Semiconductors and Semimetals, Volume 70: Recent Trends in Thermoelectric Materials Research, Part Two (Semiconductors and Semimetals)

Recent Trends in Thermoelectric Materials Research II SEMICONDUCTORS AND SEMIMETALS Volume 70

Semiconductors and Semimetals A Treatise

Edited by R. K. Willardson CONSULTING PHYSICIST

12722 EAST 23RD AVENUE SPOKANE, WA 99216-0327

Eicke R. Weber DEPARTMENT OF MATERIALS SCIENCE AND MINERAL ENGINEERING UNIVERSITY OF CALIFORNIA AT BERKELEY

BERKELEY, CA 94720

Recent Trends in Thermoelectric Materials Research II SEMICONDUCTORS AND SEMIMETALS Volume 70

Volume Editor TERRY M. TRITT DEPARTMENT OF PHYSICS AND ASTRONOMY

CLEMSON UNIVERSITY CLEMSON, SOUTH CAROLINA

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Contents

PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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LIST OF CONTRIBUTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1 Use of Atomic Diplacement Parameters in Thermoelectric Materials Research . . . . . . . . . .

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Brian C. Sales, David G. Mandrus, and Bryan C. Chakoumakos I. INTRODUCTION

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II. ELEMENTARYTHEORY OF ATOMIC DISPLACEMENT PARAMETERS . . . . . . . . III. INTERPm~TING A D P DATA . . . . . . . . . . . . . . . . . . . . . . .

1. Einstein M o d e l . . . . . . . . . . . . . . . . . . . . . . . 2. Debye M o d e l . . . . . . . . . . . . . . . . . . . . . . . . 3. Static Disorder . . . . . . . . . . . . . . . . . . . . . . . 4. Einstein and Debye Temperatures f r o m R o o m Temperature A D P Data IV. CLATHRATELIKETHERMOELECTRIC COMPOUNDS . . . . . . . . . . .

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V. ESTIMATION OF THE LATTICE THERMAL CONDUCTIVITY FROM A D P DATA

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1. Elementary Theory o f Lattice H e a t Conduction . . . . . . . . . 2. Lattice H e a t Conduction." A M o r e Realistic M o d e l . . . . .

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VI. EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . .

1. 2. 3. 4. 5.

Filled Skutterudites." LaFeaSb12 and YbFe4Sb12 . . . . . . Tl2gn Te 5 . . . . . . . . . . . . . . . . . . . . . . . LaB 6 . . . . . . . . . . . . . . . . . . . . . . . . Semiconducting Clathrates: Sr8Ga16Ge3o and BasGa16Ge3o CeRuGe 3 . . . . . . . . . . . . . . . . . . . . . . .

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V I I . SUMMARY . . . . . . . . . . . . . . . . . . . . . . . .

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REFERENCES . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 2 Electronic and Thermoelectric Properties of HalfHeusler Alloys . . . . . . . . . . . . . . . . . . . . . . . .

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S. Joseph Poon I. INTRODUCTION

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1. The Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . 2. Electronic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Thermoelectric Properties . . . . . . . . . . . . . . . . . . . . . . .

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CONTENTS

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4. Other New Intermetallic Phases . . . . . . . . . . . . . . . . . . 5. Goals o f This Chapter . . . . . . . . . . . . . . . . . . . . . .

II. EXPERIMENTALPROCEDURES . . . . . . . . . . . . . . . . . . III. UNDOPEDCOMPOUNDSWITHVALENCEELECTRONCOUNTNEAR 18 . 1. 2. 3. 4. 5. 1. 2. 3. 4.

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Transport Properties . . . . . . . . . . . . . Bandgap Features Inferred f r o m Doping Studies Impurity Band Transport Properties . . . . . . Thermoelectric Properties . . . . . . . . . . .

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V. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

REFERENCES .

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Chapter 3 Overview of the Thermoelectric Properties of Quasicrystalline Materials and Their Potential for Thermoelectric Applications . . . . . . . . . . . . . . . . . . . . . . . . . Terry M.

T r i t t , A . L . P o p e , a n d J.

Scope and Introduction . . . . . . . . . . . . . . . General Considerations . . . . . . . . . . . . . . . Analysis . . . . . . . . . . . . . . . . . . . . . Synthesis o f Stable Phases . . . . . . . . . . . . . Grain Growth . . . . . . . . . . . . . . . . . . .

IV. INTRODUCTIONTO THERMOELECTRICMATERIALS . . V. QUASICRYSTALSAS THERMOELECTRICS. 9 . . . . . . VI. THERMOELECTRICPROPERTIESOF QUASICRYSTALS . . 1. 2. 3. 4. 5.

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SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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REFERENCES .

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Chapter 4 Alexander

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Military Applications of Enhanced T h e r m o e l e c t r i c s . . . C. E h r l i c h a n d S t u a r t

I. INTRODUCTION

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Various Quasicrystalline Families . . . . . . . . . . . . . . . . . . Electrical Resistivity . . . . . . . . . . . . . . . . . . . . . . . Thermopower . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . Thermal and Electrical Transport in A l P d M n f o r Thermoelectrics . . . .

V I I . FUTtrRE DIRECTIONS AND APPROACH

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W. K o l i s

I. QUASICRYSTALS:BACKGROUND AND INTRODUCTION . . . . II. QUASICRYSTALS:STRUCTURALAND MECHANICAL PROPERTIES III. SYNTHETICMETHODSFORTHEGROWTHOF QUASICRYSTALS . 1. 2. 3. 4. 5.

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Semiconducting and Semimetallic Properties . . . . . . . . . . . Bandgap States in the V E C = 18 Alloys . . . . . . . . . . . . . Carrier Mobilities . . . . . . . . . . . . . . . . . . . . . . Magnetic Properties and Band-Structure Results . . . . . . . . . Thermoelectric Properties . . . . . . . . . . . . . . . . . . .

IV. DOPEDALLOYS

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THERMAL MANAGEMENT

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1. Biological . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Electronic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 118 118 119 121

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III. POWER GENERATION . . . . . . . . . . . . . . . . . . . . . . . . . . IV. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 5 Theoretical and Computational Approaches for Identifying and Optimizing Novel Thermoelectric Materials .

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David J. Singh I. INTRODUCTION

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II. SoME FUNDAMENTAL CONSIDERATIONS . . . . . . . . . . . . . . . . 1. The Thermoelectric Figure o f M e r i t . . . . . . . . . . . . . . . . . 2. L a t t i c e Thermal Conductivity . . . . . . . . . . . . . . . . . . III. FIRST PRINCIPLES METHODOLOGY . . . . . . . . . . . . . . . . . 1. Density Functional Calculations . . . . . . . . . . . . . . . . 2. Kinetic Transport Theory . . . . . . . . . . . . . . . . . . . IV. SKUTTERUDITES . . . . . . . . . . . . . . . . . . . . . . 1. Binary Skutterudites . . . . . . . . . . . . . . . . . . . 2. Ce-Filled Skutterudites . . . . . . . . . . . . . . . . . . . 3. La-Filled Skutterudites . . . . . . . . . . . . . . . . . . . 4. Lattice D y n a m i c s and Effects o f Filling . . . . . . . . . . . . . 5. Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . V. CHEVRELPHASES . . . . . . . . . . . . . . . . . . . . . . . . VI. f l - Z n 4 S b 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. HALE-HEUSLERCOMPOUNDS . . . . . . . . . . . . . . . . . . . . VIII. CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 6 Thermoelectric Properties of the Transition Metal Pentatellurides: Potential Low-Temperature Thermoelectric Materials . . . . . . . . . . . . . . . . . . .

128 128 131 134 134 142 146 156 162 162 166 170 172 173

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Terry M. Tritt and R. T. Littleton, I V I. PENTATELLURIDES: BACKGROUND AND INTRODUCTION

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1. Overview o f Interest in L o w - D i m e n s i o n a l Conductors in the 1980s .... 2. A n o m a l o u s Electrical Transport in H f I ' e 5 and Z r T e 5 . . . . . . . . 3. Synthesis and Structure . . . . . . . . . . . . . . . . . . . . 4. Effects o f Stress and Pressure . . . . . . . . . . . . . . . . . 5. Magnetotransport and Hall Effect . . . . . . . . . . . . . . . INTRODUCTIONTO THERMOELECTRICMATERIALS . . . . . . . . . . 1. General Description . . . . . . . . . . . . . . . . . . . . . . 2. L o w - T e m p e r a t u r e Refrigeration Applications . . . . . . . . . . . . PENTATELLURIDESAS POSSIBLE LoW-TEMPERATURE THERMOELECTRIC MATERIALS RECENT DEVELOPMENTSIN PROPERTIES OF PENTATELLURIDES . . . . . . . 1. Doping on the Transition M e t a l Site ( M x A r T e 5, M = Hf, Zr, A = Zr, 77) 2. Doping on the Chalcogen Site ( M T e s _ x C h x, M = Hf, Zr, A = Se, Sb) 3. Magnetotransport: Overview o f Recent Results . . . . . . . . . . . . 4. Thermal Conductivity o f Pentatellurides . . . . . . . . . . . . . . . 5. S u m m a r y o f Thermoelectric Properties . . . . . . . . . . . . . . . DISCUSSIONAND CONCLUSIONS . . . . . . . . . . . . . . . . . . . .

180 180 181 183 184 186 188 188 189 190 191 191 195 197 200 201 201

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VI. SUMMARY REFERENCES .

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Thermomagnetic Effects and Measurements . . . . . .

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Franz Freibert, Timothy W. Darling, Albert MigliorL and Stuart A. Trugman I. INTRODUCTION

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II. THERMOMAGNETICEFFECTS

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2. Definition o f Transport Coefficients . . . . . . . . . . . . . . . . . . 3. Electronic Refrigeration . . . . . . . . . . . . . . . . . . . . . . 4. Ideal Behavior . . . . . . . . . . . . . . . . . . . . . . .

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3. U m k e h r Effect . . . . . . . . . . . . . . . . . . . . . . .

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IV. MATERIALS SURVEY 1. 2. 3. 4.

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Basic M a t e r i a l R e q u i r e m e n t s . . . . . . . . Compensated Materials . . . . . . . . . . . B i s m u t h and B i s m u t h - A n t i m o n y A l l o y s . . Doped Materials . . . . . . . . . . . . .

V. EXPERIMENTALMEASUREMENT TECHNIQUES

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1. E x p e r i m e n t a l F u n d a m e n t a l s . . . . . . . . . . . . . . . . . . 2. I s o t h e r m a l M e a s u r e m e n t s . . . . . . . . . . . . . . . . . . . 3. A d i a b a t i c M e a s u r e m e n t s . . . . . . . . . . . . . . . . . . .

VI. SUMMARY . REFERENCES

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II. BOUNDARY IMPEDANCES . . . . . . . . . . . . . . . . . . . . . III. WIEDEMANN-FRANZ LAW AT BOUNDARIES . . . . . . . . . . . . . IV. ENERGY BALANCE EQUATIONS FOR ELECTRONSAND PHONONS OUT OF EQUILIBRIUM . . . . . . . . . . . . . . . . . . . . . . . . . V. THERMAL INSTABILITY . . . . . . . . . . . . . . . . . . . . . VI. EFFECTIVETHERMOELECTRIC PROPERTIES . . . . . . . . . . . . . .

I. INTRODUCTION

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VII. SUPERLATTICES VIII. SUMMARY . . . REFERENCES . .

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INDEX . . . . . . . . . . . . . . . . CONTENTS OF VOLUMES IN THIS SERIES .

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Preface Tremendous effort was made in thermoelectric materials research in the late 1950s and 1960s after Ioffe first proposed the investigation of semiconductor materials for utilization in thermoelectric applications. Alloys based on either the BizTe 3 or Sil_xGe x system soon became some of the most widely studied thermoelectric materials. These materials were extensively studied and optimized for their use in thermoelectric applications (solid state refrigeration and power generation; Goldsmid, 1986; Rowe, 1995) and remain the state-of-the-art materials for their specific temperature use. By the 1970s, research on thermoelectric materials had begun a steady decline and essentially vanished by the 1980s in the United States. However, since the early 1990s there has been a rebirth of interest in the field of thermoelectric materials research, and over the past few years many new classes of materials have been investigated for their potential for use in thermoelectric applications. Much of this was brought about by the need for new alternative energy materials, especially solid-state energetic materials. Many new concepts of materials, including bulk and thin-film materials, complex structures and geometry, materials synthesis, theory, and characterization have been advanced over the past decade of work. These three volumes of Semiconductors and Semimetals are dedicated to identifying the efforts of research in this past decade and preserving them in a concise and relatively complete overview of these efforts. It is hoped that this will provide future generations a significant added advantage over the current generation, who have worked hard to revive this field of research. The first two volumes are focused primarily on bulk materials, with one chapter on transport through interfaces. The first volume contains an overview of the field, including an introduction by Julian Goldsmid, who is credited with discovering the BizTe 3 materials. Volumes 69 and 70 contain reviews of theoretical, synthesis, and characterization methods and directions, as well as in-depth reviews of some of the most active areas of bulk materials research. The third volume in this series (Volume 71) is dedicated ix

X

PREFACE

primarily to low-dimensional and thin-film thermoelectric materials, including both theory and experimental work. Thermoelectric energy conversion utilizes the Peltier heat transferred when an electric current is passed through a thermoelectric material to provide a temperature gradient with heat being absorbed on the cold side and rejected at the sink, thus providing a refrigeration capability. Conversely, an imposed temperature gradient, A T, will result in a voltage or current, that is, small-scale power generation (Tritt, 1996, 1999). This aspect is widely utilized in deep space applications. A radioactive material acts as the heat source in these RTGs (radioactive thermoelectric generators) and thus provides a long-lived energy supply. The advantages of thermoelectric solid-state energy conversion are compactness, quietness (no moving parts), and localized heating or cooling, as well as the advantage of being "environmentally friendly." Applications of thermoelectric refrigeration include cooling of CCDs (charge coupled devices), laser diodes, infrared detectors, low-noise amplifiers, computer processor chips, and biological specimens. The essence of defining a good thermoelectric material lies primarily in determining the material's dimensionless figure of merit, Z T - o~26T/),, where 0~ is the Seebeck coefficient, a the electrical conductivity, 2 the total thermal conductivity (2 = 2 L + 2~; the lattice and electronic contributions, respectively), and T is the absolute temperature in kelvins. The Seebeck coefficient, or thermopower, is related to the Peltier effect by H = 0~T= Qp/I, where FI is the Peltier coetficient, Qp is the rate of heating or cooling, and I is the electrical current. The efficiency (r/) and coefficient of performance (COP) of a thermoelectric device are directly related to the figure of merit of the thermoelectric material or materials. Both q and COP are proportional to (1 + ZT) 1/2. Narrow-gap semiconductors have long been the choice of materials to investigate for potential thermoelectric applications because they satisfy the necessary criteria better than other materials. Material systems that exhibited complex crystal structures and heavy atoms, to facilitate low thermal conductivity, yet were easy to dope to tune the electronic properties, were of primary interest. Currently, the best thermoelectric materials have a value of Z T ~ 1. This value, Z T ~ 1, has been a practical upper limit for more than 30 years, yet there is no theoretical or thermodynamic reason why it cannot be larger. But recently many new materials and concepts of materials have been introduced, as you will see in the following chapters. The development of rapid synthesis and characterization techniques, coupled with much-advanced computational models, provides the ability to more rapidly investigate a class of materials for potential for thermoelectric applications. The need for higher performance energetic materials (providing alternative energy sources) for refrigeration applications such as cooling

PREFACE

xi

electronics and optoelectronics and power generation applications such as waste heat recovery are of great importance. One of the goals of the current research is to achieve Z T ~ 2-3 for many applications. Such values of Z T would make thermoelectric refrigeration competitive with vapor compression refrigeration systems and would make high-temperature materials feasible for utilization in many waste heat recovery applications, such as waste heat from automobile engines and exhaust. Over the past decade, much of the recent research in bulk materials for thermoelectric applications has revolved around the concept of the "phonon glass electron crystal" model (PGEC) developed by Slack (1979, 1995). This paradigm suggests that a good thermoelectric material should have the electronic properties of a crystalline material and the thermal properties of a glass. The "kickoff talk" given by Glen Slack in Symposium Z at the 1998 Fall Materials Research Society (MRS) was entitled "Holey and Unholey Semiconductors as Thermoelectric Refrigeration Materials" (Tritt et al., 1998). The chapters in Volume 69 such as that on skutterudites by Uher, clathrates by Nolas et al., and Chapter 1 in Volume 70 on the use of ADP parameters by Sales et al., discuss the concept of "holey" semiconductors or cage-structure materials that use "rattling" atoms to scatter phonons and reduce the lattice thermal conductivity of a material. In Volumes 69 and 70, other materials are discussed, such as the half-Huesler alloys (Poon), BiSb (Lenoir et al.), and quasicrystals (Tritt et al.) are more typical of the "unholey" materials, which have to depend on more typical scattering mechanisms, such as mass fluctuation scattering, to reduce lattice thermal conductivity in a material. The PGEC paradigm is also prevalent in much of the research focused on thin-film and superlattice materials and electrical and heat transport through interfaces. It is my strong belief that a new, higher performance thermoelectric material will be found and it will truly have a large impact on the world around us. The advances that I have seen over the past 5 or 6 years give me great optimism. However, I am always reminded just how good the BizTe 3 materials really are. The aspect of low-temperature refrigeration (T < 200 K) of electronics and optoelectronics would yield a revolution in the electronics industry. The possibility of superconducting electronics cooled below their superconducting transition by a solid-state and compact thermoelectric device is very enticing. Where will the breakthrough be? Will it be in the bulk materials, either "holey" or "unholey"? Will it be in the new exotic structures, such as superlattice or thin-film materials, or will it be in using thermionic refrigeration? In these new exotic structures we are learning much about interface scattering of the phonons as well as the electrical transport in these "confined structures." Added to this is an even greater challenge than in the bulk materials--characterizing the figure of merit of such complex geometries. Measurements on these structures have

xii

PREFACE

proven to be quite challenging. Hopefully, one or possibly more of the next generation thermoelectric materials will have been identified and discussed in one of these three volumes of Semiconductors and Semimetals. There are many possibilities and much work is left to do. I came into the field of thermoelectric materials research in 1994 while working as a research physicist at the Naval Research Laboratory, NRL, in Washington, D.C. We had decided to start a program in thermoelectric materials early that year at NRL. This program was headed by A. C. Ehrlich and included others at NRL such as David Singh, who had already been working in the field. I attended the 1994 International Conference on Thermoelectrics (ITC), which was held in Kansas City, MO. From the very first meeting I knew I had much to learn. Over the period from late 1970 until 1996, most of the research on thermoelectric materials was published and archived in the proceedings of these ITC conferences. The measurements necessary to evaluate thermoelectric materials were certainly nontrivial and the interplay of the electrical and thermal transport was indeed a challenge. At the 1994 meeting, I heard the term thermoelectrician for the first time, used by Cronin Vining, then president of the International Thermoelectrics Society. Much of the meeting was centered around BizTe 3 alloys and incremental improvements to these state-of-the-art materials, as well as more efficient design of devices based on these materials. There were talks about a new class of materials called skutterudites, which were viewed as very promising materials. The year before, 1993, Hicks and Dresselhaus had published a paper in which they predicted that much higher Z T values were possible in quantum well structures. This enhanced Z T is due to an enhanced density of states and thus higher mobility and also a higher thermopower as the quantum well width decreased from a "bulklike" term. The excitement that something new and promising might be happening in the field of thermoelectrics was apparent. Around this same time a program was developed by John Pazik at the Office of Naval Research (ONR) to investigate the possibilities of finding and developing higher performance thermoelectric materials. Then in late 1996, another new program on high performance thermoelectric materials was started by DARPA (Defense Applied Research Projects Agency), which was headed by Stumt Wolf. There were also a few programs funded by the Army Research Office (ARO), most of which were managed by John Prater. The coordination and cooperation of the ONR, ARO, and DARPA thermoelectric programs was very impressive and continues to be. The goals were lofty and still remain a challenge: "Find a material with a Z T ~ 3-4!" These D O D programs were the "heartblood" of the rebirth of research in thermoelectric materials in the 1990s in the United States. Volume 70 contains a chapter on "Military Applications of Enhanced Thermoelectric Materials."

PREFACE

xiii

Much of the work that is highlighted in these three volumes has direct ties to that original ONR program, and most were supported by one or more of the D O D programs. Without the vision of these program managers as well as DARPA, ARO, and ONR, these volumes would certainly not have been possible. I take this opportunity to acknowledge them and thank them for their support. As we underwent the "rebirth" of thermoelectrics research in the 1990s we had a distinct advantage. Many of the "great minds of thermoelectrics" such as J. Goldsmid, G. A. Slack, G. Mahan, M. Dresselhaus, and T. Harman were still very active; thus I am pleased to say that most of these researchers have contributions in these volumes. (Note: Ted Harman was invited to write a chapter but declined due to time constraints. However, some of his work on quantum dot superlattices is some of the most exciting work in the field.) Their contributions to this field of research are impressive, with some of them dating back to the "early days" of thermoelectrics in the late 1950s. The work and vision of Raymond Marlow and Dr. Hylan Lyon, Jr., of Marlow Industries and their contributions related to the rebirth of this field of research are also worth noting. Over the course of development of Recent Trends in Thermoelectric Materials Research it became apparent that the work would have to be divided initially into two and finally into three volumes. I decided to divide the volumes between two primary themes: Overview and Bulk Materials (Volumes 69 and 70) and Thin-Film/Low Dimensional Materials: Theory and Experiment (Volume 71). In the end, I think that the division of the volumes works quite well and will make it easier for the reader to follow specific areas of interest. Some chapters may seem somewhat out of place; this is due primarily to the timing of receiving manuscripts and to space constraints, and was also left somewhat to the discretion of the editor. First and foremost, I express great thanks to the authors who contributed to these volumes for their hard work and dedication in producing such an excellent collection of chapters. They were very responsive to the many deadlines and requirements and they were a great group of people to work with. I want to personally acknowledge my many conversations with Glen Slack, Julian Goldsmid, Jerry Mahan, Hylan Lyon, Jr., Ctirad Uher, A1 Ehrlich, Cronin Vining, and others in the field, as I grasped for the knowledge necessary to personally advance in this field of research. Their contributions to me and to others in the field are immeasurable. Thanks also to my many other colleagues in the thermoelectrics community. I acknowledge the support of DARPA, the Army Research Office, and the Office of Naval Research in my own research. I also acknowledge the support of my own institution, Clemson University, during the editorial and manuscript preparation process. I am truly indebted to my graduate students for their

xiv

PREFACE

contributions to these volumes, their hard work, and for the patience and understanding they exemplified during the editorial and writing process. I especially acknowledge A. L. Pope and R. T. Littleton IV for their help. A special thanks goes to my publisher, Greg Franklin, for his encouragement in all stages of the development of these manuscripts for publication. Thanks also to his assistant, Marsha Filion, for her help and contributions. I am especially indebted to my assistant at Clemson University, Lori McGowan, whose attention to detail and hard work (copying, reading, filing, corresponding with authors, etc.) really made these volumes possible. Without her dedication and hard work, I would not have been able to tackle the mountain of paperwork that went into these volumes. I also wish to acknowledge my wife, Penny, and my wonderful kids, Ben, Karen, Kristin, and Mary, for their patience and understanding during the many hours I spent on this work.

References H. J. Goldsmid, Electronic Refrigeration. Pion Limited Publishing, London, 1986. D. M. Rowe, ed., CRC Handbook of Thermoelectrics. CRC Press, Boca Raton, FL, 1995. G. A. Slack, in Solid State Physics, Vol. 34 (F. Seitz, D. Turnbull, and H. Ehrenreich, eds.), p. 1. Academic Press, New York, 1979. G. A. Slack, in CRC Handbook of Thermoelectrics (D. M. Rowe, ed.), p. 407. CRC Press, Boca Raton, FL, 1995. Terry M. Tritt, Science 272, 1276 (1996); 283, 804 (1999). Terry M. Tritt, M. Kanatzidis, G. Mahan, and H. B. Lyon, Jr., eds. Thermoelectric Materials-The Next Generation Materials for Small Scale Refrigeration and Power Generation Applications, MRS Proceedings Vols. 478 (1997) and 545 (1998). TERRY M. TRITT

List of Contributors

Numbers in parentheses indicate the pages on which the authors' contribution begins. M. BARTKOWIAK(245), Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee

BRYAN C. CHAKOUMAKOS (1), Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee TIMOTHY W. DARLING (207), Los Alamos National Laboratory, Los AlBinOS, New Mexico ALEXANDER C. EHRLICH (117), United States Naval Research Laboratory, Washington, D. C. FRANZ FREIBERT (207), Los AlBinOS National Laboratory, Los AlBinOS, New Mexico J. W. KOLIS (77), Department of Chemistry, Clemson University, Clemson, South Carolina R. T. LITTLETON, IV (179), Materials Science and Engineering Department, Clemson University, Clemson, South Carolina G. D. MAHAN (245), Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee DAVID G. MANDRUS (1), Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee ALBERT MIGLIORI (207), Los AlBinOS National Laboratory, Los AlBinOS, New Mexico S. JOSEPH POON (37), Department of Physics, University of Virginia, Charlottesville, Virginia

XV

xvi

LIST OF CONTRIBUTORS

A. L. POPE (77), Department of Physics and Astronomy, Clemson University, Clemson, South Carolina BRIAN C. SALES (1), Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee DAVID J. SINGH (125), Center for Computational Materials Science, Naval Research Laboratory, Washington, D. C TERRY M. TRITT (77, 179), Department of Physics and Astronomy, Clemson University, Clemson, South Carolina STUART A. TRUGMAN (207), Los Alamos National Laboratory, Los Alamos, New Mexico STUART A. WOLF (117), Defense Advanced Research Projects Agency, Washington, D. C.

SEMICONDUCTORS AND SEMIMETALS, VOL. 70

CHAPTER

1

Use of Atomic Displacement Parameters in Thermoelectric Materials Research Brian C. Sales, David G. Mandrus, and Bryan C. Chakoumakos SOLID STATEDIVISION OAK RIDGENATIONAL LABORATORY OAK RIDGE,TENNESSEE

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . II. ELEMENTARY THEORY OF ATOMIC DISPLACEMENT PARAMETERS

....

III. INTERPRETING A D P DATA . . . . . . . . . . . . . . . . . . .

1. Einstein M o d e l . . . . . . . . . . . . . . . . . . . . . . 2. Debye M o d e l . . . . . . . . . . . . . . . . . . . . . . . 3. Static Disorder . . . . . . . . . . . . . . . . . . . . . .

6 7 8 11

4. Einstein and Debye Temperatures f r o m R o o m Temperature A D P Data . IV. CLATHRATELIKE THERMOELECTRIC COMPOUNDS

1 4

. . . . . . . . . . .

V. ESTIMATION OF THE LATTICE THERMAL CONDUCTIVITY FROM A D P DATA

13 14 18

1. Elementary T h e o r y o f Lattice H e a t Conduction . . . . . . . . . . . . . 2. Lattice H e a t Conduction: A M o r e Real&tic M o d e l . . . . . . . . . . .

21

VI. EXAMPLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

1. 2. 3. 4. 5.

Filled Skutterudites: LaFegSb12 and Y b F e 4 S b l 2 . . . . . TlzSnTe 5 . . . . . . . . . . . . . . . . . . . . . . . LaB 6 . . . . . . . . . . . . . . . . . . . . . . . . . Semiconducting Clathrates: Sr8Ga16Ge3o and Ba8Ga16Ge3o CeRuGe 3 . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

26

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

28

V I I . SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

REVERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

I.

. . . . . . . . . . .

18

28 30 32

Introduction

Progress in thermoelectrics requires new materials, and finding new materials requires new ideas and new guidelines for materials selection. Most of the present chapter is devoted to explaining how a particular piece of crystallographic information atomic displacement parameters (ADPs) can be used to identify materials with an extremely low lattice thermal conductivity. Before launching into an exposition of this new Copyright 9 2001 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-752179-8 ISSN 0080-8784/01 $35.00

2

BRIAN C. SALES ET AL.

guideline, however, we briefly review the existing guidelines, as there are few places in which they are discussed all together. Modern thermoelectrics research dates from Ioffe's observation (Ioffe, 1957) that heavily doped semiconductors (n = 10~8-102~cm -3) make the best thermoelectrics. This observation follows from the expression for the thermoelectric figure of merit, Z = S2a/x (S = Seebeck coefficient, a = electrical conductivity, x = thermal conductivity) and the behavior of real-world materials as the carrier concentration is varied. Metals have a high a but a low S; insulators have a high S but a low a. It turns out that a carrier density of about 1019 cm -3 maximizes the quantity $2o" (known as the power factor), and such a carrier density is characteristic of a heavily doped semiconductor. The next guideline is the "10kaT rule" and concerns the size of the semiconducting energy gap. This rule states that good thermoelectrics should have energy gaps that are l OkBTop, where Top is the operating temperature. The reasoning behind this rule is as follows. Small gaps are generally good for thermoelectric performance because they lead to higher carrier mobilities. However, if the gap is too small, then the thermal excitation of minority carriers will adversely affect the figure of merit, since electrons and holes carry heat in opposite directions. An in-depth examination of this situation has been given by Mahan (1989); he found the IOkBT rule holds for direct and indirect gaps, and for both phonon and impurity scattering. The rule is also in reasonable accord with experimental data on good thermoelectrics. The theory of thermoelectrics shows that Z ~ ~(m*) 3/2 (/2 = carrier mobility, and m* = density of states effective mass) (Goldsmid, 1986). Therefore, it is desirable to maximize both m* and/~. Two important guidelines for materials result from the preceding proportionality. The first follows from the observation that m* can be increased without affecting # much if the semiconductor has several equivalent bands; therefore, good thermoelectrics are likely to be multivalley semiconductors, and crystal structures with high symmetry are required to produce several equivalent bands (Mahan, 1998; Goldsmid, 1986). The second concerns the electronegativity difference between the elements making up the thermoelectric material (Slack, 1995). The electronegativity difference is a measure of the covalency of the bonding in a material. Large electronegativity differences indicate ionic bonding, large charge transfer, and strong scattering of electrons by optical phonons. This strong scattering leads to low carrier mobilities and is one reason why oxides generally make poor thermoelectric materials. High electron mobilities, on the other hand, are found in materials composed of elements with very similar values of electronegativity. Good thermoelectrics, then, are composed from elements having small differences in electronegativity. Finding materials with favorable electronic properties is only half the story. The other half concerns finding materials with exceptionally low values of lattice thermal conductivity. A simple but useful expression for the

1

ATOMIC DISPLACEMENT PARAMETERS

3

lattice component of thermal conductivity is given by / s = 89 d, where C~ is the heat capacity per unit volume, vs is the velocity of sound, and d is the mean free path of the heat carrying phonons. Many of the guidelines for finding materials with a low lattice thermal conductivity can be understood from the preceding expression. A key guideline is to look for materials in which the average atomic weiyht is high. The origin of this rule is simple: Heavy atoms lead to small sound velocities and a correspondingly low thermal conductivity (Ashcroft and Mermin, 1976). Next, it is important to remember that mass fluctuation scatterin9 can be used to reduce the lattice thermal conductivity. The idea behind this rule is that isovalent substitutions will scatter heat carrying phonons strongly because the wavelength of these phonons is about the same as the distance between the scattering centers. Electrons, on the other hand, have a longer wavelength and will be scattered less. The value of Z T will therefore increase. Another guideline is that crystal structures with many atoms per unit cell tend to have low lattice thermal conductivities. This rule is not as well grounded theoretically (or experimentally) as the first two, but nevertheless seems to be validated by experience. One explanation for this trend is that the number of defects per unit cell tends to grow rapidly as the size of the cell increases. The amount of disorder, then, tends to be relatively greater for materials with many atoms per unit cell. Another explanation may lie in the breakdown of the concept of a phonon as the number of atoms in the unit cell grows large. Remembering that there are 3n phonon modes, where n is the number of atoms in the unit cell, it is reasonable to assume that as n grows large these modes will begin to overlap and will no longer be distinguishable. Allen and Feldman (1993) have argued that thermal transport in this situation is beginning to resemble thermal transport in a glass. Another rule is that crystal structures in which the ions are highly coordinated tend to have lower thermal conductivities than crystal structures in which the ions have low coordination. This is an empirical relationship proposed by Spitzer (1970) based on a compilation of thermal conductivity data on more than 200 semiconductors. We are not aware of any generally accepted explanation for this behavior, although it is interesting that highly coordinated ions are also involved in the reduction in thermal conductivity associated with the "rattling" cations discussed next. A guideline originally proposed by Slack (1995), involves findin9 materials in which one or more atoms per unit cell are loosely bound and "rattle" in an oversized cage. The cage is invariably constructed from many atoms that highly coordinate the rattler. Such rattlers resonantly scatter phonons and can reduce the mean free path of the heat carrying phonons to dimensions comparable to an interatomic spacing. The effect on the thermal conductivity is dramatic, as recent work on filled skutterudites (Sales et al., 1996, 2000) and germanium clathrates (Cohn et al., 1999) has shown.

4

BRIAN C. SALES ET AL.

Finally, it is important to recognize that although these guidelines describe most of the better thermoelectrics, there are some relatively good materials that do not obey the rules. A good example is N a C o 2 0 4. This material is made up of light atoms with large electronegativity differences, yet at room temperature the power factor of this compound is greater than that of BizTe 3 (Terasaki et al., 1997). The rest of this chapter is devoted to exploring the connection between a particular piece of crystallographic information, atomic displacement parameters, and lattice thermal conductivity. In the description of a new crystalline compound, crystallographers normally tabulate the room temperature atomic displacement parameter (ADP) values for each distinct atomic site in the structure. These values measure the mean square displacement of an atom type about its equilibrium position and thus comprise some of the first information that is known about a new crystalline compound. The value of the mean square atomic displacement can be due to the vibration of the atom or to static disorder. The effects that this parameter can have on various physical properties, however, have not been widely recognized. In particular, ADP values are not normally used by solid state physicists or chemists as a guide in the search for new compounds with specific properties. ADPs are regarded by many scientists as unreliable since in many of the earliest structure determinations, the ADP values would often act as repositories for much of the error in the structure refinement. In addition, crystallographers have not always reported ADP information using a consistent definition, adding further confusion as to the usefulness of ADPs (Trueblood et al., 1996). The purpose of this chapter is to illustrate that when properly determined, the ADP values can be used as a guide in the search for crystalline materials with unusually low lattice thermal conductivities. These materials are of particular interest in the design of thermoelectric compounds with improved efficiencies.

II. Elementary Theory of Atomic Displacement Parameters Atomic displacement parameters measure the mean-square displacement amplitudes of an atom about its equilibrium position in a crystal. In general there is no reason to assume that the displacements are the same in all directions, or that they bear any particular relation to the crystallographic axes. For this reason crystallographers typically report ADP information as a 3 x 3 matrix, Uij, that allows for anisotropic displacements. In the description of a new crystalline compound, crystallographers normally tabulate the room temperature ADP matrix for each distinct atomic site in the structure (Trueblood et al., 1996; Dunitz et al., 1988; Willis and Pryor, 1975; Kittel, 1968). The various ADP values thus comprise some of the first

1

ATOMIC DISPLACEMENT PARAMETERS

5

FIG. 1. Structure of SrsGax6Ge30 as determined using powder neutron diffraction (Chakoumakos et al., 2000). The large ellipsoids correspond to the motion of Sr atoms at the center of a large cage consisting of 24 atoms of Ge or Ga (randomly distributed). The view shown is along the (111) axis of the crystal.

information that is known about a new crystalline compound. Often, an isotropic ADP value, Uiso, is given for each site. Uiso corresponds to the mean square displacement averaged over all directions and is given by one-third of the trace of the diagonalized Uij matrix. Uiso is a scalar, which makes it easy to qualitatively compare the relative displacements of different atom types in the structure. Sometimes Uiso is the only ADP information given if the full Uij matrix cannot be extracted from the X-ray or neutron diffraction data set or if there are no significant anisotropic displacements. The Uij data are often expressed in crystal structure figures by drawing ellipsoids around each atom. The surface of each ellipsoid corresponds to surfaces of constant probability. Normally the 50% ellipsoid is drawn corresponding to a 50% probability of finding the atom inside the ellipsoid. The 50% probability ellipsoids can be drawn for each atom in the unit cell by a computer program such as ORTEP (Oak Ridge Thermal Ellipsoid Plots; Burnett and Johnson, 1996). An example of an ORTEP drawing of the semiconducting clathrate SrsGa16Ge3o is shown in Fig. 1 (Chakoumakos, 2000). The value of the mean square atomic displacement can be due to the vibration of the atom and/or to static disorder. In a neutron or X-ray diffraction experiment, thermal vibrations of the atoms reduce the intensity of the Bragg reflections but do not affect the width. The scattered intensity, I, of a typical Bragg peak is qualitatively given by I = I o exp[- 89

(1)

6

BRIAN C. SALES ET AL.

where I o is the scattered intensity from a rigid lattice (no vibrating atoms), (u 2) is the mean square displacement of an atom about its equilibrium position, and Ak is the magnitude of the scattering vector (which increases as the sine of the scattering angle) (Kittel, 1968). In the physics literature the exponential factor is often referred to as the Debye-Waller factor. Atoms in a crystal vibrate more at higher temperatures, which implies that (u 2) increases monotonically with temperature. The intensity of X-rays (or neutrons) scattered by a crystal is the sum of the Bragg scattering and the thermal diffuse scattering (TDS). TDS corresponds to scattering in which one or more phonons are excited. As the temperature is raised, the overall intensity from Bragg scattering decreases with a corresponding increase in TDS. ADP values can be reliably determined using powder neutron diffraction and single crystal X-ray or neutron diffraction. The analysis of neutron data is usually easier for two reasons. First, the neutron wavelength used is typically the order of 1-2 A, which is much larger than the interaction distance between the neutron and the atomic nucleus. The nuclear scattering cross section is, therefore, a scalar with no angular dependence. X-rays scatter from the electron clouds around the nucleus and the resulting atomic form factor does have an angular dependence. This means that the intensity I o (Eq. (1)) depends on angle. Since the ADP information is in the angular dependence of the scattered intensity ((u2)(Ak) 2) (see Eq. (1)), it is sometimes difficult to separate ADP information from atomic-form-factor effects. Second, for most compounds, the absorption correction for neutrons is small, but for X-rays absorption must be carefully determined to obtain good ADP values, particularly for compounds with heavy elements. In the present article ADP data are all interpreted within a harmonic approximation for the atomic potential well. For the analysis of low temperature ADP data, this approximation is adequete. However, there are several examples in the literature (Kisi and Yuxiang, 1998) where hightemperature ADP data have been used to determine the anharmonic component to the vibration. This discussion is outside the scope of this article, but the interested reader can refer to Willis and Pryor (1975) and Kuhs (1988).

III.

Interpreting ADP Data

Interpreting the meaning of the ADP information requires a model. There are two simple models of the vibrational properties of a solid that can be used to extract useful information from the ADP data: the Debye model and the Einstein model. As shown later, both of these models are useful in

1

7

ATOMIC DISPLACEMENT PARAMETERS

understanding the thermal transport and thermodynamic properties of clathratelike thermoelectric compounds. 1.

EINSTEIN MODEL

For clathrate-type compounds, in which one of the atoms is poorly bonded and rattles in an oversized cage, the simplest model for the "rattler" is that of a harmonic oscillator (also called an Einstein oscillator). In this model it is assumed that all of the rattlers vibrate independent of each other and at the same frequency (a local mode). How far this model can be applied to the averaged motion of an averaged atom in a crystal structure is an open question. Clearly the ADP values provide no information about correlations with the motion of other atoms (i.e., lattice dynamics). This local approach, however appears to work well in interpreting the ADP information from the rattler (Sales et al., 1999, 2000). The mean square displacement amplitude, (uZ), of a quantized harmonic oscillator is given by Uiso ~. ( / / 2 ) =

h/(8rc2mv)coth(hv/2kaT

)

(2)

where v is the frequency of vibration, m is the reduced mass, and h and ka are the Planck and Boltzmann constants, respectively (Dunitz et al., 1988). At high temperatures, where hv << 2kaT, Eq. (2) reduces to the classical expression Uis o =

kBT/K

= h 2 T/(4rc2mkBOE 2)

(3)

where K is the spring constant of the oscillator and the Einstein temperature of the oscillator is defined as | = hv/ka. Equation (2) is plotted in Fig. 2 for several values of | From the high temperature slope of Uiso, the Einstein temperature of the rattler can be estimated. Notice that if the temperature at which the ADP information is known (usually room temperature) is greater than or on the order of | then the room temperature ADP data and the origin (T = 0, Uiso = 0) can be used to estimate the slope, and hence | This is significant, since for a new compound usually only room temperature ADP data are available. The preceding analysis, however, assumes that the value of Uiso is all due to dynamic motion of the atom, which neglects the possibility of static disorder. Static disorder is normally only a problem in solid solutions or in compounds that contain large concentrations of defects. Static disorder tends to displace the curves shown in Fig. 2 upward by a constant amount. If the static disorder is large, the ADP data from one temperature cannot be used to estimate the Einstein (or Debye) temperature (see later discussion).

8

BRIAN C. SALES ET AL.

M a s s of Rattler = 100 a m u

0.06

. . . .

i

. . . .

!

"""-"

"

I

'

"''

'

f

'

"'"

"

I

'jill

~

. . . .

:

=50K

.

0.05

'

0.04 A

,<E r

O

0.03

2

<

0.02

150

0.01

,,

0

50

100

150

200

250

300

i,

I

K

|

~

350

T(K) FIG. 2. Calculated mean square displacement amplitude (Uiso) of a quantum harmonic oscillator with a mass of 100 amu and the Einstein temperatures (tOE) shown. At high temperatures, these curves are linear in temperature with a slope as shown.

2.

DEBYE M O D E L

In a real crystal, the vibrations of different atoms are correlated and are described by specific wavelike modes (phonons) that represent the fundamental excitations of the crystalline lattice. In general the various phonon modes fall into three acoustic branches and 3N - 3 optical branches, where N is the number of atoms in the primitive unit cell (see, for example, Kittel, 1968, or Willis and Pryor, 1975). In principle, these modes can be determined using inelastic neutron scattering or can be calculated using lattice dynamical models. Neither the experimental or theoretical approaches are easy, and as a result the lattice dynamics are only understood in detail for a few simple materials. The Debye model is the simplest attempt to account for the correlated motion of atoms in a crystal. In the Debye model all of the phonons (normal modes) are assumed to have the same velocity v, which implies a linear relationship between the frequency co and the wave vector K of each mode since v = co/K, K = 2rt/2 where 2 is the wavelength of the normal mode. The total number of modes is limited to 3Nsolid, corresponding to the correct number of normal modes. This implies an upper limit for co and K normally denoted as coD and K o. The maximum wave number, K D, is simply related

1

9

ATOMIC DISPLACEMENT PARAMETERS

to the number of atoms per unit volume, n, and is given by

(4)

K o = (6~2n) 1/3.

The Debye temperature, 19o, is defined as hogo/2ztk a. If the Debye temperature is known, the average velocity of sound is also known and vice versa, since v = ogD/K D = | Direct measurements of of sound can also be used velocity of sound given in frequently used expression

(5)

the tranverse, vt, and longitudinal, vs, velocities to estimate a Debye temperature or the Debye Eq. (5). For isotropic polycrystalline samples, a (Anderson, 1963) is

v = (1/312/v 3 + 1/v~])-1/3

(6)

Expressions relating elastic constant data to the Debye velocity of sound also are described in the literature (Anderson, 1963). The Debye model of lattice vibrations is in general much too simple, but at low temperatures (typically T < | where only long wavelength phonons are excited, the Debye model is exact. For a monatomic cubic crystal, Uiso vs T can be solved exactly within the Debye approximation (Willis and Pryor, 1965) and is given by Oiso ~. (/,/2) __ [3hZT/(4rcZmkB|174

+ 0.25|

]

(7)

and O0(x)= l/x f ~ At high temperatures (T > |

ydy

(8)

Uiso is linear in T and is given by

Uiso = [3hZ/(mka4rczoz)]T,

(9)

and at low temperatures Uis o approaches the zero point value of 3h2/ (16rtZmkB| Figure 3 illustrates how Uiso depends on temperature for Debye temperatures of 100, 200, and 300 K, and an atomic mass of 100 amu. The smaller the mean atomic mass or the lower the Debye temperature, the larger the zero point vibration. The Debye temperature can be determined from the high-temperature slope (T > | of the ADP data, as indicated in the figure. The high-temperature slope extrapolates to the origin (no

10

BRIAN C. SALES ET AL.

0.03

I

0.025

I

O D =100 K

Slope = 3h2/(mkBOo4~ 2)

0.02

A 04

o<: O

:D

I

0.015 O D =300

0.01

K

0.005

0

I

I

I

0.5

1

1.5

.....

2

T/ODeby e FIG. 3. Mean square displacement Uiso vs temperature scaled by the Debye temperature, | for a monatomic cubic solid. Uiso is shown for a solid composed of atoms with a mass of 100 amu for three different Debye temperatures. For temperatures greater than | Uiso is linear in T with a slope as indicated.

zero-point energy offset). If just the room temperature value for Uis o is used to extrapolate a slope to the origin, the error in the calculated Debye temperature is less than 10% if the actual Debye temperature is less than 600 K. If the Debye temperature is greater than 600 K, Eq. (7) can be used to self-consistently determine the Debye temperature. This is illustrated in Fig. 4. For example, if a material has a Debye temperature of 1200 K, using the room temperature ADP data will result in an error of 38% and give an apparent Debye temperature that is too low (1200/1.38 = 870 K). A natural extension of this analysis to a multielement compound is accomplished by determining Uiso for each element (or crystallographic site) and calculating the average Uiso and the average atomic mass. Although the extension of Eq. (7) to a multielement compound is plausible, it should be emphasized that the main justification for this extension is that it seems to work fairly well for many materials. As an example, for the skutterudite LaFe4Sb12, single crystal X-ray diffraction data (Braun and Jeitschko, 1980) results in an average room temperature Uiso value of (0.0165 + 0.0031 x 4 + 0.004 • 12)/17 = 0.0045 ]k 2, and an average mass per atom of 107 amu. This room temperature ADP data predicts a Debye temperature of 299 K, which can be compared to 308 K estimated from room temperature velocity of sound measurements (Sales et al., 1997). Room temperature ADP data can be used to estimate the Debye temperature and average sound velocity of any c o m p o u n d / f the amount of static disorder is small.

1

o

n (9

2.0

t._

(9

(9

-

For Room Temp ADP data: ODebye < 600 K, error __ 10%

1.8

E _ pE(9 i 1.6 0

o n"

11

ATOMIC DISPLACEMENT PARAMETERS

.

1.4

ol

._c

1.2

(9 0

1.0

0.0

0.2

0.4

WO o

0.6

0.8

1.0

FIG. 4. Plot of error that results when room temperature ADP values are used to estimate the Debye temperature of a solid, when the actual Debye temperature is greater than room temperature. (See text for details.)

3.

STATIC DISORDER

The effect of static disorder o n Uis o is qualitatively shown in Fig. 5. The curves shown in Figs. 2 and 3 are displaced upward by a constant amount. In all of the thermoelectric compounds that we have studied using neutron

0.02

'

I

Slope

=

i "

0.015

~~r~E)D~ya=

~"

/

300 K m=lO0

A

o,~

I

3h2/(mkBOD4~2)~

amu

0.01

~

Static Disorder

0.005 0

I

0

FIG. 5.

0.5

I

1"/0

I

1.5

ebye

Qualitative effects of static disorder

on

Uiso vs T.

-

12

BRIAN

C.

SALES

ET AL.

0.02

0.015

" ' ~ 0.01 .~

4

12

~v

o.oo5 [-

-./

i

0

50

.I".

,-

100

150

200

250

T(K)

300

350

FIG. 6. Atomic displacement parameters versus temperature for LaFe4Sba2, CeFe4Sb12, and YbFe4Sb12. For clarity only the rare earth ADP values are shown. The lines shown are least squares fits to the data (Sales et al., 1998).

diffraction, static disorder has only been significant for alloys in which the crystallographic sites are only partially occupied. For the stoichiometric c o m p o u n d s s u c h as L a F e 4 S b 1 2 , C e F e 4 S b 1 2 , Y b F e 4 S b ~ 2 , T12SnTe 5, a n d T l z G e T e s, t h e t e m p e r a t u r e - d e p e n d e n t A D P v a l u e s all e x t r a p o l a t e close to t h e origin. T h i s is i l l u s t r a t e d in Figs. 6 - 8 . In s o m e s t o c h i o m e t r i c c o m p o u n d s

0.06

1

0.05

;

'

I

"i

i

TI(1)

TI2SnT

..,..

0.04 o,I

0,=I: ~.' o

0.03

TI(2) -~

~

0.02

Te(2)| T(nl)

0.01

0

50

I

!

100

150

1

200

T(K)

L

i

250

300

350

400

FIG. 7. Atomic displacement parameters vs temperature for T12SnTes as determined from powder neutron diffraction. Note the larger ADP values for the T1 at site 1. These T1 atoms rattle in an oversized atomic cage. The lines shown are a least squares fit to the data (Sales et al., 1999).

1

0.05

13

ATOMIC DISPLACEMENT PARAMETERS

i

!

TI G e T e

TI(1)

2

0.04

i

-1

0.03

"<: 0

TI(2)

0.02

Te(1-4)

0.01

~

~i

Ge

i

o1! 0

!

50

100

I

150

I

1

200

T (K)

250

300

1

350

400

FIG. 8. Atomic displacement parameters vs temperature for T12GeTe 5. Note the larger ADP values for the T1 atoms at site 1. The lines shown are a least squares fit to the data (Sales et al., 1998).

(e.g., SrsGax6Ge30 ) a large amount of static disorder appears to be associated with the ability of one or more of the atom types to be displaced away from the center of the atomic coordination cage. These compounds are discussed in more detail in Section VI.

4.

EINSTEIN AND DEBYE TEMPERATURES FROM ROOM

TEMPERATURE A D P

DATA

For compounds where static disorder can be neglected and the Debye (Einstein) temperatures are less than 600 K (300 K), the foregoing analysis can be scaled into two simple expressions for | and | Oo(K) = 208/(Uiasv (A2)/O.Olmav/lO0) ~/2

(10)

120/( f [ratiso tler/O-" ,,.,.vOl mrattler/ l O0) 1/2,

(11)

and I~)E(K) --

where UiasVo(/~k2) is the weighted average of room temperature values of Uiso for each atom type in the compound given in units of A2, and mav is the average mass of an atom in the compound given in amu. Similarly, the Einstein temperature of the rattler is given by Eq. (11) with U[goT M given in

14

BRIAN C. SALES ET AL.

and the rattler mass given in amu. As shown later, Eqs. (10) and (11) are useful in the rapid screening of new compounds using data extracted from one of the many crystallographic databases. For example the Inorganic Crystal Structure Database (ICSD, produced by FIZ Karlsruhe) currently has more than 50,500 entries, and for most of the compounds the structure is all that is known (i.e., no transport data have been measured).

A2

IV.

Ciathratelike Thermoelectric Compounds

Room temperature ADP information can be used to estimate the Debye temperature and an average velocity of sound for any compound by Eqs. (10) and (5). This result is well known to crystallographers. The only restriction is that the compound should have a small amount of static disorder so that the ADP values correspond to dynamic motion rather than a static displacement of the atoms. There is a large class of promising thermoelectric compounds that contain open cages or voids in their crystal structures into which guest atoms can be added. If the guest atom is small relative to the size of the cage, the atom will be weakly bonded to the atoms comprising the cage. These guest atoms are referred to as "rattlers," since at a given temperature these atoms tend to vibrate about their equilibrium positions substantially more than the other atom types in the structure. Filling the open cages with rattlers rapidly reduces the lattice thermal conductivity (see Section V), which is desirable for a good thermoelectric material. Slack (1995) has proposed that the ideal thermoelectric material with the good electrical properties of a crystal but the poor heat conduction of a glass may be produced in these types of compounds. A clathrate compound is an inclusion complex in which molecules or atoms of one substance are completely enclosed within another compound. Ice that has trapped an inert gas such as argon can form a new crystalline structure called an ice clathrate, in which the argon resides at the center of a large cage of water molecules. In analogy with these clathrate compounds, thermoelectric compounds with weakly bound atoms (rattlers) will be referred to as "clathratelike" compounds because as a first approximation, the rattler atoms and the cage framework atoms will be treated as separate phases. Examples of clathratelike compounds are the filled skutterudites (e.g., LaFe4Sb12 ), some ternary tellurides (e.g., TlzSnTe5), semiconducting compounds with the ice clathrate structure (e.g., SrsGax6Ge3o ), and the rare earth hexaborides (e.g., LAB6). In all of these compounds, one of the atom types has a room temperature ADP value that is at least three to 10 times larger than that of the other atom types in the structure with comparable

1

15

ATOMIC DISPLACEMENT PARAMETERS

,COD) (0

k Debye

CO

k +

Einstein

k +

Hybridization

FIG. 9. Schematic illustration of the dispersion behavior of a Debye solid and a Debye solid with a localized Einstein mode. The qualitative effects on the dispersion curves of the interaction between the Einstein mode and the acoustic phonons of the solid are also sketched.

masses. For example, in the filled skutterudite LaFe4Sb12, the La ADP value is 0.0165 A 2, whereas the Sb ADP value is only 0.004 A 2 (Braun and Jeitschko, 1980). As a first approximation, clathratelike compounds are separated into two phases consisting of framework atoms and rattling atoms. Since the rattling atoms are only weakly coupled to the framework atoms, they are more appropriately treated as individual quantum harmonic oscillators (Einstein oscillators). The remaining framework atoms are treated within the Debye model. In this approximation the entire solid is composed of an Einstein mode in a Debye host solid. In a real solid, the Einstein mode will interact with the acoustic phonons of the Debye host (see Fig. 9), but it is suggested that an Einstein mode in a Debye host is a much better starting point for understanding clathrate-like compounds than just a Debye solid. The ultimate justification for this approximation, however, is that it results in predictions that are in good agreement with experiment. For a clathratelike solid, an Einstein temperature for the rattler can be determined from the ADP data. If static disorder can be neglected, the Einstein temperature can be estimated from the room temperature ADP value using Eq. (11). If static disorder cannot be neglected, the slope of the rattler ADP data vs temperature can be used to estimate an Einstein temperature (the slope will be given by hZ/(4rcZmkB~2), as in Eq. (3)). Within this same approximation, the heat capacity of a clathratelike compound will have a Debye contribution from the framework atoms and an Einstein contribution from the rattlers. At high temperatures both the Einstein and Debye models for the heat capacity Cv (at constant volume) approach the classical Dulong and Petit value of 3R or 24.93 J/deg-mol of atoms. At low temperatures, however, the Debye heat capacity decreases as T 3, while the Einstein heat capacity has a exponential decrease with temperature. The total molar heat capacity of a clathratelike compound

16

BRIAN C. SALES ET AL.

should approximately be given by cClathrate(T)

---

fCDebye(T) + (1 -- f)CEinstein(T)

(12)

where f is the fraction of framework atoms and ( 1 - f ) is the fraction of rattling atoms, and

CDebye(Z) =

x4 e x d x - (e x - - 1) 2

9Nak B

(13)

and e@E/T

(14)

:

(eOE/r_ 1)2.

As an example of this analysis, the ADP data from a partially filled skutterudite compound, T l o . z z C o 4 S b 1 2 is shown in Fig. 10 (Sales et al., 2000). In this compound, 22% of the available voids in the skutterudite structure are filled with thallium atoms. This is the maximum fraction of the voids that can be filled without compensating for the T1 charge by substitution on the Co or Sb sites. The slope of the T1 ADP data yields an Einstein temperature for the T1 atoms of 53 K. Heat capacity measurements o n T l o . z z C o 4 S b 1 2 and Co4Sb12 were made using a commercial instrument from Quantum Design. The heat capacity difference between these two

0.05 TI

0.04 t 0.03 o,~ ""o :~'- 0.02

Co

0.22

i

Sb

T 1

12

TI

,7

A

0.01

4

9

I

f'

=

Co,Sbi..

50

100

-=

!

[

150 200 T (K)

I

250

1

300

350

FIG. 10. Isotropic atomic displacement parameters versus temperature for a T10.22Co4Sb12 alloy. An Einstein temperature for the T1 atoms of 52 K was estimated from the slope of the T1 ADP data (Sales et al., 2000).

1

Tio.22C 04Sbl 2"C ~ 2 , ,, i I ! Cx2eX/(eX.1)2 C= 0.22~ = 5.5 j o u l e s / m o l e - K

4.0

A

CEinstein=

e-

3.0

n~

17

ATOMIC DISPLACEMENT PARAMETERS

x=

OE/T

E L._

O

2.0

u

O

E

(q 55 K "E" | |

1.0 -

E Q,

O

0.0

__-

--"i'nl

0

5

99

9

I

I

I

I

10

15

T(K)

20

FIG. 11. Difference in heat capacity between Tl-doped Co4Sbx2 and Co4Sb12. The T1 contribution to the heat capacity is accurately described by an Einstein contribution with an Einstein temperature of 55 K. Squares are measured data; circles are calculated.

compounds gives the contribution due to the T1 atoms. The T1 contribution is accurately described by an Einstein heat capacity with an Einstein temperature of 55 -t- 2 K (Fig. 11) and an amplitude of 0.22 3R. These results are in excellent agreement with Eq. (12) and the value of 53 K predicted by an analysis of the ADP data in Fig. 10. T12SnTe 5 is a tetragonal compound with a thermoelectric figure of merit, ZT, of about 0.6 at 300 K (Sharp et al., 1999). The ADPs for this compound from powder neutron diffraction data (Fig. 7) and from single crystal X-ray diffraction data (Agafonov et al., 1991) indicate that the T1 atoms at the center of a distorted cube of Te atoms have an unusually large displacement relative to the other atom types in the structure. Both the X-ray and neutron data ADP data give an Einstein temperature for these T1 atoms of 37 _+ 1 K. The estimated Debye temperatures for the framework atoms (all other atom types in the structure) are 169 K from single crystal, room temperature X-ray data (Agafonov et al., 1991) and 125 K from powder neutron data. This difference in estimated Debye temperatures from X-ray and neutron measurements is the largest percentage difference (30%) among the various thermoelectric compounds that we have investigated. Normally there is less than a 10% difference in the Debye temperatures calculated from X-ray single crystal vs neutron ADP data. If just the single crystal X-ray ADP data are used, the temperature of the specific heat should be approximately given by Eq. (12) with f = 7 (since only one out of eight atoms in T12SnTe 5 is treated as a rattler), and a Debye temperature of 169 K and an Einstein

18

BRIAN C. SALES ET AL.

30

r

E

20

~

15

o

E 0

10

TI2SnTe s 1

Predicted_

25

o

i

~, ~[

[Measured Data,"

i

"g"

~

I

0

50

<>

,

,

1t

~

/ / . . ';

5 l-ll

i

Measured 2

o D'

f

9

9

,

, 0 --I 5 T
100

150

T(K)

200

38 K

-] -

/

-

!I

I 15 250

4

t

300

FIG. 12. Calculated and measured heat capacity for TlzSnT %. The calculated heat capacity was determined from the published room temperature X-ray crystallographic data of Agafonov e t al. (1991). There were no adjustable parameters. The inset shows the calculated and measured heat capacity. At low temperatures, an Einstein temperature of 30 K provides a better description of the data than does the value of 38 K calculated from the room temperature X-ray data.

temperature of 38 K. There are no adjustable parameters. A comparison between the measured and predicted heat capacity is shown in Fig. 12. The agreement is surprisingly good. At temperatures below 15 K (Fig. 12 inset), however, the data indicate that an Einstein temperature of about 30 K provides a better description of the heat capacity data than does the value of 38 K determined from the room temperature ADP data.

V. 1.

Estimation of the Lattice Thermal Conductivity from ADP Data

ELEMENTARY THEORY OF LATTICE HEAT CONDUCTION

The simplest expression for the lattice thermal conductivity of a solid is given by an expression adapted from the kinetic theory of gases (see, for example, Kittel, 1968), Klattice : 1Cv Usd

(15)

1

ATOMIC DISPLACEMENT PARAMETERS

19

where Cv is the heat capacity per unit volume, vs is the velocity of sound, and d is the mean free path of the heat carrying phonons. In a more realistic treatment of lattice thermal conductivity, which is discussed in the next section, the mean free path (or relaxation time) and heat capacity depend on frequency and temperature, but for the present analysis Cv depends only on temperature, whereas vs and d are treated as scalars. In Section II it was shown how room temperature ADP data can be used to estimate the Debye temperature and the Debye velocity of sound. This analysis, which can be applied to all compounds, provides an estimate of the room temperature values of Cv and Vs that appear in Eq. (15). To estimate the lattice thermal conductivity using Eq. (15), however, requires a value for d. There have been several phenomenological expressions derived in the literature that relate the lattice thermal conductivity to the Debye temperature and the Grfineisen parameter 7 (or to the melting temperature and the Griineisen parameter). Many of these are discussed by Goldsmid (1986, pp. 73-76). Most of the formulas are related to one another through various thermodynamic relations. A typical phenomenological expression for the lattice thermal conductivity near room temperature is

2t = 8 x 10 -s

MV'/30~/(72T)

(16)

where M is the average mass of an atom (g), V is the average atomic volume (cm3), and a typical value for the Griineisen parameter is 1.8. Since all of the quantities in (16) can be obtained from the crystallography data, equations such as Eq. (16) can provide a first estimate of the thermal conductivity of any material. The estimates may be off, however, by as much a factor of 5-10. For example, if the thermal conductivity of the filled and unfilled skutterudites are analyzed using Eq. (16), both the filled and unfilled materials have about the same average mass, atomic volume, and Debye temperatures, so that the room temperature lattice thermal conductivity of Co48b12 should be about the same as that of LaFe4Sb12. The room temperature lattice thermal conductivity of LaFe4Sbx2, however, is 5 to 6 times lower than that of Co48b12. In general there is no easy way to estimate the value of d at room temperature using just crystallographic data. Hence, for most compounds there is no obvious way to estimate the lattice thermal conductivity to better than about a factor of 5. For clathratelike compounds, it has been experimentally observed by several groups (Nolas et al. 1998a; Meisner et al., 1998; Sales et al., 2000) that as relatively small concentrations of rattlers (La, Ce, or T1) are added to the skutterudite structure (Co48b12) there is an extremely rapid decrease in the lattice thermal conductivity. The mean free path, d, of the heat

20

BRIAN C. SALES ET AL.

0.06 A

0.05

E

0.04

i

0 . 0 1 + 0 . 0 4 2 5 ox ~/3

m

= .J

0.02 0.01

!

I---

0.03

0

I

m

L

0

TIxC ~

b12-ySn y & TIxC ~

I.

[

0.2

0.4

X

b12

I

1

0.6

0.8

1

FIG. 13. Variation of the room temperature lattice thermal resistivity vs the fraction of the voids filled with the rattler T1. The average separation distance between T1 rattlers varies as x- 1/3.The square (circles) refer to charge compensation with Sn(Fe). For all but the two lowest T1 concentrations, x ~ y.

carrying phonons in these compounds is determined by the various scattering mechanisms in the crystal such as acoustic phonons, grain boundaries, electron-phonon scattering, static defects, voids and "rattlers." Resonant scattering by quasi-localized "rattlers" appears to be the dominant scattering mechanism responsible for the rapid decrease in the thermal conductivity as small amounts of T1, La, or Ce are placed in the voids. This mechanism is believed to be similar to the resonant scattering described by Pohl for insulating crystals (Pohl, 1962) and by Zakrzewski and White (1992) in insulating organic clathrates. It has been demonstrated that mass fluctuation scattering is much too weak to explain the rapid decrease in thermal conductivity (Nolas et al., 1998a). The thermal resistivity of the lattice (I/thermal conductivity) at room temperature is shown in Fig. 13 as T1 is added to the voids in Co4Sb 12. Thermal resistivity is shown rather than thermal conductivity because as a first approximation, the scattering rates for different scattering processes should add (Mathiesson's rule). There is a rapid initial increase in the thermal resistance, followed by a gradual saturation of the thermal resistance as higher concentrations of T1 are added to the voids. Within experimental error, there is no clear maximum in the thermal resistance data as a function of T1 concentration, and the maximum thermal resistance occurs near complete filling for both the Fe and Sn compensated compounds. The maximum attributed to mass fluctuation scattering by Meisner et al. (1998) as a function of Ce filling is not observed in the present experiments.

1 ATOMICDISPLACEMENTPARAMETERS

21

If the T1 atoms are treated as localized Einstein oscillators, as suggested by Keppens et al. (1998), then the heat carrying phonon mean free path, d, should be a function of the distance between the T1 atoms in the crystal. The simplest estimate of the phonon mean free path is therefore the average distance between the T1 atoms. This implies that the phonon scattering from the T1 is so strong that d attains a minimum distance given by the average T1-T1 separation. The scattering of acoustic phonons by the T1 atoms should be a maximum when the acoustic phonon and rattling frequency are equal (Pohl, 1962); however, even at resonance it seems physically unlikely that d could be less than the T1-T1 separation distance. This simple argument suggests that if the role of other scattering mechanisms is minimal, the thermal resistivity should vary as x 1/3, where x is the T1 concentration (the average spacing between T1 atoms varies as x-I/3). The additional thermal resistance generated as T1 is added to Co48b12 reasonably follows a n x 1/3 behavior (Fig. 13), even though part of the thermal resistance is due to electron-phonon scattering and other scattering mechanisms (Sales et al., 2000). A plausible approximation for d in clathratelike compounds is therefore the average separation distance between the rattlers, which is known from the crystallography data. At room temperature, this argument works well for the Tl-filled skutterudites (Fig. 13). It also works for the filled skutterudites such as LaFe4Sb~2. Using the measured thermal conductivity, heat capacity, and an average value for the velocity of sound yields a mean free path of d = 9 A (Sales et al., 1997). The nearest-neighbor distance of the La atoms in LaFe4Sb~2 is 7.9 A. The real test of this hypothesis is whether this analysis gives good estimates of the room temperature lattice thermal conductivity for a variety of clathratelike compounds. As shown in Section VI, for many clathratelike systems replacing d in Eq. (15) by the average distance between the rattlers predicts a room temperature lattice thermal conductivity in suprisingly good agreement with experiment. This means that for clathratelike compounds room temperature crystallography data can be used to provide a reasonable estimate of the lattice thermal conductivity.

This is significant since a low lattice thermal conductivity is a requirement for a good thermoelectric material.

2.

LATTICEHEAT CONDUCTION: A MORE REALISTIC MODEL

A more realistic model of lattice heat conduction of a solid, within the Debye approximation, has been described by Callaway (1959) and Klemens (1958). The lattice thermal conductivity, K~attice, is given by

/~lattice =

1/3 ~o~ v2z(og,T) ~dC dco

(17)

22

BRIAN C. SALES ET AL.

with z-1(o9, T) = ~ z71(o9, T),

(18)

i

where coD is the Debye frequency, v is the Debye velocity of sound, zi is the relaxation time for the ith phonon scattering mechanism, T is the temperature in kelvins, and dC/dco is the specific heat per angular frequency. Within the Debye model, the specific heat per angular frequency is obtained from Eq. (13) by a change of variables, replacing x by hco/2nk B T. Notice that since v is a constant in the Debye model, the phonon mean free path is just d(co, T) = vz(co, T).

(19)

In a crystal various processes can scatter phonons. For the present purposes we will follow the approach described by Pohl (1962) and Walker and Pohl (1963) and consider the minimum number of scattering mechanisms that can account for the experimental data. For a solid with no resonant scattering (no rattlers), the normal scattering mechanisms are grain boundary scattering, zff 1, isotope or mass fluctuation scattering, ris-o~ (Kle-1 mens, 1958), and phonon-phonon scattering (umklapp and normal), Zv,N (Walker and Pohl, 1963). These various scattering terms are given by .CB 1 = v / L

(20)

r,is~ = VoFco4/4n = Cco 4

(21)

ZU,N-1 = B c o 2 T E - b / T .

(23)

with

and

Vo is the atomic volume, L is the average grain size, m i is the mass of the ith atom, m is the average atomic mass, and f/is the relative concentration of the ith species. Equation (23) is a phenomenological expression that accounts for both umklapp and normal phonon-phonon scattering (Walker and Pohl, 1963). These three scattering processes (Eqs. (20)-(23)) can account for the temperature dependence of the thermal conductivity of compounds with no rattlers, such as Co4Sb12. The resonant scattering by the rattlers can be phenomenologically described by a function that is

1

23

ATOMIC DISPLACEMENT PARAMETERS

proportional to the concentration of rattlers and is peaked at the Einstein frequency of the rattler,

T~eLnant(O.)) =

Aof (Co- O)E)= Ao T2

092 _

'

(24)

where A o is proportional to the rattler concentration and the particular form of the function f is taken from Walker and Pohl (1963). With regard to the clathratelike compounds, simple calculations using only Eqs. (17)-(24) do not reproduce the behavior shown in Fig. 13. The calculated thermal resistance increases linearly with the rattler concentration A o and does not saturate as is indicated in Fig. 13. One of the key ideas to understanding the data displayed in Fig. 13 is the concept of a minimum thermal conductivity or a maximum thermal resistance first proposed by Slack in 1979. In any solid, Slack proposed that it does not make sense to consider a mean free path for the heat-carrying phonons that is less than an interatomic spacing (d ~ 3 A). This hypothesis, which is born out by experiment (Cahill et al., 1992), suggests that for a crystalline compound the minimum thermal conductivity corresponds to a glass with the same composition for which d ~ 3 A = 3 x 10-8 cm. This means that there is a cutoff for the maximum scattering rate given by rma~x= V/3 X 10 .8 S-~, where the velocity of sound is given in cm/s. For materials with thermal conductivities that approach within an order of magnitude or so of the minimum value, Eq. (18) is replaced by

d(co, T) =

vr~-1(o),T)

+ dmi n

(25)

where dmi n ~ 3 A. To see if this approach produces reasonable results, the temperature dependence of thermal conductivity of the unfilled skutterudite C o 4 S b 12 ( n o rattlers) was fit using Eqs. (17) and (25). The measured Debye temperature (307 K), velocity of sound (2.93 x 105 cm/s) and grain size (10 -3 cm) were used as input parameters. The three constants (C, B, and b) in Eqs. (21) and (23) were adjusted to give a fit to the experimental data (Fig. 14). The values used were C = 1.58 x 10 -42 s 3, B - 3.87 x 10 -18 s/K, and b = 150K. (These values are in the same range as found by Walker and Pohl, 1963.) The resonant phonon scattering due to the addition of T1 rattlers was then modeled using the same parameters used to fit the Co4Sbx2 data plus a resonant term (Eq. (24)) with A o = A 7.42 x 1032 s - 3 K -2, where A is dimensionless and is proportional to the concentration of T1 rattlers. An Einstein frequency was taken from the experimental data (Fig. 11) and corresponded to an Einstein temperature of 55 K. The effect on the thermal

24

BRIAN C. SALES ET AL. 0.35 0.30

.... I

-

I

I

I

;~%

"

I

I

[

C~

0.25 0.20 ~,

0.15 0.10 0.05 0.00

9

0

I

I

I

I

!

I

50

100

150

200

250

300

350

T(K) FIG. 14. Fit to the lattice thermal conductivity of a polycrystalline CoSb 3 sample with an average grain size of 0.001 cm using Eqs. (17), (20)-(23), and (25). See text for details.

I'

I

i

s

w'l

v ~

i

a

I

!

I

J I 1 |

I

I

..... ~ ' r " r ~ q ~

t

I

i

!

s (,.

A=

0.1

0

0.01

....: ~ "..".-" 9

"

u

,

5-50

~

100-500

.... ":"

..I

0.001

resonant--

f(co-O~o)= T2J/(o~2-COo2)2 0.0001

.,, 1

;

,

~

;

,

I,II

,

!

I0

,

,

i

.lff

!

1oo

I

l

,

J

,

,,

1o00

T(K) FIG. 15. Log of lattice thermal conductivity vs log T calculated using Eqs. (17) and (20)-(25). The strength of the resonant scattering and the concentration of rattlers is proportional to the parameter A (Eq. (24)). The other parameters are the same as used to fit the Co4Sb12 data. (The triangles on the A = 0 curve are the same Co4Sb12 data shown in Fig 14.) See text for details.

1

ATOMICDISPLACEMENT PARAMETERS

25

conductivity of increasing the strength of the resonant scattering is shown in Fig. 15. In this simple model there is expected to be a dip in the thermal conductivity at a temperature between 10 and 20 K for an Einstein temperature of 55 K. Careful thermal conductivity measurements in this temperature range have not been made for the T1 filled skutterudites. A dip has been seen, however, in a related compound, SrsGa 16Ge3o (Cohn et al., 1999). The motivation, however, for calculating the lattice thermal conductivity as a function of T1 concentration was to see if the behavior in Fig. 13 could be understood. The room temperature thermal resistance from the calculated data shown in Fig. 15 is plotted in Fig. 16 vs the amplitude of the resonant scattering (the parameter A). A least squares fit of a power law to the data yield an exponent of 0.35. This exponent is relatively constant for large variations in A and is surprisingly close to the value of 89obtained by simply using the average distance between rattlers for d (Subsection 1 of this section). The calculated and measured thermal resistivity data can be compared directly if the proportionality constant between A and the T1 concentration x is determined. The room temperature thermal resistance of the Tlo.sCo4SnSb11 sample is about 50 cm-K/W. This value is obtained in the model calculation with a value of A ,~ 20. A comparison between the calculated and measured thermal resistance is shown in Fig. 17. The agreement is good.

VI. Examples In this section, ADP data are used to extract as much information as possible about the properties of several different families of clathratelike compounds. Because the vast majority of crystallographic data reported in the literature are taken with X-rays, where possible X-ray ADP data are used in the analysis. The predicted results from the analysis of the ADP data are compared to the results from a variety of different measurements.

1.

FILLEDSKUTTERUDITES: LaFe4Sb12 AND YbFe4Sbx2

Single crystal X-ray data are available for LaFe4Sb12 (Braun and Jeitschko, 1980) and YbFe4Sb12 (Leithe-Jasper et al., 1999). The room temperature X-ray data from each compound were analyzed using Eqs. (10) and (11). The estimated Debye temperatures were 299 K for LaFe4Sb12 and 238 K for YbFe4Sb12. The corresponding Einstein temperatures were 79 K for La and 62 K for Yb. Room temperature velocity of sound data (Sales et al., 1997) and low temperature heat capacity data (Gajewski et al., 1998) give Debye temperatures for LaFe4Sb~2 of 300 +_ 10 K. Low-temperature

26

BRIAN C. SALES ET AL.

160 ,~"~

140

ca

100

E 19

40

19:3

!

20 0

....

I ....

0

100

~ ....

I ....

200

i ....

300

I .....

400

500

600

R e s o n a n t Scattering A m p l i t u d e (proportional to concentration of "rattlers") FIG. 16. Calculated thermal resistance at room temperature vs the resonant scattering amplitude, A. The room temperature thermal resistance is from the calculated data shown in Fig. 15. A is proportional to the concentration of rattlers, x.

60

I

50

_

40

_

Model

I

Calculation

,

I

I

9

9

.\ Experimental

30

Data

o _J

20

i Ti x C 0 4 S b l = . y S n y & T I x C O 4 y F e y S b 12

10

0

.,

0

I

1

!

I

0.2

0.4

0.6

0.8

TI Concentration

,

1

(x)

FIG. 17. Thermal resistivity vs T1 concentration, a comparison between the model calculation and the experimental data. The thermal resistance of the Tlo.8Co4SnSb11 sample was used to determine the proportionality constant between the resonant scattering amplitude, A, and the T1 concentration x.

1 ~"

2.0

'

C :3

,4 '-

i

LaFe4Sb12-CeFe4Sb12 I

1.5

"O C O

=

0.5

0.0

,.Q >

-J

I

i"

tIi'ttlt'ii]'

0 o..

0)

i

.i

1.0

e-

27

ATOMIC DISPLACEMENT PARAMETERS

-0.5

? 1

0

5

165K _

t

I

10

15

-

.

._,'

I

,,,,.::Iii:.II 20

Energy Transfer (meV) FIG. 18. Difference in the inelastic neutron scattering data between LaFe4Sb12 and CeFe4Sb12 vs energy loss. The incident neutron energy was 30 meV and the energy resolution was 2 meV. CeFe4Sb12 was used as a reference because the neutron scattering cross section of Ce is much smaller than that of La. The difference spectra therefore reflects the vibrational density of states (DOS) associated with the La atoms. The peaks at 7 and 15 meV correspond to temperatures of 80 and 175 K (Keppens et al., 1998).

heat capacity data on YbFe4Sb12 (Dilley et al., 1998) yielded a Debye temperature of 190 K, although the magnetic contribution to the heat capacity data at low temperatures make this value uncertain. The estimated room temperature lattice thermal conductivity from the room temperature X-ray ADP data using Eq. (15) with d replaced by the rattler separation distance (about 7.9/k for La or Yb) is 0.014 W/cm-K for LaFe4Sb12 and 0.011 W/cm-K for YbFe4Sb12. These values can be compared to the measured values of 0.017 W/cm-K for LaFe4Sb12 (Sales et al., 1997) and 0.014 W/cm-K for YbFe4Sb12 (Dilley et al., 2000). The Einstein temperature for the La in LaFe4Sb12 should result in a peak in the La phonon density of states near 79 K. The La phonon density of states has been measured using inelastic neutron scattering (Keppens et al., 1998). A clear peak in the La phonon density of states was observed at 80 K along with a weaker and broader peak near 175 K (Fig. 18). Both peaks can be understood within the framework of a detailed investigation of the lattice dynamics of LaFe4Sb12 (Feldman et al., 2000; Feldman and Singh, 1996). Qualitatively the two peaks result from a hybridization process similar to that sketched in Fig. 9. A summary of ADP information from a variety of filled skutterudite phases has been reported by Kaiser and Jeitschko (1999) and Chakoumakos et al. (1999).

28

BRIAN C. SALES ET AL.

0.014

"

I

'

I '"

1

r _

j

l

0.012

_J

0.01 0.008

~i

0"006 0.004 0.002

_ , 0

I SnTe 2 5 t

I

i

I

50

100

150

200

T(K)

I

!

250

300

350

FIG. 19. Lattice thermal conductivity vs temperature for vitreous silica, T12SnTe5, and TI2GeTe5. The Wiedemann-Franz law has been used to estimate and subtract the electronic portion of the thermal conductivity. The lines through the data are guides to the eye.

2.

T12SnTe 5

The structure of T12SnTe 5 was first reported by Agafonov et al. in 1991. This compound has a room temperature value for Z T of 0.6 (Sharp et al., 1999). The compound is tetragonal with columns of T1 ions along the crystallographic c axis. There are two distinct T1 sites in the structure, and at one of the sites the T1 atoms sit near the center of a large, oversized distorted cube. The T1 ADP parameter at this site is considerably larger than for the other atoms in the structure and the T1 at this site will be treated as a rattler. From the room temperature X-ray ADP data (Agafonov et al., 1991) the Einstein temperature of the T1 rattler was estimated to be 38 K and the Debye temperature of the other atoms in the compound was 169 K. As was shown in Section IV, these values can be used to estimate the temperature dependence of the heat capacity, and the calculated heat capacity is in good agreement with the measured values (Fig. 12). The estimate of the room temperature lattice thermal condutivity from the ADP data is 0.0039 W/cm-K. The measured lattice thermal conductivity for TlzSnTe 5 is shown in Fig. 19 and at room temperature is close to 0.004 W/cm-K.

3.

LaB 6

LaB 6 is not a good thermoelectric material, but it is a clathratelike compound. LaB 6 crystallizes in a simple body-centered cubic structure with

1

ATOMIC DISPLACEMENT PARAMETERS

29

FIG. 20. Model of LaB 6 crystal structure. La atoms (large ball) sit at the center of a cube with B 6 "molecules" at each cube corner. As a first approximation, the La atoms can be treated as Einstein oscillators in a Debye solid composed of B atoms.

La at the cube center and B 6 octahedral clusters at each cube corner (Fig. 20). LaB 6 is a good metal (P3oo K = 5 #~-cm) that is used as an electron source in most high-performance electron microscopes. Because of its technological importance, much is known about the properties of LaB 6. When normalized by mass, the lanthanum ions "rattle" significantly more about their equilibrium positions than do the boron atoms. Using published ADP data (Korsukova et al., 1986) on LaB 6, the Einstein temperature of the La was calculated to be 140 K and the Debye temperature for the boron sublattice to be about 1500 +_ 200 K. The extrapolation graph shown in Fig. 4 was used because of the high Debye temperature of the boron sublattice. The temperature dependence of the heat capacity was calculated using an Einstein contribution weighted by -~ and a Debye contribution weighted by 6. The Debye temperature was adjusted to 1200 K (rather than 1500 K) to provide a better fit to the data. The agreement between the calculated heat capacity and the measured values is shown in Fig. 21. The unusual bump in the heat capacity data at about 70 K is accurately accounted for by this simple analysis using only room temperature ADP data and a 20% adjustment of the Debye temperature for the boron sublattice. Large single crystals of LaX~B6 allowed Smith et al. (1985) to map the phonon dispersion curves using neutron scattering. An unusually energy-independent phonon mode was found over most of the Brillioun zone at energies corresponding to an Einstein temperature of 150 K; a value close to the 140 K value estimated from the ADP data. This mode was attributed by Smith et al. (1985) to the independent vibration of the La atoms in LaB 6.

30

BRIAN C. SALES ET AL.

I

14 u)

E 0

12

8

m

0

E I

I

I

.i"

6

(9

D

D

.I,t"

6

11, '=

11

= 1200 K

_

i

B II

= 140 K

|

10

LaB

it#

ii w

i,"b

4

0 0

~ 0

50

100

150

i

I

200

250

300

T(K)

FIG. 21. Calculated and measured heat capacity of LaB 6. Room temperature X-ray ADP data from Korsukova et al. (1986) were used to determine the Einstein temperature for the La and a Debye temperature for the B. (See text for details.)

4.

SEMICONDUCTING CLATHRATES: S r s G a 1 6 G e 3 0 AND B a s G a x 6 G e 3 0

These compounds are cubic and have the same structure as the type I ice clathrates (Eisenmann et al., 1986). The thermoelectric properties of SrsGa16Ge3o and similar compounds were first reported by Nolas et al. (1998b) and Cohn et al. (1999). The structure can be thought of as a tetrahedral framework of Ga and Ge atoms. The framework atoms form large cages of 20 or 24 atoms that surround the Sr or Ba atoms in the structure. The alkaline earth atoms in the larger cage have unusually large ADP values. Room temperature X-ray ADP data for the Ba clathrate (Eisenmann et al., 1986) coupled with Eqs. (10) and (11) estimates an Einstein temperature of 51 K for the Ba atoms, and a Debye temperature of 274 K for the rest of the atoms in the structure. This results in a Debye sound velocity of 2.6 • l0 s cm/s, and a predicted room temperature lattice thermal conductivity of about 0.008 W/cm-K (Eq. (15), with d given by the nearest neighbor distance between the Ba rattlers). The measured lattice thermal conductivity (Fig. 22) is about 0.016 W/cm-K. Although the measured lattice thermal conductivity is about twice the predicted value, this is still better agreement than most other simple methods of estimating lattice thermal conductivity. In general it should be expected that the estimate will be lower than the measured value, since it is assumed in this simple analysis that the rattlers scatter phonons at the maximum possible rate (corresponding to d given by the rattler separation distance).

1 ATOMICDISPLACEMENTPARAMETERS

31

0.03 ~.

0.025

O U :,=

~

.

0.02 0.015

SraGa~ 6Ge3 ~

'"

_

.,~

-

O.Ol 0.005

0

50

100

150

200

250

T(K)

FIG. 22. Lattice thermal conductivity vs temperature for SraGa16Ge30 and BasGa16Ge3o crystals. The thermal conductivity of the Ba clathrate, although small, has a crystalline temperature dependence, whereas the temperature dependence of the Sr clathrate is glasslike.

There are no published single crystal X-ray ADP data for the Sr clathrate, so the neutron data reported by Chakoumakos et al. (2000) are used. ADP data for this compound are available from room temperature down to 11 K (Fig. 23). The ADP values of Sr atoms in the large cage (Sr2 in the figure) are huge relative to the other atoms in the structure. The temperature dependence of the Sr2 ADP data is unusually weak, however, suggesting a large amount of static disorder. Such a large amount of static disorder in a nominally stochiometric compound is unusual, as can be seen from comparing the ADP data in Figs. 6-8 with the data shown in Fig. 23. If just the room temperature ADP data are used, the Sr rattlers appear to have at, Einstein temperature of 44 K and a Debye temperature for all the atoms oi 180 K (without the rattler, this value would be 246 K). For this compound, however, static disorder cannot be i g n o r e d - - a t least for the Sr2 site. Takinv. the slopes of the ADP data results in a Einstein temperature of 85 K for the Sr rattler, and a Debye temperature of about 270 K and a mean sound velocity of 2.6 x l0 s cm/s. The estimated room temperature lattice thermal conductivity is 0.008 W/cm-K. If only the room temperature ADP data were used, the estimated lattice thermal conductivity would be lowered to about 0.006 W/cm-K. The measured value of the room temperature lattice thermal conductivity of Sr8Ga16Ge30 is about 0.010 W/cm-K (Fig. 22; also see Nolas et al. 1998b; Cohn et al., 1999). The large amount of static disorder at the Sr2 site is due to the tendency of the Sr atoms at this site to move or tunnel off center to one of four nearby sites located about 0.36 A from the center of the cage. The combination of tunneling, in addition to rattling, is

32

BRIAN C. SALES ET AL.

0.10

9

'

'

'

I

"

"

""

"

i

"

"

"

"

!

"

"

"'

'

I

"

"

"

"

I

"

"

'

"

SrsGal6Ge3o 0.08

~

0.06

0.04

0.02

0.00 0

50

100

150

200

250

300

T (K) FIG. 23. Temperature dependence of the isotropic atomic displacement parameters for SrsGa16Ge3o. Note the large and weak temperature dependence of the ADP values for Sr at site 2 in the structure. See text for details (Chakoumakos et al., 2000).

apparently responsible for the qualitative difference between the thermal conductivity of the Sr and Ba clathrates. The rattling of the Ba results in a low thermal conductivity, but the temperature dependence is crystal-like, whereas the rattling and tunneling of the Sr results in a true glasslike thermal conductivity (Fig. 22; see also Cohn et al., 1999; Keppens et al., 2000). 5.

CeRuGe 3

To further test some of the simple ideas discussed in this chapter, room temperature A D P information was used to identify a compound that should have a very low lattice thermal conductivity. Unusually large A D P values for one of the Ce sites were reported by Ghosh et al., (1995) for the cubic compound CeRuGe 3 (space group Pm-3n, a - 9.0061 A). The results of the refinement, which was done using both X-rays and neutrons, suggest that one Ce site contains Ce 3 § ions and the other Ce 4 § ions. A detailed study of the electron density around the various crystallographic sites concluded that the Ce 4 § ions can tunnel or move away from the site center (much like

1 0.02

1

I

CeRuGe ~'

33

ATOMICDISPLACEMENT PARAMETERS I

[

I

3

9&

0.015

L

~r-~-.Tota 9 I

0

0.01 9

Lattice

0.005

i i l i i l ~ l l l I I 9 ~iI

0

0

9

9

i

i

i

I

II

I

m l l l l -

I

9 I.

1

i

I

50

100

150

200

T(K)

,

I

250

300

FIG. 24. Thermal conductivity vs temperature for CeRuGe3. The electronic thermal conductivity was subtracted from the total using the Wiedemann-Franz law.

the Sr ions discussed earlier). Analysis of the room temperature ADP data from this compound gives an Einstein temperature for the Ce 4 § of 23 K and a Debye temperature of 128 K (or 155 K for all of the atoms except the Ce rattlers) and an average sound velocity of 1.37 • 105 cm/s. The distance between the Ce rattlers is 7.9 A, and the lattice thermal conductivity at room temperature is estimated to be 0.008 W/cm-K. The measured lattice thermal conductivity (Fig. 24) is about 0.005 W/cm-K after the electronic contribution has been subtracted using the W i e d e m a n n - F r a n z law. CeRuGe 3 has a low lattice thermal conductivity, but also has a low Seebeck coefficient and hence is not a promising thermoelectric material.

VII. Summary A new structure-property relationship is discussed that links atomic displacement parameters and the lattice thermal conductivity of clathratelike compounds. For many clathratelike compounds in which one of the atom types is weakly bound and "rattles" within its atomic cage, it is demonstrated that room temperature ADP information can be used to estimate the room temperature lattice thermal conductivity, the vibration frequency of the "rattler," and the temperature dependence of the heat capacity. X-ray and neutron diffraction crystallographic data, reported in the literature, are used to apply this analysis to several promising classes of thermoelectric materials.

34

BRIAN C. SALES ET AL. REFERENCES

Agafonov, V., Legendre, B., Rodier, N., Cense, J. M., Dichi, E., and Kra, G. (1991). "Structure of T12SnTes," Acta Cryst. C 47, 850. Allen, P. B., and Feldman, J. L. (1993). "Thermal Conductivity of Disordered Harmonic Solids," Phys. Rev. B. 48, 12581. Anderson, O. L. (1963). "A Simplified Method for Calculating the Debye Temperature from Elastic Constants," J. Phys. Chem. Solids 24, 909. Ashcroft, N. W., and Mermin, N. D. (1976). Solid State Physics. Holt, Rinehart and Winston, New York, Chapter 22. Braun, D. J., and Jeitschko, W. (1980). "Preparation and Structural Investigations of Antimonides with the LaFe4P12 Structure," J. Less Common Metals 72, 147. Burnett, M. N., and Johnson, C. K. (1996). ORTEP-III: Oak Ridge Thermal Ellipsoid Plot Program for Crystal Structure Illustrations. ORNL-6895, Oak Ridge National Laboratory, Tennessee. Cahill, D. G., Watson, S. K., and Pohl, R. O. (1992). "Lower Limit to the Thermal Conductivity of Disordered Crystals," Phys. Rev. B 46, 6131. Callaway, J. (1959). "Model for Lattice Thermal Conductivity at Low Temperatures," Phys. Rev. 113, 1046. Chakoumakos, B. C., Sales, B. C., Mandrus, D., and Keppens, V. (1999). "Disparate Atomic Displacements in Skuterudite-Type LaFe3CoSb~z, a Model for Thermoelectric Behavior," Acta Cryst. B 55, 341. Chakoumakos, B. C., Sales, B. C., Mandrus, D., and Nolas G. S. (2000). "Structural Disorder and Thermal Conductivity of the Semiconducting Clathrate SrsGa~6Ge3o," J. Alloys and Compounds 296, 80. Cohn, J. L., Nolas, G. S., Fessatidis, V., Metcalf, T. H., and Slack, G. A. (1999). "Glasslike Heat Conduction in High-Mobility Crystalline Semiconductors," Phys. Rev. Lett. 82, 779. Dilley, N. R., Freeman, E. J., Bauer, E. D., and Maple, M. B. (1998). "Intermediate Valence in the Filled Skutterudite Compound YbFe4Sb~2," Phys. Rev. B 58, 6287. Dilley, N. R., Bauer, E. D., Maple, M. B., and Sales, B. C. (2000). "Thermoelectric Properties of Chemically Substituted Skutterudites Yb~.Co4Sn~Sb12_x," J. Appl. Phys. 88, 1948. Dunitz, J. D., Schomaker, V., and Trueblood, K. N. (1988). "Interpretation of Atomic Displacement Parameters from Diffraction Studies of Crystals," J. Phys. Chem. 92, 856. Eisenmann, B., Schafer, H., and Zagler, R. (1986). "Die Verbindungen An~ m 1~TM (AI~ = Sr, Ba; " ~8~16~30 Bn~ = A1, Ga; BTM = Si, Ge, Sn) und ihre K~ifigstructuren," J. Less-Common Metals 118, 43. Feldman, J. L., and Singh, D. J. (1996). "Lattice Dynamics of Skutterudites: First-Principles and Model Calculations for CoSb3," Phys. Rev. B. 53, 6273. Feldman J. L., Mazin, I. I., Singh, D. J., Mandrus, D., and Sales, B. C. (2000). "Lattice Dynamics and Reduced Thermal Conductivity of Filled Skutterudites," Phys. Rev. B 61, R9209. Gajewski, D. A., Dilley, N. A., Bauer, Freeman, E. J., Chau, R., Maple, M. B., Mandrus, D., Sales, B. C., and Lacerda, A. H. (1998). 'Heavy Fermion Behavior of the Cerium-Filled Skutterudites CeFe4Sba2 and Ce0.9Fe3CoSbl2, '' J. Phys. Condens. Matter 10, 6973. Ghosh, K., Ramakrishnan, S., Dhar, S. K., Malik, S. K., Chandra, G., Pecharsky, V. K., Gschneidner, K. A. Jr., Hu, Z., and Yelon, W. B. (1995). "Crystal Structures and Low-Temperature Behaviors of the Heavy-Fermion Compounds CeRuGe 3, and Ce3Ru4Ge13 Containing both Trivalent and Tetravalent Cerium," Phys. Rev. B. 52, 7267. Goldsmid, H. J. (1986). Electronic Refrigeration. (Pion Limited, London). Ioffe, A. F. (1957). Semiconductor Thermoelements and Thermoelectric Cooling. Infosearch, London. Kaiser, J. W., and Jeitschko, W. (1999). "The Antimony-Rich Parts of the Ternary Systems Calcium, Strontium, Barium and Cerium with Iron and Antimony; Structure Refinements of the LaFe4Sb12-type Compounds SrFe4Sbx2 and CeFe4Sb12: The New Compounds CaOs4Sbx2 and YbOs4Sbx2," J. Alloys Compounds 291, 66.

1

ATOMIC DISPLACEMENT PARAMETERS

35

Keppens, V., Mandrus, D., Sales, B. C., Chakoumakos, B. C., Dai, P., Coldea, R., Maple, M. B., Gajewski, D. A., Freeman, E. J., and Bennington, S. (1998). "Localized Vibrational Modes in Metallic Solids," Nature 395, 876. Keppens, V., Chakoumakos, B. C., Sales, B. C., and Mandrus, D. (2000). "When Does a Crystal Conduct Heat Like a Glass?" Phil. Mag. Letters, in press. Kisi, E., and Yuxiang, M. (1998). "Debye Temperature, Anharmonic Thermal Motion and Oxygen Non-stoichiometry in Yttria Stabilized Cubic Zirconia," J. Phys. Condens. Mat. 10, 3823. Kittel, C. (1968). Introduction to Solid State Physics, 3rd ed. John Wiley and Sons, New York, pp. 69-70, p. 186. Klemens, P. G. (1958). "Thermal Conductivity and Lattice Vibrational Modes," in Solid State Physics, Vol. 7 (F. Seitz and D. Turnbull, eds.), p. 1. Academic Press, New York. Korsukova, M. M., Gurin, V. N., Lundstrom, and Tergenius, L. E. (1986). "The Structure of High-Temperature Solution Grown LAB6: A Single Crystal Diffractometry Study," J. Less-Common Metals 117, 73. Kuhs, W. F. (1988). "The Anharmonic Temperature Factor in Crystallographic Structure Analysis," Australian J. Phys. 41, 369. Leithe-Jasper, A., Kaczorowski, Rogl, P., Bogner, J., Reissner, M., Steiner, W., Wiesinger, G., and Godart, C. (1999). "Synthesis, Crystal-Structure Determination and Physical Properties of YbFe4Sb~/," Solid State Commun. 109, 395. Mahan, G. D. (1989). "Figure of Merit for Thermoelectrics," J. Appl. Phys. 65, 1578. Mahan, G. D. (1998). "Good Thermoelectrics," in Solid State Physics, Vol. 51 (H. Ehrenreich and F. Spaepen, eds.), pp. 81-157, Academic Press, New York. Meisner, G. P., Morelli, D. T., Hu, S., Yang, J., and Uher, C. (1998). "Structure and Lattice Thermal Conductivity of Fractionally Filled Skutterudites-Solid-Solutions of Fully Filled and Unfilled End Members," Phys. Rev. Lett. 80, 3551. Nolas, G. S., Cohn, J. L., and Slack, G. A. (1998a). "Effect of Partial Void Filling on the Lattice Thermal Conductivity of Skutterudites," Phys. Rev. B 58, 164. Nolas, G. S., Cohn, J. L., Slack, G. A., and Schujman, S. B. (1998b). "Semiconducting Ge Clathrates: Promising Candidates for Thermoelectric Applications," Appl. Phys. Lett. 73, 178. Pohl, R. O. (1962). "Thermal Conductivity and Phonon Resonance Scattering," Phys. Rev. Lett. 8, 481. Sales, B. C., Mandrus, D., and Williams, R. K. (1996). "Filled Skutterudite Antimonides: A New Class of Thermoelectric Materials," Science 272, 1325. Sales, B. C., Mandrus, D., Chakoumakos, B. C., Keppens, V., and Thompson, J. R. (1997). "Filled Skutterudite Antimonides: Electron Crystals and Phonon Glasses," Phys. Rev. B 56, 15081. Sales, B. C., Chakoumakos, B. C., Mandrus, D., Sharp, J. W., Dilley, N. R., and Maple, M. B. (1998). "Atomic Displacement Parameters: A Useful Tool in the Search for New Thermoelectric Materials?," in Thermoelectric Materials 1998-- The Next Generation Materials for Small-Scale Refrigeration and Power Generation Applications, Vol. 545 (T. M. Tritt, M. G. Kanatzidis, G. D. Mahan, and Hylon B. Lyon, Jr., eds.), p. 13. Materials Research Society, Warrendale, PA. Sales, B. C., Chakoumakos, B. C., Mandrus, D., and Sharp, J. W. (1999). "Atomic Displacement Parameters and the Lattice Thermal Conductivity of Clathrate-like Thermoelectric Compounds," J. Solid State Chem. 146, 528. Sales, B. C., Chakoumakos, B. C., and Mandrus, D. (2000). "Thermoelectric Properties of Thallium-Filled Skutterudites," Phys. Rev. B 61, 2475. Sharp, J. W., Sales, B. C., Mandrus, D., and Chakoumakos, B. C. (1999). "Thermoelectric Properties of T12SnTe 5 and T12GeTes," Appl. Phys. Lett. 74, 3794.

36

BRIAN C. SALES ET AL.

Slack, G. A. (1979). "The Thermal Conductivity of Nonmetallic Crystals," in Solid State Physics, Vol. 34 (H. Ehrenreich, F. Seitz, and D. Turnbull, eds.), p. 1. Academic Press, New York.) Slack, G. A. (1995). "New Materials and Performance Limits for Thermoelectric Cooling,," in CRC Handbook of Thermoelectrics (D. M. Rowe, ed.), p. 407. CRC Press, Boca Raton, FL. Smith, H. G., Dolling, G., Kunii, S., Kasaya, M., Liu, B., Takegahara, K., Kasuya, T., and Goto, T. (1985). "Experimental Study of Lattice Dynamics in LaB 6 and YbB6," Solid State Commun. 85, 16. Spitzer, D. P. (1970). "Lattice Thermal Conductivity of Semiconductors: A Chemical Bond Approach," Phys. Chem. Solids 31, 19. Terasaki, I., Sasago, Y., and Uchinokura, K. (1997). "Large Thermoelectric Power in NaCo20 4 Single Crystals," Phys. Rev. B 56, R12685. Trueblood, K. N., Burgi, H.-B., Burzlaff, H., Dunitz, J. D., Gramaacciol, C. M., Schulz, H. H., Shmueli, U., and Abrahams, S. C. (1996). "Atomic Displacement Parameter Nomenclature," Acta Cryst. A 52, 770. Walker, C. T., and Pohl, R. O. (1963). "Phonon Scattering by Point Defects," Phys. Rev. 131, 1433. Willis, B. T. M., and Pryor, A. W. (1975). Thermal Vibrations in Crystallography. Cambridge University Press, London. Zakrzewski, M., and White, M. A. (1992). "Thermal Conductivities of a Clathrate with and without Guest Molecules," Phys. Rev. B 45 2809.

SEMICONDUCTORS AND SEMIMETALS, VOL. 70

CHAPTER

2

Electronic and Thermoelectric Properties of Half-Heusler Alloys S. Joseph Poon DEPARTMENTOF PHYSICS UNIVERSITYOF VIRGINIA CHARLOTrESVILLE,VIRGINIA

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. 2. 3. 4. 5.

The Crystal Structure . . . . . Electronic Crystals . . . . . . Thermoelectric Properties . . . Other N e w Intermetallic Phases Goals o f This Chapter . . . . .

II. EXPERIMENTAL PROCEDURES

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

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

. . . . .

. . . . .

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

III. UNDOPED COMPOUNDS WITH VALENCE ELECTRON COUNT NEAR 18

. . . .

1. Semiconducting and Semimetallic Properties . . . . . . . . . . . . 2. Bandgap States in the V E C = 18 Alloys . . . . . . . . . . . . . . 3. Carrier Mobilities . . . . . . . . . . . . . . . . . . . . . . . 4. Magnetic Properties and Band-Structure Results . . . . . . . . . . 5. Thermoelectric Properties . . . . . . . . . . . . . . . . . . . . IV. DOPED ALLOYS . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Transport Properties . . . . . . . . . . . . . . . . . . . . . . 2. Bandgap Features Inferred f r o m Doping Studies . . . . . . . . . . . 3. Impurity B a n d Transport Properties . . . . . . . . . . . . . . . . 4. Thermoelectric Properties . . . . . . . . . . . . . . . . . . . . V. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES

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

I. 1.

. . . . .

37 37 39 40 41 41 42 43 43 51 53 54 57 59 60 63 64 66 71 72

Introduction

THE CRYSTAL STRUCTURE

Heusler and half-Heusler phases are among the best known intermetallic compounds. The former is represented by the general formula M~MX and the latter by MM'X, where M and M' are metals, and X is an sp metalloid or metal. For a majority of the Heusler phases, M' is either a transition metal or noble metal, and M is a transition metal, a noble metal, or a 37 Copyright 9 2001 by Academic Press All fights of reproduction in any form reserved.

ISBN 0-12-752179-8 ISSN 0080-8784//01 $35.00

38

S. JOSEPH POON

FIG. 1. Crystalstructure of the half-Heusler (MgAgAs type) phase. In the case of ZrNiSn, Zr and Sn in the rock-salt substructure are denoted by large filled and unfilled circles. Half of the cubic interstices of the ZrSn substructure are occupied by Ni (small filled circles).

rare-earth metal. The crystal structure of the Heusler phase is of the Bi3F type (space group Fm3m) and its unit cell consists of four interpenetrating face-centered cubic (fcc) sublattices. Each of the four fcc sublattices has the same unit cell size as that of the Heusler phase. There are a total of 16 atoms per unit cell. Two of the fcc sublattices, denoted by MX, form a rock-salt substructure. The other two fcc sublattices, denoted by M~ (or M'M'), occupy equivalent sites (0, 0, 0) and (89 89 89 The rock-salt substructure MX and fcc sublattices M~ are mutually displaced with respect to each other along their body diagonals by one-quarter of the unit cell. With each of the cubic interstices of MX being filled by an M' atom, the Heusler phase can be described as a "stuffed rock salt." If one of the M~ sublattices is vacant, the half-Heusler phase (space group F43m, MgAgAs-type) is formed (Jeischko, 1970). The latter phase, which is now a "half-stuffed rock salt," has 12 atoms per unit cell (see Fig. 1). Typical X-ray diffraction powder patterns of half-Heusler phases and their multinary solid solutions have been published (e.g., Evers et al., 1997; Hohl et al., 1999; Browning et al., 1999). The patterns resemble that of a

2 HALF-HEUSLERALLOYS

39

face-centered cubic lattice, that is, the reflection indices hkl are either all even or all odd. Both Heusler and half-Heusler phases are chemically diverse and are known to exist in more than 350 and 140 alloy systems, respectively (Villars and Calvert, 1991). Moreover, findings of new half-Heusler alloys continue to be reported (Section III, 1).

2.

ELECTRONICCRYSTALS

In the half-Heusler phase, the reduced coordination number of the M and X atomic sites, due to one of the M~ sublattices being vacant, has important consequences for electronic structures and measured properties. The presence of tetrahedral coordinated bonding sites reduces the overlap between d wave functions as well as that between d and sp wave functions, which gives rise to narrower bands in the energy spectra. These bonding configurations have implications for the phase stability as well as the occurrence of bandgaps and unusual magnetic states. The penchant for covalency is evident from the fact that most of the half-Heusler phases are found within a restricted range of valence electron count (VEC), which is centered at 8 or 18 electrons per formula unit. In the early 1980s, a group of half-Heusler compounds, notably NiMnSb and PtMnSb, first attracted attention for the peculiar electronic structure properties they exhibit (de Groot et al., 1983). These compounds, now widely known as half-metallic ferromagnets, are characterized by the appearance of a metallic spin-up band and a gapped spin-down band inside which the Fermi level lies (de Groot et al., 1986). The half-metallic behavior is due to the strong magnetic splitting of the energy bands. A decade ago, a group of half-Heusler compounds of formula MNiSn ( M - - T i , Zr, Hf) was found to exhibit semiconducting behavior in electrical and optical measurements (Aliev et al., 1988, 1989). A bandgap on the order of 0.1-0.2 eV was measured in these alloys. For MNiSn (Ogut and Rabe, 1995; Mahanti et al., 1999; Tobola et al., 1998) as well as other half-Heusler compounds (Pierre et al., 1997; Tobola et al., 1998), the important roles of pd hybridization and dd interaction in gap formation were shown in band-structure calculations. In fact, a semiconducting gap is obtained in the density of states (DOS) of all the systems investigated, and whether the compound is semiconducting or metallic is determined by the position of the Fermi level E v with respect to the gap (Tobola et al., 1998). For alloys with VEC of 18, E v is found to lie at the top of the highest occupied valence band. As the VEC decreases or increases, E v is seen to fall below the top of the highest occupied valence band or rise above the bottom of the lowest conduction band, which results in a p-type or n-type metallic state, respectively. In addition, spin-polarized states in several metallic phases are also observed in the calculations (Tobola et al., 1998; Kaczmarska et al., 1999).

40

S. JOSEPHPOON

The different electronic state scenarios mentioned can be seen in the DOS spectra published in the papers referenced earlier.

3.

THERMOELECTRICPROPERTIES

A good candidate thermoelectric (TE) material is preferably a narrow-gap semiconductor or semimetal with high carrier mobility and low thermal conductivity (see reviews by Mahan et al., 1997; DiSalvo, 1999; Tritt, 1999). In terms of measurable physical quantities, a dimensionless TE figure of merit Z T - (S2~r/x)T of the order of unity or larger is desired (S = Seebeck coefficient, cr = electrical conductivity, and K - thermal conductivity). The parameters S and o, which tend to manifest an inverse relationship, are strong functions of the doping level and chemical composition, which must therefore be optimized for good thermoelectric performance. It has been noted that the product factor SzG, known as the power factor, can be maximized via doping to attain the semiconductor-to-semimetal (SCSM) transition (Mahan et al., 1997). Meanwhile, the K ( = x c + ~cl, the electronic term and lattice term, respectively) of complex materials can often be modified by chemical substitutions and doping. Understanding these various effects and selecting optimization strategies can be an exceedingly difficult problem, because in complex materials there are often many possible degrees of freedom. In 1995, Slack suggested that the best thermoelectric material would behave as a "phonon-glass, electron-crystal" material (Slack, 1995). That is, the materials would have the thermal properties of a glass and the electronic properties of a crystal. There are ongoing efforts to improve the performance of the well-known TE alloys. Meanwhile, researchers have also been investigating other lesserknown compounds and exploring new alloy systems as potential TE materials. For commercialization purposes, it is desirable that the TE materials be relatively simple to synthesize and process, and that their electronic and lattice properties can be conveniently tuned and also reproduced. In addition, when used in a high-temperature environment, the materials are required to be chemically and thermally stable. Metal-rich intermetallic compounds tend to possess many of these desirable properties and may therefore serve as better alternatives to some of the traditional TE alloys. Many research groups are now actively investigating new intermetallic compounds based on rare-earth (lanthanide) metals and transition metals as prospective TE materials. Early on, several of the half-Heusler compounds HfNiSn, TiNiSn, and ZrNiSn were found to exhibit semiconducfing behavior with Seebeck coefficients as large as - 2 0 0 to -400/~V/K at room temperature (Aliev et al., 1989, 1990; Cook et al., 1996; Kloc et al., 1996). In view of their large thermopower and only moderately high resistivity, their potential as TE materials was readily pointed out (Cook et al., 1996). In addition to their

2 HALF-HEUSLERALLOYS

41

relatively simple crystal structure, the high substitutability of the sublattices, together with the "unfilled" or "open" (Uher et al., 1999a) structure of half-Heusler alloys, provide many possibilities for tuning the electronic and lattice properties via chemical modifications. Thus, the SCSM transition can be monitored easily by doping the host alloy. Also, the high lattice thermal conductivity, which is the least desirable property of these alloys, may be reduced via elemental substitution at the sublattice sites or addition into the unfilled sites to enhance phonon scattering (Hohl et al., 1997, 1999; Uher et al., 1999a, 1999b; Browning et al., 1999). There are other attributes that also make the half-Heusler alloys potentially attractive for thermoelectric applications above room temperature. The refractory-alloy phases are thermally stable at temperatures as high as 1000~ (Browning et al., 1999). The materials are relatively easy to synthesize, and multinary alloys with designed stoichiometry can be produced using a conventional arc furnace. Since the ingots can be cut using a diamond saw, their mechanical sturdiness suggests that they should be easy to process. Furthermore, their isotropic structure suggests that neither single crystals nor oriented films are required for maximizing the performance.

4.

OTHER NEW INTERMETALLIC PHASES

These range from structurally complex semiconducting intermetallic phases (Young et al., 1999) to pseudogap quasicrystals (QC) to approximants (crystalline counterparts of QC) (reviews: Poon, 1992; Berger, 1994) and to other narrow-gap (~0.2 eV) intermetallic compounds such as A12Ru with the SizTi-type structure (Volkov and Poon, 1995). Many of these intermetallic compounds, including those half-Heusler phases based on rare-earth metals (Mastronardi et al., 1999; Sportouch et al., 1999a) will be reviewed elsewhere (see Chapter 3 in this volume, as well as Section III in this chapter). Examples of QCs include the icosahedral A1CuFe and A l P d M n alloy systems, and approximants that include the A l M n S i and A l C o N i systems. These compounds, which were studied from the scientific rather than technological perspective, generally have moderate Seebeck coefficients of magnitudes up to about ~ 100 pV/K at room temperature. In the past few years, the thermoelectric properties of QC alloys, which possess low thermal conductivity of ~ 2 W/m-K, have been studied extensively (Pope et al., 1999).

5.

GOALS OF THIS CHAPTER

The semiconducting, semimetallic, and magnetic properties of halfHeusler alloys are reviewed. In particular, extensive electrical and thermal transport characterization of alloys that span a wide range of compositions

42

S. JOSEPHPOON

are discussed. Fundamental issues that involve bandgap structure, magnetic state, charge transport, and localization properties are addressed. Thermal conduction by charge and phonon in these nontraditional semiconducting phases is also discussed. These basic issues can be further studied by examining the effects due to doping and substitution on properties. Experimental results are discussed in light of band-structure calculations and transport models. Based on the discussion of electronic and lattice effects on power factor and ZT, the feasibility of using half-Heusler alloys as thermoelectric materials is evaluated. Thus, the investigative approach is designed to supplement the commonly used "synthesis and measure" approach with a fundamental basis in optimizing TE properties. This chapter is organized as following: Experimental methods are described in Section II. In Section III, electronic and thermal transport properties as well as magnetic properties of undoped compounds are discussed. Although the focus is on the MNiSn and MCoSb ( M z T i , Zr, Hf) phases and their isoelectronic multinary solid solutions, comparison is made with the lanthanide half-Heusler phases. Section IV reports on the properties of doped alloys. In Sections III and IV, comparison with band structure and transport models is made, and thermoelectric properties of both doped and undoped alloys are discussed. Section V is the conclusion.

II.

Experimental Procedures

The experimental methods described herein are those used by the author's group. Ingots of the alloys studied were prepared by arc melting appropriate quantities of the constituent elements together under an argon atmosphere. The overall purity of the starting materials was 99.97+ % for Zr-based alloys and 99.996+% for TiNiSn-based alloys. The weight loss during preparation was found to be less than 1%. The ingots' compositions were also checked by chemical analysis. Further improvement of the samples' chemical homogeneity was also attempted by melting pressed pellets of thoroughly mixed elemental powders. Both types of samples were found to exhibit almost identical properties. Two-step annealing was carried out in an evacuated and sealed quartz tube with the sample wrapped in a tantalum foil. The first annealing, carried out for 16 hours at 900~ was to homogenize the alloys, and the secondstage annealing, carried out for 10 days at 750~ for Ti alloys and 800~ for Zr alloys, was to further homogenize the alloys and also promote the ordering of the crystal structure. Annealing procedures employed by other groups ranged from 800~ for 1 week (Uher et al., 1999a, 1999b) to 800~ (Zr alloys) and 750~ (Ti alloys) for 1 to 6 weeks (Hohl et al., 1999). The phase (MgAgAs) purity of our alloys was assessed via X-ray diffraction.

2 HALF-HEUSLERALLOYS

43

Rectangular pieces of approximate dimensions of ~ 1.5 x 2 x 6 mm were cut for transport measurements. DC resistivity was measured from 4.2 to 295 K by the four-probe method. Thermopower was measured by the standard differential technique. Copper-constantan thermocouples were used to measure the hot and cold end temperatures. The copper leads of the thermocouples were used to measure the thermo-emf. Specific heat and Hall coefficient were measured from 2 to 300 K by using a commercial Physical Property Measurement System. Hall effect was measured up to 5 T. Thermal conductivity and high-temperature resistivity and thermopower measurements were performed as described elsewhere (see Chapter 2).

III. 1.

Undoped Compounds with Valence Electron Count Near 18

SEMICONDUCTINGAND SEMIMETALLIC PROPERTIES

Pure and substituted alloys are studied to elucidate bandgap features and conduction mechanism. In substituted alloys, one can examine the changes in electronic properties by varying the level and type of chemical disorder. After all, thermoelectric materials are substituted alloys that exhibit both doping and disorder effects. The first three parts of this section are devoted to undoped VEC = 18 phases and their solid solutions. Discussion of electronic structure properties as a function of VEC is featured in Subsection 4 of this section.

a.

Alloys Based on MNiSn ( M = Ti, Zr, Hf)

In Fig. 2, we present plots of normalized resistivity p(T)/p(295 K) versus temperature for a series of (Zrl_xHfx)NiSn alloys and (Zro.5Hfo.5) (Nio.vPdo.3)Sno.95 alloys. Room-temperature p values are given in the figure caption. The p(T) for ZrNiSn reported by three groups are shown in the inset to Fig. 2. It is seen that the order of magnitude of the resistivity ratio R (normalized resistivity at 4.2 K) ~ 20 measured by the author's group lies in between those measured by two other groups: R < 4 (Uher et al., 1999b) and R ~ 100 (Hohl et al., 1999). All the alloys mentioned are believed to have achieved a high degree of homogeneity (Section II). Thus, given the comparable high purity level of the alloys (99.99 +) used by the latter two groups, the large difference in the p(T) trend of ZrNiSn is rather puzzling. However, semiconducting (sc)-like p(T) at high T and "saturation" of p at low T are noted common trends in the (Zr, Hf)NiSn alloys studied. Thus, the alloys show metallic behavior as T --. 0. In particular, semimetallic (sm) conduction is observed below 100-150 K in the alloys studied by one of the

44

S. JOSEPH POON 10 3

100

0.05 102

10 ~

~,.

10

T(K)

100

100 0

50

1O0

150 200 T(K)

250

300

FIG. 2. (Top four plots) Normalized resistivity versus temperature for (Zrl_xHfx)NiSn alloys. (Bottom plot) Same for (Zro.sHfo.5)(Nio.vPdo.3)Sno.95.The p(300 K) values from top to bottom plot (x = 0.05, 0.15, 0.3, 0, and (Zr, Hf)(Ni, Pd)Sn) are 10.5, 6.5, 7.8, 5.7, and 3.8 mf~-cm (Ponnambalam et al., 1999). (Inset) p(300 K) from top to bottom plot (ZrNiSn: Hohl et al., 1999; ZrNiSn: present data; ZrNiSn: Uher et al., 1999b; (Zro.sHfo.5)NiSn: Uher et al., 1999b) are 11, 5.7, 20, and 200 mf~-cm.

groups (Uher et al., 1999b). Hereafter, alloys with p ( T ) > 100 pf~-cm and that still exhibit d p / d T >~ 0 will be characterized as semimetallic alloys. The large increase in p upon substituting only 5% of Zr with Hf is unexpected. It is quite remarkable that a value of R as large as ~ 300 is seen in a metal-based alloy. Meanwhile, the highly substituted (Zr, Hf)(Ni, Pd)Sn alloys have much smaller R of ~ 2. Taking O'4.2K as the saturation conductivity at low T, p - 1/a of (Zr, Hf)NiSn alloys are found to exhibit activated behavior at T > 100 K, and the activation energy is estimated to range from 0.02 eV (x = 0.0) to 0.05 eV (x = 0.05). The origin of these activation energies, which are smaller than the ~0.2 eV activation energy determined from p ( T ) above 300 K (Aliev et al., 1988, 1989; Cook et al., 1999), is discussed in Subsection 2. In the case of pure TiNiSn, p(T) is reminiscent of a transition from SC to SM states that occurs near 200 K (Fig. 3), similar to that reported by another group (Pierre et al., 1997). This phase is found to have an SC gap of ~0.18 eV (Aliev et al., 1988, 1989; Cook et al., 1996). As for the isoelectronically substituted Ti-alloys, p ( T ) is found to be either larger than or comparable to that of the undoped alloy over the wide compositional range of Hf (~, 50%) and Pd/Pt ( ~ 30%) substitutions. In Fig. 4, Hall coefficients (RH) measured in the temperature range 4.2-300 K for the compositions ZrNiSn and (Zro.95Hf0.os)NiSn are shown. For the ZrNiSn alloy, R n changes sign near 200 K, but the thermopower S ( T ) (Fig. 5) remains negative throughout the entire temperature range. At

2

HALF-HEUSLERALLOYS

45

104

E o I

103

::::L (3.

Q

102

101

,

,

,

,

,

,

50

100

150

200

250

300

T(K) FIG. 3. Resistivity versus temperature, plotted from top to bottom, for (Tio.96Mno.o4)NiSn, TiNiSn, and a series of TiNi(Snl_xSbx) alloys, where x = 0.005, 0.02, 0.03, 0.05, 0.1 (Poon et al., 1999).

4.2 K, R H is ~ 16 cm3/C, which indicates a small effective carrier density neff estimated to be of the order of 1018 cm -3 (holes). Without performing a two-band analysis of the carrier conduction, one can roughly estimate the room-temperature value of/'/ef f to be of the order of 1019 cm-3 (electrons). S i m i l a r nef f was r e p o r t e d for Z r N i S n a n d (Zro.5Hfo.5)NiSn p r e v i o u s l y ( U h e r et al., 1999b); h o w e v e r , R n was n o t e d to r e m a i n n e g a t i v e b e l o w 200 K. F o r the x = 0.05 alloy, the m a g n i t u d e of R n at low T is large, a n d neff is e s t i m a t e d to be ~ 1016 cm - 3 (electrons). A b o v e ~ 100 K, R n is f o u n d to exhibit a c t i v a t e d b e h a v i o r d e s c r i b e d by an a c t i v a t i o n e n e r g y ~ 0 . 0 6 eV. T h e

200

.~ 0 co E (3 "~ -'rrr

x=0.0

-200

o ~

O

-400

E

-600

O v

-800

ln,'

o ~

,'o.o5

- 1000

-1200 0

1O0

200

300

T(K)

FIG. 4. Hall coetficient versus temperature for ZrNiSn and (Zro.95Hfo.5)NiSn (Pope et al., unpublished data). Data for ZrNiSn above 100 K are additionally shown on a magnified scale.

46

S. JOSEPH POON

-100 .'~ -200 -300 -400

0

50

100 150 200 250 300 T(K)

FIG. 5. Thermopower versus temperature for (Zr, Hf)NiSn alloys. The x = 0, 0.05, 0.15, and 0.3 alloys are symbolized by 0, II, V, and 1,, respectively (Ponnambalam et al., 1999). The uppermost plot is for ZrNiSn taken from Uher et al., 1999b; the plot is very similar to that for (Zro.5Hfo.5)NiSn measured by the same group. latter is comparable to that estimated from p(T). The carrier concentration reaches ~ 3 x 1019 c m - 3 near ambient temperature. For the (Zro.sHfo.5)(Nio.TPdo.3)Sno.95 alloy, R H is small and it varies from 0.07 cma/C at 4.2 K to - 0 . 0 3 cm3/C at 295 K, yielding an effective carrier density of the order of 10 2~ c m -3. However, S ( T ) remains negative and reaches ~ - 8 0 / t V / K at 300 K. For this alloy and ZrNiSn, the saturation of resistivity and the opposite signs exhibited by R n (positive) and S (negative) at low T indicate band overlapping at the Fermi level. These experimental results, coupled with S being negative from 4.2 to 300 K and large at high T (Fig. 5), also support earlier experimental (Uher et al., 1999b) and theoretical (Mahanti et al., 1999) findings of heavy electron band masses in the MNiSn group of half-Heusler alloys. Electron mass in the range of 2 - 5 times the free electron value is obtained from transport experiments, which is in good agreement with band-structure results. Comparable large S(300 K) values have been reported by several groups (see Subsection 3 of Section I; also Hohl et al., 1999; Uher et al., 1999a, 1999b; Browning et al., 1999). However, the shape of S(T) differs quite significantly among the reported results, as illustrated in Fig. 5. In fact, the persistence of a near-zero thermopower up to 100-150 K, reported for ZrNiSn and (Zro.5Hfo.5)NiSn by Uher et al., has given strong evidence of semimetallic (or metallic) behavior at low T (Uher et al., 1999a, 1999b). In our Hf-substituted alloys, S(300 K) is found to be doubled in going from the x = 0 alloy to x = 0.3 alloy. A plausible origin of the enhanced S ( T ) observed is given in the following subsection. Table I lists the electronic transport parameters of various VEC = 18 phases.

TABLE I TRANSPORT PARAMETERS OF WELL-ANNEALED SAMPLES (SEE TEXT AND REFERENCES) OF UNDOPED V E C = 18 PHASES AND SOLID SOLUTIONS MEASURED AT AMBIENT TEMPERATURE a

Alloy ZrNiSn

HfNiSn TiNiSn (Zro.95Hfo.os)NiSn (Zro.sHfo.5)NiSn (Tio.sHfo.5)NiSn (Zro.sHfo.5)(Nio.vPdo.3)Sn NbIrSn TiCoSb (Zro.65Hfo.35)(Co , Pt)(Sn, Sb) TmNiSb ErNiSb (Ho, E r ) P d S b (Er, D y ) P d S b

Ref b

p

K

S

nef f

/A

(f~ cm)

(W/m-K)

(pV/K)

(1020 cm -3)

(cm2/Vs)

- 210 -176 -- 175 - 400 -124 - 142 -240 -280 -203 - 163 -281 -176

1 2 3 1 2 2 4 3 1 2 2 5

2 1.1 5.7 2 1.3

x 10- 2 x 10 -2 x 10 -3 • 10-1 x 10 -2 10- 2 8 x 10 -3 1 • 10 -2 2 x 10 -2 8.3 x 10 -3 4.1 x 10 -2 10 -2

17.2 8.8 8.0 12 6.7 9.3 10

3 6 5 7 8 8 9 9

4 x 10- 3 1.1 4 x 10 -2 2 x 10- 3 to 10- 2 3 • 10- 3 3 x 10- 3 1.5 X 1 0 - 3 t o 1 0 - 2 6 • 10 -4 to 2 x 10 -2

5.5

5.3 4.4 3.6

20 3.1 2.8 4 3.5 to 5.8 3.5

-80 + 176 -250 -- 100 to + 100 + 60 + 160 + 140 to + 220 + 8 0 to + 2 6 0

0.21

14.8

,-,0.45

~24

~0.06 ~0.2 0.24

,-~ 120 30 9

1

15

,-,0.05

,-~33

1.9 0.2

11 100

aThe list illustrates d a t a obtained on representative alloys. The effective carrier density neff and mobility # are estimated from Hall effect and resistivity measurements using a o n e - b a n d model. The precision of the parameters is according to that given in the references. bReferences: 1, Uher et al., 1999b; 2, H o h l et al., 1999; 3, P o n n a m b a l a m et al., 1999; 4, B h a t t a c h a r y a et al., 1999; 5, P o o n et al., 1999; 6, H o h l et al., 1998; 7, unpublished results, ~c value is for 35% occupancy of Pt and Sn in the sublattices; 8, S p o r t o u c h et al., 1999a; 9, M a s t r o n a r d i et al., 1999.

48

S. JOSEPH POON T A B L E II SPECIFIC-HEAT PARAMETERS DETERMINED FOR SEVERAL V E C = 18 HALF-HEUSLER PHASES (SEMIMETALLIC Sb(2.5% )-DOPED (Tio.sHfo.5)NiSn AND METALLIC T i C o S n INCLUDED FOR COMPARISON)

ZrNiSn

HfNiSn

TiNiSn

ZrNiSn (5%Hf)

TiCoSb

(Ti, H f ) N i S n (2.5%Sb)

TiCoSn

0D(K )

310 ~ 323 b ~ 350 c

255 a 307 b

283 ~ 417 b

336 c

419 d

350 d

348 b

7 ( m J / g - a t K 2)

0.77 a 0.30 b <0.4 c

0.5 a 0.55 b

0.23 a 0.30 b

0.11 c

0.34 d

0.3 d

2.5 b

~ et al., 1990; b K u e n t z l e r et al., 1992; c p o n n a m b a l a m et al., unpublished results; dBhatt a c h a r y a et al., unpublished results.

The coefficient 7 of the electronic term in the specific heat and the Debye temperature 0D of several alloys are tabulated in Table II. Specifc-heat results plotted in the form of C/T versus T 2 for our ZrNiSn and (Zro.95Hf0.o5)NiSn alloys are shown in Fig. 6. A small upturn below ~ 4 K is seen in the x = 0.05. The small upturn may be due to the presence of magnetic impurities or structural defects or other unknown effects. For comparison, our measurements of high-purity (99.996+) Ti-based half-

c~v 6 E 0 m4

v

E 2

0

0

J 9 0 15 0.4.7 Sx=O0 T2(K2)~ ~ e ~-" " 0.8

,

0

20

,

40

,

60

80

,

,

100 120 140

T2(K 2) FIG. 6. Specific heat data plotted in the form of C / T versus T 2 for the x - 0 a n d 0.05 alloys (Pope et al., unpublished data). The solid line through the x = 0.05 d a t a follows the expression C / T - ~ + f i T 2 + 3 T 4. (lnset) D a t a on the x = 0 sample are plotted on a m a g n i f i e d scale for clarity.

2 HALF=HEUSLERALLOYS

49

Heusler alloys also show similar small upturns. The specific-heat plot for ZrNiSn exhibits an anomalous trend at low T. Below ~ 3 K, C / T apparently decreases quite rapidly (inset), which indicates a small linear coetficient of specific heat 7 less than 0.4 mJ/g-at K 2. Previous specific-heat studies of ZrNiSn revealed a pronounced "plateau" region in the temperature range given by ~ 80 > T 2 > 30 (Aliev et al., 1990), which is not observed by us. For other alloys studied by us, specific-heat data were fit using the first three terms in the standard expression C / T = ~ + f i T 2 nt- (~T 4, where fl and 6 are are constant. Significantly smaller 0o values are seen in the Zr alloys in comparison with those based on Ti. For the MNiSn alloys measured by several groups (Table II), the low ~ values obtained are consistent with the low carrier densities found in these compounds. However, comparing ~ to their free-electron values estimated from neff (Table I; also Table III in Section IV) seems to indicate a mass enhancement factor of the order of 10 or larger for these alloys (Ponnambalam et al., unpublished). Whether these results can be accounted for by band-structure effect alone awaits further investigation.

b.

Alloys Based on M C o S b ( M = Ti, Zr, H f )

p(T) plots for MCoSb alloys are shown in Fig. 7. Except for its higher p(300 K), the trend seen in TiCoSb resembles that of TiNiSn (see also Pierre et al., 1997; Tobola et al., 1998). However, unlike ZrNiSn and HfNiSn, SM behavior in ZrCoSb (p(300 K) ~ 4 x 10-2 ~"~-cm)and HfCoSb (p(300 K)

100

Eo -1

10

j/"J

d Q.. vr

E 0 d

~

-3 9

,

9

i

o.oo7 o.oo9

Q,..

-I(K -1)

0.1 0.01

0

50

100

150 200 250 T(K)

300

FIG. 7. p versus T plots for ZrCoSb and (Til_xVx)CoSb alloys (Poon et al., 1999; also unpublished data). Bottom plot: ZrCoSb. Other four plots (from top): x = 0.005, 0.01, 0.015, and 0. Inset: Natural In p versus 1/T for x = 0.005.

50

S. JOSEPH POON

3 x 10-3 ~-cm, not shown in Fig. 7) is seen to persist up to 300 K. Also, much smaller S(300 K) values of ~ - 10 p V / K compared with ~ - 2 6 0 #V/K for TiCoSb are measured on the latter two compounds. A near-zero S(T) below ~ 6 0 K is seen in TiCoSb (Poon et al. 1999), which contrasts with the quite rapidly varying S at low T in TiNiSn (see Section IV, Fig. 12). Results derived from the specific heat study of TiCoSb are given in Table II. In an effort to achieve low thermal conductivity, we have begun to study multinary alloys based on ZrCoSb and HfCoSb (Table I). Our multinary alloys are not restricted to those with VEC -- 18. Quite surprisingly, drastic modifications of the electronic structure properties of the base compounds are observed. For example, substituting Co with Pt and Sb with Sn have resulted in both positive and negative S(300 K) up to ~ 100 laV/K in magnitude and p(300 K) ranging from ~ 2 x 10 -3 to 10 -2 ~-cm. SC behavior is seen in some of the high p alloys. Clearly, these compounds deserve our further attention.

c.

Other Phases

Since the early 1990s, new half-Heusler compounds that are not listed in Pearson's Handbook (Villars and Calvert, 1991) have continued to be found. Some examples include ZrNiBi, ZrCoBi, NbCoSb, and TaCoSb (Evers et al., 1997), NbIrSn and NbIrSb (Hohl et al., 1998), and MnNiSn (Poon et al., 1999). There are also new compounds based on the rare-earth metals. Transport properties of NbIrSn and NbIrSb have been reported (Hohl et al., 1998). The former is a p-type semiconductor with energy gap ~0.28 eV, and it exhibits p(300 K) ~ 1.1 ~-cm and S(300 K) ~ + 176 pV/K, while the latter is metallic and it exhibits p(300 K) ~ 1.5 x 10-3 ~-cm and S(300 K) + 3.5 p V / K . It is interesting to note that comparable metallic (or semimetallic) properties are seen in the NbIrSb and HfCoSb phases with supposedly SM and SC VEC values of 19 and 18, respectively. The transport properties reported on the nonlanthanide alloys can be compared with those observed in YNiSb as well as the lanthanide series LnNiSb (Sportouch et al., 1999a), LnPdSb (Mastronardi et al., 1999), and LnPdBi (Cook et all, 1996), where Ln denotes the heavy lanthanides Dy, Ho, Er, Yb, and Lu. The transport parameters of several of these alloy systems are listed in Table I. In contrast to the n-type MNiSn and MCoSb semiconductors, the latter alloys exhibit p-type Seebeck coefficients, which are in the range ~ 50-250 p V / K at ambient temperature. The high S(300 K) values observed are comparable to those seen in the former group of alloys. Apparently, the relative contributions of electron band mass and hole band mass are different in the two groups. Furthermore, the Ln alloys exhibit resistivities that are only weakly dependent on temperature (i.e., R ~ 1), in contrast to the M alloys with comparable p values. Band-structure calcula-

2 HALF-HEUSLERALLOYS

51

tions indicate that the former possess appreciably smaller gaps. For example, it is found that Eg ~ 0.28, 0.19, and 0.1 eV for YNiSb, LuNiSb (Larson et al., 1999), and LuPdSb (Mastronardi et al., 1999), respectively. Thus, taking the structural order of the samples into account, even smaller experimental gaps are to be expected.

2.

BANDGAPSTATES IN THE VEC = 18 ALLOYS

The VEC = 18 compounds that exhibit SC properties provide model systems for discussing basic bandgap properties of the half-Heusler phases. To facilitate the discussion, key features of the measured transport properties are summarized: (a) Independent of how large p(300 K) and R are, finite p values are seen at low T. In many compounds, SM behavior is noted below the temperature region 100-200 K. (b) For the n-type alloys discussed, S(T) is negative from 4.2 to 300 K and it becomes large at high T. Meanwhile, RH(T ) can be positive or negative at low T. (c) Although the situation is less well defined in other compounds, carrier conduction in the region ~ 100300 K is seen to be activated in the (Zr, Hf)NiSn solid solutions. The activation energies are estimated to be several tens of milli-electron volts, much smaller than the main gap of ~0.2 eV measured above 300 K. (d) Carrier densities at low T are estimated to range from ~ 101 s cm-3 or lower in the ternary compounds to ~102~ cm -a in the highly substituted compounds. (e) Carrier densities reach ~ 1019 cm -3 or higher at 300 K. Features (a) and (b) give direct evidence to the proposal of band overlapping. In particular, feature (b) further underscores the important role of electron band mass at the Fermi level pointed out earlier (Uher et al., 1999b; Mahanti et al., 1999). Features (c) to (e) are reminiscent of those seen in another metal-rich narrow-gap (~0.2 eV) semiconductor A12Ru (Volkov and Poon, 1995). For the latter system, measurements have revealed a small neff(4.2 K ) ~ 1016-1018 cm -3 and two activation energies of ~0.03 and 0.2 eV, with the larger being the size of the main gap. Clearly, the observed properties of the half-Heusler phases, which are based on metals, are not those expected of an intrinsic semiconductor. In other words, a semiconducting gap in the absolute sense is not attained. The results are most likely due to the fact that it is difficult, if not impossible, to achieve perfect crystallographic order in the half-Heusler alloys. Early on, site exchange between the atoms in different sublattices was pointed out (Aliev et al., 1989). As a matter of fact, the transport properties of the samples are reported to be strongly dependent on annealing conditions (Aliev et al., 1989; Uher et al., 1999b; Hohl et al., 1999). For the MNiSn compounds, the bandgaps of ~0.1-0.2 eV determined from high-T resistivity are conspicuously smaller than the gap values -~0.5 eV found in band-structure calculations (Ogut and Rabe, 1995; Tobola et al., 1998;

52

S. JOSEPH POON

Mahanti et al., 1999). One of the band-structure studies also indicates that the width of the gap depends strongly on the degree of disorder of M and Sn atoms (Ogut and Rabe, 1995). A gap as large as 0.5 eV in a perfectly ordered sample can be reduced to zero when the amount of the M/Sn site exchange reaches 15%. Not surprisingly, structural disorder has also been found to affect the magnetic properties of ferromagnetic half-Heusler phases (Subsection 4). In order to account for feature (c) noted in (Zr, Hf)NiSn, the formation of intragap states or bands in the band-overlapping region must be invoked. The bulk of the intragap band, which accommodates a carrier density of ,-~ 1019 cm -3, is separated from the main part of the conduction band by the activation energy observed. The Fermi level lies in the region where the intragap band and conduction band overlap. At low T, carrier transport takes place in the band-overlapping region where the conduction band mass is moderately heavy. As T increases, more carriers are excited to the conduction band. The enhanced S(T) observed in the (Zrl_xHfx)NiSn alloys, where x ~< 0.3 (Fig. 5), can be ascribed to the depletion in carrier density due to the formation of an intragap band below the Fermi level. For the (Zr, Hf)(Ni, Pd)Sn alloy, the overlap of intragap band and conduction band is no longer small; therefore, the notion of activation energy becomes irrelevant. Although the origin of the intragap band is not known, other bandgap models have also been considered. For example, many of the features just reported may be explained by the closing of the gap at low T, which results in the SM behavior and very small S(T) observed. As T increases, there is an opening of the gap, which leads to the SC behavior and large S(T) observed at room temperature (Uher et al., 1999b). Alternatively, a band model that involves a narrow intra-gap "vacancy band" given rise to by the vacancy sublattice has been proposed (Aliev et al., 1990). Further studies, such as measuring the optical conductivity as a function of temperature, will be needed to clarify the nature of bandgap structure.

3.

CARRIER MOBILITIES

The carrier mobilities p plotted versus T for ZrNiSn and (Zro.95Hfo.o5) NiSn are shown in Fig. 8. The room-temperature # values for several representative VEC = 18 alloys are given in Table I. In lieu of a two-band analysis of the Hall coefficient, the # values listed should be treated as order-of-magnitude estimates only. For most of the alloys studied, moderate carrier mobility values of 100-400 cm2/V-s are obtained at low T. At 300 K, the order of magnitude of p decreases to ~ 10 cm2/V-s, similar to that reported earlier for ZrNiSn and (Zro.5HFo.5)NiSn (Uher et al., 1999b). Comparable low values of p for the Ln-based half-Heusler alloys can be inferred from transport measurements (Sportouch et al., 1999a). Thus, in

2 HALF-HEUSLERALLOYS

53

400

350 .~

300

250 E o =

200 150 100 50 0

50

100 150 200 250 300 T(K)

FIG. 8. Hall carrier mobility as a function of temperature for the x = 0 and x = 0.05 alloys ( P o n n a m b a l a m et al., unpublished data).

comparison with other TE materials (Mahan et al., 1997), the half-Heusler alloys are low-mobility semiconductors. Conventional thermoelectric alloys possess # values that are at least one order of magnitude higher than those shown in Table I. The effects of low carrier mobility on TE properties is discussed in Subsection 3 of Section IV. Despite the high p and low g obtained, the carrier mobilities in halfHeusler alloys are at least one order of magnitude higher than those estimated for metallic glasses and quasicrystals (Poon et al., 1996). The latter classes of alloys are known to possess a very high degree of structural complexity. Even for the highly substituted (Zro.sHFo.5)(Nio.TPdo.3)Sn phase, # is found to be ~ 15 cmZ/V-s. Extending the comparison further, it is noted that # decreases with T in the half-Heusler alloys, but increases with T in the QCs. In fact, carrier localization has been seen in the QCs (Poon et al., 1996), but not in the high-p half-Heusler alloys listed in Table I. However, it should be pointed out that the effects of structural quality on carrier conduction in the latter alloys have yet to be investigated. In fact, we have observed hoppinglike conduction in some TiCoSb alloys that are made at slightly off-stoichiometric compositions, namely as TiCoSb~ 1.o5 alloys. The latter also exhibit very high p(300 K ) ~ 1-2 fl-cm. Clearly, for low carrier density ( ~ 10 TM cm -3 or lower) materials, whether the electronic states are extended or localized is dependent on whether the Anderson condition is fulfilled or not (Mott and Davis, 1971). In contrast, QCs and metallic glasses, although characterized by their very low mobilities, possess much higher carrier densities of at least 102~ cm -3 (Poon, 1992). Present investigation of substituted half-Heusler alloys, which possess much simpler structure than QC alloys, has shown that substitution disorder alone is not sufficient for carrier localization. These findings have helped to clarify the role of structural complexity in carrier localization.

54 4.

S. JOSEPH POON MAGNETIC PROPERTIES AND BAND-STRUCTURE RESULTS

In addition to the MnNiSb and MnPtSb VEC = 22 compounds that were found to be half-metallic ferromagnets mentioned in Subsection 2 of Section I, (de Groot et al., 1983, 1986), ferromagnetism was also observed in other metallic half-Heusler phases, such as TiCoSn (Pierre et al., 1993, 1994) and VCoSb (Terada et al., 1972). The latter were VEC = 17 and 19 phases, respectively. Extensive comparisons of measurements with band-structure calculations on the paramagnetic s e m i c o n d u c t o r - m a g n e t i c metal crossover (defined by a change in the sign of d p / d T ) in several pseudoternary phases and magnetic ground states in a series of ternary phases have been reported by Pierre et al. (1997) and Tobola et al. (1998). The VEC of these phases ranges from 17 to 19. The pseudoquaternary alloys include TiCo(Sn,Sb), Ti(Co,Ni)Sn, and (Ti,Nb)CoSn solid solutions. The ternary series include TiCoSn, TiFeSb, TiCoSb, TiNiSn, NbCoSn, VCoSb, and TiNiSb. From magnetic measurements, TiCoSn and VCoSb ( Kaczmarska et al., 1998) are "weak" ferromagnets with Curie temperatures Tc of 135 and 58 K, and saturation magnetic moments M(0) at low T equal to 0.35 and 0.18/tB, respectively. TiFeSb (VEC = 17) is metallic and exhibits Curielike susceptibility, but magnetic order is not detected down to 1.5 K. TiNiSb ( V E C - 19) is a metallic paramagnet with Pauli susceptibility Zp ~ 1.56 x 10 - 4 emu/mol. For the VEC = 18 TiCoSb, TiNiSn, and NbCoSn compounds, the first two are semiconductors while the latter is a semimetal. These alloys are paramagnetic with Zp ~ 1.7 • 10 -4, 1.3 • 10 -4, 0.53 • 10 -4 emu/mol, respectively. The K o r r i n g a - K o h n - R o s t o k e r Coherent Potential Approximation (KKR-CPA) studies performed by Tobola et al. (1998) have provided valuable insight into the influence of alloying on electronic structure properties, and several key findings emerge. The KKR-CPA calculations have shown the existence of a bandgap in the density-of-states (DOS) spectra of the seven ternary compounds investigated. The largest gap, which is about 0.9 eV, is seen in TiCoSb. The readers can refer to the many DOS spectra illustrated in the papers by Tobola et al. (1998) and Kaczmarska et al. (1999). As VEC increases from 17 to 19, the position of the Fermi level is seen to shift from being inside the valence band to inside the conduction band. Consequently, a metallic-semiconducting-to-metallic sequence of conducting states is obtained. With respect to the VEC = 18 TiNiSn and TiCoSb semiconductors, TiCoSn (VEC = 17) and TiNiSb (likewise VCoSb, VEC = 19) can be called "one-hole metal" and "one-electron metal," respectively. The Curie-Weiss behavior measured in TiCoSn (and VCoSb) can be ascribed to the high density of states (DOS) at E v. Earlier, a moderately high specific heat 7 was measured in TiCoSn ( Kuentzler et al., 1992; also see Table II). Also, with the highest local DOS being found on the Co (and V) atom, the magnetic moments are expected to reside on these

2

HALF-HEUSLERALLOYS

55

atoms. Thus, the occurrence of ferromagnetic state can be understood in light of the itinerant-electron (hole) magnetism picture. The magnetization in these alloys is found to follow an "M 2 linear in T 2'' law, which is characteristic of an itinerant ferromagnet. The same study has also resolved the puzzle of why TiCoSb (E v inside the gap) is a Pauli paramagnet while TiCoSn (high DOS at EF) is a ferromagnet. The SC-SM crossover and onset of magnetic moment observed in the pseudoternaries are found to be in good agreement with band-structure results. Despite the success of the latter model in explaining the various properties, however, some discrepancies remain. For example, the calculations predict a ferromagnetic state for TiFeSb and a fully polarized state for VCoSb, which are not observed. The authors contend that crystallographic disorder in the real systems could lead to the discrepancies, similar to those reported on observed transport properties (see Subsection 2). In fact, depending on sample preparation, VCoSb is found to exhibit a range of magnetic properties: 11 K < Tc < 58 K and 0.036/~ < M(0) < 0.18/t B (Terada et al., 1972; Kaczmarska et al., 1998). More recently, the effect of atomic disorder on the electronic structure of the half-metallic ferromagnet MnNiSn has been studied by band-structure calculations (Orgassa et al., 1999). The results indicate the presence of minority-spin states at the Fermi energy for degrees of disorder as low as a few percent. Kaczmarska et al. (1999) have investigated magnetic effects on carrier conduction in two pseudoquaternary (Til_xMnx) NiSb and (Vl_xMnx) CoSb phases formed from four ternary metallic half-Heusler phases. As in previous studies carried out by the similar group of authors, the investigation couples transport and magnetic measurements with KKR-CPA calculations. MnNiSb and MnCoSb are strong ferromagnets (Tc ~ 730 and 478 K, M ( 0 ) ~ 4.0 and 3.8/~a, respectively). MnNiSb is a half-metallic ferromagnet: Its polarized state is characterized by the majority spin 3d electrons having energies far below E v and the minority electrons having energies above EF. As a result, a local moment of ~4/~B is obtained. Near x ~ 0.4, where magnetic interactions increase rapidly, carrier localization which can be described by Mott's variable-range hopping is seen. Figure 9 shows the spectacular occurrence of insulatinglike resistivity behavior at low T for intermediate Mn contents that lie between the two metallic end phases in the (V, Mn)CoSb system. The concomitant onset of localization and magnetism at the rather high Mn contents are explained by Anderson localization of d states in these disordered alloys. Specifically, atomic and magnetic disorders operate in the valley region of the DOS constituted by the band tails of Ti and Mn in one system and V and Mn in the other, which lead to strong scattering of the carriers. Magnetic order in this intermediate regime is found to be inhomogeneous. These authors also report large magnetoresistance effects near the Curie temperature, which can also be ascribed to the scattering process suggested.

56

S. JOSEPH POON

10000 E 0

=L

Q..

1000 -

100

0

1

I

1

1

50

100

150

200

"'

I

250

300

T (K) FIG. 9. p versus T plots for (Vl_xMnx)CoSb (Kaczmarska et al., 1999). Top to bottom plot: x = 0.35, 0.3, 0.5, 0.2, and 0.7.

Summing up the various transport and magnetic studies reported, it can be seen that the interplay of orbital hybridization, Coulomb (direct and exchange) interaction, atomic disorder, and localization effects is responsible for the interesting properties observed in half-Heusler alloys. Similar effects are also known to be responsible for many of the novel properties found in the manganites and heavy-fermion compounds. The latter include YbNiSb and YPtBi (Dhar et al., 1994; Oppeneer et al., 1997). The magnetic properties of many lanthanide phases have been reviewed (Karla et al., 1998). In addition to the investigation of electronic structure properties of intermetallic phases, the band-structure approach has also been taken to study thermoelectric properties (see, for example, Chapter 5, by D. Singh).

5.

THERMOELECTRIC PROPERTIES

Several years ago, the possibility of obtaining practical Z T values at above room temperature in half-Heusler alloys was noted in some well annealed (3 weeks at 840~ followed by 2 weeks at 950~ TiNiSn samples (Cook et al., 1996). The power factor sZtr of these alloys was seen to increase from about 7.5 • 10 -4 W/m-K 2 at 300 K quite rapidly to a peak value of 2.8 • 10-3 W/m_K 2 at 750 K, while the thermal conductivity • was found to lie in the ~ 4 to 6 W/m-K range at 300 K. The TE parameters obtained thus indicated Z T to be 0.4 to 0.5 at 750 K, which was quite an impressive value for an undoped material. On the other hand, more recent measure-

2

HALF-HEUSLERALLOYS

57

ments of x of TiNiSn have resulted in values of 8-9 W/m-K at 300 K (Hohl et al., 1999; Bhattacharya et al., 1999), which indicate that Z T for pure TiNiSn is less than 0.3 at 750 K. Meanwhile, extensive studies of halfHeusler phases (Uher et al., 1999a, 1999b; Hohl et al., 1999; Browning et al., 1999) have shown that lattice thermal conductivity actually accounts for most of the x measured. Based on the room-temperature S and p values listed in Table I, SZa is seen to lie below ~10 -3 W/m-K 2 at 300 K in the undoped VEC = 18 phases. A similar range of power factor values is also found in alloys with the heavy lanthanides replacing the M elements (Cook et al., 1996; Mastronardi et al., 1999; Sportouch et al., 1999a). The MNiSn and MCoSb phases exhibit n-type TE properties. In contrast, the LnPdBi, LnPdSb, and LnNiSb phases exhibit p-type properties. Despite the low room-temperature Z T of less than 0.03 obtained in undoped MNiSn phases, the latter have been utilized as model systems for studying lattice effects on thermal conductivity. Indeed, in view of the promising Seebeck coefficients, a considerable amount of effort has been devoted to reducing the high lattice thermal conductivity, which is the least desirable property of the halfHeusler alloys. Utilizing the high substitutability of half-Heusler alloys, one may employ elemental substitution at the sublattice sites or addition into the unfilled sites to increase phonon scattering. An initial effort was made to reduce x by introducing mass fluctuation via substitution in the M sites of MNiSn phases (Hohl et al., 1997). A significant decrease of x in the pseudoquaternary solutions by as much as a factor of 2 was found (Table I). More recent studies of the pseudoternaries (Uher et al., 1999b; Hohl et al., 1999), including those with substituted Ni sites (Browning et al., 1999), have also observed substantial reductions in ~. It is also found that K increases with the improved structural order in well-annealed samples (Uher et al., 1999b). However, the lowest room-temperature x values ~ 4 W/m-K obtained are still significantly higher than those seen in compounds such as BizTe 3, skutterudites, and other phases with much more complex structures (see Ch. 3 of this volume and Chs. 4 and 5 of Vol. 69). The K(T) plots for ZrNiSn, (Zro.5HFo.5)NiSn, and (Zro.5HFo.5)(Nio.TPdo.3)Sn shown in Fig. 10 exemplify the effects of substitution with heavier elements in the M as well as Ni sublattices. It seems that K in many of the samples studied tends to increase, and in some cases quite noticeably, in the high T region (e.g., ~ 200-300 K). Whether such a trend is due to intrinsic or radiation effects will need to be clarified. Also, quantitative analysis of K(T) in light of various phonon scattering models is just beginning. Recent substitution studies have also involved the Sn sublattice. In the (Zr, Hf)(Co, Pt)(Sn, Sb) system, x(300 K) has been reduced to ~ 3 W/m-K (Fig. 10, Table I). Comparably low values of ~ can be obtained in some ternary LnNiSb (Sportouch et al., 1999a) and LnPdSb (Mastronardi et al.,

58

S. JOSEPH POON

4O 3O

v

10

IT-------_____

0

50

!

t

I

1

I

100

150

200

250

300

T (K) FIc. 10. Thermal conductivity versus temperature for alloys based on ZrNiSn. Top to bottom plot (refer to values at 300 K): ZrNiSn (Uher et al., 1999b), (Zro.sHf0.5)NiSn (Uher et al., 1999b; comparable ~c is also reported by Hohl et al., 1999), (Zro.5Hfo.5)(Ni0.vPdo.3)Sn (Browning et al., 1999), and (Zro.65Hfo.35)(Coo.65Pto.35)(Sno.35Sbo.65) (Bhattacharya et al., unpublished results).

1999) half-Heusler alloys. The mechanisms for achieving low K values in the M- and R-based alloys certainly deserve further investigation. Based on the observed trend, it is likely that ~c of the half-Heusler phases can be further reduced upon alloying the Sn/Sb site with heavier elements. Meanwhile, it is also relevant to compare the half-Heusler phases with the AgPbBiQ 3 (Q = S, Te) phases (Guseinov et al., 1972), since the latter crystallize in the NaCl-type structure. These cubic phases have lattice constants of ~ 6 A, and one of the lattice sites is occupied by the three elements Ag, Pb, and Bi in a statistically disordered fashion. Very low ~c values of about 1 W/m-K have been measured in these alloys (Sportouch et al., 1999b). Similar low tr values have been reported for the cubic LnaAu3Sb4-type phases with a more complex crystal structure (Young et al., 1999). Apart from structural complexity, the preceding studies suggest that low thermal conductivity can be realized in solid solutions that contain heavy elements, particularly in the readily realized lanthanide compounds. Using the foregoing ~c values, the room-temperature Z T values of the transitionmetal class (Hohl et al., 1999; Uher et al., 1999b; Browning et al., 1999) and lanthanide class (Sportouch et al., 1999a; Mastronardi et al., 1999) of half-Heusler alloys are found to reach ~0.06. In the more complex Ln3Au3Sb 4 phases, Z T is found to reach ~0.1 (Young et al., 1999).

2 HALF-HEUSLERALLOYS

59

IV. Doped Alloys Further studies of bandgap characteristics and carrier conduction have been carried out in doped alloys. The variation of electronic structure properties as a function of the type and level of doping have been studied. Evolution of bandgap properties, particularly with regard to the SC-SM crossover, has been of fundamental interest in the study of semiconductors (Mott and Davis, 1971). Since it is known that the power factor of a thermoelectric material is optimized in the vicinity of the SC-SM crossover (Mahan et al., 1997), another objective is then to search for effective dopants in achieving high power factor. Considering the crystallography of halfHeusler phases, the variation of properties will need to be examined for each of the three different sites in the MgAgAs structure. Systematic doping studies have been performed in several of the MNiSn and MCoSb phases.

1.

TRANSPORTPROPERTIES

a.

MNiSn ( M = Ti, Zr, Hf) Alloys

For TiNiSn, the Ti sublattice is doped with Mn and V, the Sn sublattice with Sb, and the Ni sublattice with Co. Isoelectronic substitutions of the Ti site with Hf, and the Ni site with Pd and Pt, are also performed. For some dopants, it may be necessary to extend the study to higher concentrations, and thus the issue of substitutability must be considered. In fact, VNiSn and MnNiSn are not listed as MgAgAs-phase forming systems in Pearson's Handbook of Crystallographic Data for lntermetallic Phases (Villars and Calvert, 1991). MnNiSn, but not VNiSn, is found to form the MgAgAs structure. MnNiSn is metallic with p(295 K) ~ 100 ~t~-cm. Meanwhile, the Ti site in TiNiSn can only be substituted with V up to ~ 5%. Temperaturedependent resistivity plots for the Sb-doped alloys are shown in Fig. 3. The SM behavior of the Sb-doped alloys can be seen from the plots. Semimetallic p(T) behavior is also seen in the V-doped alloys. To minimize effects due to thermal scattering at high T, the alloys' conductivities measured at 4.2 K are compared in Fig. 11, which shows p(4.2 K) versus the fraction (x) of sublattice sites being occupied by dopants. The plots reveal different dependences on dopants. Although the dependence of p(4.2 K) on V and Sb indicates a moderate to rapid increase in SM behavior, respectively, at increasing dopant content, substitution with Co shows a rather slow decrease in p(4.2 K). Previous study had shown a similar slow decrease for Xco < 0.5 (Pierre et al., 1994). Knowing that MnNiSn is metallic, it is surprising to find Mn-doped XMn < 0.07 alloys possess the largest p values

60

S. JOSEPH POON

-.._._

10-2

E o I

1 0 -3

(2.

10-4 0

5

10

15

20

X (at. % dopant) FIG. 11. p(4.2 K) of TiNiSn doped with V( 9 Sb (f-l), Co (V), and M n ( + ) ( P o o n et al., 1999). x - fraction of sublattice sites occupied by dopants (see text for site substitution).

T A B L E III TRANSPORT PARAMETERSOF WELL-ANNEALED SAMPLES OF DOPED ALLOYS BASED ON SOME OF THE VEC = 18 PHASES LISTED IN TABLE I

Alloy (Zro.98Nbo.o2)NiSn (Zro.98Yo.o2)NiS n Zr(Nio.9Feo.1)Sn ZrNi(Sno. 9 7 S b o . o 3) ZrNi(Sno. 95Ino.o 5) ZrNi(Sno.99Ino. 01) (Tio.99Nbo.o 1)NiSn (Tio. 995Vo. oo 5)C ~ TiNi(Sno. 95Sbo.o 5) TiNi(Sno.995Sbo.oos) H f N i(S no. 99Ino. 01) (Zro.sHfo.s)Ni(Sno.99Sbo.o 1) (Zro. 5Hfo. 5)Ni(Sno.99Bio.o 1) (Zro. 5Hfo.s)o.99Vo.o xNiSn (Zro. 5Hfo.5)o.99Nbo.o 1NiSn (Zro. 5Hfo. 5)0.99Ta0.01NiSn (Tio.sHfo.5)o.99Nbo.o 1NiSn (Tio.sHfo.5)Ni(Sno.975Sb0.025)

Ref.

a

a a a

b c d d

b b b

/9

K

S

(f~cm)

(W/m-K)

(#V/K)

1.0 • 10- 3 1.8x 10 -3 3.6 • 10-3 8.4 • 10 -4 3.0x 10 -2 4.2 • 10- 2 1.1 x 10-3 4.2 • 10- 2 1.6 x 10 -4 8.3 • 10 -4 5.4 • 10 -2 8 • 10 -4 4.9 • 10 -3 5.6 • 10 -3

121 +1 125 - 79 -134 - 99 -126 -420 -85 - 150 - 104 -- 150 --242 -246

#

-3)

(cm2/Vs)

- -

- -

8.1

15 10 7.4 6.6 7.7

a

1.5 • 1 0 - 3

6.8

-

a

1.0 • 10- 3 1.5 x 10 -3 6 x 10 -4

5.4 5.2 8

- 147 - 181 - 100

" H o h l et al., 1999. bUher et al., 1999b. CPoon et al., 1999. dBhattacharya et al., 1999; also unpublished results.

nef f

(102~

0.21 ~0.1

7.3 -~ 20

0.31 2.1 0.61

3.73 37.2 20.6

3.5

30

170

2 HALF-HEUSLERALLOYS

61

-50 ,,,,,,

-100 -150

=L v -200 09 -250

-300 -350

0

50

100

150

200

250

300

T(K)

FIG. 12. Thermopower of (Zro.sHfo.s)Ni(Snl_xSbx) (11) (Uher et al., 1999b). Upper plot: x = 0.01; lower plot: x = 0.005. Thermopower of TiNi(Snl_xSbx) (O) (Ponnambalam et al., unpublished data). From top to bottom: x = 0.02, 0.005, and undoped TiNiSn.

(see also Subsection 3 of this section). The latter alloys indicate SM behavior at low T (Fig. 3). Previous studies have also found efficient doping in the Sn site of (Zro.sHfo.5)NiSn (Uher et al., 1999b) and in the MSn rock-salt substructure of ZrNiSn and (Mo.sHfo.5)NiSn, where M = Ti or Zr (Hohl et al., 1999). In the efficiently doped alloys, both p and S decrease quite rapidly. Some representative results are illustrated in Table III. Figure 12 shows the dependence of thermopower on Sb content in TiNiSn (Bhattacharya et al., 1999) and (Zro.sHfo.5)NiSn (Uher et al., 1999b). The thermopower at ambient temperature is seen to change from ~ - 3 0 0 #V/K in undoped TiNiSn to ~ - 100 p V / K in the alloy doped with 2% Sb. Meanwhile, p at 300 K decreases much more rapidly, from ~ 1 0 4 pfl-cm to 400 #f~-cm (Fig. 3). For (Zro.sHfo.5)NiSn doped with 0.5% Sb, a high S(300 K) similar to that of the undoped sample is retained while p(300 K) decreases by more than one order of magnitude. Similar doping effects on p and S are observed in Sb-doped ZrNiSn (Hohl et al., 1999) and Bi-doped (Zro.5Hfo.5)NiSn (Uher et al., 1999c). In addition to electron doping, the latter groups of authors have demonstrated the effects on S ( T ) due to hole doping by using Y and In as dopants. For the MNiSn systems, the room-temperature carrier contents as estimated from R n are found to increase from --~ 1018-1019 cm -3 in the undoped alloys 0 -~ 1020 cm -3 in the doped alloys (Table III).

62

b.

S. JOSEPH POON

M C o S b ( M = Ti, Zr, H f ) Alloys

Based on chemical analysis, the compositions of nominal TiCoSb ingots are found to be TiCo~ 1.o28bo.99. Therefore, the reproducibility of results on several doped alloy systems using TiCoSb, TiCoSbl.o5, and TiCoo.98Sb1.o2 as host alloys is examined and found to be good. Doping transition metals into the three sublattices of TiCoSb produces results that are drastically different from those seen in TiNiSn. For example, substituting the Co site with 10% Ni and 1% Pt reduces p(4.2 K) by two and one order of magnitude, respectively. However, upon doping V and Mn in 0.5% of the Ti sites, p(4.2 K) is increased by nearly three orders of magnitude. The latter results are surprising since both VCoSb and MnCoSb are metallic. Other dopants studied by us have also included Sn and Te in the Sb site and Nb in the Ti site. The latter group of dopants exhibits doping trends similar to those seen in TiNiSn. Figure 13 summarizes p(4.2 K) as a function of the fraction of sublattice sites x occupied by dopants. TiCoSb doped with Sn has been previously studied with regard to the S C - S M and magnetic-state crossovers (Tobola et al., 1998; see also Subsection 4 of Section III). To further study the anomalous behavior observed in the V-doped TiCoSb alloys, we turn now to the p(T) plots shown in Fig. 7 for (Til_xVx)CoSb at low V content. Plotted on the log lo scale, p(T) for the V-doped samples displays shoulderlike features between 50 K and 100 K. The trend suggests two activation processes, with one occurring at T < 50 K and the other at T > 100 K. Hall coefficient RH(T ) for the x = 0.005 alloy is found to resemble the trend seen in (Zro.95Hfo.o5)NiSn (see Fig. 4). Below --~30 K, R H is large at ~ - 8 5 0 cm3/C and it varies little with T. At T > 30K,

102 "-'E 101 o

00

~ 10 -1 ~tc~ 10 .2 9

10 4

. . . . . 0

1

2

3

4

5

6

7

8

9

10

X (at. % dopant)

FIG. 13. p(4.2 K) of TiCoSb doped with V(V), Nb([-]), Pt(O), Mn(+), Te(V), Sn, and Ni (@) (Poon et al., 1999). x = fraction of sublattice sites occupied by dopants (see text for site substitution).

2 HALF-HEUSLERALLOYS

63

it decreases quite rapidly in magnitude and reaches ~ - 0 . 8 cm3/C near 295K. The almost constant effective carrier density of ~ 1 0 1 6 cm -3 at T < 30K suggests that the activated conduction at low T is of the nonmetallic type, possibly because of the presence of a mobility edge near the Fermi level. The latter scenario is further substantiated by the fact that the carrier mobility of ~ 2 0 cm2/V-s at 4.2 K is one order of magnitude smaller than those obtained for TiCoSb, TiNiSn, and ZrNiSn. At higher T, the concomitant rapid decrease in p and rapid increase in effective carrier density suggest that charge transport is largely taken over by the carriers that are thermally activated from a lower band to the conduction band. In fact, at T > 295 K, R n becomes positive at ~0.8 cm3/C. To examine the activated process at T > 100 K, we have plotted In p versus l I T for the x = 0.005 alloy in the inset to Fig. 7. The linear dependence above ~ 100 K yields an activation energy of ~0.05 eV. The x = 0.01 alloy also exhibits the same activation energy. At increasing V content, the signature of activated conduction at T > 100 K gradually diminishes. An S C - S M crossover is seen at x ~ 0.15. Future study will also include investigation of magnetic properties in the crossover region.

c.

Lanthanide Alloys

(Ln, Ln')PdSb (Mastronardi et al., 1999) and (Zr, Er)Ni(Sn, Sb) (Sportouch et al., 1999a) solid solutions have been studied. Unlike M-based phases, the observed properties of the Ln phases do not indicate a clear trend as a function of VEC. Apparently, because of the mixed valence nature of the rare earth elements, the alloys' VEC values are not easily defined. However, significant changes in transport properties upon alloying are observed, which seem to indicate some kind of doping effects.

2.

BANDGAP FEATURES INFERRED FROM DOPING STUDIES

The carrier transport properties reported in the previous section are now examined in light of bandgap structure. In view of the strong dependence of doping effects on host alloys, it is pertinent to review the role of the sublattice in gap formation. Bandstructure calculations performed on ZrNiSn (Ogut and Rabe, 1995; Mahanti et al., 1999), TiNiSn (Ogut and Rabe, 1995; Tobola et al., 1998), and TiCoSb (Tobola et al., 1998) have indeed shed light on this issue. For ZrNiSn and TiNiSn, the strong pd hybridization in the ZrSn and TiSn substructures is found to play an important role in the gap formation by significantly reducing band crossing near the Fermi level. However, the actual opening of the gap is due to the coupling between the Ni sublattice and ZrSn or TiSn substructure via the d

64

S. JOSEPHPOON

orbitals. A much more sensitive doping response in the TiSn and ZrNi rock-salt substructure compared to that in the Ni site would suggest that the rock-salt substructure must play a more prominent role in bandgap formation than indicated by the calculations. On the other hand, the results in TiCoSb cannot be readily related to band-structure calculations. It is plausible that the IV-IV type TiSn and IV-V type TiSb substructures possess different electronic structures, and thus the mechanisms of gap formation are different for TiNiSn and TiCoSb. In fact, it has been elaborated that the above-mentioned bandgap precursor in ZrSn is absent in the band structure of III-V type YSb: the rock-salt substructure in half-Heusler YNiSb (Mahanti et al., 1999; Larson et al., 1999). The semimetallic behavior observed at low T in all of the TiNiSn alloys, excepting the Mn-doped ones, is reminiscent of that seen in ZrNiSn alloys. In Subsection 2 of Section III, the SM trend observed at low T in the pure alloys is attributed to a bandgap structure with the Fermi level located in the region where the electron and hole bands overlap. However, different from undoped ZrNiSn and HfNiSn, which exhibit near-zero and almost temperature-independent negative thermopower at low T (Uher et al., 1999a, 1999b), the thermopower in TiNiSn is seen to decrease quite rapidly with increasing T at low T (Fig. 12). Meanwhile, R n is found to remain negative up to ambient temperature in both pure and doped TiNiSn alloys. These results indicate that as a host compound, TiNiSn exhibits stronger n-type behavior than ZrNiSn, (Zro.sHfo.5)NiSn, and HfNiSn. However, upon doping the Sn sites, the thermopowers of both TiNiSn and (Zro.sHfo.5)NiSn exhibit a distinct near-linear temperature dependence. Meanwhile, quite large room-temperature S values are retained at low doping level. As can be seen in Fig. 12, (Zro.sHfo.5)NiSn and TiNiSn doped with 0.5% Sb possess room temperature values of ~ - 200 p V / K (Uher et al., 1999b) and - 1 5 0 p V / K (Bhattacharya et al., 1999), respectively. Similar large S(300 K) values have also been reported for transition-metal-doped ZrNiSn and (Zro.sHfo.5)NiSn (Hohl et al., 1999). The near-linear S(T) dependence and the quite high room-temperature values retained in the semimetallic phases have implications on high-temperature TE properties, to be discussed in Subsection 4 of this section.

3.

IMPURITYBAND TRANSPORT PROPERTIES

To examine the results of doping Mn in TiNiSn and Mn and V in TiCoSb, we refer to the considerable amount of electronic structure calculations performed on Ti-based half-Heusler alloys that contain magnetic dopants such as Co, Mn, and Fe mentioned in Subsection 4 of Section III. It is found that Co and Mn, with their larger ionic charges, tend to retain their valence electrons upon dissolution in the host alloys (Tobola et al.,

2

HALF-HEUSLER ALLOYS

65

0 -50

~

-100 -150 -200

~" -250 -300 -350 -4O0

0

50

100

150

200

250

300

350

T(K) FIG. 14. Thermopower versus temperature for V-doped TiCoSb alloys (Poon et al., 1999). The x = 0.0, 0.005, 0.01, and 0.015 alloys are symbolized by Q, O, V, and V, respectively.

1998; Kaczmarska et al., 1999). Thus, it is plausible that doping Mn and V has led to the formation of an impurity band below the conduction band and a reduction in the effective carrier density. The electronic states in the region where the conduction band overlaps with the impurity-band tail, and where the Fermi level lies, can either be localized or extended, depending on whether the Anderson condition is fulfilled or not (Subsection 3 of Section III). This scenario of bandgap structure can provide a qualitative explanation for the low carrier density and activated conduction at low T, the activated conduction at high T, and the large thermopower of ~ - 4 0 0 #V/K in the V-doped TiCoSb alloys (Fig. 14). At low T, conduction is via activation near the mobility edge. Based on the upturn exhibited in p at low T, the hopping energy is estimated to be of the order of 0.001 eV. Meanwhile, some samples show variable-range-hopping behavior below ~ 150 K. At high T, more carriers are excited from the main part of the impurity band to the conduction band. The observation of more distinctive semiconducting features in TiCoSb is consistent with the presence of a gap that is wider, and presumably better defined, than those in other half-Heusler alloys (Pierre et al., 1997; Tobola et al., 1998). Different from TiNi(Sn, Sb), (Til_xVx)CoSb exhibit near-zero S values below ,~60 K and large negative S values at ambient temperature (Fig. 14). Meanwhile, R n measured on one of the samples (x = 0.005) is found to be positive at high T. These results again indicate band overlapping and heavy electron band mass at the Fermi level. It is noted that S(T) is enhanced significantly in the V-doped alloys. The enhancement can be ascribed to the reduction in carrier density near the Fermi level due to the formation of an impurity band in the V-doped alloys mentioned above (see discussion on S(T) enhancement due to the formation of an intragap band

66

S. JOSEPHPOON

in Subsection 2 of Section III). In view of the proximity of magnetic V d-states at the Fermi level in VCoSb, the evolution of bandgap structure and the nature of semiconductor-metal crossover in V-doped TiCoSb are naturally of fundamental interest (see also Subsection 1 of this section). 4.

THERMOELECTRIC PROPERTIES

Prospective thermoelectric materials are to be selected from alloy systems that exhibit a significant reduction in the resistivity, while retaining a reasonably large Seebeck coefficient upon doping (Tables III and I), so that a significant enhancement in the power factor can be obtained. It is based on this working criteria that several efficient dopants for enhancing SZcr in MNiSn (M = Ti, Zr, Hf) and their solid solutions are identified. Based on several studies, the most efficient dopants for these phases appear to belong to the group Va elements V, Nb, and Ta (Hohl et al., 1999) and group Vb elements Sb and Bi (Uher et al., 1999b; Poon et al., 1999; Bhattacharya et al., 1999). The highest $2o'(300 K ) values achieved are reported to vary from ~2.2 x 10 -3 W/m-K 2 (Hohl et al., 1999) to 2.8 x 10 -3 W/m-K 2 (Uher et al., 1999b; Poon et al., 1999) and to 4 • 10-3 W/m_K 2 (Bhattacharya et al., 1999). These values are three to five times larger than those found in the undoped phases (Subsection 5 of Section III). In addition to the dopant species, optimization of $2o " depends on the annealing condition and dopant concentration. Furthermore, the promising results on TE properties at high T (Subsection 5 of Section III and the following) coupled with high thermal stability indicate that these alloys are likely to be used above ambient temperature. a.

Annealing Effects on p and S

Sb is found to be an efficient dopant for TiNiSn (Figs. 3, 11, and 12). A maximum power factor of ~ 4 x 10 -3 W/m-K 2 is measured at ambient temperature. Figure 15 shows $2o " as a function of Sb content in the Sn sublattice. These power factor values, which are high, are comparable to those measured on practical TE materials (Goldsmid, 1986; also articles in C R C Handbook of Thermoelectrics, edited by D. M. Rowe, 1995; Tritt, 1999). In view of these results, measurements on Sb-doped (Tio.sHfo.5)NiSn solid solutions are carried out. The latter is selected for its lower thermal conductivity. In Fig. 16, data on p and S, with both reported at 300 K, are plotted as a function of Sb content in the Sn sublattice. The data are for samples annealed at 900~ for 14 hours (anneal-I samples) and at 900~ for 14 hours followed by 750~ for 10 days (anneal-II samples). Comparing samples treated under anneal-I and anneal-II, the percentage reduction in p is noticeably larger than the percentage change in S after the

2 HALF-HEUSLERALLOYS

67

50

%, 40 30

~ 20

8

t3

0 0.00

0.01

"

o

0

0.02

0.03

FIG. 15. Power factor at 300 K plotted versus Sb dopant content (x = fraction of Sn site doped). Upper plot is for alloys based on TiNiSn (Bhattacharya et al., 1999). Lower two plots are for alloys based on (Tio.sHf0.5)NiSn annealed at 800~ hours. (C)) and 800~ hours + 750~ days (i-q) (Poon et al., 1999).

anneal-II treatment. For the ternary TiNiSn doped with 1-3% Sb, the magnitude of S is found to increase by about 10% while p changes little after the anneal-II treatment. Long-term annealing is performed to promote structural order. For bandgap compounds, improving the structural order is expected to lead to enhanced p and S, which are indeed observed in ZrNiSn and (Zr, Hf)NiSn (Uher et al., 1999b), and are also seen in TiNiSn studied by us. In doped alloys, long-term annealing can in addition promote

1000 800 E

0 d :::L Q.

600

v

.

.

.

.

.-.0--9o0O~14h ~

.

60

j/~~-

-

v

.75

400 200

.

.

.

.

.

.

-80

.

.

-

-100

-

-120

140

0.010 0.015 0.020 0.025 0.030 0.035

FIG. 16. p and S at 300 K versus Sb dopant content (on Sn site) for the two sets of alloys based on (Tio.sHfo.5)NiSn shown in Fig. 15.

68

S. JOSEPH POON

the uniform dissolution of dopant atoms. Enhancing the uniform dissolution of dopants has the effect of decreasing p and S. Clearly, long-range atomic diffusion must be involved in achieving the desired structural state. Apparently, the combined annealing effects on the structural order and the dissolution of dopant atoms have led to the observed trends in p and S. Consequently, the sEa of both TiNiSn and (Tio.sHfo.5)NiSn are enhanced after the anneal-II treatment. Figure 15 illustrates annealing results for Sb-doped (Tio.sHfo.5)NiSn. It is seen that SErf(300 K ) reaches a maximum of -~2.6 • 10 -3 W/m-K 2 at the x - 0 . 0 2 5 composition. The power factor obtained in the solid solution system is comparable to that reported for Sb-doped (Zro.sHfo.5)NiSn (Uher et al, 1999b).

b.

Thermoelectric Figures o f Merit

Z T values at ambient temperature for the undoped half-Heusler systems are typically smaller than ~0.06 (Subsection 5 of Section Ill). In doped (Zr, Hf)NiSn alloys, these values are found to reach ,--0.12 (Hohl et al., 1999; Uher et al., 1999b). Hohl et al. (1999) first reported measurements of doped half-Heusler alloys up to 700 K. They found that S2tr increases to ,~4 x 10 3 W / m - K 2. Using x values extrapolated to high temperatures, Z T was estimated to be ~0.5 at 700 K. More recently, similar sEa and Z T values at 700 K have also been measured on Sb-doped (Zro.sHfo.5)NiSn alloys (Cook et al., 1999). Measurements of p and S on Sb-doped TiNiSn and (Tio.sHfo.5)NiSn alloys doped with x = 0.025 Sb have also been extended to 700 K to obtain the high-T power factor value (Pooh et al., 1999). The results are shown in Fig. 17. It is seen that S is doubled while p increases

0

0.8 00

-50

00_ ~00

oo o0 ~o

v' -100 > :::L -150 cO

-250

000

O0~

00

0000

OOlOO0

0

0.6

o~

eeeOeeoe

~

-200

0.7

o~

o 0

E

O

o.5 d

0000

000

O0o

0.4

E a.

0.3

100 200 300 400 500 600 700 800

0.2

T(K) FIG. 17. S and p of (Tio.sHfo.5)Ni(Sno.975Sbo.o25) measured up to 680 K (present data).

2 HALF-HEUSLERALLOYS

69

60

%, 50 ~: 40 :=L

30 20

0910 0

0

200

400

600

T(K) FIG. 18. Measured power factor versus temperature. From top to bottom plot: TiNi(Sno.95Sbo.o5) (Bhattacharya et al., 1999), (Tio.5Hfo.5)Ni(Sno.975Sbo.o25) (present data), and (Zro.sHfo.5)o.99Tao.oxNiSn (Hohl et al., 1999).

by a factor of 1.8 when T increases from 300 to 700 K. The net result is that SZa acquires a value of ~ 5.7 x 10 -3 W/m-K 2 at 700 K, more than twice its room temperature value (Fig. 18). Especially for the Ti alloys, the power factors of half-Heusler phases obtained at high T are comparable to those found in state-of-the-art TE alloys (Tritt, 1999; also papers in M R S Symp. Proc., 545 (1999) and Proc. I C T '99 (1999)). Compared with the Zr alloys (Fig. 18), the more rapid increase in S2o'(T) above ambient temperature in the Ti alloys can be attributed to the sublinear and nearly linear temperature dependence of S in the former and latter alloys, respectively (Hohl et al., 1999; also Figs. 12 and 17). The x of (Tio.sHfo.5)NiSn, which has been measured from 4.2 to 300 K, is found to have a high value of 8 W/m-K at 300 K. As a result, Z T at 300 K is only ~0.1, comparable to those reported for the doped (Zro.sHfo.5)NiSn alloys mentioned earlier. In order to estimate Z T in the high-T(300-700 K) region, we have extrapolated x(T) to 700 K using data obtained below 300 K. The electronic thermal conductivity X~l is estimated using the W i e d e m a n n - F r a n z law. Upon extracting the lattice thermal conductivity XL using the relation x L = x - ~%, we find XL to be nearly constant between 150 and 300 K. Thus, Xe is assumed to be constant above 300 K. Using the estimated K(T), we have plotted Z T versus T up to 680 K in Fig. 19. Z T at 680 K is found to be 0.46. It is interesting to note that from the S2tr and Z T versus T trends that Z T at 800 K can be easily extrapolated to a value of ~0.6. For comparison, current state-of-the-art thermoelectric materials BizTe 3, PbTe, and SiGe have Z T ~ 1 at 300, 700, and 1000 K, respectively (Tritt, 1999).

70

S. JOSEPH POON

0.5 0.4

IN

0.3 0.2 0.1 0.0

.~_ _m~r

0

''t/

,

,

,

200

400

600

800

T(K) FIG. 19. Estimated dimensionless figure of merit Z T versus temperature for the alloy shown in Fig. 17 (present data).

c.

Origin of High Power Factor and Implications for TE Properties

High $2o can in principle be achieved in a semiconductor that has a bandgap Eg larger than about 5 to 10 times kaT and a large value of Nm*3/21a. In the latter product term, N is the degeneracy of the parabolic bands, # is the carrier mobility, and m* is the carrier band mass (DiSalvo, 1999; Mahan et al., 1997). Given the fact that cubic crystal structures have the largest N value, the task is to obtain high/~ and m*. For a quite heavily doped semiconductor (technically a strong semimetal), where S ~ T/E F, with E F being the Fermi energy measured from the bottom of the conduction band, it is more appropriate to optimize the parameter Nm*2#. Assuming that the conduction is electron dominated, as inferred from the negative S(T) and RH(T ) measured, we have estimated # and m* for the x = 0.025 Sb-doped (Tio.sHfo.5)NiSn alloy. The mobility/~ obtained from RH/p is ,--30 cm2/V-s at ambient temperature. The band mass m* can be estimated from S(T) as well as the enhancement factor 7/7c = m*/me in the electronic contribution to the specific heat, where me and Ye denote freeelectron values. These estimates yield m* ~ 2.5m e and 2-3m e, respectively. Similar low carrier mobility and large electron band mass have also been noted in Sb-doped (Zr, Hf)NiSn alloys (Uher et al., 1999b). For comparison, typical TE alloys have much larger/t of several hundred cmZ/V-s at ambient temperature and m* smaller than or near rne (Goldsmid, 1986; Mahan et al., 1997). Thus, for the half-Heusler alloys, it is the moderately large effective mass that leads to the quite high SZa values. The generally low carrier mobilities obtained for half-Heusler alloys are pointed out in Subsection 3 of Section III, where it is reported that both

2 HALF-HEUSLERALLOYS

71

ternary and multinary alloys exhibit comparable low/~ values. Despite the low #, large Seebeck coefficients are obtained in many of these alloys. Meanwhile, the strong chemical disorder that is present in multinary alloys is found to be effective in reducing ~c (Subsection 5 of Section III). These findings have revealed some unique relationships between the various thermoelectric properties. The important implication is that one could then tune/~ and S quite independently of each other to optimize ZT.

d.

Preliminary Results on ( T i, Nb)CoSb Alloys

It has been pointed out that for optimal thermoelectric performance at temperature T, the energy gap Eg of a semiconductor must be at least 10 times kBT (Mahan, 1989). Thus, the presumed wider-gap (Eg ~ 0.9 eV) TiCoSb half-Heusler phases (Tobola et al., 1998) may be considered for optimal TE performance at temperatures as high as ~ 1000 K. The doping pattern in TiCoSb is different from that of TiNiSn because 4d and 5d transition metal dopants are found to be more efficient than sp elements in enhancing the SM behavior of TiCoSb (Fig. 13). Also, in view of the large thermopowers of ~ - 2 6 0 pV/K to - 4 3 0 / W / K and only moderately high resistivities of ~ 40 m~-cm observed in V-doped TiCoSb at room temperature (Fig. 14), we have begun to assess the TE properties of Nb-doped TiCoSb. For the x = 0.005 to 0.02 alloys, SZa is found to be in the range ~1.5-2.1 x 10 .3 W/m-K 2. As mentioned earlier, the stoichiometry of TiCoSb has yet to be optimized. Therefore, future studies will involve doping effects on TE properties of MCoSb (M = Ti, Zr, Hf) compounds. Present results indicate that the potential of these wider-gap half-Heusler alloys is worthy of further investigation.

V. Summary Recent studies of ternary and multinary half-Heusler phases have revealed many interesting electrical and magnetic properties in this class of bandgap intermetallic compounds. Most of the half-Heusler alloys that exhibit semiconducting behavior at elevated temperatures are found to be semimetallic at low temperatures, which is characteristic of the metal-based narrow-gap compounds studied to date. Since the chemistry of half-Heusler compounds can be conveniently modified, selective doping of the three sublattices in the crystal structure can be carried out to allow tuning of the electronic structure and lattice vibration properties. Impurity band states, previously not reported for metal-based systems, are observed in some of the doped alloys investigated. In ferromagnetic half-Heusler phases, the

72

S. JOSEPH POON

crossover from semiconductor to semimetal is found in the region where magnetism emerges. According to band-structure calculations, the ferromagnetism is of the itinerant and highly spin-polarized type. Large thermopower and moderate resistivity, with the former attributable to the existence of heavy carrier mass, are measured at and above ambient temperature. Upon doping, a high power factor comparable to those reported for the state-ofthe-art thermoelectric alloys is measured. The thermoelectric dimensionless figure of merit Z T, which is found to reach ~0.6 at 800 K, underscores the potential of half-Heusler alloys as a new class of prospective thermoelectric materials above ambient temperature. The Z T value is to be further enhanced by reducing the lattice contribution to thermal conductivity. Unlike other thermoelectric materials, half-Heusler alloys exhibit low carrier mobility. Thus, attempts made to reduce the lattice thermal conductivity are seen to have relatively less effect on the power factor. The measured properties are found to be sensitive to annealing conditions, with the latter presumably determining the underlying crystallographic order.

ACKNOWLEDGMENTS I acknowledge the contributions of many collaborators with whom I have worked in the past 2 years. My collaborators include V. P o n n a m b a l a m and Y. Xia at the University of Virginia; T. M. Tritt, S. Bhattacharya, A. L. Pope, P. N. Alboni, and R. T. Littleton of the Thermoelectric Materials Laboratory at Clemson University; and V. M. Browning and A. C. Ehrlich at the Naval Research Laboratory. The author's research is supported by the N S F Grant # D M R 97-00504. The research of Professor Terry Tritt at Clemson is supported by D A R P A / A R O 4/:DAAG55-97-1-0-267, O N R #N00014-981-0271, and O N R 4/:N00014-98-1-0444.

REFERENCES F. G. Aliev, A. I. Belogoroklov, N. B. Brandt, V. V. Kozyrkov, R. V. Scolozdra, and Yu. V. Stadnyk, Optical Properties of the Vacancy MNiSn Lattices (M = Ti, Zr, Hf), Pis'Ma. V. Zh. Eksp. Teor. Fiz., 47, 151 (1988). F. G. Aliev, N. B. Brandt, V. V. Moschalkov, V. V. Kozyrkov, R. V. Scolozdra, and A. I. Belogorokhov, Gap at the Intermetallic Vacancy System RNiSn (R = Ti, Zr, Hf), Z. Phys. B, 75, 167 (1989). F. G. Aliev, V. V. Kozyrkov, V. V. Moschalkov, R. V. Scolozdra, and K. Durczewski, Narrow Band in the IntermetallicCompounds MniSn (M = Ti, Zr, Hf), Z. Phys. B, 80, 353 (1990). C. Berger, Electronic Properties of Quasicrystals Experimental, in Lectures on Quasicrystals, edited by F. Hippert and D. Gratias. Les Editions de Physique, Les Ulis, p. 463 (1994). S. Bhattacharya, V. Ponnambalam, A. L. Pope, P. N. Alboni, Y. Xia, T. M. Tritt, and S. J.

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Poon, Thermoelectric Properties of Sb-Doping in the TiNiSnl_xSb x System, Proc. ICT '99: 18th International Conference on Thermoelectrics, edited by T. M. Tritt et at., IEEE Catalog No. 99TH8407, p. 336 (1999). V. M. Browning, S. J. Poon, T. M. Tritt, A. L. Pope, S. Bhattacharya, P. Volkov, J. G. Song, V. Ponnambalam, and A. C. Ehrlich, Thermoelectric Properties of the Half-Heusler Compound (Zr, Hf)(Ni, Pd)Sn, in Thermoelectric Materials 1998--The Next Generation Materials for Small-Scale Refrigeration and Power Generation Applications, edited by T.M. Tritt, M. Kanatzidis, H. B. Lyon, Jr., and G. D. Mahan, MRS Symp. Proc., 545, 403 (1999). B. A. Cook, J. L. Harringa, Z. S. Tan, and W. A. Jesser, TiNiSn: A Gateway to the (1,1,1) Intermetallic Compounds, in Proc. ICT '96: 15th International Conference on Thermoelectrics, IEEE Catalog No. 96TH8169, p. 122 (1996). B. A. Cook, G. P. Meisner, J. Yang, and C. Uher, High Temperature Thermoelectric Properties of MNiSn (M = Zr, Hf), Proc. ICT'99, p. 64 (1999). R. A. de Groot, F. M. Mueller, P. G. van Enger, and K. H. J. Buschow, New Class of Materials: Half-Metallic Ferromagnets, Phys. Rev. Lett., 50, 2024 (1983). R. A. de Groot, F. M. Mueller, P. G. van Engen, and K. H. J. Buschow, Recent Developments in Half-Metallic Magnetism, J. Magn. Magn. Mater., 54-57, 1377 (1986). S. K. Dhar, S. Ramakrishnan, R. Vijayaraghavan, G. Chandra, K. Satoh, J. Itoh, Y. Onuki, and K. A. Gschneidner, Jr. Magnetic Behavior of YbNiSb, Phys. Rev. B, 49, 641 (1994). F. J. DiSalvo, Thermoelectric Cooling and Power Generation, Science, 285, 703 (1999). C. B. H. Evers, C. G. Richter, K. Hartjes, and W. Jeischko, Ternary Transition Metal Antimonides and Bismuthides with MgAgAs-Type and Filled NiAs-Type Structure, J. Alloys and Compounds, 252, 93 (1997). H. J. Goldsmid, Electronic Refrigeration. Pion, London (1986) G. D. Guseinov, E. M. Godzhaev, Kh. Ya. Khalilov, F. M. Seidov, and A. M. Pashaev, Complex Semiconductor Chalcogenides, Inorg. Mater. (USSR), 8, 1377 (1972). H. Hohl, A. R. Ramirez, W. Kaefer, K. Fess, Ch. Thurner, Ch. Kloc, and E. Bucher, A New Class of Materials with Promising Thermoelectric Properties: MNiSn (M = Ti, Zr, Hf), in Thermoelectric Materials--New Directions and Approaches, edited by T. M. Tritt, M. Kanatzidis, H. B. Lyon, Jr., and G. D. Mahan. MRS Symp. Proc., 478, 109 (1997). H. Hohl, A. P. Ramirez, C. Goldmann, G. Ernst, B. Woelfing, and E. Bucher, New Compounds with MgAgAs-Type Structure: NblrSn and NblrSb, J. Phys.: Condens. Matter, 10, 7843 (1998). H. Hohl, A. P. Ramirez, C. Goldmann, G. Ernst, B. Woelfing, and E. Bucher, Efficient Dopants for ZrNiSn-Based Thermoelectric Materials, J. Phys.: Condens. Matter, 11, 1697 (1999). W. Jeischko, Transition Metal Stannides with MgAgAs and MnCu2A1 Type Structure, Metall. Trans. A, 1, 3159 (1970). I. Karla, J. Pierre, and R. V. Skolozdra, Physical Properties and Giant Magnetoresistance in RNiSb Compounds, J. Alloys Compounds, 265, 42 (1998). K. Kaczmarska, J. Pierre, J. Beille, J. Tobola, R. V. Skolozdra, and G. A. Melnik, Physical Properties of the Weak Itinerant Ferromagnetic CoVSb and Related Semi-Heusler Compounds, J. Mayn. Mayn. Mater., 187, 210 (1998). K. Kaczmarska, J. Pierre, J. Tobola, and R. V. Skolozdra, Anderson Localization of 3d Mn States in Semi-Heusler Phases, Phys. Rev. B, 60, 373 (1999). Ch. Kloc, K. Fess, W. Kaefer, K. Riazi-Nejad, and E. Bucher, Crystal Growth of Narrow Gap Semi-conductors for Thermoelectric Applications, in Proc. ICT '96, p. 155 (1996). R. Kuentzler, R. Clad, G. Schmerber, and Y. Dossmann, Gap at the Fermi Level and Magnetism in RMSn Ternary Compounds (R = Ti, Zr, Hf and M = Fe, Co, Ni), J. Magn. Magn. Mater., 104-107, 1976 (1992). P. Larson, S. D. Mahanti, S. Sportouch, and M. G. Kanatzidis, Electronic Structure of Rare-Earth Nickel Pnictides: Narrow-Gap Thermoelectric Materials, Phys. Rev. B, 59, 15660 (1999).

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G. Mahan, Figure of Merit for Thermoelectrics, J. Appl. Phys., 65, 1578 (1989). G. Mahan, B. Sales, and J. Sharp, Thermoelectric Materials: New Approaches to an Old Problem, Physics Today, 50, 42 (1997). S. D. Mahanti, P. Larson, D. Y. Chung, S. Sportouch, and M. G. Kanatzidis, Electronic Structure of Complex Bismuth Chalcogenides and Other Narrow-Gap Thermoelectric Materials, MRS Syrup. Proc., 545, 23 (1999). K. Mastronardi, D. Young, C. C. Wang, P. Khalifah, R. J. Cava, and A. P. Ramirez, Antimonides with the Half-Heusler Structure: New Thermoelectric Materials, Appl. Phys. Lett., 74, 1415 (1999). N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials. Clarendon Press, Oxford (1971). S. Ogut and K. M. Rabe, Band Gap and Stability in the Ternary Intermetallic Compounds NiSnM (M = Ti, Zr, Hf): A First-Principles Study, Phys. Rev. B, 51, 10443 (1995). P. M. Oppeneer, V. N. Antonov, A. N. Yaresco, A. Ya. Perlov, and H. Eschrig, Fermi Level Pinning of a Massive Electron State in YbBiPt, Phys. Rev. Lett., 78, 4079 (1997). D. Orgassa, H. Fujiwara, T. C. Schulthess, and W. H. Butler, First-Principles Calculation of the Effect of Atomic Disorder on the Electronic Structure of the Half-Metallic Ferromagnet NiMnSn, Phys. Rev. B, 60, 13237 (1999). J. Pierre, R. V. Skolozdra, and Yu. V. Stadnyk, Influence of Cobalt Vacancies and Sn Substitution on the Magnetic Properties of TiCo2Sn Heusler-Type Compound, J. Magn. Magn. Mater., 128, 93 (1993). J. Pierre, R. V. Skolozdra, Yu. K. Gorelenko, and M. Kouacou, From Nonmagnetic Semiconductor to Itinerant Ferromagnet in the TiNiSn-TiCoSn Series, J. Magn. Magn. Mater., 134, 95 (1994). J. Pierre, R. V. Skolozdra, J. Tobola, S. Kaprzyk, C. Hordequin, M. A. Kouacou, I. Karla, R. Currat, and E. Lelievre-Berna, Properties on Request in Semi-Heusler Alloys, J. Alloys Compounds, 262-263, 101 (1997). V. Ponnambalam, A. L. Pope, S. Bhattacharya, Y. Xia, S. J. Poon, and T. M. Tritt, Transport Properties of Half-Heusler Alloys Based on ZrNiSn, Proc. ICT '99, p. 340 (1999). S. J. Poon, Electronic Properties of Quasicrystals: An Experimental Review, Adv. Phys., 41, 303 (1992). S. J. Poon, Q. Guo, P. Volkov, and F. S. Pierce, Insulating and Semiconducting Phases of Quasicrystalline and Crystalline Aluminum Alloys, J. Non-Cryst. Solids, 205-207, 1 (1996). S. J. Poon, T. M. Tritt, Y. Xia, S. Bhattacharya, V. Ponnambalam, A. L. Pope, R. T. Littleton, and V. M. Browning, Proc. ICT '99, p. 45 (1999). A. L. Pope, T. M. Tritt, M. A. Chernikov, and M. Feuerbacher, Thermal and Electrical Transport Properties of the Single-Phse Quasicrystalline Material: A17o.sPd2o.9Mn8.3, Appl. Phys. Lett., 75, 1854 (1999). G. A. Slack, in CRC Handbook of Thermoelectrics, edited by D. M. Rowe. CRC Press, Boca Raton, FL, p. 407 (1995). S. Sportouch, P. Larson, M. Bastea, P. Brazis, J. Ireland, C. R. Kannenwurf, S. D. Mahanti, C. Uher, and M. G. Kanatzidis, Observed Properties and Electronic Structure of RNiSb Compounds (R = Ho, Er, Tm, Yb, and Y): Potential Thermoelectric Materials, MRS Syrup. Proc., 545, 421 (1999a). S. Sportouch, M. Bastea, P. Brazis, J. Ireland, C. R. Kannewurf, C. Uher, and M. G. Kanatzidis, Thermoelectric Properties of the Cubic Family of Compounds AgPbBiQ 3 (Q - S, Se, Te): Very Low Thermal Conductivity Materials, MRS Syrup. Proc., 545, 123 (1999b). M. Terada, K. Endo, Y. Fujita, and R. Kimura, Magnetic Properties of C1 h Compounds: CoVSb, CoTiSb and NiTiSb, J. Phys. Soc. Jpn., 32, 91 (1972). J. Tobola, J. Pierre, S. Kaprzyk, R. V. Skolozdra, and M. A. Kouacou, Crossover from Semiconductor to Magnetic Metal in Semi-Heusler Phases as a Function of Valence Electron Concentration, J. Phys.: Condens. Matter, 10, 1013 (1998).

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T. M. Tritt, Holey and Unholey Semiconductors, Science, 283, 804 (1999). C. Uher, S. Hu, J. Yang, G. P. Meisner, and D. T. Morelli, Transport Properties of ZrNiSn-Based Intermetallics, in Proc. ICT '97: 16th Int. Conf. on Thermoelectrics, IEEE Cat. No. 97TH8291, p. 485 (1997). C. Uher, J. Yang, and S. Hu, Materials with Open Crystal Structure as Prospective Novel Thermoelectrics, MRS Syrup. Proc., 545, 247 (1999a). C. Uher, J. Yang, S. Hu, D. T. Morelli, and G. P. Meisner, Transport Properties of Pure and Doped MNiSn (M - Zr, Hf), Phys. Rev. B, 59, 8615 (1999b). C. Uher, J. Yang, and G. P. Meisner, Thermoelectric Properties of Bi-Doped Heusler Alloys, in Proc. ICT '99, p. 56 (1999c). P. Villars and L. D. Calvert, in Pearson's Handbook of Crystallographic Data for Intermetallic Phases, 2nd ed., Vol. 1-4 (ASM International, Metals Park, OH, 1991). P. Volkov and S. J. Poon, Semiconducting Behavior of the Intermetallic Compound A1ERU, Europhys. Lett., 28, 271 (1995). D. Young, K. Mastronardi, P. Khalifah, C. C. Wang, R. J. Cava, and A. P. Ramirez, LnaAuaSb4: Thermoelectric with Low Thermal Coonductivity, Appl. Phys. Lett., 74, 3999 (1999).

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SEMICONDUCTORS AND SEMIMETALS, VOL. 70

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Overview of the Thermoelectric Properties of Quasicrystalline Materials and Their Potential for Thermoelectric Applications Terry M. Tritt and A. L. Pope DEPARTMENTOF PHYSICSAND ASTRONOMY CLEMSON UNIVERSITY CLEMSON, SOUTH CAROLINA

J. W. Kolis DEPARTMENT OF CHEMISTRY CLEMSON UNIVERSITY CLEMSON, SOUTH CAROLINA

I. QUASICRYSTALS: BACKGROUND AND INTRODUCTION . . . . . . . . . . II. QUASICRYSTALS: STRUCTURAL AND MECHANICAL PROPERTIES . . . . . . III. SYNTHETIC METHODS FOR THE GROWTH OF QUASICRYSTALS . . . . . . .

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Thermal Conductivity . . . . . . . . . . . . . . . . . . . . Thermal and Electrical Transport on AIPdMn for T h e r m o e l e c t r i c s

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Quasicrystais: Background and Introduction

Quasicrystals form a group of relatively new materials that abound in interesting and even fascinating properties, such as high mechanical strength, high hardness, low friction coefficient, low thermal conductiv77 Copyright 9 2001 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-752179-8 ISSN 0080-8784/01 $35.00

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ity, and high corrosion resistance. (Visit the Web site at http:// www.quasi.iastate.edu/for more information on quasicrystals.) Shechtman, Blech, Gratias, and Cahn first presented these materials to the world in 1984. In a search for higher strength and lighter aluminum alloys, Shechtman (in 1982) discovered a rapidly solidified (melt spun ribbon) aluminum and manganese alloy that had formed small nodules. X-ray diffraction of these nodules revealed a 10-fold diffraction pattern that is a classically forbidden crystal structure. Shechtman's logbook shows his disbelief and excitement in this data with the entry "10-fold???" (Shechtman and Lang, 1997). He knew that this was either a twinned material or something never seen before. Through experiment, Shechtman quickly ruled out the concept of this sample being a twinned crystal, but his results and insights were met with much skepticism. No one had a theoretical explanation to fit a material that did not behave as a crystal should according to the theory of Bravais vectors. As stated in the MRS Bulletin, two years after the initial discovery of tenfold symmetry, a paper was written and sent to the Journal of Applied Physics. The paper was rejected because it was believed that this discovery would be of little interest to the physics community as a whole. The paper was slightly rewritten and sent to Physics Review Letters, where it was quickly accepted and has sparked much interest in the physics community, an interest that still persists today (Shechtman and Lang, 1997). An excellent overview and introduction to the history and properties of these materials is given in a recent MRS Bulletin dedicated to quasicrystals (see Shechtman and Lang, 1997). At the time of discovery, quasicrystalline materials were metastable, but were indeed confirmed to exist. With their short lifespan, research in this field was initially focused on understanding the structure of these materials, with much less effort put forth to understand transport in these materials. In 1988 a group from the Tohoku University in Sendai discovered quasicrystals that were stable at room temperature, A1CuFe and A1CuRu. Since that time, many different stable quasicrystals have been synthesized, primarily binary and ternary materials, although recently quartenary quasicrystals have been grown (Guo and Poon, 1996; Fisher et al., 1999a). As with most new discoveries, time and concentrated research efforts were key factors in perfecting the quasicrystal growth and stability. The issue of stability and growth is addressed more completely in a later section on the growth of quasicrystals. In the early 1990s, quasicrystals were stable enough and could be grown in large enough quantities for bulk transport measurements to be performed with accuracy and reproducibility of the data. More than 100 quasicrystalline systems exist at present, which are seen to have 5-, 8-, 10-, or 12-fold symmetries, all classically forbidden. This field of study has become a "hotbed" for the scientific community with thousands of articles in the literature.

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A characteristic of quasicrystals is the low thermal conductivity values throughout a broad temperature range with values below 10 W/m-K from 2 to 1000 K. At room temperature, the thermal conductivity is typically between 2 and 3 W/m-K. This is considerably lower than metallic glasses, which are of a similar family of materials but are amorphous intermetallic materials. Thermal transport in quasicrystals is in many ways similar to that of a glass or amorphous material. It is somewhat surprising to find such low values of thermal conductivity in a material containing approximately 70% A1, since A1 exhibits thermal conductivity values more than two orders of magnitude larger than those of quasicrystals. Electrical resistivity in quasicrystals is observed to decrease with increasing temperature until the material dissociates (1000~ Quasicrystals exhibit a resistivity that is marginally metallic or in which semimetal-insulator transitions are prevalent (Guo and Poon, 1996). The magnitude of the electrical resistivity remains essentially unchanged throughout the entire temperature range, varying by less than 30%, typically. Quasicrystals have also been observed to be very sensitive to chemical composition, where slight changes in composition can severely influence transport and mechanical properties. Quasicrystals have high mechanical strength and are hard, making it difficult for cracks to propagate. One of the negative qualities of quasicrystals is the fact that they are brittle, making it difficult to use them in bulk form. However, the use of films helps to minimize and even eliminate the brittleness associated with the quasicrystals in bulk. Quasicrystals are extremely hard (800-1000 kg/mm 2) when compared to aluminum alloys (185 kg/mm2; Kang et al., 1993). The hardness can be increased by heat treatment with no microstructure changes observed at room temperature (Quasicrystals, special issue of Advanced Materials and Processes, June 1991). Quasicrystals also have a very low coefficient of friction when compared to aluminum alloys and steel (Dubois, 1993). Quasicrystals are oxidation and corrosion-resistant. Like many aluminum alloys they form a passivating layer of alumina at the surface that inhibits any further oxidation or corrosion. They also have surface energies that may make binding to these materials difficult (Rivier, 1993). In addition to these properties, quasicrystals are cheap to produce (except for the Pd alloys), with the price for the powder necessary for these applications being less than 10 dollars per pound (A1CuFe). With physical and mechanical properties of quasicrystals lying between an amorphous material and a crystalline solid, many potential applications for quasicrystals were recognized soon after their discovery. In 1998, cooking pots and pans with quasicrystalline coatings were marketed. These coatings are comparable to other nonstick coatings, with the exception that initially other nonstick coatings have a higher level of performance. However, the appeal of the quasicrystalline coating is seen in duration. Over time

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the performance of other coatings declines while quasicrystal coatings retain their initial level of performance. Quasicrystalline coatings are impervious to peeling and scratching due to the use of metal cooking utensils. The first commercial application of quasicrystals was in the early 1990s by Sandvik Steel in Sweden. They found that quasicrystalline precipitates (which impede dislocations) in their steel hardened the steel up to 3000 MPa. These steels are used in surgical instruments and more recently in some components of a wet/dry electric shaver sold by Philips Electronics. Precipitates lead to corrosion resistance as well as added strength and durability. The reader is encouraged to visit the Web page at Ames Lab on quasicrystals, which describes and links to these many applications (www.quasi.iastate.edu). Many of the potential applications and general properties of quasicrystals are discussed in several recent articles by the group at Ames Lab and in an article by Wiley Press (Goldman et al., 1996; Thiel and Dubois, 1999; Brown, 1999).

II.

Quasicrystals: Structural and Mechanical Properties

A traditional crystalline material can be completely described using a unit cell and a set of basis vectors. Recall that in 1848, Bravais, a French crystallographer, showed that there were only 14 different ways to arrange atoms in a three-dimensional space using translational and rotational symmetry, that is, the 14 Bravais lattices. From these arguments the well-known cubic, hexagonal, tetragonal, and associated structures have been described (Janot, 1997; also see for example Kittel, 1976). With these structures, if viewed in a two-dimensional space, it is easily seen that only certain angles are allowed. An integral number of unit cells must be able to completely fill all space. For example, consider a square. The angles in a square are 90 ~ angles, so with 360 ~ of freedom, four squares can fit together. Therefore, only two-, three-, four-, and six-fold symmetry exist. All other symmetries were said to be impossible and indeed were not observed. Quasicrystals are materials that exhibit a classically forbidden symmetry, for example, a crystal with fivefold symmetry. This symmetry is obvious from the diagrams shown in Figs. 1 and 2 (Goldman et al., 1996). Quasicrystalline materials are well ordered on the atomic scale, yet are aperiodic. This, of course, was part of the initial reservations concerning the existence of this new class of materials. Our previous thinking was that order and periodicity must go hand in hand. How can a material that is not perfectly periodic yield such sharp diffraction peaks in its X-ray diffraction pattern? It has been shown that stable quasicrystals exist, so how can these materials fill all space? Certainly one cannot fill a two-dimensional space with simple pentagons, as one can with squares or hexagons. However, with

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FIG. 1. Icosahedral symmetry is a property of the most-studied quasicrystalline alloys: They share the rotational symmetries of the 20-sided polyhedron called an icosahedron. They have 15 twofold, 10 threefold, and 6 fivefold rotational symmetry axes. Lines connecting edges of the icosahedron on opposite sides of the solid are twofold symmetry axes (left). Lines connecting triangular faces on opposite sides of the solid are threefold symmetry axes (middle). Lines connecting vertices on opposite sides of the solid are fivefold symmetry axes (right). (Fig. 2 in Goldman et al., 1996.)

FIG. 2. Structural differences between a crystal and quasicrystal can be seen by comparing the electron diffraction patterns of crystalline (top) and quasicrystalline (bottom) phases of an aluminum alloy. The spots are made by electrons coherently scattered from planes of atoms parallel to the beam direction. Each spot corresponds to a set of parallel planes. A crystal's symmetry can be seen by picking a motif of spots and determining how often it recurs in the pattern. Patterns along different directions in the cubic crystalline alloy have twofold (left), threefold (middle), and fourfold (right) rotational symmetry. The quasicrystal, on the other hand, exhibits twofold (left), threefold (middle), and fivefold (right) rotational symmetry. The fivefold symmetry is forbidden to crystalline materials. Another difference is that the quasicrystal has aperiodic rather than periodic translational order, as the spacing of the spots of the diffraction patterns shows. (Fig. 3 in Goldman et al., 1996.)

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the use of two different rhombohedrons, a fat and thin diamond, all space can be tiled. An Oxford mathematician, Roger Penrose, first described this tiling in 1974. Penrose hit on this tiling and rules in an attempt to send a friend a unique get-well card. The placement of the shapes is governed by complex matching rules. If the matching rules are not followed there are some places that are crystalline, some that are amorphous, and a small percentage by chance actually forming a quasicrystalline phase. Steinhardt extended this tiling to three dimensions (Steinhardt, 1997). This is not basic crystallography, but the long-range order is evident, and it is easy to see that there is no typical rotational order in this system. Recently, the International Union of Crystallographers' Commission on Aperiodic Crystals formally redefined a crystal as any substance that has an X-ray diffraction pattern with sharp, bright spots. Formerly, the textbook definition of a crystal was that it had to have periodically repeating units.

III. 1.

Synthetic Methods for the Growth of Quasicrystals

SCOPE AND INTRODUCTION

For synthetic considerations quasicrystals can be divided into two general classes, namely those that are thermodynamically stable at some temperature, and those that are only metastable. Thus far, all synthetic methods require some form of melt of the component metals to form a liquid phase followed by either direct quenching or a subsequent heat treatment after solidification. The variation of properties predominantly involves the method of solidification and the heat treatment after formation of the solid phase. All synthetic procedures to date involve the direct reaction of the metals. Thus far there have been no examples of any "chemical" transformations in the usual sense in regard to synthesis. Consequently, there are no reported procedures using either chemical or electrochemical reduction of metal oxides or halides to form quasicrystalline phases, for example. Moreover, no further chemical modifications (with the obvious exception of various annealing protocols) have been reported yet. The original work by Schechtman was conducted on metastable systems and, as such, required rapid-quench methods to trap the quasicrystalline phases (Shechtman and Lang, 1997). For a number of years, it was believed that the quasicrystalline materials were inherently unstable, and as such, could only be prepared using quench techniques. Research on the physical properties of quasicrystals was substantially enhanced by the discovery that a number of quasicrystalline phases have thermodynamic equilibrium stability at some temperatures. This was important in that it allowed for the

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preparation of good-quality bulk samples and ultimately single grains of the appropriate phases. Furthermore, it enabled investigators to use the more classical and well-understood methods to manipulate and characterize good-quality samples. In this section we review the general techniques for synthesis and single grain growth. The synthesis of several systems is discussed in more detail. Further discussion of the general aspects of quasicrystal synthesis has been presented (Ochin, 1997). The quench methods will be mentioned only briefly, since these techniques are well known from metallurgical processing. There are several other highly specialized techniques, which have been reviewed elsewhere (Janot, 1992). Also, the emphasis here is on bulk samples, so the substantial amount of work involving thin films is not mentioned (Besser and Eisenhammer, 1997).

2.

GENERALCONSIDERATIONS

In the most general sense, parent metals are melted using a variety of methods followed by heat treatment. In all cases, the highest purity metals should be used. The quasicrystalline phases often involve very oxophilic metals such as aluminum or zinc; thus, rigorous exclusion of oxygen and water during heating should be observed. During heating and any subsequent annealing, an atmosphere of inert gas should be placed over the ingot because metals such as aluminum and manganese have an appreciable vapor pressure around 1000~ leading to evaporation and resultant stoichiometric irregularities. In general, parent metals can be used as starting materials. These are mixed in the appropriate stoichiometry and processed for subsequent melting. However, for many of the aluminides, the reaction with transition metals is often very exothermic, and to prevent inconsistencies due to local heating, it is often best to "premake" binary metal aluminide phases that can be remelted with the appropriate components. When heating oxophilic metals such as aluminum or zinc at high temperatures, the choice of container is always critical. Many authors use arc-melting as the method for initial ingot formation. This is essentially a "containerless" technique in that the ingot is usually melted on a noninteracting water-cooled copper hearth. This method is particularly useful for refractory metals such as rhenium. Care should be taken to always "premelt" a zirconium ingot as an oxygen getter in the chamber before melting the ingot. Pellets should be flipped and remelted several times to ensure thorough alloying. If induction heating or resistive heating is used for the initial ingot melt, a container is necessary. In this case, quartz is not suitable in that it is too reactive at these temperatures (ca. 1000~ and will readily contaminate the product with oxides and silicon. Care should be

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TERRY M. TRITT ET AL.

taken to contain the ingot in an inert material. Alumina is suitably inert for these purposes, but the crucibles should be baked out at yellow heat before use. Of course the alumina cannot be worked and thus cannot be completely sealed. Thus, the crucible should be sealed within a quartz tube during heating. If this is done under an inert gas, the tube should be well pumped out before use and the inert gas should be well scrubbed for oxygen and water. An excellent alternative is the use of tantalum tubing, which is generally completely inert to molten metals. Furthermore it can be welded shut in an arc furnace, so the reaction can be performed in a completely sealed container. However, the Ta will react with oxygen above 500~ so the sealed ampule should be further jacketed in a sealed quartz tube under vacuum. Above 1100~ the quartz will begin to soften, and care should be taken to place an atmosphere of inert gas in the tube if it is to be reheated above ll00~ The quartz will soften and flow above 1150~ and is not useful above this temperature. Fortunately, few quasicrystals need to be heated above l l00~ to achieve melting. Tantalum suffers the additional disadvantage of being somewhat expensive. In general, most authors use an arc-melting technique for initial ingot preparation. Induction heating is a common alternative. Resistive heating is occasionally reported but is less common. It should be noted that the elemental composition of the original ingot is very important. Some quasicrystalline phase diagrams such as A1PdMn have a fairly large compositional width (Tsai et al., 1990a). Others are quite narrow. For example, slow cooling of a melt of composition A162Cuz5.5Fe12.5 provides good-quality samples of dodecahedral phase, whereas a composition of A163.5Cu24Fes2.5 leads to a classical microcrystalline phase (Janot et al., 1991, and references therein). After the melt process, the resultant ingot is typically annealed for a certain time. The annealing protocols vary widely and range from slow cooling from the melt to room temperature to rapid quenching. They can be performed near the melting point (within 100~ of m.p.) or several hundred degrees lower depending on the system. The annealing step is probably the single most important aspect of the quasicrystal synthesis, as the phase behavior is typically extremely complex and many different phases, both quasicrystalline and classical, are accessible for most compositions. The annealing ingot is considerably less reactive than the initial melt. However, it is a metallic material near its melting point and should be treated with considerable care. Again, care should be taken to rigorously exclude oxygen and water during the heat treatment. Since annealing is typically performed below 1000~ simple resistive heating is the method of choice. The annealing can generally be accomplished in alumina crucibles in quartz ampules. Quartz can be used directly if not heated above about 800~ but alumina is still preferred for its inertness.

3

3.

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85

ANALYSIS

The unusual nature of the quasicrystalline phases often requires a particular choice of analytical methods. The chemical analysis of the quasicrystalline phases is important because of the very complex phase behavior of these compounds. Many of the phases are structurally stable over a reasonable compositional range of atom percent. However, this is a subtle point and great care should be exercised in varying the chemical composition of the phases. Also, the physical properties of the materials may change dramatically with chemical composition, even though the phases appear similar. EDX is an excellent qualitative tool and has the advantage of convenience, but often suffers from lack of sensitivity and can be considered semiquantitative, at best. The two most common methods of quantitative chemical analysis are inductively coupled plasma (ICP) and electron microprobe. ICP has the advantage of providing very accurate quantitative values of the bulk sample. However, it requires a fairly large amount of sample (50-100 mg) and is a destructive technique that requires that the sample be dissolved in acids. Microprobes provide very good data as well, but the narrowness of the electron beam means that data are only provided for a very small area at a time. It is becoming apparent that the microstructure of even highquality single grains is very complex, with a variety of phases embedded in heterogeneous matrices. The localized nature of the microprobe analysis technique is significant when dealing with these complex systems. Structural information is generally provided by various diffraction techniques. The quasicrystals, in general, provide very sharp and characteristic diffraction patterns, as do their powders. Thus, powder X-ray diffraction is a convenient technique for identification of known bulk samples. Of course it should be noted that actual structural determination using X-ray diffraction is more problematic and is a specialized technique (Steurer, 1990; Cahn, 1980; Janot, 1994, Chs. 3 and 4 and references therein). Structural information can be obtained from single-phase X-ray diffraction (see, for example, Kycia et al., 1993). However, X-ray powder patterns are easily obtained and generally used to provide excellent "fingerprints" for bulk phase confirmation. Electron diffraction also provides excellent structural information, in that the 5-, 8-, and 10-fold rotation symmetry can often be readily observed in very small sections of samples (Beeli et al., 1991). Other physical properties can be determined on single grain materials using electron diffraction as well (Franz et al., 1999). Neutron diffraction is an excellent source of structural information; the structure of the A1PdMn phase in particular has been investigated in detail by this method (Janot et al., 1989; Boudard et al., 1991). Neutron diffraction has also been used to obtain magnetic structural information on the newer magnetic rare earth zinc phases REsMgz4Znso where RE = rare earth element (Charrier et al., 1997).

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Thermal analysis is also a particularly important method of characterization. The various quasicrystalline phases often have extremely complex and subtle phase behavior even over small temperature ranges; thus, careful thermal analysis is important for stability studies and characterization. When this is coupled with electron or neutron diffraction, extremely detailed structural and phase stability can be determined (Dong et al., 1991; Audier et al., 1993).

4.

SYNTHESISOF STABLE PHASES

The first isolation of a thermodynamically stable quasicrystalline phase was the preparation of A16Li3Cu, which could be isolated as an icosahedral phase after slow cooling and annealing of a melt cast film (Dubost et al., 1986). Shortly thereafter, a bulk sample of A165CuzoFe15 was prepared by annealing a melt prepared ingot just below its melting point (Tsai et al., 1987). Subsequently, a large number of other phases were found to be stable at various temperatures (Tsai, 1997). Many of these phases are still at the margins of stability, though, and are often stable only at a temperature substantially above room temperature. Thus, very careful attention must be paid to the annealing profile. Oftentimes, the desired phase is stable a few degrees below the freezing point for the ingot. Thus, a premelted ingot must be annealed at a specific temperature and then quenched through the cooled stage. Probably the most stable and generally useful phase is the A172PdzoMn 8 phase, which can be prepared readily by annealing a melted ingot about 100~ below its melting point for various amounts of time. This material has a substantial phase width and is amenable to substitution. However, even this material has an extraordinarily complex phase behavior and great care must be taken during the annealing profile and process. The observation by Tsai that many of the stable quasicrystalline phases obey the Hume-Rothery rules was a critical tool for the expansion of this field (Hume-Rothery, 1926). He observed that many stable phases had an electron to atom ration of 1.75, which led him to the design of several new phases (Tsai et al., 1990b). More importantly it allowed him to freely substitute isoelectronic metals and pairs of metals. Thus, A17oPdzoRexo can be made like the corresponding Mn compound. Also, series such as A163Cuz5TM12 (TM = Fe, Ru, Os) can be prepared. Further, pairs of metals can also be substituted keeping the e/a ratio constant. So icosahedral phases such as AlvoPdzoCr5Fe 5 can be made. A substantial list of such combinations is given in Janot's Quasicrystals: A Primer (Janot, 1992). Little is known about the mixed materials. However, given the complexity of the parent phase, it is likely that their chemistry is extremely complex.

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5. GRAINGROWTH The discovery that a number of quaiscrystalline phases are thermodynamically stable at various temperatures led to the early preparation of large single grains of some phases, particularly in the A1-Pd-Mn system (Tsai et al., 1990b; Yokoyama et al., 1991). The ability to prepare stable phases allowed for the use of classical crystal growth methodologies, such as Bridgeman and Czochralski methods, and opened the way to detailed studies of physical properties (see, for example, Audier et al., 1993). Both Bridgeman and Czochralski methods are well-understood methods of single crystal growth, and very detailed techniques have been developed to overcome almost every limitation of crystal growth. Thus, air sensitivity of the melt and its reactivity toward the parent crucibles are no longer serious obstacles to the experienced crystal grower. As such, the quasicrystals are amenable to growth via these techniques. However, both have the common limitation that they involve solidification of the phase directly from the melt. Since most quasicrystalline phases are incongruently melting, this is a legitimate concern. Nevertheless, a number of authors report the use of both Czochralski pulling and Bridgeman growth to attain relatively large samples (Kycia et al., 1993; Ishimasa and Mori, 1992; de Boissieu et al., 1992; Yokoyama et al., 1992). A very careful study of a large Bridgeman-grown sample of the A1PdMn phase has been reported (Delaney et al., 1997). The workers observed that despite the uniform appearance of the boule, single grains could only be cut from a small part of the boule. Micrographs of these grains revealed that they consisted of multiphase particles intergrown within matrices of classical alloys such as AlaPd 2. Another problem unique to the quasicrystals is their tendency to form single grains that grow by contraction of melt areas within the boule. This tendency creates macroscopic voids within the sample, leading to problems with bulk property measurements. Problems such as these are inherent in the growth of large samples of quasicrystals and lead to the question of whether true uniform large single-phase grains can be grown on these compounds. One very exciting recent development has been the use of flux techniques, developed by Fisher and co-workers, to grow very large single grains of spectacular appearance and quality (Fig. 3) (Fisher et al., 1998, 1999a,b). This technique is an important breakthrough for a number of reasons, aside from the obvious high quality and physical appearance of the samples. Almost none of the known quasicrystals, even the thermodynamically stable ones, melt congruently. Often the stability region for the quasicrystalline phase is 10-100 ~ below the freezing temperature. Thus, classical melt growth techniques such as the Czochralski and Bridgeman growth described earlier form peritectics, and often the quasicrystalline phases form as

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TERRY M. TRITT ET AL.

FIc. 3. Icosahedral (left) and decagonal (right) quasicrystals grown at Ames National Laboratory. (From Fisher et al., 1999b, d.)

outgrowths of other phases after solid formation takes place. Quasicrystals are essentially growing as annealed phases. This complexity leads to the formation of multiple phases and problematic crystal growth. The use of flux methodology allows for the use of "off-stiochioimetric" fluxes that simplify the phase formation through slow cooling. This methodology was originally described for the rare-earth-containing phases RE8Mg42Znso, first prepared by Tsai et al. (1994). The original basis for this development was the careful development of the phase diagram, which showed that the stability region for the material is not far from the freezing point and that composition of the melt need not be far from the desired elemental formula (Langsdorf et al., 1997). Armed with this information, Fisher and co-workers were able to develop an elegant methodology for the growth of large, high-quality single grains of the icosahedral phase. The presence of molten magnesium means that oxides, even alumina, are unsuitable as containers because they will react with the melt, even at the modest temperatures needed for this reaction (< 800~ The ingot had an original stoichiometry of RE:Mg:Zn of 3:51:46. Thus, the workers fabricated tantalum tubes to contain the metal charge. The vessel also contained a Ta strainer to filter the crystals. The tube was welded shut under a small partial pressure of argon and the entire assembly sealed in a quartz tube to protect the Ta. The assembly was heated in a resistance furnace to 750~ which is sufficient to melt the flux. After a soak period the temperature was lowered to 650~ and then slowly cooled to 480~ over 4 days. During this period the icosahedral phase becomes saturated in the molten flux and begins to precipitate. The lack of nucleation sites and long cooling time means that well-formed single grains of the quasicrystalline phase can grow from the flux. At 480~ the still-molten Mg/Zn flux is decanted from the large single grains, which are isolated by the Ta screen. After cooling, the grains are isolated and careful examination of the surface of the grains reveals some

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hardened flux still adhering to the surface. The quality and size of the grains is remarkable, and the authors report that the size of the grains is only limited by the size of the Ta tubing. This flux growth technique is quite general and only requires that the desired quaiscrystalline phase be soluble in an appropriate flux stoichiometry and become insoluble and remain stable at a temperature which the flux is still molten so it can be removed. To demonstrate the generality of this method, the same workers were able to use essentially the same techniques to grow large single grains of A172Ni~Co17 (Fisher et al., 1999c) and A17~Pdz1Mn s (Fisher et al., 1999d). The A1-Ni-Co flux melts at considerably higher temperature, and a flux of A177Ni~0.sCo12.5 was melted at 1250~ and cooled over 150 hr to 1000~ after which it was decanted, leading to high-quality single grains with volumes of 0.8 cm s. In this case, the flux is less reactive, allowing for the use of alumina crucibles instead of the more expensive tantalum containers. The flux was decanted from one alumina crucible to another through quartz wool to strain out the crystals. A similar method was used for the A 1 - P d - M n system, in which large, high-quality single grains were grown from a flux of composition A173Pd~9Mn s and were cooled from 875 to 830~ for 120 hr. Again the reaction can be performed in alumina crucibles filtering the flux through quartz wool. In this case Ga can be partially substituted for A1 and new mixed grains can also be made. This flux growth technique has proven to be impressively successful in a short period of time. Given the inherent limitations of the Bridgeman and Czochralski techniques described earlier, this method is rapidly emerging as the method of choice for high-quality grain growth. It should be noted that the techniques known as top seed solution growth (TSSG) has emerged as the method of choice for many noncongruently melting compounds. TSSG is combination crystal pulling and flux growth. Thus, a previously prepared seed crystal is touched to the top of a flux containing a stoichiometry that is only slightly removed from the desired composition. The seed crystal is slowly removed from the flux with the appropriate phase, one hopes, crystallizing on the surface. Of course, the flux composition is constantly changing during the growth process, but if the stoichiometry is close the changes may be small enough not to affect the crystallization process significantly. To the authors' knowledge, TSSG has not yet been reported as a technique for grain growth, but given the recent success of nearstoichiometry flux growth, it may indeed prove to be the method of choice for the preparation of large samples of high-quality grains of appropriate quasicrystals. The vast majority of quasicrystals to date involve the use of either aluminum or zinc as the predominant element. However, several other classes of elements lead to new families of quasicrystals. Not surprisingly, gallium provides the basis for several novel quasicrystals. Both icosa-

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TERRY M. TRITT ET AL.

hedral and decagonal phases have been isolated. Early examples of stable decagonal and icosahedral gallium manganese phases have been reported that are analogous to the corresponding aluminum compounds (Tartas and Knystautas, 1994). More unusual formulations are Ga5oCoz5Cu25 and Ga46Fez3Cu23Si 8. These compounds are prepared by melting ingots in quartz tubes at 1300~ followed by quenching. The quenching leads to isolation of both icosahedral and decagonal phases. Annealing at 830~ leads to loss of the icosahedral phase but further growth of the decagonal phase. DSC also indicates that the decagonal phase is indeed stable at these conditions (Ge and Kuo, 1997). Silicon-based alloys can also form quasicrystalline phases. An early example of this is the formation of Pd6oU2oSi2o. This composition readily forms glassy alloys upon quenching the melt. However, annealing the glassy alloys at 400~ for 1 week leads to formation of a stable icosahedral phase (Poon et al., 1995). An unusual example of a quasicrystalline phase is TaTel. 6, which is prepared from the reduction of TaTe 2 with elemental Ta in welded Ta or Mo tubes at 1000-1050~ for several weeks. The compound is thermodynamically stable to 1500~ but readily decomposes in air to give an amorphous compound (Conrad et al., 1998; Audier et al., 1993). Based on these few examples it is clear that quasicrystals exist in a variety of chemical systems. No doubt many further examples of new formulations will continue to emerge in different intermetallic systems in the near future.

IV.

Introduction to Thermoelectric Materials

The application and phenomenon of thermoelectric energy conversion utilizes the Peltier heat generated when an electric current is passed through a thermoelectric material to provide a temperature gradient with heat being absorbed on the cold side and rejected at the heat sink, thus providing a refrigeration capability. Conversely, an imposed AT will result in a voltage or current, that is, small-scale power generation (Goldsmid, 1986; Tritt, 1996, 1999). The essence of defining a good thermoelectric material lies primarily in determining the material's dimensionless figure of merit, Z T - ~ 2 6 T / ~ . , where ~ is the Seebeck coefficient, o the electrical conductivity, 2 the total thermal conductivity (2 = 2~. + 2E; the lattice and electronic contributions, respectively) and T is the absolute temperature in kelvins. The Seebeck coefficient, or thermopower, is related to the Peltier effect by I I = a T = Q p / I , where FI is the Peltier coefficient, Qp is the rate of heating or cooling, and I is the electrical current.* The efficiency (r/) and coefficient of performance (COP) of a thermoelectric device are directly related to the *All the "Q-terms" discussed in this paper relate to rate of heat transfer or power related to that phenomenon.

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figure of merit of the thermoelectric material or materials. Both r/and COP are proportional to (1 + Z T ) 1/2. There are a number of excellent references that discuss the aspects of the materials and the measurement, as well as thoroughly discussing the field of thermoelectric materials (Shechtman et al., 1984; Shechtman and Lang, 1997; Wood, 1988; Rowe and Bhandari, 1983; Egli, 1960). An excellent introduction to the phenomenon is given in the earlier chapters of this volume. Also given are many of the criteria and challenges in finding and investigating a new material or class of materials for potential thermoelectric applications (see chapters by Goldsmid, Kanatzidis, and Tritt and Browning in Volume 69 of this series). Semiconductors have long been the materials of choice for thermoelectric applications. The most promising materials typically have carrier concentrations of approximately 1019 carriers/cm 3. The power factor, 0~20", is typically optimized through doping to give the largest Z. High-mobility carriers are most desirable, thus yielding the highest electrical conductivity for a specific carrier concentration. In addition, to improve the figure of merit of these materials, attempts are made to lower the lattice thermal conductivity without decreasing the power factor proportionally and thus further increasing the figure of merit. Glenn Slack has described the behavior of an ideal thermoelectric material as a "phonon-glass, electron-crystal" (Slack, 1995). These materials would have poor thermal transport like a glass while maintaining the good electronic properties of a crystal.

V.

Quasicrystals as Thermoelectrics?

A question arose for us a few years ago concerning the potential of quasicrystals for thermoelectrics. Their already low thermal conductivity (2 < 3 W/m-K) and the fact that this thermal conductivity varies little with temperature, composition, or other factors led us to the investigation of these materials for potential thermoelectric applications. One of the first questions to investigate was the temperature dependence and magnitude of the electrical conductivity and thermopower in these materials. We found that the thermopower was the biggest challenge and was greatly dependent on the composition of the specific quasicrystal, the quasicrystalline family, and the sample history. All these aspects are discussed in the following section. Most of our work has centered on the A1PdMn family of materials, which also are discussed briefly in a later section. There was very recently a theoretical analysis of the potential of quasicrystals for thermoelectrics, with predictions that a figure of merit as high as 1.6 may be possible in these materials (Macia, 2000). This again gives us encouragement for these materials. The temperature dependence of resistivity in quasicrystals appears to be governed by weak localization and electron-electron interaction effects.

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TERRY M. TRITT ET AL.

Localization occurs in these materials when a small amount of transition metal is found in a nonmagnetic material. These materials may display a localized moment that is determined by Coulomb interactions and to a smaller degree spin-orbit interaction. The electrical resistivity in these systems is observed to increase as the quasicrystalline perfection increases, contrary to Matthiessen's rule (electrical resistivity increases as defects and impurities increase) (Poon, 1992). Matthiessen's rule states that P = PPHONONS + Pl PI = Pimpurities qt_ Pdefects

where the total resistivity depends on the phonons, impurities, and defects in the system. It is also interesting to note that the electrical conductivity in these systems increases with temperature to the highest temperatures of measurement (T ~ 1000 K) (Mayou, 1993). The properties of quasicrystals are observed to be very sensitive to composition. Small changes or variations in the quality or preparation of samples can greatly influence the thermal and electrical properties of these materials. One of the most promising properties of the quasicrystals for thermoelectrics is the fact that they have intrinsically low values of thermal conductivity that are essentially sample independent. Low thermal conductivity is maintained in quasicrystalline materials despite the quality or composition of the sample. Thermal conductivity from 2 to 1000 K has been observed to be below 10 W/m-K for the entire temperature range. Quasicrystals have been observed to have structural coherence up to 8000 A, giving A1PdMn the large unit cell desirable for low thermal conductivity (Bancel, 1991). Quasicrystals already exhibit favorable values of electrical resistivity for thermoelectrics (as low as p ~ 1 0 - 2 1 0 - 4 f~-cm, but can be very high for the A1PdRe system, for example) in relation to their potential as a thermoelectric material. An advantage of quasicrystals is the fact that the resistivity and thermopower can be changed through modifications of the composition without sacrificing their low thermal conductivity. The biggest challenge in these materials is in obtaining a large ~hermopower. Two basic theories or ideas exist when discussing thermopower in quasicrystals: electronic processes and band structure effects. When evaluating the electronic contributions to the thermopower in a typical material, the phonon drag thermopower, aL, is defined as CL (XL :

3ne

where CL is the lattice specific heat, n is the carrier concentration, and e is the charge of the carrier. This contribution is only valid at high tempera-

3

THERMOELECTRICPROPERTIES OF QUASICRYSTALLINEMATERIALS

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tures and accounts only for normal processes. At low temperatures the electronic contribution is proportional to T. In a quasicrystal, Umklapp processes are very important at high temperatures (Cyrot-Lackmann, 1999a; Biggs et al., 1991). These processes result in a positive contribution to the phonon drag thermopower. In many quasicrystals, at low temperature the thermopower is negative and then increases to a positive value. This positive addition to the thermopower may be explained through Umklapp processes. While providing a positive contribution to phonon drag thermopower, Umklapp processes may decrease the normal contribution and change the sign of the phonon drag contribution. The Umklapp contribution leads to thermopower values an order of magnitude larger than those observed in most metallic glasses or metals. The other theory governing thermopower in quasicrystals is based on the band structure and Fermi energy of the quasicrystal (Burkov et al., 1996; Wagner et al., 1990; Hansch, 1985). Diffusion thermopower can be defined as

T ~D ~ - -

eE F

where T is the temperature, e is the charge of the carrier, and E v is the Fermi energy. In a quasicrystal, E v is much smaller than typical values of Fermi energies in metals. Since E v is small, the diffusion thermopower is much larger than that observed in metals. These calculations are not relevant at low temperatures since electron-phonon scattering is not elastic at low and intermediate temperatures. At low temperature, the diffusion thermopower and phonon drag contribution can be extracted from the data. If ~ / T versus T 2 is plotted at low temperatures, a linear plot should result. The ~D/T intercept will give the diffusion thermopower over temperature, and the slope is proportional to the phonon drag contribution (Fig. 4).

VI.

Thermoelectric Properties of Quasicrystals

1. VARIOUS QUASICRYSTALLINE FAMILIES The icosahedral phase, which is studied in this paper, is the largest stable group of quasicrystals, having about 60 members (see Table I for some representative quasicrystals). Of these members, several can be grown into single-phase crystals large enough to permit extensive transport measurements. Icosahedral quasicrystals have fivefold symmetry. This symmetry results in clusters of 20-faced polyhedrons whose vertices meet to form

94

TERRY M. TRITT ET AL. 0.15

, . . . . . . . . . . . . . . -----y

,

= 0.1664075

. . . .

,

. . . .

- 0.00020047644x

0.1

0.05 100

200

300

T2

400

(K2)

500

600

FIG. 4. If SIT versus T 2 is plotted at low temperatures, a linear plot should result. The SD/T intercept will give the diffusion thermopower over temperature, and the slope is proportional to the phonon drag contribution.

pentagons. They are quasiperiodic in three dimensions and exhibit no periodic direction. These are primitive unit cells that have body-centered and face-centered lattices. Materials with this periodicity are denoted as i-phase materials. Stable phases that exist are i-A1LiCu, i-A1PdMn, and i-REMgZn (RE = L, Ce, Nd, Sm, Gd, Dy, Y). Octagonal quasicrystals display eightfold symmetry. They have primitive and body-centered lattices. These materials are quasiperiodic in two dimensions. There are no stable phases of these materials. Dodecagonal quasicrys-

TABLE I REPRESENTATIVE COMPOSITIONSOF QUASICRYSTALSBELONGINGTO THE ICOSAHEDRAL, OCTAGONAL, DECAGONAL,AND DODECAGONALQUASICRYSTALLINEFAMILIESa Icosahedral

Octagonal

Decagonal

A1- P d - Mn* A1- C u - Fe* AI-Mn-Zn Zn-Mg-RE* ( R E = La, Ce, Nd, Sm, Gd, Dy, Y) A1-Cu-Os* A1-Cu-Ru*

M n - Si M n - F e - Si V-Ni-Si

A1- Ni-Co* AI- C u - Co* A1-Mn-Pd* A1-Cu-Fe AI-Cu-Ni

aThe star (*) represents a stable phase.

Dodecagonal Cr-Ni V-Ni V-Ni-Si

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THERMOELECTRICPROPERTIESOF QUASICRYSTALLINEMATERIALS

95

tals have 12-fold symmetry. They have a primitive lattice and are quasiperiodic in two dimensions. Decagonal quasicrystals have 10-fold symmetry. They are quasiperiodic in two dimensions. This means that there is one periodic direction that is perpendicular to the quasiperiodic plane. These are primitive unit cells that have primitive lattices. Materials with this periodicity are denoted as d-phase materials. Stable phases that exist are d-AlNiCo, d-A1CuCo, and d-A1CuCoSi. Of these quasicrystals, A1PdMn, A1CuFe, and A1PdRe are the most commonly studied materials.

2.

ELECTRICAL RESISTIVITY

The electrical resistivity in quasicrystals has been observed to be as high as 100 mQ-cm around room temperature. The temperature dependence of the resistivity is generally negative (dR~fiT < 0) at temperatures above 100 K until the material dissociates. In fact, conductivity curves in quasicrystals all have basically the same shape displaced with differing intercepts of a4K (Fig. 5). The resistivity of A1CuFe, A1PdRe, and A1PdMn is close to that of a metal-insulator composition, although these quasicrystals are composed primarily of aluminum ( ~ 70%). Typical resistivity of A1PdMn quasicrystals as a function of temperature and composition can be seen in Fig. 6 (Mayou et al., 1993). It may be noted that there is very little

1200 1000

O

800

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250

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,=.~,rt,~A i~" 63Cu25Fe12 AI62.5Cu25Fe12.5 AI62Cu25.5 Fe12.5 600

Temperature [K]

800

1000

FIG. 5. Temperature dependence part of the conductivity: 6a(T)= a ( T ) - a4x up to 1000K for A163Cu25Fe12, AI62Cu25.sFe12.sA162.sCu25Fe12.5, and A170.sPd22Mnlo.5. (Inset) Conductivity a(T) for A1CuFe samples. (a) A163Cu25Fe12, (b) A162Cu2s.sFe12.5, (c) A162.sCu25Fe12.s, annealed at (1) 600~ (2) 800~ (From Mayou et al., 1993, Fig. 1.)

96

TERRY M. TRITT ET AL. 0

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, . . .~

i ~ ~ I ~ ~ 7 o P d 2 3 M

n7

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rr 9V

-vvvv,.,~yvvv

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300

Temperature (K) FIG. 6. Typical resistivity for A1PdMn quasicrystals with slightly different compositions. (From Akiyama et al., 1993, Fig. 2. Reprinted with permission.)

temperature dependence in the resistivity, yet wide changes in resistivity values as a function of composition. Typically the resistivity in quasicrystals has been described using the concept of weak localization and electronelectron interactions (Akiyama et al., 1993; Chernikov et al., 1993). More recently, the electrical resistivity in quasicrystals has been explained through the concept of variable-range hopping (Janot, 1996). A weak semimetal contribution is seen at higher temperature; however, the electrical conduction is believed to be dominated by a hopping mechanism. The hopping theory describes the observed behavior in quasicrystals better than weak localization and electron interaction. A1PdRe quasicrystalline alloys have resistivities that become quite large as temperature goes to zero, indicating that this quasicrystalline system may well be an insulator or heavily doped semiconductor (Akiyama et al., 1993; Pierce et al., 1993; Poon et al., 1997a). Poon contends that an insulating A1PdRe quasicrystal is consistent with an order of magnitude change in the free carrier density. Room temperature resistivity values in A1PdRe are typically ~ 10-100 m~-cm (Fig. 7) (Poon et al., 1997a). It is also observed that the resistivity in A1PdRe is affected by magnetic fields at low temperature (Bianchi et al., 1997). A1CuFe quasicrystalline alloys have a negative temperature coefficient of resistivity. Berger has found that composition affects resistivity in i-A1CuFe but not in approximate phases of A1CuFe (Berger et al., 1997). In thin films it is seen that resistivity changes appreciably through annealing the A1CuFe quasicrystal to high enough temperatures to cause the crystal to have

3

THERMOELECTRIC PROPERTIES OF QUASICRYSTALLINE MATERIALS

1,0

97

.,,

i

--

i

9

-

-,,-

-

,,-

T

-,.

!

. . . . . .

- - -

.

,.

O,B

E

1

leO

'~0.4

62~_

|

_

A

,,

.

l

__

_

,

1

.

2

r(K1

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.,

AAAAAk&AAAAAAAA.. .50"

100 ' " ----1,.~0

IA

9 9

200

9

9 9 9

9 ,,

250

9 Ae

3l

rt~ FIG. 7. Resistivity versus temperature for the i-Al7oPd2oRelo samples that reach --~1 ohm-cm at T = 0.45 K (A); also shown is a sketch of p(T) in i-A165CuzoRu15 for comparison. (Inset) log(a) versus log T. The straight line illustrates the power law appearance of data at T < 1 K. (From Pierce et al., 1993, Fig. 2.)

reproducible resistivity data (Harberkern et al., 1995). Resistivities in this system are typically 1-10 mf~-cm. Resistivity increases in A1CuFe quasicrystals as the structural quality is improved, contrary to typical observations in metals and semiconductors. Mayou notes that the density of states is nearly constant regardless of the crystal quality or composition, yet these factors have great effect on electrical transport (Mayou et al., 1993).

3.

THERMOPOWER

Thermopower in quasicrystals has not been extensively measured, and data in the literature are somewhat sparse. Thermopower values in quasicrystals are seen to vary with the quality and composition of the crystal. Thermopower in A1PdMn quasicrystals is seen to vary substantially with composition and temperature. A1PdMn with composition 70-22.5-7.5 has a thermopower that increases rapidly to 70/~V/K, whereas a similar A1PdMn quasicrystal with composition 70.5-22.5-7 has a thermopower that is slightly negative and varies little with temperature (Fig. 8) (Giroud et al., 1996). It is believed that the higher the Hall coefficient the higher the thermopower will be in this system of quasicrystals. As with resistivity in quasicrystals, thermopower is seen to vary with composition and quality.

98

TERRY M. TRITT ET AL. 8 0

i

. . . .

I

. . . .

i

. . . .

I

. . . .

t

. . . .

!

'

9

I r

~176 ,o I [

~

~

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70.5/21.5/8 ~

~

"=

~"

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.

" ~

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h

.

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.=..=

==

.=

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.

.

.

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~

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/

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.

.

.

.

.

.

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.

.

150

.

.

.

.

.

200

.

.

.

250

T (K) FIG. 8. Thermopower versus temperature for three slightly different compositions of A1PdMn quasicrystals. Thermopower is as large as + 75/~V/K for the 70-22.5-7.5 composition. In the 70.5-22.5-7 AIPdMn quasicrystal, thermopower is observed to be small and negative. Thermopower decreases with increasing temperature until approximately 300 K, when it begins to become more positive. (From Giroud et al., 1996, Fig. 3.)

Thermopower in A1CuFe is seen to vary from plus to minus 20 #V/K with composition as well as changes in annealing. Changes in the sign of the thermopower have been observed with as little as a 1% change in composition. Thermopower in A1PdRe is seen to change with annealing and composition. Maximum room temperature values of thermopower are seen to be 55/~V/K (Poon et al., 1997b). In a very similar system the thermopower is observed to be negative at low temperatures and positive for higher temperature, increasing to a value of 30/~V/K. A very distinct difference in thermopower values is observed even though there is no apparent change in the density of states. A compilation of thermopower values can be seen in Fig. 9.

4.

THERMAL CONDUCTIVITY

Thermal conductivity in quasicrystals is inherently low, almost an intrinsic property of these novel materials. Thermal conductivity, in fact, is on the order of that of an amorphous material. Janot has put forth the explanation of the poor thermal conductivity observed in these materials as being due to a reduced range of phonons due to variable-range hopping (Janot, 1996). Thermal conductivity has also been explained and seen to experimentally

3

THERMOELECTRICPROPERTIES OF QUASICRYSTALLINEMATERIALS

99

80 60

::t.

A170Pd22.5Mn7.5

40 Al7oPd20Relo 930C

g-4

o oz. o

20

-20 A162.sCu25Fe12.5810C l h

-40

0

50

100 150 200 Temperature (K)

250

300

FIG. 9. Thermopower for A1PdMn, AIPdRe, and A1CuFe quasicrystals. Thermopower in quasicrystals vary from positive to negative values. Quasicrystals even with the same base elements can have varying signs of thermopower depending on annealing conditions, quality, and composition. (Data from Harberkern et al., 1996; Giroud et al., 1996; Poon et al., 1997.)

agree with the quasiperiodic lattice scattering phonons as if there were a point defect at every atomic site (Legault, 1999). Thermal conductivity in quasicrystals is two orders of magnitude lower than in aluminum, which is somewhat surprising since many quasicrystals are composed primarily (-~ 70%) of aluminum. A1PdMn quasicrystals have thermal conductivity on the order of 1-3 W/mK at room temperature. Thermal conductivity increases with increasing temperature until a phonon saturation plateau is observed between 20 and 100 K (Chernikov et al., 1995). This plateau is observed at much higher temperatures than in amorphous materials. Above 100 K, the thermal conductivity begins to increase again (Fig. 10). Low temperature thermal conductivity (T < 2 K) is observed to increase as --~T 2, indicating scattering of phonons by tunneling states (Legault, 1999; Chernikov et al., 1995). A1CuFe quasicrystals have thermal conductivity between 1 and 3 W/mK at room temperature. Thermal conductivity in these quasicrystals behaves in much the same way as that in A1PdMn quasicrystals. Thermal conductivity at 1000 K is less than 10 K for A1CuFe quasicrystals. The temperature dependence of thermal conductivity for A1CuFe can be seen in Fig. 11 (Perrot et al., 1997). Perrot shows calculations in his paper that the Wiedemann-Franz law holds at high temperatures for these materials. He calculated the Lorentz number to be within 15% of the accepted value.

100

TERRY

.,,u,,.i,.

1 .......

,~'

-

' -

,-

M.

.~ - ~ - ,

TRITT

. . . . . . .

'"'-

ET AL.

-

. '-~,-.,

l

m~'mf

1 0 "1

QQ

,.aa =aaCt

a~

0

0 Q

10.2

10-a

9 Pd.,.~

10.4 10.1

101

10 0

11~

T(K) FIG. 10. Temperature dependence of the quasilattice thermal conductivity 2oh of icosahedral AlvoMn9Pd21 in comparison with analogous data for two amorphous materials (insulating and metallic). (From Chernikov et al., 1995; Fig. 4.)

0

.'

.

, .,.,,,

.

,

9 , .,,,,,

,

. , ,.,,j

l

m m

;>

9li i l l l i l

O

L)

9I I I l i i l /

am

9

.1

AI62CU2s.5Fe23.s

. . . . . . .

1

i

10

)

..

!

!

9 9.11

100

!

i

,

i

!

!,.-

1000

Temperature (K) FIG. 11. Thermal conductivity in an A1CuFe quasicrystal up to 1000 K is always less than 10 W/mK. Thermal conductivity at room temperature is between 1 and 3 W/mK. (From Perrot et al., 1997; Fig. 1.)

3

THERMOELECTRICPROPERTIESOF QUASICRYSTALLINEMATERIALS

101

The thermal conductivity of A1PdRe quasicrystals could not be found in the literature. We plan to measure the thermal conductivity of the A1PdRe quasicrystals in the very near future along with their other properties to evaluate their potential for thermoelectrics.

5. THERMALAND ELECTRICAL TRANSPORT ON A 1 P d M n FOR THERMOELECTRICS

After an exhaustive review of the literature in 1996, the authors of this review determined that the A1PdMn quasicrystalline system was the most appropriate to begin an investigation for potential thermoelectric applications. This was determined because the A1PdMn system has the highest reported thermopower in the literature coupled with favorable thermal conductivity and electrical resistivity. Few studies have been performed on the viability of quasicrystals for thermoelectrics, which results in incomplete data for determining Z T and thus assessing their potential as a thermoelectric material. In addition, data in the literature are not consistently reported on the same sample, resulting in data that are incomplete, since differing annealing conditions or slightly differing compositions or levels of impurities may affect the measurements. Because of the wide variability in transport properties from sample to sample, it is essential that properties be measured on the same sample in order to obtain an accurate idea of all the transport in a given material. To date, the single-phase quasicrystalline A17o.8Pdz0.9Mns.a system has shown the most promising thermoelectric properties (Pope et al., 1999). In Fig. 12 the resistivity, p, and thermopower, ~, are shown as a function of temperature. Room temperature resistivity is on the order of p = 1.5 m~-cm. The electrical resistivity is seen to increase sharply with increasing temperature, peak, and then decrease gradually with further elevation in temperature. The electrical transport is suspected to be governed by weak localization and electron interaction, although variable-range hopping has also been suggested (Akiyama et al., 1993; Janot, 1996). It is known that the resistivity will decrease with increasing temperature until the quasicrystal dissociates at T ~ 1000 K (Mayou et al., 1993). A thermopower of 85/~V/K is observed at room temperature ( T ~ 300 K). The thermopower data is some of the highest reported to date in a quasicrystalline system. These thermopower values are approximately an order of magnitude higher than those observed in a metallic glass (Howson and Gallager, 1988). As seen in Fig. 12, thermopower decreases as the temperature decreases in a "Mott diffusion" manner, as is commonly observed in metals and semimetals (Barnard, 1972). The corresponding thermal conductivity verses temperature is shown in Fig. 13. The total thermal conductivity is approximately 1.6 W/m K at room

TERRY M. TRITT ET AL.

102

1.60

I

I

I

I

80 1.55

v > 60 ~.~

~

-E

/

~ g . m

L.

9

1.45~

40

.>

0 0

E L-

~(/)9 1 40 9

r

20 I-1.35

1.30 - 0

' 50

,

0

loo

300

Temperature (K) FIG. 12. In a single-phase Al7o.aPd2o.9Mn8. 3 quasicrystal, resistivity is seen to decrease with increasing temperature above 100 K while the thermopower monotonically increases over the entire temperature range. Both of these properties are advantageous for thermoelectrics. This is the largest known room-temperature value of thermopower measured to date in a quasicrystal (Pope et al., 1999a).

.6

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

!

!

!

!

!

I

~" E

1.4

i ~i~~~4~

1.2 ~>

9

i

i

Total

9

e e e e e e e e e

1

Lattice

om

=

m

Imp,

0.8

e-

o 0

06 9

~ E

04 9

~

0.2

Elec &

0

~

9 1 4 9 1 4 9 1 4 ,9 ,

50

9

&

,

100 150 200 T e m p e r a t u r e (K)

, ....

250

300

FIG. 13. Thermal conductivity in single phase A1PdMn as a function of temperature. The total thermal conductivity is ~ 1.6 W/mK at room temperature and is composed primarily of a lattice contribution. Thermal conductivity in quasicrystals is inherently low, on the order of 2 W/mK (Pope et al., 1999a).

3

THERMOELECTRICPROPERTIES OF QUASICRYSTALLINE MATERIALS

103

temperature. The thermal conductivity is observed to decrease with decreasing temperature to a relatively flat plateau region between 50 K < T < 150 K. At this point, a small peak is observed in the thermal conductivity at low temperature. This temperature dependence is very similar to the temperature dependence of the thermal conductivity of glasslike or amorphous materials (Bianchi et al., 1998; and references therein). Both the lattice and electronic contributions (2e and 2E, respectively) are also shown in Fig. 13. The estimation of the electronic thermal conductivity, 2E, is calculated from the measured values of the electrical conductivity using the Wiedemann-Franz law (2E = trLoT, Lo = Lorentz number (Lo ~ 2.45 x 10-8 VZ/K2)).The lattice contribution is taken as the difference between the total and electronic thermal conductivity. Radiation losses, which are approximately 15% at room temperature and less than 3% below 200 K, have been estimated and corrected for in this data. From Fig. 13, it is seen that 2E is essentially temperature independent above 100 K and goes to zero at T = 0. The lattice thermal conductivity is nearly temperature independent over the measured range, exhibiting a very broad shallow minimum and a peak at around T ~ 25 K (Kalugin et al., 1996). Even with the best thermoelectric properties measured to date, the figure of merit in A17o.8Pdzo.9Mn8. 3 is still low ( Z T "~ 0.08 W/mK at T = 300 K, yet rising to a maximum of Z T ~ 0.25 W/mK at T = 550 K). Note, however, that thermal conductivity is increasing very slowly while the resistivity decreases slowly and thermopower increases almost linearly with temperature. Some high-temperature measurements have previously been performed (Pope et al., 1999b; Littleton et al., 1998). At elevated temperatures, thermopower in A17oPd22.sMnT. 5 increases linearly with temperature until approximately 300 K, where thermopower begins to saturate. Thermopower increases slowly until it exhibits a broad maximum with a value of 110/~V/K at 500 K. This is almost a 40% increase in thermopower that is achieved by taking these materials to high temperature. A similar phenomenon is seen in the electrical resistivity data. Electrical resistivity increases with increasing temperature until approximately 80 K. At this point the resistivity begins to gradually decrease with temperature. Around 400 K, the slope of the resistivity triples and electrical resistivity decreases much more quickly. Electrical resistivity continues to decrease at 600 K with no sign of saturation. An electrical resistivity of ~ 1.25 mf~-cm is observed at 600 K, exhibiting a 23% decrease over the room temperature value. Thermal conductivity data has been extrapolated to 600 K in order to compute a Z T in these quasicrystals. In order to extrapolate thermal conductivity to higher temperature, the lattice portion of the thermal conductivity was plotted, and a plateau is observed around 100 K. This plateau begins to increase at higher temperatures due to radiative losses. The lattice thermal conductivity above 100 K is held as a constant, approximately 1.1 W/m K in the A17o.sPdzo.9Mn8.3 quasicrystal. The elec-

104

TERRY M. TRITT ET AL. 0.25

0.2 0.15 ~D

~" [-.,

0.1 0.05

o

o

lOO

200

300

400

500

600

Temperature (K) FIG. 14. Z T as a function of temperature up to 600K using extrapolated thermal conductivity values. The maximized Z T at 500 K is 0.23, a substantial increase over room temperature values.

tronic portion, as calculated from the W i e d e m a n n - F r a n z law using hightemperature resistivity data, was added to the constant lattice thermal conductivity, resulting in the total thermal conductivity correcting for radiative loss and taking the measurement to high temperature. Thermal conductivity at 600 K in Alvo.sPdzo.9Mn8. 3 was extrapolated to be approximately 2.3 W/mK. With the substantial increase in thermopower and decrease in resistivity, an increase in the figure of merit is expected and is indeed seen. The power factor (~2o'T) in this single-phase sample is seen to be 0.5 W/m K at 550 K. Z T peaks at a value of 0.23 around 550 K using the extrapolated thermal conductivity to arrive at this value (Fig. 14). Obviously, there is still room for improvement; Z T needs to be over 1 in order to be competitive with other state-of-the art materials. Fortunately, there are many strategies to be employed in the quasicrystalline system in an effort to enhance thermoelectric properties. The next tactic employed to enhance thermopower was an attempt to control transport properties through an understanding of the annealing and heat treating process. The sensitivity of quasicrystals to growth conditions is also seen in the annealing conditions induced on a sample. An ingot of A1PdMn was grown and from this ingot three samples were cut. These samples had dimensions of 2 • 3 • 12 mm. The first sample was annealed for 3 hr in an 800 K furnace and then quenched, resulting in a well-formed icosahedral phase quasicrystal. The second sample was annealed for 12 hr at 800 K and then furnace cooled, resulting in many defects. The final sample was annealed for 3 hr at 800 K and then furnace cooled, resulting in a sample with two phases present. It is apparent in Fig. 15 that as temperature increases the electrical resistivity increases until approximately 40 K, where it begins to decrease linearly with temperature, as was seen

3

THERMOELECTRICPROPERTIES OF QUASICRYSTALLINEMATERIALS

105

2.8 /E" 2.6 ? 2.4 "F- 2.2 r~

2

I

,L annealed 3h quenched % ~ b , ~ .

1.8 1.6

0

9

50

100

150

200

I

....... -"1

250

300

Temperature (K) FIG. 15. Resistivityas a function of temperature for different annealing conditions. Quenching of the sample provides the lowest resistivity (Pope et al., 1999c).

before. Plots of the electrical resistivity of these samples versus t e m p e r a t u r e have basically the same shape, differing roughly by a constant. Therm o p o w e r in this system is observed to m o n o t o n i c a l l y decrease (S ~ 3 0 60 ~ V / K at 300 K) as the t e m p e r a t u r e decreases from 300 K to S ~ 0 at the lowest t e m p e r a t u r e s (Fig. 16).

70 60 5O =L

,. 3 r

40

30

20 annealed 3h quenched annealed 3h furnace cool annealed 12h furnace cool

10 0

50

1O0

150

200

250

300

Temperature (K) FIG. 16. Thermopower as a function of temperature for different annealing conditions. Annealing with subsequent quenching is seen to enhance the thermopower (Pope et al., 1999c).

106

TERRY M. TRITT ET AL.

The electrical resistivity values in the furnace-cooled samples are comparable to each other, never deviating from one another by more than 10%. There is a 20% increase in thermopower from the sample that was furnace cooled 12 hr to the sample that was furnace cooled 3 hr. The sample cooled for 3 hr corresponds to the sample with the greatest number of defects. The electrical resistivity of the quenched A1PdMn sample overall decreases by 15%. If quasicrystals are slow cooled below 600~ secondary phases begin to form with the icosahedral phases. Interestingly enough, we observe that the lowest electrical resistivity corresponds to the highest thermopower, which is a favorable result relative to the potential of these materials for thermoelectric applications. From this study it looks as if the quenched sample will be the most promising for thermoelectrics. Thermal conductivity data from these samples showed very little change from sample to sample, and thermal conductivity values were on the order of 2-3 W/mK at room temperature (Pope et al., 1999). Annealing conditions prove to be an important consideration in optimizing the figure of merit. Quasicrystalline systems do not behave like typical metals or semiconductors, resulting in a lack of intuition as to the most likely means to change properties of the base system. Of course, substituting elements or doping in crystals typically shows results consistent with being able to "tune" a material's properties, so this was attempted on quasicrystals. In an effort to enhance the thermopower and perhaps lower the thermal conductivity, the addition of Co to the parent A1PdMn system was made. Several quasicrystalline A1PdMn samples were grown with similar composition. All had similar properties, which are reported elsewhere (Pope et al., 1999d). A sample with representative measurements had a composition of AlvI.4Pdz3.1Mns. 5. Room temperature measurements were as follows: 2 ~ 1.9 W/mK, ~ ~ 16 laV/K, and p ~ 1.87 mf~-cm. The low value of thermopower was partially due to the composition of the sample as well as the quality. Several other samples were grown this time, with the addition of Co, which substituted on the Mn site. A sample of composition A 1 6 9 . 9 P d z z . s M n s . 4 C o 2 . 2 w a s synthesized. This sample also had representative values for quartenary quasicrystals with Co added. Room-temperature measurements for the Co sample were as follows: 2 ~ 2.1 W/mK, ~ 58 laV/K, and p ~ 0.94 mfl-cm. Through the addition of approximately 2% Co to the quasicrystal, it appears that the thermopower substantially increased by a factor of 2-3 while the resistivity decreased by almost a factor of 2. Thermal conductivity changed little with the addition of Co, as was expected. For a compilation of results from the addition of Co, see Table II. Figure 17 shows that thermopower power increases monotonically with temperature in the A1MnPdCo sample, whereas a small phonon drag peak is observed at low temperature in the A1PdMn sample before the thermopower begins to slowly increase with temperature. The resistivity in A1PdMn is seen to change by ~ 2 5 % over the temperature range from 10

3

THERMOELECTRIC PROPERTIES OF QUASICRYSTALLINE MATERIALS

107

TABLE II COMPARISON OF THERMOELECTRIC PROPERTIES IN A1PdMn AND A1PdMnCo Sample A171.4Pd23.1Mns. 5 A169.9Pd22.sMns.4C02. 2

~ (/~V/K)

p (mf~ cm)

2 (W/mK)

16 58

1.87 0.97

1.9 2.1

to 300 K (Fig. 18). The resistivity is seen to decrease with increasing temperature. In the A1PdMnCo sample the resistivity is seen to be essentially temperature independent with a change in value by less than 4% over the same temperature range. This data suggests that the addition of the Co has caused the thermopower and electrical resistivity to change in a manner beneficial to thermoelectrics. Preliminary measurements on other quartenary quasicrystals have been performed with similar enhancement of thermopower and reduction of resistance. A1MnPdGa has been synthesized at Ames National Laboratory. The elements Co, Re, Fe, and V have been added to A1PdMn quasicrystals at Clemson University. It has been observed that the addition of a quartenary element has not significantly changed the X-ray diffraction data (Fig. 19).

60

-e-AIPdMn ~AIPdMnCo

50

40

o o

30

~ 2o j-.

10

0

50

100 150 200 Temperature (K)

250

300

FIG. 17. Thermopower for A1PdMn and a quarternary quasicrystah A1PdMnCo. The thermopower is seen to increase by at least a factor of 3 through the addition of a fourth element (Pope et al., 1999d).

108

TERRY M. TRITT ET AL.

,

~

,

,

l

,

,

,

,

l

V

,

,

,

,

l

,

,

,

,

l

,

,

,

,

l

,

,

,

~

+AIPdMn 2.5

i2

> :,= 1.5 ,,,...

...,,

09 Q~ CC

1

0.5

9

0

,

,

,

i

50

,

i

J

i

i

,

1O0

,

,

,

l

,

150

z

,

,

!

200

Temperature (K)

i

I

250

300

FIG. 18. Resistivity is seen to change substantially in AIPdMn with the addition of Co. Resistivity in A1PdMnCo is very temperature independent from 10 to 300 K (Pope et al., 1999d).

VII.

Future Directions and Approach

Many advances have been made in quasicrystalline systems with regard to evaluating their potential for thermoelectrics. However, in order to make a viable thermoelectric material for use in devices, significant advances still need to be made, especially with respect to enhancing the thermopower in these materials. One avenue of future research will be the continued addition of elements to the parent system, A1PdMn. The addition of magnetic elements is of interest in an effort to enhance the thermopower. Typically this type of substitution yields a reduction in the mobility (because of scattering due to the spin sites), yet many times we have observed a very favorable effect, with both thermopower and electrical conductivity increasing with various dopants. The addition of these elements may also serve to reduce the already low thermal conductivity through alloy scattering. In addition to trying quartenary quasicrystalline compositions, we also hope to synthesize pentenary quasicrystals. Plans are also under way to do more comprehensive studies on the effects of damage on the thermoelectric properties of these materials. We plan to irradiate the samples and perform SEM characterization of the surface as well as complete thermal and electrical transport characterization. Damage and impurities are known to decrease resistivity, and current research indicates that as resistivity decreases thermopower may increase in these quasicrystalline systems. Once other high thermopower samples are found,

3

THERMOELECTRIC PROPERTIES OF QUASICRYSTALLINE MATERIALS

AI

Pd

70.8

109

Mn

20.9

8.3

o

. . . .

I

" ' "

"

I

. . . .

!

. . . .

t

. . . .

I

. . . .

AI Pd MnRe 70

20

8

2

.,i.,a

o

20

30

40

50

60

70

80

2O FIG. 19. X-ray diffraction data for a ternary and quarternary quasicrystal, respectively. No difference is seen in the quasicrystalline structure with the addition of Re.

high-temperature measurements will be of interest. These measurements will show the optimal operating temperature for these quasicrystalline systems. There is also the possibility that with doping, the temperature at which the Z T maximum occurs may be able to be moved. Of course, it is still not clear which composition of A1PdMn will be the best for thermoelectrics. A complete investigation of the composition space should be performed in order to determine what composition yields the best thermoelectric properties. Certainly the measurements and chemical variations we have performed indicate a very strong dependence of the electrical resistivity and thermopower on composition. This seemingly simple investi-

110

TERRY M. TRITT ET AL.

gation is actually very difficult and time consuming. Although quasicrystals only grow in a small window in composition space, there are many combinations to investigate. These compositions will all have to be grown at different times, allowing for changes in the process or changes in the purity of the materials. Small changes in the growth process lead to changes in the transport properties.

VIII. Summary As described in the previous discussion, despite the advances in quasicrystals for thermoelectrics, there is still a long road ahead if quasicrystals are to be viable for thermoelectrics. It is encouraging that theoretical predictions by Macia indicate that high values of Z T may be possible in this class of materials (Macia, 2000). Cyrot-Lackmann is also investigating quasicrystals as possible thermoelectric materials and has a patent on quasicrystals for this application (Cyrot-Lackmann, 1999b). A systematic approach in relation to doping, composition, processing, and other factors along with subsequent measurement of the transport properties will be necessary. Certainly this data coupled with ideas on how to further enhance the thermopower could greatly advance our knowledge of these materials. Quasicrystals closely match the concept that a good thermoelectric should behave as a glasslike material in relation to phonons and a metal in relation to electronic transport. A1PdMn quasicrystals have thermal conductivity that closely resembles that of an amorphous solid. The "tunability" in the electrical conductivity and thermopower allows for many compositional, impurity, damage, and additional element studies to be performed in an effort to better understand the transport in this system and optimize the figure of merit for potential thermoelectric application. Obviously, more work is required to fully address the feasibility of these materials for thermoelectric applications. However, along the way much information related to the interplay of many of the parameters to the electrical and thermal transport in these systems will be gained. Thus, a more fundamental understanding of the electrical and thermal transport mechanisms related to the quasicrystalline materials may become evident. It is the strong belief of one of us (TMT) that a new higher performance thermoelectric material will be found and it will truly change the world around us. Where will it be? Will it be in a quasicrystal? The current data tends to indicate that it probably will not be in these materials, since the hurdle of enhancing the thermopower by a factor of 4 seems too great; yet theoretically high values (ZT > 1) have been predicted to be possible. This is especially difficult without the ability to systematically dope and "tune a bandgap" as in the semiconductor thermoelements, which were described by Ioffe nearly 50

3

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111

years ago. Quasicrystals are truly a fascinating class of materials, and whether or not subsequent research efforts and time determine their feasibility (or even impracticality) for thermoelectrics, they will still retain their most unusual properties, about which we have much to learn. Most likely, there are a few surprises left in these materials (or similar classes of intermetallic materials), and probably even more applications of quasicrystals will become evident over the next few years.

ACKNOWLEDGMENTS

We acknowledge the many people who have assisted us in our understanding and investigations of quasicrystals for thermoelectrics. These include some of our collaborators, such as S. J. Poon, R. Gagnon, S. Legault, J. Olsen, M. Chernikov, P. A. Thiel, I. Fisher, and M. Feuerbacher, just to name a few. We also acknowledge the Office of Naval Research ( #N0001498-0271) and the NSF Science and Technology Initiative in the College of Engineering and Sciences for their support of the work on quasicrystals at Clemson University. Each of us acknowledges the support of our colleagues, other students, and postdoctoral associates who work in our groups, and of Clemson University during the development of this manuscript.

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S E M I C O N D U C T O R S AND SEMIMETALS, VOL. 70

CHAPTER

4

Military Applications of Enhanced Thermoelectrics Alexander C. Ehrlich UNITED STATES NAVAL RESEARCHLABORATORY WASHINGTON, D.C.

Stuart A. Wolf DEFENSE ADVANCEDRESEARCHPROJECTS AGENCY WASHINGTON, D.C.

117 118 118 119 121 122 124

I. INTRODUCTION . . . . II. THERMAL MANAGEMENT 1. B i o l o g i c a l 2. E l e c t r o n i c 3. O t h e r

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

I I I . POWER GENERATION IV. CONCLUSION . . . . .

I.

Introduction

It is indisputable that the military requirements of nation states have been a major driver of technology and the science that underlies it for hundreds if not thousands of years. Subsequently, the new technology often enters the civilian economic life of the state and changes it dramatically. Galileo is said to have developed the telescope in the early 17th century so that the Venetian navy might detect an enemy fleet before the latter detected it. In more recent times aviation, communication, medicine, and information dissemination, to mention some of the most obvious examples, have had major impetus from military research and development funding agencies. In other cases military capabilities advance rapidly by piggy-backing on what becomes, however it may have started, a rapidly evolving civilian technology. Widespread nonmilitary use of a technology can also lower costs sufficiently to make its large-scale integration into military systems feasible. Computational speed and capacity are an obvious example. Finally, there are examples of military research initiatives that are prompted by perceived civilian requirements. Thermoelectrics (TE) fits this description. The relatively recent rebirth of a vital TE research enterprise in the United States, to which Terry Tritt refers in the Preface, took place as a 117 ISBN 0-12-752179-8 ISSN 0080-8784/01

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result of an R and D initiative by the U.S. Department of Defense (DoD) led initially by the Office of Naval Research (ONR). It was followed quickly thereafter by programs of the Defense Advanced Research Projects Agency (DARPA) and subsequently other DoD funding entities. The Navy acted in response to the culmination of international negotiations on limiting the emission of greenhouse gases, which negotiations resulted in the Montreal protocols in the latter part of the 1990s, to which the United States is a signatory. In these it was agreed to eliminate the use of chlorofluorocarbons (CFCs) in air conditioning systems. CFCs are believed to be ozone-destroying gases. This commitment by the United States extended to the U.S. Navy fleet, for which air conditioning is essential. A decision was made by ONR to investigate the potential of TE systems to replace CFC- based systems in thermal management applications. Once this decision was made on the basis of environmental considerations, it was quickly realized that enhanced TE materials and devices could provide many other potential benefits. ONRs major goals became broader, and DARPAs had been from their first involvement in the program. It was not as if military applications of TE were unknown. In fact, they already existed. However, it became increasingly clear that important needs could be met if TE performance could be improved, particularly its energy efficiency. In fact, application opportunities would grow exponentially with efficiency, with, for example, area air conditioning requiring large efficiency increases to become competitive with existing technologies, whereas cooling of electronics would benefit a great deal from incremental gains. What follows is an admittedly incomplete outline of existing applications, applications achievable with moderate improvements in TE efficiency and those achievable only with major improvements in efficiency. Where appropriate, we shall also make passing references to instances of overlap with nonmilitary uses of the same technologies, because the latter are not only affected by military uses, but can also affect them by cost reduction. TE studies and applications divide themselves naturally into two areas: TE for thermal management, usually cooling (but conceivably heating, since a TE device is just a heat pump) and TE for power generation, particularly from waste heat. We further divide the thermal management application into "biological," electronic, and "other" applications and address them in that order.

II. 1.

Thermal Management

BIOLOGICAL

One area of biological use, already in service, is the direct cooling of personnel, so-called man-portable microclimate systems. This is already being done using full body coverage with a knit garment (not unlike thermal

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underwear) that incorporates relatively small-diameter plastic tubing through which a cooling fluid is forced. The cooling fluid is, in turn, temperature controlled (cooled) by a TE module. This system is useful to tank and rotorcraft crews and all personnel requiring maximally protective suits in a biologically or chemically hazardous environment. It is required also by astronauts for their extravehicular activities and is useful in the broader civilian environment by hazardous waste workers, firefighters, and others. A second example of "biological" cooling is space cooling of a large area such as a submarine or other ship. Here, thermal management by TE methods would have many benefits in addition to being environmentally "green." Unlike compression-expansion cycles for cooling, TE is equally efficient for small-scale cooling as it is for large scale. Thus, small, individual, TE-based cooling facilities could be placed in individual spaces, since there is no efficiency benefit in having a single large cooling plant with its required pumping of cooled air throughout the vessel via ductwork that consumes considerable and precious space. The latter is at a premium in most ships. Further, individual units make it much simpler to provide the needed amounts of cooling to individual areas, particularly when the thermal loading of those areas differs widely. Finally, space cooling by TE is much less acoustically noisy than more conventional systems that employ large compressors. Minimizing the acoustic signature of naval ships is highly desirable, particularly for submarines. A third broad area of biological cooling is that of refrigeration of "bioactive" materials, in particular, medication and food. Again, the good efficiency of small-scale systems and the low acoustic signature come into play here so that this application is of benefit to all the services as well as, incidentally, NASA. Refrigeration of food is not a necessity for shortduration space flights, and perhaps not even for the longer periods of occupation on the space station, but refrigeration of biological specimens used in space-based biological research is. Silent systems of refrigeration are also highly desirable here because of the difficulties the astronauts have with the acoustics inside a space vehicle.

2.

ELECTRONIC

Of much more importance to the military and much closer to wide application than area cooling are electronic applications of TE thermal management. Indeed, many weapon systems that require TE cooling are already deployed. For example, cooled infrared (IR) sensing and detecting systems and other electronic systems are incorporated in a number of operational tactical missiles. Other examples where TE cooling/temperature stabilization are extant in existing military systems include hand-carried and

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larger night vision systems employing TE-cooled focal plane arrays of short-wavelength IR detectors operating at 165-195 K; temperature-stabilized IR detectors operating in the 300-320 K range; charge coupled devices requiring temperatures approximately in the 220-250 K range; blackbodies for use in "forward looking infrared" tracking systems; temperature stabilization of optical systems; and active cooling of electronic enclosures. Much of this technology is already available in the civilian community, as evidenced by recreational as well as law enforcement use of night vision goggles and gun sights and the very recent option, being made available in some American brand automobiles, of a night vision capability communicated to the driver via a heads-up display system. Of much broader impact, and certain to grow further in importance with time both to the military and to the broader civilian community, are the performance enhancements that microprocessors and multichip modules will undergo when they can be conveniently and reliably cooled. For example, most microprocessors currently operate at approximately 100~ Cooling them by approximately 100~ can increase their speed by approximately a factor of 2 and their reliability by as much as two orders of magnitude. This can improve their economic value by factors of 2 or 3. Even a 50~ cooling will provide significant performance enhancement. It is generally believed that valuable cooling effects can be achieved with only incremental improvements in TE materials. Further, the importance of chip cooling is growing rapidly with time because of the decreasing size of the individual features on chips and the accompanying increase in power density. Eventually, passive cooling based on heat sinks and passive convective, or even forced, air circulation will not be able to meet the minimum needs of the chips, much less provide a mechanism for enhancing performance. In the short and perhaps intermediate time frame, direct thermoelectric cooling of microprocessors and multichip modules is the most likely solution to this thermal management issue. This might be done, for example, by direct mounting of the chip or module on a thermoelectric device. For the longer run, some very innovative approaches to thermal management are being explored, notably by a major DARPA program. Among these are the use of micro electromechanical systems (mems) to achieve direct fluidic or mist spraying cooling via chip integrable microchannels achieved with micromachining techniques. Even here, however, the fluidic system is usually closed cycle and itself cooled by heat rejection to the broader environment--often with the help of a thermoelectric device. A second major DARPA program titled "Cryoelectronics" seeks to evaluate and achieve the benefits to electronic systems that may be achieved by cooling some of the passive as well as active components in electronic systems, for example, communications systems. To take a specific example, the electromagnetic (em) environment in the vicinity of a naval ship is extremely dense. By this is meant that there are a large number of signals

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rather closely spaced in frequency being sent and received at the same time in the same physical space. As in the civilian cellular communications industry, signal overlap and other origin crosstalk, principally intermodulation generation originating from nonlinearities in these systems, degrades the overall performance of the system. This noise pickup can be much reduced and thus the overall system performance enormously enhanced if the Q's (the "quality factors") of the band-pass filters employed in the "front ends" of the receivers can be made much larger than they conventionally are. Conventional filters used in these systems have a Q of 20-30. Cooling them can increase the Q to values of approximately 100-200. Compact and convenient to employ (no compressors, no fluids, etc.), TE cooling modules could play a role here. There are a number of major benefits to be derived from the enhanced signal-to-noise ratio achieved in communications systems by increasing the Q of the band-pass filters. As implied in the previous paragraph, the density of communications channels that can function in the same space is enhanced. This implies a much greater interoperability. For example, it became possible to achieve reasonably good reception of TV signals rather than a "snow"-filled screen, when TV was introduced as a morale builder for sailors on long-duration assignments at sea. Secondly, the range of reliable communication is increased. Thirdly, purely "listening" functions, such as detection of an adversary's tracking and/or targeting radar system, is also enhanced. Ultimate performance can be achieved by employing high-Tc superconducting filters whose Q's can be greater than 30,000. Indeed, there is no fundamental impediment to implementing this technology at the present time. However, since one wishes to operate at temperatures no higher than between one-half and two-thirds Tc, these filters would run at about 70-80 K and require several watts of heat pumping power. As with the case of superconducting motors discussed later, these requirements cannot be met with TE cooling systems any time in the foreseeable future. TE technology could have a place here, however, if a multistage cooling system were adopted for reasons of either speed in cool-down or weight or space saving.

3.

OTHER

Finally, there are other components in these systems that could benefit from the cooling power of improved TE. For example, the filters discussed earlier are not necessarily passive elements. In many cases they can be tuned using mems (micro electromechanical systems) technology. Unfortunately, it was found that the mems response at room temperature was not entirely reproducible; that is, for the same electrical input the mems device would not return to exactly the same position. Lowering the temperature improved

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this aspect of performance significantly, which is a somewhat different kind of system enhancement using thermal management. In the applications category we call "other" we include also the growing importance of power electronics and the potential and desirable uses of high-To superconducting magnets in a number of areas. Regarding the former, there is a desire to achieve an "all-electric navy" wherein various systems currently operated with steam or other hydraulic means would be replaced by all-electric systems. For example, currently Navy planes are launched from aircraft carrier decks by high-pressure steam-driven catapults. This system has a number of problems. It is subject to steam leaks, corrosion, and safety issues. It would be desirable to replace the steam catapults by electromagnetic catapults. This would require power electronics. As another example, alternating-current synchronous motors would be preferred over current ship propulsion methods, but they, too, would require power electronics. Further, there is always a push to make everything on a ship as small as possible. This would require cooling the electronics--very plausibly by TE means. There are a number of applications for superconducting magnets that are being researched by the U.S. Navy. To cite three of these, first, there are superconducting motors for ship propulsion. In an "all-electric navy," superconducting motors have the advantage of small size for a given power. Secondly, although naval mines are becoming increasingly sophisticated, there is still an important place for minesweeping by employing powerful magnets to mimic the magnetic signature of ships. Towed by a vessel some safe distance ahead of it, such a decoy can be used to set off magnetic field-sensitive mines and thus clear a harbor, for example. Finally, there are military (as well as civilian) applications for energy storage in high magnetic fields. The superconducting motors being developed by the Navy are planned to run at a temperature of about 30 K. This might also be true for the other two applications cited earlier. Operationally, closed cycle refrigerators rather than disposable liquid refrigerants are strongly preferred. Notwithstanding low-temperature TE peaks due to Kondo anomalies, 30K is probably too low a temperature to be achieved by any foreseeable purely TE refrigerator. However, here, too, it might be advantageous to employ a two-stage system, with the upper stage being a TE stage.

III.

Power Generation

The prospects for large-scale primary power generation by TE means in the foreseeable future are poor in civilian or military settings. Here, efficiency is of primary concern and energy-competitive efficiencies by TE

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technologies would require very large improvements in high-temperature TE materials. The principal plausible role for TE in primary generation will be to enhance the overall efficiency of more conventional power plants by waste heat recovery from turbines and exhaust stacks. The driver for this has to be a lifetime cost benefit. The same must be said for partial recovery of waste heat from automobiles and trucks. Approximately two-thirds of vehicular fuel energy is "wasted" as heat with about equal portions going to the exhaust gas and radiator. It is clear that much waste heat energy is available. Potential fuel cost savings versus costs of TE energy recuperation, as well as engineering reliability, are issues. This area, civilian and military, may receive a boost from the recent arrival of highly efficient hybrid vehicles having two engines, one electric and one conventional gasoline or diesel. There is an important need in the military for electric energy at remote shelters, solid or tent, in excess of what can be reasonably supplied by batteries. For example, a small heater, cookstove, or water distillation apparatus can include a TE module to provide electric energy not only for electronics or to recharge batteries that run electronics, but also for driving a heat circulating or exhaust fan. (Obviously, even a wood-fueled stove could generate electric power in the same way.) Such self-powered gas or heating oil furnaces would also benefit a civilian community that loses its electric power but not its heating oil or gas supply in a winter storm. The potential military TE power generation uses mentioned so far overlap substantially with corresponding civilian applications. However, there are other very important military applications that have relatively few or very small-scale civilian counterparts, at least at this time. Broadly speaking, they are the powering and communications functions of unattended sensors. These sensors are found in a variety of deployments. For example, micro-air vehicles are being developed for battlefield surveillance. In this system weight is a critical factor. For this application it is anticipated that a small TE generator device will be mounted on the hot motor to power the sensors. It is often desirable to deploy sensors to monitor the movement of personnel or vehicles, such as tanks or trucks. Depending on circumstances this can be done either by their careful placement in prescribed arrays by ground personnel or, when more remote areas need to be monitored, by air drop. In very long-term deployment of ground sensors that are unattended, either because the area is inaccessible to friendly forces or because of manpower limitations, long-term low to moderate power sources are required. Batteries alone may not last a sufficiently long time. So-called "energy harvesting" becomes the power source of choice, and TE devices provide the best solution. In many situations a small TE device with one end a few inches into the ground and its other end in the air can take advantage of the temperature difference between air and ground and provide approximately 100roW, more or less continuously. Interestingly, the

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ALEXANDER C. EHRLICH AND STUART A. WOLF

temperature gradient reverses between day and night, a factor that must be engineered into the system, but of course can be. Obviously, there can be long-term sensor monitoring in cases similar to those just described, or others where relatively infrequent short bursts of high power are required, say for sensor communication with a home base. Such circumstances can be handled by using a TE module to achieve the slow but continuous charging of a high discharge rate rechargeable battery. One can also imagine the use of TE power sources for "man present" information gathering field missions. Here a TE device might be powered by a small catalytic combuster that provides the needed heat source to generate a thermal gradient without getting too hot. A source of high temperature in the field runs a high risk of detection by infrared detectors belonging to the other side.

IV.

Conclusion

For several decades the military has employed TE modules in a number of niche applications, and that continues to be the case. However, the number and importance of the niches is growing with the increasing technological sophistication of war fighting, and this is bound to continue. The United States Marines and Army Rangers go into combat carrying more weight in batteries than in munitions. Sensors will be needed for the remarkably small-caliber "smart" munitions that forward-looking research now envisions. It seems inevitable that more energy harvesting and sensor cooling will be needed. The TE knowledge base and cadre of experts developed in the scientific community by the recent surge in military-funded TE research should prove to be a national asset in meeting these needs.

ACKNOWLEDGMENTS

The authors thank Dr. Donald Gubser of the Naval Research Laboratory, Drs. Francis Patten and Robert Nowak of the Defense Advanced Research Projects Agency, Dr. John Pazik of the Office of Naval Research, and Dr. John Prater of the Army Research Office for helpful insights regarding the subject matter of this chapter.

S E M I C O N D U C T O R S A N D SEMIMETALS, VOL. 70

CHAPTER

5

Theoretical and Computational Approaches for Identifying and Optimizing Novel Thermoelectric Materials David

J.

Singh

CENTER FOR COMPUTATIONAL MATERIALS SCIENCE NAVAL RESEARCH LABORATORY WASHINGTON, D.C.

I. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . II. SOMEFUNDAMENTALCONSIDERATIONS

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

1. The Thermoelectric Figure o f Merit . . . . . . . . . . . . . . . . . 2. Lattice Thermal Conductivity . . . . . . . . . . . . . . . . . . .

III. FIRSTPRINCIPLESMETHODOLOGY. . . . . . . . . . . . . . . . . . 1. Density Functional Calculations 2. Kinetic Transport Theory . . . IV.

SKUTTERUDITES

1. 2. 3. 4. 5. V.

.

.

.

.

.

.

.

.

.

. .

. .

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

Binary Skutterudites . . . . . . . . . Ce-Filled Skutterudites . . . . . . . . La-Filled Skutterudites . . . . . . . . Lattice Dynamics and Effects o f Filling . . Prospects . . . . . . . . . . . . .

CHEVREL PHASES

.

.

.

.

.

.

.

.

.

.

.

VI. fl-Zn4Sb3 . . . . . . . . . . . . . . VII. HALF-HEUSLERCOMPOUNDS . . . . . . VIII. CONCLUDINGREMARKS . . . . . . . . REFERENCES . . . . . . . . . . . . .

I.

125 127 127 127 128 128 131 134 134 142 146 156 162 162 166 170 172 173

Introduction

Discovering a material meeting given specifications is akin to finding the proverbial needle in a haystack. F o r thermoelectrics this is further complicated by seemingly contradictory requirements high thermopower, like a semiconductor; high electrical conductivity, like a metal; and low thermal conductivity, like a glass. The bias toward complex m u l t i c o m p o n e n t compounds inherent in the need for low thermal conductivity and the fact that electrical conductivity and t h e r m o p o w e r are strong functions of doping level make the Edisonian a p p r o a c h of making and testing all candidates difficult, 125 ISBN 0-12-752179-8 ISSN 0080-8784/01

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DAVID J. SINGH

to say the least. As a result, there is strong interest in developing a fundamental understanding of how high values of the figure of merit can arise and then using this understanding plus materials specific modeling to sort through the haystack, identifying promising materials, finding trends, winnowing out poor candidates, and suggesting avenues for optimization. In a nutshell, materials are made up of electrons and atomic nuclei. Both the interactions and governing dynamics are well established in the form of electromagnetic theory and quantum mechanics. Although the resulting many-body equations are far too complex to solve exactly for any real solid, the combination of approximate methods, particularly based on density functional theory, algorithmic developments, and advances in computing technology, have made it possible to calculate quite a number of properties of materials. These include electronic structures and lattice dynamics, which are of particular importance to thermoelectric materials. Over the past few years, such calculations have become more and more commonplace and have been applied to ever more complex compounds, including a number of materials newly under investigation as potential thermoelectrics. Because the calculations are ab initio, in other words, starting from first principles, and in particular do not use fits to experimental data, they are especially useful in sorting out the microscopic physics underlying properties of complex new materials. This is because, first of all, they do not rely on experimental inputs that are often not available in this case and secondly because they do not build in the physics of the models. On the other hand, because the calculations are approximate, they cannot be viewed as a surrogate for experiment, but rather as a complementary tool. Notably, some experimental groups are starting to use band structure calculations internally to sift through various compounds (e.g. Larson et al., 1999; Young et al. 1999). The goal in thermoelectrics research is to find new materials for application. This means finding materials with low-cost, favorable mechanical, thermal, electrical interfacial, and other properties. However, the sine qua non is finding a material with high figure of merit, Z T, in the intended temperature range. This defines the maximum possible performance of a material in a device. In this chapter the focus is on ZT. The use of modern ab initio band structure techniques to understand thermoelectric materials is illustrated and the results of calculations in relation to experimental measurements, are discussed. Following general considerations and an overview of the methodology as applied to thermoelectrics, results for various classes of materials are presented in turn. The emphasis of this chapter is on novel thermoelectrics, especially skutterdites, as they have been the most extensively studied using first principles methods, although certainly useful information is still being obtained about standard materials such as bismuth telluride (Mishra et al., 1997).

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IDENTIFYINGAND OPTIMIZING NOVEL THERMOELECTRICMATERIALS 127

II. 1.

Some Fundamental Considerations

THE THERMOELECTRIC FIGURE OF MERIT

The thermoelectric figure of merit, Z T = o S 2 T/K, where a is the electrical conductivity, S is the thermopower or Seebeck coefficient, and K is the thermal conductivity, is a dimensionless combination of electrical and thermal transport parameters. In normal crystalline materials, the electrical transport is determined by the electronic structure, particularly the band structure near the Fermi energy, E v, and the carrier scattering mechanisms. The thermal conductivity consists of electronic and phonon contributions, the latter determined by the lattice vibrations and scattering mechanisms. Both the electronic and lattice components of Z T are in general strongly temperature dependent, and the electronic components are strongly dependent on doping level as well. This means that optimization of a material for thermoelectric performance is needed before its usefulness can be assessed. Experimentally, once a sample is made, it is usually straightforward to measure transport coefficients as a function of T. However, measuring as a function of doping level requires a series of samples with various carrier concentrations, which may be quite nontrivial to make. Instead, the practice has been to assume a band-structure model consistent with existing experimental data (e.g., two parabolic bands), and then fit transport data as a function of temperature to fix the parameters (effective masses, bandgaps, etc.). The resulting band structure is then used to predict the variation of transport coefficients with doping. This essentially amounts to an extrapolation of experimental data. In well-characterized materials where the appropriate band structure model is well established, this can be a very accurate, reliable approach. First principles calculations, on the other hand, do not build in expectations about the band structure model, but are subject to approximations that cannot be readily removed. As such, they are more useful for materials that are not well characterized, in the sense that the band structure model is not known a priori.

2.

LATTICETHERMAL CONDUCTIVITY

The denominator of Z T consists of the total thermal conductivity, K, which as mentioned in normal crystals consists of electronic and lattice contributions, K = K~ + KI. At first sight, it would seem that any desired value of Z T is possible in a material provided that the thermal conductivity is sufficiently lowered to make this denominator small. In reality things are not so simple. This is because the electronic part, x~ is directly related to the electronic conductivity tr in the numerator, since the same carriers that

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DAVID J. SINGH

transport charge (giving a) also carry the electronic entropy (giving Ke). This connection is expressed by the Weidemann-Franz law (actually an approximation, but a very good one for metals and doped semiconductors at ordinary temperatures), Ke = LaT, where L is the Lorentz constant. Thus, one obtains Z T = rSZ/L, where r = Ke/(Ke + /s which is necessarily less than unity. Thus, the maximum Z T that may be obtained in a material is determined by the thermopower. A value Z T = 1 requires S ~ 160/W/K with r = 1, that is, ~:l negligible. Besides underscoring the role of the thermopower in ZT, this relation provides an important motivation for seeking materials with very low lattice thermal conductivity; one would like a material with r as close to unity as possible. This is because it becomes increasingly hard to find materials with reasonable electrical conduction as one imposes the requirement of higher and higher thermopower. So to get high Z T, we seek materials with high thermopowers, and lattice thermal conductivity small compared to the electronic thermal conductivity at the temperature of interest. This leads directly to the phonon-glass-electron-crystal (PGEC) paradigm proposed by Slack (1995, 1997, and references therein), which suggests looking for materials where charge carriers are weakly scattered but heat carrying phonons are strongly scattered, so lattice thermal conduction is glasslike. The main strategy, first proposed by Slack, for achieving a PGEC is to seek a framework material with favorable electronic properties and a large empty site in its crystal structure. Then an electronically inert species filling the void may have low frequency "rattling" vibrational motions that strongly scatter heat-carrying phonons without destroying the favorable electronic properties. A less stringent generalization is the so-called designer materials approach. In this strategy, one looks for materials with a filling or other chemically adjustable site that affects the electronic and vibrational properties differently, so that the thermoelectric properties can be tuned by appropriately modifying the material. As illustrated in the section on skutterudites, band structure calculations can be very helpful not only in sorting out the electronic structure of the framework, but also in understanding how filling atoms affect the framework electronic transport and the vibrational properties.

III. First Principles Methodology 1.

DENSITY FUNCTIONAL CALCULATIONS

The main tools for modern ab initio electronic structure calculations are based on the density functional theory (DFT) of Hohenberg, Kohn, and Sham, with the local density approximation (LDA) or generalized gradient

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approximation (GGA). The density functional theory and computational implementations of it have been reviewed extensively over the past few years, and so this not undertaken here. Suffice it to say that DFT is intrinsically a ground state theory. DFT states that the total energy, Eror, of the many-body electronic system is a variational functional of the electron density, n(r), ETO r = EorT[n(r)],

where the unique density that minimizes E~vr is the actual density. The exact form of EDv r is unknown, but the functional is thought to be both highly nonlocal and nonanalytic. The Kohn-Sham version of DFT, which is the basis of the commonly used practical implementations for solids, separates out terms in the energy that are expected to be large and nonlocal, particularly the electron-ion interaction, the static electron-electron Coulomb interaction (Hartree term), and the single particle kinetic energy term. These terms can be calculated accurately, leaving the unknown part of the functional in a smaller and more readily approximated exchangecorrelation term, E~[n]: EDFT[n] = Ts[n] + EE_I[n] + EHartree[n] -t- Ex~[n].

In Kohn-Sham theory, the density is written as a sum over so-called Kohn-Sham orbitals, ~ , n(r) = ~ ~o*(r)~oi(r) where the sum is over the occupied orbitals, done according to Fermi statistics. Using this representation of n, the kinetic energy, T~[n] is straightforward and the variational property can be exploited to get mean-field-like coupled single particle equations for the orbitals, HKs[n]q~ i = e,iqgl,

with HKs[n ] = T + VE_,(r) + Vu[n](r) + V~[n](r)

and Vxc[n](r) = 6E~[n]/6n(r). These are known as the Kohn-Sham equations. The LDA consists of approximating the functional Exc[n ] by an integral of the density with a

130

DAVID J. SINGH

local function of the density,

Exc[n 3 = f d3rn(r)ex~(n(r))" GGAs are similar but include local density gradient information as well, that is, exc(n(r)) is replaced by exc(n(r), Vn(r)). Such quantities as structural parameters and vibrational frequencies are generally predicted very accurately, in many materials to within 1 and 5%, respectively, with GGAs. Total energy differences (or analytic derivatives of the energy--forces and stresses) determine these quantities. For example, to determine a phonon frequency, the total energy and forces on the atoms may be calculated for the actual crystal structure (which is in equilibrium so the atomic forces are zero) and for small lattice distortions with the symmetry of the phonon mode in question. From the variation of these with the distortion, the energy surface for this symmetry may be constructed, and the resulting dynamical matrix diagonalized to determine the phonon frequencies and eigenvectors. This is the frozen phonon method. In the context of thermoelectric materials, this approach has been applied to study phonons in skutterudites by Feldman and Singh (1996) and Feldman et al. (2000), as well as in clathrates by Dong and co-workers (Dong and Sankey, 1999; Dong et al., 1999). It has also be used to calculate specific structural parameters needed for electronic structure studies and specific phonon modes by several authors (Fornari and Singh, 1999a; Sofo and Mahan, 1998; Nordstrom and Singh, 1996). Total energies are ground state properties, so only the basic local density or generalized gradient approximations and the efficacy of the computational implementation limit the accuracy of such calculations. Band structures, upon which electronic transport properties depend, are maps of one-particle excitation energies and are intrinsically not ground state properties. In fact, the reliability of DFT calculations of band structures varies widely between material classes. For example, in group IV and III-V semiconductors, bandgaps are severely underestimated; in transition metal based compounds Kohn-Sham gaps are often close to experiment, whereas in f-electron hybridization gap materials, Kohn-Sham gaps considerably exceed experimental values. However, despite these often large errors in bandgaps, band shapes are usually more reliably predicted, so that in many cases it is possible to use Kohn-Sham band structures to analyze electrical transport properties of thermoelectric materials. This is because for the doping levels appropriate for thermoelectric applications, opposite sign carriers contribute negligibly to transport. In other words, materials are useful when doped such that the carriers are either all electrons or all holes at the operating temperature. In this case, the value of bandgap is not as important as the shape of the band edge. Nonetheless, it is important to

5

IDENTIFYINGAND OPTIMIZING NOVEL THERMOELECTRIC MATERIALS 131

keep this error in K o h n - S h a m bandgaps in mind when assessing computed results. It should also be noted that several schemes, most notably the GW method, incorporating self-energy corrections have been derived and applied to materials. The GW method, in particular, greatly improves upon K o h n Sham band structures, giving both bandgaps and band dispersions of semiconductors in much better agreement with experiment. However, at the time of this writing the GW method has not yet been extensively applied to complex materials of thermoelectric interest.

2.

KINETIC TRANSPORT THEORY

The connection between band structure calculations and electronic transport is made via kinetic (or Boltzmann) transport theory as given by Ziman (1972) and others (e.g., Hurd, 1972). This approach is valid for diffusive transport (i.e., dimensions larger than the carrier mean free path) when the semiclassical picture is valid (mean free path larger than atomic distances and certain other generally very good approximations). The x component of the Bloch-Boltzmann kinetic equation in lowest order is given by (the other Cartesian directions are similar)

ax(T) = e2 f deN(e,)v2(e)'c(e,,T)(- Of (e,)/ae). Here the integral is over the band energy, e, T is the temperature, z(e, T) is the inverse scattering rate, N(e) is the electronic density of states at energy e, f(e) is the Fermi distribution function, and the average square band velocity, vZ(e), is defined by

N(e)v2(a) =

(2tO -3

f (v2/v~)dSE

N(e)v2(e) =

(2tO -3 f (1/v~) dSE,

where the integration over dS~ is over the isoenergy surface defined by E = e. At the Fermi energy, E = EF the integral is then over the Fermi surface, v~ is the Fermi velocity, and N(e)v](e) is related to the square plasma frequency, f~2 = 4rcZN(e)v2(e). For isotropic scattering, which is often, but not always, a good approximation (see, for example, Uehara and Tse, 2000), the relaxation time, z, does not enter the expression for the Hall concentration, nH = - r 1 6 2

132

DAVID J. SINGH

with ryH = (e3/12) f deN(e)v(e) 9[Tr(m-

1) __

m - 1]. u(8)q72(8, T ) ( - S f (e)/Se),

where for simplicity the expression for cubic symmetry is given. Here m is the k-dependent effective mass tensor, which is defined as

m ~ 1 = h-1(bva/c3k~) -- h-2(O2ek/Ok~Okt3). Provided that the electronic structure is not very strongly varying on the scale of k T, the derivatives of the Fermi function in the foregoing expressions may be replaced by their T = 0 limits, that is, the delta function. This eliminates the integrals over e and thus reduces the preceding transport quantities to integrals over the Fermi surface. The Bloch-Boltzmann expression for the thermopower S is

S(T) = (e/Try(T)) f deN(e)v2(e)er(e, T)(-Sf (a)/ae) = (e r~(r ))- ~ f de~o(~, r)( - ~f (E&), where ry(e)= N(e)vZ(e)r.(e, T) as usual. At low temperature, this expression for S(T) is proportional to T and to the logarithmic energy derivative of the conductivity,

S( T) = (re2k 2 T/3ery) dry/de, evaluated at E = E v. Note that although the preceding expressions for ry, nu, and S all involve energy integrals with (-Sf(e)/&), in most cases dry(~)/de is more strongly energy dependent than ry(e), so the approximation of replacing (-Sf(e)/Se) by its low-temperature delta function limit breaks down first for the thermopower as the temperature is raised, and in practice it is useful to keep the full expression for S(T) even though the low-temperature forms may be used for the other quantities. It should also be remarked that the preceding expressions involving ry(e) contain two energy-dependent ingredients--the band structure, which determines N(e), v(e), etc., and the scattering time, fie, T), which is not directly determined by the band structure alone. In fact, for different materials and different temperatures, the importance of various contributions to r - 1can be very different, such as electron-phonon, electron-electron, point defect, or magnetic. The result is that r is generally a very strong function of both

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temperature and composition. However, for many materials, z is only very weakly energy dependent on the scale of k T at fixed temperature. Exceptions are materials with extremely sharp band structure features near the Fermi energy and Kondo systems with resonant scattering at Ev. The fact that the thermopower is related to the logarithmic energy derivative of the conductivity implies that the best thermoelectric materials should have E v in a region where the band structure and/or scattering is strongly energy dependent on a scale of k T, where Tis the desired operating temperature. In fact, the known good thermoelectrics seem to satisfy this at least as regards the band structure. On the other hand, strong energy dependence of z associated with resonant scattering is generally associated with strong scattering where the energy dependence is strongest, so in Kondo systems the resistivity is usually high when the thermopower is high, complicating the search for high-ZT thermoelectrics among such materials. In any case, when the energy dependence (at fixed composition and temperature) of ~ is negligible, ~ cancels in the kinetic transport expression for the thermopower and then S(T) can be directly obtained as a function of doping level using only the band structure. Some criteria that are useful in identifying semiconducting materials with potentially high Z T can be obtained in terms of the foregoing (see also the extensive discussion of Mahan et al., 1997, and Mahan, 1998): 1. Heavy band masses: This is because heavy masses translate into a high logarithmic derivative of the plasma frequency for a given band filling. In the constant z approximation, this directly translates into high thermopowers at a given band filling. Note, however, that this only goes so far, because heavy bands also have low Fermi velocities for a given band filling, so o may be reduced. 2. Covalent or metallic band structure as opposed to ionic bonding: This favors weak charged defect scattering and high conductivity. 3. No transition metal ions with both local moment magnetism and contributions to the band structure near EF: This is to avoid magnetic scattering. This consideration does not necessarily apply for itinerant magnets well away from their ordering temperatures. 4. Low band gaps, but not so low as to have carriers of both signs at the operating temperature (i.e., Eg ~ a few k T): This is favorable for low defect scattering of carriers for E t near the band edge in many semiconductors. This consideration does not apply in cases where the conduction band minimum and valence band maximum have quite different characters, as for example charge transfer gaps in transition metal compounds. 5. Weak electron-phonon coupling: This favors high conductivity. However, covalent bonding is normally associated with high electron-phonon matrix elements. This can be mitigated if the strongly coupled phonons have high frequencies, but again this is problematic because thermoelectric

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DAVID J. SINGH

materials need to have low values of K~normally associated with lattices that are not too stiff. 6. Multivalley band structures: For fixed Ev this yields the same Fermi velocities but higher N(EF), and therefore N(~)vZ(e), but the same thermopower. However, multivalley band structures allow more scattering channels than an equivalent single-valley band structure, reducing z. So multivalley band structures are generally thought to be advantageous for high Z T but not as advantageous as obtained assuming simple proportionality for a. 7. Band edges formed from degenerate bands or multiple bands of the same carrier sign within a few k T of the band edge: This is for the same reason as before. 8. Anisotropic band masses: This allows high average Fermi velocities due to the light mass direction(s) (so high plasma frequencies and a) at the same time as strong logarithmic derivatives. This ties in with the idea of using multivalley band structures. For example, in a cubic crystal with a band edge at F the mass is forced to be isotropic, whereas at a general point it is not (although the transport properties, which are averages over the different valleys, of course, retain cubic symmetry). In the case of more than one band within a few kT of the band edge, it may be advantageous if they have different band masses for the same reason, though the possibility of interband scattering complicates the picture. Finally, it should be noted that these are just some generally useful guidelines. Not all high-ZT materials fit even the majority of them, and besides they are partially contradictory, as are the criteria for Z T itself, such as high a and S at the same time.

IV. 1.

Skutterudites

BINARY SKUTTERUDITES

We begin with skutterudites and filled skutterudites, as to date, they are the thermoelectric materials that have been most extensively studied using first principles calculations and perhaps best illustrate the advantages of this approach applied to novel materials. Skutterudite pnictides MA3, where M is one of the transition metals, Co, Rh, or Ir, and A is P, As, or Sb (Kjekshus and Rakke, 1974), were initially suggested as potentially high-ZT thermoelectrics by Slack and Tsoukala (1994) and by Caillat et al. (1993). These skutterudites occur in a bcc Im3 structure. There are four formula units per cell and they may be regarded as derived from the simple cubic perovskite structure DMA3 (F2 denotes an

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empty site) with tilts of the octahedra so that the A-site atoms move along the cube faces to form nearly square four-membered rings. Another view of the structure is as an expanded simple cubic lattice of M atoms, 3-filled with four-membered pnictogen rings oriented alternately along the x, y, and z directions so as to preserve overall body-centered cubic symmetry, that is, M4(A4)3E], making clear the presence of a large void in the structure, though because of the ionic radii of the pnictogens and the orientation of the rings the D site is coordinated by A atoms, not M atoms as the formula might otherwise suggest. At the time the compounds were first investigated in terms of their thermoelectric potential, little was known their electronic structures. However, as noted by Slack and Tsoukala (1994), particularly IrSb 3 seemingly satisfied a number of criteria for a high-ZT phase. First of all, the large unit cell and heavy atoms suggest a low minimal thermal conductivity. Besides, should low thermal conductivity not be realized in the binary (as in fact proved to be the case), it could then be had by inserting weakly bound heavy atoms (or according to the term coined by Slack, 1997, "rattlers") into the voids to scatter heat carrying phonons. In fact, several filled skutterudites, such as LaFe4P12, were known. Second, the electronegativity difference between Ir and Sb is very small, suggesting perhaps high mobility. Third, a wide variety of stable skutterudite structure compounds and alloys are known, all forming at reasonably high temperature and all retaining the undistorted lm3 structure (Kjekshus and Rakke, 1974; Ackermann and Wold, 1977; Villars and Calvert, 1991). This wide range, from the CoP3 to CoSb3 to IrP 3 and IrSb 3, hints at possibilities for optimizing Z T by chemical modification. Finally, all these compounds were generally reported as semiconductors with moderate bandgaps of order 1 eV, although Ackermann and Wold (1977) reported finding no clean optical gap in CoAs 3 and CoSb3; a 0.45 eV optical gap was reported in CoP 3. In any case, Slack and Tsoukala (1994) found that an unoptimized sample of IrSb 3 had an attractive combination of high carrier mobility and fairly high thermopower (72 #V/K at 300 K, increasing to above 130 /tV/K at 700 K, the highest temperature measured) at a modest doping level of p = 1.1 x 1019 cm-3. Generally, skutterudite samples grown under near equilibrium conditions tend to be p-type, though n-type samples can be made. The measured thermal conductivity of 160 mW/cmK at 300 K was too high for a good thermoelectric, and alloying by replacing 50% of the Ir by Rh did not lower this nearly enough. Nonetheless, the results were considered promising, since, as mentioned, the samples were not optimized, and the presence of the void site suggested other ways in which the thermal conductivity might be lowered. Slack and Tsoukala obtained a bandgap of 1.4 eV from optical data and a band mass of 0.17 me from transport data. The implications that might be drawn from these measurements are as follows: First, IrSb 3 could be the basis of a new high-ZT material. Second,

136

DAVID J. SINGH

o

--,5

~)

-

~ --

-

_

" - - - - - ~

9

~

-10

IrSb~

-15

F

"

H

N

F

I

FIG. 1. Density functional band structure of skutterudite IrSb 3 as calculated by Singh and Pickett using the L A P W method (1994). The valence bands from - 6 to - 1 eV (relative to Ev) are mixtures of Ir d-states and Sb p-states associated with the bonding of the pnictogen rings. The lower manifold of 12 bands (starting at approximately - 1 3 eV) are pnictogen s states. Note the single degenerate nonparabolic band dispersing up from this manifold to the conduction band minimum at F.

5

~

I,~1"

-10

F

H

N

T'

P

FIG. 2. LDA band structure of CoAs3, after Singh and Pickett (1994). Note the parabolic zero gap band structure. The lowest manifold from approximately - 1 5 eV to - 9 eV are pnictogen s states.

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137

the way to achieve this would be to find a way to lower the lattice thermal conductivity without seriously perturbing the electronic structure and then optimize the doping level, p. Following these measurements, several groups (Feldman and Singh, 1996; Sofo and Mahan, 1998; Llunell et al., 1996, 1998; Fornari and Singh, 1999a, b), starting with Singh and Pickett (1994), performed density functional band structure calculations. The results were considerably at odds with the preceding picture, but consistent with the experimental measurements when analyzed in these terms. Calculated LDA band structures of IrSb 3, CoAs3, CoSb 3, and CoP 3 are shown in Figs. 1-4, respectively. These were obtained within the LDA using the general potential linearized augmented planewave (LAPW) method (Singh, 1994), which is well suited to materials with open crystal structures and low site symmetries. The corresponding total and projected electronic density of states (DOS) is shown in Fig. 5 for CoP 3 and the corresponding filled skutterudite, LaFe4PI2. The total and projected DOS for IrSb 3 is shown in Fig. 6. In all these binary skutterudites, there is a well-defined gap between manifolds of valence and conduction bands. Both of these manifolds are derived from hybridized combinations of transition metal d and pnictogen p states, reflecting the bonding character. There is a particularly noticeable bonding-antibonding separation of the pnictogen p bands, with bonding and nonbonding states in the valence bands, and antibonding combinations in the conduction bands. The strength of the transition metal pnictogen

$

V W

-10

-1 FIG. 3.

C ,Sb

H

N

r

LDA band structure of CoSb 3, after Singh and Pickett (1994).

138

DAVID J. SINGH

2.O

1.0

0,0 ILl

-1.0

-2.0

FIG. 4.

r

I1

,.o 2.O o

F

coP.,

~

i!

~

~''

0.0

4.0 G

F

Band structure of CoP 3 around the Fermi energy, after Fornari and Singh (1999a).

5.0

qlm

N

LaFe,P~

t

3.0

0.0 -9.0

4.0

EnergyleVI -3.0

0.0

3.0

FIG. 5. Total and projected electronic density of states (DOS) of CoP 3 (top) and LaFe4P12 (bottom) after Fornari and Singh (199%; see also Harima, 1998). The solid line is the total DOS on a per formula unit basis. The dashed line is the transition metal component and the light and dark shaded areas are the pnictogen p and rare earth f contributions respectively. Note the f-resonance above E F in the filled compound.

5

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139

IO0 80

Totol

I

6O 40 20

,g 4o

20

0

! !

20 0 - 1.2

Sb

L -0.8

-0.4

E (Ry)

0

0.4

FIG. 6. Total and projected electronic density of states (DOS) of IrSb3, after Singh and Pickett (1994). The DOS is on a per formula unit basis, while the Ir and Sb projections are on a per atom basis. Note the very small DOS in the region where the nonparabolic gap crossing band occurs (from - 0 . 1 to 0 Ry).

hybridization depends on the particular compound. In the Co compounds, the main transition metal d weight in the DOS (see Fig. 5) is near the top of the valence band manifold, yielding a high density of flat bands in this region. In the Rh and Ir compounds, the transition metal d states are more hybridized with the pnictogen p states, so the d component of the DOS is much smoother and parallel to the p component both in the valence and conduction bands. In contrast to the Co-derived skutterudites, there is not nearly as strong a concentration of the transition metal d character near the top of the valence band in these 4d and 5d compounds. Electronic transport properties are determined by the electronic structure only within a few k T of EF. For p-type samples this means the valence band

140

DAVID J. SINGH

edge. As seen in Figs. 1-3, this differs considerably from the previous expectation of moderate gap, parabolic band semiconducting behavior. Instead, a singly degenerate band disperses upward from the valence band manifold and crosses the gap, reaching a maximum at the F-point. As a result, the binary skutterudites are zero or narrow gap, with the shape of the band edge related to the particular pnictogen atom. The antimonides in particular have a highly nonparabolic dispersion that is effectively linear going down to very low carrier densities (Singh and Pickett, 1994; Sofo and Mahan, 1998). As noted by Singh and Pickett for IrSb 3 and emphasized by Sofo and Mahan (1998) for CoSb 3, the exact details at the band edge depend noticeably on the details of the crystal structure. This is consistent with the strong electron-phonon interaction implied by the observation of superconductivity in some of the related La-filled compounds (Meisner, 1981; Torikachvili et al., 1987; Shirotani et al., 1996, 1997; Uchiumi et al., 1999). Specifically, superconductivity has been observed in LaFe4Px2, LaRu4Px2, LaOs4P12, LaRu4Asx2, PrRu4Asl2, and LaRu4Sb12. For doping levels of thermoelectric relevance the Fermi level is far enough away from the band edge that these details are insignificant. However, it should be kept in mind that all things being equal, strong deformation potentials and the resulting large electron-phonon matrix elements reduce the carrier mobility and are therefore detrimental to ZT. CoAs 3 is found to have a zero gap with a parabolic valence band dispersion, whereas in the phosphides the gap crossing band disperses into the bottom of the conduction band manifold, yielding metallic behavior (Fornari and Singh, 1999a). Partik and Lutz (1999) have performed semiempirical tight binding band structure calculations on several skutterudite type compounds and analyzed these beginning with calculations on isolated pnictogen rings. The importance of the pnictogen rings in the phosphides was also noted by Llunell et al. (1996). They find that the inter-ring hopping is stronger in the phosphides than in the corresponding arsenides, so the former compounds have larger valence band dispersions. This is consistent with the LDA results of Fornari and Singh (1999a), showing band overlap and metallic behavior. Related to this, Watcharapasorn et al. (1999) report transport data for CoP 3 consistent with a low carrier density metal, in particular a relatively low thermopower that changes sign as a function of T. The crossover between parabolic and linear dispersion in CoSb 3 and IrSb 3 occurs within 2% of the distance from F to the zone boundary, corresponding to hole doping levels in the 3 x 10X6cm -3 range. The calculated slopes of the linear dispersing bands are ~ = -3.45 eV/,X, for IrSb 3 and c~ = - 3.10 eV/A for CoSb 3. Since, as mentioned, the experimental hole doping levels are generally higher than the crossover, the transport properties are expected to be strongly modified from standard parabolic band semiconductor behavior. Besides the different energy dependence of the DOS (quadratic vs square root) and number of carriers (cubic vs.

5

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141

power), the effective mass tensor ViVje(k), normally diagonal near a band edge, is entirely off-diagonal, corresponding to an "infinite" transport mass normal to the Fermi surface and a doping-level-dependent cyclotron mass given by k/o~. Here k is the magnitude of the wave vector and is proportional to pl/3. The Hall number yields the number of carriers in exactly the same way as for a parabolic band provided that the dispersion and scattering are isotropic, as is expected to hold in the moderately doped cubic materials. The degenerate Seebeck coefficient within the constant scattering time approximation is given by S = - (27tkgT/3e~)(rc/3p) 1/3. Because of the stronger dependence of E v on p in this quasi-linear case, the degenerate regime will hold to higher temperature than in the case of parabolic dispersion, so S will remain linear in T to higher temperature, allowing potentially higher values of ZT, albeit only at high T. However, it should also be noted that this expression has a different, weaker doping level dependence than the p-2/3 of the parabolic case. This would make optimization of Z T difficult, especially considering the fact that the conductivity will fall off (depending on the details of the scattering) with decreasing p. In any case, detailed measurements of the doping level dependence of S(300 K) on the Hall carrier concentration p in a series of samples by Morelli et al. (1995) and Caillat and co-workers (1996a, 1996c) agree remarkably well with the predicted dependence, including the first principles value of ~. Other recent transport data are also consistent with the results of the calculations, particularly the small bandgaps and nonparabolic valence band dispersion (Matsubara, 1996; Anno et al., 1998, 1999; Rakoto et al., 1998, 1999; Caillat et al., 1996a, 1996b; Tritt et al., 1996; Arushanov et al., 1997), although some of the results have been interpreted within different band structure models. The slightly different value of ~ calculated for IrSb 3 also gives quantitative agreement with Slack and Tsoukala's (1994) earlier transport data for that compound. Taken together, these data, in conjunction with the fact that the LDA band structure calculations contain no fits to the experimental data, constitute strong confirmation of the basic features of the calculated result. Why, then, were the binary skutterudites incorrectly classified as moderate gap parabolic band semiconductors until first principles calculations were done? The answer is twofold. First, since there are 16 atoms in the skutterudite unit cell, and the gap-crossing band is singly degenerate and quite dispersive, its optical signal is small and can be missed or assigned to impurities or other extrinsic effects. (But note that Ackermann and Wold, 1977, did note the absence of a clean gap in CoAs3. ) This is apparent from examining the bare densities of states of Figs. 5 and 6. Second, only a limited amount of transport data had been taken, and since it was possible to fit the data well using more than one model, as is appropriate, the simplest

142

DAVID J. SINGH

interpretations in terms of conventional parabolic models were made. More generally, as the complexity of a material increases, one measure of which, for crystals, is the number of atoms in a unit cell, the amount of experimental characterization needed to understand the electronic properties grows rapidly, as does the number of plausible band structure models. The role of first principles calculations in this case is to provide a direct window on the microscopic band structure that can be tested and refined by experiment. As mentioned, the binary skutterudites have a singly degenerate gapcrossing band with a valence band maximum at F. This band has a strong and, in the phosphides and antimonides, highly nonparabolic dispersion. This electronic structure is not favorable for p-type high-ZT thermoelectric performance. The conclusion that may be drawn from the band structure is that making a good p-type thermoelectric from the binary skutterudites (as has been done with Ce (Fleurial et al., 1996) and La (Sales et al., 1996 filling) depends on both lowering the thermal conductivity and modification of the band structure. Filling elements that are thought, either on the basis of experiment or band calculations, to modify the valence band structure by interaction with f-states are La and Ce (discussed later), as well as U, Yb, and Pr (Meisner et al., 1985; Torikachvili et al., 1987; Sekine et al., 1997; Chen et al., 1997; Dilley et al., 1998; Sekine et al., 1999; Nanba et al., 1999; Dordevic et al., 1999; Chapon et al., 1999; Shirotani et al., 1999). Of these, some La and Ce fillings are now known to produce good p-type thermoelectrics, and many of the compounds have yet to be studied in detail. The conduction band structure, on the other hand, shows several degenerate bands near the band edge at the F-point, and depending on the compound these have reasonably heavy masses for a thermoelectric. Thus, in the n-type case, modification of the band structure is not required, although suppression of the thermal conductivity is needed. This suppression has been achieved by alloying and rare-earth and T1 filling (Caillat et al., 1996a, 1996c; Morelli et al., 1997; Sales et al., 2000).

2.

Ce-FILLED SKUTTERUDITES

To date, the highest reported values of Z T in skutterudites were measured by Fleurial et al. (1996) and are in p-type CeFeaSb12 at elevated temperature. Relative to the corresponding binary, CoSb 3, p-type CeFe4Sb12 shows much reduced lattice thermal conductivity, much higher thermopower (for given p) but strongly reduced carrier mobility. Remarkably, the highest Z T sample had a measured Hall concentration p = 5.5 • 1021 c m - 3 - - e s s e n tially a metallic value. LDA band structure calculations were reported for CeFe4P12 and CeFe4Sb12 by Nordstrom and Singh (1996). They also determined the internal structural parameters u and v and corresponding Raman fre-

5

IDENTIFYINGAND OPTIMIZING NOVEL THERMOELECTRIC MATERIALS

143

quencies. Previously, CeFe4P12 and CeFe4As12 had been reported as semiconducting. CeFe4Sb12 was reported as metallic and characterized as heavy-Fermion-like (Grandjean et al., 1984; Morelli and Meisner, 1995). One possibility for explaining this behavior was that Ce is tetravelent in these compounds (or at least the p and As ones, which have smaller filling sites), thus leading to a close relationship with the then isoelectronic binary Co skutterudites. One argument in favor of this possibility is that there is a departure from the lanthanide contraction of the lattice constants of Ce-filled as compared to other rare-earth filled skutterudites, at least for the phosphides and arsenides (Grandjean et al., 1984). However, an alternate view of the Ce valency had been suggested by Meisner et al. (1985), who compared the properties of CeFe4P12 with those of UFe4Px2 and ThFe4P~2. The Ce and U compounds were found semiconducting, but the Th compound metallic. This is in contradiction to simple expectations assuming Th and Ce to be tetravalent and U trivalent. They suggested that instead it is hybridization between Ce and U f electrons and the valence bands that forms the observed gaps. This picture was also supported by electron spin resonance experiments on rare-earth impurities in CeFe4P12 (Martins et al., 1994). Lee et al. (1999) have reported XANES studies of CeRu4P12 and PrRu4Pa2 in accord with this conclusion and classify CeRu4Px2 as a hybridization gap compound (see also Long et al. (1999). The LDA band structures of CeFe4P12 and CeFe4Sb12 are shown in Figs. 7 to 9. Bandgaps at E F are apparent in both materials. These occur beneath flat bands arising from the Ce 4f states and have magnitudes of 0.34 and 0.10 eV, for the phosphide and antimonide, respectively. As in CoSb 3, the

0

-4

-6

F

H

N

F

P

FIG. 7. LDA band structure of CeFeaP12 after Nordstrom and Singh (1996). Spin orbit interactions are included. The energy zero is at the valence band maximum (denoted by the dashed line).

144

DAVID J. SINGH 0.8

J

0.4 ~ ' - - ' ~ - ~

~-'---~

-"'~-"'~

~

>. |

-0.4

-0.8

FIG. 8. (1996).

F

H

N

F

P

Blowup of the band structure of CeFe4P12 around E v, after Nordstrom and Singh

upper-lying valence bands are of hybridized transition metal d-pnictogen p character, whereas the lowest conduction bands are dominated by the narrow spin-orbit split Ce 4f states. These latter are in a gap formed by the other bands. However, a careful analysis showed that there is also substantial Ce 4f character in the upper valence bands. There is correspondingly pnictogen p and transition metal d character in the nominally 4f bands, especially near the zone boundary where the dispersion is largest. Nordstrom and Singh (1996) used an analysis of the charge density in terms of

0.8 84

0.4

-0.~

-0.8

FIG. 9. (1996).

F

H

N

F

P

Calculated LDA band structure of CeFe4Sb12 near E v, after Nordstrom and Singh

5

IDENTIFYING AND OPTIMIZING NOVEL THERMOELECTRIC MATERIALS

145

overlapping ions to determine the Ce valence. They found that the calculations could only reasonably be matched by Ce 4f 1 configurations and that best results were obtained including hybridization involving unoccupied Ce 5d states as well. In any case, the calculated results could not be interpreted in terms of tetravalent Ce. CeFe4Px2 (Figs. 7, 8) shows two parabolic bands at the F point valence maximum, whereas CeFe4Sb~2 (Fig. 9) shows only one. The highest valence band in the antimonide has mixed Ce 4f, pnicotgen p, and Fe 3d character. This heavy band (m = 2.2 me) is presumably responsible for the p-type transport in CeFe4Sb~2, although quantitative comparisons of transport properties have not been done in view of the fact that LDA calculations overestimate 4f hybridization and therefore underestimate the band masses. This overestimation also manifests itself in the fact that the calculated gaps are significantly larger than measured in the phosphide and arsenide and that the likely valence fluctuation metallic nature of the antimonide is not reproduced. Interestingly, the LDA conduction band minima show multivalley structures in these compounds with band masses in the range 10-20 me. At this time, no n-type samples of CeFe4Sbx2 or CeFe4As~2 (which has a similar, though more hybridized, band structure; Singh, unpublished; Fig. 10) have been reported, but this band structure may have interesting implications for Z T below room temperature, should highly filled (to avoid localizing these bands) n-type samples be made. Because of the greater hybridization of the arsenide, n-type conduction may be more realizable in it than in the antimonide.

0 ~---------

........ '-...................... '

> ,~-2

---'-........... ]

.... 4 6

-8

s

CeFe4As12

H

N

F

P

FIG. 10. B a n d s t r u c t u r e of CeFe4Asa2 c a l c u l a t e d w i t h i n the L S D A using the general potential LAPW method.

146 3.

DAVID J. SINGH La-FILLED SKUTTERUDITES

As mentioned, the strategy suggested by the early experimental data on the binary skutterudites is to seek modifications of them that have lower lattice thermal conductivity with minimal modification to the electronic properties. Among the modifications tried were alloying on the transition metal and/or pnictogen sites (Caillat et al., 1996c; Nolas et al., 1996a, 1996b; Tritt et al., 1996; Nolas et al., 1999, and references therein) and filling of the large voids in the skutterudite structure with heavy atoms (Morelli and Meisner, 1995; Fleurial et al., 1996; Sales et al., 1996) that might have soft anharmonic Einstein-like phonon modes and thereby reduce the thermal conductivity. Both of these approaches are successful in that they result in considerably reduced values of ~c1, and, as mentioned, in Ce(Fe, Co)48b12 and La(Fe, C0)48b12 high values of Z T have been reported. This achievement was rationalized by electron counting, although as is discussed here, the resulting picture is misleading. In any case, the argument was as follows: CoSb 3 is a semiconductor with a high power factor but poor thermal properties. LaFe4Sb12 and CeFegSbi2 are reported as metallic compounds with considerably lower thermal conductivities, presumably due to the "rattling ion" model of Slack (1997). In the highly covalent band structures of these materials, each Fe atom should contribute one less electron than a Co atom in the CoSb 3, whereas the additional trivalent La should contribute three electrons, resulting in a net deficiency of one electron per formula unit. Accordingly, the composition LaFe3CoSbx2 may be electronically similar to CoSb 3, providing the desired combination of high power factor and low ~c. As discussed earlier, there is strong evidence that Ce is trivalent in these compounds, leading to a similar argument, although there are complications due to heavy Fermion/valence fluctuation effects at low temperature. In fact, as synthesized La(Fe, Co)48b12 forms La-deficient, so that the actual Co concentration needed to obtain compensation is higher than 25%. High values of Z T approaching unity at high temperature were first reported in La-filled skutterudites by Sales et al. (1996). The problem is that first principles calculations for CoSb 3 show a small bandgap with a highly nonparabolic, quasi-linear valence band dispersion. This means that in the constant scattering time degenerate regime, S oc p-1/3. Also a oc p2/3. This is instead of the usual parabolic S oc p-2/3 and a oc p within this regime. Thus, the power factor aS 2 will be less dependent on doping level than in a conventional case and that it will be more difficult to optimize Z T by adjusting the carrier concentration p (but note that K has an electronic component proportional to aT). Thus, as mentioned, it would be preferable if the band structure of La(Fe, Co)48b12 were different from CoSb 3 as this would allow more efficient optimization. This, in fact, is the case. The calculated band structure of LaFe4Sbx2 as obtained by Singh and Mazin (1997) is shown in Fig. 11. This

5

IDENTIFYINGAND OPTIMIZING NOVEL THERMOELECTRIC MATERIALS

147

4

~_~..__~.~~------~--

__~

__~-~;

_ ~,-

_

i l

o

~

-

~

'

~

J

-2

-4

-6

r

H

N

F

P

FIG. 11. Band structure of LaFe4Sb12 as obtained by Singh and Mazin (1997). The horizontal line denotes the Fermi energy. Note the gap and the heavy Fe derived bands just below the valence band edge.

band structure shows an indirect F - N gap of 0.60 eV at a position above E F corresponding to a band filling of one more electron. This is qualitatively different from the band structure of CoSb 3. The direct gap at F is 0.76 eV. Both the conduction and valence band edges are formed from parabolic bands with hybridized F e / C o - S b character. In addition there are some rather flat (heavy mass), primarily Fe/Co d derived bands near but not at the band edge both above and below the gap. The projected electronic density of states (DOS) corresponding to this band structure is shown in Fig. 12. Although the DOS is highly covalent, it is less so than lrSb 3 and is, not surprisingly, closer in shape to CoSb 3. In particular, the Fe/Co d contribution is peaked in the region from - 1 eV to 0 eV relative to E f and is therefore concentrated in the energy region around the gap. In contrast, in IrSb 3 the Ir d contribution is spread more uniformly over the valence band region from - 6 eV to well above E F. The peaked structure of the DOS in LaFe4Sbx2 corresponds to the flat Fe/Co-d derived bands seen in Fig. 11 and results in the fact that E r lies on the edge of a very sharp decrease in the DOS. This leads to an expectation that high values of S may occur in the La(Fe, Co)4Sbx2 system, as observed and

148

DAVID J. SINGH 40

I

.~ 9 zO

-

I

,]

io 9 , o

I

LaFe 4Sb 12

3o

z

I

-6

-4

;

l :v!

-2

0

E (ev)

. Z

_

4

6

FIG. 12. Electronic DOS of LaFe4Sb12 as obtained by Singh and Mazin (1997) using the LAPW method. The DOS is on a per formula unit basis and the dotted line denotes the Co d contribution.

quantitatively determined using kinetic transport theory. However, before proceeding to discuss calculations of the transport properties for La(Fe, Co)48b12 it is helpful to discuss to what extent its band structure is similar to that of LaFe4Sb12. The degree of similarity relates nearly directly to the amount of scattering to be expected from Fe/Co disorder on the transition metal site. In order to address this issue, Singh and Mazin (1997) performed virtual crystal approximation calculations for this material with 25% and 50% replacement of Fe by Co. In these calculations they held the structure fixed at the LaFe4Sb12 structure and kept the La concentration stoichiometric. The calculations showed good rigid band behavior. This is illustrated in Fig. 13, which shows the total DOS for the three different Co concentrations, and Fig. 14, which is the band structure near E v for the semiconducting 25% Co composition. The main variation near the gap as the Co concentration is increased is due to a small upward shift of the lighter mass band forming the valence band edge at F, relative to the Fe/Co d derived flat bands. This leads to a gradual narrowing of the bandgap with increasing Co concentration. Given rigid band behavior, calculation of the transport properties as a function of doping level amounts to calculating the properties with fixed band structure for various positions of the Fermi level. For this purpose, Singh and Mazin (1997) used the virtual crystal band structure calculated with 25% Co. A blowup of this band structure near the gap is shown in Fig. 14. As in undoped LaFe4Sbx2 the gap of 0.56 eV is indirect with a valence

IDENTIFYINGAND OPTIMIZING NOVEL THERMOELECTRIC MATERIALS 149

5

40,

, '

:

30

l! ~'

20

0

/ L,'. , :J '1/: :~ " t: ;

.2

, I. ,~ , ,,.,

. l. .,'i ,:

~1'.. ,.;

'',.'

,

~ . , ..'. : :, ,: , : " ~

:

!'1

i

'"

,,

:

: P i l l ~ fl 611. : ':,li'~ ll,([l'

~"

(: I'~tlllrL/~l::

'

i

i

I"

~1 f,,

:

J;J

0

.1 E

,..,Il,.~.lil i~&

P. ,'' I

,,

'.

i

1

"

9

2

(eV)

FIG. 13. Total DOS for La(Fe, Co)48b12 with 0% (solid), 25% (dotted), and 50% (dashed) Co as obtained by Singh and Mazin (1997) using the virtual crystal approximation. Note the approximate rigid band behavior.

0.6

0.4

0.2

o .,.., LLI

-0.2

-0.4

-0.6 r

~

FIG. 14. Virtual crystal band structure of La(Fe, Mazin (1997). Note the expanded energy scale.

N

Co)48b12 with 25% Co, after Singh and

150

DAVID J. SINGH

band maximum at F. The LDA direct gap at F is 0.66 eV. The conduction band minimum is along the F - H line and coincidentally is virtual degenerate with the lowest band at N. Although these multiple minima in the zone provide a favorable arrangement for thermoelectric performance in n-type material, we note that n-type La(Fe, Co)48b12 apparently cannot be made under normal conditions without substantial La deficiency. Such nonstoichiometry would modify the band structure. However, some very n-type skutterudite compositions have been made under pressure, and this should be applicable to the present composition as well (Takizawa et al., 1999). The valence band structure is also interesting from a thermoelectric point of view, and in fact this is what determines the properties of the high Z T p-type samples reported by Sales et al. (1996). The valence band edge is derived from a singly degenerate light mass (m ~ 0.2 me) band of hybridized Sb p character, which disperses downward away from the F point. However, immediately below this band is a much flatter band (m ~ 3 me) of primarily Fe/Co-d character. With increasing hole doping, starting with undoped material, the Fermi energy initially moves into the light band. However, at a band filling of only 0.005 e/cell (1.3 x 1019 cm-3), corresponding to a Fermi energy of 0.1 eV relative to the valence band edge, the second band is reached. This heavy band shows strong onsets in N(E), v~(E), and related quantities, resulting in a strong energy dependence in o, and high values of S. Reported samples (Sales et al., 1996) show Hall numbers above 10 2~ cm-3, implying that both the light and heavy bands are active in transport. In order to understand the transport more quantitatively, calculations of transport properties as a function of hole doping away from the gap were done based on the kinetic transport expressions given earlier and the 25% Co virtual crystal band structure of Fig. 14. The resulting Hall number, carrier concentration (doping level), and thermopower at 600 K are shown as a function of Ev in Fig. 15. Over almost all of the range shown, the Hall number is lower than the doping level. This deviation, which increases with band filling, reflects deviations from parabolic band behavior. Although the presence of the heavy band is responsible for the high thermopower of this material, S at 600 K does not show any noticeable structure near - 2 6 toRy, which is the onset of the heavy band. This is because the form of the expression for S has strong contributions at energies up to 2-3 k T away from the chemical potential and because k T at 600 K is 3.7 mRy. Thus, the heavy band contributes to S over the entire range of Fig. 15. However, the carrier concentration increases sharply below the onset of the second band, reflecting its sharp dispersion. This provides a mechanism for pinning the Fermi level near the onset, providing high thermopowers even though the carrier concentration is apparently difficult to control in these materials. Although the Hall number is not reported, the sample shown by Sales et al. (1996) has S(600 K) ~ 180/tV/K. Comparing with the calculations, this corresponds to a chemical potential of - 3 0 mRy, or 4 mRy below the onset

5

IDENTIFYING AND OPTIMIZING NOVEL THERMOELECTRIC MATERIALS

300

0.06

'., ',. ',.,.

28O

9

0.04

,

.f

/

'., ',

260

'.,

e4o

\9 220

\

j.-

/

*

O.::X3

/

0.-)2 .0.~

\

\f

/J

ZOO

1110

1Q,O

0.~

j

" 9

w

"" ,

:,-~

// J

,, i

- O.04

./

.

-

,--m

J A

151

r. :'..

9 0.01. ',

- 0:31

~

,

.ma

E.Er

.-z~ ]

0..:)0

-2,

-~

FIG. 15. Hole concentration p ( + ) , Hall number (dashed), and S(600 K) (solid) as a function of Fermi level as calculated by Singh and Mazin (1997). The Fermi level is with respect to the center of the gap. The band edge is at E - Er = - 18 mRy.

of the heavy band. This matches a doping level of 0.08 holes per cell and a Hall number of 0.045 holes per cell. The Hall concentration of 1.2 x 10 20 holes per cm-3 is consistent with the reported lower end of the range of reported Hall concentrations for various samples (1 x 10 20 c m - 3 ) . However, Singh and Mazin (1997) obtained only semiquantitative agreement comparing the calculated temperature dependence of S with the reported experimental values. In particular, the agreement was not nearly as good as was obtained for binary skutterudites, particularly in view of the fact that they used the Seebeck coefficient to fix the carrier density, whereas in the binaries it was determined from independent measurements of the Hall number. The implication is that either the approximation of neglecting the energy dependence of z is inadequate, as could be the case in the light band near the onset of the heavy band, or the calculated band structure is distorted relative to the true electronic structure of the sample. The latter possibility is not readily discounted. The fact that the actual, as measured, material was La deficient must be borne in mind alongside the result that La strongly modifies the valence band structure. Specifically, L a - F e - S b hybridization determines the relative positions of the heavy and light bands, which is an important ingredient in determining the temperature depend-

152

DAVID J. SINGH

ence of S. One could go as far as to say that the deviation between the calculated and measured S(T) is evidence of the large effect of La-deficiency. Related to this, it should be noted that the thermal conductivity for this sample in the temperature range above 500 K is roughly one-third electronic and two-thirds lattice in origin. As mentioned, ~ct is strongly reduced from ~ct of CoSb 3 by La addition, but most of this effect is expected to occur at fairly low La concentrations within the rattling ion framework of Slack (1995). This is supported by measurements in the Ce(Fe, Co)48b12 on samples with different Ce concentrations (Fleurial et al., 1996). Because of the strong interaction between the valence bands and La, La vacancies should strongly scatter carriers, reducing the electrical conductivity. This is consistent with the wide range in hole mobilities (2-30 cmZ/Vs) measured for the various samples. Based on this, Singh and Mazin (1997) conjectured that samples with higher La filling, if they can be made, would have higher mobilities and thus higher values of a at a fixed band filling. Because of the Wiedemann-Franz relation, this would also lead to higher tce. In that case, the value of S would become the most important factor determining Z T at a given temperature. Then it may be quite feasible to increase Z T substantially in this material if the La concentration could be made stoichiometric near 25% Co concentration. In this case, the doping level for maximum Z T will occur at lower carrier concentrations than any of the reported samples, and additionally the temperature where Z T reaches its maximum would move down. According to the band calculations discussed earlier, the favorable thermoelectric properties of p-type La(Fe, Co)4Sb12 result from a numerical coincidence. In particular, the light "gap-crossing" band that dominates the p-type transport in CoSb 3 is pushed down because of hybridization with the higher-lying La f resonance above E v. The amount of this pushing down is just enough to bring the band maximum into close proximity to the onset of the heavier transition metal d and pnictogen p derived bands forming the top of the valence band maximum. Calculations for other La filled skutterudites (Harima, 1998; Fornari and Singh, 1999a, 1999b; Fornari, unpublished show that this close a coincidence does not occur elsewhere in the compounds LaM4AI2 , with M - Fe, Ru, Os and A = P, As, Sb. For example, as shown in Fig. 16, in LaFe4P12 the "gap-crossing" band that made CoP 3 a semimetal is pushed down far enough to open a gap, but not nearly into proximity with the other valence bands (Harima, 1998; Fornari and Singh, 1999a; see also Llunell et al., 1996). This result for LaFe4Sbx2 is in agreement with De Haas-van Alphen measurements that show two Fermi surfaces with this composition (corresponding to a hole doping p = 1 e/f.u, relative to the gap) (Sugawara et al., 1998). Besides LaFe4Sbx2, the next closest coincidence in La-filled skutterudites between the downshifted "gap-crossing" band maximum and the top of the valence band manifold occurs in LaOs4Sbl2. However, in this compound

5 IDENTIFYINGAND OPTIMIZINGNOVELTHERMOELECTRICMATERIALS 153 2.5

g

1.5

0.5

-o.s

r

i

H

-~-~-

/

N

F

p

FIG. 16. Band structure of LaFe4P12, after Fornari and Singh (1999a). Note the narrow gap that opens because of repulsion between the "gap-crossing" band and the La f-resonance above E r. The electronic DOS is shown in Fig. 5.

the "heavy" bands in the valence band manifold are more dispersive, reflecting the fact that in the corresponding binary, IrSb3, the 5d states are more hybridized and spread over the manifold. This is in contrast to the concentration of heavy Co 3d bands near the top of the manifold in CoSb3. All things being equal, this would yield lower thermopowers, but higher carrier mobility in p-type La(Os, Ir)4Sb12. On balance, this means that La(Os, Ir)4Sbx2 would probably not be as good a thermoelectric as La(Fe, C0)48b12 , especially if the latter is optimized with respect to carrier concentration and La-filling. Besides, Os compounds are undesirable for applications when alternatives are available. An interesting connection can be made between the occurrence of superconductivity in LaM4A12 and the p-type thermoelectric prospects of the corresponding doped compounds, La(M, M + 1)4A12 where M + 1 = Co, Rh, and Ir for M = Fe, Ru, Os, respectively, and A = P, As, Sb. Specifically, there appears to be a strong electron-phonon deformation potential affecting the "gap-crossing" band for phonons involving La (note the effect of La-filling), and it may well be that this band is thus responsible for the superconductivity. In the superconducting compounds, specifically LaFe4P12, LaRu4P12, LaOs4P12, LaRu4As12, and LaRu4Sb12, the "gapcrossing" band is ineffectively pushed down by the La f-resonance in the sense that with this composition most carriers are still in it. The best

154

DAVID J. SINGH

thermoelectric p-type thermoelectric prospects are in the other compounds where the "gap-crossing" band is strongly pushed down, so at the L a M 4 A I 2 composition most carriers would be in the heavier valence bands. Finally, we turn to the n-type La-filled skutterudites. Here the situation is different from the p-type case. In the binaries there are usually (note the exceptions such as CoP 3, which is metallic) already several reasonably heavy mass bands that would be active in transport. Thus, it is perhaps not entirely surprising that the addition of even small amounts of filling atoms yield materials with high Z T (Caillat et al., 1996a; Morelli et al., 1997; Sales et al., 2000). Moreover, this, in principle, has nothing to do with modification of the electronic structure by interaction with f-orbitals, as in the p-type Ce and La filled materials. This means that addition of a small amount of any filler that lowers the thermal conductivity would be as good, and one might suppose that the less electronically active the better, as this would reduce the scattering. In this regard, divalent and monovalent fillers might be better than trivalent rare earths, provided that the filling atoms have comparable vibrational properties. Also, since the role of the filling in this case is not to alter the band structure, one may focus on the thermal conductivity, using smaller rare earths than La, for example. Kaiser and Jeitschko (1999) have studied some skutterudite fillings with small divalent cations, in particular SrFe4Sb12 and CaOs4Sbx2. However, there is another avenue to high Z T among the n-type compounds. This is because the addition of a filler pulls down conduction bands near the zone boundary relative to those at the zone center. Fornari and Singh (1999a, 1999b) showed that this was just an effect of the charge

1.0

0.5

>o 0.0

r- --O.S

tLI

-1.0

-1.5

l"

H

N

1-"

P

FIG. 17. Band structure of LaRu4Sbx2 (gray line) and virtual crystal La(Ruo.vsRh0.25)4Sbx2 (black) after Fornari and Singh (1999b). Note the nearly rigid band behavior indicating weak scattering from R u - R h disorder in the alloy. The band structures are aligned at the valence band maximum. Note also the multivalley heavy mass conduction minima.

5

IDENTIFYINGAND OPTIMIZING NOVEL THERMOELECTRIC MATERIALS 155

transfer by comparing calculations for Y and La filled material. They found that, with the exception of LaFe4P12 and LaRu4P12, all the fully La filled compounds have heavy mass conduction band minima away from the F point. As mentioned, such multivalley structures are highly favorable for thermoelectric performance. Comparing the band structures of the various compounds, they focused on La(Ru, Rh)4Sb~2 as the most favorable case, as in this compound there is a coincidental degeneracy between two such pockets, in particular, one at the N-point of the zone and the other at a general point near the A line as shown in Fig. 17. The LDA bandgap is thus indirect (the valence band maximum is at F), E o = 0.16 eV. Significantly, comparison of virtual crystal calculations for La(Ruo.75Rho.zs)4Sb12 with parallel calculations for LaRu4Sb12 showed very minor differences in the conduction band structure indicating weak scattering from the R u - R h disorder in the alloy system. Using kinetic transport theory in the constant scattering time approximation, Fornari and Singh (1999b) calculated S ( T ) for various doping levels and found high thermopowers from 100 to 300 K at doping levels relevant to thermoelectric applications (Fig. 18). This combined with the expectation of low ~cI and weak alloy scatting suggests this material as a useful thermoelectric, especially below room temperature. However, there is at least one fly in the ointment. This is the fact that the result relies on a modification of the band structure due to filling with a trivalent ion, so again the issues of electron-phonon coupling and the extent

50

I

200

|

IIII

9

I

"

I

I

"

I

9

I

"

I

9

---~...

~

~

~

I

I

I

~

~

~

~

~

I 100

4.5 x 1018 4.6 x 1019

5O

0

100

200

300 400 Temperature (K)

500

600

700

FIG. 18. Calculated temperature dependence of the thermopower for La(Ru, Rh)4Sb12 at difference doping levels (in cm -3) as determined by Fornari and Singh (1999b) using kinetic transport theory and the constant scattering time approximation.

156

DAVID J. SINGH

of filling arise. The electron-phonon issue is probably less important in this case than for the p-type compounds, simply because the operating temperature is lower (in metals the electron-phonon contribution to the scattering is roughly linear in T). On the other hand, the amount of filling may be more of an issue, as obtaining high filling in skutterudites becomes more difficult as the compounds are made n-type. Takizawa et al. (1999) have shown that highly n-type skutterudite compositions can be made under pressure, but it is unclear whether there are other ambient pressure techniques that can be used.

4.

LATTICE DYNAMICS AND EFFECTS OF FILLING

At the time of this writing, the thermal conductivity, ~c, of skutterudites is less well understood from a microscopic point of view than the electronic properties. Although minimum thermal conductivity theories (Slack, 1995, and references therein; Cahill et al., 1992) indicate that they have the potential for very low values of ~h, such values are not realized in the binary unfilled compounds and alloys. For example, room temperature values ~c ~ 100 mW/cmK are reported in CoSb 3 (Caillat et al., 1996a). It is now well established experimentally that filling with rare earths and other heavy atoms drastically lowers these numbers (see Nolas, Morelli, and Tritt, 1999, for a review). Thus, the problem of thermal conductivity in skutterudites may seem to be solved--that is, the needed low ~Cl for high Z T may be obtained by rare earth filling. However, this may not be the end of the story. The values of ~h ~ 15-25 mW/cmK measured in filled skutterudite thermoelectrics are still substantially higher than the theoretical minimum thermal conductivity, and, in fact, in most samples the largest component of ~cis still ~h. This means that substantial further increases in Z T are in principle attainable if ~ct can be reduced closer to the theoretical minimum. As a result, there remains considerable interest in elucidating the vibrational properties of skutterudites, the effects of filling on them, and the mechanisms for suppression of ~Clupon filling. A first step is to identify the heat carrying modes. Feldman and Singh (1996) approached this problem by fixing a force constant lattice dynamical model for CoSb3 mainly using a large set of frozen phonon calculations. They also used some experimental infrared phonon frequencies measured by Lutz and Kliche (1982), who had previously constructed a central force constant model for this compound using the same experimental data. Not surprisingly in view of the electronic structure, which shows substantial covalency in CoSba, Feldman and Singh found that the use of noncentral bond bending terms was needed in order to match the model to the calculated phonon frequencies. From this model, Feldman and Singh (1996) were able to calculate the full phonon dispersions as well as the kinetic

5

IDENTIFYINGAND OPTIMIZING NOVEL THERMOELECTRIC MATERIALS

1.5

9

I

I

I

"

157

I

"7

E o

(I)

1.0

(D t~

0 a

0.5

tO 0t..-

0.0

,

0

-

~

50

1 O0

I

,

Frequency, cm FIG. 19. (1996).

,

150

I

200

,

i

250

-1

Phonon density of states of CoSb 3 based on the model of Feldman and Singh

transport function G(E)vZG(E). The phonon vibrational density of states (VDOS) for CoSb 3 is shown in Fig. 19; the corresponding transport function (along with LaFe4Sb12 ) is shown in Fig. 20. The contributions below ~50 cm-1 are from the acoustic phonons, whereas modes from approximately 50 cm- ~ to 175 cm-1 are dominated by Sb character. The lower energy Sb modes have mainly rigid librational S b 4 ring character, while, in order of increasing energy, the modes change to twisting (bond angle bending) motions in the rings, rigid translation motion of the S b 4 rings and bond stretching Sb motions in the rings. The Co-dominated modes occur above 230 cm-1 and are well separated by a gap from these primarily Sb modes. These high-frequency Co modes are, no doubt, irrelevant to the thermal conductivity. In this regard, Long et al. (1999) found using Mossbauer studies of Ce(Fe, Co)48b12 that these high-lying transition metal modes are weakly coupled both to the Sb framework modes and the Ce vibrations. Within kinetic transport theory, phonon contributions to the thermal conductivity are related to vZGr, where vZG is the phonon density of states weighted by the squared group velocity and z is a scattering time. The vZG factor heavily weights the low frequency part of the spectrum below 100 cm-1 and effectively suppresses any contribution from modes above 200 cm- 1 as is seen in Fig. 20. This identifies the heat carrying modes in CoSb 3 as the acoustic modes and low-lying optical modes. Both of these are heavily dominated by Sb motions, and so the key to lowering ~l must be

158

DAVID J. SINGH .2

--

1.0-

Unfilled ....... With La

/

0.80.6-

'"

0.4

:; :

0.2 0.0

"; 0

9

--

50

, 9 ," 100 150 Frequency, c ~

!

250

FIG. 20. Calculated phonon transport function, Gv~ (arbitrary units) as determined from the models of Feldman and Singh (1996), Feldman et al., (2000) for CoSb3 and LaFe4Sbl2.

scattering of Sb related phonons. Apparently, rare earth filling is effective in doing this. The simplest picture for how this might happen comes from a straightforward interpretation of the "rattling-ion" model. Within this picture, the filling atoms occupy a large void where they experience a strongly anharmonic, perhaps flat-bottomed or double well, potential within which they move incoherently and scatter low frequency phonons of the host lattice. Although there are serious theoretical challenges to working out the details of this, it is known that some very low ~cI materials have this character, especially filled semiconductor clathrates (Dong et al., 1999). Understanding what goes on in the rare-earth filled skutterudites from a microscopic point of view requires addressing the issue of how the filling atoms interact with the host lattice vibrational modes. Motivated by this, there have been a number of recent experimental studies of phonons in empty and filled skutterudites (Nolas et al., 1996b; Keppens et al., 1998; Sekine et al., 1998, 1999; Nolas and Kendziora, 1999; Dordevic et al., 1999; Feldman et al., 2000). Generally, measurements of individual phonon frequencies have shown excellent agreement with density functional calculations, most notably for the Raman active A o modes, for which the most directly comparable measurements and calculations have been done. To get a clearer theoretical understanding of the lattice dynamics, Feldman et al. (2000) used density functional frozen phonon calculations of the energy of LaFe4Sb12 and CeFe4Sb12 as a function of the rare earth position, holding fixed, at the experimental geometry, all the other structural degrees of freedom (Fig. 21). The bare Einstein frequency corresponding to this

5

IDENTIFYING AND OPTIMIZING NOVEL THERMOELECTRIC MATERIALS -I.0 20

15 I--

~5

0.0

0.5

~L

1.0 20

J

>,,

n- 10

159

-! 15

10

E

w

OL--I.0

-0.5

0.0

0.5

-JO 1.0

x (at. un.) -I .0 2Or-

A >,,

rr

E

-0.5

O0

0.5

1.0 20

15

15

10

10

w

0 -1.0

-0.5

0.0

0.5

0 1.0

x (at. un.) FIG. 21. Total energy as a function of rare earth displacement in CeFe4Sb12 (/eft) and LaFe4Sb12 (right). The points are the calculated energies; the solid line is the harmonic contribution from the curvature around 0. After Feldman et al. (2000).

distortion (this is not an observable quantity, as the rare earth motion needs to be coupled to the rest of the lattice) was calculated to be 68 cm-1 for Ce and 74 c m - ~ for La. This is squarely in the frequency range of the Sb 4 motions that contribute strongly to the thermal conductivity. The harmonic shape of the energy vs rare earth displacement apparent in Fig. 21 deserves

160

DAVID J. SINGH

comment. In particular, it should be noted that this shape is inconsistent with mechanisms that rely on interactions with strongly anharmonic soft modes or double-well type vibrations to incoherently scatter phonons and suppress ~:~.In fact, Feldman et al. (2000) found that the addition of a small quartic anharmonic term was sufficient to fit the results to better than 0.1 mRy over the entire range. On the other hand, the harmonic shape of the "bare modes" of Fig. 21 should not be interpreted as meaning that the dressed phonons involving La or Ce are harmonic in filled skutterudites. There is, in fact, evidence for substantial anharmonicity in the calculated first principles forces, for example, in the rare-earth Sb interactions that do not appear in the energy vs displacement curve of Fig. 21 because of the high symmetry (cubic anharmonic L a - S b interactions cancel for this particular distortion). Following this result, Feldman et al. (2000) constructed a full dynamical model for LaFe4Sb12 in order to elucidate what does happen upon rare-earth filling. This was done by adapting the model they had previously constructed for CoSb 3 by the addition of rare-earth related central force constants and fixing the needed terms using a set of about 40 LDA frozen phonon calculations. They found that this model was able to closely reproduce the first principles results. Aside from the addition of La related force constants, the most significant changes they found, relative to CoSb3, were a roughly 30% reduction in the two central intra-pnictogenring force constants and a 10% reduction of the C o - S b central force constants. As seen in Fig. 20 (note that Fig. 20 shows an earlier, slightly different model for LaFe4Sb12 , but there are only very small differences in the relevant below 140 cm-x range), the transport function is altered by La filling in the energy range where heat transport occurs. These changes are, however, not nearly large enough to explain by themselves the large reduction in ~c~upon La filling. Keppens et al. (1998) reported extensive neutron scattering investigations of the vibrational properties of La and Ce filled skutterudite antimonides. In their analysis, aimed at addressing the same question, that is, the character of the rare earth vibrations, they made the important observation that because the La neutron scattering cross-section is much larger than that of Ce, the difference between the LaFe4Sb12 and CeFe4Sbl2 spectra would be dominated by modes involving rare-earth motions. One of the key observations they made was that this difference spectrum has a pronounced two-peak structure. Although their analysis rests on the assumption that the lattice dynamics of the two materials are essentially the same, and Feldman et al. (2000) did find differences between La and Ce force constants based on their frozen phonon calculations, these differences of ~ 13% were too small to alter this conclusion. In the absence of a detailed model, the most natural interpretation of the two peaks, observed in the difference spectra, was in terms of two La frequencies, presumably due to strong anharmonicity, double wells, two-level systems, or some other unusual dynamics associated with the rattling ion.

5

IDENTIFYINGAND OPTIMIZING NOVEL THERMOELECTRIC MATERIALS

161

FIG. 22. Calculated inelastic neutron scattering spectral difference between LaFe4Sb12 and CeFe4Sb12 as determined by the model of Feldman et al. (2000).

To analyze and reconcile this with the practically harmonic "bare" rare-earth modes revealed by the first principles calculations, Feldman et al. (2000) used their force constant phonon models for LaFe4Sb12 and CeFe4Sbl 2. This latter model differed from that of LaFe4Sb12 through softer rare earth force constants obtained from the frozen phonon calculations for the Ce filled material. They calculated the harmonic difference spectrum given by the models and compared this with the neutron spectra as shown in Fig. 22. The result shows that the two La peak structure is present in the theory as well, even though the theory, by construction, includes no anharmonic "rattling" effects or localized La modes. What it does include is harmonic mixing. That is, the bare La mode, originally at 78 cm- 1, is shifted downward to approximately 50 cm-1 by the harmonic interaction with Sb modes around 100 cm-1. Because of this mixing, the La mode takes on some Sb character, and conversely, the Sb modes take on some La character, giving rise to the second peak in the neutron difference spectra. In other words, without the dynamical La-Sb interactions, the pure La modes would be concentrated around 70 cm-1 with a substantial number of Sb modes just above. The strong La-Sb mode hybridization pushes these groups apart and causes them to take on mixed character. The specific heat

162

DAVID J. SINGH

calculated from the model was shown to be in very good agreement with measurements, confirming the validity of the model. The implication of these results is that the suppression of Xl by rare-earth filling is subtler than simple rattling models suggest. Besides this, some features that are probably important for a detailed understanding are revealed, in particular, the strong harmonic coupling between La and Sb in comparison to the inter-ring Sb interactions. Feldman et al. (2000) reported that this apparently holds for anharmonic force constants as well, based on a limited number of calculations. Thus, the La phonons, which because of their flat dispersions are not in themselves heat carrying, interact strongly with the Sb heat-carrying modes.

5.

PROSPECTS

Experimental measurements have already demonstrated filled skutterudite samples showing Z T in excess of unity, albeit at high temperature. The remaining questions are whether this is close to the best that can be had in these compounds and whether the operating temperature for high Z T can be lowered. The understanding provided by the band structure calculations indicates that at least on a fundamental level, the answer to both questions is yes. The key issues are as follows: (1) For p-type La(Fe, Co)48b12 , the La filling needs to be increased and the carrier concentration reduced--probably by nonequilibrium synthesis or synthesis under pressure. This is also an issue for Ce-filled material. (2) For n-type material, compounds such as La(Ru, Rh)4Sb12 should be prepared with high filling and the carrier concentration optimized, again likely requiring changes in synthesis procedures. (3) Strategies for further lowering •l toward the minimal value should be explored in conjunction with optimization of the doping level, especially for n-type material, where strategies such as alloying on the rare-earth site might be effective without destroying the favorable electronic properties. It remains to be seen what can be achieved by these strategies, but the present understanding is that at the time of this writing, the limit has not yet been realized in the filled skutterudites.

V.

Chevrel Phases

Chevrel phases have been investigated as potential thermoelectrics during the past few years both experimentally (Caillat et al., unpublished; Roche et al., 1999b) and theoretically (Roche et al., 1998, 1999a, 1999b; Nunes et al., 1999). The binary Chevrels, based on the formula M6X8, with M = Mo, W and X = S, Se, Te, feature large voids in the crystal structure that can be

5

IDENTIFYING AND OPTIMIZING NOVEL THERMOELECTRIC MATERIALS

163

2.0 1.5

1.0 0.5

0.0 -0.5

-1.5 M

r'

R

X

r'

R

X

FIG. 23. Band structure of Mo6Se s in the region around the Fermi energy, after Nunes et al. (1999). The horizontal reference at 0 denotes the Fermi energy. Note the gap at a filling four electrons per formula unit higher and the heavy mass, multivalley band edge.

filled to yield a wide variety of ternary compounds. Known fillers include simple or transition metal atoms, or rare earth ions. The basic electronic structure of the binaries was established by Nohl et al. (1982) in studies aimed at understanding the high-temperature electron-phonon superconductivity in this system. These studies showed that the binaries, such as Mo6Se8, have a gap surrounded by heavy mass bands at a filling 4 electrons higher than the stoichiometric E F (see the band structure of Mo6Se s shown in Fig. 23). This and the potential flexibility in adjusting both the carrier concentration and thermal conductivity by filling the voids suggested these compounds as the basis of new thermoelectric materials. Roche and coworkers (1998, 1999a, 1999b and Nunes et al. (1999) calculated the electronic structure for various transition metal, Li, Cu, and Zn fillings. Following Nohl et al. (1982), the band structure of the barriers, such as Mo6Se8, shown in Fig. 23, is described as arising from an assembly of pseudocubic Mo6Se 8 clusters. The chalcogen atoms form a distorted Se s cube and the Mo atoms occupy this cube's face centers. As the transition metal d states and chalcogen p states are close in energy, strongly hybridized cluster orbitals result. Nunes et al. analyzed these in tight-binding terms for MosSes. They found Mo d and Se p on-site energies within a few tenths of an electron volt and a hopping amplitude, tpa~ ~ 2 eV. Of the five Mo d orbitals per atom, two participate in forming p d a bonds. Thus, of the 36 p and d derived states per cube involved in the bonding (3 p per chalcogen atom plus these two d from

164

DAVID J. SINGH

each Mo), there are 12 nonbonding p-states, 12 bonding pda states, and 12 antibonding pda states. The remaining Mo d states form 18 d bands in the gap between the nonbonding and antibonding pda bands just discussed. These orbitals hybridize with each other via direct dda hopping ~ 1.5 eV, which splits them into 12 bonding and 6 antibonding cluster orbitals. The band formation occurs through hopping between the Mo6Se 8 clusters, which is considerably weaker than the intracluster hopping, so the character of these cluster orbitals is retained in the solid. The Fermi energy in Mo6Se 8 lies in the manifold of 12 bonding dda bands at a position four electrons short of the gap between them and the six antibonding dda bands. Thus, if one assumes rigid band behavior, doping four electrons per formula unit into the system, for example by filling the voids, would make it semiconducting. The charge carriers then would have the character of the states around the gap, that is, for p-type, Mo d states along the edges of the Mo octahedra. Nunes et al., note that these Mo d bands are very sensitive to the geometry of the clusters, so small changes in geometry lead to large changes in band structure (see Fig. 24). This large electron-phonon interaction is, of course, anticipated from earlier studies on superconductivity in the Chevrels. The implication for thermoelectrics is that the carrier mobility at high temperature will be limited by electron-phonon scattering, so the materials

0.&

Mo,=,Se8, actual structure Mo~Se s cage of Ua.2MosSes

0.6

0.4 --.

0.0

oo

iiili

-0.2

.

o.-~

~ ~176 o

..

-0A

M

1-

x

F

R

X

FIG. 24. Band structure of Mo6Se 8 as calculated by Nunes et al. (1999) in its actual crystal structure (solid lines) and with the atoms shifted to their positions as reported in Li3.zMo6Se 8 (dashed lines). The large changes near the band edge resulting from the small changes in bond lengths reflect the large electron-phonon couplings in this material.

5

IDENTIFYINGAND OPTIMIZING NOVEL THERMOELECTRIC MATERIALS

165

are more likely to be high-ZT thermoelectrics at low temperature if anywhere. Furthermore, the narrow Mo d character of the bands around the gap indicates that substitutions or vacancies on the Mo sites will be strong carrier scatterers. In fact, Nunes et al. (1999) report very strong nonrigid band behavior upon transition metal substitutions, supporting this conclusion. The implication is that alloying on the Mo site to add the needed four electrons is not likely to yield a useful thermoelectric, even though semiconducting behavior may be obtained. Turning to doping by filling atoms, the crystal structure contains interconnecting interstices that form channels along the rhombohedral directions. These can be filled by small simple cations, such as Li and Zn, transition metals, and/or large cations, such as Pb. Large fillers such as Pb occupy a large interstitial site at the origin of the conventional rhombohedral cell. Small cation fillers such as Li are distributed over 12 sites arranged in two concentric sixfold rings around this large interstitial site (Cava et al., 1984; Ritter et al., 1992). Li4Mo6S 8 is a known compound with the desired band filling. Nunes et al. and Roche et al. both studied a variety of transition metal fillers. Invariably, these produced bands around E F with resulting metallic character. The exception was Cu, which was found to be monovalent with a full d shell well below E v. Of the filled compounds investigated, Cu4Mo6Se 8, Li4Mo6Se 8, and ZnzMo6Se 8 were found to be semiconducting. Furthermore, quite excellent rigid band behavior was reported upon changing the number of filling atoms, keeping the Mo6Se 8 framework structure fixed. Even extreme substitutions such as going from LiaMo6Se 8 to Li6Mo6Se 8 gave very small changes in the structure of the valence band edge as shown in Fig. 25. So one may expect that the scattering due to disorder on the filling sites, and vibrations of filling atoms, would lead to only weak carrier scattering. Moreover, these are favorable band structures for high Z T as they have heavy-mass, multivalley band edges. Nunes et al., using kinetic transport calculations, obtain thermopowers S > 150 ~V/K in the temperature range 200-250 K for Li4Mo6Se 8 with high doping levels in the range p = 0.05 to 0.10 holes per cell. This sets up the PGEC scenario of Slack. In other words, perhaps in Mo6Se 8 filling atoms can be found that (1) donate charge to make a semiconductor, (2) weakly scatter charge carriers, and (3) strongly scatter heat-carrying phonons. The most likely candidates would seem to be Li and some Cu or Zn to scatter phonons. Selwyn and McKinnon (1988) reported electrochemical phase studies of the CuxLiyMo6Se 8 system, finding that Li4Mo6Se s forms in a lower symmetry phase. However, Chevrel structure CuLiaMo6Se 8 is reported to form. To date, though, experimental efforts to dope Mo6Se 8 to the band edge using standard nonelectrochemical synthesis methods have been unsuccessful, as only lower fillings or metallic phases with different structures have been reported.

166

DAVID J. SINGH

L~MosSee

LiaMosSe 8

-0.2 -0.4 A

ul

9

-0.6

-0.8

-12

M

r

R

-X

F

R

-

X

FIG. 25. Band structure of ordered rhombohedral Li3Mo6Se 8 (solid) and hypothetical Li6Mo6Se 8 (dotted) as reported by Nunes et al. (1999).

VI.

~-Zn4Sb 3

High thermoelectric figures of merit were reported by Caillat et al. (1996d, 1997a, b) in fl-Zn4Sb a at elevated temperature. The highest measured Z T was 1.4, which is one of the highest actual measurements of Z T in any material. Whether or not fl-Zn4Sb a finds wide practical applications, understanding the origin of this high Z T value and whether it is the limit for this compound is important. From an experimental point of view, this has been complicated by the difficulty in making samples with a wide range of carrier concentrations, at least by the usual bulk synthesis techniques. Caillat et al. (1997b) do, however, report synthesis of samples with partial Cd substitution for Zn. fl-Zn4Sb3, like La(Fe, Co)48b12 appears to be a low carrier concentration metal with a very low lattice thermal conductivity. For example, the resistivity increases linearly with T over a wide range. The Hall number n u - 9 x 1019 cm-a for the reported high-ZT sample, which although small for a metal may be too large to allow analysis of the transport in terms of usual semiconductor formulas, underscoring the need for detailed band structures. The crystal structure (Villars and Calvert, 1991) is shown in Fig. 26. This structure contains pure Zn and Sb sites as well as a mixed site

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167

FIG. 26. Crystal structure of fl-Zn4Sb 3 after Kim et al. (1998) based on the Villars and Calvert (1991) handbook. White and dark gray denote the 12 Zn and 4 Sb atoms on pure sites; the positions denoted by the 6 light gray spheres are occupied by 89% Sb and 11% Zn.

occupied 11% by Zn and 89% by Sb. Taking the mixed site as a pure Sb site yields a formula Zn6Sb 5 with 22 atoms per rhombohedral unit cell. Kim et al., (1998) denoted this the stoichiometric compound (although it apparently does not exist in the Z n - S b phase diagram) and then used first principles calculations for it to determine the properties for the experimental stoichiometry, Zn4Sb3, using the rigid band approximation. They justified this a posteriori by the excellent agreement of the calculated S ( T ) with experimental data. In any case, starting with the "stoichiometric" formula Zn6Sbs, three of the Sb atoms lie on the mixed site. The remaining two Sb form Sb 2 dimers with bond length nearly exactly the twice the covalent radius of Sb on the pure site. Each of the Sb in these dimers forms three additional bonds with neighboring Zn atoms, thus taking a fourfold coordination. Both the Zn and mixed site Sb are more highly coordinated. Each Zn forms one bond with a pure site dimer Sb, one bond with another Zn, and three bonds with mixed site Sb atoms, while each mixed site Sb has six bonds to nearest-neighbor Zn. Kim et al. (1998) report that their calculated electronic structure is highly covalent with a Fermi energy lying in a manifold of Sb bonding states. Based on this, they rationalized why the compound would form with Zn substituting onto the mixed site in terms of a balance between the cost of forming breaking Z n - S b bonds and the strengthening of the covalent bonds in the system as electrons are added.

168

DAVID J. SINGH 200

..~ 150

i-energy

|u |~ J~

u Zn-~ o Sb~-p ~ Sba,Lp

]i loo Y r l,-

50

~

o.ol

Energy- E r (Ry)

~o3

FIG. 27. Calculated total and partial (projected onto LAPW spheres) electronic DOS of /~-Zn4Sb 3 after Kim et al. (1998). The total DOS is on a per unit cell basis, while the partials are per atom and are multiplied by a factor of 10 for clarity. The reference denoted "Calculated Fermi-energy" is for the stoichiometric compound; that denoted "Adjusted Fermi-energy" is for the reported doping level of Caillat et al. (1996d).

The electronic DOS for/~-Zn4Sb 3 obtained by Kim et al. within the LDA is shown in Fig. 27. As mentioned this was obtained for the "stoichiometric" composition, for which the Fermi level lies ~0.4 eV below the top of the valence band. This band edge is separated by a sizeable gap (also ~0.4 eV) from the conduction band minimum. The actual E v was determined within a rigid band approximation, by matching the calculated and measured values of n n. The rigid band approximation is expected to be very good in this material because of the strong covalency and the similar bonding of Sb and the additional Zn in this compound as implied by the existence of the mixed site. The measured n n is approximately 0.05 holes per unit cell. Matching this with their calculations of n n , Kim et al. (1998) got an upward shift of E v by 24 mRy with respect to the stoichiometric E v. This places E v 8 mRy below the bandgap and corresponds to a doping level of 0.1 holes/cell. As mentioned, n n and band filling are often not simply related in metals. In /3-Zn4Sb 3 this occurs because of the topology of the Fermi surface, which is complex, highly nonparabolic, and anisotropic (see Fig. 28, which shows the main sheet). In particular, it is nothing like the ellipsoidal shape assumed by

5

IDENTIFYINGAND OPTIMIZING NOVEL THERMOELECTRIC MATERIALS

169

FIG. 28. Calculated main Fermi surface of fl-Zn4Sb 3 after Kim et al. (1998) for the experimental doping level of Caillat et al. (1996d).

the usual semiconductor formulas. Besides, it has both hole-like and electron-like sections and varies strongly near the actual E F. The complexity of the Fermi surface near the actual EF leads to rapid variation of the Fermi velocities with energy. This results in a fortuitous enhancement of the energy dependence of a, which according to the kinetic theory, discussed earlier, leads to high thermopowers. At the actual band filling, the calculated plasma frequency is 0.22 eV, which is in the low end of the metallic regime. With this band filling, Kim et al. obtained S ( T ) in the constant scattering time approximation (averaged over direction to compare with nonoriented samples) in good agreement with experimental measurements, which strongly supports their band structure model. The temperature dependence of S is shown in Fig. 29 along with the experimental data of Caillat et al. (1996d). As may be noted, the agreement is excellent, particularly considering that only the independently measured n u was used to fix E F. Calculations at other doping levels are also plotted, illustrating the strong dependence of S on doping level in this material at temperatures where Z T is large. The experimentally measured thermal conductivity is extremely low and close to the Wiedemann-Franz value for the reported sample. Although Wiedemann-Franz depends on a number of approximations that can be violated in real materials, the clear implication of the measurements is that the lattice component of ~cis extremely low, and that for the reported sample ~ce dominates. In such a regime, Z T is roughly proportional to S 2, meaning that increasing S should lead to increases in Z T, despite the fact that ~ may decrease. The calculations show that S increases with increasing E F, and this corresponds to increasing the amount of Zn on the mixed Sb-Zn site. The

170

DAVID J. SINGH 250

~

~

'

I

'

I

I 350

,

I 450

"

I

"

I 55O

,

I

'

I 65O

i

200 4 , p =, ~ "

150

~-" 100

50 250

,

~

Temperature (K)

FIG. 29. Calculated (short dashes) and experimental (solid line with open circles) S(T) of fl-Zn4Sb 3 with the experimental (Caillat et al., 1996d) doping level as well as for doping levels of 0.15 and 0.04 holes per cell (long dashed and dot dashed, respectively) after Kim et al. (1998).

implication is that, if samples with higher Zn concentration can be made, all other things being equal, Z T will increase beyond the already high value Z T - 1.4 reported in this material.

VII. Haif-Heusler Compounds The half-Heusler compounds are based on semiconducting intermetallics, with chemical formulas M M ' M " , for example, ZrNiSn. Crystallographically, they are based on the Heusler structure M z M ' M " with one of the metal atoms replaced by a vacancy. Another view of the structure is in terms of the usual bcc lattice with an ordering of M, M', M" and vacancy onto the lattice sites keeping cubic symmetry. Initial indications that favored study of these materials as potential high Z T thermoelectrics were the wide range of possible compositions (implying flexibility to optimize the thermoelectric properties), the heavy mass, narrow gap band structures reported by Ogut and Rabe (1995) and resulting high thermopowers, and the presence of an empty site that can be filled. It should, however, be noted that unlike materials such as clathrates, the empty site in the half-Heuslers is very small, and filling with metal atoms leads just to the Heusler intermetallics, which are generally metallic high thermal conductivity compounds and often are magnetic with carrier low mobility (e.g., Endo et al., 1997). Besides, disorder in nonmagnetic Heusler compounds can lead to local moment magnetism,

5

IDENTIFYINGAND OPTIMIZING NOVEL THERMOELECTRIC MATERIALS 171

again with strong carrier scattering (Singh and Mazin, 1998). In any case, there have now been measurements of thermoelectric properties on a wide variety of half-Heusler compositions, generally focusing on heavy atom substitutions and alloying (disorder scattering) to lower the lattice thermal conductivity (Morelli et al., 1996; Kleinke, 1998a, 1998b; Slebarski et al., 1998a, 1998b; Kleinke and Felser, 1999; Browning et al. 1999; Sportouch et al., 1999; Larson et al., 1999; Mastronardi et al., 1999; Uher et al., 1999; Tobola and Pierre, 2000; Cook and Harringa, 1999). Band structure calculations have been reported for various compounds by Ogut and Rabe (1995), Slebarski et al. (1998a, 1998b), Sportouch et al. (1999), and Larson et al. (1999). Ogut and Rabe (1995) reported the first band structure studies of these materials motivated by the surprising experimental result that the intermetallics NiMSn with M = Ti, Zr, and Hf were semiconducting despite the fact that they are formed from metallic elements. They did calculations for a series of Heusler and half-Heusler compounds, Ni2TiSn, Ni2ZrSn, Ni2HfSn, NiTiSn, NiZrSn, and NiHfSn, obtaining metallic band structures for the Heuslers, as expected, and small-bandgap semiconductors with degenerate heavy mass F-point valence band maxima for the half-Heuslers. Importantly, the obtained heavy anisotropic conduction band minima at the X-point of the fcc zone in the halfoHeuslers. They associated the stability of the half-Heusler structure with that of the rock-salt MSn, M = Ti, Zr, Hf sublattice and the gap with the symmetry lowering from the Ni addition. They emphasized that the particular electron count, eight valence electrons plus a full Ni d shell, was key to semiconducting behavior. Slebarski et al. (1998a, 1998b) report calculations that yield similar band structures for NiZrSn and NiTiSn as well as X-ray photoemission studies in agreement with the calculated results, and additionally show CoZrSn as metallic. As noted, Ogut and Rabe (1995) noted that the completely filled d shell in these Ni compounds is important in making the half-Heuslers semiconducting. Slebarski et al. (1998b) also studied Ce substitution into NiTiSn, finding that Ce is trivalent in these compounds, and although this introduces interesting f-electron physics, it destroys the semiconducting gap. Larson et al. (1999) and Sportouch et al. (1999) report calculations and experiments for compounds with Sn replaced by pnictogens and trivalent substitutions for M. These have the same effective electron count and are also found to be semiconductors. The particular compounds are at the same electron count, eight valence electrons plus a full d shell, and were also found semiconducting in many cases, particularly YNiSb, LaNiSb, LuNiSb, YNiAs, and YNiBi. As in the Sn compounds, a degenerate valence band maximum at F is found along with a singly degenerate conduction band minimum at the X-point. The anisotropic mass for the conduction band of ZrNiSn is reported as ~ 10 m~ along the F - X line and ~m~ in the other two directions, in agreement with the transport measurements of Uher et al. (1999). Larson et al. (1999) also refine the explanation of Ogut and Rabe

172

DAVID J. SINGH

(1995) for the gap formation in these compounds. Although there is clear p - d hybridization in the Z r - S n derived electronic structure they report that this is less apparent in that of Y-Sb and deemphasize its role in the gap formation instead ascribing a more important role to the hybridization with Ni d states. At the present time, detailed analyses of transport properties based on these band structures have not been reported. However, the heavy mass anisotropic conduction band minima would seem favorable for high thermopowers at reasonable carrier concentrations for thermoelectric application and in fact high n-type thermopowers have been reported experimentally (Larson et al., 1999; Sportouch et al., 1999; Mastronardi et al., 1999; Uher et al., 1999). The problems in obtaining high Z T appear to be not with S but with the thermal conductivity and carrier mobility. The thermal conductivity is generally high in the ordered compounds. Although alloying can lower ~cl, attempts so far to obtain high Z T by such alloying have failed, as the carrier mobility is also strongly reduced. This is perhaps not too surprising when it is considered that the bandgap is formed by details of the electronic structure of all three atoms and in particular local bonding effects among them. This means that any disorder involving any of the three sites in the lattice will strongly affect the electronic structure around the gap and scatter electrons. It is unclear how this problem can be avoided.

VIII.

Concluding Remarks

The preceding discussion presents some examples of the use of first principles calculations to unravel the physics underlying Z T in some novel thermoelectric materials. Besides these examples, there have been applications to a handful of other compounds noteably some complex structure intermetallics: LnaAu3Sb 4 (Young et al., 1999) and clathrates (Dong et al., 1999; Dong and Sankey, 1999). However, plainly this field is in its infancy. I expect that over time the use and impact of first principles calculations in thermoelectrics research and materials science in general will continue to grow. It is hoped that in addition to the particular features of the materials in question that are elucidated by the calculations, the reader will be left with an understanding of the capabilities and limitations of the approach and how it can help in the search for new thermoelectric materials.

ACKNOWLEDGMENTS

I am most grateful for the many contributions of my co-workers at the Naval Research Laboratory, without whom this article would not have been possible. I especially thank J. L. Feldman, M. Fornari, S. G. Kim, I. I.

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IDENTIFYINGAND OPTIMIZING NOVEL THERMOELECTRIC MATERIALS 173

Mazin, L. Nordstrom, R. W. Nunes, and W. E. Pickett. I have benefited from numerous helpful discussions with T. Caillat, F. J. DiSalvo, J. P. Fleurial, R. S. Feigelson, D. Mandrus, D. T. Morelli, G. S. Nolas, B. C. Sales, G. A. Slack, J. O. Sofo, and T. Tritt. I am grateful for support from the Office of Naval Research and DARPA as well as computing grants from the D o D High Performance Computing Modernization Office.

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H. Rakoto, M. Respaud, J. M. Broto, E. Arushanaov, and T. Caillat, The valence band parameters of CoSb 3 determined by Shubnikov-de Haas effect, Physica B 269, 13 (1999). C. Ritter, E. Gocke, C. Fischer, and R. Schollhorn, Neutron-diffraction study on the crystalstructure of lithium intercalated Chevrel phases, Mater. Res. Bull. 27, 1217 (1992). C. Roche, P. Pecheur, G. Toussaint, A. Jenny, H. Scherrer, and S. Scherrer, Study of Chevrel phases for thermoelectric applications: band structure calculations on MxMo6Se 8 compounds (M = metal), J. Phys. Condens. Matter 10, L333 (1998). C. Roche, R. Chevrel, A. Jenny, P. Pecheur, H. Scherrer, and S. Scherrer, Crystallography and density of states of MxMo6Se a (M = Ti, Cr, Fe, Ni), Phys. Rev. B 60, 16442 (1999a). C. Roche, P. Pecheur, M. Riffel, A. Jenny, H. Scherrer, and S. Scherrer, Chevrel phases as good thermoelectric materials, Mat. Res. Soc. Syrup. Proc. 545, 81 (1999b). B. C. Sales, D. Mandrus, and R. K. Williams, Filled skutterudite antimonides: a new class of thermoelectric materials, Science 272, 1325 (1996). B. C. Sales, B. C. Chakoumakos, and D. Mandrus, Thermoelectric properties of thallium-filled skutterudites, Phys. Rev. B 61, 2475 (2000). C. Sekine, T. Uchiumi, I. Shirotani, and T. Yagi, Metal-insulator transition in PrRu4P12 with skutterudite structure, Phys. Rev. Lett. 79, 3218 (1997). C. Sekine, H. Saito, T. Uchiumi, A. Sakai, and I. Shirotani, Micro-probed Raman scattering study of ternary ruthenium phosphides with filled skutterudite-type structure, Solid State Commun. 106, 441 (1998). C. Sekine, H. Saito, A. Sakai, and I. Shirotani, Low-temperature Raman scattering of LaRu4P12 and PrRu4P12 with filled skutterudite-type structure, Solid State Commun. 109, 449 (1999). L. S. Selwyn and W. R. McKinnon, LixCu~.Mo6Ses--phase-behavior and electrochemistry, J. Phys. C 21, 1905 (1988). I. Shirotani, T. Adachi, K. Tachi, S. Toda, K. Nozawa, T. Yagi, and M. Kinoshita, Electrical conductivity and superconductivity of metal phosphides with skutterudite-type structure prepared at high temperature, J. Phys. Chem. Solids 57, 211 (1996). I. Shirotani, T. Uchiumi, C. Sekine, Y. Nakazawa, K. Kanoda, S. Todo, and T. Yagi, Superconductivity of filled skutterudites LaRu4Asx2 and PrRu4As~2, Phys. Rev. B 56, 7866 (1997). I. Shirotani, T. Uchiumi, C. Sekine, M. Hori, S. Kimura, and N. Hamaya, Electrical and magnetic properties of Lal_xCe~Ru4P12 and CeOs4P12 with the skutterudite structure, J. Solid State Chem. 142, 146 (1999). D. J. Singh, Planewaves, Pseudopotentials and the LAPW Method, Kluwer Academic, Boston (1994). D. J. Singh and I. I. Mazin, Calculated thermoelectric properties of La-filled skutterudites, Phys. Rev. B 56, R1650 (1997). D. J. Singh and I. I. Mazin, Electronic structure, local moments and transport in FeEVA1, Phys. Rev. B 57, 14352 (1998). D. J. Singh and W. E. Pickett, Skutterudite antimonides: quasi-linear bands and novel transport, Phys. Rev. B 50, 11235 (1994). G. A. Slack, in CRC Handbook of Thermoelectrics (D. M. Rowe, ed.), p. 407, CRC Press, Boca Raton (1995). G. A. Slack, Design concepts for improved thermoelectric materials, Mat. Res. Sor Syrup. Proc. 478, 47 (1997). G. A. Slack and V. G. Tsoukala, Some properties of semiconducting IrSb 3, J. Appl. Phys. 76, 1665 (1994). A. Slebarski, A. Jezierski, S. Lutkehoff, and M. Neumann, Electronic structure of XEZrSnand XZrSn-type Heusler alloys with X = Co or Ni, Phys. Rev. B 57, 6408 (1998a). A. Slebarski, A. Jezierski, A. Zygmunt, S. Mahl, and M. Neumann, Suppression of the gap energy in Z r -N i - S n and Ti-Ni-Sn by partial substitution of Zr and Ti by Ce, Phys. Rev. B 57, 9544 (1998b).

5

IDENTIFYINGAND OPTIMIZING NOVEL THERMOELECTRIC MATERIALS 177

J. O. Sofo and G. D. Mahan, Electronic structure of CoSb3: a narrow gap semiconductor, Phys. Rev. B 58, 15620 (1998). S. Sportouch, P. Larson, M. Bastea, P. Brazis, J. Ireland, C. R. Kannewurf, S. D. Mahanti, C. Uher, and M. G. Kanatzidis, Observed properties and electronic structure of RNiSb compounds (R = Ho, Er, Tm, Yb and Y): Potential thermoelectric materials, Mat. Res. Soc. Symp. Proc. 545, 421 (1999). H. Sugawara, Y. Abe, Y. Aoki, H. Sato, M. Hedo, R. Settai, and Y. Onuki, De Haas-van Alphen effect in RFe4P12 (R = La and Nd), J. Magn. Magn. Mater. 177-181, 359 (1998). H. Takizawa, K. Miura, M. Ito, T. Suzuki, and T. Endo, Atom insertion into the CoSb 3 skutterudite host lattice under high pressure, J. Alloys and Compounds 282, 79 (1999). J. Tobola and J. Pierre, Electronic phase diagram of the XTZ (X - Fe, Co, Ni; T = Ti, V, Zr, Nb, Mn ; Z = Sn, Sb) semi-Heusler compounds, J. Alloys and Compounds 296, 243 (2000). M. S. Torikachvili, J. W. Chen, Y. Dalichaouch, R. P. Guertin, M. W. McElfresh, C. Rossel, M. B. Maple, and G. P. Meisner, Low-temperature properties of rare-earth and actinide iron phosphide compounds: LaFe4P12, PrFe4P12, NdFe4P12, and ThFe4Pa2, Phys. Rev. B 36, 8660 (1987). T. M. Tritt, G. S. Nolas, G. A. Slack, A. C. Ehrlich, D. J. Gillespie, and J. L. Cohn, Low-temperature transport properties of the filled and unfilled IrSb 3 skutterudite system, J. Appl. Phys. 79, 8412 (1996). T. Uchiumi, I. Shirotani, C. Sekine, S. Todo, T. Yagi, Y. Nakazawa, and K. Kanoda, Superconductivity of LaRu4Xx2 (X = P, As and Sb) with skutterudite structure, J. Phys. Chem. Solids 60, 689 (1999). K. Uehara and J. S. Tse, Calculations of transport properties with the linearized augmented planewave method, Phys. Rev. B 61, 1639 (2000). C. Uher, J. Yang, S. Hu, D. T. Morelli, and G. P. Meisner, Transport properties of pure and doped MNiSn (M = Zr,Hf), Phys. Rev. B 59, 8615 (1999). P. Villars and L. D. Calvert, Pearson's Handbook of Crystallographic Data for Intermetallic Phases, 2nd ed., American Society for Metals, Metals Park (1991). A. Watcharapasorn, R. C. DeMattei, R. S. Feigelson, T. Caillat, A. Borshchevsky, G. J. Snyder, and J. P. Fleurial, Preparation and thermoelectric properties of some phosphide skutterudite compounds, J. Appl. Phys. 86, 6213 (1999). D. Young, K. Mastronardi, P. Khalifah, C. C. Wang, and R. J. Cava, LnaAuaSb4: Thermoelectrics with low thermal conductivity, Appl. Phys. Lett. 74, 3999 (1999). J. M. Ziman, Principles of the Theory of Solids, Cambridge University Press, Cambridge, U.K. (1972).

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SEMICONDUCTORS AND SEMIMETALS, VOL. 70

CHAPTER

6

Thermoelectric Properties of the Transition Metal Pentatellurides: Potential Low-Temperature Thermoelectric Materials Terry M. Tritt DEPARTMENT OF PHYSICS AND ASTRONOMY MATERIALS SCIENCE AND ENGINEERING DEPARTMENT CLEMSON UNIVERSITY CLEMSON, SOUTH CAROLINA

R. T. Littleton, IV MATERIALSSCIENCEAND ENGINEERINGDEPARTMENT CLEMSONUNIVERSITY CLEMSON, SOUTHCAROLINA

I. PENTATELLURIDES" BACKGROUND AND INTRODUCTION . . . . . . . . . . 1. Overview o f Interest in Low-Dimensional Conductors in the 1980s . . . . 2. A n o m a l o u s Electrical Transport in H f T e 5 and Z r T e 5 . . . . . . . . . . 3. Synthesis and Structure . . . . . . . . . . . . . . . . . . . 4. Effects o f Stress and Pressure . . . . . . . . . . . . . . . . 5. Magnetotransport and Hall Effect . . . . . . . . . . . . . . II. INTRODUCTION TO THERMOELECTRIC MATERIALS .

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1. General Description . . . . . . . . . . . . . . . . . . . . 2. L o w - T e m p e r a t u r e Refrigeration Applications . . . . . . . . . .

180 180 181 183 184 186 188 188 189

III. PENTATELLURIDES AS POSSIBLE Low-TEMPERATURE THERMOELECTRIC MATERIALS

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

IV. RECENT DEVELOPMENTS IN PROPERTIES OF PENTATELLURIDES . . . . . . . . .

1. 2. 3. 4. 5.

Doping on the Transition M e t a l Site ( M ~ A r T e 5, M = H f Zr, A = Zr, Ti) Doping on the Chalcogen Site ( M T e s _ x C h x, M = H f Zr, Ch = Se, Sb) . . Magnetotransport: Overview o f Recent Results . . . . . . . . . . . . . Thermal Conductivity o f Pentatellurides . . . . . . . . . . . . . . . . S u m m a r y o f Thermoelectric Properties . . . . . . . . . . . . . . . .

V. DISCUSSION AND CONCLUSIONS

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VI. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES

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190 191 191 195 197 200 201 201

203 204

179 Copyright 9 2001 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-752179-8 ISSN 0080-8784/01 $35.00

180

TERRY M. TRITT AND R. T. LITTLETON, IV

I. 1.

Pentatellurides: Background and Introduction

OVERVIEW OF INTEREST IN Low-DIMENSIONAL CONDUCTORS IN THE 1980s

During the early 1980s a considerable amount of research was being performed on low-dimensional materials. In 1954, Frohlich predicted that low-dimensional materials might provide a route to high-temperature superconductivity (Frohlich, 1954). Low-dimensional systems are also known to be susceptible to a variety of phase transitions. Systems such as charge density wave (CDW) materials were being studied intensively in hopes of finding an appropriate mechanism or route to high-temperature superconductivity, as well as for the intrinsic interest in this fascinating area of nonlinear electrical transport. Many of these low-dimensional C D W materials were based on transition metal tri-chalcogenide systems such as NbSe3 and TaS 3 (Gorkov and Gruner, 1989; Monceau, 1985). Much research was being performed on these and similar systems, such as di-selenides (e.g., NbSe2), di-sulphides (e.g., TaS2), as well as other low-dimensional systems. One of the authors of this review (TMT) worked on one such system, TaSe 3, which did not exhibit CDW behavior, but underwent a metal-semiconductor transition when exposed to an external applied uniaxial stress (Tritt et al., 1986). Many of these materials exhibited rich transport behavior, including nonlinear conductivity, large magnetoresistance, and large effects on the transport due to an applied stress and pressure, as well as many other interesting phenomena. The pentatellurides were of this class of low-dimensional materials that were being investigated for novel transport phenomena in the early 1980s (DiSalvo et al., 1981). These materials exhibit a resistive anomaly at low temperatures very similar to that in the NbSe 3 system, indicative of CDW behavior. The CDW transitions in NbSe 3 are reflected in resistive anomalies as a function of temperature, at T1 = 145 K and T2 = 59 K. Superlattice spots in the X-ray diffraction are evident, indicating the new periodicity associated with the periodic lattice displacement that drives the CDW transition, as well as the materials exhibiting a nonlinear current-voltage response due to depinning of the CDW above a threshold electric field. Much of the work on CDW materials and low-dimensional systems of this era is highlighted by the excellent review texts by Gorkov and Gruner (1989) and Monceau (1985). But to many researchers of that era, the elusive goal was to achieve the formidable barrier of a transition temperature (Tc) for superconductivity above 77 K, the boiling point of liquid nitrogen. Research on many types of materials was quickly abandoned in 1986 when high-temperature superconductivity was discovered in the cuprate materials (Bednorz and Mueller, 1986), quickly achieving Tc ~ 90 K in a YBazCU3OT_a (Wu et al., 1987). Research efforts continued on the CDW materials and other low-dimen-

6 THERMOELECTRICPROPERTIES OF TRANSITION METAL PENTATELLURIDES 181 sional systems, but with a much diminished effort. This was also the case with research efforts in the early 1980s on the low-dimensional pentatelluride materials HfTe s and ZrTe5. Many systems such as the pentatellurides were essentially forgotten as the exciting and important research on the high-Tc materials flourished. Those who were at the 1986 March APS meeting will not soon forget the excitement over these new high-Tc materials, which were so phenomenal.

2.

ANOMALOUS ELECTRICAL TRANSPORT IN H f T e 5 AND Z r T e 5

The pentatelluride materials HfTe5 and ZrTe5 were first synthesized in 1973 (Furuseth et al., 1973). Early electrical transport measurements revealed a peak in their resistivity as a function of temperature, Tp ~ 80 K for HfTe 5 and Tp ~ 145 K for ZrTe 5 (DiSalvo et al., 1981; Okada et al., 1982). This anomalous behavior was apparently the result of a phase transition that occurs in these pentatellurides. The resistivity as a function of temperature for both parent compounds, as well as various substitutions of Zr for Hf, that is, Zrl_xHfxTes, is shown in Fig. 1. The resistive anomalies are quite apparent from this figure and the possibility of doping these materials is also evident from these results. Figure 1 is taken from DiSalvo et al. (1981), where they concluded that these anomalies were associated with an electronic transition, but no evidence of C D W behavior was evident in these

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182

TERRY M. TRITT AND R. T. LITTLETON,IV

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materials. In addition, both parent materials exhibit a large positive (p-type) thermopower near room temperature that undergoes a change to negative thermopower (n-type) below the peak temperature with the zero crossing of thermopower, To, corresponding well with Tp (Jones et al., 1982). The temperature dependence of the thermopower and the electrical resistivity is

6

THERMOELECTRICPROPERTIES OF TRANSITION METAL PENTATELLURIDES 183

shown in Fig. 2 for HfTe5 and is taken from Jones et al., (1982). This shows the correlation of the thermopower and the resistivity as a function of temperature, and similar behavior is observed in the ZrTe 5 materials. These materials are currently under extensive investigation in Tritt's laboratories at Clemson University because of their potential for thermoelectric refrigeration applications at low temperatures (Littleton et al., 1998, 1999; Tritt, 1996). The current interest in these materials is primarily due to the relatively large thermopower, ~ ~ + 100/~V/K, that is evident in these materials at low temperatures, which leads to a large Peltier effect and thus promising for thermoelectric applications. The parent materials that are grown in our laboratories also exhibit the resistive transition anomalies with the peaks occurring at Tp ~ 80 K for HfTe 5 and Tp ~ 145 K for ZrTe 5. In addition, each display a large positive (p-type) thermopower (~ >~ + 125 #V/K) around room temperature, which undergoes a change to a large negative (n-type) thermopower (~ ~< - 125/~V/K) near the resistivity peak temperature. The properties of our specific materials, the growth processes, as well as the extensive doping and experimental techniques employed, will be discussed later in more detail in Section IV. Early theories suggested that the resistive anomaly evident in the pentatellurides was probably due to a CDW transition, similar to that which occurs in NbSe 3 (Gorkov and Gruner, 1989; Monceau, 1985; Coleman et al., 1985). However, the absence of distinct superlattice spots in the X-ray diffraction patterns and the absence of nonlinear conductance, both indicative of CDW materials, seemed to quickly contradict this explanation (Okada et al., 1982). Other experiments were attempted in an effort to ascertain the nature and origin of this yet-undetermined transition. Early band structure calculations indicated that these materials were semimetallic with the conduction electrons near the Fermi level originating primarily from the p-orbitals of the Te atoms (Whangbo et al., 1982). Other calculations indicated that there was a large variation of the density of states near the Fermi level (Bullett, 1982). This may be able to lead to a large thermopower in a system, as discussed later.

3.

SYNTHESIS AND STRUCTURE

Single crystals of HfTe 5 and ZrTe 5 are currently grown under conditions similar to previously reported methods (Levy and Berger, 1983). A stoichiometric ratio of the materials was sealed in fused silica tubing with iodine ( ~ 5 mg/mL) and placed in a tube furnace. The starting materials were at the center of the furnace with the other end of the reaction vessel near the open end of the furnace to provide a temperature gradient. Crystals of these materials were obtained in excess of 1.5 mm long and 100 #m in diameter with the preferred direction of growth along the a axis, as determined by face indexing. These materials are complex, long-chain

184

TERRY M. TRITT AND R. T.

LITTLETON,

IV

FIG. 3. (a) A unit cell of MTe 5 (M = Hf, Zr), along the a axis which shows the van der Waals gap that separates the individual layers. The open spheres are the transition metal (M) atoms and the cross-hatched spheres are the Te atoms. (b) A projection of a layer in MTe 5 as viewed down the a axis, showing the chains of MTe 3 pyramids. The open spheres are the transition metal (M) atoms and the cross-hatched spheres are Te atoms.

systems with 24 atoms in an orthorhombic unit cell. The structure of the pentatellurides is made up of MTe 3 (M = Hf or Zr) chains that are subsequently bridged into large 2D sheets by tellurium atoms. The sheets are then weakly bound to one another through a van der Waals gap, which is owing to the highly anisotropic nature of the bulk crystals. A diagram of the structure is shown in Fig. 3 showing both the MTe 3 chains and the van der Waals gap between the individual layers. The samples grow as long thin ribbons with the growth axis as the a axis and the thin part of the ribbon being the b axis. These materials do, in fact, exhibit anisotropic transport properties with the high conductivity axis being the growth axis (a axis). Electrical contact was made using Au wires bonded to the crystal with Au paint. The iodine vapor residual on the samples prevents using Ag paint, which forms a AgI layer on the sample, impeding good electrical contacts. Typical sample dimensions are 1-5 mm, 0.01-0.03 mm, and 0.05-0.3 mm in the a, b, and c directions, respectively.

4.

EFFECTS OF STRESS AND PRESSURE

The effect of an applied hydrostatic pressure indicated substantial changes in the magnitude of the resistive peak and thermopower in these materials (Fuller et al., 1983). This is illustrated in Fig. 4, taken from Fuller et al.

6

THERMOELECTRICPROPERTIES OF TRANSITION METAL PENTATELLURIDES 185

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(1983) showing the effect of pressure on the resistivity and thermopower when exposed to a pressure of 17 kbar. For example, in ZrTe 5 at T ~ 130 K, the resistivity is reduced by approximately a factor of 8 and the thermopower increased by a factor of 2. This was one of the facts that was most intriguing in raising the possibility of better thermoelectric properties in these materials, both the thermopower and resistivity being improved in a favorable manner. Obviously the transport properties and band structure were very sensitive to pressure and, it could be hoped, would be very sensitive upon doping these materials, in terms of both chemical pressure and electronic changes. The addition of pressure was obviously changing the band overlap and structure, affecting the population of the bands resulting in changes in the resistivity and thermopower. These pressure results eventually led to an attempt at Ti doping and substitution on the transition metal site that is discussed in a later section. Given the transport behavior of the pentatellurides and the previous effects of an applied stress on previous low-dimensional systems such as NbSe 3 and TaSe 3, Stillwell and co-workers in the late 1980s investigated the effect of an applied stress on the pentatellurides (Stillwell et al., 1989). In ZrT% the peak in resistivity was shifted to higher temperatures without substantial changes in the magnitude of the resistivity. Over the temperature range at which the peak in the resistivity occurred, the resistivity was affected very little by the applied stress. However, in HfTe 5, in contrast to

186

TERRY M. TRITT AND R. T. LITTLETON, IV 2.0

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ZrTes, the resistivity peak and the magnitude of resistivity at low temperatures, T < 20 K, were strongly affected by the applied stress. It appears that a high resistance state may be appearing at the lower temperatures with increasing stress. This behavior is illustrated in Fig. 5, taken from Stillwell et al. (1989). The high resistance state, which is evident at low temperature, also appears with Ti substitution and with the application of a magnetic field. Both of these effects are discussed in Section IV. Again, it appears that the band structure and topology of the Fermi surface are very sensitive to external parameters such as pressure or stress. This again yielded optimism in the possibilities of doping and substitutions in these materials in hopes of manipulating their transport properties for more favorable thermoelectric behavior.

5.

MAGNETOTRANSPORT AND HALL EFFECT

Early Hall measurements concurred with the thermopower measurements as to the sign of the dominant carriers in the system, with very similar temperature dependence (Izumi et al., 1982). This is illustrated in Fig. 6,

6

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showing the temperature dependence of the resistivity and Hall coefficient, from Izumi et al. (1982). They also noted that the system exhibited a large magnetoresistance. This is discussed in more detail in Section IV, where we expand on these results. They described the magnetotransport in terms of a two-carrier model in which both electrons and holes are present. Magnetic susceptibility measurements failed to elucidate any magnetic character to the observed transition (see Fig. 6 of Okada et al., 1982). The pentatellurides appear to be diamagnetic at higher temperature and exhibit some small paramagnetism at low temperature, probably due to paramagnetic impurities. Fermi surface determinations by Kamm et al. revealed a very anisotropic Fermi surface for these materials, with an ellipsoidal shape and an effective mass that was quite different in the two directions, with the effective mass along the b axis 100 times greater than that along the a axis (Kamm et al., 1985, 1987). The Fermi surface for each of these pentatelluride materials consist of three very small surfaces, two electron type and one hole type. All of the surfaces are ellipsoidal with principal axes along the unit cell axes of the orthorhombic crystal structure, with the long axis of the ellipsoid along the b axis. These are illustrated in Fig. 7, taken from Kamm et al. (1987). The Fermi surface of HfTe 5 is considerably smaller (almost a factor of 10) than that of ZrTe 5 and is thought to be the smallest Fermi surface ever determined. Later in this review, we discuss a large enhancement of the resistive peak with applied magnetic field, up to 9 T, with a change in normalized resistivity, p(9 T)/p(0), being approximately a factor of 3 in

188

TERRY M. TRITT AND R. T. LITTLETON, IV C

b

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H

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FIG. 7. Relative three-dimensional Fermi surfaces of ZrTe5 and HfTe 5. From Kamm et al. (1987), p. 1227.

ZrTe 5 and a factor of 10 in HfTe 5. These effects are very large given that there is no indication of any magnetic character to this resistive transition. These pentatellurides are certainly an extremely rich system for the investigation of novel electrical transport and magnetotransport.

II.

1.

Introduction to Thermoelectric Materials

GENERAL DESCRIPTION

As described in Volume 69, the first of this series, thermoelectric materials transfer heat via the Peltier effect when subjected to an electric current, which in turn produces a temperature gradient (Goldsmid, 2001). Heat is absorbed on the cold side and rejected at the heat sink, thus providing a refrigeration capability. An imposed AT will act conversely and result in a voltage or current, i.e., small-scale power generation (Tritt, 1996). The essence of a good thermoelectric is given by the determination of the material's dimensionless figure of merit, Z T = (~2a/2)T, where 0~ is the Seebeck coefficient, a the electrical conductivity, and 2 the total thermal conductivity (2 = 2 L + 2 E, the lattice and electronic contributions, respectively). The "power factor" of a material, a:T/p (or otZtrT), is typically optimized as a function of carrier concentration, through doping, to give the largest ZT. Semiconductors have long been the material of choice for thermoelectric applications since they were initially identified for their potential in the mid-1950s by the Russian scientist Ioffe (which led to a huge

6 THERMOELECTRICPROPERTIESOF TRANSITION METAL PENTATELLURIDES 189 research effort in that country during the 1950s and 1960s) (Ioffe, 1957). High-mobility carriers that have the highest electrical conductivity for a given carrier concentration are most desirable, and typically the most promising materials have carrier concentrations of approximately 1019 carriers/cm 3. The best thermoelectric materials have a value of Z T ~ 1 (Tritt, 1999; Mahan et al., 1997). This Z T ~ 1 has been an experimental upper limit for more than 30 years, yet there is no theoretical or thermodynamic reason why it cannot be larger. The value of Z T can be raised by decreasing the lattice thermal conductivity, 2L, or by increasing the Seebeck coefficient, ~, or the electrical conductivity, a. However, a is tied to the electronic thermal conductivity 2 E through the Wiedemann-Franz Law (2E = LoaT, where Lo = Lorentz number) and the ratio (2E/a) is essentially constant at a given temperature. Approximately 30 years ago, alloys based on the Bi-Te compounds [which are really the alloy system (Bil_xSbx)z(Tel_xSex)3] and Sl_xGex compounds were developed as thermoelectric materials for solid-state thermoelectric refrigeration and power generation applications, respectively (Goldsmid, 1986; Wood, 1988; Rowe, 1995). These materials have been extensively studied and fully optimized for their use in thermoelectric devices and are the current state-of-the-art materials. Thus, entirely new classes of compounds will have to be investigated if substantial improvement is to be realized. The need for improvement is particularly acute in the lower temperature regime, because most conventional thermoelectric materials show optimal performance well above room temperature (T > 700 K). Although there are a considerable number of applications in the hightemperature regime, there is an even greater potential for device applications for refrigeration at lower temperatures, between 80 and 400 K. The dearth of investigations in this area makes the need for the investigation of entirely new systems of materials for this regime even more acute.

2.

LoW-TEMPERATURE REFRIGERATION APPLICATIONS

A recent article emphasized that the development of low-temperature ( T < 200 K) thermoelectric devices is the most important need of new thermoelectric refrigeration materials (Mahan, 1999). Materials that provide efficient local cooling at temperatures below 200 K would greatly affect the electronics industry, since the performance of many semiconducting and other electronic devices is dramatically enhanced below room temperature. Since thermoelectric cooling of electronics requires only small-scale spot cooling, the demands on the materials and devices are not as great as for larger scale cooling applications, such as packaged refrigeration. As the field of cryoelectronics and "cold computing" grows, the need for lower temperature (100-200 K) thermoelectric materials has become more evident. The advantages of "cold computing" are discussed in an article by Sloan (1996),

190

TERRY M. TRITT AND R. T. LITTLETON, IV

where he states that "speed gains of 30%-200% are achievable in some CMOS computer processors" and that "cooling is the fundamental limit to electronic system performance." Also, a severe limitation on cellular phone technology, using superconducting narrowband spectrum dividers to increase frequency band utilization, is a reliable cooling technology. Cooling of laser diodes and infrared detectors to temperatures, 100 K < T < 200 K, would greatly improve performance and sensitivity and thus be extremely important to many technologies (Allen, 1997). Huebener and Tsuei have discussed the prospects for using existing BizTe 3 Peltier coolers for superconducting electronics (Huebener and Tsuei, 1998). However, they concluded that new materials will need to be developed to achieve solid-state cooling to temperatures on the order of 90 K for the development of superconducting electronics to become feasible. Thus, the potential payoff for the development of low-temperature thermoelectric refrigeration devices is great, and the requirement for compounds with properties optimized over wide temperature ranges has led to a much-expanded interest in new thermoelectric materials (Tritt, 1996, 1999; Sales et al., 1996; also see for example Tritt et al., 1999a).

III.

Pentatellurides as Possible Low-Temperature Thermoelectric Materials

In 1995, Slack published a paper describing the anticipated chemical characteristics of materials that might lead to an effective thermoelectric material. The successful candidate requires several specific characteristics. The compound should be a narrow-bandgap semiconductor with highly mobile carriers, whereas the thermal conductivity must be minimized. In semiconductors, the Seebeck coefficient and electrical conductivity (both in the numerator of Z T ) are strong functions of the doping level and chemical composition, which must therefore be optimized for good thermoelectric performance. Doping can also have a very profound effect on the thermal conductivity of complex materials. Understanding these various effects in complex materials is important because the factors are often related. Hence, optimization usually requires trade-offs, and there are often many possible degrees of freedom. Slack has previously stated that the most promising thermoelectric materials should behave as a phonon glass electron crystal (PGEC). The paradigm of the PGEC material is that it should behave thermally as a glass (large phonon scattering and thus low lattice thermal conductivity) and as an electronic crystal (low scattering for the electrons, thus high electrical conductivity). One of the important issues relative to the development of low-temperature thermoelectric materials is identifying mechanisms that might give high thermopower (~) at low temperatures. Possibilities include phonon drag,

6

THERMOELECTRICPROPERTIES OF TRANSITION METAL PENTATELLURIDES 191

heavy Fermion materials, Kondo systems, and materials that exhibit phase transitions, as well as quasi-one-dimensional materials. In most materials, at temperatures far from a phase transition, the electrical conductivity and thermopower are related to the electron density of states near the Fermi energy D(Ev). The conductivity is proportional to D(Ev), whereas ~ is proportional to (1/D)dD(E)/dE at E = Ev). The thermopower for metals typically goes linearly to zero as the temperature is decreased according to a "Mott diffusion" description given by

3e

~-E ~=E~

where k is the Boltzmann constant, e the electronic charge, a is the electrical conductivity, and E F is the Fermi energy. At low temperatures other contributions such as phonon drag or magnetic effects can become important. Hence, as n (or D) is increased, a typically increases while e decreases. Low-dimensional systems are known to be susceptible to van de Hove singularities (or cusps) in their density of states, electronic phase transitions, and exotic transport phenomena, which can add structure in D(E) near E F. Doping can produce very substantial effects in these types of materials and can drastically change their electronic transport. Quantum well systems take advantage of a low-dimensional character through physical confinement in thin-film structures to enhance the electronic properties of a given material (Hicks and Dresselhaus, 1993). In this paper we present results on a family of low-dimensional semimetals called pentatellurides (HfTe 5 and ZrTes) that we believe exhibit such behavior and show promise as potential new low-temperature thermoelectric materials.

IV. 1.

Recent Developments in Properties of Pentatellurides

DOPING ON THE TRANSITION METAL SITE

(MxAyTe5, M = Hf, Zr, A = Zr, Ti) The electrical resistivity (p = 1/~r) and thermopower for single crystals of the pentatelluride materials as a function of temperature are shown in Figs. 8a and 8b for HfTe 5 and ZrTe 5, respectively. Both parent materials exhibit a unique resistive transition peak, Tp ~ 80 K for HfTe5 and Tp ~ 145 K for ZrTe 5. In addition, each display a large positive (p-type) thermopower (~/> + 125 #V/K) around room temperature, which undergoes a change to a large negative (n-type) thermopower (~ < - 125/~V/K) near the resistivity peak temperature. The magnitude of this resistive anomaly is typically 3 - 7 times the room temperature value of p ~ 0.7 mr/" cm, which is comparable

192

TERRY M. TRITT AND R. T. 4

la) .~

0

.....,

50

....

2

LITTLETON,

IV

100 150 200 250 300 , .... , .... , .... , .... ,.

ZrTe 5

200 150 ~

100

~

(b) ~

50

~

-50

0

-100 -150 -200

FIG. and

8.

0

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

50

100 150 200 250 300 Temperature (K)

(a) Resistivityand (b) thermopower versus temperature for single crystals of HfT%

ZrT%.

to that of good thermoelectric materials. These materials exhibit thermopower that is relatively large over a broad range at low temperatures for both n-type (T < Tp) and p-type (T > Tp). The large values of thermopower (]e] ~ 100/~V/K) at temperatures below 250 K make these materials very interesting for potential low-temperature applications. Resistivity and thermopower measurements performed on single crystals of Hfl_xZrxTe 5, where 0 < x < 1, are shown in Figs. 1 and 9, respectively. The peak temperature determined by the resistivity versus temperature measurement is shifted to higher temperature as Hf is replaced by Zr. In each of the solid solution samples (i.e., Hfl _xZrxTe5), the resistivity exhibits a similar behavior to that of the parent compounds with a systematic temperature displacement depending on x. The thermopower of each sample also reveals a systematic shift in temperature as the Zr concentration is increased, similar to the resistivity, as shown in Fig. 9. Each concentration exhibits relatively large p-type thermopower at room temperature. As the temperature decreases, the thermopower in each sample increases until reaching a maximum. For lower temperatures, the thermopower drops sharply, passes through zero at To, and continues to decrease until reaching a maximum n-type thermopower. At even lower temperatures, the thermopower begins to rise again, going toward zero thermopower as the temperature approaches absolute zero in a relatively linear fashion, charac-

6

THERMOELECTRIC PROPERTIES OF TRANSITION METAL PENTATELLURIDES

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100 150 200 250 Temperature (K)

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FIG. 9. Thermopowerversus temperature for single crystals of Hfl_xZrxTe5.

teristic of diffusion-type thermopower. No apparent phonon drag contribution has been reported or observed at these lower temperatures. As noted, the resistance peak was first thought to be evidence of a charge density wave phenomenon; however, the origin of the anomaly in these pentatellurides, as discussed by DiSalvo et al., appears to be an electronic phase transition as opposed to a structural phase transition. Therefore, the electronic properties of this system should be susceptible to doping, which is evident with these results. The uncertainty in the concentrations of Hf or Zr does not allow for the prediction of an obvious temperature dependence with regard to the zero crossing of the thermopower or position of the resistive peak. However, there is a distinct correlation between the zero thermopower and the resistivity peak for each concentration. The thermopower of pentatelluride materials has been shown to be substantially affected by pressure as discussed in Subsection 4 of Section I. The n-type thermopower, below the peak, changes 150% or more to values of approximately - 2 4 0 p V / K in ZrTe 5 at T = 120 K and P = 12 kbar, while the resistivity decreases by a factor of 4 (Fuller et al., 1983). These trends enhance the power factor (~2o'T) by an order of magnitude. Smaller changes are observed in HfTe 5 under similar conditions; however, in HfTe 5 the resistivity is increased with pressure. Uniaxial stress measurements exhibit substantial effects in both parent materials, also discussed in Subsection 4 of Section I. These results also favor the idea of an electronic phase transition. The Ti atoms, r = 2.00 A, are substantially smaller than either Hf or Zr, both with r = 2.16A, and should produce some slight compression (or distortion) of the lattice, possibly in correlation to the applied external pressure. The Ti substitution is also isoelectronic (3d electrons instead of 4d or 5d) and hence should not directly alter the carrier concentration of the compounds. The resistivity and thermopower, respectively, are shown for

194

TERRY M. TRITT AND R. T. 150

LITTLETON,IV

50

100

200

50

100 150 200 T e m p e r a t u r e (K)

250

300

250

300

12 10

(a)

t~

8

Q.

6 Q..

4

200 150 ~ I..,

(b) ~

100 50 o -50

~

-100 -150 -200

0

FIG. 10. Resistivity (a) and thermopower (b) as a function of temperature for single crystals of Zro.90Tio.loTe 5 and Hfo.95Tio.05Te 5 compared with ZrTe 5 and HfTe 5.

the HfTe 5 and Hfo.95Tio.o5Te 5 materials in Fig. 10 (Littleton et al., 1998). This small amount of Ti substitution ( ~ 5 % ) s h i f t s the peak temperature substantially from 75 K for HfTe5 to Tp ~ 40 K for Hfo.95Tio.osTe5, but in contrast to HfTe5, the zero crossing of the thermopower occurs at a much higher temperature (To ~ 50 K) than Tp. The relative resistive peak of the Ti doped sample is nearly doubled that of the undoped resistive peak magnitude. A high resistance state appears to exist below the peak in comparison to all other pentatelluride materials studied. This is in contrast to the "metallic behavior" evident in other samples. The low temperaturehigh resistive state is similar to that observed in HfTe5 when subjected to a large applied strain (Stillwell et al., 1989). Figure 10 also shows similar resistivity and thermopower data for ZrTe 5 and Zro.9oTio.lTe 5. A nominal Ti substitution of 10% for Zr shifts the peak temperature from 140 K for ZrTe 5 to Tp ~ 110 K for Zro.9oTio.lTe5, with TO coincident with Tp. The relative resistance peak is slightly larger than that of the parent material, ZrTe 5. The strong metallic behavior is again evident below Tp, with a positive resistivity slope ( d p / d T > 0). Substitutional doping of Ti for either Hf or Zr leads to a variation of the peak temperature from 38 to 140 K, while maintaining the relatively large values of thermopower at low temperature.

6 THERMOELECTRICPROPERTIESOF TRANSITIONMETALPENTATELLURIDES 195 DOPINGON THE CHALCOGEN SITE (MTes_xCh x, M = Hf, Zr, Ch = Se, Sb) 2.

Substitutional studies are important in order to find the desired combination of elements in a compound to produce certain desirable characteristics. Such studies were performed on the Bi2Te 3 compound, which led to a series of related alloys and pseudoternary or quartenary compounds that have resulted in effectively optimized thermoelectric materials (Yim and Rosi, 1972; Yim and Amith, 1972). This optimizing process found that certain stoichiometric amounts of Se and Sb for Te aided in the enhancement of the Bi2Te 3 compounds. Antimony is an element in many thermoelectric materials, including Bi2Te 3 and of course Bil_xSb x compounds. Bismuth antimony alloys have extremely large magneto-thermoelectric figure of merit values below room temperature. These materials are discussed in detail in two review articles in this series (Freibert et al., 2001; Lenoir et al., 2001). Small amounts of antimony doped pentatellurides, shown in Fig. 11, did not seem to increase the thermoelectric properties at lower temperatures (Littleton et al., submitted). Instead, the Sb appears to completely alter the temperature dependence of both the thermopower and the electrical resistivity. The 5% nominally doped pentatellurides, HfTe4.75Sbo.25 and ZrTe4.75Sbo.25, both exhibit thermopower and resistivity values that increase linearly with temperature. The samples measured had relatively low room temperature resistivity values of ~0.48 and ,~0.36

0

....

50 , ....

100 150 200 250 300 , .... , .... , .... , .... ,.

0.5 ..... Hfi, e4.75Sbo.25 0.4 0.3 ~-'-"

o

/

.~176176 ~176176176 " .......,..""""'"

"

0.2 0.1

lla. .... i .... | .... | .... ................................

150

| ....

| ....

|.

lO0 O

o

50 ...... .........,......................1..l"b" o 0

50

100

150

200

250

300

Temperature (K) FIG. 11. Resistivity(a) and thermopower (b) as a function of temperature for single crystals of HfTe4.v5Sbo.25 and ZrTeg.vsSbo.25.

196

TERRY M. TRITT AND R. T. LITTLETON, IV 0

4

100

150

200

250

300

12a ............................ A -- Hfre , ,/'/ .... Hrre4 se~

3

(a)

50

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12b

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(b)

~ ~

0 -100

% ~

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................................ 50 100 150 200

250

300

Temperature (K) FIG. 12. Resistivity (a) and thermopower (b) versus temperature for single crystals of MTe5_xSe x (M = Hf, Zr and x = 0, 0.25).

mf~.cm for HfTe4.75Sbo.25 and ZrTe4..75Sbo.25, respectively. The HfTe4.75 Sbo.25 samples have temperature dependent values that are larger in thermopower and lower in resistivity than those of ZrTe4.75Sbo.25. The autonomous transition, which occurs in the parent materials, apparently no longer exists within these doped materials. The possibility of a dissimilar structural phase was eliminated by X-ray analysis that reconfirmed the pentatelluride structure. Results from Se substitutions on Te sites of both parent pentatelluride materials are quite promising as well (Littleton et al., 1999). As seen in Fig. 12, a 5% nominal substitution of Se for Te slightly alters the temperature dependency of Tp and To which are reduced nearly 20 K from HfTe5 to HfTe4.75Seo.25 , whereas the reduction is only a few kelvins for ZrTe 5 to ZrTe4.75Seo.25. The magnitudes of the absolute thermopower for both Se-doped materials increase approximately 20% from their parent materials. Thermopower values exceeding 200 p V / K were measured for both HfTe4.75Seo.25 and ZrTe4.75Seo.25 at temperatures of ~95 K and ~ 185 K, respectively. Another favorable effect of the Se substitution is the reduction in the resistivity. The resistivity of the parent pentatellurides decreases approximately 25% with the nominal addition of 5% Se for Te. The increase in the thermopower, combined with the decrease in the resistivity results in

6

THERMOELECTRIC PROPERTIES OF TRANSITION METAL PENTATELLURIDES

197

1.5

o ;z.,

0.5

o 0 0

50

100

150

200

250

300

Temperature (K) FI6. 13. Power factor ~2T/p(or ~2oT) as a function of temperature for HfTe4.75Seo.25 and ZrT%vsSeo.25 compared to optimized Bi2Te 3. Data on Bi2Te 3 is from Yim and Rosi (1972), p. 1132.

an enhancement of the power factor, (X2 T/p, by a factor of 2. Power factors of Se doped pentatellurides range from P.F. = ~2T/p >f 1.25 watts/mK at 150 K ~< T ~< 320 K for Hf(Tel_xSex) 5 and o~2T/p >/ 1.50 watts/mK at 225 K ~< T ~< 320 K for Zr(Tex_xSex) 5. The significance of these results is illustrated in Fig. 13, where the power factors of the Se-doped samples are compared to the power factor of the optimally doped BizTe 3, that is, [(Bil_xSbx) 2 (Tel_xSex)3]. At low temperatures, 150 K < T < 250K, the power factor of the pentatellurides significantly exceeds that of the state-ofthe-art bismuth telluride compound.

3.

MAGNETOTRANSPORT: OVERVIEW OF RECENT RESULTS

As shown in Fig. 14, the magnetoresistance is very dependent on the orientation to the magnetic field (Tritt et al., 1999b). The largest magnetoresistance occurs when the magnetic field is parallel to the b axis (B I[b), which is the axis with the weakest bond interaction. The c axis is bridged between the metal chain prisms with Te bonds, which cross-link between these metallic chains. The effect of an applied magnetic field is very anisotropic and can vary tremendously, depending on temperature and magnetic field strength. The magnitude of the peaks (180 ~ apart) can vary a small amount because of what we believe is a small Hall component due to the change in orientation. This appears, as well, in the magnetoresistance as a function of temperature at + B (as it should be for a small Hall component). The data corresponding to the effect of an applied magnetic field on the resistance as a function of temperature is shown in Fig. 15 for ZrTes. It

198

TERRY M. TRITT AND R. T. 70

,

,

~4T

---(~-- 2 T

~,

50

8

4o

'~

30

~

2O

LITTLETON,

,

IV

,

,

--0'-- 6 X

.---qF-- 8 T

'

T= 179K I

l

I

0

90

180

,

I

i

270

360

Position (r FIG. 14. R e s i s t a n c e of single c r y s t a l s of Z r T e 5 as a f u n c t i o n of o r i e n t a t i o n , qo, a b o u t the a axis at 2, 4, 6, a n d 8 T a n d T - 179 K.

appears that there is a systematic shift of the peak temperature to slightly higher temperatures with increasing field, by approximately 25 K (145 to 170 K) with a 9-T field. The most dramatic behavior is observed in relation to the magnitude of the magnetoresistance around the peak. The magnetoresistanace is relatively small at higher temperatures, T > 200 K, and increases rapidly as the temperature is lowered to near or below the peak. 600

"~

....

, ....

, ....

, ....

, ....

500

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--o-5 T

300

, ....

T

--I-I T ---e,--0 T

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100

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2

0

50

100

150

200

250

300

Temperature (K) FIG. 15. R e s i s t a n c e of single c r y s t a l s of H f T e 5 a n d Z r T e 5 as a f u n c t i o n of t e m p e r a t u r e at 0, 1, 5, a n d 9 T.

6

THERMOELECTRIC PROPERTIES OF TRANSITION METAL PENTATELLURIDES

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200

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0

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15o

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Temperature (K) FIG. 16. Normalized magnetoresistivity, crystals of HfTe 5 and ZrT%.

p(B)/p(O T), as

a function of temperature for single

The magnetoresistance then decreases and undergoes another minimum before increasing again at even lower temperatures. The resistance changes approximately a factor of 3 between 0 and 9 T for ZrTe 5 near the peak temperature. A similar behavior is observed in the HfTe 5 material, also shown in Fig. 15. In contrast to the ZrTe 5 material, only small shifts in the peak temperature are observed at low fields (B ~< 3 T) before this shift apparently saturates at higher fields. The magnitude of the magnetoresistance around the peak, at B - 9 T, is approximately an order of magnitude larger than the zero field resistance in this material. Again, the magnetoresistance is relatively small at T > 145 K and increases rapidly as the temperature is lowered near and below the peak. The magnetoresistance then decreases and undergoes another minimum before increasing again at lower temperatures. An additional difference is evident in the magnitude of the low-temperature magnetoresistance in HfTe 5. The magnetoresistance is approximately a factor of 200 at T = 2.5 K and B = 9 T and is continuing to increase in essentially a quadratic manner to our highest field values (9 T). The normalized magnetoresistance, p(9T)/p(O), is shown in Fig. 16 for both HfTe 5 and ZrTe 5. The temperature dependence of the magnetoresistance discussed previously is quite apparent. The magnetoresistance is low around room temperature and increases as the temperature is lowered, reaching a peak just below Tp and undergoing a shallow minimum before increasing at lower temperatures. This is most apparent in the HfTe5 material. This behavior is similar in structure to that observed in NbSe 3, although the magnitude of the magnetoresistance is much larger in HfTe 5. The magnetoresistance at lower temperatures (T < 25 K) as shown in Fig. 15 exhibits some very interesting behavior for HfTe 5. There appears to

200

TERRY M. TRITT AND R. T.

IV

LITTLETON,

be a higher resistive state appearing below these temperatures and with increasing magnetic field. Applied stress shows a similar effect as the applied magnetic field for T < 25 K (Stillwell et al., 1989). The application of stress does not affect the magnitude of the resistive peak by more than 20%, but below 25 K a high resistive state appears. This is very analogous to the previous studies of Ti doping in the HfTe 5 material. These results could then be related to the results from the previous stress and pressure measurements. The addition of Ti increased the peak substantially and shifted it to lower temperature (Tp ~ 40 K with 5% Ti for Hf), and a high-resistance state appeared below T ~ 25 K. Each of these parameters, stress, Ti substitution (pressure), and magnetic field, appears to have a similar effect on the low-temperature state of HfTes.

4.

THERMAL CONDUCTIVITY OF PENTATELLURIDES

A recently developed parallel thermal conductance technique (Zawilski et al., submitted) was employed to determine the thermal conductivity of pentatelluride single crystals. The temperature dependence of the thermal conductivity, 2(T), is representative of typical single crystals, with a hightemperature behavior 2 ( T ) ~ 1/T and a distinct low-temperature peak (Littleton et al., submitted; Kittel, 1976). Both HfTe 5 and ZrTe 5 samples have a thermal conductivity of ~ 4 - 5 W/m K around room temperature (Fig. 17). At lower temperatures the thermal conductivity increases to a maximum value of ~ 14 W/m K at 20 ~< T ~< 35 K. The data in the range

--o-Hfre 5 --e-ZrTe 5 lO

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t

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150

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200

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!

250

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300

Temperature (K) FIG. 17.

Thermal conductivity versus temperature for single crystals of HfTe 5 and ZrTe 5

6 THERMOELECTRICPROPERTIES OF TRANSITION METAL PENTATELLURIDES 201 50 ~< T ~< 240 K have approximately a lIT dependence, which is representative of a crystalline material, and these pentatellurides are single crystals. Below this temperature, T = 23 K, 2 decreases sharply to the lowest temperatures measured. The peak in the thermal conductivity is characteristic of single crystal data showing depletion of umklapp scattering of the phonons, where the magnitude of the peak is related to the "perfection" or lack of defects within the sample. The thermal conductivity of the single crystalline pentatellurides vary slightly in magnitude and temperature dependency from sample to sample depending on concentration, size, and sample quality. One of the challenges (other than radiation effects) in measuring the thermal conductivity of a sample of this size is accurately determining the sample dimensions, which is still a challenge.

5.

SUMMARY OF THERMOELECTRIC PROPERTIES

Both HfTe 5 and ZrTe 5 exhibit promising thermoelectric properties below room temperature. Pentatellurides possess exotic electrical behavior with a resistive transition peak and large p- and n-type thermopower above and below the transition temperature, respectively. The thermoelectric properties of the parent compounds vary greatly with temperature with resistivity values in the range ~0.6 < p < 6.0 mfl cm, thermopower of ~ - 125 < a < + 125 laV/K, and thermal conductivity ~ 2 < 2 < 20 W/m K. Substitutional doping on both and Te sites can also greatly alter the magnitude and temperature dependence of each thermoelectric property. Proper doping leads to an enhancement of the electronic properties, which is then very competitive with the power factor of existing thermoelectric materials significantly below room temperature. Materials that exhibit both large p- and large n-type thermopower are favorable for thermoelectric applications. The temperature regime in which the low-dimensional pentatelluride compounds possess both large n- and p-type thermopower makes them promising for low-temperature applications. Through controlled substitutions the temperature range at which the transition from positive to negative thermopower can be manipulated. To date, there are only a few thermoelectric material systems to address this temperature regime (Chung et al., 2000).

V.

Discussion and Conclusions

In NbSe3, there are two CDW transitions that occur at T1 = 145 K and T2 = 59 K and result in resistive anomalies similar to that observed in the pentatellurides. Naturally, the early hypothesis was that the resistive

202

TERRY M. TRITT AND R. T. LITTLETON,IV

anomalies in HfTe 5 and ZrTe 5 were due to CDW behavior. However, the absence of superlattice spots in the X-rays and the absence of any nonlinear transport indicated that this was most unlikely. The origin of the resistive anomaly and transport behavior in these materials is still an open question. For example, consider the large magnetoresistance in the pentatellurides and compare this to the NbSe 3 material. Below the temperature of the lower CDW transition in NbSe3, T2 = 59 K, the application of a magnetic field has a substantial effect on the resistance in this material. It has essentially little or no effect on the CDW transition temperature itself. Early experiments and explanations coupled with theory seemed to indicate that the magnetic field was condensing normal electrons into the CDW state. This would result in a depletion of these carriers from the normal state (by as much as a 30-50% reduction in normal carriers) and result in a corresponding higher resistance (Parilla et al., 1986; Balseiro and Falicov, 1985). It was shown, conclusively, that the number of carriers in the CDW state was not being affected by the magnetic field to within a few percent, which was essentially the resolution of the experiment (Tritt et al., 1988, 1991). It was concluded that the large magnetoresistance was due to the changing band character of the material through the CDW transition. The pentatelluride system is much more semimetallic at temperatures below the CDW transition, losing a large part of its Fermi surface ( ~ 30% of the Fermi surface at T1 and 65% of the remaining Fermi surface at T2) and more susceptible to a large magnetoresistance characteristic of narrow overlap semimetallic materials (such as Bi and Sb) (Pippard, 1989; Lovett, 1977). A similar explanation may be relevant to the anomalous magnetoresistance observed in these pentatelluride materials. Thus, it is apparent from the previous experiments (pressure, stress, and Ti doping, as well as other transition metal doping) on these pentatelluride materials that the transition and properties are very sensitive to these parameters. Structure studies do not indicate that the transition and resulting behavior are driven by any structural change in the materials. It may be that a distortion is very weak and more rotational in nature, which could result in this behavior. Some descriptions discuss the possibility that weak Jahn-Teller distortions result in the observed behavior (David Singh, personal communication). These pentatelluride materials are reported to have very small Fermi surfaces and highly anisotropic transport and band character. Small changes in the band overlap and carrier density are reflected in substantial effects on the transport properties. It could be that one of the bands is very close to the Fermi level and that small variations in energy of the system, thermal, elastic, or magnetic, can drive this band away from the Fermi level and affect the transport substantially. One of the explanations for the unusual stress dependence of the resistivity in these materials was related to the stress possibly moving one of the bands relative to the other bands and effectively emptying this band (Stillwell et al., 1989).

6

THERMOELECTRICPROPERTIES OF TRANSITION METAL PENTATELLURIDES 203

Photoemission studies to probe the density of states in these materials are currently underway and these results will be forthcoming (McIlroy et al., unpublished results). It is obvious from the data we have shown that the pentatellurides are a very rich system of materials that exhibit a range of interesting transport phenomena. The resistive transition (of a yet-undetermined origin and nature) that occurs in the pentatellurides is very sensitive to doping, stress, pressure, and, as reported here, magnetic field. It appears that the band structure of these materials should be very complicated. There is possibly one band, with very high-mobility carriers, that appears to dominate the electrical transport in these materials. More experiments are under way to further elucidate the origin of this resistive peak and to gain a further understanding of the electrical transport in these materials.

VI. Summary In summary, we have presented an overview of the transport properties of a group of materials called pentatellurides. These materials exhibit a plethora of transport phenomena as outlined and presented in this review. The transport properties of these materials make them interesting and promising for potential low-temperature thermoelectric applications. Obviously we do not have a complete theoretical picture and description of these materials. Much more work, including detailed band structure calculations as a function of temperature, is needed. In addition, very careful structural studies as a function of temperature will be necessary to a proper understanding of the role of the structure in any forthcoming theoretical description. An extensive amount of experimental data is being produced, and it is our hope that this will inspire more theoretical involvement in these materials. Understanding the origin of the anomaly will be key to the further development of these materials for potential use as a low-temperature thermoelectric material.

ACKNOWLEDGMENTS

We gratefully acknowledge our collaborations with Dr. Joe Kolis and Dr. Doug Ketchum of the Chemistry Department, who synthesized most of the materials. We also acknowledge discussions with A. Ehrlich and W. W. Fuller-Mora concerning their previous results. We acknowledge support from an ARO/DARPA grant (#DAAG55-97-1-0267) and from the research funds provided (TMT) from Clemson University. The PPMS system was purchased under an ONR-DURIP grant #N00013-98-1-0271 (ONR).

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Andrew W. Allen, Detector Handbook, thermoelectrically cooled IR detectors beat the heat. Laser Focus World 33, S 15 (1997). C. A. Balseiro and L. M. Falicov, Density waves in high magnetic fields: A metal insulator transition. Phys. Rev. Lett. 55, 2336 (1985). J. G. Bednorz and K. A. Mueller, Possible high Tc superconductivity in the B a - L a - C u - O system. Z. Phys. B 64, 189 (1986). D. W. Bullet, Absence of a phase transition in HfTe 5. Solid State Commun. 42, 691 (1982). D. Y. Chung, T. Hogan, P. Brazis, M. Rocci-Lane, C. Kannewurf, M. Bastea, C. Uher, and M. G. Kanatzidis, CsBiaTe6: A high performance thermoelectric material for low temperature applications. Science 287 1024 (2000). R. V. Coleman, G. Eiserman, M. P. Everson, A. Johnson, and L. M. Falicov, Evidence for magnetism in the low temperature charge density wave phase of NbSe 3. Phys. Rev. Lett. 55, 863 (1985). F. J. DiSalvo, R. M. Fleming, and J. V. Waszczak, Possible phase transition in the quasi-onedimensional materials HfTe 5 and ZrTe 5. Phys. Rev. B 24, 2935 (1981). Franz Freibert, Timothy Darling, A1 Migliori, and Stuart Trugman, Thermomagnetic effects and measurements. In Semiconductors and Semimetals: Recent Trends in Thermoelectric Materials Research II, Vol. 70 (Terry M. Tritt, ed.). Academic Press, San Diego, 2001. H. Frohlich, On the theory of superconductivity: The one-dimensional case. Proc. R. Soc. London Ser. A 233, 296 (1954). W. W. Fuller, S. A. Wolf, T. J. Wieting, R. C. LaCoe, P. M. Chaiken, and C. Y. Huang, Pressure effects in HfTe 5 and ZrTe 5. J. Physique C3, 1709 (1983). S. Furuseth, L. Brattas, and A. Kjekshus. The crystal structure of HfTe 5. Acta Chem. Scan. 27, 2367 (1973). H. J. Goldsmid, Electronic Refrigeration. Pion Limited Publishing, London, 1986. H. J. Goldsmid. In Semiconductors and Semimetals: Recent Trends in Thermoelectric Materials Research I, Vol. 69 (Terry M. Tritt, ed.). Academic Press, San Diego, 2001. L. P. Gorkov and G. Gruner, Charge Density Waves in Solids, Vol. 25, Modern Problems in Condensed Matter Sciences, North Holland, Elsevier Science Publishers, New York, 1989. L. D. Hicks and M. S. Dresselhaus, Effect of quantum well structures on the thermoelectric figure of merit. Phys. Rev. B 47, 12727 (1993). R. P. Huebener and C. C. Tsuei, Prospects for Peltier cooling of superconducting electronics. Cryogenics 38, 325 (1998). A. F. Ioffe, Semiconductor Thermoelements and Thermoelectric Cooling. Infosearch, London, 1957. M. Izumi, K. Uchinokura, E. Matsuura and S. Harada, Hall effect and transverse magnetoresistance in a low-dimensional conductor, HfTe 5. Solid State Commun. 42, 773 (1982). T. E. Jones, W. W. Fuller, T. J. Wieting, and F. Levy, Thermoelectric power of HfTe 5 and ZrTe 5. Solid State Commun. 42, 793 (1982). G. N. Kamm, D. J. Gillespie, A. C. Ehrlich, T. J. Wieting, and F. Levy, Fermi surface, effective masses and Dingle temperatures of ZrTe 5 as derived from the Shubnikov-de Haas effect. Phys. Rev. B 31, 7617 (1985). G. N. Kamm, D. J. Gillespie, A. C. Ehrlich, D. L. Peebles, and F. Levy, Fermi surface, effective masses and energy bands of HfTe 5 as derived from the Shubnikov-de Haas effect. Phys. Rev. B 35, 1223 (1987). C. Kittel, Introduction to Solid State Physics, 5th ed., p. 177. John Wiley and Sons, New York, 1976. B. Lenoir, T. Caillat, and H. Scherrer, An overview of recent developments for BiSb alloys. In Semiconductors and Semimetals, Recent Trends in Thermoelectric Materials Research I, Vol. 69 (Terry M. Tritt, ed.). Academic Press, San Diego, 2001.

6 THERMOELECTRICPROPERTIES OF TRANSITION METAL PENTATELLURIDES 205 F. Levy and H. Berger, Single crystals of transition metal trichalcogenides. J. Crystal Growth 61, 61 (1983). R. T. Littleton IV, T. M. Tritt, C. R. Feger, J. Kolis, M. L. Wilson, M. Marone, J. Payne, D. Verebeli, and F. Levy. Effect of Ti substitution on the thermoelectric properties of the pentatelluride materials Ml_xTixTe5 (M = Hf or Zr). Appl. Phys. Lett. 72, 2056 (1998). R. T. Littleton IV, T. M. Tritt, C. R. Feger, and J. Kolis. Transition metal pentatellurides as potential low-temperature thermoelectric refrigeration materials. Phys. Rev. B 60, 13453 (1999). R. T. Littleton IV, T. M. Tritt, J. W. Kolis, and D. Ketchum, Effect of Sb doping on the thermoelectric properties of the transition metal pentatellurides HfTes_xSb x and ZrTes_xSb x. Manuscript in progress. R. T. Littleton, IV, B. Zawilski, and Terry M. Tritt, Investigation of the thermal conductivity of pentallurides (MTes, M = Hf or Zr) using the parallel thermal conductance technique. Appl. Phys. Lett. (in press). D. R. Lovett, Narrow Gap Semimetals and Semiconductors. Pion Press, London, 1977. G. D. Mahan, Good thermoelectrics. In Solid State Physics (H. Ehrenreich and F. Spaepen, eds.). Academic Press, New York, 1999. G. D. Mahan, B. Sales, and J. Sharp, Thermoelectric materials: New approaches to an old problem. Physics Today 50, 42 (1997). P. Monceau, ed., Electronic Properties of Inorganic Quasi-One-Dimensional Compounds, Part I and Part II, D. Reidel Publishing, Academic Publishers, Boston, 1985. S. Okada, T. Sambongi, M. Ido, Y. Tazuke, R. Aoki, and O. Fujita, Negative evidences for charge/spin density wave in ZrTe 5. J. Phys. Soc. Jpn. 51, 487 (1982). P. Parilla, M. F. Hundley, and A. Zettl, Magnetic field induced carrier conversion in a charge density wave conductor. Phys. Rev. Lett. 57, 619 (1986). A. B. Pippard, Magnetoresistance in Metals. Cambridge University Press, New York, 1989. D. M. Rowe, ed., CRC Handbook of Thermoelectrics. CRC Press, Boca Raton, FL, 1995. B. C. Sales, D. Mandrus, and R. K. Williams, Filled skutterudite antimonides: A new class of thermoelectric materials. Science 272, 1325 (1996). G. A. Slack, New materials and performance limits for thermoelectric cooling. In CRC Handbook of Thermoelectrics (D. M. Rowe, ed.), p. 407. CRC Press, Boca Raton, FL, 1995. J. Sloan, Cold computing: The future of high speed processing. Superconductor Industry 9, 30 (1996). E. P. Stillwell, A. C. Ehrlich, G. N. Kamm, and D. J. Gillespie, Effect of elastic tension on the electrical resistance of HfTe 5 and ZrTe s. Phys. Rev. B 39, 1626 (1989). Terry M. Tritt, Thermoelectric run hot and cold. Science 272, 1276 (1996). Terry M. Tritt, Holey and unholey semiconductors. Science 283, 804 (1999). T. M. Tritt, E. P. Stillwell, and M. J. Skove, Effect of uniaxial stress on the transport properties of TaSe 3. Phys. Rev. B 34, 6799 (1986). T. M. Tritt, D. J. Gillespie, A. C. Ehrlich, and G. X. Tessema, Charge density wave carrier concentration in NbSe 3 as a function of magnetic field and temperature. Phys. Rev. Lett. 61, 1776 (1988). T. M. Tritt, D. J. Gillespie, A. C. Ehrlich, and G. X. Tessema, Charge density wave carrier concentration in NbSe 3 as a function of magnetic field and temperature. Phys. Rev. B 43, 7254 (1991). T. M. Tritt, M. Kanatzidis, G. D. Mahan and H. B. Lyon, Jr., eds., 1998 MRS Symposium Proceedings: Thermoelectric Materials." The Next Generation Materials for Small Scale Refrigeration and Power Generation Applications, Vol. 545 (1999a). T. M. Tritt, Nathan D. Lowhorn, R. T. Littleton IV, Amy Pope, C. R. Feger, and J. W. Kolis, Large enhancement of the resistive anomaly in the pentatelluride materials HfTe5 and ZrTe s with applied magnetic field. Phys. Rev. B 60, 7816 (1999b). M. H. Whangbo, F. J. DiSalvo, and R. M. Fleming, Electronic structure of ZrTe 5. Phys. Rev. B 26, 687 (1982).

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C. W. Wood, Materials for thermoelectric energy conversion. Rep. Prog. Phys. 51, 459 (1988). M. K. Wu, J. R. Ashburn, C. J. Torng, P. H. Hor, R. L. Meng, L. Gao, Z. J. Huang, Y. Q. Wang, and C. W. Chu, Superconductivity at 93 K in a new mixed-phase Y b - B a - C u - O compound system at ambient pressure. Phys. Rev. Lett. 58, 908 (1987). W. M. Yim and A. Amith, Bi-Sb alloys for magneto-thermoelectric and thermomagnetic cooling, Solid-State Electron. 15, 1141 (1972). W. M. Yim and F. D. Rosi, Compound tellurides and their alloys for Peltier cooling: A review. Solid-State Electron. 15, 1121 (1972). B. M. Zawilski, R. T. Littleton IV, and Terry M. Tritt, Parallel thermal conductance technique applicable to small diameter fiber-like samples. Submitted.

SEMICONDUCTORS AND SEMIMETALS, VOL. 70

CHAPTER

7

Thermomagnetic Effects and Measurements Franz and

Freibert,

Stuart

Timothy

A.

W. Darling,

Albert

Migliori,

Trugman

Los ALAMOSNATIONAL LABORATORY Los ALAMOS,NEW MEXICO

207 209 209 210 214 217 218 218 219 222 223 223 225 226 229 230 230 233 238

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . II. THERMOMAGNETIC EFFECTS . . . . . . . . .

1. 2. 3. 4.

Adiabatic and Isothermal Conditions . Definition of Transport Coefficients . . . Electronic Refrigeration . . . . . . . . Ideal Behavior . . . . . . . . . . . .

III. PHENOMENOLOGICALANALYSIS . . . . . . 1. Equations of Performance . . . . . . . 2. Microscopic Electronic Properties . . . 3. Umkehr Effect . . . . . . . . . . . . IV.

MATERIALS SURVEY . . . . . . . . . . .

1. B a s i c M a t e r i a l R e q u i r e m e n t s . . . . . . 2. C o m p e n s a t e d M a t e r i a l s

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3. B i s m u t h and B i s m u t h - A n t i m o n y 4. D o p e d M a t e r i a l s

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Alloys

1. E x p e r i m e n t a l F u n d a m e n t a l s 2. I s o t h e r m a l M e a s u r e m e n t s

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3. A d i a b a t i c M e a s u r e m e n t s SUMMARY

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V. EXPERIMENTAL MEASUREMENT TECHNIQUES

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I.

Introduction

We are primarily concerned in this chapter with transport of charge and heat in directions that are not parallel to the applied stimuli. Components of motion perpendicular to the driven direction arise via the influence of magnetic fields. These transport phenomena can provide a potentially powerful effect capable of useful refrigeration, and these phenomena can be enhanced by crystallographic anisotropy. In addition to ordinary electrical and thermal conductivity, and the Seebeck and Peltier effects present in isotropic systems in zero magnetic field, the Hall, Ettingshausen, Nernst, and 207 Copyright 9 2001 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-752179-8 ISSN 0080-8784/01 $35.00

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Righi-Leduc effects appear as a magnetic field is applied. These additional effects are names given to the phenomena that generate the various off-diagonal terms in the second-rank tensor transport coefficients for electrical conductivity, thermal conductivity, and thermopower. Complete measurements of heat and charge transport in magnetic fields can become powerful tools for exploration of both the fundamental microscopic physics of new materials and their efficacy for electronic refrigeration. In fact, the magnetic field lifts the degeneracy in the description of transport phenomena such that it becomes possible to separate electron and hole transport in a unique way. The measurement of thermomagnetic effects has become a principal characterization tool in basic materials research. The essential thermomagnetic effects (Peltier and Ettingshausen) remained only curiosities after their discoveries in the 19th century until semiconductor physics developed in association with microelectronics (Angrist, 1961; Wolfe, 1964). Electronic refrigeration progressed as a derivative of interest in semiconductors and their fascinating electrical and thermal transport properties. During this evolution, it was quickly determined that the correct measure of a material's worth in electronic refrigeration is its so-called dimensionless figure of merit ZT, because this quantity determines the maximum coefficient of performance and the maximum temperature drop across a solid-state refrigerator. The figure of merit of a material is determined purely by thermoelectric or thermomagnetic transport coefficients. The figures of merit, though very similar for Seebeck/Peltier devices and Nernst/Ettingshausen devices, have distinctly different physical meaning between the two regimes. Electronic refrigeration material studies are primarily defined as attempts to optimize this figure of merit. As such, proper measurement of thermoelectric and thermomagnetic transport coefficients is necessary and can, in fact, provide a subtle problem not widely appreciated, with serious traps for the unwary. Definitions for all the thermoelectric and thermomagnetic transport coefficients will be provided, methods of measuring these transport coefficients accurately will be described, and the transport properties of idealized thermomagnetic materials will be used to aid the reader in understanding the electronic properties that influence thermomagnetic transport. The experimental data to be discussed came primarily from research conducted during the past 50 years, when virtually every known semiconductor, semimetal, and alloy was evaluated for use as an electronic refrigeration material. It was during this period that the theoretical and experimental foundations of electronic refrigeration were laid. In cases where more detail might be sought, the reader is referred elsewhere. Although this field seems to have been dormant, the development of better electronic instrumentation, the discovery of rare earth permanent magnets, and the expanded growth in materials synthesis and fabrication has made electronic refrigeration a topic of current interest.

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THERMOMAGNETICEFFECTS AND MEASUREMENTS

209

II. Thermomagnetic Effects 1.

ADIABATIC AND ISOTHERMAL CONDITIONS

Thermoelectric and thermomagnetic transport is an irreversible thermodynamic process in which the kinetic coefficients, also known as the transport coefficients, describe the connection between the thermal and electrical currents and potentials (Callen, 1985). The concepts of adiabatic and isothermal conditions are useful in the description of these irreversible thermodynamic processes. The conventional definition of an adiabatic process is one in which no heat flows into or out of the system, and the definition of an isothermal process is one in which the temperature of the system remains constant. Although the jargon of thermomagnetic transport can be confusing, certain definitions are required and will help to separate the so-called adiabatic and isothermal measurement regimes that rapidly diverge with increasing figure of merit. When no magnetic field is present, the concepts of isothermal and adiabatic retain their conventional meanings. When measurements are made in a magnetic field, the isothermal condition is defined as the absence of a temperature gradient transverse to the magnetic field, but the presence of a transverse heat flow. The adiabatic condition in a magnetic field is defined as the presence of a transverse temperature gradient and the absence of a transverse heat flow. These conditions are important to understand because in transport measurements of thermoelectric refrigeration materials, the response of the sample to an applied electric field is the establishment of charge transport and temperature gradients parallel to that field. The temperature gradients interact with the thermopower to modify the charge current so that one does not measure the electrical conductivity, but a combined quantity, if care is not taken. It is the magnitude of the figure of merit that indicates the size and direction in which these unexpected couplings of transport effects occur. To understand this behavior and its implications, consider the following experiment on a material having large thermomagnetic transport coefficients. Immediately after the application of a longitudinal electrical field to a material with a magnetic field perpendicular to the applied electric field, no induced temperature gradients exist. These temperature gradients take time to develop. Therefore, measurements of the voltages that develop at short times provide the conventionally defined "isothermal" resistivity. After some period, transverse and longitudinal thermal gradients (neglecting the second order Joule heating effects) are created in the material via Ettingshausen and Peltier effects, respectively. These thermal gradients act to couple to other thermomagnetic effects and alter the longitudinal and transverse voltages across the sample. The resulting long-time, steady-state voltages determine the so-called "adiabatic" thermomagnetic transport

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FRANZ FREIBERT ET AL.

coefficients, which include confusing contributions from other transport processes. The extent of these contributions depends on sample geometry, experimental design, and material properties (for example, thin films deposited on electrically insulating, thermally conducting substrates should not show such behavior). The unwary researcher can be left attempting to explain erroneous electrical and thermal transport behavior.

2.

DEFINITION OF TRANSPORT COEFFICIENTS

In order to understand the thermomagnetic transport coefficient measurements and the mechanisms for electronic refrigeration, the thermal and electrical transport coefficients are defined and experimental boundary conditions are described. The phenomenology of electronic and phonon transport in solids is presented in terms of thermomagnetic effects, because this is the most general case of electrical and thermal transport. Thermomagnetic transport coefficients are used to describe linear-response transport in conductors with an electric current density (J), magnetic field (B), and thermal gradient (VT). Experimentally it is easier to electrically insulate the sample ( J = 0) and to keep its boundaries at a constant temperature (VT = 0) than it is to cancel electric fields (E = 0) or to stop heat flow (Q = 0) within the sample. Therefore, thermomagnetic coefficients are best defined by the following set of phenomenological linear transport equations

E i = ~ pij(B)J j + ~ ~ij(B)OT/c3j J

(1)

J

Qi = 2 rcij(B)Jj - Z Kij(B)c3T/63J, J J

(2)

where E i, Ji, Qi, and c~T/c~i are the ith component of the electric field, the electrical and thermal current densities, and the thermal gradient, respectively, of an anisotropic conductor. Variables j and i take on the values of x, y, and z in the laboratory right-handed Cartesian coordinate system. The antisymmetric magnetotransport coefficient tensors are designated as follows: pij(B) is the ij component of the electrical resistivity tensor, ~ij(B) is the ij component of the thermopower tensor, 7tij(B) is the ij component of the Peltier tensor, and Kij(B ) is the/j component of the thermal conductivity tensor. These thermomagnetic coefficients originate in the thermodynamic theory of irreversible processes based on the Onsager reciprocity theorem, which requires the time-reversal symmetry restrictions pij(B) = p j i ( - B ) ,

Kij(B ) = t f j i ( - B ) , ~ij(B) = ~ j i ( - B ) T ,

(3)

Of these transport coefficients, the off-diagonal components of the resistivity

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THERMOMAGNETICEFFECTSAND MEASUREMENTS

211

tensor (pij(B), i 4=j), the diagonal components of the thermopower tensor (~,(B)), the diagonal components of the Peltier tensor (rc,(B)), and the off-diagonal components of the thermal conductivity tensor (K~j(B), i 4: j) carry the sign of the majority carrier, in general (Putley, 1960). Often in theoretical discussions, the use of electrical conductivity is attractive because it simplifies the mathematics. Therefore, to be complete in this discussion the electrical conductivity tensor is defined as ~ = ~-1. Since the discovery of the thermomagnetic effects, in particular the Hall, Nernst, Ettingshausen, and Righi-Leduc effects, it has been convention to consider an explicit magnetic field dependence of the transport coefficients. In the case of the Hall effect for a free electron picture, this is justified because the electric field that builds up transverse to the electric current density and magnetic field does so to counterbalance the Lorentz force acting on the carriers in the material. Because the Lorentz force is linear in the magnetic field, the induced electric field is linear in the magnetic field, and when it is compared to the linear transport equations of Eq. (1), it can be seen that if B = Bz~, then Ey = pyxJx = R uBzJx, and so Pyx = RuB~. The quantity Ru is known as the Hall coefficient and may have implicit magnetic field dependence. Similarly, by convention c~yx = NB~, rcy~/~cyy = PB~, and Ky~/Kyy = SBz, for the Nernst, Ettingshausen, and Righi-Leduc effects, respectively. These quantities N, P, and S are known as the Nernst coefficient, the Ettingshausen coefficient, and the Righi-Leduc coefficient, respectively. Similar to the Hall coefficient, these have implicit magnetic field dependence. Experimental boundary conditions determine the direction and magnitude of applied electric currents, magnetic field, and thermal gradients with respect to the crystallographic axes of the sample. The materials considered in this writing are of orthorhombic or higher crystal symmetry with the samples shaped in the form of a rectangular prism whose edges lie in the directions of the sample principal crystallographic axes and are parallel with the laboratory x, y, z-coordinate system. Therefore, all transverse transport is due solely to magnetic field effects. Also, the magnetic field is assumed to be in the laboratory's z direction (out-of-plane) and the longitudinally applied electrical and thermal currents in the laboratory's x direction, thus fixing the direction transverse to both the magnetic field and applied currents as the laboratory's y direction, as can be seen in Fig. 1. As stated earlier, the isothermal condition is one in which no transverse thermal gradient exists (t?T/t?y = 0); however, a transverse thermal current does exist (Qy 4= 0). The adiabatic condition is one in which no transverse heat current exists ( Q y - 0); however, a transverse thermal gradient does exist (t?T/ Oy %0). From Eqs. (1) and (2) it is quite apparent that the transport equations are coupled and that care must be taken with the experimental configuration in order to determine the proper value of the thermomagnetic

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FRANZ FREIBERT ET AL.

FIG. 1. Sample orientation with respect to the electrical and thermal currents and magnetic field.

transport tensor components. The thermomagnetic transport effects for an anisotropic material, as defined in Eqs. (1) and (2), under the experimental isothermal conditions, are presented in Table I. The thermomagnetic transport effects are determined by one of two isothermal experimental cases,

TABLE I THERMOMAGNETICEFFECTS IN AN ANISOTROPIC MATERIAL UNDER ISOTHERMAL EXPERIMENTAL CONDITIONS(dT/dy - Jr - O) Experimental boundary conditions

Case 1: d T/dx = 0; Jx ~: 0

Defining quantity Ex

Jx

Ey

Jx Qx

Jx Qy

Jx

Case 2: dT/dx 4: 0; Jx = 0

Ex dT/dx Ey

dT/dx

Qx dT/dx

O~ dT/dx

Expression in transport coefficient tensor components

Pxx (resistivity)

Pyx (Hall) nxx (Peltier)

7~yx

~,~, (Seebeck)

~yx (Nernst)

-Xxx (thermal conductivity)

-- Kyx

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THERMOMAGNETIC EFFECTS AND MEASUREMENTS

213

T A B L E II THERMOMAGNETIC EFFECTS IN AN ANISOTROPIC MATERIAL UNDER ADIABATIC EXPERIMENTAL CONDITIONS (Qy -- Jr = O)

Experimental boundary conditions Case 3: d T/dx = 0; Jx # 0

Defining quantity g x

Jx

Expression in transport coefficient tensor components

Ey

Qx

m

Jx

dT/dy Jx

Ex

Jx

Pyx ~

Jx

Jx=0

Kyy 7r,yx Kxy

rtxx +

Kyy

nr~- (Ettingshausen) Kyy

Px,, +

Ey

Case 5: dT/dx r

Kyy O~yy~yx

Jx

Case 4: Qx = 0; Jx v~ 0

O~xy7ryx

Px~ +

Pyx+

axx(n~x~:,- nyx~,) + %(rCrx~x - rCx~:~x) h~xxKyy - - t~yxK xy

%~(~yxKx~- nxxK.,) + %x(nxxK.- ny~Kx~) K x x l ( , y y - K,yxl(,xy

dT/dx

7r.xxKyy t 7F,yxKxy

Jx

t( xxK, yy ~ t(,yxl(, xy

dT/dy

lr,yxKxx -- 7r,xxKy x

Jx

KxxKyy -- Kyxl(, xy

Ex

dT/dx

O~xyK,yx ~xx --

Kyy

Ey

O~yyKyx

dT/dx

Kyy

Qx dT/dx

dT/dy dT/dx

l(xyK, y x

- Gx +

~yy

KYx (Righi-Leduc)

Kyy

each case being designated by a number in Table I. Case 1 and case 2 require transient measurement techniques, where the period of the applied currents is much less than the thermal diffusivity time, so that no transverse thermal gradient develops and the experimental condition remains isothermal. The thermomagnetic transport effects for an anisotropic material as defined in Eqs. (1) and (2) under the adiabatic experimental conditions are presented in Table II. These thermomagnetic transport effects are deter-

214

FRANZ FREIBERT ET AL.

mined in one of three experimental cases, each case being designated by a number in Table II. Cases 3, 4, and 5 require direct current (DC) techniques, where the applied currents are time-independent relative to the thermal diffusivity time and the steady-state measurements are adiabatic in nature. If no magnetic field is applied, then case 3 and case 5 are identical to case 1 and case 2, respectively. Case 4 remains different from the others because there is no longitudinal thermal current within the system (Qx = 0). This phenomenological description of thermomagnetic transport in solids can be used to connect transport effects under adiabatic (indicated in Eqs. (4) and (5) by the superscripted A) and isothermal conditions by a comparison between case 1 and case 3 and between case 2 and case 5, respectively. For the thermomagnetic effects, these relations, known as the Heurlinger-Bridgman relations, are A O~xy~yx Px x - P x x = - ~ ; N,yy

A ~xx-~xx=

A 7r'yxKxy 7r,xx ~ 7"Cxx -= ~ ; Kyy

O~xyK'yx Kyy

(4)

A KxyKyx Kxx + Kxx - - ~ . K,yy

(5)

A

(6)

The equivalent transverse relations are Pyx

.

pyA __ . .

.

(XYYTT'YX. , Kyy

~yx-~yx

=

O~YYK'YX . Kyy

Because of the boundary conditions, there are no associated transverse relations for Eq. (5).

3.

ELECTRONIC REFRIGERATION

There are two electronic transport effects that can produce thermal gradients across a material with the application of an electrical current. The first effect, the Peltier effect, generates a thermal gradient in zero magnetic field dT dx

~xx T = ~Jx

(7)

Kxx

parallel to an electrical current density Jx. The associated electric field is ~x2~T Ex = PxxJx +

Jx Kxx

(8)

7 THERMOMAGNETICEFFECTSAND MEASUREMENTS

215

FIG. 2. Schematicshowing operation of a simple Peltier refrigerator. and has two components. The first term of Eq. (8) arises from the electrical resistivity and is isothermal. The second term is adiabatic and is due to the thermoelectric potential generated across the sample. An example of the behavior c a n be seen in Subsection 2 of Section V. For practical application of the Peltier effect as a refrigerator, two elements are employed, one element having electrons as majority charge carriers (n-type) and the other element having holes as majority charge carriers (p-type). The electrical current circulates around the circuit, but heat flows do not. The heat flows in the opposite direction in the p-type material as in the n-type. That is, heat flows in the direction in which the carriers move, whereas current flows in the direction in which the charge moves. The result is that heat currents, in both legs, flow either toward or away from the junction between the legs, depending on the polarity of the battery (Fig. 2). Note that if a minority carrier is present, it degrades performance by pumping heat opposite to the majority carrier. The other refrigeration mechanism is the Ettingshausen effect. It generates a thermal gradient dT ay~ T = ~J~, dy ~Cyy

(9)

normal to both the current density Jx and magnetic field B z applied to the refrigerator. The associated electric fields are 2T Ex = pxxj ~ _ ~'Y_____L_ Jx

(10)

E y - pyxJx + ~yy~x~.TJ~.

(11)

l(yy

Kyy The first term of each equation is a result of the electrical resistivity and Hall effects, respectively, and is isothermal. The second terms are electric fields produced by the Nernst (increasing Ey) and thermoelectric (decreasing Ex)

216

FRANZ FREIBERT ET AL.

FIG. 3.

Schematic showing operation of a simple Ettingshausen refrigerator.

effects and are adiabatic. An example of the behavior can be seen in Subsection 2 of Section V. If both electrons and holes are present, both carriers pump heat in a single Ettingshausen element when compared with a single Peltier element. As in the Peltier device, the electrical current circulates, however, heat pumping occurs transverse to the electrical current because of the Lorentz force, which acts in the same direction on electrons and holes because they are traveling in opposite directions so that one side is heated while the other side is cooled. In this refrigerator, majority and minority carriers pump heat (Fig. 3). If there were no minority carriers, a Hall field would be generated that exactly canceled transverse transport, reducing refrigeration to zero. As the mobility and concentration of the minority carrier approach those of the majority carriers, the Hall field tends toward zero while heat pumping tends toward a maximum. The boundary conditions described in case 4 represent two unusual scenarios for electronic refrigeration, either an Ettingshausen refrigerator with a thermal gradient parallel to the electric current or a magnetically enhanced Peltier refrigerator with a thermal gradient transverse to the electric current. This is the generally observed behavior because carrier compensation of real materials is rarely perfect. Most data on Ettingshausen refrigerators have come from experiments where the refrigerators were operated under case 4 conditions--the imperfect compensation always degrades performance. An example of the behavior can be seen in Subsection 2 of Section V. Often, however, the figure of merit for the magnetically enhanced Peltier refrigerator with minority carriers present increases for

7 THERMOMAGNETICEFFECTSAND MEASUREMENTS

217

small magnetic fields until the thermomagnetic effects overwhelm the thermoelectric effects. 4.

IDEAL BEHAVIOR

So far, the discussions of electrical and thermal transport and electronic refrigeration have only been phenomenological. In this section, the discussion turns to the properties of microscopically ideal electronic refrigeration materials. The following observations are based solely on the equations of thermomagnetic transport, which leaves for later the microscopic foundation of these observations. Based on the simple picture of charge carriers moving freely through a solid, one can derive from Eq. (7) that the largest thermal gradient that can occur from the Peltier effect is in single carrier material. Goldsmid has made estimates concerning the thermoelectric properties of an optimized Peltier material (Goldsmid, 1986). Further efforts are being made to improve the performance of Peltier materials based on altering the lattice contribution to thermal conductivity (Mahan et al., 1997). Unlike the Peltier effect, the strength of the Ettingshausen effect depends on the accuracy of compensation of electrons and holes in a material. To describe this quantitatively, it is assumed that electrons and holes do not interact and that each carrier type contributes independently to the total transport, known as the "bipolar" effect. Putley did this for a two-band conductor, a "mixed conductor," and later for a multiband conductor (Putley, 1975). For an isotropic mixed conductor having carriers of equal number and mobility (perfect compensation), an optimum Ettingshausen material would have the following behavior: 9 The total isothermal Hall effect should vanish (P~xy= -phy = Pxy), and the electrical conductivities due to each carrier type are equal (p~ = Pxx = Pxx). Therefore the total isothermal electrical resistivity should be p~x _total = (PZx + PZy)/Zpxx. 9 The total thermoelectric power should vanish (~ex = -- ~xxh= ~xx)" The Nernst effect due to each carrier type should be equivalent e = ~xy h = ~xr). Therefore, the components of the Nernst effect for (O~xy each carrier are equal, ~xy -Total = ~Zxy + O~xx(Pxy/Pxx). 9 The sum of off-diagonal elements of the magnetothermal conductivity e = - x . xh should vanish (Xxr r = Xxy)-Therefore, the total thermal conductivity should be-Total l(phonon 2 ~x,, = + 2Xxx + O~xxT/pxx, where f~phonon is the lattice contribution. Goldsmid has made estimates concerning the thermomagnetic properties of an optimized Ettingshausen material (Goldsmid, 1986).

218

FRANZ FREIBERT ET AL.

III. Phenomenological Analysis 1.

EQUATIONS OF PERFORMANCE

The usual measure of the strength of an electronic refrigeration material is its dimensionless figure of merit Z T. This quantity determines both the maximum coefficient of performance and maximum temperature depression for a solid-state refrigerator. Z T is simply the ratio of the thermomagnetic or thermoelectric voltage component to resistive voltage. The total voltage relates to the power drawn from the source, whereas the thermoelectric or thermomagnetic voltage component represents the ability of the material to transport heat and maintain a thermal gradient. For Peltier refrigeration (from Eq. (8)), 2

~xxT zxxr = ~ . PxxtCxx

(12)

For Ettingshausen refrigeration (from Eq. (10)), 2

Zxy T

- O~xyT

(13)

PxxKyy

There is a very curious property of ZxyT, not shared by ZxxT. ZxxT can have any positive value without violating thermodynamic laws, whereas ZxyT cannot exceed unity, or heat would flow opposite the temperature gradient even with zero current. This is an important performance limit for Ettingshausen coolers. Often the figure of merit has an adiabatic form that is defined in terms of adiabatically measured transport coefficients (Harman and Honig, 1967). The Heurlinger-Bridgman relations (Eqs. (4)-(6)) can be used to relate the adiabatic and isothermal figures of merit; however, the boundary conditions of the measurements must be monitored to apply this relationship correctly. It is dangerous to quote the adiabatic form of ZT, because the adiabatic Z T does not directly reflect any measure of refrigeration. Therefore, only the isothermal figure of merit will be used in further discussions. The primary measure of usefulness for a cooling process is its coefficient of performance or ~. This quantity is defined as the ratio of the rate at which heat is extracted from the source to the rate of expenditure of work. More simply, it is the total heat flow divided by the total power dissipated, A(EQ) = ~ .

vJ(~ e)

(14)

The ohmic heating must be added to the total thermal current. If the

7

THERMOMAGNETICEFFECTS AND MEASUREMENTS

219

refrigerating element is in the shape of a rectangular parallelepiped, A and V are the cross-sectional area and volume, respectively. The coefficient of performance for an isothermal Ettingshausen device is found from Eq. (14) to be = A ( g y x T J x - ~cy,d T / d y - P " x j 2 / 2 ) . V(pxx J2 + ~XxyJxdT/dy)

(15)

For a given temperature difference, the cooling power is maximized as a function of electrical current density, so that d(EQy)/dJx = 0 and the corresponding coefficient of performance for this Ettingshausen refrigerator is 1

2

(16)

= ~ Z x y T d - ( T u - Tc) " A Z.,TcT,, V '

where Zxy can be obtained from Eq. (13). The quantities T c and T n represent the temperatures of the cold and hot sides of the refrigerator, respectively. Note that if the source of heat is removed, the coefficient of performance vanishes and the maximum temperature difference across the refrigerator is __

Tc =

-~ZxyTc2.

The current that provides the maximum coefficient of performance occurs when d ~ / d J x = 0, and the corresponding coefficient of performance is (I)-- Tc[(1 '[-

ZxyTAve)I/2-(TH/Tc)-] A

(r u -rc)[(1

~

+ ZxyrAve) '/2 q- 1] V

(17)

Here, the quantity rAv e represents the average temperature of the refrigerator or ( T u + Tc)/2. To achieve greater temperature differences, serially staging multiple devices or "cascading" is necessary. A cascaded device has a characteristic pyramid shape because each stage must carry both the heat pumped and the waste heat generated by the colder stage above it to which it is connected. For all refrigeration devices, the important loss mechanisms are ohmic heating and electronic and phonon thermal conduction. For the Ettingshausen cascade, the optimum shape has an exponential cross section that approximates an infinitely staged device (Kooi, et al., 1968). Peltier refrigerators have a similar optimum pyramidal shape; however, more than six stages does not add much improvement (Goldsmid, 1986).

2.

MICROSCOPIC ELECTRONIC PROPERTIES

So far in our discussions, only the thermodynamic theory of irreversible processes and the phenomenology of electrical and thermal transport have

220

FRANZ FREIBERT ET AL.

been applied to the description of solid state refrigeration. Quantum mechanics and Fermi-Dirac statistics describe the dynamics of carriers in a conductor and enable the connection of thermomagnetic transport properties to fundamental physics (Putley, 1960; Ziman, 1960; Blatt, 1968). Using Boltzmann's theory of transport processes, each contribution from electronic concentration, electronic scattering, and electronic band structure can be separated, in an aid to understand how each component affects thermomagnetic transport behavior and ultimately electronic refrigeration. In order to study the dynamics of carriers in a crystalline solid at any level deeper than simple Newtonian mechanics, quantum mechanics must be employed. At the simplest level, the Schr6dinger equation for nearly free electrons in an infinite lattice of periodic ionic potentials V~o. having periodicity r = _+a can be used to derive some of the important physics controlling thermomagnetic transport. The SchriSdinger equation is

-

h2

--

2m

v24,(~)

+

V,o.(~)r

= ~4,(~).

(18)

Using Bloch's theorem, the wave function solutions have the form ~ ( ? ' ) = u~(~')exp(ik .?'), with energy states (e) dependent on the momentum of the electrons

e(-k) = h2{ k2 + k2 k2 ~ \2m x ~ + 2m~,}"

(19)

The periodic ionic potentials create a weak perturbation on these energy states and induce forbidden regions in the energy spectrum that occur at momentum states k = _+rc/a. The allowed energy states are known as energy bands and are separated by these forbidden regions, known as bandgaps. The region in momentum space that lies between -~z/a <<.k <<.rc/a is known as the first Brillouin zone. When the atoms occupying the lattice sites donate electrons to the energy bands, without loss of generality, their momenta can be restricted to the first Brillouin zone and under electromagnetic forces they act as though they have effective mass m* defined by

m*=

1 )-1 h-5 V~e(k') .

(20)

The simple band structure concepts introduced here are derived for the system in its ground state (T = 0 Kelvin). To understand what occurs at nonzero T, Fermi-Dirac statistics must be employed. The Fermi-Dirac distribution, which describes the probability of populating allowed energy

7 THERMOMAGNETICEFFECTSAND MEASUREMENTS

221

states for a collection of noninteracting electrons at thermal equilibrium, is f(e) = (exp[(e - ~)/k~T] + 1)-'

(21)

where ~ - ~(T) is defined as the chemical potential of the system and marks the energy at which f(e) = 1/2 at all temperatures. At T = 0 K, the greatest energy for which this function does not vanish is called the Fermi energy eF. For a system of electrons occupying an energy band where the number of possible energy states in a region de around e is given by dp(e)de, the equilibrium concentration of electrons in that region is given by f(e)r If that system is subjected to a small external force F, a short time later, a new distribution fo(e) is formed and that differs only slightly from f(e). Scattering events cause the reestablishment of equilibrium at a rate

df (e) ) dt scattering

fo(e) - f (e)

(22)

T,

where z is the relaxation or scattering time of the system. Often the mobility l~ = ez/m* is used as a connection between scattering time and effective mass. Under the applied force, each electron changes its velocity (v) by by = F/m*bt; the rate of change of the distribution function is

dr(e)) = (df (~) ) dv F dr(e) dt drift \ dv -d-[=m* dv "

(23)

In the steady state, the scattering must be balanced by the drift of the electrons, and so

dt

drift

+ \ dt

scatterino :

(24)

O.

From Eq. (22),

fo(a) = f(a) +'c

\ dv

(25)

"

If the force on the system is electromagnetic and the system is exposed to a thermal gradient, then the Boltzmann theory yields e (/y + ~ x B)Vff(e) + #Vrf(e) + m*

fo(e) - f (e)

= 0.

(26)

Neglecting higher order effects, the linearized Boltzmann equation under the

222

FRANZ FREIBERT ET AL.

relaxation approximation yields thermomagnetic transport coefficients whose magnitude contains integrals of the form

f

o

(~~

df (e__~)r

1 + (O)cT) 2 de

(27)

where coc = eB/m* is the cyclotron frequency and j can have values of 0 (for charge) or 1 (for heat) transport. To proceed further in the calculation of the thermomagnetic transport coefficients, a detailed knowledge of the electronic band structure, electronic concentration, and temperature dependence of the chemical potential and scattering processes is necessary. It is rare that such complete information exists for a material. In materials having high cyclotron frequencies, that is, few scattering events, the energy levels can be quantized in a magnetic field into Landau levels. Because the transport coefficients are direct functions of energy, these quantities can exhibit the same oscillatory type phenomena seen in the de Haas-van Alphen and Shubnikov-de Haas effects (Steele and Babiskin, 1955).

3.

UMKEHR EFFECT

The group V semimetallic elements have historically been interesting to those researching thermomagnetic transport because these systems are compensated and have extremely mobile carriers with small effective masses and long mean free paths. The most studied group V semimetal, bismuth (Bi), has a rhombohedral crystal structure with a basis and therefore could be considered a slightly distorted cubic lattice. This slight asymmetry is responsible for its highly anisotropic thermomagnetic transport properties and the Umkehr effect (Akg6z and Saunders, 1975a, 1975b; Sfimengen and Saunders, 1972). The Umkehr or "reversal" effect, first seen in Bi, occurs for certain crystallographic orientations, such that when an applied magnetic field direction is reversed a different magnitude of thermomagnetic effect is measured. This unusual behavior has been explained as a result of spacetime symmetry restrictions and can occur in any thermomagnetic transport tensor components containing terms both odd and even in magnetic field. Using the Drude model and calculated band masses and Fermi surface orientation information (Liu and Allen, 1995), the complete set of components for the electrical resistivity tensor for Bi have been calculated for magnetic fields applied along axes of symmetry. These axes have been designated as the bisectrix (~), the binary (~), and the trigonal (~'). As shown in Table III, this behavior in Bi is due solely to the 6 ~ tilt of the electron pockets with respect to the binary axis. The Umkehr effect is responsible for

7

223

THERMOMAGNETICEFFECTS AND MEASUREMENTS TABLE III

ELECTRICAL RESISTIVITY(~ cm) TENSORS FOR BISMUTH IN AN APPLIED MAGNETIC FIELDa Externally applied magnetic field B =0

Electron pockets not tilted

Electron pockets tilted 39.74 0 0

0 39.74 0

0 0 30.98

40.02 0 0

0 40.02 0

0 0 30.65

B = ( 1 tesla)~

475.92 - 11.91 45.91

11.91 518.90 - 1883.21

45.91 1883.21 3978.85

478.78 0 0

0 519.26 - 1884.52

0 1884.52 3973.65

B=(1 tesla).~

936.71 0 941.56

0 58.12 0

- 1083.03 0 2163.41

939.91 0 1007.67

0 58.13 0

- 1007.67 0 2145.85

B = ( 1 tesla)~

2552.37 -615.71 0

615.71 2552.37 0

0 0 60.51

1803.70 -285.99 0

285.99 1803.70 0

0 0 30.65

"These values were calculated from the Drude model, theoretical band masses and Fermi surface orientation, and an assumed scattering time (3) of 80.0 x 10-14 s.

the lack of antisymmetry and magnitude variation among the tensor elements when the magnetic field is applied along the bisectrix and binary axes.

IV. 1.

Materials Survey

BASIC MATERIAL REQUIREMENTS

This section serves as a review of the status and methodology of the search for the optimal Ettingshausen cooler materials. Efforts to find materials for Peltier coolers are not addressed here because this subject has been treated in great detail elsewhere (Goldsmid, 1986; Mahan et al., 1997). Magnetically enhanced Peltier systems also are not considered, except for the statement that a significant overlap exists between the types of materials sought in both cases. Ettingshausen refrigeration has a number of apparent advantages over Peltier refrigeration. An Ettingshausen device operates with an element constructed of a single material, unlike the Peltier device, which requires two materials similar in all ways except for carrier type. Ettingshausen devices depend on transverse transport, so that anisotropic properties of single

224

FRANZ FREIBERT ET AL.

crystals and layered materials can be used to optimize different transport coefficients in relevant directions; however, Peltier materials must be optimized for transport coefficients all in the same direction. Based on the previous discussion, Ettingshausen coolers gain a factor of 2 over Peltier coolers because both carrier types transport heat in parallel. The materials are not limited to semiconductors, so they may conceivably operate efficiently at much lower temperatures than semiconductor-based Peltier coolers, whose carriers become less mobile with lower temperatures. Both types of cooler may increase emciency by cascading stages, but Peltier devices are limited by material properties and junction losses. An appropriately shaped Ettingshausen element can realistically approximate an infinite-stage cascade with no junctions and much lower ultimate temperatures (O'Brien and Wallace, 1958; Harman and Honig, 1967; Kooi et al., 1968; Scholz et al., 1994). New high field permanent magnet materials such as Nd2Fe14B produce fields of just over 1 T easily and reliably. These apparent advantages are strong motives in the search for Ettingshausen materials, but none of the many materials considered has yet produced efficiencies high enough to warrant practical commercial devices. The form of the thermomagnetic figure of merit ZxyT, Eq. (10), leads naturally to a search for materials that possess a large transverse thermopower (~y) or Nernst coefficient, a small electrical resistivity (p~), and a small total thermal conductivity (Kyy), along appropriate axes. These properties, in general, cannot be independently optimized or even necessarily optimized at the same temperature and field. One such limitation, the ratio pxx~Cxx/T, known as the Lorentz number L o, should be approximately constant (153/zV/K) 2 even in anisotropic materials by the WiedemannFranz law. However, variation in anisotropy, temperature, magnetic field, materials synthesis, alloying, and impurity doping to modify band structure and scattering opens a large parameter space for exploration. Consideration of such properties as carrier density mobility and lattice thermal conductivity can narrow the search considerably. It has been shown (Bass and Tsidil'kovskii, 1957; Harman, 1963; Goldsmid, 1986) that a single-carrier material (a metal or extrinsic semiconductor) in a sufficiently large magnetic field, could have a maximum transverse thermoelectric power of about 40/~V/K. Assuming the Wiedemann-FranzLorentz law value for L o, a best ZxrT of about 0.08 can be reached. This assumption may be reasonable for metals, but its validity in a magnetic field is questionable. Nonetheless, this ZxyT is too small for practical applications because one hopes to find a material to compete with present Peltier cooler materials where ZxxT approaches unity. As noted by many authors (Harman, 1963; Harman and Honig, 1967; Goldsmid, 1986), the only materials that can reach this value of ZxyT must have combined hole and electron

7 THERMOMAGNETICEFFECTSAND MEASUREMENTS

225

conduction, with each carrier type having high mobility and equality in concentration. This strongly limits the search to semimetals or intrinsic semiconductors.

2.

COMPENSATEDMATERIALS

Compensated materials are those that behave as if they have equal numbers of electrons and holes. Number compensation can be achieved by the filling of indirect overlapping band states at ev in semimetals, or by the thermal promotion of carriers across an energy gap in undoped semiconductors. Despite this behavior, transport measurements at low magnetic fields often show these materials to be predominantly n-type or p-type. This is because the mobilities/~e and ~h, of the electrons and holes, respectively, are dependent on the respective effective masses m* and scattering times z and may be quite different. In this case, the electric field due to the Hall effect will limit the Ettingshausen thermal transport. The optimal situation would be to have carriers with high and equal mobilities and high carrier densities. As discussed earlier, this would cause the total Hall effect and Seebeck coetficient to vanish. High mobility is associated with low effective masses, that is, a high curvature of bands at the Fermi surface (Eq. (20)). In the materials exhibiting a large Ettingshausen effect, the small bandgaps or overlaps are often associated with nonparabolic bands that must be treated cautiously in theoretical considerations (Harman and Honig, 1962; Mikhail et al., 1980). Semiconductors are divided into two categories: those that are intrinsic and naturally have very small energy gaps, and those that are extrinsic and are intentionally doped with impurity states. For intrinsic semiconductors, only those with very small or zero gap energies have been suggested or considered for Ettingshausen materials. The carrier concentration at and below room temperature is too small in semiconductors with bandgaps greater than about 400 mV, such as Si, Ge, GaAs, or InP, to provide sufficient carrier concentrations. Those narrow-gap or zero-gap semiconductors (Lovett, 1977) that have been considered are BizSe 3, BizTe 3, SbzTe 3, HgSe, HgTe, CdTe, Mg2Pb, PbSe, FeSi, and Bil_xSb x (8% < x < 20%), individually and in various combined alloys. Gray tin (~-Sn) is a zero-gap semiconductor, but it is extremely difficult to produce in large single crystal or homogeneous polycrystalline form. Another interesting material, graphite, is described as a zero-gap semiconductor (Putley, 1960) or a small overlap semimetal (Charlier et al., 1992). The Fermi surface is composed of alternating hole and electron narrow ellipsoids with long axis along the hexagonal c axis sitting at each of the six

226

FRANZ FREIBERT ET AL.

vertical edges of the hexagonal Brillouin zone. The shape of the Brillouin zone gives each carrier an extremely small effective mass in the hexagonal plane. Based simply on these criteria, graphite should make a good candidate for thermomagnetic refrigeration material. This would be true were it not for the extremely large lattice contribution to the hexagonal plane thermal conductivity. This thermal conductivity is so large that it acts to reduce the thermomagnetic refrigeration figure of merit substantially (Mills et al., 1965).

3.

BISMUTH AND BISMUTH-ANTIMONY ALLOYS

Although some of these alloy semiconductors have produced the best Peltier devices, it was recognized early that the combination of ideal properties, particularly carrier density and mobility, needed for an Ettingshausen cooler could not be realized in these materials. Although the search for new materials continues, most thermomagnetic measurements are performed for fundamental electronic property characterization, and not for purposes of seeking a large value for ZT. Goldsmid has shown, based on band theory, that the optimal configuration for an Ettingshausen material is an indirect band overlap, not a direct gap. Indeed, most of the research into thermomagnetic materials over the past 50 years has been centered on the group V semimetals, particularly Bi and alloys of Bi and Sb. Bi is a semimetal with a band overlap of about 38 meV and extremely mobile carriers. The smallest electronic effective mass in Bi is about 1/103 of the electronic effective mass in most good metals. This leads to enormous sensitivity to magnetic fields in transport properties. Because the hole and electron bands both overlap the Fermi level, but do not have equal curvatures (Liu and Allen, 1995), the carriers have equal numbers, but different mobilities. Liu and Allen's band structure calculation, in good agreement with experiment, gives a range of effective masses of 0.00150.25m e for the electrons and 0.06-0.7me for the holes, depending on the directions for each carrier pocket. The small masses yield high carrier mobilities, but the absolute numbers are very low, giving Bi n-type conductivity approximately 1/104 that of Cu. This conductivity of Bi is larger than a semiconductor and does not exhibit the exponential fall-off at low temperatures shown by semiconductors. The structure is rhombohedral (R3m), although it is often approximated as hexagonal, which leads to large mechanical and transport anisotropy. The three orthogonal symmetry axes of Bi are known as the binary, the bisectrix, and the trigonal axes (Liu and Allen, 1995). A pseudohexagonal cell is often constructed to aid in the visualization of these axes. The Fermi surfaces within the Brillouin zone are small semiellipsoids or pockets. The hole pockets having a major axis along the F - T (trigonal) direction, and the electron pockets lie almost along the m

7

THERMOMAGNETICEFFECTS AND MEASUREMENTS

227

F - L directions. The small deviation angle ( ~ 6 ~ of the electron pockets (Brown et al., 1968) is responsible for the unusual magnetotransport properties discussed earlier (Uher and Goldsmid, 1974; Sfimengen and Saunders, 1972). Bi has a low melting point (271~ is easily grown in single crystal form, and cleaves easily in planes perpendicular to the trigonal axis. The Ettingshausen figure of merit for Bi has been measured (Harman et al., 1964) with low Z~yT values at high temperatures, but reaching a ZxyT of 0.4 at 85 K and 5 T. Optimal orientation of the mutually perpendicular field, current and heat pumping directions were with B along the bisectrix, J along the trigonal, and {2 along the binary axes (Goldsmid, 1963). Antimony (Sb) alloys of Bi are probably the best Ettingshausen materials yet found. Sb is isoelectronic with Bi, shares the same crystal structure, but has a different hole Fermi surface. It has, however, slightly smaller lattice constants so that it substitutes for Bi, and introduces a small decrease of the lattice constants in the alloy (Cucka and Barrett, 1962). Substituting Sb, with an atomic mass of 122, into the Bi, with an atomic mass of 209 reduces the lattice contribution to the thermal conductivity because it scatters phonons strongly. Transport data (Yim and Amith, 1972) indicate that this change is proportionately greater than the reduction in the electrical conductivity, as might be expected for isoelectronic alloying, where electronic scattering is weak. Transport, de Haas-van Alphen, and Shubnikov-de Haas measurements and theoretical calculations (Yim and Amith, 1972; Mustafaev, 1994; Brandt et al., 1972; Chu and Kao, 1970) indicate that as Sb is added, the band structure changes from a semimetal to a narrow-gap semiconductor and eventually again to an Sb-dominated semimetal. Figure 4 is a representation of the bands near ev as Sb is added up to 12 at%. The dotted lines are based on a calculation by Mustafaev and indicate the extrema of the bands. The

TL[~' ..... ~Holes ~Electr~

% Sb0%

4%

~..~.~l

L

7% 8%

12%

FIG. 4. Representationof alterations to the band structure of Bi as it is alloyedwith Sb.

228

FRANZ FREIBERT ET AL.

L- and T-point bands are indicated by parabolas. The occupied conduction band states at T = 0 K are shaded. In pure Bi, there is an overlap of 38 meV between the L- and T-point carriers, and a gap of about 15 meV at the L point. At 4% Sb, there is still a semimetallic overlap, but there is probably a zero gap at the L point. At 7~ Sb, both a direct and a much smaller indirect gap are open, making the material semiconducting. At 8% Sb, the direct and indirect gaps are equal and from 9% to 15%, the direct gap at the L-point is the smallest gap in the system at about 20-30 meV. The L-point gap structure (%) changes at the Bi-rich end as Sb is added, and the curvature of the bands invert at the 4% Sb crossover, while the T-point hole band is depressed and flattens, developing a new hole band at the H-point. This H-point hole band maximum rises in energy as the Sb alloying percentage increases, creating again an indirect-gap semiconductor at about 15% Sb, which persists until the maximum crosses the Fermi level at about 22% Sb and makes the material semimetallic. It is the ability to manipulate this band structure while maintaining high-mobility carriers at the L-points that make this the most studied alloy system for thermomagnetic devices. At low temperatures (T < 70 K) and high fields (B > 1 T), quantization of the carriers into Landau levels occurs, drastically changing the distribution of states in the band. In general, the operating regime of thermomagnetic coolers is well outside these parameters, but "high temperature" conductivity oscillations have been noted in Bi and Bil_xSb x at temperatures up to 65 K (Krasovitsky et al., 1999). In this article, Landau level effects are neglected, but there is some evidence that this may not be justified. In undoped alloys for thermomagnetic cooling, semimetallic alloys of 3 to 5 atomic percent Sb in Bi have been studied extensively (Cuff et al., 1963; Harman et al., 1964; Jandl and Birkholz, 1993; Scholz et al., 1994). In Bio.97 Sbo.o3 at 1 T, Cuff et al. found an isothermal ZxyT maximum of 0.4 at 120 K. Harman et al., working on Bio.96Sbo.o4, found a ZxyT of 0.15 at 1 T, but only at temperatures between 85 and 200 K. Higher ZxyT values may have occurred at temperatures within this range. Cuff et al. made two cooling elements with the Bio.97Sbo.o3 material, one rectangular and one with a linear taper to approximate the advantages of the infinitely staged refrigerator (O'Brien and Wallace, 1958). At 0.75 T and with minimum heat load, the rectangular block achieved a cooling of about 20 K while the tapered block cooled 35 K below a hot heat exchanger temperature of 156 K. Scholz et al. made a refrigerator of the same material, but having an optimally shaped cross-section, which reached a AT of 42 K under the same conditions. It must be noted that although these temperature drops are significant and the role of geometric staging has been verified, these A T values diminish significantly with a heat load. If between 10 and 100 A of electrical current is applied through a cross-section of approximately 1 cm 2, a separate

7 THERMOMAGNETICEFFECTSAND MEASUREMENTS Bi

229

Sb

63 I~

T1

Solid

271~ i

I ,I !

C1

,

I II |

,

C2

FIG. 5. Representationof the Bi-Sb binary phase diagram.

refrigerator is necessary to maintain the heat sink at 160 K, to dissipate the small amount of pumped heat and the great production of Joule heat. The efficiency of these units is very poor, but the convenience of an all-electronic refrigerator to operate at T c = 100 K is very appealing. There is extensive published data on the magnetotransport of the single crystal Bi-Sb system (Jain, 1959; Brandt et al., 1968, 1977; Yim and Amith, 1972; Sengupta and Bhattacharya, 1985; Kagan and Red'ko, 1991; Lenoir et al., 1996), which covers the Bi-rich Bix_xSb x alloys, their growth techniques and most of the transport properties relevant to Ettingshausen coolers. Although Sb is soluble in Bi at all concentrations, homogeneous single crystals require great care to grow because of the complication of constitutional supercooling (Lenoir et al., 1995). The phase diagram Fig. 5 shows the "boat" made by the liquidus and solidus lines in the Bi-Sb system. At a given temperature 7"1,the liquid of concentration C~ is in equilibrium with a solid of higher Sb concentration, C2. Consequently, from a given liquid concentration, the solution phase separates and various Sb concentrations will form. This may be overcome in a number of ways based on zone melting and crystallization. Creating a liquid zone of a lower Sb concentration and passing it along a solid rod of higher Sb concentration creates the correct equilibrium concentrations at the crystal growth interface. These techniques are known as "zone-leveling" (Schneider et al., 1981) in a horizontal single or multipass mode, and "traveling heater" (Lenoir et al., 1995) in a vertical mode. To defeat microscopic segregation, the growth rate must be slower than 10-5/AT cm per second, where AT is the temperature difference between the liquidus and solidus curves at the required concen-

230

FRANZ FREIBERT ET AL.

tration. There is considerable expansion on solidification for these alloys, which can easily break a volume-restraining container, for example, a glass tube.

4.

DOPED MATERIALS

Doping is the addition of small quantities of nonisoelectronic atoms to a material. The added quantity should be small enough that the overall electronic structure is not altered substantially. If an added atom ionizes in the lattice, then it may contribute an electron to the conduction band (a donor) or it may take an electron from a valence band (an acceptor), thus adding a hole. The net result is to shift the Fermi level up or down in energy. Doping is common in the narrow-gap semiconductor systems where it is used to produce p-type and n-type extrinsic conduction for thermoelectric devices. In fact, most efforts with Bi-Sb alloys for thermomagnetic devices have focused on selecting appropriate band features in the Bi l_xSb x phase diagram, and then moving the Fermi level by doping. The Bi-rich end of the Bi l_xSb~ phase diagram is n-type or electron dominated; thus, the dopants are generally acceptors from group IV, with Sn being the most widely used, although Pb has also been used (Horst and Williams, 1980; Jandl and Birkholz, 1993; Migliori et al., 1998). The carrier concentration is very low in Bi, about 3 • 1017/cm 3, which is about 1 carrier per 105 atoms. Horst suggests that with Sn doping, there is about one carrier per three atoms donated to the hole bands. This suggests that Sn doping, at the level of one part per million, will add enough carriers to significantly alter conduction in Bi and Bix_xSb x alloys, although up to several at% have been used by authors seeking strong p-type conduction for Peltier coolers. Sn doping in Bi x_~Sb~ materials provides another variable used to achieve the optimum transport characteristics. Generally, the improvements are not large, but any improvements are significant. Jandl and Birkholz tried varying the Sn doping level and found that doping 145 ppm Sn into Bio.95Sbo.o5 achieved a maximum ZxrT of 0.41 at 205 K in 1 T. They estimate (Scholz et al., 1994) that a staged Ettingshausen cooler would achieve a AT some 20 K greater under the same conditions, as their Bio.97Sbo.o3 cooler (AT= 42 K). Migliori et al., (1998) achieved ZxrT = 0.27 at 150 K in 1 T using a standard field orientation and very low levels of Sn doping in Bio.97Sbo.o3. In summary, Ettingshausen coolers operate, albeit at low efficiencies, in a temperature regime usually inaccessible to Peltier coolers. The best materials seem to be doped semimetal alloys, but there is considerable room for improvement, and a large parameter space to explore. Semiconductors, both narrow-gap and zero-gap, and graphite seem to have fallen out of the transverse thermomagnetic cooler race, but with new materials and advances in preparative techniques it is possible that some variant or modification will be found to make them viable Ettingshausen materials.

7

THERMOMAGNETICEFFECTS AND MEASUREMENTS

231

V. Experimental Measurement Techniques 1.

EXPERIMENTAL FUNDAMENTALS

There have been many references on electrical and thermal transport in solids, providing a thorough discussion of the theoretical and experimental sides of the subject. A good source of theoretical and experimental information on thermomagnetic transport coefficients can be found in Putley (1960). A closer examination of thermoelectric power of metals is discussed by Blatt et al. (1976). Hurd (1972) has thoroughly investigated the subject of Hall effect in metals and alloys. From the viewpoint of electronic refrigeration and the necessary experimental techniques for making transport measurements, Goldsmid's text (Goldsmid, 1986) provides a very sound foundation. Because much work in electronic refrigeration is aimed at cooling to cryogenic temperatures, a useful review of low-temperature experimental techniques and cryostat design can be found in Richardson and Smith (1988). The discussion here is limited to a few fundamental considerations concerning experimental design and techniques specifically for thermomagnetic transport measurements. Thermomagnetic transport measurements are crucial as characterization tools for material selection and device design. Success requires care in sample orientation for crystalline materials, especially in that the sign of the effects is important. Gerlach recognized this fact and proposed a so-called "sign convention" to aid the experimentalist in determining a voltage lead and thermocouple arrangement consistent with thermomagnetic transport phenomenology. These assignments, as seen in Fig. 6, are important to ensure consistency because the sign of thermomagnetic transport coefficients in materials having mixed conduction may change based on the nature of the transport scattering processes and the existence of phonon drag (Putley, 1960; Blatt et al., 1976). The specimen geometry and its effects on measurement must also be considered. Because the transport coefficients are defined in terms of a sample of infinite extent, it is important to understand the effect of finite extent and imperfect geometry on these measurements. At low temperature and in thin films and wires, the size and surface characteristics of a sample

Ge

-L -r~ +~

j Q

-v~

9 +Tx

-d T/dx

-dT/dy

-L

+~ +v~ FIG. 6. Schematic showing a sample with voltage probes and thermocouple oriented in agreement with Gerlach's sign convention as described in the text, Eq. (1), and Eq. (2).

232

FRANZ FREIBERT ET AL.

become important because of large mean free paths and boundary scattering, respectively (Ziman, 1960). These "quantum size effects" must also be recognized as the field of materials synthesis progresses into layered, mesoscopic, and quantum regimes (Strunk et al., 1998). From a macroscopic perspective, sample geometry should be optimized for proper measurement of thermomagnetic effects. For example, a length-to-width ratio greater than 3 is necessary to minimize the effective shorting of the thermomagnetic effects (Isenberg et al., 1948; Ertl et al., 1963). Investigations of thermomagnetic effects generally require the measurement of temperatures, temperature differences, and small voltages. Because techniques for making these measurements have been discussed elsewhere in detail, only a few brief remarks are made here. Temperature measurement and control has been simplified with the advent of automated proportional-integral-derivative (PID) and fuzzy-logic regulated commercial temperature controllers combined with well-characterized commercial temperature sensors capable of +0.05% accuracy from 1 to 325 K. Thermocouples arranged in a "differential mode" still provide the most accurate method of determining temperature differences, and those types with the largest Seebeck coefficient provide the greatest sensitivity. Although thermocouple materials have not advanced much in the past few decades, commercially available thermocouples are well characterized. In addition, the magnetic field dependence for any temperature sensor should be considered when they are being utilized in thermomagnetic transport measurements. All measurements of thermomagnetic transport coefficients involve making voltage measurements in the nanovolt to microvolt range. Making these low-level voltage measurements has become easier with the arrival of commercially available nanovoltmeters, digital lock-in amplifiers, and digital oscilloscopes. Instrumentation amplifiers and electronic filtering circuits are also easily obtainable or constructed with the aid of experienced advice (Horowitz and Hill, 1989). Finally, depending on the funds at hand, there are commercially available supeconducting magnets and physical property measurement systems. Although these systems are typically user-friendly and quickly mastered, there is no substitute for having good working knowledge of the measurements being performed so that alterations and new experimental designs can be implemented quickly and correctly.

2.

ISOTHERMAL MEASUREMENTS

The thermomagnetic transport effects are measured in one of two isothermal experimental arrangements as described in Table I and require transient current techniques, where the period of the applied currents is much less

7

THERMOMAGNETICEFFECTS AND MEASUREMENTS

233

than the thermal diffusivity time. Under these condition, no transverse thermal gradient develops and the experimental condition remains isothermal. Efforts by the authors to optimize the transport properties of Bi-Sb alloys for use in an Ettingshausen refrigerator (Migliori et al., 1997, 1998) have resulted in the use of a novel isothermal measurement technique, and its advantages and limitations are now discussed. This isothermal technique, and its advantages and limitations are now discussed. This isothermal technique has been discussed by others, but rarely implemented in practise. This technique, previously known as the Z-meter method, is a direct determination of the figure of merit of a material from the material's response to a periodic square wave electrical current. Harman first implemented these isothermal techniques on Peltier materials (Harman, 1958) and Gutherie on Ettingshausen materials (Gutherie and Palmer, 1966) in order to provide immediate determination of the isothermal figure of merit of a solid-state refrigeration material. The very simplicity of these techniques provokes the researcher into extending these methods to be used in determining thermoelectric and thermomagnetic transport coefficients as well. With additional thermocouples, it is possible to measure the resistivity tensor components, thermopower tensor components, thermal conductivity, and Peltier and Ettingshausen coefficients of the material under study simultaneously. This approach was applied by us to single crystal samples of Bi and Bi-Sb alloys, synthesized and characterized by published techniques (Lenoir et al., 1995). Single crystal grains were oriented and cut using Laue back-reflection X-ray techniques and etch pits (Lovell and Wernick, 1959; Brown et al., 1968). However, a very simple method of determining crystal orientation involved cleaving the crystal along the natural cleavage planes perpendicular to the trigonal direction. By examining the natural fractures that occur in this plane, it is easily seen that these tractures are 120 ~ apart along each of the equivalent three binary directions. The determination of the third or bisectrix direction is made consistent with a right-handed coordinate system. After orientation, the samples were finished to final width (W), height (H), and length (L) dimensions for transport measurements. If the sample was being investigated for its use in a thermoelectric device, a longitudinal themal gradient (i.e., c~T/~x 4: 0) must be allowed to exist for the longitudinal or Peltier Z-meter method to operate, and so the sample was suspended by the electrical current leads attached on either end (Harman, 1958). However, if the sample was being investigated for its use in a thermomagnetic device, no logitudinal thermal gradient should exist (i.e., c~T/Ox = O) for the transverse or Ettingshausen Z-meter method to operate. Cu blocks were attached to either end to artificially force this boundary condition. In both cases, the isothermal experimental configuration utilized an instrumen-

234

FRANZ FREIBERT ET AL.

tation preamplifier with a gain of 300 that amplified the voltage to be recorded by a digital oscilloscope. This technique was used to amplify, digitize, and record the longitudianl and transverse voltages of the sample in response in the longtudinally applied electrical current. The isothermal technique and related heat flow theory for the Z-meter method in both the longitudinal and transverse modes is well described by Gutherie and Palmer (1966); however, a basic review is given here. When an electric field is first applied to a sample, there are no thermal gradients. The longitudinal voltage measured by the oscilloscope is representative of the isothermal resistivity (cases 1 and 2). With time, steady-state thermal gradients are generated via Peltier and Ettingshausen effects that couple to the longitudinal voltage via other thermoelectric and thermomagnetic effects (cases 3 and 4). Therefore, the voltage seen by the oscilloscope has an exponential curvature whose time constant is determined by a thermal penetration time approximately given by CLZ/Ir where C is the heat capacity per unit volume of the system, ~c is the thermal conductivity, and L is the length of the sample in that direction. C/K is known as the thermal diffusivity of the material. The advantage of using this technique in determining the figure of merit for the material is that the absolute quantities of dimensions, voltages, and thermal gradients are not important in determining the effective refrigeration of a material because the technique relies only on ratios of voltage differences. The greatest limitation of this technique is that the thermoelectric of thermomagnetic voltages are dependent on the associated figures of merit ZxxT and Z~yT. Therefore, the accuracy for measurement of transport coefficients increases with the figure of merit; that is, a good electronic refrigeration material is a good candidate for the Z-meter technique. In work aimed at characterizing electronic refrigeration materials, the authors implemented the Z-meter method in both Peltier and Ettinghausen modes. If no magnetic field is applied, then the Z T derived from the Peltier method is given by ZxxT = ~xxA T~ = ~xxA Tx 2IR 2Jp~:,L

(28)

and is that for a Peltier refrigeration material (Harman, 1958). This experimental condition corresponds to case 4 with B -- 0 T. From these data (Fig. 7) and a measure of the longitudinal temperature gradient (Fig. 8), Pxx, Gx, G~, and rGx can be determined using Eqs. (3), (7), (8), and (28). However, in the Peltier Z-meter method with a magnetic field applied, the isothermal Peltier Z T cannot be isolated because the system begins operating as a Peltier/Ettingshausen hybrid refrigerator under the boundary conditions described in case 4 with B :/: 0 (Fig. 9).

7

THERMOMAGNETIC EFFECTS AND MEASUREMENTS

30

1

20

t

..--"-'"'7.

C~xATx

-

235

10 ;> :=t.

0 > -10

-20

l

-30 ,

I

5

~

I

10

,

15

20

T i m e (s) FIG. 7. Graph of longitudinal voltage as developed in time from longitudinal Z-meter method (case 4) applied to Bi at T = 140 K and B = 0 T. The electrical current was applied as an alternating square wave with a 24-s period.

80

IF- -~,~- -~-'~

40 f

[...

i r

0

I

-40

-80

,

0

I

~

4

I

8

12

Time (s) FIG. 8. Graph of the thermal difference generated across a sample as developed in time due to thermomagnetic cooling effects. The electrical current was applied as an alternating square wave with a 12-s period.

236

FRANZ FREIBERT ET AL.

150

"

1

100

(ZxyATy

50

l

>. ::I.

9

0 -50

-100 O~xxATx

-150 -200

f

2Jx0xx/L I

!

10

I ,,

!

15

20

Time (s) FIG. 9. Graph of longitudinal voltage as developed in time from longitudinal Z-meter method (case 4) applied to Bi at T = 260 K and B = 2 T. The electrical current was applied as an alternating square wave with a 16-s period.

In the case of the Ettingshausen Z-meter method, an effort to maintain the boundary conditions described in case 3 was employed to reduce the effects of a thermal gradient parallel to the electric current, by making the sample at least four times as long as it was wide (i.e., L/> 4W) and by thermally anchoring both ends of the sample to large Cu blocks. This practice greatly extends the time constant for development of the unwanted Peltier thermal gradients and increases the effective thermal mass of the sample. After the thermomagnetic component of voltage perpendicular to the current flow has stabilized, but before the thermoelectric component of voltage parallel to current flow have become significant, the electric current is reversed. An oscilloscope records the resulting voltage parallel to the current flow (Fig. 10). From this measurement, the isothermal Ettingshausen figure of merit is Z x y T = o~,,yATyL/W = o~,yATy . 2IR 2Jp,,xW

(29)

The voltage perpendicular to current flow is also recorded (Fig. 11) and exhibits exponential behavior. The thermal gradients parallel with and perpendicular to current flow were also monitored and recorded. From these data (Figs. 8, 10, and 11) and Eqs. (9), (10), (11), and (29), the transport coefficients Px,,, Py,,, O~yy, ~yx, ~Cyy, and ~yx can be determined.

7

THERMOMAGNETIC EFFECTS AND MEASUREMENTS

237

40(3 30C 200 100> ,.--, ;>

.

1

(3

-lOG -2~

r'""--f

f

-ZO0-40C~

J

I

I

1.0

I

1.5

2.0

T i m e (s) FIG. 10. Graph of longitudinal voltage as developed in time from transverse Z-meter method (case 5) applied to Bio.97Sbo.03 + 10ppm Sn at T = 140K and B = 1.5T. The electrical current was applied as an alternating square wave with a 1-s period.

20

{

1 ill

10

T >: a .

0 -5 -10 _LI

__

_

-15 -20 0.5

,

i 15.

110

, 2.0

Time (s) FIG. 11. Graph of transverse voltage as developed in time from transverse Z-meter method (case 3) applied to Bio.97Sbo.03 -t- 10ppm Sn at T = 140 K and B = 1.5 T. The electrical current was applied as an alternating square wave with a 1-s period.

238 3.

FRANZ FREIBERT ET AL. ADIABATIC MEASUREMENTS

Thermomagnetic transport measurements under adiabatic conditions fall into one of the three cases described in Table II. For each case, the applied voltages are time-independent. This approach is easy, but as seen from an examination of Eqs. (1) and (2), the transport equations are coupled and so the quantities measured are not the normally defined transport coefficients. For thermoelectric refrigeration materials, this coupling is often ignored, leading to the incorrect reporting of transport coefficient. However, for most materials, the error (dependent on Zxx T or ZxyT) is small and the thermomagnetic transport coefficients as measured are reasonably correct. To illustrate adiabatic measurements, samples of highly oriented pyrolytic graphite were obtained from a commercially available source. Single crystal grains were cut and oriented using Laue back-reflection and to cut to final width (W), height (H), and length (L). To the sample were attached differential thermocouples and voltage contacts, both perpendicular and parallel to the applied voltage. The sample was thermally anchored to the sample stage with a Cu block at one end and a 304 stainless steel block at the other end. Attached to the stainless steel block were differential thermocouples allowing the block to serve as a thermal conductivity standard against which the sample was compared. These adiabatic thermomagnetic transport coefficient measurements were made using a swept source. In this technique, the electric current or applied temperature difference is swept at a linear rate in time, while voltage measurements are recorded continuously. In accordance with linear response theory given by Eqs. (1) and (2), the thermoelectric voltages that develop are linear functions of the applied electric current (Fig. 12) or thermocouple voltage (Fig. 13), seen in an xy plot. A linear least squares fit of the measured voltages to the applied electric currents or voltages generated by differential thermocouples provided two quantities, the slope and the y-intercept. From the slope of the linear fit, the thermomagnetic transport coefficients can be determined. The y-intercept of the linear fit is a voltage offset usually caused by unwanted thermally induced voltages in the leads. Because only the slopes matter, such thermal voltages are intrinsically eliminated by this approach. Using the previously described technique, the electrical resistivity, thermopower, and thermal conductivity (Figs. 14, 15, and 16) have been determined from the oriented graphite sample described before. Here, the assumed linear behavior of thermomagnetic transport described in Eqs. (1) and (2) has provided a useful methodology by which a measurement technique can be fully explored.

7

239

THERMOMAGNETICEFFECTS AND MEASUREMENTS

35.0

I

'

I

i

I

"

I

,

I

'

I

,

I

'

I

,

30.0 >

25.0

v

20.0 O

>

15.0 10.0 5.0 0.0 -5.0

i

0.0

10.0

20.0

I

30.0

40.0

50.0

Electrical Current (mA) FIG. 12. Graph of longitudinal voltage as developed as a function of electrical current applied in a ramped fashion to the graphite sample described in the text.

40.0

'

I

'

,

'

,

'

I

,

,

'

I

,

,

'

I

,

,

'

I

,

,

"

I

,

'

35.0 >

30.0 =k

25.0

20.0 O

15.0 .,-,4

B

lO.O

,1..~

50

0.0 -5.0

i

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

J

8.0

Longitudinal T h e r m o c o u p l e Voltage (~tV) FIG. 13. Graph of longitudinal voltage as developed as a function of thermal current (plotted as longitudinal thermocouple voltage) applied in a ramped fashion to the graphite sample described in the text.

240

FRANZ FREIBERT ET AL. '

'

I

'

I

'

l

'

I

'

I

*~., 0.01

'

J

-m-- 0T

~,, "~\

IE-3

"~.~,,

""~

4

"--IT

~""~----A

--~v~v-~

~ ~ 0 ~ _ ~

-~

v

~A---~A~___

O

A

J

m~m----m--m--m--m--m--m--m--m--m--m

.

I

0

,

50

l

,

l

100

~

I

150

~

f

200

250

300

T (K) FIG. 14. The magnetic field dependent in-plane electrical resistivity of highly oriented graphite.

VI. Summary The thermoelectric and thermomagnetic transport properties of solid state systems are typically used as research tools to explore electron and phonon behavior within the material and as a measure of the usefulness of the material for potential applications. The magnetic field and temperature

'

I

'

-9- m ~ , ~ - -

I

'

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'

~m--__m~m~ . .~11~

I

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I

'

-50 ~" ;> v

-100

/

:zk

-150

--i--

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--o--

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1T - - v - - 2T --A--

-200

- - * - - - 4T

-250

,

0

I

50

i

I

100

m

I

150

,

I

200

m

I

250

300

W (K) FIG. 15. The magnetic field dependent in-plane thermoelectric power of highly oriented graphite.

7 THERMOMAGNETICEFFECTSAND MEASUREMENTS 20

9

I"

'

I

'

I

"

I

,

'

!

'

I

,

'

'1

241 '

"

"'

15

10

5

~"~

0T

--e--

0.5T

--A

IT

--T--

2T

~4T

9

0

I

50

i

I.

,

100

150

200

I

250

9

300

T (K) FIG. 16. The magnetic field independent in-plane thermal conductivity of highly oriented graphite.

dependence of the transport effects determine the performance of the material for electronic refrigeration, govern materials optimization efforts, and expose the underlying physical transport mechanisms of the material, including effects associated with carrier concentration, mobility, and scattering mechanisms. The particular complications associated with thermomagnetic transport effects have been reviewed, proper methods of measuring these transport coefficients have been discussed, and the transport properties of an idealized thermomagnetic electronic refrigeration material have been given.

REFERENCES Akg6z, Y. C., and Saunders, G. A., Space-Time Symmetry Restrictions on the Form of Transport Tensors: I. Galvanomagnetic Effects, J. Phys. C: Solid State Phys. 8, 1387 (1975a). Akz6g, Y. C., and Saunders, G. A., Space-Time Symmetry Restrictions on the Form of Transport Tensors: II. Thermomagnetic Effects, J. Phys. C: Solid State Phys. 8, 2962 (1975b). Angrist, S. W., Galvanomagnetic and Thermomagnetic Effects, Scientific American 205, 124 (1961). Bass, F. G., and Tsidil'kovskii, I. M., Theory of Isothermal Galvanomagnetic and Thermomagnetic Effects in Semiconductors, Soy, Phys. JETP 4, 565 (1957). Blatt, F. J., Physics of Electronic Conduction in Solids, New York: McGraw-Hill (1968). Blatt, F. J., Schroeder, P. A., Foiles, C. L., and Greig, D., Thermoelectric Power of Metals, New York: Plenum Press (1976).

242

FRANZ FREIBERT ET AL.

Brandt, N. B., Svistova, E. A., and Valeev, R. G., Investigation of the Semiconductor-Metal Transition in the Bismuth-Antimony System in a Magnetic Field, Soy. Phys. JETP 28, 245 (1968) Brandt, N. B., Chudinov, S. M., and Karavaev, V. G., Investigation of Gapless States in Bismuth-Antimony Alloys under Pressure, Soy. Phys. JETP 34, 368 (1972). Bradt, N. B., Semenov, M. V., and Falkovsky, L. A., Experiment and Theory on the Magnetic Susceptibility of Bi-Sb Alloys, J. Low Temp. Phys. 27, 75 (1977). Brown, R. D., Hartman, R. L., and Koenig, S. H., Tilt of the Electron Fermi Surface in Bismuth, Phys. Rev. 172, 598 (1968). Callen, H. B., Thermodynamics and an Introduction to Thermostastistics, New York: John Wiley and Sons (1985). Charlier, J.-C, Michenaud, J.P., and Gonze, X., First Principles Study of the Electronic Properties of Simple Hexagonal Graphite, Phys. Rev. B46, 4531 (1992). Chu, H. T., and Kao, Y.-H., Shubnikov-de Haas Effect in Dilute Bismuth-Antimony Alloys. I. Quantum Oscillations in Low Magnetic Fields, Phys. Rev. B1, 2369 (1970). Cucka, P. and Barrett, C. S., The Crystal Structure of Bi and Solid Solutions of Pb, Sn, Sb and Te in Bi, Acta Cryst. 15, 865 (1962). Cuff, K. F., Horst, R. B., Weaver, J. L., Hawkins, R. R., Kooi, C. F., and Enslow, G. M., The Thermomagnetic Figure of Merit and Ettingshausen Cooling in Bi-Sb Alloys, Appl. Phys. Lett. 2, 145 (1963). Ertl, M. E., Pfister, G. R., and Goldsmid, H. J., Size Dependence of the Magneto-Seebeck Effect in Bismuth-Antimony Alloys, Br. J. Appl. Phys. 14, 161 (1963). Gerlach, W., Handbuch der Physik, Vol. 13, Berlin: Springer (1928). Goldsmid, H. J., Electronic Refrigeration, London: Pion (1986). Goldsmid, H. J., The Ettingshausen Figure of Merit of Bismuth and Bismuth-Antimony Alloys, Brit. J. Appl. Phys. 14, 271 (1963). Gutherie, G. L., and Palmer, R. L., Direct Measurement of the Nernst-Ettingshausen Dimensionless Figure of Merit, J. Appl. Phys. 37, 90 (1966). Harman, T. C., Special Techniques for Measurement of Thermoelectric Properties, J. Appl. Phys. 29, 1373 (1958). Harman, T. C., Criteria for the Optimization of the Nernst Figure of Merit, Appl. Phys. Lett. 2, 13 (1963). Harman, T. C. and Honig, J. M., Galvano-Thermomagnetic Effects in Degenerate Semiconductors and Semimetals with Nonparabolic Band Shapes: II. General Theory, J. Phys. Chem. Solids 23, 913 (1962). Harman, T. C., and Honig, J. M., Thermoelectric and Thermomagnetic Effects and Applications, New York: McGraw-Hill (1967). Harman, T. C., Honig, J. M., Fischler, S., and Paladino, A. E., The Nernst-Ettingshausen Energy Conversion Figure of Merit for Bi and Bi-4% Sb Alloys, Solid State Electron. 64, 505 (1964). Horowitz, P., and Hill, W., The Art of Electronics, 2nd ed., Cambridge: Cambridge University Press (1989). Horst, R. B., and Williams, L. R., Application of Solid State Cooling to Spaceborne Infrared Focal Planes, 3rd International Conference on Thermoelectric Energy Conversion, 183 (1980). Hurd, C. M., The Hall Effect in Metals and Alloys, New York: Plenum Press (1972). Isenberg, I., Russel, B. R., and Greene, R. F., Improved Method of Measuring Hall Coefficients, Rev. Sci. Instrum., 19, 685 (1948). Jain, A. L., Temperature Dependence of the Electrical Properties of Bismuth-Antimony Alloys, Phys. Rev. 114, 1518 (1959). Jandl, P., and Birkholz, U., Thermogalvanomagnetic Properties of Sn-Doped Bi95Sb5 and Its Application for Solid State Cooling, J. Appl. Phys. 76, 7351 (1994).

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Kagan, V. D., and Red'ko, N. A., Phonon Thermal Conductivity of Bismuth Alloys, Soy. Phys. JETP 73, 664 (1991). Kooi, C. F., Horst, R. B., Cuff, K. F., and Hawkins, S. R., Theory of the Longitudinal Isothermal Ettingshausen Cooler, J. Appl. Phys. 34, 1735 (1963). Kooi, C. F., Horst, R. B., and Cuff, K. F., Thermoelectric-Thermomagnetic Energy Converter Staging, J. Appl. Phys. 39, 4257 (1968). Krasovitsky, V. B., Khotkevich, V. V., Jansen, A. G. M., and Wyder, P., "High Temperature" Oscillations of Bismuth Conductivity in the Ultra Quantum Limit, Low Temp. Phys. 25, 677 (1999). Lenoir, B., Demouge, A., Perrin, D., Scherrer, H., Scherrer, S., Cassart, M., and Michenaud, J. P., Growth of Bi 1_xSb x Alloys by the Travelling Heater Method, J. Phys. Chem. Solids 56, 99 (1995). Lenoir B., Cassart, M., Michenaud, J-P., Scherrer, H., and Scherrer, S., Transport Properties of Bi-Rich and Bi-Sb Alloys, J. Phys. Chem. Solids 57, 89 (1996). Liu, Y., and Allen, R. E., Electronic Structure of the Semimetals Bismuth and Antimony, Phys. Rev. B52, 1566 (1995). Lovell, L. C., and Wernick, J. H., Dislocation Etch Pits in Bismuth, J. Appl. Phys. 30, 234 (1959). Lovett, D. R., Semimetals and Narrow-Bandgap Semiconductors, London: Pion (1977). Mahan, G., Sales, B., and Sharp, J., Thermoelectric Materials: New Approaches to an Old Problem, Phys. Today 50, 42 (1997). Migliori, A., Darling, T. W., Freibert, F., Trugman, S. A., Moshopoulu, E., and Sarrao, J. S., New Approaches to Thermoelectric Cooling Effects in Magnetic Fields, A S M E Proceedings of the 32nd National Heat Transfer Conference, Baltimore, MD (1997a). Migliori, A., Darling, T. W., Freibert, F., Trugman, S. A., Moshopoulu, E., and Sarrao, J. S., Optimization of Materials for Thermomagnetic Cooling, Mat. Res. Soc. Syrup. Proc. 478, 231 (1997b). Migliori, A., Freibert, F., Darling, T. W., Sarrao, J. L., Trugman, S. A., and Moshopoulou, E., New Directions in Materials for Thermomagnetic Cooling, Proceedings of the Space Technology and Applications International Forum, Albuquerque, NM, 1628 (1998). Mikhail, I. F. I., Hansen, O. P., and Nielsen, H., Diffusion Thermopower of Bismuth in Non-Quantising Magnetic Fields, Pseudo-Parabolic Model, J. Phys. C Solid State Phys. 13, 1697 (1980). Mills, J. J., Morant, R. A., and Wright, D. A., Thermomagnetic Effects in Pyrolytic Graphite, Brit. J. Appl. Phys. 16, 479 (1965). Mustafaev, N. B., Energy Band Structure of Bi-Bil_xSb~ Compositional Superlattices, J. Phys. Condens. Matter 6, 2039 (1994) O'Brien, B. J., and Wallace, C. S., Ettingshausen Effect and Thermomagnetic Cooling, J. Appl. Phys. 29, 1010 (1958). Putley, E. H., The Hall Effect and Related Phenomena, London: Butterworths (1960). Putley, E. H., Galvano- and Thermo-magnetic Coefficients for a Multi-band Conductor, J. Phys. C: Solid State Phys. 8, 1837 (1975). Richardson, R. C., and Smith, E. N., Experimental Techniques in Condensed Matter Physics at Low Temperatures, Redwood City, CA: Addison-Wesley (1988). Schneider, G., Herrmann, R., and Christ, B., Crystal Growth and Electron Microprobe Analysis of Bismuth-Antimony Alloys (Bi~_xSbx), J. Cryst. Growth 52, 485 (1981). Scholz, K., Jandl, P., Birkholz, U., and Dashevskii, Z. M., Infinite Stage Cooling in Bi-Sb Alloys, J. Appl. Phys. 75, 5406 (1994). Sengupta, M., and Bhattacharya, R., Electrical and Magnetic Properties of Semimetallic and Semiconducting Alloys of Bi-Sb, J. Phys. Chem. Solids 46, 9 (1985). Smith, G. E., and Wolfe, R., Theroelectric Properties of Bismuth-Antimony Alloys, J. Appl. Phys. 33, 841 (1962).

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Steele, M. C., and Babiskin, J., Oscillatory Thermomagnetic Properties of Bi Single Crystals at Liquid He Temperatures, Phys. Rev. 98, 359 (1955). Strunk, C., Henny, M., Schonenberger, C., Neuttiens, G., and Van Haesendonck, C., Size Dependent Thermopower in Mesoscopic AuFe Wires, Phys. Rev. Lett. 81, 2982 (1998). Siimengen, Z., and Saunders, G. A., The Thermomagnetic Tensor and the Umkehr Effect in Bismuth, J. Phys. C: Solid State Phys. 5, 425 (1972). Uher, C. and Goldsmid, H. J., The Magneto-Seebeck Coefficient of Bismuth Single Crystals, Phys. Stat. Sol. (b) 63, 163 (1974). Wolfe, R., Magnetothermoelectricity, Scientific American 210, 70 (1964). Yim, W. M. and Amith, A., Bi-Sb Alloys for Magneto-Thermoelectric and Thermomagnetic Cooling, Solid State Electron. 15, 1141 (1972). Ziman, J. M., Electrons and Phonons, London: Oxford University Press (1960).

SEMICONDUCTORS AND SEMIMETALS, VOL. 70

CHAPTER

8

Heat and Electricity Transport through Interfaces M. B a r t k o w i a k and G. D. M a h a n DEPARTMENT OF PHYSICS AND ASTRONOMY UNIVERSITY OF TENNESSEE KNOXVILLE, TENNESSEE AND SOLID STATE DIVISION OAK RIDGE NATIONAL LABORATORY OAK RIDGE, TENNESSEE

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .

245

II. BOUNDARY IMPEDANCES . . . . . . . . . . . . . . . . . . . . . . . .

247

III. WIEDEMANN-FRANZ LAW AT BOUNDARIES

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

251

IV. ENERGY BALANCE EQUATIONS FOR ELECTRONS AND PHONONS OUT OF EQUILIBRIUM . . . . . . . . . . . . . . . . . . . . . . . . .

253

V. THERMAL INSTABILITY . . . . . . . . . . . . . . . . . . . . .

256

VI. EFFECTIVE THERMOELECTRIC PROPERTIES . . . . . . . . . . . . .

258

1. Effective Thermal Conductivity . . . . . . . . . . . . . . . . 2. Effective Seebeck Coe~cient . . . . . . . . . . . . . . . . . 3. Effective Figure o f Merit . . . . . . . . . . . . . . . . . .

258 261 262

V I I . SUPERLATTICES . . . . . . . . . . . . . . . . . . . . . . . . .

266

V I I I . SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . .

269

REFERENCES

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

I.

270

Introduction

Renewed interest in thermoelectric devices (Rowe and Bhandari, 1983; Goldsmid, 1986; Slack, 1995; Mahan et al., 1997; Tritt et al., 1997, 1999; Mahan, 1998) has reinvigorated the search for new compounds that exhibit superior thermoelectric properties. Another approach to improve performance of small-scale devices is to use multiple quantum wells, where heat and electricity transport is in the direction parallel to the wells (Hicks and Dresselhaus, 1993; Sofo and Mahan, 1994; Hicks et al., 1996). Here we discuss a third approach in which the transport is in the direction perpendicular to the interfaces of thin films, heterostructures, and superlattices. 245 Copyright 9 2001 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-752179-8 ISSN 0080-8784/01 $35.00

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M. BARTKOWIAKAND G. D. MAHAN

Boundary or interface effects are likely to be very important for transport in such devices. For steady-state transport in bulk, isotropic, thermoelectric materials, there are four transport coefficients that govern the phenomena: the electrical conductivity (tr), the thermal conductivities of phonons (Kp) and electrons (Ke), and the Seebeck coefficient (S). Similarly, transport across interfaces is governed by boundary impedances: the electrical contact conductivity aB, the boundary Seebeck coefficient SB, and the thermal Kapitza conductances of phonons KBp p and electrons Kse e. Moreover, as the thickness of semiconducting thin films or the period of superlattices becomes comparable with the carriers' temperature cooling length (Bass et al., 1973; Mahan, 2000), the effects of nonequilibrium between electrons and phonons become important, and one has to consider their influences on the thermoelectric properties of the device. We first review what is known about the boundary impedances and provide specific examples. There is a well-known thermal boundary resistance to the flow of heat by phonons (Peterson and Anderson, 1973; Swartz and Pohl, 1989). There have been suggestions for similar resistances to the flow of electrons through boundaries (Bulat and Yatsyuk, 1984; Anatychuk et al., 1987; Gurevich and Logvinov, 1992; Zakordonetz and Logvinov, 1997; Mahan et al., 1998). We find that the electrical components obey a law KBee = ~q~o6BT(kB/e) 2 (Mahan and Bartkowiak, 1999) similar to the Wiedemann-Franz law for bulk transport (Wiedemann and Franz, 1853; Ziman, 1960). Here T is the temperature, kB is Boltzmann's constant, and e is the electron charge. The dimensionless coefficient ~ o (Lorentz number) is between 2 and n2/3. Moreover, the boundary Seebeck coefficient SB associated with thermal excitations over a boundary barrier can be as large as (or even larger than) the bulk Seebeck coefficient S of the best thermoelectric materials (Mahan et al., 1998; Mahan and Bartkowiak, 1999). This is the basis for the recently proposed thermionic cooling and power generating devices (Shakouri and Bowers, 1997; Mahan and Woods, 1998; Shakouri et al., 1998, 1999; Mahan et al., 1998). Next, we provide the complete phenomenological equations for the heat and electricity transport in a thin-film heterostructure. Using these equations, which include all the boundary impedances and take into account that electrons and phonons can be out of nonequilibrium, we solve for the thermoelectric properties of a two-terminal device consisting of a thermoelectric of length L with metallic electrodes on each end. The equations show a new thermal instability. Thermal instabilities associated with Joule heating are well understood. The present instability does not involve Joule heating but is a thermal boundary instability. Nevertheless, it could cause the temperature of the device to rise uncontrollably when L becomes small. We calculate the effective thermoelectric properties of such a thin-film device: the effective thermal conductivity and Seebeck coefficient, as well as the efficiency and effective figure of merit. We show that, as expected, the

8

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247

boundary effects cause a reduction in the effective thermal conductivity, but they can also lead to a reduction of the effective Seebeck coefficient. As a net result, the effective figure of merit of a thin-film device can still be many times higher than that for the bulk, provided that the boundaries block the phonon heat flow more efficiently than the electronic heat flow a n d that the two subsystems are out of equilibrium. Finally, we extend these considerations to the case of superlattices. Here too the boundary effects can lead to substantial improvement of the multilayer device efficiency. The maximum degree of the enhancement is achieved for submicron thickness of the layers and depends on the properties of the superlattice materials and on the properties of the boundaries.

II. Boundary Impedances Thermal boundary resistance for the electronic heat flow have been discussed in Gurevich and Logvinov (1992) and Zakordonets and Logvinov (1997). Here we derive the full set of boundary impedances. We begin with the equation for transport of electricity (J) and heat (J e) in a bulk thermoelectric,

J=-a

(1)

+S

dV J o. = - a T S -~x -

K, dT d--x

dT K d---s

JQ = J S T K = K' -

a T S 2,

(2)

(3) (4)

where K' is the thermal conductivity at zero potential, which is different from the thermal conductivity K at zero current. The efficiency of a bulk thermoelectric device depends on the material properties through the dimensionless parameter TS2a

ZT =

TS2a =

K

(5)

Kp + K e '

called the figure of merit (Rowe and Bhandari, 1983; Goldsmid, 1986; Slack, 1995; Mahan et al., 1997; Tritt et al., 1997, 1999; Mahan, 1998). For Z T --+ c~, Carnot efficiency is obtained, but the best materials currently used

248

M. BARTKOWIAK AND G. D. MAHAN

in thermoelectric devices have Z T ~ 1 at room temperatures. By analogy, we write down similar equations for the boundary impedances: J = - aB(A V + SBAT)

(6)

JQ = -- an T S n A V - K'BA T

(7)

JQ = J S B T - KBA T

(8)

KB = K'B -- a B T S 2.

(9)

Here we have introduced the boundary impedances: trB[S/m2], SB[V/K], and KB[W/(m2K)]. This is a quite general result. The flow of current has impedances if there is a temperature difference AT across the boundary, or if there is a potential difference A V, or both. The heat current behaves similarly. An Onsager relation proves that the same Seebeck coefficient SB enters both terms. These are the general equations that describe a nonOhmic contact. We assume that the boundary impedances are due to a boundary between layers of either the same material (grain boundary) or different materials (interface in a heterostructure). The presence of the boundaries may cause appearance of electronic potential barriers of various shapes and thickness. Typical examples are: Schottky barriers at the boundaries between metals and semiconductors, double Schottky barriers at the boundaries between semiconductors when there are trapping electronic states at the interfaces, or barriers across metal to metal contacts associated with thin oxide layers. Finally, potential barriers of desired properties can be produced artificially by using molecular beam epitaxy (MBE) or chemical vapor deposition (CVD) in conjunction with bandgap engineering. Using different materials and varying the growth constraints and lattice mismatch, one can grade the barrier composition and produce heterostructures with internal barriers determined by the band edge discontinuities. In principle, electrons can either tunnel through potential barriers, or else be thermally excited over them. However, in practice, these two mechanisms may not be easy to distinguish. For example, in case of a Schottky barrier of parabolic shape, electrons with sufficiently high energy can easily tunnel through the barrier near its top, when it is very thin. In such cases thermionic emission and tunneling are coupled, and it is necessary to describe the transport using a unified general approach (Fonash, 1972). Heat across the junctions is carried both by electrons and phonons. The phonon contribution is described by the Kapitza thermal conductor KBp p (Peterson and Anderson, 1973; Swartz and Pohl, 1989). Theoretical description of the phonon thermal boundary conductance between two solids have mainly been based on the acoustic mismatch theory (Little, 1959). In this theory, the two media are regarded as two elastic continua and a perfect

8

HEAT AND ELECTRICITY TRANSPORT THROUGH INTERFACES

249

junction between the two is assumed. A phonon that is incident on the interface has a certain probability of being reflected or transmitted, but does not scatter. Another model uses diffuse mismatch to explain the behavior of rough interfaces at higher temperatures (Swartz and Pohl, 1989). It is assumed that all phonons are diffusively scattered at the interface, that is, an incident phonon is reflected and transmitted into all phonon states with the same energy. Experimental data of the phonon Kapitza thermal conductance are usually somewhere between the values predicted by these two limiting models. Typically, at room temperatures, KBp p are on the order of 106 to 109 W/(m2K) (Swartz and Pohl, 1984; Goodson et al., 1995; Lee and Cahill, 1997; Stoner and Maris, 1993). Theoretical calculation (Chen, 1998) has provided an improved understanding of the mechanisms controlling the phonon transport across boundaries and in superlattices. Very little can be said in general about the electronic contribution to the heat transfer across the junction, because it strongly depends on the mechanisms of electronic transport through the boundary. However, since electrons carry both heat and electricity, it seems obvious that the thermal electronic conductance KBe e is related to the electrical boundary conductivity a B. Indeed, as discussed in the next section, these electrical components obey a Wiedemann-Franz-type law (Mahan and Bartkowiak, 1999). The situation becomes more complicated when phonons on one side of an interface are coupled directly to electrons on the other side. An example of such phenomenon was provided by a series of experiments of the heat transport across the interface between diamond and several metals (Stoner et al., 1992). In some cases, the measured boundary thermal resistance was in reasonable agreement with the prediction of the acoustic mismatch model. However, for the case of very soft metals, such as lead, for which the phonon mismatch with diamond is even greater, the measured heat flow was up to 100 times larger than expected from the acoustic mismatch model. This discrepancy has been explained by taking into account the direct energy transfer between electrons at the metal temperature and joined vibrational interfacial modes at the diamond temperature (Huberman and Overhauser, 1994). Therefore, we will explicitly consider three kinds of thermal boundary conductances: (1) the usual phonon Kapitza conductance KBp p, which describes heat transfer between phonon subsystems on both sides of the interface, (2) the electronic boundary conductance KBee, which describes the heat carried across the boundary by electrons, and (3) the mixed conductances KBe p and KBp e corresponding to the direct coupling between electrons (phonons) on one side of the boundary and phonons (electrons) on the other side, as in the case of lead and diamond just described. The total thermal boundary conductance is thus defined as K B = K~pp + K~p~ + KBe p "1- gBe e.

Electrical boundary conductivity a B is determined by details of the potential barriers at the junctions and on the transport mechanism. A1-

250

M. BARTKOWIAKAND G. D. MAHAN

though in the description of heat and electricity transport in heterostructures presented hereafter, we consider aB as ohmic, in general, it can depend on the current density flowing through the barrier. In macroscopic thermoelectric devices, electrical contact resistivity PB = 1lab between the semiconductor and metal electrodes has been reported typically between 10 -9 and 1 0 - S f ~ m 2 (Rowe and Bhandari, 1983). This is in case when standard techniques of making a junction (such as soldering or hot pressing) are used. Of course, the presence of the resistance at the contacts reduces the efficiency of the device, as there is an extra Joule heating at the junctions. This effect must be taken into account when designing a thermoelectric device. The condition PB << L/a, where L is the length of the semiconductor, is usually used to ensure that contact resistance does not degrade the device performance. If the junction contact resistivity was indeed 10 -9 ~ m 2, using this criterion for a sample with the bulk conductivity a of order of 105 (f~m)-1 would lead to the conclusion that the sample size must be significantly larger than 0.1 mm. Here we are interested in heterostructures with semiconducting layers of much lower thickness. However, samples with micron or submicron thickness have to be produced using thin-film growth techniques (such as evaporation, MBE, CVD, or laser ablation), and the contact resistances are expected to be much smaller. In fact, as reported by Venkatasubramanian and Colpitts (1997), in thin-film short-period Bi2Te3/ Sb2Te 3 and Si/Ge CVD-grown superlattice structures, the boundary resistance between layers was so low that it could not be measured (i.e., the resistivity of the structure in the direction perpendicular to the interfaces was about the same as that in the direction parallel to the interfaces). Moreover, the structure exhibited quite ohmic behavior even for very large current densities. The usual thermoelectric (Seebeck and Peltier) effects, though making frequent references to the junctions between two materials, are related to differences in their bulk properties and are not dependent on the detailed nature of the junctions. The boundary Seebeck coefficient SB introduced previously, on the other hand, is related to the thermoelectric effects that are specific to the junctions. Such boundary thermoelectric effects for tunneling through oxide barriers between metals were first derived and measured by Smith et al. (1980). Since then, they have attracted little attention, despite the fact that such effects may play an important role in the development of more efficient thermoelectric devices. A more general thermodynamic analysis of interfacial transport and more details of the thermoelectric effect at tunneling junctions were given by Johnson and Silsbee (1987). Within the context of thermionic cooling and power generating devices, besides tunneling, we have considered boundary impedances related to the thermal excitations of electrons over a barrier (Mahan et al., 1998; Mahan and Bartkowiak, 1999). We discuss some of these results in details in the following section.

8

HEAT AND ELECTRICITY TRANSPORT THROUOH INTERFACES

III.

251

Wiedemann-Franz Law at Boundaries

Let us first consider boundary impedances due to a thin boundary that can be modeled as a potential barrier with a rectangular shape. If the barrier is thin enough, the electrons can tunnel through it with a higher probability than they can be thermally excited over it. For simplicity assume that the junction is symmetric, with the same material on both sides. We follow the general derivation as used in the tunneling Hamiltonian (Mahan, 2000), to get for the case in which there is a voltage V

j

d3k 3 VzJ-[nz(e, ) - nF(e + eV)] J = 2e " (2~) de = 2

d3k

(2rt)~ Vz(e - [ 2 ) Y [ n F ( g ) -- nF(C, -}- eV)].

(10)

(11)

The difference in Fermi functions can be expanded to first power in the voltage,

nF(e) -- nF(e + e V) = e V

-- de,I"

(12)

The tunneling probability 3- is small and is assumed to be dominated by an exponential dependence upon the energy ~ of the electron as measured from the chemical potential: ~ ( e ) = ~(#)exp{A~), where the constant A = 1/x//EdUo, E d = hZ/2md 2, and where d is the barrier thickness. Tunneling occurs only if A k B T << 1. Otherwise, the electron is thermally excited over the barrier and one uses the corresponding results for thermionic emission discussed later. This inequality sets the maximum value of d for tunneling to occur. It is now easy to evaluate the preceding expressions for the currents and derive the boundary resistances e2m

a B - 27r2Ah3 Y SB =

KBe e = ~

-~Ak~r

~r

K B e e -- ~O00"BY

(13) (14)

(15)

(16)

252

M. BARTKOWIAKAND O. D. MAHAN

with

~o =--~

1 --~(AkBT) 2 .

(17)

The thermal boundary conductivity is given by the boundary form of the Wiedemann-Franz law (Wiedemann and Franz, 1853; Ziman, 1960) with the Lorentz number 50o close to the quantum bulk Lorentz number, 7~2/3, by our assumption of tunneling ( A k B T << 1). For the case of a metal-oxidemetal junction, the boundary Seebeck coefficient given by Eq. (14) can be estimated explicitly. Using a typical value for A of 3 eV-1, one finds at T = 300 K that SB ~ 20 ~V/K. As compared with bulk Seebeck coefficients of the best thermoelectric materials, this is not a very high value, but neither is it one that can be simply ignored. These results for tunneling junctions are in agreement with those presented by Smith et al. (1980) and by Johnson and Silsbee (1987). Next, consider the case that the barrier is too thick for tunneling. In this case, the dominating transport mechanism is thermionic emission. We earlier derived the boundary impedances for the thermal excitations of electrons over a barrier (Mahan et al., 1998). They are given in terms of Richardson's equation, where A* ~ 120 A/(cm 2 K2), and T is the average temperature in the barrier: (18)

JR = A*T2e-V~ e JR aa = k a T

(19)

1

(20)

SB =~--~ [Uo + 2kaT] KBe e -- 2kBJ------~n= 2%T

.

(21)

e

Here Uo is the barrier height, as measured from the chemical potential. Both Richardson's equation and the preceding results for boundary impedances are valid only when U o is large enough (say, at least 2 k a T ). Equation (21) is again a boundary form of the Wiedemann-Franz law. The constant 5~ = 2 in this case, since the electrons which are thermally excited over the barrier obey Maxwell-Boltzmann statistics at these high excitation energies. It is important to note that SB is not the difference of the bulk Seebeck coefficients on the two sides of the junction. It is a separate contribution. For the case of the barrier height optimized for the highest efficiency of a single-barrier thermionic refrigerator, U o ~ 2 k a T (Mahan et al., 1998;

8

HEAT AND ELECTRICITY TRANSPORT THROUGH INTERFACES

253

Mahan and Woods, 1998), one gets SB ~ 340/~V/K. This is just a rough estimation, but it indicates that the boundary Seebeck coefficient can be very large and must be taken into account whenever thermionic emission may be an important mechansm of transport across boundaries. Since an electron will tunnel through thin barriers and will be thermally activated over thick barriers, these two examples are generally representative for most physical situations. On the other hand, cases for which barriers cannot be described as rectangular and for which tunneling and thermionic emission are present simultaneously (e.g., Schottky barriers) are still to be worked out. Yet another problem of energy balance in the barrier region and the related problem of partially diffusive transport (Shakouri et al., 1998) has to be addressed for barrier thickness approaching the electron energy relaxation length (which is on the order of a micrometer). This usually applies to barriers produced artificially by using band offset engineering. In this case, however, it is more reasonable to consider the barrier region as a separate segment rather than as a boundary.

IV.

Energy Balance Equations for Electrons and Phonons Out of Equilibrium

In quasielastic scattering of electrons by phonons, energy relaxation time is much longer than momentum relaxation time (Bass et al., 1973; Mahan, 2000). Consequently, the characteristic electron cooling length 2 can be much longer than the electron mean free path. According to the data of Zakordenets and Logvinov (1997), 2 is on the order of a micrometer, at least for some thermoelectric materials. Similar estimation follows from Monte Carlo simulations for GaAs (Shakouri et al., 1998). Because the thermal boundary resistances of electrons and phonons are not the same, these systems can have different temperatures at the boundary and require a distance 2 to equilibrate. In other words, within the distance of the order of from the boundaries the average energy of electrons (or their temperature T~) may differ from the lattice (phonon) temperature Tp. This situation is schematically illustrated in Fig. 1. Let us consider a double-heterojunction structure of a semiconductor film of the thickness L = 2a sandwiched between two identical metallic or semiconducting substrates. The thin film extends - a < x < a and is connected to a cold reservoir at x = - a with one boundary, and to a hot reservoir at x - a through another boundary. In general, the stationary distribution of temperature in a conductor or a semiconductor heated by a flowing current of density J is described by the Domenicali equation (Domenicali, 1954). To deal with the electron and phonon subsystems out of equilibrium, this equation must be modified (Bulat and Yatsyuk, 1984;

254

M. BARTKOWIAK AND G. D. MAHAN

T

he

Thp

7'

i/T

KJI -a

p,e _...,, 7"

-a+k

0

- l

a-~

a

FIG. 1. Schematic temperature distribution (top panel) and heat flux flows (bottom panel) in a double-heterojunction structure working as a refrigerator.

Anatychuk et al., 1987; Gurevich and Logvinov, 1992; Zakordonets and Logvinov, 1997). For the essentially one-dimensional geometry of the considered system, the coupled equations for the heat balance between electrons and phonons take the form d2Te __ K e d x 2 = p j 2 _ P ( Te _

d 2Tp

-- K p d x 2 --

Tp),

P(T~ - Tp).

(22)

(23)

Here p = 1 / a is the bulk electrical resistivity, and the parameter P represents the strength of the electron-phonon interaction. A formula for P for metals was given by Allen (1987). For doped semiconductors, dimensional analysis gives the approximate formula P ~ nekBCO o where n e is the density of

8

HEAT AND ELECTRICITY TRANSPORT THROUGH INTERFACES

255

electrons and coD is the Debye frequency. We regard all quantities such as p, P, Ke, p as constants. Equations (22) and (23) are solved in the standard way (Bulat and Yatsyuk, 1984) by defining a "center-of-thermal-conductivity" temperature KT~tc = K e T e + KpTp, where K = Ke + Kp. The solutions are

X

re = Ta + Tb -L +

pJZ(a2

x2 ) nt- p j 2

2K

1

6ktr 2 + - i ~

(A cosh(Kx) + B sinh(Kx))

(24) x pJZ(aZ-x2) Tp = T,, + Tb -L + 2K

pj2 KK 2

b (A cosh(~cx) + B sinh(Kx)). 1 +----~ (25)

Here tc2 = P ~ K KeK p

~=~ e Kp

(26)

Inverse of the parameter tc can be interpreted as the electron-phonon cooling length 2 discussed earlier. The two differential equations introduce four unknown constants A, B, Ta, Tb. They have to be determined by the boundary conditions for the flow of heat for electrons and phonons at the boundaries. In general, these boundary conditions must include all the interracial phenomena discussed in Section II. In particular, one has to take into account the energy that electrons gain in the electric field across the boundary, that is, one has to consider Joule heating at the boundary. It occurs wherever there is a current J and voltage difference A V = (pBJ -t- SBAT), and the heating is JAV. The boundary electrical resistance is PB = 1laB. We assume that half of this heat ends up on each side of the boundary. For example, if heat is flowing from left to right, then there are three heat currents: the amount (JQL) approaching from the left, the boundary current (JeB), and the amount (JeR) departing on the right. The relationship between them is JeB = JO_L + 89 BJ + SB A T)

(27)

JQR = JeB + -}J(PBJ + SBAT)

(28)

JQR = JeL + J(PB J + S~AT).

(29)

The last equation is obtained by adding the first two. The last equation is used to calculate the heat entering or leaving the device. The boundary Joule heating has not been included in previous theories, although it makes an important contribution. These equations, along with the definition (8),

256

M. BARTKOWIAKAND G. D. MAngy

provide all of the needed information to match the flow of heat and current at the boundary and to write down the general boundary conditions. For the phonon subsystem, we have

-Kp\

dx J+a : + K . p p [ T p ( + a ) - Th/cp] +_ KBpeETp(W_a) - Th/ce].

(30)

Since close to the boundaries in the substrates as well, electrons and phonons can be out of equilibrium, in general, we assume different phonon temperatures T~p(Thp) and electron temperature T~e(Tne) in the cold (hot) substrate. Moreover, this choice allows us to consider direct heat transfer from the electron subsystem in the substrates to the phonon subsystem in the sample, as discussed in Section II. Boundary conditions for the electron subsystem are somewhat more complicated, as additionally they must take into account the bulk Peltier (or Seebeck) effect and the boundary Joule heating:

(dre)

S Ze ( -Jr-a ) a - K e \-~x /l

+a

J -+ -~ [ p BJ +_ S B ( Th /ce - Te( _+ a))]

= jSB Te(+-a) + Th/ce

2

-T- KBee[Th/ce - Te( + a)] -T- KBep [ Th/cp -- Te ( "-~a)]. (31)

These boundary conditions are schematically illustrated in Fig. 1. They provide four linear algebraic equations that have to be solved for the four unknown variables A, B, Ta, Tb in Eqs. (24) and (25). This can be done analytically, but the resulting general expressions are extremely long and complicated, so that we do not present them here.

V.

Thermal Instability

Next we show the existence of a possible thermal instability associated with the boundary impedances. This can be done in a simple fashion without analyzing in detail the general solutions for A, B, Ta and Tb. For simplicity, assume that electrons and phonons in the substrates are in equilibrium (Tcp = T~e = T~, Thp = The = Th), and that there is no direct heat transfer between electron and phonon subsystems across the boundary (KBpe -- KBep = 0). Write the equations determined by the boundary conditions as M Y = C where Y = (A, B, Ta, Tb) is a vector, M is a matrix, and C are the source terms that contain the known constants (material and

8

HEAT AND ELECTRICITY TRANSPORT THROUGH INTERFACES

257

boundaries parameters, T~, Th, and J). The determinant of the matrix M has the form det M

=

D o - - j s.2D 2

(32)

J Js = K-ee

-- SB),

(33)

where the constants Do, D2 are both positive. So the determinant is zero when the current has a value given by

Js = --

/•0

-.

(34)

2

When the determinant vanishes, the quantities A, B, T,, Tb diverge. Clearly, this signifies an instability. Our first thought was that it was a thermal runaway caused by Joule heating. However, Joule heating is caused by the terms p j2 or pBJ 2, which are not in the determinant, so that the instability is not caused by Joule heating. It seems to be a new type of thermal instability. The cause of the instability is revealed by the dimensionless parameter j~. It depends on an interesting combination of bulk parameters (e.g., S) and interface parameters (e.g., KBee, SB). The numerator ~ J(S - SB) is the rate (JS) at which heat is sent toward the interface by the current minus the rate (JSB) at which it is pumped through the boundary by the boundary thermoelectric effect. The difference is the rate at which it accumulates at the boundary. The denominator contains the thermal boundary conductivity KBe e for electrons, which measures the ease with which this energy diffuses through the boundary. Low values of KBe e make it hard for the energy JS to get through the interface. So the instability is due to inability of the system to transport through the interface the heat coming at it. Energy accumulates in the system, which causes overheating. Since every material has these coefficients, this instability could exist at every interface. However, in thermoelectrics the Seebeck coefficient is relatively large, which makes it occur at lower current densities. It is interesting to examine how the instability varies with the length L = 2a of the system. The ratio Do/D 2 is of order unity in both limits of a/2 >> 1 and a/2 << 1. So this ratio has no great variation with sample length. Nevertheless, the instability is more important for short devices. This comes from the current density J. The voltage scale of a thermoelectric device (Rowe and Bhandari, 1993; Goldsmid, 1986) is set by A V ~ S A T ~ IR, which is independent of thickness. However, the resistance R = pL*/(Area), where L* - L + 2apB. Thus, if one chooses the current density J to operate at the value of maximum efficiency, it has the approximate value of

258

M. BARTKOWIAKAND G. D. MAHAN

JM ~ SAT/pL*, which increases as L* becomes smaller. Short devices have smaller resistances and take larger currents to operate at maximum efficiency. Then values ofjs are larger and one approaches the instability point. In actual practice, the boundary form of the W i e d e m a n n - F r a n z law prevents the occurrence of the instability. Note that js depends upon KBe e, whereas JM depends upon PB. The constraint on the product KBeeP B -~oT(kB/e) 2 prevents JM from reaching the singular point given by js ~ 1. Our computer solutions only found an instability when we took values of KBe e to be larger than proscribed by the boundary form of the WiedemannFranz law.

VI. Effective Thermoelectric Properties Efficiency of a bulk thermoelectric device is determined by the figure of merit Z T (Eq. (5)) of the thermoelectric material. For the heterostructure considered here, the device includes the boundaries and, at least when the thickness of the film is small, its efficiency is influenced by the boundary effects. The same applies to other apparent (measured) thermoelectric properties of the heterostructure--they are altered by boundary phenomena. Here we define and calculate the effective thermoelectric properties of the whole device. We consider the substrates as external terminals of our device and assume that they are in internal equilibrium, so that T~p = T~e = Tc and Thp = The ~-- r h.

1.

EFFECTIVETHERMAL CONDUCTIVITY

The effective thermal conductivity of the device at zero current, K*, can be calculated directly from the information on the heat conduction contained in the boundary conditions (30, 31). If we turn off the current, and let the heat flow freely from the hot reservoir to the cold one, then the heat flux across the sample can be calculated a s Joh(J = 0 ) , where JQh, the total heat flow from the hot junction, is given by the sum of the phonon and electron contributions, JQh -- JQhp -'[- JQhe

= Kp \-~xJ~ - STe(a)J + Ke \ - ~ x ] a -- -2 [PBJ + SB(T h -- Te(a)) ].

(35)

When the heterostructure is working as a refrigerator, what matters is the rate of heat removed from the cold source JQc, which again can be

8

HEAT AND ELECTRICITY TRANSPORT THROUGH INTERFACES

259

calculated from Eqs. (30, 31) as JQc : JQcp + JQce

(dre)

q- S Te ( - a )J - K e ~k-~X / _ a

= -Kp\dxJ_a

J 2 [PBJ -- S B ( T c - Te(--a))]"

(36)

Of course, Jo.h = -- Jo.c when J = 0. We can now apply the general definition of the thermal conductivity and calculate K* as K*

-

LJQh(J

= 0) .

(37)

L-Z The result is K* = K

a(KKn&c coth(atc) + KBeKBp (1 + 6) 2) 0 + K b x coth(aK)(K + aKB) + KKBpb 2'

where KBe , KBt,, K B and |

(38)

are defined as KBe : KBe e + KBep

(39)

KBp : KBp p -Jr- KBp e

(4o)

KB = KBp p + Knv e + KBe p -'F KBe e

(41)

| = Kt~e(K + a(1 + 6)2Knp).

(42)

Formula (38) takes into account the boundary phenomena, the fact that both electrons and phonons carry the heat, and the fact that they can be out of equilibrium. The effective thermal conductivity depends on the thickness of the film and is less than K. The reason for this is well known: the thermal boundary resistances reduce the thermal conductivity (Swartz and Pohl, 1989; Goodson et al., 1995; Lee and Cahill, 1997; Stoner and Maris, 1993; Chen, 1997, 1998; Lee et al., 1997; Venkatasubramanian, 2000; Simkin and Mahan, 2000). The fact that electrons and phonons are out of equilibrium causes the appearance of the length scale 2 = 1/~: in Eq. (38). Only when the thickness of the film is much longer that 2, i.e., a• ~ oo, does one recover the standard expression for the effective thermal resistivity, 1

K*

1 -

~

Kp + K e

1 ~

aK B

.

(43)

260

M. BARTKOWIAKAND G. D. MAHAN

~176 0.4 , ~ / ~ . , /

9

0~ ~,,'/ 0.0 1.0

10" W/(m'K)

~ = 10. W/(m,Ki '

I

..... .

~

'

gs"

.

-.

~

.........

o.V

1

o.o / 4.0

,

,

---

. . . . T --

J

3.0 ~

(c)

1.0

o.o ............:...... i 0

0.2

i

i

0.4

L [~m]

0.6

~ ..........

1

0.8

1

:

FIG. 2. Effective t h e r m a l c o n d u c t i v i t y (a), Seebeck coefficient (b), and figure of merit (c) of a BizTe 3 thin film. T h e results, n o r m a l i z e d by the c o r r e s p o n d i n g values for bulk BizTe 3, are presented as a function of the thickness of the film for several values of the electronic t h e r m a l b o u n d a r y c o n d u c t i v i t y KBe.

It is seen that as the thickness of the film L = 2a ~ 0, then also K* ~ 0. In fact, the same conclusion follows also directly from Eq. (38). However, for 2 > 0, this is true only when both Kse and KBp are finite. Otherwise, if for example KBe ~ oo and KBp was finite, phonons would transfer their energy to the electrons, which in turn would carry the heat across the junction, so that the phonon thermal boundary resistance 1/(aKnp) alone could not stop the heat conduction. These results are illustrated in Fig. 2a for the case of a BizTe3 thin film, where the effective thermal conductivity K* of the film relative to the bulk thermal conductivity of BizTe 3, K = 2 W/(mK), is plotted as a function of the film thickness for several values of the electronic thermal boundary conductivity KBe. Here, the phonon thermal boundary conductivity is taken a s KBp= 5 • 1 0 6 W/(m2K) and the cooling length as 2 = 0.56 #m (Zakordonets and Logvinov, 1997). The other bulk thermoelectric parameters of

8

HEAT AND ELECTRICITY TRANSPORT THROUGH INTERFACES

261

Bi2Te 3 are typically S = 230 #V/K, tr = 1.1 • 105 (f~m)- 1, Kp = 1.5 W/ (mK), and g e = 0.5 W/(mK), which lead to Z T = 0.88 at T = 300 K (Rowe and Bhandari, 1983; Goldschmid, 1986; Slack, 1995). As long as both KBe and KBp are finite (which is indeed true in reality), effective thermal conductivity can be made arbitrary small just by reducing the thickness of the semiconducting sample. Unfortunately, as discussed hereafter, this does not mean that the effective figure of merit can be made arbitrary large. 2.

EFFECTIVESEEBECK COEFFICIENT

According to the general definition of the Seebeck coefficient (Eqs. (1) and (6)), it can be calculated from the total voltage drop across the device A V for an open circuit, AV S* = ~ .

L-Z

(44)

For the heterostructure considered here, A V is the sum of the voltage across the semiconducting thin film due to the bulk Seebeck effect and the voltages across the boundaries produced by the boundary Seebeck effect, A V = SBETh -- Te(a)] + SETe(a ) - T e ( - a ) ] + S B E T e ( - a ) - T~], (45)

so that the effective Seebeck coefficient of the device can be written as S* = SB + Z*(S - SB)

(46)

E * = Te(a) - Te(-a).

(47)

These simple results lead to an interesting general conclusion. It may seem that the boundary Peltier and Seebeck effects work in pretty much the same way as the bulk ones. Moreover, since both S and SB can be either positive or negative and ISBI can in principle be as large as ]S], one could expect that the boundary thermoelectric effects can either amplify or weaken the bulk ones, depending on a specific combination of the signs and values of S and S~. This, however, is not true. Since E*~< 1 (for J = 0, the temperature difference Te(a) - T e ( - a ) cannot be larger than Th - T~), Eq. (46) implies that aS*] ~< max(]S], ]SB]). In other words, bulk and boundary thermoelectric effects cannot work in parallel (or, rather, in series). In fact, the highest effective Seebeck coefficient is obtained when only one type of thermoelectric effects is p r e s e n t - - t h e one with the higher Seebeck coefficient--whereas the influence of the other type is minimized or eliminated.

M. BARTKOWIAKAND G. D. MAHAN

262

Using the solution (24) and the boundary condition (31) for the case of J = 0, we arrive at the final expression E* =

O + aKKBbK coth(a~:) - KKBp6 O + KfK coth(aK)(K + aKB) + KKBpb 2"

(48)

Just as in the case of the effective thermal conductivity, one can verify that as the thickness of the film L = 2a ~ 0, then also Z* ~ 0, except for the case of isothermal boundary conditions for the electronic subsystem (i.e., when KBe ~ ~ ) , for which E* = 1. This behavior is not caused by the fact that electrons and phonons are out of equilibrium. Taking the limit 2 ~ 0 leads to E* =

aKB K +aKn'

(49)

but does not alter the preceding conclusion. Blocking the electronic heat transfer at the boundaries causes the temperature distribution in the thin film to flatten (i.e., Te(a) - Te(-a ) decreases), and consequently the apparent bulk Seebeck effect is reduced. This becomes more important as the thickness of the film decreases, since this is when the heat transport is dominated by the boundary thermal resistances. Then, according to Eq. (46), S* ~ SB, and we are left with the boundary thermoelectric effect alone. The effective Seebeck coefficient for a BizTe 3 thin film as a function of the film thickness is plotted in Fig. 2b. Here we have assumed that the boundary Seebeck coefficient SB is negligible, so that E* = S*, and we have used the same values of the other thermoelectric and boundary parameters as in the calculation of the effective thermal conductivity of the preceding subsection. It is seen that when the electronic thermal boundary conductivity KBe ~ ~ , that is, when electrons can cross the boundary without any resistance, S* = S. Otherwise, S* ~ 0 for L ~ 0.

3.

EFFECTIVEFIGURE OF MERIT

According to Eq. (5), as L ~ 0, the effective figure of merit of the device, Z*, should follow the behavior of S*2/K *. Using Eqs. (38) and (48) one can show that S*2/K * ~ O, except for the case of isothermal electronic boundary conditions. This, in general terms, explains why reducing the thickness of the semiconducting sample and making the effective thermal conductivity small does not necessarily lead to a device with large effective figure of merit. However, we need to be more rigorous, as there is still one more factor in the expression (5) for Z T - - e l e c t r i c a l conductivity. Moreover, it is not obvious that using Eq. (5) and simply replacing the bulk thermoelectric

8

HEAT AND ELECTRICITY TRANSPORT THROUGH INTERFACES

263

parameters by the effective ones is justified. The meaning of Z* defined in this way would not be quite clear. Therefore, instead of trying to define properly the effective electrical conductivity (for example, in a way similar to that of Gurevich and Logvinov, 1992), we use a more straightforward method. Just as in the standard theory of thermoelectricity (Rowe and Bhandari, 1983; Goldsmid, 1986; Slack, 1995), we calculate the efficiency of our heterostructure device applied both as a power generator and as a refrigerator. Then we maximize the efficiencies with respect to the current density flowing through the devices, and calculate the maximum efficiency (~max) of a power generator and the coefficient of performance (COP) of a refrigerator. These two quantities are expressed in the standard form as

(Dmax

Th-T ~ y-1 Th ? + Tc/Th,

T~ y - T h / T c C O P - - Th_ T~ 7 + 1

(50)

with y=N/1 +zTh+

T~

.

(51)

Finally, knowing (/}max and COP, we solve Eqs. (50) and (51) for Z and call it Z*. In general, the formulas for the efficiency including boundary phenomena are too complicated to have an analytical expression for (/)max and COP. Therefore, we calculate Z* numerically separately for a power generator and for a refrigerator. Results for these two cases always appear to be approximately the same, providing an independent check of the numerical procedures. Z* calculated in this way still depends on Th - Tc (although very weakly), so it is not quite universal. However, it provides a convenient way to represent the results and to judge the device performance. Figure 2c shows the effective figure of merit of the same thin-film Bi2Te 3 device as that described in the preceding subsections. We take KBp = 5 • 106 W/(m2K), and we assume that there are no boundary thermoelectric effects (S~ = 0) and that there is no electrical boundary resistance (pn = 0). For the case of nonisothermal boundary conditions for the electron subsystem (for finite Kse ), the latter assumption is artificial, as P8 and KBe must satisfy the boundary Wiedemann-Franz law. However, here we want to single out the effects of the thermal electronic boundary conductivity KBe, and we will consider the effects of PB separately. It is clear from Fig. 2c that (as expected from the corresponding data for S*) the presence of electronic thermal resistance at the boundaries considerably limits the maximum Z* that can be achieved. In fact, for KBe small enough, there is no enhancement of the figure of merit at all. Moreover, the degradation of the device performance due to the boundary electrical thermal resistance becomes

264

M. BARTKOWIAKAND G. D. MAHAN

particularly important as the thickness of the film decreases. On the other hand, for large K ~ , as the thickness of the film decreases, one can expect a significant enhancement of the effective figure of merit. The maximum enhancement of the figure of merit is obtained for isothermal electronic boundary conditions (Kse = oo). Note that for this particular case, the assumption of PB = 0 is justified. We simply let electrons flow through the boundaries without any resistance, that is, we assume that the electronic contacts are perfect. In this case, the calculations can be performed analytically, and the result is simply $2o z* =

(52)

K*'

where K* can be calculated from Eq. (38) by taking the limit KBe ~ o0. Clearly, the obtained enhancement of Z* is caused entirely by the reduction of the effective thermal conductivity. It is interesting to examine it in more detail. We express the final result for K* in terms of the effective thermal resistivity, 1

1

1

K * = Kp 4- K e -1- aKBp(1 + ~)2 + aK6tc coth(aK) "

(53)

It is the sum of the bulk thermal resistivity of the sample (given by the first term in Eq. (53)) and the boundary contributions (represented by the second term). The boundary thermal conductivity in the denominator of the second term is the sum of the phonon contribution aK~p(1 + 6) 2 and the electronic contribution aK6x coth(ax). The fact that the latter is finite is somewhat surprising, because, according to our assumptions, the electrons can carry heat across the junction quite freely (KBe = ~). However, not only can the phonon heat pass the boundary directly (because of finite KBp), but also it can flow via the electron subsystem. Part of the phonon energy is transferred to the electron subsystem because of the electron-phonon coupling and is taken across the junction by the electrons. The electronic contribution to the boundary thermal conductivity can be minimized by reducing the size of the sample, but it becomes dominating as a becomes large compared to the cooling length 2 = 1Ix. This result is easy to understand. Because of the nonequilibrium, the energy transfer between electrons and phonons is very small at a distance shorter than 2 from the boundaries, so that the extra phonon heat flow across the junction is eliminated. The obtained result that the Kapitza phonon resistances at the junctions cause a reduction of the effective thermal conductivity of small samples and leads to the enhancement of Z* may seem obvious at the first sight. However, as shown here, the mechanisms involved are in fact fairly

8

HEAT AND ELECTRICITY TRANSPORT THROUGH INTERFACES

265

complicated. The electron and phonon subsystems must be out of equilibrium at a finite distance from the junction in order for the phonon thermal boundary resistance to have any effect at all. (Note that when the cooling length 2--.0, the second term in Eq. (53) vanishes, i.e., the boundary resistance disappears!) Otherwise, phonons, being in equilibrium with electrons, would transfer their energy to the electron subsystem, which in turn would carry the heat across the junction, so that the Kapitza resistance would play virtually no role in the heat conduction. The maximum enhancement of the figure of merit, Z*ma x ~-- Z(1 + 1/6), is obtained from Eqs. (52) and (53) by taking the limit a ~ 0, that is, when the thickness of the film tends to zero. An interesting point here is that this is true for any finite value of Ksp. Note that since 6 = Ke/K p, Z*ax = Z(Kp + Ke)/Ke = S 2 o ' / K e , that is, it corresponds to the case when the heat back flow mediated by phonons is eliminated completely. Nonzero electronic thermal boundary resistance, according to the boundary Wiedemann-Franz law, is always accompanied by the boundary electrical resistance PB, which inevitably degrades the performance of the device. However, the influence of PB on the effective figure of merit is not critical. It is important for the case of small KBe (because then pn is correspondingly high), but in this case the performance of the device is degraded anyway because of the reduction of the effective Seebeck coefficient. On the other hand, when KBe is large enough to get an enhancement of Z*, the presence of p~ that satisfies the boundary Wiedemann-Franz law does not change the results very much. This is illustrated in Fig. 3, where the ratio Z*/Z for our BizTe 3 thin film device is plotted for the case of KBe = 108 W / ( m Z K ) and for PB chosen to satisfy the boundary

2.0 1.5

II

'.~Ii/ 0.5 ~

o.ol ......... 0

0.2

OB = 0

0.4

L [l.tm]

0.6

w-F

0.8

1

FIG. 3. Enhancement of the figure of merit as a function of the thickness of a BizTe 3 film for the boundary electrical resistivity PB satisfying the boundary Wiedemann-Franz law (solid curve), and for p n - - 0 (dashed curve). The electronic thermal boundary conductivity is taken as Kne = 108 W/(mZK).

266

M. BARTKOWIAKAND G. D. MAHAN

Wiedemann-Franz law. The dashed curve, presented here for comparison, corresponds to the vanishing boundary electrical resistance. The general conclusion that follows from the presented analysis is that three conditions must be satisfied simultaneously in order to get a significant enhancement of Z* as the thickness of the film decreases: (1) There must be high phonon Kapitza resistivity 1/K~p at the boundaries, (2) the thermal electronic boundary resistivity 1/Kse and the associated electrical boundary resistivity PB must be as small as possible, and (3) the electron and phonon subsystems must be out of equilibrium and the cooling length 2 of the material should be as long as possible.

Vll.

Superlattices

It is relatively straightforward to extend the approach introduced in Section IV to the case of a superlattice. Electron and phonon temperature distributions in each layer (segment) of the superlattice are still described by the Domenicali equations (22) and (23), and their solutions have the form of Eqs. (24) and (25). Also, the boundary conditions for each segment are still given by Eqs. (30) and (31). However, phonon and electron temperatures of the cold and hot reservoirs (T~p, Thp, rce , The ) for a given segment are determined by the solutions of the Domenicali equations in the neighboring segments, as illustrated schematically in Fig. 4. Of course, this introduces self-consistency into the problem, and one ends up with a set of 4N linear equations (where N is the number of layers in the superlattice) for the electron and phonon temperatures at the left and right boundaries of each segment. These equations have to be solved numerically, and the solutions are used to determine the global distributions of the temperatures in the superlattice, as well as its effective thermoelectric properties. In the latter

FIG. 4. Schematicdistributions of electron and phonon temperatures in a superlattice.

8

HEAT AND ELECTRICITY TRANSPORT THROUGH INTERFACES

267

calculation, we use the same methods as those applied in the preceding section to a double-heterojunction structure. We calculate the effective thermal conductivity, the effective Seebeck coefficient, and the effective figure of merit of various superlattices. As an example, let us first consider a 400 ~ / 2 0 0 ~ BizTe3/Sb2Te 3 superlattice with 11 layers, similar to those described by Venkatasubramanian and Colpitts (1997) and Venkatasubramanian (2000). Values of the thermoelectric parameters of Bi2Te 3 used in the numerical calculation have been listed in Subsection 1 of Section VI. For Sb2Te 3, we take S = 90 #V/K, a = 3.3 x 105 (f~m)-~, Kp - - 0.4 W/(mK), and g e = 1 W/(mK), so that Z T at 300 K = 0.58 (Rowe and Bhandari, 1983; Goldsmid, 1986; Slack, 1995). The cooling length of Sb2Te 3 is 2 = 1 pm (Zakordonets and Logvinov, 1997). We assume that interface scattering of phonons at the boundaries between Bi2Te 3 and Sb2Te 3 leads to a considerable phonon thermal boundary resistance, so that KBp p = 5 x 106 W/(m2K). The electronic subsystems, on the other hand, are assumed to be in very good contact across the boundaries, in agreement with experimental data of Venkatasubramanian and Colpitts (1997). We take KB~e = 109 W/(m2K), and PB satisfying the boundary W i e d e m a n n - F r a n z law. Finally, we have assumed that there is very little direct energy transfer between electrons on one side of the interface and phonons on the other, Knp e = KBe p = 1 0 3 W/(m2K), and that the boundary Seebeck coefficient is very small, SB = 1 pV/K. The device is sandwiched between reservoirs that are assumed to be in internal equilibrium. The cold reservoir is kept at Tc = 295 K and the hot reservoir at Th = 305 K. With these assumptions, one can calculate the effective thermoelectric properties of the superlattice as K* = 0.8 W/(mK), S* = 204/~V/K, and Z * T at 300 K = 1.75. Thus, the superlattice exhibits a considerable enhancement of the figure of merit due to the reduction of thermal conductivity. The effective Seebeck coefficient, although reduced as compared with bulk Bi2Te3, remains larger than that of Sb2Te 3. Distributions of the temperatures in the superlattice for the current density J = 5 x 108 A/m 2, for which the device applied as a refrigerator works at the maximum efficiency, are shown in Fig. 5. It is seen that electrons and phonons are strongly out of equilibrium everywhere. Surprisingly, the electronic temperature gradients in the Sb2Te 3 layers appear to be pointing in the wrong direction, so that the bulk Peltier effect in these layers actually pumps heat in the direction opposite to the desired directions. Nevertheless, the device remains quite efficient, exhibiting Z * T = 1.75. Finally, in Fig. 6, we present the calculated effective thermoelectric parameters of another Bi2Te3/Sb2Te 3 superlattice with 55 layers, for which the layers are all of the same thickness L. The results are plotted as a function of L. They exhibit general features similar to those of the Bi2Te 3 thin film considered in the preceding section and presented in Fig. 2. The exception is the effective Seeback coefficient. As the thickness of the layers

268

M. BARTKOWIAK AND G. D. MAHAN

304

302

300

298

/

/'---

> ,

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0.1

0.4

.

O.5

.

.

.

' "

.

0.7

06

x [/.tm]

FIG. 5. Distribution of the electron temperature (solid curves) and the phonon temperature (dashed curves) in a superlattice working as a refrigerator at maximum efficiency.

2.0"

'

1.51 ~

K(gi2Tea) = 2 W/(mK)

~_ 1.0

'

'

'

!

(a)

b2Te3) = 1.4 W/(mK)

~" 0.5 .

0.0

.

i

.

.

.

w

.

.

.

|

leO ~

i

(b)

170 S(Bi2Tea) = 230 I~V/K

160

S(Sb2Te3) = 90 u.V/K

150

' 9

'

'

r

-

'

L

(c)

1.4 1.2 I~11.0

8

0.60"8I~../ZT(Sb2Tea ) =0.58 |

0.0

,

i

0.5

110 L [gm]

115

210

FIG. 6. Effective thermal conductivity (a), Seebeck coefficient (b), and figure of merit (c) as a function of layer thickness L for a L/L BizTe3/SbzTe 3 superlattice with 55 layers.

8

HEAT AND ELECTRICITY TRANSPORT THROUGH INTERFACES

269

increases, the effective Seeback coefficient approaches the average of the bulk coefficients of BizTe 3 and SbzTe 3. On the other hand, as L decreases, the nonisothermal electronic boundary conditions cause a less rapid reduction of the effective Seebeck coefficient of BizTe 3 layers than that of SbzTe 3 layers. As a result, S* exhibits a weak maximum for a submicron thickness of the layers. The results for the effective thermal conductivity are consistent with experimental data of Venkatasubramanian (2000). However, the experimentally observed minimum of K* for a very small superlattice period 2L 50 A cannot be explained using our model. This phenomenon is related to the crossover between the particle and wave-interference types of heat transport in periodic lattice structures (Simkin and Mahan, 2000), which is not taken into account in the phenomenological description presented here.

VIII. Summary In summary, we have introduced the equations that describe the boundary impedances of a thermoelectric. These equations are quite general and apply to all kinds of boundaries. We have given examples of these impedances for the cases of tunneling and thermionic emission, and found that the electrical components obey a boundary form of the Wiedemann-Franz law. Estimation of the corresponding boundary Seebeck coefficients indicate that S~ for thermionic emission can be as large as bulk Seebeck coefficients of good thermoelectric materials. However, we have shown that the boundary and bulk thermoelectric effects cannot be combined to enhance the effective Seebeck coefficient of a heterostructure or a superlattice. On the contrary, whenever there is a finite electronic thermal resistance at the junctions, the boundary effects reduce the effective Seebeck coefficient as the thickness of the semiconducting layers decreases. On the other hand, as expected, the presence of thermal boundary resistances at the interfaces causes reduction of the effective thermal conductivity. These two effects are combined in the effective figure of merit of the device, and, depending on values of the boundary impedances, may lead either to its enhancement or to its reduction. We have found that the maximum enhancement can be achieved for submicron thickness of the layers when the boundaries block the phonon heat flow more efficiently than the electronic heat flow and when the two subsystems are out of equilibrium. Electrical boundary resistances at the junctions degrade the performance of the device, but its influence on the effective figure of merit is not very important as long as p~ satisfies the boundary Wiedemann-Franz law. Finally, in solving the equations for the transport of heat through a device with a boundary at each end, we find a new type of thermal instability associated with the boundary. This instability should be present in short thermoelectric devices.

270

M. BARTKOWIAK AND G. D. MAHAN

ACKNOWLEDGMENTS

Research support is acknowledged from the University of Tennessee, from Oak Ridge National Laboratory managed by Lockheed Martin Energy Research Corp. for the U.S. Department of Energy under contract DEAC05-96OR22464, and from a Research Grant No. N00014-97-1-0565 from the Applied Research Projects Agency managed by the Office of Naval Research.

REFERENCES P. B. Allen, Theory of thermal relaxation of electrons in metals, Phys. Rev. Lett. 59, 1460 (1987). L. I. Anatychuk, L. P. Bulat, D. D. Nikirsa, and V. G. Yatsyuk, Influence of size effects on the properties of cooling thermoelements, Fiz. Tekh. Poluprovodn. 21, 340 (1987) [Soy. Phys. Semicond. 21, 206 (1987)]. F. G. Bass, V. S. Bochkov, and Yu. G. Gurevich, Current-voltage characteristics of bounded semiconductors (review), Soy. Phys. Semiconductors 7, 1 (1973). L. P. Bulat and V. G. Yatsyuk, Thermal effects at boundaries of solids, Fiz. Tekh. Poluprovodn. 18, 615 (1984) [Soy. Phys. Semicond. 18, 383 (1984)]. G. Chen, Size and interface effects on thermal conductivity of superlattices and period thin-film structures, J. Heat Trans.--Transactions ASME 119, 220 (1997). G. Chen, Thermal conductivity and ballistic-phonon transport in the cross-plane direction of superlattices, Phys. Rev. B 57, 14958 (1998). C. A. Domenicali, Stationary temperature distribution in an electrically heated conductor, J. Appl. Phys. 25, 1310 (1954). S. J. Fonash, Current transport in metal semiconductor contacts - - a unified approach, Solid State Electron. lfi, 783 (1972). J. Goldsmid, Electric Refrigeration (Pion, London, 1986). K. E. Goodson, O. W. K~iding, M. R6sler, and R. Zachai, Experimental investigation of thermal conduction normal to diamond-silicon boundaries, J. Appl. Phys. 77, 1385 (1995). Yu. G. Gurevich and G. N. Logvinov, Thermo-emf and thermoelectric current in unipolar semiconductors with finite dimensions, Fiz. Tekh. Poluprovodn. 26, 1945 (1992) [Soy. Phys. Semicond. 26, 1091 (1992)]. L. D. Hicks and M. D. Dresselhaus, Effect of quantum-well structures on the thermoelectric figure of merit, Phys. Rev. B 47, 12727 (1993). L. D. Hicks, T. C. Harman, X. Sun, and M. S. Dresselhaus, Experimental study of the effect of quantum-well structures on the thermoelectric figure of merit, Phys. Rev. B 53, R10493 (1996). M. L. Huberman and A. W. Overhauser, Electronic Kapitza conductance at a diamond-Pb interface, Phys. Rev. B fi0, 2865 (1994). M. Johnson and R. H. Silsbee, Thermodynamic analysis of interfacial transport and of the thermomagnetoelectric system, Phys. Rev. B 35, 4959 (1987). S.-M. Lee and D. G. Cahill, Thermal conductivity of Si-Ge superlattices, Appl. Phys. Lett. 70, 2957 (1997). S.-M. Lee, D. G. Cahill, and R. Venkatasubramanian, Thermal conductivity of Si-Ge superlattices, Appl. Phys. Lett. 70, 2957 (1997). W. A. Little, Can. J. Phys. 37, 334 (1959). G. D. Mahan, Good thermoelectrics, in Solid State Physics, Vol. 51, ed. H. Ehrenreich and F. Spaepen (Academic Press, 1998), p. 81.

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271

G. D. Mahan, Many-Particle Physics, 3rd ed. (Plenum, 2000). G. D. Mahan and M. Bartkowiak, Wiedemann-Franz law at boundaries, Appl. Phys. Lett. 74, 953 (1999). G. D. Mahan and L. M. Woods, Multilayer thermionic refrigeration, Phys. Rev. Lett. 80, 4016 (1998). G. D. Mahan, B. Sales, and J. Sharp, Thermoelectric materials: New approaches to an old problem, Phys. Today 50, 42 (1997). G. D. Mahan, J. O. Sofo, and M. Bartkowiak, Multilayer thermionic refrigerator and generator, J. Appl. Phys. 83, 4683 (1998). R. E. Peterson and A. C. Anderson, The Kapitza thermal boundary resistance, J. Low Temp. Phys. 11, 639 (1973). D. M. Rowe and C. M. Bhandari, Modern Thermoelectrics (Reston, 1983). A. Shakouri and J. E. Bowers, Heterostructure integrated thermionic cooler, Appl. Phys. Lett. 71, 1234 (1997). A. Shakouri, E. Y. Lee, D. L. Smith, V. Narayanamutri, and J. E. Bowers, Thermoelectric effects in submicron heterostructure barriers, Microscale Thermophys. Eng. 2, 37 (1998). A. Shakouri, C. LaBounty, J. Piprek, P. Abraham, and J. E. Bowers, Thermionic emission cooling in single barrier heterostructures, Appl. Phys. Lett. 74, 88 (1999). CRC Handbook of Thermoelectrics, ed. D. M. Rowe (CRC Press, 1995). M. V. Simkin and G. D. Mahan, Minimum thermal conductivity of superlattices, Phys. Rev. Lett. 84, 927 (2000). A. D. Smith, M. Tinkham, and W. J. Skocpol, New thermoelectric effect in tunnel junctions, Phys. Rev. B 22, 4346 (1980). J. O. Sofo and G. D. Mahan, Thermoelectric figure of merit of superlattices, Appl. Phys. Lett. 65, 2690 (1994). R. J. Stoner and H. J. Maris, Kapitza conductance and heat flow between solids at temperatures from 50 to 300 K, Phys. Rev. B 48, 16373 (1993). R. J. Stoner, H. J. Maris, T. R. Anthony, and W. F. Banholzer, Measurements of the Kapitza conductance between diamond and several metals, Phys. Rev. Lett. 68, 1563 (1992). E. T. Swartz and R. O. Pohl, Thermal boundary resistance, Rev. Mod. Phys. 61, 605 (1989). T. M. Tritt, M. G. Kanatzidis, H. B. Lyon, Jr., and G. D. Mahan, eds.,Thermoelectric Materials--New Directions and Approaches, Vol. 478 (Materials Research Society, 1997). T. M. Tritt, M. G. Kanatzidis, G. D. Mahan, and H. B. Lyon, Jr. eds., Thermoelectric Materials 1998-- The Next Generation Materials for Small-Scale Refrigeration and Power Generation Applications, Vol. 545 (Materials Research Society, 1999). R. Venkatasubramanian, Lattice thermal conductivity reduction and phonon localizationlike behavior in superlattice structures, Phys. Rev. B 61, 3091 (2000). R. Venkatasubramanian and T. Colpitts, Enhancement in figure-of-merit with superlattice structures for thin-film thermoelectric devices, in Thermoelectric Materials--New Directions and Approaches (T. M. Tritt, M. G. Kanatzidis, H. B. Lyon Jr., and G. D. Mahan, eds), Vol. 478. Materials Research Society, 1997. G. Wiedemann and R. Franz, Ann. Phys. 89, 497 (1853). V. S. Zakordonets and G. N. Logvinov, Thermoelectric figure of merit of monopolar semiconductors with finite dimensions, Fiz. Tekh. Poluprovodn. 31, 323 (1997) [Semiconductors 31, 265 (1997)]. J. M. Ziman, Electrons and Phonons (Clarendon, Oxford, 1960), p. 260.

This Page Intentionally Left Blank

SEMICONDUCTORS AND SEMIMETALS, VOL. 70

Index

Adiabatic conditions, 209-210 Adiabatic measurements, 238-239 AgPbBiQ3, 58 A1CuFe, 78, 96-99 AICuRu, 78 A1PdMn, 97, 99-109 A1PdRe, 98, 101 Annealing doped alloys and effects of, 66-68 quasicrystals, 84 Arc melting, 83-84 Atomic displacement parameters (ADPs) applications, 25, 27-33 background information, 2-4 Debye model, 8-11 clathratelike thermoelectric compounds, 14-18 Einstein and Debye temperatures from room temperature, 13-14 Einstein model, 7 elementary theory of, 4-6 filled skutterudites, 14, 25, 27 interpreting, 6-14 lattice thermal conductivity estimations from, 18-26 rare earth hexaborides, 14, 28-30 semiconducting compounds with ice clathrate structure, 14, 30-32 static disorder, 7, 11-13 summary of, 33 ternary tellurides, 14, 28

BasGa16Ge3o, 30-32 Bandgap features inferred from doping studies, 63-64 Bandgap states in VEC, 51-52 Binary skutterudites, 134-142 Biological cooling, 118-119 Bismuth and bismuth-antimony alloys, 226-229 Bloch-Boltzmann kinetic equation, 131 for thermopower, 132 Bloch's theorem, 220 Boltzmann constant, 7 Boltzmann transport theory, 131-134, 220, 221-222 Boundary impedances, 247-250 Bragg reflections, 5-6 Bridgeman method, 87 Brillouin zone, 220

Carrier mobilities, 52-53, 63 Ce-filled skutterudites, 142-145 CeRuGe 3, 32-33 Chalcogen site, doping on, 195-197 Charge density wave (CDW), 180, 201,202 Chemical vapor deposition (CVD), 248 Chevrel phases 162-166 Clathratelike thermoelectric compounds, 14-18

273

274 Coefficient of performance (COP), 263 Cryoelectronics, 120-121, 189-190 Crystal structure, of half-Heusler alloys, 37-39 Curie-Weiss behavior, 54 Czochralski method, 87

Debye model, 8-11 Debye temperatures from room temperature, 13-14 Debye-Waller factor, 6 Defense Advanced Research Projects Agency (DARPA), 118, 120 de Haas-van Alphen effects, 222, 227 Density functional calculations, 128-131 Density functional theory (DFT), 128-129 Density-of-states (DOS), 54 Diffusion thermopower, 93-94 Doped alloys. See Half-Heusler alloys, doped Doping, 230 on chalcogen site, 195-197 on transition metal site, 191-194 Drude model, 222

EDX, 85 Einstein model, 7 Einstein oscillator, 7 Einstein temperatures from room temperature, 13-14 Electrical resistivity, 95-97 Electrical transport, 181-183 Electron diffraction, 85 Electronegativity, 2 Electronic crystals, of half-Heusler alloys, 39-40 Electronic military applications, TE and, 119-121 Electronic refrigeration, 214-217 Electron-ion interaction, 129 Electron microprobe, 85 Energy gaps, 2 Ettingshausen effect, 207, 211,215-216, 217

INDEX Ettingshausen refrigeration, 218, 219 materials surveyed, 223-230

Fermi-Dirac distribution, 220-221 Fermi-Dirac statistics, 220 Fermi distribution function, 131 Fermi surface, 187-188, 222 Figure of merit, thermoelectric, 68-69, 127, 208, 218, 236, 247, 262-266 Filled skutterudites, 14, 25, 27 Fluctuation scattering, 3 Flux techniques, 87-89 Frozen phonon method, 130 Fuzzy logic, 232

Generalized gradient approximation (GGA), 128-129, 130 Grain boundary scattering, 22 Graphite, 225-226 Gruneisen parameter, 19 GW method, 131

H

Half-Heusler alloys, 170-172 crystal structure, 37-39 electronic crystals, 39-40 experimental procedures, 42-43 intermetallic compounds, 41 summary of, 71-72 thermoelectric properties, 40-41 Half-Heusler alloys, doped, 58 annealing, effects of, 66-68 bandgap features inferred from doping studies, 63-64 figures of merit, thermoelectric, 68-69 impurity band transport properties, 64-66 lanthanide alloys, 63 MCoSb, 62-63

275

INDEX MNiSn, 60-61 thermoelectric properties, 66-71 transport properties, 60-63 Half-Heusler alloys, undoped bandgap states in VEC, 51-52 carrier mobilities, 52-53 discovery of new, 50 magnetic properties and band-structure results, 54- 56 MCoSb, 49-50 MNiSn, 43-49 semiconducting and semimetallic properties, 43-51 thermoelectric properties, 56-58 Hall concentration, 131-132 Hall effect, 186-188, 207, 211, 215, 225 Harmonic oscillator, 7 Hartree term, 129 Heat and electricity transport background information, 245-247 boundary impedances, 247-250 energy balance equations, 253-256 figure of merit, 262-266 Seebeck coefficient, 261-262 summary of, 269 superlattices, 266-269 thermal conductivity, 258-261 thermal instability, 256-258 Wiedemann-Franz law, 251-253, 263, 265-266 Heurlinger-Bridgman relations, 214, 218 Heusler compounds, 37-38 HtTe 5 effects of stress and pressure, 184-186 electrical transport in, 181-183 synthesis and structure, 183-184

Icosahedral phase, 93 Impurity band transport properties, 64-66 Inductively coupled plasma (ICP), 85 Inorganic Crystal Structure Database (ICSD), 14 Instability, thermal, 256-258 Isothermal conditions, 209-210 Isothermal measurements, 232-237 Isotope scattering, 22

Kapitza thermal conductor, 248-249, 264265 Kinetic transport theory, 131-134 Kohn-Sham equations, 129 Kohn- Sham gaps, 130-131 Kohn-Sham theory, 129 Kondo systems, 191 Korringa-Kohn- Rostoker Coherent Potential Approximation (KKR-CPA), 54-55

LaB 6, 28-30 LaFe4Sbl 2, 25, 27 La-filled skutterudites, 146-156 Landau levels, 222 Lanthanide alloys, 50, 57, 63 Lattice dynamics and effects of filling, 156-162 Lattice thermal conductivity, 127-128 Lattice thermal conductivity estimations, from atomic displacement parameters elementary theory of heat conduction, 18-21 model of heat conduction, 21-26 Linearized augmented planewave (LAPW) method, 137 Linear transport equations, 210 LnNiSb, 50, 57 LnPdBi, 50, 57 LnPdSb, 50, 57, 63 Local density approximation (LDA), 128, 129-130, 137, 142, 143, 145 Lorentz force, 211,216 Low-dimensional conductors, development of, 180-181 Low-temperature refrigeration applications, 189-190

M Magnetic properties and band-structure results, 54- 56 Magnetotransport, 186-188 overview of, 197-200

276 Man-portable microclimate systems, 118-119 Mass fluctuation scattering, 22 Mathiesson's rule, 20, 92 Maxwell-Boltzmann statistics, 252 MCoSb, 49, 50, 57, 62-63 Micro electromechanical systems (mems), 120, 121 Military applications, thermoelectrics and biological cooling, 118-119 conclusions, 124 electronic, 119-121 power generation, 122-124 research in, 117-118 thermal management, 118-122 MnCoSb, 55 MNiSn, 43-49, 57, 60-61 MnNiSb, 54- 55 MnNiSn, 50, 55, 60 MnPtSb, 54 Molecular beam epitaxy (MBE), 248 Mott diffusion, 191 Mott's variable-range hopping, 55

NbCoSb, 50, 71 NbCoSn, 54 NbIrSb, 50 NbIrSn, 50 NbSe 3, 180, 201 Nernst effect, 207, 211, 215 Neutron diffraction, 85 n-type La-filled skutterudites, 154-156

Office of Naval Research (ONR), 118 Onsager reciprocity theorem, 210, 248 ORTEP (Oak Ridge Thermal Ellipsoid Plots), 5

Paramagnetic semiconductor-magnetic metal crossover, 54

INDEX Peltier heat/effects, 90, 207, 214-215, 216217, 250 Peltier refrigeration, 218, 234 Pentatellurides, thermoelectric properties of background information, 180-188 doping on chalcogen site, 195-197 doping on transition metal site, 191-194 effects of stress and pressure, 184-186 electrical transport, 181-183 low-dimensional conductors, development of, 180-181 as low-temperature thermoelectric materials, 190-191 magnetotransport, overview of, 197-200 magnetotransport and Hall effect, 186188 recent developments in, 191-201 summary of, 201- 203 synthesis and structure, 183-184 thermal conductivity of, 200-201 Phonon drag thermopower, 92, 94, 190 Phonon glass electron crystal (PGEC) paradigm, 128, 190 Phonon-phonon scattering, 22 Planck constant, 7 Power factor, 2, 40, 70-71 Power generation, applications, 122-124 Proportional-integral-derivative (PID), 232

Quantum mechanics, 220 Quasicrystals (QCs), 41 applications, 79-80 discovery and background information, 77-80 future directions and approaches, 108110 structural and mechanical properties, 80-82 summary of, 110-111 as thermoelectrics, 91-94 Quasicrystals, synthetic growth of analysis, 85-86 general issues, 83-84 grain growth, 87-90 introduction, 82-83 synthesis of stable phases, 86

277

INDEX Quasicrystals, thermoelectric properties of electrical resistivity, 95-97 families of, 93-95 thermal and electrical transport on A1PdMn, 101-109 thermal conductivity, 99-101 thermopower, 97-99 Quench methods, 82, 83

Rare earth hexaborides, 14, 28-30 Rattlers, 14 Rattling, 3, 7, 128 Refrigeration of bioactive materials, 119 electronic, 214-217 low-temperature, applications, 189-190 Resonant scattering, 20, 22, 23-26 Righi-Leduc effects, 208, 211

Schrodinger equation, 220 Seebeck coefficient, 225, 246, 248, 250, 261-262, 267, 269 Seebeck effect, 207 Semiconducting and semimetallic properties, undoped alloys, 43-51 Semiconducting compounds with ice clathrate structure, 14, 30-32 Semiconductors, intrinsic and extrinsic, 225 Semiconductor-to-semimetal (SC-SM) transition, 40, 41, 59 Shubnikov-de Haas effects, 222, 227 Single particle kinetic energy term, 129 Skutterudites binary, 134-142 Ce-filled, 142-145 filled, 14, 25, 27 La-filled, 146-156 lattice dynamics and effects of filling, 156-162 prospects, 162 Sr8Gal6Ge30, 30-32 Static disorder, 7, 11-13 Static electron-electron Coulomb interaction, 129

Superconducting magnets and motors, 122 Superlattices, 266-269

TaCoSb, 50 Tantalum tubing, 84 TaSe 3, 180 Tellurides, ternary, 14, 28 Thermal analysis, 86 Thermal conductivity, 98-101,258-261 of pentatellurides, 200-201 Thermal diffuse scattering (TDS), 6 Thermal instability, 256-258 Thermoelectric figure of merit, 68-69, 127, 208, 218, 236, 247, 262-266 Thermoelectric materials introduction to, 90-91, 188-189 low-temperature refrigeration applications, 189-190 pentatellurides as possible lowtemperature, 190-191 quasicrystals as, 91-94 Thermoelectric properties, of half-Heusler alloys, 40-41 of doped, 66-71 of undoped, 56-58 Thermoelectric properties, of pentatellurides background information, 180-188 doping on chalcogen site, 195-197 doping on transition metal site, 191-194 effects of stress and pressure, 184-186 electrical transport, 181-183 low-dimensional conductors, development of, 180-181 as low-temperature thermoelectric materials, 190-191 magnetotransport, overview of, 197-200 magnetotransport and Hall effect, 186188 recent developments in, 191-201 summary of, 201-203 synthesis and structure, 183-184 thermal conductivity of, 200-201 Thermoelectric properties, of quasicrystals electrical resistivity, 95-97 families of, 93-95

278

INDEX

Thermoelectric properties (Continued) thermal and electrical transport on A1PdMn, 101-109 thermal conductivity, 99-101 thermopower, 97-99 Thermoelectrics military applications, 117-124 theory of, 2-4 Thermomagnetic effects and measurements adiabatic conditions, 209-210 adiabatic measurements, 238-239 background information, 207- 208 bismuth and bismuth-antimony alloys, 226-229 compensated materials, 225-226 doped materials, 229-230 electronic refrigeration, 214-217 equations of performance, 218-219 experimental techniques, 230-232 ideal behavior, 217 isothermal conditions, 209-210 isothermal measurements, 233-236 materials surveyed, 223-230 measurement techniques, 230-238 microscopic electronic properties, 219222 phenomenolgocial analysis, 218-223 summary of, 240-241 transport coefficients defined, 210-214 Umkehr effect, 222-223 Thermopower, 92, 93-94, 97-99 Bloch-Boltzmann expression for, 132 TiCoSb, 54, 55, 62, 63, 64, 65, 66, 71 TiCoSn, 54, 55 TiFeSb, 54, 55 TiNiSb, 54 TiNiSn, 50, 54, 57, 59, 61, 62, 63, 64 TlzSnTe 5, 28 Transport coefficients, defined, 210-214 Transport properties, 51 doped half-Heusler alloys and, 59-63 impurity band, 64-66

U Umkehr effect, 222-223 Umklapp processes, 93

Undoped alloys. See Half-Heusler alloys, undoped

Valence electron count (VEC), 39 bandgap states in, 51-52 van de Hove singularities (cusps), 191 van der Waals gap, 184 VCoSb, 54, 55, 66 VNiSn, 59

W Wiedemann-Franz law, 99, 128, 169, 189, 251-253, 263, 265-266, 267

X-ray diffraction, 85

YbFe4Sb12, 25, 27 YbNiSb, 56 YNiSb, 51 YPtBi, 56

Z-meter method, 233-236 fl-Zn4Sb 3, 166-170 ZrCoBi, 50 ZrNiBi, 50 ZrNiSn, 43-49, 61, 63, 64 ZrTe 5 effects of stress and pressure, 184-186 electrical transport in, 181-183 synthesis and structure, 183-184

Contents of Volumes in This Series

Volume 1

Physics of III-V Compounds

c. Hilsum, Some Key Features of III-V Compounds F. Bassani, Methods of Band Calculations Applicable to III-V Compounds E. O. Kane, The k-p Method V. L. Bonch-Bruevich, Effect of Heavy Doping on the Semiconductor Band Structure D. Long, Energy Band Structures of Mixed Crystals of III-V Compounds L. M. Roth and P. N. Argyres, Magnetic Quantum Effects S. M. Puri and T. H. Geballe, Thermomagnetic Effects in the Quantum Region W. M. Becket, Band Characteristics near Principal Minima from Magnetoresistance E. H. Putley, Freeze-Out Effects, Hot Electron Effects, and Submillimeter Photoconductivity in InSb H. Weiss, Magnetoresistance B. Ancker-Johnson, Plasma in Semiconductors and Semimetals

Volume 2

Physics of III-V Compounds

M. G. Holland, Thermal Conductivity S. I. Novkova, Thermal Expansion U. Piesbergen, Heat Capacity and Debye Temperatures G. Giesecke, Lattice Constants J. R. Drabble, Elastic Properties A. U. Mac Rae and G. W. Gobeli, Low Energy Electron Diffraction Studies R. Lee Mieher, Nuclear Magnetic Resonance B. Goldstein, Electron Paramagnetic Resonance T. S. Moss, Photoconduction in III-V Compounds E. Antoncik and J. Tauc, Quantum Efficiency of the Internal Photoelectric Effect in InSb G. W. Gobeli and L G. Allen, Photoelectric Threshold and Work Function P. S. Pershan, Nonlinear Optics in III-V Compounds M. Gershenzon, Radiative Recombination in the III-V Compounds F. Stern, Stimulated Emission in Semiconductors

279

280

CONTENTS OF VOLUMES IN THIS SERIES

Volume 3

Optical of Properties III-V Compounds

M. Hass, Lattice Reflection W. G. Spitzer, Multiphonon Lattice Absorption D. L. Stierwalt and R. F. Potter, Emittance Studies H. R. Philipp and H. Ehrenveich, Ultraviolet Optical Properties M. Cardona, Optical Absorption above the Fundamental Edge E. J. Johnson, Absorption near the Fundamental Edge J. O. Dimmock, Introduction to the Theory of Exciton States in Semiconductors B. Lax and J. Ca. Mavroides, Interband Magnetooptical Effects H. E Fan, Effects of Free Carries on Optical Properties E. D. Palik and G. B. Wright, Free-Carrier Magnetooptical Effects R. H. Bube, Photoelectronic Analysis B. O. Seraph& and H. E. Bennett, Optical Constants

Volume 4 N. N. D. A. R. L. N. R.

Physics of III-V Compounds

A. Goryunova, A. S. Borschevskii, and D. N. Tretiakov, Hardness N. Sirota, Heats of Formation and Temperatures and Heats of Fusion of Compounds AraBv L. Kendall, Diffusion G. Chynoweth, Charge Multiplication Phenomena IV. Keyes, The Effects of Hydrostatic Pressure on the Properties of III-V Semiconductors W. Aukerman, Radiation Effects A. Goryunova, F. P. Kesamanly, and D. N. Nasledov, Phenomena in Solid Solutions T. Bate, Electrical Properties of Nonuniform Crystals

Volume 5 Infrared Detectors H. Levinstein, Characterization of Infrared Detectors P. W. Kruse, Indium Antimonide Photoconductive and Photoelectromagnetic Detectors M. B. Prince, Narrowband Self-Filtering Detectors I. Melngalis and T. C. Harman, Single-Crystal Lead-Tin Chalcogenides D. Long and J. L. Schmidt, Mercury-Cadmium Telluride and Closely Related Alloys E. H. Putley, The Pyroelectric Detector N. B. Stevens, Radiation Thermopiles R. J. Keyes and T. M. Quist, Low Level Coherent and Incoherent Detection in the Infrared M. C. Teich, Coherent Detection in the Infrared F. R. Arams, E. W. Sard, B. J. Peyton, and F. P. Pace, Infrared Heterodyne Detection with Gigahertz IF Response H. S. Sommers, Jr., Macrowave-Based Photoconductive Detector R. Sehr and R. Zuleeg, Imaging and Display

Volume 6

Injection Phenomena

M. A. Lampert and R. B. Schilling, Current Injection in Solids: The Regional Approximation Method R. Williams, Injection by Internal Photoemission A. M. Barnett, Current Filament Formation

CONTENTS OF VOLUMES IN THIS SERIES

281

R. Baron and J. W. Mayer, Double Injection in Semiconductors W. Ruppel, The Photoconductor-Metal Contact

Volume 7

Application and Devices Part A

J. A. Copeland and S. Knight, Applications Utilizing Bulk Negative Resistance F. A. Padovani, The Voltage-Current Characteristics of Metal-Semiconductor Contacts P. L. Hower, W. W. Hooper, B. R. Cairns, R. D. Fairman, and D. A. Tremere, The GaAs Field-Effect Transistor M. H. White, M OS Transistors G. R. Antell, Gallium Arsenide Transistors T. L. Tansley, Heterojunction Properties

Part B T. H. R. R.

Misawa, IMPATT Diodes C. Okean, Tunnel Diodes B. Campbell and Hung-Chi Chang, Silicon Junction Carbide Devices E. Enstrom, H. Kressel, and L. Krassner, High-Temperature Power Rectifiers of GaAsl_xP x

Volume

8 Transport and Optical Phenomena

R. J. Stirn, Band Structure and Galvanomagnetic Effects in III-V Compounds with Indirect Band Gaps R. W. Ure, Jr., Thermoelectric Effects in III-V Compounds H. Piller, Faraday Rotation H. Barry Bebb and E. W. Williams, Photoluminescence I: Theory E. W. Williams and H. Barry Bebb, Photoluminescence II: Gallium Arsenide

Volume

9

Modulation Techniques

B. O. Seraphin, Electroreflectance R. L. Aggarwal, Modulated Interband Magnetooptics D. F. Blossey and Paul Handler, Electroabsorption B. Batz, Thermal and Wavelength Modulation Spectroscopy I. Balslev, Piezopptical Effects D. E. Aspnes and N. Bottka, Electric-Field Effects on the Dielectric Function of Semiconductors and Insulators

V o l u m e 10

Transport Phenomena

R. L. Rhode, Low-Field Electron Transport J. D. Wiley, Mobility of Holes in III-V Compounds C. M. Wolfe and G. E. Stillman, Apparent Mobility Enhancement in Inhomogeneous Crystals R. L. Petersen, The Magnetophonon Effect

282

CONTENTS OF VOLUMES IN THIS SERIES

Volume 11

Solar Cells

H. J. Hovel, Introduction; Carrier Collection, Spectral Response, and Photocurrent; Solar Cell Electrical Characteristics; Efficiency; Thickness; Other Solar Cell Devices; Radiation Effects; Temperature and Intensity; Solar Cell Technology

Volume 12

Infrared Detectors (II)

W. L. Eiseman, J. D. Merr&m, and R. F. Potter, Operational Characteristics of Infrared Photodetectors P. R. Bratt, Impurity Germanium and Silicon Infrared Detectors E. H. Putley, InSb Submillimeter Photoconductive Detectors G. E. Stillman, C. M. Wolfe, and J. O. Dimmock, Far-Infrared Photoconductivity in High Purity GaAs G. E. Stillman and C. M. Wolfe, Avalanche Photodiodes P. L. Richards, The Josephson Junction as a Detector of Microwave and Far-Infrared Radiation E. H. Putley, The Pyroelectric Detector--An Update

Volume 13

Cadmium Telluride

K. Zanio, Materials Preparations; Physics; Defects; Applications

Volume 14

Lasers, Junctions, Transport

N. Holonyak, Jr. and M. H. Lee, Photopumped III-V Semiconductor Lasers H. Kressel and J. K. Butler, Heterojunction Laser Diodes A Van der Ziel, Space-Charge-Limited Solid-State Diodes P. J. Price, Monte Carlo Calculation of Electron Transport in Solids

Volume 15

Contacts, Junctions, Emitters

B. L. Sharma, Ohmic Contacts to III-V Compounds Semiconductors A. Nussbaum, The Theory of Semiconducting Junctions J. S. Escher, NEA Semiconductor Photoemitters

Volume 16 Defects, (HgCd)Se, (HgCd)Te H. Kressel, The Effect of Crystal Defects on Optoelectronic Devices C. R. Whitsett, J. G. Broerman, and C. J. Summers, Crystal Growth and Properties of Hg 1_ xCdxSe alloys M. H. Weiler, Magnetooptical Properties of Hgl_xCdxTe Alloys P. W. Kruse and J. G. Ready, Nonlinear Optical Effects in Hgl_xCdxTe

Volume 17

CW Processing of Silicon and Other Semiconductors

J. F. Gibbons, Beam Processing of Silicon A. Lietoila, R. B. Gold, J. F. Gibbons, and L. A. Christel, Temperature Distributions and Solid Phase Reaction Rates Produced by Scanning CW Beams

CONTENTS OF VOLUMES IN THIS SERIES

283

A. Leitoila and J. F. Gibbons, Applications of CW Beam Processing to Ion Implanted Crystalline Silicon N. M. Johnson, Electronic Defects in CW Transient Thermal Processed Silicon K. F. Lee, T. J. Stultz, and J. F. Gibbons, Beam Recrystallized Polycrystalline Silicon: Properties, Applications, and Techniques T. Shibata, A. Wakita, T. W. Sigmon, and J. F. Gibbons, Metal-Silicon Reactions and Silicide Y. I. Nissim and J. F. Gibbons, CW Beam Processing of Gallium Arsenide

Volume 18 Mercury Cadmium Telluride P. W. Kruse, The Emergence of (Hgl_xCdx)Te as a Modern Infrared Sensitive Material H. E. Hirsch, S. C. Liang, and A. G. White, Preparation of High-Purity Cadmium, Mercury, and Tellurium IV. F H. Micklethwaite, The Crystal Growth of Cadmium Mercury Telluride P. E. Petersen, Auger Recombination in Mercury Cadmium Telluride R. M. Broudy and V. J. Mazurczyck, (HgCd)Te Photoconductive Detectors M. B. Reine, A. K. Soad, and T. J. Tredwell, Photovoltaic Infrared Detectors M. A. Kinch, Metal-Insulator-Semiconductor Infrared Detectors

Volume 19 Deep Levels, GaAs, Alloys, Photochemistry G. F. Neumark and K. Kosai, Deep Levels in Wide Band-Gap III-V Semiconductors D. C Look, The Electrical and Photoelectronic Properties of Semi-Insulating GaAs R. F. Brebrick, Ching-Hua Su, and Pok-Kai Liao, Associated Solution Model for Ga-In-Sb and Hg-Cd-Te Y. Ya. Gurevich and Y. V. Pleskon, Photoelectrochemistry of Semiconductors

Volume 20

Semi-Insulating GaAs

R. N. Thomas, H. M. Hobgood, G. W. Eldridge, D. L. Barrett, T. T. Braggins, L. B. Ta, and S. K. Wang, High-Purity LEC Growth and Direct Implantation of GaAs for Monolithic Microwave Circuits C. A. Stolte, Ion Implantation and Materials for GaAs Integrated Circuits C. G. Kirkpatrick, R. T. Chen, D. E. Holmes, P. M. Asbeck, K. R. Elliott, R. D. Fairman, and J. R. Oliver, LEC GaAs for Integrated Circuit Applications J. S. Blakemore and S. Rahimi, Models for Mid-Gap Centers in Gallium Arsenide

Volume 21

Hydrogenated Amorphous Silicon Part A

J. I. Pankove, Introduction M. Hirose, Glow Discharge; Chemical Vapor Deposition Y. Uchida, di Glow Discharge T. D. Moustakas, Sputtering I. Yamada, Ionized-Cluster Beam Deposition B. A. Scott, Homogeneous Chemical Vapor Deposition

284

CONTENTS OF VOLUMES IN THIS SERIES

F. J. Kampas, Chemical Reactions in Plasma Deposition P. A. Longeway, Plasma Kinetics H. A. Weakliem, Diagnostics of Silane Glow Discharges Using Probes and Mass Spectroscopy L. Gluttman, Relation between the Atomic and the Electronic Structures A. Chenevas-Paule, Experiment Determination of Structure S. Minomura, Pressure Effects on the Local Atomic Structure D. Adler, Defects and Density of Localized States

Part B J. I. Pankove, Introduction G. D. Cody, The Optical Absorption Edge of a-Si: H N. M. Amer and IV.. B. Jackson, Optical Properties of Defect States in a-Si: H P. J. Zanzucchi, The Vibrational Spectra of a-Si: H Y. Hamakawa, Electroreflectance and Electroabsorption J. S. Lannin, Raman Scattering of Amorphous Si, Ge, and Their Alloys R. A. Street, Luminescence in a-Si: H R. S. Crandall, Photoconductivity J. Tauc, Time-Resolved Spectroscopy of Electronic Relaxation Processes P. E. Vanier, IR-Induced Quenching and Enhancement of Photoconductivity and Photo luminescence H. Schade, Irradiation-Induced Metastable Effects L. Ley, Photoelectron Emission Studies

Part C J. I. Pankove, Introduction J. D. Cohen, Density of States from Junction Measurements in Hydrogenated Amorphous Silicon P. C. Taylor, Magnetic Resonance Measurements in a-Si: H K. Morigaki, Optically Detected Magnetic Resonance J. Dresner, Carrier Mobility in a-Si: H 7". Tiedje, Information about band-Tail States from Time-of-Flight Experiments A. R. Moore, Diffusion Length in Undoped a-Si: H W. Beyer and J. Overhof, Doping Effects in a-Si: H H. Fritzche, Electronic Properties of Surfaces in a-Si: H C. R. Wronski, The Staebler-Wronski Effect R. J. Nemanich, Schottky Barriers on a-Si: H B. Abeles and T. Tiedje, Amorphous Semiconductor Superlattices

Part D J. I. Pankove, Introduction D. E. Carlson, Solar Cells G. A. Swartz, Closed-Form Solution of I - V Characteristic for a a-Si: H Solar Cells I. Shimizu, Electrophotography S. Ishioka, Image Pickup Tubes

CONTENTS OF VOLUMES IN THIS SERIES

285

P. G. LeComber and W. E. Spear, The Development of the a-Si: H Field-Effect Transistor and Its Possible Applications D. G. Ast, a-Si: H FET-Addressed LCD Panel S. Kaneko, Solid-State Image Sensor M. Matsumura, Charge-Coupled Devices M. A. Bosch, Optical Recording A. D'Amico and G. Fortunato, Ambient Sensors H. Kukimoto, Amorphous Light-Emitting Devices R. J. Phelan, Jr., Fast Detectors and Modulators J. I. Pankove, Hybrid Structures P. G. LeComber, A. E. Owen, IV. E. Spear, J. Hajto, and W. K. Choi, Electronic Switching in Amorphous Silicon Junction Devices

Volume 22

Lightwave Communications Technology Part A

K. Nakajima, The Liquid-Phase Epitaxial Growth of InGaAsP IV. T. Tsang, Molecular Beam Epitaxy for III-V Compound Semiconductors G. B. String,fellow, Organometallic Vapor-Phase Epitaxial Growth of III-V Semiconductors G. Beuchet, Halide and Chloride Transport Vapor-Phase Deposition of InGaAsP and GaAs M. Razeghi, Low-Pressure Metallo-Organic Chemical Vapor Deposition of Gaxlnl_xASPl_ r Alloys P. M. Petroff, Defects in III-V Compound Semiconductors

Part B J. P. van der Ziel, Mode Locking of Semiconductor Lasers K. Y. Lau and A. Yariv, High-Frequency Current Modulation of Semiconductor Injection Lasers C. H. Henry, Special Properties of Semiconductor Lasers Y. Suematsu, K. Kishino, S. Arai, and F Koyama, Dynamic Single-Mode Semiconductor Lasers with a Distributed Reflector IV. T. Tsang, The Cleaved-Coupled-Cavity (C 3) Laser

Part C R. J. Nelson and N. K. Dutta, Review of lnGaAsP InP Laser Structures and Comparison of Their Performance N. Chinone and M. Nakamura, Mode-Stabilized Semiconductor Lasers for 0.7-0.8- and 1.1-1.6-/~m Regions Y. Horikoshi, Semiconductor Lasers with Wavelengths Exceeding 2 #m B. A. Dean and M. Dixon, The Functional Reliability of Semiconductor Lasers as Optical Transmitters R. H. Saul, T. P. Lee, and C. A. Burus, Light-Emitting Device Design C. L. Zipfel, Light-Emitting Diode-Reliability T. P. Lee and T. Li, LED-Based Multimode Lightwave Systems K. Ogawa, Semiconductor Noise-Mode Partition Noise

286

CONXENXS OF VOLUMES IN THIS SERIES

Part D Capasso, The Physics of Avalanche Photodiodes P. Pearsall and M. A. Pollack, Compound Semiconductor Photodiodes Kaneda, Silicon and Germanium Avalanche Photodiodes R. Forrest, Sensitivity of Avalanche Photodetector Receivers for High-Bit-Rate LongWavelength Optical Communication Systems J. C. Campbell, Phototransistors for Lightwave Communications

F. T. T. S.

Part E S. Wang, Principles and Characteristics of Integrable Active and Passive Optical Devices S. Margalit and A. Yariv, Integrated Electronic and Photonic Devices T. Mukai, Y. Yamamoto, and T. Kimura, Optical Amplification by Semiconductor Lasers

Volume 23 R. C. G. R. R. D. D. D. R. R.

Pulsed Laser Processing of Semiconductors

F. Wood, C. W. White, and R. T. Young, Laser Processing of Semiconductors: An Overview W. White, Segregation, Solute Trapping, and Supersaturated Alloys E. Jellison, Jr., Optical and Electrical Properties of Pulsed Laser-Annealed Silicon F Wood and G. E. Jellison, Jr., Melting Model of Pulsed Laser Processing F Wood and F W. Young, Jr., Nonequilibrium Solidification Following Pulsed Laser Melting H. Lowndes and G. E. Jellison, Jr., Time-Resolved Measurement During Pulsed Laser Irradiation of Silicon M. Zebner, Surface Studies of Pulsed Laser Irradiated Semiconductors H. Lowndes, Pulsed Beam Processing of Gallium Arsenide B. James, Pulsed CO 2 Laser Annealing of Semiconductors T. Young and R. F Wood, Applications of Pulsed Laser Processing

Volume 24

Applications of Multiquantum Wells, Selective Doping, and Superlattices

C. Weisbuch, Fundamental Properties of III-V Semiconductor Two-Dimensional Quantized Structures: The Basis for Optical and Electronic Device Applications H. Morkoc and H. Unlu, Factors Affecting the Performance of (A1,Ga)As/GaAs and (A1, Ga)As/InGaAs Modulation-Doped Field-Effect Transistors: Microwave and Digital Applications N. T. Linh, Two-Dimensional Electron Gas FETs: Microwave Applications M. Abe et al., Ultra-High-Speed HEMT Integrated Circuits D. S. Chemla, D. A. B. Miller, and P. W. Smith, Nonlinear Optical Properties of Multiple Quantum Well Structures for Optical Signal Processing F Capasso, Graded-Gap and Superlattice Devices by Band-Gap Engineering W. T. Tsang, Quantum Confinement Heterostructure Semiconductor Lasers G. C. Osbourn et al., Principles and Applications of Semiconductor Strained-Layer Superlattices

CONTENTS OF VOLUMES IN THIS SERIES

Volume 25

287

Diluted Magnetic Semiconductors

Iv. Giriat and J. K. Furdyna, Crystal Structure, Composition, and Materials Preparation of

Diluted Magnetic Semiconductors W. M. Becker, Band Structure and Optical Properties of Wide-Gap A~I~Mn~BIv Alloys at

Zero Magnetic Field S. Oseroff and P. H. Keesom, Magnetic Properties: Macroscopic Studies T. Giebultowicz and T. M. Holden, Neutron Scattering Studies of the Magnetic Structure and

Dynamics of Diluted Magnetic Semiconductors J. Kossut, Band Structure and Quantum Transport Phenomena in Narrow-Gap Diluted

Magnetic Semiconductors C. Riquaux, Magnetooptical Properties of Large-Gap Diluted Magnetic Semiconductors J. A. Gaj, Magnetooptical Properties of Large-Gap Diluted Magnetic Semiconductors J. Mycielski, Shallow Acceptors in Diluted Magnetic Semiconductors: Splitting, Boil-off, Giant

Negative Magnetoresistance A. K. Ramadas and R. Rodriquez, Raman Scattering in Diluted Magnetic Semiconductors P. A. Wolff, Theory of Bound Magnetic Polarons in Semimagnetic Semiconductors

V o l u m e 26

Compound Semiconductors and Semiconductor Properties of Superionic Materials

III-V

Z. Yuanxi, III-V Compounds H. V. Winston, A. T. Hunter, H. Kimura, and R. E. Lee, InAs-Alloyed GaAs Substrates for

Direct Implantation P. K. Bhattacharya and S. Dhar, Deep Levels in III-V Compound Semiconductors Grown by

MBE Y. Ya. Gurevich and A. K. Ivanov-Shits, Semiconductor Properties of Supersonic Materials

Volume 27 High Conducting Quasi-One-Dimensional Organic Crystals E. M. Conwell, Introduction to Highly Conducting Quasi-One-Dimensional Organic Crystals I. A. Howard, A Reference Guide to the Conducting Quasi-One-Dimensional Organic

Molecular Crystals J. P. Pouquet, Structural Instabilities E. M. Conwell, Transport Properties C. S. Jacobsen, Optical Properties J. C. Scott, Magnetic Properties L. Zuppiroli, Irradiation Effects: Perfect Crystals and Real Crystals

Volume 28

Measurement of High-Speed Signals in Solid State Devices

J. Frey and D. Ioannou, Materials and Devices for High-Speed and Optoelectronic Applications H. Schumacher and E. Strid, Electronic Wafer Probing Techniques D. H. Auston, Picosecond Photoconductivity: High-Speed Measurements of Devices and

Materials J. A. Valdmanis, Electro-Optic Measurement Techniques for Picosecond Materials, Devices,

and Integrated Circuits. J. M. Wiesenfeld and R. K. Jain, Direct Optical Probing of Integrated Circuits and High-Speed

Devices G. Plows, Electron-Beam Probing ,4. M. Weiner and R. B. Marcus, Photoemissive Probing

288

CONTENTS OF VOLUMES IN THIS SERIES

Volume

29 Very High Speed Integrated Circuits: Gallium Arsenide LSI

M. Kuzuhara and T. Nazaki, Active Layer Formation by Ion Implantation H. Hasimoto, Focused Ion Beam Implantation Technology T. Nozaki and .4. Higashisaka, Device Fabrication Process Technology M. Ino and T. Takada, GaAs LSI Circuit Design M. Hirayama, M. Ohmori, and K. Yamasaki, GaAs LSI Fabrication and Performance

Volume 30 Very High Speed Integrated Circuits: Heterostructure H. Watanabe, T. Mizutani, and A. Usui, Fundamentals of Epitaxial Growth and Atomic Layer Epitaxy S. Hiyamizu, Characteristics of Two-Dimensional Electron Gas in III-V Compound Heterostructures Grown by MBE T. Nakanisi, Metalorganic Vapor Phase Epitaxy for High-Quality Active Layers T. Nimura, High Electron Mobility Transistor and LSI Applications T. Sugeta and T. Ishibashi, Hetero-Bipolar Transistor and LSI Application H. Matsueda, T. Tanaka, and M. Nakamura, Optoelectronic Integrated Circuits

V o l u m e 31

Indium Phosphide: Crystal Growth and Characterization

J. P. Farges, Growth of Discoloration-free InP M. J. McCollum and G. E. Stillman, High Purity InP Grown by Hydride Vapor Phase Epitaxy T. Inada and T. Fukuda, Direct Synthesis and Growth of Indium Phosphide by the Liquid Phosphorous Encapsulated Czochralski Method O. Oda, K. Katagiri, K. Shinohara, S. Katsura, Y. Takahashi, K. Kainosho, K. Kohiro, and R. Hirano, InP Crystal Growth, Substrate Preparation and Evaluation K. Tada, M. Tatsumi, M. Morioka, T. Araki, and T. Kawase, InP Substrates: Production and Quality Control M. Razeghi, LP-MOCVD Growth, Characterization, and Application of InP Material T. A. Kennedy and P. J. Lin-Chung, Stoichiometric Defects in InP

Volme 32 Strained-Layer Superlattices: Physics 7". P. Pearsall, Strained-Layer Superlattices F. H. Pollack, Effects of Homogeneous Strain on the Electronic and Vibrational Levels in Semiconductors J. Y. Marzin, J. M. Gerhrd, P. Voisin, and J. A. Bruin, Optical Studies of Strained III-V Heterolayers R. People and S. A. Jackson, Structurally Induced States from Strain and Confinement M. Jaros, Microscopic Phenomena in Ordered Superlattices

Volume 33 Strained-Layer Superlattices: Materials Science and Technology R. Hull and J. C. Bean, Principles and Concepts of Strained-Layer Epitaxy W. J. Schaff, P. J. Tasker, M. C. Foisy, and L. F Eastman, Device Applications of Strained-Layer Epitaxy

CONTENTS OF VOLUMES IN THIS SERIES

289

S. T. Picraux, B. L. Doyle, and J. Y. Tsao, Structure and Characterization of Strained-Layer Superlattices E. Kasper and F. Schaffer, Group IV Compounds D. L. Martin, Molecular Beam Epitaxy of IV-VI Compounds Heterojunction R. L. C,unshor, L. A. Kolodziejski, A. V. Nurmikko, and N. Otsuka, Molecular Beam Epitaxy of II-VI Semiconductor Microstructures

V o l u m e 34

Hydrogen in Semiconductors

I. Pankove and N. M. Johnson, Introduction to Hydrogen in Semiconductors H. Seager, Hydrogenation Methods I. Pankove, Hydrogenation of Defects in Crystalline Silicon W. Corbett, P. De6k, U. V. Desnica, and S. J. Pearton, Hydrogen Passivation of Damage Centers in Semiconductors S. J. Pearton, Neutralization of Deep Levels in Silicon J. I. Pankove, Neutralization of Shallow Acceptors in Silicon N. M. Johnson, Neutralization of Donor Dopants and Formation of Hydrogen-Induced Defects in n-Type Silicon M. Stavola and S. J. Pearton, Vibrational Spectroscopy of Hydrogen-Related Defects in Silicon A. D. Marwick, Hydrogen in Semiconductors: Ion Beam Techniques C. Herring and N. M. Johnson, Hydrogen Migration and Solubility in Silicon E. E. Hailer, Hydrogen-Related Phenomena in Crystalline Germanium J. Kakalios, Hydrogen Diffusion in Amorphous Silicon J. Chevalier, B. Clerjaud, and B. Pajot, Neutralization of Defects and Dopants in III-V Semiconductors G. G. DeLeo and W. B. Fowler, Computational Studies of Hydrogen-Containing Complexes in Semiconductors R. F. Kiefl and T. L. Estle, Muonium in Semiconductors C. G. Van de Walle, Theory of Isolated Interstitial Hydrogen and Muonium in Crystalline Semiconductors

J. C J. J.

V o l u m e 35

Nanostructured Systems

M. Reed, Introduction H. van Houten, C W. J. Beenakker, and B. J. van Wees, Quantum Point Contacts G. Timp, When Does a Wire Become an Electron Waveguide? M. Biittiker, The Quantum Hall Effects in Open Conductors W. Hansen, J. P. Kotthaus, and U. Merkt, Electrons in Laterally Periodic Nanostructures

V o l u m e 36 D. A. A. O. D.

The

Spectroscopy of Semiconductors

Heiman, Spectroscopy of Semiconductors at Low Temperatures and High Magnetic Fields F. Nurmikko, Transient Spectroscopy by Ultrashort Laser Pulse Techniques K. Ramdas and S. Rodriguez, Piezospectroscopy of Semiconductors J. Glembocki and B. F. Shanabrook, Photoreflectance Spectroscopy of Microstructures G. Seiler, C. L. Littler, and M. H. Wiler, One- and Two-Photon Magneto-Optical Spectroscopy of InSb and Hgl_xCdxTe

290

CONTENTS OF VOLUMES IN THIS SERIES

Volume 37

The Mechanical Properties of Semiconductors

A.-B. Chen, A. Sher and W. T. Yost, Elastic Constants and Related Properties of Semiconductor Compounds and Their Alloys D. R. Clarke, Fracture of Silicon and Other Semiconductors H. Siethoff, The Plasticity of Elemental and Compound Semiconductors S. Guruswamy, K. T. Faber and J. P. Hirth, Mechanical Behavior of Compound Semiconductors S. Mahajan, Deformation Behavior of Compound Semiconductors J. P. Hirth, Injection of Dislocations into Strained Multilayer Structures D. Kendall, C. B. Fleddermann, and K. J. Malloy, Critical Technologies for the Micromachining of Silicon I. Matsuba and K. Mokuya, Processing and Semiconductor Thermoelastic Behavior

Volume 38

Imperfections in III/V Materials

u. Scherz and M. Sche~ter, Density-Functional Theory of sp-Bonded Defects in III/V Semiconductors M. Kaminska and E. R. Weber, El2 Defect in GaAs D. C. Look, Defects Relevant for Compensation in Semi-Insulating GaAs R. C. Newman, Local Vibrational Mode Spectroscopy of Defects in III/V Compounds A. M. Hennel, Transition Metals in III/V Compounds K. J. Malloy and K. Khachaturyan, DX and Related Defects in Semiconductors V. Swaminathan and A. S. Jordan, Dislocations in III/V Compounds K. W. Nauka, Deep Level Defects in the Epitaxial III/V Materials

Volume 39

Minority Carriers in III-V Semiconductors: Physics and Applications

N. K. Dutta, Radiative Transitions in GaAs and Other III-V Compounds R. K. Ahrenkiel, Minority-Carrier Lifetime in III-V Semiconductors T. Furuta, High Field Minority Electron Transport in p-GaAs M. S. Lundstrom, Minority-Carrier Transport in III-V Semiconductors R. A. Abram, Effects of Heavy Doping and High Excitation on the Band Structure of GaAs D. Yevick and W. Bardyszewski, An Introduction to Non-Equilibrium Many-Body Analyses of Optical Processes in III-V Semiconductors

Volume 40

Epitaxial Microstructures

E. F Schubert, Delta-Doping of Semiconductors: Electronic, Optical, and Structural Properties of Materials and Devices A. Gossard, M. Sundaram, and P. Hopkins, Wide Graded Potential Wells P. Petroff, Direct Growth of Nanometer-Size Quantum Wire Superlattices E. Kapon, Lateral Patterning of Quantum Well Heterostructures by Growth of Nonplanar Substrates H. Temkin, D. Gershoni, and M. Panish, Optical Properties of Gal _~InxAs/InP Quantum Wells

CONTENTS OF VOLUMES IN THIS SERIES

Volume 41

291

High Speed Heterostructure Devices

F. Capasso, E Beltram, S. Sen, A. Pahlevi, and ,4. Y. Cho, Quantum Electron Devices: Physics and Applications P. Solomon, D. J. Frank, S. L. Wright, and F. Canora, GaAs-Gate Semiconductor-InsulatorSemiconductor FET M. H. Hashemi and U. K. Mishra, Unipolar InP-Based Transistors R. Kiehl, Complementary Heterostructure FET Integrated Circuits T. Ishibashi, GaAs-Based and InP-Based Heterostructure Bipolar Transistors H. C. Liu and T. C. L. G. Sollner, High-Frequency-Tunneling Devices H. Ohnishi, T. More, M. Takatsu, K. Imamura, and N. Yokoyama, Resonant-Tunneling Hot-Electron Transistors and Circuits

Volume 42

Oxygen in Silicon

F. Shimura, Introduction to Oxygen in Silicon W. Lin, The Incorporation of Oxygen into Silicon Crystals T. J. Schaffner and D. K. Schroder, Characterization Techniques for Oxygen in Silicon W. M. Bullis, Oxygen Concentration Measurement S. M. Hu, Intrinsic Point Defects in Silicon B. Pajot, Some Atomic Configurations of Oxygen J. Michel and L. C. Kimerling, Electical Properties of Oxygen in Silicon R. C. Newman and R. Jones, Diffusion of Oxygen in Silicon T. Y. Tan and W. J. Taylor, Mechanisms of Oxygen Precipitation: Some Quantitative Aspects M. Schrems, Simulation of Oxygen Precipitation K. Simino and I. Yonenaga, Oxygen Effect on Mechanical Properties W. Bergholz, Grown-in and Process-Induced Effects F. Shimura, Intrinsic/Internal Gettering H. Tsuya, Oxygen Effect on Electronic Device Performance

Volume 43

Semiconductors for Room Temperature Nuclear Detector Applications

R. B. James and T. E. Schlesinger, Introduction and Overview L. S. Darken and C. E. Cox, High-Purity Germanium Detectors A. Burger, D. Nason, L. Van den Berg, and M. Schieber, Growth of Mercuric Iodide X. J. Bao, T. E. Schlesinger, and R. B. James, Electrical Properties of Mercuric Iodide X. J. Bao, R. B. James, and T. E. Schlesinger, Optical Properties of Red Mercuric Iodide M. Hage-Ali and P. Siffert, Growth Methods of CdTe Nuclear Detector Materials M. Hage-Ali and P Siffert, Characterization of CdTe Nuclear Detector Materials M. Hage-Ali and P. Siffert, CdTe Nuclear Detectors and Applications R. B. James, T. E. Schlesinger, J. Lund, and M. Schieber, Cdl_xZnxTe Spectrometers for Gamma and X-Ray Applications D. S. McGregor, J. E. Kammeraad, Gallium Arsenide Radiation Detectors and Spectrometers J. C. Lund, E Olschner, and A. Burger, Lead Iodide M. R. Squillante, and K. S. Shah, Other Materials: Status and Prospects V. M. Gerrish, Characterization and Quantification of Detector Performance J. S. lwanczyk and B. E. Part, Electronics for X-ray and Gamma Ray Spectrometers M. Schieber, R. B. James, and T. E. Schlesinger, Summary and Remaining Issues for Room Temperature Radiation Spectrometers

292

CONTENTS OF VOLUMES IN THIS SERIES

Volume 44 II-IV Blue/Green Light Emitters: Device Physics and Epitaxial Growth J. Han and K L. Gunshor, MBE Growth and Electrical Properties of Wide Bandgap ZnSe-based II-VI Semiconductors S. Fujita and S. Fujita, Growth and Characterization of ZnSe-based II-VI Semiconductors by MOVPE E. Ho and L. ,4. Kolodziejski, Gaseous Source UHV Epitaxy Technologies for Wide Bandgap II-VI Semiconductors C. G. Van de Walle, Doping of Wide-Band-Gap II-VI Compounds--Theory R. Cingolani, Optical Properties of Excitons in ZnSe-Based Quantum Well Heterostructures ,4. Ishibashi and A. K Nurmikko, II-VI Diode Lasers: A Current View of Device Performance and Issues S. Guha and J. Petruzello, Defects and Degradation in Wide-Gap II-VI-based Structures and Light Emitting Devices

Volume 45 Effect of Disorder and Defects in Ion-Implanted Semiconductors: Electrical and Physiochemical Characterization H. Ryssel, Ion Implantation into Semiconductors: Historical Perspectives You-Nian Wang and Teng-Cai Ma, Electronic Stopping Power for Energetic Ions in Solids S. T. Nakagawa, Solid Effect on the Electronic Stopping of Crystalline Target and Application to Range Estimation G. Miiller, S. Kalbitzer and G. N. Greaves, Ion Beams in Amorphous Semiconductor Research J. Boussey-Said, Sheet and Spreading Resistance Analysis of Ion Implanted and Annealed Semiconductors M. L. Polignano and G. Queirolo, Studies of the Stripping Hall Effect in Ion-Implanted Silicon J. Stoemenos, Transmission Electron Microscopy Analyses R. Nipoti and M. Servidori, Rutherford Backscattering Studies of Ion Implanted Semiconductors P. Zaumseil, X-ray Diffraction Techniques

Volume 46 Effect of Disorder and Defects in Ion-Implanted Semiconductors: Optical and Photothermal Characterization M. Fried, T. Lohner and J. Gyulai, Ellipsometric Analysis A. Seas and C. Christofides, Transmission and Reflection Spectroscopy on Ion Implanted Semiconductors A. Othonos and C. Christofides, Photoluminescence and Raman Scattering of Ion Implanted Semiconductors. Influence of Annealing C. Christofides, Photomodulated Thermoreflectance Investigation of Implanted Wafers. Annealing Kinetics of Defects U. Zammit, Photothermal Deflection Spectroscopy Characterization of Ion-Implanted and Annealed Silicon Films ,4. Mandelis, A. Budiman and M. Vargas, Photothermal Deep-Level Transient Spectroscopy of Impurities and Defects in Semiconductors R. Kalish and S. Charbonneau, Ion Implantation into Quantum-Well Structures A. M. Myasnikov and N. N. Gerasimenko, Ion Implantation and Thermal Annealing of III-V Compound Semiconducting Systems: Some Problems of III-V Narrow Gap Semiconductors

CONTENTS OF VOLUMES IN THIS SERIES

Volume 47

293

Uncooled Infrared Imaging Arrays and Systems

R. G. Buser and M. P. Tompsett, Historical Overview P. W. Kruse, Principles of Uncooled Infrared Focal Plane Arrays R. A. Wood, Monolithic Silicon Microbolomcter Arrays C. M. Hanson, Hybrid Pyroelectric-Ferroelectric Bolometer Arrays D. L. Polla and J. R. Choi, Monolithic Pyroelectric Bolometer Arrays N. Teranishi, Thermoelectric Uncooled Infrared Focal Plane Arrays M. F. Tompsett, Pyroelectric Vidicon T. W. Kenny, Tunneling Infrared Sensors J. R. Vig, R. L. Filler and Y Kim, Application of Quartz Microresonators to Uncooled Infrared

Imaging Arrays P. W. Kruse, Application of Uncooled Monolithic Thermoelectric Linear Arrays to Imaging

Radiometers

Volume 48

High Brightness Light Emitting Diodes

G. B. Stringfellow, Materials Issues in High-Brightness Light-Emitting Diodes M. G. Craford, Overview of Device issues in High-Brightness Light-Emitting Diodes F. M. Steranka, A1GaAs Red Light Emitting Diodes C. H. Chen, S. A. Stockman, M. J. Peanasky, and C. P. Kuo, OMVPE Growth of AIGaInP for

High Efficiency Visible Light-Emitting Diodes F. A. Kish and R. M. Fletcher, AIGaInP Light-Emitting Diodes M. W. Hodapp, Applications for High Brightness Light-Emitting Diodes I. Akasaki and H. Amano, Organometallic Vapor Epitaxy of GaN for High Brightness Blue

Light Emitting Diodes S. Nakamura, Group III-V Nitride Based Ultraviolet-Blue-Green-Yellow Light-Emitting

Diodes and Laser Diodes

Volume 49

Light Emission in Silicon: from Physics to Devices

D. J. Lockwood, Light Emission in Silicon G. Abstreiter, Band Gaps and Light Emission in Si/SiGc Atomic Layer Structures T. G. Brown and D. G. Hall, Radiative Isoclectronic Impurities in Silicon and

Silicon-Germanium Alloys and Supcrlatticcs J. Michel, L. V. C. Assali, M. T. Morse, and L. C. Kimerling, Erbium in Silicon Y. Kanemitsu, Silicon and Germanium Nanoparticlcs P. M. Fauchet, Porous Silicon: Photolumincsccncc and Elcctrolumincsccnt Devices C. Delerue, G. Allan, and M. Lannoo, Theory of Radiative and Nonradiative Processes in

Silicon Nanocrystallitcs L. Brus, Silicon Polymers and Nanocrystals

Volume 50

Gallium Nitride (GaN)

J. I. Pankove and T. D. Moustakas, Introduction S. P. DenBaars and S. Keller, Mctalorganic Chemical Vapor Deposition (MOCVD) of Group

III Nitridcs W. A. Bryden and T. J. Kistenmacher, Growth of Group III-A Nitrides by Reactive Sputtering N. Newman, Thermochemistry of III-N Semiconductors S. J. Pearton and R. J. Shul, Etching of III Nitrides

294

CONTENTS OF VOLUMES IN THIS SERIES

s. M. Bedair, Indium-based Nitride Compounds A. Trampert, O. Brandt, and K H. Ploog, Crystal Structure of Group III Nitrides H. Morkoc, F HamdanL and A. Salvador, Electronic and Optical Properties of I I I - V Nitride based Quantum Wells and Superlattices K. Doverspike and J. I. Pankove, Doping in the III-Nitrides T. Suski and P. Perlin, High Pressure Studies of Defects and Impurities in Gallium Nitride B. Monemar, Optical Properties of GaN W R. L. Lambrecht, Band Structure of the Group III Nitrides N. E. Christensen and P. Perlin, Phonons and Phase Transitions in GaN S. Nakamura, Applications of LEDs and LDs I. Akasaki and H. Amano, Lasers J. A. Cooper, Jr., Nonvolatile Random Access Memories in Wide Bandgap Semiconductors

V o l u m e 51A

I d e n t i f i c a t i o n o f Defects in Semiconductors

G. D. Watkins, EPR and ENDOR Studies of Defects in Semiconductors J.-M. Spaeth, Magneto-Optical and Electrical Detection of Paramagnetic Resonance in Semiconductors T. A. Kennedy and E. R. Glaser, Magnetic Resonance of Epitaxial Layers Detected by Photoluminescence K. H. Chow, B. HittL and R. F Kiefl, pSR on Muonium in Semiconductors and Its Relation to Hydrogen K. Saarinen, P. HautofiirvL and C. Corbel Positron Annihilation Spectroscopy of Defects in Semiconductors R. Jones and P. R. Briddon, The Ab Initio Cluster Method and the Dynamics of Defects in Semiconductors

V o l u m e 51B

Identification of Defects in Semiconductors

G. Davies, Optical Measurements of Point Defects P. M. Mooney, Defect Identification Using Capacitance Spectroscopy M. Stavola, Vibrational Spectroscopy of Light Element Impurities in Semiconductors P. Schwander, W. D. Rau, C. KisielowskL M. Gribelyuk, and A. Ourmazd, Defect Processes in Semiconductors Studied at the Atomic Level by Transmission Electron Microscopy N. D. Jager and E. R. Weber, Scanning Tunneling Microscopy of Defects in Semiconductors

Volume 52

SiC Materials and Devices

K. Jiirrendahl and R. F Davis, Materials Properties and Characterization of SiC V. A. Dmitriev and M. G. Spencer, SiC Fabrication Technology: Growth and Doping V. Saxena and A. J. Steckl, Building Blocks for SiC Devices: Ohmic Contacts, Schottky Contacts, and p-n Junctions M. S. Shur, SiC Transistors C D. Brandt, R. C Clarke, R. R. Siergiej, J. B. Casady, A. W. Morse, S. Sriram, and A. K Agarwal, SiC for Applications in High-Power Electronics R. J. Trew, SiC Microwave Devices

CONTENTS OF VOLUMES IN THIS SERIES

295

J. Edmond, H. Kong, G. Negley, M. Leonard, K. Doverspike, W. Weeks, A. Suvorov, D. Waltz, and C. Carter, Jr., SiC-Based UV Photodiodes and Light-Emitting Diodes H. Morkof, Beyond Silicon Carbide! III-V Nitride-Based Heterostructures and Devices

Volume 53 Cumulative Subject and Author Index Including Tables of Contents for Volume 1-50

Volume 54

High Pressure in Semiconductor Physics I

W. Paul, High Pressure in Semiconductor Physics: A Historical Overview N. E. Christensen, Electronic Structure Calculations for Semiconductors under Pressure R. J. Neimes and M. I. McMahon, Structural Transitions in the Group IV, III-V and II-VI Semiconductors Under Pressure A. R. Goni and K. Syassen, Optical Properties of Semiconductors Under Pressure P. Trautman, M. Baj, and J. M. Baranowski, Hydrostatic Pressure and Uniaxial Stress in Investigations of the EL2 Defect in GaAs M. Li and P. Y. Yu, High-Pressure Study of DX Centers Using Capacitance Techniques T. Suski, Spatial Correlations of Impurity Charges in Doped Semiconductors N. Kuroda, Pressure Effects on the Electronic Properties of Diluted Magnetic Semiconductors

Volume 55

High Pressure in Semiconductor Physics II

D. K. Maude and J. C. Portal, Parallel Transport in Low-Dimensional Semiconductor Structures P. C. Klipstein, Tunneling Under Pressure: High-Pressure Studies of Vertical Transport in Semiconductor Heterostructures E. Anastassakis and M. Cardona, Phonons, Strains, and Pressure in Semiconductors F. H. Pollak, Effects of External Uniaxial Stress on the Optical Properties of Semiconductors and Semiconductor Microstructures A. R. Adams, M. Silver, and J. Allam, Semiconductor Optoelectronic Devices S. Porowski and I. Grzegory, The Application of High Nitrogen Pressure in the Physics and Technology of III-N Compounds M. Yousuf Diamond Anvil Cells in High Pressure Studies of Semiconductors

Volume 56

Germanium Silicon: Physics and Materials

J. C. Bean, Growth Techniques and Procedures D. E. Savage, F. Liu, V. Zielasek, and M. G. Lagally, Fundamental Crystal Growth Mechanisms R. Hull, Misfit Strain Accommodation in SiGe Heterostructures M. J. Shaw and M. Jaros, Fundamental Physics of Strained Layer GeSi: Quo Vadis? F. Cerdeira, Optical Properties S. A. Ringel and P. N. Grillot, Electronic Properties and Deep Levels in Germanium-Silicon J. C. Campbell, Optoelectronics in Silicon and Germanium Silicon K. Eberl, K. Brunner, and O. G. Schmidt, Sil_yCy and Sil_x_rGexCr Alloy Layers

296

CONTENTS OF VOLUMES IN THIS SERIES

Volume 57

Gallium Nitride (GaN) II

K J. Molnar, Hydride Vapor Phase Epitaxial Growth of III-V Nitrides T. D. Moustakas, Growth of III-V Nitrides by Molecular Beam Epitaxy Z. Liliental-Weber, Defects in Bulk GaN and Homoepitaxial Layers C. G. Van de Walle and N. M. Johnson, Hydrogen in III-V Nitrides IV. G6tz and N. M. Johnson, Characterization of Dopants and Deep Level Defects in Gallium Nitride B. Gil, Stress Effects on Optical Properties C. KisielowskL Strain in GaN Thin Films and Heterostructures J. A. Miragliotta and D. K. Wickenden, Nonlinear Optical Properties of Gallium Nitride B. K. Meyer, Magnetic Resonance Investigations on Group III-Nitrides M. S. Shur and M. Asif Khan, GaN and AIGaN Ultraviolet Detectors C. H. Qiu, J. I. Pankove, and C. Rossington, III-V Nitride-Based X-ray Detectors

Volume 58

Nonlinear Optics in Semiconductors I

A. Kost, Resonant Optical Nonlinearities in Semiconductors E. Garmire, Optical Nonlinearities in Semiconductors Enhanced by Carrier Transport D. S. Chemla, Ultrafast Transient Nonlinear Optical Processes in Semiconductors M. Sheik-Bahae and E. W. Van Stryland, Optical Nonlinearities in the Transparency Region of Bulk Semiconductors J. E. Millerd, M. ZiarL and A. PartovL Photorefractivity in Semiconductors

Volume 59

Nonlinear Optics in Semiconductors II

J. B. Khurgin, Second Order Nonlinearities and Optical Rectification K. L. Hall, E. R. Thoen, and E. P. Ippen, Nonlinearities in Active Media E. Hanamura, Optical Responses of Quantum Wires/Dots and Microcavities U. Keller, Semiconductor Nonlinearities for Solid-State Laser Modelocking and Q-Switching A. Miller, Transient Grating Studies of Carrier Diffusion and Mobility in Semiconductors

Volume 60

Self-Assembled InGaAs/GaAs Quantum Dots

Mitsuru Sugawara, Theoretical Bases of the Optical Properties of Semiconductor Quantum Nano-Structures Yoshiaki Nakata, Yoshihiro Sugiyama, and Mitsuru Sugawara, Molecular Beam Epitaxial Growth of Self-Assembled InAs/GaAs Quantum Dots Kohki MukaL Mitsuru Sugawara, Mitsuru Egawa, and Nobuyuki Ohtsuka, Metalorganic Vapor Phase Epitaxial Growth of Self-Assembled InGaAs/GaAs Quantum Dots Emitting at 1.3/zm Kohki Mukai and Mitsuru Sugawara, Optical Characterization of Quantum Dots Kohki Mukai and Mitsuru Sugawara, The Photon Bottleneck Effect in Quantum Dots Hajime ShojL Self-Assembled Quantum Dot Lasers Hiroshi Ishikawa, Applications of Quantum Dot to Optical Devices Mitsuru Sugawara, Kohki MukaL Hiroshi Ishikawa, Koji Otsubo, and Yoshiaki Nakata, The Latest News

CONTENTS OF VOLUMES IN THIS SERIES

Volume 61

297

Hydrogen in Semiconductors II

Norbert H. Nickel Introduction to Hydrogen in Semiconductors II Noble M. Johnson and Chris G. Van de Walle, Isolated Monatomic Hydrogen in Silicon Yurij V. GorelkinskiL Electron Paramagnetic Resonance Studies of Hydrogen and HydrogenRelated Defects in Crystalline Silicon Norbert H. Nickel Hydrogen in Polycrystalline Silicon Wolfhard Beyer, Hydrogen Phenomena in Hydrogenated Amorphous Silicon Chris G. Van de Walle, Hydrogen Interactions with Polycrystalline and Amorphous Silicon-- Theory Karen M. McNamara Rutledge, Hydrogen in Polycrystalline CVD Diamond Roger L. LichtL Dynamics of Muonium Diffusion, Site Changes and Charge-State Transitions Matthew D. McCluskey and Eugene E. Hailer, Hydrogen in III-V and II-VI Semiconductors S. J. Pearton and J. W. Lee, The Properties of Hydrogen in GaN and Related Alloys J6rg Neugebauer and Chris G. Van de Walle, Theory of Hydrogen in GaN

Volume 62

Intersubband Transitions in Quantum Wells: Physics and Device Applications I

Manfred Helm, The Basic Physics of Intersubband Transitions Jerome Faist, Carlo SirtorL Federico Capasso, Loren N. Pfeiffer, Ken IV. West, Deborah L. Sivco, and Alfred Y. Cho, Quantum Interference Effects in Intersubband Transitions H. C. Liu, Quantum Well Infrared Photodetector Physics and Novel Devices S. D. Gunapala and S. V. Bandara, Quantum Well Infrared Photodetector (QWlP)Focal Plane Arrays

Volume 63

Chemical Mechanical Polishing in Si Processing

Frank B. Kaufman, Introduction Thomas Bibby and Karey Holland, Equipment John P. Bare, Facilitization Duane S. Boning and Okumu Ouma, Modeling and Simulation Shin Hwa Li, Bruce Tredinnick, and Mel Hoffman, Consumables I: Slurry Lee M. Cook, CMP Consumables II: Pad Francois Tardif, Post-CMP Clean Shin Hwa Li, Tara Chhatpar, and Frederic Robert, CMP Metrology Shin Hwa Li, Visun Bucha, and Kyle Wooldridge, Applications and CMP-Related Process Problems

Volume 64

Electroluminescence I

M. G. Craford, S. A. Stockman, M. J. Peanasky, and F A. Kish, Visible Light-Emitting Diodes H. ChuL N. F Gardner, P. N. Grillot, J. IV. Huang, M. R. Krames, and S. A. MaranowskL High-Et~ciency AIGalnP Light-Emitting Diodes R. S. Kern, IV. G6tz, C. H. Chen, H. Liu, R. M. Fletcher, and C. P. Kuo, High-Brightness Nitride-Based Visible-Light-Emitting Diodes Yoshiharu Sato, Organic LED System Considerations V. BuloviO, P. E. Burrows, and S. R. Forrest, Molecular Organic Light-Emitting Devices

298

CONTENTS OF VOLUMES IN THIS SERIES

Volume 65

Electroluminescence II

V. Bulovid and S. R. Forrest, Polymeric and Molecular Organic Light Emitting Devices: A Comparison Regina Mueller-Mach and Gerd O. Mueller, Thin Film Electroluminescence Markku Leskelii, Wei-Min Li, and Mikko Ritala, Materials in Thin Film Electroluminescent Devices Kristiaan Neyts, Microcavities for Electroluminescent Devices

Volume 66

Intersubband Transitions in Quantum Wells: Physics and Device Applications II

Jerome Faist, Federico Capasso, Carlo Sirtori, Deborah L. Sivco, and Alfred Y. Cho, Quantum Cascade Lasers Federico Capasso, Carlo Sirtori, D. L. Sivco, and A. Y. Cho, Nonlinear Optics in Coupled-Quantum-Well Quasi-Molecules Karl Unterrainer, Photon-Assisted Tunneling in Semiconductor Quantum Structures P. Haring Bolivar, T. Dekorsy, and 17. Kurz, Optically Excited Bloch Oscillations Fundamentals and Application Perspectives

Volume 67

Ultrafast Physical Processes in Semiconductors

Alfred Leitenstorfer and Alfred Laubereau, Ultrafast Electron-Phonon Interactions in Semiconductors: Quantum Kinetic Memory Effects Christoph Lienau and Thomas Elsaesser, Spatially and Temporally Resolved Near-Field Scanning Optical Microscopy Studies of Semiconductor Quantum Wires K. T. Tsen, Ultrafast Dynamics in Wide Bandgap Wurtzite GaN J. Paul Callan, Albert M.-T. Kim, Christopher A. D. Roeser, and Eriz Mazur, Ultrafast Dynamics and Phase Changes in Highly Excited GaAs Hartmut Haug, Quantum Kinetics for Femtosecond Spectroscopy in Semiconductors T. Meier and S. W. Koch, Coulomb Correlation Signatures in the Excitonic Optical Nonlinearities of Semiconductors Roland E. Allen, Traian Dumitric& and Ben Torralva, Electronic and Structural Response of Materials to Fast, Intense Laser Pulses E. Gornik and R. Kersting, Coherent THz Emission in Semiconductors

Volume 68

Isotope Effects in Solid State Physics

Vladimir G. Plekhanov: Elastic Properties; Thermal Properties; Vibrational Properties; Raman Spectra of Isotopically Mixed Crystals; Excitons in LiH Crystals; Exciton-Phonon Interaction; Isotopic Effect in the Emission Spectrum of Polaritons; Isotopic Disordering of Crystal Lattices; Future Developments and Applications; Conclusions

Volume 69

Recent Trends in Thermoelectric Materials Research I

H. Julian Goldsmid, Introduction Terry M. Tritt and Valerie M. Browning, Overview of Measurement and Characterization Techniques for Thermoelectric Materials

CONTENTS OF VOLUMES IN THIS SERIES

299

Mercouri G. Kanatzidis, The Role of Solid-State Chemistry in the Discovery of New Thermoelectric Materials B. Lenoir, H. Scherrer, and T. Caillat, An Overview of Recent Developments for BiSb Alloys Ctirad Uher, Skutterudites: Prospective Novel Thermoelectrics George S. Nolas, Glen A. Slack, and Sandra B. Schujman, Semiconductor Clathrates: A Phonon Glass Electron Crystal Material with Potential for Thermoelectric Applications

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