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This is to certify that the
dissertation entitled
Electronic Structure of Narrow Gap Semiconductors:
Understanding Gap Formation and
Thermoelectric Properties

presented by

Paul Melvin Larson

has been accepted towards fulfillment
of the requirements for

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MS U i: an Affirmative Action/Equal Opportunity Institution 0- 12771

 

 

 

 

 

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MAY BE RECALLED with earlier due date if requested.

 

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DATE DUE

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ELECTRONIC STRUCTURE OF NARROW GAP SEMICONDUCTORS;
UNDERSTANDING GAP FORMATION AND THERMOELECTRIC
PROPERTIES

By

Paul Melvin Larson

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillmentof the requirement for the degree of

DOCTOR OF PHILOSOPHY
Department of Physics & Astronomy

2001

 

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ABSTRACT

ELECTRONIC STRUCTURE OF NARROW GAP SEMICONDUCTORS:
UNDERSTANDING GAP FORMATION AND THERMOELECT RIC
‘ PROPERTIES

By

Paul Melvin Larson

Electronic band structure calculations are invaluable theoretical tools to under-
stand structural, transport, and optical properties of materials. We have used this
tool in the search for new high performance thermoelectric materials, which are usu-
ally narrow-gap semiconductors. We have studied the electronic structures of, these
systems both to understand which properties of the band structure are most impor-
tant for thermoelectric prOperties and the nature of the gap formation.

Narrow-gap semiconductors lie between metals and wide-gap semiconductors, so
understanding the nature of the gap formation is very important. The small band gaps
in the systems we have studied generally arise from hybridization between different
hands. We have used the local density approximation (LDA) and the generalized
gradient approximation (GGA) within density functional theory (DFT). These have
been implemented using the full-potential linearized augmented planewave (FLAPW)
method within the WIEN97 package. This state-of-the—art method is among the most
accurate methods for calculating-the electronic structure of solids.

We have studied four classes of compounds. These include the half-Heusler com-

 

 

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pounds, the ternary Zintl-phase compounds, the simple chalcogenides, and the com-
plex chalcogenides.

The ternary half-Heusler compounds, considered having a stuffed N aCl structure,
show promising thermoelectric properties. The band gap formation is understood by
starting with the semi-metallic binary N aCl compounds from which they are formed.
Adding the transition (or noble) metal atom causes a strong p-d hybridization near
the Fermi energy which Opens up the band gap. This hybridization also leads to
highly anisotropic effective masses at the conduction band minimum which are found
in the best thermoelectric materials. Similar band gap formation is found in the
ternary Zintl-phase compounds which are considered a stuffed Th3P4 structure. The
band gaps in these ternary compounds are larger than experiment, unlike what one
usually finds in LDA/GGA calculations. We explain this discrepancy by the noting
that the position of the d-levels are too high in energy in LDA/GGA calculations
which increases the hybridization near the Fermi energy and gives a larger gap.

The best known thermoelectric materials belong to the class of simple chalco-
genides, including Bl2T€3 for room temperature (300 K) and PbTe for high tempera-
ture (700 K) applications. In contrast to the previous materials, here the relativistic
effects are important and the band gaps are found to agree very well with experiment.
We also find that the band structure of these materials show large band degeneracy
and highly anisotropic effective masses at the band extrema as well as a narrow band
gap, ideal for good thermoelectrics. We have searched for these properties in complex
chalcogenides BaBiTe3, CsBi4Te6, and KzBlgsela. We have found that the best ther-
moelectric of these three, CsBi4Te6, has the highest anisotropic effective mass ratio,

similar to what we had found for BigTe3.

DEIDICATION

To my parents

iv

ACKNOWLEDGMENTS

I would like to acknowledge the immeasurable support from my advisor, Professor
S.D. Mahanti. I would also like to acknowledge Professor Mercouri G. Kanatzidis
in Chemistry and Dr. Tim Hogan in Engineering for collaborations on the thermo-
electricity project. Especially helpful have been Dr. Duck-Young Chung and Dr.
Sandrine Sportouch. I would like to acknowledge the collaborations with the photoe-
mission experimentalists Dr. Rong Liu, Viktoria Greanya, and Wayne C. Tonjes.

I would finally like to acknowledge funding from DARPA Grant No. DAAG55-97-
0184.

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TABLE OF CONTENTS

LIST OF TABLES .................................................................................... xi
LIST OF FIGURES ................................................................................. xiii
LIST OF SYMBOLS OR ABBREVIATIONS ......................................... xxi
CHAPTER 1.
INTRODUCTION ........................................................................................ 1
CHAPTER 2.
THERMOELECTRICIT Y ....................... . .................................................... 4
2.1. GENERAL PRINCIPLES .................................................... 4
2.2. MAXIMIZING COP AND EFFICIENCY .......................... 11
2.3. ZT FROM BOLTZMANN TRANSPORT
EQUATIONS ............................................................. 14
2.4. GUIDANCE FOR BAND STRUCTURE ........................... 17
CHAPTER 3.
FIRST-PRINCIPLES AB INI T10 ELECTRONIC
STRUCTURE CALCULATIONS .............................. 22
3.1. SOLVING THE DIRAC EQUATION ................................. 23
3.1.1. FULLY-RELATIVISTIC
ON E-PARTICLE EQUATIONS ....................... 23
3.1.2. SCALAR RELATIVISTIC APPROXIMATION ........ 24
3.2. DENSITY FUNCTIONAL THEORY (DFT) ..................... 27

3.2.1. DERIVATION OF KOHN-SHAM (KS)
' EQUATIONS ................................................... 27

vi

3.2.2. PHYSICAL PROPERTIES OF DFT

CALCULATIONS ........................................... 32
3.3. METHODS OF CALCULATION (DFT) ............................ 33
3.3.1. PLANEWAVES AND ATOMIC ORBITALS ............. 34

3.3.2. LINEARIZED MUFFIN TIN ORBITAL
METHOD WITHIN THE ATOMIC SPHERE
APPROXIMATION (LMTO-ASA) ......................... 35

3.3.3. FULL—POTENTIAL LINEARIZED
AUGMENTED PLANEWAVE METHOD

(FLAPW)) .............................................................. 38
3.4. OTHER METHODS OF CALCULATION
(NON-DFT) ............................................................ 40
3.4.1. HARTREE—FOCK(HF) METHOD ............................ 40
3.4.2; GW METHOD... ......................................................... 41
CHAPTER 4. '
HALF-HEUSLER COMPOUNDS .............................................................. 44
4.1 GAP FORMATION AND ELEMENTAL
REPLACEMENTS .................................................. 46
4.1.1. GAP FORMATION IN YNiSb AND ZrNiSn ........... 46

4.1.2. ELEMENTAL REPLACEMENTS: CHANGING
THE PNICTIDES ................................................... 55

4.1.3. ELEMENTAL REPLACEMENTS: CHANGING
THE TRANSITION METAL .................................. 57

4.1.4. ELEMENTAL REPLACEMENTS: ROLE OF
FILLED, UNFILLED AND PARTIALLY-
FILLED f-LEVELS ................................................. 59

4.1.5. ELEMENTAL REPLACEMENTS OF NiO
WITH AuH/AgH/Cu1+ ......................................... 64

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4.1.6. ELEMENTAL REPLACEMENTS OF N10
WITH COI‘/Rh1‘ ................................................... 67

4.1.7. ELEMENTAL REPLACEMENTS OF N10
WITH Fez" /Ru2‘ .................................................... 69

4.1.8 ELEMENTAL REPLACEMENTS OF Ni0
WITH Mn3‘ ............................................................ 70

4.2. STRUCTURAL STABILITY OF HALF-
HEUSLER COMPOUNDS ..................................... 72

4.2.1. TOTAL ENERGY OF THE TERNARY
SYSTEMS IN DIFFERENT
CONFIGURATIONS ............................................... 72

4.2.2. ANALYSIS OF THE STABILITY OF
THE TERNARY COMPOUNDS BASED

ON STABILITY OF BINARY PAIRS ...................... 82
4.2.3. ENERGETICS OF ANTI-SITE DEFECTS ............... 87
4.3. THERMOELECTRIC PROPERTIES OF
HALF-HEUSLER COMPOUNDS ........................... 94
CHAPTER 5.
TERNARY ZINTL PHASE COMPOUNDS .............................................. 99

5.1.. CRYSTAL STRUCTURE AND ZINTL CONCEPT ........ 100

5.2. ELECTRONIC STRUCTURE OF ZINTL PHASE
(STUFFED Th3P4) COMPOUNDS ............ 103

5.2.1. GAP FORMATION AND ROLE OF
d-HYBRIDIZATION .................................... 103

5.2.2. COMPARISON OF Zr3Ni3$b WITH
HALF-HEUSLER COMPOUND ZrNiSn ..... 104

viii

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5.2.3. UNUSUAL DISCREPANCY BETWEEN THE
BAND GAPS FROM THEORY

(LDA/GGA)AND EXPERIMENT ............... 108
5.2.4. COMPARISON OF. ngNi38b4 WITH Hf3Ni33b4
AND Ti3Ni3Sb4 ............................................. 110
CHAPTER 6.
SIMPLE CHALCOGENIDES............................................; ...................... 114
6.1. PbS, PbSe, PbTe ............................................................... 114

6.1.1. BAND STRUCTURES OF PbS, PbSe, AND PbTe 114

6.2.2. THERMOELECTRIC PROPERTIES OF
LEAD SALTS ............................................. 122

6.2. BigTeg, Sb2T63, Bi28e3, AND RELATED SYSTEMS ........ 123

6.2.1. BizTeg CRYSTAL STRUCTURE ............................... 123
6.2.2. BAND STRUCTURE AND CAP FORMATION
IN Bi2T€3 ........................................................ 126
6.2.3. BAND STRUCTURE AND CAP FORMATION
OF BI2T€3 ANALOGS ................................... 133
A. 312883 AND SbgTBg ................ 133

B. Bi2Te28e, BizTegs,
BigSegTe AND SngezTe 136

C. Bi283 ...................................... 140
6.2.4. EFFECTIVE MASSES OF THE BAND
EXTREMA OF BI2T€3 ................................. 143
6.2.5. COMPARISON OF THE BAND
STRUCTURE WITH
PHOTOEMISSION EXPERIMENTS ........... 148

CHAPTER 7.

COMPLEX CHALCOGENIDES .............................................................. 164
7”]. BaBiTe3 ................................... . ......................................... 165
7.1.1. GENERAL FEATURES OF THE BAND
STRUCTURE OF BaBiTe3 ......................... 165
7.1.2. EFFECTIVE MASSES AND THERMOELECTRIC
PROPERTIES OF BaBiTe3 ......................... 172
7.2 CSBI4T€6 ............................................................................ 175

7.2.1. CRYSTAL STRUCTURE AND OBSERVED
PROPERTIES ........ - ..................................... 175

7.2.2. GENERAL FEATURES OF THE BAND
STRUCTURE AND GAP FORMATION

OF CSBI4T€6 ................................................ 179

7.2.3. EFFECTIVE MASSES AND THERMOELECTRIC
PROPERTIES OF CSBI4T€5 ........................ 181
7.3. K231884413 ........................................................................... 187
7...31 a-KgBi38e13....... ........................................................ 188
7...32 fi-KgBigsela ............................................................... 193

CHAPTER 8
CONCLUSIONS ........................................................................................ 201
BIBILIOGRAPHY .................................................................................... 206
X

LIST OF TABLES

Table I. 8-Electron Rule of Elemental Covalent Crystals[64] ................

Table II. Experimental and calculated equilibrium lattice
parameters and total-energy differences for half-Heusler

systems (GGA) .....................................................................

Table III. Muffin-tin radii of half-Heusler systems ................................

Table IV. Total-energy differences for NinSn and
different binary systems with fixed lattice parameters
calculated using the pseudopotential LDA

[Ogut and Rabe[59]] and using LAPW within LDA and
GGA .....................................................................................

Table V. Relaxed equilibrium lattice parameters and total-energy
differences for NinSn and different binary subsystems

..... 46

..... 76

..... 80

83

calculated using LAPW within LDA and GGA ......................... 85
Table VI. Energetics Of Site-switching on NaCl substructure of

NinSn supercell (GGA) ........................................... 89
Table VII. Energetics of site-switching on Ni sublattice

of NinSn supercell (GGA) ........................................................ 92
Table VIII.'Calculated reciprocal effective masses of ngNi38b4,

Hf3NI3Sb4, and ZI'3Pt3Sb4 ......................................................... 113
Table IX. Experimental energy gaps for PbS, PbSe, and PbTe

as a function of T (For experimental references,

see Dalven[101]) ......................................................................... 116
Table X. Comparison Of calculated components of the

reciprocal effective mass tensor with experiment[122]

(a; = [ml/mgr1 and at = [mt/me]‘1) ....................................... 123

Table XI. Lattice constants in atomic units (a.u.) and u and 12
parameters for BI2T€3 analogs in the combined
representation of the unit cell with hexagonal lattice
constants and rhombohedral atomic coordinates used
in these calculations (Bi at :l:(u, u, 11), Tel at i(v, v, v),
and Te2 at (0, 0, 0) in BI2T93. For details see Wyckoff[63]) ..... 134

Table XII. Theoretical and experimental values of the effective
mass tensor parameters associated with the conduction
band minima and valence band maxima for Bi2T€3 .................. 146

Table XIII. Comparison of the photoemission and calculated DOS
peaks for (a) BI2T€3 (b) BizTegse, (c) Bigseg, and
(d) Bi283. For a, b, and c the theoretical reference
energies are chosen such that the highest valence band peak
positions coincided with experiment.[149] For (I, the theoretical
reference energies are chosen so that the top two theoretical
bands lie equidistant from the highest experimental peak .......... 160

Table XIV. Theoretical values of inverse effective mass
parameters associated with the conduction and
valence band extrema of BaBiTe3 ............................................... 174

Table XV. Effective mass associated with the valence and
conduction band extrema along three princiapl axes
and the angles these axes make with a set of three
suitably chosen orthogonal axes (see text) ................................ 183

Table XVI. Components of the reciporal effective mass tensor
for a-KzBigsem ........................................................................... 192

xii

LIST OF FIGURES

Figure 1. Schematic Of a two-leg thermoelectric device. Current
travels from the right to the left, up through the n-type leg
and down the p-type leg to produce cooling on the
upper surface[1] ........................................................................... 5

Figure 2. Reduced coefficient of performance nmu/anamot (5 ¢,)
for various values of ZT[17] .......................................................... 9

Figure 3. Reduced efficiency emu/£00m“ (E 77,.)
for various values of ZT[17] ...................................... 10

Figure 4. Plot of S, a, n, and ZT for an idealized semiconductor

as a function of carrier concentration, n. S decreases with
11 while a and I»; increase. The maximum ZT occurs near

n = 1019/cm3l2] ........................................................................... 13
Figure 5. Plot of ZT vs. B parameter (Equation 23)[24] ......................... 16
, Figure 6. Conduction and valence bands of HgTe (left) and CdTe

(right). Upon alloying (Cd1_ngxTe) the bands cross[18] ........... 18
Figure 7. Semiconducting gap Opening due to p—d hybridization ............. 19
Figure 8. Off-axis gap formation due to spin-orbit hybridization ............ 20

Figure 9. Overlapping atomic spheres surround each atom of a
crystal in the ASA[50] .................................................................. 35

Figure 10. Partitioning of the unit cell into atomic spheres (I) and
an interstitial region (II)[53]. ........................................................ 38

Figure 11. Comparison of characteristic direct and indirect
LDA, CW, and experimental energy gaps for all
semiconductors for which GW calculations have been
reported (For references see Aulbur[31])............ .......................... 43

xiii

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Figure 12. Two different perspectives of the cubic half-Heusler
structure (YNiSb) Left: Ni. atoms inserted in a YSb
NaCl-type lattice. Right: Y atoms inserted in a N iSb
ZnS-type lattice .......................................................................... 45

Figure 13. FCC Brillouin zone .......... , ...................................................... 45

Figure 14. Comparisons Of the Ni (1 orbital characters in YNiSb
and .YNigsb. The Size of the circles is proportional to the
orbital character ........................................................................... 48

Figure 15. Electronic structure Of ZrNiSn and YNiSb ............................ 49

Figure 16. Comparison Of the different orbital characters of ZrSn
(a) Sn p and (b) Zr (1, and YSb (c) Sb p and ((1) Y (I. The Size
of the circles is proportional to the orbital
character ....................................................................................... 51

Figure 17. Comparison Of the different orbital characters Of ZrNiSn
(a) Sn p, (b) Zr d, and (c) Nid, and YNiSb (d) Sb p, (e) Y d,
and (f) Ni d. The Size Of the circles is proportional tO the
orbital character ........................................................................... 53

Figure 18. The electronic structures of ZrHeSn and ZrNiSn using
the same lattice parameter ......................................................... 54

Figure 19. Comparison of the electronic structures Of YNiAs,
YNiSb, and YNiBi. The size of the circles is proportional
to the pnictide s orbital character ............................................... 56

Figure 20. Comparison of the electronic structures Of YNiSb,
YPdSb, and YPtSb. The Size Of the circles is proportional
to the metal 3 orbital character ................................................... 58

Figure 21.’ Electronic structures of (a) YNiSb, (b) FCC LaNiSb,
(c) LuNiSb, and (d) hexagonal LaNiSb. The Ni d bands and
La and Lu f bands are Shown ...................................................... 60

Figure 22. Electronic structures of YbSb (a) without and (b) with 801.
The corresponding DOS (c) without and (d) with 801 are
also given ..................................................................................... 61

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. Figure 23. Electronic structures of YbNiSb (a) without and (b) with SOI.
~ The corresponding DOS (c)'without and (d) with 801 are
also given ...................................................................................... 63

Figure 24. Electronic structures of semimetallic MgCuBi
and MgAgAS ................................................................................. 65

FigUre 25. Electronic structures of LuAuSn and LuNiSb. The Ni (1
bands lie closer to E: than the Au ,d bands. The size Of the
circles is proportional to the orbital character ............................. 66

Figure 26. Electronic structures of TiCoSb and TiNiSn. The Co d
bands lie closer to E; than the Ni (1 bands. The size of the
circles is proportional to the orbital character ............................. 68

Figure 27. Electronic structures Of TiRth ............................................. 69

Figure 28. Electronic structures of NbFeSb and YNiSb. Note the
difference in curvature near the X point in the LCB .................... 70

Figure 29. Calculated DOS of half-metallic MnNiSb for (a) up and
(b) down spin states .................................................................... 71

Figure 30. Half-Heusler crystal structure (a) XY Z, (b) ZYX,
and (c) YXZ ............................................................................... 74

Figure 31. Total energies as a function of lattice parameter for the
three different configurations of NiTiSn (the lowest energy is
set to zero). ............................................... - ................................... 75

Figure 32. Total energies as a function of lattice parameter for the
three differentconfigurations of NinSn (the lowest energy is
. set to zero) .................................................................................. 75

Figure 33. The electronic structures of (a) XYZ, (b) ZYX, and
(c) YXZ configurations of NiTiSn ............................................... 77

Figure 34. The electronic structures of (a) XYZ, (b) ZYX, and
(c) YXZ configurations Of NinSn.......... ..................................... 78

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(c) YXZ configurations Of NinSn. The size Of the circles
overlying band structure is directly prOportional tO the
strengths of the Ni d orbital character ....................................... 81
Figure 36. Crystal structures of (NiMP)4 supercell for
(a) Ni4(Zr38n)(Sn3Zr), (b) (Ni3Sn)Zr4(Sn3Ni), and
(c )(Ni3Zr)(Zr3Ni)Sn4 ................................................................... 88

Figure 37. Electronic structures of (a) Ni4Zr4Sn4,
(b) Ni4(Zr38n)(Sn3Zr), (c) (Nigsn)Zr4(Sn3Ni),

and (d)(Ni3Zr)(Zr3Ni)Sn4 ............................................................ 90
Figure 38. Crystal structures of (NiMP)4 for rearrangements

of the Ni lattice ........................................................................... 91
Figure 39. Electronic structures of (a) Ni4Zr4Sn4 - 38a,

(b) NI4ZI’4SH4 - 38b, and (b) NI4ZI‘4SH4 - 38C ............................. 93
Figure 40. Electronic structures of TthSb. Note the

Off-axis LCB ................................................................................ 97
Figure 41. Electronic structures of ThNiSn. Note the LCB

at a general point ........................................................................ 98
Figure 42. (a) Crystal structure Of Y3Sb4 and (b) Y3Ni3Sb4 ............... 101
Figure 43. BCC Brillouin zone. ........................................................... 101

Figure 44. The (a) Zr (1 and (b) Sb p orbital characters
0f ZI‘3Sb4 .................................................................................... 105

» Figure 45. The (a) Zr (1, (b) Sb p, and (c) Ni d orbital characters
Of ZI‘3NIng4 ............................................................................... 107

Figure 46. Electronic structures of (a) Zr3Ni3Sb4, (b) Hngi3Sb4', and
(C) ZI‘3Pt3Sb4 .............................................................................. 111

Figure 47. Electronic structures of (a) PbS, (b) PbSe, and PbTe, using
the experimental lattice parameters (corresponding to 4 K) ...... 117

 

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Figure 48. Total energy Of PbS as a function of lattice parameter (in an.)
obtained in GGA. The points were plotted using the computer
program KALEIDAGRAPH. [120] The experimental value IS given
by the arrow ............................................................................... 120

Figure 49. Electronic structures Of (a) PbS, (b) PbSe, and PbTe, using
the relaxed lattice parameters ..................................................... 121

Figure 50. Crystal structure of Bi2T€3 along the XZ plane Showing the
quintuple layer leaves of Te-Bi-Te-Bi—Te separated by a
Van der Waals gap ...................................................................... 124

Figure 51. (a) Rhombohedral and (b) hexagonal representations Of
the unit cell of BI2T83. In (b) the Te atoms are Open circles
and the Bi atoms are shaded circles ........................................... 125

Figure 52. Brillouin zone of BI2T€3 (corresponding to the '

rhombohedral unit cellthe Bi atoms are Shaded circles ............... 125
Figure 53. Band structure of B_i2Te3 without SOI ................................. 128
Figure 54. Band structure Of BI2T€3 with SOI ...................................... 128

Figure 55. Band structure of BI2T€3 with SOI. Included are
(a) total band structure and the orbital characters Of the bands:
(b) Bi p, (c) Tel p, and (d) Te2 p. The Size Of the circles
overlying band structure is directly proportional to the
strengths Of the orbital character ................................................ 131

Figure 56. Total and partial DOS for Bi2Te3 with SOI. Included are
(a) total DOS partial DOS: (b) Bi p, (c) Tel p, and (d) Te2 p..132

Figure 57. Band structure Of Bi2Se3 (a) without and (b) with 801 ....... 135
Figure 58. Band structure of szTe3 (a) without and (b) with $01 ...... 136

Figure 59. Band structures Of BigTezSe (a) without and (b) with SCI
and BigTe2S (c) without and (d) with SOI ................................. 137

Figure 60. Band structure Of SbgTeQSe (a) without and (b) with SOI..139

Figure 61. Band structures Of ‘BigsegTe (a) without and (b) with S01
and Sb2Se2Te (c) without and (d) with 301 ............................... 140

xvii

p.
p. ‘1“.

l.,’..,

I.

r“ .
5 AL...

.5.

b‘” -
Ad...

 

 

 

......

'-~A-

F‘.‘"

-L ..

...,1

.y"

Figure 62. (a) Orthorhombic unit cell of Bi283 (shown in the a-b plane)
= and its (b) corresponding~Brillouin zone............~ ............................ 141

Figure 63. Band structure of Bigsg (a) without and (b) with 801 ........... 142
Figure 64. Fitting of the energy band near the conduction band

minimum to a parabola for BI2T83 using the computer

program KALEIDAGRAPH[120] ......................................................... 145

Figure 65. Positions Of the electron and hole pockets lying on the
reflection plane of the Brillouin zone for Bi2Te3 for

(a) CBM and (b) VBM .................................................................. 147

Figure 66. Valence band photoemission (UPS) and conduction band
inverse photoemission (IPES) for Bi2S3, BigSeg, BizTEzse, and BI2T€3
crystals. UPS and IPES Spectra for each crystal are connected at Ep.
Energies are referred to Ep. (courtesy of Yoshifumi Ueda).[149] ..... 151

Figure 67. Total DOS for (a) BI283, (b) Bi28e3, (c) BizTegse, and
(d) BI2T83. The arrows indicate the positions of the photoemission
peaks in the work by Ueda et al.[149] ............................................. 152

Figure 68. Photoemission spectra for BI2T€3 by Greanya et al.
Peaks 1-6 are indicated by arrows. For the purpose of
comparison, the energy reference (zero of energy) was set
such that the lowest binding energy peak in the spectra
coincides with the position Of the corresponding peak in
the data of the previous UPS experiment by Ueda et al.[149] ......... 156

Figure 69. Photoemission spectra for Bi2Se3 by Greanya et al.[139]
Peaks 1-9 are indicated by arrows. For the purpose Of
comparison, the energy reference (zero of energy) was set
such that the lowest binding energy peak in the Spectra
coincides with the position of the corresponding peak in
the data of the previous UPS experiment by Ueda et al.[149] .......... 157

Figure 70. Total DOS for (a) BI2T83 and (b) Bi2Se3Se. The arrows
indicate the positions of the photoemission peaks seen by
Greanya et al.[139] For the purpose Of comparison, the energy
reference (zero of energy) was set such that the lowest
binding energy peak in the spectra coincides with
the position of the corresponding peak in the data of
the previous UPS experiment by Ueda et al.[149] ............................ 158

xviii

Figure 71. Projection Of the crystal structures in the yz plane of
(a) orthorhombic BaBiTe3 and (b) rhombohedral BI2T83. The
Bi / Te blocks are highlighted in the shaded area for BaBiTe3 ...... 167

Figure 72. Brillouin zone of BaBiTe3 .......... 167
Figure 73. Infrared absorption spectra of BI2T€3 and BaBiTe3.

The semiconductor gaps are indicated in the Spectra .................. 168
Figure 74. Band structure of BaBiTe3 without SOI .............................. 170
Figure 75. Band structure Of BaBiTe3 with 801 ................................... 170
Figure 76. Monoclinic crystal structure of CsBiITeG ............................. 176

Figure 77. (a)Triclinic crystal structure and (b) Brillouin zone
of CsBi4Te5 used in this calculation ............................................ 177

Figure 78. Comparison of ZT for the best current doping of CSBI4T€5
and optimally-doped Bi2_bexTe3 alloy. The peak for CsBi4Te6

(ZT~0.8) occurs at 225 Kelvin ..................................................... 178
Figure 79. Band structure Of CsBi4Te5 (a) without and (b) with SOI...180

Figure 80. Quasi-two dimensional Sheets contributing to the hole
transport overlying the crystal structure of CsBi4Te6 ................... 186

Figure 81. Projection of the crystal structure of a-KgBlgselg viewed
down the b—axis. BI2T€3-, CdIg-, and Sbgseg-type building
blocks in the structure are highlighted by the shaded areas ......... 189

Figure 82. Brillouin zone Of a-KzBiSSem ................................................ 190
Figure 83. Band structure of a-KgBigSelg (a) without and

(b) with $01 ................................................................................. 191
Figure 84. Crystal structure of fi-KzBiSSem .......................................... 194
Figure 85. Brillouin zone of fl-KgBigSem ............................................... 195

Figure 86. Band structure Of ,B-K2B138813 — Configurationl
(a) without and (b) with 801 ...................................................... 196

xix

Figure 87. Orbital characters Of (a) .Bi8 and (b) Se10 overlying the
band structure of fi-KgBigSem - Configurationl with SOI. The
size of the circles overlying the. band structure is
directly proportional to the strength Of the orbital character ......... 197

Figure 88. Band structure Of ,B-KgBigSe13 - Configuration2
(a) without and (b) with SOI .......................................................... 198

Figure 89. Band structure of fl-KgBiSSelg - Configuration3
(a) without and (b) with SOI .......................................................... 199

 

.E.

 

~h .H ‘ \\ v. Q fll. . Via. v A .. .mL
.1. _ .... ... WI T. G nlo 2... 21 U. l J 1K LL
. T. E E

LIST OF SYMBOLS OR ABBREVIATIONS

07,- = [m,-/m,..]’1 = Reciprocal effective mass associated with m,-
a,,- = [m/ mg];-1 2 Effective inverse mass tensor
7 = Band degeneracy factor

n = Thermal conductivity

6 = Energy or efficiency

7) = Coefficient of performance

mph —-— Phonon thermal conductivity

u = Chemical potential

p = Electrical resistivity

a 2 Electrical conductivity

A“ = LDA/GGA band gap correction

a.u. = Atomic unit (0.529177 A)

A 2 Electron affinity

ARPES 2 Angle resolved photoemission SpectrOSCOpy
ASA = Atomic sphere approximation

BCC 2: Base centered cubic

BZ = Brillouin zone

c = Speed of light

CBM = Conduction band minimum

COP = Coefficient of performance

E 2 Electric field

e = Electron charge

eV = Electron-volts

DFT = Density functional theory

DOS = Density of states

E = Energy

Ep = Fermi energy

E90,, = Band gap

FCC = Face centered cubic

FLAPW = Full—potential linearized augmented planewave
G = Reciprocal lattice vector

GGA = Generalized gradient approximation

h = Planck’s constant/277

HF = Hartree-Fock

HVB = Highest valence band

I = Ionization energy

J = Electric current

k = momentum vector

k3 = Boltzmann constant

K 2 Thermal conductance or Kelvin degrees
K = k + G

KS = Kohn-Sham

LAPW = Linearized augmented planewave
LCB 2 Lowest conduction band

LDA = Local density approximation

LMTO = Linearized muffin-tin orbital

me = Bare electron mass

m,- = Band effective mass in direction i
MTR = Muffin-tin radius

OR = Ogut & Rabe

P = Power

PES = Photoemission Spectroscopy

Q 2 Thermal current

quuicr = Peltier heat

R 2 Electrical resistance

S = Seebeck coefficient

T =, Temperature

TB 2 Tight-binding

TDOS = Total density of states

SOI == Spin-orbit interaction

UPS 2 Ultraviolet photoemission Spectroscopy
VBM = Valence band maximum

KC 2 Exchange-correlation

XPS 2 X-ray photoemission spectroscopy
ZT = SZO‘T/K. = Thermoelectric figure of merit

xxii

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CHAPTER 1
INTRODUCTION

Technological needs have fueled a. search for new thermoelectric materials. These
materials, which can be used in either solid state refrigeration or power generation,
produce a temperature gradient from an applied electric current or, conversely, an
electric current from an applied temperature difference. These devices have no moving
parts, produce no waste products to be released into the environment, and are scalable
to any size needed. They have been used as power generation in deep space probes
and as cooling units in portable refrigerators. The problem of using these materials
on a wider scale is that their coefficient of performance (or efficiency) is much lower
than current technologies.[1, 2, 3]

AS will be discussed in detail in the next chapter, the coefficient Of performance
(or efficiency) of a thermoelectric device is determined by the dimensionless quantity
ZT given by

(1)
ZT : SQUT/I'i,

where S is the Seebeck coefficient, 0' is the electrical conductivity, T is the average
absolute temperature, and n is the thermal conductivity. The highest values of ZT
are found to be about 1 and are seen in doped narrow-gap semiconductors having a
large Seebeck coefficient and electrical conductivity but a low thermal conductivity.
Finding a material with ZT z 3 would make thermoelectric devices competitive with
Several. existing technologies for cooling and power generation, but any improvement

0f ZT would lead to immediate applications.[1, ’2, 3]

The search for new thermoelectric materials requires both experimental (synthesis -
. and measurement) and theoretical studies, especially electronic structure calculations.[4]
Thermoelectric properties depend on subtle features Of the electronic structure near
the Fermi energy (Ep), so electronic structure calculations provide a method to pre-
dict and understand new thermoelectric materials- The Optimum materials usually
have band gaps on the order of 10kBT (where T is the Operating temperature), de-
generate bands near Ep, and anisotropic transport Of the doped system.[1, 2, 3, 4]
In this dissertation, we will discuss the band structure of four general classes of
narrow-gap semiconductors to understand both the formation of their semiconduct-
ing gaps and their thermoelectric properties. These materials are synthesized based
on the guiding principle Of combining Simple structures, such as cubic or rhombohe-
dral, to form more complex structures with a more glass-like thermal conductivity
and yet maintaining the high electrical conductivity Of a simple crystal[5] (electron
crystal/phonon glass[6]). The first Of these classes of materials is the half-Heusler
. compounds formed by stuffing transition metal ions into the pockets Of a NaCl
lattice.[7, 8, 9] This class provides numerous elemental replacements Of individual
atoms which can affect the electronic structure. The half-Heusler compounds are
closely related to the next class, the ternary Zintl phase compounds, where Ni atoms
are stuffed in a more complex Th3P4 lattice in order to produce lower values of n.[10]
The third class consists of the simple chalcogenides, the best known bulk thermoelec-
tric materials for almost 40 years, including PbTe at high temperatures and BI2T€3
at room temperature.]11, 12] The final class consists Of complex structures formed
from combinations of units with the crystal structure of the cubic PbTe and the
rhombohedral BI2T83, such as BaBiTe3[11, 13], CsBi4Te6]14, 15], and K2BiSSe13I16].

Understanding the electronic structure of these materials gives us an insight into

which features of the electronic structure, such as band degeneracies, Size of band
gap, and anisotropic effective masses, are most important for high values of ZT. We
hope to Show that electronic structure calculations provide an invaluable theoretical

tOOl to understand the properties of thermoelectric materials.

CHAPTER 2.
THERMOELECTRICITY

2.1. GENERAL PRINCIPLES

Thermoelectricity is a transport. property which relates the thermal gradient to the
electrical current in a material. Thermoelectric devices can be used either to trans-
form an induced electric current into a temperature gradient in a material, used for
refrigeration, or transform an induced temperature gradient into an electric current,
used for power generation.[l, 2, 3, 4]

Figure 1 shows a schematic of a two-leg thermoelectric device which can be used
for heating or cooling. The device consists of an n-dOped sample, where the carriers
are electrons, and a p—doped sample, where the carriers are holes. AS the current
enters the device from the right, the Slightly more energetic electrons (for a typical
doping, 1019/cm3, E~0.1 eV) travel against the current, thereby heating the bottom
of the device and leaving the top cooler. AS the current goes down the left leg, the
slightly more energetic holes (E~0.1 eV) travel in the direction of the current, again
heating the bottom of the device and leaving the top cooler. An equilibrium is reached
as phonons transmit the heat back through the lattice from the hot end to the cold
end. A large temperature difference can be best maintained if the system has a low
thermal conductivity.[17, 18]

The main disadvantage Of thermoelectric devices is their low coefficient of perfor-
mance (efficiency) for use in refrigeration (power generation) compared with more
conventional technologies, such as Freon cooling. The best thermoelectric devices

have only about 1 / 3 the efficiency of Freon refrigerators. Thermoelectric devices fill a

 

3-1.
HTIL J

 

 

f")

 

 

niche market for cooling computer chips and‘infrared detectors as well as their largest

use in automobile beverage coolers which plug into the car’s cigarette lighter.[‘19]

Figure 1. Schematic Of a two-leg thermoelectric device. Current travels from
right to left, up through the n-type leg and down the p-type leg

to produce cooling on the upper surface.]1]

 

 

 

 

 

 

 

 

 

If we consider no coupling between these currents, the electrical and thermal cur-
rents depend On their sources Of the electric field and thermal gradient, respectively,
(2)
J 2 0E,
Q = - KVT,
(3)

where J is the electrical current, Q is the thermal current, E is the electric field,

 

1“

‘FL

.‘.

 

Ty
kl

”'L’.
A‘ .

 

 

 

and VT is the temperature gradient. In any real material, the. electrical current flow
causes Slightly more energetic electrons to move from one end of the material to the
other, producing a temperature gradient. Conversely, a temperature gradient induces
a movement of electrons Opposite the direction of this gradient. In order to explain
this interrelation, one defines a quantity S known as the Seebeck coefficient (also
called the thermoelectric coefficient or the thermopower) as a coupling term between
the thermal and electrical currents produced and the sources of the electric field and
the temperature gradient[17, 18, 20, 21], i.e.,
(4)
J = OIE - SVT]
Q : (O’TS)E - KVT
(5)
The coefficient of performance can be derived for a thermoelectric material by
studying the rate at which heat is removed only from the cold reservoir (at temper-
ature TC). This rate can be written as a sum Of three terms. The first is the Peltier
cooling due to the current in the material
(6)
QPeltier = STCJ-

The second term is the Joule heating, written as a negative quantity since it adds

heat.
(7)
quule . I ' %JR2

Here R is the resistance rather than the resistivity (R = (a A/L)‘1 where L is

the length and A the area of the device). The third term is the heat due to thermal
transport due tO both electrons (or holes) and phonons which move from the hot
reservoir to the cold reservoir. This is also written as negative Since it adds heat to
the cold resevoir.
(8)
thermal = - K AT

Here K is the total thermal conductance rather than the thermal conductivity (K r:
K. A / L where L is the length and A the area of the device) and AT is the temperature
difference across the device. The total heat removed from the cold reservoir is then[17]
(9)

(1c 2 QPeltz‘er + (1.1er + QThermat = STCJ - %JR2 - K AT.

Integrating both sides Of Equation 4 , we can get a relationship for the voltage in
the system.
(10)
V 2 SAT + JR

The power necessary to remove qC from the cold reservoir is the product Of the
voltage and the current.
(11)
P 2 VJ => (SAT + JR)J :> SJAT + J2R

The coefficient of performance (GOP) is defined as the ratio Of the heat removed

and the power necessary to remove the heat, a quantity which may be larger than 1.

n = qC/P = (STCJ - 51R2 - KAT)/(SJAT + PR)

It is inconvenient for the GOP to depend on the current. For given system param-
eters (S, It, R), the optimum COP can be found by defining m = J R/ S so that
(13)
1) = (mTC - ém2 - (KRAT/S?))/(mAT + m2)

Maximizing 77 with respect to m (361% : 0) solves for the In which maximizes the
GOP in Equation 13 to Obtain
(14)
Tim... = (Tc/ AT) {[01 - (Th/Tell/(w + 1)}.

where the Carnot efficiency (TC/ AT) is modified by the factor {[w - (Th/TC)]/(w

+ 1)}, where
w : (1 + ZT)%,

the factor ZT = SzaT/n (defined in Chapter 1). This modifying. factor approaches
1 as w ——> 00. A graph of nmax/nCamo, (545,.) asa function Of ZT for different values
of the ratio Th/Tc is shown in Figure 2 (T;, is the temperature Of the hot end of the
device).[17] A conventional Freon compressor refrigerator, in order to maintain an air
temperature of 270 K, must have the two Sides of the compressor cool from 330 K to

250 K. These refrigerators, with Th/TC ~—- 1.3, have n,,,,,,,/ 770mm, (276,) : 0.26.]21]

Figure 2 Reduced coefficient of performance 17m” Maw,“ (2715,.) for various

values of ZT.]17]

0.30
0.28
0.26
0.24
0.22
0.20
0. l8
0. l6
0. I4
0. l2
0. I0
0.08

Reduced Coefficient of Performance (M )

0.02

  
  

“0.3 T’400‘K

8"0.3 T'300‘K

0.3 T'200.K

2 4 l0 l2“

A Similar derivation for the efficiency, 6, Of a thermoelectric power generator yields
(16)
5mm: 2 P/Qh:,(AT/Th) {(w ' Ill/[w + (TC/Th)”,

where P is the power output and qh is the rate Of heat Observed at Th. Again the

 

.-..

P
'o'uln'nl
\

 

 

Carnot value (AT/Th) is modified by a factor {(02 - 1)/[w + (Tc/Th)”. This factor
approaches 1 as w -> 00. A graph of emu/coamm (577,.) as a function of ZT is Shown
in Figure 3.]17] The typical efficiency in a temperature range from 300 K to 900 K of
a good thermoelectric device made out Of PbTe is about 19% while that of a modern

central power steam engine is about 40%.]21]

Figure 3 Reduced efficiency Emu/600",“ (517,.) for various values of ZT.]17]

 

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2.2 MAXIMIZING COP AND EFFICIENCY

From our previous discussions, it is iS evident that to maximize the COP (effi-
ciency) of a thermoelectric refrigerator (power generator), one must maximize the
dimensionless quantity ZT, known as the thermoelectric figure of merit. The largest
value Of ZT found experimentally for bulk systems is on the order of 1 (for Bl2T€3
alloys at room temperature), but must be about equal to 3 to have a comparable
COP to a conventional compressor refrigerator.[1, 2]

In order to maximize ZT at a given T, one must not only have a large Seebeck co-
efficient, but the electrical conductivity must be largewhile the thermal conductivity
remains small. This is difficult Since most materials with large electrical conductivi-
ties, such as metals, also have large thermal conductivities, while systems with a small
thermal conductivities, such as insulators, also have small electrical conductivities.]2]
One must be careful to realize that, in the case of metals, the Wiedemann-Franz
law states that there is a limiting relationship between the thermal conductivity and
the electrical conductivity (where k3 is the Boltzmann constant and e the electron

charge)
- 2 k‘ 2
3': 33‘ (E) T

which means that the thermal conductivity can be reduced only SO far before
reducing the electrical conductivity. All of thematerials which we will discuss have
a larger thermal conductivity than given by this limiting relationship.[22]

To bypass the limitations of the Wiedemann-Franz law, one needs a material where

the electrons travel freely as if through a crystal, to increase the electrical conductivity,

11

_ .
tI-O v o

9-4 u'n
-

i

’ .
'rv- 7‘.
M. ...A

 

, _
41...:
l
Pit-">-
- .. ..
O» '

Ailf \
’ V
1. . 1. .
IJ~I
a)”; 1‘
4“

 

‘r,
1" '
‘«."/‘.,

»,.

14’-
11.1,",

‘41

 

(0’ should be small enough SO that 14 is dominated by phonons instead of electrons),
but phonons are scattered as if through a glass, to reduce the thermal conductivity.
Slack’s electron crystal/ phonon glass concept[6, 23] has been central in finding new
thermoelectric materials, although different groups have used different methods to
achieve this goal. One tries to use systems with complex unit cells since high thermal
conductivity is found for systems containing light elements, forming densely packed
crystals, and formed largely through covalent bonds. Lower thermal conductivity is
found for systems containing heavier elements with more ionic bonding.[20]

The best thermoelectric materials tend to be narrow-gap semiconductors. The
electrical conductivity (0) and electron thermal conductivity (14:) are increasing func-
tions of carrier concentration (n).

(18)
a : 11811.
rt 2 03;; (EEVT 2110113;- (if-VT
(19)

The Seebeck coefficient (S) is a more complicated function of carrier concentration
(11), but increases for decreasing n and becomes very small (~EEBF—T 80 IIV/K) for
metallic systems where n reaches 1022/cm3.[21] Calculations of the electrical resistivity
(equal to the reciprocal of the electrical conductivity). Seebeck coefficient, thermal
conductivity, and ZT are Shown in Figure 4 for an idealized semiconductor as a
function of carrier concentration.[2]

It can be seen that S decreases with increasing carrier concentration while a and
1c increase. A maximum for ZT = SzaT/K is found near 1018 carriers/cm3, which iS

the concentration associated with doped, narrow gap semiconductors.]2]

12

Figure 4 Plot Of p (2 1/0), S, It, and ZT for an idealized semiconductor as a
function of carrier concentration, 11. S decreases with n while a and

K. increase. The maximum ZT occurs near n :- 1019/cm3.[2]

 

RESISTIVITY p (1110 cm)
Insulator
0
Metal

 

 

Insulator
0
Metal

200*-

CARRIER ENTROPY S OLV/K)

 

 

K

electronic + Klattice

\ .

. Klauice

Insulator
Metal

20. .'
0. 000.0000...
4 I l I I

THERMAL CONDUCTIVITY
x (mW/cm K)

 

 

.0
on t— O
I
O

.0
0‘
I
Insulator
Metal

.0
.5
I

DIMENSIONLESS FIGURE
OF MERIT ZT

.0

N

I
O

9 e
o n I I I 4 a
It I? II I? .‘0 ll 2.‘

10 10 10 10 10 10 10
CARRIER CONCENTRATION r: em")

 

 

 

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For semiconductors, one must in general consider transport Of both electrons,
with Seebeck coefficient 8., and electrical conductivity 0.3, and holes, with Seebeck
coefficient 8,, and electrical conductivity 0),. The total Seebeck coefficient is

<20)

where S, is negative and 8;, is positive. In order for the hole and electron transport
not to cancel each other due to intrinsic electron-hole excitations, but to allow for
a large enough electrical conductivity (by doping), the band gap of the material
Should be on the order Of 10 k 3T.[1, 2, 3] Further, if there are a multiply-degenerate
band extrema, such as when one lies at a general point in the Brillouin zone, the
thermopower adds up as before for each extrema. For the same carrier concentration
in N identical pockets, the carrier concentration for each pocket is lower. As seen in
Figure 4, a lower carrier concentration corresponds to a larger Seebeck coefficient in
each pocket which gives a larger total Seebeck coefficient. Therefore, degenerate band
extrema also are expected to increase the Seebeck coefficient for a material.]11, 24]

Scattering, due-to impurities, grain boundaries, etc., greatly affects a and K, but
has a much less of an effect on S. S = limATAO%-¥- which makes this quantity fairly
insensitive to morphology while a and 14 are very sensitive to the direction which they

are measured.[21]
2.3. ZT FROM BOLTZMANN TRANSPORT EQUATION

Since S, a, and n, which control the .ZT parameter, are transport coefficients.

one can solve the Boltzmann transport equations in order to better understand their

14

r l“
‘T' 1I
L..4L4.
, .
4--yy<
h A...
'5'» v
'4 4
ya...
I
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.. ...
'7'?
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Fifi/l1.
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L4" .
‘U’Il'.
,1
9
.1‘1 ,

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<1...

 

 

behavior as a function Of material parameters. The details of the derivation can be
found elsewhere.]24] Using the electronic dispersion relation for a Single anisotropic
parabolic band in the energy independent relaxation-time approximation as a function

of the momentum k,- and effective masses m,- (50 is an Offset energy)

 

(21)
712162 7.21.3, 62k?
8(k33’ ky’ k3) : 2mr Zmy + 27712 + 80
ZT can be written in terms of Fermi-Dirac integrals F,, where
(532)
.. 2k T 3 1 ..fd-
F1 : F1(C ) : 47l- (7%)?(mxmylnz):.foooetrT-C‘T+l
and (‘2 C/kBT where C is the chemical potential so that ZT is[24]
(23)
5F3
gun-12% *l2F %
_ 2
ZT — 25F§ ’
1 7
B+ZF3775T11
, 2
Where the quantity B appears in the denominator of Equation 23 is given by
(24)
2k T 3 1 k2 T11.
B : "371,7 7(7§—)2(m1mym2)2—h—7

with band degeneracy ’7, mobility in the direction of transport 11,, phonon thermal
conductivity mph, and effective masses m,- (the other terms are fundamental constants).
As B increases, ZT also increases, as shown in Figure 5.]24]

Hicks and Dresselhaus pointed out that to increase B, one must reduce aph, the

phonon contribution to the thermal conductivity, as expected, but also maximize the

15

 

'7"

‘ C

..‘~

0

l‘f'r "

‘-Aa.:, '
‘pJ.

 

t
f .
luf'" .,
A“
t
tn

 

 

 

product Of the effective masses multiplied by the mobility in the direction Of transport
(here chosen to be along x). However, the mobility is inversely proportional to the

effective mass parallel to the direction of transport.|24]

(25)
Hz CX —

Figure 5 Plot Of ZT vs. B parameter (Equation 23).]24]

2

10 4-44

 

‘l

10r

F100:

N

-1

‘l01

 

 

-2
10 -3 2 - U 1 2

 

This implies that the effective mass in the direction of transport Should be small
to maximize the mobility, but the effective masses perpendicular to the direction Of
transport should be large. AS already mentioned, one wants a large degeneracy (7)
Of the band extrema for both the conduction and valence bands to increase S. While
the value Of 7 has a maximum of 48 in cubic systems, this effect does not propor-
tionately increase ZT by a factor Of 48 but has a more subtle effect on increasing the
thermopower.[11] Mahan and Sof0[25] have also predicted that a good thermoelec-
tric should have a narrow distribution of carriers with a high carrier velocity in the

direction of the applied field, which again can be Obtained in anisotropic systems.

16

 

.5.—

 

4
744,...

' o
v- ... Pt

 

 

 

 

 

 

 

2.4. GUIDANCE FOR THE BAND STRUCTURE

In order to search for new materials with high ZT, electronic structure calculations
cannot give us directly the transport coefficients such as thermopower or thermal and
electrical conductivity. However, several properties which appear to be important
for good thermoelectric materials can be discovered from electronic structure cal-
culations. We look for systems with a semiconducting band gap on the order of
IOkBTII, 2, 3], degenerate band extrema[11, 24], and anisotropic effective masses at
the band extrema[24]. Electronic structure calculations of the best known thermo-
electric materials can be used to see if the band gaps are about IOkBT, to check
if the high ZT arise's frOm degenerate band extrema, to calculate the effective mass
anisotropy by fitting a the calculated energies to second order polynomials, and use
the information about the wavefunctions to understand the transport in these materi-
als. Understanding how these properties lead tO a large ZT in these known materials
will allow for making predictions in the most promising new thermoelectric materials.

In order to find systems with band gaps on the order of 10k 3T (~0.25 eV for room
temperature applications), we should not look for semiconductors with large charge
transfers since the gaps in these systems are generally too large (i.e. Gal’As”: Gap
2 1.52 eV[29]). Therefore, it is best to look at semiconductors whose gaps form
through some sort of hybridization mechanism.

Hybridization may also be key in forming the anisotropic effective masses associ-
ated with these materials. Let us consider the band structure Of Hg1_ICdee alloy.
CdTe is a typical semiconductor while HgTe is semimetallic with an inverted band

gap where the 3 level is energetically beneath the p level at the I‘ point (Figure 6).

17

 

1 \
I"? 9 F

I In; L
‘ —

'nn »
'1‘ 1‘
.3..u,
d

W“ ‘
f“
‘4“: ,
i
I”. :h '.
‘u ,.\.;.h
u ‘ vla
I .

,
113' "'7
‘I,J._L
1..
.
.,. .
he \L'r
L
' a ”-41“

 

 

 

 

The more dispersive 3 levels lie above and below Ep with less a dispersive p level
lying between. Changing the value Of x can finely tune the band gap by shifting the

position Of the p level with respect to the S level.[19]

Figure 6. Conduction and valence bands of HgTe (left) and CdTe (right). Upon
alloying (Cd1_IHgITe) the bands cross.[19]

"97' We

\

V
’
’
‘—
’
‘
\
‘
s
\
fl fl

This logic can also be applied to systems with narrow d levels. Consider a semimetal-

\
\
s
‘
S
x» “’
I
’
I
’
’
’
’

lic system where the overlapping bands lie at different positions in the Brillouin zone.
Addition of a d level tO this semiconductor can produce hybridization between the d
and more dispersive p orbitals. The resulting hybridization mayproduce a narrow-
gap semiconductor where the hybridized p-d level may have d-like character and be
non-dispersive in one direction of k-space but more p-like and dispersive in another
direction in k—space. This satisfies two criterion for good thermoelectric materials,
formation of a narrow-gap semiconductor and highly anisotropic effective masses.

This will be the reasoning behind looking at the half-Heusler compounds.[7]

18

[)1-1‘1u‘z I--\I

s
C
¢

...]
‘7. ‘p-

 

 

 

 

 

Figure 7. Semiconducting gap Opening due to p—d hybridization.

 

 

m
”0-.--..
...-...

 

E, ---->

 

1\/
‘ : \

Energy (eV)
Energy (eV)

 

 

 

 

 

 

 

 

The schematic of p-d hybridization (Figure 7) Shows two p-bands which both cross
the Fermi energy at different positions. After hybridization with an added non-
dispersive d-band, a gap Opens in the Spectrum. For Simplicity, the details of the
d-bands have been left out Of the picture.

The presence Of Off-axis band extrema can come about due to strong Spin-orbit
hybridization. Consider the band structure for a narrow-gap semiconductor before
the inclusion of spin-orbit interaction. While there may be several off-axis extrema,
consider the semiconducting gap forming at the I‘ point. It is well known that the
spin-orbit interaction generally reduces the band gap, so a rigid shift may cause a

hybridization of the bands forming the gap, leaving the Off-axis extrema as the band

19

l

l

 

l ' A V . A I
Q ‘ ‘ ‘8 .1] v 0 I
. . .
I. 1 r .. . Ira Ly.
..- .l. . 1.0 . rnu
rd I <14 . MWA. w ‘1 :4
. . .
1‘ Y M I la .,
..I. :5 SM 1...-..

 

 

extrema (Figure 8). This is the reasoning behind our study of BizTe3 and related

materials. [1 1]

Figure 8. Off-axis gap formation due to spin-orbit hybridization.

 

 

 

 

0.. E, ---->

o
c:

>
Fri

'11

Energy (eV)
"“3: 2;:

......"

Energy (eV)

 

 

 

 

 

 

 

 

 

 

Optimum characteristics in the band structure may be a result of the crystal
structure itself. Layered structures, such as those seen in Bi2Te3[11], BaBiTe3[11],
and CsBi4Te6[15] suggest that the transport (effective masses) Should be highly
anisotropic. It is, therefore, important to compare the calculated anisotropy found in
the band structure with experiment.

The different physical reasons discussed above will be shown to be important for

the gap formation of promising thermoelectric materials. An important focus of this

20

work is to study the electronic structure of known thermoelectric materials as well as
promising new thermoelectric materials in order to understand which features. such
as a narrow band gap, anisotropic effective masses, and off-axis band extrema, are
indeed seen in the electronic structure calculations. With this information it will be

then possible to predict which are the best new thermoelectric materials.

In order to be considered for thermoelectric applications, a new system must be
comparable with the previously studied materials. The best room temperature ther-
moelectric materials for nearly 40 years are BI2T€3 alloys which have ZT~1. The best
high temperature thermoelectric materials are PbTe with ZT~0.8 at about 700 K and
Si-Ge alloys with ZT~1 at about 1200 K. CsBi4Te5 has a maximum ZT~0.8 at 225
K[26] and Bi-Sb alloys have ZT~0.6 at about 100 K, lower due to the the reduced value
of T (in ZT) and the lower electron mobility at low temperatures.[3] High values of
ZT have also been reported in PbTe-Pb(Se,Te) quantum-dot superlattices by Harman
et al.[27] and in p—type BI2Te3/Sb2T83 superlattice films by Venkatasubramanianl28].

21

CHAPTER 3.
FIRST-PRINCIPLES AB INITIO
ELECTRONIC STRUCTURE CALCULATIONS

Electronic structure calculations are a valuable tool to understand the structural,
transport, and Optical prOperties Of materials. Recent improvements in calculational
methods and computer speeds have made these calculations not only a more powerful
predictive tool but also have allowed for larger and more complex systems to be
studied.[30]

The basis Of electronic structure calculations lies in the nature of electrons in
solids. Electrons interact through strOng Coulomb repulsions which makes it seem
difficult at first to describe them as distinct entities. However, electrons in solids
do not act as “bare” particles, rather surrounding themselves with positively charged
polarization clouds. The electron and screening cloud form a “quasiparticle” which
weakly interacts with other quasiparticles in the solid. Despite the strong interactions
between the bare particles, the weak interaction of the quasiparticles allows for an
effective single particle approximation, particularly for low energy properties where
they can be treated as acting in the mean field Of the other quasiparticles in the
solid. Such an approximation appears to be valid in a large class Of solids (simple
metals, semiconductors, and insulators) but not in systems with strong electron-
electron interactions and weak screening, such as in magnetic insulators.[31]

Electronic structure calculations involve solving a single particle Dirac equation
within a single-particle approximation. This involves forming a one-particle potential

and technical methods for solving the Single-particle equations.]31] We will discuss

22

(.29

 

 

these aspects separately.
3.1. SOLVING THE DIRAC EQUATION

3.1.1. FULLY—RELATIVISTIC ONE-PARTICLE EQUATIONS

Assuming that a single particle description is valid, one still has to solve the
Dirac equation for an electron because the full relativistic effects applied to electronic

structure are included in this equation[32, 33]. The Dirac Hamiltonian is given by

(26)
HD 2 ca ' p + (13 - 1)mec2 + V(r)
where a and 13 are 4 x 4 matrices
(27)
0 a I O
a z 5:
a 0 0 —I

0;, (Ty, and 0,, being Pauli spin matrices. p is the four-component momentum,
me is the free-electron mass, 0 is the speed of light, and V(r) is the effective single-
particle potential. The eigenvectors, \II, of the Dirac Hamiltonian can be written as
four component spinors consisting of two component spinors, <I> (the large component)

and X (the small component).

(‘28)

This leads to a set of coupled equations

23

cta- Plx = 7..- - We
(“(0' p)<I* : (E - V + 2mec2)x
(30)
Solving for X in Equation 29 and inserting this into Equation 30 yields an equation
for (I) alone,
(31)

. E—V _ / r
27:2.(0' p)(1+ 2mecg) 1 (0' p)<I> + V<I>= 5(1)

 

 

3.1.2. SCALAR RELATIVISTIC APPROXIMATION

Equation 31 is still the exact Dirac equation. This equation serves as the basis for
relativistic electronic structure calculations which have been increasing in popularity
in recent years.[34] However, for valence electrons in solids, where the relativistic

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

(32)

 

and use the relationships

pV = V p - ihVV
(U'VV) (U'P) = (VV'P) +iU' lVaPl
(311
so that Equation 31 can be written as a linear, second-order differential equation

in <1).

24

 

 

 

—v
[(1 2111602)}: 2711 j \](I) 4171:: ’(VVVQ) ' 411137:2 (fwd) : 5(1)

In a spherically symmetric potential, V(r), this equation can be simplified further.

 

 

(35)
2 4 2 .
.I’_ ;_ P _ 5 611/ 6 1 d_l__’1 __
I 2712,, + \ Smile? 4771302 dr 07‘ + 2711262 dr 1*“ 5)”) — 5(1)

The first two terms are the nonrelativistic Schroedinger Hamiltonian while the
third and fourth terms are the massvelocity and Darwin terms, respectively. The
last term is the spin-orbit interaction (SOI). Due to this SOI term, 1 and s are not
good quantum numbers while their sum, j : l + s, is a good quantum number. In

practice one solves a new equation which is Equation 36 without the last term]32. 33]

(37)

 

[2%. + V 8m:c2 ’ 4771;721:3196 I 5‘8

called the scalar relativistic Schroedinger equation where I and s are still good
quantum numbers. However, spin-orbit effects are important, especially in heavier
elements (which are present in most thermoelectric materials with high values of
ZT[4]). The first thing that is done is to write 65 in terms of radial functions (f(r) and

g(r)) multiplied by Spherical Harmonics, Ylm.
(38)

9011/1...
-'if(7‘)Yzm

The scalar relativistic equations are then solved with this choice Of (,6. To include

801, one method is to double the entire basis set to accommodate H80

25

,r-

1's 0
H ._ h_i_lil
>0 0

50 ”— fllcz dr 1'

where M is the relativistic mass M : me + 5%. This approach is usually com-
putationally expensive. An alternative, called the second variational method, does
not involve doubling the size of the basis. Rather, spin up and spin down states are
considered separately using the eigenvectors of the scalar relativistic calculation, but
for a finite number of states near Ep. This reduces the number of basis sets involves

in the calculation of the energy bands in the presence of SOI.

.,_ 9T(7‘)Yzm ,,_ 91(rlem
69 — 99‘-
-ifl(7‘)Yzm —ifl(7“)Yzm

The states denoted in Equation 40 are used as the new basis set for the expansion
of the wavefunction (I) in Equation 36. The total Hamiltonian including SOI can be
solved with less effort since the number of qbl and (1)1 in the space of orbitals of the
low lying bands (reduced vectors space) are usually much less than in the original
eigenvalue problem.[32, 33, 35]

One shortcoming of the second variational method was found in the case of metal-
lic Pb where there were significant differences between the results using this method
and a fully-relativistic calculation. The Pb 612% state strongly mixes with the 6.9%.
the latter coming very close. to the nucleus where relativistic effects are strong. The p
states used as basis functions in the second variational treatment do not approach the
nucleus like the 3 states, since the good quantum numbers in the scalar relativistic

calculation are l and s, not j. The strong mixing of the 6p% and 63% orbitals should

26

N

n.‘

 

.\

 

also in principle affect the band structures of Bi, Hg, and T1, though these effects
have not been explicitly studied[36]. However, the second variational method has
found to be quite successful in predicting the band structures of PbTe[37, 38, 39] and
Bl2T€3[11, 12, 13, 40], so the applicability of the second variational method to semi—
conducting compounds containing heavy atoms has not yet been critically examined.
In this dissertation, second variational procedure for spin-orbit will be used due to
the simplicity of its implementation and we will check its reliability by comparing
our calculations with published fully-relativistic calculations (i.e. solving the Dirac

equation) wherever available.

3.2. DENSITY FUNCTIONAL THEORY (DFT)

3.2.1. DERIVATION OF KOHN-SHAM (KS) EQUATIONS

A major theoretical and computational advance in the electronic structure of solids
was made by Pierre Hohenberg and Walter Kohn[41] and later by Walter Kohn and
Lu Sham[42] with the advent of density functional theory (DFT). This theory states
that the ground state electronic properties of a material are determined solely through
its electron density.[30] The Hohenberg-Kohn proof starts from a Hamiltonian for M

nuclei and N electrons with the form

(41)

j>l ]T','—7‘j]

E T + V’Coul + lev'cxt(r)a

--N12 N N__1_ N M 2,3
He " X31:1 5 V: + 2i=1 2 ‘ E32:1 >3A=1 IRA-7‘1]

where the first term, T, is the kinetic energy, the second, Vow, is the electron-

electron Coulomb repulsion, and the third, sglvmm, is the nuclear-electron at.-

1 Y

A‘.

r—..,
l

 

 

traction (Z A is the atomc number and RA the nuclear position). The Hamiltonian
depends on the positions of the nuclei which create an external potential, Vert(r), on
the electrons. The essence of DFT is the fact that the terms T and V00”) are the
same in all N-electron systems, and the external potential, which differs from system
to system, can be written as a single-particle function. The universality of T and
V Co”, are the basis of the universal functional Fln] (using wavefunctions \11)

(42)

F[n] Z m17L¢,2_,n(r)<‘I/ ] T + VCoul ] \IJ>,

where the Kohn-Sham total energy can be written as
(43)
EKS[n] = F[n] + fVm(r) n(r) dr

The Kohn-Sham energy must be minimized under the constraint N z fn(r) dr

(44)

6IEKslnI - u (N - n(r) dr>1 = 0

While it is a Lagrange multiplier-in Equation 44, the functional derivative defines

the chemical potential, MN, in DFT.
(45)
Wln(r)=no(r) = #-N
In order to obtain the Kohn-Sham equations, the kinetic energy operator, TOIn],
is written separately from EK5]n].
(46)

Toln] I minwzdn(,)<\ll ] $372 I \IJ>

28

It is customary to separate F [11] into a Hartree piece where the Hartree energy,
EH[n] (derived from the Hartree potential, VHIHD, is
(47)
'EHlnl = flfflfiEL/TZ dl‘ dr’ 2 éfVHU‘) (11‘,
so that the difference between F[n] and T0]n] is EH]n] and a remainder, EICIn],
which is called the exchange-correlation energy.
(48)
F[n] — T0]n] : EH]n] + Erc]n]

The exchange-correlation potential, VIC]n], is derived from Em]n]
(49)
6Elcl7z] _

6n(r) ]n(r>=no(r> = Vacclnl

In order to satisfy the fermion nature of the electrons, the total wavefunction must
be antisymmetric, and to accommodate this requirement, it is convenient to write the

total wavefunction, \II, as the Slater determinant of single particle wavefunctions, C1,.

(50)
C11 C12 - - CW
(21
_ l
\I} —'\7;:\—',-! £th
LCNI - - - CNNd

 

 

so that the total density is a sum of the squares of these one-electron wavefunctions]43]

29

IMP) =1 Zi.k|C{i(r)l2

This choice of \1' makes it possible to calculate Toln] and the minimization over all

\Il such that W]2 -—> n(r) gives rise to the following one-particle equations (known as
the Kohn—Sham equations)

(52)

l-%V2 + VNe + VH "r Viclnll 5.4?) = 5%. 1i-(r) E HA'sC?k(r)

In Equation 52, -%V2 is the single particle kinetic energy, VNe is the nuclear-
electron attraction, V H is the Hartree potential, and Vgcln] is the exchange-correlation
(XC) potential. The single particle states, &(r), are called the Kohn—Sham orbitals
and efk are the Kohn-Sham energies. The XC potential depends on the electron
density, n(r), but the density is obtained from the solutions of the one-particle
Schroedinger equations, 5c.[43, 44] Since the XC potential is a part of the Hamil-
tonian, it affects the wavefunctions, and the wavefunctions determine the charge
density and, hence, the XC potential. This means that the electron density and KC
potential must be solved self-consistently.[35, 43, 44]

The XC potential includes correlation effects, which keep electrons of opposite
spin from coming too close to each other, and exchange effects, arising from the
fermion nature of electrons. VNe and VH are local, i.e. (VNef)(r) :2 VNe(r)f(r) and
(VHf)(r) : VH(r)f(r), but the XC potential is inherently non-local, i.e. (Vuf)(r) :
f ch(r,r’)f(r’ )dr’ . One of the most commonly used exchange-correlation potentials is
the local density approximation (LDA) where VIC' depends only on the density n at

position r.[44]

30

a
4 1
‘D

. 4

n. .

 

.31

{h
.i‘ y‘
,- ,
.

V.‘
I.“ ,v
.

A
1' .
I.” ‘.

LDA : ch(nl(r)i n](r)) : f(IlT(I‘), n](r))

LDA does not always do a good job in finding the ground state properties of ma-
terials. An important example of this is that LDA finds that ferromagnetic FCC
Fe has a lower energy than BCC Fe by about 6 mRy, contradicting experimental
evidence.[45, 46] Some of the non-local nature of the XC potential can be incorpo-
rated by including derivatives of the charge density in what is called the generalized
gradient approximation (GGA). This approximation is developed by expanding the
XC potential to second order in gradients of the density (Vn) and cutting off spurious
long-range contributions-in real space. This leads to a fairly complicated expression
which favors a less homogeneous electron density than LDA.[47, 48]

(54)
GGA : mom). ni(r)) = mum. mm, mm, mm)

In many cases, GGA improves the values of calculated properties compared to
experiment, such as cohesive energies, total energies, and work functions.[46] While
LDA predicted the wrong ground state for Fe, GGA correctly predicts that BCC
Fe has a lower energy than FCC Fe by about 11 mRy.[45] However, GGA tends to
overestimate lattice parameters while LDA tends to underestimate them. Most of the
improvements of GGA over LDA are modest, and sometimes GGA does not explain
properties of some materials as well as LDA, so special care should be taken that

spurious results do not occur due to the choice of the XC potential.[29, 46]

31

3.2.2. PHYSICAL PROPERTIES OF DFT CALCULATIONS

In DFT, the Kohn-Sham eigenfunctions, (fur), solved in Equation 52 have no
rigorous physical meaning; they are simply functions calculated in an intermediate
step to solve for the charge density in Equation 51. The eigenvalues found in equation
52 are not single-particle excitation energies, although they are often compared to
experimental results as if they were. Surprisingly, DFT results often agree very well
with experiments. However, one should not overlook some of the general results of
DF T which have direct physical interpretation.[31]

The ionization energy, I, is the energy required to remove an existing electron from
a solid while the electron affinity, A, is the energy gained by adding an electron to a
solid. These can be identified in an N-electron system as[31]

(55)
I = EM - EN ‘ A = EN - EN“

where EM is the ground-state energy of an M-electron system. In a metal, one can
associate (Equation 45) the chemical potential, pN, or the Fermi energy, Ep, with
the highest occupied Kohn-Sham orbital. In a semiconductor one has[31]

(56)

I : ‘ENJV A I 'NNH : '5N+1,N+1a

and the band gap, Egap, is given by the difference of I and A,]31]

Egap = I - A = 5N+1,N+1 - €N,N = Axc + 5N.N+1' €N,N-

In the above, 5N,M is the MM Kohn-Sham eigenvalue of an N-particle system and

32

 

 

 

 

A“ : 5N+1,N+1- e N, N+1 is the energy difference between the highest occupied .Kohn-
Sham orbital in the (N + 1)-electron system and the lowest unoccupied Kohn-Sham
orbital in the N-electron system. This difference arises from a discontinuous jump in
the XC potential, VIC, when adding an electron.[31]
(58)
Arc = ch,N+1(r) - \f'xc,~(r)

This difference, A“, is positive and usually position independent, so the XC po-
tential changes only by a constant. This is the source of the “band-gap problem” in
DFT where the band gaps can be 0.5 -2 eV smaller than experiment[31]. In most of
the band gap calculations in this dissertation, we will ignore this correction and only

use E a: EMA/+1 - EMA; as an estimate of the band gap.

3.3. METHODS OF CALCULATION (DFT)

There are many methods to solve the Kohn-Sham equations in a solid. In or-
der to solve for 5c(r) self-consistently, one must start with a reasonable trial func—
tion which satisfies the crystalline periodicity and appropriate boundary conditions.

Planewaves and atomic orbitals form complete basis sets which are often used in
several methods to expand the wavefunctions[35, 49]. To simplify the numerical
alculation, several methods, such as the linearized muffin-tin orbital method in the
omic sphere approximation (LMTO-ASA), make approximations to the intera-tomic
tentiaI, V(r) [50, 51]. The method which we have used, the full potential linearized
newave (FLAPW), makes no approximations to the shape of the potential and

arates the space into two- regions, spheres about the atomic positions and an inter-

33

 

 

 

p.
.1» F
4

 

stitial region. The wavefunctions are expanded in atomic orbitals inside the spheres
and in planewaves in the interstitial region.[35, 51, 52, 53] We will briefly discuss these

expansions and see how they are implemented in LMTO-ASA and FLAPW.

3.3.1. PLANEWAVES AND ATOMIC ORBITALS

Planewaves form a complete basis, easily programmable, and describe well nearly-
free electron states. The most general type of expansion in a planewave basis can be
written as follows in terms of the reciprocal lattice vectors G. (cG(k) are expansion
coefficients)

(59)
((1.) :eik'r Ea c(;(k) eiG‘r

The major strength of the planewave expansion is the ability to expand in the
:omplete basis of planewaves to achieve greater improvements in approximating the
wavefunctions. This also turns out to be the major weakness, since it may take from
everal thousand to several million planewaves to adequately represent the localized
;ates associated with the atomic orbitals near the atomic positions.[35]

Atomic orbitals and planewaves prove to be the most important basis set for ex-
Lnsion of the wavefunctions within DFT, but the method of calculation used is often
.med after the method to approximate the potential. Tight-binding calculations are
8 most direct application of using atomic orbitals. but require a large number of
pansion coefficients since the atomic orbitals cannot describe the regions far from
2 atomic positions.[22] The pseudopotential method (used with both atomic orbital

i planewave expansions) is a very successful scheme to reduce the time necessary

34

H I

.II

 

Noi;

in the calculations. In this method the potential associated with the core electrons is

replaced by a function which reproduces the full-electron result when solving for the

valence electrons self-consistently.[35, 49, 54]

3.3.2. LINEARIZED MUFFIN-TIN ORBITAL METHOD WITHIN
THE ATOMIC SPHERE APPROXIMATION (LMTO-ASA)

An important and popular method of solving the Kohn-Sham equations within
DFT is the linearized muffin-tin orbital method within the atomic sphere approxima-
tion (LMTO-ASA) which arose from the previously used cellular method.[22] In this
method, approximations are made to the interatomic potential, V(r), which drasti-
cally reduce the computational time while still giving reliable results. The solid is

separated into two regions, overlapping spherical regions around the atomic positions

ind the region outside of these spheres(Figure 9).

Figure 9. Overlapping atomic spheres surrounding each atom of a crystal

in the ASA.[50]

 

35

The total potential is kept the same'as the actual potential inside the atomic sphere
radius, ras, but becomes a constant value, VMTZ, outside the spheres.
(60)
V(r): { V(r) r < rag

7 \
{ V MTZ r 2 I‘as

The wavefunctions are defined in terms of atomic orbitals (o1(E,r)) and power law
terms.
(61)
C,m(E, r) : i’Yf"(r) { ¢1(E,r) + pl(E) (r/ras)l r <‘ rag
{ (ras/r)’ r > ras

The term p,(E) is required to keep the solution continuous and differentiable in all

space where

(52)
_ DI(E]+I+1 _ 7' id;
ME) — D:(E)-l WE) — Elam—m...) ar' m...

This method has solutions for the region outside of the spheres which extends into
1e other spheres, as can be seen by Equation 61. The correct one-center expansion in
re sphere is in terms of atomic orbitals for r < ras is (aft, are expansion coefficients),

(G3)
emu-«3, r) = 23,... a); i‘Y{"(r) ¢,(E,r).

The tails from the expansions for all of the other spheres (r > r03) must cancel

Ede the sphere for Equation 63 to be the correct expansion.

36

(54)
sz(E, 1') = 2,... afrlfr i'Ysz‘) ME) (F/ras)’

The expansions outside the spheres (Equation 64) must be cancelled by Sfiw‘l'm
(65)

2.... ail; [2(21+ 1) p.033) a, 5.....- spmwj = o

is called the structure matrix which contains the information about

where Sfi‘m

flan
the atomic positions given by[50]
(66)

Sf’,m’.l,m ‘-= gt',m'.z,m 2R¢0 9'“? (:fiillflfllv 4” il errinfill‘

where gumghm are Gaunt coefficients, 1” :‘ l’ + l, and m” = m’ - m. Finding
:he eigenfunctions (and eigenvalues) of the Schroedinger equation within this method
‘equires solving for the tail cancellation in Equation 65. However, p,(E) explicitly
lepends upon energy. The linearization in LMTO comes about by expanding these
mctions about specific energies near the center of the bands as in Equation 67. This
: done by introducing linearization energies, E,, at the middle of the bands so that
...(r) does not explicitly depend on E.[51]
(67)
sz(l‘) = i’an(r) { Arm¢z(Ez,r) + PIIEI) (r/I‘as)l F < I‘as

{ (rue/r)l r :> [as

37

I
.

3.3.3. FULL-POTENTIAL LINEARIZED AUGMENTED PLANEWAVE
METHOD (FLAPW)

The method which we have used, the full-potential linearized-augmented planewave
(FLAPW) method, incorporates several of the aspects of the methods described be-
fore. Similar to the LMTO-ASA method, the space is separated into two regions, non-

overlapping spheres about the atomic positions and the regions between the spheres

(Figure 10).

Figure 10. Partitioning of the unit cell into atomic spheres (I) and an interstitial

region (II).[53]

 

II

 

 

 

The wavefunctions are expanded with atomic orbitals inside the spheres and planewaves
in the interstitial region between the spheres, combining the expansions of the tight-
binding and planewave methods. Like the pseudopotential method, the electrons arc-
separated into core and valence states, but the core states are not replaced but an-

kept within the self-consistent cycle and affect the final band structure.[35]

38

LAPVV is a linearized version of the augmented planewave (APW) method. In the
LPVV method, the wavefunctions are expanded in terms of basis functions which de-
)end explicitly on the energies (Equation 68). Region I is inside the spheres while Re-
gion II is the interstitial region (with volume V and expansion coefficients Cj’m(k,.))]52]

(68)

Region I =5 Cfi'lreE) = z31m fm(K)¢1(F,E) Yzm(r)
Region II :> (fir) : VlveiK'f

where K = k + G. In order that the basis functions do not depend explicitly on
energy (so that ‘11 in H\II = E‘II is linear in energy), the radial parts of the wavefunction
are expanded about specific linearization energies (E) which lie near the center of
the bands, as in the LMTO method. In order to have the function and the derivative
equal at the sphere boundaries, the function and its derivative are included in the

expansion (for expansion coefficients Afm and B?,,,).]51, 53]

(69)

Region I => Cfnlr) I EzrnlAf’mézfiaEI) + BfméIIIaErlermIT)
Region II :> ka(r) : fab",

where was.) z {femangzfl The full-potential LAPW (FLAPW) method
has been put in a convenient collection of programs which we have used, WIEN97I55].
This package contains SOI as a second variational method and solves for the full
potential with no approximations to shape. Both the LDA of Perdew and Wang|47|
and the GGA of Perdew, Wang, and Ernzerhofl48] has been used for the KC potential.

To further improve the convergence, the LAPW basis set can be expanded to

include a radial function at a second energy (Em), called a local orbital (LO).

39

‘1

u'

._H

(70)
Region 1 => (3,0) = szlAi’mabKnEhz) + B?,..<151(I,E1.z) + Cfin¢z(r,E2.z)IYz-m(r)

Since there are no approximations to the potential or core electrons, the F LAPW
method is a very useful method to obtain highly accurate electronic structures. How-
ever, this progress comes at a price of longer times for convergence compared to the
pseudopotential or LMTO-ASA methods. This means that large systems, containing
perhaps 100’s of atoms, cannot be achieved using the level of accuracy in LAPW

methods due to the problem of N 3 scaling with the total number of e1ectrons.[35]

3.4. OTHER METHODS OF CALCULATION (NON-DFT)

There are other electronic structure methods beyond DF T which can either be
used separately or in conjunction with DFT methods in order to better describe
correlation effects within the material. Some of these methods have similar time and
size requirements as most DFT methods while others, though improving the quality

of the results, are so computationally demanding that they can only be used only for
simple systems.

3.4.1. HARTREE-FOCK (HF) METHOD

One of the earliest methods is the Hartree-Fock (HF) method where the N-particle

wavefunction, \II, is written, as the Slater determinant of the individual single particle
wavefunctions.[56]

40

 

 

 

C11 C12 - « CiN
C21
‘I’Hp :JTVT det
_ CN1 - - - CNN _

where C.(rj) E (0 describes the jth electron occupying the orbital (.(r). These
wavefunctions are used to solve the Schroedinger equation (Equation 36). Unlike in
DFT where the wavefunctions were merely a construction to produce the electron
density, the wavefunctions in HF have physical meaning. In HF according to Koop—
man’s theorem[56], the ionization energy is I : ~5N,N and the electron affinity is A :
- EMA/+1, so the band gap is simply E90,, : I - A : amt/+1 - EMN with no correction
as in DFT by way of the Kohn-Sham eigenvalues. The results of HF tend to overes-
timate the band gaps in contrast to LDA/GGA which underestimates the band gap.

The overestimation of the band gaps in HF is usually associated with the omission of

correlation effects.[44]

3.4.2. GW METHOD

Both LDA and HF give poor account of excitation energies in solids, particularly
he band gaps in semiconductors. In recent years, a Green function approach for the
[uasiparticle excitations which was originally developed for interacting homogeneous
~lectron gas problem has been applied to real solids. An approximate version, callml
3W method has been proven to be quite successful for a large class of semiconductors.

.‘he single particle Green function, G(r,r’;E), which contains the information about

41

F

quasiparticle excitations is given by

G‘1(r,r’;E ) = G61(r,r’;E) - 2mm).

where G51(r,r’;E) is the reciprocal of the Green function for a reference system
and E(r,r’;E) is the self-energy correction. The GW method gets its name from
calculating the self energy (E(r,r’;E)) as the product of a dressed Green function
(G(r,r’;E - 1.12)) and the dynamically screened Coulomb interaction (W(r,r’;w)). In
this approximation, the self-energy correction E(r,r’;E) is given by[31, 57, 58]
(73)
E(r,r’;E) = 237;] do) e—i6“G(r,r’;E - w) W(r,r’;w)

Starting either from the Kohn-Sham or HF eigenvalues and eigenfunctions, one can
construct the self energy, 23(r,r’;E). The GW method produces values of gaps (Egap)
and band dispersions for a wide range of semiconductors which are much closer to ex-
periment than LDA/GGA (see Figure 11. Note Eé‘gf < Egg it: E525). Unfortunately.
this method is computationally very expensive, with most results found on fairly sim-
ple monatomic or diatomic crystalline systems. While this approach is quite useful

in understanding the gap structure of semiconductors, it cannot be readily applied to

the complex systems which we have studied in this dissertation.]31, 57, 58]

42

Figure 11. Comparison of characteristic direct and indirect LDA, GW, and
experimental energy gaps for all semiconductors for which GW

calculations have been reported (For references see Aulbur[31]).
-2 0 2 4 6 8 10

Si

Ge [mm gap ]OV]

   
  
  

3C SiC
LiCl
fl-C3N4
BN
BP
BAs
AlAn
AIP
AlAs
Ale
GaN
GaP
GaAs
GaSb
InP
InSb
AlosGaosAS
Ino.53(330.47A8
GaAsonN 0.25
ZnO
ZnS
ZnSe
Zn
CdS

M80
Mn
NiO
CaCqu
LigO
ZIO:
SD02

43

CHAPTER 4.
HALF-HEUSLER COMPOUNDS

Half-Heusler compounds are among the most common and versatile intermetallic
materials. These compounds can be semiconducting (such as in the M NiSn (M :
Ti, Zr, Hf))[7, 59], metallic (such as in M CoSb (M = V, Nb))[60], and even may dis—
play half-metallic ferromagnetic behavior (M MnSb (M : Co, Ni)).[61] Some of these
systems also display heavy-fermion characteristics (MYbBi (M 2 Pt, Ni))[62] The
semiconducting half-Heusler systems are interesting from a thermoelectric perspective
due to the large power factors (PF : 820) observed in several of these semiconduc-
tors (M NiSn (M 2 Ti, Zr, Hf)).[61] An interesting feature of these semiconductors is
the nature of the gap formation. The full-Heusler alloys with two Ni atoms/ formula
unit (such as M Nigsn (M 2 Ti, Zr, Hf)) are weakly semi-metallic as are the binary
compounds M Sn where the Ni atoms are completely removed. This means that the
inclusion of the Ni atom has a subtle effect in opening the gap, an effect which will
be explored in detail in this dissertation.[7, 59]

The half-Heusler structure (174-3m) can be described as a filled NaCl structure
(of YSb in Figure 12) where transition metal atoms are inserted in the octahedrally-
coordinated interstitial sites. In this description, Ni atoms can be considered as guest
atoms in a Y3+Sb3“ network. In the full-Heusler compounds, the Ni atoms occupy all
of the tetrahedral holes in the lattice. An alternate way to describe the half-Heusler
structure is to consider this as a filled adamantane, [NiSb]3", ZnS zinc-blend-type
lattice in which the Y3+ occupy the octahedral holes in a diamond lattice.[63] The
FCC Brillouin zone used for the electronic structure calculations of the half-Heusler

compounds is given in Figure 13.

44

Figure 12. Two different perspectives of the cubic half-Heusler structure (YNiSb)
Left: Ni atoms inserted in a YSb NaCl-type lattice.

Right: Y atoms inserted in a NiSb ZnS—type lattice.

 

 

Y
Y( Sb a Sb
/ Y
V - void
F] A Sbspace
if
N10 .9 “'

 

 

 

 

Figure 13. FCC Brillouin zone.

 

 

45

 

T l.

4.1. GAP FORMATION AND ELEMENTAL REPLACEMENTS

4.1.1. GAP FORMATION IN YNiSb AND ZrNiSn

The half-Heusler crystal structure includes many constituent compounds due to the
large number of elements which can fOrm the NaCl substructure and the transition (or
noble)metal atoms which can go into the octahedrally-coordinated pockets. In order
to find narrow-gap semiconductors for thermoelectric applications[25], it is necessary
to know which combination of elements may produce semiconductors. In systems
with s—p bonding there is the well known 8-electron rule which states that the valence
electron count (EC) should equal 8 as seen in several semiconductors, where the binary
compound can be written as ANBB‘N where element A has N valence electrons and

element B has 8-N. Typical examples are given in Table I.[64]

Table I. 8 Electron Rule of Elemental Covalent Crystals[64]

 

 

 

 

 

 

Elements in Semiconductor AB A Electron Count: N B Electron Count: 8-N
A) & B) C, Si, Ge, a-Sn 4 4
A) P, As, Sb; B) Al, Ga, In 5 3
A) S, Se,Te; B) Zn, Cd, Hg 6 2
A) Br, I; B) Cu, Ag, Au 7 1

 

 

 

 

 

A similar electron-rule has been suggested for ternary systems which have filled (1
shells. One can imagine placing atoms with filled d shells into known binary semicon-
ductors to form ternary semiconductors, assuming that the d orbitals interact very

weakly with the s-p bonds. The filled d shell adds 10 valence electrons to the bi-

46

 

’
,-
.
“4L.
.
'r
.'-'
k e
' o
t
I?"
.. ‘
‘v .
o
t.
1
l n
y.

 

A..

 

 

#y semiconductor, which follows the 8 EC rule, to yield the 18 EC rule for ternary
npounds.[65] This explanation has been applied to a wide variety of ternary sys-
ns, including half-Heusler compounds. All known half-Heusler compounds which
e semiconductors have EC 2 18, while those known systems where EC # 18 are
rmi-metallic, half-metallic, or metallic.[60, 61] To understand the gap formation of
he half-Heusler compounds, one must determine whether EC 2 18 is sufficient for
.he formation of the band gap or if it is only a necessary component.

One of the first semiconducting half-Heusler compounds which has been stud-
ied extensively for thermoelectric applications was ZrNiSn.[59, 66] This material has
EC 2 18 and a simple charge balance argument suggests the valences should be
Zr4+Ni°Sn4". In the ternary compound ZrSn forms a NaCl lattice with the Ni atoms

entering in an FCC configuration within the octahedrally-coordinated pockets. This
material forms a semiconducting gap of about 0.17 eV[59, 66], in the right range for
thermoelectric applications.[1, 2, 3] An obvious avenue to look for new semiconductors
is to find new half-Heusler compounds where the valence of the atoms in the NaCl

structure is changed without changing the transition metal atom so that EC 2 18.

One such system is YNiSb, where EC 2 18 and a simple charge balance argument

should yield Y3‘V’NiOSb3’, which will be discussed here.[7, 8, 67, 68, 69]

Before investigating the similarities and differences between ZrNiSn and YNiSb, it
is important to understand the differences between the full-Heusler (F m-3m) YNizsb
and half-Heusler (F-43m) YNiSb. According to the nominal valence count, i.e., Y3+,
Ni", and Sb3‘, both of these systems should be semiconducting, but, as one can
see in Figure 14, the full-Heusler YNisz is metallic while the half-Heusler Y NiSb

is semiconducting, the same results seen in ZrNigsn and ZrNiSn.[59] The d-bands in

the half-Heusler compound are about the same width in both compounds. The Ni d

47

itals interact with Y d orbitals the in the half-Heusler compound (Figure 14) while
5e bands simply cross in the full-Heusler compound. This appears to be due to

higher symmetry of the full-Heusler compounds which does not allow for the
oridization of the flat Ni d bands with the more dispersive Y d bands. A periodic
ray of vacancies within the half-Heusler lattice, increasing the interaction of the Ni

and Y d orbitals, appears to be necessary for the formation of the semiconducting

ap .]70]

Figure 14. Band structure and comparison of the Ni d orbital characters in YNiSb

and YNizsb. The size of the circles is proportional to the orbital

 

 

   
 
  
 
   

  
 
  
        
 
 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

character.

(207 0 YNiSb Ni d (b)7 o YNiZSb Ni (1

A 1 . .o'. f] A j 1‘12" '. 3.

> If ,: > >93: ~".'./:‘ \" e . '

3 l ' , 3 Efiffifi ' ~f -- ‘1'

>3 >~ :r—(‘W - / ' :'

glee EF gloo‘ .... E1:
‘3 L12 4'!“— ‘5‘— :—

 
    

  

aw»
-.

....‘0.

w

 

\

'7'w L A 1“ A XZWIK 'w L A 1“ A XZWK

 

 

 

 

 

 

 

 

The electronic structure calculation of ZrNiSn (along with that of HfNiSn and
TiNiSn) have been performed previously using a pseudopotential method by Ogut &
Rabe[59, 66]. We have performed FLAPW calculations in these materials and in a new

set of compounds, YNiSb and related systems.[7, 9] Our FLAPW calculations agree

48

scale, the electronic structures of these two materials look very similar (Figure 15).
The valence band maximum (VBM) forms at the I‘ point while the conduction band
minimum (CBM) is at the X point. The dispersion about this minimum is highly
anisotropic, very flat along XF but much more dispersive along XW (Figure 15).]7, 8]
Photoemission studies of ZrNiSn show that the Ni d peak appears approximately 2 eV

below Ep[71] as predicted by our and previous calculations[7, 11‘, 59]. However, the

 

TB-LMTO-ASA band structure for ZrNiSn provided by Slebarski et 01. appears to be
incorrect, the VBM forming along either I‘ M or I‘K instead of at the f‘ point.[7, 11. 59]

Figure 15. Electronic structure of ZrNiSn and YNiSb.

(a)7 0 ZrNiSn (b)7 0 YNiSb

7 .

 

 

.l

G
1y
I

 

 

 

is

so

 

 

 

.9
c:

   

 

 

Eneggy (eV)

(
l/ \ K
IN
EngelsHeV)
.’
l
"' \MK

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1." A XZWIK "w L A 1

E“
;>

‘he half-Heusler compounds ZrNiSn and YNiSb contain parent binary compounds
1 and YSb, respectively, as part of their N aCl sublattice. These compounds should
miconducting according to a nominal valence count (Zr4+Sn4‘and Y3+Sb3‘) and

" = 8 rule[64], but our calculations, performed at the ternary’s equilibrium

e parameter (Figure 16), show semi-metallic behavior, violating the EC : 8

49

 

 

rule. The number of states at Ep for YSb is found to be 0.19 states/eV, a factor
of 2.5-3 smaller than the earlier LDA results.[72] Ogut and Rabe (OR)[59] pointed
out the significance of p-d hybridization along the FX direction on the energy gap of
ZrSn. They argued that a strong k-dependent hybridization between Zr d” and dyz
and Sn p. and 1),, along (001) is one of the two reasons behind the gap formation at
Ep. The strength of the Sn p and Zr d orbitals are shown in Figure 16a, and 16b, the
strengths of the orbital characters being proportional to the size of the circles. The
band labeled 2’ starts out with Sn p character at the F point and changes over to Zr
a’ character at the X point. The band labeled 2 shows just the Opposite behavior,
evidence of strong k-dependent p-d hybridization between bands 2 and 2’, as pointed
)ut by OR]59]. In YSb, however, this Sn p-Zr d hybridization is very weak, as seen in
‘igure 16c and 16d, yet the addition of Ni to form the half-Heusler structure opens
very similar band gap. Therefore, it appears that, contrary to the suggestion of
R]59], strong p-d hybridization is not a prerequisite for the formation of the band
p in the half-Heusler compounds. Rather, as we argue in this dissertation, the

esence of Ni d orbitals plays a significant role in the gap formation.]7]

50

 

 

Figure 16. Comparison of the different orbital characters of ZrSn (a) Sn p and
(b) Zr d, and YSb (c) Sb p and ((1) Y d. The size of the circles is

proportional to the orbital character.

ZrSn YSb

(a) 7.- Slip

 

 

  
  
  

 

 

 

 

 

S3
<'.>_ _

 

 

 

 

 

Energy (eV)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

z ( ’7‘w A A zW'k

n ZrNiSn the Ni d orbitals affect the band structure in two ways. The hybridized
r-Zr d band with a large dispersion (labeled 1 in Figure 17a, which starts from
I" point with F12 symmetry at ~0.5 eV and goes to ~-2.5 eV at the X point
X4: symmetry) in ZrSn hybridizes strongly with the Ni (1 orbitals (Figure 17c)

and become the lowest conduction band (also labeled 1 in Figure 17a, along FX with
X3 symmetry at the X point). In addition, the strongly hybridized Sn p-Zr d band
(labeled 2 in Figure 17a, which starts from the F point at ~0.0 eV and goes to ~01
eV at the X point) gets slightly modified. The energy of this band at the X point
drops below that at the F point by about ~0.5 eV. The net effect is the formation of

an indirect gap semiconductor with a gap of 0.50 eV, which agrees very well with the

previous calculation. [59]
The situation appears to be different in YNiSb. The lowest Y d band, which has

a downward dispersion from F to X (labeled 1 in Figure 17d) strongly hybridizes
with the Ni d band and somewhat weakly with the Sb p band. This hybridization
pushes the CBM at the X point above the VBM at the I‘ point. The net result is the
formation of an indirect narrow gap semiconductor, but in this case the gap is 0.28
eV, almost a factor of two smaller than that of the isoelectronic compound ZrNiSn. In
spite of the basic differences in the nature of bonding, the electronic band structure

in the neighborhood of the Fermi energy is remarkably similar in ZrNiSn and YNiSb.

(Figure 17)]7]

52

 

Figure 17. Comparison of different orbital characters between ZrNiSn (a) Sn p,
(b) Zr d, (c) Ni (1, and YNiSb (d) Sb p, (e) Y d, (f) Ni (1. the size of

the circles is proportional to the orbital character.

ZrNiSn YNiSb

       
  

 

 

 

 

 

 

 

(a) Sn p (d) Sb 9
7 . 7.0 ‘T, 7,,
A A 3".“ ”.0 It“. .3 vL-j. 3"
> > ..-:::.. ,.- .“'..::.~--
0 o :..--"““"~‘3m i. my
v v 1 ." ". {I
>5 >.. 1::
2.” W 3 M13
0 s 0.0‘ . 1:
c: t: d—zfi"
m m 3min...“ oooooo “8:: .........
1 1
43
-7. . ‘

 

 

 

 

 

 

 

 
 

A
0"
V

>3
o
0..

 

 

.3
c:

 

 

 

 

 

 

Energy (eV)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

, ...,“ .....
_ _ 1
(c)7‘0.hi1d j .7 -. (f) 7.04N1d - .......
is. 5:2?" " ~ 52 :2“ " r”
I! f ‘ h“
>, - Ni >‘ 3
an E 3000 E
000 F0 F
G C:
m Lu
...... —i
............. . “mo-«...
'7'0w L A . A xzwx ‘7'0w L A r A xzwx

 

 

 

 

 

OR stated that the gap formation in ZrNiSn was due to the both strong Sn p-Zr

d hybridization and the reduction of the crystal symmetry when a neutral Ni atom
enters half of the octahedrally-coordinated sites of ZrSn.[59] In order to test this
idea, we have placed He atoms in the Ni positions of ZrNiSn to form the hypothetical
compound ZrHeSn. He will not contribute any orbitals to hybridize with ZrSn, but
will reduce the crystal symmetry from N aCl to half-Heusler. No gap exists in ZrHeSn
(Figure 18) since He has no d orbitals to hybridize with the Zr d orbitals. Thus the
presence of Ni (1 orbitals to hybridize with Y/ Zr (1 orbitals, rather than only lowering
the symmetry produced by adding Ni to the N aCl sublattice to form the half-Heusler

structure]59], is necessary for the formation of the small band gap in ZrNiSn and

YNiSb.

Figure 18. The electronic structures of ZrHeSn and ZrNiSn using the same

lattice parameter

 

 

 

ZrHeSn ZrNiSn
(a) (b)
7.0 7
\ / \J/ \
S‘ 9
d) .I' 0)
is A A a
33 0.0 HP 5 0.
c: s:
LU LL]

 

 

 

 

 

 

 

 

.' =1
w 1- A 1“ A xzwx

54

 

4.1.2. ELEMENTAL REPLACEMENTS: CHANGING THE PNICTIDES

It is well known that thermoelectric properties depend sensitively on the details
of the electronic structure near Ep. The half-Heusler structure allows for a wide
variety of elemental replacements which may affect these subtle features of the band
structure while maintaining a semiconducting gap. We will discuss several of such
replacements.

The first of these modifications which we will discuss is the change of the pnictides
from As to Sb to Bi. Band structure calculations for YNiAs, YNiSb, and YNiBi show
that all three are narrow-gap semiconductors (Figure 19). The band structures near
Ep remain very similar, but the effects of SOI become more prominent in going from
As to Bi. This is seen in the band gaps which decrease from 0.53 eV (YNiAs) to 0.31
eV (YNiSb) to 0.13 eV (YNiBi) in this series. The reduction in the band gap appears
to correlate with the lowering of the pnictide .5 band (Figure 19), lying above EF,
and moving closer to Ep near the F point in going from As to Bi. In YNiAs, the As
.9 orbital is nearly degenerate with the Y d orbitals at the F point but move down
further in YNiSb and YNiBi. As these pnictide 3 levels become lower at the F point,

the band along FX becomes flatter and the gap decreases.

Figure 19. Comparison of the electronic structures of YNiAs, YNiSb, and YNiBi.

The size of the circles is pr0portiona1 to the pnictide s orbital character.

(a) 7 ‘ YNiAs As 5

 

Energy (eV)

A

U

v
>1

YNiSb Sb 8

Energy (eV)
0

I
.‘3

A
0
v
\I

YNiBi Bi 8

.....

Energy (eV)

56

4.1.3. ELEMENTAL REPLACEMENTS: CHANGING THE TRANSITION METAL

The second type of elemental replacement we discuss is the change of the tran-
sition metal from Ni to isoelectronic Pd and Pt. In YNiSb, the Ni d orbitals lie
approximately 2 eV below Ep while in Y PdSb and YPtSb they are lower in energy,
approximately 4 eV below Ep (Figure 20). Therefore, we would expect that the
band gaps for YPdSb and YPtSb would be smaller than that for YNiSb since the
Y d-Pt / Pd d hybridization in YPdSb and] YPtSb would be weaker than the Y d-Ni
d hybridization in YNiSb. However, the band gaps change from 0.31 eV (YNiSb)
to 0.15 eV (YPdSb) to 0.21 eV (YPtSb), so that the Pd compound conforms to our
expectations but the Pt compound does not. A second effect is also seen, stronger
relativistic splitting at the valence band maximum (VBM) at the F point. The SOI
splitting increases from 0.13 eV in YNiSb to 0.24 eV in YPdSb to 0.64 eV in YPtSb.
A third effect is also present, which was also seen in the substitution of the pnictides,
where the transition metal valence s orbitals move down closer to E F. It is important
to note that the gap becomes direct at the F point in YPtSb while the CBM is at
the X point. Combination of these effects in YPtSb, including greater splitting of the
VBM which moves down this peak, and the CBM moving from the X point to the F

point due to the lowering of the Pt 3 band, explain the lower Pt d levels do not lead

to a reduced band gap.

57

Figure 20. Comparison of the electronic structures of YNiSb, YPdSb, and YPtSb.

The size of the circles is proportional to the transition metal 3 orbital

character.

(a) 7. YNiSb Ni 5

Energy (eV)
9

 

A
U
v

.“

YPdSb Pd 3

Energy (eV)
9

A
O
v .

."

Energy (eV)

 

58

4.1.4. ELEMENTAL REPLACEMENTS: ROLE OF FILLED, UNFILLED,
AND PARTIALLY FILLED f-LEVELS

A third type of elemental replacement comes from changing Y to a rare-earth
element. It is well known that LDA/GGA calculations have difficulty with highly
localized partially-filled f -shells, where the Fermi energy (Ep) is always placed in the
middle of the very narrow f -band. Corrections such as LSDA+U can be introduced
to partially correct for this in an ad hoc manner.]73] We have avoided this problem by
replacing Y (Figure 21a) with atoms with either filled f-levels (Lu) or empty f-levels
(La). The empty f-levels in LaNiSb lie approximately 3 eV above the Fermi energy,
and have little effect on the band structure near the Fermi energy. (Figure 21b) On
the other hand, the filled f -levels in LuNiSb, which lie at approximately -4.5 eV and
-6 eV below Ep with a SOI splitting of 1.52 eV, push up the conduction band states
slightly to reduce the band gap in this material. (Figure 21c) This can be seen in the
band gaps going from 0.31 eV (YNiSb) to 0.40 eV (LaNiSb) to 0.19 eV (LuNiSb).[7]
The observation that the empty Lu f -levels do not affect the band structure as much
as the filled La f -levels maybe due to greater f—d hybridization in the Lu compound
or a lack of splitting of the La f—levels in the conduction band states which are not
treated correctly in DFT[31].

The experimental structure for LaNiSb is actually known to be hexagonal instead
of half-Heusler. The difference in the coordination of the Ni d orbitals and differ-
ent hybridization between the Ni (1 and La d orbitals lead to a metallic instead of

semiconducting behavior in this compound (Figure 21d).[69]

59

Figure 21. Electronic structures of (a) YNiSb, (b) FCC LaNiSb, (c) LuNiSb, and
(d) hexagonal LaNiSb. The Ni d bands and La and Lu f bands are

      
 

 

 

 

 

 

 

 

 

 

 

shown.
(a) 7 YNiSb (c) 7 LuNiSb
3., s
>~ >
5 5
1‘5 d 15
f
-7 -7
(b) 7OLaNiSb-fcc (d) 7 LaNiSb-
A E k A 9
g ‘_ ‘ __ ’{élti’e-Laf 8
v ‘
am} ‘7 ’7 E1: 2'3 f
s I .. a)
15 1 42.-:25 fl ”‘3
‘mw . A r A z 7

 

 

Although LDA/GGA is not adept for partially-filled, narrow-band systems, it has
shown some success in handling f-electron systems with one electron (Ce) or one hole
(Yb). In YbNiSb, Yb has one hole in the f-shell and is most likely a mixed-valence,
heavy-fermion system.[68] Similar attempts to study the band structure of mixed-
valence systems have been successfully made in CCR112812.]74] The only previously
published electronic structure of YbNiSb[67] (aside from our own[7]) was carried out
at the scalar relativistic level within LDA using LMTO-ASA. Unfortunately, these

results differ significantly from that of a closely related system, YthBi[62], and the

60

 

calculated band structure[67] appears strange with several flat portions, implying a
lack of convergence in the band structure calculation.

The metallic behavior of YbNiSb can be understood using the same technique as in
the semiconducting compounds YNiSb and ZrNiSn by first studying the parent NaCl
binary compound YbSb. YbSb displays a strong f—p hybridization near E p although
the majority of the f -1evels are below Ep (Figure 22a). The SOI is quite strong for
the f-levels and the splitting between fg and f; is about 1.25 eV (Figure 22b). The
f-levels also become flatter as seen in the increase in the density of states (DOS) with

SOI (Figure 22c, d).

Figure 22. Electronic structure of YbSb (a) without and (b) with SOI. The

corresponding DOS (c) without and (d) with SOI are also given.

(a)7 0 YbSb - No SOI

 

 

 

 

 

 

 

 

 

 

 

 

 

A ’

> /

3

>5

$300.0 ¥. \ aEF

G ‘7.

Lu x “~Ybf
'7'0 7‘ A x z w K

(b) 7.0 f

9

8,

36

s Be...

m *z
‘7'0 ' f‘ A x z W K

 

 

 

 

 

 

 

61

(c)

(d)

500

400

states/eV
00
8

N
O
O

500

400

states/9V
(JD
8

N
O
O

YbSb - No SOI

 

 

 

 

 

.l l l 141
-13-1 2-11 -10 -9 -8

YbSb - SOI

57-635-21-352-105

Energy [eV]

 

r

T

L

 

 

Energy [eV]

 

1 1 1 M 1 1 1 I m
-13-12-11-1O -9 —8 -7 -6 -5 -4 -3 -2 -1 0

1

 

lA—W
2345678

Insertion of Ni in the lattice to form YbNiSb places the Ni d orbitals below the f

62

levels, which get pushed up in energy. As a result, there is a partial charge flow from
the Yb f levels into the Sb p bands. (Figure 23a) Yb f levels become partially occupied
and YbNiSb acquires a heavy fermion character much like YthBi, the splitting
between fg and f; remaining about 1.25 eV (Figure 23b), in agreement with the
calculations for YthBi.[62] The total density of states (TDOS) (Figure 23d) at the
Fermi level, N (e p), is 12.6 states / eV, 77% of which is of Yb f character. In comparison,
the corresponding numbers for YthBi are 4.5 states/eV and 60%, respectively.]62]

It should be emphasized that if we do not include SOI (Figure 23c), N(€F) reduces
to about 6.9 states/eV and the f character is also reduced to about 61%. Therefore,
SOI should play an important role in the low-energy properties (the linear term in

the heat capacity and the transport properties) of this system.

Figure 23. Electronic structure of YbNiSb (a) without and (b) with SOI. The
corresponding DOS (c) without and (d) with SOI are also given.

(a) 7 YbNiSb - No SOI

Energy (eV)

 

(b) 7 YbNiSb - SOI

Energy (eV)

 

63

 

(c) 10 YbNiSb - No SOI .

 

  

 

 

 

 

 

 

 

8 "' -4
> 6 +- .
Q
U)
2 4
(U
775 4 ~ A
2 i- - 1
~13-12-11-1o-9 -e -7 -6 -5 -4 -3 -2 -1 e 1 2 3 '4 5 6 7 8
Energy [eV]
(d) 10 YbNiSb - SCI
8 L .
E 6 ' ‘
B
2
(U
1'5 4 h- -
2 - .
-1'5-12-11-10-9 3-7 -6 -5 -4 -3 -2 i 1 2 3 4 5 6 7 u

 

 

-1 15
Energy [eV]

4.1.5 ELEMENTAL REPLACEMENTS OF NiO WITH AuH/AgH/Cu1+

More semiconducting half-Heusler phases can be found if the transition metal is
changed from Ni to another transition or noble metal with a valence other than 0
(+1, -1, +2, -2, etc.) with a corresponding change in the valence of the elements in

the NaCl substructure so as to maintain EC 2 18. When changing Ni to Cu (Ag,

64

 

Au), the valence goes from Ni0 to Cu+1(Ag+1, Au“). Curiously, MgCuBi is the only
On containing and MgAgAs the only Ag containing half-Heusler compounds which
might be semiconductors, but no gap opens in either of these materials (Figure 24).
It is clear that the EC 2 18 rule does not work in these cases (Mg: N = 2, Cu: 1’
= 11; Bi: N = 5; Mg: N = 2, Ag: N = 11; As: N = 5).]65, 70] It is important
to note the lack of empty d valence orbitals in Mg compared to Zr, Y, La, Lu, etc.,
which were found to be the predominant factor in the gap formation of YNiSb and
ZrNiSn.[7] It seems that the EC 2 18 rule is a necessary but not suflicient condition

for the opening of a semiconducting band gap.

Figure 24. Electronic structure of semi-metallic MgCuBi and MgAgAs.

 

   

 

 

(b) MgAgAs
7.0: v \
.. 2N 9
a. a r-
e i /\

EngODE— \, EF
m :

/ /

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.0‘ -7.0
WLAFAXZWK WLAFAXZWK

There are several potential semiconducting half-Heusler compounds containing
Au. We will compare one of interest, Lu3+Au1+Sn4‘ to the previously studied Ni
compound Lu3+Ni°Sb 3’. The Au 0! orbitals appear at approximately 6.0 eV below

Ep (Figure 25a) compared to the Ni d orbitals which appear at approximately 2.0

65

eV below Ep (Figure 25b). Consequently, the hybridization of the Au d levels with
the d orbitals of Lu is much smaller compared to that with the Ni (1 levels. The Au
(1 contribution to the CBM at the X point in LuAuSn (Figure 25a) is much smaller
than the Ni 11 contribution to the CBM at the X point in LuNiSb (Figure 25b). The
strength of the Au (1 or Ni (1 characters in Figure 25 are proportional to the size of
the circles. The weaker hybridization leads to a reduction in the band gap, becoming

slightly negative in LuAuSn compared to 0.19 eV in LuNiSb (Figure 25).

Figure 25. Electronic structure of LuAuSn and LuNiSb. The Ni d bands lie
closer to E p than the Au d bands. The size of the circles is

proportional to the orbital character.

 

 

     
  

 

 

 

 

 

 

 

 

 

(a)7n LuAuSn Au (b)7n LuNiSb Ni d
:-. 333:3". .‘: Iv I"'In
> ~ > a .-
8 8 .
>1 >1 I
felon EF :‘300-“3 . Er:
t: G
LI.) LL)
I)
h
l ........ \ _’ ~mc
10 .. .. ..

 

 

 

 

 

 

 

 

,1. ’7'“. '
WLAFAXZWK WLAFAXZWK

66

4.1.6. ELEMENTAL REPLACEMENTS OF NiO WITH Col’/ R111“

Replacing Ni0 by Col" and Rhl‘ has the opposite effect of replacing Ni0 by Au1+ .
We will consider a representative compound, Ti4+Col’Sb3‘, whose band structure can
be compared with that of Ni-containing Ti4+Ni°Sn4‘. The band structure of TiNiSn
is roughly the same as that of ZrNiSn with nearly the same band gap, 0.50 eV (Figure
26b).[59] The calculated band gap for TiCoSb is significantly larger, 1.08 eV (Figure
26a). The band structure for TiCoSb is different from all the semiconducting half-
Heusler systems discussed so far. Its VBM moves from the F point, still predominantly
of Sb 1) character, to the L point, primarily of Co d and Zr d character, as is the case
for the CBM at the X point. The C0 d level not only has a significant contribution
below Ep, around -1 eV, but also above Ep around 2 eV. The flat Co (1 levels lying
about 1.5 eV below E 1: along WL push up the hybridized Co d and Ti d orbitals at
the L point above the Sb p levels at the F point. The greater hybridization of the
Ti (1 orbitals with Co (1 (Figure 26a) compared to Ni d orbitals (Figure 26b) near E p
leads to an increase in the band gap.

This relationship between the position of the transition (or noble) metal d orbitals
and the band gap could be modeled by an effective Hamiltonian. The size of the band
gap for a generic half-Heusler system could be predicted by knowing, for instance,
the position of the Ni 01 orbitals in relation to the Zr (1 orbitals in ZrNiSn, and by
comparing the calculated values of the band gaps. While we will not do this, our
results suggest that such a modeling would be useful in further understanding the

band gaps of the half-Heusler compounds.

67

Figure 26. Electronic structure of TiCoSb and TiNiSn. The Co (1 bands lie
closer to E; than the Ni d bands. The size of the circles is

proportional to the orbital character.

 

(a)7 0 TiCoSb Co (1 (b)7 TiNiSn Ni d

S‘ 9

8 3

56 is

'6 FFEFB 0. O F
r: 1:

L1.) L1.)

 

 

 

-.0 - - -7.
W'LAFAXZWK WLAFAXZWK

Unlike TiCoSb, in Ti4+Rh1“Sb3’ the VBM moves back to the the F point from the
L point as in Ti4+Ni°Sn4’, with an even larger gap, 1.16 eV (Figure 27). The band
along WL, which was pushed up to form the maximum in T iCoSb, is not pushed up
as much in TiRth. The CBM also changes significantly in TiRth. In both of these
materials, there are two bands near the CBM, a flat hybridized band of Rh d and Ti
d and a more dispersive unhybridized Ti (1 band. In TiCoSb, this more dispersive
band is 0.25 eV below the less dispersive band at the X point while in TiRth the
flatter band is below the more dispersive band by about the same amount. This
change in the dispersion in the conduction band should have an important effect on

the thermoelectric properties of the electron doped system.

68

Figure 27. Electronic structure of TiRth.

0 TiRth

Energy (eV)

 

4.1.7. ELEMENTAL REPLACEMENTS OF NiO WITH Fe2'/Ru2‘

The last elemental replacement we will discuss is changing NiO to Fez" and Ru2‘.
Again we will compare Y3+Ni°Sb3" with a representative material, Nb5+Fe2’Sb3‘.
As in TiRth, the LCB consists of two bands, a more dispersive unhybridized Fe (1
band and a flatter hybridized Fe d-Nb d band. The more dispersive Fe (1 band lies
below the flatter hybridized Nb d and Fe d band, now by about 1 eV. As in TiCoSb,
Fe (1 orbitals in N bFeSb split at about -0.5 eV to push up the bands between WL,
producing a VBM at the L point. The calculated band gap is 0.53 eV, larger than

the value of 0.28 in YNiSb.

69

Figure 28. Electronic structure of NbFeSb and YNiSb. Note the difference in

curvature near the X point in the LCB.

(a)7. NbFeSb (b)7‘ YNiSb

 

Enefirgy (eV)

Energy (eV)

7 -7.
WLAFAXZWK WLAFAXZWK

4.1.8. ELEMENTAL REPLACEMENTS OF Ni0 WITH Mn3‘

Finally, the logical extension is to try Mn3“ as the transition metal in the half—
Heusler structure. This does not yield any new semiconductors, since Mn often has
a mixed valence. However, half-metallicity has been predicted in MnNiSb, where
Mn forms part of the NaCl substructure rather than replacing the transition metal
Ni atoms.[75] Half-metallic systems have a metallic majority spin (here up) state
with a semiconducting minority (here down) spin state (Figure 29). Subsequent
photoemission and transport measurements have found that the polarization, unlike
the 100% which should exist in a truly half-metallic system, is only about 50%.]76] In
the presence of weak Mn-Sb disorder (even on the order of 5%) the semiconducting
gap seen in the minority spin channel disappears and the system becomes completely

metallic (though still ferromagnetic)[75], consistent with nearly 10% Mn-Sb disorder

70

which is seen experimentally.]76] -
Figure 29. Calculated DOS of half-metallic MnNiSb for (a) up and (b) down
spin states.

(a) . .
aneb -.Sem.Ue

 

120 ~

100 —

80~ —

states/6V

 

 

 

-11-1o-9-6-7-6-5-4-3-2-1o12 3 4 5 6 7 8 910
(b) . Energy [eV]
14o MnNiS.b-.Sem.°9we

 

 

 

 

 

 

 

120 - i -1
100 - a
3 80 ~ .1
75
.22
g 60 ~ «
a, ,
4O - -
20 . 1' .
1 L 1 1 1 1 1 mm
4140-9 -8 -7 -6 -5 -4 -3 —2 -1 0 1 2 3 4 5 6 7 8 9 10
Energy [eV]

71

4.2. STRUCTURAL STABILITY OF HALF-HEUSLER COMPOUNDS

4.2.1. TOTAL ENERGY OF THE TERNARY SYSTEMS IN DIFFERENT
STRUCTURAL CONFIGURATIONS

We have studied elemental replacements as important for the gap formation, but
the positions of the atoms in the unit cell of the half-Heusler compounds also play an
important role in the semiconducting gap formation and structural stability. There
has been some confusion regarding the nature of the ordering of the atoms within the
unit cell and which ordering gives the lowest energy. Ogut and Rabe (OR) have stud-
ied the stability of ZrNiSn, using LDA and the pseudopotential plane-wave method,
within three possible configurations. They correctly identified the configuration of
atoms with the lowest energy, the same configuration seen experimentally.]59] They,
however, concluded that the inclusion of Ni atoms into the octahedrally coordinated
pockets of the NaCl-type basis structure ZrSn had energetically very little effect on
the stability of the final compound ZrNiSn. This would imply that Ni atoms essen-
tially enter the above pockets as neutral atoms and do not take part in any sort of
bonding with the neighboring Zr and Sn atoms. However, in the previous section we
showed that hybridization of Ni atoms played a significant role in the gap formation.
The effect of Ni atoms was not just simply to lower the local symmetry compared to
the binary compounds, thereby leading to additional hybridization between Zr and
Sn orbitals, forbidden in the absence of Ni, as envisaged by OR.[7, 9]

A recent paper by Ishida et al.[77] analyzed the structural stability of a series of
half—Heusler compounds (without reference to the results of OR), using the LMTO-

ASA method and LDA, and found that the lowest-energy configurations for most

72

members studied were different from those seen experimentally.]66] This has led us
to reevaluate the question of structural stability of the half-Heusler semiconductors
using FLAPW.]11] To this end we have investigated the energetics of the same six
systems studied by Ishida et al.[77]: NiTiSn, NinSn, NinSn, NiSch, NiScSn, and
NiTiSb.

Within this section only, the possible configurations of the constituent atoms inside
the unit cell can be written in the same shorthand notation used by Ishida et al.[77]
For a half-Heusler compound NiMP (M 2 Ti, Zr, Hf, Sc; P 2 Sn, Sb), this notation
denotes the Ni atom as X, the M atom as Y, and the P atom as Z. The order of X, Y,
and Z tells the relative positions of the atoms within the cell. The first letter refers to
the atoms within the octahedrally coordinated pockets located at the position (7}, i,
A) while the second refers to the atom at position (%, %, %) and the third to the atom
at position (0, 0, 0). Therefore, a configuration designated by X YZ has the Ni atom
in the octahedrally coordinated pocket with the M and P atoms forming the NaCl
substructure (written as NiMP) while YXZ exchanges the position of the Ni and M
atoms within the cell (written as M NiP) (Figure 30). These different configurations
can also be distinguished by the Greek nomencl’atures (a, B, 7) where (1 refers to

the X YZ configuration, B refers to the Z YX configuration, and '7 refers to the YXZ

configuration, respectively.]59]

73

Figure 30. Half-Heusler crystal structures (a) XYZ, (b) ZYX, and (c) YXZ.

XYZ YXZ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) (b) (C)
O M',X 0 R2 0 MY

The total LAPW-GGA energies of one system of interest, NiTiSn, for these three
different configurations of the constituent atoms in the unit cell as functions of the
lattice parameter are given in Figure 31 (the lowest energy is set to zero).

Due to symmetry, interchanging the positions of the atoms comprising the NaCl
substructure without changing the position of the atom within the octahedrally coor-
dinated pocket gives the same energy within the convergence prescribed by the self-
consistent iterations. The configuration which has the lowest energy is XYZ where the
Ni atoms lie in the octahedrally coordinated pockets. The energy difference between
this lowest-energy structure (XYZ) and the other two structures is about 2 eV/ unit
cell. The equilibrium lattice parameters for the YXZ and ZYX configurations also
differ by about 0.2 A from that of the XYZ. The XYZ structure is the most compact,
and there is a small energy difference between the YXZ and ZYX arrangements which
is less than 0.1 eV/ unit cell. These results hold for all of the systems studied, includ-
ing NinSn (Figure 32), whose results are consistent with the earlier pseudopotential

calculationl59].

74

Figure 31. Total energies as a function of lattice parameter for the three

different configurations of NiTiSn (the lowest energy is set to zero).

 

 

 

 

 

 

 

 

 

 

c XYZ
o ZYX
o YXZ
NiTiSn
2.5, .. -..-.,--
i ' 1 1
2‘ * 332.333.33.138 .
A 1.5’ 1
E i 1
S l .
L.
g . l
L” 0.5] ‘ ‘
f I
OL °Onne::¢nn0
I .
-0.5l1Lm1111 111;.1..1_11-..
5.8 5.9 6 6.1 6.2 6.3

Lattice Parameter (A)

Figure 32. Total energies as a function of lattice parameter for the three

different configurations of NinSn (the lowest energy is set to zero).

 

 

 

 

 

 

 

 

 

 

c XYZ
O ZYX
o m
NinSn
Savfifiy ku rfivafivvv'v
i a °§ooooo1i°°°i
2.5 1 o
i 000000000 j
1 I
2, .
s I 1
1
9, 1.5, ,
> L ' 1
2') l
m 11 .
C , 1
LU » 1
0.5_ .
. 4
.05l..-11.-.-111m11.-m.1.-1111- l
6 6.1 6.2 6.3 6.4 6.5 6.6

Lattice Parameter (A)

75

 

Our LAPW-GGA results for the six different compounds investigated by Ishida
et al.[77] are listed in Table II. The first column lists the systems (XYZ) and their
experimental lattice parameters (in a.u.), where available. The next three columns
give the calculated equilibrium lattice parameters of the three types of atomic ar-
rangements while the final three columns give the energy differences (in eV) between
these different configurations. The configuration for each compound which Ishida
et.al.[77] found to have the lowest energy is italicized. The results of LAPW-GGA
differ dramatically from those of LMTO-ASA LDA, but these LAPW-GGA results
for NinSn agree very well with those obtained with the pseudopotential LDA.[59]
In order to see whether these differences were caused by LDA/GGA, both LDA and
GGA were tested for NinSn. The differences between these two methods showed

energy differences only on the order of 1-2%.

Table II. Experimental and calculated equilibrium lattice parameters and

total-energy differences for half-Heusler systems (GGA).

 

 

 

 

 

 

 

 

 

 

System Exp. (XYZ) XYZ (a) ZYX (l3) YXZ (7) ZYX-XYZ YXZ-XYZ YXZ-ZYX
(a.u.) (a.u.) (a.u.) (a.u.) (eV) (eV) (eV)
NinSn 11.187 11.261 11.516 11.595 2.03 2.12 0.09
NinSn 11.463 11.572 11.955 12.150 2.43 2.67 0.24
NinSn 11.546 11.607 12.042 12.168 2.30 2.68 0.38
NiSCSb 11.442 11.607 11.942 12.107 1.66 2.39 0.74
NiSCSn NA 11.690 12.080 12.121 2.23 2.19 -0.01
NiTiSb 11.187 11.291 11.589 11.663 1.51 1.37 -0.14

 

 

 

 

 

 

 

 

76

 

Figure 33. The electronic structures of (a) XYZ, (b) ZYX, and (c) YXZ

configurations of NiTiSn.

(a) N iTiSn
3.0 2“.

 

 

\ pi...

 

.0
o 1

Energy (eV)
\\
{i

 

." o
..
... o
o

 

l k l

 

 

 

 

3.0 _

(b) 3.0 SnTiNl

 

 

 

 

 

(>
f:
o
r
>
*1
M
K
x‘

 

 

 

 

.9
g
4

Energy (eV)
1
M1 .
K r

 

 

 

 

 

 

(C) 3.0

 

 

I
I.
o

l .l

 

Energy (eV)

 

 

 

 

 

 

 

 

 

Figure 34. The electronic structures of (a) XYZ, (b) ZYX, and (c) YXZ

configurations of N inSn.

(a) 3 o N inSn

 

    

 

4_n_ .1. L 1
.
.....
.....

  
 

.0

 

Energy (eV)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

78

Comparison of the electronic structure in these three configurations shows a drastic
effect. The electronic structures for N iTiSn in the XYZ (NiTiSn), ZYX (SnTiN i), and
YXZ (TiNiSn) configurations are given in Fig. 32. As can be seen. the lowest energy
configuration, NiTiSn, is a semiconductor with the V BM at. the F point and the
CBM at the X point, as seen in the previous calculations.[77] In SnTiNi, the lowest
conduction band (LCB) along I‘X drops to form a zero-gap semiconductor. In T iNiSn,
the LCB moves below Ep while the highest valence band (HVB) shifts away from the
F point to the K point and moves above Ep to form a metallic system.[ll]

These results also hold for the other semiconducting systems studied (NinSn,
NinSn, NiSch). The results for the different configurations of NinSn in Figure 34
agree with the pseudopotential calculations by OR.[59]

It is unclear from the paper by Ishida et al.[77] which lattice constants were used
in their calculations. Also, the values of the energy differences between the different
configurations were not given and the nature of the electronic states of these half-
Heusler compounds in configurations other than XYZ were not discussed. They
claimed that their results were sensitive to the muffin-tin radii (MTR) chosen, so
the same MTR was kept the same for each configuration of the individual systems,
given in Table III. In view of our present calculations, their suggestion that these
half-Heusler systems can exist in configurations other than XYZ is not likely. Also it

would disagree with several experiments on these materials.[66]

In order to understand the reason for the gap reduction in the ZYX and YXZ
configurations, we have investigated the orbital character of the band structure at
the relaxed lattice positions for the three configurations. A significant difference is
seen in the position of the Ni d orbitals in each of these configurations (Figure 35).
In the XYZ configurations, where Ni lies in the octahedrally coordinated pockets, the

Ni d orbitals lie about 2 eV below Ep. In the ZYX configuration, where Ni forms

79

part of the NaCl substructure with Ti and Sn is in the octahedrally coordinated
pockets, the Ni d orbitals play a more prominent role in the valence band with less
importance in the lowest conduction band which moves down to reduce the gap. In
the YXZ configuration, where Ni forms in the NaCl substructure with Sn and Ti is
in the octahedrally-coordinated pockets, the Ni d orbitals hybridize less with the Ti
d orbitals and become prominent both above and below Ep, the system becoming

metallic (Figure 35).

Table III. Muffin-tin radii of half-Heusler systems.

 

 

 

 

 

 

 

 

System Muffin-tin radii (a.u.)
NiTiSn .' ‘ 2.33
NinSn i 2.30
NinSn 2.36
NiSch 2.31
NiScSn 2.37‘
NiTiSb 2.24

 

 

 

 

A few words should be said about NiTiSb and NiScSn, the two metallic half-Heusler
structures which we have studied. As in the semiconducting cases, the lowest energy
configuration is the XYZ, the same as seen experimentally. This means that the
stability of this configuration need not be due to the formation of a band gap in the
system. However, the YXZ has a lower energy than the ZYX in these two cases. In
comparison, when XYZ was semiconducting, ZYX was a zero-gap semiconductor while

the YXZ was metallic, while in these two systems all configurations are metallic.|11]

80

Figure 35. The electronic structures of (a) XYZ, (b) ZYX, and (c) YXZ
configurations of NinSn. The size of the circles overlying the

bandstructure are directly proortional to the with the Ni d

character at each point.

(a) N iTiSn Ni (1

éaeéfia

 

 

 

 

 

 

 

 

 

 

'3'0r A XZWQL A r )2 K x
(b) 3 0 SnTiNi Ni d

 

..

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Energy (eV)

 

 

81

4.2.2. ANALYSIS OF THE STABILITY OF THE TERNARY COMPOUNDS
BASED ON THE STABILITY OF THE BINARY PAIRS

Since we have found that the XYZ configuration has the lowest energy, we will
follow the analysis by OR[59] who studied the energetics of the binary compounds
which form in three different configurations in order to see which types of bonds
lead to the stability of the ternary compounds. The half-Heusler compounds can be
considered either as a NaCl lattice with atoms placed in half of the octahedrally-
coordinated pockets or as a ZnS lattice with atoms placed within the voids of this
network (Figure 12). The lowest energy configuration should in principle lower the
interaction energy between all three components in the structure, so it is worthwhile
to understand how each pair of atoms within the lattice interact in order to form the
lowest energy configuration. In NinSn (XYZ), Nin forms a diamond structure while
ZrSn for a NaCl structure while in ZrNiSn(YXZ) ZrSn and Nin both form diamond
structures. In the calculation of OR[59], first the energies of the binary compounds
were studied at a “fixed” lattice parameter for all three ternary configurations and
were then compared to energies obtained at “relaxed” lattice parameters. We will
show that a better understanding of the energetics arises from using the relaxed
lattice parameters appropriate for these different configurations.

The energy differences between different configurations of the binary compounds
are given in Table IV using LDA and the fixed lattice parameter, as was used by
OR.[59] In the top third of the table, the results of the pseudopotential calculation
of OR, in the middle third our LAPW-LDA results at the fixed lattice parameter.
and in the final third our LAPW-GGA results at the fixed lattice parameter. The

first column gives the binary system studied, the next three columns gives the crystal

82

structure within the XYZ, ZYX, or YXZ configuration of the half-Heusler system,

and the energy differences are tabulated in the last three columns, first between the

half-Heusler systems and then between the binary systems from which it is composed.

Table IV. Total-energy differences for N inSn and different binary systems

with fixed lattice parameters calculated using the pseudopotential

LDA [Ogut and Rabe[59]] and using LAPW within LDA and GGA

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

NinSn XYZ ZYX YXZ ZYX-XYZ YXZ-XYZ YXZ-ZYX
OR [59] Pseudopotential (LDA)
11.546 a.u. 2.28 eV" 2.85 eV° 0.57 eV°
Diamond-NaCl Diamond-NaCl Diamond-diamond
ZrSn NaCl Diamond Diamond 2.03 eV 2.03 eV 0 eV
N aCl-diamond Diamond-diamond Diamond-NaCl
Nin Diamond NaCl Diamond 0.26 eV 0 eV -0.26 eV
Diamond-diamond NaCl-Diamond NaCl-diamond
NiSn Diamond Diamond NaCl 0 eV 0.23 eV 0.23 eV
NinSn XYZ ZYX YXZ ZYX-XYZ YXZ-XYZ YXZ-ZYX
LAPW (LDA)
11.546 an. 2.63 eV 3.29 eV 0.66 eV
Diamond-NaCl Diamond-NaCl Diamond-diamond
ZrSn NaCl Diamond Diamond 2.03 eV 2.03 eV 0 eV
NaCl-diamond Diamond-diamond Diamond-NaCl
Nin Diamond NaCl Diamond 0.50 eV 0 eV -0.50 eV
Diamond-diamond NaCl-Diamond NaCl-diamond
NiSn Diamond Diamond NaCl 0 eV 0.76 eV 0.76 eV

 

83

 

 

NinSn XYZ ZYX YXZ ZYX-XYZ YXZ-XYZ YXZ-ZYX

 

 

l
l

1.pr (GGA)

 

1 1 .546 a.u. 2.60 eV

 

 

 

3.25 eV 0.65 eV
Diamond-NaCl Diamond-NaCl Diamond-diamond
ZrSn NaCl Diamond Diamond 2.33 eV 2.33 eV 0 eV

 

NaCl-diamond Diamond-diamond Diamond-NaCl

 

 

 

 

 

 

 

 

 

 

 

 

Nin Diamond NaCl Diamond 0.22 eV 0 eV -0.22 eV
Diamond-diamond NaCl-Diamond N aCl-diamond
NiSn Diamond Diamond NaCl 0 eV 0.51 eV 0.51 eV

 

“Energy differences for ternary compounds calculated by OR[59] computed at relaxed

lattice parameters

Overall, the calculated energy differences between the different configurations ob-
tained using pseudopotential-LDA and LAPW-LDA agree fairly well (Table IV). The
largest energy difference is seen for the ZrSn configuration in a NaCl over a diamond
structure. However, there still are differences between the methods. The difference
in the energies between the NaCl and diamond structures in Nin is found to be 0.50
eV in LAPW compared to 0.26 eV in the pseudopotential method. Also, the energy
difference between the NaCl and diamond structures in NiSn is found to be 0.76 eV
in LAPW compared to 0.23 eV in the pseudopotential method. As has been noted
before, this pseudOpotential calculation missed a significant contribution to the CBM
above Ep which came from the Ni (1 orbitals. Comparison of the pseudopotential-
LDA and LAPW-GGA finds that these actually have better agreement compared
to the pseudopotential-LDA and LAPW-LDA. The difference between Nin in the
NaCl and diamond phases is only 0.22 eV in the LAPW-GGA compared to 0.26 eV
in the pseudopotential calculation and for NiSn it is only 0.51 eV in the LAPW-GGA

84

 

compared to 0.23 eV in the pseudopotential calculation.[59]

Table V. Relaxed equilibrium lattice parameters and total-energy differences for

NinSn and different binary subsystems calculated using LAPW
within LDA and GGA.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

NinSn XYZ ZYX YXZ ZYX-XYZ YXZ-XYZ YXZ-ZYX
LAPW (LDA)
11.546 3.0." 11.429 8.0. 11.735 8.0. 11.928 a.u. 2.50 eV 2.98 eV 0.48 eV
Diamond-NaCl Diamond-NaCl Diamond-diamond
ZrSn NaCl Diamond Diamond 1.99 eV 1.89 eV -0.10 eV
NaCl-diamond Diamond-diamond Diamond-NaCl
Nin Diamond NaCl Diamond 1.08 eV 0.72 eV -0.37 eV
Diamond-diamond NaCl-Diamond NaCl-diamond
NiSn Diamond Diamond NaCl 0.32 eV 1.49 eV 1.17 eV
NinSn XYZ ZYX YXZ ZYX-XYZ YXZ-XYZ YXZ-ZYX
LAPW (GGA)
11.546 a.u.." 11.261 a.u. 12.042 a.u. 12.168 a.u. 2.30 eV 2.68 eV 0.38 eV
Diamond-NaCl Diamond-NaCl Diamond-diamond
ZrSn NaCl Diamond Diamond 1.78 eV 1.71 eV -0.07 eV
NaCl-diamond Diamond-diamond Diamond-NaCl
Nin Diamond NaCl Diamond 1.16 eV 0.68 eV -0.47 eV
Diamond-diamond NaCl-Diamond NaCl-diamond
NiSn Diamond Diamond NaCl 0.36 eV 1.30 eV 0.95 eV

 

 

 

 

 

 

 

“Experimental value.[59]

These energetics are found to be dependent also on the lattice parameters chosen.

85

 

 

Table V is set up in the same way as Table IV except that the lattice parameters for
the minimum energy configurations for the three different arrangements (XYZ, ZYX,
YXZ) are used in the calculation of the energetics of the ternary and binary systems

in our LAPW calculations.

For NinSn the energy difference between the XYZ and ZYX (YXZ) configuration
for LDA is 2.50 eV (2.98 eV)/unit cell and for GGA is 2.30 eV (2.68 eV)/unit cell,
whereas the energies of the ZrSn substructure originating from the XYZ and ZYX
(YXZ) configurations for LDA is 1.99 eV (1.89 eV)/unit cell and for GGA is 1.78
eV (1.71 eV) / unit cell, about three-fourths of the energy gain found in the ternary.
The energy difference between the XYZ and ZYX structures for Nin is 1.08 eV for
LDA and 1.16 eV for GGA, 30-50% less than that for ZrSn. Similarly, the energy
difference between the XYZ and YXZ structures for N iSn is 1.49 eV for LDA and 1.30
eV for GGA, each about 25% less than that for ZrSn. Comparing these numbers to
those obtained at fixed lattice parameters, it is clear that the relaxation has a strong
effect on the relative energies of the binary compounds. For example, comparing the
NaCl and diamond phases of Nin and ZrSn, for GGA the Nin energy difference is
only M of that for ZrSn using the fixed lattice parameters but becomes 65% that
of ZrSn using the relaxed parameters. Similar values are found for both LAPW-LDA
and LAPW-GGA. Also, we have showed that the LAPW-LDA and LAPW—GGA give
similar results to the previous pseudopotential calculation at fixed lattice parameters,

so the difference in the analysis is due to the effects of lattice relaxation.[59]

The energy differences for NiTi and NiSn in the incorrect configuration using the re-
laxed parameters showed that they were an appreciable part of the relaxation energy,
much more than obtained using the fixed lattice parameters. Also, the pseudopo-
tential calculation by ORI59], which attributed so much to having TiSn in the NaCl

configuration as the main lowering in energy in the half-Heusler structure, missed the

86

important contribution of the Ni (1 orbitals in the CBM which are so important to
the gap formation. Therefore, in the structural stability of the half-Heusler system,
the Ni d orbitals play an important, although secondary role to the configuration of

TiSn, in the structural stability of the half-Heusler system.
4.2.3. ENERGETICS OF ANTI-SITE DEFECTS

Anti-site defects within NiMP compounds have been used to explain why the
calculated band gaps of the half-Heusler systems are larger than the experimental
values[81], an unusual property since LDA usually underestimates the band gaps
of semiconductors. OR performed a series of calculations within the virtual-crystal
approximation for Ni(Sn1-erx)(Sanr1_3) for a few values of z lying between a: r: 0
and a: = 0.5. They found that the band gap indeed decreased with increasing a: to
produce a semi-metal when x = 0.15.[59]

This argument for the band gap reduction is based on x-ray diffraction studies
done at low temperatures (<250 K) which found anti-site disorder in the NaCl sub—
lattice of ZrNiSn (ZrSn) but no disorder in the Ni sublattice after annealing.[78] The
temperature range in which these materials are expected to be used for thermoelectric
applications is anywhere from 300 to 700 K[66], where the structure has not been fully
investigated. In fact, a Ni-defect phase was detected before annealing with different
lattice parameters a; and a“ which disappears after annealing.[79] Therefore, defects
in the Ni sublattice and within the MP framework and their effect on the electronic
structure need to be investigated.

We have investigated the effects of Ni and Zr-Sn disorder in the simple cubic
supercell (NinSn)4. This allows for changing the positions of the 12 atoms (4 Ni,

4 Zr, and 4 Sn) to see the effects of disorder within the lattice either by exchanging

87

their positions or moving atoms from their ideal positions to one of the 8 octahedrally-
coordinated sites (4 of which are occupied in the ideal structure). Configurations of

these atoms with disorder in the ZrSn sublattice are shown in Figure 36.

Figure 36. Crystal structures of (NiMP)4 supercell for (a) Ni4(Zr38n)(Sn3Zr),
(b) (Ni3Sn)Zr4(Sn3Ni), and (c) (Ni3Zr)(Zr3Ni)Sn4.

N‘Wfl'XI'J-W W33X503W0 (MW

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) (b) (C)

We find that exchanging the positions of Sn/ Zr atoms with the Ni atoms or other
Sn/ Zr atoms within the supercell (to produce a different configuration) produces a
fairly substantial increase in the energy. Using the relaxed parameters and atomic
radii used for the lowest energy configuration, the total energy differences are given in
Table VI. Switching one Ni and one Zr (Sn) atom increases the energy by a fairly large
amount, 1.71 (1.98) eV/ FCC unit cell while switching one Zr atom with one Sn atom
has a smaller energy increase (1.25 eV / FCC unit cell). This is in agreement with our
calculations for the structural stability of MP NaCl and the greater stability of Ni
within the octahedral pockets of MP. In order to understand the effects of Ni when
switching the Zr and Sn positions, we have also calculated the difference in energy

in switching the position of one Zr and one Sn atom in Ni4Zr4Sn4 and compared

88

that with the energy change when the same positions where switched in Zr4Sn4. The
energy difference was 1.25 eV / F CC unit cell in Ni4Zr4Sn4 compared to 1.10 eV/F CC
unit cell in Zr4Sn4, the Ni contributing 0.15 eV/unit cell due to the changed Zr/Sn

environment.

Table VI. Energetics of site-switching on NaCl substructure of NinSn
supercell (GGA).

 

 

 

 

 

 

 

 

 

 

System Energy - energy Ni4Zr4Sn4 (eV) AE/ FCC unit cell (eV)
Ni4Zr4Sn4 0 0
Ni4(Zr38n)(Sn3Zr) 4.99 1.25
(Ni;,Sn)Zr4(Sn3Ni) 6.82 1.71
(Ni3Zr)(Zr3Ni)Sn4 7.93 1.98
Zr4Sn4 0 0
(Zr;;Sn)(Sn3Zr) 4.39 1.10

 

 

 

 

 

In each of the cases where the positions have been changed, the band structure
becomes metallic (Figure 37). OR claimed that the band gap closed for 15% Sn-Zr

disorder[59], whereas we have studied only 25% disorder in the supercell.

89

Figure 37. Electronic structures of (a) Ni4Zr4Sn4, (b) Ni4(Zr3Sn)(Sn3Zr),
(c) (Nigsn)Zr4(Sn3Ni), and (d) (Ni3Zr)(Zr3Ni)Sn4.

(b)

    

(C) 5.0

 

 

 

 

 

9 /
8
is
i-0.0
2 EF
m
l I
-5.0 I -5.0
r z MTR A FAXZMTR s x

A different scenario for anti—site defects in half-Heusler is for the Ni atoms, within
the octahedrally-coordinated pockets, to lie outside an FCC configuration by exchang-
ing positions with the unoccupied sites. In order to investigate this possibility, the
total energies of several rearrangements of the Ni sublattice have been calculated,
again using the same lattice parameter and atomic radii for the relaxed FCC cell.

(Figure 38) The first configuration is obtained by moving one Ni atom into one of the

90

four empty sites; the second configuration is obtained by moving two Ni atoms into
empty sites to form an eclipsed arrangement; and the third configuration is obtained

by moving two Ni atoms into empty sites to form a staggered configuration.

Figure 38. Crystal structures of (NiMP)4 supercell for rearrangements of the Ni

lattice.

. ff.

M 4 l .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) (b) (c)
0M 0 P o M

The energy increases above the half-Heusler structure of these three configura-
tions are much smaller than those associated with the rearrangements of the NaCl
substructure by about a factor of 2 (060.8 eV/FCC unit cell compared to 1.3—2.0
eV/ FCC unit cell), shown in Table VII.

91

Table VII. Energetics of site-switching on Ni sublattice of NinSn supercell

 

 

 

 

 

 

(GGA).
System Energy - energy (N i4Zr4Sn4) (eV) AE/ FCC unit cell (eV)
Ni4Zr4Sn4 0 0
Ni4Zr4Sn4 - 35a 2.37 0.59
Ni4Zr4Sn4 - 35b 2.66 0.67
Ni4Zr4Sn4 - 35c 3.14 0.78

 

 

 

 

 

These systems are also metallic, but there are fewer bands crossing Ep compared
to systems with disorder in the NaCl sublattice for the three configurations given
in Figure 38 (Figure 39). The reduced formation energy and less metallic behavior
seems to imply the configurations containing disorder in the Ni lattice would be
preferred over those with disorder in the NaCl substructure, contradicting what is

seen experimentally.[78]

The results of this analysis do not agree with experimental measurements which
find that there is Zr/ Sn disorder in NinSn but no disorder of the Ni atoms out of an
FCC configuration.[78] It is possible that Ni defects have been annealed out of these
samples while the Zr-Sn defects, perhaps due to large energy barriers, are quenched

at low temperatures.

92

Figure 39. Electronic structures of (a) Ni4Zr4Sn4 - 38a, (b) Ni4Zr4Sn4 - 38b,

and (c) Ni4Zr48n4 - 380.

(a)

 

(b) 5.0

  

 

 

 

 

 

 

 

 

 

 

:W‘ V N If." / V ‘x
9 >
3 /
>~ _ "ii:;:.._‘\
$00.0 f \ E]:
:5
\ \ figs:
—-— ——f
'5"r ‘2 MTR ‘A 1‘sz inn 8 x

 

 

 

 

(c) 5.0

Energy (eV)
.0
O

 

The Zr/ Sn disorder in NinSn is believed to be due to the similarity of the atomic
radii of Zr (0.160 nm) and Sn (0.158 nm) and has been suggested as the mechanism to
reconcile the differences between the band gaps found experimentally from that in the
electronic structure calculations of the ordered NinSn.[81] The effect of Ni disorder
within the octahedrally-coordinated pockets should also reduce the band gaps of
the half-Heusler compounds. The experimental value for the band gap (0.19 eV) for
NinSn is much smaller than the value obtained from electronic structure calculations

(0.51 eV). The electronic structure calculations performed by OR[59] in the virtual-

93

crystal approximation found that the band gap decreased to zero for about 15% Zr/Sn
anti-site disorder. On the other hand, the atomic radii of Ti (0.147 nm) and Sn (0.158
nm) are different enough that anti-site defects should be extremely rare in N iTiSn.[79]
However, the experimental value for the band gap in this compound (0.12 eV[7'9]) is
also much smaller than that of the ordered NiTiSn obtained in electronic structure
calculations (0.51 eVI59]). Since one system displays anti-site disorder on the NaCl
sublattice while the other does not, while the calculated and experimental values
are nearly the same, it seems unlikely that anti-site disorder can completely explain
the band gap difference in these materials. One cannot blame LDA/GGA which
usually underestimates rather than overestimates the band gaps in binary or covalent
semiconductors. We will return to this question after discussion our calculation of

the other ternary compounds in the Zintl phase.

4.3. THERMOELECTRIC PROPERTIES OF HALF-HEUSLER COMPOUNDS

An understanding of the band structure of the half-Heusler compounds allows for
study of their thermoelectric properties. Half-Heusler compounds have been studied
extensively for their thermoelectric properties.[59, 80] With the considerable variety
of changes seen in the band structure in the semiconducting half-Heusler phases, it
is necessary to understand what properties would be best for thermoelectric applica-
tions. In order to maximize ZT —- Son/n, it is necessary to have a large S and a and
a low re. From the derivation of ZT from the Boltzmann transport equations, the B
parameter (Equation 24 in Chapter 2) gives insight about how to increase ZT.[2~1| As
stated before, to increase B, which then increases ZT, one needs to look for systems

with low mph, multiply degenerate band extrema to increase 7, and highly anisotropic

94

effective masses, small effective masses in the direction of transport in order to max-
imize and large effective masses perpendicular to the direction of transport.[24]

Several of the half-Heusler systems seem to be ideal candidates for electron-doped
thermoelectric materials based on the concept of anisotropic effective masses since
there is a very flat band along the F X direction which becomes more dispersive along
XW. Recently, Uher et al.[81] have studied the transport properties of ZrNiSn and
related compounds. They have estimated the effective mass of the states near the
bottom of the conduction band to be about 2-3me, where me is the free-electron
mass. We have calculated the effective masses associated with the conduction band
minimum (at the X point) and find the effective mass along I‘ X to be about 10mg,
whereas along the other directions are about me. This gives an average effective
mass of 10%me which is consistent with Uher et al.’s findings.[81] Similar effective
mass values have been found in YNiSb where the heavy electron mass along the FX
direction is a factor of 2 larger. In contrast, the hole effective masses are not as large
and nearly isotropic.

Experimentally, these systems display a large S (~-40 to -250 uV/ K) and large a
(0.1-10/mQ-cm) in the electron-doped regime. This gives a large power factor (PF)
(Son~0.8-1.4 W/mK).[80] This is consistent with our calculation for a system with
highly anisotropic effective masses. Unfortunately, I: is very large in these materi-
als (~10 W/mK compared to 1.5 W/mK for BizTe3), leading to values of ZT of
around 0.03 near room temperature (RT) and 0.29 (TiNiSn), 0.26 (ZrNiSn), and 0.22
(HfNiSn) obtained at 700K using the RT values for the thermal conductivity.[66, 81,
82,83]

In order to reduce n and further increase the power factor, many studies of ZrNiSn,

HfNiSn, and TiNiSn alloys have been performed.[66, 81, 82, 83] Doping on the Ni site

95

shows little improvement of PF, but doping of the Sn site by Bi and of an alloyed site
of Zr, Hf, and Ti by Nb and Ta show marked improvements in PF by as much as an
order of magnitude. However, the phonon thermal conductivity is not substantially
reduced, leading to ZT at RT which is nearly an order of magnitude smaller than that
of B12T83.]66, 81, 82, 83] One possibility to reduce rt is to use this materials in mul-
tilayer systems which have been shown to improve ZT in PbTe-PbSeTe quantum-dot
superlattices[27] and in p—type B12T€3 / Sb2T€3 superlattice films]28], mainly through
reduction of n.

The effective masses in YNiSb are larger and more anisotropic than those in Zr—
NiSn, but these improvements are not sufficient to expect dramatic improvements in
ZT. One of the reasons for the high thermal conductivity is the higher symmetry of
the half-Heusler crystal structure. This high symmetry can be used to our advantage
by increasing the band degeneracy factor, 7, to a maximum value of 24 provided the
extrema occur at a general point in the Brillouin zone.

In ZrNiSn and YNiSb, the CBM lies at a high symmetry point X where '7 : 3.
The degeneracy factor, and hence S, can be increased if the minimum were to lie at a
more general point in the Brillouin zone (BZ).[7] This feature was found in TthSb
(Figure 40) with the CBM moving to halfway between 1" and X, which increases its
degeneracy factor to 'y = 6. The VBM at the W point is nearly degenerate VBM at.
X and I‘. We find a gap of 0.68 e\' "rhich is larger than measured value of 0.44 eV for
the isostructural URth.[84]

96

Figure 40. Electronic Structure of TthSb. Note the off-axis LCB.

TthSb

7.0

Energy (eV)
C
O

 

-7.0

This off-axis CBM was also seen in ThNiSn. The previous calculation[10], without
the inclusion of SOI, showed a band structure very similar to that of TthSb with
the CBM forming halfway between 1" and X, but here the SOI is very important. The
inclusion of SOI reduces the band gap to less than zero, which is smaller than the
measured value of about 0.06 eV]85], but the CBM has shifted closer toward the X
point which increases the degeneracy factor to 'y = 12 (Figure 41). Unfortunately,

this material has not been studied yet for its thermoelectric properties.

97

Figure 41. Electronic structure of ThNiSn. Note the LCB lies at a general point.

ThNiSn
7.0

VA
5

 

\_

    

 

 

 

 

 

 

 

 

Energy (eV)
9
O

 

 

 

-7.o /\ - .
W L A l" A X Z W K

 

In summary, the half-Heusler compounds are still interesting for their potential

thermoelectric properties provided that their K can be reduced by a factor of 5 to 10.

98

CHAPTER 5.
TERNARY ZINTL PHASE COMPOUNDS

As discussed in detail in the last chapter, the half-Heusler structure has been de-
scribed as a stuffed NaCl structure where transition metal ions form an FCC lattice
within the pockets of a NaCl structure (Figure 12).]7, 59] With different chemical con-
stituents within the structure, half-Heusler compounds can display semiconducting[7,
59], half-metallic[75, 76], or metallic behavior[62, 67, 74]. The narrow gap semicon-
ductors of this class have been studied for thermoelectric applications.[4] While these
materials have respectable values for their thermopower (S) and electrical conductivity
(0)]4, 80] which lead to a large power factor (PF 2 820), due to their simple, isotropic
crystal structure, their thermal conductivity, n, has been fairly large.[4, 66, 81, 82, 83]

Since the thermoelectric efficiency of a material depends on the dimensionless
thermoelectric figure of merit, ZT= S20T/Iill, 2, 3], attempts to reduce the thermal
conductivity in half-Heusler compounds have proven only moderately successful since
the power factor has been observed to also reduce with alloying.[4, 66, 81, 82, 83]
The large values of K. in the half-Heusler alloys have proven the major difficulty in
using these materials for thermoelectric devices. A similar class of materials, Zintl
phase compounds in the structure Y3Au38b4, contain several of the same elements
as the half-Heusler compounds, have similar bonding and gap formation, but, due
to their cage-like structure and complex units cells, have a low n.[4] Due to their
potentially high ZT values, a thorough understanding of this new class of potential
thermoelectric materials is therefore needed through electronic structure calculations,

and is the topic of this chapter.

99

5.1. CRYSTAL STRUCTURE AND ZINTL CONCEPT

Zintl ions and Zintl phases have been studied for over a century, but their properties
are not widely known.[86] Zintl ions are clusters or aggregates of “polar” intermetallic
ions with strong charge transfer between elements. Zintl phases form out of Zintl ions
arranged to produce polar or salt-like compounds which include binary or ternary
compounds formed between alkali or alkali-earth elements and the main-group ele-
ments from group IV on. Most of these compounds are expected to be semiconducting
or semimetallic due to the inherent charge balance necessary for their formation.[87]

Zintl phase compounds of the form Y3Au3Sb4 are understood as a stuffed Th3P4
structure. Cubic Y 3Sb4 (Th3P4 structure) derives from doubling the unit cell of BCC
Sb (2 Sb atoms/unit cell) in all three directions to produce a Sb supercell (16 Sb
atoms/ unit cell). The Sb atoms are then displaced slightly along the 111 direction of
the supercell to produce two different types of Sb polyhedra: regular disphenoidsl88]
and distorted tetrahedra. The Y atoms enter the center of the disphenoids and are
8-fold coordinated to Sb to form Y35b4 (Figure 42a). The noble (or transition) metal
atoms, such as Au, enter the distorted Sb4 tetrahedra in Y38b4 to form the ternary
compound Y3Au3Sb4 (Figure 42b). After Au is added, Y is 4-fold coordinated to Sb
and 4-fold coordinated to Au, Au is 4-fold coordinated to Sb and 4-coordinated to Y,
and Sb is 3-fold coordinated to Au and 4-fold coordinated to Y. In the half-Heusler
compounds such as YNiSb, Ni is 4-fold coordinated to Y and 4-fold coordinated to Sb,
and the N i-Sb and Ni-Y distances are the same. However, in the Y3Aung4 structure,
due to the distortions of the tetrahedra, the different coordinations of the elements
are at slightly different distances.]89] The BCC Brillouin zone used in the electronic

structure calculations for the Zintl phases is given in Figure 43.

100

Figure 42.(a) Crystal structure of Y38b4 and (b) crystal structure of Y3Ni38b4.

The large open circles are Sb, the small open circles are Y, and the

filled circles are Ni.

 

 

 

 

 

Figure 43. BCC Brillouin zone.

 

101

Zintl phases and half-Heusler compounds share many similarities including the
formation of their crystal structure from parent binary structures and the nature of
the opening of their band gaps.[90, 91, 92, 93] Both of these structures contain a
backbone compound which is predicted to be either semiconducting or semimetallic
(such as ZrSn or Zr3Sb4 (isostructural to Y3Sb4)) where transition (or noble) metal
atoms enter into the octahedrally-coordinated pockets of this parent compound. In
both the half-Heusler and Zintl phases, Ni is octahedrally coordinated to 4 Zr and 4
Sn (Sb) atoms, while 4 Ni atoms form a tetrahedra about the Zr atoms. However,
there is a fundamental difference between these two structure types in regard to the Ni
coordination with Sn (Sb). In the half-Heusler phase, Ni forms tetrahedra about Sn,
but in these Zintl phases there are only 3 Ni atoms about the Sb positions, even though
the corresponding Ni-Sb—Ni angle (114.88°) is close to the tetrahedral angle. This
difference in local coordination may be the reason certain noble or transition metals
appear to prefer one structure over the other. While Ni prefers to form semiconductors
in the half-Heusler structure, Cu forms more stable semiconducting compounds in this
Zintl phase configuration for R3Cung4 (R 2 Y, La, Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho,
Er).[93] In fact, some of the U-containing compounds can be in either phase (UNiSn
and U3Ni38n4, UPtSn and U3Pt38n4, URth and U3Rh3Sb4).[93]

As discussed in Chapter 2, the search for new thermoelectric materials involves
finding new narrow-gap semiconductors.[1, 2, 3] Several semiconducting members of
this class of Zintl compounds have been found and studied experimentally.]lO, 89,
90, 91, 92, 93, 94] Th3Ning4]10] has been found both experimentally and theoret-
ically to be a narrow-gap semiconductor. Both U3T3Sb4 (T = Ni, Pd, Pt)[90] and
Ce3Pt3Bi4[91] are heavy-fermion, narrow-gap semiconductors with magnetic moments
on the U and Ce sites, respectively. However, these heavy-fermion, narrow-gap semi-
conductors do not have large thermoelectric coefficients at room temperature.]95]

In this dissertation we will present our investigations of the electronic structure

102

of three Zintl phase compounds of the Y3Au38b4 structure type, namely Zr3Ni3Sb4,
Hf3Ni38b4, and Zr3Pt33b4. The reasons for studying these systems is (i) to understand
the role in the gap formation of Ni / Pt d hybridization in comparison with the half-
Heusler compounds, such as ZrNiSn, and (ii) to understand the electronic origin of

their thermoelectric properties.

5.2. ELECTRONIC STRUCTURE OF ZINTL PHASE (STUFFED Th3P4)
COMPOUNDS

5.2.1. GAP FORMATION AND THE ROLE OF d-HYBRIDIZATION

The gap formation in the half-Heusler compounds Y3+NiOSb3’and Zr4+Ni0Sn4‘ de-
rives from a charge transfer between the elements of the NaCl substructure combined
with the strong interaction of the Ni d and Y/ Zr (1 orbitals.[7, 59] Similarly, in the
Zintl compounds the charge transfer in the binary parent compound (Zr4+)3(Sb3')4
to form the ternary (Zr4+)3Nig(Sb3‘)4 should be important in understanding the
gap formation in these materials. The only previous electronic structure study of
Zr3Ni38b4, Hf3Ning4, and Ti3Ni38b4, is the calculation of the DOS using extended
Huckel calculations[94], with their inherent limitations.

Zr3Ni3Sb4, Hf3Ni38b4, and Zr3Pt38b4 were identified by Wang et al.[94] to have the
same crystal structure as the better studied Y3Au35b4, the first compounds in this
structure type to not contain Y or an f -electron element. As discussed for Y3Au3Sl>_..
in Zr3Ni3Sb4 Ni has 4 Zr and 4 Sb neighbors, Zr has 4 Ni and 4 Sb neighbors, and Sh
has 4 Zr and 3 Ni neighbors. The shortest Zr-Sb distance is 2.9144 A, the ShOI‘tOsl
Zr-Ni distance is 2.7746 A, and the shortest Ni-Sb distance is 2.4858 A.[94]

103

In their analysis of Zr3Ni3Sb4, Wang et al.[94] claim that, in a Zintl-phase concept,
Sb is considered a nonmetallic element since it is the most electronegative component.
They claim the observed short Zr-Ni distance, 2.7746 A, produces a strong covalent
bonding between Zr and Ni, so that Zr3Ni3 forms the covalent network which then
has a predominantly ionic bonding with the interpenetrating Sb atoms. This picture
is consistent with their extended Huckel calculation which showed a reduced charge
on Zr. However, this description disagrees with Jingtai et al. [96] who studied similar
metallic rare-earth (RE)-copper Zintl phases, RE3Cung4. They claim that, due
to the shorter Cu-Sb distance (2.630 A compared to the Cu-RE distance of 2.868
A), Cu38b4 forms the covalent network, and the interpenetrating RE atoms, in the
distorted octahedra of Cu3Sb4, have ionic bonding with the Cu38b4 covalent network.
As in Zr3Ni38b4, the electronegativity of Sb is larger than that of the rare-earth
atom.[96] Both Wang et al.[94] and Jingtai et al.[96] agree that there should be a

strong charge transfer between Zr and Sb in Zr3Ni3$b4.

5.2.2. COMPARISON OF ngNi3Sb4 WITH THE HALF-HEUSLER
COMPOUND ZrNiSn

Due to the similarities between the half-Heusler and Zintl phase compounds, we
find the gap formation of these two classes of materials to be very similar. Let us
briefly review the gap formation in the half—Heusler compound ZrNiSn (for details see
Chapter 4). The gap formation in the half-Heusler compounds has been understood
by first studying the parent binary compound ZrSn and then investigating the effects
of the inclusion of the Ni (1 levels in the gap formation in the ternary compound. In

ZrNiSn, there is a strong Ni d-Zr d hybridization in the conduction band near the

104

X point which is important for the formation of the band gap.[7, 59] An important
question to ask is whether it is possible to understand the gap formation of Zr3Ni3Sb4
by studying what effect the Ni (1 orbitals would have on semimetallic Zr38b4. Using
the experimental value of the lattice constant of the ternary compound[7], we have
calculated the band structure of Zr38b4 (Figure 44). It can be seen that the electronic
structure of Zr38b4 is slightly semimetallic with a conduction band crossing Ep near
the F point and a valence band crossing above Ep along FH and along FN (Figure
44). This is consistent with the self-consistent APW calculation for Th3Sb4, which is
a known existing compound.]10] As can be seen, the majority of Zr (1 bands lie above
Ep, the majority of the Sb p bands below, consistent with the view that these are
mostly polar salts in the Zintl phase.[87] The important question is how the inclusion

of Ni changes the bonding within this network.

Figure 44. The (a) Zr d and (b) Sb p orbital characters of Zr3Sb4.

(a) Zr3Sb4 Zr (1
2.0 v

_.A ,-, 4'?- .

     

 

 

 

 

 

105

 

 

 

 

 

 

 

 

 

In Th3Ni3Sb4[10], it has been found that the Th d-Ni d hybridization at the I‘ point
was much stronger than the Sb p-N i d hybridization and that the former hybridization
leads to the formation of the semiconducting gap. This is true even though the Ni-Sb
distance (2.644 A) is smaller than the Ni—Th distance (2.874 A).]10] Therefore, the
interatomic distance is not a clear indicator of the local bonding, as is commonly
used. We also find that, when Ni is added to Zqu.; to form Zr3Ni3Sb4, the gap
formation is similar to that of the half-Heusler compounds, when Ni is added to ZrSn
to form ZrNiSn. The strengths of the orbital characters in ngNi38b4 (Figure 45)
show that the Ni (1 hybridizes more strongly with Zr d than with Sb p. The Zr d-Ni
d hybridization at the 1" point is much stronger than the Sb p-Ni d hybridization
and leads to the formation of the semiconducting gap in Zr3Ni3Sb4. Again, the N i-
Sb distance (2.4858 A) is smaller than the Ni-Zr distance (2.7746 A). The physics
of the gap formation in the half-Heusler and Zintl-phase compounds is very similar:
the introduction of Ni atoms into the semi-metallic binary compound (ZrSn, YSb,
or Zr3Sb4) results in a strong Ni d-Y/ Zr d hybridization which opens a narrow band
gap. Since Zr, Sb, Sn, and Niare not heavy elements, the inclusion of 801 has only

a small effect on the band structure, such as minor splitting of degenerate bands and

106

a small reduction of the band gap, but with no change in the shape of the LCB or
HVB.[7]

Figure 45. The (a) Zr d, (b) Sb 1), and (c) Ni (1 orbital characters of Zr3Ni38b4.

(a) Zr3Ni3Sb4 Zr (1
2.0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

107

 

 

WI]

int

,3,

 

The similarity between the gap formation in these Zintl-phase compounds and the
half-Heusler compounds also can be seen when changing the transition metal atom
from Ni to Au or Cu. In the half-Heusler compounds, such as LuAuSn, the reduced
Au d-La d hybridization compared to the Ni d-La d hybridization in LuNiSb led to a
reduction of the band gap in the former material (note that for charge compensation
that Sb —+ Sn if Ni -—» Au). Weaker Au/Cu d-Y d hybridization has been seen in
Y3Au38b4 and Y3Cu3Sb4.]89] Zr3Ni38b4 has a band gap of 0.39 eV compared to 0.23
eV for Y3Cu33n4 and Zr3Pt38b4 has a band gap of 0.78 eV compared to 0.61 eV in
Y3Au33n4.]89] In these compounds, the Au and Cu d levels lie about 5 eV below the
VBM[89], while the Ni (1 levels lie only about 2 eV below and Pt d levels lie only
about 34 eV below the VBM[7, 59] Previous calculations of the band structure of
La3Au3Sb4 found that the gap was 0.17 eV, half of that for Th3Ni38b4. This difference
was attributed directly to the reduced Au d-La d hybridization due to the lower Au
(1 levels in the former compound.[10] Reduced Zr/Y/La d-Cu/Au d hybridization
compared to Zr/ Y/ La d-Ni / Pt d hybridization leads to a reduction of the band gap

in both the half-Heusler and Zintl phase compounds.

5.2.3. UNUSUAL DISCREPANCY BETWEEN THE BAND GAPS FROM
THEORY (LDA/GGA) AND EXPERIMENT

As in the half-Heusler structure, there seems an unusual discrepancy between the
experimental and theoretical band gaps, the theoretical values being much larger
than the experimental values[7, 10, 59], the opposite of what is usually found in
LDA/GGA calculations.[31] The situation seems even more confusing due to the

comparisons of the electronic structure calculations of the binary compounds Th3X4

108

 

'il.

a

'1‘

(X = P, As) with the ternary Tthi38b4. The calculated band gap for the binary
system Th3P4 by Takegahara et al.[10] gives 0.007 eV compared to the experimental
value of 0.40 eV, whereas in Th3As4 they find 0.05 eV compared to the experimental
value of 0.43-0.44 eV. For Th3Sb4 they find a negative gap (semimetal) compared to
the experimental value of 0.48 eV. The reason for this discrepancy was attributed the
inadequate treatment of the XC potential in LDA/GGA. The trend for the band gaps
in the ternary compounds is just the opposite. The calculated value of the gap for the
ternary T h3Ni3Sb4 is 0.36 eV compared to the experimental value of 0.07 eV. These
calculations were performed without the inclusion of SOI which can be important for
heavy atoms like Th and are likely to play an important role in changing the band
gap]10]

The discrepancy of finding a larger band gap in a band calculation than found
experimentally has been seen also in the half-Heusler compounds. Ogut & Rabe[59]
tried to explain the observed smaller values of the band gap in ZrNiSn by invoking
disorder in the ZrSn lattice. Similar arguments have been put forward to explain
the observed smaller gap in these Zintl-phase compounds.[lO] However, our supercell
calculations for the half-Heusler compounds[11] (Chapter 4) showed that this type
of disorder is found to be energetically unfavorable compared to Ni disorder. Also,
while disorder of the ZrSn lattice has been seen in ZrNiSn[78], there is no experimental
evidence of disorder in these Zintl-phase compounds, where the Zr and Sn sites are not
equivalent. Therefore, we will try to understand this dramatic discrepancy between
theory and experiment from a different perspective.

The observed discrepancy between the experimental and theoretical band gaps may
lie in the uncertainty in the positions of the Ni (1 orbitals in different calculations.

As seen in Chapter 4, when the transition metal at orbitals lie closer to Ep, such as

109

in TiCoSb[80], the Ti d-Co d hybridization is stronger, leading to a larger band gap
in the material. It is well known that LDA/GGA calculations place the d orbitals
of transition metals too high in energy while other methods, namely Hartree-Peck,
place these (1 orbitals too low in energy.[97] Only a small shift, less than 0.5 eV, in
the position of the d orbitals changes the gap from 0.51 eV in TiNiSn[59] to 1.08
eV in TiCoSb[80]. Shifting of the d levels down reduces the band gaps. In the
Zintl phases, we find that Zr3Ni33b4 has a band gap of 0.39 eV compared to 0.23
eV for Y3Cu38n4 and Zr3Pt38b4 has a band gap of 0.78 eV compared to 0.61 eV in
Y3Au38n4.[89] In these compounds, the Au and Cu d levels lie about 5 eV below the
valence band maximum[89] while the Ni d levels lie only about 2 eV below the valence
band maximum.[7, 59] The position of the Ni d level[97] may be the source of the
overestimation of the band gap in the half-Heusler and Zintl phase compounds, not

disorder between Zr and Sn/Sb.[10, 59]

5.2.4. COMPARISON OF Zr3Ni3Sb4 WITH Hf3Ni3Sb4 AND Zr3Pthb4

Zr3Ni38b4, Hf3Ni3Sb4, and Ti3Ning4 are isoelectronic Zintl-phase compounds, the
first compounds in the Y3Au38b4 structure type not containing Y or an f-electron
atom. The electronic structures of all three compounds show small semiconducting
gaps (Figure 46). We find that the effect of SOI is to lower the band gap from 0.58
eV to 0.51 eV in Hf3Ni3Sb4, from 0.45 eV to 0.39 eV in Zr3Ni3Sb4, and from 0.86 eV
to 0.78 eV in Zr3Pt3Sb4. Though extended Huckel calculations rarely do a good job
in predicting band gaps, the previous calculation for Zr3Ni38b4 (without SOI) gave a

gap of 0.57 eV, consistent with our results.[94]

110

Figure 46. Electronic structures of (a) Zr3Ni38b4, (b) Hf3Ni3Sb4, and
(C) ZI‘3Pthb4.

(a) 2 o Zr3Ni3$b4

 

(b) Hf3Ni3Sb4
2.0

Energy (eV)

no.
ooooo
'0
I

2.

(c)2.0 Zr3Pt3Sb4

.0
0

Energy (eV)

111

The electronic structures of Zr3Ning4, Hf3Ni3Sb4, and Zr3Pt3Sb4 are remarkably
similar with valence band maxima forming along the same directions in k-space. The
HVB in each of the compounds is apparently multiply degenerate with maxima along
FN, PH, and PP (Figure 46). These bands arise primarily from Hf/ Zr d-Ni/Pt d
hybridization, with some Sb p hybridization. In each case, the effect of SOI is to
remove degeneracies from the bands and reduce the observed band gap by about
0.07 eV. As in the half-Heusler systems, the bands near Ep arise due to a strong
hybridization of the three metallic elements.

A closer study of the effective masses of these materials shows that there are only
two VBM along PH and PP in Zr3Ni38b4, and Hf3Ni3Sb4, the supposed maxima
along I‘N in these two compounds, which has a slightly lower energy than the other
two, is really a saddle point with a minimum along the z-direction. The calculated
reciprocal values of the effective masses (0,- = [mi/marl) are given in Table VII.
These effective masses have not been measured experimentally. The CBM at the I‘
point is doubly degenerate, one band more dispersive than the other. The values of
the reciprocal effective masses are very similar in the three compounds, except that

the Pt compound has heavier CBM effective masses and no saddle point along FN.

112

Table VIII. Calculated reciprocal effective masses of ngNing4, Hfgl 'i;,Sb4,

and Zr3Pthb4.

 

 

 

 

 

 

 

 

 

 

a,- -——]m,o,/me]‘1 Zr3Ni38b4 Hf3Ni3Sb4 Zr3Pt3Sb4
CBM(I‘) a: : ay 2 02 1.46 1.53 0.07
OzI = (1,, = 02 0.72 0.57 0.08
VBM (FN) a: = 0,, 0.21 0.27 0.50
a: -0.04 -0.23 0.54
VBM(I‘H) a: 2 0y : (1,, 0.17 0.19 0.12
VBM (PP) aI = 0;, 0.47 0.40 0.26
a, 1.26 1.24 0.88

 

 

 

 

 

 

Zr3Ni38b4, Hf3Ni3Sb4, and Zr3Pt3Sb4 have a low 5 due to the complex unit
cells.[94, 95] Our calculations also show anisotropic effective masses, though much
less anisotropic than the half-Heusler compounds. As pointed out for Y3Aung4
and Y3Cu3Sb4[89], the multiply degenerate VBM away from the high symmetry
points leads to a large thermopower in these systems.[ll, 24] The different values
for the band gaps allows for shifting the temperature range at which the materials
are useful. These Zintl-phase compounds have promising power factors, like the half-
Heusler compounds, however with a much lower K. due to the more complex crystal
structure.[94, 95] Further experimental studies of these materials for their thermo-

electric properties is warranted.

113

CHAPTER 6.
SIMPLE CHALCOGENIDES

Both half-Heusler compounds (Chapter4) and Zintl-phase compounds (Chapter 5)
are potential candidates for new thermoelectric materials. However, the best currently
available bulk thermoelectric materials belong to a class called simple chalcogenides.]4]
These include alloys of BIQT€3 with B12883 and Sb2T€3 for room temperature (~300K)
and PbTe for higher temperatures (~600K) applications.[3, 25] The search for new
thermoelectric materials has focused on using these materials as building blocks
for synthesizing novel materials with complex structures which will have high ZT
values.[4]

While there are general expectations about what one should look for in a good
thermoelectric material (Chapter 2), studying the electronic structure of known ther-
moelectrics allows one to see which mechanisms are the most important. Also, before
studying the electronic structure of complex chalcogenides, we will discuss the physics
of gap formation in simple chalcogenides which have been extensively studied in the
past. We will present our FLAPW results and then compare with the previous calcu—
lations which were performed using different methods, and to understand the features

which are found in the electronic structure of the best known thermoelectric materials.
6.1. PbS, PbSe, PbTe
6.1.1. BAND STRUCTURES OF PbS , PbSe, AND PbTe

PbS, PbSe, and PbTe, commonly referred to as the lead salts, are compounds in

114

the NaCl structure type with 2 atoms/ FCC unit cell. The lead salts have been used
in many applications, including near-infrared photoconductive detectors[98], photo-
conductive and photovoltaic conductors[99], and p—n injection lasers]100]. PbS, PbSe,
and PbTe are also good thermoelectric materials near 700 K with a maximum ZT
around 1.[3, 25] These materials have also been used in quantum dots and multilayer
systems which show even higher values of ZT, mainly due to reductions in n.[4, 27, 28]

The lead salts are among the few materials where the temperature coefficient
(Ea—5%?) is positive, which means that the semiconducting gap increases with tem-
perature. Almost all other elemental or binary compound semiconductors exhibit
negative values of the temperature coefficient. In fact, all three compounds have
about the same temperature coefficient, (fa—$33 z 4.7 x 10’4 eV/ K (Table IX). This is
related to the negative pressure coefficients (ea—g?) for all three systems and positive
volume coefficients of thermal expansion, (1. Therefore, as the temperature increases,
the lattice expands, acting like a negative pressure (6%:g3), which then increases the
band gap (215%12 > 0). Another unusual feature is that the band gap decreases going

from PbS to PbSe but increases going from PbSe to PbTe (Table IX). Generally, there

is a monotonic decrease of Egap with increasing atomic number.[101]

115

Table IX. Experimental energy gaps in eV for PbS, PbSe, and PbTe as a function

of T. (For experimental references, see Dalven [101])

 

 

 

 

 

 

 

 

T(K) PbS PbSe PbTe

4.2 0.286 :l: 0.003 0.165 2}: 0.005 0.190 :1: 0.002

12 0.286 0.145 0.187

77 0.307 i 0.003 0.176 :t 0.005 0.217 i 0.002
300 0.41; 0.42 0.27; 0.29 0.31; 0.32
373 0.44 0.31 0.34

 

 

 

 

 

 

PbS, PbSe, and PbTe have been extensively studied in the past, and their electronic
structures have been calculated by several authors using different methods. These
include tight-binding (TB) calculations[102], the augmented planewave method]103,
104], the k'p method]105, 106], the Green function method]107], the orthogonalized
planewave (OPW) method]108], the empirical pseudopotential method]109, 110, 111,
112, 113, 114, 115], the relativistic LMTO method]116, 117, 118], and the FLAPW
method including SOI[37, 38, 39]. Early calculations[102, 103, 104, 105, 106, 107,
108, 109, 110, 111, 112, 113] correctly predicted the position of the direct gap at the
L point in the BZ. However, many of these calculations included the band gap as an
input parameter. The later papers[37, 38, 39, 115] note that, even after including
SOI, the LDA/GGA band gaps are too small compared to experiment, which has
been noted in many other semiconductors.[31]

Our F LAPW calculation of the band structures of these three compounds (includ-
ing SOI) at the experimental lattice parameters (corresponding to T = 4 K) shows
that the band gaps are much smaller than experiment, 0.10 eV for PbS, 0.01 eV for
PbSe, and 0.09 eV for PbTe, though the trend in the bands gaps going from PbS to

116

PbTe is reproduced.

Figure 47. Electronic structures of (a) PbS, (b) PbSe, and (c) PbTe using

the experimental lattice parameters (corresponding to 4 K).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

117

Several calculations of these materials have tried to improve the band gaps for bet-
ter agreement with experiment. All of these have been zero temperature calculations,
so the attempts have been to reproduce the experimental values of 0.29 eV for PbS,
0.17 eV for PbSe, and 0.19 eV for PbTe.[119] One of these calculations uses a one-
dimensional multipole expansion for SOI to yield 0.24 eV for PbSe]115]. In a similar
vein, another calculation expands the SOI using a Slater-Koster approach in order to
fit the gap in PbTe[38]. Another adds a constant potential to the conduction-band
states in an FLAPW calculation to obtain the experimental values of the gapsl37].
There is an ab initio relativistic full-potential LMTO calculation which yields 0.069
eV for PbS, 0.141 eV for PbSe, and 0.31 eV for PbTe[117]. Finally, one group uses
the lattice parameters appropriate to the minimum energy configuration within an
FLAPW-GGA calculation to yield values of the gap of 0.26 eV for PbSe and 0.15 eV
for PbTe[39].

It is clear that some of these approaches seem more reasonable than others.
Though the band gaps do not agree very well with experiment, the fully relativis-
tic calculation]117] seems the most reasonable, since it is known that including SOI
in a second-variational procedure has problems with Pb 612% states in metallic Pb]36].
The one-dimensional multipole expansion[115] and Slater-Koster expansion[38] of SOI
also seem reasonable approaches, but are unduly complicated for this problem. The
addition of a constant potential to the conduction-band states[37] seems rather ad
hoc, especially within an ab initio FLAPW approach.

The method to obtain the gap which seems to make the most sense (in terms of
computational efficiency) is calculating the band structure using a second variational
approach for SOI after finding the minimum energy lattice parameters[39]. DF T

calculations are only approximate solutions since the XC potential is not exactly

118

known[31]. Therefore, one should choose a reasonable model of the system to study,
which can be done by first finding the minimum energy configuration of a crystal.
Also, it is known experimentally that the band gap increases with increasing lattice
parameter]101], so an increase in the theoretical lattice constant is expected to also
increase the calculated band gap. The minimum energy configuration for each system
in GGA was found by increasing the lattice parameters which gives good agreement
with the experimental band gaps. In LDA, the minimum energy configuration for
each system was found by reducing the lattice parameters which further reduces the
calculated band gaps.[39] We will try to reproduce the published results[39] using
GGA to better understand the effects of lattice relaxation.

The total energy was calculated within GGA for PbS, PbSe, and PbTe, and is
plotted as a function of lattice parameter in Figure 48 for PbS. We have found the
total energy has a minimum with lattice parameters approximately 1-2% larger than
the experimental value. The equilibrium lattice parameters changing in PbS from
11.204 a.u. to 11.422 a.u., in PbSe from 11.573 a.u. to 11.725 a.u., and in PbTe
from 12.176 a.u. to 12.376 a.u. The arrow in Figure 48 gives the position of the

experimental lattice parameter.

119

Figure 48. Total energy of PbS as a function of lattice parameter (in a.u.)
obtained in GGA. The points were plotted using the computer
program KALEIDAGRAPH[120]. The experimental value
is given by the arrow.

PbS
0.03 -

 

 

0.025 _"

 

 

0.02 3-3.. . - _.

 

0.015 ;~-—--—-»

0.01 ’

 

Energy (eV)

 

_ o. . 1
0.005 . ; . o ‘

 

0’ 4 QALQQ

 

 

3 1
A A— L 1 _.L L L A 1 A L

-OOOO5’A..1..-.i. A.
11.2 11.25 11.3 11.35 11.4 11.45

 

Lattice Parameter (a.u.)

Electronic structure calculations performed at these larger lattice parameters do
not change the shape of the bands appreciably, but the band gaps increase to nearly
the experimental values for the three systems (Figure 49), in PbS to 0.26 eV compared
to the experimental value of 0.29 eV, in PbSe to 0.08 eV compared to the experimental
value of 0.17 eV, and in PbTe to 0.17 eV compared to the experimental value of 0.19
eV.[119] These calculations suggest that it might be important to calculate the energy
gaps using equilibrium lattice parameters instead of experimental ones. This is also
a case, such as in the determining the minimum energy configuration between BC C '
Fe and FCC Fe]45], where GGA better reproduces the properties of the system than

LDA. We find that GGA increases the band gaps of these materials close to the

120

experimental values while the previous LDA calculation[39] reduced the gaps even

further.

Figure 49. Electronic structures of (a) PbS, (b) PbSe, and (c) PbTe using the

relaxed lattice parameters from GGA and including SOI.

PbS

“\ /‘ ‘x/ \

3.0

 

1 1 1 1
I...-

 

Energy (eV)

 

 

 

 

 

 

 

. \ \
- 3 _ 0 I i:- _. A ]—\= , ’.

 

 

 

 

 

 

 

 

 

 

Energy (eV)

 

 

 

 

 

 

 

121

6.1.2. THERMOELECTRIC PROPERTIES OF THE LEAD SALTS

Most of the electronic structure calculations on the lead salts focused on the prob-
lem of correctly predicting the band gaps instead of using this information to better
understand the thermoelectric properties of these materials. Due to the increase of
the band gap with temperature[101], the band gap is on the order of about 10kBT
at 700 K[l, 25, 3, 4], estimated to be about 0.5 eV for a recent estimation of ZT in
these materials[19]. If we assume that the CBM and VBM remain at the L point,
the degeneracy of the band extrema is small, which is insufficient to explain the large
ZT at high temperatures.[ll, 24] One must, therefore, look at the anisotropy of the
effective masses.

The effective masses in PbS, PbSe, and PbTe are known to be highly nonparabolic

and temperature dependent[101, 121] as well as highly anisotropic[122]. As seen in
Table X, the effective mass along the (100) direction (defined as longitudinal) is much
different from that of the (111) direction (defined as transverse).[122] The effective
masses have been calculated (using the band structure for the relaxed lattice), but

the reciprocal values are written to compare with B12T€3 in the next section.

The agreement between the theoretical and experimental values looks reasonable.
The largest anisotropy is found in PbTe, which is the best thermoelectric material
of the three.[19] According to the mass dependence of the B parameter (Equation 24
of Chapter 2), higher ZT should occur in systems with large band degeneracy and
highly anisotropic effective masses.[11, 24] The mechanism for the high ZT in PbTe

seems to be the highly anisotropic transport mass with only a small enhancement

from the band degeneracy factor.

122

 

 

Table X. Comparison of calculated components of the reciprocal effective mass

 

 

 

 

 

 

 

tensor with experiment[122] (a, = [mt/me]—1 and a, = [mt/marl).
] PbS PbSe PbTe
] Experiment Theory Experiment Theory Experiment Theory
] Conduction a; 7.25 13.53 16.39 18.58 30.30 24.48
] at 8.13 9.55 10.31 10.06 4.24 5.45
] Valence a; 6.70 12.89 16.13 16.41 33.33 18.89
] at 6.13 7.91 8.33 8.19 2.39 3.55

 

 

 

 

 

 

 

 

 

6.2. B12T83, B12883, Sb2T83, .AND RELATED SXISTEIVIS

The best room temperature thermoelectric materials are alloys of B12T€3 with
Bi28e3 and SbgTeg. While the thermoelectric properties of these materials were opti-
mized nearly 40 years ago]123], there is no accepted theoretical explanation of their
high observed ZT values. The effects of SOI in these materials is very important to

their gap formationlll, 40, 124], but only one other band structure calculation has
been performed where SOI has been included self-consistently]11, 40]. Understand-
ing the features of the band structure relevant for the high ZT in these materials is
therefore important in the search for new thermoelectric materials, several of which

contain portions of the B12T83 crystal structure in their complex unit cells.[4]

6.2.1. BIQT€3 CRYSTAL STRUCTURE

The crystal structure of BI2T€3 is rhombohedral with the space group D3,, (R-3m)
with five atoms in the unit cell. These atoms form five atom layers called “quintuple-

layer leaves” of Te-Bi-Te-Bi-Te along the z direction with primarily ionic and covalent

123

bonding within the layer with Van der Waals bonding between the layers.[63]

Figure 50. Crystal structure of B12T63 along the XZ plane showing the quintuple

layer leaves of Te-Bi-Te-Bi-Te separated by a Van der Waals gap.

'9
f"

The five atoms in the unit cell reduce to three inequivalent atoms identified at Tel,
Te2, and Bi as indicated in Figure 51. The corresponding Brillouin zone is given in

figure 52.

124

Figure 51. (a) Rhombohedral and (b) hexagonal representations of the unit cell

of BlgTEg. In (b) the Te atoms are open circles and Bi are shaded circles.

 

 

 

Figure 52. Brillouin zone of B12T83 (corresponding to the rhombohedral unit cell).

2
“(Tm-mu)

 

 

 

 

 

125

6.2.2. BAND STRUCTURE AND GAP FORMATION IN BIQT€3

The electronic structure of Bl2T€3 has been studied extensively in the past[40,
124, 125, 126, 127, 128, 129, 130], but there have been few studies of the electronic
structure of analogs containing either Sb or Se[40] or those containing both Se and

Te. Some of these materials are found to be good thermoelectrics while others are

not. Since the lattice contribution to re depends mostly on the rhombohedral crystal

structure, which is the same in all of the compounds, the differences in the ther-
moelectric properties must be electronic in origin and should be seen in the band
structure. Before discussing the electronic structure of these structural analogs, let’s
first understand the electronic structure of B12T63.

The importance of relativistic effects, especially that of SOI, has been noted in
each of the previous calculations. However, several questions remain to be resolved
in the band structure of Bi2Te3. For example, Thomas et al.[124] explained the

physics of the gap formation by incorporating SOI, but were unable to incorporate

it self-consistently into their ab initio LAPW calculation. Later, Mishra et al.[40]

did include SOI self-consistently in their ab initio LMTO calculation, but they did

not explore the nature of the bands that hybridized (leading to uncrossing) near

Ep. More recently, Youn et al.[130] used the LAPW method with a different XC
functional and a different cut-off energy between the valence and core states to find
the CBM at a different location in k-space than the previous calculations[ll, 40].
While the degeneracy of the hole pockets in the previous calculations agreed with
the de Haas-van Alphen experiments[131, 132, 133, 134], only Youn et al.[130] found
agreement with the degeneracy of the electron pockets, a result which we could not

epeat using the WIEN97 program.[55] A careful study of the electronic structure of

126

BlgT€3 is therefore needed to understand the formation of the semiconducting gap

and the thermoelectric properties.
SOI affects the band structure of B12T€3 significantly. Before SOI is included, the
band structure shows a direct gap semiconductor at the F point with a band gap of
0.37 eV. (Figure 53) This is about twice the value reported by Thomas et al.[124]
who used a different LDA XC potential. There are two additional peaks in the HVB
(along the F a and ZF directions) and two valleys in the LCB (also along the l" a
and ZF directions) which have an energy slightly lower than the VBM and slightly
higher than the CBM, respectively. An orbital analysis reveals that the LCB and the
HVB arise primarily from an admixture of Bi p and Tel p bands. The conduction
band is predominantly Bi p in nature whereas the valence band is predominantly
Tel p. Note that the Tel atoms are in the layers that straddle the Van der Waals
gap. Te2 p states lie primarily below the HVB although there is a small amount of
Te2 p character in the HVB. In the presence of SOI, the band structure near EF is
dramatically changed. The main effect of SOI is to move the conduction band bottom
down relative to the top of the valence band, due to the different SOI of Bi and Te,
so that the bottom of the conduction band crosses the top of the valence band at F
(Figure 54). The resulting hybridization, which leads to an uncrossing of the bands,
opens a new gap which is indirect. This point was made by Thomas et al.[124] in
their tight-binding parameterized calculations. They pointed out, however, the need
for a complete self-consistent calculation including SOI to understand the subtle gap

structure of Bi2T€3.]11]

127

Figure 53. Band structure of Bl2T83 without SOI.

 

BiZTe3 - NO SOI

 

LJIIL4IIJ

 

 

Energy (eV)
C
O

 

JngLl l l l l

 

 

 

 

 

-1.0 -'
U a

 

 

 

1.0

 

3]-

l l .L l 1 l l
e<~
. IT‘S?»
K
N ‘0
K"
\.

 

 

Energy (eV)
2

l l l l l l l

 

 

 

 

 

-l.0

 

 

 

128

 

Including SOI keeps the shapes of the off-axis peaks virtually unchanged, but
moves them closer by approximately 0.50 eV. The strong hybridization occurring
near I‘ depresses the HV B peak to such an extent that the two valence band side
peaks now have higher energy and form the VBM (Figure 54). (This hybridization
near F is much stronger than for that of the side peaks in the valence and conduction
bands, so the overall shift is not rigid.) The two VBM lie on the yz plane of the B2,
which is a plane of reflection symmetry. They will be denoted as 01 (along Pa) and
v2 (along ZF). Similarly the hybridization caused by the SOI shifts the CBM away

from the F point to a point between I‘ and Z, denoted as CO. The net result is an
indirect-gap semiconductor with a band gap of 0.13 eV, in very good agreement with
the previously published LMTO-ASA calculation[40] of 0.11 eV. In addition to this
minimum, there are two higher-lying minima, c1 (along Pa) and c2 (along ZF). The

gap between 221 and CI is 0.27 eV and that between v2 and C? is 0.23 eV.

Thomas et al.[124] have carried out optical measurements in Bi-doped B12T€3 and
find an indirect gap of 0.15 d: 0.02 eV and a direct gap of 0.22 :l: 0.02 eV, which
agrees rather well with our values of 0.13 eV and 0.19 eV, respectively. Shubnikov-
de Haas and de Haas-van Alphen measurements[131, 132, 133, 134] of both p. and
n-doped Bi2T€3 give six-fold degenerate valence and conduction band extrema. The

VBM 22.? appears to be the one seen in experiment. A recent calculation by Youn et
al.[130] shows that, using a FLAPW method and the Hedin-Lundqvist form of the
XC potential] 135] for 60 special k-points and a cut-off energy between the valence and
core states of -1.8 Ry, the band which we have called c2 moves down by ~0.10 eV,
with little difference in the rest of the band structure, to become the CBM. This point
has the six-fold degeneracy for the conduction band seen experimentally, whereas our
calculation does not, but their band gap has decreased to 0.06 eV. (It is interesting
to note that we have better agreement with the band gap while Youn et al.[130] find

better agreement with the degeneracy of the CBM.) They further find that the true

129

extrema lie slightly off of the high symmetry lines which are plotted in Figure 54.]130]
Using the WIEN97 program[55] and the identical input parameters, the result of a six-
fold degenerate CBM could not be reproduced. As is well known, DFT calculates the

occupied valence band states well, but does not do as good a job for the unoccupied
conduction band, especially on the energy scale of ~0.1 eV.[31] It is disconcerting,
however, that these results could not be reproduced with our program.

The orbital characters of the LCB and HVB in the presence of SOI are given in
Figure 55 where the size of the circles is proportional to the orbital character. The Bi
p, Te1 p, and Te2 p bands are all involved in different amounts in the bonding near
E p, but the LCB has a larger amount of Bi p character than Te1 p or Te2 p character,

and the HVB has a larger Te1 p character than Bi p or Te2 p character.
The orbital character also can be seen in the DOS of the Bi p, Te2 p, and Tel
p partial densities of states (Figure 56) lying above and below Ep. Te 1 lies on the
Van der Waals gap, bonded to one layer of Bi atoms below (or above) and weakly
to a Te1 atom across the gap, while Te2 is bonded to two layers of Bi atoms, one

above and the other below (Figure 50). The result is the stabilization of the Te2 p

bands which lie primarily below Ep and contribute relatively less to the conduction

band. The n-type transport is primarily through hybridized Bi p and Tel p orbitals

(contributing to the LCB) and p-type transport through hybridized Te1 p, Bi p, and

Te2 p orbitals (contributing to the HVB). This is consistent with the results of carrier

scattering lifetime measurements[124] and a previous calculation of electronegativity
of each of the constituent atoms.[136] Finally, the Tel, Te2, and Bi valence 3 states
lie approximately 10 eV below E p and do not play any significant role in the bonding

and transport properties.[11]

130

 

Figure 55. Band structure of BIQT€3 with SOI. Included are (a) total band

structure and the orbital characters of the bands: (b) Bi p, (c) Tel p,

and (d) Te2 p. The size of the circles overlying the band structure

is directly proportional to the strength of the orbital character.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 
   

 

 

 

 

 

(8)1'0.V\J \V v (b)l.BiP .
go-Ojv 2 EF§0'0 EF
m f/M/V m ]
-1.0 WC/N 411/] 4,. A té u i
U a 1‘2 F L U a 1'2 F F 1.
(Ch. Telp fl (d)10T62% v
. I] v ' ' (I: V
f: 51% : '*
5.? ; 1%: \J
250-0: EFEO'OA BF
whiff] II An

 

 

131

Figure 56. Total and partial DOS for BigTe3. Included are (a) total DOS and

(a)

slates/9V

(b)

(C)

(d)

statesleV

states/eV

stales/eV

partial DOS: (b) Bi p, (c) Tel p, and (d) Te2 p.

Total

 

d

 

 

 

 

 

 

 

- Eros-r9); [é )-

   

 

 

 

 

 

 

 

- E-ne.rg)—/ [é ]

-1 O

1

5345—6—1'

 

 

   

-6 -5 -4 -:3 - -1
Energy [a ]

132

  

 

O

 

1 2 3 4 5 6

6.2.3. BAND STRUCTURE AND GAP FORMATION OF BigTe3 ANALOGS
A. B12803 AND Sb2T03

The gap formation in Bl2T€3 is drastically affected by strong SOI of the Tel p and
Bi p orbitals. The SOI effects of Bi and Te cannot be separated since both are heavy
elements and the LCB and HVB are strongly hybridized. We expect that the effects
of SOI are larger for Bi than for Te, so that the electronic structures of 812883 and
Bl2T€3 will have more similarities than those of Sb2T83 and BIQT63. However, the
opposite is true. This difference is important for understanding the thermoelectric
preperties, since Bl2T83 and Sb2T83 have been widely used as thermoelectric materials
but Bi28e3 has not.]4] The lattice constants used in the calculations within this section

are given in Table XI.
The present calculation for B12893 (Figure 57) agrees very well on the global scale
with the previous LMTO-ASA calculation.[40] Our LAPW-GGA calculations give a
gap of 0.32 eV, slightly larger than the previous calculated value of 0.24 eV.[40], but
both results are close to the experimental value of 0.2-0.3 eV.]137] The differences
between our and the previous calculation[40] is that we find a larger lowering of the
valence band peak at the F point (after inclusion of SOI) so that this peak has nearly
the same energy as the two secondary peaks along UaI‘ and ZF (Figure 57), whereas
the F point peak was found to have a higher energy in the previous calculation.[40]
In fact, in our calculation what appears to be a peak in the valence band at the
F point is really a saddle point with a maximum only in the F Z direction. While
electron-doped Bi28e3 is known to have a minimum at the F point, in agreement

with our and the previous calculations[40], hole-doped BigSeg has not been studied

133

experimentally.]138] It will be interesting to check the prediction of the theoretical
results in this regard for two six-fold degenerate hole pockets at general points of the

BZ and one saddle surface near the F point.[139]

Table XI. Lattice constants in atomic units (a.u.) and u and 1) parameters
for Bl2T€3 and its analogs in the combined representation of the unit
cell with hexagonal lattice constants and rhombohedral atomic
coordinates used in these calculations. (Bi at 2t(u, u, u),
Tel at :l:(v, v, v), and Te2 at (0, 0, 0) in BigTe3.
For details see Wyckoff.[63])

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

System Lattice constants (a.u.) u v
a c

BigTe3 8.2836 57.6121 0.3990 0.2080
812893 7.8197 54.1220 0.3990 0.2060
Sb2T83 8.0313 57.4477 0.4000 0.2110
BigTegSe 8.0880 56.4272 0.3961 0.2117
BigTegS 8.1560 56.7107 0.3920 0.2120
SbgTeQSe 7.914 56.5730 0.3942 0.7854
Bi2Se2Te 7.9710 55.5960 0.3985 0.2115
SngegTe 7.7570 55.7530 0.3953 0.2146

 

134

 

Figure 57 . Band structure of Bi28e3 (a) without and (b) with SOI.

 

 

 

 

 

(a) BiZSe3 - NO 501 (b) BiZSe3 — SOI
“0 J V ' .= 1'0. 1’; \ \ §\

9 ‘ 9 : (‘2

e e : ° .

>3 >3 \

$00.0 EF £3900] E]:

a . c: ,

LL] LU .

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

The only previous electronic structure calculation of Sb2T83 was a tight-binding
(TB) study which only calculated the DOS.[140] We find that the electronic structure
of Sb2T83 more closely resembles that of BI2T83 than B12383. The VBM of szTe3
lies along UaI‘ and ZF with the CBM lying along [‘2 with higher lying states along
UaI‘ and ZF (Figure 58), very much like that in Bl2T93. Before the inclusion of SOI,
Sb2T83 is a zero—gap semiconductor at the I‘ point with secondary peaks along UaI‘
and ZF (Figure 58a). The inclusion of SOI causes the HVB to come closer relative to
the LCB (and overlap) (Figure 58b) as in BIQT83. This produces an uncrossing of the
bands with an opening of a new, larger gap of about 0.14 eV. While Sb p is mostly
above BF and Te p mostly below Ep, a strong hybridization of the Sb p and Tel p

occurs along I‘Z, similar to B12T83.l11] The Te2 p orbitals do not appear to play any

significant role near E p, similar to what was found in BigTe3.[11]

135

Figure 58. Band structure of szTe3 (a) without and (b) with SOI.

 

 

     
      

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) n SbZTe3-NO SOI (b) 0 SbZTe3 - SOI
1J3,/
9 :V 9 WM
8 : 3 : '=
>3 >1 ‘
q9300.0 Epgon‘
c: . :. c: I
: 1
: N 1
1.0‘ A W 1.0‘
U 17‘ 1.. U I7

 

 

 

B. BigTeQSe, BlzTegs, SbgTegse, BigSegTe, AND Sb28€2T€

There has been no theoretical or experimental studies of the band gaps in BizTegse,
BIQTGQS, Sb2T€2S€, Bi2Se2Te, or SbgsegTe. We have carried out electronic structure
calculations in a series of such compounds to see how the gap structure changes
when one or two of the three Te atoms are replaced by Se or S. Band structures
of Bi2(Te,Se,S)3 generally resemble Bl2T€3 or Bi2Se3, depending which element is in
greater abundance in the system. We find the physics of gap formation is the same in
all these systems but the final band structures differ due to the size of the band gap
before including SOI and the relative strengths of the Bi/ Sb and Te/ Se/ S SOI. The

crystal systems differ from that of BigTeg, Bi2Se3, or szTe3 by changing the atom at
the Te2/Se2 site which we did not find to play an important role in the gap formation

in BigTe3.

The unit cells of BigTegSe and BizTe2S differ from that of B12T83 by replacing one

136

of the three Te atoms by a Se or S atom, respectively, this atom going into the Te2
site.]l4l] Before the inclusion of SOI, the band gap in BigTe2Se is 0.10 eV and in
BIZTQQS is 0.32 eV, the gaps forming at the F point with secondary peaks along UaF
and ZF (Figure 59a,c). The inclusion of SOI causes the HVB to come closer relative
to the LCB (and overlap) as in Bi2Te3 and Sb2T83. This produces an uncrossing of
the bands with an Opening of a new indirect gap. The HVB forms along UaF and
ZF, while the bottom of the conduction band forms along I‘Z (Figure 59b,d). This
more closely resembles the band structure of Bl2T83 than Bi2Se3.[11, 40] While Bi
p is mostly above E; and Te p mostly below Ep, a strong Bi p-Tel p hybridization
occurs along FZ, similar to BigTe3.[11] The Se2 p and S2 p orbitals do not appear to

play any significant role near E p, similar to what was found for Te2 p in B12T83 and

Se2 p in Sb2T83.]11]

Figure 59. Band structures of BizTegSe (a) without and (b) with SOI and
Bi2Te28 (c) without and (d) with SOI.

 

 

 

 

 

(a)l 0 BiZTeZSe - No 801 (b)10 BiZTeZSe - SOI

3 : ' 3 i l.

53 I a 1 J

3.3 0.0 BF 30.0 EF
G t:

m m \r

/)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

137

 

 

(c) l n BiZTeZS - No SOI (d)l 0 BiTe2S - SOI

 

 

 

 

 

w: \ :\/ \/ /\
e 2 0
53 : :3 i
30.0, 330.0, EF
t: . =3 .
... ,5. ... . v,
1.0‘

 

 

 

 

 

 

 

 

L“

A: ,
Lin [‘2 I: F

 

 

 

 

While only one atom in the unit cell of SbgTegse differs from that of Sb2T83,
specifically the atom at the Te2 site[141] which does not strongly hybridize with Te1
or Sb near E p at the I‘ point in either compound, the band structures are significantly
different.(Figure 58, 60). Before the inclusion of SOI, SbgTe2Se has a band gap of
0.03 eV. The VBM and CBM have a maximum at the I‘ point along Pa and F2, with
secondary peaks along UaI‘ and ZF, but this point is a local minimum along FF and

FL(Figure 60a). The inclusion of SOI causes the HVB to come closer relative to the
LCB so that the two bands overlap, drastically changing the bands near the I‘ point.
The CBM moves away from the 1" point, similar to that in BigTeg or SbgTeg, but
remaining closer to F. Due to the smaller SOI of Sb and Se in the Te2 site (and
different hybridizations of Te1-Se2 compared to Te1-Te2), the VBM remains at the
F point (Figure 60b), similar to Bi28e3, and the band gap increases to 0.31 eV. Here,
the atom occupying the Te2 site plays a role in the gap formation by changing the
band gap before inclusion of SOI (0 in Sb2T83, 0.03 in SbgTe2Se) so that the degree
of overlap of the LCB and HVB differs after including SOI, which then change the

final band structures.

138

Figure 60. Band structure of SbQTBQSe (a) without and (b) with SOI

 

 

  

 

 

 

 

 

 

 

(a)10 SbZTeZSe - N0 501 (b)10 SbZTCZSe - SOI
3 ‘ - 3 : t/ (
>5 >3 ‘
$00.0 EF $0.0: E1:
6 = 'l. f
. U a I. Z I3 F l

 

 

 

 

 

 

 

 

4.0'

 

 

 

 

 

The gap formation of Bi2Se2Te and Sb2Se2Te is far simpler than those in the
previously studied systems. The unit cells of BigSegTe and szsegTe differ from
those of BiZSe3 and (hypothetical rhombohedral) Sb2Se3, respectively, by replacing
one of the three Se atoms by a Te atom, this Te atom going into the Se2 site.[141]
Before the inclusion of SOI, the band gap in Bi2Se2Te is 0.56 eV and in SbgsegTe
is 0.18 eV, both of these gaps forming at the F point with secondary peaks along
Uaf‘ and ZF (Figure 61a,c). The inclusion of SOI causes the HVB to come closer
relative to the LCB but they do not overlap so the gap remains direct (Figure 61b,d),
decreasing to 0.06 eV in Bi2Se2Te and to 0.05 eV in SbQSegTe. Again, changing the
atom at the Se2 site increases the band gap before inclusion of SOI so that the LCB

and HVB do not come close enough to overlap in these two systems.

139

Figure 61. Band structure of BigSegTe (a) without and (b) with SOI and
Sb2Se2Te (c) without and (d) with SOI

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) BiZSeZTe - NO 501 (b) BiZSeZTe - SOI
1.0. , ,. \ 1.0
2W 1 w /\
9 - l 9 l .
e : e s‘
a ' >~ ‘1
5600‘ EF $33000 EF
c: j r: .
m . Lu ./
1.0‘ .11; W\V\/\ \ A,
U - U a 1“ z 1.: r 1-
(c) SbZSCZTe - N0 SOI (d) SbZSCZTe - SOI
1.0x 7 1.0, \ \/ \
E M % 300M
1% ‘ E. a '/
20.0. EF E00. EF
m : m 1 q _,:‘
s 1 1 ,, VT
l‘OU a x r 2" P r 1- ”U a ' 1‘2 1= 1' 1-
C. 81283

While S lies in the same column as Se and Te, Bi2S3 forms in a different crystal
structure than B12T83 and Bi2Se3. Bi283 has been studied for thermoelectric appli-
cations in the past. While it was found to have promising values of thermopower
and thermal conductivity, its electrical conductivity was found to be too small.[142]

Due to its larger band gap, it has also been investigated as a prospective material

140

for photovoltaic converters.[l43] While this material has been studied extensively
experimentally[143], its band structure has not been inveStigated yet.

The crystal structure of Bi283 is orthorhombic in the space group V},6 (anm) with
twenty atoms in the unit cell.[63] The orthorhombic unit cell and its corresponding
BZ are shown in Figure 62. The crystal structure consists of sheets of atoms parallel
to the z-axis with each S surrounded by three Bi atoms and each Bi atom surrounded
by three S atoms.[63] These twenty atoms reduce to five inequivalent atoms identified
as Bil, Bi2, S1, S2, and S3. The lattice parameters (a, b, e) are 11.15 A, 11.30 A,
and 3.981 A, the atomic positions taken from Wyckoff. [63]

Figure 62. (a) Orthorhombic unit cell of Bi2S3 (shown in the a-b plane) and its

(b) corresponding Brillouin zone.

 

 

141

 

 

 

 

 

 

 

 

The electronic structure of Bi2S3 shows a large number of bands within a few
eV of Ep. Before the inclusion of SOI, an indirect gap of 1.55 eV forms between the
CBM at S and the VBM along FX (Figure 62). After the inclusion of SOI, the HVB is
practically unchanged, but the CBM moves from S to a point along I‘X (Figure 63) to
form a nearly direct band gap of 1.24 eV. The band gap has been measured by several
groups to be between 1.2-1.7 eV[142, 144], one of the most recent measurements giving

a value of 1.3 eV[142].

Figure 63. Band structure of Bi2S3 (a) without and (b) with SOI.

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) 20Bi253- N0 801 (b) 2,, Bi2S3- SOI f
' ' 5d Vfl’VQWI f
:N ‘> :
s C’QC 9 ’5?“ III xv
Q) 'J O : I, -:‘~ "." 3 ~
v ,. v ,7 .V7 f"
a a “I '
30. EFBO'O T.‘- ‘£1:
E. t:
.5?" “3.3? \.
'r zr ers RU r ZT rxs RU

 

 

 

 

142

 

 

In order to understand the electronic structure of this material near the gap region,
an orbital analysis of the bands near Ep was performed. We find that Bil and Bi2
p orbitals contribute primarily to the bottom of the conduction band while 81, S2,
and S3 p orbitals contribute to the top of the valence band. While there is some
hybridization between these orbitals, it is much less that that between Bi p and Te/ Se
p in B12T83 and Bi28€3.[11, 40] Thus, Bi283 is more ionic. These differences for Bi283,
with respect to Bi2T€3 and Bi28e3, may be due to both the different crystal structure

of Bi283 compared to BI2T€3 and Bi28e3 and the different electronic affinities of S, Se,

and Te.
6.2.4. EFFECTIVE MASSES OF THE BAND EXTREMA OF Bi2T€3

In the last two sections of this chapter, we will test the validity of our band struc-
ture calculations (aside from comparing the band gap values which has already been
done) by comparing them with several experiments on these materials. In this section,
the orientation of the carrier pockets and effective masses of the doped Bi2T83, which
determine the properties close to E1: ('~0.1 eV), will be compared. These properties
are important to understand the thermoelectric properties of these materials. The
next section will compare the calculated DOS to results of photoemission studies over
a larger energy range (~10 eV) in Bi283, BigSeg, BizTegse, and BigTeg.

Shubnikov-de Haas and de Haas-van Alphen experiments have found six-fold de-
generate CBM and VBM in BI2T63. While these experiments cannot locate the posi-
tion of the extrema in k-space, the number of extrema and the effective mass tensor

associated with these extrema have been repeatedly studied in BigTe3.[131, 132, 133,

134] The effective mass tensor is defined as[22]

143

_ _ 9215
mm, 211 Z 012' Z ’3 zlolmkjl

 

where me is the free electron mass and 27m is Planck’s constant. In most cases,
the off-diagonal terms of the mass tensor are small, so diagonal elements of the mass
tensor, an, accurately give the inverse of the effective mass parameters m, (i = 1, 2,
3). In the general case, however, the mass tensor has to be diagonalized to obtain the
effective masses associated with the principal directions. We fit the calculated E vs
k along different orthogonal directions to parabolas and obtained the coefficients 0,,
All of our calculated extrema in BI2T€3 lie on planes of reflection symmetry (the
z axis is the trigonal axis, the y axis is the bisectrix axis, and the yz plane is one of
the three reflection planes) the energies near the top of the HVB and the bottom of
the LCB can be fitted using the equation
(75)

2me/h2 5k: anki + ayy k: + azzkf + 2(yyzkykz

where a constant term has been omitted and k is measured from the extrema. Note

that the de Haas-van Alphen measurements find sixfold-degenerate hole and electron
pockets lying on the reflection planes (containing the trigonal and bisectrix axes).
The cross sections of these pockets in the yz plane are ellipses whose major axes
make an angle 6 with the y axis (the bisectrix axis). The angle 6 is given by[131, 133]
(7G)

6 : éarctaleQyZ/(Qu ' 013131)]

The effective masses for BigT€3 were obtained by fitting values of the energy close to

144

different LCB minima and HV B maxima to Equation 75 while moving along suitably
chosen directions in the BZ, (Figure 64). The energy dispersion curves near the
LCB minima and HVB maxima of Bi2T€3 are known to display strong nonparabolic
behavior[131, 132, 133, 134], but a fair approximation to a parabolic band can be
made if points are chosen close to the band extrema.

Theoretical values of the coefficients 01,,- are given in Table XII along with their
experimental values. They agree quite well with the previous LMTO calculation[40],
though using the values in their paper, 0 should be approximately 40° not 35° as
reported in their paper. The curvature in the x direction is obtained using values very
close to the HVB maximum and LCB minimum and, as we can see, the theoretical
values are about a factor of 3 too large compared to experiment. For higher doping, i.e.

for larger k1, the effective mass decreases due to nonparabolic effects which improves

the agreement with experimental values.

Figure 64. Fitting of the energy band near the conduction band minimum to a

parabola for BIQT€3 using the computer program KALEIDAGRAPH[120].

 

0.09
0.085 Y I MO + M1'x + Ma'x' + ma?
0 0.069625
T' m 0
3 M2 124.67
"a 0.08 6 0.99923
‘5
E” 0.075
3
0.07
. 0.065
-0.015 -0.01 -0.005 O 0.005 0.01 0,015

145

parameters associated with the conduction band minima and valence

band maxima for BI2T83.

Table XII. Theoretical and experimental values of the effective mass tensor

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘ Conduction band c0 61 c2 Experiment[131] (i10%) J
5,, = [rm/m..|,-;,,1 20.25 112.75 95.84 46.9
6,, = [m/me ,7; 3.97 2.87 3.56 5.92
5,, = [m/m. .31 3.75 5.49 6.20 9.50
6,, = [m/me ,7; 2.93 2.54 2.39 4.22
0 —43.9° 313° 327° 335°
Valence band 01 02 Experiment[133] (i10%)
an = [m/me]; 107.51 90.05 32.5
ayy : [m/m,.,]_;yl 3.97 2.91 4.81
an = [m/me g; 5.54 5.60 9.02
ayz 2 [m/me 37,} 2.76 2.34 4.15
9 37.1° 30.0° 31.5°

 

 

 

 

 

 

 

The positions of the electron and hole pockets CI, 62, 01 , and 02 on the reflection
planes as well as the angle 9 associated with these pockets are accurately reproduced
compared to experiment (Figure 65). The angle 6 of the electron pocket associated
with the CO minimum is nearly 90° away from that seen in experiment. This, along
with the more recent calculation[130], suggests that 60 does not correspond to a

conduction band extrema seen experimentally.[131, 132, 133, 134]

146

Figure 65. Positions of the electron and hole pockets lying on the reflection

plane of the Brillouin zone for BI2T€3 for (a) CBM and (b) VBM.

(a) z

 

 

 

 

 

 

 

F
A 43.9
r , r
\
\ \ , L
\ \ éu'
\
2 (U)
a
(h) z
A
2 far (A)
N ' -------- F

 

 

As already stated, the nature of the bands near Ep play an important role in
the the observed large room temperature values of ZT in these materials. First, in
applications, B12T€3 is alloyed with Bi28e3 and Sb2T83 to reduce re. However, the

large S and a values must have an electronic origin. The B parameter (Equation

147

24 in Chapter 2)]11, 24] suggests highly anisotropic effective masses and large band
degeneracy will produce high ZT. We find that in BI2T63, the effective masses, while
fairly small, are highly anisotropic. Unlike the half-Heusler alloys, these masses are
much closer to those found in other doped semiconductors. Further, there is a 6-fold
degeneracy in the VBM and CBM which enhances the thermopower. We find that in
Bi2863 the VBM and CBM have a lower degeneracy which can explain the lower value
of ZT seen in this material compared to BI2T€3. The combination of anisotropic (but
not large) effective masses and highly degenerate band extrema, coupled with the low

s, are the reason for the large ZT values seen in BI2T€3. These features should be

investigated in other thermoelectric materials.[11, 12, 13]

6.2.5. COMPARISON OF THE BAND STRUCTURE WITH
PHOTOEMISSION EXPERIMENTS

Photoemission experiments allow for a testing the band structure calculations over

a larger energy scale. Photoemission spectroscopy (PES) is a direct probe of a ma-
terial’s occupied states. In these experiments, photons incident on a sample result
in ejection of electrons which are then collected and energy analyzed. In the re—
verse experiment, inverse photoemission spectroscopy (IPES), electrons are incident
on the sample while photons are ejected and studied to analyze the unoccupied states.
Angle-resolved PES (ARPES) allow for a comparison with the calculated E(k) vs.
k. On the other hand, an angle-integrated PES spectrum gives information about
the occupied DOS modified by energy dependent matrix elements between initial and

final states. [145]

Angle—resolved photoemission on BI2T€3[146] and BigSe3[147] have been performed

148

recently in order to compare the experimentally determined E(k) vs k with our band
structure. Greanya et al.[146] studied the band dispersions of p- and n-doped BI2T€3
along two k-space directions, F Z and UaI‘, the directions which one should find 01,

60, and c1. Their measured overall width of the valence band states of 5-5.5 eV

for both systems agreed very well with the current calculation.[ll] They found the
six-fold VBM along UaI‘, consistent with the current calculation[11] and Shubnikov-

de Haas and de Haas-van Alphen experiments[131, 132, 133, 134] The position of
the maxima was found not at the calculated position, however, which may be due

to anti-site doping of Bi into the Te sites[146] known to play a large effect in the
band structure near Ep[11]. Their doping was insufficient to find the CBM. They
also found that the dispersion along FZ, through the Van der Waals gap was much
smaller than predicted[11], though this might be due to electron-phonon interaction,
electron-electron interaction, anti-site defects, and the limited penetration depth of
the photons into the material along this direction. Greanya et al.[147] also studied
n-doped Bi28e3 along FZ, FaU, FF, and I‘ L to find a single VBM at the F point
with a close lying local maximum about 0.1 eV below along FF. Greater dispersion
is seen along F Z than in Bi2T83, though these bands are still fairly flat. Again, the
differences between the experiment and the band calculation may be due to anti-site
defects and electron-electron Iinteractions.[147]

Several angle-integrated PES studies have been performed on Bi2X3 (X : Te, Se,

S) systems in the past[139, 148, 149, 150]. Two fairly recent photoemission studies
will be compared with our calculations for Bi2T83, BigTeQSe, 1312383, and B1283. We
find that the positions of the peaks in our DOS agree rather well with the measured

PBS, and, in general, increases in the energy resolution of the PES spectrum tends

to improve the agreement between experiment and our calculations.

149

While there have been several sets of photoemission data performed on BI2T93-
related systems, most of these have been x-ray photoemission (XPS) measurements
with a low energy resolution which makes comparison with the calculated DOS unre-
liable. Debies ct al.[148] studied BI203, Bi283, Bi28e3, and BI2T83, where the energy
resolution was not explicitly specified, but was fairly low. Nascimento et al.[150]
studied Bigse3 with an energy resolution of 0.8 eV. Ueda used both PES and IPES
measurements to study the filled and empty states of Bi283, Bi2Se3, BigTegse, and
Bi2T€3 with an energy resolution of 0.2 eV for the PES experiments and 0.58 eV for
the IPES experiments. More recently, Greanya et al.[139, 146, 147] used PES studies

to understand BiQSe3 and BI2T€3 using an energy resolution of 0.08 eV.

Figure 66 gives the PES spectra and the positions of the dominant spectra found
in the experiment of Ueda et al.[149] for Bi283, BiZSe3, BigTeQSe, and BigTe3. In Ueda
et al.’s[149] experiments, Ep is maintained by electrical contact between the sample
and substrate on which it is mounted.[145] In the electronic structure calculations,
Ep, which should be somewhere in the middle of the band gap, can be anywhere
between the VBM and the CBM. Since we do not know the experimental value of
EF, its exact position depending on doping, etc., we will compare our theoretical
results with experiment by rigidly shifting the theoretical DOS such that the highest
valence band peak coincides with that seen in the experiment. The positions of the
peaks obtained by Ueda et al.[149], are numbered cI-c3 in the conduction band and
111-072 (n = 3 for BigTeg, n = 5 for Bi28e3 and BigTe2Se , and n = 4 for B1283) in the

valence band, and these numbers are placed in the calculated DOS plot (Figure 67),
CI and 01 lying closest to E p.

150

Figure 66. Valence-band photoemission (UPS) and conduction band
inverse-photoemission (IPES) for BI283, Bi28e3, BigTeQSe, and BI2T€3
crystals. UPS and IPES spectra for each crystal are connected at E p.

Energies are referred to Ep. (courtesy of Yoshifumi Ueda).[149]

 

 

 

‘1' I 1 I I I I T I I 1 I I ! U T V I
UPS; : IPES
hv=21.2 eV ' . nave-9.4 eV
A ’\ l -l | °. ' :l/ '-.\/-
E 1.... '. 3'253
:1 ‘. —-——-——L° ' I .1 fi fi/
g 4“.ka 31.. k/\\/
; 1 .BIQSeg
:5 P‘ Tb. I
5 \\ I ’3’". | fur“
E 1 3978259
l‘s ‘ ll ’ fl.‘ 2
\ \~ //'\/; '_ i :l’ a
V 5121123
1 l l 1 l 1 l 4% L 1 l l l L L

 

-1 O -5 O 5 10
Energy (eV)

 

Figure 67. Total DOS of (a) Bi283. (h) Bizseg, (c) BigTeQSe, and (d) BizTEg.

The arrows indicate the positions of the photoemission peaks seen

in the work by Ueda.[149]

 

 

 

 

 

 

(3:235 BIZSS
v!1
30 v _
20 v.3 ‘12 ‘32 V ~
“2‘ V V C.‘ V
1 5 v ..
1 O "
5 1 -4
91 - 3 — : — — z — — - — - o - - : 3 O
(b) Energy [eV]
7 BIZSea
T fi' V # V V v2 V fl ‘7 V f V V V I 1 fi’
6 ' ‘1‘
vi4 v v ‘
5 - V V c1 c‘2 a
4 V V V
(r) b \15
8 3 _ V .4
2 ~ d
1 I. —l
91 2 - : - .. z - - .. .. - o - : : 2 0
Energy [eV]
(C) Bl2Te2Se

DOS

3
V'O-‘Nwhmmimm

DOS

IOJNQhUIm‘JDQ

 

 

0-9 -‘a :73 -‘5 -2: -33

    

Energy [eV]
BlZTea

 

 

 

      

      

A

 

 

-\. A . g. . .
O-Q -8 -7 —6 -5 -4 -:3 -2

-‘1 *6 “i 2
Energy [eV]

 

 

~‘2-‘10f23456’78—‘9—‘To

 

A close look at Figure 67 shows that the double-peaked structure near the VBM
is found in the PES of all three compounds, Bi2T83, BigTegSe, and BiQSeg, while
Bi283, which has a completely different crystal structure, displays a single prominent
peak in this experiment. The separation between these two peaks is 0.3-0.4 eV, very
close to that seen in the calculated DOS. The DOS peak near the VBM obtained
in the previous LMTO-ASA calculation had only one peak[40], in contradiction to
these experiments.[139, 149] Also, this double-peak is found to'come primarily from
Te1/Sel and Te2/Se2 p states. It appears that the narrow width of these two peaks
is consistent with the picture preposed by Ueda et al.[149] who ascribed these to lone
pair states of Sel and Se? in BigSeg. A similar picture must be true for Bi2T€3 and
BigTegse which have similar local coordination geometry.

For Bl2T€3 the energy separation between the HVB peaks is found to be 0.5 eV' in
the measurement[149] compared to 0.3 eV found in our calculation.[139] There is only
one peak seen at -3.5 eV. For the IPES results, within the experimental resolution
(0.56 eV) the conduction band peak c1 corresponds to the calculated peak positions
at 0.5 and 1.0 eV while c3 at 3.4 eV lies slightly below the calculated peak position
at 2.9 eV. Matrix element effects and noise affect the heights and positions of the
PES peaksll45], but while these have not been included in the DOS calculation, the
agreement between the theoretical DOS peak positions and experiment is fairly good.

For BigSeg, the separation of the double-peaked structure is 0.5 eV compared to

the calculated value of 0.4 eV. The calculated peak at -2.9 eV may correspond to M.
shifted by -0.5 eV, while peaks at -3.8 eV and -4.6 eV may be seen as the single pmk
v4 . The calculated peak at -5.4 eV agrees well with U5. The two lowest peaks in tin-
conduction band at 0.8 eV and 1.3 eV lie within the experimental resolution of c2 and

c3, respectively.[149] For Bi28e3, the overall agreement between the theoretical DOS

153

and experimental PBS (and IPES) is very good.

In BigTeQSe, the separation between the valence band peaks v1 and 122, the double-
peak structure at the top of the PES spectra[149] is 0.5 eV compared to the calculated
value of 0.4 eV. The peak predicted at -2.9 eV may be peak 213 shifted by -0.5 eV, the
peaks at -3.9 and -4.6 eV may correspond to the single peak v5 at -4.2 eV, and the
peak predicted at -5.3 eV correspond to '05 . As regards to the conduction band, three
peaks (c1, c2, and c3) are seen experimentally (0.7, 2.5, and 3.7 eV), two of which
agree well with our calculated DOS peaks (0.7 and 3.7 eV).[149|

Finally, for Bi283, which has a different crystal structure than the above three
compounds, the general structure of the calculated DOS (Figure 66) agrees very well

with the PES spectra[149]. Experimentally one sees a dominant valence band peak
below E p, while our calculation predicts a double peak with a separation of 0.7 eV with

a fairly smooth background between -3.5 and -7 eV. In order to facilitate comparison
between theory and experiment, the energy reference (Ep) will be set so that v1 lies
between the calculated double peak structure. It is surprising that the double peak
structure could not be resolved in experiment in spite of the resolution of 0.2 eV for
the valence band. Perhaps matrix element effects can explain this discrepancy. The
three rather broad peaks seen in the experiment ('02, v3, and 214) are seen in our
calculated DOS as a single broad peak. The overall agreement with the conduction
band is also fairly good. Several features in the calculated DOS agree well with the
positions of the measured peaks c1 , 62, and c3, the lowest predicted conduction peak

lying within the energy resolution of peak c1.[139]

More recently Greanya et al.[139, 146] have carried out high resolution angle-
integrated PES experiments on Bi2T€3 and Bi2383. We will next compare our the-

oretical results with their experiment. The photoemission spectrum for Bi2T€3 by

154

Greanya ct al.[139, 146] is shown in Figure 68, that for BizSeg in Figure 69. The
PES data was taken using a synchrotron radiation source which allows for the effects
of matrix element and atomic cross-sections to be examined by measuring spectra at
several different photon energies. As can be seen, six peaks (numbered 1-6 and shown
by the arrows in Figure 68) were found in the valence band for Bi2T€3 and nine peaks
for BigSeg (numbered 1—9 and shown by the arrows in Figure 69), several of these
peaks seen at certain photon energies but not at others. These measurements did not
have an energy reference, so the bands were rigidly shifted so that the topmost peak

agreed with the previous calculation, as had been done with the theoretical DOS.[139]

155

 

 

Figure 68. Photoemission spectra for BigTe3 by Greanya et al.[139]. Peaks
1-6‘ are indicated by arrows. For the purpose of comparison, the energy
reference (zero of energy) was set such that the lowest binding energy
peak in the spectra coincides with the position of the corresponding

peak in the data of the previous UPS experiment by Ueda et al.[149]

 

     

 

 

 

 

l I
BizTe3
Multiple Photon Energies

A

c1)
3::

G

:3
e
<'.
v

>5
3:

m

C

G)

H

C
r-4

l9eV
l l l
-8 —6 -4 -2 0

Binding Energy (eV)

156

 

Figure 69. Photoemission spectra for Blgse3 by Greanya ct al.[139]. Peaks
1-9 are indicated by arrows. For the purpose of comparison, the energy
reference (zero of energy) was set such that the lowest binding energy
peak in the spectra coincides with the position of the corresponding

peak in the data of the previous UPS experiment by Ueda et al.[149]

 

 

-1

23 eV_
1.0?
'E
:3
.6
H
:9 21 eV
3‘ ‘l
”'m‘
c:
o
E 20 eV

 

19g.

 

 

l l 17 eV

—

 

-4
Energy (eV)

The positions of the peaks found in the PES study of Bi2T€3 and Bi28e3 by Greanya
et al.[139, 146] are shown in Figure 70 over our calculated DOS plots.

157

Figure 70. Total DOS of (a) Bl2T€3 and (b) BiQSeg. The arrows indicate the
positions of the photoemission peaks seen in Greanya et al.[139].
For the purpose of comparison, the energy reference (zero of energy)
was set such that the highest valence band DOS peak coincides with
the position of the corresponding peak in the data from the

experiment of Ueda et al.[149]

 

 

 

 

 

 

(a)7 r 832793 .
< .2 .1
6 6 5 4 \f "’
t t t
5 r 3 -<
4 _ V J
U)
o
0 3 . -
.. \j .
1 .- .J
0 * ‘ * a ‘3 to. ‘1 V
"B '7 '6 '5 Eneréy [eV] ' ' '
(b)7 Bi23e3 .
r ' 5' 4 g 1
e. l i (Y i i
8 7 6
5 . E E =
V V V
m 4 9 ,r’V
O E
D 3 >- V '1

 

 

 

The double-peak structure at the VBM for Bl2T€3 has an energy separation of 0.27
eV, in much better agreement with the calculated value of 0.3 eV. The positions of
the next three peaks in this experiment.[139, 146] (peaks 3, 4, and 5 ) agree very well
with the calculated positions at -2.1, -2.6, and —3.9 eV, respectively. Experimental
peak 6 is slightly below a DOS peak while the DOS peak at -5.3 eV is not seen in
this experiment.[139, 146]

For Bigseg this experiment[139] find nine peak positions compared with the experi-
mentally determined eight, more than the five found by the previous experiment[149].
The double peak structure has a separation of 0.36 eV compared to the calculated
value of 0.4 eV. The calculated peak at -2.9 eV corresponds well to peak 3. The
experimentally observed peaks 4 and 5, seen at —2.64 and -3.08 eV, respectively, lie
symmetrically on either side of the calculated peak at -2.9 eV. Peak 6 (-3.52 eV) lies
about 0.3 eV above the calculated peak at -3.8 eV. Peaks 7, 8, and .9 clearly corre-
spond to the calculated peaks at -4.4, -4.6, and -5.4 eV, respectively.[139] For both
Bi2T€3 and Bi2Se3, improvements in the energy resolution[139, 149] lead to better
agreement of the experimental PBS and theoretical DOS peaks. Further increasing
the energy resolution of the experiments and addition of matrix element effects into
the calculation should further improve this agreement.

Finally, we summarize the peak positions found in the theoretical DOS and those
in different PES measurements in Table XIII. The first columns of Table XIII give
the peak positions for both the valence and conduction bands by Debies et al.[148],
Ueda et al.[149], and Nascimento et al.[150], in the systems where they have been
reported. In BigSeg and Bl2T93, the results of Greanya et al.[139] have been reported
in the second to last column. The last column (with the heading “DOS”) presents the

positions of the peaks obtained from our electronic structure calculations. Clearly

159

the agreement between theory and experiment improves when uses the experimental

data with the best resolution.

Table XIII. Comparison of the photoemission and calculated DOS peaks for

(a) Bi2Te3, (b) BigTegSe, (c) BiQSe3, and (d) BIQS3. For a, b, and c
the theoretical reference energies are chosen such that the highest
valence band peak positions coincided with experiment.[149]

For d, the theoretical reference energies are chosen so that the top
two theoretical bands lie equidistant from the highest

experimental peak.[149] ~

Bl2T€3

 

Valence Band Valence Band Valence Band DOS

 

 

 

 

Debies[148] Ueda[149] Greanya[139]
-1.2 eV -l.1 eV -1.10 eV -1.1€V
-1.6 eV -1.37 eV -1.4 eV

 

-2.07 eV -2.0 eV

 

-2.66 eV -2.5 eV

 

 

 

 

 

 

 

 

 

 

 

 

 

-3.4 eV -3.5 eV -3.77 eV -3.8 eV
-4.51 eV -4.2 eV
~52 eV
Conduction Band

Ueda[149] DOS

0.7 eV 0.5 eV

1.7 eV 1.0 eV

3.4 eV 2.9 eV

 

160

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

BigTegSe
Valence Band DOS
Ueda|149]
-1.1eV -1.1eV
-1.6 eV -1.5 eV
5-2.0 eV
-2.9 'eV
-3.5 eV -3.9 eV
-4.2 eV -4.6 eV
-5.3 eV
Conduction Band DOS
Ueda[149]
0.7 eV 0.7 eV
1.4 eV
1.8 eV
2.5 eV 3.1 eV
3.7 eV 3.7 eV

 

 

161

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Bi2Se3
Valence Band Valence Band Valence Band Valence Band DOS
Nascimentoll50] Debies[148] Ueda[149] Greanyal139]
—1.1 eV) -1.10 eV -1.1 eV
—1.6 eV -1.6 eV -1.46 eV -1.5 eV
-2.1 eV -2.03 eV -2.1 eV
-2.64 eV -2.9 eV
-3.08 eV
-3.3 eV -3.52 eV -3.8 eV
-4.4 eV -4.4 eV -4.2 eV -4.26 eV -4.4 eV
-4.66 eV ~4.6 eV
-5.4 eV -5.62 eV -5.4 eV
Conduction Band Conduction Band DOS
Nascimentol150] Ueda[149]
1.0 eV 0.9 eV
1.4 eV
2.5 eV 2.1 eV
3.5 eV 3.8 eV 3.1 eV

 

 

 

 

 

162

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

81283
Valence Band Valence Band DOS
Debies[148] Ueda[149]

-2.2 eV -2.2 eV -1.9 eV
-2.6 eV
-3.4 eV -3.5 eV
-3.8 eV
-4.1 eV
-4.6 eV -4.5 eV
-5.0 eV
-5.7 eV -5.7 eV
-6.7 eV

Conduction Band DOS

Ueda[149l -

0.4 eV

0.9 eV 0.9 eV

1.5 eV

2.5 eV 2.7 eV

4.0 eV 4.1 eV

 

 

 

163

 

CHAPTER 7.
COMPLEX CHALCOGENIDES

Technological needs have driven a search for new thermoelectric materials.[4] Since
bulk materials are more easily amenable for fabrication in electronic circuitry and
superconducting devices, classes of materials such as skutterudites[23], half-Heusler
alloys[7, 11, 81], clathrates[151], and pentatelluridesl152] are being studied both ex-
perimentally and theoretically for their potential thermoelectric properties. We have
already looked at half-Heusler alloys, Zintl phases, and simple chalcogenides in previ-
ous chapters, but the best new thermoelectric materials are likely to be skutterudites,
clathrates, or complex chalcogenides.[4] These systems appear to have large carrier
effective masses, high carrier mobility, large degeneracy of the valence and conduc-
tion band extrema, and low lattice thermal conductivity, all of which are necessary
to enhance the thermoelectric figure of merit, ZT = S20T/K.

Different approaches have also been attempted to look for new thermoelectric ma-
terials. Hicks and Dresselhaus[24] argued that systems displaying large anisotropy or
reduced dimensionality (quantum confinement) in their charge transport can also
exhibit a significant enhancement in ZT. Mahan and Sofo[25] predicted than in
good thermoelectric materials the energy distribution of the carriers should be nar-
row and have a high carrier velocity in the direction of the applied field. Conse-
quently, nanofabrication of one- and two-dimensional arrays of materials have been
attempted, and high values of ZT have been reported in PbTe-PbSeTe quantum-dot
superlattices by Harman et al.[27] and in p—type B12T93/Sb2Te3 superlattice films by
Venkatasubramanian[28]. The major source of improvement for the enhanced ZT in

these superlattice systems appears to be in their reduced thermal conductivity (K)

164

rather than an increase in the power factor (PF 2 S20).

Since the best bulk materials for thermoelectric applications are simple chalco-
genides, specifically PbTe and BlgTe3 alloys[2, 3], complex chalcogenides provide a
promising avenue for searching for new thermoelectric materials. Complex chalco-
genides with large unit cells containing weakly bound atoms or molecules, called
“rattlers”, have been studied to reduce the phonon contribution to K. without affecting
S and a. The basic principle underlying the synthesis of new thermoelectric mate-
rials by Dr. Mercouri Kanatzidis’ group in the Chemistry Department at Michigan
State University is to synthesize complex bismuth chalcogenides to reduce n while
maintaining the high values of S and a of BigTe3. This is done by combining portions
of existing structures such as PbTe (FCC) and B12T€3 (rhombohedral) to form new,
more complex structures with enhanced thermoelectric properties.[5] Three represen-
tatives of these systems, BaBiTeg, CsBi4Te6, and K213188613 contain highly layered
and anisotropic crystal structures with weakly bound Ba, Cs, and K atoms, respec-
tively. These cations act as “rattlers” off which phonons are expected to scatter to
reduce m, but allowing for electron transport within the layers to maintain a large
S and 0. Electronic structure calculations of these materials should lead to a bet-
ter understanding of the transport and electronic contributions to the thermoelectric

properties of these materials.
7”]. BaBiTe3
7.1.1. GENERAL FEATURES OF THE BAND STRUCTURE OF BaBiTe3

BaBiTe3 belongs to a class of compounds which are derived from the BIQTG3 crystal

165

structure.[153] The Bi/Te building blocks from BIQT€3 are arranged side by side in
layers connected by Te atoms. The heavy Ba atoms lie between the layers.[153, 154,
155] The structure of BaBiTe3 also contains Teg’ (Te-Te) units which serve to connect
the Bi / Te blocks into layers. The unit cell of BaBiTe3 is more complicated than that
of B12T83. Its crystal structure is orthorhombic rather than rhombohedral, the space
group being P212121 with 40 atoms in the unit cell [A = 4.47 A, B = 16.53 A, C =
17.89 A][153, 154], but internal symmetries reduce the number of inequivalent atoms
to ten: Ba1-2, Bil-2, and Te1-6.~ The crystal structure of BaBiTe3 and, for comparison
with that of the parent compound, BlzT€3 (projected onto the yz plane) are shown
in Figure 71. The underlying local BigTe3-like structures (shown by the shaded area)
connected by Te5—Te6 chains existing in BaBiTe3 can easily be seen in Figure 71. The
Te2 and Te4 atoms lie on the edges of the layers and are bonded to only one layer of
Bi atoms while the Tel and-Te3 atoms are bonded to two layers of Bi atoms. The
Te1 atom has, however, a low Bi coordination and is likely to behave more like Te2
and Te4. In B12T83, the Tel atom is bonded to one layer of Bi atoms and lies near
the Van der Waals gap, whereas the Te2 atom is bonded to two layers of Bi atoms.
As a consequence, the Te2 p bands in B12T€3 have a lower energy and appear away
from E p.[11] Therefore, one would expect that in BaBiTeg, Te1, Te2, and Te4 p bands
would act more like the Tel p bands in B12T83, contributing to the top of the valence
band DOS near Ep, whereas Te3 p bands would act more as Te2 p bands in BigTeg,
stabilized several eV below Ep with little contribution to the DOS near E p. Also, Bi
p states will contribute primarily to the conduction band. Since Te5 and Te6 have no
counterparts in B12T83, the bands associated with these atoms will be new features

in BaBiTe3.[11]

166

Figure 71. Projection of the crystal structures inthe yz plane of (a)
orthorhombic BaBiTe3 and (b) rhombohedral B12T63. The Bi/ Te
blocks are highlighted in the shaded area for BaBiTe3.

 

 

 

 

 

 

 

 

 

167

Electronic structure calculations were performed both with and without spin orbit
interaction (SOI). The band structure of BaBiTe3 is much more complicated than
that of B12T€3 due to the larger number of atoms in the unit cell. We will focus,
therefore, on the region close to E p. The bands are plotted along the high symmetry
directions of the orthorhombic Brillouin zone, shdwn in Figure 72.

Before discussing the theoretical band structure in detail, we wish to discuss
the room temperature diffuse reflectance spectroscopy (Figure 73) for B12T€3 and
BaBiTeg. These experiments in BaBiTe3 reveal absorptions at 0.42 eV (which is the
intercept found by extending the linear portion to the axis) and at 0.28 eV (where
the spectrum extends to zero after the linear portion). This absorption spectrum in
BaBiTe3 is unusual in the sense that it has two values. This was not seen in B12T€3

(Figure 73) which has one at the band gap value of 0.13 eV.

Figure 73. Infrared absorption spectra of BigTe3 and BaBiTeg. The

semiconductor energy gaps are indicated in the spectra.

BaBi'l'e3

Absorption

 

 

 

- E" ~0.42eV
E ~o. 1 3eV 8

IIIIJJIlllllllll‘llljlllllllllll

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Energy (eV)

 

 

Due to the atomic arrangement inside the unit cell and the presence of the Ba2+
ions, the orbital characters of the HVB and LCB in BaBiTe3 are distinctly different
from those seen 111.812TC3. As a result, the effect of SOI is not as dramatic. The
energy bands without SOI centered in the range of ~2 eV about Ep are given in
Figure 74. The VBM occurs at a point between F and Z (this HVB in the FZ
direction is extremely flat) while the CBM lies at the Z point. The band gap at Z is
calculated to be 0.45 eV, in good agreement (but see below for a discussion when SOI
is included) with the larger of the two band gaps found experimentally, 0.42 eV. This,
however, cannot explain a second smaller gap of 0.28 eV also seen experimentally.]153]
Our results agree rather well with a previous unpublished LMTO calculation which
also did not include SOI.’[156]

Introducing SOI has some broad general effects, such as lifting the degeneracy of
several bands and moving the CBM and VBM closer together (Figure 75). However, it
has little effect on the position of the VBM, while the CBM moves to a point between
F and Z to produce a nearly direct gap. A careful analysis of our results shows the
VBM is at (0, 0, 0.325) and the CBM is at (0, 0, 0.3625), in units of the primitive
reciprocal lattice vectors. The new band gap is about 0.26 eV, in good agreement with
the smaller experimental value of 0.28 eV.]153] There exist several other indirect band
gaps in the Spectrum, such as the 0.34 eV gap between the VBM at F and the CBM
between F and A. As pointed out before, the direct band gap at the F point is 0.47
eV, which is comparable to and may be identified with the larger of the experimentally
observed band gaps of 0.42 eV.]153] The unusual shape of the experimental absorption
spectrum near the threshold of the smaller band gap (0.28 eV) may result from the
forbidden pep transitions when Ak = 0. More accurate measurements and a detailed

calculation of the optical absorption spectrum will clarify this issue.

169

Figure 74. Band structure of BaBiTe3 without SOI.

1.2
1.0
0.8
0.6
0.4
0.2
0.0

Energy (eV)

-0.2
-0.4
0.6
-0.8

-I.O

 

Figure 75. Band structure of BaBiTe3 with SOI.

l .2
l .0
0.8

0.6

cc
chin

Energy (eV)
3 s:

 

170

The orbital characters of the bands in the neighborhood of Ep have been care-
fully analyzed, but will not be shown to save space. It is observed that the main
contribution in the LCB near the CBM is associated with the Bil p, Te5 p, and Te6
p states while the Tel p, Te2 p, and Te4 p states contribute most strongly to the
VBM. This is in agreement with the analogy made between the bonding found in
BlgTeg and BaBiTe3 with respect to the position of the atoms within the layerslll].
The bands contributing to the DOS near E p consist of Te and Bi p bands, the Ba 5p
bands lying about 13 eV below and the Ba 5d and 63 bands lying several eV above
Ep. Therefore, Ba 5p, 5d, and 6s orbitals contribute very little to the states near
Ep, the Ba2+ ion acting mainly as a spacer. The interesting case is that of the Te5
and Te6 atoms which connect the Bi/Te blocks together through the Te—Te chains
along the z axis and have no analogy in BizTe3. As mentioned before, the Te 5 p
and Te6 p states contribute to the LCB along with the Bil p and Bi2 p orbitals, pri-
marily the former. The crystal structure of BaBiTe3 can be considered as [Bi4Te10]8"
segments separated by Ba2+ ions and connected by Te5-Te6 chains. The previous
extended Huckel calculation assigned a 3+ oxidation state to Bi with an electronic
configuration of 6326p0, the two remaining 63 electrons acting as a “lone pair”. A
strong hybridization between the “lone pair” states was found with the atoms in the
Te5-Te6 chains for the lowest conduction band. Our band structure calculation finds
these Te5-Te6 orbitals playing an important role in the LCB, in agreement with the
“lone pair” (20*) description.[154]

The nature of the charge transport in this anisotropic material depends on the
dispersion of the bands along different directions. In the conduction band, the dis-
persion along the FY direction, perpendicular to the Ba2+ layers, is very small while

that along the FZ direction, i.e., through the Te5-Te6 necks is appreciable. One also

171

sees that there is a large dispersion along the F A direction which should be similar
to that seen in the'xy plane of BigTe3.[11] Thus, n-doped BaBiTe3 can be treated as
quasi-two—dimensional system with respect to charge transport. The valence band has
what appears to be similar dispersions in several directions, so requires more careful
analysis.

Since the inclusion of SOI does not cause as drastic a change in the band structure
as in B12T€3I11] (Chapter 4), the position and degeneracy of the calculated CBM
should be more reliable than in B12T83. However, several off-axis minima, whose en-
ergies lieabove the CBM by about 0.1 eV, will also contribute to the charge transport

and effectively increase the band degeneracy, thereby leading to a higher ZT value.

7.1.2. EFFECTIVE MASSES AND THERMOELECTRIC PROPERTIES OF
BaBiTe3

As has been discussed in previous chapters, the thermoelectric properties of a ma-
terial can be understood by studying the B parameter (Equation 24 in Chapter 2).
Increasing B, by way of a large band degeneracy, increases the figure of merit, ZT.
Unlike BigTeg, the lower symmetry of the more complicated unit cell of BaBiTe3 im-
plies that the degeneracy factor of the band extrema, 7, is reduced. Also, in BaBiTe3
the Bi / Te blocks are connected through Te5-Te6 atom chains. Orbital analysis of the
band structure shows that they contribute significantly to the CBM along with the
Bi atoms. Therefore, while electrons can flow through the layers, the Te5-Te6 chains
will block the holes from traveling through the Tel, Te2, and Te4 layers. It is there-
fore necessary to study the effective masses of the band extrema to understand how

anisotropic the charge transport will be for both the holes and the e1ectrons.[17, 18]

172

Since the mobility, p, is inversely proportional to the effective mass in the direction
of transport, in order to increase B (and therefore ZT), one would like to find systems
with small effective masses in the direction of charge transport to increase )1. and large
effective masses perpendicular to the direction of transport (Equation 24 in Chapter
2). The off—diagonal terms of the mass tensor (Equation 74 in Chapter 6) must vanish
due to the orthorhombic symmetry, so diagonal elements of the mass tensor, a,,-, give
the inverse of the effective mass parameters m,, (i = 1, 2, 3). These effective masses
for BaBiTe3 were obtained by calculating values of the energy close to different LCB
minima and HVB maxima while moving along suitably chosen directions in the BZ.
The 01,-,- values are given in Table XIV.

All the band extrema in BaBiTe3 lie along the F Z direction and hence are two-fold
degenerate. Large values of (1,,- correspond to small effective masses which give a
large mobility in that direction. A large mobility is found for the electrons in the XZ
plane, corresponding to charge transport through the Bi / Te layers. This is similar to
what is seen in B12T€3 and is consistent with the description of the Te5-Te6 chains
connecting the Bi / Te blocks and contributing to the LCB. The hole transport, on the
other hand, a large mobility is found in the XY plane, perpendicular to the Bi/ Te
layers. For the holes are unable to move through the Te5-Te6 chains, leading to their
charge transport perpendicular to the Bi / Te layers. However, the values of cm in the
X and Y directions are a factor of 2-3 smaller than those in the conduction band,

implying that the mobility of the electrons is better than that of the holes.

173

Table XIV. Theoretical values of inverse effective mass parameters associated

with the conduction and valence band extrema of BaBiTeg.

 

Conduction band rz
an = [m/me]; 11.61
am, : [m/me ,7; 2.79
on = [In/me g.) 13.16
Valence band FZ
an 2 [m/me]; 7.61
ayy = [m/meh‘é,1 5.40

z 2: [m/me ;:1 2.89

 

 

 

 

 

 

 

 

 

 

 

(1

~
~

 

In comparison with the effective masses of BlzTeg (Table XII), the first important
point is that these inverse effective mass parameters for BaBiTe3 are much more
symmetric. For the CBM, am for B12T93 is calculated to be 112.75me’l compared
to ayy of 2.87m; 1, a ratio of almost 40. The ratio in BaBiTe3 is only about 4.
Similarly for the VBM, a” for B12T€3 is calculated to be 107.51m;1 compared to ayy
of 3.97m; 1. a ratio of about 25, The ratio in BaBiTe3 is only about 1.5. Also, in
B12T€3 am, < an, the dispersion through the Bi/ Te network stronger than through
the Van der Waals gap for each extrema. However in BaBiTeg, am, > an for the
valence band, the hole transport blocked by the Te5-Te6 chains through the Bi/Te
network.[11]

The larger band gap, the more isotropic effective masses, and the reduction in the
mobility of the hole transport through the Bi/ Te network due to the Te5-Te6 chains
explains why BaBiTe3 is not as promising a thermoelectric material as B12T63, even

though the complex unit cell reduces Ii. These calculations suggest that attempts to

optimize BaBiTe3 should aim at preparing solid solutions of the type BaBi1_beITe3,

174

so that the Te sites are not disrupted. Given that the Te states are dominant near
the VBM, Sb substitution at the Bi sites should have minimal effect on the hole
mobilities. Any substitution, however, at the Te sites (e.g., Se) is expected to greatly
degrade the hole mobilities. On the other hand, attempts to optimize BaBiTe3 as an
n-type materialare expected to be more challenging because the CBM has substantial
contribution from Bi and Te (Te5-Te6) states. Therefore, any substitution of either

Bi or Te will likely have serious negative effects on electron mobilities.

7.2. CsBi4Te6
7.2.1. CRYSTAL STRUCTURE AND OBSERVED PROPERTIES

CsBi4Te5, like BaBiTeg, belongs to a class of compounds which are derived from
the BigTe3 crystal structure.[153] The unit cell of CsBi4Te6 is monoclinic with space
group 0.2/m containing 88iatoms/unit cell [A 2 97.425 a.u., B 2 8.264 a.u., C 2
27.424 a.u., (3 2 101.438°]. This unit cell is shown in Figure 76 where the different
atoms in the cell have been numbered. It has a layered anisotropic structure with
anionic [Bi4Te6] laths connected through Bi-Bi bridges to form two-dimensional slabs,
and these Bi / Te slabs are separated by layers of Cs+ ions. Bi-Bi bonds have not been
found in chalcogenide compounds before. The axis of the laths (normal to the paper)
is the direction of the highest charge 'mobility and is usually referred to as the needle
axis.[26, 155] There is a gross structural resemblance of this compound with BigTe3
and BaBiTeg where the Bi/Te slabs are separated by a Van der Waals gap and a

Ba++ ion layer respectively.[11]

175

.u.I. I.II.IIII. III.I.II.II II
.u.u ...-....y-.: ...-....:..:.w .I.

. . . . .o. .o.o
O 0.0 O OIIOOwo $03.0 I

a a, o o g :3 .05 ommhomao $2.3 omowflm».

a... ...-.- 3...- 55 5.3 3... ...-
II I.y
.I.I .I.II.II.II .III.II.II .I

 

 

. Monoclinic crystal structure of CsBi4Te6.

o o .o o o o o o o o .0

Figure 7

176

In order to reduce computer memory used in the electronic structure calculations,
_ we have used an equivalent triclinic unit cell (Space group: P-I) 'with 44 atoms/ unit
cell was used in our electronic structure calculation [A 2 49.233 a.u., B 2 49.233 a.u.,

C 2 27.424 a.u., a 2 101.438°, fl 2 101.438°, 7 2 9.693°]. This crystal structure and

its corresponding Brillouin zone are given in Figure 77.

Figure 77. (a) Triclinic crystal structure and (b) Brillouin zone of CsBi4Te5 used

in this calculation.

 

 

 

(a) A\
B O.
00.000 ' oOO'O.
Bi
0
oTeQ OO .
. CS. 00 O O
I
00000000 . 0.90.0
C

 

 

177

CsBi4Te6, as seen in Figure 78, has a larger figure of merit (ZT~0.8 at 225 Kelvin)
than conventional optimized Bi2_IbeTe3 alloys at these temperatures.[26] Doping
studies of this material shows that the hole-doped samples show a much larger ZT than
the electron-doped samples.[158] The thermal conductivities of these two materials
are comparable, which suggests that the improved ZT for CsBi4Te6 is electronic in
origin. Therefore, a careful analysis of the electronic structure of this material is

necessary to understand the origin of the high ZT for this system.

Figure 78. Comparison of ZT for the best current doping of CsBi4Te6 and
optimally-doped Bi2_1beTe3 alloy. The peak for CsBi4Te6
(ZT~0.8) occurs at 225 Kelvin.

1.0 1.0

 

0.8 0.8

D E]
. U .
CsBI4Tee E] El

[3 El

El
I
0.6 0.6

0.4 0.4

0.2 0.2

lTrllrIIITIIIIITlnl
El
lllllllllllllllllll

E]
O

 

 

IlllllllllllllILllJllJ4 00

0.0 I l l .
00 150 200 250 300 350

50

O

Temperature

178

7.2.2. GENERAL FEATURES OF THE BAND STRUCTURE AND GAP .
FORMATION OF CsBi4Te6

Electronic structure calculations show that CsBi4Te5 is a narrow-gap semiconduc-
tor with a band gap of approximately 0.04 eV (Figure 79), smaller than the measured
value of 0.05-0.11 eV [26] (usually seen in DFT calculations[31]). In the absence of SOI
(Figure 79a), the band gap is 0.37 eV. The introduction of SOI (Figure 79b) produces
an almost rigid shift of the conduction band toward the valence band. Similar shifts
are also seen in BaBiTe3 and B12T83[11], but in BizTe3 the shift is so large that new
hybridization gaps form, thereby giving completely different positions for the valence
and conduction band extrema. In CsBi4Te6, the top of the valence band occurs at
the F point while the bottom of the conduction band occurs at a general point in
the Brillouin zone (denoted as C‘) (C‘ 2. (0.88125, 0.88125, 0.175) in terms of the
reciprocal lattice vectors.) In addition, there are several local conduction band min-
ima appearing slightly above this point (~01 eV above), along F Z and RV directions
in the Brillouin zone, which may contribute to the charge transport. By looking at
the equation for the B parameter (Equation 24 of Chapter 2), where increasing B
increases the figure of merit, ZT, a large band degeneracy, '7’, can give large values of

ZT.]ll]

179

Figure 79. Band structure of CsBi4Te6 (a) without and (b) with SOI.

CsBi4Te6 - No S01

(21)

 

oVi ....

. . w:...H.\n... .....

A; fig

 

 

 

 

 

 

 

 

 

 

A>ov .Awaocm—

 

CSBl4T66 - SOI

(b)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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0.
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180

7.2.3. EFFECTIVE MASSES AND THERMOELECTRIC PROPERTIES OF
CSBI4T€5

The 0.04-0.11 eV band gap in CsBi4Te6 follows the 10k3T criterion for a good
thermoelectric[1, 2, 3, 4] because S is expected to peak at 225 K, though it had been
believed that only semimetals were good thermoelectrics below room temperature.]19]
While multiple conduction band extrema may be a reason for anticipating a large ZT
value in the electron-doped CsBi4Te6 (large thermopower values in n-doped systems
have not yet been seen[158]), the single maximum of the valence band at the F
point requires another explanation for the large observed values of ZT in the hole-
doped compound. In order to explore whether anisotropy in the charge transport
(confinement idea of Hicks & Dresselhaus[24]) may play a role, we have computed
the effective carrier masses along different directions and have analyzed the detailed
nature of the electronic states (orbital character and parentage, i.e, Bi p or Te p,
etc.) associated with the valence band maxima (holes) and conduction band minima
(electrons).

Considering the effective mass dependence of the B parameter (Equation 24 of
Chapter 2), we see that since the mobility, pm is inversely proportional to the effec-
tive mass in the direction of transport, B depends on the effective mass parameters
as m—Hm—T 2 Rm. For an isotropic system, R 2 1. For an anisotropic system we
can assume the smallest effective mass along the x direction, the direction of charge
transport, while one or both of the effective masses associated with the other direc-
tions are larger.. Considering two cases (i) m, 2 my 2 mz/w, (ii) In; 2 my/w 2 m: .c.
w > 1, we find that the ratio R is larger than the isotropic case by a factor x/J for (i)

and by a factor w for (ii). (When the three masses are different one has to generalize

181

this slightly.) Thus the effective mass anisotropy can significantly affect ZT.

In CsBi4Te6 the effective masses along three specific directions, i.e., the needle
growth direction (FX), the direction of the Bi/Te slabs (FV), and through the Cs+
atom layer (F Z) were computed. Since the system is triclinic, the full reciprocal mass
tensor must be computed and diagonalized (Equation 74).]22] In most cases, the off-
diagonal terms of the mass tensor are small, so diagonal elements of the mass tensor,
or“, accurately give the inverse of the effective mass parameters m,, (i 2 1, 2, 3).
In the general case, however, the mass tensor has to be diagonalized to obtain the
effective masses associated with the principal directions. We fit the calculated E(k)
vs k along different orthogonal directions to parabolas and obtained the coefficients
02'2”-

The effective masses for CsBi4Te5 were obtained by calculating values of the energy
close to different LCB minima and HVB maxima while moving along suitably chosen
directions in the BZ using the equation

(75a)

2me/h2 5k: amkf. + a,,, k5 + aukg + 2a,,yk1ky + 2ayzkykz + mum,

where a constant term has been omitted. The three orthogonal directions lying
closest to] the three directions mentioned above will be denoted as X, Y, and Z,
respectively. The effective masses obtained from the eigenvalues of the inverse mass
tensor are given in Table XV. Due to the observed large number of hybridized bands
near Ep, the band extrema rapidly become nonparabolic, so that the effective mass
approximation is not expected to hold at large carrier concentrations. However, up
to carrier concentrations consistent with the optimum doping of this system (~3 x

1018/cm3[26, 158]), the agreement between the calculated dispersions and parabolic

182

fit was found to be excellent.

Table XV. Effective masses of CsBi4Te6 associated with the valence and
conduction band extrema along three principal axes and the angles

these axes make with a set of three suitably chosen orthogonal axes

 

 

 

 

 

 

 

 

 

(see text).

Valence Band (F) Angle Ina/me
X (needle) 12.37° 0.02

Y (Bi-Bi bonds) 0.97° 0.16
Z (Cs layer) 12.35° 0.09

Conduction Band (C’)

X (needle) 12.38° 0.04

Y (Bi-Bi bonds) 3.17° 0.47
Z (Cs layer) 11.98° 0.16

 

 

 

 

 

The corresponding eigenvectors define different directions in k-space. For both the
electrons and the holes, the lightest effective mass direction makes an angle of about
12° with the needle axis while that with the intermediate mass direction also makes
an angle of 12° with the axis perpendicular to the Cs+ atom layers. The direction of
the heaviest carriers lie within 1-3° of the Y axis, i.e., parallel to the Bi-Bi bonds. The
heaviest valence band mass is a factor of 50 larger than the lightest valence band mass
while the heaviest conduction band mass is only a factor of 10 larger than the lightest
conduction band mass. Thus the valence band holes are more spatially restricted
than the electrons, particularly along the direction of the Bi-Bi bonds. These results
suggest that the effective hole transport takes place not along the Bi/ Te slabs but

in planes nearly perpendicular to the slabs containing the needle axis. A similar

183

situation was found in BaBiTeg, but the degree of anisotropy was much smaller.[11]
Also the effective masses in BaBiTe3 were a factor of 5-6 higher leading to a lower
overall mobility (lower conductivity). The charge transport in B12T€3 is generally
along the layers with much less dispersion through the Van der Waals gap while a
much stronger charge transport is seen in hole doped CsBi4Te5, perpendicular to the
Cs+ atom layer and in the needle growth direction, but not along the Bi / Te network.
The effective masses in the transpOrt directions are, however, comparable to those of
CsBi4Te6.[11]

To understand the nature of the electron and hole states near the respective band
extrema further we have analyzed the wave functions in some detail. When the
strengths of the orbital character associated with these states are analyzed for dif-
ferent atoms, it is seen that the valence band charge density forms a quasi-two di-
mensional layer lying almost perpendicular to the [Bi4Te6] slabs while the conduction
band charge density is more three—dimensional, consistent with our effective mass
calculations. We find that the states corresponding to the CBM are mostly of Bi
character, especially Bi7 and Bi8, the atoms forming the Bi-Bi bond which connects
the Bi/Te laths together, a bonding not before seen in chalcogenides. These arise
from the anti-bonding state of the Bi-Bi pair, the bonding state lying below the va-
lence band maxima. This may be understood in terms of the stabilization of the
donor electron from Cs by this Bi-Bi bond which is necessary for the formation of a
semiconductor.[26] (The Cs atoms act primarily as electron donors with little contri-
bution near the top of the valence band or the bottom of the conduction band).

We find the states corresponding to the valence band maxima consist of Bi (Bil,
B16) and Te (Te1, Te3, Te6, Te8) (Figure 76) atoms, the charge density associated with

these atoms forms corrugated, two-dimensional sheets separated by approximately 13

184

O
A (Figure 80). There are very few atoms lying between the sheets with contributions

to these states (which would have allowed for transport in the direction of the Bi-Bi
bonds), consistent with the large effective mass found in this direction. The Bi-Bi
bonds, connecting the Bi/ Te network together, have their anti-bonding states in the
conduction band and their bonding states deep in the valence band with only a small
contribution to the states near the valence band maxima. This implies that hole
transport through the Bi-Bi bonds will be blocked in this direction, explaining the
large effective masses and two-dimensional sheet-like transport.[l5]

Greanya et al.[159] recently performed an ARPES study of CsBi4Te6 to find the
CBM at the F point, in agreement with our calculation. While they found both dis-
persions along FV and F Z to be much flatter than that in the FX direction, they were
only able to obtain a few experimental points in the two nondispersive directions due
to the needle-like morphology. They were also able to estimate a band gap of 0.05 eV,
consistent with our calculation[15] and the previous experiments[26]. These results
give further confidence in our analysis of the electronic structure of this complex novel

material.

185

Figure 80. Quasi-two dimensional sheets contributing to hole transport

overlying the crystal structure of CsBi4Te5.

   

O O

t .
'0!»
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(A.

 

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186

 

To summarize our results for CsBi4Te6, systems with different structures have been
synthesized to exploit low dimensional transport which result in enhanced ZT.]27, 28,
160] The vast majority of these attempts, however, revolve around nanofabrication
of quantum wells or multilayer films. CsBi4Te6, on the other hand, quite naturally
displays quasi-two dimensional hole transport with regions of the crystal of several
A width participating very little in the charge transport. These “insulating” regions
are sandwiched between conducting regions, leading to structures similar to quantum
wells or multilayers, but in a _b_ullg material. The source of this anisotrOpic hole trans-
port arises from the Bi-Bi bonds, which are not seen in other bismuth chalcogenides.
It is possible to take advantage of this novel quantum architecture in other Cs—Bi-Te
compounds where the Bi-Bi bonds are closer together or further apart. While the
conduction band has several degenerate minima, the transport in this band is much
more isotropic. However, large values of ZT are still possible for the electron-doped
systems when several of these local minima get occupied. BaBiTe3, which has a strong
structural similarity to CsBi4Te5, does not show as large an anisotropy as the latter
compound. Furthermore the effective mass along the optimum direction (needle axis)
is a factor of 10 larger in BaBiTe3 than in CsBi4Te5. This may explainthe low ZT

values observed in BaBiTe3.[ll]
7H3 K213188813

The third member of the complex Bi-Te—Se systems which has great potential
for high thermoelectric performance is fl—KgBigSelgll6]. K2BlgS€13 forms in two dis-
tinct crystal structures, O-K2B138913 and fi-KzBigSem, both of which are stable phases

within a limited temperature window. These two systems are formed of similar build-

187

ing blocks, BigTe3—, NaCl-, Cd12—, and Sngeg-type rod fragments, which arrange in
the same stoichiometrybut with different architecture[16]. The inclusion of alkali
atoms stabilizes the covalent bonding in the Bi-chalcogen framework with cages. The
K atoms occupy the positions inside these cages as K+ ions and can act as “rattlers”[5],
consistent with the concept of an electron crystal/ phonon glass.[6]

In order to determine the figure of merit, ZT 2 SazT/n, in these compounds,
S, a, and K. have been measured as a function of temperature. The highest room
temperature (RT) values for a in a4KgBigSe13 [is 2 S/cm compared to fi-KzBigSelg
with 250 S/cm.[16] The largest RT value of S for both a- and ,B-K2B188813 is —200
pV/ K, which means that the carriers are electrons. The thermoelectric properties
of [3-K2Bi8Se13 are believed to be enhanced by the disordered Bi-K sites within the
lattice which reduces the phonon thermal conductivity (n); a value of 1.28 W/m'K
has been seen compared to 1.31 W / m'K for B12T€3 alloys. This yields a value of 0.22
for ZT in unoptimized fi-KgBigSe13.[l6] Values of n for a-K2BigSe13, with no disorder,

have not been measured due to the low a values.[161]
7W3]. Q-KgBlgselg,

The crystal structure of a-KgBigSelg is triclinic with space group P-I [A 2 26.108
a.u., B 2 22.858 a.u., C 2 7.872 a.u., a 2 89.98°, fl 2 98.64°, '7 2 87.96°][16]. This
crystal structure is given in Figure 81 with its corresponding Brillouin zone in Figure

82.

188

 

Figure 81. Projection of the crystal structure of a-K2B138813 viewed down the

b-axis. BigTe3-, Cd12-, and szseg-type building blocks in the

structure are highlighted by the shaded areas.

 

 

 

 

 

 

 

189

Figure 82. Brillouin zone for a-K2B188813.

k3

 

The band structure of a-KzBi38e13 with and without SOI is given in Figure 83.
The effect of SOI is to shift the conduction band down relative to the valence band
and thereby decrease the gap values. Both with and without SOI, a semiconductor
forms with the VBM at the F point and the CBM at the X point. When SOI is
included the calculated band gap is 0.47 eV (Figure 83b) compared to the measured
value of 0.76 eV.[161] There are however direct gaps at the X point of 0.74 eV and at
the V point of 0.72 eV. V i

190

Figure 83. Band structure of a-KgBigSelg (a) with and (b) without SOI.

(a);0 K2Bi88613- No SO

     
 
 
    

  

 
  

 

 

 

 

 

 

 

 

 

 

 

psi/’15:?
> 1.0 /
<1)
v
is
330.0 EF
:5
3?“ / t»: \
'1.0 x? F‘. .' A". .“\\
ref 2-2 .123: .. ”In.
7’: ngf --g' I," "Q! eff-‘2:
zen .12222fid {ex
-2 . ham»: Met-aim. 4
F X U Z R F Z

“”20 K2Bi8$el 3 - ' SOI

 

'o

Eneggy (eV)

l
....A
I

The low a and high S are consistent with a wide-gap semiconductor found in this
system. The thermoelectric properties in this material may also be influenced by
anisotrOpic effective masses, which can lead to a high ZT.]24, 11] The effective masses
of the band extrema in a-KgBi38e13 were calculated assuming that the angles a and
'7 are 90°. In this approximation, the effective masses were obtained by calculating

values of the energy close to different LCB minima and HVB maxima while moving

191

along suitably chosen directions in the BZ. These values were then fitted using the
equation
(75b)

2 _ .2 . .
2me/h 5k — amkx + ayy k; ‘t‘ azzkg + Zarzkxkz,

where a constant term has been omitted. Generally, the off-diagonal terms are
small so the reciprocal of the diagonal terms, 0,, (i 2 1, 2, 3), 'can be considered
the effective mass parameters. As can be seen (Table XVI) the effective masses
of the valence band maxima are fairly isotropic. We find that the SngB3 fragments
connecting the Cdlz layers together, consisting of Sel-7 and hybridized Bi4 p orbitals,
having a strong contribution to the HV B. On the other hand, these Sb28e3 fragments
do not contribute strongly to the LCB, which consist mainly of Bil-3 p orbitals. This
leads to a more 2D transport of the electrons within the CdIZ layers, similar to what
is seen in the layers of BlgT€3.[11, 59] This suggests better thermoelectric properties

from the electron-doped system compared to the hole-dOped system.

Table XVI. Components of the reciprocal effective mass tensor for CY-KgBIgSQm

 

mv/me mc/me

a” 2 [m/me];1 1.05 5.56

1:

ayy 2 [m/me ,7; 1.44 1.09

ozzz 2 [m/me g; 1.19 7.69
01M 2 [m/me],;,1 0.71 2.94

 

 

 

 

 

 

 

 

 

 

192

7...?)2 fi-KgBlgsem

While having the same stoichiometry as a-KgBigSelg, x3-K2Bi38e13 is isostructural
to K2B18813[162] in space group P21/m and is formed of BigTe3-, NaCl-, and Cdlg-
type infinite rod-shaped blocks to form a much more 3-D structure.[l6] The a form
is slightly less dense than the 6 form, because in the a form the Bi atoms are in
octahedral or higher coordination geometry compared to the ,6 form where the Bi
atoms are found in a trigonal-pyramidal geometry. The B12T€3 and Cdlg rods in the
6 form are arranged side-by-side to form layers perpendicular to the c-axis where the
NaCl units connect the layers to build a 3-D framework with "tunnels filled with K+
ions[l6]

The monoclinic crystal structure of ,B-KgBlgsem is closely related to that of Sr4BiGSe13,
i.e., one replaces two Sr2+ atoms with two K+ atoms and the remaining two Sr2+ atoms
with Bi3+ atoms[163]. This leads to a mixed occupancy of Bi and K atoms within the
crystal structure. The K2 position is fully occupied by K+ atoms and Bil-7 are fully
occupied by Bi3+atoms. However, the site K1 / B9 has 62% K and 38% Bi occupancy
while the K3/Bi8 site has 62% Bi and 38% K occupancy.[16] A very similar type
of disorder is found in the isostructural compound K2B13813, but in this compound
the disorder is between K1/Bi9, K3/Bi8, and K2.[162] The K1 /1319 site has three
bonds ranging from 2.85 to 3.07 A and five longer ones from 3.44 to 3.70 A, while the
K3/Bi8 site has three bonds from 2.84 to 3.04 A and five longer ones from 3.31 to
3.68 A.[16] However, the two sites are not. equivalent to each other. The Bi8/ K3 site
lies at the corner of one of the NaCl blocks while the Bi9/K1 site lies along the side
of the block next to Se9 which lies on the corner. Electronic structure calculations of

fi-KgBigSelg should in principle incorporate the Bi9/K1 and Bi8/ K3 disorder. The

193

.5:

, C, 2 34.804 a.u.

cture [A 2 33.056 a.u., B 2 7.914 a.u.

clinic crystal stru
90.49°] is given in

mono

Figure 85.

ponding Brillouin zone in

Figure 84 and the corres

Of ,B-KgBlgselg .

Figure 84. Crystal structure

CdIz-type

 

 

Figure 85. Brillouin zone of fi-KzBigSew.

k7.

 

 

 

 

Our electronic structure program[55] cannot treat sites as partially occupied, such
as in the virtual crystal approximation[164]. Therefore, we will perform preliminary
band structure studies to understand the effects of disorder. Different configurations
will be studied with the extreme occupancy of the K1/Bi9 and K3/Bi8 sites. As
already mentioned, the site K1/B9 has 62% K and 38% Bi occupancy while the
K3 / Bi8 site has 62% Bi and 38% K occupancy.[16] The first arrangement, which will
be called Configuration 1 will place K at the K1/Bi9 site and Bi at the K3/Bi8 site.
Configuration 2 will have these positions reversed, Bi at the K1/Bi9 site and K at
the K3/Bi8 site.

Configuration 1 seems closest to the structure seen experimentally.[16] Both before
and after the inclusion of SOI, Configuration 1 yields a semi-metallic or metallic
behavior (Figure 86) with some very flat bands along FY and ABF, in the plane

perpendicular to the needle axis (defined in Figure 84).

195

Figure 86. Band structure of fi-KgBigSelg - Configuration 1 (a) without and

(b) with SOI.

(a) K2Bi8$el3 - No SOI

  
 

 

(b) KZBi8Sel - SOI
1.5
91
3
0.5

 
   
   

These flat regions existing at Ep ar'e inccinsistent with the measured band gap of
0.59 eV.[16] An orbital analysis of these bands shows that they correspond to Bi8 p
and Se9 p states. (F igure 87) Se9 lies on.th‘e edge of the NaCl block next to the K
atom in the disordered Bi9/K1 site. Also, the other bands crossing Ep correspond to

the Bi atom in the Bi8/K3 site. It appears that the Se9-K1 bonding is too weak to

196

form a bond so that the Se9 orbitals float to Ep to form a semi-metallic or metallic

behavior.

Figure 87. Orbital character of (a) Bi8 and (b) Se9 overlying the
band structure of ,B-KQBlgselg - Configuration 1 with SOI.
The size of the circles overlying the band structure is
directly proportional to the strength of the orbital character

(a) KZBi8Sel 3 Bi8 tot

Energy (eV)

 

(b) 2 o K2Bi8$el3 Se9 tot

 

 

 

 

 

 

 

 

 

_\
l.5 "§\
\——a
10 /<>l\
’ Ir"
E 0.5
>1
an 00 EF
2
LU ‘0" —
-l.
ll
-2.
r Y C Z D E A B r Z

Configuration 2 has Bi is in the Bi9/K1 site and K is in the Bi8 / K3 site. The total

197

energy of this configuration is 1.49 eV / formula unit lower than that of Configuration
1. Before the inclusion of SOI, there is a band gap 0.38 eV, the V BM along the very
flat bands along ABF and FY and the CBM along AB. While there is a band gap, the
HVB is unusually flat, so that the bonding is very weak in some directions. After the
inclusion of SOI, the band gap disappears and the band structure of Configuration 2

looks very similar to that of Configuration 1.

Figure 88. Band structure of fil-KzBiBSelg,~ - Configuration 2 (a) without and
(b) with SOI.

(a) K2Bi88e13 - No SOI

     
  
   

.(b) ' K2Bi8$e13 — SOI
1.5
91
:3

0.5

198

Figure 89. Band structure of B-KgBiSSelg - Configuration 3 (a) without and
(b) with SOI.

(a) ~K2Bi8$el3 - No SOI
2.0

 

 

 

 

 

 

 

 

 

 

 

 

 

(b) K2Bi8Sel3 - SOI
2.0

 

 

 

 

 

 

 

 

 

 

’\
1.5 /:t
A ”—9)—
?) 1.0 ...»or—
30.5
5 0.0 ~ EF
{50-5 \
- l .0
- 1.5
-2.0
F Y C Z DE AB F Z

A third configuration has been tried which differs from Configuration 2 by ex-
changing Bi on the Big/K1 site with K on the K2 site. This type of disorder has been
seen in K2Bl8813 but not in fi-KgBiSSelg. This configuration has a slightly higher
energy than Configuration 2, by 0.03 eV/formula unit. Before the inclusion of SOI.

the CBM is at the A point while the VBM is between A and B to produce a gap of

0.35 eV. After the inclusion of SOI, the band gap decreases to 0.04 eV, smaller than
the experimental value of 0.59 eV[16]. The HVB is much more dispersive along ABP
and F Y meaning the bonding along these directions is stronger.

While the Bi-K disorder in fi-KzBiSSem has been suspected to reduce. r416], this
disorder also has a profound effect On the band structure near Ep. We have found
either narrow-gap semiconducting or semi-metallic behavior depending on the position
of the atoms on the disordered sites on the lattice and the calculated gap is much
smaller than the meaSured value of 0.59 eV[16]. In order to better understand the
band structure of this material, the band gap should .be measured as a function of
temperature to see if the low temperature value differs from the room temperature
value (as in the lead salts[101]). Also, though the virtual crystal approXimation[164]
generally reduces the band gaps in semiconductors[59], such a calculation should be
performed since the ratio of Bi/ K atoms in the disordered sites is likely to have a

profound effect on the band structure.

200

CHAPTER 8.
CONCLUSIONS

In this dissertation we have investigated the electronic structure of a wide variety
of systems including an extensive study of the half-Heusler compounds and simple
chalcogenides along with a detailed study of more complicated structures such as
Zintl phases and complex chalcogenides. We have investigated the subtle properties
of the gap formation in these semiconductors and used these results to understand
their thermoelectric properties.

The focus of this dissertation is two-fold. First, we have used our calculations to
understand which properties in the band structure are the best for thermoelectric
properties and which materials display these properties. We have found that systems
with highly anisotropic effective masses (though nOt large values) and multiply de-
generate band extrema have the best thermoelectric properties, as predicted by the
B parameter (Equation 24 in Chapter 2). The anisotropic effective masses seem even
more important as they may describe reduced dimensionality in the charge transport
in bulk materials which was found important in CsBi4Te6. [15] These are valuable
criteria for finding new thermoelectric materials using band structure calculations.

Our analysis of a number of systems explained why certain materials were bet-
ter thermoelectrics than others. We found that the six-fold degenerate VBM and
CBM in Bl2T€3 and Sb2T€3 explain their improved thermoelectric performance com-
pared to Bigseg with CBM and VBM at the I‘ point. Also, we found that the highly
anisotropic masses in Bl2T83 and CsBi4Te5 enhance their thermoelectric properties
While the more isotropic effective masses in BaBiTe3 lead to a less promising thermo-

electric material. We have also predicted that the anisotropy in the CBM effective

201

masses in the half-Heusler compounds would make the n-doped systems better for
thermoelectric applications than the p—doped systems.

The second focus of this dissertation is to understand the gap formation of these
narrow-gapsemiconductors. Most of these semiconductors are formed from hybridiza-
tion of different bands which lead to their small gaps. After the presentation of the
numerous band structures which have been given in this dissertation, it is important
to give a short critique of the methods we have used and how reliably we can trust
these results. These include ground state properties, such as the total energies and
band dispersions, and excited state properties, such as the band gaps.[30, 41]

The structural properties have been shown to be found correctly in DF T in many
systems, such as in BCC Fe where GGA correctly finds this has a lower energy than
FCC Fe while LDA did not.[45, 46] Our calculations found that in half-Heusler systems
containing Ni, disorder of the Ni sublattice is energetically favorable to disorder of
the NaCl substructure[11]. Experimentally one doesn’t see Ni sublattice disorder
at low T, instead higher energy defects are seen in the NaCl substructure (Zr-Sn
disorder in ZrNiSn)[78]. Further theoretical and experimental study of disorder in
the half-Heusler compounds is necessary to understand this result.

Relaxation of the lattice parameters within the electronic structure calculation also
is valuable for obtaining reliable results. DF T is in general exact, but LDA/GGA
are only approximate since the KC potential is treated approximatelyl3l]. Therefore.
a model crystal structure, such as relaxing the lattice parameters, often gives band
gaps and total energies in better agreement with experiment. In PbS, PbSe. and
PbTe, the lattice relaxation nearly reproduced the experimental band gaps[115]. The
effects of lattice relaxations were not as profound in the other simple chalcogenides

or half-Heusler systems which we have tried, while lattice relaxation on large systems

202

becomes nearly impossible due to the number of configurations and time necessary
to find the minimum energy structure. We strongly believe that lattice relaxation is
necessary to obtain an accurate band structure.

The band dispersions (effective masses) are also ground state properties within
DF T, so these should be predicted correctly because k is a good quantum number.[30,

. . -1 _ __ -2 (92E .
41] The effect1ve mass tensor 13 defined as [m],j — aij — h [81:61: , ].[22] In most cases
1 J

 

when the correct orthogonal directions in the crystal are chosen, the off-diagonal terms
of the mass tensor are small, so diagonal elements of the mass tensor, afi, accurately
give the inverse of the effective mass parameters mg (i = 1, 2, 3). The effective masses
of the semiconductors GaAs, InAs, and AlAs have been studied extensively both ex—
perimentally and theoretically. A good agreement is found between the most of the
calculated values and the measured values[165] which confirms the belief that DFT
calculations do a good job in measuring these ground-state properties. The large cal-
culated effective masses of the half-Heusler compound ZrNiSn[7] agrees well with the
measured value.[81] Even in the case where the bands are known to be nonparabolic,
such as in BigTe3[11] and the lead salts[122], reasonable agreement can be found
between theory and experiment. Since we found that anisotropic effective masses
were very important for high values of ZT, finding systems With highly anisotropic
effective masses is an effective use of electronic structure calculations in finding new
thermoelectric materials.

One of the most well known problems in LDA/GGA is that semiconducting band
gaps are usually found to be smaller than experiment. This is not surprising since
the Kohn-Sham energies are related to the band gap by Egap = Arc + E N, N+1 - EN‘N
where A“ 2: €N+1.N+1 - eN‘N is the shift of the exchange-correlation energy when

adding an electron.[31] There is no way to estimate A“, so the LDA/GGA band

203

 

LDA/GGA

gaps which have been reported are simply Egap

= 8N,N+1 - EN‘N. The band
gap is actually an excited state property because both I and A in Egap : I - A are
both excited state properties. Therefore, it is not surprising that the band gaps
in LDA/GGA do not agree with experiment. In most group IV, III-V, and II-V'I
semiconductors, such as Si, Ge, InSb, GaAs, etc., the predicted band gap is smaller
than what is seen experimentally.[Bl, 57, 58] The CW method explicitly adds and
removes an electron to compensate for this problem and gives much better agreement
between the theoretical and measured values for, the band gaps.[31]

The semiconductors discussed above are either strongly covalent or strongly ionic
compounds. In contrast to these compounds, we find that the band gaps in BaBiTe3[11],
Bl2T€3l11, 40, 130], and CsBi4Te5[15] agree very well-with the experimental values.
In the best thermoelectric materials which we have studied, the effects of SOI are
very important. This may be a serious problem since LDA/GGA calculations within
the scalar-relativistic method using second-variational treatment of SOI have been
linked to poor prediction of the positions of the bands compared to full relativis-
tic calculations for the 6p metallic elements Hg, Tl, Bi, and Pb.[36] However, the
second-variational treatment of SOI has found good agreement in PbTe[37, 38, 39]
and Bi2T€3.[11, 12, 13, 40] Is it possible that there is a happy cancellation taking
place between these two approximations of SOI and LDA/GGA which leads to a bet-
ter than expected agreement between experiment and theory? This point needs to
be investigated further using fully-relativistic DFT calculations.[34]

The band gaps of some semiconductors are overestimated in LDA/GGA calcu-
lations. This is seen in the half-Heusler compounds, such as ZrNiSn With 0.50 e\'
compared to the measured value of 0.12 eV[7, 59]. This is difficult to reconcile with

Equations 57 and 58 since A“ should be positive and GW calculations give larger

204

band gaps than LDA. Therefore, another explanations must be introduced. In Zr-
N iSn, it is claimed that virtual crystal calculations predict that 15% Zr—Sn disorder
leads to a closing of the band gap in this material.[59] However, TiNiSn has the same
structure and nearly the same predicted and measured band gaps as ZrNiSn. How-
ever, while anti-site disorder has been observed in ZrNiSn, no such disorder has been
observed in TiNiSn, which makes the argument difficult to accept at best.[8, 66]

The above discrepancy is more evident in the Zintl-phase compounds where the
gap formation is similar to that in the half-Heusler compounds. The calculated band
gaps for the binary parent compound T h3Sb4 by Takegahara et al.[10] gives a negative
gap compared to the experimental value of 0.48 eV. The calculated value of the gap
for the ternary Tthi3Sb4 is 0.36 eV compared to the experimental value of 0.07
eV.[IOI Since disorder on the parent binary compounds substructure is expected to
reduce the gap in the ternary compound, the discrepancy of finding a smaller band
gap for the binary compound and a larger value for the ternary compound becomes
harder to fathom. We suggest another possible mechanism discussed below.

It is well known that LDA/GGA. calculations place the d orbitals of transition
metals too high in energy while other methods, namely Hartree-Fock, place these
d orbitals too low in energy.[97] As explained in greater detail in Chapter 5, shift-
ing of the d levels down reduces the band gaps while moving them closer to Ep
increases the gaps.) The Co—containing half-Heusler compounds all had larger gaps
than the Au-containing half-Heusler compounds. The lack of disorder in TiNiSn and
the discrepancy between the gaps in the ternary Zintl phases and in their binary
parent compounds make this a more believable explanation than reduction of the
gaps through disorder. This should be studied further using GW[31] calculations and
LSDA+U[73] within DFT calculations which better handle narrow d- and f-bands.

205

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