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'2 307

PhD.

This is to certify that the
dissertation entitled

LOCAL STRUCTURE STUDY OF NEW
THERMOELECTRIC MATERIALS

presented by

HE LIN

has been accepted towards fulfillment
of the requirements for the

degree in Physics

 

 

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42/5; 56

Date

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LIBRARY
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2/05 p:/ClRC/DateDue.indd-p.‘l

LOCAL STRUCTURE STUDY OF NEW THERMOELECTRIC MATERIALS
By
He Lin

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy
2006

ABSTRACT

LOCAL STRUCTURE STUDY OF NEW’ THERMOELECTRIC MATERIALS
By
He Lin

The atomic. pair distribution function (PDF) technique is used to study the local
structure of new thermoelectric materials. The PDF is obtained via F01.1rier transfor-
mation of powder total scattering data including the important local structural in-
formation in the. diffuse scattering intensities underneath. and in—between. the Bragg
peaks. Having long been used to study liquids and amorphous materials, the PDF
technique has been recently successfully applied to highly crystalline materials owing
to the advances in modern X-ray and neutron sources and computing power.

Devices based on the theri‘noelectric effect hold promise for solid—state cooling and
power generation but developments in materials properties are required for widespread
application. Recently, the importance of naiiometer-scale structures has been recog-
nized for improving the thermoelectric properties of materials. In this work we couple
the power of the PDF method for giving nano—scale structural information with the
need to characterize nanostructure in a number of promising novel thermoelectric
materials.

Promising Nanoclusters are found to exist in the PbTe based materials with
unusual thermoelectric properties. For two series of material AngmeTem+2 and
(PbTe)1_,,(PbS)J1C we verify the bulk nature of such nanoclusters and also determine
the average chemical composition and nature of these nanoclusters. Traces of nan-
oclusters are also found in Ag1_ISnSb1+ITe-3 series materials.

This information is important in the effort to understand the relationship between

the existence of nanoclusters and the exceptional thermoelectric properties.

To my parents, brother, cousinry, friends, and all those who once

helped me To my family, friends, and all those who once helped me

iii

Acknowledgments

When this moment come. all that happened in the last six and half years come
to my mind—excitemeiits in research. trivial things in life. But first of all that stands
out is this grateful feeling towards these special people. I need to thank them all, but
the words are just so limited now.

Above all. I need to express my great. appreciation to my advism‘, Professor
Simon Billinge. Such unlimited understanding and patience— he has been the source
of strength for me. Simon’s optimistic, positive attitude of life alwz-iys cheers me up
and gives me the confidence in any case. His deep insight in science and great skills
in both research and people have. made study and work an thorough enjoyment. No
spurious words to thank Simon for the tremendous help he once gave me.

I am grateful to my thesis committee members, S. D. Mahnti, Chong-Yu Ruan,
C.-P. Yuan, M. G. Kanatzidis. My highest respect to all of them. My Ph.D. research
benefits a lot from collaborations with Eric Quarez, J. Androulakis, Hoang Khang.
Our cooperations have been extremely efficient and smooth due to mutual interest in
science.

I need to thank all of current and former post-doctors and students in our gr01.1p._
My research will not succeed without their help. Their unselfish spirit and contri-
bution make our research work smooth and fast. Here I need to list them all: Emil
Bozin, Pavol Juhas, Gianluca Paglia, Ahmad Masadeh, Hyunjeong Kim, Moneeb
Taiseer Shatnawi, Mouath G. Shatnawi and Xiangyun Qiu. All of my success and
happiness also belong to you guys.

I would like to thank Debbie Simmons and Cathy Cords who make my study
here much easier with their considerate help. I also acknowledge help from Didier
Wermeille, Doug Robinson for help with X-ray experiments.

My grandparents, my parents. my brother, my cousins, all of my other family

members, I can not achieve this without your support. This time belongs to you.

iv

The work at MSU l:)enefited from support from National Science Foundation
through NIRT grant DMR-0304391. X-ray data were collected at the GIDD beam-line
Advanced Photon Source (APS). Use of the APS is supported by the US. DOE under
Contract No. VV-3l-109-Eng-38. The MUCAT sector at the APS is supported by the
US. DOE under Contract No. VV~74()5-E11g~82.

Contents

1 Introduction

1.1 Physics of thermoelectric materials ...................
1.2 Chronology of TE effect (.liscoveries ...................
1.2.1 The Seebeck effect. ........................
1.2.2 The Peltier effect .........................
1.2.3 The Thomson effect ........................
1.2.4 The Kelvin relationship .....................
1.3 Engineering aspects of TE physics .................. g . .
1.3.1 Simple overview of the Seebeck effect ..............
1.3.2 TE power generation .......................
1.3.3 Utilizing Peltier cooling .....................
1.3.4 Using Thomson effect to predict Seebeck coefficient . . . . . .
1.4 Evolution of TE materials ........................
1.4.1 Metals ...............................
1.4.2 Semiconductors ..........................
1.4.3 Pioneer work of A. F. Ioffe ....................
1.4.4 Further developments and current efforts ............
1.5 Engineering of novel TE materials - principles and examples .....
1.6 Thesis layout ...............................

The Pair Distribution Function Method

2.1 Importance of the local structure ....................

2.2 The atomic pair distribution function (PDF) technique ........
2.2.1 History and current status of the PDF method .........
2.2.2 Definition of the PDF and its variants .............

2.3 Rapid acquisition PDF experiments and PDF data analysis . . . . . .
2.3.1 Utilizing image plate detector for rapid data acquisition . . . .
2.3.2 The RA-PDF experimental procedure ..............
2.3.3 Data processing using FIT2D and PDFgetX2 .........
2.3.4 Structural refinement using the PDFF IT program .......
2.3.5 Handling the multiphase problem ................

3.1 Introduction ................................
3.1.1 Background of AngmeTe,,,+2 system .............
3.2 iVIotivation .................................

vi

cocooo-qqmeeecowwmi—w-a

£9

10
10
14

16

16
17
18
19
23
23
24
25
29
30

3 Study of the structure of new thermoelectric material AngmeTem+2 32

32
32
33

3.3 Experimental details ........................... 34

3.3.1 Sample preparation ........................ 34

3.3.2 High energy x-ray diffraction experiments ........... 34

3.3.3 Mtxleling ............................. 38

3.4 Results ................................... 43
3.5 Discussion ................................. 47
3.6 Surmnary ................................. 49

4 Thermoelectric material Ag1_xSnSb1+ITe3 50
4.1 Interesting physics in the Ag1_xSnSb1+xTe3 system .......... 50
4.2 PDF study of the Ag1-xSnSb1+ITe3 system .............. 51
4.2.1 Introduction ............................ 51

4.2.2 Experimental Details ....................... 53

5 Thermoelectric PbTe/PbS system 62
5.1 Physics of the (PbTe)1_z(PbS)1. system ................. 62
5.2 Experimental methods .......................... 66
5.3 Modeling .................................. 68
5.4 Results ................................... 70
5.5 Summary ................................. 81

6 Concluding Remarks 85
6.1 PDF analysis of new thermoelectric materials ............. 85
6.2 Alternative approaches and future work ................. 86
6.2.1 Alternative approaches ...................... 86

6.2.2 Future Work ............................ 88

6.2.3 Obtaining PDFs using other source and detector ........ 88
Bibliography 90

vii

List of Figures

1.1
1.2
1.3
1.4

1.5
1.6

1.7
1.8

2.1
2.2

2.3
2.4

3.1

3.2

3.3

3.4

Seebeck effect. ...............................
Thermoelectric power generation ....................
Peltier Cooling ..............................
Seebeck coefficient, electrical conductivity, and power factor vs. carrier
concentration ...............................
Thermal conductivity decrease in Skutterudites by the introduction of
various scattering mechanisms ......................
ZT for various p—type thermoelectric materials .............
ZT for n-type thermoelectric materials .................
ZT for various PbTe based thermoelectric materials ..........

Experimental setup for the RA-PDF experiment. ...........
Two dimensional contour plot of Ni data from the Mar345 Image Plate
Detector ................................. ,.
Three panels figure, I (Q), F (Q), and C(r) of an analyzed Ni data.

Structural refinement of Ni data using the PDFFIT program .....

(a) The raw data diffraction pattern observed on the image plate. (b)
F (Q) and (c) C(r) for the AgOBGPblngTego sample. In the Fourier
transform, Qmax was set to 26.5 A“. .................
0(7) and DG(1‘) (compared to PbTe) for samples with different m
value. Magenta curve is for PbTe, blue curves are for sample
Ag0,85Pblgste-20, green for sample AngmeTeH, red for
AngngTeg. ...............................
The unit cells for different models are shown here. (a) is the PbTe
major phase. In all plots Te is shown as red atoms and Pb as green.
(b) Chemically disordered AngTe2 in N001 and chemically disordered
Anggste4 in N C20. (c) Chemically ordered AngTe2 in N COI and
partially chemically disordered AngQSbTe4 in N C21. ((1) Chemically
ordered AngngTe4 in model N C22 resulting in 2-fold supercell along
one crystal axis. In all models the Te (red) sublattice is not changed.
(a) PDF from the homogeneous H model for sample AngsSbTeg .
The line with empty circles is the data, the solid line is the calcu-
lated curve from the fitting and the line offset below is their difference.
(b) Chemically disordered case of model N C20 for AngGSbTeg. Line

attributions are the same as in (a). ...................

viii

11
12
12

13
15

25

26
27
28

36

37

41

44

3.5

4.1

4.2
4.3
4.4
4.5

4.6

4.7

5.5

A HRTEM image of a region of a sample of Ag0,5(~,Pb158bTe20. The
four smaller pictures at the side are the amplified pictures for different
(lattice) local region and their fourier transformed images [1] .....

(a) High res(_)luti(_)n TERI image of Ag(),55SiiSl)1,1_.-_,Te3 which SlIOWS a
name—structured region of the crystal. The name-structures have ge—
ometrical dimensions of 3-30 11m and are dispersed throughout the
crystal evenly. (b) A close—up view of an embedded nanocrystal. (c)
The interface region between the embedded nanocrystal and the ma-
trix showing a high degree of coherency. ((1) Dark field imaging over
a limited area in the crystal that Shows the compositional variations
between the matrix and the embedded nanocrystal ..........
Raw diffraction data on the 2D image plate from AgSnngTe4

F(Q) and C(r) for all samples in the Ag1_ISnSb1+xTe3 series. .
C(7) and AG (r) (compared to Sn4Te4) for samples with different chem—
ical compositions ..............................
Lattice parameters from the single-phase refinements for samples with
different chemical compositions in the Ag1_ISnSb1+ITe3 series. . . . .
Thermal factors for the cation site from single—phase refinement for
samples with different chemical composition in the Ag1_xSnSb1+xTe;3
series. ...................................
Thermal factors for the Te site from single-phase refinements for sam—
ples with different chemical composition in the
Ag1_xSnSb1+xTe3 system. ........................

HRTEM pictures showing multi—scale phase separation happening in
some part of sample. ...........................
Equilibrium phase diagram for PbTe—PbS ................
In-lab powder X—ray diffraction of PbTe75%—PbS25% sample with in—
dices showing two phases are present of roughly PbTe and PbS com-
position. .................................
Raw x-ray powder diffraction data from the 2D detector for the :1: =
0.50 (PbTe)1_a,(PbS),c sample. Data from the (a) unquenched and
(b) quenched samples are shown for comparison. The 1-D integrated
powder diffraction patterns obtained from these data are shown in
Figure 5.5(a) and on an expanded scale in Figure 5.7. The white circle
in the center of each 2D difractogram represents a shadow from the
beam-stop ..................................
Experimental (a) F (Q) and (b) C(r) for all unquenched samples. In
the Fourier transform, Qmax was set to 26.0 A“. The data are offset
for clarity. The compositions of the (PbTe)1_x(PbS)I samples are in-
dicated in panel (a). From top to bottom: :1: = 1.00 (green), :1: = 0.75
(yellow), 2: = 0.50 (magenta), :r. = 0.25 (blue), :16 = 0.16 (cyan), and
:1: = 0.00 (black). .............................

ix

48

52
54
55

56

57
58

59

63
64

65

68

5.6

Cf!
\l

5.8

5.9

5.10

5.11

5.12

Representative refinen'ients of the PbTe data using (a) Rietveld and
(b) PDF approaches. Symbols represent data, and solid lines are the
model fits. The difference curves are offset for clarity ..........
The low-Q diffraction patterns of all the (PbTe)1_I(PbS),,.samples stud-
ied, where F(Q) = Q(S(Q)—1). From top to bottom: :1: = 1.00 (green),
11‘ = 0.75 (yellow), :r = 0.50 (light and dark magenta), :r = 0.25 (blue),
:1: = 0.16 (cyan), and :r = 0.00 (black). The data corresponding to the
quenched :1: = 0.50 sample (light magenta) is superimposed on top of
that of the unquenched sample (dark magenta) without being offset.
The other data are offset for clarity. Vertical dashed lines indicate po—
sitions of several characteristic Bragg peaks in the endmember data to
allow for easier comparison. .......................
Experimental PDFs for various (PbTe)1_,,.(PbS)I samples on expanded
scale. The PDFs, from top to bottom correspond to :r = 1.00 (green),
:1: = 0.75 (yellow), :1: = 0.50 (magenta), quenched :r = 0.50 (bright
magenta), :1: = 0.25 (blue), 1‘. = 0.16 (cyan), and :1: = 0.00 (black). The
data corresponding to the quenched at = 0.50 sample (light magenta) is
superimposed on top of that of the unquenched sample (dark magenta)
without being offset. The other data are offset for clarity. Vertical
dashed lines indicate positions of a few selected characteristic PDF
features of the endmembers for easier comparison ............
Representative refinements of the :1: = 0.50 sample data using (a) Ri-
etveld and (b) PDF approach. Symbols represent data, and solid lines
are the model fits. The difference curves are offset for clarity ......
PDFs of converged models for (a) :1: = 0.00 and (b) :1: = 1.00
(PbTe)1_x(PbS)x samples. Comparison of the data for (c) quenched
and (d) unquenched :r = 0.50 samples (open symbols) with the solid
solution (c) and mixture ((1) models (solid lines), respectively. See
text for details. Vertical dashed lines indicate positions of selected
PDF features characteristic for the endmember compositions, for eas-
ier comparison. ..............................
HRTEM images of (a) a: = 0.16 and (b) quenched :1: = 0.50
(PbTe)1_x(PbS)x samples. ........................
Representative refinements of the x = 0.16 sample data using (a) Ri-
etveld and (b) PDF approach. Symbols represent data, and solid lines
are the model fits. The difference curves are offset for clarity. . . . . .

71

72

74

75

78

83

84

List of Tables

3.1

3.2

3.3

4.1

4.2

5.3

5.4

5.5

Results from PDFFIT for the AngoSbTeo sample. n = A—R’ere is
the ratio of atom numbers in PbTe phase to whole sample, no is the
expected ratio calculated from the chemical stoichiometry (see text for
details). Uamm are the displacement parameters for atoms on different
sites .....................................
Results from PDFFIT for the AngoSbTeo sample. TL 2 eV—Nl-Te is
the ratio of atom numbers in PbTe phase to whole sample, no is the
expected ratio calculated from the chemical stoichiometry (see text for
details). Uamm are the displacement parameters for atoms on different
sites. ....................................
Results from model N C3o for three different m-members. n = fifi—‘V—fl
is the ratio of atom numbers in the PbTe phase to whole sample, no
is the expected ratio calculated from chemical stoichiometry. Uatom is

the thermal factor for atoms on different site. .............

Refinement result from the single—phase model to each of the samples
in the Ag1_xSnSb1+xTe3 system ......................
Comparison of results from one-phase and two-phase refinements for
Ag0,85SHSb1_15T€3 831111318. ........................
Result from physically wrong model W ..................

45

45

47

57

60

Refinement results from PbS and PbTe compared with literature values. 71

Refinement results for two-phase fitting. “Rietveld” and “PDF” refer
to Rietveld and PDF fits, respectively, where the composition of the
two phases was fixed to PbTe and PbS. n and no refer to the refined
and expected (based on stoichiometry) phase fractions for the PbS-rich
phase ....................................
Refinement results from both PDF and Rietveld for the quenched 50%
sample from a homogeneous solid-solution model. ...........
Rietveld and PDF refinement results from three different models for
the PbTeog4Sojo sample: model A is solid solution model, model B is
a simple two-phase mixture of PbTe and PbS. n and no refer to the
refined and expected (based on stoichiometry) phase fractions for the
PbS-rich phase. . ............................
Model C is a mixture of pure PbTe phase plus a solid solution of
composition PbTeoo-PbSoo. .......................

xi

76

79

80

Chapter 1

Introduction

1.1 Physics of thermoelectric materials

Thermoelectrics (TE) are an important class of technological materials that are capa-
ble of converting heat flow into electrical current, and vice versa [2, 3, 4, 5]. Modern
TE materials are special types of semiconductors that, when coupled, function as a
“heat pump”. By applying a low voltage DC power source, heat. is moved in the
direction of the current (+ to -,)° Usually, they are used for thermoelectric modules
Where a single couple or many couples (to obtain larger cooling capacity) are com-
bined. One face of the module cools down while the other heats up, and the effect
is reversible. Thermoelectric cooling allows for small size and light devices. high reli-
ability and precise temperature control, and quiet operation. Disadvantages include
high prices and high operating costs, due to low energy efficiency [6].

It can be shown that the maximum efficiency of a thermoelectric device, used
either to convert heat to electricity or to remove heat using electric power, is the
DI‘Oduct of the Garnet efficiency and a device dependent term that is a function of
the Parameter Z T, where T is temperature and Z is the thermoelectric figure of merit.
The materials that are of interest for TE applications are poor thermal conductors

but at the same time good electrical conductors, 1.6., they maximize the TE figure-

of— merit.

ZT= SET. (1.1)

h‘. p

 

where S, p, and K denote thermoelectric power, electrical resistivity, and thermal
conductivity, respectively [6]. There is a large need for higher performance materials
than those that currently exist. Much of the new materials research in the TE field
is focused on finding materials with high thermoelectric figure of merit Z.

Early TE materials were BIQTP3 [7] and Si-Ge [7] systems. More recently, the focus
on new materials (.levelopment was shifted to skutterudites[8, 9] and clathrates [10],
superlattice structures [11, 12, 13, 14], and low-dimensional [15] and disordered sys-
tems [16, 17]. Currently the best. TE materials are artificial multilayered semiconduct-
ing alloys with low phonon thermal conductivity and large electronic mobility [18, 11].
The misfit—layered oxides like Ca3Co4Og accomplish a similar effect in naturally as-
sembled crystals that play a dual role of being a “phonon glass” and an “electron
crystal” [7]. They attract new interest as candidates for high-temperature TE appli—
cations.

This thesis represents an attempt to understand the role that local structure plays

in newly emerging and promising classes of TE materials based around PbTe.

1.2 Chronology of TE effect discoveries

1.2.1 The Seebeck effect

Seebeck (1770-1831) is famous for the discovery of the first thermoelectric effect [19].
The Seebeck coefficient is defined as the open circuit voltage, V, produced between
two points on a conductor where a uniform temperature difference, (1T of 1K exists

between those points. More precisely,

(iv = SdT, (1.2)

or equivalently

E = SVT, (1.3)

where S is called the Seebeck coefficient or thermoelectric power.

1.2.2 The Peltier effect

In 1834 Peltier [20] described thermal effects at the junctions of dissimilar conductors
when an electrical current flows between the materials. Peltier himself failed to fully
understand the implications of his findings and it wasn’t until four years later that
Lenz concluded that there is heat absorption or generatiOn at the junctions depending
on the direction of current flow (see [21]). The Peltier coefficient II represents how
much heat current is carried per unit charge through a given material. When two
different materials are connected to each other at two junctions and a current is
flowing through the circuit. due to the different II values, heat will transfer from one

junction to the other: one junction cools off while the other heats up.

1.2.3 The Thomson effect

In 1851, Thomson [22] (later Lord Kelvin) predicted and observed experimentally
the cooling or heating of a homogeneous conductor resulting from the flow of an
electrical current in the presence of a temperature gradient, dT/drr. This is known as
the Thomson effect. Any current-carrying conductor, with a temperature difference
between two points, will either absorb or emit heat, depending on the material. If a
current density J is passed through a homogeneous conductor, heat production per

unit volume, q, is the usual Joule heat. plus the Thomson heat:
q = p.12 — anT/dx. (1.4)

Here, p is the resistivity and it. is the Thomson coefficient.

1.2.4 The Kelvin relationship

The three thernmelectric effects above are related by the Kelvin relationships (see [23]),
assumed to be valid for all thermoelectric materials. The first. Kelvin (Thomson)
relation connects the Peltier coefficient II, Seebeck coefficient S and the absolute
temperature T

11 = 5.1:. (1.5)

The second Kelvin (Thomson) relation relates the Thomson coefficient 11. with the

derivative of the Seebeck coefficient with regard to the temperature

11. = T—'—-. (1.6)
(

1.3 Engineering aspects of TE physics

1.3.1 Simple overview of the Seebeck effect

In TE materials there are free carriers which carry both charge and heat. The simplest
picture one can think of is that of a gas of charged particles. If the molecules are not
charged, when such a gas is placed in a box with a temperature gradient, where one
side is cold and the other one is hot, the gas molecules at the hot end will move faster
than those at the cold end. The faster hot molecules will diffuse further than the cold
molecules do, and so the cold end will have a higher density. The density gradient will
cause the molecules to diffuse back to the hot end. In the steady state, the effect of
the density gradient will exactly counteract the effect of the temperature gradient so
there is no net flow of molecules. If the molecules are charged, the buildup of charge
at the cold end will also produce a repulsive electrostatic force (and therefore electric
potential) to push the charges back to the hot end. The electric potential produced

by a temperature difference is known as the Seebeck effect. and the proportionality

Hot
n —- type P " type

 

 

Voltage

 

 

ee h+h+
— +

Cold

Figure 1.1: Illustration of the Seebeck effect[reproduced with the permission of Dr.

Jeff Snyder] [24]

 

 

 

 

constant is called the Seebeck coefficient. If the free charges are positive (the material
is p—type), positive charge will build up on the cold end which will have a positive
potential. Similarly, negative free charges (n—type material) will produce a negative

potential at the cold end. This is schematically illustrated in Figure 1.1.

1.3.2 TE power generation

If the hot ends of an n-type and a p—type material are electrically connected, and
a load is connected across the cold ends (Figure 1.2) the voltage produced by the
Seebeck effect will cause electric current to flow through the load, generating the

electric power [6]. Then the following will hold

v = 55 T, (1.7)

 

i
AT
RL V

 

Figure 1.2: Schematic of thermoelectric power generation[reproduced with the per-
mission of Dr. Jeff Snyder] [24]
and the resistance is given by

Rload x (1-8)

L
a A’
where a is the electrical conductivity, L/A is the ratio between the length and the

cross section area of the TE elements. This further brings

V2 A

P=IV=—z s2 6T2— 1-9
R 0 L ( )
820 is a property of a material known as the thermoelectric power factor. For efficient
operation, high power must be produced with a minimum of heat Q. The thermal

efficiency is then defined as
,, _ 5
Q.

It can be seen then that to have a high efficiency thermoelectric generator two require-

(1.10)

ments must be met: large 6T to increase the Carnot factor, and a large thermoelectric

figure of merit Z T. Increasing Z T is the focus of all contemporary TE materials de-

velopment and increasing (5T represents a goal of generator design.
The thermoelectric figure of merit, ZT, characterizes the efficiency of a TE ma-

terial at Operating temperature T. Z is a. property of a material defined as

8‘2
Z = —— 1. 1
p < 1 >
where
— l (112)
p — a. .

Here, S is the Seebeck coefficient. of the material (measured in microvolts/ K), a is
the electrical conductivity of the material and n is the total thermal conductivity of

the material.

1.3.3 Utilizing Peltier cooling

If an electrical current is introduced in a TE material by an external electric potential
(Figure 1.3) then the heat. can be forced to flow from one end to the other. The
coefficient of performance and the maximum temperature drop that can be achieved

is again related to the efficiency of a TE material through the TE figure of merit ZT.

1.3.4 Using Thomson effect to predict Seebeck coefficient

The second Kelvin (Thomson) relation provides a connection between the Thomson

coefficient 11 and the derivative of the Seebeck coefficient with respect. to temperature:

(15
= T—. 1.13
H H ( )
By measuring the change in bulk heating as the current direction is reversed for

fixed temperature gradient, we can determine the temperature derivative of the ther-

mopower, and therefore compute the value of S at high temperatures, given its low-

Heat Flow

 

 

 

 

Figure 1.3: Peltier Cooling[reproduced with the permission of Dr. Jeff Snyder] [24]

temperature value.

1.4 Evolution of TE materials

It was in 1909 [25] and 1911 [26] when Altenkirch showed that good thermoelectric
materials should possess large Seebeck coefficients, high electrical conductivity and
low thermal conductivity. A high electrical conductivity is necessary to minimize
the Joule heating, whilst a low thermal conductivity helps to retain heat at the
junctions and maintain a large temperature gradient. These three properties were
later embodied in the so—called figure-of—merit, Z. Since Z varies with temperature, a

useful dimensionless figure-of-merit can be defined through Z T, as mentioned earlier.

1.4. 1 Metals

Although the properties favored for good thermoelectric materials were known, the
advantages of semiconchictors as thermoelectric materials were neglected and research
continued to focus on metals and metal alloys in the early TE days. These materi-
als however have a constant ratio of electrical to thermal conductivity (W'idemann-
Franz-Lorenz law) so it is not possible to increase one without increasing the other.
Metals best suited for TE applications should therefore possess a high Seebeck co-
efficient. Unfortunately most metals possess Seebeck coefficients in the order of 10

microvolts/ K, resulting in generating efficiencies of only fractions of a percent [7].

1.4.2 Semiconductors

It was during the 1920’s that. the development of synthetic semiconductors with See-
beck coefficients in excess of 100 microvolts/ K increased interest in TE field. At that
time it was not apparent that the semiconductors were superior TE materials due to
their higher (and hence more favorable) ratio of the electrical conductivity to thermal

conductivity, compared to that of metals.

1.4.3 Pioneer work of A. F. Ioffe

As early as 1929 when very little was known about semiconductors, Abram F edorovich
Ioffe (1880—1960) showed that a thermoelectric generator utilizing semiconductors
could achieve a conversion efficiency of 4 percent, with further possible improvement
in its performance. By the 1950’s, Ioffe and his colleagues [27] had developed the the-
ory of thermoelectric conversion. which forms the basis of all modern thermoelectric

theories.

1.4.4 Further developments and current efforts

By the late 1950’s and early 1960’s, a large number of semiconductor materials were in-
vestigated. Some of the nuiterials emerged with Z T values significantly higher than in
metals or metal alloys. Research focused on developing materials with high figure-of-
merit values over relatively narrow temperature ranges, because no single compound
semiconductor can exhibit a uniform high figure-of—merit over a wide temperature
range. Of the great number of materials investigated, those based on bismuth tel-
lurium, lead telluride and silicon-germanium alloys emerged as the best for operating
temperatures of about 450 K. 900 K and 1400 K respectively [3].

In more recent days the research on developing high performance TE materials
is focused in three main directions [3]: (a) superlattices and nanowires aiming at
increasing S, and reducing K, (b) utilizing nonequilibrium effect aiming to decouple
electron and phonon transport, and (c) bulk nanomaterial synthesis.

One of the more promising avenues in synthesizing bulk materials with enhanced
TE properties is based around PbTe class of materials that appear to contain nano—
inhomogeneities whose role for TE properties is not fully understood. Investigation
of the local structure of materials based on lead telluride, and the role that it plays

in the TE properties is the subject of this thesis.

1.5 Engineering of novel TE materials - principles
and examples

From the definition of the TE figure of merit provided earlier one can conclude that
a material with a large thermoelectric power factor and therefore high Z T, needs to
have a large Seebeck coefficient (found in low carrier concentration semiconductors
or insulators) and a large electrical conductivity (found in high carrier concentration

metals). The TE power factor exhibits maximum somewhere between metal and

10

s

 

     
     
 
    
    

  

 

 

sv..,......rT1,rrrft..14]10w0
1200 Power Factor _ Conduit/Why 11-1
A I \ 0 = ""
§ az/p : azo ’ p d 8000 g
3 1000 . a
‘-' Seebeck '
U
s ; E
o ’ Semimotalor " 40m :-
0 HoavityDop-od ‘ 5
"g 400 Semiconductor Metal : Q
8 « 2000 ’5
,5,» 200 - g
0 - to
17 18 19 20 21 22

Log (Carrier Concentration)

Figure 1.4: Seebeck coefficient, electrical conductivity, and power factor vs. carrier
concentration [reproduced with the permission of Dr. Jeff Snyder] [24] [Most images
in this thesis are presented in color]

semiconductor (Figure 1.4). Good TE materials are typically heavily doped semicon-
ductors or semimetals with carrier concentrations between 1019 and 1021 carriers/m3.
To ensure that the net Seebeck effect is large, there should only be a single type of
a carrier. Mixed n-type and p—type conduction will lead to opposing Seebeck effect
and low thermopower (defined here as absolute value of Seebeck coefficient).

By having a band gap large enough, n-type and p-type carriers can be separated,
and doping will produce only a single carrier type. Thus good thermoelectric materials
have band gaps large enough to have only a single carrier type but small enough to
sufficiently high doping and high mobility (which leads to high electrical conductivity).

A good TE material also needs to have low thermal conductivity. Thermal conduc-
tivity in these materials arises from two sources of heat transport. Phonons traveling
through the crystal lattice transport heat and lead to lattice thermal conductivity.
The charge carriers (electrons or holes) also transport heat and lead to the electronic

thermal conductivity. The electronic term is related to the electrical conductivity

11

 

~' ' ' ' ' 'Sea'tterihgllllecfiahfsmsr '
80 [ CoSb3

 

electron-phonon
M-site disorder .
' Sbsite disorder 1
6° ' Filling j

1 doped 003133

- 'l
40- . .

f M ‘
20" .. A n ‘ -l

 

 

v

Thermal Conductivity (10'3 W/cm K)

 

 

 

FeszTe CeFeBCoSbm
0 . . I . . . a 1 4 . k. 1 L . A . l . . . . n . A . A
100 200 300 400 500 600
Tamperature(°C)

Figure 1.5: Thermal conductivity decreases in Skutterudites by the introduction of
various scattering mechanisms. [reproduced with the permission of Dr. Jeff Sny-
der] [24]

 

1.4 _ . . - . ..
1.2;
1.03-
0.8E-
0.63-
0.43.

zT

  

(Pb.Sn)Te

 
 

 

 

 

0.2 i
0 o : . P.b‘l.-e n L 1 L - 1 J A l
i 0 200 400 600 800 1000

Temperature (°C)

Figure 1.6: ZT for various p—type thermoelectric materials [reproduced with the
permission of Dr. Jeff Snyder] [24]

12

 

d-L
N5

"VI'I'U

.0.
O

   
 

CD
'I'U

AIAAAJA

Figure of Merit zT

uninnmlnn

 

 

o ‘200‘ 400 600 800 1000
Temperature (°C)

 

Figure 1.7: ZT for various n-type thermoelectric materials [reproduced with the per-
mission of Dr. Jeff Snyder] [24]

through the Wiedeman-Franz law
.6, = Lo T (1.14)

where the Lorenz factor, L, depends slightly on the details of the band structure but

for thermoelectric materials does not vary from the free electron gas value of

n2k§ _8 2
3e? 2 2.45 x 10 V/K (1.15)

 

by more than a factor of 2.

Thus the greatest opportunity to enhance ZT is to minimize the lattice thermal
conductivity. This can be achieved by increasing the phonon scattering by introducing
heavy atoms, disorder, large unit cells, clusters and rattling atoms.

The ideal thermoelectric material is then one which is an ”electron crystal - phonon

glass” [7] where high mobility electrons are free to transport charge and heat but the

13

phonons are disrupted at the atomic scale from transporting heat [28].

Using these principles. a variety of high Z T materials have been developed. Many
of these materials have an upper temperature limit of operation. above which the
material is unstable. Thus no single material is the best. for all temperature ranges,
and different materials should therefore be selected for different applications based
on the desired temperature range of operation. This leads to the use of a segmented
thermoelectric generator.

Z T for various p—type and n—type thermoelectric materials are shown in Figures 1.6
and 1.7. Novel TE materials based around PbTe show great promise for operating
temperatures at or slightly above the room temperature, and are therefore of partic-
ular interest. Recent advances have seen dramatic improvements in Z T by relatively
small compositional changes, as illustrated in Figure 1.8 [29, 30. 31, 17]. These mate-
rials appear to exhibit. nanoscale phase separation, or nanosize domains, as observed
by high resolution imaging experiments, which could be an important structural in—
gredient necessary to enhance the TE figure of merit. However, the exact nature of
these nanosize features (e.g. whether the inhomogeneities are bulk property of the
samples or are just a surface effect) and their role for enhancement of the TE prop-
erties is not fully understood. This work represents a local structural study aimed to

address some of these issues in this promising class of TE materials.

1.6 Thesis layout

This thesis is organized as follows. Next chapter provides a brief introduction into
the atomic pair distribution function (PDF) technique and the rapid acquisition PDF
experimental approach used in this study. Chapter 3 presents detailed local structural
results on the high performance family of TE materials Ag‘PbmeTem+2. This is
followed by the results obtained for Ag1_xSnSb1+xTe3 described in Chapter 4. Local

structural perspective of the phase separation and TE properties in (Pl)T€)1._.r(PbS)x

14

 

2.5 I U I I I Y I I I I l I I I I I r t I I I I I I 1 I

1.5

   
   

 

CsBi Te 3‘2 a
4 o »

1 14114 I l l

0.5

I I U U l U I I T l U I I I I I t l I l I U I I

  

 

 

 

l L l l l l J kl l l l l l l l l l l l

0 200 400 600 800 1 000 1 200 1 400

 

Temperature, K

Figure 1.8: ZT for various PbTe based thermoelectric materials including recently
discovered novel materials with dramatic ZT improvements compared to those shown
in Figures 1.7 and 1.6. [reproduced with the permission of Prof. Kanatzidis]

are given in Chapter 5. Finally, in Chapter 6 concluding remarks are accompanied
with comments on possible future research avenues for the local structural studies of

this class of novel TE materials, which concludes the thesis.

15

Chapter 2

The Pair Distribution Function
Method

2.1 Importance of the local structure

The structure plays an important role with regard to properties of modern materi-
als. Advances in science and technology critically depend on our understanding and
utilization of these properties. Many industrially important materials have complex
structures which are usually difficult or impossible to study using conventional crys-
tallographic method [32]. A significant fraction of the thermoelectric materials men-
tioned in Chapter 1 fall in this category. The structure of the three families of PbTe
based thermoelectric materials investigated in this work and discussed in the following
chapters can not be studied completely and accurately using conventional crystallog-
raphy due to the limited structural coherence of the nanoscale features observed by
imaging methods. Conventional crystallographic structural studies in such cases are
often augmented by a local structural probe, such as extended x-ray absorption fine
structure spectroscopy (EXAFS) [33], nuclear magnetic resonance (NMR) [34], or the
atomic pair distribution function (PDF) technique [35].

The local structure plays an important role in many functional materials [35, 36].

16

It represents a description of the atomic neighborhood typically on a lei‘igth-scale
of several nanometers or shorter. Crystalline materiz-ils with well established long
range order can be studied using conventional crystallography. In this case, the
average structure obtained from conventional crystallography is the same as the local
structure. However, structure of a large number of interesting functional materials
does not possess long range. periodicity. In such cases the real local structure usually
deviates, sometimes quite dramatically, from the average structure obtained using
conventional crystallographic methods. Understanding the underlying structure is
important for understanding the material properties and for materials engineering.
Local structural probes such as NMR [34] and EXAF S [33] play an important
role in revealing the local structure of a material. However, these methods can only
detect local environments up to approximately 5 A or less. To study the structures

of complex materials on a length scale up to nanometers or even higher, different

experimental methods need to be used, such as the atomic PDF method.

2.2 The atomic pair distribution function (PDF)
technique

The atomic pair distribution function (PDF) technique is a total scattering based
local structural method, that provides structural information on various length scales.
Unlike crystallography (eg. the Rietveld method) that uses structural information
solely from the Bragg peaks, and disregards the rest of the diffraction pattern through
a background function, total scattering methods use both Bragg peak and the diffuse
scattering information. In this way both information about periodic structure and
local deviations from periodicity are taken into account and treated on an equal
footing. In recent years, the PDF method has demonstrated great power by being

successfully used for solving various structural problems of wide range of functional

17

materials [35, ‘36] and gained appreciable attention from material science, solid state

physics and chemistry communities.

2.2.1 History and current status of the PDF method

The PDF method was initially used to study structure of liquid and amorphous
materials [37, 38, 39. 40]. In the past decade or so, it has been applied for the
study of crystalline solids as well. The advent of synchrotron based radiation sources,
such as pulsed spallation neutron sources and x-ray synchrotron sources, that provide
high—intensity short-wavelength probes, has made the PDF method accurate and
reliable. The PDF data analysis and modeling of crystalline materials are also highly
computationally demanding and have become practical with the appearance of high-
speed computers. In the past. five years the PDF has also been utilized to study the
structure of nanocrystalline materials [41], or materials with significant. amount of
disorder where nanometer scale plays an important role [42].

The PDF analysis applied to highly crystalline materials, such as Si or Ni, yields
the structure in quantitative agreement with the average structure obtained through
conventional crystallographic methods [43]. Complementary to the average structure
analysis, Billinge et al. and BoZin et al. have applied the PDF technique exten-
sively to study the nature of the local disorder and nanometer scale inhomogeneities
in otherwise well crystalline materials such as colossal magneto—resistive mangan-
ites and high temperature superconducting cuprates [44, 45, 46, 47, 48]. In cases
when the long range crystallinity fades out gradually, such as in nano—crystalline and
non-crystalline materials, the reciprocal space approach taken by the conventional
crystallographic methods becomes less and less effective. The Bragg peaks become
less and less sharp and there are fewer of them in the diffraction pattern which be-
comes rather featureless in such cases. making the conventional analysis rather hard if

not impossible. However, this has little effect on the features obtainable through the

18

 

 

Ill"

6»:

PDF technique where no long range periodicity is assumed. For example, Petkov et al.
have shown that. local and intermediate length-scale structures can be obtained from
PDF data. when a certain level of local atomic order is preserved [49, 50, 51, 52]. As
for non-crystalline materials such as liquids, glasses. and amorphous materials, only
the very local structure persists and the angstrom scale becomes the only meaningful
length-scale. In such cases, the PDF technique represents the method of choice for
structural investigations [53. 54, 55]. At present, with the advent of high flux instru-
ments and with advancements in detector coverage, PDFS with significantly higher
real space resolution have been achieved [56, 57, 58]. Complex modeling schemes,
such as reverse Monte Carlo (RMC) and atomic pair potential based regression a1-
gorithms, are becoming popular tools in efforts for solving the structure of complex

materials [59, 60, 61, 62, 63].

2.2.2 Definition of the PDF and its variants

The atomic pair distribution function (PDF), C(r), is a one-dimensional pair cor-
relation function that gives the probability of finding atomic pairs separated by a
distance T. To obtain the PDF for a given structure, an atom z' is selected, and its
environment is inspected to look for its neighbors. A peak is assigned for every atom
j found at the position corresponding to the inter-atomic distance rij, and zero ev-
erywhere else. Then the same procedure is repeated for all possible choices of atoms
2', and the resulting functions are then averaged out to obtain the total C(r) of the

structure. In what follows, a series of definitions of various PDF variants is provided.

 

0(7‘) = 909(7‘) = 47,]VT2 Z Z 5(7‘ - run), (21)

19

Where Do is the number density in the system of N atoms. (5 is a Dirac delta function.
The function p(r) is called the atomic pair density function (PDF). The function g(r)
is called the atomic pair distribution function, also abln‘evie‘ited as PDF.

The PDF, g(r), is related to the radial distribution function (RDF), R.(r), by
R(r) = 47r‘r2g(r). The radial distribution function has a useful property that the
quantity R(r)dr gives the number of atoms in an annulus of thickness (11‘ at distance
r from another atom. The coordination number, or the number of neighbors, NC is
then given by NC = f, 712 R(r)(lr where 7‘1 and r2 are the lower and upper r-limits that
define the RDF peak corresponding to the coordination shell in question.

Under the kinematic approximation, the sample scattering amplitude is,

mo, 1‘) = ((17) Z bueiQR"(tl (2.2)

where bll is the scattering amplitude of the atom V, and (b) is the average value over
all atoms. However, only the intensity of the diffracted beam, which is directly related
to the square of the magnitude of \II(Q), i.e. [@(QHZ, can be measured. This is well
known as the phase problem.

The coherent differential cross section is shown below:

d0c(Q) _ (’1)?

do — N

 

1 - _
\II(Q)I2 = N Z bubyezQuzu R”) (2.3)
V41

where N is the number of atoms in the whole sample.
Total scattering structure function is related to the measured intensity in the

following way:

_ dac(Q)
_ do

 

+ (b)2 - (b2) (2-4)

and

20

 

3(Q) = 2 (2-5)

Typically S (Q) depends on both the amplitude and the direction of Q. For
samples of finely powdered crystallites, the scattering from the ensemble is isotropic.
S (Q) will then only depend on the magnitude of the wave vector Q. This is known
as powder averaging. The PDF can be directly obtained via. a. sine Fourier transform
of S(Q):

C(r) = Z /00 Q[S(Q) — 1] sin QrdQ (2.6)

7" 0

On the other hand the following holds:
C(T) = 47r7‘po(9(7') — 1) (2.7)

C(r) is usually called the reduced pair distribution function, while g(r) is the pair
distribution function. {)0 is the number density which is a constant for a specific
structure, and represents number of atoms per unit cell volume.

As mentioned, the PDF analysis of powder diffraction data differs from the conven-
tional crystallographic method in the way the diffuse scattering intensities in-between
and underneath the Bragg peaks are treated. Conventional crystallographic analysis
uses only the Bragg peaks which arise from the long range order of a structure. The
PDF technique takes into account both the Bragg peaks and the diffuse scattering
intensities which come from local deviations from the long range ordering. This is
evident in the above equations. Therefore, the PDF method is capable of probing
both the local and the average structure. This advantage over conventional analy-
sis gives the PDB wider range of applicability. In the case of crystalline materials
with disorder, the PDF technique and conventional crystallography are complemen-

tary, as the former provides additional information about the local structure. In the

21

case of nano-crystalline and non—crystalline materials, with few or no Bragg peaks,
cryst.allography fails, and the PDF is the method of choice.

The PDF peaks in the low-r region contain the local structural information. Thus
the real space resolution is an important factor when neighboring peaks are to be
resolved. The atomic thermal motions broaden the PDF peaks. Termination effects
due to finite measurement range give uncertainties in r value and also further broaden
the peaks [64]. Therefore, to obtain high real space resolution it is usually required
to obtain data over an extended Q range (2 25.0 A”). and to collect. data at low
temperature when possible. The PDF peak width and area reveal further details
about the local distortions and number of neighboring atoms as well. The average
structural information can be obtained from the PDF structural refinement using the
least square regression fitting approach implemented in the program PDFF IT [65, 66].
The comparison between the calculated PDF based on the average structural model
and the experimental PDF is a. good way to check for the possible existence of local
disorder. If the disorder is present, the average model is used as a starting point to
build various potential models of the distorted structure, and these are then attempted
to obtain a better fit.

Lattice vibrations can affect the PDF by modifying the Bragg peaks. This is
described by the Del‘)ye-VValler factor. Real experimental measurements can only
scan a finite region of Q-space. This introduces a cut-off in Q space, and thus induces
fluctuations in the PDF signal. Background signal can also introduce fluctuations.
The interested reader can look for quite detailed description of the technical issues of
the PDF method in the literature. Excellent source is, for example, a book by Egami

and Billinge [35].

22

2.3 Rapid acquisition PDF experiments and PDF
data analysis

The PDF measurements using the conventional x—ray experimental setup involving go—
niometers and point detectors are typically very slow and usually take more than eight
hours for collecting a. dataset on a sample at. a single temperature, even when using
third generation synchrotron x-ray sources. This presented abottleneck preventing
the PDF technique from widespread application in areas such as nano—materials, for
example, where only small quantities of specimens are available. Recent development
of the rapid acquisition PDF (RA-PDF) method reduces the data collection time by
three to four orders of magnitude [67]. This new approach in collecting data is briefly
described below, followed by a brief description of program PDFgetX2 [68] used to
obtain the PDF from raw x-ray powder diffraction data. The program PDFFIT2 will
be introduced further as well, used to extract the structural information from the

experimental PDFs.

2.3.1 Utilizing image plate detector for rapid data acquisi-
tion

The crucial part. of the RA—PDF setup is the detection part. In RA-PDF experiment,
an image plate (IP) detector is used, rather than a conventional solid state detector.
This 2D data collection greatly reduces the data collection time, and opens new
possibilities on the experimental front.

One layer of very small crystalline grains of photo-stimulable phosphor mixed
with organic binders acts as a detecting medium. BaF(Br,I):Eu2+ is used as a photo—
stimulable phosphor. This material is capable of storing a fraction of the absorbed
x-ray energy, and emitting photo-stimulated luminescence (PSL) later when stimu—

lated by visible laser light. The x-ray irradiation results in a certain number of En”

23

and F pairs proportional to the absorbed x-ray energy. The F centers are caused
by the absence of halogen anions from their designated positions in the lattice. The
energy trapping state of F centers is meta-stable with a long lifetime. However, the
trapped electrons can be easily excited by visible light (2 2 eV) to return to the
conduction band. One of the processes occuring is the recombination of Eu3+ with
one election to Eu“, along with the emission of a photon of 3.2 eV energy (blue
light). A red He-Ne laser is usually used for the process of read—out. Its wavelength
(632.8 nm) is considerably separated from the PSL wavelength (39011111). A conven-
tional high-quantum efficiency photo—multiplier tube (PMT) is used to collect the
photo-stimulated photons. The signal is then amplified and digitalized to be pro—
cessed by computers. The remaining F centers in the phosphor after read-out can be
further erased by exposing to visible light.

More technical det ails and extensive quantitative description on the characteristic
of an IP detector can be found in [69, 70, 71, 72, 73, 74, 75, 76]. Dynamic range and
linearity of the IP response, spatial resolution, active area size, and energy dependence
are important technical issues for an IP as an x—ray area detectors. The standard

procedure to get high quality data from RA-PDF experiment is described further.

2.3.2 The RA-PDF experimental procedure

The diffraction experiments for this study were performed at the 61D-D beam line
at the Advanced Photon Source (APS) located at Argonne National Laboratory, Ar-
gonne, IL (USA). High energy x-rays were delivered to the experimental hutch using a
double bent Laue monochromator capable of providing a flux of 1012 photons/ second
and operating with x-rays in the energy range of 80-130 keV. Typical RA—PDF ex-
perimental setup is shown in figure 2.1.

A Mar345 image plate camera, a round disk with a diameter of 345 mm, was

mounted orthogonal to the beam path, with the beam centered 011 the IP. The sample-

24

 

Figure 2.1: Experimental setup for the RA-PDF experiment. See text for details

to—detector distance can be determined by calibrating with a silicon standard sample
[67] using the Fit2D program package [77].

Powder samples were carefully ground using a mortar and pestle and sieved to
obtain fine powders. The powders were packed in a hollow flat aluminum plate sample

container with cylindrical hole 5 mm in diameter, and sealed with kapton tape.

2.3.3 Data processing using FIT2D and PDFgetX2

All raw 2D data were integrated and converted to intensity versus 20 format using
the Fit2D program package [77], where 20 is the angle between the incident and scat-
tered x-rays. Figure 2.2 shows an example 2D diffraction pattern obtained from a
Ni powder sample [67], used routinely in RAPDF experiments for calibration pur-
pose. Multiple data sets for the same sample were combined using the same program.
Data for the empty container were also'collected and subtracted from the sample

data during the correction step. Standard corrections for multiple scattering, polar—

25

800 1000 [200 I400 1600 1800 2000 2200 2400 2600

Rows

 

800 1000 1200 I400 1600 1800 2000 2200 2400 2600
Columns

 

300 1000 3000 10000 30000
Intensity

Figure 2.2: Two dimensional contour plot from the Mar345 Image Plate Detector.
The XRD data are from nickel powder measured at ambient conditions. The wave-
length of the radiation was 0.1270 A. The concentric circles represent intersections of
different scattering cones with the area detector (Debye-Scherrer rings). The sample
was contained in a flat plate, 1.0 mm thickness, irradiated volume 0.25 mm3, beam
size 0.5x 0.5 m2. The small dark area in the center of the image is a shadow cast
by the beam stop assembly [67].

ization, absorption, Compton scattering and Laue diffuse scattering were applied to
the integrated data to obtain the reduced total structure function F (Q), as described
in detail in Refs. [67, 35]. Data correction and processing utilized the PDFgetX2
program package [68]. Figure 2.3 demonstrates the data processing path from I (Q)

to C(r) for Ni powder data shown in Figure 2.2, as obtained from A. S. Masadeh.

26

 

1

(a) _

l

I(au)
0 510152025

Tl'l‘I'I'T

lIlI

 

 

_ (b) _'

 

 

 

 

0
AN
I - -
sa- -
k. _ -
0
at . l . 1 . l . l . r . l
4 8 12 16 20 24 28
-1
Q (ll )
m_ I r I I I [fl [ T I I I I r -‘
H_ .4
m
CERF]: (c) _
on: co— —
v r— —q
‘5 °_
©_ _
l- I I I I I I I I I I I I I I

 

 

 

4 8 12 16 20 24 28
r (A)
Figure 2.3: (a) The corrected experimental intensity of the Ni raw data shown in
Figure 2.2, (b) the experimental reduced structure function F (Q) = Q(S (Q) — 1) of

data in (a), (c) the experimental C(r) obtained by Fourier transforming the data in
(b) with Q"... of 30.0 11-1. Credit: A. s. Masadeh.

27

 

 

 

 

 

.0 ' I ' I ' I ' I ' I _
N
o -
N
e -I
.l
[he -
d
It) -I
.l
O
n .
| I l . l . I . l . I
5 10 _1‘5 20 25
Q (8 )
I ' I ' I '
o I -
'- ; ‘
l : I "
{I‘m ' i _
5 s , ‘ 2 ‘.
0 . I E I; ‘
"l’ q
' ‘WW
I I I I I l I

 

 

10
r (3)
Figure 2.4: (top) Experimental F(Q) of Ni powder, and (bottom) experimental G(r)

(symbols) with the fit of the structural model (solid line) and the difference curve
underneath. Credit: H. J. Kim

28

PDFgetX2 is a GUI driven program used to obtain a pair distribution function
from an x—ray powder diffraction data set. In the prograi’n PDFgetX2, a. user-friendly
graphical user interface (GUI) has been built to fz—icilitate user iI'Iteracti(_)ns with data
processing software. Standard corrections [35] such as various background subtrac-
tions, sample absorption. 1')(.)larizati()11, and Compton intensities are available. Oblique
incident angle correction and empirical energy dependence of the detection efficiency
are also implemented for RA—PDF setup. Standard uncertainties due to finite count-
ing statistics are estimated and propagated in all steps. The S (Q) data sets also
contain the Faber-Ziman coefficients for all partial structure factors as additional

columns. Example data processing is shown in Figure 2.3 as mentioned above.

2.3.4 Structural refinement using the PDFFIT program

Structural information was extracted from the PDFs using a full-profile real-space
local-structure refinement method [43]. The program PDFF IT [65] has been used
to fit the experimental PDFs, as well as an updated version of the program, PDF-
FIT2 [66], that is also available at the present time. An example fit, performed by
H. J. Kim, is provided for the Ni powder data in Figure 2.4.

The PDFFIT program allows for multiple data-sets to be co-refined and is also
capable of handling multiple phases. Starting from a given structure model and given V
a set of parameters to be refined, PDF FIT searches for the best structure that gives
the best agreement between calculated PDF and the experimental PDF data. The
residual function (Rw) is used to quantify the agreement of the calculated PDF from

a model to the experimental data:

 

 

RIL‘— \/Z::21W( V i)[Gobs(Tz l—G'colch‘ill;2 ' (28)
Zi=1w(ri)Gobs(ri)

The weight w(r ,) is always set to unity in all the models attempted in this study.

29

2.3.5 Handling the multiphase problem

The ability to refine n'Iultiple phases in PDFFIT2 is an important feature extensively

used in this study, since many of the samples used involve more than one phase.

The detailed derivation of two-phase PDF is shown in the following series of equa-

tions. Generalization to the problems involving more than two phases can be obtained

straight forwardly.

G(r) =

 

b [:25
g(r)= 47r ——\/——r [)0 Z Z[<b r —Tij)]

CT") = $1.: 23- [33:20 ( Toll

= 47TT/)09(7")

Gill“) ‘ 47l7'I00
[\lr 2i Zj [<bbb>26(r _Tij)l — 47f7'p0
a: Z... 53 [El—“3:34 —rmn>] +— w: 231.32%"

 

(2.10)

‘qull “ 4717300

 

30

2 r
<<bb>>2 Afr—vmrzm Zn l<bmbb >2 Tim-"ll
I’ b b
+<<bb:: ‘fV—l- 1—-Vpr 2p zq [<bp :26(T —TPQ)] — 47”.“)
n b N
<<bb>>22 33997710,“ ((5:22 170119;)(7‘) _ 47T7‘p0
(2.11)
Nm + Np = N (2.12)
1‘", < bm >2 1”,, < 1),, >
—————— m 7‘ + '7‘ 4717‘ 2.13
< b >2 g ( ) < b >2 Jp( ) film/Pm + flip/pp ( )

iv? n

-———— 2.14
N3 + N, ( l

4,1 m I

Nm, Np are the numbers of atoms in each phase, N is the number of atoms in the
whole system. < bm >, < bp > is the average scattering power for each phase and
< b > is the one for the whole system. p0 is the number density of atoms for the whole
system, pm and p" are. the number densities of atoms for each phase. Equation 2.14
shows in'Iportant result that PDF contribution from each phase is weighted by the
squared averaged scattering power from each phase multiplied by the number of atoms

in each phase.

31

Chapter 3

Study of the structure of new

thermoelectric material

AngmeTem+2

3. 1 Introduction

3.1.1 Background of AngmeTem+2 system

Pure PbTe is a. narrow band n-type indirect bandgap semiconductors with the band-
width of only 0.31 ev at room temperature. The highest ZT is 0.36 at 300 kelvin.
W'ith optimized doping, ‘n-type doped PbTe can reach 0.84 at 648 kelvin while p-type
can reach 0.7 at 698 kelvin.

Compounds in the series based on composition AgPl)meTem+2 can exhibit excep—
tional thermoelectric properties [17]. They are promising for electrical power genera-
tion and in the temperature range 600 to 900 kelvin, they are expected to significantly
outperform all other reported bulk thermoelectric systems. The dimensionless ther-
moelectric figure of merit, Z T [78]. of the m ~ 18 composition material was found to

reach 1.7 at. 700 kelvin, compared to the highest observed ZT of only 0.84 for PbTe

32

at. 648 kelvin in n-doped material [79. 80]. This is a. surprisingly large enhancement
in ZT for the addition of just 10% per formula-unit of silver and antimony ions. It is
clearly of the greatest importance to trace the origin of the ZT enhancement.

A recent theoretical analysis showed that resonant structures form in the DOS
near E f in the presence of ordered Ag and Sb atoms in the matrix and in the nan-
oclusters observed in HRTEM [81. 82]. The calculations used gradient corrected
density functional theory and assumed different structural models for the clusters,
since details of their structure and chemical ordering are not known. This type of

DOS resembles that. of the "best thermoelectric material” predicted earlier [82, 18].

3.2 Motivation

High resolution transmission electron microscopy (HRTEM) images from these ma-
terials indicate the presence of nanosized domains of a Ag-Sb-rich phase endotaxially
embedded in the PbTe matrix [29]. An interesting possibility is that these nan-
oclusters are key components in the ZT enhancement. The HRTEM images show
the clusters are randomly distributed through the matrix and are not long-range or-
dered. Randomly distributed nano—scale clusters which strain the lattice might be
expected to increase phonon scattering and reduce the thermal conductivity which
would enhance Z T provided the electrical conductivity was not degraded to a greater
degree. An additional enhancement in Z T is possible if the material has an increased
electronic density of states (DOS) at the Fermi-level, E f.

The composition and atomic arrangements within the nanoclusters is a challenging
topic since the clusters are not periodically long-range ordered. They are dispersed
inside a matrix and cannot be studied crystallogra1.)hically. A probing method sensi-
tive to local structure is needed such as the atomic pair distribution function (PDF)
analysis of x-ray powder diffraction data [35]. Detailed description on PDF method

can be found in Chapter 2. Recently, the PDF was successfully used to study chemical

33

short-range ordered clusters randomly embedded in a parent. matrix [83], in analogy
with the present situation. Here we use PDF to study a series of compounds in the
AngmeTe,,,+-2 series with m = 6, 12 and 18. For compz-Irison we also studied the
end member compound, PbTe, the m z 00 member of the series. The resulting PDFs
have sufficiently high quality to see a structural signature of the nanoclusters, even
in the PbTe—rich m = 18 compound. These differences were sufficiently large to allow
models of the local structure to be differentiated, confirming the existence of the clus-
ters in the bulk, and narrowing down their composition and the atomic arrangement

in the clusters.

3.3 Experimental details

3.3.1 Sample preparation

Ingots with nominal compositions AngsSbTeg, AnglngTeMand Ag0,86Pb18SbTe20
were synthesized by Eric. Quarez in the laboratory of Prof. Mercouri Kanatzidis by
annealing, in quartz tubes under vacuum, mixtures of Ag, Pb, Sb, and Te elements
at 1000 °C for 8 h. This was followed by a fast cooling to 850 °C for 1 h, slow cooling
to 800 °C for 12 h, and then cooling to 400 °C for 12 h. This method of cooling

produces more consistent samples.

3.3.2 High energy x—ray diffraction experiments

X—ray diffraction measurements were made on the AngmeTem+2 series of materials
with m = 6, 12, 18 and 00 at room temperature using the recently developed rapid
acquisition pair distribution function (RA-PDF) approach [67] at. the MU-CAT 6-ID-
D beam-line at the Advanced Photon Source (APS) at Argonne National Laboratory.

X—ray powder diffraction samples were prepared by carefully grinding the com-

pounds in a mortar and pestle and sieving through a 400-mesh sieve. The pow-

34

ders were packed into hollow flat aluminum plate sample containers with a radius of
0.25 cm and thickness of 1.0 mm, sealed between thin Kapton films.

The. x-ray energy used was 87.005 keV (A = 0.14248 A). The data were collected
using a circular image plate (IP) camera IN'Iar345, 345 mm in diameter. The camera
was mounted orthogonally to the beam path with a sample—to—detector distance of
208.86 mm which was determined by calibrating with a silicon standard sample [67].

In order to avoid saturation of the detector, each n’Ieasurement was carried out by
multiple exposure to the x-rays. Each exposure lasted 10 seconds, and each sample
was exposed five times to improve the counting statistics. An example of the raw
data on the image plate is shown in Figure 3.1(a).

All raw 2D data were integrated and converted to intensity versus 26 format us-
ing the Fit2D program package [77], where 26? is the angle between the incident and
scattered x-rays. Data sets for the same sample were combined using the same pro-
gram. Data for the empty container were also collected and subtracted from the
sample data during the correction step. Standard corrections for multiple scattering,
polarization, absorption, Compton scattering and Laue diffuse scattering were ap-
plied to the integrated data to obtain the reduced total structure function F (Q), as
described in detail in Data correction and processing utilized the PDFgetX2 program
package [68]. An example of the F (Q) for the m = 18 sample is shown in Figure
3.1. Sine Fourier transformation of F (Q) gives the atomic PDF, G(r), according to
G(r) = g 3:: F (Q) sin(Qr) dQ, where Q is the magnitude of the scattering vec-
tor. The good statistics in the high-Q region of the data (Figure 3.1(b)) allowed a
Qmax = 26.5 A‘1 to be used which gives high-quality PDFs with good resolution. This
is evident in Figure 3.1(c) where G (r) is plotted for the sample Ag0,86Pb18SbTe20.

The G(r) data for all samples are plotted in Figure 3.2 on top of each other. The
difference curves plotted below are the differences between the different m-value PDFs

. and pure PbTe. The difference curves show thctuations that are much larger than

35

 

   

 

 

ILIJHIIIII
5 10152025

 

 

—1
(2(4 )
__ I I r I I l I l I l I l I l I l I l I _,
‘0.— c _
N
A - —I
N
' I
50
U — -
“3.
(\I2_ _
[ I l I l I l I l I l I l I l I l I l I ‘

 

 

 

2 4 6 81012141618
1‘03)

Figure 3.1: (a) The raw data diffraction pattern observed on the image plate. (b)
F (Q) and (c) G(r) for the .Ag0_x(‘,Pblgst€20 sample. In the Fourier transform, QM,
Was set to 26.5 A“.

36

 

 

G (13(2)

LSD—haw AVA AAVAA M A A AAA/f
WW WI

1 I i I I l l l i l I l l
2 4 6 8 1 O 12 14 16 18
r (A)
Figure 3.2: G(r) and DG(r) (compared to PbTe) for samples with different m value.

Magenta curve is for PbTe, blue curves are for sample AgogepbmeTezo, green for
sample Anglgste14, red for Ang6SbTe8.

 

 

 

 

 

 

the estimated random errors on the data and therefore have a real origin, encoding
the local structural differences between the AngmeTem+2 and PbTe compounds.
The fluctuations in the difference curves are highly correlated between the different
m—values, growing in amplitude from m = 18, 12 to 6, as expected. This suggests
that the local structures in each case are similar and gives some confidence that the
results from lower fn-value compounds can give insight about higher m-members.
It also gives us confidence that the smaller ripples in the difference curve from the
m = 18 compound have a real structural origin. m = 12 and 2.5 to compare with

m = 6 are shown also.

37

3.3.3 Modeling

Structural information was extracted from the PDFs using a full-profile real—space
local-structure refinement method [43] analogous to Rietveld refinement [32]. We
used an updated version[84] of the program PDFFIT [65] to fit the experimental
PDFs. PDFF IT allows for multiple data-sets to be refined and can also handle
multiple phases. Starting from a given structure model and given a set of parameters
to be refined, PDFF IT searches for the best structure that. is consistent with the

experimental PDF data. The residual function (Riv) is used to quantify the agreement

of the calculated PDF from model to experimental data:

 

RID : \/Z:=1W(T£[G0bs(ri)2— GC(IIC(Ti)l2 (31)
Zizl “'(7‘1')Gobs(ri)

Here the weight w(7',-) is set to unity.

In this modeling we took advantage of the ability to refine multiple phases in
PDFFIT. We searched for domains of Ag and Sb rich material embedded in the
PbTe matrix. Provided we fit the PDF over a range of I" that is much less than
the particle diameter, it is a good approximation to model the data as being made
up of two distinct phases. This neglects cross-terms; i.e., atom pairs where one
atom is in one phase and the neighboring atom is in the other phase. However, our
experience suggests that these terms are small and a reasonable and simple starting
point is to neglect these terms and model the phases as distinct (i.e., incoherent).
The HRTEM images suggest that the nano—cluster domains have diameters of the
order of a few nanometers and our fitting is carried out over a range up to 20 A.
Thus, some inconsistencies in the fits in the high-r range should be attributable to
the neglected cross-terms. This approximation can be removed in the future, but only
at the expense of having to fit the data with very large models. The success of the

. current modeling seems to suggest that this is not warranted at this point.

38

In PDFF IT, each phase in the nnllti-phase mixture has its own scale-factor that
is refined. This scale factor reflects both the relative phase-fraction of the phases. but
also any differences in the scattering power of the two phases. which depends on the
respective compositions of the phases. In Chapter 2. we derived the mathematical
equation defining PDF in multiphase. Here we present the equations that allow us
to extract phase fractions from the refined scale factors of the phases. In PDFFIT,

the total PDF G(r) is defined as a sunnnation of the different phases as follows:

G;(Tk) = fsBsU‘UZLprGKUl (3-2)

where fS is the overall scale factor and BS is an experimental resolution factor for data
set .9. The sum is over the different. structural phases, p, in a multiphase refinement

and Gp(7'k.s) is the model PDF for a single phase p. The weighted abundance of

(b >2 i ' -
(5;, 3,3 where (bp) and (b) are the averaged scattermg

 

each phase is given by fp 2
factors for phase p and the whole sample, respectively, and NI) and N are the total
atom number for phase p and the whole sample. We can easily calculate 1133 from the
stoichiometry of phase 1) and the whole sample. After refinement we extract %1’- from
the weighted scale factor and then compare it to the calculated one to see whether
the refinement. result is self-consistent with the known stoichiometry. For example,
let’s suppose we use two phases PbTe and AngbeTeHg to model Ang,,,SbTem+2

[7 ‘ . . _ A-N‘r ) 1 iv
. We can set up the f(_)llow1ng two equations to get. —f{2111 and 1“:

N1. LN! N e
_ = -1“.— + _ (3.3)
2:1: + 4 m(21: + 4) 2m
and
N1 + pre = N. (3.4)

. . N . .
Since :1: and m are known we can extract the expected ratio 13 for comparison with

39

the value obtained from the I“(.‘ll11<‘1'll€llt.

To test this procedure. we used a sample made by mechanically mixing PbTe
and AngTeg powders with atom number ratio of 1:3 and carried out a two-phase
refinement. The refined value of if was 0.69 compared to the expected values of 0.75.
This suggests that we can obtain the phase fractions to an accuracy at. the 10% level.

Each data-set (for the finite-m cases) was modeled with a sequence of models.
Model H is a single. phase homogeneous model of the correct average composition.
Models N C0”, N C 1,,. NC 2”, NC 3,, and NC4n are two-phase models that test for
the presence and nature of nanoscale clusters in the material (‘NC’ refers to nano-
cluster). In all the NC models, the first phase is always a pure PbTe component. The
second phase comes from the embedded nanoclusters where we have tried different
models varying their composition and chemical ordering. The number after the NC,
‘0’, ‘1’, ‘2’, ‘3’ or ‘4’, refers to the increasing Pb component in the second phase
as will be explained in more detail later. The integer index n refers to a different
chemically ordered variant of each nanocluster model, where n increases when the
chemical ordering in the special variant increases.

In solid solution model H, one homogeneous phase is defined in which the dopant
Ag, Sb atoms randomly occupy the Pb sublattice. The cubic symmetry of the PbTe
matrix is retained, and thus only one lattice parameter is refined. These models have
four refinable structural parameters and two experimental parameters for a total of
six refinable parameters. The PbTe structure is shown in Figure 3.3(a).

In the case of N C0”, a two-phase model is applied. The major phase is still PbTe.
The chemical component of the second phase is the same as bulk AngTe2 [85]. In
this model there are no Pb atoms inside the minor phase. For the minor phase
of this model we tried both a chemically disordered cluster model NCOO with a
cubic unit cell and Ag, Sb atoms distributed randomly on the lead sublattice (Figure

. 3.3(b)) and a tetragonal unit cell with Ag and Sb atoms chemically ordered on the Pb

40

0
9)
Q.
.J
(.2.

 

 

 

 

 

 

 

Ill "l . .
e“ .‘fT—w—lif er
. II! .
e eat-a .
eeaea ec‘re‘cfic‘v
.pb .Te IAg/Sb(NCOo)
Ag/Sb/Pb(NC2°)
(c’fi’é’cfi’o‘ié‘ (”eases
e" smite 6" a fit): e
all Lalo, eases
Ii ii Ill 0" 6” "‘7‘ It" 0”
3 wt, ._._ a V
’ae'“““l" aIiaIaeoAg
C
L. 6‘ o" o“ 6* a” .a .-_—.a_—.~ .s
.IAMNCOI) AISbINcoa as. ‘3 0?" ‘3 6“"
l AQISMNCZI) LPbINc2,)

Figure 3.3: The unit cells for different models are shown here. (a) is the PbTe major
phase. In all plots Te is shown as red atoms and Pb as green. (b) Chemically disor—
dered AngTe2 in N 0'01 and chemically disordered AngQSbTe4 in N C20. (c) Chem-
ically ordered AngTez in N001 and partially chemically disordered AngngTe4
in N 02,. (d) Chemically ordered AngQSbTe4 in model N C22 resulting in 2-fold
Sllpercell along one crystal axis. III all models the Te (red) sublattice is not changed.

41

sublattice sites (N CO]. Figure 3.3(c)). These models have nine and eleven structural
parameters, respectively. resulting in eleven and thirteen total refinable parameters.

The model NC?" also contains two phases. the major phase is still PbTe while
the minor phase contains atoms with the chemical composition of AngngTe4. In
this model, we also tried various possible chemical ordering possibilities for the minor
phases, which can be totally chemically ordered ( N C22), partially chemically ordered
(N C 21) and totally chemically disordered (NC20). In the totally chemically ordered
case, the unit. cell contains 16 atoms forming 4 layers as shown in Figure 3.3(d).
There are two types of layer. Ag. Sb and Te atoms form one type of layer and Pb, Te
atoms form the second type. The two types of layer intersect with each other. The
two lattice parameters in the plane of the layer are the same, but the lattice parameter
in the perpendicular direction is approximately doubled. The resulting symmetry is
refined as tetragonal. This model has twelve structural parameters and fourteen total
refinable parameters. In the N C21 variant, Ag and Sb atoms distribute randomly
in their plane but do not substitute on the Pb or Te sites (Figure 3.3(c)). In the
N C20 case, (Figure 3.3(b)) Pb, Sb and Ag atoms distribute randomly on the metal
sublattice of the whole minor phase resulting in a cubic structure. In both of the two
latter cases, there are only eight atoms in the unit cell. These models have eleven
and nine structural. and thirteen and eleven total refinable parameters, respectively.

Models N C 10, N C30 and N C40 are almost the same as model N C20 except
that the chemical compositions of the minor phase are AngSbTeg, AngngTe5 and
Ang4SbTe5, respectively. The modeling of the different N C2,, models indicated that
the PDF was not sensitive to the degree of chemical ordering in the minor phase and
the results for chemically ordered or partially ordered cases of models N C In, N C 3n
and N C4,, are not presented here.

All refinements were performed over the range of PDF from 2.85 A to 20 A. The

. PbTe end member compound was fit with a homogeneous model H and two-phase

42

models N C 1,, and NC 2,,. All models were fit to the m. = 6, 12 and 18 datasets.

3.4 Results

First we consider the pure PbTe end-member cornImund. The homogeneous model H,
as expected, fit reasonably well resulting in an Rw = 0.086. Displacement parameters,
Um, for Te and Pb atoms are 0.013 A2 and 0.029 A2 respectively and the lattice
parameter is 6.47 A. Refining the two—phase model NC 0,, and N C2,, to the PbTe data
did not. result in an improvement in Ru! despite the greater number of parameters.
The scale factor for the second, non-physical, phase becomes very small (smaller
than 0.3 percent), and the displacement parameters in this phase also become very
large, indicating that the fit is attempting to eliminate the second phase. The result
from the two phase refinement shows that the PDF is able to distinguish single from
two—phase behavior.

We now turn our attention to the m = 6 compound that has the largest volume
fraction of second phase in it. First this was fit with the homogeneous model H. The
fit is poor as shown in Figure 3.4(a), with Rm 2 0.212.

Significantly better fits were obtained from the two-phase models (Table 3.1, Ta-
ble 3.2 and Figure 3.4(b)) with Rw = 0.0724 from the chemically disordered model
N C20. The refined values are shown in Table 3.1 and Table 3.2. Similar results were
obtained from the chemically disordered models. This analysis strongly suggests that
the Ag and Sb clusters are present in the bulk of the material and are not an artifact
of the TEM measurement.

We now wish to differentiate between the different composition two—phase mod-
els N COn—N C4". In terms of fit to the data and Rw, all four models performed
comparably well, both in the chemically ordered and disordered states. The refined
. parameters that produce these good fits allow us to differentiate somewhat between

the models. In particular, the refined phase fractions for the two phase refinements

43

 

 

 

 

I I I l I l I I I I I I I l I I III
a
m— : () —
: ,1
A
CI ‘ I‘
I
m- '. —
I
.AAAAAAAM._A AA AAA A
I
I l I l I ILILJ I I I I I I I I I
5 ' I ' I ' rrrfl ' I ' I ' I ' I '
u :- cl
(b)
N— 4
01
N_ _.
I
m-.‘/\A AMA A AVA
v vvv v' ‘vv vvwv vv
I II I I I I I I I I I I l I I I l I

 

 

 

2 4 6 8 10 12 14 16 18
r (3)
Figure 3.4: (a) PDF from the homogeneous H model for sample AngeSbTeg . The
line with empty circles is the data. the solid line is the calculated curve from the

fitting and the line offset below is their difference. (b) Chemically disordered case of
model N C20 for AngGSbTeg. Line attributions are the same as in (a).

44

'

Table 3.1: Results from PDFFIT for the AngSSbTeB sample. n = Alli-V91“ is the ratio
of atom numbers in PbTe phase to whole sample, no is the expected ratio calculated
from the chemical stoichiometry (see text for details). Unto," are the displacement
parameters for atoms on different sites.

 

 

 

 

 

 

 

 

model H NCOO N C01 NC 10
PbTe Ru. 0.22 0.066 0.065 0.070
n/no — 0.276/0750 0257/075 0321/0625
a — 6.41 6.41 6.41
Urn. ~ 0.0291 0.0255 0.0299
U pb -~ 0.0295 0.0307 0.0297
Phase 2 a 6.33 6.22 6.21 6.22
c - — 6.24 —
Up. 0.080 0.0480 0.0488 0.0415
U pb 0.061 ~ ~ 0.08444
UAg 0.061 0.0782 0.0382 0.0844
U 5% 0.061 0.0782 0.203 0.0844

 

 

 

Table 3.2: Results from PDF FIT for the AngeSbTeg sample. n = flfim is the ratio
of atom numbers in PbTe phase to whole sample, no is the expected ratio calculated
from the chemical stoichiometry (see text for details). anm are the displacement

parameters for atoms on different sites.

 

 

 

 

 

 

 

 

 

 

N020 NC21 N022 NC30 NC40
PbTe Ru, 0.072 0.070 0.075 0.070 0.071
n/no 0358/0500 0383/0500 0372/0500 0.376/0325 0404/0250
a 6.41 6.41 6.41 6.41 6.41
UTe 0.0285 0.0305 0.0297 0.0299 0.0299
U [)5 0.0320 0.0298 0.0311 0.0294 0.0293
Phase 2 a 6.22 6.23 6.226 6.219 6.220
c - 6.19 12.40 - -
UTe 0.0704 0.0409 0.0325 0.0384 0.0377
U pb 0.0550 0.0852 0.0875 0.08724 0.0879
UAg 0.0550 0.0852 0.0875 0.08724 0.0879
U51, 0.0550 0.0852 0.0875 0.0872 0.0879

 

45

 

can be compared with the values that should be obtained based on the overall chem-
ical composition of the material. As can be seen in Table 3.1 and Table 3.2, the
N00,, and NC 1,, models significantly underestimate, and NC4,, significantly overes-
timates, the phase fraction. The N02,, and N03,, compositions give phase fractions
much closer to those stoichiometrically expected, with N03,, giving the best agree-
ment. This is strong evidence that. the nanoclusters contain significant amounts of
Pb atoms and are not pure AngGSbTC-g .

The refinements suggest that the average composition of the nanoclusters is
“AngngTe5”. However, it is unlikely that the real clusters have this composi-
tion since it is not possible to construct an ordered model with this composition by
interleaving Ag/ Sb and Pb layers on the Pb sublattice; it is necessary to have a
layer with Ag/ Sb mixed with Pb. As we discuss below, this is not expected on the-
oretical grounds. It could come about due to the presence of anti-phase boundaries
between Pb regions and Ag/ Sb regions, in analogy with the Na3BiO4 material stud-
ied previously [83], though it seems unlikely that this can occur within an individual
nanocluster. From this point of view, it seems more likely that clusters with composi-
tions of AngQSbTe4 and Ang4SbTe5 coexist in the matrix yielding, on average, the
observed “Ang3SbTe5” composition. It should also be noted that some uncertainty
exists in the two—phase modeling, especially taking into account the fact that we are
modeling coherently embedded nanoclusters approximated as an incoherent mixture.
The strong result is that significant Pb content exists in the nanoclusters but there
is probably some uncertainty on the precise value.

We investigated the chemical ordering within the nanoclusters by focusing on the
N02,, model that lends itself to rational chemically ordered models. Refinements
of the chemically disordered and partially ordered variants of models N 02,, yielded
comparable fits to the chemically ordered fits, with comparable values of refined

parameters (Table 3.1 and Table 3.2) suggesting that. the current PDF measurements

46

Table 3.3: Results from model N 03o for three different m-members. n.

NV. -
—_L.I;_LLIS

\

the ratio of atom numbers in the PbTe phase to whole sample, no is the expected
ratio calculated from chemical stoichiometry. Unto", is the thermal factor for atoms

on different site.

 

 

 

 

 

 

 

 

 

AngoSbTeo AngnSbTeM AgogonISSbTego PbTe

PbTe Ru. 0.072 0.066 0.074 0.086
n/no 0376/0325 0571/0643 0693/0750 -

a 6.41 6.43 6.45 6.47

UT, 0.029 0.0246 0.0214 0.013

U pb 0.032 0.0280 0.0262 0.029
Phase 2 a 6.22 6.26 6.29 -
UT, 0.0371 0.0779 0.0826 -
UH, 0.0815 0.0876 0.0691 -
UM 0.0815 0.0876 0.0691 -
Ugo 0.0815 0.0876 0.0691 -

 

 

alone are not sensitive enough to differentiate the chemical ordering within the minor
phase.

Finally, we note that similar results were obtained when the m = 12 and m = 18
samples were refined in the same way. The results for the chemically disordered
“Angoste5” model are presented in Table 3.3. The refined phase fractions nicely
track the nominal composition giving us good confidence that the two-phase modeling
is giving physically meaningful results and that nanoclusters of average composition

close to Ang3SbTe5 are present.

3.5 Discussion

The success of models N02,, and N03,, verify that the TEM observations of nan-
oclusters reflect a bulk average property of this material. These models also provide
evidence for the chemical composition of the minor phase and give a hint to the
chemical distribution of Ag, Sb and Pb atoms in the minor phase, although little
information is available about the degree of chemical ordering.

In Figure 3.5 we show a HRTEM image that suggests that clusters are present

47

 

Figure 35: A HRTEM image of a region of a sample of AgooonwaTego. The four
smaller pictures at the side are the amplified pictures for different (lattice) local region
and their fourier transformed images [1]

that result in a doubling of the lattice parameter in the second phase, though not
all clusters show this behavior. This is consistent with the partially or fully ordered
model N02,, variants, n = 1, 2, which alternate Pb and Ag/ Sb layers on the metallic
sublattice. The fully chemically ordered case in model N 022 was found to be the
stable configuration in a coulomb lattice-gas Monte Carlo simulation study of the
ground state of this system as a function of m [86]. Thus, we believe that clusters
with the totally chemically ordered form in model N 022 (Figure 3.3(d)) are present
as nanoclusters in the large m compounds. This may not be the unique form of the
nanoclusters, and indeed, not all the nanoclusters evident in the TEM images show
this cell doubling. They presumably form by a nano—phase separation of constituents
accompanied by an imperfect and defective ordering and there appears to be con—
siderable spatial disorder of the chemical constituents; though the nanoclusters are
coherently endotaxially embedded in the matrix, they are not well ordered. Incorpo-
rating more Pb in the nanoclusters allows the system to balance its desire to phase
separate, with maintaining a degree of lattice matching to keep the nanoparticles
embedded in the matrix without incoherent interfaces. The refined lattice parame-

ters for the nanocluster phases are smaller than the matrix: 6.23-6.29 A compared

48

to 6.41 A for the strained matrix and 6.47 A for relaxed PbTe. Both the chemical
inhomogeneities and the inhomogeneous lattice strain are likely to increase phonon
scattering. The size of the nanoclusters may also be important in making this scat-
tering mechanism effective. It was claimed in [81] that the short-range nature of the
local chemical ordering can broaden out. Fermi-surface resonances which plays an im-
portant role in thermopower enhancement. More recent theoretical calculations on
electronic conductivity and Seebeck coefficient suggest that the power factor is only
slightly modified in the Angn,SbTe,,,+2syst.em [87]. The main reason for observing
enhanced Z T in this system comes from a reduction of the lattice thermoconductivity
due to the presence of nanoclusters [88, 87]. This is consistent with people’s intuition
on such system and also our result. Presumably, the size of the nanoclusters, their
exact composition, the atomic ordering within them and their concentration with
PbTe will be a sensitive function of the preparation conditions.

Adding Pb atoms in the minor phase greatly improves the result of the refinement.
The reason is that Pb atomic number is much larger than the atomic numbers of Ag,

Sb and Te and its scattering factor is quite different from those of the other three.

3.6 Summary

In this structural study based on the PDF method, we verified that in the bulk
material of AngmeTem+2, nanoclusters of a minor phase containing Ag, Pb, Sb and
Te atoms form in the matrix of PbTe. We give evidence showing that the chemical
composition of the minor phase is most likely between AngQSbTe4 and Ang4SbTeo.
We propose a structure for the minor phase based on PDF, TEM and theoretical

considerations.

49

Chapter 4

Thermoelectric material

Agl—xsnSb1+xTe3

4.1 Interesting physics in the Ag1_,SnSb1+,Te3 sys-
tem

Theoretically there is no upper limit for ZT but, physically, electron conductance
a, Seebeck coefficient S and thermal conductance K. depend on each other. This is
why it is difficult to find high ZT materials. Traditionally, heavily doped (~ 1019
carriers/c1113) semiconductors provide the best compromise in getting high Z T [2]. All
Inaterials used in current thermoelectric devices are heavily doped semiconductors.
When the doping exceeds 1020 carriers/cm3, it generally leads to a very small
Seebeck coefficient, S < 60uV/ K [30]. Agoo5SnSb1, 15Te3 is unusual in that it shows
an almost metallic carrier concentration(~ 5 x 102’cm‘3) but also possess a large
Seebeck coefficient of the order of ~ 160,11V/ K at 600K [30]. Agog5SnSbL15Te3 is a
non-stoichiometric derivative of AgSnSbTeo, which is formed from the combination
of two narrow band-gap semiconductors, AngTe-z and SnITe4, both of which adopt

the rock salt (NaCl) structure. AgoooSnSbHoTeo exhibits a large thermoelectric

50

power response. The carriers are holes that have an anomalously l‘ieavy mass and
exhibit other anomalous properties such as a quadratic temperature dependence of
the resistivity over a very wide range of T [30]. There are many poorly understood
aspects of this material and its high ZT, not least the fact that it is p—type, rather
than an n~type semiconductor at all [30].

As with the other thermoelectric materials studied in this thesis work, HRTEM
indicates that AgogoSnSbuoTeo is a nano—structured composite rather than a solid
solution [30] as seen in Figure 4.1. This nano—structure is evident throughout the
crystal and the material is thought of as a type of bulk nano—composite [30]. The
nano—regions have a structure that is significantly modified from the matrix (Fig-
ure 4.1(b)) and coherently embedded (Figure 4.1(c)). These compositional fluctu-
ations at the nanoscopic level are similar to those of AngmeTen,+2 system (as

described in Chapter 3).

4.2 PDF study of the Ag1_,SnSb1+,Te3 system

4.2.1 Introduction

The composition and atomic arrangements within the nanoclusters is a challenging
topic since the clusters are not periodically long-range ordered, but dispersed inside a
matrix, and cannot be studied crystallography. A method sensitive to local structure
is needed such as the atomic pair distribution function (PDF) analysis of X-ray pow-
der diffraction data [35]. As discussed in chapter 3, this approach was successfully
used to study chemical short-range ordered clusters randomly embedded in a parent
matrix [1, 83], in analogy with the present situation. Here we report a PDF study of
the compounds Ag1_,,.SnSb1+,Te3. For comparison we also studied the end member

compounds, AngTeQ, and AgSnQSbTe4.

51

 

Figure 4.1: (a) High resolution TEM image of Ago85SnSbuoTe3 that shows a nano-
structured region of the crystal. The nano—structures have geometrical dimensions of
3 - 30 nm and are dispersed throughout the crystal evenly. (b) A close-up View of an
embedded nanocrystal. (c) The interface region between the embedded nanocrystal
and the matrix showing a high degree of coherency. (d) Dark field imaging over a
limited area in the crystal that shows the compositional variations between the matrix
and the embedded nanocrystal. (Scales for (a) 10 nanometers, (b) 5 nm, (c) 20 nm,
((1) 10 nm) [reproduced from [89] with permission from M. G. Kanatzidis]

52

 

 

 

4.2.2 Experimental Details
Sample Preparation

SII4T€4 and AngTe2 are the natural end-member compositions for the Ag 1-,SIISb1+,Te3
system. Since a nano—phase separation is observed in the HRTEM image for Agoo5SnSb115Te3
sample (Figure 4.1), we want to see which compound in the phase diagram of this sys-
tem might be the possible nano—phase observed. Accordingly we considered a number
of compositions across the phase diagram: with at = 0 (AngTeg), 0.2 (AgQSHSb2T65),
0.25 (AggSnngoTeg), 0.333 (AgSnSbTeo), 0.5 (AgSn2SbTe4) and 1.0 (Sn4Te4).

The Ag1_,,SnSb1+,,Te3 samples were produced by John Androulakis in the group
of Prof. M. G. Kanatzidis by mixing appropriate stoichiometric ratios of high purity
Ag, Sn, Sb and Te. The initial charge was sealed in evacuated fused silica tubes,
heated at 1000°C, held there for 4 days and then slowly cooled to room temperature.

Data from the same crystal batch were used.

High energy x-ray diffraction experiments

X-ray diffraction measurements were made on the Ag1_,SnSb1+,Te3 series of mate-
rials (Sn4Te4, AgoosSnSblooTeo, AngTeg, AgSnSbTeo, AgSnngTe4, AgosnSb3Te7,
AgQSnSbgTeo, Ag3sn28b3T88, Ag4Sn38b4Te11) at room temperature using the RA-
PDF approach described in Chapter 2 at the MU-CAT 6-ID-D beam-line at the
Advanced Photon Source (APS) at Argonne National Laboratory.

X-ray powder diffraction samples were made by carefully grinding the compounds
in a mortar and pestle and sieving through a 400-mesh sieve. The powders were
packed into hollow fiat aluminum plate sample containers with a radius of 0.25 cm
and thickness of 1.0 mm, sealed between thin Kapton films.

The x-ray energy used was 87.005 keV (A = 0.14248 A). The data were collected
. using a circular image plate (IP) camera Mar345, 345 mm in diameter. The camera

was mounted orthogonally to the beam path with a sample-to—detector distance of

53

 

Figure 4.2: Raw diffraction data on the 2D image plate from AgSngste4. Note the
excellent powder statistics.

208.86 mm which was determined by calibrating with a silicon standard sample [67].

In order to avoid saturation of the detector, each measurement was carried out by
multiple exposure to the x-rays. Each exposure lasted 10 seconds, and each sample
Was exposed five times to improve the counting statistics. An example of the raw
data on the image plate is shown in Figure 4.2. As described in Chapter 3, Fit2D
and PDFgetX2 program are used to process the data. Good statistics in the high-Q
region of the data allowed a Qma, = 26.0 A‘1 to be used which gives high-quality
PDFs with good resolution.

The F (Q) and G(r) for all samples are plotted in Figure 4.3(a) with the corre-

' ‘ Sponding G(r) functions in Figure 4.3(b). The data quality are good as evidenced by

54

 

(n)

30

    

 

AngTeZ
AgZSnSb2Te5
Agz Sn2
AgZQSnZSbSITeB 3
of‘ A30 Sbl Sn'I'e3
AgOB5Sb115SnTe3 5
U
AgsIISb'rea o i
'4
: ° 4
AngSnZTM '
Sn'I'e S-‘
o '
I I I l I

 

 

 

0|:—

15

10 _1,'> 20 25 10
NA) r(l)

Figure 4.3: F (Q) and G(r) for all samples in the Ag1_,.SnSb1+,.Te3 series.

the good signal-noise of the diffraction data and the low amplitude of fluctuations in
the unphysical region of G(r) below 1' = 2 A in all the curves of Figure 4.3(b).

The G(r) data for all the samples are plotted in Figure 4.4 on top of each other
with difference curves below, where the difference curves are defined as A00”) =
G(r) — GsnTe(7‘)- The difference curves show fluctuations that are much larger than
the estimated random errors on the data and therefore have a real origin, encoding

the local structural differences between the Ag1_,SnSb1+,Te3 and Sn4Te4 compounds.

Results

We would like to know whether phase separation is a bulk property for the
Ag1_,SnSb1+,Te3 system and the chemical composition and structure of the nan—
odomains. Accordingly we would like to search for Ag— and Sb—rich domains embed-
ded in the SnTe matrix. Because the atomic numbers of the cations in the system are

very close to each other (Ag 47, Sn 50, Sb 51), the x—ray PDF method we have used

55

 

j l I l I

I Sn'I'e --—4

“SbSnZI'M -—«
A3085Sb115SnT03 *—

Ag298n28b31'l'08 0—‘

r I Agzsnszres .—

“‘ .Tez ——

10

I-
t»

D

’

" .-
p

I

\

i

G (A'z)
O
C I
‘I
I
s
\-
l ‘; -
- ’
\
‘
) \ _

 

 

5 10 15
r (A)

Figure 4.4: G(r) and A0(r) (compared to Sn4Te4) for samples with different chemical
compositions.

is not sensitive to the chemical ordering of the system. However, by simple modeling
and combining the result with theory, we can still get a clue of structure, especially
atomic distortions, of such a system. This was done by studying different compo-
sitions across the Ag1_,SnSb1+,Te3 phase diagram, which might be line-compounds
and participating in the nano-phase separation that is observed in Ago_85SnSb1,15Te3.

We fit a simple single-phase model to data from all the samples in the
Ag1_,SnSbl+,Te3 system. Fitting range is from 2.75 A to 20 A. Model H0 is a
single phase homogeneous model of the correct average composition. In the model,
the structure is NaCl like. Tellurium atoms occupy the Cl‘ sites, while all the cations
are distributed homogeneously on the Na+ sites. In Table 4.1, we list the Ru, and
refined isotropic thermal factors on each site. The resulting lattice parameters and
thermal factors for different samples are plotted as a function of x in Figures 4.5, 4.6

and 4.7.

56

 

71

 

 

 

 

 

 

 

R41! UTe (A2) brSnAng (A2)
SnTe 0.097 0.0149 0.0262
AgSn-szTeo 0.081 0.0230 0.0395
AgSnSbTeg 0.088 0.0254 0.0433
AgggSDQSbngeg 0.097 0.0271 0.0494
AggSnSboTeg, 0.090 00259 0.0455
AngTe2 0.111 0.0294 0.0510

 

 

 

 

Table 4.1: Refinement. result from the single-phase model to each of the samples in
the Ag1_,SnSb1+xTeo system.

 

 

 

 

 

v.
o . , . , . , . , .
A
«1:
v
5.. «a
Q) G
4.3
E
(U
I...
“'5 N.
‘1,»
a)
0
OH
4..)
4..)
.3
'1‘
I I I L 4 I I I
0 0.2 0.4 0.6 0.8
Sn ratio

Figure 4.5: Lattice parameters from the single-phase refinements for samples with
different chemical compositions in the Ag1_,,SnSb1+,.Te3 series.

In each case, the homogeneous solid-solution model does a reasonable job fitting
the data with small Ros. Reasonable, though slightly large, atomic displacement
factors are obtained on the anion site, but significantly enlarged ones are seen on the

cation site. Note, that we are not using an ordered model, even in AngTeg. The

57

 

0.05

 

 

 

Q) T I O I O' I I I I T T I
4.)
"-4
V)
g . O .
13' Q o
as
O 3
C.‘ o'” O J
OH
I...
o
4.)
o
(0
q...
'8
E 8" i
O
f...
o
.c: c
E L I L L I I I I L I L I ¥J_ I J I I I
O 0.1 0.2 0.3 0.4 0.5 0.8 0.7 0.8 0.9
Sn ratio

Figure 4.6: Thermal factors for the cation site from single-phase refinement for sam-
ples with different chemical composition in the Ag1_,SnSb1+,Te3 series.

large ADPs presumably reflect the fact that the bond-lengths of Ag and Sb to Te are
different but this is not allowed in the homogeneous model.

Lattice parameters obtained from one phase refinement are shown in Figure 4.5
with the Vegard’s law behavior shown. The refined value has negative offset from
the linear behavior of Vegard’s law [90, 91]. Vegard’s law is an empirical rule which
says that a linear relation exists, at constant temperature, between the crystal lattice
constant of an alloy and the concentrations of the constituent elements. Vegard’s law
applies when the solute and solvent have similar bonding properties. Deviations from
Vegard’s law are the norm rather than the exception and so the behavior observed
here is not unusual.

In a solid solution system the thermal factors are expected to show a. parabolic

58

 

0.05

 

 

 

r I I I T I I I I I I
(D
5..
c...
0 <-
q_ _
Li C
O
4.)
O L
(0
q...
I—I
(U ('3
a -
O
8
CD 0
CD
ES. 0 o .
O
8. I 4 I I I m I L I I I 4 L 4* I m I 4
00 0.1 0.2 0.3 0.4 0.5 0.8 0.7 0.8 0.9
Sn ratio
Figure 4.7: Thermal factors for the Te site from single—phase re-

finements for samples with different chemical composition in the
Ag1_,SnSb1+,Te3 system.

behavior with a peak at around 50%. This is qualitatively observed on the anion site
(Figure 4.6, with a couple of outliers (Agoo5SnSb145Te3 and AggSnng3Te3). The
anion sites have significantly smaller ADPs (Figure 4.7) and do not show parabolic
behavior. These results are slightly surprising as in similar semiconductor alloy sys-
tems [92, 57] the greatest disorder is seen on the opposite sublattice to that which is
being alloyed, whereas in this case the large ADPs are seen on the mixed ion sublattice
with little disorder on the Te sublattice.
Thermal factors in cation sites for two samples (Ago_85SnSb1_15Te3 and AgoSnngoTeo)

appear different from others. They do not seem to follow the parabolic relation. This
might mean that the structures for Ago_8oSnSb1_15Te3 and AgoSnosboTeo samples are

not solid solution. It was already found in AgSnngTe4 sample that phase separation

59

 

 

 

 

 

 

 

 

model Ho model N 00o
1?, 0.112 0.116
SILITP4 n/no — 002/0333
a, (A) 6.135 6.216
UT, (AZ) 00256 0.088
Um... (A?) 00494 0.024
AngTe-2 a (A) — 6.129
UT, (A?) — 0.0269
UAW (A?) — 0.0613

 

 

Table 4.2: Comparison of results from one-phase and two—phase refinements for
AgoooSnSbHoTeo sample.

 

 

 

 

R... 0114

a (A) 6.135
(1,, (11?) 0.0258
How-0,, (A?) 00402

 

 

 

 

Table 43: Result from physically wrong model IV.

does appear. So it is quite important. to get HRTEM image from Ag38n2Sb3Te8 sam-
ple. It is possible that AgoSnngoTeo sample might be the line compound where

chemical ordering is formed in this material. Then the nano—phase observed in
Agog5SnSb1ooTe3 sample might actually be AgoSnoSboTeo .

To explore the possibility of nano—scale phase separation we tried a number of
two-phase models. Because the PDF is not sensitive to chemical ordering in this case
we only tried the totally disordered case of the minor phase where n. = 0.

Next we compare the results from single— and two-phase models to the
AgogsSnSbusTeo sample. The results are summarized in Table 4.2. R.,,, from both
models are reasonably good the refined values look physically reasonable.

The refined values are shown in Table. 4.2. We can see there is not much difference
in the physical parameters between the two different models. The refined phase factor
from the two phase model is quite unphysical. It looks like the compound is just

AngTegbut this is impossible. An explanation is that our technique is insensitive to

the chemical composition. To verify this, we tried a model IV with a totally incorrect

60

strfichiometry. We use only a single-phase model with the structure of AngTeo. We
still get a rea.sonal_)le refinement, result as shown in Table 4.3. This proves that our
x-ray PDFs are very insensitive to chemical composition. This presents a difficulty
when fitting two—phase models and so we did not pursue this further.

The amount of additional information furnished by PDF in the case of the
Ag1_,SnSb1+,Teo was rather limited due to the similarity in x-ray scattering length
of the elements involved in the alloy. This work could be pursued further in the
future by combining x-ray data with neutron diffraction and PDF data, and with

complementary techniques such as EXAFS.

61

Chapter 5

Thermoelectric PbTe/PbS system

5.1 Physics of the (PbTe)1_,(PbS), system

The thermoelectric properties of PbTeoMSooo are reported to be far superior to that
of PbTe [89] . The highly desirable properties of PbTe, such as its high electron
mobility, are retained, but a much lower lattice thermal conductivity(039 W/mK,
only 28% that. of the PbTe at room temperature) is obtained. The system shows
a ZT ~ 1.28 at 660 K that, it is expected, can be further enhanced by optimized
doping.

A recent HRTEM study reveals spinodal decomposition happens in the PbTe
composition [89]. A naturally mane—structured lattice with compositional fluctuations
with a characteristic wavelength of about 2-5 mm was found. The HRTEM image is
reproduced in Figure 5.1.

We can see that nano—phase separation happens on three length scales. The peri—
odic compositional fluctuations are characteristic of spinodal decomposition [93].

The phase diagram for the (PbTe)1_,,(PbS)Jr system was obtained around 50 years
ago [94, 95] and it indicates that phase separation occurs over a wide range of compo-
sitions in this system. It is reproduced in Figure 5.2. This was verified in the recent

study by Androulakis et al. [89] and in our work, described below. For example, Fig—

 

Figure 5.1: HRTEM pictures showing multi-scale phase separation happening in some
part of sample. Scales are 20 nanometers in (a) and (b), and 10 nanometers in (c)
and (d). [Reproduced from [89] with permission]

ure 5.3 reproduced from Reference [89] shows phase separation in a PbTe75%-PbS25%
sample.

It is known that unequilibrum states of materials can be created by special heat
treatment which does not allow the system to reach those equilibrium states shown
in the phase diagram. In these state, the system can show interesting behavior in
nano—scale. Theoretical and experimental studies suggest that materials that show
nano—phase separation appear to be promising in getting a high ZT [15, 17, 96, 88].

The material with composition PbTeoMSooo shows a very low room temperature

63

 

   
  

 

 

 

  

 

“.00 ’“"T""""‘""'P‘ 'v-v‘r-rW-r'r-vw“ w-ow ‘r ‘7‘ r~v~-r fi-rh-r-rw—T- -(-'v-r"r"r-9‘T‘r—v—r-
o C 1 h T 'I "r f,
1000 l L E
1 F
l r
1 I
1 F
900 a /;
3 E
9 1
I IPbS,PchI :
000 ‘5 L
l I-
’ :
i i
l I
1 I-
700 ', E“
i > o
g i : e-a
D-o j _. a:
600 i (PbS) + (PbTe) -
3 E
500 4, I.
i I
l .'
l, .r
400 'T'W’.w*' "r—nfi-V'PFV'W‘PI'U'I fi-vwv—vw-r- - f
I-PFH'W "PY-I Wn .'_ ‘
0 IO 20 30 20 50
PbS
“.% Te .- PbTe

Figure 5.2: Equilibrium phase diagram for PbTe-PbS [94]

lattice thermal conductivity of 0.4 VV/m K [89]. This value is only 28% of that
observed in the PbTe system, which is remarkable given that the two are isostructural
and PbTeo,34So,1o has only 16 At. ‘70 of S substituted on the Te site. Understanding
the origin of this remarkable reduction in K. for a small doping change should give
important insights into the thermoelectric problem.

Early studies on the (PbTe)1_,,(PbS),, system showed that phase separation oc-
curs at low temperature over almost the whole doping range [94, 95]. A miscibility
gap exists over a wide range of composition and extends almost up to the melting
point of the alloy. There are no known intermediate compounds and the phase sepa-

ration occurs into phases which are almost pure PbTe and PbS over the whole alloy

64

 

 

 

 
  
 

   

 

 

 

  

 

 

 

 

1axm0 . ,i . , . , . , ., ,r T , .
Iamxm - 5: 5 g’ 4
O 1 N
- a . Y
6 3 2*
140000- ._ j\ 1
' E 8 3 . \M
,g 120mXI, §§ g _
S r {1’ 42 4'3 44 -
-é 100mx1, 2 2nwhthw9 .
3 L ‘
fig ammo -
'7) r g; .
5 axmot- g, A
g I- : A A* .,
3" 8 "S _[
W‘ I: S E.“
* '5; 2 .1»
amxm - n. 2 I_ff_ n
n. 0.
0 -1
l 4 I I l I I I l I l I
20 25 30 35 40 45 50 55

26(degs)

Figure 5.3: In-lab powder X-ray diffraction of PbTe75%-PbS25% sample with indices
showing two phases are present of roughly PbTe and PbS composition. Reproduced
from [89] with permission

range. Theoretical work [95] supports such a picture and the calculated phase di-
agram using a thermodynamic model agreed with the previous experimental data.
Earlier work [97, 98, 99] suggested a smaller range for the miscibility gap in the phase
diagram and this discrepancy was attributed to the subtle difference in chemical pro-
cessing [94] and quenching rate. A high resolution transmission electron microscopy
(HRTEM) study of the interesting PbTeog4Sooo composition revealed that spinodal
decomposition happens in this system [89]. What is apparent from the TEM images
is, first, that phase separation appears to be happening on different length-scales

in PbTeoMSooo and, second, that naturally forming striped nanostructures due to

65

spinodal (‘lecomposition are evident in portions of the sample. Here we investigate
this question further using bulk diffraction probes of the average and local atomic
structure. We address two questions. First. can we confirm that the nano—scale phase
separation is a bulk property and can we characterize the average chemical com—
position and structure of the spinodal domain? We have also extended the study to
other compositions in the phase diagram to see how these effects evolve with changing
composition.
The atomic pair distribution function analysis of x-ray diffraction data is a useful
method for studying nano—phase separated samples [36, 35]. In the PDF approach
both Bragg and diffuse scattering are analyzed and it yields the bulk average local
atomic structure. Recently it was successfully used to study the thermoelectric mate—
rial, Ag,Sbe,,,Te,,,+2, where silver and antimony rich nano—scale clusters were found
to be coherently embedded in the PbTe matrix as a bulk property [1].

We have used both PDF and Rietveld methods to study the (PbTe)1_,,(PbS)I
system. We find phase separation occurring over the whole composition range. Re-
finements from both Rietveld and PDF methods show that the :1: = 0.25, 0.5, and
0.75 samples are macroscopically phase separated into phases that are almost pure

PbS and PbTe. This does not. happen in the important 16% PbS doped sample. How-
ever, taking all the evidence together we suggest that the 16% sample is a nanoscale
mixture of a PbTe rich phase with a partially spinodally decomposed phase of nom—

inally 50% composition. Such a phase was stabilized and observed in a quenched
.Z‘ = 0.5 sample in this study. This offers the opportunity in the future for engineering
nano— and micro-structures with favorable thermoelectric properties by controlling

the thermal history in these materials.

66

5.2 Experimental methods

The materials were made by John Androulakis in the laboratory of professor M. G.
Kanatzidis. Powder samples in the (PbTe)1_,,.(PbS)I series were made with different
compositions: :r = O, 0.16, 0.25, 0.50, 0.75 and 1. The samples were produced by mix-
ing appropriate ratios of high purity elemental starting materials with a small molar
percentage of Pblg, an n-type dopant. The initial loads were sealed in fused silica
tubes under vacuum and fired at. 1273 K for 6 h, followed by rapid cooling to 773 K
and held there over a period of 72 11. One :r = 0.5 sample was also quenched rapidly
to room-temperature. More details of sample synthesis can be found elsewhere [89].
Finely powdered samples were packed in flat plates with a thickness of 1.0 mm
sealed between kapton tape windows. X-ray powder diffraction data were collected
using the rapid acquisition PDF (RA—PDF) method [67], which benefits from very
high energy x-rays and a two—dimensional detector. The experiments were conducted
using synchrotron x—rays with an energy of 86.727 keV (A = 0.14296 A) at the 6—ID-D
beam line at the Advanced Photon Source (APS) at Argonne National Laboratory.
The data were collected using a circular image plate camera (Mar345) 345 mm in
diameter. The camera was mounted orthogonally to the beam path with a sample-
to—detector distance of 210.41 mm.

In order to avoid saturation of the detector, each room temperature measurement
was carried out in multiple exposures. Each exposure lasted 5 seconds, and each
sample was exposed five times to improve the counting statistics. Two representative
2D diffraction images for unquenched and quenched PbTeoi5SO_5 samples are shown
in Figure 5.4(a) andi(b), respectively. The excellent powder statistics, giving uniform
rings, are evident. All the samples yielded similar quality images. The 2D Data sets
from each sample were combined and integrated using the program FIT2D [77] before

further processing.

Data from an empty container were also collected to subtract the container seat-

67

 

Figure 5.4: Raw x-ray powder diffraction data from the 2D detector for the :c = 0.50
(PbTe)1_I(PbS)Jr sample. Data from the (a) unquenched and (b) quenched samples
are shown for comparison. The 1—D integrated powder diffraction patterns obtained
from these data are shown in Figure 5.5(a) and on an expanded scale in Figure 5.7.
The white circle in the center of each 2D difractogram represents a shadow from the

beam-stop.

tering. The corrected total scattering structure function, S (Q), was obtained using
standard corrections [35, 67] with the program PDFgetX2 [68]. Finally, the PDF was
obtained by Fourier transformation of S (Q) according to G(r) = % IO "M Q[S(Q) —
1] sin(Qr) dQ, where Q is the magnitude of the scattering vector. A Qmu = 26.0 A“
was used. Setting Qmar to a lower value results in lower real-space resolution and
higher-amplitude termination ripples and a higher Qmar introduces excessive statis-
tical errors in the data since at higher values of Q the signal to noise ratio becomes

unfavorable, as apparent in Figure 5.5(a). Figure 5.5, shows F(Q) = Q(S(Q) — 1)

and G(r) for all the samples. The good statistics and overall quality of the data are

apparent in Figure 5.5(a). The low spurious ripples at low-r in the G(r) functions
are also testament to the quality of the data [100]. Note that G(r) has been plotted

all the way to 7' = 0 in these plots, which is a stringent test of this.

68

 

I I I l l I

F (21“)
2
—. 3;:
m ow
01 o
N N
10 20
G (Ii—2)

 

 

 

 

 

53
"ll WWW/hr
W” PbTe
O O
I I 1 l J 1 l 4 l l l n
o 10 20 5 10 15
Q (3‘1) 1‘ (A)

Figure 5.5: Experimental (a) F (Q) and (b) G(r) for all unquenched samples. In the
Fourier transform, Q"m was set to 26.0 A“. The data are offset for clarity. The
compositions of the (PbTe)1_I(PbS)I samples are indicated in panel (a). From top
to bottom: :1: = 1.00 (green), 1; = 0.75 (yellow), :1: = 0.50 (magenta), a: = 0.25 (blue),

I = 0.16 (cyan), and :1: = 0.00 (black).
5.3 Modeling

Both PDF (using the PDFfit2 program [66, 65]) and Rietveld [32] (using the TOPAS
program [101]) refinements were carried out on the system. The models used in the
fits are described below.

One of the main outcomes of this study is to determine the phase composition
of the phase-separated sample as a function of doping. When phase separation is
long~ranged, Rietveld refinement can be used to estimate the relative abundance of
the phase components [102, 103, 104, 105, 106, 107].

Phase segregation can also be determined from the PDF [1, 108]. In PDFfit2,

each phase in a multi—phase fit has its own scale-factor in the refinement. The scale

69

factor reflects both the re.lative 1_)liase-f1'action of the phases and the average scattering
power of each phase. which depends on the chemical ccnnpositions of each phase. The
comersion from scale-factor to atomic-fraction is done using the equations derived in

Ref. [1].

For each sample, we tried different. models. The structure is of the rock-salt

type. space group Fin-3m. First we start from a homogeneous (solid solution) model
where the anions are assumed to randomly distributed on the sites of the anionic
sublattice. In this model, S atoms substitute the Te site randomly without breaking
the symmetry. The only structural parameters refined are the lattice parameter and
the thermal factors.

The next model we tried was a simple two—phase model in which a phase separation
into a PbTe—rich and PbS-rich phase was assumed. The phase diagram for this system
shows a miscibility gap at low temperature over a wide composition range [94, 95].
The two phases that coexist have compositions rather close to the pure end-members
and there is very limited solid solubility. Based on this, and in an effort to keep the
modeling as simple as possible, we modeled the phase separation as a mixture of pure
PbTe and PbS; however, allowing the lattice parameters to vary as would be expected

if the phases were not the pure end-members. The parameters that were allowed to
vary in these fits were lattice parameters, thermal factors and phase specific scale
factors which reflect the relative abundance of each phase. More complicated phase

separated models were also tried where the composition of the phases was varied as

described below.

5.4 Results

First we carried out PDF and Rietveld refinements on the undoped end-members of
the series, PbS and PbTe. The level of agreement of Rietveld and PDFfit refinements

can be seen in Figure 5.6 and Table 5.1. These fits give a baseline for the quality

70

 

 

I (a.u.)

 

 

 

 

 

 

 

 

 

 

to
N'— (b) 7
(‘3‘ ' A .
on: O A
c: of V .
(\i— .J
l .. vAv/‘x. A [\A. A ATAW'A A A— MU va ‘
J m l a I 1 L r l l 1 L md 1 l m
2 4 6 8 10 12 14 16 18
r (A)

Figure 5.6: Representative refinements of the PbTe data using (a) Rietveld and (b)
PDF approaches. Symbols represent data, and solid lines are the model fits. The

difference curves are offset for clarity.

Table 5.1: Refinement results from PbS and PbTe compared with literature values.

 

 

 

 

 

Literature[Ref. [109]] RietVeld PDF

R.” - 0.03994 0.0852
am, ( ) 6.4541(9) 6.4776(3) 6.465(3)
Um, (A?) 0.0204(3) 0.033(5) 0.032(4)
UTe (A?) 0.0141(2) 0.009(9) 0.014(4)

12,, - 0.04377 0.0820
anS (A) 5.9315(7) 5.9460(3) 5.940(3)
Um. (A?) 0.0163(3) 0.023(3) 0.0185(5)
Us (A?) 0.0156(5) 0.018(4) 0.030(5)

 

 

 

 

71

of the fits for materials without disorder. The fits are acceptable and the refined
parameters are in reasonable agreement with literature values for PbTe, though out-
side the estimated errors. This refiects in part that the samples were different, and
also the fact that systematic errors are not accounted for in the error estimates. The

PDF and Rietveld refinements are also only in semi-quantitative agreement. The pa-

rameter estimates were made on the same data—sets but using different methods and,

again. systematic. errors are not accounted for in the error estimates. Even in these

nominally pure materials the refined thermal factors are rather large [110], which is

in agreement with previous work [109].

Now we consider the chemically mixed systems. The existence of phase separation
can be qualitatively verified in our samples by looking at the diffraction patterns in
Figure 5.7. The top curve is PbS and the bottom curve is PbTe and the vertical
dashed lines are at the positions of the main Bragg-peaks of these phases. Despite
the low-resolution of the data, a characteristic of the RAPDF measurement [67], for
compositions :r = 0.25, 0.50 and 0.75 a coexistence of PbS and PbTe diffraction
patterns is clearly evident as the diffraction patterns are qualitatively recognizable
as a linear superposition of the end-member patterns. Diffraction peaks appear at

precisely the positions of the end-member Bragg-peaks. The same is true for the
annealed :1: = 0.5 sample (dark magenta). On the other hand, the quenched :1: =
0.5 sample has a diffraction pattern that resembles the PbTe pattern but shifted
significantly to the right. This is what would be expected for a solid-solution, rather
than phase separated, sample suggesting that quenching the sample suppresses phase
separation.

The situation is slightly less clear for the :6 = 0.16 sample which resembles closely
the pure PbTe diffraction pattern. The effects of phase separation would be difficult
to see in this case because of the small PbS component. However, careful inspection
of the curve indicates that the main peaks are shifted to the right, in analogy with

the quenched x = 0.5 sample. We thus conclude that this sample is a solid-solution
on the macro-scale probed in a diffraction pattern.

We would like to consider evidence in the local structure for phase separation.

The PDFs of the data in Figure 5.7 are shown in Figure 5.8 arranged in the same way

and with the same colors as they are in Figure 5.7. The samples that are macroscopi-

 

F (231—1)
15 30 45 60 75

 

0

 

 

 

2 3 4
Q (4‘1)

Figure 5.7: The low-Q diffraction patterns of all the (PbTe)1_I(PbS)zsamples studied,
where F(Q) = Q(S(Q)—1). From top to bottom: :6 = 1.00 (green), 1‘ = 0.75 (yellow),
2: = 0.50 (light and dark magenta), 1: = 0.25 (blue), ct = 0.16 (cyan), and a: = 0.00
(black). The data corresponding to the quenched :5 = 0.50 sample (light magenta)
is superimposed on top of that of the unquenched sample (dark magenta) without
being offset. The other data are offset for clarity. Vertical dashed lines indicate
positions of several characteristic Bragg peaks in the endmember data to allow for

easier comparison.
cally phase separated (x = 0.25, 0.5 (annealed) and 0.75) also show phase separation
tendencies in the local structure as expected, the curves having the qualitative ap-
pearance of a mixture of the end-member PDFs.
The behavior of peaks in the PDF in solid—solutions has been discussed previ-
ously [92, 57]. The nearest neighbor peaks retain the character of the end-members.

albeit with a small strain relaxation. However. peaks at higher—r, from the second-

73

 

G (8‘2)

 

 

 

 

Figure 5.8: Experimental PDFs for various (PbTe)1_I(PbS)I samples on expanded
scale. The PDFs, from top to bottom correspond to :5 = 1.00 (green), :r = 0.75
(yellow), .’L‘ = 0.50 (magenta), quenched x = 0.50 (bright magenta), a: = 0.25 (blue),
1' = 0.16 (cyan), and :c = 0.00 (black). The data corresponding to the quenched
.1: = 0.50 sample (light magenta) is superimposed on top of that of the unquenched
sample (dark magenta) without being offset. The other data are offset for clarity.
Vertical dashed lines indicate positions of a few selected characteristic PDF features

of the endmembers for easier comparison.
neighbor onwards, appear broadened because of inhomogeneous strain in the sample
but are peaked at the average position expected for the virtual crystal model. The
x = 0.5 (quenched) and :1; = 0.16 samples follow this behavior even on the 1 nm
length-scale suggesting that they are solid-solutions even on the local scale.

To investigate the phase separation phenomenon more quantitatively, we carried

out two-phase refinements for the macroscopically phase separated samples on both

74

 

I (a.u.)

 

 

 

 

 

 

 

 

 

 

T j 1 I I I I
l0
N'— (b) 7
A ” .
0'1 ’
o \
s » r
i- v 1
care
I Am AvAA ._ A - AA _ M
.. 'V V‘V V w w" V Vv Vq
a l 1 l 1 l 1 l 1 l 1 l 1 l 1 l 1 l 1

 

2 4 6 8 10 12 14 16 18
r (A)

Figure 5.9: Representative refinements of the x = 0.50 sample data using (a) Rietveld
and (b) PDF approach. Symbols represent data, and solid lines are the model fits.
The difference curves are offset for clarity.

the diffraction data and the PDF. Figure 5.9 shows representative fits from the :c =
0.50 sample. The refined parameters are reproduced in Table 5.2. In the table the
n and no refer to the refined fraction of the sample in the PbTe phase and the
expected fraction based on the stoichiometry and assuming phase separation into
pure PbTe and PbS, respectively. The two-phase fits using pure PbS and PbTe as
the two phases are good (“Rietveld” and “PDF” columns in the table), as indicated
by the low residuals that are comparable to the end-member fits. The refined atomic
displacement parameters (ADPS) are also in good agreement with the end-member
refinements, though the refinements of this parameter are somewhat unstable on the
PbS phase when it is the minority phase as it does not contribute strongly to the
scattering in that case. The result that relatively large ADPs are needed on the Pb

Site in PbTe and on the S site in PbS are reproduced in the two—phase fits of the

Table 5.2: Refinement results for two—phase fitting. “Rietveld” and “PDF” refer to
Rietveld and PDF fits, respectively, where the composition of the two phases was fixed
to PbTe and PbS. n and no refer to the refined and expected (based on stoichiometry)

phase fractions for the PbS—rich phase
25% 50% 75%

Rietveld PDF Rietveld PDF Rietveld PDF
R1,. 0.03427 0.118 0.0468 0.151418 0.03385 0.0996
n/no 019/025 020/025 0.50/0.50 049/050 071/075 080/075
C 6.4669(3) 6.446(3) 6.4418(3) 6.414(3 6.4301(3) 6.415(3)
(

(

 

 

 

 

 

)

A2) 0.037(6) 0.040(4) 0.041(6) 0.040( ) 0.040(7) 0.040(5)
UTe A?) 0.015(6) 0.016(4) 0.0052(6) 0.019( ) 0.033(7) 002(4)
apb5(A) 5.9768(3) 597(1) 5.9841(3) 5.953(4) 5.9738(3) 5.956(3)
UPb (A?) 0.044(8) 0.027(5) 0.034(7) 0.025( ) 0.024(6) 0.023(3)
U501?) 0.073(8) 003(5) -0.0027(7) 0.031( ) (3)

 

 

 

 

 

 

 

 

 

0.0065(6) 0.029

 

phase separated samples.

The lattice parameters of the PbTe in the phase separated samples are consistently
shorter than for the pure material, and they are consistently longer for the PbS
phase component. This effect is real and reflects the fact that the phases in the
phase separated samples are actually solid-solutions with finite amounts of S in the
PbTe and Te in the PbS phase, respectively. We can make a rough estimation of
the composition of the phase separated phases by considering their refined lattice
parameters and assuming that Vegard’s law [90, 91] is obeyed in the vicinity of the
end-member compositions. In this case, the formula for the lattice parameter in the
solid solution of composition PbTe(1_y)Sy is (1,, = g(apre) + (1 —y)apbs. Thus, we can
estimate the compositions of the solid-solutions in the phase separated phases from
the Rietveld refined lattice parameters. We find that in the :c = 0.25 phases, y = 0.94
for the PbS rich phase and y = 0.05 for the PbTe rich phase. This verifies that the
composition of the phases in the two-phase mixture are indeed very near PbTe and
PbS. The values determined from the :1: = 0.5 and 0.75 samples give the same result
with the estimated composition of the PbTe-rich phase as y = 0.895 and that of PbS

y = 0.03. This suggests that the solid solubility limit is in the region 3-10% at both

ends of the phase diagram.

 

The powder diffraction data are relatively insensitive to small changes in chemical
composition of the particular phases [108] which explains the good fit to the data
with the endmember PbS and PbTe compositions, albeit with modified lattice pa-
rameters. However, for completeness we have carried out two-phase refinements to
the phase separated data using the nominal compositions for the two phases that were
determined above. The fits were comparable to those where the composition of the
two phases were limited to pure PbTe and PbS but with more physical ADPs being
refined on the PbS component.

The agreement of the refined with the nominal composition, n/no, is best. in the
r = 0.50 sample in both the PDF and Rietveld data. It is less good, though acceptable
for the 0.25 and 075. Due to the relative insensitivity to chemical composition we
expect rather large error bars on these quantities and don’t ascribe significance to
the differences. The agreement between the Rietveld and PDF results shows that the
phase separation is macroscopic since we get the same result in both the local and
average structures.

We now consider the samples that appear from the qualitative analysis of the
data to be solid-solutions: :1: = 0.5 (quenched) and a: = 0.16. In Figure 5.10 we
consider the :r = 0.5 sample. In this figure, model PDFs of the undoped endmembers
are reproduced for reference and the positions of their main peaks are marked. The
quenched data are shown as grey symbols in the cuves (c) and the annealed data in
the curves ((1). The magenta lines are simulated PDFs. In (c) the simulated PDF
is from a homogeneous solid-solution virtual-crystal model with the right nominal
composition and lattice parameter. It agrees well with the data. In (d) the simulated
PDF is a linear combination of the PbTe and PbS PDFs. In each case the ADPs of
the simulations have been adjusted to give the best agreement with the data. The
simulations fit rather well indicating that this picture of phase separation (annealed)

vs solid solution (quenched) is a good explanation of the bulk behavior for the :1: = 0.5

77

 

G (4‘2)

 

 

 

 

910

r (A)

Figure 5.10: PDFs of converged models for (a) .’L‘ = 0.00 and (b) a: = 1.00
(PbTe)1_1(PbS)I samples. Comparison of the data for (c) quenched and (d) un-
quenched :c = 0.50 samples (open symbols) with the solid solution (c) and mixture (d)
models (solid lines), respectively. See text for details. Vertical dashed lines indicate
positions of selected PDF features characteristic for the endmember compositions, for

easier comparison.
sample. Quantitative refinement results for the quenched 50% sample are reproduced
in Tab. 5.3 The fits are good with low Rw’s and reasonable refined parameters. The
refined lattice parameter is between the end-member values as expected and the ADP
on the lead-site is further enlarged from the endmember values as expected due to
disorder in the alloy.
In the quenched :c = 0.5 sample the solid-solution is not thermodynamically sta-

ble but is metastably trapped by the rapid quench. The quench is very successful

78

Table 5.3: Refinement results from both PDF and Rietveld for the quenched 50%
sample from a homogeneous solid-solution model.

Rietveld PDF
Ru. 0.047 0.163
a (A) 6.2571(4) 6.217(3)
Um, (A?) 0.055(5) 0.062(3)
U783 (A?) 0.017(5) 0.054(3)

 

 

 

 

 

 

 

at suppressing phase separation as discussed above. However, it is not completely
successful, as TEM images of the quenched :r = 0.5 sample indicate that the sample
has compositional modulations, as shown in Figure 5.11(b). The striped nature of
these modulations suggests that there is an arrested spinodal decomposition taking
place in the 50% doped sample, that would result in sinusoidal compositional modu-
lations about the nominal 50% composition. The amplitude of the modulations are
not known, but the good agreement of the homogeneous solid-solution model to the
PDF and Rietveld data suggest that the variation in composition around the nominal
50% is not too large.

Thus we understand the quenched 50% sample to be close to an ideal metastable
solid solution, but with an arrested spinodal decomposition that gives rise to nano-
scale compositional modulations.

Of greater interest from both a technological and scientific viewpoint is the behav-
ior of the a: = 0.16 sample that shows especially good thermoelectricity. As discussed

above, the diffraction data in Fig 5.7 suggests that the sample is macroscopically a
solid solution even though it lies outside the range of solid solubility suggested by the
phase diagrams [94, 95] and inferred from the composition of the PbTe-rich phase of
the phase—separated compositions in our own refinements (25%, 50%, 75% sample).
We tried fitting two-phase and homogeneous models to both the diffraction and
PDF data. The results are shown in Table 5.4 and Table 5.5 with representative fits
shown in Figure 5.12. As expected from the qualitative analysis of the data discussed

above, the single—phase solid—solution model (model A) provides acceptable fits to the

79

 

model A model B
R iet vcld P DF R iet veld PDF

 

 

 

 

1?“. 0.04047 0.1209 0.05180 0.121
n/no ~ — 014/016 0037/016
PbTe. a, (A) 6.4264(5) 6.403(3) 6.4233(4) 6.403(24)

U,,b(A?) 0.047(5) 0.047(3) 0.035(6) 0.035(3)

 

 

 

 

 

 

UTE (A2) 0.0061(6) 0.019(3) 0. 023(6 (6) 0.029(4)
second phase a (A) -- — 5.900(1) 5.942(4)
Um, (A?) — - 0.018(8) 0.021(6)
(15% (A? ) — 0.8013( ) -0.0024(6)

 

 

 

Table 5.4: Rietveld and PDF refinement results from three different models for the
PbTeoMSoJo sample: model A is solid solution model, model B is a simple two-phase

mixture of PbTe and PbS. n and no refer to the refined and expected (based on
st oichiometry) phase fractions for the PbS-rich phase.

 

 

 

 

 

modelC
Rietveld PDF
Ru, 0.03068 0.114
n/no 0.31/0.32 024/032
PbTe a (A) 6.4203(4) 6.4163

( )
Upb (A?) 0.028(6) 0.036(4)
UT, (A?) 0.016(6) 0.025(5)
second phase a. (X) 6.1673(3) 6.255( )
( )

(

 

UprAf) 0.253(8) 0.064
Usre (A?) 0.253(8) 0.0706)

 

 

 

 

 

 

Table 5.5: Model C is a mixture of pure PbTe phase plus a solid solution of compo-
SlthIl PDT€0,5-PDSO.5.

data. The refined lattice parameters are shorter than pure PbTe. According to the
Vegard’s law analysis, the refined lattice parameter gives a nominal composition for
this sample of 0.14 (Rietveld) / 0.12 (PDF), in reasonable agreement with the actual
composition. Enlarged ADPs are found on the Pb sublattice with smaller ADPs on
the Te lattice, as was the case for the PbTe end-member. As expected for a solid-
solution, the ADPs are enlarged with respect to PbTe.

For completeness, we also tried the simple model of phase separation into pure
PbTe and PbS end-members. The results appear in Table 5.4 as model B. The

Rietveld fit is significantly worse as measured by Rm. In the case of the PDF fit the

80

Ru. is comparable but the refinement reduced the phase fraction of the second phase
and adjusted the lattice parameter of the majority phase, moving the refinement back
towards the solid-solution result.- This refinement also returned unphysical negative
atomic displacement factors on the minority phase. The solid-solution model is clearly
preferred over full phase separation from the bulk diffraction measurements.

The TEM images from the 16% sample (Ref. [89] and Fig. 5.11(a)) suggest that
it is two—phased, with one phase being homogeneous and the other resembling the
quenched :r = 0.5 sample. with arrested spinodal decomposition. A model that simu-
lated this situation was successful compared to the PDF data, as shown from model C
in Table 5.5. This model assumed that the nominally 16% sample is phase separated
into regions that are pure PbTe and regions that resemble the quenched 50% sample,
i.e., they are nominally :7: = 0.5 solid-solutions but. also exhibitng spinodal decom-

position as suggested by the TEM images. Thus, model C is a phase separation into
pure PbTe and a solid solution of composition PbTeo,5So_5. This model gives the
lowest Rw’s for fits to the 16% compound in both the Rietveld and PDF refinements.
The phase fractions were free to vary but refined to values that are close to 'those
expected. The lattice constants refined to reasonable values. The majority phase
lattice constant was close to that of the PbTe rich phase in the two-phase refinements
in Table 5.2. In the case of the minority phase, the lattice constant lay between pure
PbTe and PbS consistent with a nominal 50% composition. The ADPs are slightly
large in the PbTe-rich phase but physically reasonable. In the minority phase the
ADPs are unphysical in the Rietveld refinement suggesting that this parameter is not
well determined in the refinement. However, in the PDF refinement they are more
reasonable, but very large. This is perfectly consistent with the fact that this minority

phase itself actually has a compositional variation due to the spinodal effects.

81

5.5 Summary

This work confirmed the phase separation tendency of the PbTe/PbS system. It also
showed that phase separation can be effectively, but not. completely, suppressed by
quenching at 50% composition, where a partial spinodal decomposition appears to be
taking place, at least in a portion of the sample.

However. the main result is an improvement in our understanding of the state
of the technologically promising 16% sample. Measurements of the bulk average
structure, and the bulk local structure, indicate that it is not phase separated into
PbTe—rich and PbS-poor endmembers like the other similarly processed samples in
the series. The best explanation of all the data at hand is that this sample prefers a
phase separation into a PbTe-rich phase and a phase that is nominally 50% doped,

but which has a partial spinodal decomposition reminiscent of the quenched 50%
sample. Such a nano—scale phase separation is thought to be important in producing
the very low lattice thermal conductivity n that is observed in this material [89].
Interestingly, in this case the effect appeared not after a quench, but after an anneal,
suggesting that it might be the thermodynamically preferred state, though this needs
to be investigated further.
The other important observation from this work is that quenching is very im-
portant in determining the phase separation and resulting nano—scale microstructure.
This suggests that in this system it may be possible to engineer Ii, and therefore Z T

in the bulk material by appropriate heat treatments. This is a promising route for

future research.

 

Figure 5.11: HRTEM images of (a) a: = 0.16 and (b) quenched x = 0.50
(PbTe)1-x(PbS)I samples.

83

 

 

 

I (a.u.)

 

 

 

 

 

 

 

 

 

 

I m
l0
:6 - (b) 4
A ' .
‘l’
«I: o A
v - V -
o .5
N " —4
1 l 1 l 1 l 1 l 1 J 1 l 1 l 1 l 1 l 4
2 4 6 8 1 O 1 2 14 1 6 1 8
r (A)

Figure 5.12: Representative refinements of the a: = 0.16 sample data using (a) Ri-
etveld and (b) PDF approach. Symbols represent data, and solid lines are the model

fits. The difference curves are offset for clarity.

84

Chapter 6

Concluding Remarks

6.1 PDF analysis of new thermoelectric materials

We set out to determine the local structure of novel thermoelectric materials, us-
ing the atomic pair distribution function (PDF) method [35] and combining other
experimental methods.

The goal of this study was to understand the origin of the unusual thermoelectric
properties in these novel materials from the structural aspect, and to provide the
feedback people in materials engineering need to synthesize new materials with Z T
suitable for industrial applications.

A common feature is found in these studied materials, which is the presence of
nanoclusters in bulk material. In the AngmeTem+2 system where high ZT was
found, nanoclusters with average chemical composition AngmeTem+2 were found
embedded in the PbTe matrix. In the (PbTe)1_,.(PbS),r system where the extremely
low thermal conductivity was found, nanoclusters with average chemical composi-
tion of PbTe were found in the representative material PbTe75%-PbS25%. In the

Ag1_ISnSb1+xTe3 system, although the current PDF data are not sufficient in draw-
ing a conclusive result about the chemical composition of the nanoclusters, the trace

of nanoclusters is found by studying the HRTEM pictures and by the preliminary

85

:3,

Q‘- am hi” m
1

analysis of PDF data. Although very strong arguments like ”these nanoclusters di-
rectly cause such unusual thermoelectric properties’ can not. be clearly made at this
current. moment, we do expect that there must be a close. connection between these
nanoclusters and these unusual thermoelectric properties.

A deep and precise understanding of how, physically, such nanostructures give
rise to these unusual transport behavior, especially in a quantitative way, needs much
further work. Computer simulations are needed to understand the transport phenom-
ena and their relationship to the nanoscale structures we have studied here. With
the emergence of high performance computing and daily-increasing computing power
of PCS, we look forward to seeing fruitful results from theory and simulation in the
future in these interesting new materials.

Experimentally, many further detailed research work needs to be done. How to
control the cluster size? How cluster size affect the figure of merit Z T? Of course
more systematic studies need to be carried on in these systems. There is a still very
long way to go for thermoelectric material study. This thesis only shows a path or a
direction to go.

PDF played a central role in our study of such nanoscale structure problem. Its
power is shown in our analysis procedure. Qualitative results from PDF are easily
and intuitively drawn directly from the data, with quantitative modeling providing

valuable additional information. We expect to see more applications of the PDF

method in the study of such nanoscale structure problems.

6.2 Alternative approaches and future work

6.2. 1 Alternative approaches

In this subsection, other experimental methods helpful in solving the structure of

these materials are listed. These are complementary experimental methods and are

86

supposed to provide complementary information on material structures that PDF is

not sensitive to. These methods are either already tried or could be tried later.

EXAFS

Extended X-ray Absorption Fine Structure (EXAF S) method was tried on

AgogonloSbTego sample. EXAF S is the oscillating part of the X-ray Absorption

Spectrum (XAS) that. extends to about 1000 eV above an absorption edge of a. par-
ticular element of a sample. It gives us the nature of the neighboring atoms sur-
rounding the selected central atom (their approximate atomic number). EXAFS is
a complementary method for PDF, since PDF is not very sensitive to the chemical
ordering especially when the atomic numbers of the compositional elements are close
to each other. It is difficult to distinguish different. models using the PDF method in
which the only difference between models comes from the substitution of atoms with
near atomic numbers . As described in Ref. [1], we give an estimation of the average
chemical composition for the second phase, but we cannot totally decide the chemical
ordering in the second phase. We tried EXAFS to solve this problem. Although
theoretical calculations are very complicated for EXAFS, the computer software for
approximate calculation is already available. Data for AgogonlngTego sample was
collected. K edges for Ag and Sb element were tried. Detailed analysis can be found
in the report [111]. The result from EXAFS is not conclusive for this specific mate-
rial. One reason is that Sb, Ag have very close atomic numbers (less than 5). The
other is that theoretical calculations involving more shells are needed to distinguish
the different models. However, experimental EXAFS signal can only go as far as 5 A,

thus the signal from experiment is not enough to distinguish the different chemical

ordering models we tried.

87

 

 

 

Anomalous X-ray scattering

Another promising method to study the chemical ordering is the anomalous X-ray
scattering method. The exjj)erimcnt uses an x-ray energy near the resonant edge
of the selected element in the sample. The scattering factor of an atom is nearly
independent of the energy of x—rays except in the vicinity of the absorption edge of
the atom where it depends rather strongly on the X-ray energy E. By carrying out the
measuren‘ient at two or more energies in the vicinity of the edge of the element. a, the

scattering power f(,.(Q) of the (1 atoms change but those of the other constituent ions

 

don’t. The idea is to measure two energies, one near the absorption edge, the other

far from the edge. The difference in specific peaks of the two PDFs thus derived will

show the contributions from this special element and thus provide information on the
distribution of the element in the materials. This provides information on chemical
ordering which is usually not very sensitive in the regular PDF method. This approach
has been demonstrated [112] but difficulties in carrying out the measurements make it
far from commonplace. However, it is a promising approach for the Ag1_$SnSb1+xTe3
project. More detailed description, including math formula, can be found in the
book [35]. This method is a quite promising experimental technique, and will certainly

gain more and more broad application in wide field of material science if it is further

developed.

6.2.2 Future Work

6.2.3 Obtaining PDFs using other source and detector

PDF data with higher Q space resolution are needed for the project Ag1_ISnSb1+xTe3
as described before in Chapter 4. Two types of experiment can be done for such a
purpose. One is to use the solid state detector in an x-ray experiment and scan to

wider diffraction angles. The other is to use a neutron source. These experiments

88

take longer than the RAPDF measurements to collect data and we only do them now
when there is sound reason to conduct such experiment. that is when they provide
needed informaticm which other experiment like RAPDF can not provide. It is not
completely clear that higher resolution is warranted in this case.

The other simple way to determine whether there is two phase separation hap—
pening or nanophase exists in bulk material is to determine the size of the clusters.
To determine the size of the clusters. it would be good to consider the PDF over
a wider r-range. Q resolution affects the height of the PDF signal and determine
how far we can go in the 1" range and this would require a new set of experiments
with PDFs determined from higher resolution diffraction data. Using the refinement
results obtained from the low 7‘ range and consecutively evolve into the high 7‘ range,
we can approximately estimate in which length scale the PDF refinement give un-
physical result and that length scale will be around the cluster size. This consequent

refinement. method is already implemented in the new PDFGUI [66] program.

89

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