Engineering of Thermoelectric Materials for Power Generation Applications

Dissertation

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Vladimir Jovović

Graduate Program in Mechanical Engineering

The Ohio State University

2009

Dissertation Committee:

Joseph P. Heremans, Advisor

Walter R. Lempert

Igor V. Adamovich

Vish Subramaniam

Vladimir Jovović

2009

Abstract

The efficiency with which thermoelectric devices for power generation convert heat into electricity is governed by the quality of thermoelectric materials which is characterized the nondimensional figure of merit, zT. In this work, we develop two new highzT material systems. First we prove experimentally that the modification of the density of states can be used to successfully increase zT of PbTe from 0.8 to 1.5 at 725K. This is achieved by doping PbTe by Tl. In this work we experimentally investigate this alloy system and other group IVVI compound semiconductors doped with group III or rare earth elements. Experimentally measured properties are used to calculate electronic properties of materials: Fermi energy and density of states effective mass, among others.

We observe pinning of Fermi energy level in IVVI:III systems and an increase of effective mass in PbTe:Tl and PbSeTe:Tl, thus resulting in increased thermoelectric efficiency.

We also identify rocksalt IVVI 2 compounds as a class of materials with intrinsically minimum thermal conductivity. We focus on identifying electronic structure of representative of his class, AgSbTe2, by measuring de Haas – van Alphen oscillations

in magnetic field. From the measured electronic structure we calculate optimal carrier density develop methods for doping this material. The result is an increase in figure of merit from 0.5 to 1.3 at 400K.

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Acknowledgments

The most important contributor in completing this dissertation was certainly my advisor

Joseph Heremans. I was lucky to be his graduate student; he guided me through the fundamentals of solid state physics and helped me comprehend thermoelectricity and showed incredible patience while reading my texts. His enthusiasm and drive were great motivation to complete large number of projects. I would also like to acknowledge help of Joseph West in building thermoelectric laboratory and quickly starting early experiments. I would also like to thank my family for support and my laboratory partners

S.J. Thiagarajan, M. Nielsen and C.M. Jaworski for their help.

This work is the result of collaborative effort of number of research groups around the world; Dr. J. Snyder at Caltech, D. Khokhlov at Moscow State University, A.

Nikorici of Moldova Academy of Sciences, T. Story, Z. Golacki at Institute of Physics of

Polish Academy of Sciences and K. Kurosaki at Osaka University and their students and collaborators.

In the end I would like to acknowledge financial support of BSST and Dr. Lon

Bell, the Department of Mechanical Engineering, and The Ohio State University for awarding me with Presidential Fellowship for this work.

Vita

2002 – 2005 MS in Mechanical Engineering, UNIVERSITY OF NEW HAMPSHIRE, Durham, OH USA 1997 – 2002 Diploma in Mechanical Engineering, UNIVERSITY OF NOVI SAD, Novi Sad, Serbia

Publications

M. Murata, D. Nakamura, Y. Hasegawa, T. Komine, T. Taguchi, S. Nakamura, V. Jovovic, and J. P. Heremans, “Thermoelectric properties of bismuth nanowires in a quartz template” Appl. Phys. Lett . 94 , 192104 (2009),

M. Murata, D. Nakamura, Y. Hasegawa, T. Komine, T. Taguchi, S. Nakamura, C. M. Jaworski, V. Jovovic, and J. P. Heremans “Mean free path limitation of thermoelectric properties of bismuth nanowire”, J. Appl. Phys . 105 , 113706 (2009),

J. Sootsman, V. Jovovic, C. Jaworski, J.P. Heremans, Jiaqing He, V. P Dravid, M. Kanatzidis, “Understanding Electrical Transport and the Large Power Factor Enhancements in CoNanostructured PbTe” Mater. Res. Soc. Symp. Proc., San Francisco CA, 2008

J.P. Heremans, V. Jovovic, E.S. Toberer, A. Saramat, K. Kurosaki, A. Charoenphakdee, S. Yamanaka, G.J. Snyder, “Enhancement of Thermoelectric Efficiency in PbTe by Distortion of the Electronic Density of States”, Science 321 , 554 (2008)

V. Jovovic, J.P. Heremans, “Doping optimization of the thermoelectric properties of AgSbTe 2” Journal of Electronic Materials, Proceedings to International Conference on Thermoelectrics , Corvallis, Oregon 2008

D.T. Morelli, V. Jovovic, J.P. Heremans, “Intrinsically Minimal Thermal Conductivity in Cubic IVVI2 Semiconductors”, Phys. Rev. Lett. 101 , 035901 (2008)

V. Jovovic and J. P. Heremans, “Energy Band Gap and Valence Band Structure of AgSbTe 2”, Phys. Rev. B 77, 245204 (2008)

V. Jovovic, S. J. Thiagarajan, J. P. Heremans, T. Komissarova, D. Khokhlov, and A. Nicorici, “Low temperature thermal, thermoelectric, and thermomagnetic transport in indium rich Pb 1−x Sn xTe alloys”, J. Appl. Phys. 103 , 053710 (2008)

V. Jovovic, S. J. Thiagarajan, J. P. Heremans, D. Khokhlov, T. Komissarova, and A. Nicorici, “HighTemperature Thermoelectric Properties of Pb1xSnxTe:In”, edited by T.P. Hogan, J. Yang, R. Funahashi, and T. Tritt, Mater. Res. Soc. Symp. Proc . 1044, U04 09, Warrendale, PA, 2007

V. Jovovic, S. J. Thiagarajan, J. West, J. P. Heremans, T. Story, Z. Golacki, W. Paszkowicz , V. Osinniy, “Transport and magnetic properties of dilute rareearth–PbSe alloys”, J. Appl. Phys. 102 , 043707 (2007)

S. Joottu Thiagarajan, V. Jovovic, J. P. Heremans, “On the enhancement of the figure of merit in bulk nanocomposites ”, Phys. Stat. Sol. (RRL) 1, No. 6, 256–258 (2007)

Fields of Study

Major Field: Mechanical Engineering

Table of contents

Abstract ...... ii Acknowledgments...... iv Vita...... v List of Figures ...... ix List of Tables...... xx 1 Introduction...... 1 1.1 Electron entropy and efficiency of thermoelectric devices...... 2 1.2 Brief overview on current progress in development of bulk thermoelectric materials ...... 7 1.3 Research objectives...... 11 2 Measured material properties and measurement techniques...... 13 2.1 Electrical conductivity...... 13 2.2 Hall coefficient...... 15 2.3 Seebeck coefficient ...... 17 2.4 Nernst effect ...... 19 2.5 Thermal conductivity ...... 20 2.6 Measurement of transport properties and estimated errors ...... 23 2.6.1 Electrical resistivity...... 24 2.6.2 Seebeck coefficient ...... 25 2.6.3 Thermal Conductivity ...... 27 2.6.4 Hall Coefficient and Nernst Coefficient...... 28 3 Band Structure Models...... 30 3.1 Method of Four Coefficients Single Carrier Systems...... 36 3.2 Two Carrier Conduction...... 44

vii

4 Modification of Electronic Density of States in IVVI Semiconductors ...... 47 4.1 Introduction...... 47 4.2 Effects of alloying IVVI alloys with rare earth elements ...... 52 4.2.1 Alloying PbSe with Ce, Pr, Nd, Eu, Gd and Yb ...... 53 4.2.2 Alloying Pb 1xSn xTe with Nd ...... 66 4.3 Dilute alloys of IVVI compounds with group III elements ...... 81 4.3.1 Alloying Pb 1xSn xTe with In...... 85 4.3.2 Effects of doping PbTe, PbSe xTe 1x and Pb 1xSn xTe with Tl ...... 101

5 Anharmonically bonded IVVI 2 semiconductors with minimum thermal conductivity...... 121

5.1 Thermal conductivity of IVVI 2 alloys...... 122

5.2 Electronic structure of AgSbTe 2 ...... 130

5.3 Doping and optimization of thermoelectric properties of AgSbTe 2 ...... 145 Conclusions...... 159 References ...... 162

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List of Figures

Figure 1 Seebeck effect; charge separation is supported by maintaining a temperature gradient across the thermoelectric material...... 3

Figure 2 Simple thermoelectric generator consisting of two legs made from p and ntype semiconductors...... 4

Figure 3 a) Temperature Seebeck and b) Temperatureentropy diagram showing thermodynamic cycle in which simple "ideal" TE device operates...... 5

Figure 4 Dependence of device conversion efficiency to materials figure of merit, zT.....7

Figure 5 Stateoftheart commercial p and ntype thermoelectric materials have maximum zT<1.2. 5...... 9

Figure 6 Thermoelectric properties as a function of carrier concentration in commonly used narrow gap semiconductors...... 11

Figure 7 Geometry of a sample...... 14

Figure 8 Geometry in which Seebeck and Nernst coefficients are observed...... 18

Figure 9 (a) Sample configuration and (b) standard flowthrough cryostat used for measurement of transport properties...... 23

Figure 10 Measurement of Seebeck coefficient...... 25

Figure 11 Forming of bands from discrete atomic energy levels by broadening through interatomic coupling...... 30

Figure 12 Energy wave vector relation in onedimensional lattice. Left is multizone representation and to the right an equivalent reduced zone representation...... 32

Figure 13 Density of states, FermiDirac distribution and carrier density for (a) ntype and (b) intrinsic semiconductors at T>0K...... 33

Figure 14 Pisarenko plot showing dependence of Seebeck coefficient to carrier density in

2carrier conduction region and in regions where nondegenerate and degenerate statistics can be applied...... 36

Figure 15 (a) Brilluoin zone with major crystalographic directions and named points. Γ point is center of the zone. (b) Electrons and holes are distributed in eight half pockets at

L points. Heavy holes are distributed at Σ points and are not shown here...... 48

Figure 16 Energy dependence of density of state for atom energy level E R, hybridized with band. The Fermi energy level E F is positioned in the vicinity of this level...... 51

Figure 17 Lattice constant and Pauli electro negativity (X) of RESe alloys as compared with PbSe...... 54

Figure 18 Magnetic susceptibility of PbCeSe, PbNdSe, PbPrSe, PbYbSe, PbEuSe and

PbGdSe samples used in this study. Lines in left figure are added to emphasize linear 1/T

law...... 57

Figure 19 Electrical resistivity and Hall coefficient at the zero field for PbSe:RE alloys.59

Figure 20 Seebeck coefficient and Transverse Nernst coefficient at the zero field...... 59

Figure 21 Thermal conductivity of PbSe:RE samples measured static heater and sink method. Solid lines represent total thermal conductivity and dashed lines calculated electronic component e. Electronic component is calculated using free electron Lorentz number...... 60

Figure 22 Transport properties: carrier density, mobility scattering coefficient and effective density of states mass of dilute PbSe:RE alloys...... 61

Figure 23 Free carrier density vs concentration of RE atoms in PbSe:RE alloys. Dashed line represents monovalent donor. Europium alloy is omitted as Eu in PbSe:Eu is intrinsic...... 62

Figure 24 Pisarenko plot shows in solid line Seebeck coefficient of PbSe as a function of carrier density at room temperature. For comparison resulting Seebeck coefficients are plotted for all six RE alloyed samples...... 65

Figure 25 Location of valence and conduction band edge in Pb 1xSn xTe alloys as a fuction of Sn concentration at 4K. Location of indirect Σ band is shown for ilustration and it is not to scale...... 67

Figure 26 Xray diffraction data for several horizontal cross section in the ingot of

(PbTe) 94 (NdTe) 4. Insert shows shift in the position of the peak indicating change in Nd concentration verticaly through ingot...... 69

Figure 27 Xray diffraction data for (PbTe) 80 (NdTe) 20 indicates second phase separation on the top of the sample. Second phase is circled and it is mostly Nd 2Te 3...... 70

Figure 28 Transport properties of PbTe and SnTe alloyed with 4, 6 and 20% of Nd...... 71

Figure 29 Electron mobility and carrier density for PbTe and SnTe alloyed with 4, 6, and

20% of NdTe. SnTe based sampes are ptype and PbTe ntype semiconductors...... 71

Figure 30 Magnetic susceptibility vs temperature. Data is used to determine exact concentrations of Nd ions in PbSnTe matrix...... 73

Figure 31 Resistity, Seebeck, Transverse Nernst and Hall Coeficeints of Pb 1xSn xTe aloys with ~1.5% Nd...... 74

Figure 32 Carrier density and mobility for electrons and holes in PbSnTe:Nd 1.5% ...... 76

Figure 33 Doping eficinecies for samples with <40% Sn...... 76

Figure 34 Effective mass and scatering coeficinet for PbTe, PbSe, Pb 20 Sn 80 Te and

Pb 30 Sn 70 Te all alloyed with ~1.5%Nd...... 78

Figure 35 (a) Fermi energy at 80K as measured from the band gap edge shown in blue for conduction and red for valence band; black dashed line is added to guide the eye. (b)

xii

Activation energy as extrapolated to 0K as a function of Sn concentration shown in reference to valence and conduction band edges. Dashed line is added to guide the eye..

...... 79

Figure 36 Temperature dependent Fermi energy for PbSnTe:Nd samples...... 80

Figure 37 Pisarenko plot at 300K showing S(n) for PbTe ploted in solid line against

S(300K,n) for PbSnTe:Nd aloys in this study...... 81

Figure 38 Location of indium impurity level in Pb 1xSn xTe as a function of x is shown in dashdotdash line. Figure also indicates relative position of bottom of conduction and top of valence band at 4K and location of hole band. At 4K band edge is at

170meV from the valence band edge...... 83

Figure 39 Measured thermomagnetic and galvanomagnetic properties as function of temperature for set of samples with different Sn concentrations and 0.4 to 1%In...... 86

Figure 40 Measured electrical conductivity, Seebeck, Hall and transverse Nernst coefficients for samples with 15 and 18% Sn and indium concentrations ranging from 0.3 to 6%...... 87

Figure 41 Relative magnetoresistivity as a function of magnetic field at temperature of

80K. Samples can be identified using Table 4...... 90

Figure 42 Magnetic field dependence of Seebeck coefficients measured at temperature of

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80K. Symbols and alloys correspond to those listed in Table 4. Solid lines are plotted to guide the eye...... 90

Figure 43 Hall voltage as a function of magnetic field at temperature 80K plotted for alloys listed in Table 4 with corresponding symbols...... 91

Figure 44 Transverse Nernst voltage as a function of magnetic field at temperature 80K plotted for alloys listed in Table 4 with corresponding symbols. Solid lines are added to guide the eye...... 91

Figure 45 Fermi energy (a) and carrier density (b) of Pb 1xSn xTe:In samples at 80K. dashed lines are inserted to guide the eye...... 94

Figure 46 Mobility of majority carriers and effective mass of measured Pb 1xSn xTe:In samples. Dashed lines are added to guide the eye. In measured mobility of samples with

<1% In we can notice trend in which mobility reaches maximum at x=18%...... 95

Figure 47 a) Scattering exponent and b) Pisarenko plot showing dependence of Seebeck coficent and carrier density for measured Pb 1xSn xTe:In samples at 80K...... 96

Figure 48 Temperature dependence of a) electrical conductivity and b) the Seebeck coefficient, c) low field Hall coefficient and Nernst coefficients of Pb1xSnxTe:In samples, with x=0, 15, 18, 22 and 30%. The Hall coefficient of the x=30% sample changes sign, and is shown as an inset on a linear scale. Samples for which no lines are

xiv

drawn were those for which no single, temperatureindependent scattering exponent λ could fit through all data points, presumably because λ is temperaturedependent, which as not accounted for in the model. The Nernst coefficient of the x=22 and 30% samples changes sign and is shown on a linear scale...... 98

Figure 49 Fermi energy level as a function of temperature plotted relative to the conduction and valence band of the x=0, 15, 18 and 22% sample, and on the right panel for the x=30% sample. The zero point for the energy scale is defined at midgap. Solid lines show temperature dependent position of the valence and conduction band edge.

Colors correspond to different Sn concentrations...... 99

Figure 50 Temperature dependence of resistivity (a), thermopower (b), thermal conductivity (c) and figure of merit zT (d) for: Pb 0.99 Tl 0.01 Te (open and closed symbols), Pb 0.98 Tl 0.02 Te (open and closed symbols), Pb 0.59 Sn 0.40 Tl 0.01 Te (open and closed symbols) and Pb 0.58 Sn 0.40 Tl 0.02 Te (open and closed symbols)...... 104

Figure 51 Hall coefficients (a) and transverse Nernst coefficients (b) of Pb 0.99 Tl 0.01 Te,

Pb 0.98 Tl 0.02 Te, Pb 0.58 Sn 0.40 Tl 0.02 Te ...... 105

Figure 52 Mobility (left) and resistivity (right) plotted against carrier density at 400K for

PbTe samples alloyed with 1% thallium (dashed line) and samples alloyed with 2% of thallium (solid line)...... 107

Figure 53 Density, Hall mobility and Nernst mobility of Pb 0.99 Tl 0.01 Te, Pb 0.98 Tl 0.02 Te and

Pb 0.58 Sn 0.40 Tl 0.02 Te...... 108

Figure 54 Pisarenko plot shows S(p) for set of PbTe:Tl samples. Solid line represents calculated S(p) for pure PbTe assuming the known band structure and acoustic phonon scattering. Crosses are used to show number of PbTe Tl samples, ( ) Pb 0.99 Tl 0.01 Te, ( )

Pb 0.98 Te 0.02 Te and ( ) Pb 0.58 Sn 0.40 Tl 0.02 Te samples corresponding to those in Figure 53.

...... 109

Figure 55 Temperature dependence of Fermi energy and effective mass for Pb 0.98 Tl 0.2 Te sample...... 110

33 Figure 56 Energy gap of PbTe 1xSe x alloys as function of Se concentration shown relative to the mid gap. Lines represent position of valence and conduction bands at L points. Schematic representation of position of heavy Σ point band is shown only for end point concentrations PbTe and PbSe. 36 ...... 113

Figure 57 Galvanomagnetic and thermomagnetic properties of four Pb 0.98 Tl 0.02 Te 1xSe x alloys with x=0, 0.05, 0.1 and 0.2...... 115

Figure 58 (a) Carrier density, (b) Hall and (c) Nernst mobility of Pb 0.98 Tl 0.02 Te ( ),

Pb 0.98 Tl 0.02 Te 0.95 Se 0.05 ( ), Pb 0.98 Tl 0.02 Te 0.9 Se 0.1 ( ) and Pb 0.98 Tl 0.02 Te 0.8 Se 0.2 ( )...... 116

Figure 59 (a) Pisarenko plot showing S(p) for set of PbTeSe samples alloyed with Tl. For

xvi

reference same plot contains number of points for Pb0.98Tl0.02Te samples. (b) Figure of merit shows decrease with increasing Se concentration...... 118

Figure 60 Total thermal conductivity of Pb0.98Tl0.02PbTe 1xSe x alsoys (a). Lattice component of thermal conductivity calculated at 300K and at 600K (b). Solid black line

3 represents literature vales of lattice thermal conductivity of PTe 1xSe x alloys...... 120

Figure 61 Powder Xray diffraction data of AgSbTe 2, cubic and hexagonal AgBiSe 2.

Inset shows ordered rock salt structure of IVVI 2 in which these elements preferentially crystalize. 105 ...... 124

Figure 62 (a) Total thermal conductivity of low doped AgSbTe 2, AgBiSe 2 in cubic and hexagonal form, AgInTe 2 and PbTe. Dashed line represents calculated minimum thermal conductivity in AgSbTe 2. (b) Specific heat at constant pressure of AgSbTe 2 and AgBiSe 2.

...... 126

Figure 63 Illustration of Normal and Umklapp scattering mechanisms...... 127

Figure 64 Electrical resistivity, thermopower, transverse zero field Nernst and zero field

Hall coefficients of AgSbTe 2...... 133

Figure 65 Longitudinal and transverse (Hall) magnetoresistance of AgSbTe 2 at selected temperatures...... 134

Figure 66 Longitudinal and transverse (Nernst) magnetoseebeck of AgSbTe 2 at 85, 205,

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305 and 405K...... 134

Figure 67 Partial electronic properties: conductivity, mobility and carrer density of holes and electrons in AgSbTe 2...... 138

Figure 68 Position of Fermi energies of holes and electrons in metallic and semiconducting materials...... 139

Figure 69 Dashed lines are partial electron and hole transverse Nernst and Seebeck coefficients. Solid line stands for total calculated N and S. Symbols ( ) are measured zero field values shown here for reference...... 141

Figure 70 (a) magnetic susceptibility of AgSbTe 2 measured in <111> crystallographic direction. (b) Normalized values of Fourier transform of measured data...... 142

Figure 71 Calculated figure of merit of AgSbTe2 as a function of carrier density and temperature...... 146

Figure 72 Measured (a) thermal diffusivity and (b) calculated and measured thermal conductivity of undoped. Static heater and sink method was used to measure thermal conductivity in temperature range 80 to 300K on undoped sample (solid line) and sample doped with excess Ag (dashed line)...... 149

Figure 73 Electrical resistivity and Seebeck of doped AgSbTe 2 materials. Solid lines are added to guide the eye. Block (a) and (b) are resistivity and Seebeck coefficients of

xviii

samples doped with group I and V elements and (c) and (d) are samples doped with group

III elements. Stoichiometric samples are always included as a reference...... 150

Figure 74 Zero field transverse Nernst coefficient and Hall Coefficient of undoped

()AgSbTe 2 sample and materials doped with 2%AgTe ( ), 1% NaSe 0.5 Te 0.5 ( ),

1%NaTe ( ), 1.5%TlTe( ), 1%BiTe( ) and 1% excess Pb( ). Solid lines are added to guide the eye...... 151

Figure 75 Zero field transverse Nernst coefficient and Hall Coefficient of undoped

()AgSbTe 2 sample and those doped 1.5%TlTe( ), 1%BiTe( ) and 2% GaTe( ).. 151

Figure 76 Figure of merit of AgSbTe2 based alloys doped with Ag, Na, Bi, Pb, Ga, In and Tl and that of reference undoped AgSbTe 2 alloy...... 156

Figure 77 Effects of temperature cycling on thermopower and electrical conductivity of

AgSbTe 2...... 157

Figure 78 Figure of merit of undoped AgSbTe 2...... 158

Figure 79 Figure of merit of comercial and research alloys including alloys developed using method of modification of density of states, PbTe:Tl, and by utilizing anharmonic atomic bonds, Na and Tldoped AgSbTe 2...... 161

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List of Tables

Table 1 Summary of magnetic susceptibility measurements on PbSe:RE alloys...... 57

Table 2 Overview of alloys of PbTe:Nd and SnTe:Nd used in solubility study...... 68

Table 3 Matrix of PbSnTe:In samples analyzed in this study...... 85

Table 4 List of symbols used to denote Pb1xSnxTe:In samples and measured zerofiled

Seebeck and electrical resistivity all at 80K...... 89

Table 5 Calculated Fermi energy and density of states effective mass at 80K for

Pb 0.98 Tl 0.02 Te 1xSe x alloys ...... 116

Table 6 Calculated and measured properties of Fermi energy surface ...... 143

Chapter 1: Introduction

Thermoelectric energy conversion is an allsolidstate technology used in heat pumps and electrical power generators. Scalability, high power density, reliability, and silent operation are some of the main advantages of thermoelectric (TE) generators.

Unfortunately they are compensated by the relatively low efficiency of commercially available materials, limiting the use of the technology to niche applications for the past seventy years. High energy costs and the need for increased fuel efficiency which would result in reduced greenhouse gas emission have led to a renewed interest in the field.

Thermoelectric generators can potentially convert waste heat in a variety of applications, including automotive exhaust and solar concentrators. Simplicity of design and lack of maintenance renders TE generators ideal for small scale power generation (<1kW).

However, to fully benefit from all the advantages of TE generators, it is necessary to improve the efficiency with which the TE material converts heat into electricity. In 1947 gas powered commercial thermoelectric generators operated with efficiencies of one half of a percentage as reported by M. Telkes.1 Today, devices like those launched in Cassini space mission operate with 7% efficiency but full commercial success of TE technology is expected when overall conversion efficiency reaches 20%.35

1.1 Electron entropy and efficiency of thermoelectric devices

Devices used for power generation operate by utilizing phenomenon observed by

Thomas J. Seebeck in 1821;2 when a temperature gradient is established in a material, as shown in Figure 1, conducting carriers tend to “condense” in the colder region establishing an electrical potential differential between the two ends of the material. The ratio of the voltage difference to the temperature gradient is defined as a Seebeck coefficient:

V S = − T 1

If a charge carrier behaves as a free electron in a system where we do not observe irreversible processes, then following observations hold: Electrochemical potential, e, is equal to the product of free electron charge, q, and potential difference between two points:

2 e = q ⋅ V .

3 Electrochemical potential, as described in Eq. 2, equals Gibbs free energy, g= e, g = u + pv −Ts . 3

Here, u is internal energy, p is a pressure and v and s are specific volume and entropy respectively. Finite changes in electron free energy would correspond to finite changes in electrochemical potential resulting from potential difference dV : qdV = du + pdv + vdp −Tds − sdT . 4

2 At constant pressure, p, formulation described in Eq. 4 simplifies to qdV=sdT. By comparing this result with the definition of the Seebeck coefficient, Eq. 1 we can see that the Seebeck coefficient of a free electron equals the ratio of the electron entropy and charge,

S = s . q 5

In materials where dominant charge carriers are electrons, Seebeck coefficient has positive sign, while those populated by holes have negative Seebeck coefficient.

+ + + + + + + + + + + + + + + + + + +

∆T - ∆V +

Figure 1 Seebeck effect; charge separation is supported by maintaining a temperature gradient across the thermoelectric material.

A simple TE device used for power generation is depicted in Figure 2. The device is built using two materials, for example n and ptype semiconductors. The legs are connected in such a way that when applied, heat flux Q flows in parallel through both.

Result of a heat flow through material with finite thermal conductivity is a temperature gradient ∇T and an accompanying Seebeck effect. The two legs are electrically connected in series by a metallic bridge at the hot end of device. The potential difference

across the cold ends of the device will be equal to the sum of the potential differences in the p and ntype legs, V=V p+V n, both caused by Seebeck effect. An external load, with resistance R, is connected across the cold ends. Power delivered by this device is then

W=V 2/R and the current I=V/R.

HEAT, Q in HOT

_ _ + + _ + _ + + _ _ ∇T + _ _ + + _ + _ + _ +

COLD R I

Figure 2 Simple thermoelectric generator consisting of two legs made from p and ntype semiconductors.

We can observe that this device is operating between two constant temperatures TH and

TC and generating power output W. We will also assume that each of the described processes is reversible and that a device operates in a thermodynamic cycle between two isothermal reservoirs at temperatures T H and T C with electrons as working fluid. This justifies the earlier statement that TE devices are allsolidstate devices which simply transform heat into electricity. If we add the additional condition that electrons in this device behave as the electron gas or free electrons we can state that Seebeck coefficients in both legs are constant and independent of temperature, we will call this an “ideal TE

material”. This device is shown in a temperatureSeebeck diagram in Figure 3 (a).4 By recalling Eq. 5, we can see that this cycle can be easily represented in a more familiar temperatureentropy diagram as shown in Figure 3 (b). Using this information we can conclude that thermoelectric device built using “ideal thermoelectric” materials with

Seebeck coefficients independent from temperature, operates in a reversible thermodynamic Carnot cycle, with efficiency ηC:

TH −TC ηC = TH 6

a) b) T T H T H

TC TC s=S/q

S S s s n p n p

Figure 3 a) Temperature Seebeck and b) Temperatureentropy diagram showing thermodynamic cycle in which simple "ideal" TE device operates.

However, materials with free electrons as charge carriers can not exist and the efficiency of TE devices is limited by irreversible processes, mainly due to finite thermal conductivity of materials which result in heat flux through the device and Joule heat generation. Maximum device efficiency can be achieved by optimizing the impedance of the external load as shown for example by Ryan and Stevens: 6

 S 2σ  1+  T  −1  κ  . η =ηC  S 2σ  T 7 1+  T  + H   T  κ  C

The efficiency of thermoelectric generators is limited to a fraction of their Carnot efficiency, ηc, determined by the thermoelectric material figure of merit , zT :

S 2σ zT = T . 8 κ

A material’s zT is nondimensional number, which is a function of the thermoelectric power or Seebeck coefficient, S, electrical and thermal conductivities, σ and κ respectively, and the absolute temperature T. It can be observed from equation 7 that the efficiency with which a TE generator operates depends mostly on the figure of merit zT .

For this reason, TE materials are crossreferenced by comparing their zT ’s. Figure 4 shows this relation. We can see that if we were to reach zT =5 thermoelectric devices could operate with up to 50% of Carnot‘s efficiency, but such zT is still quiet a way in the future. Values of about 2 are within a reach of researchers today.

0.6 T= 600K 400K 200K

0.4 Carnot η /

max 0.2 η

0 0 1 2 3 4 5 ZT

Figure 4 Dependence of device conversion efficiency to materials figure of merit, zT .

1.2 Brief overview on current progress in development of bulk

thermoelectric materials

The standard commercial alloys used for thermoelectric cooing today are based on Bi 2Te 3 and of general composition (Bi 1xSb x)2(Se yTe 1y)3. They have zT optimized near 300K and operating range 0 to 100 oC. For generation, materials with zT optimized around 750K are used. Typically these are the IVVI compound semiconductors based on PbTe and Pb 1 xSn xTe. For the last four decades, zT has been limited to one in all temperature ranges,5see Figure 5. This would correspond to maximum of 15% of Carnot’s efficiency in the power generation applications, Figure 4. Figure 5 shows an overview of state of the art materials in use today. 5 Any thermoelectric module consists of many p and ntype legs which are connected electrically in series and thermally in parallel, in a similar way

as the single pair shown in Figure 2. The electrical load can be selected such that the device yields a maximum efficiency:6

1+ 5.0 Z(TH +TC ) −1 η = ηC . 9 TH 1+ 5.0 Z()TH +TC + TC

We denote the device figure of merit ZT , by using capital “Z” . This figure of merit includes all losses which take place on the device level such as those caused by electrical and thermal resistances in the contacts and incompatibilities of materials used in segmented modules. 7 Device ZT can be measured by evaluating the overall device efficiency, ideally we can assume that the device efficiency is close to the material’s efficiency zT . While Eq. 7 is derived for small differences in temperature between the cold and the hot side of the module, the nature of power generation is such that this is not the case. In an ideal case, the material figure of merit, zT, would be high throughout the temperature range TH to TC.

Figure 5 shows that no one material has this property. Today, high efficiency power generation modules are segmented with such materials that the peak zT remains high throughout the temperature range.

a) b) 1.4 1.4 TAGS SiGe Sb 2Te 3 Bi 2Te 3 1.0 CoSb 1.0 3 CeFe Sb PbTe 4 12 PbTe SiGe zT zT 0.6 0.6

0.2 0.2

0 200 400 600 800 1000 0 200 400 600 800 1000 T ( oC) T ( oC)

Figure 5 Stateoftheart commercial p and ntype thermoelectric materials have maximum zT<1.2.5

Recent progress in thermoelectric materials has primarily decreased the denominator of zT (Eq. 8) by creating materials with nanometerscaled morphology to dramatically lower the thermal conductivity by scattering phonons. Quantumdot superlattices,8 or nanodot superlattices,9 to use newer nomenclature, have reached values of zT = 2 above room temperature. This work certainly marks the beginning of a second renaissance in the field of thermoelectricity by providing evidence that highzT material can be prepared. The results were obtained on thin films which are incompatible with highvolume and power generation applications. Subsequent efforts have also been successful 10 in bulk thermoelectric semiconductors where a maximum zT of 1.7 was reached in

AgPb mSbTe 2+m systems (LAST, here m=10 and 18 ). In this material, improvements are again achieved by reduction in thermal conductivity, κ. Progress in thermal conductivity

reduction was shown most recently and dramatically in silicon nanowires, 11,12 where a reduction of κ enables a material that was not considered a good thermoelectric to reach zT=0.5 11 at room temperature, a value comparable to those of commercial thermoelectric materials. Thermal conductivity shows a reduction with a decrease in nanowire diameter.

This is in agreement with calculations of Ref. [13] which predicts a decrease in thermal conductivity with a decrease in the phonon mean free path. As nanowires are 1 dimensional solids only phonons with a mean free path smaller than the wire diameter can exist.

Unfortunately, at least in bulk materials, there is a lower limit to the lattice thermal conductivity imposed by wave mechanics: the phonon mean free path cannot become shorter than the interatomic distance. 14 As a consequence, the minimum thermal conductivity of PbTe is about 0.33 W/mK at 300 K (compared to that of 1.9 in similarly doped bulk PbTe). Value of κ=0.33W/m.K was reported as calculated indirectly from zT on nanodot structures 8, result contested by Cahill et al. 15 by directly measuring κ of same structures. Even though lower values have been seen for interfacial heat transfer, 16 progress beyond this point in bulk materials must come from the power factor P=S 2σ, the numerator of Eq. (1) and in particular the Seebeck coefficient.

1.3 Research objectives

It is the aim of this work to explain and illustrate a new approach in engineering the thermoelectric properties of semiconducting materials by using the atomic properties of materials. In the past, two methods were used to increase the thermoelectric figure of merit. A common approach, originating from the fact that good thermoelectric materials are semiconductors, is carrier concentration optimization. Qualitatively, all three TE properties S, σ and κ are functions of carrier concentration as shown in Figure 6.

Dependence of Seebeck coefficient to the carrier density, S(n) , is known as a “ Pisarenko relation ” and its physical origins will be discussed in Chapter 3 after introducing details of electronic structure. A carrier concentration can be selected such that it optimizes the ratio of all three properties and maximizes the zT . The doping level optimizes zT at one specific temperature and it has to be tailored to the desired operating temperature range.

Conventional p and ntype PbTe are optimized by doping with Bi or Na 17 respectively.

zT |S| σ intensity

κ carrier concentration Figure 6 Thermoelectric properties as a function of carrier concentration in commonly used narrow gap semiconductors.

A second approach proposed as a theory by Hicks and Dresselhaus 18 and experimentally confirmed on Binanowires by Heremans et al. 19 relies on size quantization effects which strongly increases the Seebeck coefficient at a given carrier density. The third approach is to limit the phonon mean free path and reduce the lattice component of thermal conductivity. This approach is successful as long as the phonons mean free path reduction is significantly larger than that of the electron mean free path.

This dissertation introduces a new approach in engineering the properties of thermoelectric materials. The focus is on intrinsic properties of materials which can be modified by the proper selection of constituent materials at the atomic level. This document describes two classes of ideas in which zT is enhanced by reengineering electronic properties of materials, namely by (1) Modifying Electronic Density of States , and (2) by creating Atomic Level Anharmonicities in Ternary Semiconducting Alloys .

Chapter 4 will describe the first approach illustrated by experimental results, namely analysis of PbTe, PbSe, and SnTe based systems alloyed with rare earth (RE) and group

III elements. Using the second method, it is possible to select materials which already have thermal conductivities close to minimum possible and benefit from their electronic structure. The second approach has been successfully demonstrated 20,21,22 and it will be described in Chapter 5 and illustrated by the performance of AgSbTe 2.

Chapter 2: Measured material properties and measurement techniques

In this chapter we describe basic effects that are considered in analyzing electronic structure of thermoelectric materials in this document. Only general definitions and discussion of the effects will be presented. All the effects described in this chapter can be derived using generalized forces and fluxes and irreversible thermodynamics as shown by

Angrist23 for example. These effects are described using simple geometries shown in

Figure 7 for illustration; these are similar to those used to perform actual measurements.

Measurement techniques will be described at the end of this chapter.

2.1 Electrical conductivity

The simplest way to observe this phenomenon is to apply an electric field to material, for example electric field in the xdirection Ex, shown in Figure 7 a). Each electron will be move under effect of force Fe=qEx while drifting through the conductor with velocity v.

By considering Newton law we can write that average drift velocity depends on electron mass m and average time between collisions, scattering time τ.

τ = qEv 10 x m

13 For n electrons populating unit volume of semiconductor, each charged by charge q drifting with average velocity v in the direction of electric field Ex, current flux j, will be j = nqv . 11

a) b) B w z w th th

E E x V + L y - z

x Ix Ix V + -

Figure 7 Geometry of a sample

Electrical conductivity is defined as a transport property, in a general case a tensor, r r describing proportionality of electron flux j to the electric field E, j = σE . From equations (10) and (11) we can see that electrical conductivity, in the case of a one dimensional field Ex, can be written as:

j qτ σ = = nq 12 Ex m

At this point it is common to introduce the definition of electron mobility as a ratio of the drift velocity to the accelerating field, ≡ v = qτ , and rewrite Eq. 12 as: E x m

σ = nq 13

14 For practical purposes we do not measure fields and fluxes but rather flows and potential

differences, such as electrical potential V, ( Ex = dV dx = V / L )and electrical current

I=j .th .w. It is also common to introduce electrical resistivity ρ=σ 1. Following the nomenclature in Figure 7 a) we can write that electrical resistivity can be measured as:

Ex V w⋅ th w⋅ th ρ ≡ = = R , 14 j I L L where R≡V/I is the 4wire resistance of the sample.

2.2 Hall coefficient

This transport property is observed when electron flux is passing through the sample which is placed in a magnetic field. This phenomenon will be described using the configuration shown in Figure 7 b), where magnetic field is in zdirection and electron flux in xdirection. Process will be described as a 4step process:

1. Electrons are passing through the sample in the direction of the electric field Ex.

2. When an external magnetic field is applied in the zdirection electrons are deflected

under the effect of the Lorentz force, FL=qv xB=qvB z. Result is a charge buildup at

the right side of sample shown in Figure 7 b).

3. Electrical neutrality will be restored by Hall field, EH, which will result in force FH

acting on the electrons. FH=qE H exactly compensating for the Lorentz force.

4. The stationary state is achieved; and net flux of carriers is restored and remains only

in the direction of external electric field Ex.

We can use the definition of electron flux, j, in terms of mean free drift velocity, v, in addition to Eq.11, and write Hall field as:

1 E = B × j . 15 H nq

The property describing proportionality between Hall field and the cross product between electron flux and magnetic field is Hall coefficient, RH. In the general case, this value is also tensor, but in this text we will treat it as a constant. In this simple model, we are assuming isotropic material and one dimensional electric flux.

1 R = . 16 H nq

Hall coefficient can be easily measured using the configuration shown in Figure 7 b) where electrical current I and transverse voltage VH are measured on a sample with the known geometry.

EH VH th ⋅ w RH = = = Rt ⋅ th 17 jBz I w

Here, Rt denotes transverse Hall resistance VH/I .

Comparing equations 13 and 17 we can observe that it is possible to calculate two important electronic properties of materials simply by measuring electrical resistivity and

Hall coefficient. Hall coefficient will allow us to calculate carrier density n for electrons or p for holes: n,p=R /q. H 18 Negative Hall coefficient (following convention in Figure 7 b) in single carrier system

indicates that dominant carriers are electrons with density n, while positive RH indicates material populated by holes of density p. Having calculated carrier density it is possible also to determine electron mobility from Eq. 13 as:

RH = . 19 ρ

Note on customary units; ρ is in m, RH in m/T. From this it follows that is in 1/T which is equivalent to commonly used units m 2/V .sec or 10 4 cm 2/V .sec.

2.3 Seebeck coefficient

This transport property was already introduced in the first chapter and it was explained that it can be measured as a ratio of a potential difference, V, and a temperature difference, T, between two ends of the material. Strictly speaking, Seebeck coefficient is a transport property and it can be a tensor. It relates electrical fields and temperature gradients which are established in a material when heat flux, Q, passes through it,

E=S.∇T.

a) b) B w z w th th

Ex Ey Vs L z TC y

x Qx Qx V + N -

Figure 8 Geometry in which Seebeck and Nernst coefficients are observed.

Figure 8 depicts a geometry in which heat flux is in the xdirection and it establishes the electric field in same direction. This is a one dimensional case in which the Seebeck coefficient is defined as a constant:

− dV Ex dx Vs S = − = = 20 dT dT T −T dx dx H C

From this expression, we see that measurement of the Seebeck coefficient does not depend on the geometry of a sample; it is simply a property of the material related to the electron entropy as shown in Eq. 5. For experimental evaluation of the Seebeck coefficient, it is sufficient to measure temperatures TH and TC in addition to the potential difference, VS, between two points at which these temperatures are measured.

2.4 Nernst effect

The Nernst coefficient is a transport property that is the thermal equivalent to Hall coefficient and it is observed when heat flux passes through a material which is in a magnetic field, B. To describe this phenomenon, we will use a configuration in which heat flux is a one dimensional and perpendicular to the magnetic field as shown in Figure

8 b). In this case, electrons are diffusing through the material with diffusion velocity, vD, in the direction opposite to the applied heat flux, Qx. These electrons are then deflected by Lorentz force like in the case of Hall effect. Equilibrium is established by the electric field Ey which can be calculated by measuring Nernst voltage, V. In this configuration

Nernst coefficient is:

E = ⋅ B dT . y z dx 21

For practical purposes, to calculate , we need to determine sample geometry, measure magnetic field, transverse (Nernst) voltage V and temperatures TH and TC.

dV E dy V L = y = = B dT B dT B (T −T ) w 22 z dx z dx z C H

2.5 Thermal conductivity

Thermal conductivity is the last of thermal transport properties that will be introduced in this chapter. Thermal conductivity is a thermal transport coefficient (in the general case a tensor) which relates heat fluxes to temperature gradients: v r q = −κ∇T . 23

This is familiar form of Fourier’s law and it can be further simplified in the case of one dimensional heat flux as shown in Figure 8. Here, heat flux is qx=Q x (w .th) and results in a temperature gradient in the same direction:

dT q = −κ . x dx 24

In most solids it is possible to neglect radiative heat transfer through the material and analyze thermal conductivity, κ, as a sum of lattice thermal conductivity, κL, and electronic thermal conductivity κe.

κ = κ +κ total L e 25

Electronic thermal conductivity κe

If we once more observe electrons in a solid as free particles or an electron gas we can see that under temperature gradient dT/dx, electrons diffuse through the material. For example electrons at two different temperatures T and T+dT have energies 3/2k BT and

3/2k B(T+dT) . If we assume heat flux in only one direction, electrons will diffuse towards

lower energy region under the influence of thermodynamic force F = 3 k dT . Tx 2 B dx

Electron of mass m will drift under effect of force, FT, with average drift velocity uT,

u F = m T . From this simple analysis, we see that average drift velocity becomes: Tx τ

3 τ u = k T . T 2 B m 26

Now, we can calculate that total heat flux of n electrons carrying heat 3/2k BT with average drift velocity uT to be:

2  3  2 dT  τ  qx =   k B nT   .  2  dx  m  27

If we assume that average electron scattering times are the same as those developed when electrons drift under effects of electric field we can combine equations 27 and 12 to write:

2  3  2 dT  σ  q = k nT   . x   B  2   2  dx  q  28

By comparing the previous equation with Fourier’s law we see that we can write:

2 2  3   kB  κ e =     σ ⋅T = L0 ⋅σ ⋅T ,  2   q  29

Lo is known as the free electron Lorentz number. This simple derivation shows that electron gas should have electrical and thermal conductivities proportional to each other, this is known as the WiedemannFranz law.

Deviation from this law was observed early on, for example by Dunaev who measured

thermal conductivity of lead sulphide as a function of temperature and impurity concentration. 3 As observed in some of the materials presented in this text Lorentz number, L, is close to the free electron Lorentz number, L0, only in degenerately doped semiconductors and metals. In general we can write that L/L 0 ≠ 1, exact values for

Lorentz number can be calculated using procedures outlined in the next chapter.

Lattice thermal conductivity κL

As originally proposed by Debye in 1914 lattice thermal conductivity is a result of thermal motion or oscillations of particles in solids. Oscillations of individual particles or collections of particles in a solid are equivalent to propagation of sound waves through a solid. Heat carrying waves, phonons, travel with speed of sound between two ends of sample. Following Debye model lattice thermal conductivity can be then represented as:

1 κ = C ⋅ v ⋅ l . L 3 v Φ 30

Here, Cv is the specific heat per unit volume, v is the average phonon velocity and lΦ is average phonon free mean path.

In addition to these properties we can define other galvanomagnetic and thermomagnetic properties which were not used in analysis of materials in this document such as: v r Peltier coefficient Π: relates heat flux and charge flux: Q = Πj . Π is related to

Seebeck coefficient through Kelvin’s relation S= Π.T.

Ettingshausen coefficient P: relates temperature gradient and electric flux in the

presence of magnetic field dT = P(B ⋅ j ) . dy z x

RiggiLeduc coefficient Ξ: relates transverse and longitudinal temperature

gradients in magnetic field dT = Ξ B ⋅dT dy ( z dx)

2.6 Measurement of transport properties and estimated errors

Electrical resistivity, Seebeck, Nernst and Hall coefficients are measured on the samples cut into parallelepipeds with dimensions w x th x L as shown in Figure 7 and Figure 8.

Majority of measurement is performed in standard flowthrough cryostats in the temperature range 77 to 650K and in magnetic fields from 1.4 to +1.4T. For illustration, one of the actual samples and cryostats are shown in Figure 9. In addition to these instruments commercially available instruments are used to measure thermal diffusivity and specific heat, for example. Following paragraphs will outline some of the techniques used to measure these properties and the main sources of the measurement errors.

Figure 9 (a) Sample configuration and (b) standard flowthrough cryostat used for measurement of transport properties. 23

2.6.1 Electrical resistivity

Electrical resistivity was measured using the standard 4wire AC and DC methods. In this method sample resistance was measured by passing direct or alternating current of value

I(Amp) through the electrodes placed on the top and bottom of the sample and measuring the voltage drop V across two probes attached to the side of the sample. Configuration is identical to that shown in the Figure 7 (a) and resistance is calculated as R=V/I . In order to insure that the current flux lines are parallel between the voltage contacts samples are long with small cross section. Usually L/th=6 and distance between contacts is L=3 .th .

Electrical resistivity is calculated following the equation 14. When using DC method it is required to correct for extraneous voltages which are recorded as a consequence of the

Seebeck voltages produced by the temperature gradients. By measuring two voltages for currents passed in two opposite directions it is possible to eliminate this error. Depending on the sign of the Seebeck these voltages would add or subtract depending on the direction of the current. Using of AC currents has the same effect.

Average sample has L=4±0.2mm , w=2±0.05mm and th=1±0.05mm and we can use these dimensions to estimate geometric errors. Neglecting measurement errors in measuring current and voltage potentials as they are much smaller than geometric errors we can estimate that in average error in measuring electrical resistivity equals:

w ⋅th ρ = R → L ε (ρ) = ε (R) 2 + ε (w) 2 + ε (th) 2 + ε (L) 2 ≠ 0% 2 + 5% 2 + 2% 2 +1% 2 ≈ ± 5.5 %

2.6.2 Seebeck coefficient

Since Seebeck coefficient is the property that does not depend on sample geometry it can be measured as a ratio of the voltage drop over the temperature drop. For practical reasons we use following geometry shown in Figure 10. Samples have geometry driven by

Qin

Th Cu

Con Tc

VS VTc Tb VTh

Figure 10 Measurement of Seebeck coefficient.

resistivity measurement. Temperature differential is established by passing the heat from the top of the sample using 120 or 300 strain gauges as heaters. This strain gauge can be seen in Figure 9 (a) for example. Heat flux is selected in such a way that temperature differential between two thermocouples does not exceed 1 to 3 degrees, or 1% of absolute sample temperature. Samples are thermally anchored to the cold finger of the cryostat

seen in Figure 9 (b). Temperature differential is measured as difference T=ThTc, using two copperconstantan thermocouples. Voltages VTh and VTc are measured across thermocouple leads as shown in figure above. To calculate actual temperatures Th and Tc we measure Tb as a temperature of sample base and cryostat’s cold finger; Tb is measured

VTh using calibrated Cernox thermometer. Th is calculated as Th = + Tb . Here, SCuCon (Tb)

SCuCon is Seebeck coefficient of copperconstantan thermocouple as a function of

38 temperature . Seebeck voltage VS is measured by connecting voltmeter across the copper leads of top and bottom thermocouple. Measured voltage is sum of Seebeck voltage in sample plus Seebeck voltage in the copper leads. Thus, V used in equation 20 to

. calculate Seebeck coefficient is V=V SSCu (T b) (T hTc).

To ensure that all four ends of thermocouple wires that are attached to the base are at temperature Tb and that Th and Tc are measured exactly on the sample thermocouples are made of thin wires, dTC =0.0025mm . This diameter minimizes heat flow through the wires while still allows manageable handling of the thermocouples.

By carefully attaching couples to the sample and applying measurement technique which allows measurements of voltage and temperature differentials at exactly same points on the sample, minimizes errors in measuring Seebeck coefficient. Estimated measurement error is at the order of 3%, and mostly due to noise in the voltages.

2.6.3 Thermal Conductivity

At temperatures less than 300K thermal conductivity is measured by using the static heater and sink method. Heat flux is passed into the sample using static heater (strain gauge shown in Figure 9 (a) ). Heat flow is measured by monitoring current and voltage

. . on the heater, Qin =Vheater Iheater . The heat flux qin =Q in /(w th) establishes temperature gradient which is measured using thermocouples as discussed in previous paragraphs, dT/dx=(ThTc)/L. Based on these measurements it is possible to calculate thermal conductivity as:

q V I L κ = in = heater heater dT w⋅ th T − T 31 dx h c

The main source of error is radiation heat transfer: not all Qin passes through the sample.

These losses are proportional to T3. Radiative heat exchange was minimized by the use of three radiation shields to values estimated below 0.4mW/K at 300K. To reduce the radiative heat flux errors the thermal conductivity data were taken on samples with a larger crosssection (usually 3x3x2 mm) in order to maintain the samples’ thermal conductance to around 4mW/K. For this reason static method can be reliably used only for measurements below room temperature and, at 300K, will give more than 10% error.

If we neglect radiative losses the accuracy is the same as for the resistivity.

High temperature ( T>300K) total thermal conductivity is measured using the transient thermal diffusivity method. Using this method κ is measured by measuring sample density γ, specific heat at constant pressure Cp and thermal diffusivity D.

κ=γ .C .D p 32 All of these properties are measured using commercially available instruments.

Density is measured only at room temperature by measuring buoyant force on standard

Archimedes attachment to the Mettler Toledo XM100 analytical scale. Specific heat is measured using the Thermal Analysis Q200 differential scanning calorimeter with manufacturer’s declared error of ±5%. Thermal diffusivity is measured using Laser Flash

3000, instrument manufactured by the Anter Corporation. Measurement error in this case does not exceed 10%. In order to ensure accuracy of the measurement density and the thermal diffusivity are measured on polished disks with 10mm diameter and thickness of

1mm to 2mm. Specific heat is measured on samples with the thickness 0.2 to 0.3 mm weighting 2 to 30mg. From the equation 32 we can see that total error in measuring thermal conductivity using this method reaches 10% 2 + 5% 2 + 2% 2 ≈ 11%

2.6.4 Hall Coefficient and Nernst Coefficient

To measure Hall ( RH) and adiabatic Nernst ( a) coefficients we simply measure Hall resistivity or Nernst voltage as a function of magnetic field, B, and calculate slope as B goes to zero. To calculate exact values we use equations 17 and 22. All materials analyzed in this text have cubic symmetry. The cubic symmetry of the samples implies that no Umkehr effects are expected; therefore the results can be deduced from the data taken in both field polarities. The transverse effects (Hall and Nernst coefficients) are

subject to Onsager relations and extracted as the components that are odd with field. 24

That is, slopes are calculated both in positive and negative field and values subtracted in order to remove even terms. Origin of errors is again mainly in measuring sample’s geometry; for RH and a errors are estimated to 2% and 7% correspondingly.

R R = th → H B 2 2 2 2 2 2 ε (RH ) = ε (R) + ε (B) + ε (th) ≈ 0 + 0 + 2% = 2%

V L a = → Bz (TC − TH ) w 2 2 2 2 2 2 2 2 2 2 ε ( a ) = ε (V ) + ε (B) + ε (T ) + ε (L) + ε (w) ≈ 0 + 0 + 0 + 2% + 7% ≈ 7%

25 The measured transverse Nernst coefficient a is the adiabatic one ; the data reported are the isothermal transverse Nernst coefficient which is calculated from the adiabatic one as in reference [25].

Chapter 3: Band Structure Models

This chapter briefly outlines the mathematical tools used to model the semiconductor band structure. These models are used to relate measured galvanomagnetic and thermomagnetic properties with transport properties. Discussion is based on band theory of solids. Atoms are organized in the periodic structures. Bands originate from discrete energy levels of atoms in lattice. These are broadened by interaction of neighboring atoms and their behavior depends on the spacing of the atoms and on the energy level from which they originate. As indicated in Figure 11 bands originating from the inner shells are narrow and less effected by neighboring atoms.

plevel Band

Forbidden gap slevel

Electron Energy Band

Atomic spacing a0

Figure 11 Forming of bands from discrete atomic energy levels by broadening through interatomic coupling.

30 Energy levels of electrons on outer shells are broader and they might overlap with energy levels originating from other levels. Number of available levels in every band is integer multiple of number of atoms in the solid. All energy levels are filled each with one electron with spin up and one with spin down state. Details of electron wave nature are shown in literature, for example by Ziman in reference [26]. Without going into details we will just state that it is common to refer to energy dependence to wave vector, k, rather than interatomic spacing. Due to periodic nature of lattice structure discontinuities in allowed energy levels occur when the electron energy wave vector reaches end of

Brillouin zone. For example in a onedimensional lattice that is when kx=nπ/a , where a is a lattice constant and n an integer value. Energy dependence on kxvector is shown in the

Figure 12. Here, we will only define two important parameters: wave group velocity

1 ∂E v = which is actual velocity of electron as wave packet and electron effective mass h ∂k m*, that is apparent electron mass which is accelerated with acceleration a=dv/dt by

−1  ∂ 2 E  external force F, m* = h 2   . We can see that group velocity reaches its maximum  2   ∂k  when bands are filled up to inflection point shown in the Figure 11, at the same time effective mass changes sign, i.e. from positive to negative and we have transition from conduction to valence band. Energy gap Eg is energy area with no electron states, band gap is energy difference between bottom of the conduction and top of the valence band, shown in Figure 11.

31 E(k) E(k)

Inflection point

3π/a 2π/a π/a kx π/a 2π/ 3π/ π/a kx π/a a a

Figure 12 Energy wave vector relation in onedimensional lattice. Left is multizone representation and to the right an equivalent reduced zone representation.

h2 2 kx Near kx, the energy is parabolic function of k: E = ; near the inflection point we 2m*

 E  h2k 2 have what we call “ nonparabolic ” bands and E1+  = x Number of available   *  Eg  2m energy states in solid between energies E and E+dE is determined by the density of states

(DOS) function g(E ), in 3dimensinal bulk material this relation is a square root of energy and shown in Figure 13 for illustration. FermiDirac statistics determines probability that electron will occupy state with energy E:

1 f (E) = .  E − EF  1+ exp  33  k BT 

Here, EF is Fermi energy or energy level up to which bands are filled, also shown in

Figure 13.

a) E E E E n Ec

0 1 g(E) f0(E) Carr. Conc. (n or p)

b) E E E

Ec n E

E v p

g(E) f0(E) Carr. Conc. (n or p)

Figure 13 Density of states, FermiDirac distribution and carrier density for (a) ntype and (b) intrinsic semiconductors at T>0K

Now using these concepts we can define electron number density as:

∞ n = g(E) f (E)dE . ∫−∞ 34

Graphical representation of this result is shown in Figure 13. Here, we can see that depending on the actual position of EF and temperature we can either have one or more types of conducting carriers, Figure 13 (b) shows electrons and holes, n and ptype carriers.

In the electric field X? electrons with energies between E and E+dE contribute to the total electrical conductivity with:

v (E) σ (E) = g(E)q DRIFT = g(E)q(E) , X 35 which is analogous to Eq. 12. and uses same definition of mobility. Then total electrical conductivity is integral over all available energies E: 26

∞  ∂f  σ = σ (E)− dE ∫ 36 −∞  ∂E 

Mott 27 shows the Seebeck coefficient to be:

k 1 ∞ (E − E )  ∂f  S = B σ (E) F  dE ∫ 37 q σ 0 kBT  ∂E 

In degenerate statistics, f(E)=1 for E>E F and f(E)=0 for E

π 2 k 1  dσ (E)  S = B ()k T   3 q B σ dE 38   E=EF

Using the Eq. 35 we can write this in the form:

2 π kB  1 dg(E) 1 d(E) S = kBT  +  . 3 q g dE dE 39  E=EF

 E − EF  If we apply nondegenerate statistics, i.e. when f (E) = exp−  to Mott’s relation,  kBT 

Eq. 37, we get that the Seebeck coefficient depends on density of states effective mass, scattering coefficient λ and carrier density:

  2   2πm* k T  k  3 ()d B  S = B λ + 3 + ln h  . q  n  40      

Scattering coefficient, λ, is used in modeling relaxation time τ in bulk 3D solids as a simple power function of energy with λ,

τ (E) ∝ E λ− 2/1 . 41

This approximation is only valid for elastic collisions, so it is assumed that the states of the particles before and after collision are at the same energy level. For collisions with acoustic phonons, λ=0; for neutral impurity scattering λ=1/2; for ionized impurity scattering, λ=2. 32

Expression in Eq. 40 was derived by Pisarenko and first time referenced by Ioffe in reference [55]. There he also calculates the numeric values of constants in the PbTe assuming acoustic phonon scattering and bottom of the band effective mass. With this, the Pisarenko relation for electrons or holes PbTe at 300K becomes:

S(n) = 477 −172ln(n) ( V/K) . 42 This is valid in the nondegenerate statistics and it is shown in Figure 16 as a solid black line. Tail of dependence in Figure 16 is governed by the Eq. 39, while low carrier density end corresponds to picture in which we observe conduction by both electrons and holes, as seen in Figure 13 (b), which we call bipolar conduction. We can find many experimental data to confirm this dependence, a very nice example being work of

Airapetyants, Rudnik et.al. on the structure of the valence band of PbTe. 28

two carrier nondegenerate statistics degenerate conduction statistics 600

400 V/K) S ( S 200

0 1x10 16 1x10 17 1x10 18 1x10 19 1x10 20 n(cm3)

Figure 14 Pisarenko plot showing dependence of Seebeck coefficient to carrier density in 2carrier conduction region and in regions where nondegenerate and degenerate statistics can be applied.

3.1 Method of Four Coefficients - Single Carrier Systems

We will first focus on case in which Fermi energy is either in conduction band

(see Figure 13 a) or in valence band and thermal energy ( kBT) is small such that system is entirely dominated by one type of carriers, either holes or electrons. In this case the number of properties we measure is equal to the number of unknown transport properties.

Electrical conductivity, Seebeck, NernstEttingshausen, and Hall coefficients are related

* to the transport properties: Fermi energy EF, carrier mobility , effective mass m , and energy scattering exponent λ. We name this the method of four coefficients 29,30 and use it

as the fundamental analytical tool through this work. We will distinguish 3 fundamental cases: I. Near the inflection point in Figure 12 the energy to momentum relation is not parabolic but we keep full FermiDirac statistics Eq. 33, II. Fermi energy is deep in the band (degenerate i.e. f(E)=0 for E>E F and f(E)=1 for E

2 3/1 * * * 3/2 43 m = (mL ⋅ mT ) A ,

with the factor A describing number of electron or hole pockets in the Brillioun zone.

I onparabolic, nondegenerate band

Assuming a nonparabolic, single carrier and nondegenerately doped system it is possible to use a simplified form of the equations relating four transport properties and four measured coefficients as derived in Ref [31]. Using a nonparabolic dispersion law relating energy E and wave number k, Eq. 44, one can relate the four measured transport properties with the four unknown coefficients.

h 2 k 2 h 2 k 2  E  γ (E) = L + T = E1+  , * *   44 2mL mT  Eg 

Carrier density of holes or electrons ( p or n) is:

2/3 * ∞ (2m k BT ) 2/3  ∂f  p, n = γ ()x − 0 dx . 2h 3 ∫ 45 3π 0  ∂x 

1 Here, we use normalized energy x=E/k BT and corresponding derivative dx = dE . f0 k BT

1 is a FermiDirac distribution function f 0 (x) = with xF=E F/k BT. 1+ exp(x − xF )

Electrical conductivity and Seebeck coefficient are defined as in Reference [32]:

2/3 * 2 ∞ 2/3 (2m kBT ) e γ ()x  ∂f  σ = τ ()x − 0 dx , 46 2h3 * ∫ 3π mα 0 γ ′()x  ∂x 

∞ 2/3  γ (x)  ∂f 0    xτ ()x − dx  ∫ γ ′ x ∂x k B  0 ()    S = − xF , 47 e  ∞ γ ()x 2/3  ∂f    τ ()x − 0 dx  ∫ ′  0 γ ()x  ∂x  

where, f0 is the Fermi distribution function. Transverse properties: Hall and Nernst

Ettingshausen coefficients can also be related to transport integrals:

∞ 2/3 ∞  γ (x)  ∂f   2/3  ∂f    τ 2 x − 0 dx γ x − 0 dx ∫ 2 ()   ∫ ()    3K()K + 2 1  0 ()γ ′()x  ∂x   0  ∂x   RH = 2 2 , 48 ()2K +1 ne  ∞ γ ()x 2/3  ∂f    τ ()x − 0 dx  ∫ ′   0 γ ()x  ∂x  

∞ 2/3  γ (x) 2  ∂f 0    xτ ()x − dx  ∫ 2 ∂x k B  0 ()γ ′()x    = − S + xF . 49 R σ e  ∞ γ ()x 2/3  ∂f   H  τ 2 ()x − 0 dx  ∫ 2  0 ()γ ′()x  ∂x  

* mL Here, K = * is effective mass anisotropy coefficient, ratio of longitudinal and mT transverse effective masses. In PbTe and PbSebased materials anisotropy is small and the prefactor to Hall coefficient is approximately 1 within 5%. Equations (46) to (49) can be applied to describe single carrier system with nonparabolic band. These four

* equations with four unknowns can be solved for m , E F, and λ. The latter two are in

(46)(47) implicit as energy dependence of relaxation time τ(E) defined by Eq. (41). To verify that results are valid for all calculations we can backsubstitute the calculated parameters into full integral model, Reference [29, 30]. Good agreement in calculated and measured S, σ, R H and , provides confirmation that the model is valid.

For practical reasons it is also interesting to calculate the actual Lorentz number which is used to determine electronic component of thermal conductivity

κ e = LσT . (29)

As mentioned in previous chapter, Lorentz number depends on the material’s electronic properties as described:

 ∞ 2/3 ∞ /3 2 2  γ ()x 2  ∂f0   γ ()x  ∂f0   2  x τ ()x − dx  xτ ()x − dx    k  ∫ γ ′()x  ∂x   ∫ γ ′()x  ∂x    L = B 0 − 0 . 50    ∞ /3 2  ∞ /3 2    e  γ ()x  ∂f  γ ()x  ∂f   τ ()x − 0 dx  τ ()x − 0 dx    ∫ γ ′()x  ∂x   ∫ γ ′()x  ∂x     0  0  

* Once the four parameters m , E F, and λ are fitted to (46) to (49) Eq. 50 we can be used to estimate the temperature dependence of the Lorentz number and with that the

electronic component of thermal conductivity. Only then we can accurately compute the lattice component of the thermal conductivity as κL=κtotal κe.

II onparabolic, degenerately doped

As most samples measured in this study are degenerately doped we can use the Bethe

Sommerfeld expansion to greatly simplify the fits as shown by Thiagarajan in reference

[31]. As a crude reference we can use this approach when EFEC>(34)k BT, where Ec is band edge energy at k x=0 in Figure 12 . Electron density n or the hole density p are given by:

2/3 (2m*γ (E)) n = . 51 3π 2h 3

Electrical conductivity σ = ρ1 and the Hall coefficient are simply given by familiar forms:

σ = nq(EF ) , 52

3K(K + )2 1 R = , H 2( K + )1 2 nq 53 where in PbSe and PbTebased materials due to low anisotropy prefactor of the Hall coefficient can be neglected and RH is approximated with RH = 1/ne , expression identical to that of the free electron model. The Seebeck S and isothermal transverse Nernst coefficients are then:

π 2 k  γ (' E ) γ (" E ) S = ± B k T ()λ +1 F − 2 F , B   54 3 e  γ (EF ) γ (' EF ) 

π 2 k  1  γ (' E ) γ (" E ) = B k T λ −  F − 2 F B   55 3 e  2  γ (EF ) γ (' EF ) 

From the nonparabolic dispersion relation we can use calculated derivatives γ′(E) and

γ″(E) from Eq. (44) and solve for n or p, E F and λ, transport relations.

EF γ ′(EF ) = 1+ 2 Eg 56

2 γ ′′(EF ) = E g 57

By this we obtain four explicit relations that are function of measured parameters and can be solved in following sequence:

3K(K + )2 1 n or p = 2 , 58 2( K + )1 RH e

(E ) = ne , F σ 59

2/1  2   k k T   k k T  k − A + π 2 B B  +  A + π 2 B B  + 2π 2 B A      e Eg e Eg e 60 Eg      E = , F 2 A and

 4k T  a + B  3  k B π Eg + 2EF  R σ  2 1 1 H  E  λ = +  e 3  E + F  . 2 k T  2E  F E  61 B 1+ F  g   E   g   

Here A is lump parameter A=Sa/(σR H).

Solving set of equations 46 to 49 is computationally demanding task with diverging solutions. By eliminating integral form using the assumption of degenerately doped system we greatly simplify the calculations. Unfortunately we must impose limit that

EF>(34)k BT in order to apply model described by equations 58 to 61.

III Parabolic band

In this text we will also make use of assumption of parabolicity of the band,

lim(Eg ) → ∞ . In this case the transport properties are easiest to write using the transport

17 integrals. For parabolic bands are given as a function of the Fermi integral Fj(xF):

∞ x j dx F (x ) ≡ . j F ∫ 62 0 1+ exp(x − xF )

Once more we make use of reduced Fermi energy notation, given by xF ≡ EF k BT . In the case of quadratic dependence of energy to momentum and power law dependence of scattering to energy, thermal properties are:

5  + λ F3 (x )  k (2 ) +λ F S = − B  2 − x  and 3 F 63 e ()2 + λ F1 +λ (xF )   2 

5 5  + 2λ F3 (x ) + λ F3 (x ) k (2 ) +2λ F (2 ) +λ F = B  2 − 2  . 3 3 64 e ()2 + 2λ F1 +2λ (xF ) ()2 + λ F1 +λ (xF )  2 2 

Electrical properties are still defined using 52 and 53

σ = nq(EF ) , 52

3K(K + )2 1 R = , H 2( K + )1 2 nq 53 while carrier mobility can be written as:

π F0 (xF ) = 0 . 2 F1 (xF ) 65 2

Factor 0 is temperature independent mobility constant. Carrier density in parabolic band is defined as:

* (2 m k BT) n, p = F1 (x ). 2 3 F 66 π h 2

From 62 to 64 we can calculate n (or p), , EF and λ by solving 4 equations with equal

* number of unknowns and than use these values to determine m and 0 from 65 and 66.

By applying this methodology it is possible to diagnose the increase in energy dependence of density of states or corresponding n(E)/dE as indicated by an increase in the calculated effective density of states mass m*. m* can be compared with well documented literature 33 values for IVVI semiconductors. Same calculations provide us with values of scattering coefficient λ. An increase in energy dependence of scattering simply reflects to an increase in energy exponent in Eq. (41). In case of degenerately doped IVVI semiconductors λ=0.5 to 1, depending on the scattering mechanism. An increase in λ over unity would be an indication of resonant state scattering as for example observed in PbTe with nanoinclusions of Pb reported by Heremans.

3.2 Two Carrier Bipolar Conduction

The term intrinsic conduction is used to denote semiconductor in which we observe thermally activated carriers, Figure 13. For example if Fermi energy is positioned in the conduction band then electrons from the highest occupied state can reach lowest empty conduction band when T>Eg/2k B. This behavior is shown in Figure 13 (b) and it is commonly observed in many narrow gap semiconductors and it is one of characteristics of materials we analyze in this work. In more general case minority carriers can originate from a band with the same carrier sign for example in Bi 2Te 3 we can observe activation of holes from the deeper valence band when temperatures are increased. In this case we can use more general terms: two carrier conduction or bipolar conduction. In this general case all electrical and thermal currents through the material are sum of the contributions from the separate bands. With this, properties become function of partial properties of the band i.e. electrical conductivity becomes sum of individual conductivities of type 1 and type 2 carriers. 25

σ = σ 1 + σ 2 = n1q1 + n2 q 2 , 67

2 2 n11 + n2 2 RH = , 68 n2 2 + n2 2

S σ + S σ S = 1 1 21 2 and 69 σ

( σ + σ )(σ )+ (S − S )(R σ − R σ )σ σ = 1 1 2 2 1 2 H1 1 H 2 2 1 2 . 70 σ 2

Intrinsic conduction is specific case in which type 1 carriers are electrons and type 2 carriers are holes. In this case we assume positive sign for carrier densities p and n and corresponding mobilities. Partial Seebeck coefficients of electrons and holes have negative and positive sign respectively. Same convention is used for partial Nernst and

Hall coefficients. With this 67 to 70 becomes:

σ = σ e + σ h = nqe + pq h , 71

2 2 − n e + p h RH = , 72 ne + p h

Seσ e + Shσ h S = and 73 σ e + σ h

( eσ e + hσ h )(σ e + σ h )+ (S h − Se )(RHhσ h − RHeσ e )σ eσ h = 2 . 74 ()σ e + σ h

To model each partial property we can use either one of the models described previously for the single carrier conduction band. Therefore, we have system of four equations with eight unknowns. This is underdetermined problem and we will show two different approaches how to solve this problem, one in Chapter 4 and one in Chapter 5. One observation we can make at this point is that in the intrinsic system absolute value of S reaches its maximum value as number of minority carriers increases. This is due to opposing sign of partial Seebeck coefficients Se and Sh i n the numerator of equation 73.

The effect of bipolar conduction on the carrier density dependence S(n) is shown in

Figure 14. For the same reason Nernst coefficient described by Eq. 74 changes the sign and crosses zero at similar temperatures. Exact temperature at which crossing occurs

depends on the details of the band structure, but we use these two observations as an indication of intrinsic conduction.

Chapter 4: Modification of Electronic Density of States in IV-VI Semiconductors

4.1 Introduction

In this approach, the basis for the enhancement of thermoelectric properties is the

MahanSofo theory,34 which describes systems in which there is a local increase in the density of states g(E) over a narrow energy range (ER), as shown schematically in Figure

16. Group IVVI semiconductors are selected since they are well documented systems

35 and PbTe and Pb1xSnxTe are used as high temperature TE materials. Lead chalcogenides and their alloys with Snchalcogenides crystallize in the rocksalt structure with large number of intrinsic defects in the metal or chalcogen sublattice.36 These impurities act as donors or acceptors resulting in carrier concentrations36 of 1018 to

1020cm3, and only in very carefully prepared stoichiometric samples defect densities can be minimized resulting in densities of 1016cm3 carriers.33 As discussed for example in

Ref. [36] all lead chalcogenides are direct Lpoint conductors at temperatures <450K and have similar energy gaps (at 4K PbTe=187meV, PbSe=145 and PbS 283meV). Locations of Lpoints in Brillouin zone are shown in Figure 15. Electrons and holes fill eight ellipsoids and since these are shared with neighboring zones degeneracy coefficient A in the Eg. 43, is 8/2=4. Gap opens above 100K with temperature at the rate of 0.4meV/K

47 resulting in approaching of heavy hole band at Σ point to the valence band edge at the same rate. These materials are strongly non parabolic and the increase in band gap with temperature is followed by proportional decrease in what is already small effective mass.

a) b) <001>

Σ ΣΣΣ

Γ X Γ <010>

<100> X L L

<111>

Figure 15 (a) Brillouin zone of PbTe with major crystalographic directions and named points. Γ point is center of the zone. (b) Electrons and holes are distributed in eight half pockets at L points. Heavy holes are distributed at Σ points and are not shown here.

Mahan and Sofo indicated that a material with a deltashaped or Lorentzian shaped density of states function could have a zT much larger than that of TE materials today. This increase originates from an improved S. In a similar manner, Ravich 37 describes the effects localized states have on the scattering of electrons in a solid, and names them resonant scattering states. 48 If we observe Eq. 8 which defines figure of merit zT and note that electrical conductivity is a product of the energy dependent carrier density n(E), the carrier charge q, and the energy dependent mobility (E) we get that:

zT = (S 2n) q . 75 κ L +κe

The above equation shows that any increase in the Seebeck coefficient not accompanied by significant decrease in carrier density and mobility results in an improved zT . In the classical treatment of semiconductors, this is achieved by changing the carrier concentration n(E). Here, we show that we can improve S at a given carrier concentration by modifying the electronic density of states.

The Seebeck coefficient given by the Mott expression (Eq. 39)

2 π k B  1 dg(E) 1 d(E) S = k BT  +  , 3 q g dE dE 39   E=EF

and effective density of states mass can be used together to derive Pisarenko expression.

However, here we will use the effective density of states mass for 3D solid (illustrated using Figure 16) to reformulate Mott relation:

3 2m* 2 g(E) = ( ) E . 76 4π 2h3

With this Mott relation can be reformulated to read:

2 2 * π kBT 1 dg(E) 1 dτ (E) 1 dm  S =  + − *  . 77 3 q n dE τ dE m dE  EF

Therefore, a larger the effective mass at a given E means a larger density of energy states at that energy level. Equation 77 shows that there are two mechanisms that can increase

Mechanism (1) an increased energydependence of g(E) , for instance by modifying the electronic structure of the material which would result in a localized increase in the energy dependence of the density of states g(E), (our theory).

Mechanism (2) in which S is enhanced due to an increased energydependence of (E) , for instance by a scattering mechanism that strongly depends on the energy of the charge carriers, (Ravich’s theory). 36

EF g(E)

Figure 16 Energy dependence of density of state for atom energy level E R, hybridized with band. The Fermi energy level E F is positioned in the vicinity of this level.

Equation 77 is the basis of the MahanSofo theory, provided that the semiconductor has a Fermi energy EF properly aligned with respect to the position of the energy of the broadened resonant level in the band, a situation shown in Figure 16. The

* concept can also be expressed using the notion of effective mass m d, as shown for degenerate semiconductors: 38

2 2 3/2 8π kBT *  π  S = md   . 78 3qh2  3n 

Since zT also depends on the carrier’s group velocity via the electrical conductivity, the value of EF that maximizes zT is somewhat different from the value that maximizes S and m*34 . Hence, one of challenges in designing TE materials on the basis of MahanSofo theory is to obtain the right balance between an increase in effective mass and a decrease

in group velocity, the former as measured by S(n) and the latter as measured by carrier mobility.

An increase in the energy dependence of electron scattering in materials is the basis of mechanism (2). This mechanism is described by Ravich who also shows that the resulting increase in S is not sustainable at higher temperatures where acoustic scattering starts to dominate and (E)~E 1/2 . It is not possible to distinguish these two mechanisms by observing only S(n) at 300K, shown in Figure 14 for conventional PbTe. To diagnose which mechanism is in effect it is necessary to evaluate electronic properties and specifically λ and m* separately. This work uses the method of four coefficients (pg.

30).for this purpose.

4.2 Effects of alloying IV-VI alloys with rare earth elements

Early on in this work we made a survey of dilute IVVIRE alloys (RE = rare earth atoms). 39 PbTe and PbSe are the materials of main interest here since their electronic properties are well documented. 33,25,51 The temperature dependence of the band structure is well defined and the energy gap of PbTe can be finely tuned by alloying with Sn or Se.

The MahanSofo theory predicts that largest thermopower and therefore zT enhancement can be expected if a delta or Lorentzian shaped density of state function is formed in the vicinity of Fermi energy level. Exact position of EF must be such that product of drift

velocity and Seebeck coefficient is maximized since at the point where S=S max drift velocity, and therefore electrical conductivity, is zero. Deep 4f energy levels of RE atoms could be expected to form very sharp resonant states in IVVI alloys. The property of IVVI alloys to change energy gap on alloying with Sn and Se can be used to change the position of these levels relative to the Fermi energy in the band.

4.2.1 Alloying PbSe with Ce, Pr, Nd, Eu, Gd and Yb

As discussed by Heremans and Jovovic 39 the best known use of rareearth alloys with the IVVI compounds are infra red laser diodes.40 The origin of this work is

41 discovery and use of Pb 1xEu xTe alloys which were developed as a largergap semiconductor almost latticematched to PbTe and used in heterojunctions. MahanSofo propose that the 4 f levels of rareearth elements introduced as impurities to IVVI alloys could form favorable impurity bands and hybridize with the bands of host material. This is expected to improve S if the Fermi energy of the alloy reaches the 4 f level, as explained. We study here a number of materials in order to determine where this condition might be achieved.

Most of the REtellurides, selenides and sulfides crystallize in the NaCl structure, 42 as does PbSe. The relation between the lattice constants 42 of the various rare earth and lead selenides are summarized in Figure 17.

La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu 6.2 Pb 1.8 PbSe ) Å (

t 6 1.6 n a ) t s V n e ( o

1.4 X e c i t t 5.8 a L 1.2

5.6 1 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 Z

Figure 17 Lattice constant and Pauli electro negativity (X) of RESe alloys as compared with PbSe.

Except for Eu, the most rareearths studied here are trivalent atoms. By substituting for divalent lead in PbSe, we expect that those rareearths will act as donors. The Pauling electronegativity X of the rareearth atoms, as given by Allred 43 and summarized in

Figure 17, is very different from that of Pb. Electronegativity differences X = XRare Earth

44 – XPb dominate alloys scattering , following a formula

C 300 = , 79 Alloy 4X 2 x 1( − x) T where x is the atomic concentration of the rareearth elements and C is a parameter that depends on the dielectric constant of the material and the resulting electrostatic screening of the rareearth atom’s dipole moment in the lattice. Lead salts have very high dielectric constants, C ≈ 7.5 cm 2 / V .sec. Therefore very large alloy scattering must be added to the

other scattering mechanisms, specifically phonon and ionized impurity scattering. This will result in electron mobilities that are decreased by alloy scattering. We still expect to benefit from the increase in effective mass that will result from a hybridization of 4f levels.

Materials used in this study are Pb 1xCe xSe, Pb 1xPr xSe, Pb 1xNd xSe, Pb 1xEu xSe, Pb 1 xGd xSe, and Pb 1xYb xSe alloys. Samples are prepared by T. Story, Z. Golacki, W.

Paszkowicz and V. Osinniy at Institute of Physics of Polish Academy of Sciences. These samples properties are compared to those of pure PbSe (AlfaAesar 99.9999% 6N pure) for reference. All samples are polycrystalline and prepared by melting high purity components and annealing. The alloys were prepared by the Bridgeman method. The exact details of annealing schedules are not known.

In order to determine the quality of samples and concentrations of RE elements we report magnetic susceptibility data. In addition we perform a full set of galvanomagnetic and thermomagnetic measurements. Following the previously described method of four coefficients we deduce the carrier’s transport properties, specifically density, mobility, densityofstates effective mass and scattering exponent.

Experiment

Analysis by Xray diffraction, performed in Poland, revealed a rocksalt crystal lattice for all alloys. The presence of second phase inclusions in trace amounts was detected in the PbCeSe and PbPrSe crystals. All samples are ntype, except for the Pb 1

xEu xSe alloy and the reference PbSe sample, which are ptype.

The magnetic susceptibility χ was measured using a SQUID magnetometer in the temperature range of 5 to 120 K and results are shown in Figure 18. Measurements were performed by Tomasz Story and Viktor Osinniy on 1050mg samples cut in the neighboring slices from what was later used to measure galvanomagnetic and thermomagnetic properties. Standard 1/ χ vs. T plots are given. It is easy to recognize the expected CurieWeiss (CW) behavior. From CW law (Eq.80), we can see that magnetic susceptibility is proportional to 1/T. Coefficient of proportionality includes effective

Bohr magneton number ( p) which is tabulated for different elements depending on their

45 ground state and Bohr magneton B. The slope of 1/ χ vs. T is then simply proportional to the number of ions per unit volume /V . Experimental points presented in Figure 18 are corrected for the diamagnetic contribution of PbSe lattice. From these we can see that

Pb 1xEu xSe has perfectly regular CW behavior with small CW temperature. Using

80,we can calculate the concentration of Eu 2+ ions, x=0.089 (i.e. 8.9 at. %).

1 2 p 2 χ = B 80 3 V kBT

Second sample with regular CW behavior is Pb 1xGd xSe with x=0.040 for Gd3+ ions.

3+ Sample Pb 1xNd xSe has quite regular CW behavior with x=0.023 for Nd ions and negative CW temperature. Estimate of concentration of Yb ions is difficult due to irregularity of traces and it is estimated with 50% error to be x=0.01. Pb 1xCe xSe has non

CW behavior expected for Ce 3+ ions under the effect of cubic crystal field. 46 Calculated concentration of Ce in this sample is x=0.030. Pb 1xPr xSe: has CW behavior with high antiferromagnetic CW temperature. Magnetic inclusions and XRD analysis confirm the presence of second crystal phase rich in Pr and for this reason no composition estimate is possible here. Summary of these results is shown in Table 1.

Figure 18 Magnetic susceptibility of PbCeSe, PbNdSe, PbPrSe, PbYbSe, PbEuSe and PbGdSe samples used in this study. Lines in left figure are added to emphasize linear 1/T law.

3+ Pb 1x Ce xSe Ce donor x=0.03 High antiferromagnetic temperature 3+ Pb 1x Pr xSe Pr donor x=? High antiferromagnetic temperature 3+ Pb 1x Nd xSe Nd donor x=0.023 Pure CW behavior 2+ Pb 1x Eu xSe Eu donor x=0.089 Pure CW behavior 3+ Pb 1x Gd xSe Gd donor x=0.04 Pure CW behavior 3+ Pb 1x Yb xSe Yb donor x=0.01 Less clear CW behavior, error 50% Table 1 Summary of magnetic susceptibility measurements on PbSe:RE alloys

After determining structure and composition we perform galvanomagnetic and thermomagnetic measurements. As described in Chapter 2 small parallelepipeds were cut from the samples and their transport properties measured in a conventional flow cryostat

from 80 to 380 K, in magnetic fields of 1.3 to 1.3 Tesla. For this set of samples the galvanomagnetic measurements were taken using D.C. method

The resistivity ρ as even function in magnetic field following Onsager’s relations is deduced from the value of the resistive voltage averaged over both directions of current and reported at zero magnetic fields. We observe very small presence of the magnetoresistance in the REalloyed samples. Magnetoresistance can be caused by geometrical effects 47 and for that reason it is not discussed here. Small magnetoSeebeck effect was observed but again it is not reported since it can also originate in geometrical effects.48,30 The transverse Nernst voltages are measured by reversing the sign of the magnetic field and taking differences in transverse voltages a procedure allowed by the cubic symmetry of PbSe.49 The lowfield condition B<<1 is fulfilled throughout the measurement range, and the Hall and Nernst coefficients are calculated as the slopes of the Hall resistance and Nernst voltages against the magnetic field to yield the Hall ( RH) and Nernst ( a) coefficients. The galvanomagnetic data, resistivity and Hall coefficients at zero fields as a function of temperature are presented in Figure 19; Thermomagnetic data are presented in Figure 20 and the thermal conductivity in the figure Figure 21.

1 4 1x10 1x10 Pb1-xEuxSe Pb1-xCexSe 2 Pb1-xYbxSe 1x10 5 1x10 Pb1-xPrxSe

3 Pb1-xNdxSe 1x10 m/T) Pb Gd Se m) 1-x x 6

1x10 PbSe | ( | ( 4 ρ

1x10 H |R 1x10 7 1x10 5

1x10 6 1x10 8 100 200 300 400 100 200 300 400 T(K) T(K)

Figure 19 Electrical resistivity and Hall coefficient at the zero field for PbSe:RE alloys.

4 6x10 Pb1-xEuxSe Pb1-xCexSe 0 4x10 4 0x10 Pb1-xYbxSe Pb1-xPrxSe Pb Nd Se 2x10 4 1-x x Pb1-xGdxSe 4x10 6 PbSe S(V/K) 0x10 0 N (V K) N / T (V

2x10 4 8x10 6 4x10 4 100 200 300 400 100 200 300 400 T(K) T(K) Figure 20 Seebeck coefficient and Transverse Nernst coefficient at the zero field.

The thermal conductivities are measured using static heater and sink method as described in Chapter 2. The electronic component of thermal conductivity can be roughly estimated by using the free electron Lorentz number and Eq. 29, and subtracted from total thermal conductivity to calculate lattice component as shown in Eq. 25. Thermal conductivity of all samples is dominated by the lattice contribution, except for Pb 1xCe xSe and Pb 1xNd xSe, and to a lesser extent Pb 1xGd xSe. In these cases the electrical conductivities are large and reflect to the large electronic components of thermal

conductivity. Estimated values of the lattice thermal conductivities for all samples are not significantly different from that of the pure PbSe, (~1.8W/mK at 300K). 3

8.0 Pb1-xEuxSe Pb1-xCexSe Pb1-xYbxSe Pb Pr Se 6.0 1-x x Pb1-xNdxSe Pb1-xGdxSe PbSe 4.0 (W/Km) κ

2.0

0.0 100 200 300 400 T(K)

Figure 21 Thermal conductivity of PbSe:RE samples measured static heater and sink method. Solid lines represent total thermal conductivity κ and dashed lines calculated electronic component κe. Electronic component is calculated using free electron Lorentz number.

Morelli et al. 50 predict reduction in thermal conductivity in materials where crystal field splitting is observed, see Table 1. However, in Pb 1xCe xSe, the lattice thermal conductivity is not reduced as predicted although Story and Golacky claim to observe this phenomenon. This is probably because the crystal field splitting energy (360 K) is too high compared to both the Debye temperature (for PbSe: 144156K 51 ) and the measurement temperature range.

Having temperature dependence to ρ, R H, S and we can use method of four coefficients to compute n (or p), λ and m* at every temperature using Eq. (51) to (54). We are assuming nonparabolic dispersion of the electrons and holes. For these calculations we use temperature dependence of Eg of the pure PbSe at the Lpoint of the Brillouin zone. 33,51 The density of states effective mass reflects the facts that in degeneratelydoped

* 2/3 * 2 * PbSe, the Fermi surface are 4 ellipsoids of revolution and we have m =4 [( m T) m L

]1/3 . At each temperature and for each sample, Equations (51) to (54) are fitted to the experimental data and the results are given in Figure 22.

21 100 200 300 400 100 200 300 400 1x10 10000 Pb1-xEuxSe Pb Ce Se 1x10 20 1-x x

) Pb Yb Se

3 1-x x 19 1000 ) m 1x10 s Pb1-xPrxSe

c ( V

18 Pb Nd Se /

p 1-x x

, 1x10 2

n 100 Pb Gd Se 17 m 1-x x c

1x10 ( PbSe 1x10 16 10 0.6 1.0

0.4 ) 0.5 e λ (m *

0.0 0.2 m

0.5 0.0 100 200 300 400 100 200 300 400 T(K) T(K)

Figure 22 Transport properties: carrier density, mobility scattering coefficient and effective density of states mass of dilute PbSe:RE alloys.

The reference sample and PbSe:Eu are the only two ptype samples. Europium is divalent in PbSe:Eu and it increases the band gap, as previously established.52 Rareearth elements Ce, Pr, Nd, and Gd act as donors. PbSe:Yb appears to has a mixed behavior. It is interesting that it has a lattice constant that is not aligned with the other trivalent rare earths, Figure 17. The efficiencies of the rareearths as dopants are shown in Figure 23:

Ce and Nd have doping efficiency close to one. This is indication that they are somewhat less than monovalent donors at the concentrations in which they are in the samples. The doping efficiency of Gd, Pr and especially that of Yb, are much lower, and indication of possible mixedvalence behavior.

1x10 21

Nd Ce

20 1x10 Gd ) 3 m c (

Pr p , n 1x10 19

Yb 1x10 18 1x10 20 1x10 21 RE Atomic Concentration (cm3)

Figure 23 Free carrier density vs. concentration of RE atoms in PbSe:RE alloys. Dashed line represents monovalent donor. Europium alloy is omitted as Eu in PbSe:Eu is intrinsic.

The mobility of all rareearth doped samples is reduced with respect to that of electrons in PbSe. Comparing the mobility of the Ce, Gd, and Nd with that of pure PbSe and lightly doped PbSe:Yb we can conclude that the mobility levels for these samples are not unexpected for ionized impurity scattering at that doping level. We do not need to invoke alloy scattering for these samples, possibly because the mobility is already strongly lowered by ionized impurity scattering. The Yb containing sample has a much higher mobility, but it is still more reduced than expected and here alloy scattering may be a contributor. Sample with lowest values of mobility is Prcontaining, here most of the reduction originates in the presence of the second phase. It is reported in reference [53] that the mobility in Pb 1xEu xSe decreases dramatically with x, and the present result is consistent with that. The scattering coefficient of all samples remains between one and zero which is expected if scattering is dominated by acoustic phonons, as in most semiconductors. The exception is for the Prcontaining sample at lower temperatures, which may be related to nonuniformity. The fact that the dopants in the present samples are paramagnetic ions has no obvious effect on the scattering mechanisms.

The measured densityofstates hole effective mass at the band edge at 300 K reported here is 0.24 me for the pure PbSe sample, while a calculation using the hole

54 masses of Preier gives 0.23 me. The electron densityofstates mass calculated at 300 K from reference [54] is 0.30 me . Calculated masses are shown in fourth panel of Figure

22 and we can see that the electrons in the Ce and Yb alloys have similar masses. The

52 hole mass in Pb 1xEu xSe is similar to that in PbSe. Kanazawa et al. measure energy gap

of Pb 1xEu xSe using spectroscopic methods as Eg(x) =E g (PbSe) +2.16 x . For our sample at 300 K, Eg ≈ 0.50 eV. Expected proportional increase in mass is not observed. We observe different temperature dependence presumably due to the fact that the energy gap of Pb 1xEu xSe may have different temperature dependence than that of PbSe. However, we have no data to support this assumption. The electron densityofstates mass of the Gd containing sample is similar to that of PbSe at 300 K, but increases at low temperature.

The Ndcontaining sample has a much heavier electron mass than PbSe. The data on the

Pr sample are again affected by the inhomogeneities in the sample and are unreliable.

In conclusion; except perhaps for Pb 1xNd xSe, the Fermi level is not near the 4 f level of the rareearth donor, or the 4 f level of the rareearths does not hybridize with the bands of PbSe. This claim can be supported by observing the thermoelectric properties.

We plot the Seebeck coefficient as a function of carrier density for both n and ptype samples (a “Pisarenko plot” used in the thermoelectric literature 55 ) in Figure 24.

1000 , PbEuSe

" PbYbSe

) PbSe K / V 100 (

| 2 PbPrSe S | PbGdSe > % PbNdSe M PbCeSe

10 1x10 17 1x10 18 1x10 19 1x10 20 1x10 21 n,p (1/cm3)

The full line is calculated with the band parameters 54 for PbSe. It can be observed that reference sample falls directly on the line while others samples do not. The increase in

Seebeck coefficient follows observed increase in effective mass as it can be predicted by

* relation between S and m , Eq. 41. One alloy that shows significant increase is Pb 1 xNd xSe with an S more that twice of that predicted for similarly doped PbSe. However, the decrease in mobility outweighs the increase in Seebeck coefficient and zT remains small for all of these alloys.

Measured magnetic, galvanomagnetic and thermomagnetic properties of dilute

Pb 1xCe xSe, Pb 1xPr xSe, Pb 1xNd xSe, Pb 1xEu xSe, Pb 1xGd xSe, and Pb 1xYb xSe alloys, show that the trivalent rareearths act as donors in PbSe. This indicates that that the rareearth

atom substitutes for Pb 2+ and doping efficiency decreases as the atomic number increases.

Eu 2+ ion in PbEuSe alloys does not change the charge balance but increases the energy gap without increasing the effective mass. For the thermoelectric applications of these alloys the most promising rare earth element is Nd. For this reason we move to investigate effects of alloying PbSnTe with Nd.

4.2.2 Alloying Pb 1-xSn xTe with Nd

Since PbSe:Nd demonstrates the largest relative improvement of almost 100% in Seebeck coefficient we use galvanomagnetic and thermomagnetic measurements to explore the effects of doping Pb 1xSn xTe alloys with Nd. Both PbTe and NdTe form rock salt structures and at room temperature have comparable lattice constants of 6.443 and

6.249Å, respectively. 33 Alloying PbTe with Sn reduces the lattice constant, for example

33 Pb 0.7 Sn 0.3 Te has a= 6.408Å. Most interesting feature of Pb 1xSn xTe alloys is that with increase in Sn concentration ( x) band gap changes in such way that for x=0 direct gap at

51 L point is Eg≈190meV while with x= 34% alloy becomes zero gap conductor. At room temperatures this inversion occurs at higher tin concentrations. This dependence is shown here in Figure 25 Figure 38 and it is measured by laser emission 56 and luminescence. 57

Strictly speaking, electron and valence band minimum and maximum are both positioned at same Lpoint of the Brillouin zone. In PbTe wave functions of the electrons correspond to the L 6 and holes to L 6+ points. Isoenergetic surfaces are 4elipsoids extended in <111> directions. In SnTe L 6+ state is above L 6 which corresponds to inversion of electron and

hole bands. This extraordinary feature will be referred to several times in this text since it can be used to investigate behavior of donors or impurity bands as a function of band gap.

L E 80 C

) x (Sn%) V

e 0 m ( 10 20 30 40

E 40

80 EV Pb1xSnxTe band edge 120

Figure 25 Location of valence and conduction band edge in Pb 1xSn xTe alloys as a fuction of Sn concentration at 4K. Location of indirect Σ band is shown for ilustration and it is not to scale.

Experiment

Materials studied here are Pb 1xSn xTe:Nd alloys with x taking values between 0 and 100%. Samples were prepared by melting 6 purity elements in vacuum at temperatures 100 degrees above melting point. Prior to rapid cooling, the molten materials were rocked. This process was followed by annealing at 500 oC for 7 days.

In the preliminary study we determine solubility of Nd in these alloys. This study was performed on samples with x=1 and x=0, shown here in Table 2.

type ominal d concentration (PbTe):(NdTe) N 2% 4% 6% 20% (PbTe):(Nd 2Te 3) n 6% (SnTe):(NdTe) p 2% 4% 6% Table 2 Overview of alloys of PbTe:Nd and SnTe:Nd used in solubility study.

This study was based on analysis of xray diffraction data. It was observed that samples which are not rocked * will not have uniform composition through the ingot. As an illustration we show one of the samples xrays in Figure 26.

* Rocking is process of shaking molten material sealed68 in ampoule prior to cooling to improve mixing of materials.

450 2 400 10mm 3 (PbTe) 97 (NdTe) 4 350

300 4 500 250 cps XRD OLT066 .dat 1 200 XRD OLT066 .dat 2 400 150 XRD OLT066 .dat 100 3

50 XRD OLT066 .dat 5 300 27.2 27.3 27.4 27.5 27.6 27.7 2θ cps 200

100

0 20 25 30 35 40 45 50 55 60 65 70 2θ

Figure 26 Xray diffraction data for several horizontal cross sections in the ingot of (PbTe) 94 (NdTe) 4. Insert shows shift in the position of the peak indicating change in Nd concentration verticaly through ingot.

This sample was prepared with 4% Nd substituted for Pb. It is observed that Nd substitution is not uniform and that a larger concentration of Nd atoms is in the bottom of the sample. This is even more pronounced in the samples with higher concentrations of

Nd, for example 20% shown in the Figure 27. In this case top of the sample shows presence of Nd 2Te 3 rich phase. Attempt was made to alloy PbTe with Nd2Te 3, however this alloy has higher melting point and did not dissolve in PbTe after heating to 1050 oC for 3 hours.

300 XRD OLT068 .dat 1 250 XRD OLT068 .dat 3 XRD OLT068 .dat 5 200 XRD OLT068 .dat 7 1

150 3 cps 15mm 5 100

50 7

0 20 25 30 35 40 45 50 55 60 65 70 2θ

Figure 27 Xray diffraction data for (PbTe) 80 (NdTe) 20 indicates second phase separation on the top of the sample. Second phase is circled and it is mostly Nd 2Te 3.

To completely evaluate performance of these materials we measure four properties as shown in Figure 28. First we observe that SnTe based alloys are ptype and PbTe based alloys are ntype. Electrical resistivity of SnTe alloys is two orders of magnitude smaller than that of PbTe. Material with nominal 20 atomic percent of NdTe is highly non uniform and sample used in this study is that from the bottom. No attempts were made to exactly determine Nd concentration in this alloy. Samples with 6% NdTe and with 3%

Nd 2Te 3 show presence of a second phase. It can be observed that sample with 6% NdTe exhibits ambipolar conduction; Seebeck coefficient changes sign two times and it is positive between 120 and 320K. Nernst coefficient and Hall coefficient follow same behavior. For this reason (PbTe) 94 (NdTe) 6 has highest values of electrical resistivity.

Using simple parabolic model it is possible to estimate carrier density and electron mobility in these alloys, for example by using equations 18 and 19. These results are

1x10 -6

-4 1x10 0x10 0

-6

) -1x10 Legend: m m)

-6 (PbTe) (NdTe) -5 94 6

( -2x10 ( 1x10 H (PbTe) (NdTe)

ρ 96 4 R -3x10 -6 (PbTe)80(NdTe)20? (PbTe)97(Nd2Te3)3 -4x10 -6 (SnTe)94(NdTe)6 -6 1x10 (SnTe)96(NdTe)4 -5x10 -6 100 200 300 400 100 200 300 400 500 -4 T (K) -6 T (K) 1x10 4x10

2x10 -6 0x10 0 ) T) . K / 0

V 0x10 (

S -4

-1x10 N (V/K -2x10 -6

-2x10 -4 -4x10 -6 100 200 300 400 500 100 200 300 400 500 T (K) T (K)

Figure 28 Transport properties of PbTe and SnTe alloyed with 4, 6 and 20% of Nd.

21 1x10 Legend: 1x10 3 (PbTe)94(NdTe)6 (PbTe)96(NdTe)4

20 (PbTe)80(NdTe)20? s)

. 1x10 ) (PbTe) (Nd2Te3) 3 97 3

/V 2 2 1x10 m (SnTe) (NdTe) c 94 6 (

(cm p (SnTe) (NdTe)

96 4 , n 1x10 19 1x10 1

1x10 18 100 200 300 400 500 100 200 300 400 500 T (K) T (K)

Figure 29 Electron mobility and carrier density for PbTe and SnTe alloyed with 4, 6, and 20% of NdTe. SnTe based samples are ptype and PbTe ntype semiconductors.

shown in Figure 29 and indicate that SnTe although it has high concentration of Nd atoms remains heavily ptype while Nd dopes PbTe ntype. It is also observed that mobility of Nd 2Te 3 doped alloys are reduced as compared to those of NdTe doped alloys.

Conclusion of this introductory study is that the maximum concentration of Nd which results in uniform samples is that of 2% NdTe. Following these findings we prepare set of Pb 1xSn xTe:Nd which covers the entire range of Sn concentrations from 0 to 100%. The exact concentrations of neodymium in these alloys were determined by

SQUID magnetometery using Vibrating Sample Magnetometer (VSM) attachment to

Physical Properties Measurement System (PPMS) commercial instrument made by

Quantum Design. The temperature dependence of the magnetic susceptibility was measured in a DC magnetic field of 1kOe. After subtracting the constant diamagnetic

58 background of Pb 1xSn xTe, the inverse magnetic susceptibility shows linear dependence to 1/T obeying the CurrieWeiss law. From this temperature dependence, shown here in

Figure 30, we can deduce the concentration of paramagnetic Nd atoms in each of the alloys. Results are summarized in Figure 30, and indicate concentrations of Nd from 1.2 to 1.5%. A measurement of magnetic susceptibility as a function of magnetic field χ(H) from 0 to 50kOe confirms these are paramagnetic samples and that Nd atoms do not form clusters which would have ferromagnetic signature.

In the second experiment, galvanomagnetic (electrical resistivity and Hall voltage) and thermomagnetic properties ( S and Nernst voltage) are measured on small parallelepipeds (~1.5x1x7 mm 3) cut from bulk polycrystalline samples.

6 1.6x10 PbTe :Nd 1.3%

Pb0.9Sn0.1Te :Nd1.4% 6 Pb Sn Te :Nd1.2% 1.2x10 0.8 0.2 Pb0.7Sn0.3Te :Nd1.5%

8.0x10 5 [gr/emu]

χ 1/ 4.0x10 5

0.0x10 0 0 40 80 120 T [K]

Figure 30 Magnetic susceptibility vs. temperature. Data is used to determine exact concentrations of Nd ions in PbSnTe matrix.

All measurements are performed from 80 to 420K and in magnetic fields of 1.9 to +1.9T in a standard flow cryostat. Additional galvanomagnetic measurements were performed in the range from 2 to 80K and magnetic fields of 7 to +7T in AC transport configuration made for QDPPMS. In both cases electrical resistivity was measured using the standard

4wire AC method. Here we report only the zero field values of resistivity (Figure 31) although strong magnetoresistances were observed in all samples. Once again we report only zero field thermopower in Figure 31; the magnetoSeebeck was small and did not saturate in the fields available here. The Hall coefficient RH, shown in same figure is the zerofield slope of Hall resistance as a function of magnetic field. Measured adiabatic

Nernst voltages are used to calculate the isothermal NernstEttingshausen coefficient as a zerofield slope in magnetic field.

1x10 4 0 160

40 120 1x10 5 m] V/K] V/K]

80 80 [

[ [ ρ S S 1x10 6 120 40

1x10 7 160 0 1x10 5 12 PbTe :Nd

8 Pb0.9Sn0.1Te :Nd Pb Sn Te :Nd 1x10 6 0.8 0.2 Pb Sn Te :Nd 4 0.7 0.3 m/T] Pb0.6Sn0.4Te :Nd [ V / T .K] VT /

| | Pb Sn Te :Nd 0.4 0.6 H 0 [ 7 |R Pb0.3Sn0.7Te :Nd 1x10 Pb0.2Sn0.8Te :Nd 4 SnTe :Nd

1x10 8 8 0 100 200 300 400 0 100 200 300 400 T [K] T[K] Figure 31 Resistivity, Seebeck, Transverse Nernst and Hall Coefficients of Pb 1xSn xTe alloys with ~1.5% Nd.

Discussion All samples with less than 70% of Sn are ntype and up to 420K we do not observe any intrinsic behavior; this is indicated by negative Seebeck, Hall and Nernst coefficient. Sample Pb 30 Sn 70 Te:Nd 1.5% has positive Nernst coefficient at T>400K; Nernst coefficient is the property which first reflects onset of intrinsic conductivity. That is, if we were to continue measuring Seebeck at higher temperatures it would expect to observe saturating S due to presence of thermally activated holes. Samples with more than 80% of Sn are ntype. It can also be observed that samples with <70% Sn have

Seebeck coefficients that are very similar, the consequence of similar doping levels. As

shown in Figure 31, in the measured temperature range S shows a constant increase indicating an absence of thermally activated secondary carriers across the band gap.

Electrical resistivity rises with x, as alloy scattering becomes the dominant effect at x=70%Sn and then starts to drop again. Temperature dependences of the Hall coefficients with constant high temperature values shown are a sign of low temperature activated behavior with a saturation limit. The low values of the Nernst coefficient signify that there is no enhancement in scattering.39 Using the four measured transport properties ρ, S, R H and , we can calculate four independent parameters using the method of four coefficients. Following the procedure outlined in Chapter 3 with the assumption of

* degenerate nonparabolic band structure, we calculate n, E F, , λ and m .

Samples with x<0.6 show an increase in electron concentration with T but indicate saturation at high temperatures. With increasing x the saturation temperature decreases indicating that the Nd donor level approaches the bottom of the band. However, the saturation electron concentrations show no significant dependence on x. For x>0.6 the carriers are no longer just the electrons and Hall effects denotes ( pn) or ( np). Values of

RH, Figure 32, are much smaller and show different temperature dependence.

1x10 20 1x10 5 PbTe :Nd

Pb0.9Sn0.1Te :Nd

Pb0.8Sn0.2Te :Nd 4 1x10 Pb0.7Sn0.3Te :Nd Pb Sn Te :Nd

] 0.6 0.4 ] s . 3 Pb0.4Sn0.6Te :Nd V /

19 3 2

[cm Pb Sn Te :Nd 1x10 1x10 0.3 0.7 m c

[ Pb Sn Te :Nd

0.2 0.8 n, p n, SnTe :Nd 1x10 2

1x10 18 1x10 1 0 100 200 300 400 0 100 200 300 400 T [K] T [K]

Figure 32 Carrier density and mobility for electrons and holes in PbSnTe:Nd 1.5%

Doping efficiencies, defined as ratio of trivalent Nd atoms to carrier concentrations, can be determined by combining SQUID and Hall data, Doping efficiencies are 1020% for all samples with less than 40% Sn as shown in Figure 33 and increase with increase in temperature. Samples with higher Sn concentrations are not shown here and they have doping efficiencies of <5%.

0.3 PbTe :Nd

0.25 Pb0.9Sn0.1Te :Nd

Pb0.8Sn0.2Te :Nd Pb Sn Te :Nd 0.2 0.7 0.3

0.15

DoppingEff. 0.1

0.05

0 0 100 200 300 400 T [K]

Figure 33 Doping efficiencies for samples with <40% Sn

Electron mobilities are comparable to previously reported values for Pb 1xSn xTe alloys of similar doping level, 17 though only hole mobilities are known for x>30%. To our best knowledge ntype doping in materials with x=30% was achieved only up to 2x10 19 cm 3 by stabilizing the Fermi level in the forbidden band by alloying with 1.5% of Ga. 59 It is important to point that by alloying with Nd we increase the range in which PbSnTe remains ntype up to Pb 30 Sn 70 Te, a result never before reported. Additionally, in SnTe:Nd alloy we observe strong temperature dependence to carrier density which reaches

2x10 19 cm 3 at 420K. Tin telluride always has ptype conduction due to large number of

Sn vacancies, usually on the order of 10 20 cm 3.37

Calculated scattering exponents exhibit limited temperature variation in the range 80 to

420K, and decrease from λ= 0.9 (x=0) to λ=0.2 (x=0.6) and then it goes back to λ=1 indicating mostly acoustic phonon scattering as x reaches ~70%. This is consistent with observed maximum in electrical resistivity shown in Figure 31. Our calculations do not show any increase in effective mass m*. Figure 34 shows temperature dependence of effective mass and scattering coefficient for selected set of samples in which the condition that EF>3k BT was satisfied in the entire temperature region.

Figure 34 Effective mass and scattering coefficients for PbTe, PbSe, Pb 20 Sn 80 Te and Pb 30 Sn 70 Te all alloyed with ~1.5%Nd

With calculated effective mass and carrier concentrations it is possible to determine the position of Fermi level ( EF). Position of conduction band edge and EF are shown relative to the band edge and show as positive for ntype and negative for ptype samples in the

Figure 35 at T=80K . It can be observed that in all of these systems the Fermi energies are deep in the conduction band for samples with x<50% thus resulting in high electron concentrations, never achieved before for alloys with x>30% . For the samples with x=50 to 70% samples have Fermi energy at the bottom of the band or in the gap. Ptype

SnTe:Nd has E F deep in the valence band. Figure 35 shows temperature dependences of these energies indicating that only samples with <40% Sn have Fermi energies which remain the conduction band. The activation energies of electrons Ea from donor level calculated from temperature dependence of carrier concentration using n~exp(E a/k BT) are much smaller than the energy gap Eg in the samples with x=0, 10 and 20% and a

degenerate with the conduction band of the x= 30% to 70% samples. These energy levels are shown in Figure 35 (b) and we infer they correspond to the Nd 4f level acting as a donor from forbidden gap. When referred to the mid gap, this level can be considered Sn concentration independent.

a) E (80K) 200 F

EC ] V e

m 0 [ 0 20 40 60 80 100 E

EV xTin Concentration [%] 200

b) 200 EC 100 Ea(0K) ] V e

m 0 [ 0 20 40 60 80 100 E

100 xTin Concentration [%]

EV 200

Figure 35 (a) Fermi energy at 80K as measured from the band gap edge shown in blue for conduction and red for valence band; black dashed line is added to guide the eye. (b) Activation energy as extrapolated to 0K as a function of Sn concentration shown in reference to valence and conduction band edges. Dashed line is added to guide the eye..

300 PbTe :Nd

200 Pb0.9Sn0.1Te :Nd

Pb0.8Sn0.2Te :Nd ]

V 100 Pb0.7Sn0.3Te :Nd e

m Pb Sn Te :Nd [ 0.6 0.4

F 0 E Pb0.4Sn0.6Te :Nd

Pb0.3Sn0.7Te :Nd 100 Pb0.2Sn0.8Te :Nd SnTe :Nd 200 100 200 300 400 T [K]

Figure 36 Temperature dependent Fermi energy for PbSnTe:Nd samples.

In IVVI compounds, impurity levels created by Te vacancies or halogens are always completely ionized and merge with the bottom of the conduction band, not imposing any temperature dependence to the carrier concentration. 17 In this work we show that alloying for purpose of doping with Nd results in the formation of a distinctive tin concentration independent level anchored to the mid gap energy level of the Lpoint.

Calculated effective masses show no increase which would be an indication of hybridization of the sp band and RE 4f level. The measured thermopower remains at the levels previously measured in Pb 1xSn xTe of appropriate carrier concentration, as shown in Pisarenko plot in Figure 37. 17

300 PbTe :Nd Pb Sn Te :Nd 250 0.9 0.1 Pb0.8Sn0.2Te :Nd

200 Pb0.7Sn0.3Te :Nd

Pb0.6Sn0.4Te :Nd V/K)

150 Pb0.4Sn0.6Te :Nd

Pb0.3Sn0.7Te :Nd |S| ( |S| 100 Pb0.2Sn0.8Te :Nd 50 SnTe :Nd

0 1x10 18 1x10 20 n (1/cm3)

Figure 37 Pisarenko plot at 300K showing S(n) for PbTe plotted in solid line against S(300K,n) for PbSnTe:Nd alloys in this study.

4.3 Dilute alloys of IV-VI compounds with group III elements

Doping of IVVI compound semiconductors is simply achieved by replacing cations which are group IV elements with 6 electrons in outer shell. By substituting metal

(Pb,Sn) for halogens (I, Cl) which have 7 electrons we get donor action. Alkali metal

(Na,Li) can also substitute for group IV element and act as acceptors making materials p type. Doping with group V elements is very interesting since Bi and Sb have very different behavior. Bismuth substitutes for Pb and it is used as donor. Sb can be used to substitute either for cation or anion and it has either donor or acceptor action. 60 The introduction of group III elements such as In, Ga and Tl in IVVI alloys results in the formation of impurity levels, Cd has similar behavior. 37,61 Unlike the halogen or alkali

elements which have clearly defined donor or acceptor action, group III elements form impurity levels that can have different location in the band and depending on the position of these levels they act either as donors or acceptors. Thallium makes PbTe ptype which corresponds to behaving as the acceptor. At the low temperatures In is an donor in PbTe and an acceptor in SnTe. Gallium however, has a tendency to form two states depending on concentration, 62 and it can be either in conduction band or in the forbidden gap. These impurity levels have been observed mostly by optical measurements and are carefully reviewed for example by Khokhlov et. al. in Ref. [61] and Ravich and Kaidanov in [37].

Although they all share common characteristics such as pinning of Fermi energy level, each of these impurities related levels have distinguishable properties. These will be briefly addressed here.

Energy levels associated with In in PbTe are in at ~70meV 37,63 as measured from the bottom of the conduction band in PbTe. However if PbTe is alloyed with SnTe this energy level shows strong dependence on Sn concentration, x, in such way that it resides in a forbidden band when x reaches ~22%.64 The In impurity energy level is in valence band for all alloys with x> 27%. These results are based on measuring optical properties such as absorption and emission spectra 37 and are summarized in Figure 38, where we show the position of impurity band relative to the valence and the conduction band edges.

200

Indium impurity level at 4K

) 100 V

e EC m (

x (Sn%)

E 0 10 20 30 40 E V Pb Sn Te band edge L 1x x 100

We mention that impurities pin Fermi energy level, this results in a fixed value of carrier concentration for given x regardless of the In concentration.37 In the case of In, pinning was further confirmed by doping of PbTe with additional iodine; the carrier concentration will stay independent of iodine concentration as long as In concentration is larger than that of iodine. In PbTe it will remain at a constant level of ~7x10 19 cm 3. With iodine concentration exceeding In concentration, carrier density rapidly grows. 65 Therefore, In can not be used to dope PbTe since any significant concentration of In will result in only one carrier density. At 22% of tin, the Inlevel is in the midgap showing properties

66 interesting for infrared sensors. Study of Nemov et al. on Pb 0.78 Sn 0.22 Te with less than

3% In shows halffilled InTe band. The Fermi level is then stabilized at the impurity level

positioned within kBT under the bottom of the conduction edge. If concentration of In is increased to reach as high as 5%, the samples show overlap of impurity level and conduction band, indicating broadening of this level. From this data authors conclude that the energy derivative of density of states, dg(E)/dE becomes negative as the gap between impurity states and conduction band disappears. This type of interaction indicates that impurity level is actually forming a resonant state, i.e. that it’s electronic levels couple to the band structure of the host solid Pb 0.78 Sn 0.22 Te.

Thallium in lead chalcogenides has similar effects but it’s energy level is positioned in the valence band. Measurement of low temperature Hall effects on samples simultaneously doped with Tl and Na show that 0.5% of Tl stabilizes carrier density at

~5x10 19 cm 3 and that this concentration can not be exceeded until Na concentration reaches ~2 atomic percents.36,67 Carrier density and therefore chemical potential remain constant in this region but density of states changes with double doping with Na and Tl, showing local maximum when Na concentration reaches 0.6%. 68 These estimates were made by measuring electronic component of specific heat which is proportional to

36, 37 temperature and density of states γ, C e= γ T . Position of thallium level in various IV

VI compounds was determined by measuring absorption spectra. In PbSe 69 Tllevel is at

20meV below edge of valence band and in PbSe 70 at 260meV. Both studies were performed at room temperatures.

4.3.1 Alloying Pb 1-xSn xTe with In

Experimental results of Nemov 66 and first principle studies of Hoang and Mahanti 71 indicate that In impurity levels will interact with the conductivity band and form resonant states. If such state exists in the vicinity of the Fermi level, it will result in a sharp, delta shaped feature in the density of states. Following the theory of Mahan and Sofo 34 this will then improve thermoelectric figure of merit, zT . To investigate the possibility that this can be achieved we study set of alloys with composition Pb 1xySn xIn yTe where x=0 to 0.3 and In concentrations y do not exceed 6 atomic %. For shorter nomenclature we will denote these samples as Pb 1xSn xTe:In y%. Table 3 lists 18 chemical compositions used in this study. In all of these samples indium is substituting for Pb. By measuring transport properties we quantitatively determine the interaction between indium impurity state as indicated by effective mass and the position of Fermi level.

Concentration of Sn in PbSnTe:In alloys 0% 5% 10% 15% 18 % 22 % 30 % 38 % 0.5 1 1.5 1 2 0.01 0.5 1 1 3 1.5 0.1

In In 0.75 6 1 0.4 [%] [%] 0.75

concentration 0.3 Table 3 Matrix of PbSnTe:In samples analyzed in this study.

Experimental procedure

All samples listed in Table 3 were prepared using the Bridgeman method and provided by D. Khokhlov and A. Nikorici of Moscow State University and Moldova

Academy of Sciences, respectively. Small parallelepipeds were cut from samples and

etched in hydrobromic acid to remove surface impurities. The transport properties were measured in a conventional flow cryostat from 80K to 420K at discrete values of magnetic field ranging from 2T to 2T.

200 1x10 5 ) m )

K 0 W / V / 1 m m) ( 4 ( -200 V/K)

1x10

/ S s ( S

σ (1 σ -400

1x10 3 -600 100 200 300 400 100 200 300 400 T (K) T (K) -3 1x10 2x10 -5

0x10 0

) -4 T -5 Sample Structure: 1x10 ) / -2x10 K . m

T PbTe:In 0.5% W m/T)

/ -5 ( V/K) -4x10 V Pb90Sn10Te:In 1%

( | ( H -5 -5 Pb85Sn15Te:In 1% H N

R 1x10

N( -6x10 | R Pb82Sn18Te:In 1% -8x10 -5 Pb78Sn22Te:In 0.4% Pb70Sn30Te:In 0.5% 1x10 -6 -1x10 -4 100 200 300 400 100 200 300 400 T (K) T (K)

Figure 39 Measured thermomagnetic and galvanomagnetic properties as function of temperature for set of samples with different Sn concentrations and 0.4 to 1%In.

-100 1x105

-200 ) ) K m

-300 / W 1x104 V m) m / ( 1 V/K) ( / -400

S ( s S σ (1 σ -500

1x103 -600 -4 100 200 300 400 100 200 300 400 1x10 T (K) 2x10-5 T (K)

0x100 )

T Sample Structure: -5 / ) -2x10 K m Pb Sn Te:In 1% . 85 15

W Pb85Sn15Te:In 3%

/ -5 ( m/T) -4x10 V Pb Sn Te:In 6% V/K) 85 15

( Pb Sn Te:In 0.3% H

( 82 18 -5 R N

H -5 | -6x10 Pb82Sn18Te:In 0.75%

1x10 N ( R Pb82Sn18Te:In 1% -8x10-5 Pb82Sn18Te:In 1.5% Pb82Sn18Te:In 2% -1x10-4 100 200 300 400 100 200 300 400 T (K) T (K)

Figure 40 Measured electrical conductivity, Seebeck, Hall and transverse Nernst coefficients for samples with 15 and 18% Sn and indium concentrations ranging from 0.3 to 6%.

Figure 39 and Figure 40 show the temperature dependences of zerofield electrical conductivity, Seebeck, Hall and transverse Nernst coefficients. For clarity, data is presented as a function of Sn concentration in Figure 39 and function of In concentration in Figure 40. In general, we can observe that with increase in Sn concentration Seebeck coefficient and electrical conductivity increase, indicating a decrease in the number of carriers. This is compatible with reported indium energy dependence as a function of x.

By observing In dependence, (Figure 40) we can see that with increase in In concentrations electrical conductivity drops, which is indication of decrease in electron mobility. We also observe an increase in Seebeck coefficient for samples with large indium concentrations (3 and 6%) which is consistent with Nemov’s observations. 66 At this point, we also note that Seebeck coefficients of all samples show two carrier conduction behavior, as seen from the temperature dependences. The Seebeck coefficient reaches maximum as number of thermally excited secondary carriers increases. This claim is supported by fact that Nernst coefficient changes it sign at same temperatures where S(T) reaches turning point. To perform detailed analysis we can not directly apply method of four coefficients described in Chapter 3 since this tool is specifically developed for single carrier systems. For this reason we start our analysis by observing only data at 80K as discussed by Jovović et al. in reference [31].

Discussion

We first observe magnetic field dependence of galvanomagnetic and thermomagnetic properties at 80K. Electrical resistivity and Seebeck coefficients at zero fields are listed in Table 4.

In Symbol Composition conc. ρ(B=0) ( m) S(B=0) ( V/K) XXX PbTe:In 0.5% 3.41E06 66.21 ‚‚‚ Pb 90 Sn 10 Te:In 0.75% 2.79E06 74.56 lll Pb 90 Sn 10 Te:In 1.5% 3.95E06 93.18 OOO Pb 85 Sn 15 Te:In 1% 1.08E05 177.6 """ Pb 85 Sn 15 Te:In 3% 2.72E05 272.82 >>> Pb 85 Sn 15 Te:In 6% 4.48E05 239.35 +++ Pb 82 Sn 18 Te:In 0.3% 3.09E05 147.65 ,,, Pb 82 Sn 18 Te:In 0.75% 4.14E05 285.78 ... Pb 82 Sn 18 Te:In 1% 1.71E04 264.29 /// Pb 82 Sn 18 Te:In 1.5% 2.04E04 313.45 444 Pb 82 Sn 18 Te: In 2% 3.51E04 300.02 $$$ Pb 78 Sn 22 Te: In 0.01% 3.16E04 518.20 &&& Pb 78 Sn 22 Te:In 0.1% 5.86E04 339.28 ZZZ Pb 78 Sn 22 Te: In 0.4% 7.86E04 529.67 ??? Pb 70 Sn 30 Te: In 0.5% 4.44E04 290.69 Table 4 List of symbols used to denote Pb1xSnxTe:In samples and measured zerofiled Seebeck and electrical resistivity all at 80K.

a) b)

2 4

1.6 3 ) ) 0 0 ( 1.2 ( ρ ρ ) ) 2 B B ( 0.8 ( ρ ρ(Β) ρ ρ(Β)

1 0.4

0 0 0 1 2 0 1 2 B (T) B (T)

Figure 41 Relative magnetoresistivity as a function of magnetic field at temperature of 80K. Samples can be identified using Table 4.

a) b) 4x10 4 1x10 4

2x10 4 2x10 4

0x10 0 K) K) / / 4 V V 3x10 ( 4 ( S 2x10 S

4x10 4 4x10 4

6x10 4 5x10 4 0 1 2 0 1 2 B (T) B (T)

Figure 42 Magnetic field dependence of Seebeck coefficients measured at temperature of 80K. Symbols and alloys correspond to those listed in Table 4. Solid lines are plotted to guide the eye.

a) b)

1x10 2

1x10 3 1x10 4

1x10 4 m) m)

( 5

1x10 ( | | 1x10 H H |R 1x10 6 |R

7 1x10 1x10 6

1x10 8 0 1 2 0 1 2 B (T) B (T)

Figure 43 Hall voltage as a function of magnetic field at temperature 80K plotted for alloys listed in Table 4 with corresponding symbols.

a) b) 1x10 5 0x10 0

5x10 6

0x10 0 1x10 5

V/K) 5x10 V/K) ( ( N N 1x10 5 2x10 5

1x10 5

2x10 5 3x10 5 0 1 2 0 1 2 B (T) B (T)

Figure 44 Transverse Nernst voltage as a function of magnetic field at temperature 80K plotted for alloys listed in Table 4 with corresponding symbols. Solid lines are added to guide the eye.

Magnetoresistivity is defined as Mρ≡(ρ(B)ρ(0))/ρ(0) and shown in Figure 41 and (a) data is divided in such way that panel (a) represents different Sn concentrations, namely

0, 10, 15, 18, 22 and 30% and panel (b) different In concentrations in two Pb 1xSn xTe alloys, with x = 0.15 and 0.22. Magnetic field dependence of resistivity can be reconstructed using the zero magnetic field values of resistivity ρ(B=0) listed in Table 4.

Presented data show that increasing Sn concentration will lead to increase in Mρ. The same trend can be observed when we add more In with the exception of the alloys with x

= 0.15 and 1% In. Data in Figure 41 show that at the maximum field used in this test (2T)

Mρ reaches values larger than 2. Even reaching Mρ=4 it still does not saturate. The largest values of magnetoresistance are observed in samples with In concentrations of 1 and 3%. MagnetoSeebeck is defined as MS=|[S(B)S(B=0)]/S(B=0)|. Data are presented in Figure 42, (a) as a function of Sn concentration and (b) as a function of In concentration. Data indicate similar behavior as Mρ. We observe an increase in MS as the

Sn concentration increases. Maximum MS is observed in samples with In level in energy gap (Pb 78 Sn 22 Te:In, sample: ZZZ). However, unlike the resistivity, the Seebeck saturates at magnetic fields above 1.5T at values MS < 1.4 . The sample with largest MS has its In level in the forbidden band, and also shows the largest value of RH (Figure 43) and

(Figure 44). Nernst coefficient crosses from positive to negative value (Figure 44), behavior typical of materials with large number of minority carriers.

We can also observe that the Hall coefficient, indicated by the slope of the curves in

Figure 43, does not significantly change with increase in indium concentration. This

result indicates that in increase in indium concentration does not change the position of

Fermi energy.

Using the method of four coefficients and procedure outline in the Chapter 3 for degenerate, nonparabolic semiconductors (equations 5861) we can use the experimental

Seebeck coefficient, electrical conductivity σ = ρ 1, NernstEttingshausen, and Hall coefficients to calculate important fundamental properties: Fermi energy EF, carrier mobility , effective mass m*, and energy scattering exponent λ. We can use this procedure only at 80K since, as we have described earlier and as we will show in following paragraphs, it is only at this temperature that we can neglect contributions of minority carriers. To verify these results we use full set of equations which use Fermi integrals and calculated band parameters and make no assumptions regarding band degeneracy.29,30 The equations are shown in Chapter 3, Eq. 46 through 49. Good agreement between the recalculated and measured values of S, σ, R H and is achieved and validates the quality of our model.

As previously discussed by Jovovic et al. [31] the Fermi energy (Figure 45) closely follows the position of the In level observed using optical methods. This has a linear dependence on Sn concentration, shown in Figure 38. For example, in the case of

PbTe:In at 4K, the In level is at 70meV in the conduction band and it will move with temperature at the rate of 0.3meV/K. 37 This would bring it to 47meV at 80K versus

EF=52meV as calculated by our model. As previously indicated, position of Fermi energy does not depend on In concentration. For example samples with 18% of Sn show

very small differences in calculated EF’s when the In concentrations is changed from 0.3 to 2%. The same trend is observed in calculated carrier density, shown in the same figure.

This indicates that indium impurity pins Fermi level at the location which only depends on Sn concentration, at least as long as there is less than 6% In. It is also important to notice that if each In 2+ was to donate at least one electron the carrier densities would far exceed those measured here.

a) 60 In 0.5% In 1% 40 ) In 1% V

e In 3% m 20 ( In 6% (meV) xx (Sn%)

F F In 0.3% E E 0 In 0.75% 10 20 30 In 1% -20 In 1.5% In 2% 1x10 19 b) In 0.01% In 0.1% In 0.4% 1x10 18 In 0.5% ) 3 ) m 3 c 17

/ 1x10 1 (

n n,p (cm 1x10 16

1x10 15 0 10 20 30 xx (Sn%)(Sn% )

Figure 45 Fermi energy (a) and carrier density (b) of Pb 1xSn xTe:In samples at 80K. Dashed lines are inserted to guide the eye.

The calculated values of mobility of these carriers are shown in the Figure 46 (a). If we focus on samples with low In concentrations we can observe a local maximum in

mobility for samples with 18% of Sn. At this point In level and Fermi energy cross into the forbidden band resulting in the freezing of carriers. The effective mass calculated using four coefficients method provides information about the energy dependence on momentum at the Fermi energy. We can see that effective mass, Figure 46 (b), linearly decreases as x reaches 22%. This is consistent with the measured mobility since mass is inversely proportional to mobility. However, we must point out that in the four parameter method we determine the bottom of the band mass m0 from nonparabolic model

 E  m(E) = m ⋅γ (' E) = m ⋅1+ 2  . Effective mass follows ratio E /E which reaches 0 0   F g  Eg  zero at x=22% ( EF=0 at this concentration Figure 45).

0.3 a) In 0.5% b) In 1% 0.25 In 1% 100000 In 3% 0.2 In 6% e m

) In 0.3% e / V 0.15 *

/ In 0.75% 2 m

m In 1% c 10000 0.1 ( m*/m (1/T) In 1.5%

In 2% m 0.05 In 0.01% In 0.1% 1000 0 In 0.4% 0 10 20 30 0 10 20 30 In 0.5% x (Sn%) x (Sn%)

Figure 46 Mobility of majority carriers and effective mass of measured Pb 1xSn xTe:In samples. Dashed lines are added to guide the eye. In measured mobility of samples with <1% In we can notice trend in which mobility reaches maximum at x=18%.

The observed variations in effective mass, except in sample with 18% Sn, are very small and cannot be considered definite proof of hybridization between In level and

band. They are more consistent with the nonparabolic nature of the bands. Also, the large effective mass of sample with 1% In and 15% Sn, shown in the Figure 46 (b), could be a consequence of larger noise in the data for this sample and can not be considered significant. The calculated scattering exponents λ are shown in Figure 47 (a) and clearly indicate that the scattering is primarily due to neutral impurities and optical phonons. At temperature of 80K, we do not observe any resonant scattering as reported at temperatures below 60K in Ref. [72]. Having no increase in effective mass or scattering coefficient indicates that In level does not effect the dispersion relation. Following Mott’s relation we do not expect increase in Seebeck coefficient. Experimental data summarized in Figure 47 (b) support this prediction; the Pisarenko plot indicates that at 80K there is no significant deviation from the expected values of S(n) of bulk Pb 1xSn xTe. Seebeck coefficients as a function of carrier density follow the theoretical (Pisarenko) curve; here we use curve for PbTe 55 since there is no significant difference between this and curves for Pn 1xSn xTe when x<20%

In 0.5% 1.6 a) b) 1000 In 1% In 1% 1.2 In 3% In 6% )

0.8 K In 0.3% / In 0.75% V/K) l V 100 λ

m In 1%

0.4 (

S ( S In 1.5% S In 2% 0 In 0.01% In 0.1% -0.4 10 In 0.4% 15 17 19 0 10 20 30 1x10 1x10 1x10 In 0.5% x (Sn%) n (1/cm3)

Figure 47 a) Scattering exponent and b) Pisarenko plot showing dependence of Seebeck coefficient and carrier density for measured Pb 1xSn xTe:In samples at 80K 96

Analysis of data presented in Figure 39 and Figure 40 can be extended beyond

80K but by taking in consideration effects of minority carriers. In these twocarrier systems, 25 four measured properties are sum of partial properties of holes and electrons given by equations 71 to 74.

In general we talk about this in Chapter 3 as a problem with eight unknown parameters. In this case we can use only four available measured properties to determine these eight unknowns at the each temperature: the electron and hole Fermi energies, their mobilities, and their scattering exponents. This system is underdetermined and it must be simplified. Based on calculated values at 80K we can conclude that density of states effective mass follows bottom of the band mass of Pb 1xSn xTe. By selecting to use the

51 * * literature values for the electron and hole densityofstates effective masses m e and m h, which are given as a function of Sn concentration and the temperature we can reduce number of unknown parameters. Also, we can assume that independently of temperature both electrons and holes are scattered using the same mechanism. To further reduce the number of free parameters, we will allow only a very small deviation of partial mobilities

17 from e =2 h. With these constrains, the system of equations 46 to 49 and 71 to 74 becomes determined and the partial electron and hole properties can now be calculated using the transport integrals at each temperature. Free parameters are the Fermi energy and the electron and hole mobility while all other are constrained. Computations are performed only on selected set of samples with ~1% In and results are summarized in

Figure 48. Again to verify our model data points are plotted over calculated values shown as full lines.

6 a) 1x10 b) 400 0% Sn 0% Sn 15% Sn 15% Sn 18% Sn 18% Sn 200 22% Sn 1x10 5 22% Sn 30% Sn 30% Sn

0 m] .

4 1x10 V/K] [1/ S S [ σ 200

1x10 3 400

1x10 2 600 100 200 300 400 100 200 300 400 T [K] T [K] 5 4 1 4 3x10 1x10 1x10 0% Sn 2.0x10 d) 0% Sn c) 15% Sn 15% Sn 18% Sn 18% Sn 4 22% Sn 2 1.2x10 30% Sn 1x10 22% Sn 1x10 5 30% Sn 2x10 5 4.0x10 5 3 1x10 ] 1 6 K m/T]

1 1x10 5 T 100 200 300 . 5 [ 4.0x10 1x10 m H

4 [

R 1x10 a N 100 200 300 400

5 1x10 0x10 0 100 200 300 400 T [K] 1x10 6 100 200 300 400 1x10 5 T [K] Figure 48 Temperature dependence of a) electrical conductivity and b) the Seebeck coefficient, c) low field Hall coefficient and Nernst coefficients of Pb1xSnxTe:In samples, with x=0, 15, 18, 22 and 30%. The Hall coefficient of the x=30% sample changes sign, and is shown as an inset on a linear scale. Samples for which no lines are drawn were those for which no single, temperatureindependent scattering exponent λ could fit through all data points, presumably because λ is temperaturedependent, which as not accounted for in the model. The Nernst coefficient of the x=22 and 30% samples changes sign and is shown on a linear scale.

200 120

80 160

40 120 ] ] T [K]

V 100 200 300 V e e m

m 0 [ [

E E 80

-40

0% Sn 40 15% Sn 30% Sn 18% Sn -80 22% Sn

100 200 300 400 T [K] 0 -120 Figure 49 Fermi energy level as a function of temperature plotted relative to the conduction and valence band of the x=0, 15, 18 and 22% sample, and on the right panel for the x=30% sample. The zero point for the energy scale is defined at midgap. Solid lines show temperature dependent position of the valence and conduction band edge. Colors correspond to different Sn concentrations.

Calculated positions of the Fermi energy levels with respect to the band edges are shown in Figure 49. Here we see that Fermi level crosses into the gap near 350K for

PbTe:In. As we increase the Sn concentration, the temperature at which this crossover occurs decreases: for x=15 and 18% the transition is at approximately 100K. Samples with 22 and 30%Sn have Fermi energies which remain in the forbidden gap for the entire temperature range. The results for x=0% as well as for x=30% suggest that the indium level is temperature independent and does not follow band edge but rather stays anchored to some temperature independent level.

Takaoka and Murase in their work 73 also analyze similar samples with

21%< x<35%, and observed that the Fermi energy can cross into the gap at a temperature that depends on x. Unfortunately, Takaoka and Murase use only the Hall coefficient and the resistivity in their study. They obtained very large values for the ratio of electron to hole mobility, which are in contradiction with the previously observed properties of

PbSnTe 33 and with the ratio of effective masses of holes and electrons. Also, they offer to reader to select one of two solutions: one with the Fermi energy in the gap and another with the Fermi energy in the valence band. We confirm that the former is the correct solution.

Two trends are apparent in thermal conductivity: alloying with Sn results in

17 decrease in thermal conductivity as expected in Pb 1xSn xTe; and, for all Sn concentrations, increase in In concentration results in decrease in thermal conductivity, probably due to the increase in the number of scattering centers. Effects of alloy scattering do not result in significant decrease in the thermal conductivity and these remain at the levels commonly observed in PbSnTe. 3 The very strong temperature dependence of the Fermi energy renders these materials unsuitable for thermoelectric applications. Presented results identify a band structure picture that is different at practical operating temperatures from what it is at the cryogenic temperatures where most experiments are carried out. In these samples calculated zT does not exceed 0.2 at all measured temperatures.

100

4.3.2 Effects of doping PbTe, PbSe xTe 1-x and Pb 1-xSn xTe with Tl

Alloying PbTe and PbSnTe with Tl.

Thallium has the same pinning effect on the Fermi energy as Indium. 61 We will show that thallium in PbTe has ability not only to pin Fermi energy but to, modify the electronic density of states in such way that overall thermoelectric efficiency increases. When Tl is

74 used as a dopant EF is pinned at a level 100meV below the top of the valence band. We explore the possibilities that this level: (1) hybridizes with the valence band and deforms the DOS, (2) interacts with heavy holes band or (3) that it perhaps forms new conducting band. For any of these effects to be of use in designing thermoelectric materials position of the Tlenergy level must have its temperature coefficient either of the opposite sign as for In or at least have the higher temperature at which it crosses into the gap. Here we show that in the case of PbTe:Tl, the Tl level remains in the valence band at least up to

600K. The literature on this topic is not without controversy when it comes to doping with Tl: a report by Kaidanov et al. 75 showed a room temperature S=120 V/K at a carrier concentration of p=1.16x10 19 cm 3, practically on the known Pisarenko curve for conventional PbTe for which we calculate S=125 V/K. This data point would indicate that there is no change in normally observed band picture of PbTe. However; Heremans et al 19 in 2005 publish same value S=120 V/K measured on the sample doped with same amount of thallium (10 19 cm 3) but this time carrier density in the sample is 1x10 20 . These controversial and promising results and experience with PbSnTe:In were certainly the reasons to prepare and measure PbTe:Tl.

101

A first set of samples was prepared in Japan at Professor Ken Kurosaki’s laboratories by using powder metallurgy techniques. Compositions of samples are selected in such way that we try to reach maximum solubility of Tl in PbTe. According to Rustamov 76 that would correspond to 2% at room temperature or approximately 5% at 200 oC, unfortunately Chami 77 shows that at room temperatures there is practically no solubility of Tl in PbTe matrix and at 500 oC there is only about 2% solubility. From the indium work 20 described in previous paragraphs and a literature review on Tl 37 we knew that as little as 0.001% of group III element is enough to pin Fermi level. However, for example in the case of Pb 78 Sn 22 Te:In more than 5% of In is needed to observe hybridization. For this reason we try to reach the solubility limit and prepare PbTe with 1% and 2% of Tl.

Samples were prepared by melting stoichiometric amounts of Pb, Te and Tl 2Te under high vacuum. Melt was rocked in furnace at 1000 oC in order to ensure homogeneity.

Samples were furnace cooled to 530 oC and annealed for one week; estimates of cooling rates were not made. After annealing, material was ground into powder and hot pressed at

o 530 C under ArH2 gas mixture for two hours into 3mm thick disks with radius of 12mm.

Material properties were then evaluated at two separate sites. Tests in the temperature range 2K to 420K were performed at the Ohio State University: these include measuring

Seebeck, Hall and Nernst coefficients, resistivity and specific heat. Thermal conductivity, resistivity, Seebeck and Hall coefficients in the temperature range 300K to 670K were measured at California Institute of Technology in Dr. G. Jeffery Snyder’s laboratory. In order to confirm our findings we prepared a second set of samples at OSU by omitting

102

the grinding and pressing step. In this case samples were melted and rocked at 1050 oC in vacuum sealed quartz ampoules. Molten materials were cooled to 700 oC and annealed for seven days. These samples were only tested in 80 to 420K temperature range at The Ohio

State laboratories and they confirm results observed on materials prepared using powder metallurgy.

Figure 50 shows summary of measured properties for representative samples prepared at Osaka University and measured at OSU and Caltech, part of these results were discussed by Heremans et. al. in reference [78]. Two samples of each of the compositions Pb 0.99 Tl 0.1 Te, Pb 0.98 Tl 0.2Te and Pb 0.58 Sn 0.40 Tl 0.2 Te were first measured in temperature range 300 to 670K at Caltech. These measurements were performed on

12mm disks with thicknesses 1.7 to 2mm. Resistivity and Hall coefficient (Figure 51) were measured using Vander Pauw geometry in the plane of the disk. Thermopower was measured in the cross plane direction using method described in Chapter 2 by sweeping temperature with rate 0.1deg/min from 300K to 670 and then back to 300K. These three measurements were followed by measurement of thermal diffusivity using Netzch laser flash system. By using literature values of specific heat 51 and the density of PbTe we calculate thermal conductivity, Eq. 32.

Following the thermal conductivity tests the samples were cut in parallelepipeds and resistivity, Seebeck, Hall and transverse Nernst coefficients were measured in temperature range 80 to 420K. Figure 51 shows summary of measured Hall and transverse Nernst coefficients. Thermal conductivity was also measured in this

103

temperature range using static heater and sink method described in Chapter 2. In order to check the isotropy of the samples prepared by using hotpress and anneal method we measure the thermal conductivity and Seebeck coefficient in two directions: inplane and crossplane. Measured properties are within 3% of experimental error and are used to verify uniformity of these samples.

a) b) 1.00x10 4

300 ) ) 200 m V/K (

( ρ S

Pb99Tl1Te100 5 100 1.00x10 Pb98Tl2Te100

Pb59Tl1Sn40Te100

Pb58Tl2Sn40Te100 0 0 200 400 600 800 0 200 400 600 800 T [K] T [K]

6 2 c) d) 5

4 1.6

) 3 1.2 K . zT W/m 2 κ ( 0.8

0.4

1 0.9 0 0 200 400 600 800 0 200 400 600 800 T [K] T [K]

Figure 50 Temperature dependence of resistivity (a), thermopower (b), thermal conductivity (c) and figure of merit zT (d) for: Pb 0.99 Tl 0.01Te (open and closed symbols), Pb 0.98 Tl 0.02Te (open and closed symbols), Pb 0.59 Sn 0.40 Tl 0.01Te (open and closed symbols) and Pb 0.58 Sn 0.40 Tl 0.02Te (open and closed symbols).

104

a) 1.00x10 6 b) 3

2.5

2 Τ) . /Κ m/T) 7 V

1.00x10 1.5 ( ( H a a R Ν 1 Pb99Tl1Te100 Pb98Tl2Te100 0.5 Pb58Sn40Tl2Te100 1.00x10 8 0 0 200 400 600 800 0 100 200 300 400 500 T [K] T [K]

Figure 51 Hall coefficients (a) and transverse Nernst coefficients (b) of Pb 0.99 Tl 0.01 Te, Pb 0.98 Tl 0.02 Te, Pb 0.58 Sn 0.40 Tl 0.02 Te

Data shown in Figure 50 and Figure 51 is compilation of OSU and Caltech measurements. Data in Figure 50 include several additional samples measured only at

Caltech. We can observe small discrepancies in measured Hall coefficient in all samples between two laboratories; and a discrepancy is also observed in the measured resistivity, thermopower and Hall coefficient in samples with 40% tin. We also notice that, on the

1% Tl sample, the lowtemperature resistivity lies above that of the hightemperature measurements in the region of overlap and that the reverse is true for the Seebeck coefficient. The main sources of discrepancies are small experimental errors which are mainly related to sample’s geometry. Note that there are significant differences in geometries used at Caltech (disks) and at the Ohio State University laboratories (small crosssection parallelepipeds respectively). In addition to the discrepancy due to the geometrical errors it is possible that small differences originate in sample’s

105

inhomogeneity. This has little influence on zT since the power factor S2σ is the same for both samples in the range of temperature overlap. By comparing a number of samples prepared with PbTe and 1 or 2% of Tl we can see that Seebeck coefficients at all temperatures have same results, deviations around median values are within declared measurement error. However, much larger deviations are observed in a resistivity. In general, we cannot conclude that the resistivity depends on Tl concentration but rather that it is only dependent on the observed number of free carriers. From the measured Hall coefficient and the resistivity we can calculate mobility and carrier densities (Eq.’s 58 and 59) for larger number of samples prepared using powder metallurgy approach and solid state reactions, Figure 52. We see that mobilities of samples with 1% Tl are 1020% larger than those of samples with 2%Tl and that they follow similar dependence on carrier density. Carrier density varies in the range 5x10 19 to 1.2x10 20 independent from Tl concentration. We conclude that carrier density is very sensitive to sample preparation, probably due to sample being nonstoichiometric. The mobility decreases with an increase in carrier density and Tl concentration has little effect on mobility. Resistivity

19 shows clear minimum at 6x10 carriers for Pb 0.98 Tl 0.02 Te, which constitutes the optimum doping level.

The thermal conductivity of measured PbTe samples is ~2.4W/mK at 300K, which is the

79 result expected of bulk PbTe, and 1.4W/mK in Pb 60 Sn 40 Te. Here we can conclude that alloying with Tl has no effect on thermal conductivity. The figure of merit is calculated from available data and shown in Figure 50. Maximum figure of merit on these samples

106

reaches 1.6 at 670K. When compared to conventional ptype PbTe doped with Na (Figure

5) we see that zT is doubled.

80 12

10 Pb Tl Te 60 99 1 100 Pb Tl Te 8 98 2 100 ) ) s . cm .

40 6 (m (cm2/V ρ Pb98Tl2Te100 4 20 Pb Tl Te 2 99 1 100

0 0 4.0x10 19 8.0x10 19 1.2x10 20 4.0x10 19 8.0x10 19 1.2x10 20 p (cm3) p (cm3)

Figure 52 Mobility (left) and resistivity (right) plotted against carrier density at 400K for PbTe samples alloyed with 1% thallium (dashed line) and samples alloyed with 2% of thallium (solid line).

In order to understand the origins of this increase we will focus only on two PbTe samples alloyed with 1 and 2% of Tl. These are samples whose Hall and Nernst coefficients are shown in Figure 51. The Hall coefficient RH (Figure 51) and resistivity

(Figure 50 a) are used to determine the carrier density and mobility as a function of temperature, this is shown in Figure 53. The Nernst coefficient a, which is positive for all our samples, is expressed in units of mobility, by dividing it by the freeelectron thermoelectric power, kB/q. All samples show very similar Nernst and Hall mobility.

Electron mobilities are much smaller than those observed in PbTe, for example PbTe

107

alloyed with similar atomic concentrations of Nd has a four times larger , Figure 32.

1.00x10 21 1000 1000 a) b) c) ) s 1.00x10 20 100 100 . /V ) 2 s . ) cm 3 /V ( 2 b cm (

cm q/k p . (

19 =Ν

1.00x10 10 10 Ν Pb99Tl1Te100 Pb98Tl2Te100 Pb58Sn40Tl2Te100

1.00x10 18 1 1 0 200 400 600 800 0 200 400 600 800 0 100 200 300 400 500 T [K] T [K] T [K]

Figure 53 Density, Hall mobility and Nernst mobility of Pb 0.99 Tl 0.01 Te, Pb 0.98 Tl 0.02 Te and Pb 0.58 Sn 0.40 Tl 0.02 Te.

The local maximum observed in the resistivity at 200 K is present in every sample prepared by powder metallurgy; it is attributed to a clear minimum in mobility, however, its origin was not determined. In these samples the thermal conductivity is not decreased compared to conventional PbTe, but the electrical conductivity is smaller than that of conventional PbTe. Therefore, the very large increase in zT can be attributed only to the increase in the Seebeck coefficient. For illustration in Figure 54 we show not only the

Pisarenko plot S(p) for 3 samples shown in Figure 53 but for all other measured PbTe:Tl samples. We can observe a clear tendency: all PbTe based samples have similar Seebeck coefficient which appears independent of carrier density. This behavior contradicts that predicted by Pisarenko relation for simple bands and was observed earlier in In and Nd in this text for example. The thermopower shows a 24 times increase as compared to that expected for conventional PbTe. It is this increase in S that results in corresponding increase in zT. Effect is larger as carrier density increases. We see that largest zT is

108

observed in high carrier density materials.

300

250

200 )

V/K 150 (

S 100

0 1.0x10 18 1.0x10 19 1.0x10 20 p (cm3)

Following procedures outlined in Chapter 3 but using a parabolic model we calculate the Fermi energy EF and the scattering exponent λ. From the Fermi energy and carrier density we calculate the density of states effective mass. The assumption of parabolic dependence of energy and momentum is source of large error in calculations since it does not hold for bands distorted by the effect of a resonant level. For this reason calculated effective mass is only used for qualitative diagnostic purposes and it does not

109

relate to actual band property. It is used to quantify the relative increase of the density of states at the Fermi energy in PbTe:Tl compared to that of pure PbTe. Four parameter analysis shows that scattering coefficient is λ≈0 leading us to conclude that the acoustic phonon scattering is dominant mechanism. We performed the analysis only on one sample with 2%Tl. Calculated position of Fermi energy and effective mass are shown in

Figure 55.

0.06

0.05 ) V e (

F 0.04 E

0.03 2 e /m * 1 d m

0 0 100 200 300 400 500 T(K)

Figure 55 Temperature dependence of Fermi energy and effective mass for Pb 0.98 Tl 0.2 Te sample.

Calculations show that at room temperature the Fermi energy is at roughly 60meV in the valence band. Optical measurements position Tl level at 100meV 37 and this is within thermal activation energy kBT(300K) from EF level. Same figure shows the

110

* 80 calculated values of md , which is four times larger than those of the pure PbTe. The temperature at which the resistivity shows a maximum also corresponds to the temperature above which the mass is relatively constant. However, this does not help us to determine the physical origin of the maximum in ρ. By calculating the scattering coefficient and effective mass we can conclude that Tl increases thermopower by changing the g(E)/dE dependence in the Mott relation, Eq. 77 this corresponds to stating that Mechanism I described in the introduction to this chapter is responsible for the increase in S(n) . The significance of this result is that we identify change in density of state distribution function that is temperature independent and indicated by the corresponding increases in effective mass and related increase in S, Eq. 78. If as suggested by Ravich resonant state was introducing new scattering mechanism and increasing related scattering coefficient we would expected its effect to decrease with increasing temperature where acoustic and optical phonon scattering becomes dominant.

This would prevent us from the use of the materials which have this mechanism in high temperature applications such as electrical power generation.

One prepared set of alloys with 40% Sn and 1 or 2% of Tl does not show same effect in increase in zT as pure PbTe samples. Although electrical conductivity is larger due to larger mobility (Figure 53) these samples exhibit maximum zT=0.4. These values are

70% smaller than those in PbTe:Tl and show interesting behavior; relatively constant zT in wide temperature range above 450K. Pb 60 Sn 40 Te is an alloy with near zero energy

33 gap. We can see that thermopower of Pb 60 Sn 40 Te:Tl samples are much smaller than

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those of PbTe:Tl and that it reaches maximum at 450K. This is an indication of bipolar carrier conduction (electrons and holes) in which partial Seebeck coefficients of holes and electrons cancel each other following Eq. 73. Electrons are thermally activated across the very narrow gap. For this reason, the point corresponding to Seebeck coefficient of

Pb 0.58 Sn 0.40 Tl 0.02 Te in Pisarenko plot falls below expected values, Figure 54.

Alloying PbTe 1xSe x with Tl.

In order to avoid bipolar conduction we avoid alloying PbTe with Sn and investigate effects of alloying PbTe 1xSe x with Tl. Energy gap in PbTePbSe alloys remains relatively constant, independent of Se concentration. Position of Σ point band is more sensitive to Se content and it moves from 170 to 300meV 36 with x=0 to x=100.

Ravich and Nemov 36 report that maximum carrier density in PbSe:Tl alloys reaches up to

2.5x10 20 cm 3. In addition Ioffe 3 has shown that alloying PbTe with Se reduces thermal conductivity, as shown for example in Figure 60. By combining these two, increase in carrier concentration and decrees in thermal conductivity, a further increase the overall figure of merit zT could be expected from alloying PbTe:Tl with PbSe. However we observe that with the increase in Se concentration the carrier density drops and that alloying has little effect on the high temperature thermal conductivity, which is already limited by normal phonon to phonon scattering.

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120

6 80 L L6 ) PbSe Te V x 1x e 40 m (

g E

d 0 e z i l

a 0 20 40 60 80 100

m x%Se r 40 X o n L6+ 80 L6+ 170meV 300meV 120 Σ Σ

PbTe PbSe

33 Figure 56 Energy gaps of PbTe 1xSe x alloys as function of Se concentration shown relative to the mid gap. Lines represent position of valence and conduction bands at L points. Schematic representation of position of heavy Σ point band is shown only for end point concentrations PbTe and PbSe. 36

In this study we observe properties of the limited number of Pb 0.98 Tl 0.02 Te 1xSe x alloys with x=0, 5, 10 and 20%. We do not cover entire range and prepare only alloys with 5, 10 and 20%Se. Materials were prepared by substituting 2 atomic percent of Pb with Tl to maintain stoichiometry. Samples are prepared by mixing 6N purity Pb, Te and Se with appropriate amount of 5N purity metallic Tl in quartz ampoules and sealing them in ~10 6

Torr vacuum. Materials were heated to 1000 oC, heated for 4 hours, rocked, and then rapidly cooled to 500 oC for a 7 day anneal.

Testing on these samples was performed at the Ohio State University laboratories on polycrystalline samples. Thermopower and resistivity were measured in temperature range 80650K and zero field Hall and transverse Nernst coefficients in the range 80

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420K. Tests were performed on prismatic samples following the procedures outlined in

Chapter 2. Low temperature and high temperature tests were performed on same pieces and results are summarized in Figure 57. Thermal conductivity was calculated as a product of measured thermal diffusivity, density and specific heat, total thermal conductivity is shown in Figure 60. We can observe that with increase in Se concentrations, the temperature dependence of the electrical resistivity changes. Low temperature resistivities of samples with higher Se concentrations are smaller than those of reference Pb 0.98 Tl 0.02 Te. Unfortunately, the opposite is true at high temperatures, here effects of alloy scattering dominate and samples with 20% Se have 45 times higher resistivity. High temperature thermopowers of Se alloys are 1015% larger. Measured

Hall coefficients are constant in measured temperature range and show increase with Se concentration. From this data we can calculate carrier densities and observe decreasing trend in carrier densities from 6x10 19 to 2x10 19 cm 3 as Se concentration reaches 20%.

Transverse Nernst coefficient shows larger temperature dependence. Low temperature

Nernst coefficient is larger for highSe samples. We see that is high for 10 and 20% Se samples in the regions where electrical resistivity is low and that all 4 samples reach same values of at 400K, where resistivities are identical.

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1.00x10 3 400

300 1.00x10 4 ) ) m V/K 200 (

( ρ S 1.00x10 5 100

1.00x10 6 0 0 200 400 600 800 0 200 400 600 800 T [K] T [K] 1.00x10 6 2.5

2 Τ) . 1.5 /Κ m/T) 7 V

1.00x10 ( ( H a

R 1 Pb98Tl2Te100 Ν

Pb98Tl2Se5Te95

Pb98Tl2Se10Te90 0.5

Pb98Tl2Se20Te80

1.00x10 8 0 0 100 200 300 400 500 0 100 200 300 400 500 T [K] T [K]

Figure 57 Galvanomagnetic and thermomagnetic properties of four Pb 0.98 Tl 0.02 Te 1xSe x alloys with x=0, 0.05, 0.1 and 0.2.

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1.00x10 20 1000 1000 a) b) c) ) s 100 100 . /V ) 2 s . ) cm 3 /V ( 2 b cm ( cm q/k p . ( =Ν

10 10 Ν

1.00x10 19 1 1 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 T [K] T [K] T [K]

Figure 58 (a) Carrier density, (b) Hall and (c) Nernst mobility of Pb 0.98 Tl 0.02 Te ( ), Pb 0.98 Tl 0.02 Te 0.95 Se 0.05 ( ), Pb 0.98 Tl 0.02 Te 0.9 Se 0.1 ( ) and Pb 0.98 Tl 0.02 Te 0.8 Se 0.2 ( ).

* Se concentration Eg EF(80K) m /m e(80K) L/L 0 (meV) (meV) ( ) 300K 650K Valence (0%) 190 0.14 0% 190 42 0.87 0.89 0.80 5% 194 59 0.47 0.90 0.77 10% 195 52 0.28 0.86 0.75 20% 198 55 0.29 0.87 0.70

Table 5 Calculated Fermi energy and density of states effective mass at 80K for Pb 0.98 Tl 0.02 Te 1xSe x alloys

The Nernst coefficient is sensitive to the nature of scattering, but also to electron mobility. If we compare electron mobility calculated from measured Hall coefficient and resistivity with Nernst mobility we will see that these two take similar values throughout the temperature range; this can easily be observed by comparing panels (b) and (c) in

Figure 58. Further, by applying method of four coefficients and assuming nonparabolic degenerate bands we calculate that at 80K scattering exponents in all four samples take

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values close to 0.5. This indicates the dominant scattering mechanism is acoustic phonon scattering. Based on observed temperature dependences of Nernst and Hall mobilities we can assume that this conclusion is valid in the entire temperature range. Further calculations can be made to evaluate the position of Fermi energy and effective mass.

The summary of these results is shown in Table 5; we can observe that Fermi energy levels remain relatively constant or slightly increase with x and that increase in Se concentration is followed by strong reduction in effective mass. Since effective mass is proportional to S we would expect it to decrease as predicted by 49. However, it can be observed from the attached Pisarenko plot in Figure 59 (a) that the Seebeck coefficient remains constant for all alloys but that carrier density starts decreasing. By adding Se to

Pb 0.98 Tl 0.02 Te we start approaching the Pisarenko line of conventional PbTe and the effect of the relative increase in thermopower is reduced. The increase in resistivity and decrease in the carrier density cause the figure of merit to drop as x increases. In Figure

59 we can see that at 650K zT= 1.38 for Pb 0.98 Tl 0.02 Te, and it keeps increasing with T. For

Pb 0.98 Tl 0.02 Te 0.8 Se 0.2 we reach what is maximum zT= 0.8 because the minority electrons start decreasing S at a lower temperature than in sample with less Se. The two other samples fall in between these two values.

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1.6 Pb98Tl2Te100

Pb98Tl2Se5Te95 b)

Pb98Tl2Se10Te90 1.2 Pb98Tl2Se20Te80

150

a) T

z 0.8 100 V/K) 0.4 S( 50

0 0 2x10 19 4x10 19 6x10 19 8x10 19 0 200 400 600 800 p (cm 3) T [K]

Figure 59 (a) Pisarenko plot showing S(p) for set of PbTeSe samples alloyed with Tl. For reference same plot contains number of points for Pb0.98Tl0.02Te samples. (b) Figure of merit shows decrease with increasing Se concentration.

To evaluate effects of alloy scattering in these alloys we measure thermal conductivity using static heater and sink method and transient method in the temperature range 300

650K. These results were used to calculate zT’s and they are summarized in Figure 60. At this point we will focus on evaluating lattice component of thermal conductivity by subtracting electronic contribution from total value of κ, for example by using Eq. 29:

κ e = LσT . 29

In the general case we can use Eq. 50 to evaluate electronic contribution using free

2 2 electron κe Lorentz number (L0=π /3(k B/q) ) and experimental values of electrical conductivity σ=ρ 1. Using this approach one may grossly overestimate the electronic contribution of thermal conductivity in particular at high temperatures. Following Ravich

[17] and using nonparabolic Kane model we can simplify Eq. 50 to write:

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2 2  k   2L λ+1  1L λ+1   L =  B   −2 −  −2   50  q   0L λ+1  0L λ+1      −2  −2   Here, L is transport integral defined by Kolodziejczak81 as

m k ∞ 2 ()E−E  E   2E   e F  nL m = E n  E +  ⋅1+  dE . 81 k  ()E−E 2  ∫  E T   E T  F 0  g ()  g () ()1+ e  Using equations 46 and 50 and calculated Fermi energies, acoustic phonon scattering and energy gap values for various alloys we can evaluate temperature dependence of Lorenz number. In Table 5 we show values of L/L 0 at 300 and 630K. These values are use to deduce lattice components of thermal conductivity shown in Figure 60. We can see that at room temperature the thermal conductivity significantly drops as we add Se. At room temperature alloy scattering is a dominant mechanism; this is consistent with literature values. 3 At elevated temperatures such as 630K thermal conductivity does not exhibit this strong dependence on alloying. At this temperature phonon – phonon scattering dominates and contribution of alloy scattering is of lesser importance.

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6 2 a) b) Pb98Tl2Te100 5 Pb Tl Se Te 98 2 5 95 1.6 300K Pb98Tl2Se10Te90 300K

4 Pb98Tl2Se20Te80 ) ) 1.2 K K . . 3 W/m W/m ( (

L κ κ 0.8 2 630K

0.4 1

0 0 0 200 400 600 800 0 20 40 60 80 100 T [K] x (%) PbSe Te x 1x

Figure 60 Total thermal conductivity of Pb0.98Tl0.02PbTe 1xSe x alloys (a). Lattice component of thermal conductivity calculated at 300K and at 600K (b). Solid black line 3 represents literature vales of lattice thermal conductivity of PTe 1xSe x alloys.

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Chapter 5: Anharmonically bonded I-V-VI2

semiconductors with minimum thermal conductivity

Group IVVI2 compounds are a second class of “semiconductors” analyzed in this text.

Group V element can be P, As, Sb or Bi, the group VI element S, Se or Te, and the group

I element can be Cu, Ag or Au,82 or an alkali metal like Li or Na.83 Although all of these combinations are possible only few of these, mainly sulfur based, compounds preferentially form rocksalt structure. We also see that for example CuBiTe2, CuSbTe2 and AgBiTe2 form Bi2Te3 – like, hexagonal structures. Structure of IVVI2 compounds is

84 very similar to that of IVVI’s, for example AgSbSe2, AgSbTe2, AgBiS2, AgBiSe2 and

82 CuBiSe2 all crystallize in the rocksalt structure. That is if we substitute the group IV cation in IVVI alloy with one group I and one group V element we will have a material that has similar properties but with a unit cell which is two times larger. AgSbTe2 lattice constant is a/2=0.6076 nm, very close to lattice constant of PbTe, and we will use this material as a representative for the class of IVVI2 compound semiconductors.

In this class of semiconductors, we focus on using the anharmonicity of the chemical bonds that drives the phononphonon Umklapp and Normal processes to reduce the lattice thermal conductivity.55 By this we can reach the same low thermal conductivity as in nanostructures, but avoid costly production of nanostructures.

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We create thermoelectrically relevant materials that are easy to manufacture in large quantities. We also note that unlike most materials used in thermoelectric power generation, Figure 5, AgSbTe 2 is leadfree and thus environmentally friendly. Since little is known of electronic structure of these materials work is divided in three parts: (1) demonstration of minimal thermal conductivity in rocksalt IVIVI 2 compounds; (2) investigation of electronic structure of AgSbTe 2 as one of representative materials and (3) doping study.

5.1 Thermal conductivity of I-V-VI 2 alloys

85 In 1959 Hockings reports that at the room temperature thermal conductivity of AgSbTe 2 is extremely small, on the order of κ = 0.63 W/mK. Dudkin and Ostranica the same year

86 report similar values measured on other IVVI 2 compounds. Alloys of AgSbTe 2 and

87 88 90 AgBiTe 2 with PbTe, SnTe and GeTe are wellknown thermoelectric materials. Since

1961 until late 1990’s AgSbTe 2 was known as the bulk ptype thermoelectric material with the highest figure of merit zT , which reaches 1.3 at 720 K, 89,90 as reported by Rosi

[90], but materials were not used since their preparation was not well controlled. The

10 recently prepared AgPb mSbTe 2+m was a first bulk alloy that exceeds this limit. From a

87,88 metallurgical point it can be viewed as solution of AgSbTe 2 with PbTe, although this

91 view is oversimplified since both AgPb mSbTe 2+m and (AgSbTe 2)1x(GeTe) x alloys are reported to contain nanoscale inclusions.92 Here we focus on pure ternary alloys in order to avoid confusion between nanostructuring and anharmonic bonds as a mechanism that imposes limit on the thermal conductivity.

As a design rule we use the fact that octahedral coordination in the rocksalt semiconductors has a high degree of anharmonicity, which suppresses lattice thermal conductivity κL by about a factor of 4 compared to tetrahedrallybonded semiconductors.

Examples of this contrast are IVVI compounds we talked of in Chapter 4. Although

PbTe can not match electronic properties of classic IIIV semiconductors such as GaAs,

InAs and InSb 93 it has lattice thermal conductivity is an order of magnitude lower. This in turn leads to much larger zT values. AgSbTe 2 in particular has a much higher anharmonicity than PbTe and a lattice which is two times larger. The result is a lattice thermal conductivity which is significantly smaller and, as we will show, it is at the amorphous limit, minimum possible in a bulk solid. We demonstrate that this is entirely due to the phononphononprocesses and that this property is intrinsic to all IVVI 2 compounds crystallizing in the rocksalt structure.

92,94 Structural studies of AgSbTe 2 identify a complicated microstructure in which large grain have a chemical composition approximated by Ag 22 Sb 28 Te 5 with silver telluride separating on grain boundaries several microns wide. In his study Armstrong 92 was able to identify secondary phases as Ag 2Te and Sb 2Te 3 by performing xray analysis. For our study we prepared several ingots of AgSbTe 2 and AgBiSe 2 using conventional solidstate chemistry similar to the technique used for PbTe. Charges of stoichiometric amounts of

6N purity Ag, Sb, (or Bi) and Te (or Se) were sealed in quartz ampoules under high vacuum. All materials were then heated to 800 oC where they remained for 6 hours. This

122123

was followed by slow cooling at the rate of 0.5deg/min to 500 oC. Samples were then annealed at these temperatures for 10 days and then cooled to room temperature at the rate of 3deg/min. When prepared using this procedure AgSbTe 2 crystallizes in rock salt structure and AgBiSe 2 takes hexagonal structure. To prepare rock salt AgBiSe 2 we rapidly cool samples to 600 oC from melt, anneal for 24 hours, and than rapidly cool in water to room temperature. Here we include measurements of AgInTe 2, a IIIIVI 2 chalcopyrite (a0=0.6406 nm, c0=1.256 nm) in order to contrast the thermal conductivity of several materials. Powder xray diffraction was performed on all samples to verify purity and structure of materials. As shown in Figure 61 in cubic AgBiSe 2 and AgSbTe 2 we do not observe presence of secondary phase. Shift in a peak position between

AgBiSe 2 and AgSbTe 2 is due to change in lattice constant.

Bi or Sb Te or Se Intensity

AgBiSe2 (hex.) AgBiSe2 (cubic) AgSbTe2 20 30 40 50 60 70 2θ

Figure 61 Powder Xray diffraction data of AgSbTe 2, cubic and hexagonal AgBiSe 2. Inset shows ordered rock salt structure of IVVI 2 in which these elements preferentially crystalize. 104

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The ingots were polycrystalline, with easy to observe crystallites exceeding several mm on the side. The thermal conductivity data we discuss in this section were taken using static heater and sink method described in Chapter 2. Since we suffer from radiation losses we limit our measurement to temperature range 80 to 350K.

Measured total thermal conductivities are shown in Figure 62. We can immediately notice that AgSbTe 2 and cubic AgBiSe 2 have smallest values of thermal conductivity, four times smaller than that of PbTe, and half of hexagonal AgBiSe 2. All selected samples were with relatively low electrical conductivities such that at this point we do not make correction for electronic component of thermal conductivity. Following discussion on electronic properties of AgSbTe 2 we will go in the greater detail on explaining very interesting structure of electronic contribution to thermal conductivity in this system. At this point following the work of Morelli et. al. and reference [95] we only focus on explaining the origins of low thermal conductivity.

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10.0 1x10 0 a) b) 1x10 1

) 1x10 2 K . g /

1.0 J (

(W/mK) P 3 C κ 1x10

AgInTe2 AgSbTe2 AgBiSe2 hex. AgBiSe 1x10 4 2 AgSbTe2 (cubic) AgBiSe2 cubic 0.1 1x10 5 100 200 300 400 1 10 100 1000 T(K) T (K)

First we will only briefly describe two phonon scattering mechanisms: Normal and

Umklappscattering. 17 To illustrate these we will refer to Figure 63. First, phonons can not extend beyond Brillouin zone that is their length is limited to 2πb, where b is width of reciprocal lattice shown schematically as a sphere in Figure 63. If two phonons of momentum q1 and q2 interact in Brillouin zone and their sum q3< πb we would observe formation of new phonon q3= q 1+ q 2. In this case momentum is preserved and such interaction does not limit thermal conductivity to the first order. We call this a normal scattering process. If q1 and q2 are such that their sum is larger than πb resulting phonon would extend beyond first Brillouin zone which is forbidden by the selection rule. Instead we will form equivalent vector, phonon with momentum q3+ 2 πb= q1+ q 2. This is called

126

an “Umklapp process.” In this case momentum is not preserved and we have a resistive behavior which reduces heat flow trough materials. The normal processes contribute to reduction in thermal conductivity since they create phonons with larger q that are more likely to undergo Umklapp processes. In 1914 Debye presented theory showing that thermal resistance of materials was due to anharmonicity of atomic vibrations. 17

Normal Umklapp

φ1 φ1

φ 2π b φ3 3 φ3 φ2 φ2

Figure 63 Illustration of Normal and Umklapp scattering mechanisms.

We can see in Figure 62 (a) that the lattice thermal conductivities of AgSbTe 2 and

AgBiSe 2 are less than 0.7 W/mK, and almost constant as a function of temperature. This is consistent with literature values reported for AgSbTe 2 at room temperature, 0.63

86 W/mK. The total thermal conductivity of AgInTe 2 is more than double that of AgSbTe 2, and like that of PbTe shows strong temperature dependence. The behavior of PbTe and

AgInTe 2 is characteristic of Umklappprocess scattering that intensifies as temperature increases. At higher temperatures we excite short wave length phonons. Since they have

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larger momentum, q, we are increasing probability of having Umklapp scattering mechanism and by that increase resistive behavior. We can recall from previous chapter sequence of Pb 0.98 Tl 0.02 Se 1xTe x samples and their lattice component of thermal conductivity Figure 60 (b). At room temperature we see strong effects of alloy scattering but as temperature increases to 650K dependence on Se concentration disappeared since

Umklappscattering started to dominate. We see same temperature dependent (1/T) effect of Umklapp scattering here in AgInTe 2 but not in IVVI 2’s.

Our result match literature value leading us to conclude that the low lattice thermal conductivity of the AgSbTe 2 samples is not limited by the defect structures but rather by an intrinsic mechanism. As discussed by Morelli 95 the intrinsic thermal conductivity of a solid in a temperature range in which heat is conducted only by acoustic phonons, and in which there are only interactions among the phonons by anharmonic Umklapp processes, is given by: 96

Mθ 3δn 3/1 κ = A a . L γ 2T 82

In this equation n is the number of atoms in the primitive unit cell, δ3 is the volume per atom, θa is the Debye temperature of acoustic phonons, M is the average mass of the atoms in the crystal, and A is a collection of physical constants ( A ≈ 3.1x10 6 if κ is in

W/mK, M in amu, and δ in Angstroms). Grüneisen parameter, γ, is a direct measure of the anharmonicity of the bonds, and defined as: 97

βBVm γ = . 83 CV 128

This value can be calculated by measuring the volume thermal expansion coefficient β, isothermal bulk modulus B, the specific heat at the constant volume CV, and calculating

98 82 molar volume Vm. For AgSbTe 2, we have: n = 4, and find B, β, from the literature. We measure specific heat Cp in the temperature range from 2 to 340K (Figure 62 (b)) for both AgSbTe 2 and for AgBiSe 2 and from these we calculate Debye temperature θa and

. high temperature limit of specific heat. For AgSbTe2 θa=125K and CP(T→∞) =0.205J/g K

. and for AgBiSe 2 θa=139K and CP(T→∞) =0.21J/g K, these values are in good agreement for

99 literature values for AgSbSe 2. From these we can calculate that γ ≈ 2.15 , and result is similar for both materials. For PbTe γ ≈1.5, 100,51 (this is a material with already large

Grüneisen parameter and anharmonicity) we see that in IVVI 2 materials we have 70% larger values. As a consequence we see relative decrease in thermal conductivity which can be calculated form Eq. 82; for PbTe at room temperature we calculate that

101 κL=2.8W/mK (n=2 and θa=120K ) while for AgSbTe 2 we obtain κL=0.68W/mK. Both calculated values are in good agreement with measured properties, Figure 62 (a) A rough

l 1 l estimate of the phonon mean free path from the formula κ L = 3 CV v and the data given above (the sound velocity v=4800 m/s is related to the modulus and the density and measured by Wolfe et. al. 102 ) shows that it is on the order of l ≈ 0.6 nm. This roughly corresponds to the interatomic distance at 300 K. This minimum thermal conductivity is shown as dashed line in Figure 62 (a).

We conclude that anharmonic model holds and that in the IVVI 2 semiconductors, the

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lattice thermal conductivity is limited to its minimum possible value by intrinsic phonon phonon processes alone. The temperaturedependence of the lattice thermal conductivity given in Eq. 82 follows a 1/ T law, which is observed for AgInTe 2 but not for AgSbTe 2.

This is consistent with the conclusion above: the lattice thermal conductivity reached its lowest limit already below room temperature, because the phonon mean free path is limited to the interatomic distance and cannot decrease further.

The difference in the thermal conductivity between the IIIIVI 2 and the IVVI 2 semiconductors is a direct consequence of the difference in chemical bonding discussed by Zhuze [82]. All of the electrons in the outer shell of group IIIIVI 2 compounds

3 s p participate in sp bond formation. On contrary in IVVI 2 alloys external V and VI electrons do not; they form electron cloud of relatively large radius. It is that these outer shell electrons introduce nonlinearity in phonon dispersion relation and by that increase anharmonicity.

5.2 Electronic structure of AgSbTe 2

103 By 1957, rocksalt AgSbSe 2 and AgSbTe 2 were synthesized and identified as narrowgap ( ∼0.2 eV) semiconductors. A galvanomagnetic and thermoelectric study of

Wernick 102 revealed what appeared to be contradictory properties: some samples of

AgSbTe 2 have positive Hall coefficients while others have negative one. However, they would all have a positive Seebeck coefficient. This is in contradiction with our discussion

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in Chapter 2 where we show that both Hall and Seebeck effect take on a sign of the charge carriers. From metallurgical point of view, these materials are hard to prepare and this might be a reason that fundamental studies of electronic properties of these materials were non existent. The first study to our knowledge was that of Hoang and Mahanti who

104 calculated band structure of AgSbTe 2. These calculations show AgSbTe 2 to be a semi metal with a small overlap between valence and conduction bands. The authors observe that the lowest energy structure is a rock salt structure with Ag and Sb atoms ordered in the metallic sublattice. Electrons reside in the pockets at the Lpoint of the Brillouin zone and the holes at the Xpoints. Although authors show overlap between these isoenergetic surfaces we will note that the calculations used the density functional theory are not meant to accurately estimate energy gaps. More recently band structure calculations

105 observe similar electronic structure in cubic AgBiTe 2, but here we focus only on

AgSbTe 2 and leave other IVVI 2 alloys to be analyzed at some later time.

To add to our understanding of the electronic structure of AgSbTe 2 we perform analysis on the polycrystalline samples and measure galvanomagnetic and thermomagnetic properties. Since these materials form a large crystalline structure we also use single crystal slices to measure quantum oscillations in magnetic susceptibility and resistivity to determine the crosssections of the hole Fermi surface. The de Haas – van Alphen effect is observed as oscillation of the magnetic susceptibility in a strong magnetic field. This effect is due to oscillations of density of states and in degenerate semiconductors they are proportional to 1/B measurements of these periods were used to

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determine effective masses.

When measuring galvanomagnetic and thermomagnetic properties we follow procedures outline in Chapter 2. AgSbTe 2 has cubic structure just like PbTe so when measuring properties in the magnetic filed we can apply all symmetry arguments as in previous chapter. The transverse Nernst and Hall coefficients are extracted as the components that are odd with field. Longitudinal magnetoresistance and magnetoseebeck effects are extracted from the relevant even components. Purity of materials is verified by powder xray diffraction. Tests were performed on the sample shown in Figure 9 (a) with dimensions 1x1x7.5mm. The electronic transport properties were taken from 77 to

400 K in magnetic fields from 2 to 2 Tesla. Results are summarized in Figure 64. We can immediately notice metallic electrical resistivity indicated by a increase proportional to the temperature. Hall and Seebeck coefficients which have opposing signs, as reported by Wernick. 103 In a single carrier conduction system both would have same sign so we start our analysis with the assumption of two carrier conduction system as described in

Chapter 3. In this case we will support our analysis by measuring magnetic field dependencies of longitudinal and transverse magnetoresistance and magnetoseebeck, shown in Figure 65 and Figure 66.

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1.0x10 4 320

280 5.0x10 5 240 m) V/K)

3.0x10 (

ρ 200 S ( 2.0x10 5 160

1.0x10 5 120 100 200 300 400 100 200 300 400 T (K) T (K) 18 1.0x10 6

16 2.0x10 6

14 m/T) 6

V/K.T) . 3.0x10 12 N ( RH ( 4.0x10 6 10

8 5.0x10 6 100 200 300 400 100 200 300 400 T (K) T (K)

Figure 64 Electrical resistivity, thermopower, transverse zero field Nernst and zero field Hall coefficients of AgSbTe 2.

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0.04 0

80K 0.03 2 300K

) 80K m) B 100K ( 0.02 R ( xy M

ρ 200K 200K 4 0.01 300K 100K 0 6 0 0.4 0.8 1.2 1.6 2 0 0.4 0.8 1.2 1.6 2 B (T) B (T)

Figure 65 Longitudinal and transverse (Hall) magnetoresistance of AgSbTe 2 at selected temperatures.

0.05 20 205 0.04 85 16 305 0.03 12 V/K) S 205 405 M 0.02 ( 8 305 xy

S 85 0.01 4 405 0 0 0 0.4 0.8 1.2 1.6 2 0 0.4 0.8 1.2 1.6 2 B(T) B (T) Figure 66 Longitudinal and transverse (Nernst) magnetoseebeck of AgSbTe 2 at 85, 205, 305 and 405K.

Longitudinal magnetoresistance shown in Figure 65 as Mρ ≡ ρ(B /) ρ(B = )0 − 1, is measured with current lines and electric field lines parallel and perpendicular to magnetic

field lines, transverse component is ρ xy ≡ E y (Bz /) jx following for example notation in

134

Figure 7. Zero field slope of ρxy with respect to field is actual Hall coefficient shown in

Figure 64, R ≡ lim (∂ρ / ∂B ). Longitudinal and transverse magnetoseebeck follow H Bz →0 xy z same definitions; zero field slope of transverse Seebeck coefficient S xy is shown in

Figure 64 as a transverse Nernst coefficient, ≡ lim (∂S / ∂B ). Bz →0 xy z

Procedures outlined in Chapter 3 can be used only if we have single carrier conduction system. In this case when we have both holes and electrons zero field properties are not sufficient to determine mobilities, Fermi energies, densities and scattering coefficients.

Unlike in the case of PbTe where we could follow literature values to reduce the number of unknowns, in AgSbTe 2 we do not have this possibility. However, magnetic field dependencies (Figure 65 and Figure 66) can be used as an additional parameter. Here, we will use Mρ(B) and ρxy (B) to determine partial mobilities and carrier densities of holes and electrons, and MS(B) and Sxy (B) to determine scattering mechanism and position of

Fermi energy. In the lack of better way we will assume simple parabolic bands both for holes and electrons. By this we can define partial Hall coefficients of electrons and holes as RHe =1/ne and RHh =1/pe and neglect the anisotropy of isoenergetic Fermi surfaces.

Degeneracy of these surfaces will therefore be included in density of states effective

* masses m d.

Following classic text of Putley [25], we can write the longitudinal and transverse component of electrical resistivity are the diagonal and offdiagonal elements of the equivalent conductivity elements σxx (B Z) and σxy (B Z):

135

σ xx ρxx = 2 2 and σ xx +σ xy 84

σ ρ = xy . xy 2 2 85 σ xx +σ xy

Each of the species of carriers partially contributes to conduction as shown in Eq. 71, when magnetic field, B, dependence is added we have:

nee peh σ xx = 2 2 + 2 2 and 1+ e B 1+ h B 86

ne 2 B pe 2B σ = e + h xy 2 2 2 2 87 1+ e B 1+ h B

We can substitute these expressions in Eq. 84 and 85 to get magnetic field dependencies of magnetoresistance and transverse resistivity. By performing Taylor expansion on these, around B= 0 we get:

ρ (B) Mρ(B) ≡ xx −1 = AB2 + CB4 ρ xx )0( 3 3 2 2 2 ()ne + ph ()()ne + ph − − ne + ph A = 2 ()ne + ph

5 5 3 3 2 2 2 ()ne + ph ()()()ne + ph − ne + ph + 2 − ne + ph C = 2 ()ne + ph 2 2 2 2 3 3 ne ph (e + h + 2eh )[()− ne + ph − 2()ne + ph ()ne + ph ] − 4 ()ne + ph

88 and:

136

3 ρ xy (B) = RH B + DB − n 2 + p 2 R ≡ lim ()∂ρ / ∂B = e h H Bz →0 xy z ne + p h 4 4 2 2 2 2 3 3 ()− ne + p h ()()− ne + p h [ − ne + p h − 2()ne + p h ()ne + p h ] D = − 2 − 4 ()ne + p h ()ne + p h

We see that number of unknown parameters in these two equations reduces to four and we can perform “bestfit” analysis. Magnetic field dependencies allow us to calculate what are parameters A, C and D; from direct measurements shown in Figure 64 we have

ρ =ρxx (B=0) and R H so we can calculate partial carrier densities n and p and partial mobilities e and h. Results of these calculations are shown in Figure 67, from these we can see that mobility of electrons exceeds that of holes for 3 orders of magnitude. This is much different than what is observed in PbTe where this ratio is 2. 33 This is indicative of holes being distributed in a heavy mass band and electrons in a light mass band as calculated by Hoang. 104 Carier densities are also much different but their ratio is inverted.

This is a ptype material with about 5x10 19 cm 3 holes and ~1x10 17 cm 3 electrons.

Corresponding partial electrical conductivities are also shown in same figure. Now we can resolve the mystery of “anomalous Hall coefficient” observed by Wolfe and

Wernick 102 ; sign of Hall coefficient in this two carrier conduction system is negative due to larger mobility of minority carriers which appears as a squared term in expression for

RH.

137

2.0x10 4 10000 e 19 p 4 1x10 ) 1.5x10 )

1 1000

σ 3

h /Vs) m 2

1 4 1.0x10 100 17 n ( (cm 1x10 n,p (cm n,p σ σ 3 5.0x10 h σe 10

0.0x10 0 1x10 15 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 T(K) T (K) T (K)

Figure 67 Partial electronic properties: conductivity, mobility and carrier density of holes and electrons in AgSbTe 2.

To calculate EF and Eg we will make use of measured thermomagnetic data. In twocarrier systems, the Seebeck and transverse Nernst coefficients are given by Eq. 73 and 74. Partial Seebeck and Nernst (Eq. 64 and 65) coefficients are given as functions of the Fermi energy in the conduction and valence bands, EFe and EFh. They are defined using Fermi transport integrals, Eq. 63.

Following paragraphs closely follow reference [22]. We assume that acoustic phonon scattering, λ = 1/2, is the dominant mechanism at all temperatures. This will allow us to reduce the number of unknown parameters. EFe and reduced Fermi energy

. ξe≡E Fe /k B T for electrons is measured from the bottom of the conduction band and EFh

. and ξh≡E Fh/k B T for holes from the top of the valence band. Now we must consider two possibilities, shown in Figure 68:

1. material is a semimetal, the conduction and the valence band overlap each other

by a small overlap energy Eo:

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E E = −(E + E ), ξ = −ξ − o Fe Fh o e h 90 k BT

2. material is a semiconductor with an energy gap Eg, and we have:

E E = −E + E ,ξ = −ξ + g Fe Fh g e h 91 k BT

We can see that mathematically Eg=Eo and we can use the relations (90) and (91) when determining if material is metal or semiconductor by observing the sign of Eo. We also further reduce number of unknown parameters by relating EFh and EFe and assuming that

Eo does not have any temperature dependence.

Semiconducting metalic

Eg Eo EFe EFh EFh E E F F

Figure 68 Position of Fermi energies of holes and electrons in metallic and semiconducting materials.

Now we can use equations 73 and 74 for zero field S and described in Chapter 3 in discussion of two carrier systems and define partial contributions of electrons and holes using integral forms 63 and 64 as a function of EFh and EFe . For partial electrical

139

conductivities of electrons and holes we use those shown in Figure 67. From these we calculate at each temperature, EFe(T), and one parameter adjusted for all temperatures, Eo or Eg. Less importance was given to the Nernst coefficient than to the Seebeck coefficient since this is the transport coefficient that is most sensitive to the scattering exponent, λ. Our results show positive Eg, material has small energy gap of 7.6 meV.

Numerical error on this calculation is at least 50% of this value but calculations do indicate presence of small gap. From physical point of view thermal energy kBT for

T=100 K is large enough to erase the practical importance of this gap. Temperature dependence of Fermi energies EFe and EFh is sufficient to calculate partial contributions

Se, S h, e,and h. When we calculate total values of S and from Eq. 73 and 74 we get very good match in S while calculated deviates since it is sensitive to temperature dependence of scattering coefficient. Calculated hole density (result shown in the Figure

67) and calculated Fermi energy EFh can be used to determine the densityofstates

* effective mass m d using Eq. 51. From EFh (T=0K) = 15 meV and the hole density

19 3 * extrapolated to 0 K of 3x10 cm we get m d = 2.2 ± 0.6 me, where me is the free electron mass.

140

40 400 Sh 20 200 0 V/K) Nh V/K) 20 S ( S ( Ne 0 S 40 e

200 100 200 300 400 100 200 300 400 T(K) T(K)

Figure 69 Dashed lines are partial electron and hole transverse Nernst and Seebeck coefficients. Solid line stands for total calculated and S. Symbols () are measured zero field values shown here for reference.

When same procedure is applied to electrons we get a density of states effective mass of

∼ 0.02 me. Error in this calculation is probably large since we use EFe calculated from partial Se which contributes only about 20% to total thermopower. However we see that there is an agreement with calculated partial mobilities since the ratio of

* * m electron/m hole≈e/h, shown in Figure 67.

Having calculated carrier densities and mobilities of holes and electrons we can focus on mapping the Fermi surface crosssection at EF by measuring magnetic susceptibility. De Haas van Alphen oscillations in the magnetic susceptibility were measured in a SQUID magnetometer at 5 K. Sample is the singlecrystal piece cut from the main ingot. Crystal axis <111> was oriented in parallel with the magnetic field.

Magnetic susceptibility was measured in the range 3.5 to 5T, Figure 70(a).

141

8x107

M(emu) 0x10

8x107 3.5 4 4.5 5 B(T) b) 1 0.8 B 0.6 0.4 A intensity 0.2 0 0 20 40 60 80 100 120 1/[1/B] (T)

Figure 70 (a) magnetic susceptibility of AgSbTe2 measured in <111> crystallographic direction. (b) Normalized values of Fourier transform of measured data.

Detailed analysis of De Haasvan Alphen oscillations on this sample was given by

Jovovic and Heremans in reference [22] and is repeated here for completeness. Fourier transform of magnetic susceptibility data periodic in 1/B identifies two frequencies

1/[1/B]a = 25.9 and at 1/[1/B]b = 62.0 Tesla, shown here in Figure70(b) and marked as A and B respectively. From analysis of galvanomagnetic data we know that material is ptype therefore these two periods correspond to oscillations of holes in quantized orbits on the Fermi surfaces. From these we can calculate crosssectional area of Fermi Surface,

45 AF, in direction perpendicular to <111>:

142

2πq 1 A = F h /1[ B] 92

17 2 From these we get two values corresponding to AFA = 2.47x10 m and AFB =

5.93x10 17 m2.

Pocket A Pocket B Mass (m e): * 106 m XG 0.109 ±0.003 0.144 ±0.004 Ab initio * 106 m XW 0.332 ±0.004 1.36±0.02 Ab initio * 106 m d per pocket 0.21 ±0.01 0.57 ±0.02 Ab initio * m C <111> 0.25 0.52 Experiment

Surface ( 10 17 m2): 106 AF(EF=15meV) 3.02 6.2 Ab initio

AF exp. 2.47 5.93 Experiment Table 6 Calculated and measured properties of Fermi energy surface

Band structure calculations of Hoang 106 provide us with great details of electronic structure of ordered AgSbTe 2. The two isoenergetic pockets are centered near the Xpoint of the Brillouin zone. At EF=15meV, value measured on our samples these two pockets are filled while the third pocket centered on the ' point partially along the XΓ axis is 70 meV below the Fermi energy and it is empty. Calculated effective masses along the XW and XΓ directions are given in Table 6. Using Fermi surface measured from oscillations in magnetic susceptibility and shape of Fermi surface calculated by Hoang 106 we estimate

* * the density of states effective mass m dA and m dB corresponding to two periods in

143

Figure 70 (b). The Fermi surface crosssections normal to the <111> axis of each set of three (A) and (B) pockets are degenerate, and can be approximated by an ellipse with a

* 17 cyclotron mass m C given in Table 6. These are calculated from equation 51 and:

3 hπ 2  3p  2 /1[ B] =   K 2q  π  93

Here N is number of pockets and K is degeneracy factor which will not be separately calculated and it will be included in calculated effective masses. From the

* * calculated values of the densityofstates mass m dA and m dB of each pocket we calculate a total densityofstates mass for the valence band as

* 3/2 * 2/3 * 2/3 3/2 md = 3 (mda + mdb ) ≈ 5.1 ± 2.0 me . The accuracy of the De Haasvan Alphen measurements is superior to that of the transport measurements, and the cyclotron masses

* (m d = 1.7 ± 0.2 me.) correspond quite well to the calculated masses. These are also in good agreement with masses calculated from Seebeck coefficients (2.2 me). Using the

Fermi energy ( EFh (T=0K) = 15 meV) deduced from the Seebeck coefficient, it is finally possible to calculate the crosssections AFA and A FB of the two pockets in the sample measured. These crosssections compare very well to the ones deduced from the De

Haas van Alphen measurements, see Table 6, providing confirmation of the entire picture for the valence band.

144

5.3 Doping and optimization of thermoelectric properties of

AgSbTe 2

AgSbTe 2 is a semiconductor with a very narrow energy gap (∼7 meV) with highly mobile electrons that dominate the Hall measurements. Holes have 9 times larger effective mass of electrons hence they will dominate the thermoelectric power. Silver is known as a very mobile atom in crystal structure of PbTe and similar behavior can be expected in I

19 3 VVI 2 alloys. The high density of excess holes ( ∼5x10 cm ) is consistent with vacancies on the metal sites (probably Ag in our samples), but it could arise from antisite defects (Sb or Te sites). The low lattice thermal conductivity and the large densityof states hole mass make the material a good ptype thermoelectric semiconductor. We can use calculated effective mass, Fermi energy and lattice thermal conductivity of

2 κ=0.7W/m .K to calculate zT = T ⋅ (S σ ) as a function of carrier density. We ()κ L + L0σT are using simple parabolic model for S(n,T) and σ(n) and assumption of single carrier conduction, equations 63 and 52, these are certainly unrealistic idealizations in this case.

Results are summarized in Figure 71 and indicate the optimal carrier densities of high

10 20 cm 3.

145

2 Temperature

1.5 823K 800K 750K 700K 650K T

zT 600K

z 1 550K 500K 450K 400K 0.5 350K

0 1x10 20 1x10 21 np(cm (m3)3)

Figure 71 Calculated figure of merit of AgSbTe2 as a function of carrier density and temperature.

Experiment

Following these calculations we prepare set of p and ntype doped alloys. The majority of this data and conclusions are reported by Jovovic et al. in reference [20]. In principle, since Ag is a monovalent metal atom and a major constituent, ptype doping is only conceivable by changing the stoichiometry on the metal sublattice. We can do this by increasing the concentration of the group I 1+ metal over the group V 3+ metal. Second possibility would be to change the stoichiometry between the metals and the chalcogen.

Third is to use divalent atoms (Pb 2+ , Bi 2+ ) to dope the material ptype if they were to substitute for the group V 3+ metal. However, there is no mechanism which would allow

146

us to control that they do not substitute equally for the group I 1+ metal. This would result in amphoteric behavior; in fact, they might even become donors if they substituted preferentially for Ag. We study here doping strategies involving excess Ag, Na and Tl and contrast these results to materials doped with Bi and Pb, Ga and In. All samples were prepared following same procedure as that used for stoichiometric AgSbTe 2. All starting materials are 6N purity or higher. When doping with Ag, Na, Tl, In, G and Bi we compensate by adding equal amount of Te. In addition to using pure Na we use Na 2Se and compensate for excess metal by adding Te. Lastly when doping with Pb we do not compensate with excess Te. For this reason we will refer to these dopants as AgTe,

NaSe 0.5 Te 0.5 , NaTe, TlTe ,InTe, GaTe and Pb and note that this is only abbreviated notation and that it does not describe structure of these materials. We noticed that materials alloyed with Na have significantly larger crystallite size up to 78 mm 3, while

Pb alloyed samples for example show much smaller grains of 12 mm 3. Purity of samples and structure was measured by Xray diffraction. We notice that samples doped with excess Ag show presence of the second phase and we assume that it corresponds to

Ag 2Te phase at grain boundaries. We do not observe phase separation in other samples or in the samples used in the previous studies.

Thermal conductivity data in the temperature range 80300K were taken on several samples with crosssections of ~9 mm 2 using the static heater and sink method. In

Figure 72 (b). we report only two sets of thermal conductivity data measured using the

5 static method. The solid line is measured on the undoped AgSbTe 2 of resistivity 7x10

147

m at the room temperature and the dashed line refers to the sample doped with 2 at.% of excess Ag over Sb. As shown earlier the lattice thermal conductivity of these materials is κL=0.61W/mK and that the total electronic component of thermal conductivity ( κe) is

h e 22 dominated by the ambipolar ( κambi ) term κe= κambi+ κ e+ κ e in the undoped materials.

Effects of radiative heat transfer are significantly larger at T>300K and in this range we apply the transient diffusivity method. Since small changes in chemical composition do not affect specific heat for Cp we use data measured on stoichiometric AgSbTe 2, shown in Figure 62 (b). Thermal diffusivity was measured on 10 mm diameter discs, thickness

1.2 mm in Anter FlashLine 3000 system, shown in Figure 72 (a). Total thermal conductivity was calculated using Eq. 32. For these calculations we use single temperature independent value of density of 6.852 g/cm 3, value is measured at room temperature on AgSbTe 2 at Thermal Properties Research Laboratories ( TPRL ).

148

70 1 a) b)

60 0.8 4 ]

50 K . m /

/sec] 10 x AgSbTe 2 W 0.6 2 [ + 2%AgTe κ

[cm 40 +1%NaSe0.5Te0.5

α +1%NaTe +1%Pb +1.5%TlTe +2%InTe +2%GaTe 30 0.4 300 400 500 100 200 300 400 500 T [K] T [K]

Measuring electrical properties on several cuts form ingots indicates that doped samples show increased level of nonuniformity. To ensure accuracy of calculated zT galvanomagnetic and thermomagnetic properties were measured on prismatic samples cut from of the ingots immediately next to samples used for thermal conductivity measurements. Tests were performed in temperature range 80 to 400K in order to avoid the observed phase transition at 417K. Effects of this phase transition on the electronic properties will be discussed at the end. Zero magnetic field ρ and S are shown in Figure

73. Hall resistivities and adiabatic transverse NernstEttingshausen voltages are measured in field 1.5 to 1.5T . Calculated Hall coefficient R H and isothermal Nernst coefficients are reported in Figure 74 for samples doped with group I and group V elements and separately in Figure 75 for samples doped with group III elements.

149

3 400 AgSbTe 1.0x10 a) 2 b) + 2%AgTe 5.0x10 4 +1%NaSe0.5Te0.5 +1%NaTe

+1%Pb 4 300 2.0x10 ]

4 m V/K] 1.0x10 [

ρ S[ 5 200 5.0x10

2.0x10 5

100 1.0x10 5 100 200 300 400 100 200 300 400 T [K] T [K]

3 400 AgSbTe 1.0x10 2 d) c) +2%GaTe 4 +2%InTe 5.0x10 +1.5%TlTe

300 2.0x10 4 ]

4 m V/K] 1.0x10 [

ρ S[ 5 200 5.0x10

2.0x10 5

100 1.0x10 5 100 200 300 400 100 200 300 400 T [K] T [K] Figure 73 Electrical resistivity and Seebeck of doped AgSbTe2 materials. Solid lines are added to guide the eye. Block (a) and (b) are resistivity and Seebeck coefficients of samples doped with group I and V elements and (c) and (d) are samples doped with group III elements. Stoichiometric samples are always included as a reference.

150

100 200 300 400 100 200 300 400 2x10 8 AgSbTe2 0.1 8 1x10 + 2%AgTe 5x10 9 +1%NaSe0.5Te0.5 0 0x10 0 +1%NaTe 5x10 9 +1%Pb

T] 8 . 0.1 1x10

8 m/T] V/K 2x10 [ 0 4 0x10 H R N [ 6 8 1x10 2x10 6 12 3x10 6

16 6 4x10

20 5x10 6 100 200 300 400 100 200 300 400 T [K] T [K] Figure 74 Zero field transverse Nernst coefficient and Hall Coefficient of undoped ()AgSbTe 2 sample and materials doped with 2%AgTe ( ), 1% NaSe 0.5 Te 0.5 ( ), 1%NaTe ( ), 1.5%TlTe( ), 1%BiTe( ) and 1% excess Pb( ). Solid lines are added to guide the eye.

100 200 300 400 100 200 300 400 8 4x10 AgSbTe2 0.1 0x10 0 +2%GaTe 0 +2%InTe 4x10 8 +1.5%TlTe 0.1 8 8x10

T] 0.2

. 1x10

7 m/T] V/K 0.3 2x10 [ 0 4 0x10 H R N[ 6 8 4x10 8x10 6 12 1x10 5

16 5 2x10

20 2x10 5 100 200 300 400 100 200 300 400 T [K] T [K]

Figure 75 Zero field transverse Nernst coefficient and Hall Coefficient of undoped ()AgSbTe 2 sample and materials doped 1.5%TlTe( ), 1%BiTe( ) and 2% GaTe( ).

151

Discussion

Properties of materials doped with AgTe, NaSe 0.5 Te 0.5 , NaTe, TlTe, InTe, GaTe and Pb are compared to those of stoichiometric AgSbTe 2. We notice some consistencies in doping IVVI 2 and IVVI alloys with group III elements and for that reason these will be discussed separately. Stoichiometric AgSbTe 2 is a two carrier conduction system and we observe that even upon efficient doping ptype with Na or Tl all prepared AgSbTe 2 based materials remain two carrier conduction systems. Although it is theoretically possible to calculate partial electronic properties for all materials prepared for this study, like we did

22 for undoped AgSbTe 2, we limit our analysis only on observing total electronic properties. Following paragraphs discuss some of these.

Adding excess Ag to sample would result in changing electron balance due to excess of group I element over group V. We observe that when excess Ag is 2% both ρ and S decrease , Figure 73 (a) and (b). This does correspond to increase in hole concentration. Transverse properties RH and both decrease, Figure 74. Here we only provide an intuitive explanation for this observation. The Hall coefficient decreases because balance of low mobility ptype carriers and high mobility electrons tips towards larger number of holes. This will have effect on the sign of the Hall coefficient turning it positive and matching that of S,. Doping with AgTe has a similar effect on which is now dominated by the partial Nernst and hole mobility. Thermal conductivity is slightly larger than that of undoped alloy due to the increase in the electronic contribution to κ,

152

Figure 72 (b). As a result the overall zT of silver doped alloy is smaller than that of the reference sample, Figure 76.

Sodium is the second group I element selected for doping. This was performed by using either metallic Na or Na2Se. In order to maintain ratio of metal and chalcogen atoms in case of Na2Se we compensate by additional Te. For this reason we track this doping as material doped with NaSe0.5Te0.5. Na2Se was selected as a dopant to avoid sodium oxide which readily forms on metallic Na even when stored in glove box with less than <1ppm of O2. We can see in that NaSe0.5Te0.5 doped sample has lowest electrical resistivity of all samples prepared. This is contrasted with relatively high resistivity of sample doped with metallic Na, explained by presence of oxides. At temperatures above 150K The Seebeck coefficients for both Na samples are larger than that of reference sample, Figure 73 (b). Both RH and of these materials indicate diminishing influence of minority electrons. The Hall coefficient is smaller for

NaSe0.5Te0.5 but it remains negative in both cases. The Nernst coefficient is small and changes sign at T>340K, for NaSe0.5Te0.5 sample as shown in Figure 74. Nernst coefficient of metallic Na doped sample remains negative throughout. Thermal conductivity of this alloy is larger than that of the reference sample, Figure 72(b). The figure of merit of AgSbTe2:NaSe0.5Te0.5 reaches a value of zT=1.2 at T=400K, this is highest value we measure on all doped samples and it is comparable to the that reported by Rosi at 700K.90 We will point that these two numbers can not be directly compared since Rosi’s result is measured at much higher temperatures. Doping with NaTe gives

153

zT =0.8 which is still significantly larger than figure of merit of undoped sample.

Doping with group V element was attempted to show contrast between ptype and ntype doped materials. One sample was prepared with BiTe and a second only with excess Pb. Transverse properties of both samples do not change compared to those of

20 stoichiometric AgSbTe 2. Longitudinal properties of Bidoped sample follow those of

AgSbTe 2, while for Pbdoped sample but we see an increase in S which is compensated by increasing ρ Figure 73. We conclude that Pb probably substitutes on the Ag sites while Bi substitutes for Sb and is electrically inactive. Overall zT remains at the same level as that of reference sample.

In this text we dedicated Chapter 4.3 to explain use of group III elements in IVVI alloys. By extension we assume that AgSbTe 2 follows properties of mother alloy PbTe than we can expect that In and most likely Ga in AgSbTe 2 stabilizes in conduction band and Tl in valence band. Here we prepare alloys with 1.5% TlTe and 2%GaTe and In TE and we observe their doping effects.

Thallium appears to stabilize carrier density in valence band. We can immediately notice that this is the only sample for which Hall coefficient remains positive in the entire temperature range. Nernst coefficient becomes negative at T=300K, indication of two carrier conduction. The temperature dependence of electrical resistivity is much different when compared to other samples; it appears semiconducting with characteristic temperature decay. Indium and gallium appear to be ntype dopants. Indium doped sample takes Hall coefficient value which is negative and larger than that of

154

stoichiometric sample while RH is relatively smaller in case of the Gadoped samples.

Qualitatively, the same behavior is observed in Nernst coefficient. From Eq. 79 and 80 we can see that this corresponds to increased number of negatively charged carriers in indium and decrees in electron density in Ga. It is possible to conclude that In stabilizes carrier density deeper in conduction band above EF of stoichiometric sample and Ga has similar effect but at lower energies, however both are still in conduction band. Definite confirmation of this assumption would require further study. However, since the conduction band of AgSbTe 2 has a small effective mass, the ntype material is not promising thermoelectric and such a study was not conducted. We summarize by observing thermal conductivity for these 3 samples and see that it remains at expectedly low levels. Figure of merit of group III doped IVVI 2 follows predicted; Tl is ptype dopant and increases zT to the point that it reaches unity at 400K while doping with In and Ga has negative effect on zT , see Figure 76.

In summary, we prove the postulate that the thermoelectric efficiency can be improved by doping the material ptype because it has heavy holes. Monovalent group I elements commonly used as ptype dopants in PbTe and Tl, prove to be acceptors in

AgSbTe 2. Bi and Pb have weak donor or amphoteric behavior. In and Ga are donors and have negative effects on figure of merit. The overall zT of both Nadoped and Tldoped

AgSbTe 2 can reach values in excess of 1 at 400 K , and remain higher than 0.8 near and above room temperature and we could conclude that in this temperature range AgSbTe 2 perhaps shows largest efficiencies of all bulk materials, see Figure 5. unfortunately

155

temperature range in which we can use AgSbTe 2 is limited by phase transitions which

107 occur at T F1 =418K and T F2 =633K.

1.2 AgSbTe2 +2%AgTe +1%NaSe Te 1 0.5 0.5 +1%NaTe p-type 1%BiTe 0.8 doping 1%Pb 2%GaTe 2%InTe T 0.6 z AgSbTe 2 1.5%TlTe

0.4 n-type doping 0.2

0 100 200 300 400 T [K]

Figure 76 Figure of merit of AgSbTe2 based alloys doped with Ag, Na, Bi, Pb, Ga, In and Tl and that of reference undoped AgSbTe 2 alloy

To determine effect of these phase transitions we simply measure S and ρ of stoichiometric samples in 3 temperature cycles, first and second in the range 350 to 550K and third 350 to 650K. We observe that both thermopower and resistivity significantly decreases after first cycle. Phase transition at 418K is endothermic process and it is reflected on measured properties, they show large peak. Intensities of these peaks are much smaller during second cycle and completely disappear in the third cycling performed from 350 to 650K.

156

0.0002 Temperature range 350-550K cycle 1 340 Temperature range 350-550K cycle 2 Temperature range 350-650K cycle 3 320

4 300 1.0x10 m] V/K] 280 AgSbTe2 [ ρ

S[ 350-550K 260 350-550K 5E005 350-650K 240 TF1 TF2 TF1 TF2 220 3E005 300 400 500 600 300 400 500 600 T [K] T [K]

Figure 77 Effects of temperature cycling on thermopower and electrical conductivity of AgSbTe 2.

In this sample decreases in S and ρ are such that overall power factor does not change as we cycle temperature. Figure 78 shows that maximum zT we measure reaches 1.15 at

600K and that it remains unchanged with cycling. Due to 418K phase transition any possible use of AgSbTe 2 is limited to T<400K since we observe that doped samples do not recover highzT and that curve shown in Figure 78 is exception rather than a rule.

Recent work of Nielsen indicates that 418K phase transition can be removed by preparing thermodynamically stabile offstoichiometric Ag xSb yTe 2 and quaternary alloys of Ag 1

108 xNa xSbTe 2.

157

1.2

0.8 T z

0.4

0 200 400 600 T [K]

Figure 78 Figure of merit of undoped AgSbTe 2.

158

Conclusions

The potential of renewable energy sources is still vastly unexplored. Excluding hydropower, only 3% of energy in developed countries comes from renewable resources.

The potential of thermoelectric materials has been recognized in early days of this field; thermoelectric materials can be used to generate electricity directly from the solar or geothermal energy. Although the efficiency of a thermoelectric generator cannot compete with large coal, gas, and nuclear power generation systems, it can be used to increase the efficiency of these systems. In small scale energy applications, the impact on fuel consumption and vehicle efficiency is significant. Thermoelectric materials are affordable and devices are simple to build and maintain. However this potential has not been used due to the limited efficiency of thermoelectric materials. Although steady progress has occurred since the 1950’s, a three to fourfold increase in figure of merit is required before thermoelectric generators become commercially viable.

The two approaches used in this work to improve zT are completely new as we use nature of chemical bonds and interactions at the atomic level to enhance thermoelectric efficiency. The work on PbTe:Tl represents the first successful attempt to modify electronic structure of a material to enhance figure of merit zT following Mahan

Sofo theory.34

159 To date, all work focusing on electrons was based on changing carrier concentration by doping or on using sizequantization in nanostructures. This work provides experimental evidence that trace elements can significantly modify electronic structure of materials and enhance its figure of merit, zT . It is a goal of this thesis and related work to provide evidence that this new “tool” can be used by the thermoelectric community in process of designing next generation of TE materials.

Techniques demonstrated here open new and unexploited venues for enhancement of zT . Two material systems of particular interest are AgSbTe 2 for low temperature applications and PbTe:Tl as a representative material for power generation. Today we can claim that several bulk material systems are crossing zT >1 barrier and have performance which will enable development of commercial products. It is our hope that methods described in this dissertation will be used to develop next generation of thermoelectric materials presumably in conjunction with such technologies as nanostructuring.

160 2.0 nano ntype PbAgSbTe Kanatzidis, Science (2004) p-type PbTe:Tl )

y (OSU, 2008) c

n 1.5 p-type AgSbTe2 e i (OSU, 2008) c i f f

e Research Bi2Te3 alloy s

e 1.0

r PbTe alloy u Commercial s a CoSb3 e

m BiSb alloy ( 0.5

z SiGe alloy

0.0 0 100 200 300 400 500 600 700 800 T(K)

Figure 79 Figure of merit of commercial and research alloys including alloys developed using method of modification of density of states, PbTe:Tl, and by utilizing anharmonic atomic bonds, Na and Tldoped AgSbTe 2.

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