materials by Venkatasubramanian et al. [98], where room

30

temperature ZT values of ~2.4 and ~1.4 have been obtained in p-type Bi2Te3/Sb2Te3 and
n-type BigTe3/Bi2Te2.83$e0_17 SL thin films respectively. Unlike BigTe3 alloy systems
where the charge and energy transport take place in planes parallel to the Van der Waals
layers (defined as in-plane), in these SL systems charge transport takes place in a direction
normal to this plane (defined as the cross-plane). For certain values of m and n, one finds
a dramatic change in thermal conductivity which leads to an increased value of ZT. A
fundamental question is how SL structures affect the electronic properties. Experimental
value of the mobility anisotropy pep/pip was found to depend on the superlattice period
and values of m and n. In Fig. 4.1, Hep/Hip is shown as functions of the thickness d of
the Bi2Te3 and SbgTe3 atomic layers. One “quintuple layer” (see section 4.2) consist of
five atomic layers and has a thickness of ~10A for both BIQTC3 and SbgTeg. In order
to understand how band gap and effective mass anisotropy depend on these quantities I
have carried out electronic structure calculations in bulk BizTeg and Sb2Te3 and three SL

systems, m=n=1 (11), m=1, n=2 (12), and m=2, n=4 (24).

4.2 Model Structures and Geometry optimizations for

(Bi2T63)m(Sb2T€3)n SL

The SL structures consist of multiple “quintuple layer” of BigTe3 and SbgTe3, with each
“quintuple layer” containing 2 Bi/Sb and 3 Te atomic layers. As has been discussed in sec-
tion 3.2 the crystal structure of BigTeg and Sb2Te3 can be described in hexagonal symmetry
with (P-3ml) space group and contains three “quintuple layers”. Since experimental deter-

31

_.L
In.

 

1.05 r

0.35 _
. Bi2Te3 ~ 10A

0 Bizre3 ~ 20A
0 20 40 60

dszTea (A)

Figure 4.1: Hole mobility anisotropy in (BigTe3)m(Sb2Te3)n SL. [98]

Mobility anisotropy, uj/pin-p.ane
O
\1
I

 

 

 

minations of the structures of (Bi2Te3)m(Sb2Te3)n SL are not available, one has to assume
some starting structures for electronic structure calculations. For (1 l) BigTe3)1(Sb2Te3)1
SL, a hexagonal structure with (P3ml) space group symmetry was used, with each unit
cell containing one BigTe3 layer and one Sb2Te3 layer (Fig. 4.2(a)). The inversion sym-
metry of the bulk systems can not be preserved in this structure because there are only two
“quintuple layers” in a unit cell. The (12) BigTe3)1(Sb2Te3)2 SL on the other hand has the

same space group symmetry (P-3ml) as the bulk and the inversion symmetry is preserved

(Fig. 4.2(b)).

The initial lattice parameters were taken as averages of bulk Bi2Te3 and SbgTe3 lattice

32

parameters, which are 3(11) = b(11) = (a3 + aS)/2, C(11) = (CB + C5)/2 for the (11) SL,
a(12) = b(12) = (a3 + a5)/2, C(12) = (C3 + 2c5)/3 for the (12) SL, and a<24) = 3(12)’ C(24) =
2c(12) for the (24) SL (Fig. 4.2(c)). Here (a 3, CB) and (a S, c5) are the lattice parameters of
bulk BigTe3 and Sb2T63 respectively. For each SL structure, the initial atomic coordinates
in the “quintuple layers” are assumed to take the same fractions as in the bulk Sb2T63. In
bulk BigTe3 and Sb2T€3 two different types of Te atoms exist. The Te atom lying in the
Van der Waals gap and bonded to one layer of Bi(Sb) atoms and one layer of Te atoms is
Tel, and the Te atom which is bonded to two layers of Bi(Sb) atoms is Te2. So in the initial
structure, the Bi-Tel and Bi-Te2 bond lengths are assumed equal to the Sb-Tel and Sb-TeZ
bond lengths respectively. The SL geometries were then optimized starting from the initial
lattice structures. As a first step, global lattice parameter optimizations (volume and c/a
ratio) are performed, and then the internal atomic coordinates are relaxed until the forces

on the atoms are below 1.0 mRy/a.u. = 0.0136 eV/A.

For bulk Bi2Te3 and SbgTe3, only volume optimizations were performed. The values
of optimal volume Vopt correspond to 4% and 6% increase compared to the experimental
values Vex), for BigTe3 and SbgTe3 respectively. The internal atomic coordinates are then
relaxed while keeping Vopt and (c/a)0pt ratios fixed. The fractional changes of the atomic
positions from their initial values are found to be rather small. The starting values (which
are the experimental values) of the Tel-Bi, Te2-Bi, and Tel-Tel bond lengths in BigTe3
are 3.117A, 3.227A, and 3.572A respectively. The volume optimization increases these
values to 3.158A, 3.269A, and 3.619A. In SbgTe3, the starting values of the Tel-Sb, Te2-
Sb, and Tel-Tel bond lengths are 2.974A, 3.168A, and 3.75A, whereas the corresponding

33

 

(a)

 

 

 

 

N001]

 

 

 

k2

 

J I
..
on o.
I
0

Figure 4.2: Models for (BigTe3)m(Sb2Te3)n SL structures of (a) (l l), (b) (12), and (c)
(24). Hexagonal Brillowin zone ((1).

values after volume optimization are 3.032A, 3.23A, 3.823A. The effect of relaxation is to
increase all the bond lengths; the amount of increase range from 0.04-0.07A, the largest

relaxation takes place for Tel-Tel bond length in SbgTeg.

For the (l 1) SL, Vopt and (c/a)opt ratio are 6.95% and 1.0% larger than their initial
values. For the (12) SL, the Vopt and (c/a)opt ratio are 6.0% and 0.5% larger than the ini-
tial ones. The volume increase for the SL are found to be larger than in the bulk Sb2T63
and Bi2Te3, indicating more relaxation effects for SL. In (11)SL after volume and inter-
nal (atomic) optimizations the Tel-Sb, Tel-Bi, Te2-Sb, Te2-Bi and Tel-Tel bond lengths
are 3.03921, 3.072A, 3.196A, 3.278A, and 3979/31, whereas the corresponding values for

34

(12)SL are 30421, 3.081A, 32025., 3274/1, and 3.913.. There is very good agreement be-
tween the bond lengths of the (11) and (12)SL after optimizations. As expected there

are differences between bulk and SL structure bond lengths with the largest difference of

~0.08A.

4.3 Electronic Structure of (BioTe3)m(Sb2Te3)nSL

Electronic transport properties of doped semiconductors depend sensitively on the nature of
energy states which are close to the chemical potential (~0.2eV above and below the chem-
ical potential). Therefore the details of the band structure (positions of the band extrema,
degeneracy of local minima and maxima and dispersion of the bands) near the energy gap
are very important in electronic transport. The valence band maximum (VBM) in bulk
BigTe3, bulk Sb2Te3 and (11), (12) SL is six-fold degenerate and is located in the MFA
plane, but the position of VBM in this plane and the energy dispersion near VBM are dif-
ferent in bulk and SL systems (Fig. 4.3). Along I‘K direction the valence band of (11)SL
has an energy dispersion which is very different than that of bulk Bi2Te3 and SbgTeg, with
a second VBM which may also contribute to transport. This suggests that by creating
(BigTe3)m(Sb2Te3)n SL one can not only engineer the hole effective mass and hole mo-
bility anisotropy of the bulk BigTe3 and SbgTeg without changing the degeneracy factor,
but also can introduce additional near by energy maxima (local) which can contribute to
transport.

The conduction band minimum (CBM) in bulk BigTe3 before volume optimization is
six-fold degenerate, being located in the MFA plane. It is necessary to include relativistic

35

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.3: Band structure of (a) bulk BigTe3, (b) bulk Sb2Te3, (c) (11) SL, and (d) (12)
SL.

36

p1/2 local orbitals to get this minimum. [48] However after volume optimization this mini-
mum is shifted up in energy and a new CBM appears along the PA direction and is two-fold
degenerate (Fig. 4.3(a)). According to the Haas-van Alphen experiments, both VBM and
CBM in BigTe3 are six-fold degenerate, [39, 84, 40]. It has proved to be extremely diffi-
cult to reproduce the six-fold degeneracy of the CBM because the details of the electronic
structure seem to be extremely sensitive to the calculation procedure, and only two pub-
lished result has been able to reproduce the six-fold degeneracy of CBM under restricted
conditions. [102, 48] The CBM in SbgTe3 is located along the FA direction and is two-
fold degenerate (Fig. 4.3(b)), whereas the CBM in both (11) and (12) SL is located in the
MFA plane with six-fold degeneracy (Fig. 4.3(c),(d)). We were not able to find Haas-van
Alphen experiments for Sb2Te3, in order to compare with our theoretical results. Large de-
generacy factors significantly enhance the thermopower for a given concentration since the
concentration for each electron or hole pocked is reduced and the thermopower increases
with decreasing concentration. [57] The dispersions of the energy bands near CBM in both
the SL are very similar, suggesting that the electron effective mass and electron mobility
anisotropy may not be too sensitive to the detail characteristics of (BigTe3)m(Sb2Te3)n SL

systems.

37

4.4 Effective Masses in Bluk BIQTC3, SbgTe3 and

(Bi2Te3)m(Sb2Te3)n SL

The carrier effective masses, which determine the transport properties and the B factor (see
Eq. 1.2), were calculated after obtaining the VBM and CBM positions in the MFA plane
of the Brillouin zone. This plane is one of the three reflection planes, which in real space
coincides with yz plane where z-axis is the trigonal axis and y is the bisectrix axis. The
cross section of electron and hole pockets in yz plane are ellipses whose major axes make
an angle 0 with the y-axis. Therefore the energies near the band extrema can be fitted using

the Eq. 4.]:

2m m
739-5: $123+ 123+ k§+
Ii. mm myy mzz my;

me me me

 

 

 

kykz (4.1)

where mi,- are the effective masses, ki is the magnitude of the reciprocal space vector along
the i-th orthogonal direction (i=X.y.Z) and me is the free electron mass. The cross term
mfg/my,z is smaller than mfg/myy and me/mzz for bulk BigTe3 and SbgTeg. We did not
calculate this term since we are interested only in the effective mass along the orthogonal
directions. Since the energy band gaps are larger than 0.1eV, we expect Eq. 4.1 to give
a reasonable estimation of the effective mass near the band extrema. The hole effective
masses for bulk BigTeg, SbgTe3 and SL before and after volume optimizations are given in
Table 4.1. The effect of the volume optimizations is to increase the hole effective masses
by no more than 0.1% in BigTe3 and 0.2% in SbgTe3. The effect of the global and internal
optimizations on the (11)SL is to increase the effective masses by ~0.4% (except for myy

38

which is decreased by ~0.7%), whereas for the (12)SL, the effective masses increase by
0.4-0.5%. As expected, the effects of the relaxations on the (l 1) and (12)SL are more
important than in the bulk Bi2Te3 and Sb2T63, because for the SL we started from average
structures for the SL. The values of the direct and indirect energy gaps are also shown in
Table 4.1.. The theoretical values of the energy gaps for bulk systems are smaller than those
given by experiments due to the GGA(LDA) problem of underestimating the gap values. [2]
The value of the indirect gap in 312T63 is ~0.06eV and the corresponding experimental
value is 0.15eV, whereas in SbgTe3 the indirect gap is ~0.12eV and the corresponding
experimental value is 0.3eV. The values of the indirect gap in (l l), (12), and (24)SL are
~0.06, ~0.07, and ~0.03 respectively. Although band gaps have not been experimentally
measured in SL systems, we expect the experimental band gaps to be a factor of 2 larger
than our calculated values within GGA. For BigTe3 the experimental hole effective masses
are also given for comparison. [39, 40] The main discrepancy is for mm, the theoretical

value is a factor of 2 smaller than experiment.

To estimate the carrier mobility ratio and compare with experiment [98], a simplified
assumption is made, namely the relaxation time is assumed to be isotropic. The carrier
mobility ratio in this case is equal to the ratio of the inverse effective masses. We define the

anisotropy of cross-plane mobility to in-plane mobility as:

 

5‘93 — "‘22 (4.2)

 

The results of mobility anisotropy ratio for the VBM are also given in Table 4.1. The calcu-
lated values of the mobility ratio for BigTeg and SbgTe3 obtained after volume relaxation

39

Table 4.1: Valence band maximum VBM hole effective masses and mobility anisotropy.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

M myy IKE ,llcp pop exp. Direct Indirect
me me me 142p 112p gapleV) gap(eV)
BIQTC3 0.016 0.173 0.1 17 0.25 0.124 0.059
BIQTC3 global 0.016 0.189 0.128 0.23 0.122 0.059
optimized
BIQTC3 0.031 0.208 0.1 l 1 0.49 0.33 [12] 0.22 [91] 0.15 [91]
exp. [39, 40]
SbgTe3 0.036 0.094 0.067 0.78 0.134 0.091
Sb2T63 global 0.041 0.112 0.082 0.73 0.178 0.119
optimized
Sb2T63 exp. — — — 0.5 [31] 0.3 [5]
(1 l)SL 0.06 0.177 0.074 1.21 0.083
(11)SL global 0.059 0.142 0.092 0.91 0.083
optimized
(l l)SL internally 0.085 0.106 0.104 0.91 0.88 [98] 0.056
optimized
(12)SL 0.029 0.106 0.074 0.62 0.027
(12)SL global 0.039 0.129 0.094 0.64 0.045
optimized
(12)SL internally 0.042 0.148 0.1 l 0.6 1.02 [98] 0.067
optimized
I (24)SL II 0.031 IO.112 I 0.061 I 0.8 I 0.98 [98] I I 0.026 I

 

are 0.23 and 0.73, whereas the measured experimental values are 0.33 and 0.5 respectively.
For the (l l), (12), and (24)SL the calculated values are 0.91, 0.6, and 0.8 and the cor-
responding measured values are 0.88, 1.02, and 0.98. The differences in the calculated
and measured experimental values may be because of the errors in values of the effective
masses given by GGA calculations or also to the simplified assumption of the isotropic

relaxation time used to obtain Eq. 4.2.

The results for the CBM are given in Table 4.2. As in the case of the VBM the effect
of the lattice relaxation is to increase the electron effective mass. The volume optimization

40

Table 4.2: Conduction band minimum CBM electron effective masses and mobility

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

anisotropy.
110.”. mW KILL: Itcp pep exp. Direct Indirect
me me me mp mp gapleV) gap(eV)
BigTe3 0.017 0.153 0.143 0.21 0.1 16 0.059
BigTe3 global 0.017 0.167 0.145 0.21 0.108 0.059
optimized
BigTeg 0.021 0.169 0.105 0.36 0.29 [44] 0.22 [91] 0.15 [91]
exp. [39, 40]
Sb2Te3 0.145 0.436 0.287 0.76 0.091
Sb2T63 global 0.148 0.53 0.354 0.65 0.119
optimized
SbQTC3 exp. —— — — 0.3 [5]
(1 l)SL 0.067 0.105 0.337 0.24 0.083
(11)SL global 0.132 0.119 0.471 0.27 0.083
optimized
(l l)SL internally 0.238 0.138 0.662 0.26 0.056
optimized
(12)SL 0.08 0.108 0.34 0.27 0.1 16 0.027
(12)SL global 0.082 0.119 0.479 0.2 0.121 0.045
optimized
(12)SL internally 0.1 19 0.143 0.512 0.25 0.166 0.067
optimized
I (24)SL II 0.474 I 0.431 I 6.27 I0.07 I I J 0.026 I

 

increases the electron effective masses in BigTe3 and szTe3 by no more than 0.1% and
0.2% respectively. The maximum increase in the values of the effective masses introduced
by global and internal optimizations in the (1 l) and (12)SL are 2.6% and 0.5% respec-
tively. The calculated values for the mobility ratio in BigTe3 and Sb2Te3 are 0.21 and 0.65,
whereas in the (11) and (12)SL these values are 0.26 and 0.25. As anticipated from the
band structure results (conduction band energy dispersion), in contrast to the hole mobility,
the electron mobility ratio is less sensitive to different (312T63)m(Sb2TC3)n SL.

41

4.5 Summary

Electronic structure calculations in different SL’s show that both VBM and CBM are six-
fold degenerate. This large degeneracy significantly increases the figure of merit ZT. [57]
The effect of lattice relaxation is to increase the values of the effective masses and energy
gaps. The calculated values of the hole mobility ratio Hep/Hip in the SL are in good agree-
ment with the measured experimental values. [98] The values for (1 l) and (24)SL are close
to unity and they are higher than those of the bulk BigTeg and Sb2Te3. Our calculations
show that the hole mobility ratio is very sensitive to the different types of SL, suggesting
that the valence band can be engineered through (BigTe3)m(Sb2Te3)n SL structures. On the
other hand, the values of the electron mobility ratio in SL are in between the values of the
bulk BigTe3 and Sb2T63 and they are less sensitive to the different types of SL, suggesting
that electron mobility ratio may not be engineered very easily through (BigTe3)m(Sb2Te3)n
SL structures. Our calculations indicate that the formation of SL structures does not de-
teriorate the electronic contribution to ZT (compared to bulk). Thus the large values of

observed ZT come from reduced phonon thermal conductivity.

42

Chapter 5

Electronic Structure of Complex

Ternary System K2B18Se13

5.1 General Introduction for K2B18Se13

In the previous chapter, I discussed the idea of constructing superlattices of different periods
to enhance the thermoelectric figure of merit ZT. Another approach to increase ZT, which
has been very successful, is to chemically synthesize new complex (ternary and quaternary)
narrow band gap semiconductors containing Bi, Te, Se atoms with different arrangements
of Bi-Te-Se blocks referred as quantum architectures. These complex chalcogenides with
their large unit cells containing weakly bonded atoms called "rattlers", have been studied to
reduce the phonon thermal conductivity mph without drastically affecting their power factor

PF (S20). [35]

The ZT values of a thermoelectric material increase monotonically with the dimension-

43

less parameter B described in Chapter 1 and given in Eq. 1.3. B is directly proportional
to the anisotropy factor R=[/\;z/(/\N)\yy)]1/2, of the effective mass parameters and to the
degeneracy A,» of the electron or hole pockets. These have to be maximized in order to have
high B values and consequently high ZT. In a solid it is very difficult to maximize both
the degeneracy 7 and R, because 7 can have high values in systems with high symmetries
(i.e. cubic, hexagonal), whereas R can be maximized in systems with low symmetries (i.e.
triclinic, monoclinic). Several new ternary compounds with low symmetry have been syn—
thesized with a view to increase ZT. One such compound is CsBi4Te6 which is the best

known thermoelectric for T<200K. [10]

Another promising complex chalcogenide with low symmetry is KgBi88e13, where
one tries to achieve large R, has been synthesized recently and its thermoelectric proper-
ties have been investigated. [9] This compound occurs in two distinct phases denoted as
a—K2Bi85e13 and B —KgBigSe13. These two phases represent an example where similar
buildings blocks combine to give compounds with the same stoichiometry but different ar-
chitectures. Whereas the a-phase is a wide band gap semiconductor (band gap ~0.76eV)
and not a good thermoelectric, the B-phase is a narrow gap semiconductor and shows great
promise for room and high temperature thermoelectric. One of the fascinating character-
istics of the complex Bi-chalcogen systems is that they are either charge compensated (Bi
is 3+ and X=Se/I‘e are 2-) or contain electron or hole-doped Bi/I‘e frameworks. However
the systems are highly inhomogeneous with regards to the local Bi-X, Bi-Bi, and X-X
bondings (can be covalent, ionic or van der Waals). This feature shows up directly in their
crystal structures and local coordination. For example in BaBiTe3 we have covalent Te-Te

44

bonds [1 l] and in CsBi4Te6 we have covalent Bi-Bi bonds. [10] The latter is in fact quite
unusual for Bi-chalcogen systems. The differences in local bonding and coordination from
one atom to another in the unit cell give rise to dramatic differences in the nature of the
states near the top of the valence band and bottom of the conductions bands, states respon-
sible for charge and energy transport in thermoelectric materials. As a result these complex
Bi-chalcogen systems show a wide range of thermoelectric behavior. [10, 11] Electronic
structure calculations in CsBi4Te5 and BaBiTe3 have been performed earlier by Larson er.
al.. In this thesis I focus on K2Bi88e13, trying to understand the difference in electronic

structure of the a- and B- phases.

5.2 Crystal Structure of K2BigSe13

K2Bi8Se13 forms in two distinct phases, a—KgBigSe13 (triclinic with space group P-1) and
B—KgBigSelg (monoclinic with space group P 21/m). [9] The a-phase contains BizTe3-
, CdIg- and Sb2Se3-type rod fragments parallel to the c-axis (Fig. 5.1) whereas the B-
phase possesses an architecture made up of BigTe3-, Cd12-, and NaCl-type rod fragments
(Fig. 5.2). The CdIg-type and BigTe3-type rods in a- and B- phases are arranged side by
side to form layers perpendicular to the y-axis with tunnels filled along the c-axis with K+
cations. Whereas in the B-phase the NaCl-type rod fragments connect the layers to build
a 3-dimensional framework, in the a-phase the small Snge3-type rod fragments make a
weaker connection between the layers. The inclusion of alkali atoms stabilizes the covalent
bonding in the Bi-chalcogen framework. The width of the BigTeg- and NaCl-type rods
in the B-phase is three Bi polyhedra, while the width of Cdlz-type rod is only two Bi

45

.polyhedra. The dimensions of these building blocks define the structural characteristics
of each structure type in these materials. Overall, the B-phase is slightly more dense than
the a-phase, because in the latter 25% of the Bi atoms are found in a trigonal pyramidal
geometry, while in the former all Bi atoms are in an octahedral or greater coordination

geometry. [9]

The a-phase has no structural disorder. On the other hand the B-phase has two sites
with very similar coordination environment, which have K/Bi disorder. The Bi8/K3 site
(in the BigTe3-block) contains 62% Bi and 38% K while the Kl/Bi9 site (in the NaCl-
block) contains 38% Bi and 62% K (see Fig. 5.2). Recent better refinement gives 50%l50%
occupancy for these two mixed sites. The high coordinate disordered Bi/K atoms and Cdlg-
type channels serve to stitch the fragments together. Since Bi3+ and K+ have similar sizes,
this similarity in the coordination environment of the Bi8/K3 and K1/Bi9 sites explains the
disorder between Bi and K atoms. Due to the K/Bi disorder the experimental positions
of the atoms at the mixed sites determined by X-ray powder diffraction are some average
positions making the atomic distance of K1-Se9 (2.85A) and K3-Se4 (2.84A) too short as
compared to the atomic distances between K2-Se sites (where there is no disorder) ranging
from 3.28-3.71A. [9] This large difference is due to the fact that when Bi is present at the
disordered site it will move closer to the S64 and Se9 atoms thereby making the average
distance between the disordered site and the Se shorter. Consequently there will be local
relaxations of the K+ ions at the disordered sites.

46

 

 

 

0 Se5

c 3 f

 

 

 

   

 

 

 

. .6 ' I I o. o ’ I I :
. :1!" 9 9'42? 9 2'.
A O ’ ’ . o o I .

Figure 5.1: Projection of the crystal structure of a—K2B188e13 viewed down the c-axis
(z-axis). Bi2Te3-, Cdlg- and Sngeg- building blocks in the structure are highlighted by
the shaded areas. 47

Bi Tea t e
2 yp NaCl type

       
   

 

 

 

 

 

 

Figure 5.2: Projection of the crystal structure of B—K2Bi88e13 viewed down the c-axis
(z-axis). BigTe3-, CdIg- and NaCl- building blocks in the structure are highlighted by the
shaded areas. 48

5.3 Electronic Structure and Effective Mass

Electronic structure calculations were performed using both the local density approxima-
tion (LDA) and the generalized gradient approximation (GGA) of Perdew, Burke and Emz-
erhof [70] for the exchange and correlation potential. Before discussing the results of cal-
culations, I will present the details of the optimization (atomic positions) procedure. Full
optimization calculations including the shape and volume optimization and internal relax-
ation of all the atoms in the unit cell require a large amount of computational time for this
complex compound (with 92 atoms in the unit cell). Therefore, only the volume optimiza-
tion for the a-phase and the Supercell model of the B-phase (to be discussed later) have
been performed. To see the effect of internal optimization (where positions of atoms in-
side the unit cell are allowed to relax) on the band structure, a simpler configuration of the
B-phase was chosen. In this configuration we ignore the mixing of K and Bi atoms at the
mix sites (resulting in a unit cell with smaller number of atoms) and allow for relaxations
of both the K and Bi atoms along with their neighbouring Se atoms (Se4 and Se9). For all
the calculations spin-orbit interaction ($01) was included. Since this partially optimized
calculation did not significantly change the band structure results, all other calculations for

more complex unit cells were carried out only with volume optimizations.

5.3.1 O—K2B188813

The unit cell parameters of a—KgBi88e13(triclinic, space group P-l with 23 atoms/ unit
cell) are: a=26.108 a.u., b=22.858 a.u., c=7.872 a.u., a=89.98°, B=98.64°, y=87.96°. [61]
The corresponding Brillouin zone is shown in Fig. 5.3. The band structures along differ-

49

 

 

 

 

Figure 5.3: Brillouin zone of a—K2B188e13 Figure 5.4: Brillouin zone of B —K2Bi8Se13

ent symmetry directions at optimal volume Vopt are shown in Fig. 5.5. This compound is
found to be an indirect band gap semiconductor. The effect of the 801 is to shift the con-
duction band down relative to the valence band and thereby decrease the band gap. Such
reduction in the band gap caused by the 301 have been seen in other complex ternary sys-
tems, CsBi4Te6 [51] and BaBiTeg. [50] The optimal volume Vopt for GGA calculations
is 6% larger than experimental volume Vamp, whereas for LDA calculations Vopt is 2%
smaller than Vamp. This is consistent with many previous calculations within LDA and
GGA. [79, 88, 46] The values of the band gap without and with SCI for LDA/GGA cal-
culations are given in Table 5.1. Excepting for GGA without SOI, the band gape values
changed by less than 3%. The band gap values range between 0.37-0.47eV compared to
the experimental value of 0.76eV. There can be two sources for this discrepancy; one is
usual LDA/GGA band gap problem [2] and the other is that the measured gap is optical
and hence a different gap than the indirect one was measured.

50

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 
   

 

 

 

 

            
       
 
   

 

 

 

 

 

 

 

 

 

 

       

(a) \ V (b) . ,1" i
. \
10 10.
A A N
> I/ > ' ’\\
3 e .
a I :13 - /
L000 E1: Ln00 Br
1 f \x
N 4% \N 57"/\
> *6 vv
-1.0. YIA -10 e- .
FX UZ R PX UZ R V 1“ z
.. n: (d) . / - \y
tal
a. / -
i» I
1;; /
8
Be 1110.0 is;
1 ~_/
\1 ‘\\’\‘J F/\ \
at \\ I)" QE“ . i
-1.() MIA\:2.-/Mut€\\\ -1.0 5‘;
PX UZ R V F Z M UZ R V r Z

Figure 5.5: Band structure of a—K2Bi83e13 at Vopt (a)LDA without SOI, (b)LDA with
SCI, (c)GGA without SCI, and (d)GGA with SCI.

51

Table 5.1: Energy band gap values in eV corresponding to experimental volume Va“, and
optimal volume Vapt for a—K23188613.

 

 

 

 

 

 

LDA GGA Experiment [61]
Verp I vapt Vecrp I Vopt
without SC] 0.71 0.71 0.82 0.65 0.76
with 801 0.37 0.38 0.47 0.46

 

 

 

 

 

 

 

 

Thermoelectric properties of this material are influenced by anisotropic effective masses
(the R factor), which can lead to a high ZT value. [50, 51] The effective masses for both
LDA and GGA at Vopt were obtained by calculating the values of energy close to the
conduction band minimum (CBM) and the valence band maximum (VBM) while moving
from the extremum points along the three orthogonal directions. We calculated energies
for I0 values up to: 0.0625101,” 0.0625k2y, and 0.03125k3z where klx, ka and kgz are the
magnitudes of the projections of the reciprocal lattice vectors I01, I02, and I53 along the x, y
and 2 directions respectively. The energy values were fitted to a quadratic polynomial. In

eneral e~ can be ex anded about an extremum oint as :
8 k P P

 

2m . .
72551: = Zine/n; w = m, z (5.1)
232'
Air“ : me , (5.2)
mij

where )1,- j are the effective mass parameters (inverse of Aij is the effective mass tensor) and
me is the free electron mass. In the present case the angles a and 7 are close to 90° whereas
the angle B differs appreciably from 90°. Thus we can assume Amy and AW to be small and
drop them from the expansion. The four remaining Aij’s are given in Table 5.2. As can be

52

Table 5.2: Effective mass parameters at Vow for a—KgBigselg

 

 

 

 

 

 

 

 

LDA GGA

VBM LCBM VBM | CBM

A... 1.78 6.16 1.31 5.57
AW 0.98 1.15 0.99 1.16
12, 1.29 9.29 1.15 8.56
An 0.71 0.13 0.84 0.25

 

 

 

 

 

 

 

seen in Table 5.2, the effective mass parameters of the VBM are fairly isotropic whereas
those of the CBM are anisotropic with the smallest effective mass along the z-direction
(c-axis). The differences in the effective mass parameter values within LDA/GGA are less
than 36% and as expected (the effective mass tensor depend on the wave functions, whereas
the energy in density functional theory depend on the charge density) these values are larger

than the differences in the energy gap values which is ~21% at Vopt.

Orbital character analysis reveals that Sel—7 and BM atoms (see Fig. 5.1 for the num-
bering of different atoms in the unit cell) contribute to the highest valence band (HVB).
This suggests a nearly 3-dimensional hole transport (in GGA which is supposed to be
better, Aij vary between 0.84-1.31). On the other hand, Sb28e3-type fragments do not
contribute strongly to the lowest conduction band (LCB), which consists primarily of
Bil-3 p orbitals. This leads to a nearly 2—dimensional charge transport of the electrons
confined to the BigTeg- and CdIz-type layers (large variations in Aij from 0.25-8.56),
suggesting better thermoelectric properties for the electron-doped systems through large
R=[/\Lug/(Am)tyy)]1/2 factor defined in Chapter 1. [50]

53

5.3.2 ,8 - K2Bi88813

The unit cell parameters of ,l'fi—KgBigselg (monoclinic with space group P 21/m with 46
atoms/unit cell ) are: a=33.055 a.u., b=34.886 a.u., c=7.946 a.u., and y=90.49°. [9] The
corresponding Brillouin zone is shown in Fig. 5.4. A proper electronic structure calcu-
lation for this compound should in principle incorporate the Bi8/K3 and Kl/Bi9 mixed
occupancy. Ab initio electronic structure calculations for disordered systems are still un-
der development stage (KKR-CPA). [63] Therefore we have chosen two different ordered
configurations with extreme occupancy of the Bi8/K3 and K1/Bi9 sites: Configuration I
contains Bi8 atom at the Bi8/K3 site and K1 atom at the K1/Bi9 site, whereas Configura-

tion II contains K3 atom at the Bi8/K3 site and Bi9 atom at the Kl/Bi9 site.

Both before and after the inclusion of the SOI, Configuration I yields a semi-metal with
several very flat bands along the FY and ABI‘ directions of the Brillouin zone, (see Fig. 5.6)
i.e. in the plane perpendicular to the z-axis also called the needle axis. Single crystal
conductivity measurements show either a semi-metallic or a narrow-gap semi-conducting
behavior. [9] Although the observed transport behavior appears to be consistent with band
structure results, the measured optical gap of 0.59eV does not appear to be consistent with
these results. An orbital analysis shows that the top of the "valence band" consists of Se9 p
and Se10 p bands (Fig. 5.6 a, b) while the lowest two conduction bands have Bi8 p and Se4
p hybridized orbital character (Fig. 5.6 c, d). Se9 lies at the edge of the NaCl- fragment
next to the K1 atom occupying Kl/Bi9 site whereas the Se4 lies at the edge of the Bi2Te3-
fragment next the Bi8 atom at the Bi8/K3 site. In this configuration Se9 atom has two
K1 atoms as nearest neighbors whereas Se4 atom has two Bi8 atoms as nearest neighbors.

54

 

 

(b).

 

 

 

 

 

Energy (eV)
Energy (eV)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

"

I ..

4'
()g

 

CZDE ABr z

 

fl
~25} "<

1.0 :. 1.0...
(C) ,e . (d)
.20 e.g.?! .

   

 

 

~59

Energy (eV)
Energy (eV)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

~:. .:-
O ~ ‘ .~ .
O. 0.4}. .O:.:.:.o . '94::
a... o r ...o of fl. 0
'0... III}. ’0'}. 3“”. I. as}.
. o. o o o . o“... '3: ”:4 “*3... . o
39:“.‘3 ”.0 a. I I“... III“...
0 . .0 I... .0 '0‘ I“ E“ I d I":::' .. ..
.0040. I. ‘I '0'...“ I ‘II‘I~I‘.:I.:'
on. ° I 1 ." 'a'“ I. I I ' v o'
I. . M I‘ I 0 ' I. '0
at. . l. o “a. ‘00...
.0 If? I‘ ”2"~$ :I at; ‘4'

 

1.0 1.0 ..,....
rY czne A131“ z W CZDE ABI‘ z

Figure 5.6: Band structure of B —KgBi8Se13-Confi guration I with SCI. Orbital character of
(a) Se9 p, (b) SelO p, (c)Bi8 p, (d) Se4 p. The size of the circles is directly proportional to
the strength of the orbital character.

55

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

  

 

 

 

 

 

 

 

0.0: .0 I‘ I: I. '2'... :5 I" I01 0. '0‘...
(a) . I 3“ I “o, 1" 1:1 ”my“. I 1 2'?“ (b) ,. ..W m‘ I' ..M. Ii. :3:
. :2“. a" a. ‘ o . .0 II“: . . .. ”.0. (J ,0 o 0‘ .‘ 1' . . :2: 1
.. . P40 000‘... ..... 1.4 M...
A I ' I ' . om. .:. I: :“w'f. ,o A . .'. M 2: 3...... .° .4
8 00 P.” flag... 0 0- .. . 3 M.. . u .0 .
>‘ o 0" “on I 2 f... o >‘ 0 A K I ’ . f“ o
o .' I) c f 0 I o .. I) o I Po .
OD .’ 0 . ’ .. g o .0 .1 a“... ’4
s 000 .- . . It. .7 . EF 0) 0.0-1 ' ”WEE:
: r 1‘3. .. I " a . I o :0?
LL] m I‘L- .1. J
g 0. 2'» _ .: ..0... .. D. o o.
.0 32:31! . , "9.2:. I.
i ' ~ I“ . we...“
, .. i' .:.... 5.. l. '}.::'o
l ”:2“. .1 ”:0. ~ D. :3.
5 1' ,r - '«4 .
-1 0 B - . ..

. -- .0 u ‘ i --
FY FY CZDE ABI" Z
Figure 5.7: Band structure of B—KgBigSelg-Configuration I with 801 for partially opti-
mized calculations. Orbital character of (a) Se9 p, (b) Se4 p. The size of the circles is
directly proportional to the strength of the orbital character.
The hybridization of Se4 p and Bi p states lowers the energy of Se4 p states, whereas the
hybridization of Se9 p states with Bi p states is reduced since Se9 atom has two K1 atoms

as nearest neighbors. As a result the Se9 p orbitals are not very well stabilized in energy

and they float to the Fermi energy to give a semi-metallic behavior.

Since the experimental positions of the atoms at the mixed sites are some average po-
sitions, we performed volume optimization and internal relaxations (carried out only for
the configuration I) of the atoms at mixed sites and their nearest neighbors (Se4 and Se9
atoms) in order to see the impact of the relaxation effects on the band structure results.
Vopt for GGA calculations correspond to ~4% increase from Val-p. The internal opti-
mization calculation increases the K1-Se9 atomic distance to 3.21A from 2.85A whereas
the Bi8-Se4 atomic distance of 2.88A remains almost unchanged. In spite of the above

56

changes, the optimized calculations (Fig. 5.7 a, b) show the same semi-metallic behavior
with band structure results very similar to those of the unoptimized calculation (Fig. 5.6
a, d). Therefore, our calculations for other configurations (discussed next) did not include

internal relaxations.

In the case of Configuration 11, before the inclusion of 801, there is a direct gap of
0.38eV which appears promising vis — (‘1 — vis experiment. But after inclusion of 801,
the gap disappears and the band structure looks very similar to that of the Configuration
1. Orbital character analysis shows that in this case the top of the "valence band has" Se4
p orbital character (instead of Se9 in Configuration 1) (Fig. 5.8 a), while the lowest two
conduction bands have Bi9 p and Se9 p (instead of Bi8 p and Se4 p) hybridized character
(Fig. 5.8 b). In this configuration Se4 atom has two K3 atoms as nearest neighbors. As
a result Se4 p orbitals now float up to the Fermi energy giving rise to a semi-metallic

character.

Since both the configurations with extreme occupancy of Bi and K atoms at the mixed
sites (Configuration I and 11) show semi-metallic behavior, the observed semiconductor
behavior of ,B-KgBigsem has to originate from the mixed site occupancy. In order to in-
corporate the mixed occupancy we have chosen a lxlx2 supercell (92 atoms/cell) with an
alternative occupancy of K and Bi atoms at the mixed sites. In this ordered superlattice
model both Se9 and Se4 atoms have one K and one Bi atoms as nearest neighbors. The
LDA/GGA band structure results show that the system is an indirect gap semiconductor
(Fig. 5.9). The effect of 801 is to decrease the energy band gap as has been seen in many
Bi, Te, and Se narrow band gap semiconductors. [50, 51] Vopt for LDA calculations cor-

57

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(3 (b) [N
9 9 '-.
e e a
>» >~ .—-0\
DD 00
E . EF E EF
m m
I
-l 0 ~ -1.0
DE ABF Z DE ABF Z

 

 

Figure 5.8: Band structure of B—KgBigSem-Configuration II with 801. Orbital character
of (a) Se4 p, (b) Bi9 p.

respond to ~4% decrease from Vamp, whereas Vopt for GGA calculations correspond to
~5% increase from V617,. The VBM occurs at the A point (located at [0.5, 0.5, 0] in frac-
tions of the primitive reciprocal lattice vector lengths) whereas the CBM is located at [0.4,
0.55, O] for LDA and at [0.375, 0.575, 0] for GGA inside the Brillouin zone. The values of
the energy band gap are given in Table 5.3 which range from 0.56—0.66eV without SCI and
0.32-0.44eV with $01. As before the effect of volume relaxation on the band gap is quite

small. The experimentally measured optical gap is 0.59eV. [9]

Orbital character analysis reveals that the states near the bottom of the CBM consist of
Bi9 and Bi8 p orbitals strongly hybridized with the Sc and Bi p orbitals of atoms close to
the mixed sites whereas the states near the VBM have mostly hybridized 8610 and BB p
orbital character. The states near the VBM have small contribution from Se9 p orbitals and

58

 

 

 

 

 

 

 

 

 

‘ V, ,’4

4.. ‘ A.
Imffiwr

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_ ...._/..:_s.
, 5%,? Y

 

   

 

 

 

 

V.

2...... ...., ... r

 

 

 

 

 

 

 

 

 

 

 

FY CZDE ABF Z

 

 

 

0

-1.0

Figure 5.9: Band structure of B—KgBigSe13-Supercell at Vopt (a)LDA without SOI,

(b)LDA with SCI, (c)GGA without SCI, and (d)GGA with SCI.

59

Table 5.3: Energy band gap values in eV corresponding to experimental volume Vexp and
optimal volume Vopt for B —K2Bi88e13-Supercell.

 

 

 

 

 

LDA GGA Experiment [9]
Vexp l Vopt Verp l Vopt
without 801 0.58 0.56 0.63 0.66 0.59
with 801 0.33 0.32 0.39 0.41

 

 

 

 

 

 

 

 

 

no contribution from Se4 p orbitals since the latter states are better stabilized in energy. Bi
atoms at mixed sites stabilize the p orbitals of Se4 and Se9 atoms by lowering their energy.
Therefore, this alternative K/Bi order at mixed sites along the c-axis (needle axis) is crucial

for the gap formation.

In order to better understand the nature of the states near the CBM and VBM we plot
the charge density in planes parallel to the z-axis containing atoms which have the highest
contributions to these states. The charge densities were calculated for the A point in the
Brillouin zone within a small energy window of 0.2eV around CBM and VBM. The charge
density for the CBM is given in Fig. 5.10 (a) and (b) whereas that for the VBM is given
in Fig. 5.10 (c) and (d). In the case of CBM the charge is distributed around Bi and Se
atoms, away from the K atoms, suggesting that the charge transport occurs mostly along
the c-axis through Bi-Se framework since in the plane perpendicular to the z-axis charge
density around the K atoms is extremely small. The plot of charge density through the same
plane as in Fig. 5.10 (b) for the VBM states (Fig. 5.10 c) shows that the atoms included
in this plane have very small contributions to these states. The states near the VBM are
antibonding, mostly localized around SelO, Bi3, Se3, and Se9 atoms (Fig. 5.10 d). The
charge density plots reveal that the electron and the hole transport regions are separated in

60

 

 

 

 

Figure 5.10: Charge density of B—KgBigSe13-Supercell in planes parallel to the c-axis
corresponding to (a), (b) CBM states and ( c), (d ) VBM states. The charge density is
represented by closed lines from low charge density regions to high charge density regions

4

in steps of 5x10‘ electrons/A3.

space.

The effective masses associated with the two LDA/GGA CBM at [0.4,0.55,0]l[0.375.
0.575, 0] (called CBMl) and at the A point (called CBM2) in the Brillouin zone together
with the effective masses of VBM at the A point were calculated by dropping the Amy
term in Eq. 5.1, since the angle 7 is very close to 90°. The results are given in Table 5.4.
As seen in the a-phase, the LDA/GGA differences of the effective mass parameters are
larger than the differences in the energy band gap (~28%) at Vopt. The effective mass
calculations show strong anisotropy in both the hole and electron effective masses with the
smallest effective mass along the needle axis. As has been discussed before the value of
27‘ increases with the dimensionless parameter B defined in Chapter 1 and the B factor is
directly proportional to the fraction R=[Azz/()\mAW)11/2 of the effective mass parameters.

61

Table 5.4: Effective mass parameters at Vopt for fi—KgBigSelg-Supercell

 

 

 

 

 

 

 

 

 

 

 

 

LDA GGA
VBW CBM] [CBM2 VBM LCBMI |CBM2
Am 3.66 1.63 5.35 2.75 2.22 9.3
M 0.73 1.08 0.62 0.33 0.9 2.31
A22 8.77 5.15 11.86 7.62 6.01 12.37
(WP/2 1.81 1.71 1.89 2.9 1.74 0.76

 

 

 

 

 

The values of R are also shown in Table 5.4. These values are greater than unity (except
CBM2 within GGA) suggesting that fi—KgBigSem has a great potential for both p and
n-type thermoelectric. The high effective mass anisotropy in the vicinity of the VBM and

CBM explains the strong charge transport anisotropy found in this system. [45]

5.4 Summary

Electronic structure calculations show that the a—KzBigSelg is an indirect band gap semi-
conductor with a LDA/GGA band gap of 0.38eV/0.46eV at the optimized volume Vopt.
Direct band gaps of 0.65eV at the V point and 0.75eV at the X point in the Brillouin zone
are found, in good agreement with the observed optical gap of 0.76eV. [61] On the other
hand in B —K2Bi88e13, the atoms at the mixed sites are found to be very important in de-
termining the electronic properties of this material. When the mixed sites are treated as
fully occupied by K or Bi atoms (Configurations I and II) the system is a semimetal. In-
corporation of mixed occupancy (through a supercell structure) results in an indirect gap
semiconductor with a LDA/GGA band gap of 0.32eV/0.4leV at Vopt. Although the in-
direct gap (relevant in transport) has not been measured experimentally, we find two flat

62

bands along the FY direction separated by ~0.49eV for LDA and ~0.53eV for GGA. This

is consistent with the observed direct gap of 0.59eV (optical). [9]

It is of interest to discuss the effect of disorder between K and Bi atoms at the mixed
sites on the electronic band structure of 13—K2Bi88e13. It is possible, that in the real system
there exist regions close to Configuration I and II, which may create in-gap states. These
states will then contribute to charge transport raising the value of electrical conductivity.
This may be a plausible explanation for the higher observed value of electrical conductivity
(0:250 S/cm) in 13—K2Bi88e13 as compared to the low value of electrical conductivity

(0:2 S/cm) in a—KgBigSelg.

The ability of the mixed K/Bi sites to achieve various degrees of ordering has been
deduced from samples of B—KgBigSelg, with different preparation and cooling history.
Rapidly cooled samples exhibit a high enough K/Bi disorder to mask the presence of the
band-gap (due to the in-gap states) and they also exhibit very high room temperature elec-
trical conductivity (more than 1200 S/cm). Slowly cooled samples on the other hand (or
annealing the rapidly cooled samples) have considerably fewer defects associated with the
K/Bi disorder that result in readily observable band-gap of 0.59eV (spectroscopic) and sig-
nificantly reduced electrical conductivity. Therefore a full exploration of the thermoelectric
properties of this system will have to involve a careful control of the degree of disorder in
the mixed K/Bi sites coupled with judicious doping. In addition, our calculations suggest
that p—type samples could also be very interesting in that regard since they give a very high
anisotropy factor R=[/\zz/(/\m/\yy)]1/ 2 associated with the valence band maximum. To
date, p-type doping in fi—KgBigSelg has not been observed but the electronic structure

63

results give plausible justification to pursue ways to achieve it.

Thermoelectric properties of 13—K2Bi88e13 are also enhanced by the existence of some
degree of K/Bi disorder which reduces the lattice thermal conductivity mph. A value less
than 1.28 W/mK for the total thermal conductivity has been seen [9] compared to 1.3]
W/mK for BigTe3_,,,SeJr alloys. The strong anisotropy in the electron effective mass com-
bined with the low value of rap}, make [3—K2Bi88e13 to be an excellent candidate for room

and high temperature thermoelectric materials.

Chapter 6

Electronic Structure of AngmeTe2+m
(LAST-m), Complex Quaternary

Systems

6.1 General Introduction

A novel class of quaternary compounds AngmeTe2+m denoted as LAST-m have at-
tracted considerable attention during last two years because of their large figure of merit
ZT (ZT~l .2 for LAST-12 and ~2.2 for LAST-18 at 800K). [30] These compounds form
in the rock-salt structure (Fm3m symmetry) similar to the binary PbTe, where Ag and Sb
occupy Pb sites. One expects that by replacing two divalent Pb atoms by a monovalent Ag
atom and a trivalent Sb atom, one will maintain the charge compensation and the resulting
system will be a narrow band gap semiconductor. By controlling the Ag and Sb concen-

65

 

Figure 6.1: Electron diffraction pattern showing a lowering of the crystal symmetry from
cubic to orthorhombic. They correspond to (a and c) [1 1 116, (b) [I 1 216 and (d) [231k zone
axis in face center cubic (fcc) symmetry (ac=6.4A). Superstructure spots can be indexed in
orthorhombic symmetry according to an ~9.1A, b0 ~15.8A and c0 ~22.3A unit cell. The
corresponding zone axis are (a and c) [00110, (b) [01210 and (d) [lOI]o. Indexations are
given for both symmetries.

tration and their spatial ordering one can manipulate the bond strength and consequently
their transport properties. Furthermore, having high symmetry structures, LAST-m systems
are expected to maximize the degeneracy of the bands 7 and by replacing Pb with Ag(Sb)

atoms one expects to reduce the lattice thermal conductivity mph, quantities which enter in

the B factor defined in Chapter 1.

There are clear experimental evidences from single crystal X-ray diffraction (Fig. 6.1)

66

IIIOIIOOo

.
vo'oooooiov'

~000900000

I
.
C
0
.
.
.
O
.
.
C
.
.
c
. I

booooooooui».

1
l

09.60.“...

.i-
'rll
' §
1
9

l
- I

.i.§'.‘--.‘

.‘ ‘Q’

O n '

 

Figure 6.2: High resolution electron microscopy image from a LAST-l8 sample (a), and

6
mm.
.mk
.ww
W...“
mm
mm
ms
bn
Mm
A m
eh
hd
t]
)
mm
M[
am

t
X f
.mo
am
mo m
Tm
%H.
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Om
C p
mm
m
mm
fm
mm
om
.mm
me

e
t
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um
swam
tet
m m a
fuck
eod
.htm.m

and high resolution electron microscopy (Fig. 6.2) that LAST-m systems are not simple
solid solutions between AngTe2 and PbTe phases. Rather, they exhibit microscopic inho-
mogeneities rich in Ag-Sb embedded into a PbTe matrix. These systems are example where
quantum confinement (QC) concept can be achieved in bulk by controlling the formation
of Ag-Sb microstructural arrangements. In addition, it is not correct to view the Ag and Sb
atoms as dopants because they are present in large stoichiometric proportions ( l Ag(Sb)
atom in 12 or 18 Pb atoms).

Given the close structural relationship of AngmeTe2+m to PbTe, a fundamental
question is how the PbTe electronic structure gets modified by extensive substitution with
Ag and Sb atoms and their microstructural arrangements. In order to address this ques-
tion, I have carried out careful and systematic electronic structure calculations in AngTe2
and PbTe the two end phases, Ag(Sb) single impurity in PbTe, Ag-Sb pair in PbTe, Ag-Sb
chain, plain, and cluster microstructural arrangements in PbTe. The primary focus is to
isolate some of the generic electronic structure features associated with the replacement of

Pb by Ag and Sb and see how they influence the transport properties.

6.2 Electronic Structure of AngTe2

AngTe2 forms in the rock-salt structure (unit cell parameter a=6.06A) similar to PbTe.
The nature of chemical bonding in AngTe2 and PbTe is mixed ionic-covalent. With de-
creasing ionicity one expects the rock-salt structure to become unstable toward a rhombo-
hedral phase. In fact, a cubic to rhombohedral phase transition occurs at room temperature
with ordering of Ag and Bi in AgBng and AgBiSez. For AngSe2 and AngT62 early

68

experimental studies indicated that the disordered cubic structures persisted even at room
temperature. [21] Recent careful experiments suggest long range complex ordering of Ag
and Sb. [72] But the exact crystal structure is not known. In the absence of a detailed struc-
tural information from experiment we have taken several ordered structures and compared

their energies using ab initio DFT.

In the simple ionic model AngTe2 is charge balanced Ag+Sb+3Te2‘2. For the
cations, the fluctuating charge from the average +2 charge (Pb+2) is -l for Ag and +1
for Sb. We use the magnetic analogy (Ang and Sb=.L) for defining different ordered struc-
tures. These structures are antiferromagnetic (AF) type I, H, IIIA and IIIB (Fig. 6.3A,B,C,
and D). In addition to these “antiferromagnetic” orders, other structures were chosen and
named type l-AngTe2 (Fig. 6.3E), type 2-AngTe2 (Fig. 6.3F). Type 2-AngTe2 consists
of alternating layers of Ag and Sb perpendicular to the [111] cubic direction. Since the
type 2-AngTe2 can be reduced to a smaller hexagonal unit cell with the c-axis parallel to
the [111] cubic direction (Fig. 6.30), the latter was used in this calculation. Also models in
a 2x2x1 hexagonal unit cell consisting of mixed Ag/Sb layers perpendicular to the c-axis

have been considered and are denoted as type 3-AngTe2 (Fig. 6.311).

Let’s discuss the general features of the density of states (DOS) to see whether these
compounds are narrow band gap semiconductors, semimetals or metals. DOS for type
I AF structure shows semimetallic character with a pseudogap near the Fermi energy
(Fig. 6.4A). The Ag d orbitals are located below the Fermi energy and they are fully occu-
pied (Fig. 6.4B). In the (-5.0eV, 5.0eV) energy range the Ag s orbitals hybridize with Te
p orbitals (Fig. 6.4C) . This suggests that Ag atoms are in an oxidation state which is less

69

 

 

 

Figure 6.3: Unit cell lattices of ordered structures: (A) type I, (B) type II, (C) type IIIA,
and (D) type IIIB AF order; (E) type l-AngTeg, (F) type 2-AngTe2, (G) type 2-AngTe2
in hexagonal symmetry, and (H) type 3-AngTe2. For the sake of clarity Te atoms are not
included.

70

(D
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J

.5

 

 

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(AF T total DO§_ .(BF - partial DOS: A91 d_.

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—L.-L
O

 

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-5 o 5 1b 30 -5
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80.8 .
00.6 t
0.4 .
0.2 u

 

 

 

 

 

 

   

16

<16" -5 o 5
Energy [eV]

-5 o ' "
Energy [eV]5

Figure 6.4: Density of states (DOS) for type I AF model structure.

than +1. The Te and Sb s states are very well stabilized in energy between -12eV and -8eV.
These states do not mix with the p states of Te and Sb atoms (Fig. 6.4C,D). The Te and Sb
p states, which lie in the range -5eV and OeV hybridize with each other, indicating a strong
covalent interaction between Te and Sb atoms. The overlap of the p states of Te and Sb

gives a semimetallic band structure.

The DOS results for all other ordered structures show similar semimetallic behavior
with low DOS near the Fermi energy. This is a result of the hybridization between the p
states of Te and Sb atoms, which are very close in energy. In Fig. 6.5 we show only the to-
tal DOS of these different structures. The rapid increase in the DOS near the Fermi energy
(both above and below) suggests that these systems will show large thermopower when the

71

chemical potential moves away from the minimum with doping.This observation is con-
sistent with the large room temperature experimental value of thermopower S=26611VIK
measured by Noda er. al. [64] Noda et. al. also measured electrical conductivity 0 and
thermal conductivity K3. These values are 147 S/cm and 1.07 W/mK respectively. The
room temperature ZT for AngTe2 is ~0.29. In comparison, the best room temperature
thermoelectric are BigTe3 alloy (ZT=1.14) and (BigTe3)7,,(Sb2Te3)n superlattice systems
(ZT=~1.4 and ~2.4).

Even if all the different structures are semimetals with a pseudogap at the chemical po-
tential it is important to find out which structure has the lowest energy. To compare the total
energy Etot of different ordered structures, volume optimization and force minimization up
to 1mRy/a.u were performed. The total energies per formula unit f.u. (f.u.=AngTe2) are
given in Table 6.1.

The lowest energy E0 correspond to type l-AngTe2 order (see Fig. 6.5D). Etot of all
the ordered structures studied are within the 0-0.2eV/f.u. energy range. These energies,
except for the type l-AngTeg, are very close (0.131-0.203eV). This suggest that AngTe2

can develop disorder without much difficulty at finite temperatures.

Table 6.1: Total energies of AngTeg ordered model structures.
[ Type H atoms/cell I Eta/f.u. (Ry) | Etot'EO (eV)]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I AF 8 -50789.51675 0.161

11 AF 64 -50789.51637 0.166

IIIA AF 16 -50789.519 0.131

IIIB AF 16 -50789.51658 0.164
l-AngTe2 32 E0=-50789.5286 0.0

2-AngTe2 l2 -50789.5137 0.203

3-AngTe2 48 -50789.5178 0.147

 

 

72

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

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Figure 6.5: Total DOS of:(A) type II, (B) type IIIA, and (C) type IIIB AF structure; (D)
type l-AngTeg, (E) type 2-AngTe2, and (F) type 3-AngTe2.

73

Experimental measurements in AngTe2 give rather intriguing results. Electrical con-
ductivity and Hall coefficient measurements in single crystal AngTeg by Gochev er.
al. [22] in the temperature range 300-6OOK show a temperature dependence not typical
for semiconductors. There is a minimum in the conductivity at T~500K. Fitting the data to
an exponential in the range 500-600K gives an energy gap Eg~0.2eV. Energy gaps rang-
ing between 0.27-0.35eV have been obtained in earlier studies. Later measurements by
Elsayed et. al. [17] in the range 600-7OOK gives Eg~0.218eV for the solid state. Our the-
oretical calculations show that the different structural models we have investigated give a
semimetal with a pseudogap near the chemical potential. Furthermore, the near proximity
in energy of the different structures suggest there will be structural defects. These will give
rise to localized states near the chemical potential which will control transport properties,
particularly at low T. Careful low T (T<3OOK) experiments of a and S are needed to sort
things out. Whether one can understand these unusual electrical properties of AngTe2

within a pseudogap picture needs further investigations.

6.3 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. 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

74

 

 

 

 

 

 

 

 

 

 

 

 

 

>-

1‘1:
"y

Figure 6.6: Face centred cubic (fcc) Brillowin zone of PbTe.

these compounds. [67, 60, 97, 99, 1, 47] Wei and Zunger [99] have recently carried out
extensive electronic structure calculations using local density approximation (LDA) [29]
as implemented through the linearized augmented plane-wave (LAPW) method. [85] They
have argued that the above anomalous features can be ascribed to the occurrence of the Pb
s-band below the top of the valence band, setting up coupling and level repulsion at the
L point. Albanesi er al [I] 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) [70]. They found a large dielectric constant
for PbTe in agreement with experiment. [83, 89] I have carried out extensive all electron
LAPW calculations within both LDA and GGA to confirm the findings of Wei and Zunger,
to analyze the covalency between Pb and Te (by calculating the effective ionic charge)
and to understand the bonding and antibonding nature of the electronic states near the

fundamental gap.

The band structure results show that PbTe is a direct band gap semiconductor with a
small gap value of ~0.09eV at the experimental volume Vexp. The volume optimization

75

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Figure 6.7: Band structure of PbTe at V3“, along different directions in the fcc Brillouin

zone. Orbital character of : (a)

Pb p and (b)Te p. The size of the circles is proportional

with the orbital character. (c)Band structure of PbTe at Vopt.

 

0.4 r
0.2 -
0 ,
-0.2 -
-0.4 -
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-0.8 -
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35 o '5
ENERGY (eV)

Figure 6.8: -COHP as a function of energy for PbTe without including spin-orbit interaction

(SCI).

76

increases the gap to ~O.l9eV (Fig. 6.7c). The GGA(LDA) optimal volume vopt corre—
spond to 5% increase(decrease) in Verp. The gap occurs at the L point in the fcc Brillouin
zone (Fig. 6.6). Orbital analysis shows that Te and Pb p states hybridize suggesting co-
valent interaction between Pb and Te atoms(Fig. 6.7a,b). In order to find the ionicity of
Pb and Te atoms we have computed the electronic charge around each atom according to
Bader’s “Atoms in Molecules” theory. [3] According to this theory the electronic charge is
calculated inside a volume centered around each atom. For each atom, the charge densities
along different directions are calculated in order to find the points with minimum charge
density. Then, the volume around each atom is found by connecting the points with min-

‘0'64, which is much

imum charge density. The results show an ionic state of Pb+0'64Te
smaller than that of the pure ionic model Pb+2Te‘2. Therefore in PbTe covalent interac-
tions play an important role. The reduction in formal charges from 2(-2) to O.64(-O.64) is
consistent with the large dielectric constant of PbTe. A crystal orbital Hamilton popula-
tion (COHP) analysis has been found to be a bond—detecting tool for solids and molecules.
COHP partitions the band structure energy (in terms of the orbital pair contributions) into
bonding, nonbonding and antibonding energy regions within a specified energy range. [15]
In Fig. 6.8, -COHP is plotted as a function of energy. Positive values of -COHP describe
bonding energy regions whereas negative values describe antibonding energy regions. As
can be seen in Fig. 6.8 in the (~1,0)eV energy range (states near the top of the valence
band), the electrons occupy antibonding states. Both the valence band maximum (VBM)

and conduction band minimum (CBM) states at the L point are antibonding in nature (Note

that in Fiq. 6.8, we have arbitrarily chosen Fermi energy E F to be at the top of the valence

77

band).

6.4 Effect of Ag-Sb Microstructural Arrangements on

Electronic Structure of LAST-m systems

6.4.1 Defects in PbTe

Defects (shallow and deep) in semiconductors are known to profoundly alter their elec-
tronic structure near the band gap and control their transport properties. Unlike shallow
impurity levels, which are produced by long ranged Coulomb potential, deep levels are
produced by the short-range atomic-like defect potential. [69] It is believed that shallow
impurities primarily control the magnitude and type of conductivity (by controlling the
type of carriers, electrons or holes, and their concentration) whereas deep defect levels
primarily control the charge-carrier life-time. The detailed understanding of shallow de-
fects in common semiconductors traces back to the classic work of Kohn and Luttinger
and can be regarded as basically understood. [41, 68] However the problem of defects in
narrow-gap semiconductors, particularly the physics underlying deep defect states is far
from understood. LAST-m systems are particularly interesting because here both shallow
defects caused by long-range Coulombic potential, deep defects formed by short range
interactions and mixture of these may coexist.

The theory of deep defects in semiconductors in general and narrow band-gap semi-
conductors in particular has a long history. For substitutional defects in PbTe Lent er.
al. [53] presented a simple chemical theory of s- and p- bonded substitutional impurities.

78

This theory naturally gave deep defect states near the fundamental band gap and also pre-
dicted resonant levels further away from the gap. The central idea of this theory is that
due to large static dielectric constant of (~103) [l], the Coulombic forces are screened out
and local bonding considerations dominate the impurity state formation. The resulting de-
fect states are referred to as deep defect states. Their location and character is determined
mainly by the short range central cell defect potential. Because the fundamental band gaps
of IV—VI semiconductors are so small (~02 eV), most of these defect energy levels can
lie any where vis-(‘z-vis the fundamental band gap. A careful ab initio electronic structure

calculation is therefore needed to understand these defect states quantitatively.

To understand the basic physics underlying these deep defect states we present a simple
chemical argument given by Lent et. at. If we start from pure PbTe and consider Cs-row
impurities at the Pb site we can think of a Pb ”defect” at the Pb-site which has an s-level
in the valence band and p-levels in the conduction band. The conduction band is primarily
Pb p-like and the Pb 5 states lie in the valence band. Making the defects more electropos-
itive i.e. going from column-IV to Column-0, drives these defect levels up in energy until
eventually the s-level crosses into the conduction band (between Column-O and Column-I)
to form the deep levels of the vacancy in the conduction band far above the conduction
band minimum (CBM). Similarly by making the defects more electronegative (going from
Column-IV to Column-VH) drives the energy levels down, so that the p—levels ultimately
drop into the valence band. Lent et. al.’s calculations indicated that the defect p-levels
cross the gap to the right of Column-VIII. They showed that Bi (trivalent) impurity gave
a resonant p-like state (more than 0.5 eV above the CBM) which was partially occupied.

79

In contrast Cs impurities gave s-like deep defect states near the top of the valence band,
which were partially occupied. In the following subsections we see how ab initio calcula-
tions which are self-consistent and incorporate the screening effects properly, quantitatively

justify some of these empirical findings.

6.4.2 Models for Ag-Sb Microstructural Arrangement

First, calculations for isolated Ag and Sb impurity atoms in the PbTe lattice were performed
in order to obtain a clear picture of their individual role in modifying the electronic struc-
ture of PbTe (Fig. 6.9a). These calculations were carried out using supercell models and
the size of the supercell was chosen to minimize the impurity-impurity coupling within
available computing resources. Then both Ag and Sb were introduced to simulate stoi-
chiometries relevant to those of LAST-m compounds. Given that the exact crystal structure
is not known, several plausible microstructural models were examined, all of which in-
volved long range ordering of the atoms. In one model Ag and Sb atoms were placed in
monolayers (Fig. 6.9b,c). In another model Ag and Sb atoms were placed along infinite
chains running parallel to a crystallographic unit cell axis (e.g. c-axis)(Fig. 6.9d). In yet
a third arrangement, the atoms were placed in the center of a 3x3x3 supercell to create a
“AngTe2 nanodot” embedded in a PbTe matrix (Fig. 6.9e). The chain and cluster models
are in qualitative agreement with experimental transmission electron microscopy (TEM)
observations of Ag-Sb ordering. [30] Although these arrangements capture much of the
crystal physics in these materials, there are many more that could be considered which
nevertheless should result in similar general conclusions.

80

 
      
  
    
 

  
   

     
 
    
       

      
         

  

  
 

       
  
 

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33939759993: ‘. ’
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Figure 6.9: Unit cell models for (a) single Ag impurity atom in Ang31Te32, (b) Ag-Sb
layer perpendicular to [001] direction in AngPblgTego, (c)A Ag-Sb layer perpendicular
to the fcc [111] direction in AngPblOTelg, (d) Ag-Sb chain parallel to the [001] direction
in AngPb30Te32, and (e) Ag-Sb cluster in AngPblsTelg. For the reason of clarity we
show only Pb fcc lattices with Pb in gray, Ag in white, and Sb in black colors.

To model the isolated Ag(Sb) and Ag-Sb pair impurities, 2x2x2 supercells with 64
atoms have been constructed. For the isolated case, Ag(Sb) were chosen at the origin of the
supercell with a separation of two lattice constants (12.924A) between the Ag(Sb) atoms
(Fig. 6.9a). For the Ag-Sb pair, two arrangements have been considered (not shown in
Fig. 6.9), one where the Ag and Sb are far apart (Sb at the origin and Ag at the center of
the supercell) with a separation distance of ~11.19A and the other where the Ag and Sb
are as close as possible (Sb at the origin and Ag at the next nearest neighbor site of Sb)

81

with a separation distance of ~4.57A. For the structure where Ag-Sb layers are separated
by several Pb layers two cases were considered, one where the Ag-Sb layer is normal to
the [001] direction in a lxle supercell (4O atoms/cell) with the Ag-Sb layer located in
the 2:0 plane (Fig. 6.9b) and the other where the Ag-Sb layer is normal to the [11]] di-
rection (Fig. 6.9e). Since the face centered cubic (fcc) unit cell can be viewed along the
[1 ll] direction as a hexagonal unit cell, for the second arrangement a 2x2x2 hexagonal
supercell (48 atoms/cell) was used with the Ag-Sb layer perpendicular to the c-axis which
is the [l l 1] direction of the fcc unit cell. To model the chains a 2x2x2 supercell was used,
where the Ag-Sb chains are oriented parallel to the [001] direction (Fig. 6.9d) and sepa-
rated by 12.924A. Finally, for the “AngTe2” clusters a 3x3x3 supercell (216 atoms/cell)
has been constructed, where the cluster consists of six Ag-Sb pairs located at the center
of the supercell with a minimum separation distance between two clusters of ~12.924A

(Fig. 6.9e).

6.4.3 Electronic Structure of Ag-Sb Microstructural Arrangements

The total DOS for isolated Ag atoms is shown in Fig. 6.10a. It can be seen that Ag intro-
duces states near the top of the PbTe valence band (VB). Partial DOS analysis shows that
these states consist mostly of p orbitals of the six nearest neighbor Te (Te2) atoms of Ag
(Fig. 6.10b). These states are resonant with the VB and extend into the PbTe gap region.
On the other hand the isolated Sb impurity introduces resonant states near the bottom of the
PbTe conduction band (CB) (Fig. 6.10c) which extend nearly ~0.75 eV into the CB start-
ing from the CB bottom. It also dramatically increases DOS near the CB bottom. Sb and

82

 

 

90 .(5) ‘ ' ' ' toial 0'03 _'_1 2 (b) partial oos:'Te2 p ('A nn) '__
80 r PbTe total DOS ..... « partial DOS: e4 p __
. - PbTe partial DOS: Te p ..... ‘

  

 

 

 

 

   

 

 

 

 

  
  

-1.5 -1 -0.5 0 0.5 1 1.5 2 -1O -5 O 5 10
Energy [eV] Energy [eV]
90p(c)' ' ' Ttotal Dds—V 2 (d)' partial Do'.:'Te2p('Sb nn)'__
80. PbTe? total DOS ..... 4 pa ial DOS: Te4p._
70. f .

15- PbTe p rtial DOS: Te p ..... ‘

 

4

 

 

 

 

 

A

$2 -1‘.5 51 -o.5 '6' 05 i 1.5 ‘-10 -5 o 5
Energy [eV] Energy [eV]

 

 

to

Figure 6.10: Total DOS of (a) Ag and (c) Sb single impurity atom in PbTe. For comparison
the total DOS of bulk PbTe is shown in dashed line; (b) Partial p DOS of Te2 (Ag nearest
neighbor), Te4 (away from Ag impurity), and Te in bulk PbTe; ((1) Partial p DOS of Te2
(Sb nearest neighbor), Te4 (away from Sb impurity), and Te in bulk PbTe.

its Te nearest neighbors (Te2) atoms have the highest contribution to these resonant states
(Fig. 6.10d). The Sb p states hybridize with Pb p and Te p states in the range (-O.25,2.5)eV.
The interaction through hybridization between the Sb p and Te2 p resonant states with the
Pb p states dramatically change the CB bottom Pb p states, which are pushed down in

energy into the PbTe gap (see discussion of the band structure).

Bader analysis of the electronic charge for the single Ag impurity show ionic states of
Ag+0°23 and Te2-0'56, whereas for Sb single impurity ionic states of Sb+0‘47 and Te2-0°58
for the Ag(Sb) impurity and its nearest neighbors Te atoms. [3] These ionic states are very
different from those assumed in the pure ionic model (Ag+1, Sb+3 and Te-2) suggesting

83

 

 

 

 

 

 

 

 

 

Figure 6.11: VB electronic charge density plots in plane perpendicular to [001] direction in
the (-l,0.25)eV energy range for (a) Ag single impurity, (b) bulk PbTe, and (c) difference
between Ag single impurity and bulk PbTe . CB electronic charge density plots in plane
perpendicular to [001] direction in the (-0.5,2)eV energy range for (e) Sb single impurity,
(b) bulk PbTe, and (c) difference between Sb single impurity and bulk PbTe . The charge
density is represented by closed lines from low charge density regions (0 electrons/A3) to
high charge density regions (0.5 electrons/A3) in steps of 0.01 electrons/A3.
84

a strong covalent interaction between Sb and Te atoms. Also this analysis suggests some
covalent interaction between Ag and Te atoms and screening of the Ag+ ion. We also
have plotted the valence band charge density in the (-l,O.25)eV energy range for the Ag
single impurity, bulk PbTe and the difference between these two electronic charge densities
(Fig 6.11a,b,c). Similarly for the Sb single impurity we have plotted the conduction band
charge density in the (-0.5,2)eV energy range in comparison with the conduction band
charge density of bulk PbTe and the difference between these valence charge densities
(Fig 6.11d,e,f). Ag impurity in PbTe acts as p-type dopant which give the Fermi energy
bellow the top of the VB (Fig. 6.10a), whereas Sb impurity acts as n-type dopant which
gives the Fermi energy above the bottom of the CB (Fig. 6.10c). The energy ranges for
the charge density plots are relative to the Fermi energies of the Ag and Sb impurities
in PbTe. The charge density plots clearly show that the valence(conduction) electronic
charge density associated with the Ag(Sb) single impurity is mostly localized on the Ag(Sb)
impurity atom and its nearest neighbors Te atoms, consistent with the picture that these are
deep defect levels. Both the VB states in the (-1,0.25)eV energy range and CB states in the

(-O.5,2)eV energy range are antibonding states.

Results for isolated Ag-Sb pairs are consistent with the Ag(Sb) single impurity results
in the sense that Ag introduces new states near the top of the VB, whereas Sb introduces
new states near the bottom of the CB (Fig. 6.12). The latter interacts with the Pb p states
near the bottom of the CB, thereby decreasing the PbTe gap. Both cases (Ag-Sb far apart
and Ag-Sb next nearest neighbors, not shown in Fig. 6.9) show semiconducting behavior
with a very small gap and a more rapidly increasing DOS near VB and CB extrema as

85

 

 

90*(a)' ' ' ' 'totaerO'S_—‘ 90~(b)' ‘ F ' 'totaTDOS_<

   

 

 

 

 

 

 

 

80. PbTe total DOS ..... . 80. PbTe total DOS ..... l
70. 70. l
(060. 0360' ,
50 . 50
840. 840.
30. 30.
20. 20.
10. . , 10- . .
O . . J "-. . .401... 4 . O . . . °'- . "
-2 -1.5 -1 -O.5 O 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Energy [eV] Energy [eV]

Figure 6.12: Total DOS of (a) Ag-Sb pair impurity with the Ag-Sb distance of ~1 1.19A,
and (b) Ag-Sb pair impurity with Ag-Sb distance of ~4.57A, in PbTe. For comparison the
total DOS of bulk PbTe is shown in dashed line.

compared to the DOS of PbTe. The specific features of the DOS in the gap region are
very different for these two cases. Total energy comparison for AngPb30Te32 shows that
these two structures are very close in energy, the case when Ag-Sb atoms are far apart has
a slightly lower energy by ~20meV/(unit cell), indicating a covalency induced repulsion

between Ag and Sb atoms.

It is interesting to compare the DOS results for different layered structures of Ag—Sb
(Fig. 6.9b,c). When the Ag-Sb layer is perpendicular to the [001] direction, the states
associated with the Ag-Sb layer completely fill the PbTe gap giving a semimetallic behav-
ior (Fig. 6.13a), whereas the Ag-Sb layer perpendicular to [111] direction shows a semi-
conducting behavior (Fig. 6.13b). This indicates that the electronic structure of LAST-m
systems and consequently their electronic properties are very sensitive to the microstruc-
tures. The Ag-Sb chain model shows semiconducting behaviour (Fig. 6.13c). This chain
model has the same stoichiometry (AngPb30Te32) as the Ag-Sb pair impurity models.
Total energy comparisons show that the chain model has a lower energy by 0.2eV/(unit
cell) than the Ag-Sb pair impurity models, suggesting that Ag-Sb chain orderings along

86

 

 

'total' DOS __ ‘

60 (a). ' F ' 'totaerO'S_ 70r(b)'
- PbTe total DOS ..... .

50. PbTe total DOS ..... . 6O

4 f 50“
(1)40“-
830
20.
10

   

O
l
o

 

 

 

 

o
0“.
A

o . . ."-.z z .
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

AAA,

0 . . . ....... ' 'z
-2 -1.5 -1 -o.5 o 0.5 1 1.5 2

 

 

 

 

Energy [eV] Energy [eV]
901(0) 1 ' ' ’ 'totaT 005.4 300. (d7 ‘ ' imam s_.
80* " PbTe total DOS ----- . 250 PbTe total D S .....

 

60. 200 ,
85o ..- 8
040." 0150

30. 100.

20.

10’ 50.

 

 

 

 

 

 

 

 

L

0 - . . i ...-:' -
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Energy [eV]

 

 

o . . . L - .
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Energy [eV]

Figure 6.13: Total DOS of (a) Ag-Sb layer model perpendicular to [001] direction, (b) Ag-
Sb layer model perpendicular to [111] direction, (c) Ag-Sb chain model, and (d) Ag-Sb
cluster model. The total DOS of bulk PbTe is shown in dashed line.

[001] directions are favorable microstructures. This is consistent with some of the results
of electron crystallographic studies which indicate the presence of Ag-Sb chains in the
crystal. [30] The DOS results for the “AngTe2” cluster model show also semiconductor

behavior (Fig. 6.13d).

Next, we would like to see how the PbTe band structure (egvsg) is affected by the
microstructural arrangements of Ag-Sb pairs. To see these differences we also plotted the
PbTe band structure in the simple cubic (sc) Brillouin zone given in Fig. 6.14a. The band
structures for bulk PbTe, Ag-Sb chain model, Ag-Sb pair impurity with the Ag-Sb distance
of ~11.19A, and Ag-Sb layer model perpendicular to the [11 1] direction are shown in
Fig. 6.15. In sc Brillouin zone the CBM and the VBM of bulk PbTe occur at the I‘ point,

87

 

Figure 6.14: (a) Simple cubic Brillouin zone, (b) hexagonal Brillouin zone.

 

 

     
   
 
   

 

(a) 1.n ., V (b)1 n
A ‘ \/ A
> . J >
3 . 3
is - a
5 0.0 13;: '5
= . ::
LL] LL}

 

 

 

 

—1.o£

A
)e
'<.=

 

 

 

Energy (eV
0
o

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

' R
(e)1.n ..... . . ..... (01.0, . ,
E . :;' | - E j I
>5 >5 ,
$00.0 0°20 1
m r1100 E
«a: F
a" ’\
-l. '- -
0- f‘ LM FA DHK F H A

Figure 6.15: (a) Band structure in simple cubic Brillouin zone (see Fig. 6.14a) of (a) bulk
PbTe, (b)-(d) Ag-Sb chain model; orbital character of (b) Sb p, (c) Te5 p (Sb nearest neigh-
bor), and (d) Te6 p (Ag nearest neighbor), (e) Ag-Sb pair impurity with the Ag-Sb distance
of ~11.19A. Band structure in hexagonal Brillouin zone (see Fig. 6.14b) of (f) Ag-Sb layer
model perpendicular to the [l l 1] direction.

88

which is the same as the L point in the fcc Brillouin zone. For the Ag-Sb chain model
and the Ag-Sb pair impurity model with the Ag-Sb distance of ~1 1.19A, the CBM and
VBM occur at the I‘ point. For Ag-Sb layer model perpendicular to the [111] direction,
the CBM occurs at the F point, while the VBM occurs along the I‘ K direction giving an
indirect narrow band-gap semiconductor. Comparison of the chain model band structure
(Fig. 6.14b) with that of PbTe (Fig. 6.14a) reveals that a new Sb resonant fiat band appears
at ~O.6eV in the CB of PbTe. This new resonant band hybridize with the Pb p bands near

the bottom of CB, which are pushed down in energy and having a flatter energy dispersion.

6.4.4 Summary

Ab initio electronic structure calculations give a clear picture of the deep defect states in
PbTe when nominally divalent Pb ion is substituted by Ag (monovalent) and Sb (trivalent)
ions. Supercell calculations for isolated Ag and Sb impurities and different microstructural
arrangements of Ag-Sb show a generic feature; when Sb atoms replace Pb atoms, Sb hy-
bridizes with neighboring Te atoms giving a very flat resonant band in the CB and forming
strong covalent interactions. When Ag replaces Pb, the p states of Te which are the near-
est neighbors of Ag are strongly perturbed. Therefore the electronic structure of LAST-m
compounds depend sensitively on these Te p states perturbed by the Ag(Sb) atoms. These
deep defect states appear near the top of the valence band (for monovalent Ag) and overlap
with the PbTe conduction band (for trivalent Sb). In the latter case they are clearly resonant

states.
We find that the details of the DOS near the band gap of LAST-m compounds depend

89

sensitively on the microstructural arrangements of Ag-Sb pairs in PbTe. The nature of
the states near the top of the VB and bottom of the CB in these quaternary compounds is
substantially different from those in PbTe. The common feature of these Ag-Sb arrange-
ments is that they have a more rapidly increasing DOS near the gap as compared to bulk
PbTe. Therefore, it is important to find out how these features in the DOS affect the elec-
tronic transport coefficients. It is well accepted that resonant structures in the DOS near
Fermi energy, created by quantum size effects [27, 26, 28], superlattice engineering [98]
or chemical means [58, 35, 36] are very desirable features because they could enhance the

thermoelectric figure of merit ZT. This will be the subject of our studies in the next chapter.

90

Chapter 7

Electronic Transport in PbTe and

AngmeTe2+m (LAST-m)

7.1 General Introduction

Electronic structure calculations of AngmeTe2+m (LAST-m) systems, described in the
previous chapter, have revealed that these systems have an enhanced density of states
(DOS) near the gap as compared to the bulk PbTe due to the appearance of distinct res-
onant states. Therefore, it is important to know how these features in the DOS affect the
electronic transport properties such as electrical conductivity 0 and thermopower S, the two
quantities which define the power factor PF (PF=0S2). Also, it is important to understand

why LAST-m systems show better thermoelectric properties than PbTe.

In order to answer the above questions and also to see how well the theory can de-
scribe quantitatively the transport coefficients, I have performed finite temperature elec-

91

tronic transport calculations in PbTe and LAST-m systems. These calculations were per-
formed using Boltzmann transport equations and nonparabolic Kane model for the energy
dispersion eEst. [75] Also, the energy dispersion given by ab initio electronic structure

calculations have been used in some transport calculations.

7 .2 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. [104] This equation states that in a steady state situation, at
any point in space F and for any value of the electronic wave vector 1:, the net rate of change
of f E0", t) is zero, where f EU", t) is the probability that an electron(hole) of wave vector 1?

is at point 7" in the crystal. Then:

[50,-5%] + [ah-5G1] + [8530?] = 0 ’ (7.1)
at diff. at field. at scatt.

where the different contributions to the change are due the diffusion, external fields and

 

 

 

scattering. As described above eq. 7.1 is valid for the steady-state, not for equilibrium
(fl-50", t)=f§(r", t)) which holds in the absence of external fields. f2“) is the equilibrium
single particle Fermi distribution function, the F dependence comes through the local tem-
perature T0"). It is given by:

92

1
ffgm = w,
(’11) (4—) + 1

KBTW)

(7.2)

 

where (I? is the energy, )1. is the chemical potential, KB is the Boltzmann constant. For
simplicity we will ignore the spin index and assume that each state I: 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 F at time t is equal to those in

the neighborhood of r" — trig at time zero:

fg(f‘, t) = fg(f’- tffg, 0) . (7.3)

The rate of change of the distribution function due to diffusion can be written as:

 

 

 

[8f]; (1") VT . (7.4)

J (h f f. at 8r(T) 8T
Using the definition of Fermi distribution function the derivative with respect to tempera-

ture in Eq. 7.4 can be rewritten as:

 

8T -_ 2K T2 =
[exp (he B”) +1] B

and then the rate of change of the distribution function due to diffusion is given by:

6~-u
.. 6:17p _._ _. ~—
%_ (1537) ‘k “ (jg) ___€k “ (75)
60 T ’ °
k

93

 

 

0t 06 ~ T

0 .. . a -. .. — .
diff. k

By analogy to Eq. 7.3 we can apply Liouville’s theorem in 1.5-space:

f]:(F~t) :f-'

k—frcfi 0) . (7.7)

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

wave vector k is given by:

m? = (2133' . (7.8)

The velocity of the electron in the state I: is given by:

r r 86’? (7 9)
l “ = —: . .
k 0k
In Eqs. 7.8 and 7.9, e is electronic charge, m is free electron mass, and h = h/27r with )2

being the Planck constant. The rate of change of the distribution function due to the field

can be written as:

 

 

 

 

] 2 flat)? z -Eafrfir : £81? EM: (.802) E
field at 8k 66,-; 0k ’1. 66,-; 36,: k
(7.10)

 

[afgm

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

94

approximation the rate of change of the distribution function due to scattering is given by:

 

= , (7.11)
T-' 7":

[0.733(7)] __fk_f,(;) __9_g
an”. k

k
where g]: is the difference between the steady-state and equilibrium distribution functions,

and TE is the relaxation time.

Using the Eqs. 7.6, 7.10, and 7.1 1, the the Boltzmann transport equation (Eq. 7.1) can

be written as:

 

 

k -» k k —+
g]; = e (— #) vgrgE + (— 4) ng~(—VT) . (7.12)
86k 86 k T
In order to find the expressions for electrical conductivity tensor (3’ and thermopower

tensor (3, the microscopic and macroscopic Ohm’s laws have to be used, which relate
the current density to the applied electric field and temperature gradient. The microscopic

Ohm’s law is given by:

.. 1 a 1 _,
JZV:ef*E=VZ€ggI-E , (7.13)
k is
since
Zef97~=o (7.14)
a k k ,
k

where V is the volume of the crystal in real space. From Eq. 7.12 and Eq. 7.13, the current
density can be expressed as:

95

 

 

 

 

The macroscopic Ohm’s law in the presence of an external electric field E and temperature

gradient VT is given by:

.1“: WE + ‘6’?(—VT) = 59(12‘ + ‘§’(—v:r)) . (7.17)

Direct comparison of the two Ohm’s laws in Eqs. 7.16 and 7.17 gives the expressions for

the electrical conductivity and thermopower tensors:

 

 

<——> __ e :( 1:) ~ ~
0 —— — — T-o ~ ~ (7.18)
V I? 66,-; k k
‘?=(‘5+)'1‘X , (7.19)
where (:4) is given by:
<——> e afg)
A = — — (6~—p)7’~"~"~ . (7.20)
.. k k k
VT 1: ( 86k

Analogous to the current density equations, the microscopic and macroscopic heat den-
sity equations for fQ can be written as:

96

_. 1 4 1 5
JQ = ‘— Z fflrg —;1.)1:’-co = T, Zgl-Cokg — p)v~ (7.21)
['5
; WET}? + firm—VT) , (7.22)

where 170) is the electronic thermal conductivity tensor at constant E field (i=0). Direct
comparison of the two heat density equations gives the expression for the electronic thermal

conductivity tensor at constant E field (E =0):

 

1 (9f: 2 -
+730 2 W Z — 66’: (6"; — 10211-51 gr}; . (7.23)
E I:

It should be pointed out that one is sometimes interested in constant T (j =0) electronic
thermal conductivity which requires solving simultaneously Eqs. 7.17 and 7.22 for (E)

with the constraint i=0. This thermal conductivity is usually referred as (7361. [58]

7.3 N onparabolic Kane Model for Energy Dispersion

In this section I will discuss a simple model for the band structure of PbTe. In narrow band-
gap semiconductors (i.e. lead chalcogenides) the energy of an electron, measured from
the band edge, is comparable with the energy gap Eg between the conduction and valence
bands. Because of this, the dependence of the energy on crystal momentum is nonquadratic
and the effective masses are functions of the energy. The nonparabolic Kane model was
introduced to describe the nonparabolic dependence of energy on the crystal momentum
E. [75] In this model, the longitudinal effective mass ml and the transverse effective mass

97

 

2:22: \/ \/ \

$190.05 BF

W LAFAXZWIK

 

 

 

 

 

 

 

.5 Eng
\1
/V

 

Figure 7.]: Band structure of PbTe in face centered cubic Brillouin zone.

mt depend only on the interaction between the lowest conduction band and the highest
valence band and the contributions of other bands are assumed to be small. In Fig. 7.1
the band structure of PbTe is shown. The conduction band minimum and the valence
band maximum in PbTe occur at the L[1/2,l/2,1/2] point in units of the primitive face
centered cubic reciprocal lattice I: vectors. The longitudinal effective mass ml is defined
along the LP direction in the Brillouin zone, whereas the transverse effective mass mt is
defined along the perpendicular direction to LP. For the simple parabolic model the energy

dispersion can be expressed as:

2 2,132 k2
9:?— —t-+—-’— , (7.24)
k 2 mt ml
where c; is the energy, k, and kt are the magnitudes of the longitudinal and transverse

components of I: (measured from the L point). In the Kane model the energy dispersion is
given by:

98

6*. 1‘12 21-2 k2
2 1 ..L- Z __ __t. __l_ , 7.25
6k ( + Eg) ‘2 (mt + 772,) ( )

In this model the effective masses and the mass anisotropy coefficient of electrons and holes
are equal. The constant energy surfaces are ellipsoids and 772) and mt have the same energy

dependence.

7.4 Transport Coefficients in the Kane Model

In order to calculate the transport coefficients we have to compute different quantities ap-
pearing in Eqs. 7.18, 7.19, and 7.23, which include the canier velocity it"; and the relax-
ation time TE. For this, we change the discrete sums over 1: into integrals, the carrier veloc-
ity 17,;- is calculated using the Kane model for the energy dispersion, and the relaxation time
expressions for different scattering mechanisms are obtained within Kane model. These

expressions are given in the section 7.5.

PbTe and LAST-m systems have cubic symmetry. Therefore, the components 0”,, and

S #V of the electrical conductivity and thermopower tensors can be express as:

0"“; = (Spy/0' , (7.26)

5111/ = 6111/5 , (7.27)

where a and S are related to the trace of ‘5‘) and ? tensors and they are given by:

99

1
a = 3727*? , (7.28)

1
s = 37.”? . (7.29)
H
T7“?) and Tr S are related to 17" g'figzr% terms.

The Kane model energy dispersion Eq. 7.25 contains two types of effective masses and

in order to simplify the calculations the fallowing substitutions can be made:

2 "It 2
kt = Tn—szé , (7.30)
k? = Ell/9;? , (7.31)
7nd

Using the new variables k; and k2, Eq. 7.25 can be written as:

 

6 I h2 ’12
1 J2- : (2H2 112 E —k'2 . 7.32
ek,( + E9) 2m; t + 1 ) 2m, ( )

In terms of the new variable k’ the expressions for (E) and ? are:

 

2 6f~
H _. 6_ .. k .. an
a —- V Z: 661;, Tk’vk vk (7.33)
k,
E = (‘3)1 (71+ , (7.34)

H
where A is given by:

 

0f~
H — L _ k, _, _ _,.-:_,-.'_,
A — i'T Z ( (961;?) ((k’ ")Tk "171 ' (7'35)

a

’6’

Changing the sums into integrals, Eq. 7.34 for thermopower becomes:

 

(7.36)

Using Eq. 7.32, the infinitesimal unit volume in reciprocal space (13117 can be expressed as:

3

 

 

 

1 I 1
.. 22m 2 6 2 2e
3 I _ I2 I_ d
(1k — 47rk dk — 47r ’23 [6 (1+ E_g)] (1+ Ti?) (16 , (7.37)
where we have dropped the suffix for the carrier energy 61:].
The carrier velocity defined in Eq. 7.9 is given by:
1 a h 15"
#19:"062—7 2. . (7.38)
h 61;, 7nd 1 + E5

--0

Using the above equation, the 1127:11 ~ -17 ~ term in the transport equations can be expressed

k’ k’

as:

E
2 _ 2 6' (1 + Fr?)
2).. _ . (7.39)
k, mil 1 26 2
( + 2.7)
Substituting Eqs. 7.37 and 7.39 into Eq. 7.36 and using Eqs. 7.27 and 7.29, the ther-

mopower along the x-axis Sm (51,sz ) becomes:

101

__._1.2°(—%2)<e—E) [41+ < )]2 (1.2)—1....

9
021—2161215212 ‘

(1 + 26 )—1 T (16
where the relaxation time 76 is assumed to depend only on energy. Since 5 depends on the

(7.40)

 

ratio of the two integrals (Eq. 7.36) the spin and band degeneracy 2) factors cancel out in

Eqs. 7.36 and 7.40.

Taking into account the spin degeneracy and the band degeneracy "y, the expression for

the electrical conductivity (59 can be rewritten as:

 

 

 

 

 

 

k,
(7.41)
and using Eqs. 7.37 and 7.39, Tr‘B‘> can be expressed as:
2(2m;)% 0° 0f 6 g 26 -1
T =2 —— 1 —— 1 — d . 7.42
.. .. W] ( ..)[e(+E,)] (+E,) . 1

In order to compare the electrical conductivity with the measured experimental values along
a specific crystallographic direction (e. i. x-axis), the component a” has to be calculated.
on (033:0) is given by:

102

1 3
2 2 (2771121)2 00 (9f 6 9 26 _1
, : —"3 — 1 1 d ' 7'4
a” 3 " «217,3 f0 06 6 + 13., + E9 7‘ E ( 3)

The concentration of carriers 71 can be written as:

 

_ 27 _ 22,2 V d3]?
"‘VZfP—VW/ (27-2) ’ (7'44)
kl exp +1

and using Eq. 7.37 the concentration 77. can be expressed as:

 

1
22 2 2° [2(1+2.-)]"’(1+%§)22
errp (rel-B +
The expression 7m?” is defined as:
mist/2 = ’ymg/2 , (7.46)

where md is called density of states effective mass. The relation between the density of
states effective mass and longitudinal and transverse effective masses can be deduced using

the Eqs. 7.30, 7.31 and the following equation:

d3]? = d3}? , (7.47)

which sets the equality between the infinitesimal volumes in reciprocal space. Then the
density of states effective mass can be expressed as:

103

2 I 2 2 11?
712,, = /3md = 7'3 (Input) . (7.48)

The transport coefficients can be expressed in terms of generalized Fermi integrals:

n m _ 00 0f n f m 26 I
L, (,tt,Eg) —/0 (—E) (6) [6 (1+E)] (l-l-E—g) d6 . (7.49)

In the case of constant relaxation time 76:70 independent of energy 6, the expressions for

thermopower, electrical conductivity and concentration have a simplified form:

3
KB 11431011139)

 

 

5:132: = _ 3 - fl 1 (7-50)

6 0 2

L_1(/-‘aEg)
l
2 2(2mfi)270 0 g

a” = 376 W L_1(,u,Eg) , (7.51)

3

1 (2mg)? 0 g

n = 37 ”2’13 L0 (u, Eg) . (7.52)

Note that in this limit Sam is independent of the relaxation time and systems with large

differences in on can have nearly the same 5'”.

7 .5 Relaxation Times in the Kane Model

Different scattering mechanisms of charge carriers in lead chalcogenides have been inten-
sively studied in many papers. [75, 77, 76, 103, 20] It is found that the dominant scattering

104

mechanisms are by point defects and by thermal phonons and 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 _<_ 5x1018 cm—3, scattering by Coulomb potential of the
vacancies dominates, whereas for high carrier densities in 2 1019 cm—3, the Coulomb po-

tential gets screened out and scattering by the short range potential of vacancies dominates.

As the temperature increases, the relative importance of the charged vacancies in scat-
tering mechanisms decreases as the scattering by thermal phonons increases. For tempera-
tures above 300K the scattering by acoustic phonons and optical phonons (both polar and

deformation potential) have the main contribution to the relaxation time.

Calculations for the transport coefficients are made at temperatures above 300K , at
which all scattering mechanisms which are important in PbTe can be described in relaxation
time approximation. The expressions for the dominant contributions to the relaxation time
taking into account the band nonparabolicity in PbTe within the Kane model, have the

following forms [77, 103, 20]:

I) Deformation potential of acoustic phonons (Ta):

105

 

 

 

 

T0,0(T)
Ta 2 g 2
6 _ _—
(1+ 215:5) ((1 4) B]
4 _ 75; (1 - Ka)
6
(1+ 217;)
B 815; (1+ 155) K
— 2
3 (1 + 2159—)
27rh4C
7'0,(1(T) Z l 3
EECKBTdeo)?

where EaC is the acoustic deformation potential coupling constant for the conduction band,

C, is a combination of elastic constants, K0 is the ratio of the acoustic deformation po-

tential coupling constants for the valence and conduction bands K a = Eav/Eac with the

values Ka=1 for n-type PbTe and K a=l .5 for p-type PbTe, and mdo is the density of states

effective mass for a single ellipsoid (7:1).

11) Deformation potential of optical phonons (To):

106

 

 

 

 

1
‘2 ‘2
T()()(T) (6 + ET)
7., — J (7.57)
(1+ 225;) ((1 —.~1)2 — B]
f 1—1\
A _ F“ 0) , (7.58)
(1 +2155)
8 6 1+ ‘ K
E’ E‘ o
B _ 9 ( 9)2 (7.59)
3 (1 + 2E2?)
2f'12a2 fiw 2
mm = M 0) 3 , (7.60)
nEgCKBTdeO) 2

where a is the PbTe lattice constant, p is the PbTe density, wo is the frequency of the optical
phonons, K 0 is the ratio of the optical deformation potential coupling constants for valence
and conduction bands, K0 2 ECU/E00, which are taken to be the same as for acoustic

phonons, K a.

111) Polarization scattering by optical phonons (71,0):

712 (6+i-25)%F_1

 

 

TI”: 1 , (7.61)
2 2 -1_ —1 6
e (27nd0) KBT(€oo 50 ) 1+2E§
2 6 1+ ‘
F=1—61n(1+6—1)— EA E52)[1—26+2621n(1+6'1)] , (7.62)
(1+2ig)
6 = (2kr0)’2 , (7.63)

where 50 and 500 are the static and high frequency perrnittivities, k is the carrier wave

107

vector and 7’0 is the screening length of the optical phonons. The parameters k and 7‘0 are

given by:

2
k2 _ 27!),(10 (f '1' 16:3)

 

 

 

 

 

— h, , (7.64)
5 2 3 1
226 m
-2 _ d 0 2
_ —— L . 7.6
T0 fihBE‘OO ( 1 ) ( 5)
IV) Short range deformation potential of vacancies (7v):
62 —?
T0,U(T) (6 ‘1' E5)
TU : 2 (7.66)
6
(1+ 2E5) [(1— A) — B]
6 (1 _' Ky)
A = E; , (7.67)
(1 + 2%)
825— (1 + 15—) Kv
B Z 9 g 2 3 (7.68)
3 (1 + 213)
7r1‘i4
T0,U(T) _ 1 a (769)
UgcmdowmdoflNv

where Nv is the vacancy density, K v is the ratio of the short range deformation potential
coupling constants of vacancies for valence and conduction bands, K v = va/ch, which

are taken to be the same as for acoustic phonons, K0.

IV) Coulomb potential of vacancies (TC):

108

2 . 1 62 %
50(2md0)2 (6 + 13—)

 

TC : 2 2 9 , (7.70)
7" f —- 6
7r(Ze ) 31211110 +5) 5/(1+€)1(1+2E§)
€- : (2k7‘v)2 , (7'71)

where Z 6 is the vacancy charge, and rv is the screening radius of the vacancy potential

given by:

 

_ 47re2
7‘. 2 = 50 007) , (7.72)
2%( )g 2 5
_ me M H

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

The total scattering relaxation time 76 is expressed as:

1 1 1 1 1 1
—=—+—+—-—+—+— . (7.74)
Tf Ta To Tpo TU TC

We find that only the first three terms in the above sum contribute at temperatures above
300K. The values of the relaxation times at room temperature in the 1-200meV energy
range are: 70:0.15-2.9x10“123, 70:0.18-2.0><10‘12s, rpo=0.1-o.19x10-123, Tv=2.4-
44.7x10‘12s, and 70:5.2-1216.0x10‘12s; 7,) and T0 are orders of magnitudes larger
than Ta, 70 and 7pc. The parameters used to calculate the relaxation times were taken from
Ref. [103] and they are given in Table 7.1. Experimentally one finds the energy gap E9 and

109

Table 7.]: Parameters used to calculate the relaxation times for PbTe and LAST-m systems
at 300K.

 

[ Parameter H Unit of measurement 1 Value ]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ml/m 0.24
50 400
500 32.6
C, N/m‘2 071,11011
hwo eV 0.0136
(1 A 6.461
p g/cm3 8.24
Z 0.14
Eac eV 15
Eoc eV 26
Um; ergs cm3 3x10‘34
Ka, K0, K.) for n-type 1.0
K0, K0, K v for p-type 1.5

 

the density of states effective mass md to be temperature dependent. [94, 101] This comes
from strong electron-phonon coupling. In our transport calculations we have incorporated
the T-dependent E9 and md using experimental data. As it will be discussed later in detail,
we found that the temperature dependence of md was very important to have good agree-
ment with measured values of the transport coefficients. Experimentally it is found that Eg
increases linearly with temperature for T $400K and above 400K, it remains constant. [94]

The temperature dependence of Eg(T) is given by:

E9 = 0.19 + (0.42 x 10—3)T, forT g 400K , (7.75)

E9 = 0.356, forT > 400K . (7.76)

The temperature dependence of md (see Eq. 7.48) comes primarily through the T-

110

dependence of the transverse effective mass mt. which is also taken from experiment [101]:

T1 = 0.02459 + (8.669341 x 10‘5)T , (7.77)

m

where m is the bare electron mass.

7.6 Transport Coefficients Using ab initio Electronic

Structure

In order to calculate the concentration 71, the electrical conductivity an and thermopower
Sm using ab initio electronic structure results, the sums in Eqs. 7.44, 7.18, and 7.19 have

to be generalized over the spin and band index:

 

 

2
n— Ngv Zfdnifi) , (7.78)
17,311:
0 —' 262 2 ——if— 7‘ v (n-E)v (71"?) (779)
"WW - Bdrm/3) «71.8 w ., , -

S 1 21113]? (_fngfi) [6012313) — p] 76(niJ-Cf)vm(ni,bum-(mi, E)
xx 2 "—

6T 3 .. ~ ’
Zn“: (—m) Te(ni,g)1)m(ni,k)vm(ni,k)

 

(7.80)

where n, is the band index, and N I? is the number of E points chosen in the Brillouin zone.

For relaxation times T€( we use the same energy dependence as given by the Kane

7113’?)
model (see section 7.5). In the Eqs. 7.79, and 7.80 we have dropped the suffix 601,, 1.6) of

111

the distribution function fan. E)’
.2,

-o
‘

Instead using the Eq. 7.9 for the velocities ’17(n,-, k), we calculate 301.17, I?) using the

—o

momentum matrix elements 501,-, k):

_. .4 _.

_. _. 1 , -* 9 .
12(ni, k) = 7731201,, k) = +- <u2(ni, k)|p 01012:, k)> , (7.81)

m

which are available in the optical properties package of WIENZK program. [6]

7.7 Results and Discussions

7.7.1 PbTe Using Kane Model

In order to calculate electrical conductivity am and thermopower Sm for a given concen-
tration n at different temperatures, Eqs. 7.40, 7.43, and 7.45 have to be solved simultane-
ously, the common link being the chemical potential 11. We use the Mathematica program
to solve these coupled equations.

As mentioned in the section 7.5 of this chapter, we have taken into account the tem-
perature dependence of the energy gap E9 and density of states effective mass mm which
are given in Eqs. 7.75, 7.76 and 7.77. At high concentrations, 71 ~ 1019cm_3, which are
required for a good thermoelectric, the n or p-type PbTe can be described using a single
band model since at these concentrations contributions to transport comes primarily from
a single type of carriers. First we give the results for n-type PbTe.

At concentration n = 5 x 1019cm‘3, using the values given in Ref. [103] for the
acoustic and optical deformation potential coupling constants for the conduction band

112

(EaczlSeV and Em.=26eV), we underestimate the electrical conductivity at room tem-
perature (Fig. 7.2a). The scattering from optical phonons is too strong. In order to fit
the electrical conductivity data at room temperature we have to decrease the value of E00
to ISeV (Fig. 7.2b). The results for electrical conductivity, thermopower and power factor
PF=0S2 using both sets of values for the deformation potential coupling constants are given
in Fig. 7.2. Clearly, a reduced value E00 gives a better fit to the high T data. After taking
into account the temperature dependence of E9 and md we obtain excellent agreement with

experiment for electrical conductivity and good agreement for thermopower.

In is interesting to explore how significant are the temperature dependence of E9 and
md in the entire temperature range, a study which has not done before. For this we cal-
culate a, S and the power factor 820 for a constant md using the values m, 2 024771
and mt = 0.024m for the longitudinal and transverse effective mass respectively. The
results for both the sets of deformation potential coupling constants are given in Fig. 7.3.
For constant md, the electrical conductivity is overestimated whereas the thermopower is
underestimated. Also the temperature dependence of thermopower is not well described.
Therefore the temperature dependence of md is very important to have a good agreement
with experimental values of electrical conductivity and thermopower. It is important to note
that one can have fortuitous good agreement for the power factor as shown in Fig. 7.3(e)
even when a and S are not very well reproduced individually. This suggests that in order
understand correctly the temperature dependence of the power factor in PbTe systems, the
temperature dependence of the electrical conductivity and thermopower have to be studied
separately and understood.

113

(a)

 

17500

15000
E12500
310°”
"7500.
b 5000.

2500.

s
O
..

 

..
...
..oo.oooo.o
.

 

 

260

400
Temerature TCK)

600 800

 

(c) 0*

-501
3 -100
m —150

-200»

 

 

 

o 200 400

A . 1.
600 800

Temperaturo T(K)

 

3

mmavzkio
MN
0 9

H
9'

10’

 

4

 

 

200 400

Power FICCOI‘

6710 800

Temperature '1'(K)

 

 

 

 

 

 

 

 

 

 

 

 

(b) .4 - .1 1.
25000 n.
I
20000 1
0

315000 ‘0

29 ‘~

1 10000 x,

.D .Q
5000. ‘\..

“o...
0’. A g T......A......‘
0 200 400 600 800
Tomorature'lxx)

(d) 0r ..~ - a 4
.50 ..‘:‘o

E “..-

‘3-100 ‘~:

Q.‘.
0) ‘8
-150 5‘.
0%..
400: + - ‘ . ‘~.4
0 200 400 600 800
Tomperaturo'l‘cx)

(1) _fi -
g 25 "a". .."..
£20, o" . “i
a '0' O
“15» '0'

O o'
4.110» ,O '
a x
In 5» o'
u ,o
0»; 1 - - _-
g 0 200 400 600 800
n. Temperature'rat)

Figure 7.2: Temperature dependence of (a)E1ectrical conductivity, (c)Thermopower, and
(e)Power factor for n-type PbTe at concentration 72. = 5 x 1019cm"3 obtained using
Eac=156V and Eoc=26eV values of the deformation potential coupling constants; Tem-
perature dependence of (b)Electrical conductivity, (d)Thermopower, and (f)Power factor
for n-type PbTe at concentration 71 = 5 x 1019cm‘3 using Eac=15eV and Eoc=15eV. The-

oretical values are shown as dashed lines and the experimental values are shown as black
points from Ref. [16] and as gray points from Ref. [76].

114

(a)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2500 7 '. 4 (b) ' ‘.
1 300001 '.
20009: i * 25000. ‘
0‘ ‘.
5150001 '. 1820000» ‘.
3 “ 315000 “s
VIOOOO’ .‘s ‘ V ’ “\
‘2 . b10000» ‘ ~.
5000’ ...........Q. 5000’ . .............
. 9.... . O...
o - . .' ’. .. o » . . .’ 0. o
0 200 400 600 800 0 200 400 600 800
Teapot-31:11:01.1!) Tonporaturo'rat)
(c) °"~-... ' ' ’ ‘16) °"-. * ' '
-501 “an... 1 ‘5°’ "‘m.
. .. ...0.. ..........
E400» . "avg-100» o "a“;
. . O . . O
"' -150 , . m -150
( 0 0
40°- - - . - 4°01- - - g -
0 200 400 600 800 0 200 400 600 800
Towel-ature Tat) Temperature Tat)
"1325, rm. (“E40 ...-
g .. 0"... . 3‘ ,Ooo.‘..‘
& 20» ..ov ‘ 5 30 .o' '
5,15 0." S, ,x' . 0
8 x 82°: . ..
U 10 ’0. «U '0'
g 0' 2 10 '0' O
In 5 '0' ‘0
u ,x u ,.
E o 200 400 600 800 5 ’0 200 400 600 500
a. Twat-ature rm 0. Temperature Tat)

 

 

 

Figure 7.3: Temperature dependence of (a)E1ectrical conductivity, (c)Thermopower, and

(e)Power factor for n-type PbTe at concentration 71 = 5 x 1019cm_3

obtained using

Eac=15eV and Eoc=26eV values of the deformation potential coupling constants; Temper-
ature dependence of (b)Electrical conductivity, (d)Thermopower, and (f)Power factor for

n-type PbTe at concentration 71 = 5 ><1019cm‘3

using coupling constants values Eac=15eV

and Eoc=15eV. Theoretical values obtained using constant md are shown as dashed lines.
Experimental values are shown as black points from Ref. [16] and as gray points from

Ref. [76].

115

 

 

 

 

 

 

 

 

(a) , r w
17500L :
|
15000l g
12500l a
$10000 H
“'7500. \.
t95000. ‘~,:
2500. “mg...
0 AA 4 A f...‘.fcon.
0 200 400 600 800
Temperature'lxx)
(c) 0 a... 5 ' '
- . "a.
g: 50 ”fl
3400. *‘c,
.Q“
“450 "5..
o .‘u.
-200' A A A_ A 1;
o 200 400 500 800
Temperature'flx)

 

 

 

 

 

 

 

(h) f f
25000» I.
I
20000» t
|
‘ 0
515000» ‘.
‘ \ s
m s
. ”10000 ‘s‘
. b x‘
5000» '5...
0‘...-
[A A A 4...........
O 200 400 600 800
Tenperatureflx)
(d) 0" ~ 5 * 5
‘ -50’ ...‘:o
E x.-
‘3-100» “3.
0..
\
m-ISO» ‘3..
0“.”
—200»A A A A A 1;
0 200 400 600 800
Temperature'rat)

Figure 7.4: Temperature dependence of (a)Electrical conductivity, and (c)Thermopower for
n-type PbTe at concentration n = 5 x 1019cm‘3 using Eac=15eV and Eoc=26eV values
of the deformation potential coupling constants; Temperature dependence of (b)Electrical
conductivity, and (d)Thermopower for n-type PbTe at concentration 77. = 5 x 1019cm'3 us-
ing coupling constant values Eac=15eV and Eoc=15eV. Theoretical values obtained using
temperature dependent E9 in the OK-9SOK range are shown as dashed lines. Experimental
values are shown as black points from Ref. [16] and as gray points from Ref. [76].

We have considered also the case where the energy gap Eg increases linearly with the

temperature in the whole OK-950K temperature range (instead of Eqs. 7.75 and 7.76). The

results for the electrical conductivity and thermopower are given in Fig. 7.4. We can see that

By temperature dependence is not very crucial to have good agreement with experiment at

this high concentration, but it is important at low concentrations when carriers from the

valence band can be excited into the conduction band across the band gap. For this case

two band model (conduction and valence) has to be employed to calculate the transport

coefficients.

To see whether the agreement between a, S and S20 and experiment obtained with

116

 

 

parameter values Eac=15eV and Eoc=15eV (values chosen to match the value of a at
300K for the concentration 12. = 5 x 1019cm_3) persist at different concentrations, we
have calculated these quantities at two other concentrations: n = 2.4 x 1019cm"3 and
72 Z 1 x 1020cm”3. The results are shown in Fig. 7.5. For n = 2.4 x 1019cm’3,
the room temperature electrical conductivity is slightly underestimated, whereas for n =
1 x 1020cm_3 this value is slightly overestimated. Over the entire 300K-950K temperature
range the agreement for electrical conductivity with experiment is quite good, suggesting
that Eac=15eV and Eoc=15eV values of the deformation potential coupling constants de-

3 concentration range. The

scribe well the scattering mechanisms in the 1 — 10 x 1019cm-
thermopower is overestimated for both the concentrations; as a consequence the power

factor is also overestimated.

7.7 .2 PbTe Using ab initio Electronic Structure

In order to calculate the transport coefficients of n-type PbTe using ab initio density func-
tional results the sums given in Section 7.6 were evaluated over the band index and I: points.
The carrier velocities were calculated using the momentum matrix elements. It was found
that a very dense mesh of irreducible I: points (more than 15000) have to be taken in the
Brillouin zone in order to have good convergence for the values of the transport coefficients.
This will require a tremendous amount of computational time and computer resources in
the case of complex LAST-m systems, but for PbTe such calculations are feasible. For the

relaxation time the same expressions were used as in the nonparabolic Kane model.

The calculated values for the electrical conductivity and thermopower at n = 5 x

117

 

 

 

 

 

 

 

 

 

 

(b)

 

 

 

 

 

 

 

 

 

 

35000 I'-
30000 '.
5000 E
820000 “.
~15000 \
010000 “‘8-
-.,..A 5000 ‘~~.,_._
-....___.. . w-...._.__.
0 200 400 600 800 3 200 400 600 800
Temperature '1‘!!!) Temperature Tat)
(c) ° -,
.50 ‘5‘.
E “"
g-1oo ‘32.
v §‘..
“L150 ‘~:~:
V.
—200 ‘~.
0 200 400 600 800 0 200 400 600 800
Temperature '1'(K) Temperature Tat)
mg .- (f) .
03° .“-~. 02° °
' .‘5‘
$25 I". e &15 e
320 ,v E
315 ,o’ 310
‘610 ‘6
I! 0 fl 5
h 5 ,' In
H ,t n
8 ° :2 °
0 0 200 400 600 800 o 0 200 400 600 800
0. Temperature Tat) 9. Temperature Tat)

Figure 7.5: Temperature dependence of (a)E1ectrical conductivity, (c)Thermopower, and
(e)Power factor for n-type PbTe at concentration 72 = 2.4 x 1019cm‘3; Temperature de-
pendence of (b)Electrical conductivity, (d)Thermopower, and (f)Power factor for n-type
PbTe at concentration n = 10 x 1019cm'3. Theoretical values obtained using tempera-
ture dependent E9 and md and using Eac=15eV and Eoc=15eV values for the deformation
potential coupling constants are shown as dashed lines. Experimental values are shown as
black points from Ref. [16] and as gray points from Ref. [76].

118

1019cm‘3 are shown in Fig. 7.6. One expects that ab initio energy dispersion should
give better agreement with experiment as compared to the Kane model. In contrast the
agreement is not as good. The electrical conductivity is overestimated (Fig. 7.6a) and the
thermopower is underestimated (Fig. 7.6b). This is due to the fact that in ab initio calcu-
lations the temperature dependence of the density of states effective mass md can not be
incorporated in the calculation of chemical potential using expression for concentration n
(see Eqs. 7.78 and 7.45). For the same n, the ab initio results (Eq. 7.78) give a higher value
of the chemical potential ,0 than that given by the Kane model (Eq. 7.45). This results in
an increase in the electrical conductivity and decrease in the thermopower at a given tem-
perature. To prove that not incorporating the temperature dependence of md in theoretical
calculations is the reason of disagreement between the ab initio results and experiment, we
have calculated the transport coefficients in the Kane model using constant values for md in
the expression for concentration 72 (Eq. 7.45), but temperature dependent md in the expres-
sions of relaxation times. These results are shown in Fig. 7.6a,b as continuous lines and
they clearly show the overestimation of electrical conductivity and underestimation of ther-
mopower. The zero temperature ab initio electronic structure calculations can not correctly
describe the finite temperature transport coefficients in PbTe particularly when the band
structure parameters such as E9 and md are strongly T-dependent. A finite temperature ab
initio density functional theory is necessary to describe the transport properties of PbTe and
LAST-m systems properly. Clearly this is not possible at the present time. Therefore in our
transport calculations for the LAST-m systems we use only the energy dispersion given by

the Kane model.

119

 

 

 

 

 

 

 

 

 

 

a , -
(35000 . (b) o ‘

20000» -50r

15000» g

E 3400’

in

‘10000 « 0)

b -150. .
5000. .
0AA A A A A A .200: - - -

0 200 400 600 800 0 200 400 600 800
Temperature Tat) Temperature Tat)

Figure 7.6: Temperature dependence of (a)Electrical conductivity and (b)Thermopower for
n-type PbTe at concentration 77. = 5 x 1019cm‘3 and using Eac=15eV and Eoc=15eV
values of the deformation potential coupling constants. Theoretical values obtained using
ab initio energy dispersion and temperature dependence of E9 and md are shown as dashed
lines. Theoretical values obtained using nonparabolic Kane energy dispersion with constant
md in the expression of n are shown as continuous lines. Experimental values are shown
as black points from Ref. [16] and as gray points from Ref. [76].

7.7 .3 LAST-m Using Modified Kane Model

To calculate the transport properties of LAST-m systems, we make use of the ab initio band
structure results to construct an effective nonparabolic Kane model. Transport coefficient
calculations were performed for the chain model of LAST-30 systems (Figs. 6.9d) through
a qualitative analysis of the band structure results of the chain model and bulk PbTe. The
band structure of PbTe and chain model in the simple cubic (sc) Brillouin zone are given in
Figs. 6.15a,b. In sc Brillouin zone of PbTe, the conduction band minimum and the valence
band maximum occur at the I‘ point which is same as the L point in the face centered cubic
(fcc) Brillouin zone. The lowest conduction band, which contributes to the transport in the
fcc Brillouin zone is one-fold degenerate and the multiplicity of L point is four, whereas in
sc Brillouin zone the lowest conduction band is four-fold degenerate and the multiplicity

of P point is one.

More specifically, the lowest conduction band in sc Brillouin zone along the F X di-

120

rection is four-fold degenerate, whereas in the FR direction the first conduction band is
one-fold degenerate and the second conduction band is three-fold degenerate. In the case
of chain model these bands split and the first conduction band has mostly Sb orbital char-
acter (Fig. 6.15b). The lowest conduction band for the chain model has a flatter energy
dispersion compared to PbTe. To simulate the chain model for the lowest conduction band,
we have taken a heavier density of states effective mass mfihain, whereas for the other three
conduction bands we have used the same md as for PbTe. Although we recognize that this
approach can not quantitatively describe the chain model of LAST-30 systems since md for
the other three conductions bands are not exactly the same as those of PbTe, the approach
can qualitatively describe the transport properties in all LAST-m systems. This is a con-
sequence of the electronic structure results, which show that all the Ag-Sb microstructural
arrangements have an enhanced conduction band density of states (heavier md) near the

band bottom as compared to PbTe (see subsection 6.4.3).

We have considered two values for the effective mass mfIhai" of the lowest conduc-
tion band. One is mfihain=15md and the other is mfihain=2.5md, where md is the density
of states effective mass of PbTe. The results for concentration 17. = 5 x 1019cm"3 (us-
ing Eac=15eV and Eoc=15eV values of the deformation potential coupling constants) are
given in Fig. 7.7. The chain model for mghain=15md has a lower electrical conductivity
and a higher thermopower compared to PbTe. As a result the power factor shows a small
increase in the OK-SOOK temperature range (Fig. 7.7a,c,e). Increasing the value of mfihai"
even more (mfihai"=2.5md), the electrical conductivity of the chain model is further de-

creased, whereas the therrnopower is increased. This gives an enhanced power factor in

121

the OK-SOOK temperature range (Fig. 7.7b,d,D. These qualitative transport calculations
show that LAST-m systems in the OK-SOOK temperature range have a small increase in the
power factor compared to PbTe. The enhancement of the power factor increases with the

aha!” values.

increasing m

We have calculated the transport coefficients for a higher concentration (72 = 1 x
1020cm—3) for wrigha'i"=2.5md. These results are shown in Fig 7.8. The power factor
for this concentration shows an enhancement in the 0K-700K temperature range. This sug-

gest that the temperature range for which the power factor of LAST-m systems show an

enhancement relative to PbTe increases with concentration.

7.8 Summary

Transport calculations using Kane model for energy dispersion in PbTe show that the tem-
perature dependence of the density of states effective mass md is very important to correctly
describe the temperature dependence of the electrical conductivity and thermopower. Tak-
ing into account this temperature dependence and that of the energy band-gap, we see that
the electrical conductivity has a strong T ‘3 temperature dependence (in agreement with
experiment). [75]

Calculations for PbTe using ab initio electronic structure results for the energy disper-
sion overestimate the electrical conductivity and underestimate the thermopower at a given
temperature. This is a consequence of the fact that in ab initio electronic structure results
the temperature dependence of md can not be taken into account in the calculation of the
chemical potential. This increases the chemical potential values for a given 12 and tem-

122

 

 

   

 

 

 

 

 

 

0 200 400 600 800 ‘ 0* 200 400 607) 800
Temperature!!!) Temperature“
(c) 0" ' * ' ' we) 0" f ' '

 

 

a -100»

 

   

 

 

 

 

 

 

 

       

 

 

 

 

 

 

”’ -150»
400*. A A A A 1. A A A A A I...
0 200 400 600 800 0 200 400 600 800
Temperature Tat) Temerature Ta!)
(.)3 . Y Y .3 a (f) ob f
“‘ 25A o...‘. ‘N 25’ O...‘.. .
gzo . Q‘ E20 . Q‘A
3 15» 8 15)
u 10’ U 10’
2 2
In. 5. In. 5»
u 0 H 0
§ ’0 200 400 600 800 § 0 200 400 600 800
9. Temperature rat) a. Temperature Tm

Figure 7.7: Transport coefficients at concentration n = 5 x 1019cm"3 using temperature
dependent E9 and rm; and Eac=15eV and Eoc=15eV for the deformation potential cou-
pling constants. Temperature dependence of (a)Electrical conductivity, (c)Thermopower,
and (e)Power factor for mflhain=l .Smd; Temperature dependence of (b)Electrical conduc-
tivity, (d)Thermopower, and (f)Power factor for mflhain=2.5md. Theoretical values for
bulk PbTe are shown as dashed lines and for the chain model of LAST-30 as continuous

lines. Experimental values are shown as black points from Ref. [16] and as gray points
from Ref. [76].

123

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) - . - . f (b) 0»'
20000» ‘,A .25.
I
15000) 'i, « ’50’
E . E -75
£10000 3400
0 01-125»
5000. —150.
-175»
0 200 400 600 800 0 200 400 600 800
Temperature Tat) Temperature Ta!)
(CYE -
0120.
<
&
315.
310.
u
8
In 5'
H
§ 0 200 400 600 800
p. Tempereture'l‘at)

Figure 7.8: Transport coefficients at concentration n = 1x1020cm_3 using temperature de-
pendent Eg and md and Eac=15eV and Eoc=15eV values of the deformation potential cou-
pling constants. Temperature dependence of (a)Electrical conductivity, (c)Thermopower,
and (e)Power factor for mfihai"=2.5md. Theoretical values for bulk PbTe are shown as
dashed lines and for the chain model of LAST-30 as continuous lines. Experimental values
are shown as black points from Ref. [16] and as gray points from Ref. [76].

perature. This effect enhances the electrical conductivity and reduces the thermopower.
Therefore, the zero temperature ab initio density functional theory (DFT) can not describe
correctly the transport properties in PbTe when rm; and E9 are known to depend strongly on
T. This may also be true for the LAST-m materials. A finite temperature DFT is necessary

to capture the correct electronic transport in PbTe and related materials.

The qualitative transport calculations for LAST-m systems (chain model of LAST-30)
show a small enhancement of the power factor relative to PbTe. This enhancement in-
creases with increasing md values. The temperature range, for which this enhancement of

the power factor is achieved, increases with increasing n.

124

The transport calculations reveal that the enhanced density of states (enhanced 7714)
near the band gap of LAST-m systems as compared to that of PbTe seen in the electronic
structure calculations can give rise to an increase in the power factor. However this increase
is not large enough to increase ZT by a factor of 2 (as seen experimentally). Therefore, the
reason why LAST-m systems have better ZT compared to PbTe is a combination of a small
enhancement in the power factor and perhaps a strong reduction in the phonon thermal

conductivity due to the formation of microstructures (seen in electron microscopy). [72]

125

Chapter 8

General Summary and Future

Directions for Research

Electronic structure calculations reveal that the effect of quantum confinement QC result-
ing from surfaces on the electronic structure of binary layered Bi28e3 and BigTe3 is the
appearance of surface states. Our calculations clearly show the presence of finite in-gap
surface states and a suppression of the top valence band antibonding states. These sur-
face effects are a consequence of the strong influence of the interlayer bonding on the bulk
electronic structure of these compounds, showing an important covalent contribution to the
interlayer bonding besides the Van der Waals contribution. Our calculations are in very

good agreement with scanning tunneling spectroscopy (STS) results. [95]

Electronic structure results in (BigTe3)m(Sb2T63)n superlattices (SL) show that the
formation of SL give rise to a small increase in the electronic contribution to the figure
of merit ZT as compared to bulk B12T63 and SbgTe3. The calculated values of the hole

126

mobility ratio “(w/[lip in the SL are in good agreement with the experimental values. [98]
Our calculations show that the hole mobility ratio is very sensitive to the different types
of SL, whereas the electron mobility ratio is less sensitive. This suggests that the valence

band can be engineered through (BigTe3)m(Sb2Te3)n SL structures.

Electronic structure calculations of complex ternary K2Bi88e13 system show that its
electronic structure and consequently the thermoelectric properties are very sensitive to
the difference in the two crystal structures a—KgBi88e13 and ,B—KgBiSSelg of this sys-
tem. a—KgBigSelg has no structural disorder, whereas ,B-K23188613 has two sites with
K/Bi mixed occupancy. While a—KgBigSe13 is an indirect band gap semiconductor,
fi—KgBigSelg shows semimetallic or semiconductor behavior. In B—KgBigSe13 the atoms
at the mixed sites are found to be very important in determining the electronic properties
of this material. When the mixed sites are treated as fully occupied by K or Bi atoms the
system is a semimetal. Incorporation of mixed occupancy results in an indirect gap semi-
conductor. Both phases of K2Bi88e13 have low symmetry where one tries to achieve high
anisotropy R=[Azz/(/\m/\yy)]1/ 2 ratios involving the effective mass parameters Aij (see
Chapter 1). Effective mass calculations show that 6—KgBiSSe13 has large R values com-
pared to a—KgBiSSelg. The high R values in B—KgBigSem combined with a low value of
the phonon thermal conductivity (due to K/Bi disorder) make ,B-K23188613 an excellent

candidate for room and high temperature thermoelectric material.

Electronic structure results of the complex quaternary LAST-m systems reveal that their
electronic structure is very sensitive to the Ag-Sb microstructural arrangements. The com-
mon feature of all these arrangements is that they have a more enhanced density of states

127

(DOS) near the gap as compared to PbTe. This feature is desirable because it could en-
hance the ZT. Transport calculations using nonparabolic Kane model for energy dispersion
in bulk PbTe show that the temperature dependence of the density of states effective mass
md is very important to have good agreement with experiment for electrical conductiv-
ity a and thermopower S. The calculations for PbTe using ab initio electronic structure
results for the energy dispersion overestimate 0 and underestimate S at a given temper-
ature. This is a consequence of the fact that in ab initio electronic structure results the
temperature dependence of md can not be taken into account. Transport calculations for
LAST-m systems show a small enhancement of the power factor relative to PbTe is pos-
sible. Although increase in the DOS can enhance the thermopower S, it tends to reduce
electrical conductivity 0. However, the net effect is a small increase in the power factor.
Therefore, we believe that the major contribution to the enhanced ZT of LAST-m systems
compared to PbTe comes from a strong reduction in the phonon thermal conductivity due

to the formation of microstructures (as seen in transmission electron microscopy). [72]

Electronic structure calculations within density functional theory (DFT) offer varieties
of tools to determine the stability of compounds, their atomic geometries, their band struc-
ture 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 new potential thermoelectric materials. In order to characterize the thermoelec—
tric properties of a material, transport calculations for a, S, and n have to be performed.

128

Unfortunately, to perform such calculations one has to make approximations whose valid-
ity is not fully under control. By this it is meant that we do not have an estimate of the
magnitude of the made errors. Such an approximation is the rigid band model, where the
electronic band structure of doped semiconductors (requirement for a good thermoelectric)
is assumed to be the same as for undoped semiconductors. Another approximation is to
assume that the electronic band structure at finite temperature is the same as that at OK. In
this thesis, we show that this approximation can not be made for PbTe and LAST—m com-
pounds and particularly for compounds with strong electron-phonon coupling. For such
systems finite temperature DFT calculations are required. The other approximation is the
relaxation time approximation made in describing the electronic scattering mechanisms.
Future calculations of transport coefficients should improve upon this. Finally, we mention
the problem of LDA/GGA in underestimating the gap band. [2] For small band gap semi-
conductors, which most good thermoelectrics are, this could be a problem in calculating
the transport coefficients. Although methods like GW approximation have been developed
in recent years, their applications to study the transport properties of complex materials is
still in its infancy. One expects considerable theoretical effort to be put in this direction
in the future. Therefore, we believe that for understanding transport properties of complex

systems a very close interaction between theory and experiment is necessary.

129

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