Semiconductors and Semimetals, Volume 71: Recent Trends in Thermoelectric Materials Research: Part Three (Semiconductors and Semimetals)

Recent Trends in Thermoelectric Materials Research 111 SEMICONDUCTORS AND SEMIMETALS Volume 71

Semiconductors and Semimetals A Treatise

Edited by R. K. Willardson CONSULTING PHYSICIST 12722 EAST 23R0 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 III SEMICONDUCTORS AND SEMIMETALS Volume 71 Volume Editor TERRY M. TRITT DEPARTMENT OF PHYSICS AND ASTRONOMY CLEMSON UNIVERSITY CLEMSON, SOUTH CAROLINA

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Contents PREFACE . . . . . . . LIST OF CONTRIBUTORS .

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XV

Chapter 1 Quantum Wells and Quantum Wires for Potential Thermoelectric Applications . . . . . . . . . . . . . . . . .

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M. S. Dresselhaus, Y.-M. Lin, T. Koga, S. B. Cronin, O. Rabin, M. R. Black, and G. Dresselhaus I. II. III. IV. V.

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PROOF-OF-PRINCIPLE STUDIES . . . . . . . . . . . . . . . . . . . . . . THE CONCEPT OF CARRIER POCKET ENGINEERING . . . . . . . . . . . . . . APPLICATION TO SPECIFIC 2 D SYSTEMS . . . . . . . . . . . . . . . . . . . 1. P b T e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. SiGe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Bi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. NANOWlRES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction to Nanowires . . . . . . . . . . . . . . . . . . . . 2. Structure and Synthesis o f Bismuth Nanowires . . . . . . . . . . . . 3. Electronic Structure o f Nanowires . . . . . . . . . . . . . . . . . 4. Doping o f Bi Nanowires . . . . . . . . . . . . . . . . . . . . . 5. Semi-Classical Transport M o d e l f o r Bi Nanowires . . . . . . . . . . 6. Temperature-Dependent Resistivity o f Bi Nanowires . . . . . . . . . . 7. Magnetoresistance o f Bi Nanowires . . . . . . . . . . . . . . 8. Seebeck Coefficient o f Bi Nanowires . . . . . . . . . . . . . . 9. Thermal Conductivity . . . . . . . . . . . . . . . . . . . . 10. Raman Spectra and Optical Properties . . . . . . . . . . . . . 11. Comparison between Bi and Sb Nanowires . . . . . . . . . . . . VII. SUMMARY REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 3 8 11 17 17 26 44 54 54 54 58 68 70 83 96 103 107 109

111 114 115

vi

CONTENTS

Chapter 2 Thermoelectric Transport in Quantum Well and Quantum Wire Superlattices . . . . . . . . . . .

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123

D. A. Broido and T. L. Reinecke I. INTRODUCTION

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II. SEMIQUANTITATIVE THEORY OF THE POWER FACTOR

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III. QUANTITATIVE THEORY OF THE POWER FACTOR . . . . . . . . . . . . . .

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1. Quantum Well Superlattices . . . . . . . . . . . . . . . . . . . . . 2. Quantum Wire Superlattices . . . . . . . . . . . . . . . . . . . . . 3. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . IV. LATTICE THERMAL CONDUCTIVITY AND THE FIGURE OF MERIT

135 139 140

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V. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES

Chapter 3

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152 153

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Thermionic Refrigeration . . . . . . . . . . . . . .

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G. D. Mahan I. INTRODUCTION

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157 160

II. VACUUM DEVICE . . . . . . . . . . . . .

III. ONE-BARRIER SOLID-STATE DEVICE . . . . .

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IV. MULTILAYER DEVICES . . . . . . . . . . .

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V. WHY BALLISTIC?

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VI. DISCUSSION . . . . . . . . . . . . . . .

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

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Chapter 4 Phonon Blocking Electron Transmitting Superlattice Structures as Advanced Thin Film Thermoelectric Materials . . . .

175

Rama Venkatasubramanian I. INTRODUCTION

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II. Low-TEMPERATURE HETEROEPITAXY OF B i E T e a - S b 2 T e 3 SUPERLATTICES

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III. IN-PLANE CARRIER TRANSPORT IN l i E T e a - S b E T e 3 SUPERLATTICES . . . . .

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IV. PHONON TRANSPORT IN B i E T e a - S b 2 T e 3 SUPERLATTICES

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V. MEASUREMENTS OF CROSS-PLANE THERMAL CONDUCTIVITY

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VI. LATTICE THERMAL CONDUCTIVITY IN SUPERLATTICES . . . . . . . V I I . MEAN FREE PATH REDUCTION IN SUPERLATTICES V I I I . DIFFUSIVE TRANSPORT ANALYSIS

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IX. PHONON REFLECTION AT SUPERLATTICE INTERFACES

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X. EQUIVALENCE BETWEEN DIFFUSIVE TRANSPORT AND LOCALIZATION .

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XI. K L AND /MFP OF ULTRA-SHORT-PERIOD SUPERLATTICES . . . . . . . . . . X I I . LOCALIZATION-LIKE BEHAVIOR IN S i - G e

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SUPERLATTICES . . . . . . . . .

X I I I . CROSS-PLANE CARRIER TRANSPORT IN l i E T e a - S b 2 T e

3 SUPERLATTICES

X I V . ADIABATIC PELTIER EFFECT IN THIN FILM THERMOELEMENTS

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X V . DIFFERENTIAL COOLING IN BULK AND SUPERLATTICE THERMOELEMENTS .

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X V I . SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . .

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

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Chapter 5

P h o n o n T r a n s p o r t in L o w - D i m e n s i o n a l

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Structures

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203

G. Chen I. INTRODUCTION

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II. PHOYONS IN BULK AND Low-DIMENSIONAL MATERIALS . . . . . . . . . . . 1. Phonon Thermal Conductivity in Bulk Materials . . . . . . . . . . . . . 2. Phonon Dispersion in Nanostructures . . . . . . . . . . . . . . . . . . III. THIN FILM THERMAL CONDUCTIVITY MEASUREMENT TECHNIQUES . . . . . . . 1. Microsensor Methods . . . . . . . . . . . . . . . . . . . . . . . . 2. Optical Pump-and-Probe Methods . . . . . . . . . . . . . . . . . . . 3. Optical-Electrical Hybrid Methods . . . . . . . . . . . . . . . . . . IV. ANALYTICAL TOOLS . . . . . . . . . . . . . . . . . . . . . . . . . 1. Lattice Dynamics and Phonon Dispersion Analysis . . . . . . . . . . . 2. Boltzmann Transport Equation . . . . . . . . . . . . . . . . . . . 3. Boundary Conditions f o r B T E . . . . . . . . . . . . . . . . . . . . 4. Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . 5. Molecular Dynamics Simulation . . . . . . . . . . . . . . . . . . V. THERMAL CONDUCTIVITY OF NANOSTRUCTURES . . . . . . . . . . . . . 1. Thermal Conductivity o f Single-Layer Thin Films . . . . . . . . . . 2. Thermal Conductivity o f Superlattices . . . . . . . . . . . . . . . 3. Thermal Conductivity o f One-Dimensional Structures . . . . . . . . 4. Heat Conduction in Nanoporous and Mesostructures . . . . . . . . VI. PHONON ENGINEERING IN NANOSTRUCTURES . . . . . . . . . . . . . VII. CONCURRENTELECTRON-PHONON MODELING . . . . . . . . . . . . VIII. SUraMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Bi2Te3 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

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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, AT, 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 = ~2tTT/~, where ~ is the Seebeck coefficient, a the electrical conductivity, ~ the total thermal conductivity (~ = '~L + 2E; the lattice and electronic contributions, respectively), and T is the absolute temperature in kelvins. The Seebeck coetficient, or thermopower, is related to the Peltier effect by H - ~ T = Qp/I, where H 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 figure of merit of the thermoelectric material or materials. Both ~/ and COP are proportional to (1 + Z T ) 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 Bi2Te 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

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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 Bi2Te 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 Stuart 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

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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

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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. R. BLACK (1), Department of Electrical Engineering and Computer

Science, Massachusetts Institute of Technology, Cambridge, Massachusetts D. A. BROIDO (123), Department of Physics, Boston College, Chestnut Hill, Massachusetts G. CHEN (203), Mechanical and Aerospace Engineering Department, University of California, Los Angeles, California S. B. CRONIN (1), Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts G. DRESSELHAUS (1), Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts M. S. DRESSELHAUS(1), Department of Physics and Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts T. KOGA (1), Department of Applied Physics, Harvard University, Cambridge, Massachusetts Y.-M. LIN (1), Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts G. D. MAHAN (157), Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee O. RABIN (1), Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts T. L. REINECKE (123), Naval Research Laboratory, Washington, D.C. RAMA VENI~TASUBRAMANIAN (175), Research Triangle Institute, Research Triangle Park, North Carolina XV

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SEMICONDUCTORS AND SEMIMETALS, VOL. 71

CHAPTER

1

Quantum Wells and Quantum Wires for Potential Thermoelectric Applications M. S. Dresselhaus, Y.-M. Lin, S. B. Cronin, O. Rabin, M. R. Black, and G. Dresselhaus MASSACHUSETTSINSTITUTEOF TECHNOLOGY CAMBRIDGE, MASSACHUSETTS

T. Koga HARVARD UNIVERSITY

CAMBRIDGE, MASSACHUSETTS

I. II. III. IV. V.

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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PROOF-OF-PRINCIPLE STUDIES . . . . . . . . . . . . . . . . . . . . . . THE CONCEPT OF CARRIER POCKET ENGINEERING

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APPLICATIONTO SPECIFIC 2 D SYSTEMS . . . . . . . . . . . . . . . . .

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1. P b T e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. SiGe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Bi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

VI. NANOWlRES . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Introduction to Nanowires . . . . . . . . . . . . . . . Structure and Synthesis o f Bismuth Nanowires . . . . . . . Electronic Structure o f Nanowires . . . . . . . . . . . . . Doping o f Bi Nanowires . . . . . . . . . . . . . . . . . Semi-Classical Transport M o d e l f o r Bi Nanowires . . . . . . Temperature-Dependent Resistivity o f Bi Nanowires ..... Magnetoresistance o f Bi Nanowires . . . . . . . . . . . . Seebeck Coefficient o f Bi Nanowires . . . . . . . . . . . . Thermal Conductivity . . . . . . . . . . . . . . . . . . R a m a n Spectra and Optical Properties . . . . . . . . . . . Comparison between Bi and Sb Nanowires . . . . . . . . .

. . . . . . . .

. . . . . . . .

44 54 54 54 58 68 70 83 96 103 107 109 111

VII. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I.

17

114 115

Introduction

Evaluation of new materials including low-dimensional materials for thermoelectric applications is usually made in terms of the dimensionless thermoelectric figure of merit Z T where T is the temperature (in degrees Copyright 9 2001 by Academic Press

All rights of reproduction in any form reserved. ISBN 0-12-752180-1 ISSN 0080-8784/01 $35.00

2

M.S.

DRESSELHAUS ET AL.

Kelvin) and Z is given by 82o-

Z = ~

(1)

where S is the thermoelectric power or Seebeck coefficient, a is the electrical conductivity, and K--

K e -~ Kph

(2)

is the thermal conductivity, which includes contributions from carriers 0%) and from the lattice (~Cph).Equation (1) emphasizes the importance of a large S for high thermoelectric performance (or high ZT), where S denotes the voltage generated by a thermal gradient. Large values of Z T require high S, high a, and low tc. Since an increase in S normally implies a decrease in a because of carrier density considerations, and since an increase in o- implies an increase in the electronic contribution to ~c as given by the WiedemannFranz law, it is very difficult to increase Z in typical thermoelectric materials. The best commercial 3D thermoelectric material is Bio.5Sbl.5Te 3 in the Bi/(l_x)SbzxTe3(l_y)Se3y family with a room temperature Z T ~ 1. It is believed that if materials with Z T ~ 3 could be developed, many more practical applications for thermoelectric devices would follow. Early work on low-dimensional thermoelectricity focused on general theoretical models using the simplest kind of calculations, such as for a quantum confined low-dimensional electron gas (Hicks and Dresselhaus, 1993a). The results of these simple calculations implied that significant enhancement in Z T could indeed be attained within the quantum well of a superlattice when the quantum well was made of a good bulk thermoelectric material, the quantum well width was sufficiently small, and the optimum carrier concentration was present. Even greater enhancement in Z T was predicted for a good thermoelectric material when prepared as a 1D quantum wire (Hicks, 1996; Hicks and Dresselhaus, 1993b). Reduced dimensionality [as occurs in quantum wells (2D) or quantum wires (1D)] has been considered as one approach for increasing ZT, because of the many advantages provided by such systems (Dresselhaus et al., 1997a). Generic advantages of low dimensional materials include: (1) enhancement of the density of states near the Fermi energy E v, leading to an increase in the Seebeck coefficient; (2) opportunities to exploit the anisotropic Fermi surfaces in multivalley cubic semiconductors; and (3) opportunities to increase the boundary scattering of phonons at the barrier-well interfaces, without as large an increase in electron scattering. Although isolated quantum dots (0D systems) do not provide a conduction path, coupled quantum dots might provide adequate conduction paths for carriers but less effective heat conduction paths for phonons. The study of quantum dots for thermoelectric applications has not yet been systematically pursued, but this approach does offer significant promise for thermoelectric applications.

1

QUANTUMWELLS AND QUANTUMWIRES

3

In addition to the use of a good bulk thermoelectric material for the quantum well or quantum wire, low-dimensional thermoelectricity benefits from the judicious choice of the barrier material as one that satisfies the following criteria: (1) low thermal conductivity to minimize barrier heat conduction, which would decrease the overall Z T compared to the Z T of the quantum well material alone; (2) lattice matching and a similar thermal expansion coefficient as the quantum well material (these properties are needed to provide sharp interfaces that do not degrade the carrier mobility and electronic properties of the quantum well material as the well width decreases); (3) a sufficiently large bandgap, band offset, and barrier layer thickness to confine carriers within the quantum well; and (4) low interdiffusion of the barrier material into the quantum well region, which could give rise to loss of the superlattice structure, increased electron-impurity scattering and uncontrolled carrier doping in the quantum well region. Using an alloy material in the barrier region lowers the thermal conductivity through the phonon-random impurity scattering mechanism. Low dimensionality also enables some materials that do not exhibit a high Z T in 3D to show a high Z T value in lower dimensions (Hicks et al., 1993a, 1993b, 1993c). For example, bulk bismuth is a semimetal for which the contributions from electrons and holes to the Seebeck coefficient S are of opposite sign and almost cancel each other, so that the net S is very small, although the contributions to S from electrons and holes are large, individually. Low-dimensional systems form subbands for the directions where quantum confinement occurs, and the resulting lowest quantized subband in the conduction band lies above the 3D conduction band extremum, and correspondingly the highest quantum subband in the valence band lies below the 3D valence band extremum. Furthermore, as the size of the quantum well (2D), quantum wire (1D), or quantum dot (0D) decreases, the lowest conduction subband increases in energy, and the highest valence subband decreases in energy. Therefore, the band overlap between the valence and conduction bands, which is responsible for the semimetallic behavior of materials such as bismuth, will vanish at some critical confinement size de, where the semimetal-semiconductor transition occurs. In the semiconducting regime, single carrier transport can be achieved, thereby enabling effective use of these materials for thermoelectric applications. In this chapter, we summarize the present status of efforts to enhance Z T by using low-dimensional systems, focusing on accomplishments to date, present directions, and future trends for research in the field. II.

Models

Because of the constraints imposed by carrier confinement in a lowdimensional system and the need for optimization of the carrier density for achieving a high Z T, theoretical modeling has played a significant role in the development of the field of low-dimensional thermoelectricity.

4

M.S.

DRESSELHAUS ET AL.

Early work employed the simplest possible model for thermoelectricity in 2D quantum well structures (Hicks and Dresselhaus, 1993a, 1993b, 1993c; Dresselhaus et al., 1997a, 1997b). More specifically it was assumed that the electrons in the valence and conduction bands are in simple parabolic energy bands and that the electrons occupy only the lowest subband of the quantum well. The electronic dispersion relations for a 2D system are then given by h27r2 h 2 k~2 h2 ky2 #2o(kx, ky) = ~ + ~ + 2m~d 2

(3)

where d w is the width of the quantum well; and mx, my, and mz are the effective mass tensor components of the constant energy surfaces. It is further assumed that the current flows in the x direction and that quantum confinement is in the z direction. The corresponding relation used for a square 1D quantum wire of width d w is h2k~2 h2~:2 h27~2 #~o(k~) = 2-~m~-~ 2 m y d~~~2, m r v~ d

(4)

where the current flow is also along the x direction, and quantum confinement occurs in the y and z directions. All the thermoelectric transport coefficients (Hicks, 1996; Ashcroft and Mermin, 1976; Sun and Bolton, 1996), which are in general tensors, can be calculated by (Hicks, 1996; Hicks and Dresselhaus, 1993a, 1993b, 1993c; Hicks et al., 1993) (~ = ~f(0)

S -- - -

(1) ~

(5)

(6)

( ~ o ( 0 ) ) - 1. o~o(1)

where

d3k

~.(k)v.(k)v.(k)(~.(k)

- 0 ".

(8)

Here, the sum is over the n carrier pockets, and the integral over k-space involves d3k when the integration is over three-dimensional k-space, where the three-dimensional (3D) density of states factor under the integral sign is given by d3k/4g 3. To extend the theory into lower dimensional transport,

l

QUANTUMWELLSAND QUANTUMWIRES

5

we replace the 3D density of states factor by appropriate expressions. For example, in the case of a two-dimensional (2D) system, the density of states factor becomes dZk/27rZdw, where dw is the thickness of the quantum well, so that

d2k

z,,v(k)v.,~(k)v,,~(k)(e,,~(k) - ()"

o(~) 2D __ e 2 E f

(9)

tl,V

where the index v is the subband index. Similarly, for the one-dimensional (1D) density of states, we have dk/lrd 2, where dw is the quantum wire thickness, which yields

z,,v(k)v,,,~(k)v,,,~(k)(e,,,v(k) - ()~

(10)

n,Y

Because of the factor (1/dw) in Eq. (9) and the factor (1/d 2) in Eq. (10), we can expect larger density of states factors as the quantum well or wire thickness decreases, and as the dimensionality decreases. Although we have written the &ct') tensors for 2D and 1D systems in Eqs. (9) and (10) in the same tensor form as in the 3D case, the ~o~,) tensors in lower dimensional systems are themselves reduced to lower dimensionalities. In some cases, the ~o~,) tensors reduce to scalars, because the velocity vector v,,v is confined to an axis determined by the transport direction and therefore becomes a scalar. In this case, the anisotropy of the material is described by the quantum confinement of the anisotropic band structures. This issue is further discussed in terms of models for specific materials. From the integrals in Eqs. (8), (9), and (10), follow the simplest generic expressions for the thermoelectric figure of merit for 3D, 2D, and 1D systems. Using Eqs. (1), (2), and (5) through (8), we obtain a generic expression for Z3DT:

Z3DT ._

[(5F3/2/3F1/2) _ ~,]2 3F1/2 v _ (25FZ3/z/6F1/2) 1/B3D + ~F5/2

(11)

where the dimensionless quantity B3D , which is dependent on materials parameters, is given by

B3D = ~

l (2kBY~ 3/2 (mxmymz) 1/2 k2Tlax ~ h2 ]

(12)

eKph

and the Fermi-Dirac-related integral F i denotes

F, = Fi(~*) =

f o exp(x xidx - (*) + 1

(13)

6

M . S . DRESSELHAUS ET AL.

where x is a dimensionless energy, x = E/k~T, and (* is a dimensionless electrochemical potential (* = (/knT. Equation (11) is then solved numerically. For a given value of B3D, the reduced chemical potential (* = (/k~T is varied to maximize the value of Z3DT. The corresponding generic expression for Z2DT is given by [(2F1/Fo) - (*]2F o

ZzDT =

1/B2 D +

(14)

3F 2 --4F2/Fo

where the dimensionless quantity B2D is given by (15)

/z k2 T~x .

eKph For a specific value of the quantity B2D , the thermoelectric figure of merit within the quantum well Z2DT is optimized by varying the chemical potential or the doping level of the system. The higher the BED value, the higher is the optimal ZEDT value (Hicks, 1996). Therefore, the quantity BED provides a guideline for selecting good thermoelectric materials and for designing optimum quantum well thermoelectric superlattices. For a generic quantum wire, the ZIDT within the quantum wire becomes (Hicks, 1996): Z,DT=

[(3F'/z/F-'/2) -- (*]2 89 5 1/BID + 2F3/2 -(9F2/2)/(2 F -

(16) 1/2)

where the dimensionless quantity B1D is given by

2 (2k.ql,,

(mx) 1/2 k2T#x

(17)

eKph For a specific value of the quantity B1D , the figure of merit within the square quantum wire Z1DT is optimized by varying the chemical potential of the system. The higher the BiD value, the higher is the optimal Z1DT value (Hicks, 1996). Therefore, the quantity BID provides a guideline for selecting good thermoelectric materials and for designing optimum quantum wire thermoelectric materials. In the case of quantum confined systems, explicit results for Z2DT (quantum well) or Z1DT (quantum wire) within the quantum well or wire were obtained, using one of the following highly simplified approaches for estimating the phonon contribution to the thermal conductivity/~ph: (1) /~ph is conservatively approximated using 3D experimental data; (2) /Eph is taken from 3D experimental data for small phonon mean free paths (lL < dw),

1 QUANTUM WELLS AND QUANTUM WIRES 15.0

-

.

-

.

-

.

-,

.

v..

BIzTe, ZT

.

7

_

.

10.0

10

5.0

0.0

2D

9

i

10

.,

2O

9

,

30

-

40

.

.

.

.

.

5O

6O

dw (h) FIG. 1. Calculateddependence of ZT within the quantum well or within the quantum wire on the well or wire width dw for a BizTe3-1ikematerial at the optimum doping concentration for transport in the highest mobility direction. Also shown is the ZT for bulk (3D) BizTe3 calculated using the corresponding 3D model (Hicks, 1996; Hicks and Dresselhaus, 1993b).

while

l L --

dw is used in the kinetic theory determination of ~Cphgiven by CvVlL /'r

=

(18)

3

for l L > dw, where Cv and v are, respectively, the heat capacity and the velocity of sound of the quantum confined material; (3) ~:ph is obtained from a more detailed treatment of the scattering of phonons by the interface between the quantum confined region and the barrier region (Sun and Bolton, 1996). Using these simple assumptions, a substantial enhancement was calculated for Z T within the quantum well for 2D systems having small quantum well widths relative to their corresponding bulk values. To make these calculations more useful, we show in Fig. 1, the enhancement of Z T within a BizTe 3 quantum well as a function of dw, and an even greater enhancement in Z T is predicted for BizTe 3 when prepared as a 1D quantum wire (Hicks, 1996; Hicks and Dresselhaus, 1993b). The results of Fig. 1 suggest that a good thermoelectric material in 3D might be expected to exhibit even higher Z T values in reduced dimensions. To make a fair comparison between 3D and lower dimensional materials, all Z T values in Fig. 1 are given for the optimum carrier concentration (i.e., the most favorable placement of the Fermi level for a given geometry for the low-dimensional BizTe 3 material). However, the quantum confinement lengths required for enhanced Z T are quite small for a BizTe3-1ike material (see Fig. 1) especially in comparison

8

M . S . DRESSELHAUS ET AL.

to the basic structural unit of Bi2Te3, which is ~ 10 ,~ in the c direction. By selecting thermoelectric materials with smaller effective masses, it is possible to increase the length scale over which an enhanced Z T can be achieved. The larger length facilitates the preparation of good interfaces from a materials science point of view. For example, for actual PbTe-based superlattices, the interface properties begin to be degraded for dw < 1.7 nm (Harman et al., 1998; Hicks et al., 1996; Sun, 1999). In early experimental studies, PbTe was chosen as a quantum well for thermoelectric applications because of its desirable thermoelectric and materials science properties. Regarding its thermoelectric properties, PbTe has a reasonably high room temperature Z T in bulk form ( Z T ~_ 0.4), reflecting the high carrier mobility, multiple anisotropic carrier pockets, and low thermal conductivity that can be achieved under isoelectronic alloying with Sn and Se. In addition, calculations indicated that carrier confinement and enhanced Z T could be achieved for quantum well widths <100,~ (Hicks et al., 1996). High-mobility quantum well superlattices can be prepared with PbTe as the quantum well material and Pbl_xEuxTe as the barrier material (Yuan et al., 1993; Springholz et al., 1993), with wellcontrolled and stable interfaces, using epitaxial growth techniques, such as molecular beam epitaxy. Good lattice matching and similar thermal expansion coetficients (Springholz and Bauer, 1993) across the interfaces and relatively rapid growth rates permitted the preparation of PbTe-PbEuTe superlattices with ~ 100 superlattice periods (Harman et al., 1996). Bismuth doping of the Pbo.927Euo.ov3Te barrier region resulted in n-type PbTe quantum wells. Modulation doping of the superlattice yielded quantum wells with carrier concentrations that could be varied to levels greater than 1019 electrons/cm 3 within the quantum well, and Hall mobilities comparable to and in some cases greater than that in the best 3D films prepared in the same apparatus with a comparable doping level (Harman et al., 1996). Quantum confinement could be achieved for quite a low Eu concentration of x = 0.073, where the band offsets for the conduction and valence bands were calculated to be 171 and 140meV, respectively, for a bandgap of 630 meV in the barrier regions and 319 meV in the quantum wells at 300 K (Yuan et al., 1993).

III.

Proof-of-Principle Studies

Early studies of low-dimensional thermoelectricity focused on the demonstration of proof-of-principle of enhanced Z T within a quantum well structure using simple theoretical models, comparisons between theory and experiment, and comparisons between the low-dimensional (2D) and 3D Seebeck coefficient, all comparisons being carried out under optimum

1

QUANTUM WELLS AND QUANTUM WIRES

400

9

edw=20 A

iI

'300

o~,,

4

200

100 . . . . . .

0

'

'

i

1 0 ~8

.

.

.

.

.

.

.

.

nil,, , (cm -3)

i

.

.

.

.

.

1 0 '~

FIG. 2. Seebeck coefficient of n-type bulk PbTe and (111) oriented PbTe MQWs as a function of Hall carrier concentration at 300 K. The experimental results for bulk PbTe and for the (111) oriented MQWs taken from Harman et al. (1996) are plotted with various symbols according to their quantum well thickness dw. The solid line shows an empirical fit to the Seebeck coefficient as a function of carrier concentration for bulk PbTe given by S(laV/ K) = - 4 7 7 + 175 log~o(n/10 ~7 cm -3) (Harman et al., 1996). The points for quantum well samples with dw in the range 15-25 ,/k show an enhanced Seebeck coefficient.

doping conditions. Proof-of-principle studies were first carried out for the PbTe-Pb~_xEuxTe superlattice (Hicks et al., 1996) and more recently for the strain-relaxed Si-Sio.TGeo. 3 superlattice (Sun et al., 1999a). The first proof-of-principle study was done on PbTe-Pb~_~EuxTe quantum well superlattices (x -_-0.073) (Hicks et al., 1996), where PbTe was the quantum well material and P b l _ x E u J e (x ~- 0.073) was the barrier material. With this materials system (Yuan et al., 1993; Springholz et al., 1993), high-quality superlattice samples with more than 100 superlattice periods were prepared using the molecular beam epitaxy (MBE) technique (Harman et al., 1997). At the heart of that study is the demonstration of the enhancement of the Seebeck coefficient, as shown in Fig. 2 for quantum well widths dw 1.5 < dw < 2.5 nm (Harman et al., 1996). This enhanced Seebeck coefficient is directly related to the increased density of states at each subband edge (Section I). Furthermore, good agreement was obtained between experimental measurements of S2n as a function of carrier density and of quantum well thickness (Hicks et al., 1996) and the corresponding calculations for the PbTe-Pb~_xEuxTe superlattice, using only literature values for the band parameters. No adjustable or fitting parameters were employed in the comparison between experiment and theory (Hicks et al., 1996). The reason for making the comparison between theory and experiment for S2n rather than S2a is to test the validity of the theoretical model in terms of intrinsic phenomena rather than phenomena sensitive to materials processing conditions that more strongly influence the carrier mobility. The proof-of-principle study on n-type PbTe was carried out using large barrier widths d b >> dw, where d b ~ 400/k, and small quantum well widths

10

M . S . DRESSELHAUS ET AL. 100

,

,

_

,,

,l,,,

,

60 E o % ,. "-',

%

40

t.-

20

i

1

,

d

2

i

3 (nm)

,

i

4

i

5

FIG. 3. The composition between experimental data for S2n vs quantum well width dw and the theoretical curve at optimal doping level to maximize Z2DT for the optimum thermoelectric figure of merit for a strain relaxed Si-Sio.7Geo.3 quantum well superlattice of 15 periods at room temperature (Sun, 1999).

Z3DT w a s in fact small because of the small fraction of the sample that contributed to S. The demonstration of proof-of-principle in the strain relaxed SiSio.TGeo. 3 superlattice system is shown in Fig. 3. These strain-relaxed Si-Sio.vGeo. 3 2D superlattices were grown on an insulating SOI (silicon on insulator) substrate to minimize contributions to the Seebeck coefficient from the substrate, followed by a thin ( ~ 2000 A) graded Si 1 _xGe x buffer layer to relax the lattice strain (Sun et al., 1999a). For the proof-of-principle studies, superlattices were grown with 15 superlattice periods, having quantum well widths between 10 and 50]k alternating with 300A of Sio.TGeo. 3 barrier layer. The measured S2n at 300K were compared to model calculations based on the well-established band structures of Si and Sii_xGe~ alloys, using only literature values and no adjustable parameters. This comparison between the model calculation and measurements for S2n provides clear confirmation that reduction of the size of the quantum well results in an increase in SEn and that the model calculation has a similar dependence on quantum well width as the experimental points, when taking account of experimental uncertainties in the data (Sun et al., 1999a). The systematic discrepancy in Fig. 3 between the theoretical modeling and the experimental data was attributed to the following three reasons: (1) the measured Seebeck coefficient for the various superlattice samples (15 periods) is larger than the Seebeck coefficient for a single period by about 10% due to substrate corrections; (2) the assumption of a uniform carrier distribution leads to an underestimate of the carrier density by 5-20%; and (3) transmission electron microscope (TEM) studies reveal that the samples d w ,~ 20 ,~, so that the

1

QUANTUMWELLS AND QUANTUM WIRES

11

have a wavy in-plane microstructure, and have thinner layers than measured by a surface profilometer (Sun et al., 1999a). These proof-of-principle studies were valuable in establishing the general validity of the models for low-dimensional thermoelectricity, which indicated that higher Z T values could be achieved by exploiting the higher density of states, which increases the power factor S2a, and by taking advantage of the interface boundary scattering of the superlattice to reduce the phonon thermal conductivity Kph more than the electrical conductivity a. Now that the basic concepts behind low-dimensional thermoelectricity have become widely accepted, the research focus has shifted to more detailed studies of specific quantum well systems and to using low-dimensional superlattices to enhance the 3D thermoelectric figure of merit of the entire superlattice Z3DT, including both the quantum well and barrier regions, as discussed in Section IV.

IV.

The Concept of Carrier Pocket Engineering

The process by which low-dimensional superlattices of given constituents are designed to optimize their 3D thermoelectric properties has been called "carrier pocket engineering" (Koga et al., 1998b, 1999a, 1999b, 1999c). In this approach, the large barrier widths used in the proof-of-principle studies (see Section II) are greatly reduced to become comparable to the quantum well widths, so that the electron wave functions are no longer confined to the quantum wells, and the carrier confinement conditions considered in Section II are relaxed. The design parameters that are used in this process include: (1) the layer thicknesses of the quantum well and barrier, dw and dB, respectively; (2) the growth direction of the superlattice [such as the (001) or (111) directions], which is selected to maximize the density of states near the Fermi level; (3) the composition and/or lattice constant of the substrate, which is selected to control the strain of the quantum well and barrier constituents, in order to maximize the density of states near the Fermi level. The possibility of using not only the quantum well regions but also the barrier regions to contribute to the Seebeck coefficient and to Z3DT has also been considered within the framework of the carrier pocket engineering concept. For example, in the GaAs-Gal_xAlxAs (x > 0.4) superlattice system, carriers in the F-point and L-point pockets of the GaAs (quantum well) layers contribute to S, while carriers in the X-point pockets in the Gal_xAlxAs (barrier) layers of the superlattices also contribute to S (Koga et al., 1998b). In the implementation of the carrier pocket engineering calculations, a Kronig-Penney model is used for matching the wavefunctions and their derivatives across the quantum well-barrier interfaces, and for calculating

12

M.S.

DRESSELHAUS ET AL.

the bound state energy levels. Since the particle current at the barrier-well interfaces has to be conserved, we can write the following boundary conditions on the wave functions and their derivatives across the interface:

kI'/a "- kIJb,

1 dWa m~ dz

=

1 dWb m b dz

(19)

where a and b, respectively, denote the quantum well and barrier regions on either side of the interface. From these boundary conditions and assuming square well potentials, the Kronig-Penney energy dispersion relation becomes: (Q/mb) 2 -- (K/ma) 2 2(Q/ma)(K/m a)

sinh(Qb) sin(Ka) + cosh(Qb) cos(Ka) = cos k(a + b) (20)

where K and Q in the quantum well and barrier, respectively, are given by K-"

x/2maE/h,

Q = x/2mb(U -- E)/h

(21)

and U is the barrier height, a is the quantum well thickness, b is the barrier layer thickness, m, is the effective mass for the quantum well layer, mb is the effective mass for the barrier layer, and E is the energy. Assuming that we have quantum confinement along the z axis, the z component of the effective mass tensor is obtained from the cross section of the ellipsoidal constant energy surface for the quantum well and barrier layer materials, whereas the in-plane components of the effective mass tensor for the 2D quantum well are obtained by the projection of the 3D constant energy surfaces on the plane of the quantum well. Explicit applications of the carrier pocket engineering concept have been carried out thus far for GaAs-Gal_xAlxAs (Koga, 1998b) and Si-Sil_xG % superlattices (Koga et al., 1999b, 1999c). The carrier pocket engineering concept was first applied to the GaAsGa~_xAlxAs system because the electronic band structure was well known and the materials science for the fabrication of controllable superlattices was well established, although bulk GaAs itself is not a good thermoelectric material (Z3DT =0.14 at 300K). Using the carrier pocket engineering approach, a large enhancement in the thermoelectric figure of merit for the whole superlattice, Z3DT, was predicted for short period GaAs-A1As superlattices relative to ZaDT for bulk GaAs (Koga et al., 1998b). In calculating the energy dispersion relations for short period GaAs-GaA1As superlattices along the superlattice axis (z axis) within the context of the cartier pocket engineering concept, carrier pockets at each high symmetry point (F, X, and L points) in the Brillouin zone were considered. A plane wave approximation was used for the wave functions, and boundary conditions of the

1

QUANTUM WELLS AND QUANTUM WIRES

13

Kronig-Penney model were applied (Kittel, 1986): kI'/GaAs = ~I'/A1As and the GaAs-A1As interface, where qJGaAs and klJAIAs denote the electron wavefunction in the GaAs and AlAs layers, respectively, and mz(GaAs) and mz(AiAs) are the z components of the effective mass tensor for the GaAs and AlAs layers, respectively. The conduction band offsets at each symmetry point are calculated using the experimentally determined A E c / A E 9 = 0.68 (Adachi, 1985), where A E c is the conduction band offset at the F point, and AE0 is the difference in the direct energy bandgap at the F point between GaAs and AlAs. The in-plane components of the effective mass tensor are calculated by projecting the 3D constant energy surface for the quantum well material onto the plane of the quantum wells. It is noted that, for band offsets calculated in this way, the F- and L-point quantum wells are formed within the GaAs layers and the X-point quantum wells are formed within the AlAs layers, so that the carriers that are located in different carrier pockets in reciprocal space can be physically separated in real space, thereby minimizing the intervalley scattering. This reduction in scattering is also favorable for achieving high Z3DT. The band parameters used in the carrier pocket engineering model for GaAs-A1As superlattices are summarized in Table I (Koga et al., 1998b). The transport coefficients (a, S, and Ke) for the whole superlattice are calculated using textbook equations in terms of the ~2060(')function defined in Eq. (9) (Ashcroft and Mermin, 1976) and using the simplest possible model of a constant relaxation time approximation and a parabolic energy

(dqflGaAs/dZ)/mz
TABLE I BAND PARAMETERSAT THE F, X, AND L SYMMETRYPOINTS FOR GaAs AND AlAs (SEE TEXT) Band parameter [mJm]cans a [mJm]cans" [mJm]AIAs a [ml/m]klafl Nb U o (eV) c /2 (cm2/V 9s) a Offset energy (eV) e

F point 0.067 O.150 1 1.084 3000 0

X point

L point

0.23 1.3 O.19 1.1 3 0.242 180 0.234

0.0754 1.9 0.0964 1.9 4 0.132 950 0.284

aTransverse mt and longitudinal m~ mass data taken from Adachi (1985). bNumber of equivalent carrier pockets. cPotential barrier height (data taken from Adachi, 1985). aBulk carrier mobilities for the quantum well material at the F, X, and L points (data from Madelung, 1987a). eThe offset energy denotes the energy of the bottom of the quantum wells at the X and L points relative to the F point of bulk GaAs (Koga et al., 1998b).

14

M . S . DRESSELHAUS ET AL.

dispersion relation (Hicks, 1996). For superlattice samples that have an energy dispersion along the z axis, 5~ ) for the whole superlattice is evaluated by

~ d~

(22)

where p(e) is the density of states as a function of energy for the whole superlattice for the pertinent subband and P ZD is the density of states for the pertinent quantum well, assuming a single subband in an infinite potential well with no band-broadening effects where P ZD is given by (m t(2 D)mt(2 D)) 1/ 2 P2D =

(23)

~dwh2

The ~:FSLG~functions for the whole superlattice are calculated by summing the 5~ ) contributions over all subbands. Using detailed calculations for the lattice thermal conductivity for a GaAs-A1As superlattice as a function of dw (Chen, 1997), the calculations for the S, a, and K are combined to calculate the Z3DT for the whole superlattice. For an equal thickness (d w = dB) superlattice at the optimal doping level, Z3DT has been calculated, first assuming that only the F point subband is occupied. The results of this calculation are plotted in Fig. 4 normalized to

15-

-

I0 CD

AE (meV)

v

%5

0

0

L

l

,

t

40

,

I

d w (A)

,

t

80

.,.

I . ,_

120

FIG. 4. Plot of the normalized Z 3 D T for a GaAs-A1As superlattice with d w - d B and carriers only in the F point pocket, at the optimum doping concentration. For the superlattice, Z 3 D T is normalized to that of bulk GaAs calculated for T - 300 K. The inset shows the density of states for electrons as a function of energy relative to the conduction band edge (AE) for selected superlattices ( d w - 20 [solid line], 40 [short dashed line], and 80 [long dashed line] A, respectively) as well as for bulk GaAs (dash-dotted line). See Table I for the band parameters used (Koga e t al., 1998b).

1

QUANTUM WELLS AND QUANTUM WIRES

15

Z3DT for bulk GaAs (ZT)bulk, which is calculated in a similar way at the optimum doping level. The density of states for electrons near the band edge is shown in the inset of Fig. 4 as a function of energy for dw = 20, 40, and 80 A and for bulk GaAs, showing the increase in the density of states as the quantum well width decreases. While (ZT)bulk for bulk GaAs is as small as 0.0085 at the optimum carrier concentration, Z3DT for the (20 A-20 A) F point GaAs-A1As superlattice is more than 0.1 at the optimum carrier concentration, which is more than a 10 times enhancement in Z3DT relative to the value for bulk GaAs. The assumption that only the F point is occupied with carriers at the optimum doping concentration is valid for GaAs-A1As superlattices with dw > 30 A or for GaAs-Al~_~GaxAs superlattices with x < 0.4 (Koga et al., 1998b). Because of the low effective mass for the F point electrons in GaAs (see Table I), the reduction of dw results in a rapid increase in the energy of the lowest F point subband energy, allowing occupation of other carrier pockets in the Brillouin zone for dw <<.30A. Figure 5(a) shows the resulting calculated density of states for the subbands at the F, X, and L points of the Brillouin zone for a (001) oriented (20 A-20 A) superlattice. By growing a superlattice in the (001) orientation, an energy splitting between the longitudinal (X~) and the two transverse pockets (X t) is introduced for the three X-point pockets that are equivalent in the 3D Brillouin zone. For d w = 20 A = d~ in GaAs-A1As superlattices, the F, X t, and L subbands lie very close in energy, and the energy of the Xl point subband edge is about 50 to 60meV lower than that of the other subbands. Even though the

10

......

, ...........

(a) (001) SL ] - - -

T-3OOK,///

o

(:3

,

I~(~

jJ__J~ -

c~ -

~T. . . . . . . . (b) (111) SL

-

---r~l)'" Ir' - - ' I J-------I 0.25 0.30 0.35 0.40

E(eV)

~'('~!l JI "'---J-~'-/'-I)0.25 0.30 0.35 0.40 0.45

E(eV)

FIG. 5. Density of states for electrons in GaAs-A1As superlattices as a function of energy relative to the F point of bulk GaAs calculated at the F point (short-dashed line), X point (long-dashed line), and L point (solid line): (a) a (001) oriented GaAs (20A)-A1As (20A) superlattice, and (b) a (111) oriented GaAs (20 A)-AIAs (20 A)superlattice. The degeneracy of each type of cartier pocket is indicated in parentheses, and the subscripts ! and t denote ellipsoids oriented in the longitudinal superlattice growth direction or in the transverse direction that is normal (oblique) to the growth axis (Koga et al., 1998b).

16

M . S . DRESSELHAUS ET AL. ' ....

1

'

0.4

i:.." T=3OOK

/

0.2

/ ,/"

I'o/

N

'

./

.." .'"

/-... . // .." / / /,/ s ""

/

'

"

/

1

~..:<.. ,x | \,\j ,.----

--..

\]

."

..." .. , , ~ ' r

, / , 7 "~~ /

0.1

/~,";;/.///~

,,,/ /'/

0.05 __

/

,

t

/,

.. / " // """~ I , 10 TM

,

r point SL (20Ay(2OA) (001) SL for (20Ay(2oA) (001) SL for - - - (001) SL for (20A)/(30A) , , , , l ( 111 ) SL, for, J ~ 10 TM 10

(3oAy(2oA) (2oAy(2oA),

Carrier Concentration (cm -3) FIG. 6.

Estimated ZaDT as a function of carrier concentration for a (001) oriented

(20A-20 A) superlattice (dotted line), (001) oriented (30A-20A) superlattice (short-dashed line), (001) oriented (20 A-30 A) superlattice (long-dashed line), and (111) oriented (20 A-20 A) superlattice (long dash-short dash line). The solid line denotes Z3DT calculated for a (20 A-20 A) r point superlattice (Koga et al., 1998b).

mobility for the X point carriers is rather low (see Table I), it is nevertheless found that for all the structures considered (Koga et al., 1998b), the highest Z3DT occurs for the (20 A-20 A) superlattice for the (001) orientation (see Fig. 6). The results in Fig. 6 further show that, at low carrier concentrations ( < 2 • 1018 cm -3) the highest Z3DT is obtained only if the F-point subband is occupied. With increasing carrier concentration, and under optimal doping, Z3DT can, however, be increased by further increasing the carrier concentration and occupying other carrier pockets in the Brillouin zone, yielding Z3DT ,~ 0.41 for n ~ 6 • 1019 cm-3 using a (001) growth direction for the superlattice as shown in Fig. 6. Almost as high a Z3DT is predicted for the GaAs (20 ,/k)-A1As (20 A) (111) superlattice, but at a somewhat lower doping concentration (Koga et al., 1998b). The relative subband energies for the F, X, and L cartier pockets are very sensitive to the GaAs and AlAs layer thicknesses of the superlattice, as well as the superlattice growth direction, as shown in Fig. 6. Such plots are useful for the design of short-period superlattices to maximize Z3D T. At present, it is not clear whether the heavy doping condition (n > 1019 cm-3) suggested by the carrier pocket engineering model of Fig. 6 is attainable in an actual GaAs-A1As superlattice. If there is an upper bound on the doping level, or if high doping levels rapidly reduce the carrier mobility, then modified versions of this carrier pocket engineering model with additional constraints must be used to design a superlattice for optimum thermoelectric performance. For the best superlattice design in Fig. 6, an enhancement in Z3DT of

1 QUANTUMWELLS AND QUANTUMWIRES

17

48 is predicted relative to that for bulk GaAs ( Z T ~ 0.0085). Here a factor of 6 enhancement is attributed to the reduced thermal conductivity from preferential interface scattering of phonons, a factor of ~ 2 to optimization of the power factor from the F-point electrons. A further factor of 4 is attributed to contributions from L-point carriers within the GaAs layers and X-point carriers within the AlAs layers, since the energy of the X-point conduction band in AlAs lies below the X-point conduction band in GaAs within the GaAs-A1As superlattice (Koga, 2000).

V. 1.

Application to Specific 2D Systems

PbTe

The early experiments to investigate the possibility of using low-dimensional materials for the enhancement of their thermoelectric properties were carried out on the PbTe-PbEuTe superlattice system (Hicks et al., 1996). The primary reason for the selection of this system for this exploratory work was that the superlattice technology for the PbTe-PbEuTe system was well developed, and that detailed modeling of the transport properties could be carried out because of the extensive literature on the electronic band structure of this system (Ravich et al., 1970; Dornhaus et al., 1985). Thus PbTe-PbEuTe superlattices provided an ideal system for the initial proofof-principle studies. The multiplicity and anisotropy of the carrier pockets are important factors for good thermoelectric materials. This is generally true in both 3D bulk and 2D M QW (multiple-quantum-well) systems, although there are certain distinctions between 3D and 2D systems, as described later. In 3D bulk systems, a good thermoelectric material such as BizTe 3 has multiple anisotropic constant energy surfaces (carrier pockets) that have a small effective mass in one direction and large effective masses in the other two directions. A small effective mass usually leads to a high carrier mobility in that direction, whereas large effective masses in the other two directions lead to a high density of states. The multiplicity m of the carrier pockets directly introduces a multiplicative factor to the density of states. Since the key idea of having enhanced thermoelectric properties in 2D MQW systems is to deliberately make the effective mass along the quantum well growth axis (z direction) very large, while keeping the carrier mobility high in the plane where transport occurs, good materials for testing these anisotropy ideas should have at least one large effective mass component (which ensures a large density of states) and one small effective mass component in the x y plane (which leads to a high mobility in the plane). It is to our advantage in making MQW structures to utilize modeling to choose the quantum well growth direction that optimizes the expected thermoelectric properties

18

M . S . DRESSELHAUS ET AL.

(Hicks and Dresselhaus, 1993a). PbTe provides an attractive model system for studying the effects of low-dimensionality on the thermoelectric properties for the following reasons: (1) bulk PbTe itself is a reasonably good thermoelectric material ( Z T ,~ 0.4 at 300 K), (2) the muliplicity and high anisotropy of the constant energy surfaces allow us to explore the effect of the extra degree of freedom associated with the quantum well width or quantum wire diameter on the thermoelectric properties as discussed, and (3) well-established band parameters necessary for theoretical modeling are readily available in the literature. It has been theoretically predicted and experimentally demonstrated that an enhanced thermoelectric figure of merit Z can be achieved in (111) oriented n-type PbTe-Pb~_xEuxTe MQWs (x ~ 0.073) (Harman et al., 1996; Hicks et al., 1996). Z T ~ 1.2 has been obtained within the quantum well at room temperature in this system (Harman et al., 1996).

a.

Bulk Properties o f P b T e and Related Materials

The lead chalcogenides crystallize in a cubic (NaCl-type) rock-salt structure with m3m symmetry. The unit cell is face-centered cubic and the coordination number for all the atoms is six. A summary of the important structural and electronic band parameters of the lead salts is given in Table II. The lead chalcogenides are usually classified as polar semiconductors with mixed ionic-covalent chemical bonds. Since electrons are considered to travel mainly in the cation (Pb) sublattice, while holes are considered to travel mainly in the anion (chalcogen) sublattice (Ravich et al., 1970), the substitution of Eu for Pb is expected to result in a large reduction in electron mobility in the Pbl_xEuxTe alloys, but a smaller decrease in the hole mobility. Investigations of the carrier scattering in lead chalcogenides have shown that carriers are scattered both by the optical and acoustic phonons, and the role of acoustic scattering is very important from relatively low (--~ 5 K) to relatively high ( > 400 K) temperatures (Ravich et al., 1970; Yu et al., 1971a, 1971b). All the lead chalcogenides, including PbS, PbSe, and PbTe as well as the related Pbl _xEuxTe alloys, have their conduction band minima at equivalent (111) L-points in the Brillouin zone. The resulting constant energy surfaces are highly anisotropic. For example, the effective mass component for PbTe along the (111) axis of the constant energy ellipsoid is about 10 times larger than the perpendicular component (see Table II), although the anisotropy is lower for PbSe and yet lower for PbS. The large increase in the energy bandgap E o of PbTe for a small addition of Eu provides a convenient method for producing a barrier material with a sufficiently large barrier height to achieve significant quantum confinement, while maintaining good lattice matching (Yuan et al., 1993, 1997; Ueta, 1997).

TABLE II STRUCTURAL AND ELECTRONICBAND PARAMETERSOF THE LEAD SALTS Property

PbS

PbSe

PbTe

Reference

Lattice constant (300 K) (A) Thermal expansion (300 K) a (10-S/K)

5.936 2.027

6.124 1.94

6.462 2.04

Melting point (K) Density (300 K) (g cm -3)

1384 7.597

1355 8.26

1197 8.219

Bulk modulus - V d P / d V (10 6 Ncm 2) E 9 (4 K) (meV)

286

145

3.9 188

E o (300 K) b (meV)

410

278

311

(mlel/mo) (4 K) ~

0.105

0.070

0.220

(m~/mo) (4 K) ~

0.080

0.040

0.020

(m~l/mo) (4 K) c (m~/mo) (4 K) ~ (3Eo/(3P (10 -6 eV cm 2 kg -1)

0.105 0.075

0.068 0.034

0.236 0.0246 -8

Dalven (1969) Novikova and Abrikosov (1964) for S, Se; Houston et al. (1968) for Te Harman and Melngailis (1974) Harman and Melngailis (1974) for S, Lippmann et al. (1971) for Se, Miller et al. (1981) for Te Nimtz (1983) Mitchell et al. (1964) for S, Harman and Melngailis (1974) for Se, Preier (1979) for Te Zemel et al. (1965) for S, Schlichting (1970) for Se, Preier (1979) for Te Cuff et al. (1964) for S, Se; Ichiguchi et al. (1980) for Te Cuff et al. (1964) for S, Se; Ichiguchi et al. (1980) for Te Cuff et al. (1964) for S, Se; Preier (1979) for Te Cuff et al. (1964) for S, Se; Preier (1979) for Te Ravich et al. (1970)

aLinear thermal expansion coefficient at 300 K. bE o --- 630 meV for Pbo.927Euo.o73Te (Hicks et al., 1996). C(m~l/mo) and (m~e/mo), respectively, denote the band edge effective mass components If and 2_ to the [111] direction for electrons in PbS, PbSe, and PbTe, and likewise for the L-point holes under the transcription e ~ h.

20

M . S . DRESSELHAUS ET AL.

Our interest in the electronic structure of PbTe, for example, is directed toward the use of PbTe as a quantum well material under quantum confinement conditions, where the electron energies can differ greatly from the band-edge electron energy in bulk PbTe. For this reason, electron energy dispersion relations for the L-point carriers in PbTe must be considered over a large energy range. Over such a large energy range nonparabolic effects become important. In this general framework, the energy dispersion relations E(k) for pure PbTe and Pbl _xEuxTe alloys at the L point in the Brillouin zone are derived from k . p perturbation theory and are given by (Hicks, 1996; Yuan et al., 1994):

9

-2r~

+ 2m---~l-

2m'~

mo (mo

2m'[

E(k) +

(24)

mo

where the z axis is along a (111) direction; Pt and Pl are the transverse and longitudinal components of the momentum matrix element, respectively; E(k) is the energy of the electron measured from the middle of the energy bandgap E 0 as a function of wave vector measured with respect to the L point in the Brillouin zone; and m~, m~, m~', and my are the components for the far band effective mass tensor, where the superscripts c and v denote the conduction and valence bands, respectively; and the subscripts t and l denote the transverse and longitudinal components of the effective mass tensor, respectively, with respect to the (111) direction. The parameters used in Eq. (24) are summarized in Table III for bulk PbTe and Pb~_=Eu~Te (Yuan et al., 1994). The band-edge mass tensor and the density of the electronic states as a function of energy are then calculated from the energy dispersion relation given by Eq. (24), and the components of the band-edge effective mass tensor are obtained by evaluating the second derivative of Eq. (24) at the L point in the Brillouin zone where k~ = kr = k= = 0: m o _ 1 c32E - z

my

--

h 2 63k2

( 2P~ =

+

- \moE o

mo) 4-

-~

(25)

Here the + and - signs are for the conduction and valence band, respectively; v stands for t (transverse) or l (longitudinal); 2 stands for c ~2 is (conduction band) or v (valence band); mo is the free-electron mass; m~ the pertinent band-edge effective mass tensor component; and explicit values for m~" ~ calculated using Eq. (25) are given in Table IV. Values for the effective mass parameters thus obtained for electrons and holes at the band edges are given in Table II at 4 K.

1 QUANTUM WELLS AND QUANTUM WIRES

21

TABLE III BAND PARAMETERSOF BULK PbTe AND Pb~_xEuxTe USED IN EQ. (24) (Yuan

et al., 1994)

2p2mo (eV)

Pt/PI

m~/mo

m~t/mo

m'[/mo

m'[/mo

6.02 8.23

3.42 3.86

0.060 0.060

0.505 0.505

0.102 0.102

0.920 0.920

PbTe Pbx_xEuxTe

Since transport measurements for thermoelectric applications are carried out as a function of temperature, it is important to note that although the form of the dispersion relations in Eq. (24) is applicable over a wide temperature range, the values of the band parameters appearing in Eqs. (24) and (25) show quite a large temperature dependence. Systematic experimental studies of the temperature dependence of the L-point energy bandgap of Pbl_xEU,,Te on temperature and Eu concentration x yield (Yuan et al., 1997) Eo(x, T) = 189.7 + 0.48. T(K) 2

1 - 7.56x + x. 4480 (meV) T ( K ) + 29

(26)

while the temperature dependence of the effective mass components at the band edge are given by (Dornhaus et al., 1985) m r = 0.024 + 3.15 • 10-5T m

ml

(27)

= 0.25 + 3.33 • 10-4T.

m

At 300 K, Eq. (26) predicts that E o = 321 meV for bulk PbTe and that E o = 635 meV for Pbx_xEuxTe with x ~ 0.09. The principal components for the band-edge effective mass tensor and for E o at 300 K are listed in Table

TABLE IV COMPONENTS FOR THE BAND-EDGE EFFECTIVE MASS TENSOR FOR BULK PbTe AND Pbl _xEuxTe (x ~ 0.09) AT 300 K CALCULAXEDUSING EQ. (25) (Koga, 2000)

PbTe Pb 1- xEux Te

~ lmo

~ lmo

~ lmo

~ lmo

0.0282 0.0338

0.279 0.35

0.0350 0.0439

0.372 0.511

22

M . S . DRESSELHAUS ET AL.

IV for PbTe and for Pbo.91Euo.o9Te and their temperature dependences are plotted in Koga (2000). The measured relation between the lattice constant and the energy bandgap E 0 for PbTe-related crystals has been used to give the dependence of E o on the Eu concentration in Pbl_xEuxTe as part of a systematic and thorough study of various (chemical, structural, optical, electronic, and magnetic) properties of MBE-grown Pbl_~EuxTe alloys as a function of Eu content x (0 < x < 1) (Ueta, 1997), leading to the predicted dependence of the bandgap and effective mass parameters on temperature and Eu concentration, as discussed. Also of importance to the preparation of multiquantum-well structures is the lattice mismatch between PbTe and Pbl_xEuxYe alloys, which is as small as 0.4% for x ~ 0.1. The small lattice mismatch, the wide energy bandgap of Pbl_~EuxTe (E o ~-635 meV for x -~ 0.09) relative to that for pure PbTe, the very low electron mobility in Pbl_xEu~Te, and the good thermal stability of the PbTe-Pbl_xEu~Te superlattice, make Pbl_xEuxTe an excellent material for the barrier layers of PbTe MQWs because of the large degree of the quantum confinement, the negligible electrical conduction in the barrier layers, the large potential barrier height in the barrier layers, and the small carrier mobility in the barrier layers.

b.

Multi-Quantum- Well Superlattices o f Pb Te

In this section, we summarize the progress that has been made on the study of PbTe quantum well structures for thermoelectric applications. Multi-quantum-well n-type PbTe structures were first studied experimentally and theoretically to establish proof-of-principle that enhancement of the Seebeck coefficient and of the quantity SZn (proportional to the power factor) could be achieved within the quantum well relative to the same thermoelectric material in bulk form (see Section III) (Hicks et al., 1996). In this early work, the samples were based on (111) n-type (bismuthdoped) PbTe quantum wells separated by Pbl_xEuxTe barrier layers (nominally x = 0.073), grown on appropriate buffer layers and a B a F 2 substrate (Harman et al., 1996). For these proof-of-principle studies, comparisons between experiment and theory were made only at room temperature and the simplest possible approach was used for the calculations, which were based on the constant relaxation time approximation. Assuming that the 2D quantum well layer is parallel to the xy plane and the current flow is in the x direction, we can write the thermoelectric transport coefficients for transport along the x direction, using the x components of the ~'~') tensor given by Eq. (9). The transport tensor

1

QUANTUMWELLSAND QUANTUMWIRES

23

components of ~qot,) can be calculated as

~'()x)

-

~qo(o) = Dx[F0]

(28)

~t'tx~2 = Dx (k , r )[ 2F ~ - (*Fo]

(29)

Dx(k,r)z[3F2 - 4 ( * F , + (*2Fo]

(30)

where (* = ~/kBT is the reduced chemical potential. The quantity Dx, which depends on the band parameters of the material, is given by

Dx=~-dw\h2,]

mxx

(31)

where dw is the quantum well width and the Fermi-Dirac related function F i (for i = 0, 1, 2, ...) is given by Eq. (13). Equations (5) to (7) can now be used to calculate the electrical conductivity a, the Seebeck coefficient S, and the electronic contribution to the thermal conductivity ~cr

e (kBT' ~ O"= ~ w \ - ~ J (mxmy)X/2ltx(F~ S=

k"( 2F1 - e

K e :

\~--o

(32)

) -

~*

k2T(knT'~(m~my)l/2/a 3F 2 ndwe \ h 2 j x

(33)

Fo ]

(34)

where the mobility is given by/~x = ez/mx. From Eqs. (14), (15), and (32) to (34), ZzDT can be calculated for a specific value of the quantity BZD which is characteristic of specific thermoelectric materials. We can then optimize ZEDT by changing the chemical potential of the system. These relations within the constant relaxation time approximation were used in the early proof-of-principle studies on PbTe quantum well multilayers (Hicks et al., 1996). Subsequently, a number of improvements were made to these model calculations, including consideration of the effect of carrier tunneling through thin barriers (Broido and Reinecke, 1997; Mahan, 1997a, 1997b), and considerations of explicit scattering mechanisms that are important for PbTe, such as acoustic phonon deformation potential scattering and polar optical phonon scattering (Koga et al., 1998a, 1999d). These improvements to the model calculations enabled the proof-of-principle to be established over a wide temperature range (Koga et al., 1998a, 1999d).

24

M . S . DRESSELHAUS ET AL.

Several groups noted that (100) PbTe-Pba_xEu~Te superlattices should provide greater enhancement of ZZDT relative to (111) PbTe-Pb~_xEuxTe and to bulk PbTe (Koga, 2000; Dauscher et al., 1997; Scherrer et al., 1999). As a cubic material, bulk PbTe should have Z T independent of crystal orientation. The advantage of the (100)PbTe orientation for the superlattice growth is that in this case, all the (111) carrier pockets in the superlattice are equivalent and therefore can yield a higher density of states at the superlattice band edge. Other groups have carried out thermoelectric measurements on PbTebased superlattices and related lead salt systems (Sipatov et al., 1999; Beyer et al., 1999) and have also found enhanced Z T in PbTe quantum wells compared to bulk values, each group providing new insights and extensions of the experimental work and theoretical modeling. One group has looked into the whole family of lead salt superlattices in an attempt to determine which lead salt-barrier combinations are most promising (Sipatov et al., 1999). Their results thus far indicate that the PbS-EuS superlattice system shows an enhancement in power factor over bulk values for PbS. Experiments have also been carried out on p-type PbTe-Pbx_~EuxTe superlattices using thin BaF z layers in the barrier regions. These results showed even greater ZzDT enhancement (Harman et al., 1997; 1999a) than for their n-type counterparts. In the case of the p-type samples, it is believed that contributions from a second set of valence band carrier pockets with heavy effective masses contributed significantly to increasing the enhancement of ZzDT for these samples (Algaier, 1961; Sitter et al., 1977). Also of interest are experimental results by Harman et al. (1996), showing Hall mobilities (ktnall) for the PbTe-Pbl_xEuxTe MQW samples that are higher than those for similarly grown bulk PbTe using molecular beam epitaxy techniques at comparable carrier concentrations nnall. The highest #nail found in their MQWs (1420 cmZ/V, s at nnall ~ 1019 cm-3) is very close to the generally accepted bulk value ( ~ 1700 cmZ/V, s for nHall < 1018 cm-3) for undoped samples. Recently, research on PbTe-based superlattices has focused on optimizing Z3oT, which has led to increased focus on short period superlattices. The first system investigated from this standpoint was the (111) PbTe-Te superlattice system by Harman, using only a few monolayers of Te to form the second constituent. The large observed enhancement in Seebeck coefficient and in Z3DT for this system (Harman et al., 1999b) led to the synthesis of (111) PbSeo.98Teo.o2-(111) PbTe superlattices, which showed even larger Z 3 o T values (Harman et al., 1999b). The growth of these superlattices was carried out by molecular beam epitaxy on a BaF 2 substrate at a rapid growth rate of 0.8 to 10 #m/hr to yield single crystal superlattice films, with a thickness of 5 to 70/~m and a mirror-like surface. A typical PbSeo.98Teo.oz-PbTe superlattice film sample consisted of 7235 periods, each period 9.93 nm thick, leading to a film thickness of 71.5/~m, with an

1 QUANTUMWELLS AND QUANTUM WIRES

25

equivalent alloy composition PbSeo.13Teo.sv. With this superlattice system, it is believed that quantum dots form in the PbSeo.98Teo.o2 interface region and the host lead salt material provides good conduction paths between the quantum dots, which have a very high density of states at the Fermi level. At the same time, the quantum dots provide very effective scattering centers for phonons but do not scatter electrons as strongly. Evidence for the quantum dot formation is provided by growth mechanism studies in the PbTe-PbSe system (Springholz, 1998), which show that nanocrystals with well-established facets grow at the superlattice interfaces, using similar growth conditions as were used by Harman in his molecular beam epitaxy growth of these PbTe-based superlattices (Harman et al., 1999a, 1999c, 2000; Springholz et al., 1998; Holy et al., 1994). It was found that the ( l l l ) P b T e Te and (lll)PbSeo.98Teo.o2-PbTe superlattices yield values of the Seebeck coefficient ISI (ranging from ISI - 80/~V/K for n = 4.0 x 1019 cm-3 to ISI 203 #V/K for n = 5.8 x 1018 cm -3 at 300 K) that are larger than those for the corresponding bulk PbTe ( I S I - 22/~V/K for n = 4.0 x 1019 cm-3 and I S I - - 1 6 8 # V / K for n = 5.8 x 1018cm -3 at 300K). The values for the observed Hall carrier mobility for the PbTe-Te superlattices are found to be in the same range as, but slightly smaller than, those for the corresponding PbTe bulk (thick) films. Combining measured values for S (Seebeck coefficient) and a (electrical conductivity), the resultant values for S2a and Z3DT~ estimated using conservative bulk values for ~Cph from the literature (Kph = 2 W/m" K at 300 K), are found to be enhanced by 23 and 25%, respectively (Harman et al., 1999a, 1999c, 2000), for these quantum dot superlattices relative to their bulk counterpart materials. It was found experimentally that the optimum values for S2a and for Z3DT occur at slightly higher carrier concentrations for the PbTe/Te superlattices (/'/opt '~ 1 x 1019 cm -3) than for the corresponding PbTe bulk (thick) films (hopt ~ 6 x 1018 cm-3) (Harman et al., 1999a, 1999c, 2000). Experimental optimization showed that the maximum Z3DT o c c u r s at a quantum dot superlattice period of about 13 nm. While values of Z3DT = 0.9 were obtained at 300 K, temperature-dependent measurements of Z3DT for a PbSeo.98Teo.o2-PbTe quantum dot superlattice showed increases in Z3DT with increasing T, reaching a value of Z3DT ~ 2 at 550 K as shown in Fig. 7, the highest experimental Z3DT value ever reported for a thermoelectric material of any type (Harman et al., 1999b). This large value for Z3DT is believed to be due to an increased S arising from the delta-function-like density of states of the quantum dots, and a decreased ~Cpharising from the increased scattering of phonons by the quantum dot interfaces and by the random Se atoms on the Te sites of the superlattice sample. No theoretical model has yet been developed to account for these observations or to make predictions for further enhancing Z3DT. Many experimental parameters remain to be optimized, including the quantum dot quality, size, and density; the deposition temperature; the growth rate; the equivalent alloy

26

M.S.

DRESSELHAUS

ET

AL.

composition PbTeSe, or the addition of Sn to make PbSnTeSe and optimization of these quaternary alloy compositions. Thus it is believed that Z3DT> 2 should be possible with this low-dimensional system through both experimental and theoretical optimization strategies.

2.

SiGe

The study of Sil_xGe x alloys as possible materials for thermoelectric power generation was undertaken as early as 1954 by Ioffe and Ioffe. In bulk form, Si I _xGex is a promising thermoelectric material for high-temperature ( ~ 1000 K) applications (Mahan, 1997a; Slack and Hussain, 1991; Vining, 1995). It was first used in space in the SNAP-10A nuclear reactor, and has been the exclusive choice for radioisotope thermoelectric generators (RTGs) launched by the U.S. space program since 1976 (Wood, 1988). In addition to having attractive thermoelectric and physical properties, bulk Sil_xGe x devices can operate at temperatures up to about 1300 K without significant degradation. In keeping with the principle that low-dimensionality can be used to enhance the thermoelectric performance of good 3D thermoelectric materials, the Si-Si~_xGe~ quantum well system has been studied for its thermoelectric properties because it is not only an attractive system for demonstrating proof-of-principle, but also has the potential for use in

2.5

-

-

,

-

'

. . . .

T=525

/,

.,

2.0

0

e 9

b

o ~e

~.s-

o

:1

01 .,,,,,.

LI.. Ub,,

T.594

/9

'f.O

."

9

m

9t l

9 I 9

I

0

o

"

I

10

t

T-459

~

9 I

u

U 0 ,,m.

E o.s 6 .,C d[

0 3OO

,

4OO

I

500 Temperature

ii

600

(K)

FIG. 7. Thermoelectric figure of merit vs temperature for an n-type PbSe0.98Teo.o2-PbTe quantum dot superlattice structure showing the highest experimental Z3DT reported to date (Harman et al., 1999b).

1

QUANTUM WELLS AND QUANTUM WIRES

27

thermoelectric device applications. Other reasons for studying the thermoelectric properties of Si-Si~_~Ge~ superlattices include: (1) the Si~_~Ge~ alloy is already a good thermoelectric material, and there is a good chance that Sil_xGe x in a superlattice configuration may provide even better thermoelectric properties than Sil_xGe x alloys; (2) the reduction of the lattice thermal conductivity Kph due to the boundary scattering of phonons and the properties of threading dislocations has been studied extensively in Si-Ge superlattices (Lee et al., 1997; Chen et al., 2000); and (3) the effect of uniaxial lattice strain at the Si-Ge interfaces (Rieger and Vogl, 1993; Van de Walle, 1989) provides an additional degree of freedom that can be used to control the conduction band offsets for the miniband formation in a superlattice. By carefully designing the Si-Sil _xGex superlattice structures, we expect this system to have sufficiently good thermoelectric performance to be interesting for possible device applications at room temperature and above, and the system is expected to be compatible with Si-based microelectronics.

a.

Bulk Si and Ge Properties

Si and Ge both crystallize in the diamond structure. The conduction band of Si is characterized by six minima at equivalent A points located at (2re/a)(0.85, 0,0) along the (100) axes of the Brillouin zone, while the surfaces of constant energy for n-type Si are ellipsoids of revolution with their major axes along (100), as shown in Fig. 8 (Kittel, 1986). In contrast, Ge has four electron carrier pockets at the four equivalent L points rc/a(111)

kz

k~

FIG. 8. The six symmetry-related carrier pockets of n-type Si in its Brillouin zone. The long axes are directed along (100) directions. The six pockets are labeled with a, b, c, d, e, and f.

28

M . S . DRESSELHAUS ET AL. TABLE V STRUCTURAL AND ELECTRONIC BAND PARAMETERS FOR

Si AND Ge Band parameter ao [,/k]a m,/m b mJm b Band minima Nc /2e [ c m 2 / V " s] a #h [ c m 2 / V " S] d

Xph [W/cm- K] e AE si-Ge [eV]Y cll [1011 dyn cm-2] 9 ciz [1011 dyn cm-2] g c44 [1011 dyn cm-2] ~ ~,Au l e V I h =L b. .~u [eV]h

Si

Ge 5.6579 0.082 1.59 L 4 3600 1800 0.60

5.4307 0.19 0.92 A 6 1350 480 1.313 0.25 16.577 6.393 7.962 9.16 16.14

12.853 4.826 6.680 9.42 15.13

aLattice constant from Madelung (1987b). bEffective mass parameters from Madelung (1987b). CNumber of equivalent valleys. nBulk carrier mobilities for electrons and holes at 300 K. Data taken from Madelung (1987b). eValue of the lattice thermal conductivity at 300 K from Lee et al. (1997). YConduction band offset between the average position for Si A valleys and the average position for Ge L valleys. The literature values for AEsi-6e are between 0.15 and 0.35eV (Rieger and Vogl, 1993), depending on the hydrostatic component of the strain in the superlattice. 9Elastic constant at 300K from McSkimin and Andreatch (1963, 1964). hDeformation potential parameters at the A and L point extrema.

in the Brillouin zone. Each ellipsoid of revolution is characterized by its transverse and longitudinal effective masses, which are m~ = 0.1905 mo and roll = 0.9163 too, respectively, for Si, where mo is the free-electron mass, and m• = 0.082 m o and roll = 1.59 m o, respectively, for Ge (Ashcroft and Mermin, 1976). In Fig. 8, the carrier pockets for Si are labeled with a, b, c, d, e, and f, which is convenient for discussing the anisotropy of the superlattice material in the presence of quantum confinement. The holes for both Si and Ge are in a single approximately spherical carrier pocket at the F point of the Brillouin zone. Some pertinent parameters describing the structure and properties of Si and Ge are given in Table V. In forming a superlattice between Si and Ge, the large difference in lattice constants gives rise to a lattice mismatch of 4.1%, which must be considered in designing the thickness of the quantum well and barrier regions to minimize the density of misfit dislocations.

1

b.

QUANTUM WELLS AND QUANTUM WIRES

29

S i - S i s _ ~ G e x Superlattices

In forming a 2D superlattice of Si-Six _xGex, it is relatively difficult to get quantum confinement for electrons in the Si layer in the conduction band because of the small conduction band offsets. Si s _xGex forms a continuous series of solid solutions with gradually varying Ge concentration x, which can vary from 0 to 1. The indirect bandgap E0,in d between the conduction band extremum at the A points and the valence band extrema at the F point decreases with x up to a value x = 0.85 (Braunstein et al., 1958). Thus, for Si-Six_xGe ~ quantum well superlattices grown on a Si substrate, the band structure forms a type I band alignment, as shown in Fig. 9(a) and the quantum well is located in the Si 1_ xGex regions of the superlattice for both n-type and p-type Si-Six_~Ge x superlattices. To form quantum wells for electrons inside the Si layers, a superlattice structure with type II band alignment [Fig. 9(b)] must be fabricated. This band alignment is achieved experimentally by growing a relaxed Sis_xGe ~ buffer layer on top of the Si substrate. The relaxed Si~ _~Ge~ buffer layer

(a) Si EtC.

Ev

Sis_.Ge.

Si

i

I

1

I

Six_~Ge~ I

Si

,., J

[ (b)

Si

Sil-zGez

Si

I

Sil_xGez

J

Si

l

Ev FIG. 9. (a) Schematic bandedge diagram of a type I Si-Si 1_xGex quantum well superlattice grown on a Si substrate where the quantum well is formed in the Sil_xGe ~ regions of the sample both for an n-type and a p-type superlattice. (b) Schematic bandedge diagram of a type II Si-Six_xGe ~ quantum well superlattice grown on a relaxed Si~_~Ge~ substrate, where it is seen that quantum wells now are formed in the Si region of the sample (Sun, 1999).

30

M . S . DRESSELHAUS ET AL.

effectively increases the lattice constant of the substrate so that it is slightly larger than that of Si. The subsequent Si layers are therefore under tensile strain. Due to the tensile strain in the Si layers, quantum wells for electrons are now formed in the conduction band (Abstreiter et al., 1985; Kasper et al., 1987; Ismail, 1992). Most of the Si-Si~ _xGe x superlattices that have thus far been grown for thermoelectric applications are of this strain-relaxed type II structure.

c.

Thermoelectric Figure of Merit for a 2D Quantum Well

The simplest estimate for Z2DT for Si quantum wells is obtained for a model assuming an infinite barrier height. This is followed by the model calculations for barrier heights characteristic of an optimized Si-Si~_xGex superlattice. In the various calculations, Si quantum wells are taken to be in the xy plane, the current flow is taken to be in the x direction, and z is the quantum confinement direction. The Si-Sil_xGex quantum well structures are modeled in terms of the 2D dispersion relation given by Eq. (3), assuming a parabolic approximation for the energy bands near the band extrema. Because there are six anisotropic electron carrier pockets along the six equivalent (100) directions in the Brillouin zone, the effect of anisotropy and the contributions from all six electron carrier pockets of the Si quantum wells need to be considered. Therefore, the value of BZDin Eq. (14) becomes

(35) where in the case of Si quantum wells ~=2

(mll~ 1/2 \rail

+ 2

(m~_~l) 1/2

+ 2

(36)

thereby accounting for the anisotropy of the six different ellipsoids in the Si conduction band, in which m• and mll are, respectively, the transverse and longitudinal effective mass components of electrons in the conduction band. As a first approximation to the issue of carrier scattering, a constant relaxation time r independent of wave vector k or of the energy E(k) is assumed, where z is determined by (Krishnamurthy et al., 1986) =

3#,

e(1/mll + 2/ml)

(37)

in which/t, is the measured electron mobility. The ~ factor in Eqs. (35) and

1

QUANTUMWELLS AND QUANTUM WIRES

31

(36) for B2D reflects the number of electron carrier pockets and their anisotropy, indicating the desirability of having many anisotropic carrier pockets to get high ZzDT values. For example, for n-type Si, the value of is 7.19, which is about 20% larger than the number of carrier pockets (6). The value BZD in Eq. (35) is determined by the intrinsic properties of Si and the width of the quantum well dw. For a given value of BZD, the reduced chemical potential (* in Eq. (14) must then be optimized to yield the maximum value of ZzDT within the quantum well. In the case of quantum well systems, (* may be varied both by doping as well as by changing the quantum well thickness dw. Reduced dimensionality thus provides an additional degree of freedom for increasing ZzDT above the value characteristic of bulk materials. In a quantum well structure, since phonons can be scattered from the interfaces, the phonon thermal conductivity may be reduced relative to the bulk value, which may be estimated from the kinetic theory formula given by /s = 1Q vl, where for Si, l = 282 A is the phonon mean free path, C~ = 1.658 J K - ~ cm-3 is the lattice heat capacity, and v = 8.4332 x 105 cm/s is the velocity of sound. If the layer thickness dw is greater than 282 A, then layering does not seriously affect the phonon mean free path l, and ~Cphshould then be similar to its bulk value except for a reduction of ~Cphdue to explicit interface scattering. However, for dw < 282 A, then l and ~Cph are limited by phonon scattering from the interfaces and a rough estimate for Kph is obtained by setting l = dw. Detailed calculations for ~Cph(T) have been carried out for Si-Si~_ xGex superlattice structures and are described in Chapter 5 of this volume. Superlattice structures with many periods of quantum wells and barriers are needed for making thermoelectric devices with sufficient cooling or generating power. For model calculations, a band offset U in the conduction band between the Si layer and Si x_ xGex layer was taken as 100 meV, which is a typical value when x in the buffer layer is chosen to be appropriate for producing a strain-relaxed superlattice, and a barrier width large enough to ensure good quantum confinement of the electrons in the Si layers. Using the simplest model sketched above for a Si-Si~ _xGex superlattice with a large barrier width relative to the quantum well width dw, and a small band offset for a strain-relaxed type II superlattice, it was shown that a quantum well is formed in the silicon region and a quantum barrier is formed in the Si~-xGex region. When the six electron ellipsoidal pockets for Si are confined in quantum wells, four of them (transverse pockets) become fourfold degenerate subband levels (pockets labeled by a, b, c, and d in Fig. 8), and the other two (longitudinal pockets) become twofold degenerate subband levels. Because the z component effective mass for the transverse pockets is smaller than that for the longitudinal pockets, the transverse subband levels tend to lie higher in energy than the longitudinal subband levels.

32

M . S . DRESSELHAUS ET AL.

For the relatively small barrier height of U = 100meV, more than one subband needs to be taken into account for modeling the transport properties. For a multiple subband model, the transport tensor is a linear combination of contributions from each transport subband [-see Eq. (9)]. Or equivalently, the overall electrical conductivity and Seebeck coefficient can be calculated from

= E

(38)

v

and S = ~ v o(v)StV)

Ev ~(v)

(39)

where it is seen that the subbands with higher electrical conductivity are more heavily weighted in calculating the Seebeck coefficient. To model the thermal conductivity for Si-Si I _ xGex, the interface scattering effect has to be considered explicitly (Chen et al., 2000; Chen, 1997). A

2.5

dw= dw = dw= dw =

2.0

25A 50 A 75A 100 A

f

/

/ / / / /

1.5

/ /

h-a N

/ / / /

1.0

/

/ J

f

0.5

s

j

f

s

o

"" "

m '~" "', ~

~''

%

f f

0.010

1

.

.

.

.

~ 021.

.

.

.

.

.

.

.

.

.

10~3'

......

i 0 ~4

n2o (cm -2) FIG. 10. The 2D thermoelectric figure of merit within the Si quantum wells for the Si-Sio.vGeo. 3 superlattice structure for four different quantum well thicknesses dw as a function of electron sheet carrier density at T = 300 K, and taking the specularity coefficient p = 1 (Sun, 1999; Sun et al., 1998).

1 QUANTUM WELLS AND QUANTUM WIRES

33

2.5

2.0

T = 300 K

1.5 I-. N

O

1.0

0.5

0.0

0

,

I

20

I

l

4O

60

8O

100

d w (A) FIG. 11. The optimal 2D thermoelectric figure of merit Z2o within the Si quantum wells for the Si-Sio.7Geo. 3 superlattice structure plotted as a function of quantum well width, dw, taking the specularity coetficient p - 1 (Sun, 1999; Sun et al., 1998).

parameter p is introduced to characterize the specularity of the interface scattering, where p - 1 refers to pure specular scattering, while p = 0 refers to pure diffusive scattering (Chen, 1997). These calculations show that the reduction in thermal conductivity in the 2D Si-Sil _xGex system is significant even in the case of pure specular interface scattering (p = 1), which is the most conservative approach for describing •ph in 2D superlattices and this reduction in the thermal conductivity becomes more prominent as p decreases from 1 to 0. Combining the power factor modeling, described earlier (Sun, 1999), with the thermal conductivity Ke + Kph, using the W i e d e m a n n - F r a n z law to obtain Ke and the model calculation for Kph (Chen, 1997) with p = 1, the thermoelectric two-dimensional figure of merit ZzDT within the quantum well has been obtained for the Si-Sil_xGe x superlattice and the results are plotted as a function of the carrier density per atomic Si layer r/zD in Fig. 10, and as a function of quantum well width dw in Fig. 11. We see that ZZD T within the quantum well can reach as high as 2.0 at room temperature for a superlattice with a quantum well width of 25 ,/k. These results imply that the Si-Sil_xGe ~ superlattice system has a very good potential for thermoelectric applications, even at room temperature.

34

d.

M . S . DRESSELHAUS ET AL.

Temperature-Dependent Behavior

Since the Si-Six_xGe x system is aimed at high temperature operation (well above 300 K), the investigation of the temperature-dependent thermoelectric performance of Sil_xGe x quantum well systems at elevated temperatures becomes important. Since the power factor for Sil_xGe~ materials generally increases with increasing T above room temperature (Slack and Hussain, 1991; Vining, 1995), the power factor as well as the thermoelectric figure of merit within the quantum well are expected to show even greater enhancement above 300 K. Since the value of ZzDT for quantum wells is very sensitive to the carrier mobility/~,, the temperature dependence of the electon carrier mobility has been considered explicitly using an empirical relation for bulk Si in the temperature range between 300 and 1000 K /~, = 2.11 • 105T -1

(40)

using existing experimental data (Li, 1977; Smith, 1978). Plots of Z2DT

as

a

1.5

1.0

% N

900K

0.5

0.01011

'

.

.

.

.

.

.

.

I

1012

.

.

,

,

,

,

,,,I

1013

n~ (cm-2)

.

.

.

.

.

.

.

i

1014

.

.

.

.

.

.

.

.

10 is

FIG. 12. The calculated Z2DT v e r s u s carrier density at various temperatures (300, 400 . . . . . 900 K) for Si-Si 1_ xGex superlattice structures with a quantum well width dw = 50 ]k. The electron mobility is determined empirically as/~. = 2.11 • 10ST -1 cm2/Vs (T in K) for n-type bulk Si with carrier concentration of 1018 c m -3, appropriate for thermoelectricity applications (Sun, 1999; Sun et al., 1998).

1 QUANTUMWELLS AND QUANTUMWIRES

35

2.0

1.5

\

\

\

1000 K

....

700K

--\

~ 1.0 N

-- -- -

500 K ,,

300 K

\ "x \ N

0.5

0.0

,

50

I

,

1 O0

150

dw

(A)

200

FIG. 13. The optimal Z2DT v e r s u s layer thickness dw at various temperatures for a Si quantum well. The electron mobility is determined empirically as #. = 2.11 • 105T- 1 cm2/Vs (T in K) for n-type Si with a carrier concentration of 1018 c m - 3 appropriate for thermoelectricity applications (Sun, 1999; Sun et al., 1998).

function of q u a n t u m well width at elevated temperatures are shown in Figs. 12 and 13 as a function of nzD and q u a n t u m well width dw, respectively. These results suggest that ZzDT for Si-Sil_xGex q u a n t u m wells become more favorable at higher temperatures. Measurements of the temperature dependence of the mobility for actual q u a n t u m wells are needed to improve the reliability of this model. Furthermore, material science issues may cause problems at high temperatures for superlattice structures, insofar as the Si-Sil_xGex interface may become diffusive, and thermal excitations of electron and hole pairs may reduce the Seebeck coefficient. These effects need to be studied experimentally.

e.

Proof-of-Principle Study

Proof-of-principle studies were carried out on a superlattice of 15 periods of Si-Sio.TGeo. 3 where the Si q u a n t u m well widths were varied from 1 to 5 nm and the Sio.7Geo. 3 barrier regions were very wide (30nm) so as to ensure good q u a n t u m confinement (Sun, 1999; Sun et al., 1999b). The superlattices were grown on a 100-nm-thick Sio.7Geo. 3 buffer layer, which

36

M . S . DRESSELHAUS ET AL.

was on top of 100 nm of a graded buffer layer Six_xGe x (0 ~< x < 0.3) all on a SO1 (silicon-on-insulator) substrate, which contained 200nm of Si on 360 nm SIO2. This substrate arrangement for the superlattice-buffer layers limited their contribution to the measured Seebeck effect to less than 15%, as was shown by measurements of S and a on matched samples with 5, 10, and 15 superlattice periods (Sun et al., 1999c). To dope the superlattice n-type, a 6-doping process with Sb was used in the middle of the Sio.TGeo. 3 barrier layer regions to yield approximately the optimum Sb concentration or placement of the Fermi level. To test the validity of the model transport calculations, a comparison of the quantity S2n (proportional to the power factor) was made between the measurements and the model calculations, using no fitting parameters (Sun et al., 1999b). In this proof-of-principle study, the calculated electron wavefunctions were allowed to penetrate the barrier regions, but the optimization procedure focused on the enhancement of Z2o T within the quantum well. The resulting comparison between experiment and theory presented in Fig. 3 shows an increase in S2n as the quantum width decreases, and qualitative agreement between experiment and theory is obtained. In Section III, three reasons and their corresponding correction factors are given to account for the systematic discrepancy between the theoretical modeling results and the experimental data. When all three correction factors are considered, quite good agreement is achieved between the theoretical modeling and the experimental proof-of-principle study for the Si-Six_xGe x superlattices, thereby providing support for the thermoelectric model calculations within the quantum well (Sun et al., 1999b). These encouraging results for the basic validity of the modeling calculations for quantum confined systems suggested the extension of these modeling calculations, which had initially been directed to optimize Z2DT within the quantum well, to now consider the whole superlattice. In this next phase of the modeling effort, carrier quantum confinement in the quantum well was relaxed both for the quantum well and barrier geometry, and the carrier concentration was optimized to maximize Z3o T, and the thermoelectric figure of merit for the entire superlattice was optimized with respect to all available variables. This optimization scheme, called "carrier pocket engineering" has been successfully applied in a preliminary way to the Si-Sil_xGe x superlattice system (Koga et al., 1999b), as described in the next section.

f

Carrier Pocket Engineer&g of Si-Ge Superlattices

The carrier pocket engineering concept provides a systematic method for optimizing the geometry and the structure of the superlattices (such as the layer thicknesses, the superlattice growth direction and the carrier concentration) to produce the maximum three dimensional Z3DT (Koga et al.,

1

QUANTUMWELLS AND QUANTUM WIRES

37

Band diagram for Ge and Si

O.lleV

l

I

e

10.54eV .....

E=O

( . 1 5 - . 3 5 eV)

0.66 eV

1.11 eV

~

GeSi

~

FIG. 14. Schematic band diagram to show the relative energies between the various conduction band minima for bulk Si (right) and Ge (left). The actual values for the conduction band offset AEsi-Ge (defined in the figure) range between 0.15 and 0.35 eV, according to the magnitude of hydrostatic component of the lattice strain (Koga, 2000).

1998b), and this concept has been applied to Si-Ge superlattices (Koga et al., 1999b; Koga, 2000).

Bulk Si and Ge have their conduction band minima at the A and L points in the FCC Brillouin zone, respectively, and the number of the equivalent sites at these high symmetry points are 6 and 4, respectively. A theoretical framework for the design of Si-Ge superlattice samples, that could be used eventually for practical applications, has been established based on the optimal use of these carrier pockets to maximize the power factor for the whole superlattice (Koga et al., 1999b). Samples to test the resulting theoretical framework have been prepared and comparisons have been made between experiments and theory to test the validity of the carrier pocket engineering model (Koga, 2000). The results of this effort are summarized in this section. Figure 14 shows a schematic band diagram to illustrate in a comparative way the various conduction band structures for bulk Si and Ge. We find that Si and Ge have the lowest conduction band minima at the A point (which lies between F and X in the Brillouin zone) and the L point in the Brillouin zone, respectively. When an interface is created between Si and Ge, the value for the conduction band offset between Si A point and Ge L point (denoted by A E Si-Ge) in the absence of the uniaxial strain ranges between 0.15 and 0.35 eV, depending on the hydrostatic component of the lattice strain in the superlattice (Rieger and Vogl, 1993). It is, however, found that the final results of the calculated values for the thermoelectric transport coefficients are rather insensitive to the value of A E Si-Ge used in the calculations. Of great importance to the carrier pocket engineering process is the uniaxial strain due to the lattice mismatch at the Si-Ge interfaces, which

38

M.S.

DRESSELHAUS ET AL.

leads to a splitting of the conduction bands that are degenerate in the absence of strain (Van de Walle, 1989). These splittings (denoted as AEca or AE~) are expressed with respect to the average band position, which is shifted only by the hydrostatic component of the strain. When the uniaxial strain along the (001) direction is applied to a Si or Ge crystal, as is realized in each layer of Si and Ge in a (001) oriented S i - G e superlattice, the energy of the L-point valley extrema is not affected in this case but the conduction band minima at the A-point (called the A valleys) split according to (Van de Walle, 1989), AEOOl = ~="(~ 2--A • -AElOO,OlO _ -

1-a

-~=,,(E

(41)

611)

~ -

(42)

Ell)

where the superscripts to AEc denote the alignment direction for the pertinent valley, --,'~ais the strain deformation potential for the A valley, and El(Eli) is the component for the lattice strain tensor perpendicular (parallel) to the interface. Similarly, under uniaxial strain along the (111) direction, as is realized in (111) oriented Si-Ge superlattices, the energies of the A-point valleys are not affected, while the energy of the L-point valleys splits according to (Van de Walle, 1989), AEcX11 = ~-~t, 2=L" El and

AEci i 1'] 1]'1]-i =

-

(43)

Ell)

2,-L • - ~=~(~

~ll)

.

(44)

The values for the lattice strains e• and Ell a r e calculated, assuming that the lateral lattice constant (11 to the interfaces) for the strained layer is equal to that of the substrate, where a linear interpolation scheme is utilized to determine the lattice constant for a Si s _ xGex substrate via E• = (a• - ao)/a o, and Ell = ( a l l - ao)/ao. Here, ao is the lattice constant for the layer in the absence of lattice strain; all and a j_ are the lattice constants parallel and perpendicular to the interfaces, respectively, for the strained layer; and all is taken as the lattice constant for the substrate (Van de Walle, 1989), with a• calculated according to

aoI' o a,ao ')} where D = 2(c12/Cll ) and D = 2(Cll + 2C~2 -- 2 c 4 4 ) / ( c l l + 2C12 -}- 4C44 ) for the (001) and (111) oriented superlattices, respectively. The values for the elastic constants c~, c12, c44, and other parameters that are used in the present calculations are summarized in Table V. Using the KrSnig-Penney model and the band parameters in Table V, the thermoelectric transport coefficients in Eqs. (38) and (39) are calculated.

QUANTUMWELLSANDQUANTUMWIRES

1

(a)

0.29e

Lsi

~'1A__: ~~-

on (001)SiosGeos , ."ST" AIIGe

"

/

~

~

='"

",' A Gc

0.66ev

.... ~'~2--~

EGo/0.36eV

39

Gel z.x~

A - -

. . . . .

"._._

u.llev

. _ .

A... T o.14eV

E=0 --*----. . . . . ( I 3- . . . . . . . . . . . ~" * As' \[A" / on (001)Si o.23ev

on (111 )Sio.sGeo.S IY2s, ,"'~'Ti-G~: ;~o'-I'

(b) Lsi

___

i' .19eV _' Lsi~,."

"

0"~_/~Ge

I ] 0.45eV (

!

0.36e~1

/ i

hI'''''''

:LGe

-

i

:A-.: *'---~;" E=0

I (~

~176 "Ge

". "". . . .

on (111)Si

FIG. 15. (a) Conduction band offset diagram for a (001) oriented Si-Ge superlattice at the A~176(black solid line), A1~176176176 (light dash-dotted line), and L (black dashed line) points in the Brillouin zone. The left, middle, and right diagrams denote the band offsets for the unstrained layers for a superlattice grown on a (001) Sio.sGeo.5 substrate, and for a superlattice grown on a (001) Si substrate, respectively. (b) Conduction band offset diagram for a (111) oriented Si-Ge superlattice at the L 111 (black solid line), L Ill'Ill'Ill (light dash-dotted line), and A (black dashed line) points in the Brillouin zone. The left, middle, and right diagrams denote the band offsets for the unstrained layers, for a superlattice grown on a (111) Sio.sGeo.5 substrate, and for a superlattice grown on a (111) Si substrate, respectively (Koga et al., 1999b; Koga, 2000).

T h e effect of the uniaxial lattice strain on the position of the A valley m i n i m a calculated for the (001) o r i e n t e d S i - G e superlattices g r o w n on (001) Sio.sGeo. 5 a n d on (001) Si substrates is s h o w n in Fig. 15(a). W h e n the superlattice is g r o w n on a (001) Sio.sGeo. 5 substrate, the Si a n d G e layers experience a tensile a n d a c o m p r e s s i v e stress, respectively. Therefore, the energy for the Ge(Si)A valley is shifted u p w a r d ( d o w n w a r d ) a n d the Ge(Si)A 1~176176176 valley is shifted d o w n w a r d (upward), which m a k e s the effective barrier height larger (smaller) for the q u a n t u m well derived from the A~176 ~~176176176 valley ( K o g a et al., 1999b, 1999c; K o g a , 2000). T h e resulting density of states for electrons s h o w n in Fig. 16(a) is calculated for a (001) o r i e n t e d Si(20 A ) - G e ( 2 0 A) superlattice g r o w n on a (001) Sio.sGeo.5

TM

40

M . S . DRESSELHAUS ET AL.

2.0 0.3 ..,-~ ....-~ ....-~...

....

,--.,

,>

t~ i

E 1.0 ,o

o

,_.__.

03 0a

._1

! ,

0, V

'i " ~ '

,,.,,.1

1 ~" ' |

.

- o.oo,,..,o..,.o,,,,o..,.o., i -I n [cm-a] [ (a)

0.0

~-

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

i,X,,s,i, i ,,

t

7,. 2.0 ~

~o 1.0

~" a

0.0

I ~176

t,,, to)

-0.5

b Ik_Si;

"

"/

,~

I

o,F"

I L~

__ _r-"-~"

0 Energy [eV]

---

0.5

FIG. 16. Density of states for electrons in S i - G e superlattices as a function of energy relative to the A point of bulk Si calculated at the A (dark solid and dark dash-dotted curves), and L (light solid and light dash-dotted curves) points in the Brillouin zone for: (a) a Si(20A)-Ge(20A) superlattice grown on a (001) oriented Sio.5Geo. 5 substrate, and (b) a Si(15 A ) - G e ( 2 0 A) superlattice grown on a (111) oriented Sio.sGeo. 5 substrate. The calculated value of Z3DT as a function of the carrier concentration is shown in the inset of each figure. See Fig. 15 for the meaning of the subband symbols (Koga et al., 1999b; Koga, 2000).

substrate, where the thicknesses for the Si and Ge layers (20 A) are chosen so that the resulting ZT is maximized. Figure 15(a) shows that the quantum wells for the A and L valleys are formed in the Si and Ge layers, respectively, although Fig. 16(a) shows that the L point carriers make a negligible contribution to the transport, because the energy for the L point subband edge is very high (,-~200 mV above the subband derived from the A 1~176176176 valleys). As shown in the inset to Fig. 16(a), the resultant Z3DT calculated with this superlattice structure is 0.24 at 300 K, which is rather small for a good thermoelectric material, although it is much larger than the corresponding Z3DT for bulk Si (Z3DT = 0.014 at 300 K). Growth of the Si-Ge superlattice on a (001) Si substrate further enhances ZaDT because the effective barrier height for the quantum wells derived from the A~176valley will be larger due to the compressive stress on the Ge layer [-see Fig. 15(a)]. Thus, the A~176176176176176 valley in the Ge layer is shifted to higher (lower) energy because of the uniaxial strain along the (001) direction, while the Si

1

QUANTUMWELLS AND QUANTUM WIRES

41

A valleys are left degenerate (since the Si layer is unstrained). With this design of the superlattice, the subband levels associated with the AT M valley and the A1 oo,olo valleys, respectively, stay very close to each other in energy, because the large effective mass along the (001) direction (confinement direction) for the AT M valley is compensated by the large barrier height for the quantum wells. The resulting Z3DT calculated for this superlattice is 0.78 at 300 K, which represents more than a factor of three enhancement relative to the corresponding ZaDT for the superlattice grown on a (001) Sio.5Geo.5 substrate, though there may be serious problems associated with the actual fabrication of such a superlattice sample because of lattice mismatch (Koga et al., 1999b, 1999c; Koga, 2000). Increased ZaDT is also expected for a Si-Ge superlattice grown in the (111) direction [see the middle diagram in Fig. 15(b)], where the subbands derived from the A valleys of Si and Ge remain degenerate due to symmetry. The resulting Z3DT for the (111) oriented Si(15 A)-Ge(20 A) superlattice grown on (111) Sio.5Geo. 5 is calculated to be 0.98 at 300 K [see Fig. 16(b)], which is a factor of 4 enhancement relative to the Z3DT calculated for the (001) oriented Si(20 A)-Ge(20 A) superlattice grown on a (001) Sio.5Geo.5 substrate. An even larger Z3DT is expected if the superlattice is designed such that the subbands derived from the A valley and the L ~11'111'111valleys stay very close to each other in energy, which is conceptually realized by

Structure for (001) Si(20 ,~,)/Ge(20 ,A,) SLs grown on Si-on-insulator (SOl) substrate

b:~l ~~fi~_b:~ S~i_~I /

/

Sb~_., '

SiIZ0AI

Ge (20/1,) Ge (20 A) Si~176 (0.3Bm)

" . Sil_xGe x graded buffer layer

x: 0~0.5 (ll.tm) (001) SOl (1800 A Si)

I-

insulator

(3800

}

• oo periods undoped

[unintentionally doped "-- I by the residual Sb

Limpurities

,~ S i O 2 )

(001) Si substrate

FIG. 17. A schematic diagram of the sample structure for strain-relaxed Si(20 A)-Ge(20 A) superlattices grown on (001) oriented SOI substrates. For the proof-of-principle studies of the carrier pocket engineering concept, four superlattice samples of this type were prepared, with different dopant concentrations (denoted by n +) in the range between 1 x 10 TM and 2 x 1019 cm -3. An SOI substrate is composed of 1800 A of a (001) oriented Si layer on top of a 3800 A SiO2 layer, which in turn is grown on top of a (001) oriented Si substrate. This 3800-A SiO 2 layer is expected to provide electrical insulation between the superlattice film and the (001) oriented Si substrate underneath the 3800-A SiO2 layer (Koga, 2000).

42

M . S . DRESSELHAUS ET AL.

FIG. 18. TEM cross-sectional micrographs for a (001) oriented Si(20 A)-Ge(20 ]~) superlattice grown on a SiGe graded buffer layer on top of a (001) oriented Si substrate, showing (a) a wide cross-sectional area of the sample and (b) a high (atomic)-resolution image corresponding to (a) (Koga, 2000; Sander and Gronsky, 2000).

growing a Si(15 A)-Ge(40 ]k) superlattice on (111) oriented Si [see the right-hand diagram in Fig. 15(b)]. Since the Ge layers in this superlattice are compressively strained, while the Si layer is unstrained, only the Ge L-point valleys are split into a L ~ valley (higher in energy) and L 111'111'ffi valleys (lower in energy). The resulting Z3DT calculated for this structure is 1.25, which is a factor of 5 enhancement relative to the Z3DT for the Si(20 A)-Ge(20 A) superlattice grown on (001) Sio.sGeo. 5, though fabrica-

1

QUANTUM WELLS AND QUANTUM WIRES i ,

400

[

i

n = 5 x 1 0 ' * e m -3 ( T h e ,

.... -----

9

'1

43

'

ionized im~nb/)

n= 7x10 ~* r "~ 0 " h ~ . k ~ i z e d irrlpurity) ~ " n=-l.SxlO'gcm = (Theo. ionized i m p u r ~ . ~ .

,--., 300

200 =B

0

0 JL194 o JL197

.,~g~J j ~ " -

100

""

0

100

200

T [K]

300

400

FIG. 19. Seebeck coefficient as a function of temperature measured for three samples prepared in accordance with Fig. 17:JL194 (open circles), JL197 (open diamonds), and JL199 (open triangles). These experimental measurements are compared with results of a semiclassical transport model, for n = 5 x 10 ~8 cm -3 (dot-dashed curve), 7 x 10 x8 cm -3 (short-dashed curve) and 1.5 x 1019 c m - 3 (long-dashed curve), without the use of any fitting parameters. The origin for the marked increase in the value of ISI above 300 K has not been convincingly explained (Koga, 2000).

tion of this superlattice would be very difficult because of lattice mismatch at the interfaces. Experimental proof-of-principle studies for the carrier pocket engineering concept have been carried out on (001) oriented Si(20 ]k)-Ge(20 A) superlattices with 100 superlattice periods grown on a (001) SOI substrate, as shown in Fig. 17. The SOI substrate consists of a thin (1800-]k) layer of (001) Si grown on 3800A of SiO 2. The SiO 2 layer provides electrical isolation from the (001) Si substrate, which could otherwise contribute to measurements of the Seebeck coefficient of the sample because of the large Seebeck coefficient of silicon. The lattice strain of the Si-Ge superlattice is relaxed by a 0.3-pm-thick Sio.5Geo. 5 buffer layer grown on a thickened (1 #) graded buffer layer of Si 1_xGex, where x is varied from 0 to 0.5. The high quality of the sample was verified by TEM images of the superlattice, as shown in Fig. 18. Measurements of the Seebeck coefficient vs T for a series of Si(20 ]k)-Ge(20 A) samples with different nominal carrier concentrations are shown in Fig. 19 in comparison to theoretical calculations of S(T). These theoretical calculations, which are expected to be valid in the temperature range 180 < T < 300 K used no adjustable parameters, and assumed that the scattering is dominated by that from ionized impurities and from acoustic phonons. The measurements were made on samples that were homogeneously doped with Sb throughout the superlattice region. The fit between theory and experiment provides experimental confirmation of the basic carrier pocket engineering concept. A better fit to the S(T) data might be achieved by also considering scattering from neutral defects that are predominantly associated with interface boundary scattering. A fit to the

44

M . S . DRESSELHAUS ET AL.

.....

7

o

JL193

,T~

%

"

JL1690~ / JL167~,,O

~

~

300 K

JL194

o-". . . . . . . . "-,,

~, X

(aftersubtrae~on) - - - S.=~.(homo.doped)

- - - - ' S,.,,, (homo. doped + "c~

010 ~a

'

'

i

~

i

i

II

1/~90

n [cm -31

I

I

''

....

1

O20

FIG. 20. Measured thermoelectric factor S2t/ as a function of carrier concentration at 300 K, measured for three different samples, together with a comparison to the calculated results of semiclassical transport models. The dash-dotted curve was obtained using Matthiessen's rule 1/Ztot(E ) - 1/Zimp(E ) + 1/Zac(E ) + l/text to take into account the extrinsic scattering mechanisms that are present in actual superlattice samples, while the solid curve was obtained including only ionized impurity scattering and longitudinal acoustic phonon deformation potential scattering without the use of any fitting parameters (Koga, 2000).

temperature dependence of S Z n w a s also made for the same set of samples as in Fig. 19, and the results are presented in Fig. 20, showing a good fit between theory and experiment without the use of any fitting parameters (Koga, 2000). 3.

Bi

Bismuth is a very attractive material for low-dimensional thermoelectricity (see Section V.3.a) because of the very large anisotropy of the three ellipsoidal constant energy surfaces for electrons at the L point in the rhombohedral Brillouin zone (see Fig. 21) (Isaacson and Williams, 1969), and the high mobility of the carriers for the light-mass electrons (Saunders and Siimengen, 1972). In addition, bulk bismuth has carriers with very long mean-free paths (Zhang et al., 1998a) for electronic transport and heavymass ions, which are highly effective for scattering phonons (see Table VI). Bismuth can also be alloyed isoelectronically with antimony to yield a high mobility alloy with highly desirable thermoelectric properties (see Section V.3.b). From the known bismuth-antimony phase diagram discussed in Section V.3.b (Brandt et al., 1971), it is possible to select the Sb concentration x so that the Bil_xSb x alloy is semiconducting and to select x even more restrictively so that the lowest conduction band and the highest valence band are both at the L point in the bismuth Brillouin zone (see Fig. 21). Since both L-point electrons and holes have very similar and highly

1 QUANTUMWELLS AND QUANTUM WIRES

had,*l , < t

45

] trigonal(z)

"~A~, "

....

r

~

eh~'tronp~wket(A)

. . . . ---L

Idnary (7.~..,.. -,'''~

Jill01 FIG. 21. The Fermi surfaces of Bi, showing the Brillouin zone with the fifth band hole pocket about the T point and the three sixth-band L-point electron pockets labeled A, B, and C. For quantum wells with their confinement direction, or for nanowires having their wire axes, in the bisectrix trigonal plane, the mirror plane symmetry of the bulk bismuth structure results in the crystallographic equivalence of the L-point carrier pockets B and C. However, the L-point carrier pocket A is not equivalent crystallographically to carrier pockets B or C.

anisotropic constant energy surfaces, which give rise to high mobility carriers, the Bil _xSb x alloys provide good opportunities for preparing high Z T n-type and p-type low-dimensional bismuth-based thermoelectric materials. (See Section V.3.b). As a bulk material, semimetallic Bi has a low Seebeck coefficient S because of the approximate cancellation between the contributions to S from the electron and hole carriers, since for a two-carrier system Eq. (39) can be written

S :

G e S e at. ffhSh

(46)

(7 e -1- t7 h

where O'e, lYh, Se, and Sn, respectively, denote the electrical conductivity and Seebeck coefficient for the electrons and holes. For the case of bismuth, the mobility of the electrons is much larger than that of the holes so that S tends to be weakly negative. It was recognized early that Bi could be a good thermoelectric material if the hole carriers could be removed (Gallo et al., 1963); however, no reliable mechanism was proposed to remove the hole carriers in pure bismuth. On the other hand, it was recognized long ago that Bil_xSb x alloys, when properly doped and oriented, could be among the best presently available thermoelectric materials (Smith and Wolfe, 1962;

TABLE VI BISMUTH PARAMETERS

4~

Property

Bulk

Mass density (g/cm 3) (300 K) Melting point (K) (1 atm) Velocity of sound (105 cm/s) (4.2 K) Velocity of sound (105 cm/s) (300 K) Phonon mean free path a (nm) (77 K) Thermal conductivity (W/mK) (300 K) Lattice constant b (,/~) Compressibility ( M b a r - a) Bulk modulus ( M b a r - 1 ) Young's modulus (dyn/cm) Volume coeff of thermal expansion (K -1)

9.8 544.4

L-point bandgap (meV) (0 K) Plasma frequency (cm-1) (2 K) Work function (eV) Debye temperature (K) Static dielectric constant Carrier density c (10 x7 cm-3) (77 K) Carrier density (10 xv cm -3) (300 K)

Trigonal

Binary

Bisectrix

2.02 1.972 14.7 6.0 c = 11.862 1.82

2.62 2.540 15.2 9.8 a =4.5460 0.62

2.7 2.571 14.8 9.8 a = 4.5460 0.62

2.12 • 1011

3.10 • 1011

3.10 • 1011

0.326 • 10- 3 3.965 • 10- 5 13.6 158+3 4.22 112 84 4.4 2.4

105

105

Reference White (1972) Samsonov (1968) Lin (2000) Eckstein et al. (1960) Lin (2000) Gallo et al. (1963) Schiferl and Barrett (1969) White (1972) White (1972) Gopinathan and Padmini (1975) White (1972) Verdun and Drew (1974) Boyle and Brailsford (1960) Suhrmann and Wedler (1962) Fischer et al. (1978) Gerlach et al. (1976) Saunders and Sfimengen (1972) Saunders and Siimengen (1972)

aFrom the relation x = Cvvsl/3. Along [01i2] and [1071], l = 15.3 and 15.7 nm, respectively. bThe rhombohedral angle for Bi is ~ = 57~ ' and sublattice parameter u = 0.237 as compared to ct = 60 ~ and u = 0.25 for a cubic system. For Sb the lattice constants are a = 4.308 ~ c = 11.274 ~ ct = 57~ ', and u = 0.233. CThe carrier density at 4 K for Bi is 2.7 x 1017/cm 3, for Sb is 3.7 • 1019/cm 3, and for As is 2 • 102~ 3.

1 QUANTUMWELLS AND QUANTUM WIRES 8.0

9

w'~

-

T

-

\

'

T

-

T

47

-

Y = 300 K n-type trigonal direction

6.0 1D 4.0

2.0

0.0

0

i

I

i

i

,

I

,

,

i

r

30

d (nm) FIG. 22. Calculated thermoelectric figure of merit ZT for 1D and 2D Bi systems in the trigonal direction as a function of dw, the width of the quantum wire (1D) or quantum well (2D) (Sun, 1999).

Yim and Amith, 1972), especially in the liquid nitrogen temperature range (near 77 K). Low dimensionality, however, offers an opportunity to overcome the problem of the low Seebeck coefficient in pure bismuth. As the quantum well (or wire) width decreases, the band edge for the lowest subband in the conduction band rises above that for the highest subband in the valence band, thereby inducing a semimetal-semiconductor transition (Zhang et al., 1998a). If the 2D (or 1D) bismuth system is then doped to the optimum doping level, a large enhancement in ZzDT (and even more enhancement in Z1DT ) should be possible as the quantum well (or wire) width is decreased, as shown in Fig. 22. In this calculation, an infinite barrier and no carrier tunneling were assumed. Of particular interest is the observation that significant enhancement in ZzoT (or ZIDT) occurs for relatively large quantum well (or wire) widths. For example, at 300 K, Fig. 22 predicts Z2DT > 1 for a 50-A Bi quantum well when quantum confinement is in the trigonal direction. As discussed in detail in Section VI, the relatively low melting point of bismuth makes it an attractive material for preparing nanowires (see Table VI) (Hansen, 1972), where Z~DT > 1 is expected for a nanowire oriented along the trigonal crystalline direction and having a diameter < 10nm (see Fig. 22). The large values of Z2DT and Z~DT in Fig. 22 stem from the highly anisotropic Fermi surface of Bi. Values for the band parameters of bulk bismuth, which give rise to the high Fermi surface anisotropy, are given in Table VII. Since these band parameters are highly sensitive to temperature, in Table VIII we include the best presently

48

M . S . DRESSELHAUS ET AL. TABLE VII THE BAND STRUCTURE PARAMETERSOF BULK Bi AT T ~< 77 K

Parameters

Notation

Band overlap a Bandgap at L point Electron effective mass tensor elements at the band edge for L(A) pocket b T-point hole effective mass tensor elements at the band edge Electron mobility c at 77 K in units x 104 cm2/V s

Ao

Hole mobility c at 77 K in units x 104 cmZ/V s

Value

/~1 /.t2

- 3 8 meV 13.8 meV 0.00119 m o 0.263 mo 0.00516 mo 0.0274 m o 0.059 mo 0.059 mo 0.634 mo 68.0 1.6

1/3 /~4 ]~hl "-~h2 /2h3

38.0 -4.3 12.0 2.1

EoL me1 me2 me3 me4 mhl mh2 mh3

Reference Isaacson and Williams (1969) Verdun and Drew (1974) Heremans and Hansen (1979)

Isaacson and Williams (1969)

Saunders and Sfimengen (1972)

Saunders and Siimengen (1972)

aThe band overlap is 177.5 meV for Sb and 356 meV for As. bThe tilt angle of the L-point electron ellipsoids are 6.0 ~ - 4 ~ - 4 ~ for Bi, Sb, and As, respectively. The tilt angle of the H-point hole "ellipsoids" are 53 ~ for Sb and 37.5 ~ for the major As hole ellipsoid. CThe form of the effective mass tensor and the mobility tensor are assumed to be the same for a given carrier pocket.

available values for the temperature dependence of many of the band parameters listed in Table VII. If Bi-based materials become important for thermoelectric applications, better values for the temperature dependence of the bismuth band parameters are likely to be needed. The quantum well widths suggested by Fig. 22 should in general be achievable experimentally for multi-quantum-well superlattices. We discuss in the following subsection the present status of research on 2D Bi quantum well structures for thermoelectric applications, while Section V.3.b reviews the corresponding progress with Bil_xSb x alloys, and Section VI reviews progress with Bi nanowires.

a.

Bi Quantum Wells

The simplest possible theoretical model starts from Eq. (3) and neglects nonparabolic electronic energy band effects for the strongly coupled L-point electron bands and the tilting of the constant energy surfaces with respect to the principal axis (Isaacson and Williams, 1969; Heremans and Hansen, 1979). Explicit expressions for Z z D T a r e given in Hicks et al., (1993) in terms

1

QUANTUM WELLS AND QUANTUM WIRES

49

TABLE VIII TEMPERATURE DEPENDENCE OF SELECTED BULK Bi BAND STRUCTURE PARAMETERS

Parameters

Temperature dependence

Band overlap (meV) (Gallo et al., 1963) m 0 --

-38

(T < 80 K)

- 3 8 - 0 . 0 4 4 ( T - 80)

}

+4.58 x 1 0 - 4 ( T - 8 0 ) 2

(T > 80 K)

(47)

- 7.39 x 1 0 - 6 ( T - 8 0 ) 3 Direct bandgap (meV) (Vecchi and Dresselhaus, 1974) L-point electron effective mass components a (Heremans and Hansen, 1979; Vecchi and Dresselhaus, 1974)

EoL = 13.6+ 2.1 • 10-3T+ 2.5 • 10-4T 2

[me(T)]ij =

(48)

[me(O)]ij (49) 1 - 2.94 • 10- 3T + 5.56 • 10- vT 2

"me(0 ) is the effective mass tensor at T = 0 K.

of the effective mass parameters for electrons and holes for this simple model for the Bi electronic structure. Calculations of Z z D T as a function of the quantum well width d w (see Fig. 22) indicate that 2D superlattices based on Bi quantum wells should be highly desirable for 2D thermoelectric applications (Hicks et al., 1993). Following this early calculation, which assumed a barrier with an infinite barrier height and assumed complete confinement of the wave function within the quantum well, Broido and Reinecke refined this simple model by assuming finite barrier heights and widths. These modifications showed some reduction in Z T for 2D Bi quantum wells, but still indicated substantial promise for Bi quantum wells as a thermoelectric material (Broido and Reinecke, 1995). The theoretical effort has since been extended further in the context of carrier pocket engineering, with Bi taken as the quantum well material and PbTe (or more generally Pb I _xEuxTe) as the barrier material (Koga, 2000). Carrier pocket engineering calculations to optimize Z3DT for the Bi( l l l ) P b T e superlattice system have been carried out for a Pbx_xEu~Te barrier material in an effort to enhance the carrier confinement effect by increasing the values for the band offset (Koga, 2000). To simplify the calculation, equal band offsets for the L-point valence and conduction bands were assumed, a simplified parabolic energy band model was introduced, with effective mass parameters appropriate to the Fermi level of bulk bismuth at 80 K and for Pbl_xEuxTe (see Section V.l.a), which were used to approximate the Bi-Pbl_~EuxTe superlattice band parameters at 300 K. The thermal conductivity for the superlattice was approximated by ~ph -- 1.5 W/mK, taking account of the quantum well and barrier materials and the interface between them. This simple calculation showed Z3DT to

50

M . S . DRESSELHAUS ET AL.

first increase with decreasing quantum well width until some optimum value dw(opt) primarily due to the increased density of states at the Fermi level. On further decrease of dw, tunneling effects across the barrier become increasingly important, thereby again reducing Z3DT for small dw. An optimum Z3DT = 2.3 was thus predicted at 300 K for dw = 3 nm, a EuTe (x -- 1) barrier thickness of dB = 5 nm, and an n-type carrier concentration of 8.6 x 1017 cm -3 (Koga, 2000). Under the assumptions used, Z3DT was found to decrease with decreasing x (or decreasing band offset) and with decreasing T. In this work, the lowest Z3DT-- 1.2 was obtained for x = 0.1 (a band offset of 250 meV) and T = 77 K, where carrier pocket engineering yielded preliminary design parameters of dw = 5 nm, dn = 10 nm, and a carrier density of 2.2 x 10 x7 cm-3. Thus if a barrier material for Bi could be found that would prevent Bi interdiffusion across the superlattice interfaces, 2D bismuth superlattices could have high potential for thermoelectric applications. The carrier pocket engineering calculations for Bi quantum wells were inspired by the earlier synthesis of compositionally modulated Bi/PbTe multilayers, with (n/m) layers of their respective constituents grown on a cleaved muscovite mica substrate that induces the growth of Bi in the trigonal (111) direction and PbTe in the (111) direction, with near lattice matching along (27.0), which is normal to (111), with d(2~0)= 2.272 A for Bi and 2.280A for PbTe (Shin et al., 1984a). A wide range of Bi-PbTe multilayer thicknesses were grown from (6/6) to (16/16) periods and from (2/12) to (6/12), where (n/m) denotes the number of atomic planes per modulation wavelength of bismuth (n) and PbTe (m), respectively. X-ray diffraction characterization of these samples showed a strong (111) texture (Shin et al., 1984a). Transport measurements were made along the binary and bisectrix directions of these multilayer samples (Shin et al., 1984b). The resistivities at 4 K were about 103 to 104 times larger than in bulk Bi and depended strongly on the multilayer period (n/m). The results were interpreted in terms of 2D weak localization theory (Shin et al., 1984b) and subsequent work in a magnetic field showed the carrier-carrier interaction effect to be even more important than 2D weak localization (Shin et al., 1984c) at low temperatures ( < 10 K) (Shin et al., 1984b). A very weak linear temperature dependence of the resistivity was also observed at higher temperatures ( > 50 K) (Shin et al., 1984b), indicative of strong interface scattering. Hall coefficient measurements showed a very strong dependence on n, the number of Bi atomic planes per modulation wavelength, as n increased to about n = 4 or 5, after which the carrier density changed from n-type to p-type and became very high. Keeping the number n for Bi constant and increasing the number m of PbTe planes to above 12 changed the Hall coefficient from p-type to n-type, with a large increase in R n occurring above m = 15. From these measurements, it is concluded that

1

QUANTUMWELLSAND QUANTUMWIRES

51

strong interdiffusion of Bi occurs into the PbTe regions for short period Bi-PbTe superlattices (Shin et al., 1984c). Subsequent attempts to grow Bi thin films and multi-quantum-well structures encountered difficulties in obtaining high-quality layers suitable for thermoelectric applications (Casian et al., 1997, and references therein). In most cases the initial growth of a thin Bi film is characterized by island formation and island growth, which result in polycrystalline films of small grain size. Further diffusion of bismuth into the barrier layer and the unintended doping result in a lack of control over the carrier concentration and in the appearance of anomalous thermoelectric properties despite an apparently good superlattice structure (Hicks, 1996; Harman, private communication). CdTe(lll), Si, and B a F 2 have been used for substrates, and CdTe barrier layers have also been used with some success (Dauscher et al., 1997; Yoshida et al., 1993; Cho et al., 1999b). Despite the effort that has gone into fabricating 2D Bi quantum well structures, the major impediment to demonstrating enhanced thermoelectric performance in Bi quantum wells is the absence of a good barrier material for the synthesis of reliable quantum wells with good interfaces. Thus, despite the encouraging results provided by theoretical models, the progress in developing 2D Bi quantum structures for thermoelectric applications has been disappointing thus far.

b.

Bil

xSbx Quantum Wells

Bismuth and antimony both crystallize in the s a m e A 7 structure with space group R3m (Wyckoff, 1964) and Bi I _xSbx alloys form a solid solution over the entire composition range x. The lattice constants and structural parameters a, c, and ~ (see Table VI) obey Vegard's law over the composition range x ~< 0.3 (Cucka and Barrett, 1962), which is the range of predominant interest to low-dimensional thermoelectricity. The deviation of the rhombohedral angle ~ from 60 ~ increases as x increases, as expected from Table VI. In antimony, the electron pockets are at the L point of the Brillouin zone, while holes are at the H points (see Fig. 21). The departures from cubic symmetry are more pronounced in Sb than in Bi based on the values of their structural parameters; and consequently Sb has a larger carrier density than Bi. The L, T, and H hole carrier pockets all differ greatly with respect to their effective mass tensors and their density of states masses (Lenoir et al., 1996). The substitution of Bi atoms by Sb atoms in the Bi lattice has a major effect on the electronic band structure of Bi, as shown in Fig. 23, which summarizes the dependence of the band structure of Bil_ xSbx alloys. Three features of particular interest include: (1) the crossing of the (L point) L s m

52

M . S . DRESSELHAUS ET AL.

Antimony

Bismuth

,i

E H

>

~o

J

SM

,

t

SC

]SM

t

SM ' semimetal SC : s e m i c o n d u c t o r

FIG. 23. Schematic diagram for the energy bands near the Fermi level for Bi l_xSb x alloys as a function of x at low T (<~ 77 K) (Lenoir e t al., 1996).

and L A energy bands at x = 0.04; (2) the semiconducting range of Bi I _xSbx between 0.07 < x < 0.22, delineated by the crossing of the L-point conduction band with the T-point hole band at x = 0.07 and with the H-point hole band at x = 0.22; and (3) the range between 0.09 < x < 0.16 where the highest lying valence band is at the L point, giving rise to the possibility of p-type carriers that are mirror images of the L-point electron carriers, which have such excellent thermoelectric properties. The thermoelectric properties of Bil_xSb x alloys have been studied by various authors since the early work of Jain (Smith and Wolfe, 1962; Lenoir et al., 1996; Jain, 1959; Yasaki, 1968). These studies show a strong dependence of the thermoelectric behavior on alloy composition x, temperature, orientation, and sample purity, with the thermopower being highest along the trigonal direction and at low temperature (30 K) for x = 0.095. Regarding the thermal conductivity, the large mass difference between Bi and Sb (atomic weights 209 and 122, respectively) gives rise to very effective point defect scattering for phonons, resulting in a large decrease in the thermal conductivity ~ at low T, where thermal transport is predominantly by phonons, while at room temperature, where ~ce becomes important relative to Xph, the effect of alloying is less important. Combining the results for S, a, and ~c, it is found that alloying Bi with Sb enhances Z3DT, with values of Z3DT= 0.41 at 80K and 0.33 at 300K for an x = 0.095 sample with transport occurring along the trigonal direction. The ZaDT for transport in the plane perpendicular to the trigonal direction is lower (e.g., Z3DT = 0.24

1 QUANTUMWELLS AND QUANTUMWIRES

53

at 80 K) up to room temperature. The largest reported Z3DT for a Bil_xSbx bulk alloy is Z3DT = 0.88 at 80 K in a magnetic field of 0.13T (Yim and Amith, 1972). The galvanomagnetic and thermoelectric properties of this alloy composition has been studied extensively (Goldsmid and Volckmann, 1997; Sakurai et al., 1997; Cho et al., 1997). The transport properties of Bil_xSb ~ films (1 #m thick) on CdTe(lll)B substrates (where B denotes a Te plane at the interface) were studied in the range 0 ~< x ~< 0.183, showing that the semimetal-semiconducting transition occurred at lower Sb concentrations (x < 0.035) for the thin films as compared to bulk material (see Fig. 23) (Cho et al., 1999a). Also the maximum bandgap shifted from x = 0.15 in bulk Bi~_xSbx to x = 0.09 in the films, where a maximum bandgap of 40 meV was measured as compared with ~ 20 meV for the bulk material. Consequently, larger Seebeck coefficients were obtained ( ~ 113/~V/K) for an x = 0.088 film than for the corresponding bulk sample. Also of significance is the large shift of the temperature where the maximum power factor was obtained from 80 K in the bulk material (Lenoir et al., 1996) to 250 K for the film (Cho et al., 1999a). The shift of the semimetal to semiconductor transition to lower x values and the shift of the temperature where Z T is a maximum was attributed to strain in the Bi~ _xSbx film due to lattice mismatch between the film and the CdTe(lll)B substrate. This strain effect thus provides a means to control the thermoelectric properties of the films, as discussed in Section IV in connection with carrier pocket engineering of superlattices. The effect of n-type and p-type doping with Sn has been studied in thin Bio.91Sbo.o9 films (1 #m thick) (Cho et al., 1999a); this work follows earlier studies on Bio.ssSbo.12 bulk material (Brown and Silverman, 1964). Both studies were carried out near the maximum bandgap of their respective systems, and high doping levels were obtained (5 x 1020 cm -3 for n-type and 1 x 1021 cm -3 for p-type). Since undoped Bio.91Sbo.o9 shows a Hall coefficient that is weakly n-type, a sign change in the Hall coefficient is observed as a function of temperature since thermal excitation of carriers produces electrons that are more mobile than the holes. The heavily doped samples, however, did not show high power factors, presumably due to the strong carrier scattering at these high impurity levels (Cho et al., 1999a). Another system of interest is that of Bi-Sb superlattices. Recently Bi-Sb superlattices of high quality were grown by MBE on CdTe(111)B (Cho et al., 1999a), on BaF2(lll ) and on Si(lll). Interestingly, samples grown on BaF 2 showed very low negative thermopowers (possibly due to the cancellation of positive and negative contributions of holes and electrons), while samples grown on Si showed a significant unexplained enhancement in S at elevated temperatures ( ~ 300/~V/K at 350 K) (Yoshida et al., 1993). BiSb superlattices show promise as a thermoelectric material around and somewhat below room temperature. For higher temperatures, its performance becomes unsatisfactory due to degradation of the internal superlattice structure.

54

M . S . DRESSELHAUSET AL.

VI.

Nanowires

1. INTRODUCTION TO NANOWIRES The advantage of one-dimensional materials for thermoelectric applications was clearly apparent from early work in the field of low-dimensional thermoelectricity (Hicks and Dresselhaus, 1993b). These early papers were theoretical because the feasibility of synthesizing arrays of high-mobility thin wires of promising thermoelectric materials had not yet been developed. Only recently has the technology for the synthesis of quantum wire arrays embedded in insulating templates been developed. This technology was initially developed by chemical engineers working in the petroleum production industry in connection with catalysts and molecular sieve materials. Nanowires are of particular interest for thermoelectric applications because in the limit of small diameters, where nanowires show strong quantum confinement effects, the quantum wires are expected to show greater enhancement in Z T compared to quantum wells of the same confinement size dw. Furthermore, the geometry of the quantum wires allows for heat flow along the wires and could be particularly useful for special device applications. Also for some materials that are attractive for thermoelectricity, such as bismuth, efforts to develop good quantum well barrier materials have thus far not been successful, although good barrier materials for quantum wires (in the form of polymers or anodic alumina) have been successfully synthesized (Huber et al., 1995). Thus there has been considerable incentive to study thermoelectric phenomena in nanowires. Thus far, bismuth is the dominant thermoelectric quantum wire material that has been fabricated and studied for thermoelectric applications, although some success has been demonstrated with the fabrication of quantum wires from antimony (Heremans et al., 2000a), Bi2Te 3 (Stacy, private communication; Lieber, private communication) and Si (Lee et al., 1999). However, except for the case of Bi and Sb, little attention has been given to the thermoelectric properties of these wires. Ballistic transport in thin metallic wires has been studied more generally for many years (Murayama, 1994; Giordano, 1988; Beutler and Giordano, 1998; Datta, 1995). In this section, the structure, characterization, and thermoelectric-related properties of bismuth nanowires are reviewed.

2.

STRUCTURE AND SYNTHESIS OF BISMUTH NANOWIRES

Arrays of hexagonally packed parallel bismuth nanowires, 7-110 nm in diameter and 25-65 Ftm in length, have been prepared (Lin, 2000; Zhang et al., 1998b, 1999b). These nanowires are embedded in a dielectric matrix of

1

QUANTUMWELLS AND QUANTUM WIRES

55

FIG. 24. Cross-sectional view of the cylindrical channels of 65 nm average diameter of an anodic alumina template, shown as a TEM image. The template has been mostly filled with bismuth, and the TEM image was taken after the top and bottom sides of the sample had been ion milled with 6 KV Ar ions (Zhang, 1999; Zhang et al., 2000).

anodic alumina, which, because of its array of parallel nanochannels, is used as a template for preparing the Bi nanowires (see Fig. 24) (Zhang et al., 1998b). The bismuth is confined to these nanochannels and the bismuth does not diffuse into the anodic alumina matrix. The Bi nanowires are highly oriented with a common crystallographic direction along the wire axis. The anodic alumina templates, having an array of parallel and nearly cylindrical channels, are produced by anodizing aluminum substrates in weak acid solutions (Zhang et al., 1998b, 1999a). To achieve a better regularity of the pore ordering and to improve the pore size distribution, a two-step anodization process has been adopted for the preparation of anodic alumina templates. In the two-step anodization process, the anodic alumina layer was completely removed after the first anodization to produce an A1 substrate with a self-textured surface, which resulted from the curvature of the interface between the AlzO3 barrier layer and the underlying A1 substrate. The textured A1 substrate was then re-anodized under the same conditions as the first anodization process, so that ordered pores (see Fig. 24) could grow with the aid of the textured surface (Li et al., 1998; Masuda et al., 1997). Scanning electron microscopy (SEM) images of the top surfaces of the anodic alumina templates are shown in Fig. 25 after the two-step anodization process for pore diameters of 44 and 18 nm, and here it is seen that better ordering is achieved after the two-stage anodization process and for larger diameter pores. There are two principal ways that have been demonstrated for filling the nanochannels, one from the liquid phase under modest pressure (Zhang et

56

M . S . DRESSELHAUS ET AL.

FIG. 25. SEM images of the top surfaces of porous anodic alumina templates after the second anodization step (see text) in: (a) 4 wt% oxalic acid (H2C204) and (b) 20 wt% HzSO 4. The average pore diameters of (a) and (b) are 44 and 18 nm, respectively (Lin, 1960).

al., 1998b) or under high pressure (Huber et al., 1999b), and the other from the vapor phase (Heremans et al., 1998). By both filling methods, arrays of essentially single crystal Bi nanowires are produced (Zhang et al., 1998b; Heremans et al., 1998; Demske et al., 1999). Electrochemical methods have also been used to produce Bi nanowires in various porous membranes (e.g., mica or polymer films) (Piraux et al., 1999; Liu et al., 1998; Rabin, unpublished). The structural properties of the fabricated Bi nanowire arrays (Zhang et al., 1998b) have been investigated by various characterization techniques, such as x-ray diffraction (XRD), scanning electron microscopy, transmission electron microscopy, and selected-area electron diffraction (SAED) (Sam-

1

QUANTUMWELLS AND QUANTUM WIRES

57

FIG. 26. TEM micrograph of two long free-standing Bi nanowires with ~ 23 nm diameters. The inset shows a high-magnification image taken from the indicated position on the long wires (Zhang, 1999; Zhang et al., 2000).

sonov, 1968; Zhang et al., 1998b, 1999b). The common orientation of the Bi nanowire axes was confirmed by high-resolution electron microscopy (HREM) and SAED studies on free-standing Bi nanowires (see Fig. 26), which were prepared by dissolving away the anodic alumina template in a special acid solution (Zhang et al., 1999b). The free-standing wires were found to have nearly uniform diameters (within 10%) along the long wire lengths. XRD experiments show that nearly all ( ~ 90%) of the wire axes in a nanowire array are oriented along the same crystallographic direction (Zhang et al., 1998a, 1999b), the most common being the [1011] direction in the hexagonal coordinate system (or [0, 0.949, 0.315] in the Cartesian system) for nanowire diameters > 60 nm. For nanowire diameters ~<50 nm, the dominant growth direction is [0112] (or [0, 0.834, 0.552] in the Cartesian system). The individual wires contain a number of grains with different in-plane orientations, but with a common orientation along the wire axis (Sander et al., 2000). The grain sizes along the length of the wire can be as small as the wire diameter, but most grains are much larger. Figure 27 shows the XRD pattern of 52-nm Bi nanowire-Al20 3 composite with a reflection plane perpendicular to the wire axis, and the insert in Fig. 27 shows the SAED pattern from the same sample. This wire diameter was chosen to demonstrate the diffraction pattern obtained for wire diameters in the intermediate range, where wires in the template grow with axes either along [-1011] or [0112]. The XRD pattern in Fig. 27 shows that the [-1011] phase is the dominant phase for 52-nm-diameter nanowires, although the [0112] phase is also well represented in the nanowire array. The SAED pattern shows that a given wire has a common crystallographic axis, which is either along the [1011] or [0112] direction (Zhang et al., 1999b; Sander et al., 2000). In the XRD experiments, all the strong diffraction peaks are found to be close to the peak positions of a polycrystalline Bi standard, showing that the rhombohedral crystal structure of bulk Bi is also preserved in the small diameter Bi nanowires [at present there is confirmation of this result down to ~ 23 nm (Lin, 2000; Zhang et al., 1998b)]. m

58

M . S . DRESSELHAUS ET AL.

FIG. 27. XRD pattern for the anodic alumina-Bi nanowire composites. The average wire diameter of the Bi nanowires is about 52 nm. The insert in the figure shows the SAED pattern taken from the same sample. The two experimental results indicate that the Bi nanowires are highly crystalline and possess a preferred growth orientation (Lin, 2000; Zhang, 1999).

3.

ELECTRONIC STRUCTURE OF NANOWIRES

To model the electronic structure, we assume, as a first approximation, the simplest possible model for an ideal 1D quantum wire, where the carriers are confined inside a cylindrical potential well bounded by a barrier of infinite potential height (Hicks and Dresselhaus, 1993b). An extension of this simple approach provides a reasonable approximation for a Bi nanowire embedded in an alumina template, in view of the large bandgap of the anodic alumina template (3.2 eV), which provides excellent carrier confinement for the embedded quantum wires. Due to the small electron effective mass components of Bi, the quantum confinement effects in Bi nanowires are more prominent than for other wires with the same diameter. Since the electron motion in the quantum wires is restricted in directions normal to the wire axis, the confinement causes the energies associated with the in-plane motion to be quantized, and the energy of the lowest energy subband will be raised by approximately h2 AE

~

~

, 2 mpdw

where m* is the in-plane effective mass of the electrons and

(50)

dw is the wire

1 QUANTUMWELLS AND QUANTUMWIRES

59

diameter. Since electron motion is allowed only along the wire axis, the electrons are expected to behave like a 1D electron system, with a dispersion relation that has the form

E n m ( k l ) = ~3nm -!

h2k2 2m~'

(51)

where e,m represents a quantized energy level labeled by two quantum numbers (n, m), k I is the wave number of the electron wave functions traveling along the wire axis, and m~' is the dynamical effective mass for electrons moving along the wire. For materials with a highly anisotropic electronic energy band structure such as Bi, the mass m~, which determines the subband energies ~,,,, can be very different from the mass mr', which characterizes the motion along the wire. In the nanowire system, the quantized subband energy e,m and the transport effective mass m~' along the wire axis are the two most important band parameters that determine almost every electronic property of this unique 1D system. Due to the highly anisotropic electron and hole pockets, however, the calculation of the band structure in Bi nanowires has been very challenging. In the first theoretical calculation for the Bi nanowires, the quantized energy levels were evaluated by using a cyclotron mass approximation (Zhang et al., 1998a, 2000). In this approximation, the in-plane mass perpendicular to the wire axis m* was approximated by the cyclotron effective mass given by det Me~ 1/2 m* = \~,.Me.]j

(52) A

where M~ is the effective mass tensor for the electrons (or holes), I is the unit vector along the wire axis, and the transport effective mass was approximated by A

A

m~' ~ I. M~. l

(53)

In the earliest models, parabolic carrier pockets were assumed for both electrons and holes. With these approximations, the quantized energy levels could be readily derived by solving the 2D Schr6dinger equation with circular boundary conditions (Zhang et al., 1998a). However, the cyclotron effective mass approximation is an oversimplification of the in-plane effective mass, and instead of m* = x//mlm2 for the cyclotron effective mass, where m 1 and mz are the two principal mass components in the plane normal to the wire axis, the quantized energy states e.m are to a much closer

60

M.S.

DRESSELHAUS ET AL.

approximation determined by m*, given by 1

1

m~

m1

t

1

(54)

m2

The discrepancies between m* and m* can be very significant, if the values of ml and m2 are very different, as they are for bismuth. Subsequently, an improved model to describe the electronic states in Bi nanowires was developed, which was based on the square wire approximation for wires oriented along the three principal axes (Sun et al., 1999b, 1999d). The square wire boundary condition greatly simplified the eigenvalue problem for solving the Schr6dinger equation and enabled an analytical solution for the quantized energy levels. In this approximation, the transport effective mass along the wire axis was derived as

m* = (f. M ; '. h

(5 5)

for a parabolic carrier pocket. With this approximation, it was possible to explore the importance of nonparabolic effects and the temperature dependence of the electronic structure and of the predicted transport properties. Although the square wire approximation provided a better solution to the in-plane effective masses and quantized energy states than the previous cyclotron effective mass approximation, further improvements were needed to describe the actual circular wires used in the experiments, and their proper symmetries (Lin, 2000). Recently, calculations of the band structure of 1D Bi quantum wires have been carried out that explicitly take into account the cylindrical wire boundary conditions and the anisotropic carrier effective masses in Bi (Lin, 2000). In addition, the nonparabolic features of the L-point conduction band and the temperature dependence of the various band parameters are also included to provide a more accurate model for the electronic structure of the Bi nanowires. For an infinitely long circular wire with a diameter dw, the z' axis is taken to be parallel to the wire axis with the x' and y' axes lying on the cross-sectional plane of the wire. Since the wires are allowed to be oriented along an arbitrary direction with respect to the crystallographic directions, the inverse effective mass tensor of one of the carrier pockets in the wire coordinates (x', y', z') has the general form

= M -1 =

0~21

0~22

o~2

~31

0~32

~3

(56)

1 QUANTUM WELLS AND QUANTUM WIRES

61

where ~j O~ji. Explicit values for the components of the effective mass tensor M for the T-point holes and L-point electrons are given in Table VII. Since there is only one carrier pocket for holes at the T point, Eq. (56) is sufficient for describing the T-point holes. However, for the L-point electrons, Eq. (56) describes only carrier pocket A (see Fig. 21), and rotations of Eq. (56) by _+2n/3 around the trigonal axis are needed to obtain the effective mass tensors for the B and C electron carrier pockets. For the T-point holes, parabolic energy bands are assumed, and the Schr6dinger equation is simplified and given by =

2

el l t?x,-----5+ ~22~--~ u =

E

2m33,/

u

(57)

Equation (57) has solutions u(x', y') that satisfy the boundary condition u(r'= d w / 2 ) = 0, yielding the eigenvalues of u(x', y') in Eq. (57) that are quantized 2 2

h kz,

(58)

E,,m(kz,) = e.,,m -t 2m33

where e,m is the eigenvalue of Eq. (57) corresponding to the band-edge eigenstate at kz, = 0 labeled by the quantum numbers (n, m). Here m33 = ~:'-M. ~:'

(59)

is the effective mass component along the wire axis, and the in-plane effective masses are m x, ~

t~ 111

-- ( 3 ~ ' - M - 1 " 2

') -

1

(60)

mr, = c~221= (9" M-'-~')-1 in the x' and y' directions, respectively. The eigenvalue e,m has an analytic expression only when ~11 = c~22, but in general, the values of e,m must be solved numerically (Lin et al., 2000). This is true for both the T-point holes and for the L-point electrons. The electron states in Eq. (58) are split into many subbands with band edges at E = e,m, and each subband behaves like a one-dimensional electron system in the z' direction with transport effective mass mz, = m33. The L-point bands, however, are highly nonparabolic due to the strong coupling between the L-point conduction and valence states. The quantized subband energy levels for the nonparabolic L-point electrons can be described by the Lax model

62 Enm(kz)

M.S.

DRESSELHAUS ET AL.

-

E~

F ff-~

1

+4Enm(kz)EoL

(61)

where the band-edge energy of each subband is given by

Snm--Enm(kz=O)=

EaL ~--~/ 2

F

4~'nm

1 + Eo---~

= (7.m - 1) E~

2

(62)

and 2 2

~ h k~ E,m(kz) = ~,m + 2Fnz

(63)

Here ~nm is the primitive subband energy for parabolic bands derived from the eigenvalues of Eq. (57) for the L-point electrons, and ffh is the primitive effective mass along the wire axis corresponding to Eq. (59) and

~nm ~

k// 4~"m 1 + Eo---~ .

(64)

The 1D dispersion relation Enm(kz) of each subband for the Bi nanowire is more complicated than for the case of bulk bismuth (Lin et al., 2000; Lax and Mavroides, 1960). For energies near the subband edge, however, it is a good approximation to express the full dispersion relation in Eq. (61) in terms of a Taylor's expansion as h2k 2 E"m(k~) ~ e"m + 2m*.m

(65)

where

m*z,nm=

1 -Jr-~

fflz --" Ynmfflz

(66)

is the transport effective mass along the wire for the corresponding subband, which takes nonparabolic effects into account. As indicated in Eq. (66), the transport effective masses m'z,,,, are different for every subband, and they increase as the subband edge energy increases. It should be noted, however, that in deriving the nonparabolic dispersion relation, there is a far band

1

QUANTUM WELLS AND QUANTUM WIRES

63

TABLE IX CALCULATED EFFECTIVE MASS COMPONENTS OF EACH CARRIER POCKET FOR DETERMINING THE

BAND STRUCTUREOF Bi NANOWIRESAT 77 K ALONG THE INDICATED CRYSTALLOGRAPHIC DIRECTIONS BASEDON THE EFFECTIVEMASS PARAMETERSOF BULK BISMUTHGIVEN IN TABLE VII Mass component e- pocket A

mx, mr ,

e- pocket B

mx, mr ,

e- pocket C

m x,

m z,

m z,

mr , m~, Hole pocket

m x,

mr , m z,

Trigonal

Binary

Bisectrix

[0152]

[1051]

0.1175 0.0012 0.0052 0.1175 0.0012 0.0052 0.1175 0.0012 0.0052 0.0590 0.0590 0.6340

0.0023 0.2659 0.0012 0.0023 0.0016 0.1975 0.0023 0.0016 0.1975 0.6340 0.0590 0.0590

0.0023 0.0012 0.2630 0.0023 0.0048 0.0666 0.0023 0.0048 0.0666 0.6340 0.0590 0.0590

0.0029 0.0012 0.2094 0.0016 0.0125 0.0352 0.0016 0.0125 0.0352 0.1593 0.0590 0.2349

0.0024 0.0012 0.2542 0.0019 0.0071 0.0526 0.0019 0.0071 0.0526 0.3261 0.0590 0.1147

The z' direction is chosen along the wire axis. Calculations are done for Bi nanowires oriented along the three principal axes (trigonal, binary, and bisectrix) and the [0112] and [ 1 0 i l ] directions (which are preferential growth directions for Bi nanowires) and correspond to (0, 0.8339, 0.5519) and (0, 0.9503, 0.3112) in Cartesian coordinates, respectively. All mass values in this table are in units of the free electron mass, mo (Lin, 2000).

contribution (hZk2/2mo) that is usually neglected when applying k.ff perturbation theory (on which the Lax model is based) to bulk bismuth, due to the very small effective masses at the 3D band edge for strongly coupled bands. For the Bi nanowires where the 3D L-point energy bands split into many subbands, however, the far band contributions are not negligible for higher subbands where the effective masses at the subband edge are no longer small compared to the free-electron mass mo. We therefore write the more complete expression to relate m*z , m n to fizz: 1 m*z ~,mtl

=

1 mo

~

1

(67)

~ n m lTl'lz

From the band structure parameters for bulk Bi (Table VII), values for mx,, mr,, and m z, (or ~hz) at 77 K for Bi nanowires for the three principal

crystallographicB axes (trigonal, binary, and bisectrix directions), and for the preferential [0112] and [1011] growth directions are given in Table IX. It should be noted that rex, and mr, can be interchanged without affecting any physical results. Equations (66) and (67) should be used to include nonparabolic effects into the entries in Table IX for the electrons, while the

64

M . S . DRESSELHAUS ET AL. 100

~\ \

80

j

60

~

_ J 4L.o

20 0

-/

-20 -40

.....

I .......

s

,

0

50

100

Wire diameter (nm)

150

=38

= 13.6

200

FIG. 28. The subband structure at 77 K of Bi quantum wires oriented along the [0112] growth direction, showing the energies of the highest subbands for the T-point hole carrier pocket, the L-point electron pockets (A, B, and C), as well as the L-point holes. The zero energy refers to the conduction band edge in bulk Bi. As the wire diameter d w decreases, the conduction subbands move up in energy, while the valence subbands move down. At dc = 49.0 nm, the lowest conduction subband edge formed by the L(B, C) electrons crosses the highest T-point valence subband edge, and a semimetal-semiconductor transition occurs (Lin, 2000).

parabolic approximation of Eq. (65) should be used to describe the small energy excursions from k=, = 0 for the dispersion along the wire axis. Since the nanowires with smaller ( < 40 nm) wire diameters tend to be oriented along the [0112] direction, the calculated subband structure of Bi quantum wires oriented along the [0112] directions are shown in Fig. 28 at 77 K. For the [0112] wires, the degeneracy at the L point is lifted, resulting in two inequivalent groups of carrier pockets: a single electron pocket A and two electron pockets B and C with symmetry and band parameters the same as each other but different from those of pocket A. The L-point electron pocket A has smaller mass components (m~,, my,) in the quantum confined direction than the electron pockets/3 and C (see Table IX). Therefore, the electron pocket A forms a higher energy conduction subband, while the electron pockets B and C form a twofold degenerate subband at a lower energy (see Fig. 28). It should be pointed out that since electron pocket A has a larger mass component (m~,) along the wire axis than pockets B and C, the dispersion relation of the L(A) subband has a smaller curvature (see Fig. 28). The band edge of the lowest subband of the L(B, C) electrons increases with decreasing wire diameter dw, while the highest subband edges m

1

QUANTUM WELLS AND QUANTUM WIRES

80

|

>

c~ "0 c:

"

1

i

40 20

..... ,

............ (a) Trigonal (b) [0112] (c) [101-1]

, '.. '~,"'.. '\ , ".... '\ ,"-

t~

LU

'

i,{, , \

60

E

i~

i't

65

---

(d) Bisectrix

(e) Binary

, \ . ,".....,. '\ '~x~"...

~

'-..

0

\

: ' , ~",....

"\

"'.., ...

-20 -40

0

. . . .

v 50

. . . . . . . . . 1 O0

Wire Diameter(nm)

-38 150

FIG. 29. Calculated bandgap energy between the lowest electron subband and the highest hole subband of Bi nanowires oriented along the (a) trigonal, (b) [0112], (c) [1011], (d) bisectrix, and (e) binary directions as a function of wire diameter. The nonparabolic effects of the L-point electron pockets have been taken into account, and the cylindrical wire boundary condition is used in the calculations. The values of d<, the critical wire diameter, where the semimetal-to-semiconductor transition occurs, are 55.1, 39.8, 48.5, 50.0, and 49.0nm for nanowires oriented along the trigonal, binary, bisectrix, [0112], and [1011] directions, respectively. The curves for the bisectrix and [10T1] directions are too close to one another to be resolved.

of the T-point and L-point holes move downward in energy. At d w < 49.0 nm, the energy of the lowest L-point conduction subband edge exceeds that of the highest T-point valence subband edge, indicating that these nanowires are semiconducting. The subband structure of the 1D Bi quantum wires is strongly dependent on the wire orientation. For the trigonal wires, the subband energies of the three L-point electron pockets remain degenerate for all wire diameters, and the critical wire diameter for the semimetal-semiconductor transition at 77 K occurs at dc = 55.1 nm. The bisectrix wires and the [1011] wires have the same symmetry as the [0112] wires, and therefore, the three wire orientations (bisectrix, [1011], and [0112]) possess similar subband structures (see Fig. 28) with slightly different critical wire diameters. For binary wires, the electron pocket A forms a lower energy conduction subband than the electron pockets B and C, which form a twofold degenerate subband at a higher energy for the binary wire orientation, in contrast to Fig. 28. The calculated indirect bandgap E'o between the highest T-point hole subband edge and the lowest L-point electron subband edge at 77 K is shown in Fig. 29 for five different wire orientations. The critical wire diameters, where the indirect bandgap vanishes E'o = 0, are 55.1, 39.8, 48.5, 49.0, and 48.7 nm for

66

M . S . DRESSELHAUS ET AL. .

.

.

.

,

.

.

.

.

,

.

.

.

.

,

.

.

.

.

,

.

.

.

.

,

.

.

.

.

60 E e-

"E" 50 ~

" ~ \

"......

E ._~ 40

D . i

..... bi~

N 3o

--'--

m O ~

~3 2o 10

. . . .

0

~

50

"'-. "~? .....

"",_ ~(," ....

b i s e c ' t r i x

101T2] [101-11 . . . .

~,

i

. . . .

1O0

,

. . . .

150

,

. . . .

200

"~".,.

"-, ~ ,

9

250

9

9

300

T e m p e r a t u r e (K) FIG. 30. Calculated critical wire diameter d c for the semimetal-semiconductor transition as a function of temperature for Bi nanowires oriented along various directions (Lin, 2000).

B

nanowires oriented along the trigonal, binary, bisectrix, [0112], and [1011] directions, respectively (Lin, 2000). It should be noted that since dc is determined by the band structure of Bi, which is strongly temperature dependent for T > 80 K, the critical wire diameter dc for the semimetal-semiconductor transition will also be temperature dependent at high temperatures (T > 80 K). The temperature dependence of d~ has been calculated using the best available values for the temperature dependence of the bismuth band parameters, which are given in Table VIII. Figure 30 shows the resulting calculated critical wire diameters d~ for the semimetal-semiconductor transition in Bi nanowires as a function of temperature for different wire orientations. Since the band overlap Ao between the L-point conduction band and the T-point valence band, and the electron effective mass components both increase with temperature (see Table VIII), a smaller wire diameter is required to achieve the semimetal-semiconductor transition at higher temperatures. At 300 K, the best current estimates for the critical wire diameters d~ are 15.4, 11.2, 13.6, 14.0, and 13.6 nm for nanowires oriented along the trigonal, binary, bisectrix, [0112], and [1011] directions, respectively. Based on the band structure derived for 1D Bi quantum wires, the carrier concentration for undoped Bi nanowires has been calculated by adjusting the Fermi level so that the number of holes is equal to the number of electrons. Figure 31 shows the calculated total carrier densities of Bi nanowires oriented along the [0112] growth direction as a function of temperature for different wire diameters. Since the critical wire diameter dc is temperature dependent, three different types of temperature dependences of the carrier densities are predicted for undoped Bi nanowires, depending

1 QUANTUMWELLS AND QUANTUM WIRES

10 ~9

bulK~"'~"

67

I ~- I ~ /

018 E1 o v

"g t-

1

017

121 ~-

016

o 1 0 ~5

1014

1 O0

Temperature (K)

200

300

FIG. 31. Calculated total carrier density (electrons and holes) of Bi nanowires oriented along the [0112] direction as a function of temperature for different wire diameters--10, 40, and 80 nm, in comparison to that for bulk Bi. The carrier density of 10-nm Bi nanowires has a temperature dependence similar to that of a narrow-gap semiconductor, while 80-nm nanowires behave like a semimetal. The carrier density in 40-nm Bi nanowires has a semiconductor-like temperature dependence at low temperatures (T ~<170 K) and a semimetallike temperature dependence at high temperatures (T > 170 K) (Lin, 2000).

on the wire diameters. For 10-nm Bi nanowires, which are always in the semiconductor regime up to 300 K (see Fig. 30), the carrier density increases exponentially with temperature up to 300 K, where n is predicted to be 1.2 x 101 s cm-3. On the other hand, for 80-nm Bi nanowires, which remain in the semimetallic regime even down to 0 K, the carrier density is expected to have a similar temperature dependence to that of bulk Bi, and the carrier density at 300 K is predicted to be 8.2 x 1018 cm -3. The smaller carrier density of the 80-nm nanowires compared to bulk Bi (n = 1.34 • 1019 c m - 3 ) arises because of the smaller band overlap between the conduction band and the valence band in Bi nanowires as compared to bulk Bi. As for the 40-nm Bi nanowires, the semimetal-semiconductor transition temperature is prem dicted to be around 170 K for a [0112] nanowire (see Fig. 30). Thus, for T <~ 170 K, 40-nm Bi nanowires are in the semiconductor regime, and the carrier density drops significantly with decreasing temperature, while for T > 170 K, the nanowires are in the semimetal regime, and the carrier density has a similar temperature dependence in comparison to bulk Bi. It should also be noted that for semiconducting wires, the slope of the curves in Fig. 31 is approximately proportional to the bandgap between the conduction band and the valence band, which increases with decreasing T

68

M . S . DRESSELHAUSET AL.

for 40-nm Bi nanowires below 170 K. Therefore, the slope of the curve for 40-nm Bi nanowires in Fig. 31 decreases with increasing temperature T.

4.

DOPING OF Bi NANOWIRES

For intrinsic Bi nanowires, equal numbers of electrons and holes are expected, whether in the semimetallic or semiconducting state. In thermoelectric applications of bismuth, however, it is necessary to control the Fermi level so that (1) the transport phenomena are dominated by a single type of carrier only (i.e., electrons or holes) and (2) the electrochemical potential is placed to achieve the optimum ZT. In addition, to optimize the efficiency of thermoelectric devices, it is essential to obtain a high Seebeck coefficient S. Since the contributions from holes and from electrons have opposite signs with regard to the Seebeck coefficient, however, the magnitude of S in pure Bi is usually very small, although individual contributions from electrons or holes can be quite significant. Therefore, it is expected that Bi nanowires can be a very promising thermoelectric material if the Fermi level can be adjusted properly so that only electrons or only holes contribute to S and the electrochemical potential is set to maximize Z T (see Section V.3). Since Bi is a group V element, the Fermi level can be increased by introducing a small amount of a group VI element, such as Te, which acts as an electron donor in Bi. Group IV elements such as Sn or Pb, on the other hand, act as electron acceptors in Bi, and can be used to synthesize p-type Bi. The effect of n-type doping of bulk bismuth (Noothoven Van Goor, 1971) and of thin films to a concentration of 5 • 1019 cm -3 (Heremans, et al., 1988) has been studied and of p-type doping of bulk Bi (Noothoven Van Goor, 1971) and to a level of 8400 ppm Sn in 1-/~m-thick Bio.91Sbo.o9 alloy films (Cho et al., 1999a). N-type Bi nanowires have been studied more intensively because of the very small electron effective mass and the highly anisotropic Fermi surface of the electron carrier pockets in Bi, which result in higher Z T values. Most of the synthesis studies have involved Te as the n-type dopant (Cho et al., 1999a; Zhang et al., 1999b; Heremans and Thrush, 1999a; Heremans, 1999). From the results of XRD experiments, it has been found that the crystal structure of Bi is not changed by introducing a small amount of Te (see Section VI.2). Thus, we assume that the electronic band structure of Te-doped Bi is the same as pure Bi in the spirit of a rigid band approximation, except for the presence of a Te donor energy level located below the conduction band edge by E d. The ionization energy Ed required for releasing donor electrons from Te atoms can be estimated using the Bohr hydrogen-like model (Heremans et

1

QUANTUM WELLS AND QUANTUM WIRES

69

al., 2000a; Ketterson, private communication; Lin, 2000), E d ~ 13.6 x m*/m~ E2

(eV)

(68)

where m* is the effective mass at the conduction band edge, mo is the free-electron mass, and e is the dielectric constant (e ~- 100) of Bi in the low-frequency limit (see Table VI). The value for the effective mass m* is taken as the average value of the electron effective mass tensor, and at low temperatures, m*/m o ~_ 0.002. With this large value for e and small value for m*, we obtain a very small value for E d E d ~- 3.20 x 10- 3 meV

(69)

with a very large effective Bohr radius

a~ = 0.5 (]k) x

E m*

~ 2.5 #m

(70)

so that the energy required to ionize the Te atom in Bi is very small. It should be noted, however, that the results obtained in Eqs. (69) and (70) are valid only for bulk materials. In Te-doped Bi nanowires, where the wire diameter dw is smaller than the effective Bohr radius a~, the ionization energy will be increased due to the confinement of donor electrons to the vicinity of the impurity atom. Since the Coulomb potential energy is inversely proportional to the distance between two charged particles, a rough estimation of the ionization energy Ed(~o) in nanowires can be made by multiplying the ionization energy Ed(aO) in bulk materials [Eq. (69)] by the ratio between the effective Bohr radius and the wire diameter. Thus, for 40-nm Te-doped Bi nanowires, an estimate for the ionization energy would be

a~ Ed(1D ) '~ Ed(3D ) •

dw

'~ 0.2 m e V .

(71)

At very low temperatures where the thermal energy kBT <~ EdtlO), the donor electrons will freeze out, and the freeze-out temperature is Tfo_ 2 K for 40-nm Te-doped Bi nanowires. For the temperatures of interest (ranging from 4 to 300 K), we can assume that all the donor atoms are ionized in 40-nm Te-doped Bi nanowires and that each Te atom donates one electron to the conduction band.

70

M . S . DRESSELHAUS ET AL. 1020

1019

E 1018 m -3

.-t"

"o 1017

--/~

Nd=2 "73x10'` cm-3

.--

~,~, ~ N d = l O tTcm -3

i,-

O 1016

Nd=O 10 Is

, 0

~ 1 O0

,

J 200

,

Temperature (K)

300

FIG. 32. Calculated total carrier density (electrons and holes) of 40-nm Bi nanowires oriented along the [0112] direction of various dopant concentrations as a function of temperature (Lin, 2000).

Figure 32 shows the calculated carrier densities of 40-nm Bi nanowires oriented along the [0112] direction as a function of temperature for several different donor concentrations N d. At low temperatures where the intrinsic carriers are overwhelmed by the carriers from the Te dopants, the total carrier density is essentially constant and is approximately equal to the dopant concentration Nd. As the temperature rises, the number of intrinsic carriers will increase exponentially, and will eventually exceed the number of dopant carriers above a certain threshold temperature Tth, so that the carrier concentration will again be dominated by the intrinsic carriers in this higher temperature regime. This threshold temperature Tth for the extrinsicintrinsic transition increases with dopant concentration Ne, as shown in Fig. 32. For N d = 10 ~7, 2.73 x 10 ~8, and 7 x 1018 cm-3, the estimated threshold temperatures are Tth --~ 85, 200, and 250 K, respectively. Thus the temperature of operation must be carefully taken into account when designing Bi nanowire systems for thermoelectric applications.

5.

SEMI-CLASSICAL TRANSPORT MODEL FOR Bi NANOWIRES

The thermoelectric-related transport coefficients of Te-doped Bi nanowires can be derived from the simple semi-classical model summarized in Section II, which is based on the Boltzmann transport equation. For a

1 QUANTUM WELLS AND QUANTUM WIRES

71

one-band system, the important thermoelectric-related transport coefficients, such as the electrical conductivity a, the Seebeck coefficient S, and the thermal conductivity Xe are derived from Eqs. (5) to (8) (see Section II). In a one-band quantum wire, Eq. (8) is replaced by

1D =

7r2d2

-- ~

z(k)v(k)v(k)[E(k)

- Er ~

(72)

in which E(k) is the electronic dispersion relation, z(k) is the relaxation time, E I is the Fermi energy, and f ( E ) is the Fermi-Dirac distribution function 1 f (E) = 1 + e ~E-~')/kBT

(73)

Since the numerical calculation of Eq. (72) requires knowledge of the k dependence of the relaxation time r(k), and since from fundamental principles of scattering mechanisms the calculation of r(k) is usually very complicated, we use a simple first approximation, known as the constant relaxation time approximation, to simplify the calculations of the thermoelectric properties of materials. In this formalism, z(k) = r is taken to be constant in k, and in energy, and ~ can be related to the carrier mobility # along the wire by eT

/t

m*

(74)

where m* is the transport effective mass along the wire, and # can, in principle, be obtained from experimental measurements. Thus, the integration of Eq. (72) can be carried out readily as long as the dispersion relation E(k) is known. For a one-band system described by a parabolic dispersion relation, Eq. (72) becomes &t,(o) 1D = D[ 89 1/2] ~(kBT)D[3F1/2 - 89 1/2] 1D "- [ _ ( k B T ) D [ a F 1 / 2 _ 89 1/2

~1)

(75) (for electrons) (for holes)

~q:'~2)o= (kBT) 2D [5F3/2 -- 3~*F1/2 + 89* 2 F - 1/2]

(76) (77)

where D is given by 16e

1/2

(78)

72

M . S . DRESSELHAUS ET AL.

and F i, given by Eq. (13), denotes the Fermi-Dirac related functions, with fractional indices i = - 89 !2, 3 , . . . . The reduced chemical potential ~* is defined as (, =

,~(Es - e~e~ T [(e~~

(for electrons) (for holes)

Ey)/kBT

(79)

where e~o) and e~o) are the band edges for electrons and holes, respectively. For Bi quantum wire systems, there are many 1D subbands due to the multiple carrier pockets at the L points and the T point, and the quantum confinement-induced band splitting also forms a set of 1D subbands from each single band in bulk materials. Therefore, when considering the transport properties of real 1D nanowire systems, contributions from all of the subbands near the Fermi energy should be included. In a multiband system, the ~(')'s, defined by Eq. (8) for bulk materials and by Eq. (72) for the case of quantum wires, should be replaced by the sum 6o~) ~total = Z~ 2~ of contributions from each subband (label by i), and the transport coefficients ~r, S, and Xe then become O'total-- E Gi

(80)

i

St~

=

1 Fr(2) KTe't~ : C ~

(81)

Ei~ ~-'i (Ti

L L~t~ --

(i(1) ~2--I ~'x-'t~ I ~ / L~total _1

(82)

where ~r/ and Si are the electrical conductivity and the thermopower corresponding to each subband, respectively. In Bi nanowires, the sums in Eqs. (80) to (82) include subbands associated with electron pockets A, B, and C, as well as contributions from T-point holes and L-point holes (see Section VI.3). Another physical quantity of interest in thermoelectric applications is the lattice thermal conductivity •L, which, together with the electronic thermal conductivity Ke, determines the total thermal conductivity of the system. From kinetic theory, the thermal conductivity of phonons is given by K,~ = - ~ c . v t

(83)

where Cv is the heat capacity per unit volume, v is the sound velocity, and I is the mean free path for phonons. We note that, for an ideal quantum wire system embedded in a host material with a large bandgap, the electron wave functions are well confined within the quantum wire, and they can only

1 QUANTUM WELLS AND QUANTUM WIRES

73

travel along the wire axis. The host material that confines electrons cannot confine the phonon paths, however, and thus, because of acoustic mismatch, phonons will be scattered when they move across the wire boundary. This increased boundary scattering of phonons in the quantum wire system will decrease the phonon mean free path l as well as the lattice thermal conductivity along the wire. The simplest approximation to model the lattice thermal conductivity in the quantum wire system is to replace the phonon mean free path I in Eq. (83) by the wire diameter d w if dw < I in the bulk material. It should be noted that for dw << l, the lattice thermal conductivity is expected to decrease dramatically, more so than the decrease in the electrical conductivity. This reduction in the lattice thermal conductivity is one of the reasons for the expected enhanced thermoelectric performance in low-dimensional systems. a.

Optimization

of ZIDT through

Doping at 77 K

In applying the general results of this subsection to calculate ZIDT for Bi nanowires, the anisotropy of the electronic structure results in anisotropic effective mass tensors, and other related quantities, as discussed in Subsection 3 of this section. For example, the mobility tensor for each carrier pocket for Bi is also highly anisotropic. For the L-point electron pocket A (see Fig. 21), the mobility tensor has the form

],le(A) ~-

t

#el

0

0 t

0

Pe2

Pe4

0

]Ae4 /Lie3/

(84)

and the mobility tensors for electron pockets B and C can be derived by a rotation of Pe~a) by _ 120 ~ about the trigonal axis. For the T-point holes, the mobility tensor has the form

Ph =

t!1 ~ ~ #hl

0

(85)

flh3

The values for these mobility tensor elements at 77 K are listed in Table X (Saunders and Siimengen, 1972). Since the mobility tensors are anisotropic, the mobility #7 for carriers traveling along the wire will depend on the wire orientation, and/~7 is given by #7 -- (l'" jl/- 1. ~)-1

(86)

74

M . S . DRESSELHAUS ET AL. TABLE X

VALUESOF THE MOBILITYTENSORELEMENTSFOR ELECTRONAND HOLE POCKETSOF Bi AT 77 K (Saunders and Sfimengen, 1972)

Pel

Pe2

Pe3

Pe4

Phi

Ph3

6.8 x 103

1.6 • 104

3.8 x 103

- 4 . 3 • 104

1.20 x 103

2.1 x 104

The mobility values are given in units of cm 2 V - 1 s- 1

where l is the unit vector along the wire axis, which follows from the general definition of the carrier mobility in terms of ~t = ez/m* and from Matthiessen's rule summing 1/zi for each scattering process i. The lattice thermal conductivity in bulk Bi is also anisotropic, and has the form

X L ~---

tKL,0 0

0

KL,_I -

0

0 t

(87)

KL,II

where •s and Ks are the thermal conductivities parallel and perpendicular to the trigonal axis, respectively. By extrapolating the experimental data for xs measured between 100 and 300 K (Gallo et al., 1963), the lattice thermal conductivity tensor elements at 77 K are estimated as Ks = 13.2 (W/mK) and ~cL,II= 9.9 (W/mK), respectively (Gallo et al., 1963). For Bi nanowires oriented in directions other than the three principal axes, the lattice thermal conductivity along the wire is then given by

KL,~"~- l'" K"L "7--- COS20KL,L -~- sin20KL, jl

(88)

where 0 is the angle between the wire axis and the trigonal axis. The phonon mean free path l in Bi nanowires is estimated by the heat capacity Cv, sound velocities v, and the thermal conductivity tcL,~via Eq. (83). At 77 K, the value for the heat capacity of Bi is measured as Cv _~ 1.003 (jK-1 cm-3) (White, 1972). The measured sound velocities v of Bi at 1.6 K (Walther, 1968) and 300 K (Eckstein et al., 1960) along selected directions are listed in Table XI, in which the interpolated values for v at 77 K are also given. The calculated phonon mean free path l of bulk Bi at 77 K is listed in Table VI for B Bi crystals oriented along the three principal axes, and also along the [1011] and the [0112] directions. As the wire diameters become smaller than the phonon mean free path calculated in bulk Bi, the scattering at the wire boundary becomes the dominant scattering process for phonons, and the phonon mean free path in these nanowires is approximately limited

1

75

QUANTUM WELLS AND QUANTUM WIRES TABLE XI

THE SOUND VELOCITIES /) OF Bi ALONGTHE THREE PRINCIPAL AXES v(105 cm/s) T (K)

1.6 77 300

Trigonal

Binary

Bisectrix

0, ~ ,

[0152]

[ 10-11]

2.02 2.01 1.972

2.62 2.60 2.540

2.70 2.67 2.571

2.15 2.13 2.082

2.26 2.24 2.18

2.58 2.45 2.375

The values of 77 K are interpolated from the experimentally measured results at 1.6 K (Walther, 1968) and 300 K (Eckstein e t al., 1960).

by the wire diameter, or l ~- d w. Thus, the lattice thermal conductivity KL in Te-doped Bi nanowires will decrease significantly as the wire diameter decreases below dw <<. 15 nm (see Table VI). Using the general formalism presented in Subsection 5 of this section for S, tr, ~ce, and the preceding discussion on KL and procedures to account for the multiple carrier pockets and their anisotropy, the thermoelectric figure of merit ZIDT has been calculated. Figure 33 shows the calculated ZIDT for n-type Bi nanowires oriented along the trigonal axis at 77 K as a function of the dopant concentrations for three different wire diameters. We note that the value of ZIDT for a given donor concentration increases dramatically with decreasing wire diameter dw (see Fig. 23 also), and the maximum ZIDT

6 5 4

% N- 3

2 1

~

. . . . .

iO"'

. . . . .

iO

. . . . .

Dopant Concentration

ir

10 20

(cm -3)

FIG. 33. Calculated ZIDT for Te-doped Bi nanowires oriented along the trigonal axis at 77 K as a function of Te dopant concentration for three different wire diameters (Lin, 2000).

76

M.S. DRESSELHAUSET AL.

,00o . ~N~-I.0

/ ~ ........... '~

\

[lOiq]

0000 '

0 Dopant Concentration (cm-a)

FIG. 34. Calculated Z1D T at 77 K as a function of Te dopant concentration N d for 10-nm Te-doped Bi nanowires oriented along different directions: trigonal, binary, bisectrix, [1011-1, and [0112] (Lin, 2000).

for each wire diameter occurs at an optimized donor concentration Nd(opt) , which increases somewhat as the wire diameter decreases. For 5-nm Bi nanowires oriented along the trigonal axis at 77 K, the maximum ZIDT at 7 7 K is about 6, with an optimized electron concentration Nd(opt)"~ 1018 cm -3. The value of Z1DT also strongly depends on the wire orientation due to the anisotropic nature of the Bi band structure and of the thermal properties of Bi. Figure 34 shows the calculated figure of merit Z1DT at 77 K as a function of donor concentration N d for 10-nm Bi nanowires oriented in different directions. For 10-nm Bi nanowires at 77 K, the trigonal nanowires have the highest optimal ZID T, which is about 2.0, while bisectrix wires have the lowest optimal Z1DT ~_ 0.4. This optimal Zlo T increases as the wire orientation is varied from the bisectrix axis closer to the trigonal axis, as shown for [1011] and [0112] nanowires (see Table XII), which, respectively, make angles of 71.9 ~ and 56.5 ~ with respect to the trigonal axis. The optimum carrier concentrations Ndtopt) and the corresponding Z1DT of n-type Bi nanowires at 7 7 K are listed in Table XII for various wire diameters and orientations. Figure 35 shows the calculated optimal Z~DT at 77 K as a function of wire diameter for n-type Bi nanowires oriented along the three principal axes and the [1011] and [0112] directions. It should be noted that, although binary wires have a smaller critical wire diameter dc for the semimetal-semiconductor transition than bisectrix wires (see Fig. 30), 10-nm diameter binary wires have a higher optimal Z~DT than bisectrix wires for the same wire diameter. The dependence of the optimal Z~DT on the wire orientation can be qualitatively explained by a simple argument considering the dependence of ZID T on the effective mass in a 1D system. First, we note that the optimal Fermi energy for the maximum Z~DT

1

77

QUANTUM WELLS AND QUANTUM WIRES TABLE XII

THE OPTIM-CMDOPANT CONCENTRATIONSNdlopt) (IN 10 is cm-3) AND THE CORRESPONDING Z1DT OF n-TYPE AND p-TYPE Bi NANOWIRESAT 77 K FOR VARIOUS WIRE DIAMETERSAND ORIENTATIONS (Lin, 2000) 5 nm Wire orientation Trigonal n-type p-type Binary n-type p-type Bisectrix n-type p-type [1011] n-type p-type [0112] n-type p-type

10 nm

40 nm

Ndtopt)

Z1DT

Nd(opt)

ZID T

Ndtopt)

Z1DT

0.96 0.96

6.36 6.36

0.81 12.9

2.0 0.72

0.38 6.2

0.31 0.17

0.35 0.79

3.68 1.78

0.28 10.3

1.14 0.16

0.56 7.9

0.13 0.05

4.1 0.74

2.21 0.32

1.78 0.19

0.40 0.40

4.97 0.50

0.03 0.07

3.21 1.04

2.69 1.16

1.57 0.43

0.51 0.19

2.57 0.63

0.04 0.05

2.07 2.59

3.41 2.46

1.33 0.58

0.70 0.18

2.73 0.75

0.06 0.03

is usually below but very close to the lowest conduction subband edge (Sun et al., 1999d), as discussed in Subsection 3 of this section. Therefore, for semiconducting Bi nanowires with Fermi energies close to the optimal Fermi level, the system can be approximately described by a one-band model at low temperatures, in which the thermal energy k~T is much smaller than the bandgap and adjacent subband separations. Since the Seebeck coefficient in a one-band system is almost independent of the band structure and is determined by the position of Fermi energy only, the dependence of Z1DT on the carrier effective mass is only influenced by the electrical conductivity cr and the electronic contribution to the thermal conductivity •e" In this Fermi energy range, however, ~e is usually small due to the low carrier densities, and the total thermal conductivity is dominated by the lattice thermal conductivity KL. For a one-dimensional system, the carrier density n is proportional to ~ - g and to the degeneracy g of the lowest energy subband, so that the electrical conductivity, which is written as a = ne2z/m *, is proportional t o gm *-1/2. Therefore, the optimal Z1DT is roughly proportional to Am*-1/2 in a one-dimensional transport system. The relative values of Am*-1/2 for Bi nanowires oriented along the trigonal, binary, and bisectrix directions are calculated to be 1:0.69:0.19, respectively, which agree quite well with the relative values of the optimal Z IDT of 1:0.57:0.2 for 10-nm Bi nanowires in these three directions (see Table XII).

78

M . S . DRESSELHAUS ET AL. 4

....

,

,

n-type

3

Iiti

bisectrix [lOl

~1,1,.

77 K

1]

\ /

)I,I\A ~,~ '..

1

")'\ ~""" ~

',:~\'.. ..... x \

,x..,\ 0

,

+

0

10

/

~

/

/

binary / tdgonal

/

"..

............... ~

20

30

diameter

(nm)

40

50

60

FIG. 35. Calculated Z1DT at 77 K as a function of Te dopant concentrations N d for 10-nm Te-doped Bi nanowires oriented along different directions: trigonal, binary, bisectrix, [10il], and [0112] (Lin, 2000).

As a comparison, the optimum acceptor concentration Na(opt) and the corresponding Z IDT for p-type Bi nanowires are calculated and listed in Table XII for various wire diameters and orientations. Compared with the results in Table XII for n-type Bi nanowires, we note that p-type Bi nanowires in general have a much lower ZID T, and the asymmetric behavior is discussed in the following subsection.

b. Effect of the T-Point Holes on the Thermoelectric Properties of Bi Nanowires In Bi nanowires, due to the presence of T-point holes, which dominate the transport phenomena for the holes in most situations, the transport properties exhibit an asymmetric behavior for n-type and p-type Bi nanowires. The T-point holes, which are also anisotropic, have a larger effective mass component (~0.634 m0) along the trigonal direction, and smaller effective mass components (~0.059 too) for the binary and the bisectrix directions (see Table VII). The different anisotropy of the T-point holes from the L-point carriers further complicates the dependence of the thermoelectric properties on the wire orientation. To reveal the effect of the T-point holes on the thermoelectric properties of Bi nanowires, we first discuss Z xDT for the simplest case, that of trigonal Bi wires, in which the three electron pockets at the L points become degenerate. The solid curves in Fig. 36 show the calculated ZIDT at 77 K as

(a)

0.5

.

.

,

. . . .

,

Eh(i.) (~

-

_

.

,

Eh(T)(0)

E*.'(I .)(0)

0.4

% N

0.3

0.2

,'

,'

!~ ~oT-~nthol~s

/

!,,

~ ,. /

,'/-xi

:,'/i

', .'

i

0.1

\

; :

00

(b)

.'~ \

,7i

/

-50

\

i.'/i I

\

I

\

\

/: :

0 Energy (meV)

l?h(l.)(o)

\

50

F....h( T,(0 }

l:T'e(l.J _ (oj:

i I

I N

I I

1

f t

i

~

no T-pointholes

I I

: i

I

i

I I

: :

/

.~.. \ ' ~ %oo - 2 0 0 -loo ' o

~;o

Energy (meV)

(c)

e

,

,

J.

,

_

2oo (oi

7 6

5

%

N-

4 3 2 1

060o

-400

-2'oo'

6

Energy (meV)

'

2oo

" 4oo

FIG. 36. Calculated Z1DT a s a function of the chemical potential at 77 K for: (a) 40-nm, (b) 10-nm, and (c) 5-nm Bi nanowires oriented in the trigonal direction. The zero of energy refers to the conduction band edge in bulk Bi, while o(o) C'h(T) and ~.(o/ denote the highest subband edges of the T-point holes and the L-point holes, respectively, and the lowest subband edge of the L-point electrons is labeled by e~,). Shown by the solid line is the calculated ZIDT when both the T-point holes and the L-point holes are present, and the dashed line shows the Z~DT when it is assumed that there are no T-point holes. For 5-nm Bi nanowires, the two curves (solid and dashed) coincide with each other, showing that for very small nanowire diameters, the T-point holes have negligible effect on the transport properties (Lin, 2000).

80

M . S . DRESSELHAUS ET AL.

a function of Fermi energy for 40-, 10-, and 5-nm trigonal Bi nanowires, respectively, where the zero of energy refers to the conduction band edge of the T-point electrons in bulk Bi. The highest subband edges of the T-point holes and the L-point holes are denoted by ~o) C-'h(T) and oto) C'h(L)~ respectively, and the lowest subband edge of the L-point electrons is labeled by ~(o) C,e(L). The calculated Z1DT for 40- and 10-nm Bi nanowires usually has two or more extreme values: the extremal Z1DT with the higher Fermi energy corresponds to n-type nanowires, while the extremal Z~DT with a lower Fermi energy corresponds to p-type nanowires. The curves for 40- and 10-nm nanowires show an asymmetric behavior for p-type wires compared to the n-type wires, as expected [see Figs. 36(a) and 36(b)]. However, the 5-nm Bi nanowires exhibit a symmetric behavior for the n-type and the p-type nanowires [see Fig. 36(c)], as discussed in the following. To understand the effect of the T-point holes, Z~DT has also been calculated assuming that there are no T-point holes for the corresponding Bi nanowires, and the result for that case is also sketched in Fig. 36 by the dashed lines, showing the symmetry of Z~DT between the n-type and the p-type cases, due to the symmetric band structures of the L-point electrons and the L-point holes. For 40-nm trigonal nanowires, where the highest subband edge of the T-point holes C'h(T) o~o) is only about 15meV below the lowest conduction subband edge o~o) the optimized Fermi energy for n-type Bi nanowires is ~ located within the conduction band, and is not far from the valence band edge compared to the thermal energy keT ( ~ 6 meV). Therefore, the presence of T-point holes (solid curve) partly cancels the contribution of the electrons to the thermopower S and the optimal Z~DT is slightly reduced with respect to the dashed curve, which neglects the T-point holes [see Fig. 36(a)]. We also note that although both the dashed and the solid curves move down as the Fermi energy decreases from the optimized value toward the valence band, the solid curve, which includes the contributions from the T-point holes, decays faster due to the increasing influence from the T-point holes as Ev decreases. The adverse influence of the T-point holes on Z xDT for the n-type Bi nanowires becomes less significant as the wire diameter decreases, so that the T-point valence band is pushed farther away from the conduction subband edge owing to the quantum confinement effect. As shown in Figs. 36(b) and 36(c), the T-point holes have essentially no effect on Z~DT for 5- and 10-nm n-type Bi nanowires. For the p-type Bi nanowires, however, the T-point holes play a more important role in determining the behavior of ZIDT than for their n-type counterparts. For p-type Bi nanowires with wire diameters of 40 and 10 nm, there are two ZIDT extrema [see the solid curves in Figs. 36(a) and 36(b)], where the smaller Z~DT extremum occurs near the subband edge o~o) C'h(T) of the T-point holes, and the larger Z~DT extremum is located near the subband edge ~or0) of the L-point holes. The two extrema in Z~DT are mainly influenced by the T-point holes and the L-point holes, respectively. In the

1 QUANTUMWELLS AND QUANTUMWIRES

81

trigonal nanowires, the T-point holes have a large effective mass component along the transport direction, so that the T-point holes by themselves will have a small value of ZID T, and are therefore undesirable for p-type thermoelectric applications. For example, for 10-nm Bi nanowires, as shown in Fig. 36(b), ZIDT has an extremum of only ~0.14 when the Fermi energy oto~ , in which case the conduction is dominated by T-point holes, is near C.h(T) and in this case, Z1DT is much smaller than the optimal value of Z IDT ~ 2 for n-type wires. In addition, even though the L-point holes are favorable for p-type thermoelectric materials, just like the L-point electrons, the presence of the heavy-mass T-point holes will reduce the overall thermoelectric performance significantly. As shown in Figs. 36(a) and 36(b), the values of the optimal ZxDT for E F near the subband edge of the L-point holes are only about 0.17 and 0.7 for 40- and 10-nm p-type nanowires, respectively, which would otherwise be ~0.32 and 2, if there were no T-point holes. For 5-nm Bi nanowires, however, the thermoelectric performance is essentially independent of the presence of the T-point holes, as shown in Fig. 36(c), where the solid curve coincides with the dashed curve. The dramatically different behavior in the Z~DT for 5-nm Bi nanowires relative to 10or 40-nm wires is due to the crossing of the edge of the T-point hole band and the L-point hole subband edge for the trigonal Bi nanowires with very small wire diameters. For Bi nanowires oriented along the trigonal direction, the T-point holes have the smallest effective mass components (~0.059 mo) in the quantum confined directions (the binary and the bisectrix directions), while the carriers in the L-point pockets have one of the largest effective mass components (~0.11 mo) in the confined directions. Since band shifting due to the quantum confinement effect is approximately proportional to the inverse of the largest mass component in the confined directions, the subbands of the T-point holes will move downward faster than the L-point holes as the wire diameter decreases. In addition, the nonparabolic band structure also tends to reduce the band shifting of the L-point holes because of the increasing effective masses away from the band edge. Therefore, at a certain wire diameter (~< 6 nm), the highest subband edge of the T-point holes oto) Ohtr) crosses the highest subband edge of the L-point holes OhtL),~ and the highest valence subband is formed by the L-point holes. The suppression of the T-point holes with respect to the L-point holes provides us with nearly symmetric conduction and valence bands, so that for dw < 6 nm the influence of the T-point holes is negligible for the range of Fermi energies of interest [see Fig. 36(c)]. Figure 37 shows the calculated ZIDT as a function of acceptor dopant concentration at 77 K for trigonal Bi nanowire, with different wire diameters, and the dashed curves show the results for the corresponding wires assuming that there are no T-point holes. The results for 5-nm p-type Bi nanowires are depicted in the inset to the figure for clarity. We note that with the presence of the T-point holes, the acceptor dopant concentration

82

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9

9

9

' ' ' 1

DRESSELHAUS ET AL. '"

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"

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. . . .

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.

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.

no T-point holes

2.0

......., ...... ~ ..... .~ .... _,

/

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///

I

\~ \

10 nm - - - - - " \ ~\~\\ 40 nm

~'~/'~ /

L

000, '

10 le 10 lr 10 '8 Acceptor concentration ( 1/cm 3)

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FIG. 37. CalculatedZIDT for p-type Bi nanowires oriented along the trigonal axis at 77 K as a function of acceptor dopant concentration for three different wire diameters. The dashed curves represent the results assuming there are no T-point holes. The inset shows the corresponding results for the 5-nm nanowires where the T-point holes do not contribute significantly at 77 K (Lin, 2000).

N, required for the optimal Z loT for 10- and 40-nm Bi nanowires is much higher than for their n-type counterparts (see Fig. 33), and the optimized N, are ,-~0.96 x 1018, 1.3 x 1019, and 6.2 x 1018 cm -a for 5-, 10-, and 40-nm nanowires, respectively. For Bi nanowires oriented along the binary and the bisectrix directions, the T-point holes have identical subband structures due to symmetry. In these two wire orientations, the T-point holes have one of the smallest effective mass components along the transport direction, and the largest mass component in this case lies in the confined plane. Since the thermoelectric performance for each band is roughly proportional to (m*)-~/2, where m* is the effective mass along the transport direction, the T-point holes are more favorable for thermoelectric applications for binary and the bisectrix wires rather than for trigonal wires (Lin, 2000). However, unlike trigonal wires, the band crossing of o(o) C'h(T) and ~(o) C'h(L) is not observed for the binary and bisectrix wires for wire diameters down to 5 nm. This is because of the smaller band shifting for the T-point holes owing to the heavier mass in the confined directions for binary and bisectrix wires. Therefore, for 5-nm binary and bisectrix wires, the T-point holes still play a crucial role in determining the overall Zlr)T in the p-type range (Lin, 2000) 9 For binary wires, the band crossing of o(o) .(o) will occur for C,h(L) and C'h(T) wire diameters slightly below 5 nm, so that n-type and p-type binary

1 QUANTUMWELLS AND QUANTUM WIRES

83

nanowires with ultrasmall wire diameters are expected to have symmetric performance. As for bisectrix wires, such a band crossing is very unlikely because the very small effective mass component of the L-point pockets (~0.0023mo) in the confined directions will move the band edge of the L-point holes down much faster than that of the T-point holes under the D quantum confinement effect. Bi nanowiress oriented along the [1011] and [0112] directions have subband structures similar to that of the bisectrix wires. A summary is given in Table XII of results for ZIDT and the optimum doping concentrations at 77 K for p-type Bi nanowires for three nanowire diameters and for various crystal orientations of the nanowire axis. In conclusion, the calculated results show that the T-point holes reduce the thermoelectric performance of the n-type Bi nanowires with larger wire diameters (/> 40 nm) significantly, while for smaller wire diameters, where the bandgap is large compared to that of kBT, the adverse influence of the T-point holes on Z 1DT for the n-type Bi nanowires become negligible. As for p-type Bi nanowires, the T-point holes are even more undesirable, and a significant enhancement in Z1DT can be achieved if the T-point holes can be removed or suppressed below the L-point holes. One possible approach to suppressing the T-point hole contribution is to introduce an appropriate amount of antimony (Sb) into bismuth to form Bil_xSbx alloys. It is predicted that for 0.075 < x < 0.18, the T-point holes will lie below the L-point valence band edge in bulk Bi, so that the superior properties of the L-point holes can be utilized. In addition, the suppression of the T-point holes can also be accomplished in Bi nanowires through the quantum confinement effect for some wire orientations. As shown in Fig. 36(c), for trigonal Bi nanowires with very small wire diameters (~< 6 nm), oto) C , h ( T ) will cross oto) C'h(L) due to the quantum confinement effects, and no doping with Sb is needed.

6.

TEMPERATURE-DEPENDENT RESISTIVITY OF Bi NANOWIRES

Measurements of the temperature dependence of the resistance R(T) of Bi nanowire arrays have been carried out on samples prepared both from the liquid phase by pressure injection (Zhang et al., 1998b) and by vapor phase deposition (Heremans et al., 1998), yielding results consistent with each other. Due to the geometric limitations, the first attempt to study the temperature dependent properties of Bi nanowires was made with a twoprobe measurement. Although an absolute value of the resistivity cannot be derived through this two-probe method, the temperature dependence can be examined by normalizing the resistance to that at a common temperature, for example, R(300 K). Figure 38 shows the temperature dependence of the resistivity R(T)/R(300 K) of Bi nanowire arrays of various wire diameters prepared by the vapor deposition process.

84

M . S . DRESSELHAUS ET AL.

2.0

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-

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-

n"

.

+ 4-+++++

9 9 ooo ~ 70 nm

0.5

B u l k Bi

0,0

1

10

T (K)

lO0

FIG. 38. Experimental temperature dependence of the normalized resistance for Bi nanowire arrays of various wire diameters prepared by the vapor deposition method, in comparison with the corresponding data for bulk Bi. The measurement of the resistance was made while the Bi nanowires were in their alumina templates using a two-probe measurement technique (Heremans et al., 2000b).

As shown in Fig. 38, the temperature dependence of the resistance of Bi nanowires is very different from that of bulk Bi, and is very sensitive to the wire diameter. At high temperatures (T > 70 K), the resistance of all nanowire arrays shown in this figure increases with decreasing temperature. When T < 70 K, the resistance of the smaller diameter Bi nanowires (7-48 nm) in the semiconducting regime continues to increase with decreasing temperature, while the resistance decreases with decreasing temperature for the nanowire samples of larger diameters (70 and 200nm) in the semimetallic regime. The general trend observed here is consistent with other measurements on Bi nanowires in this wire diameter range prepared by pressure injection (Zhang et al., 2000), and with previous results on single-crystalline Bi wires of larger diameters (d >f 200 nm) (Brandt et al., 1977; Gurvitch, 1980). It is also instructive to plot the data for R ( T ) / R(300 K) vs T on a linear plot, as shown in Fig. 39.

1

QUANTUM WELLS AND QUANTUM WIRES

85

2.0 I I o [~

D~_~

40 nm

1.5 v

8

1.0

f

O0

200 nm

ft. bulk Bi

0.5

0.0 0

=

1 0

0

' T

200

'

300

(K)

FIG. 39. The two-point resistance of Bi nanowire arrays prepared by the pressure injection method. The resistance measurements were normalized to room temperature and plotted versus temperature. Samples of 40- and 200-nm wire diameter are shown in comparison to bulk Bi. Bulk data taken from Heremans et al. (1998, 2000b).

The striking difference in the temperature dependence of the resistance between Bi nanowire arrays and bulk Bi can readily be explained qualitatively, and a simple physical argument is given here in terms of the temperature dependence of the carrier density n(T) and mobility #(T). For both Bi nanowires and bulk Bi, n(T) increases with increasing temperature, while #(T) decreases. However, for Bi nanowires with larger wire diameters that are semimetallic (e.g., 70 and 200 nm), the increase in the carrier density with increasing temperature is much slower than that in the semiconducting wires with smaller diameters (e.g., dw <~48 nm), especially at low temperatures ( < 1 0 0 K , see Fig. 31). For smaller diameter nanowires (<48 nm in Fig. 38), the increase in n(T) outweights the decrease in #(T) when the temperature increases, and therefore the resistance drops. On the other hand, for bulk Bi or Bi nanowire arrays with larger wire diameters (70 and 200nm in Fig. 38), the temperature dependence of #(T) becomes more important due to the weaker T dependence of n(T) in the semimetallic regime, and therefore, the resistance increases with increasing temperature for T < 100 K. The carrier density of the 7-nm semiconducting Bi nanowires increases by many orders of magnitude ( ~ 108) when the temperature increases from 100 to 300K, while that of the 70-nm nanowires also increases by about 16 times. However, since the mobility of bulk Bi decreases by a factor of 13 from 100 to 300 K, the increase in n(T) overwhelms that of #(T), and the resistance of Bi nanowires (7-70nm) decreases with temperature for T > 100 K.

86

M . S . DRESSELHAUS ET AL.

Based on the model for the electronic structure for Bi nanowires (see Section VI.3) and the transport model (developed in Section VI.6 for the more general case of a doped Bi nanowire system), the normalized temperature dependent resistance R ( T ) / R ( 3 0 0 K) for the 70- and 36-nm Bi nanowires has been calculated (Lin, 2000a), and the results are shown by the solid curves in Fig. 40, exhibiting trends consistent with the experimental results in Fig. 38. Calculations for the wire diameters of 70 and 36 nm are particularly interesting, because these wires represent two different types of Bi nanowires: semimetallic and semiconducting, respectively. In the modeling of Fig. 40, some assumptions were made to take into account the discrepancies between an ideal 1D Bi quantum wire and a real Bi nanowire. First, in a perfect single crystalline Bi quantum wire, there is no scattering at the wire boundary because the electron (or hole) wave function and the local carrier density vanish at the wire boundary as a result of the assumed ideal infinite-potential interface. Within the Bi nanowire, possible scattering mechanisms are electron-phonon and electron-electron interactions. However, in a real Bi nanowire sample, the boundary conditions are far from ideal, and the energy barrier at the boundary is finite instead of infinite. Furthermore, real Bi nanowires may have a higher defect level in a thin layer at the boundary than in the interior of the nanowires due to the various surface conditions at the Bi-A120 3 interface. Therefore, the electrons will experience substantial boundary scattering, due to the finite amplitude of the electron wave functions at the boundary. This boundary scattering effect has been observed in magnetoresistance measurements (see Section VI.7) (Zhang et al., 1998a, 2000; Heremans et al., 2000b). In addition to the scattering at the wire boundary, the electrons can also be scattered at grain boundaries within real Bi nanowire samples, for which a domain size of the order of the wire diameter has been observed (Sander et al., 2000). Since the domains in Bi nanowires possess the same crystal orientation along the wire axis, however, the small-angle scattering at the grain boundary would be expected to be a minor scattering mechanism in determining the transport properties of a real Bi nanowire. Another striking difference between ideal Bi nanowires and real Bi nanowire samples is the uncontrolled impurities that act as dopants in Bi nanowires. For ideal semiconducting Bi quantum wires, the carrier density should decay exponentially with decreasing T at low temperatures, and the resistance should correspondingly increase dramatically. Instead, even for the 7-nm Bi nanowires, the measured resistance increases slowly and steadily with decreasing temperature (see Fig. 38). This effect can be attributed to the uncontrolled impurities in the Bi nanowires, which give rise to a finite carrier density at low temperatures. The uncontrolled impurities will not only alter the carrier density, but will also decrease the carrier mobility by ionized impurity scattering.

1 QUANTUM WELLS AND QUANTUM WIRES

87

The effect of each scattering mechanism mentioned can be characterized by a scattering time z. The total scattering time, ~tot, in a real Bi nanowire can be approximated by adding the scattering rates in accordance with Matthiessen's rule (Ashcroft and Mermin, 1976) 1

1

=

"~tot(T)

~ bulk(T)

1

+

1

+ ~

"~boundary

(89)

"gimp(T)

in which "/Sbulk is the total relaxation time in bulk Bi, and 27boundaryand l~imp are the relaxation times for boundary scattering (including wire and grain boundary scattering) and ionized impurity scattering, respectively. Note that boundary scattering and ionized impurity scattering are only important at low temperatures ( < 1 0 0 K ) , and that phonon scattering becomes the dominant scattering mechanism at higher temperatures ( > 100 K). The relaxation time for ionized impurity scattering is approximately proportional to T 3/2 (Brooks, 1955), while boundary scattering is much less temperature dependent, and for simplicity "/Sboundary w a s assumed to be a constant, independent of temperature. Since the mobility Ft is proportional to the relaxation time z, the approximation used to find the mobility, considering all of these scattering effects, was 1 ]2tot(T)

=

-

1

#bulk(T)

+ ~

1

~boundary

+

1

(90)

]2imp(T)

in which #bulk(T) can be found in the literature (Vecchi and Dresselhaus, 1974), ~boundary was assumed to be constant in T, and ~imp ~" T3/2" In the curve for the 70-nm Bi nanowires in Fig. 40, the two mobilities flboundary and ~imp were fitted to 50 m 2 V - 1 s- 1 and 1.0 • T 3/2 m E V - 1 S - 1, respectively. As for the 36-nm Bi nanowires, the mobility terms were fit to ~boundary ~ 33 m 2 V - 1 s- 1 and ]2imp ~ 0.2 X T 3/2 m 2 V - x s- 1, and the carrier density, due to the presence of uncontrolled impurities, was fitted t o N i m p ~ 5 • 1016 cm -3, which amounts to less than 100 impurity atoms in the Bi nanowires per 1/~m in length. Since the 70-nm Bi nanowires are semimetallic at low temperatures, however, the carrier density contribution due to the small amount of uncontrolled impurities has an insignificant effect, and this effect is neglected in the modeling, taking account only of the effect of these uncontrolled impurities on scattering carriers. We also note that flboundary is smaller for the 36-nm Bi nanowires than for the 70-nm nanowires, due to their smaller wire diameter. As the wire diameter decreases, the contribution of the #imp term decreases more rapidly than that of the flboundary term. Regarding the main contribution of the last two terms to Eq. (90), the term -1 ~boundary dominates o v e r ~i~nlp for 77 K and above, and the ]~imp -1 term is

88

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DRESSELHAUS ET AL. .

.

.

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.

i

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,

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FIc. 40. The calculated temperature dependence of the resistance for Bi nanowires of 36 and 70 nm, using a semiclassical transport model (Lin, 2000b).

relatively larger for the small diameter nanowires. The normalized resistance R(T)/R(300 K) curves in Fig. 40 show general trends for the temperature dependence of the normalized resistance of the 36- and 70-nm Bi nanowires, consistent with the experimental results in Fig. 38 for the actual nanowire arrays, showing strong evidence that the different temperature dependences of R(T)/R(300 K) for Bi nanowires with different wire diameters are predominantly due to the quantum confinement-induced semimetal-semiconductor transition, which occurs when the wire diameter in Fig. 38 decreases below 50 nm. The effect of crystal quality can be accounted for in the same transport model by the value of ~boundary in Eq. (90). Instead of a nonmonotonic behavior for semimetallic Bi nanowires as shown in Fig. 38, R(T) is predicted to show a monotonic T dependence at a higher defect level. The dashed curve in Fig. 40 shows the calculated R(T)/R(300 K) for 70-nm wires with increased boundary scattering (jUboundary '~' 6 m 2 V- 1 s- 1), exhibiting a monotonic T dependence, similar to that of Bi nanowires prepared by electrochemical deposition (Hong et al., 1999), which is likely to produce polycrystalline nanowires. Generally speaking, for samples with many grain boundaries the 1/'t'boundary term is large, leading to a qualitatively different temperature dependence and a lower overall mobility. The differences in the slopes of the temperature dependence of the low temperature resistance (T < 10 K) also provide experimental evidence for the semimetal [large nanowire diameter and (c~R/~T) > 0] to semiconductor [small nanowire diameter and (c~R/c~T) < 0] transition in Bi nanowires, as shown in Fig. 41 (Heremans et al., 1998). Note that because of the very small geometric sizes of the smallest nanowires, the number of carriers in a sample is quite different from that in a bulk material by many orders of magnitude. For a typical carrier density of 1018 cm -3, the number of carriers in a 40-nm nanowire would be only 1200 per micrometer in length, and the number decreases quickly as the

1

QUANTUM WELLS AND QUANTUM WIRES

1.05

* 48 l-

3e.~

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.

9 ; ":,',...:...,.:

Wirediameter 70rlm

~

T (K)

10

FIG. 41. The temperature dependence of the zero-field resistance for Bi nanowires of different diameters, normalized to the resistance at 10 K (Heremans et al., 1998).

wire diameter decreases. For example, for the 10-nm wire that we discussed in Section VI.3, the number of electrons would be a factor of 16 smaller. Therefore, a more detailed study of the transport properties of Bi nanowires with small wire diameters should more explicitly take into account the small number of carriers in the wires.

a.

Four-Point Resistivity Measurement of Individual Bi Nanowires

Although the two-point resistance measurement in the Bi filled templates has the great advantage of simplicity, an absolute value for the resistivity cannot be determined because the number of wires contributing to the conduction is unknown and because of the contribution of contact resistance. The ability to remove the Bi nanowires from the anodic alumina templates and to prepare free-standing Bi nanowires indicates a workable strategy for quantitative measurement of the resistivity p(T) (along with other temperature-dependent transport properties) for Bi nanowires. To determine the absolute resistivity, the nanowires were removed from the alumina template, and electrodes were attached to an individual nanowire in order to perform a four-point transport measurement. The resistivity can then be obtained by the cross-sectional area and the length. The objective of this work, which is still at an early stage, is to measure the resistivity of a single Bi nanowire as a function of wire diameter, temperature, and

90

M . S . DRESSELHAUS ET AL.

FIG. 42. Left: SEM image of a 70-nm Bi nanowire with four electrodes attached to the nanowire. The circle on the large left electrode is a reference point used to find the nanowire and to attach electrodes to it by a lithographic process. Right: Optical image of a larger area of the lithography pattern. The nanowire lies at the center of the "X" pattern (Cronin et al.,

2ooo).

magnetic field. Although the two-point resistance measurements of Bi nanowire arrays in the alumina template can reveal the T dependence of the nanowire resistivity, such measurements only give a qualitative description of the unique electronic structure and other transport properties, due to limited knowledge of the contact resistance and the number of wires to which ohmic contacts have been made. To carry out the four-point resistivity measurements, electrodes were patterned on top of a single Bi nanowire on a Si substrate coated with a thin SiO 2 layer, as shown in Fig. 42 (left), using an electron beam lithography technique (Cronin et al., 1999, 2000). Electrodes consisting of 1000-.3,thick gold layers with a 50-]k-thick adhesion layer of chromium were used, and the processing of these electrodes followed a standard "lift off" method. Beause of the small size of the nanowires and the low melting point of Bi (~271~ care has to be taken to prevent burn-out of the nanowire by static discharge. Such a discharge was prevented by keeping the sample shorted at all times, except when the transport measurements were made (Cronin et al., 1999, 2000). The right image in Fig. 42 shows a large area of the pattern that is written through use of an electron beam lithography technique. The entire pattern consists of four large bonding pads (300 x 300/tm) and four leads that extend inward, forming an "X" shape connecting the large bonding pads to the four electrodes shown in the SEM image of Fig. 42. Along the perimeter of the pattern (not shown), the bonding pads are shorted to each other by thin conducting strips to prevent static discharge through the Bi nanowire. From the slope of the I - V plots in the linear regime, the resistivity is determined.

1 QUANTUMWELLS AND QUANTUMWIRES

91

The first report on four-point resistivity measurements on Bi nanowires was made for a 70-nm-diameter Bi nanowire at 300 K (Cronin et al., 1999), and the results showed the resistivity to be approximately six times greater than that of bulk Bi. From theoretical modeling, the band shifts due to quantum confinement are expected to have only a small effect on the carrier density at room temperature for a 70-nm Bi nanowire, so that the increase in resistivity is attributed to increased scattering at the wire boundaries and grain boundaries within the wire that are not present in highly crystalline bulk Bi. The presence of grains within the Bi nanowire, with a size comparable to the wire diameter (Sander et al., 2000) is expected to result in grain boundary scattering, which contributes to the increase in the resistivity of the thin Bi nanowires. Although the smallest diameter individual Bi nanowire that has thus far been measured by a four-probe method has a diameter of 70 nm, it is believed that the measurement technique can be extended to wires of even smaller diameter, perhaps by an order of magnitude or more.

b.

Temperature Dependent Resistivity of Te-Doped Bi Nanowires

To optimize Z1D T, doping of the nanowires is needed to control the electrochemical potential (or Fermi level) by achieving the optimal doping level (as discussed in Section VI.5.a). Thus far, only one experimental study of the transport properties of doped Bi nanowires has been performed and this was carried out on n-type wires using Te as a donor dopant. The Te-doped Bi nanowire samples were prepared by the pressure injection of liquid Bi containing Te dopant at a known concentration of Te. Before the transport measurements, the nanowire arrays were thermally annealed at 200~ under vacuum for 8 hr to release the residual high stress within the nanowires. Figure 43 shows experimental results for the T dependence of the resistivity for an undoped Bi nanowire sample and for three Te-doped Bi nanowire samples, with Te dopant concentrations of 0.025, 0.075, and 0.15 at. %, respectively, where the quoted concentrations are based on the ratio of the Bi and Te atoms that were introduced into the quartz tube for producing the Bi-Te alloy for the melt injection process. The wire diameters for the three doped and the undoped Bi nanowire samples are all about 40 nm. From Fig. 43 we can see that the T dependence of the resistivity for all four samples exhibits a qualitatively similar behavior, but distinctly different in detail for different Te concentrations. For all four samples, the resistance increases as the temperature decreases. However, the curve for the undoped Bi nanowires shows the strongest temperature dependence, with R(4 K) being ~ 2.3 times greater than the resistance at 300 K, and the T dependence of the resistance weakens as the doping concentration N d

92

M . S . DRESSELHAUS ET AL.

2.4 o

2.2

o o o un...o,..e......dnffRi

0.025at% Te

o o

A

A

A

2.0

O n

v

ob-

1.8

rr 1.6 I--

6

0.075 at% Te

o

0 O0 0 0 0 0 0 0 0 0 0

o

0.15 at% Te 1.4

O

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i

i

i

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i

i

,

I

i

i

.

.

.

.

10

.

1

,

l

100 Temperature (K)

FIG. 43. Temperature dependence of the resistance for 40-nm undoped and Te-doped Bi nanowire arrays for various doping concentrations. The resistance of each nanowire sample is normalized to R(270 K) (Lin, 2000a).

increases. In contrast, for the sample with the highest doping concentration (0.15at. %Te), the resistance is almost constant from 2 K up to about 100 K, while the resistance of the undoped Bi nanowire sample decreases monotonically with increasing T over the whole temperature range measured (2-300 K). The very different behavior of the resistance versus T for various doping concentrations can be qualitatively explained in terms of the difference in the donor concentration N d within the Bi nanowires. According to the theoretical calculations in Section VI.3, the 40-nm Bi nanowires will undergo a semimetal-semiconductor transition as T is lowered below T ~_ 170 K. Thus, the intrinsic carrier density will drop significantly with decreasing T, as T decreases below 170 K (see Fig. 31). Therefore, the T dependence of the carrier density is strongly perturbed by the presence of Te dopants, especially at low temperatures. For Te-doped Bi nanowires, the carrier density is approximately equal to the Te dopant concentration Nd up to a certain threshold temperature Tth, which increases with Nd, as shown in Fig. 32. At high temperatures (>~ 200 K), electronphonon scattering is usually the dominant scattering mechanism for carriers, and this scattering mechanism is only weakly dependent on the doping level.

1 QUANTUMWELLS AND QUANTUMWIRES

93

Thus at high temperatures, the resistance for the samples in Fig. 43 with different Te doping levels will exhibit a similar T dependence, since for all samples in Fig. 43 a similar T dependence is predicted for the carrier concentrations n(T), as shown in Fig. 32, and a similar T dependence is expected for the carrier mobilities ~(T) at high temperatures (T >i 200 K). On the other hand, in the low temperature regime, ionized impurity scattering plays a more important role in decreasing the carrier mobility for the Te-doped Bi nanowires compared to that of undoped Bi nanowires. However, since the measured l/(T) for most cases is expected to be less T-dependent than n(T), the overall (normalized) resistance R(T)/R(270 K) at low temperatures decreases with increasing N d, as shown in Fig. 43. Thus, Te-doped nanowires are expected to have a temperature-dependent resistance similar to that of the intrinsic Bi nanowires at high T, and to behave like an extrinsic semiconductor at low T. The exact donor concentration in the Bi nanowires, which is not known at the present time, is likely to be smaller than the Te concentration in the metal melt due to the possible segregation of Te atoms when the alloy solidifies. Therefore, to a first approximation, we assume that about 10% of the Te dopant prepared in the metal melt is present in the final nanowire product, and we assume that each Te atom will donate one electron to the conduction band when ionized. With this assumption, the donor concentrations Nd for the 0.025, 0.075, and 0.15at. % Te-Bi alloy samples are 6.67 x 10 ~7, 2.0 X 1018, and 4.0 x 1018 cm- 3, respectively, in the final nanowire products. Based on the measured resistance in Fig. 43 and the calculated carrier density, we can obtain the temperature dependence of the average mobility l/ave(T) of the Bi nanowires as shown in Fig. 44 for the T dependence of l/ave - ~ for the three Te-doped Bi nanowires, in comparison to that of the undoped Bi nanowire sample. Since the total mobility is approximately determined through Eq. (90), the inverse mobility is more appropriate for relating the experiments to actual scattering mechanisms. An expression similar to Eq. (90) for the average mobility l/doped(T) of the Te-doped Bi nanowires can be written as 1

=

1

+

l/doped(T) l/undoped(T)

1 l/defect(T)

+

1

l/imp(T)

(91)

in which l/undoped is the average mobility of the undoped Bi nanowire with the same diameter, 1/l/imp is the contribution to 1/l/doped due to the increased ionized impurity scattering in doped wires, and 1/I/ira p is the contribution to 1/l/doped due to the possibly higher defect concentration that might be present in Te-doped Bi nanowires. We note that at high temperatures ( > 200 K), the experimentally related curves for the three Te-doped Bi nanowire samples in Fig. 44 all have a temperature dependence similar to that of the undoped Bi

94

M . S . DRESSELHAUS ET AL.

18

[-.

-3

:::L Q r-'t"q =t.

..., T 2-6

Nd,= 0 0

,

,

100

1 200

Temperature

,

,

, 300

(K)

FIG. 44. Calculatedtemperature dependence of ]Aav - e1 for 40-nm Te-doped Bi nanowires with different dopant concentrations, compared to 40-nm undoped Bi nanowires. The calculation is based on the measured temperature-dependent resistance in Fig. 43 and the calculated temperature dependent carrier concentration, which is similar to the curves shown in Fig. 31. The dashed curve and the solid curve are fitting curves for the mobility of the undoped Bi nanowire sample for T > 100 K (Lin, 2000).

nanowire sample, indicating that 1//~undoped makes the major contribution to 1///dope d for the doped samples at high temperatures. The average mobility of undoped Bi nanowires can be well fitted by a flundoped '~ T-2.9 dependence for T > 100 K, as shown by the dashed line in Fig. 44. The general trend of the curves for Te-doped Bi nanowires can be qualitatively explained by the temperature dependence of ]Aundoped, ]Aimp, and fldefect. Since the scattering effect due to defects is similar to the neutral impurity scattering, which is essentially temperature independent, fldefect may be treated as T independent compared to other scattering mechanisms. As the temperature decreases, the contribution of ~undoped -1 drops rapidly, and the contributions of ~imp - 1 and Pdefect -1 start to dominate, depending on the Te dopant concentrations. Since ~imp - 1 generally increases with decreasing temperature, we see a m i n i m u m of ~doped -1 for Te-doped Bi nanowires. In addition, because the ionized impurity scattering and the scattering due to defects both increase with Te dopant concentrations, ~imp - 1 -[- ~/defect - 1 will make a larger contribution to /~aoped - 1 at low temperatures for the Bi nanowires with higher Te dopant concentrations (see Fig. 44), in agreement with experiment.

1

QUANTUMWELLS AND QUANTUM WIRES

95

Note, however, that the T 3/2 dependence for the mobility ~imp, which is a common temperature dependence for ionized impurity scattering in most materials, can hardly account for the temperature dependence of the total mobilities of Te-doped Bi nanowires in Fig. 44, which shows a much weaker temperature dependence than T -3/2 at low temperatures. The reasons for the anomalous T dependence of flimp for Te-doped Bi nanowires is explained as follows. First, we recall that the scattering time "timp for the ionized impurity scattering is proportional to m * 2 v 3 where v is the carrier velocity and m* is the effective mass (Brooks, 1955). For an ordinary material with a parabolic dispersion relation E ~ k2/m *, we obtain z ~ m * l / 2 E 3/2. The T 3/2 dependence of ~imp is then recovered by averaging (Zimp) with an appropriate distribution function. However, for Bi that has a nonparabolic dispersion relation for electrons, each electron subband in Bi nanowires is also nonparabolic [see Eq. (61)], and the group velocity of electrons in the (n, m) subband is calculated as

l c3E"m(k)- ~ V"m(k ) = -h Ok - X/ - - ~

1+

7L~zE.,~- ~/2 -2-~

(92)

J

in which 7,,, and E,m(k ) are defined in Eqs. (64) and (61), respectively. We note that, for electrons lying very close to the subband edge, Eq. (92) can be written as the familiar relation V,m = hk/m*m, while for electrons far from the subband edge, the second term in the bracket of Eq. (92) becomes negligible, and V,m ~-- w/2EoL/fn~, which is independent of the electron energy and the subband index. This very different relation between the group velocity v and the electron energy E for Bi nanowires will result in a different temperature dependence for ]Aimp in Te-doped Bi nanowires than for ordinary materials. In Te-doped Bi nanowires, the number of electrons is significantly increased due to the Te doping, so that the Fermi energy moves into the conduction band. Those electrons that are responsible for the transport properties will have an approximately energy-independent group velocity v ~- x/2EoL/~z, as discussed earlier in the chapter. It then follows that the relaxation time for ionized impurity scattering rimp, which is still proportional to m*Zv3, would be approximately energy independent. Therefore, the temperature dependence of Pimp for Te-doped Bi nanowires should be much weaker, consistent with the results in Fig. 44, which show an almost constant mobility for doped samples at low temperatures (T < 100K). However, although///imp ~- e ( Z i m p ) / m * ~ m*- 1/2E3~2 doesn't have an explicit T dependence, it is, in fact, temperature dependent because EoL(T ) and m*(T) in Bi are highly T dependent at higher temperatures (T > 80 K). From Table VIII, which gives the temperature dependence of the band structure parameters of Bi, both the direct bandgap EoL and the electron effective mass m* increase with increasing T, and the overall value of

96

M . S . DRESSELHAUS ET AL.

m*-1/2E3~2 and ]-/impalso increase almost linearly with T in the intermediate temperature range (100 K < T < 250 K). At low and intermediate temperatures (T < 200 K), ]-/imp -x is the dominant term in determining the total j//doped~ -1 while at high temperatures (T > 200 K), k/undoped -1 becomes more important. Therefore, a minimum in //doped -a is observed around 200 K for the more heavily doped Bi nanowires (see Fig. 44). Note that in addition to the resistance measurements discussed in the preceding section, there are many other experiments, such as magnetic and optical measurements, that can reveal further information about the transport properties of the unique Bi nanowire 1D system. For example, the low-field magnetoresistance measurements can be used to evaluate the carrier mobility, which, in turn, can provide a more accurate estimation of the carrier densities.

7.

MAGNETORESISTANCE OF Bi NANOWIRES

Because of the inherent one-dimensional geometry of nanowires, certain conventional measurements, such as the Hall effect, which are traditionally carried out to determine the carrier density, cannot be performed. Magnetooscillatory effects cannot be used in many cases to determine the Fermi energy because of wire boundary scattering (which makes it difficult to satisfy OgcZ>> 1), and optical measurements on the Bi-anodic alumina samples to determine the plasma frequency are largely dominated by contributions from the host alumina template, and even single nanowire measurements of the absolute resistivity are quite challenging, as described in Section VI.6.a. Therefore, determining the effects of doping and annealing Bi nanowires often cannot be assessed by conventional means. Magnetoresistance (MR) measurements provide an informative technique for characterizing Bi nanowires because these measurements yield a great deal of information about electron scattering from wire boundaries, the effects of doping and annealing on scattering, and localization effects in the nanowires (Heremans et al., 1998). Figure 45 shows the longitudinal magnetoresistance (B parallel to the wire axis) for 65- and 109-nm Bi nanowire samples at 2 K. In the low-field regime, the MR increases with B (positive MR), up to some peak value, B,,, beyond which the MR becomes a decreasing function of B (negative MR). This behavior is typical of the longitudinal MR of Bi nanowires in the diameter range 45 to 200nm (Zhang et al., 1998a, 2000; Heremans et al., 1998, 2000b) and can be understood on the basis of the classical size effect of the nanowire. The MR of wires with diameters smaller than 40 nm, also discussed in this section (Heremans et al., 1998, 2000b), shows a strong dependence on B. The peak position B,, moves to lower B field values as the wire diameter increases, as shown in Figs. 45(b) and 45(c) (Zhang et al.,

1 0.16

QUANTUM WELLS AND QUANTUM WIRES

(a.)

0.14

I--E

0.12

97

(b.)

4 3 2

0.10

~'0.08

0

ft.

,~ 0.06

~ 0.04

20 40 60 80 100 T (K)

3

E2

0.02 0.00

m

I

-0.02

-0.04

,

0

1

,

,

,

,

2

3

4

5

a (T)

0 0.00

0.01 l/d,

(nm "1 )

0.02

FIG. 45. (a) Longitudinal magnetoresistance, AR(B)/R(O), at 2 K as a function of B for Bi nanowire arrays with diameters 65 and 109 nm before thermal annealing. (b) The peak position B,. as a function of temperature for the 109-nm-diameter Bi nanowire array. (c) The peak position B,. of the longitudinal MR at 2 K as a function of 1/d w, the reciprocal of the nanowire diameter (Zhang et al., 2000).

2000), where B,, is seen to vary linearly with 1/dw. The application of a longitudinal magnetic field produces helical motion of the electrons along the wire, and above some critical field, approximately B,,, the radius of the helical motion will become smaller than the radius of the wire, causing a decrease in the wire boundary scattering, and giving rise to a negative magnetoresistance c~R/c~B < 0. However, for low fields, B ~< B,,, the magnetic field deflects the electrons causing increased scattering with the wire boundary, thereby giving rise to an increase in resistance or a positive magnetoresistance, which is common to most crystalline solids (Ashcroft and Mermin, 1976; Kittel, 1986). The condition for Bm is given by B m ~ 2 c h k v / e d w where k v is the wave vector at the Fermi energy (Piraux et al., 1999). In summary, for B ~< B,,, the cyclotron radius is larger than the wire radius and we have a positive MR, while for B >~ B,,, the cyclotron radius is smaller than the wire radius, and we have a negative MR. This phenomenon, called the classical size effect for the magnetoresistance, provides much insight into the scattering of electrons in Bi nanowires. The peak position, B,,, is found to increase linearly with temperature in the range 2 to 100 K, as shown in Figs. 45(b) and 45(c) (Zhang et al., 2000). As T is increased, phonon scattering becomes important and therefore a higher magnetic field is required to reduce the resistivity associated with

98

M . S . DRESSELHAUS ET AL.

boundary scattering sufficiently to change the sign of the MR. Likewise increasing the grain boundary scattering (Piraux et al., 1999) also increases the value of B m at a given T and wire diameter. Application of a transverse magnetic field does not show a significant reduction in wire boundary scattering, and therefore the transverse MR is always positive (Zhang et al., 2000). Thermal annealing of undoped Bi nanowire samples causes a significant decrease in the magnitude of the magnetoresistance as well as a decrease in the peak position, B,, (Zhang et al., 2000). This behavior indicates that prior to annealing, the scattering at defects and impurities is dominant over scattering at the wire boundary, even at a low temperature (2 K). The observed decrease in MR on annealing indicates that the Bi nanowires become purer after thermal treatment, as one would expect. Bi nanowires doped with Te have been fabricated and characterized, as discussed in Section VI.4. The longitudinal MR of Te-doped samples shows no peak in the MR (as can be seen in Fig. 45 for undoped samples), and instead, the longitudinal MR of Te-doped samples is found to be a monotonically increasing function of magnetic field (positive MR) in the magnetic field 0 ~< B ~ 5.4 T at 2 K (Zhang, 1999; Zhang et al., 2000). The disappearance of the negative MR is attributed to a change in the dominant scattering mechanism from wire boundary scattering (which can be reduced by applying a B field) to magnetic field-independent ionized impurity scattering from the Te dopant ions. Annealing the Te-doped samples yields MR behavior that is in striking contrast to that of the undoped samples described above. On annealing, an increase in the MR of the Te-doped samples is observed (Zhang et al., 2000). This indicates that the dopants are being pushed out of the nanowire to the wire boundary, thereby increasing the role of boundary scattering and decreasing the role of charged impurity scattering. For Te doped samples with dw < 40 nm, the longitudinal MR monotonically increases as B 2 and shows no peak (Heremans et al., 2000b), indicating that B, in the measured range (up to 5.4 T), is too low to reduce the cyclotron radius below the wire radius. Consequently, increasing the magnetic field in this field range always leads to increased boundary scattering. In addition to the longitudinal magnetoresistance measurements, transverse magnetoresistance measurements (B perpendicular to the wire axis) have also been performed on Bi nanowire array samples, where a monotonically increasing B 2 dependence over the entire range 0 ~< B ~< 5.5 T is found for all Bi nanowires studied thus far (Zhang et al., 2000; Heremans and Thrush, 1999a; Heremans, 1999). This is as expected, since the wire boundary scattering cannot be reduced by a magnetic field perpendicular to the wire axis. The negative MR observed for the Bi nanowire arrays above Bm shows that wire boundary scattering is a dominant scattering process for the longitudinal magnetoresistance, thereby establishing that the mean free path is larger than the wire diameter and that the Bi nanowires have high crystal quality.

1

QUANTUM WELLS AND QUANTUM WIRES

99

Also encouraging for thermoelectric applications are the results on the high-field classical size effect in the longitudinal magnetoresistance (Zhang et al., 1998a; Zhang, 1999), showing that the defect and impurity levels in the nanowires are sufficiently low so that the wire diameter is comparable to or smaller than the carrier mean free path (Cronin et al., 1999), and ballistic transport can occur in the nanowires in a high longitudinal magnetic field. The ability of the electronic structure and transport models for Bi nanowires to account for the dependence of the classical size effect in the magnetoresistance on temperature, magnetic field, nanowire diameter, and annealing conditions (Zhang et al., 1998a, 2000; Zhang, 1999) is important for predicting the behavior of Bi nanowires in the smaller diameter range, well below 10 nm, where enhancement in Z 1 D T is expected (Sun et al., 1999e). Previously reported results for the low-field magnetoresistance (Zhang et al., 1998a) lend direct confirmation for the general features of the model for the electronic structure of Bi nanowires presented in Section VI.3, which is used for calculating the transport properties, yielding estimates for the thermoelectric figure of merit (Sun et al., 1999e). Studies of localization effects in the Bi nanowires (Heremans et al., 1998) show that these effects only become measurable below 5 K (Zhang et al., 2000; Heremans et al., 1998). Although localization effects increase in importance with decreasing T, even at 2 K, they are quite small in magnitude for single crystal Bi nanowires (Zhang et al., 2000), though they are very important in disordered polycrystalline thin Bi wires and films (Beutler and Giordano, 1988; Beutler et al., 1987). In the low-magnetic-field regime, the magnetoresistance of Bi nanowires fits the parabolic field approximation A R ( B ) / R ( O ) = ABZ/po, where Po denotes the resistivity at zero magnetic field. The magnetoresistance coefficient, A/po, plotted in Fig. 46 as a function of temperature, is roughly proportional to the square of the mobility. For bulk Bi, A l p o at B = 1 T is

1

.... D

~',

9

9

0.1

o Q. <~ 0 . 0 1

0.001

9

&

9109-nm 96 5 - n m

I . 1 100 20O Temperature (K)

300

FIG. 46. Longitudinalmagnetoresistance coefficient, Alp o (T- 2) is plotted on a logarithmic scale, as a function of temperature for 109-nm (diamond points) and 65-nm (triangle points) diameter Bi nanowire arrays (Zhang et al., 1998a).

100

M . S . DRESSELHAUS ET AL.

found to decrease by seven orders of magnitude as the temperature is increased from 4 to 300 K. However, the MR coefficients of the Bi nanowires are found to vary more weakly with temperature, as shown in Fig. 46. This behavior can be understood by considering the important role of wire boundary scattering in the Bi nanowires. In bulk Bi, the mobility decreases by ~ 3.5 orders of magnitude over the temperature range 4 to 300 K, due to phonon scattering. In the nanowires, wire boundary scattering plays a dominant role in reducing the mobility at low temperatures so that the decrease in mobility due to phonon scattering is less prominent in the temperature dependence of the mobility and hence of the magnetoresistance coefficient A l p o. The detailed difference in the temperature dependence of the magnetoresistance coefficient A/po for the 65- and 105-nm-diameter nanowires is that only one subband contributes to the magnetoresistance for the 65-nm nanowire while two subbands contribute for the 109-nm nanowire, because of the smaller energy separation between subbands for the larger diameter nanowires (Zhang et al., 1998a). By applying a magnetic field, a transition from a 1D localized system, which is characteristic of low magnetic fields, to a 3D localized system can be induced as the magnetic field is increased. The effect of this transition can be seen in Fig. 47, where the longitudinal magnetoresistance is plotted for Bi nanowire arrays of various nanowire diameters in the range 28 to 70 nm for T < 5 K (Heremans et al., 1998). In these curves, a subtle steplike feature is seen at low magnetic fields, and this feature is independent of temperature and of the orientation of the magnetic field, and depends only on wire diameter. The corresponding transverse magnetoresistance curves (Heremans et al., 1998), also show a step at the same magnetic field strengths. The lack of dependence of the magnetic field of the step on temperature and magnetic field orientation indicates that the phenomenon is not related to the effective masses, which are highly anisotropic in Bi, but rather is related to the magnetic field length, L n = (h/eB) 1/2, which is the spatial extent of the wave function of electrons in the lowest Landau level, and L n is independent of the effective mass. Setting Ln(Be) equal to the diameter d w of the nanowire, defines a critical magnetic field strength, Be, below which the carrier wave function is confined by the nanowire boundary (the 1D localization regime), and above which the wave function is confined by the magnetic field (the 3D localization regime). This calculated field strength, Be, is indicated in Fig. 47 by vertical lines for the appropriate nanowire diameters, and these calculated Be values provide a good fit to the steplike features in the MR curves shown in Fig. 47. The physical basis for this phenomena is associated with localization of a single magnetic flux quantum within the nanowire diameter (Heremans et al., 1998). Localization effects are expected to be smeared out due to the finite distribution of nanowire diameters and also nonuniform diameters of the nanowires along their

1 QUANTUMWELLS AND QUANTUM WIRES

101

1.012 .85K 3.0 K

1.006

BLI1A 70 nm

4.0 K

1.004

1,000 "

1.008

rn v n" --~

t,O04

1.38K 2.0 K 4.0 K

48 ~

n1,000 1.020

O.M T

I

1.010

36rim

1.1111t

0.87T

1.3e K

28nm 1.000

0

1

2

3

4

5

B03 FIG. 47. Longitudinal magnetoresistance as a function of magnetic field for Bi nanowires with 28, 36, 48, and 70 nm diameters. The vertical bars indicate the critical magnetic field Bc at which the magnetic length 1998).

L n = ( h / e B ) 1/2

equals the nanowire diameter (Heremans

et al.,

lengths within the templates. Therefore localization effects are expected to be more noticeable in single nanowire measurements, although such measurements have not yet been done. The last magnetic field characterization technique discussed in this section is the Shubnikov-de Haas (SdH) quantum oscillatory effect. SdH oscillations, in principle, provide the most direct measurement of the Fermi energy and carrier density for the Bi nanowire system. To observe SdH oscillations, however, the magnetic field is applied parallel to the nanowire axis, and the electrons must complete at least one cyclotron orbit without being scattered. Thus the cyclotron radius must be smaller than the wire radius and the mean free path to observe the SdH effect. SdH oscillations occur when the quantized Landau levels pass through the Fermi energy as the magnetic

102

M . S . DRESSELHAUS ET AL. A

E

o.oI

I '

'

0 v

"

-1

'

!

~" --' " ,

'

'Io

g

O.O0-

m

.

.k' ~" -0.01

2

0 r''!

0

0

"i

9

-r

5

r

!

3

B~ i

i

10

i

4 i

I

]

15

i

5 !

t

~

I

20

1/13 (1KesJa)

FIG. 48. (a) The oscillatory magnetoresistance for the magnetic field parallel to the nanowire axis of an array of parallel undoped Bi nanowires 200 nm in diameter embedded in an anodic alumina template after the background MR has been subtracted. (b) Fourier transform of the oscillatory part of the magnetoresistance, showing two well-defined SdH periods, the 4.2 T - 1 period being identified with the heavy electron cyclotron orbit and another period at 9.25 T -1 identified with T-point holes (Heremans and Thrush, 1999a; Heremans, 1999).

field is increased. By determining the period of the SdH oscillation (periodic in l/B), the position of the Fermi energy can be determined by the relation

1 = mcEev [1 + E}/Eo] A(1/B) hq

(93)

where A(1/B) denotes the SdH period; mc is the cyclotron mass; E,~ is the electron Fermi level; and E o is the L-point gap, where nonparabolic effects are explicitly considered for electrons, while for the holes, the nonparabolic term Eh/Eo can be neglected. Figure 48 shows SdH oscillations reported for an undoped Bi nanowire sample with a 200 nm wire diameter, which was found to be slightly n-type, due to uncontrolled impurities. Also measurements of SdH oscillations were made on Te-doped 200-nm-diameter Bi nanowires, also showing two different periods at 14.6 T-1 (identified with light mass electron orbits) and at 19.5T -1 (identified with heavy mass electron orbits) (Herernans and Thrush, 1999a; Heremans, 1999). For small nanowire diameters, large magnetic fields are required to produce cyclotron radii smaller than the wire radius. However, for very large B fields, all Landau levels will have passed through the Fermi level

1

QUANTUMWELLS AND QUANTUM WIRES

103

and no oscillations can be observed. This is especially important because of the small effective masses of Bi, which result in a large Landau level spacing. Observing SdH oscillations in doped samples is also difficult because impurity scattering reduces the mean free path, requiring high B fields to satisfy the requirement that carriers complete a cyclotron orbit prior to scattering. Although SdH oscillations provide the most direct method of measuring the Fermi energy and carrier density of Bi nanowire samples, this technique may, however, not work for smaller diameter nanowires nor for nanowires that are heavily doped, because it may be difficult to obtain well-defined SdH oscillations for such samples.

8.

SEEBECKCOEFFICIENT OF Bi NANOWIRES

Thus far, there have been very few measurements on the Seebeck coefficient of Bi nanowires (Huber and Calcao, 1997), although the recent achievement of reliable measurements on 200-nm Bi nanowire arrays (Heremans and Thrush, private communication) is encouraging, and corroborates extensive prior studies of the thermoelectric properties of bulk single crystals (Brandt et al., 1977; Gurvitch, 1980; Beutler et al., 1987; Costa-KrS.mer et al., 1997; Garcia et al., 1997; Brandt et al., 1987). Improvement in the measurement technique for the Seebeck coefficient of Bi nanowires is still needed to extend the measurements to the smaller nanowire diameters of interest for possible thermoelectric applications (Sun et al., 1999e). To evaluate the potential of Bi nanowires for applications as a thermoelectric material, the Seebeck coefficients must be determined experimentally. However, the very small thicknesses ( ~ 5 0 Ftm) of the Bi nanowire array samples make these measurements difficult and potentially inaccurate, because of the difficulty of making thermocouples small compared to the sample thickness. In addition to the importance of the Seebeck coefficient to thermoelectric performance, the Seebeck coefficient can provide useful information about the electronic structure and transport properties of Bi nanowires. As mentioned in Section VI.7, direct measurements of the carrier density by the Hall effect are not possible and measurements of the absolute resistivity are difficult and expensive. Consequently, it has been difficult to assess the effects of doping Bi nanowires quantitatively. The Seebeck coefficient, in conjunction with some modeling, can provide valuable information about the position of the Fermi energy, as well as to verify theoretical models. The Seebeck coefficient [defined by Eqs. (6) and (8)], unlike the resistivity, is intrinsically independent of sample size and the number of nanowires contributing to the signal, because S depends on the ratio of 5~ ~~ on temperature, and is expected to depend on wire diameter. The alumina

104

M . S . DRESSELHAUS ET AL.

template containing an array of Bi nanowires therefore provides a convenient package for measuring the Seebeck coefficient of Bi nanowires. Two techniques for measuring the Seebeck coefficient of Bi nanowire arrays are described in the literature, one using a differential thermocouple arrangement (Heremans and Thrush, 1999a), and another, in which the thermocouples are mounted in direct electrical contact with the sample (Cronin et al., 1999) to measure the temperature difference, AT, across the Bi nanowire sample. The thickness of the thermocouples used in both measurements was 12.5/~m, which is comparable to the 50-ym sample thickness. Because of the relatively large thickness of the thermocouples, the measured AT is actually the temperature difference across both the sample and the thermocouples, and therefore, the measured AT overestimates the true sample A T. Thus, the measured Seebeck coefficient will be a lower limit of the true Seebeck coefficient of the sample. Despite the small thickness of the samples, large A T's (in excess of 10 K) are achievable (Heremans and Thrush, 1999a; Cronin et al., 1999) because of the low thermal conductivity

i~++

'

++. ~_~+~~ ~o

"

-10

~,

-20

~, ~

.--=

,~

_

'

"A ~

;\ ~

"§

%

-

s0ope=

0%\

l

+

",\

"

~

/

+*'~

9\

',\ \ ~

_

i l

',\ \ ~ ,\ \ %

l [

'

" 9

~ 1o.v~" : r

\~

"

+ sewnple2(Woocr=)

9

-:-.".-

,\

I

Bi n a n t e s 200 nm d~memr 9S a m ~ l (~aint)

- ./

+

WOo +

,~

-

-30

I

...... ,,,,,~.... v

+-I

+++~I

I 1

"..'~.

-"~.i

I I -0.5pV/K 2 ,,-

-40

~

--

Bulk Bi 1 . [010] -50

0

,

...~

I 100

9.. . . . .

.

...... I

I,, 200

i

q*

300

T (K)

FIG. 49. The temperature dependence of the Seebeck coefficient of two undoped Bi nanowire arrays and one Te-doped Bi nanowire array. All three samples have Bi nanowire diameters of 200 nm. The results for the Seebeck coefficient for the Bi nanowires are compared to that of bulk Bi along the bisectrix direction indicated by the dashed curve (Heremans and Thrush, 1999a; Heremans et al., 2000b).

1

105

QUANTUMWELLS AND QUANTUM WIRES

of the alumina template ( ~ 1.7 W/mK; Borca-Tasciuc and Chen, private communication). The Seebeck co~,tficient of bulk Bi is low ( - 5 0 to - 1 0 0 #V/K) because of the presence of both electrons and holes, whose contributions to S tend to cancel each other. Figures 49 and 50 show the measured Seebeck coefficients of 200- and 40-nm Bi nanowires, respectively. One immediately notices the low magnitudes for these Seebeck coefficients. As mentioned earlier, the measurement technique underestimates the Seebeck coefficient. For the 200-nm nanowires, the band overlap is not expected to deviate from the bulk value, and therefore 200-nm Bi nanowires contain approximately the same number of holes and electrons as bulk Bi. We thus attribute the low values of S to the same cancellation between electron and hole contributions that occurs in bulk Bi and the underestimate of the measurement technique for measuring S. However, for the 40-nm diameter wires, the band structure is expected to be significantly different; in particular, we expect the band overlap between the valence and conduction bands found in bulk Bi to be absent in these nanowires, which instead have a small bandgap, appropriate to the semiconducting state, thus potentially providing a higher Seebeck coefficient. However, as shown by the calculations presented in Fig. 51, S is large when the chemical potential is near the band edge, but S is small when the chemical potential is far from the band edge. The chemical potential can be varied by adding dopants that increase the

0

i~

l-zl

9

'

!

9

'i

9

" '

rq........[] 0.075 '/o Te doped 0 ....... 9O. 15% Te doped

CI~"~

-10

>

=L

(/) -20

-30

J

I

100

i

Temperature(K)

I

200

i

300

FIG. 50. The temperature dependence of the Seebeck coefficient of two 40-nm Te-doped Bi nanowire arrays with different doping concentrations (Lin, 2000).

106

M . S . DRESSELHAUS ET AL. 200

|

i

!

Valence Band Edge

.....

Conduction Band Edge

100 >

:zt.

r (..) (g 0

ro 0

03

!

-100

I I I i

-200

O0

,

I

-50

,

II

*

,

L

0 50 Chemical Potential (meV)

,

100

FIG. 51. The calculated Seebeck coefficient for a 40-nm-diameter Bi nanowire with transport along the [0152] direction at 77 K, considering nonparabolic dispersion relations for the conduction band and a circular wire cross section (Lin, 2000).

electron (or hole) concentration. The presence of Te dopants in Bi nanowires moves the chemical potential to lie within the conduction band, thus explaining the low measured values of the Seebeck coefficient. The theoretical optimum Z1DT is predicted to lie close to the conduction band edge (for n-type), as indicated in Fig. 51. The challenge that now presents itself is how to control the chemical potential ( sensitively enough to optimize the thermoelectric properties, and how to measure ( precisely enough for the optimization of Z1DT. From Figs. 49 and 50, we notice a striking difference in the effect of doping for nanowires of different diameters (40 and 200nm). On the addition of Te dopant, the magnitude of S is increased for the 200-nmdiameter samples, as shown in Fig. 49. However, the magnitude of the Seebeck coefficient is decreased on doping of the nanowires with a diameter of 40 nm, as shown in Fig. 50. These results appear to be in contrast with each other. However, as we look at the difference in the band structure of these two samples with different diameters, the reasons for this contrasting behavior become clear. For the 200-nm-diameter nanowires, there is essentially no shift in the band-edge energy relative to bulk bismuth. Therefore, at T - 77 K (for example), the nanowires are semimetallic with a band overlap of 38 meV. We calculate S as a weighted sum of contributions from the hole and electron bands (S e and Sh), weighted by the hole and electron conductivities (ae and ah). The addition of Te dopant causes an increase in

1 QUANTUMWELLS AND QUANTUMWIRES

107

the electron conductivity and hence an increase in the magnitude of the negative Seebeck coefficient, since electrons dominate over holes in the transport phenomena of Bi nanowires. For the 40-nm-diameter nanowires, however, the band edges are shifted appreciably due to quantum confinement effects. The 40-nm nanowire is predicted to be semiconducting at low temperatures, with a calculated bandgap of about 8 meV at 77 K. Figure 51 shows the calculated Seebeck coefficient for 40-nm-diameter Bi nanowires oriented along the [0112] direction (the same orientation as the wires in the sample of Fig. 50), plotted as a function of the chemical potential. Also indicated in the figure are the conduction and valence band edges. It should be clear from this graph that the S of a lightly doped sample (where the chemical potential is expected to lie just inside the conduction band) is larger in magnitude than the S of a heavily doped sample (where the chemical potential lies further in the conduction band), thus explaining the contrasting behavior that is observed for the 200- and 40-nm-diameter nanowire samples. In conclusion, present measurements of the Seebeck coefficient are qualitative, at best. More measurements on different wire diameters, and different doping concentrations must be made to form a conclusive picture. The general challenge will be in controlling the chemical potential precisely enough to observe the enhanced thermoelectric properties predicted by theory.

9.

THERMALCONDUCTIVITY

Since reliable values for Z T require detailed knowledge of the thermal conductivity, direct measurements of the temperature-dependent thermal conductivity x(T) and gaining an understanding of the science behind ~c(T) in Bi quantum wires are essential for progress in making useful lowdimensional devices from Bi nanowires (Chen et al., 2000). Thermal conductivity calculations of 1D nanowire systems (Chen et al., 1999) suggest large reductions in the lattice thermal conductivity for Bi nanowires relative to their bulk counterparts because of the strong phonon scattering at the nanowire-host interfaces. From an experimental standpoint, substantial progress has been made with developing techniques for reliable measurements of the thermal conductivity of Bi nanowire arrays, with particular emphasis having been given to the 3o~ technique, both along the nanowire direction and perpendicular to it (Chen et al., 2000). Room temperature measurements of the thermal conductivity of unfilled alumina templates with 100-nm pores have been made along the pore direction, yielding a value of ~ 1.7 W/cm K (BorcaTasciuc and Chen, private communication). Preliminary thermal conductivity measurements on CoSb 3 (skutterudite) nanowires indicate a lattice

108

M . S . DRESSELHAUS ET AL.

Bulk Bt: Blsectdx direction

0.01000

10.00

+ + + ~ 0.00100

E l.3szS

"

0.00010

0.10 * Sample 1 (Ag-paint) + Sample 2 (Wood's) & Doped Bi:Te

0.00001

1

10

T (K)

100

0.01

FIc. 52. The temperature dependence of the thermal conductance of three 200-nm Bi nanowire samples with (left ordinate), in comparison to the temperature dependence of the thermal conductivity of bulk Bi and Wood's metal (right ordinate) (Heremans, 1999; Heremans and Thrush, 1999b).

thermal conductivity lower than that of a skutterudite alloy and suggest that phonon scattering is dominated by boundary scattering at the interface between the nanowire and the anodic alumina template (Borca-Tasciuc and Chen, private communication). Preliminary temperature-dependent thermal conductance measurements on 200-nm Bi nanowires in an anodic alumina template were carried out using a two-probe method in the 2- to 300-K range (see Fig. 52) showing a T" dependence at low temperature (1.2 < n < 1.4) (Heremans, 1999; Heremans and Thrush, 1999b). Because of the uncertainty in the number of Bi nanowires that were connected to the two probes, results for the thermal conductance rather than for the thermal conductivity were reported for the three nanowire samples. The two samples with the highest thermal conductance were undoped and the sample with the lowest thermal conductance was doped n-type with Te to a concentration of 5 • 101S/cm3. The magnitude of the exponent n in the low-temperature conductance and the presence of a knee in ~c(T) near 20 K (which was attributed to the phonon mean free path in the Bi nanowire being equal to the nanowire diameter) was interpreted to indicate that the Bi nanowires contribute significantly to the thermal conductance of the Bi nanowire-alumina composite samples, but

1 QUANTUMWELLS AND QUANTUM WIRES

109

that the overall contribution to the thermal conductance was dominated by that from the anodic alumina template.

10.

RAMAN SPECTRA AND OPTICAL PROPERTIES

Preliminary measurements have been carried out on both the Raman spectra (Pimenta, private communication) and the optical properties (Zhang, 1999; Black et al., 2000) of Bi nanowires, and preliminary modeling studies of the optical properties have also been undertaken (Black et al., 2000; Huber et al., 1999a). Early interest in the optical properties of Bi nanowires was aroused by very preliminary measurements of the dependence on the nanowire diameter of the optical absorption of arrays of Bi nanowires within their alumina templates, showing a semimental-semiconductor transition to occur at 300 K at a wire diameter between 56 and 22 nm (Zhang et al., 1998a). It was found that the Raman spectra for the nanowires and for bulk bismuth were stronger for 647-nm laser excitation than for 514 nm. The Raman spectra for bulk Bi (see Fig. 53) show two peaks at 71 and 98 cm -1 with E 0 and A10 symmetries. The Raman spectra for the bismuth nanowires also show two peaks at the same frequencies, nearly independent of the wire diameter, consistent with a crystal structure that is common to both the nanowires and bulk bismuth. At the present time, it is not possible to comment on the relative intensities of the two features in the Raman spectra in Fig. 53 for the nanowires relative to the bulk crystal, because of the sensitivity of the peak intensities to the scattering geometry of the experiment. Optical studies potentially provide information on the energy for transitions between subbands in the valence to the conduction bands, as well as for the plasma frequency and the dielectric constant. The experiments are, however, complicated by the small diameters of the Bi nanowires relative to the wavelength of light. Because of the large anisotropy of the Bi electronic band structure, there are great advantages to making optical measurements on Bi nanowires aligned within the anodic alumina templates. This strategy, however, necessitates subtraction of the contribution of the alumina templates to the frequency-dependent dielectric function. This subtraction process must be carried out within the effective medium approximation or some other suitable approach, since the wire diameters are small compared to the wavelength of the light. The theoretical framework for carrying out the subtraction process using the effective medium approach has been considered in terms of the Maxwell-Garnett approximation (Black et al., 2000). Corrections to this approximation have also been considered for such nanowire systems to account for dynamical polarization effects (Huber et al., 1999a).

110

M . S . DRESSELHAUS ET AL.

bulk

_.

50

9

.

I

_..

100

1..

,

j

I

150

2(X)

.L

250

Raman shift (crn-1)

FIG. 53. The Raman spectra taken at 514.5-nm laser excitation for three Bi nanowire samples with diameters of 25, 60, and 100 nm (Pimenta, private communication).

Large diameter nanowires (dw >> 200nm) are expected to have a dielectric function Gl(co) + ie2(co) similar to that of bulk bismuth for which the static dielectric constant is G~(0)= 84 parallel to the trigonal direction and G]-(0)= 105 for the perpendicular direction (Boyle and Brailsford, 1960). Preliminary room temperature measurements on the reflectivity ~(co) and transmission Y-(co) have been carried out in the photon energy range 500 < hco < 4000cm-1 on both an empty template and a bismuthfilled template with a diameter dw = 40 nm (Black et al., 2000). From these ~(co) and Y-(co) measurements, the complex dielectric function G((D)-'-Gl(CO ) "~-iG2(co ) was determined for the host anodic alumina template Ghost(co) and for the composite alumina-Bi filled sample Gcomposite(co). The frequency-dependent dielectric function for the Bi nanowires Gmetal(co) can then be determined at each frequency, using Maxwell-Garnett effective medium theory:

Gc~176 Gcomposite +

-- Ghost -" f

gGhost

G m e t a l - Ghost

(94)

Gmetal + KGhost

where f is the volume fraction of the composite sample that consists of Bi

1

QUANTUMWELLS AND QUANTUM WIRES

111

nanowires in the alumina template and K is a screening parameter, which was simply taken to be K = 1 for cylindrical wires (Black et al., 2000) and f was determined from an SEM photograph of the cross section of the filled nanowire composite sample. From the resulting ~2(0~) data, oscillators at ~o ~_ 1000 and 1650cm-1 were identified, and these oscillations were tentatively attributed to optical transitions from van Hove singularities near the valence band edge to corresponding singularities in the density of states near the conduction band edge. The two oscillators were identified with the different band edges for the A carrier pocket and the degenerate B and C carrier pockets at the L point of the Brillouin zone for [1011] bismuth nanowires (see Section VI.3) (Lin, 2000; Black et al., 2000). Detailed optical studies as a function of frequency, temperature, and nanowire diameter are needed to clarify the band-edge states for Bi nanowires, to study the semimetal-semiconductor transition in detail, and to clarify the dependence of the static dielectric constant and plasma frequency on wire diameter.

11.

COMPARISONBETWEEN Bi AND Sb NANOWIRES

In this section, we comment on why Bi nanowires are much more attractive for thermoelectric applications than Sb nanowires or other members of the group V family of semimetallic elemental or alloy materials, other than the Bi 1_ xSbx alloys discussed in Section V.3.b. The availability of alumina templates (see Section VI.2) prompted the examination of other nanowire materials in the group V semimetallic family, such as antimony, for their potential as low-dimensional thermoelectric materials. The vaporphase deposition technique, based on filling the pores of an alumina template (see Section VI.2), was used for the preparation of Sb nanowires, following a similar approach as had been used to prepare Bi nanowires from the vapor phase (Heremans et al., 2000b). An x-ray diffraction pattern of the Sb nanowires showed no clearly resolved diffraction peaks, from which it was concluded that the Sb nanowires grown in this way are either amorphous or polycrystalline (Heremans et al., 2000a). Bulk Sb is also a semimetal, with 3.74 • 1019 cm-3 electrons and holes, situated in quasi-ellipsoidal pockets in the Brillouin zone, as shown in Fig. 54. The effective masses of Sb along the principal axes of the ellipses are given in Table XIII in relation to Bi and As, the other two elemental group V semimetals (Dresselhaus, 1971; Windmiller, 1966). The electron effective masses are 5 to 90 times heavier in Sb than in Bi, and the electron density is two orders of magnitude larger (see Table VI). As a result of quantum confinement effects, the energy of the lowest-lying conduction subband edge in Sb must increase much more than in Bi and the highest-lying valence subband edge must also decrease much more in Sb than in Bi in order for

112

M . S . DRESSELHAUS ET AL.

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l\l

K [11201

.,,.....~~o .. [1210]

ei~

\%

[lO~O1

LI[I~ E

Bisectrix

0/

cyl

FIG. 54. The three L-point electron carrier pockets and the six H-point hole carrier pockets of antimony in the rhombohedral Brillouin zone showing the main symmetry directions and the mirror plane normal to the binary axis.

a semimetal-semiconductor transition to occur. This implies that the observation of the semimetal-semiconductor transition, which is observed in Bi as the wire diameter is decreased below about 50 nm, is expected to occur at much smaller diameters in Sb nanowires. Thus, the semimetalsemiconducting transition in antimony nanowires is expected to occur at too small a diameter to be of practical interest for thermoelectric applications, and As nanowires, if they could be fabricated, would be of even less practical interest for thermoelectric applications. Two-probe temperature-dependent resistance measurements R(T)/ R(300 K) were made on arrays of Sb wires within alumina templates, with

TABLE XIII BAND PARAMETERSFOR THE ELECTRON AND HOLE CARRIERS IN THE GROUP V SEMIMETALS AT LIQUID HELIUM TEMPERATURES Bismuth

m 1,

m 2, m 3, EF

Nv a

Arsenic

Antimony

Units

Electrons

Holes

Electrons

Holes

Electrons

~ holes

7 holes

mo mo mo meV

0.00119 0.266 0.00228 27.2 3

0.064 0.064 0.69 10.8 1

0.093 1.14 0.088 93.1 3

0.068 0.92 0.050 84.4 6

0.135 1.52 0.127 202 3

0.106 1.56 0.089 154 6

0.046 0.016 - 1.82 21 6

"Number of carrier pockets.

1

QUANTUM WELLS AND QUANTUM WIRES 1.2

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'~' ' ' ' ' ' 1

'

-

1.o

'

' '''"1

113

'

Sb nanowtru d = I0 nm. sample 2

- - J l i l l

/

t

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M

0.8

d=l!

cx:z~ r'l

clg

~. Ix

o.6

d

0.4 -

d, ooo

0.2

1

ooo

9 9 ~

i

i

t

~

|

'

"

f Jill

i

10

T (~q

l

l

i

l IttJ

,i

100

FIG. 55. Two-probe temperature dependence of the normalized resistance R(T)/R(300 K) for Sb nanowire arrays with diameters in the 10 < dw < 200 nm range aligned in their alumina templates (Heremans et al., 2000a).

wire diameters in the range 10 < dw < 200nm (see Fig. 55). The results show a clear dependence of the normalized R(T)/R(300 K) on the nanowire diameter, though R(T) remains metallic down to a diameter of 10 nm, as expected from the large band overlap of 178 meV in Sb (see Table VII). The differences between the R(T)/R(300 K) results for the two Sb nanowire array samples with the same nominal 10-nm diameter are attributed to different distributions of wire diameters in the 10-15 nm range for the two samples. Of particular interest is the very different temperature dependence of R(T)/R(300 K) for the Sb nanowires (see Fig. 55) relative to the behavior in the Bi nanowires shown in Fig. 38. A comparison of the behavior of Bi and Sb nanowires in Figs. 38 and 55 shows that R(T)/R(300 K) for the Sb nanowires over the whole range of wire diameters 10 < dw < 200 nm is more closely related to the behavior of bulk Bi than to that in Bi nanowires. Measurements of the longitudinal and transverse magnetoresistance of antimony nanowires as a function of magnetic field and temperature reveal a number of properties similar to those previously discussed for Bi nanowires (see Section VI.7), such as the classical size effect leading to a negative magnetoresistance at high magnetic field (Zhang et al., 2000), localization effects seen as a steplike increase in the magnetoresistance when one magnetic flux unit is contained in the nanowire cross section (Heremans et al., 1998), and the quadratic field dependence of the magnetoresistance at low fields and its connection to carrier mobility (Zhang et al., 2000).

114

M . S . DRESSELHAUS ET AL.

Of particular interest for thermoelectric applications are the properties of Bi~_xSbx nanowires for x < 0.25, corresponding to the semiconducting or small-band-overlap semimetallic regime. Thermoelectric studies of these Bil_xSbx nanowire systems as a function of nanowire diameter, temperature, and carrier density await future investigation.

VII. Summary In this chapter, the predictions of an enhancement in the thermoelectric figure of merit of low-dimensional material systems relative to their corresponding bulk counterparts, as well as the present state of experimental confirmation of these predictions, are reviewed. Progress with specific quantum well, quantum wire, and quantum dot materials systems, such as the lead salts, Si-Ge, and bismuth is discussed. To date most of the effort has gone into proof-of-principle studies, though actual demonstration of the highest thermoelectric figure of merit (Z3DT) of any material to date has been seen in the low-dimensional system PbSeo.98Teo.oz-PbTe, where the enhancement is attributed to quantum dot formation associated with the interface between the PbTe and PbSeo.98Teo.02 . In this chapter, particular attention is given to a discussion of the structure and properties of bismuth quantum wires, which are still at an early state of research. Bismuth nanowires, however, offer significant promise for practical applications, because they can be self-assembled and are predicted to have desirable thermoelectric properties when they have wire diameters in the 5- to 10-nm range. Though temperature-dependent resistance measurements have been carried out for Bi nanowires in this diameter range, reliable thermoelectric measurements have not yet been reported. The introduction of low-dimensional concepts into the field of thermoelectricity has stimulated new approaches to thinking about better thermoelectric materials, new strategies for achieving higher Z3D T, and new application areas for thermoelectrics, such as thermal management of integrated circuits. It would be fair to say that the introduction of lowdimensional concepts into the thermoelectrics field has injected a large increase in interest and attention to thermoelectric materials and phenomena by the more general scientific community. It is, however, too soon to assess the eventual impact of these low-dimensional concepts on the eventual use of thermoelectricity for practical applications.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the helpful discussions with Prof. Jean-Paul Issi, Dr. Joseph Heremans, Ted Harman, and Prof. Gang Chen. They are also thankful to many other colleagues for their assistance with the

1

QUANTUM WELLS AND QUANTUM WIRES

115

preparation of this chapter. The authors are grateful for support for this work by the U.S. Navy Contract No. N00167-98-K-0024, the MURI program subcontract PO No. 0205-G-7Al14-01 through UCLA, DARPA contract No. N66001-00-1-8603, and by NSF grant No. DMR 98-04734.

REFERENCES Abstreiter, G., H. Brugger, and T. Wolf, Phys. Rev. Lett. 54, 2441 (1985). Adachi, S. J. Appl. Phys. 58, R1 (1985). Algaier, R. S., J. Appl. Phys. 32, 2185 (1961). Beyer, H., A. Lambrecht, J. Nurmus, H. B6ttner, H. Greissmann, A. Heinrich, L. Schmitt, M. Blumers, and F. V61kein, in The 18th International Conference on Thermoelectrics: ICT Symposium Proceedings, Baltimore, Institute of Electrical and Electronics Engineers, Inc., Piscataway, NJ, 1999. Beutler, D. E., and N. Giordano, Phys. Rev. B 38, 8 (1988). Beutler, D. E., et al., Phys. Rev. Lett. 58, 1240 (1987). Ashcroft, N. E., and N. D. Mermin, in Solid State Physics, chapter 14, p. 271 (Saunders College Publishing, New York, 1976). Black, M. R., Y.-M. Lin, M. S. Dresselhaus, M. Tachibama, S. Fang, O. Rabin, F. Ragot, P. C. Eklund, and Bruce Dunn, in Nanophase and Nanocomposite Materials III." MRS Symposium Proceedings, Boston, December 1999, edited by S. Komareni, J. C. Parker, and H. Hahn, Materials Research Society Press, Pittsburgh, PA, 2000. Borca-Tasciuc, T., and Gang Chen (private communication). Boyle, W. S., and A. D. Brailsord, Phys. Rev. B 120, 1943 (1960). Brandt, N. B., E. A. Svistova, and M. V. Semenov, Soy. Phys. JETP 32, 238 (1971). Brandt, N. B., D. V. Gitsu, A. A. Nikolaeva, and Y. G. Ponomarev, Soy. Phys. JETP 45, 1226 (1977). Brandt, N. B., D. B. Gitsu, V. A. Dolma, and Y. G. Ponomarev, Soy. Phys. JETP 65, 515 (1987). Braunstein, R., A. R. Moore, and F. Hereman, Phys. Rev. 109, 695 (1958). Broido, D. A., and T. L. Reinecke, Appl. Phys. Lett. 67, 1170 (1995). Broido, D. A., and T. L. Reinecke, Appl. Phys. Lett. 70, 2834 (1997). Brooks, H., Adv. Electron. Electron. Phys. 7, 158 (1955). Brown, D. M., and S. J. Silverman, Phys. Rev. 136, 290 (1964). Casian, A., I. Sur, A. Sandu, H. Scherrer, and S. Scherrer, in Sixteenth International Conference on Thermoelectrics: Proceedings, ICT "97, p. 442, Institute of Electrical and Electronics Engineers, Inc., Piscataway, NJ, 1997. Chambers, R. G. Proc. R. Soc. London A 202, 378 (1950). Chen, Gang, J. Heat Transf: Trans. ASME 119, 220 (1997). Chen, Gang, S. G. Volz, T. Borca-Tasciuc, T. Zeng, D. Song, K. L. Wang, and M. S. Dresselhaus, in Thermoelectric Materials-- The Next Generation Materials for Small-Scale Refrigeration and Power Generation Applications." MRS Symposium Proceedings, Boston, Vol. 545, edited by T. M. Tritt, H. B. Lyon, Jr., G. Mahan, an M. G. Kanatzidis, pp. 357-368, Materials Research Society Press, Pittsburgh, PA, 1999. Cho, S., A. DiVenere, G. K. Wong, J. B. Ketterson, and J. R. Meyer, J. Appl. Phys. 85, 3655-3660 (1999a). Cho, S., Y. Kim, A. DiVenere, G. K. Wong, J. B. Ketterson, and J. I. Hong, J. Vac. Sci. Technol. A 17, 2987-2990 (1999b). Cho, S., A. DiVenere, G. K. Wong, and J. B. Ketterson, in Sixteenth International Conferrence on Thermoelectrics: Proceedings, ICT '97, Dresden, Germany, edited by Armin Heinrich and Joachim Schumann, p. 188, Institute of Electrical and Electronics Engineers, Inc., Piscataway, NJ, 1997. IEEE Catalog Number 97TH8291; ISSN 1094-2734.

116

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Costa-Kr~imer, J. L., N. Garcia, and H. Olin, Phys. Rev. Lett. 78, 4990 (1997). Cronin, S., Y.-M. Lin, T. Koga, X. Sun, J. Y. Ying, and M. S. Dresselhaus, in The 18th International Conference on Thermoelectrics: ICT Symposium Proceedings, Baltimore, p. 554, Institute of Electrical and Electronics Engineers, Inc., Piscataway, NJ, 1999. Cronin, S. B., Y.-M. Lin, T. Koga, J. Y. Ying, and M. S. Dresselhaus, in Molecular Electronics: M R S Symposium Proceedings, Boston, December 1999, edited by S. T. Pantelides, M. A. Reed, J. Murday, and A. Aviram, Materials Research Society Press, Pittsburgh, PA, 2000. Cucka, P., and C. S. Barrett, Acta. Crystalloyr. 15, 865 (1962). Cuff, K. F., M. R. Ellet, C. D. Kuglin, and L. R. Williams, Proc. 7th Int. Conf. Phys. Semicond., p. 677 (1964). Dalven, R., Infrared Phys. 9, 141 (1969). Datta, S., in Electronic Transport in Mesoscopic Systems (Cambridge University Press, Caqmbridge, 1995). Dauscher, A., B. Lenoir, O. Boffoue, X. Devaux, R. Martin-Lopez, and H. Scherrer, in Sixteenth International Conference on Thermoelectrics. Proceedings, ICT '97, pp. 429-433, Institute of Electrical and Electronics Engineers, Inc., Piscataway, NJ, 1997. Demske, D. L., J. L. Price, N. A. Guardala, N. Lindsey, J. H. Barkyoumb, J. Sharma, H. H. Kang, and L. Salamanca-Riba, in Thermoelectric Materials--The Next Generation Materials for Small-Scale Refrigeration and Power Generation Applications." M R S Symposium Proceedings, Boston, Vol. 545, edited by T. M. Tritt, H. B. Lyon, Jr., G. Mahan, and M. G. Kanatzidis, p. 209, Materials Research Society Press, Pittsburgh, PA, 1999. Dornhaus, R., G. Mimtz, and B. Schlicht, Narrow-Gap Semiconductors, Springer Tracts in Modern Physics, Vol. 98 (Springer-Verlag, Berlin, 1985). Dresselhaus, M. S., J. Phys. Chem. Solids, Suppl. No. 1 32, 3-33 (1971). Dresselhaus, M. S., T. Koga, X. Sun, S. B. Cronin, K. L. Wang, and G. Chen, in Sixteenth International Conference on Thermoelectrics." Proceedings, ICT '97, Dresden, Germany, edited by Armin Heinrich and Joachim Schumann, pp. 12-20, Institute of Electrical and Electronics Engineers, Inc., Piscataway, NJ, 1997a. IEEE Catalog Number 97TH8291; ISSN 1094-2734. Dresselhaus, M. S., X. Sun, S. B. Cronin, T. Koga, G. Dresselhaus, and K. L. Wang, in Thermoelectric Materials--New Directions and Approaches: M R S Symposia Proceedings, San Francisco, Vol. 478, edited by T. M. Tritt, M. G. Kanatzidis, H. B. Lyon, Jr., and G. D. Mahan, p. 55, Materials Research Society Press, Pittsburgh, PA, 1997b. Eckstein, Y., A. W. Lawson, and D. H. Reneker, J. Appl. Phys. 31, 1534 (1960). Fischer, P., I. Sosnowsk, and M. Szymanski, J. Phys. C 11, 1043 (1978). Gallo, C. F., B. S. Chandraselkhar, and P. H. Sutter, J. Appl. Phys. 34, 144 (1963). Garcia, N. et al., in Nanowires, edited by P. A. Serena and N. Garcia, pp. 243-250 (Kluwer Academic, The Netherlands, 1997). Gerlach, E., P. Grosse, M. Rautenberg, and W. Senske, Phys. Status Solidi (b) 75, 553 (1976). Giordano, N., Phys. Rev. Lett. 61, 2137-2140 (1988). Goldsmid, H. J., and E. H. Volckman, in Sixteenth International Conference on Thermoelectrics: Proceedings, ICT '97, Dresden, Germany, edited by Armin Heinrich and Joachim Schumann, p. 171, Institute of Electrical and Electronics Engineers, Inc., Piscataway, NJ, 1997. IEEE Catalog Number 97TH8291; ISSN 1094-2734. Gopinathan, K. K., and A. R. K. L. Padmini, Solid State Commun. 16, 817 (1975). Gurvitch, M., J. Low Temp. Phys. 38, 777 (1980). Hansen, M., in Constitution of Binary Alloys (McGraw-Hill, New York, 1958). Harman, T. C. (private communication). Harman, T. C., and I. Melngailis, Appl. Solid State Sci. 4, 1 (1974). Harman, T. C., D. L. Spears, and M. J. Manfra, J. Electron. Mater. 25, 1121 (1996). Harman, T. C., D. L. Spears, D. R. Calawa, S. H. Groves, and M. P. Walsh, in Sixteenth International Conference on Thermoelectrics." Proceedings, ICT '97, Dresden, Germany,

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edited by Armin Heinrich and Joachim Schumann, p. 416, Institute of Electrical and Electronics Engineers, Inc., Piscataway, NJ, 1997. IEEE Catalog Number 97TH8291; ISSN 1094-2734. Harman, T. C., M. P. Walsh, and D. L. Spears, in The 40th Electrical Materials Conference, Charlottesville, VA, 1998; J. Electron. Mater. 27, 7 (1998). Harman, T. C., D. L. Spears, and M. P. Walsh, J. Electron. Mater. Lett. 28, L1 (1999a). Harman, T. C., P. J. Taylor, D. L. Spears, and M. P. Walsh, in The 18th International Conference on Thermoelectrics." ICT Symposium Proceedings, Baltimore, p. 280, Institute of Electrical and Electronics Engineers, Inc., Piscataway, NJ, 1999b. Harman, T. C., M. P. Walsh, and D. L. Spears, J. Electron. Mater. Lett. 28 L1 (1999c). Harman, T. C., P. J. Taylor, D. L. Spears, and M. P. Walsh, J. Electron. Mater. Lett. 29, L1 (2000). Heremans, J. P., Thermal Conductivity 25, 114-121 (1999). Heremans, J., and O. P. Hansen, J. Phys. C: Solid State Phys. 12, 3483 (1979). Heremans, J., and C. M. Thrush (private communication). Heremans, J., and C. M. Thrush, Phys. Rev. B 59, 12579 (1999a). Heremans, J., and C. M. Thrush, in Proc. 25th International Thermal Conductivity Conference, Ann Arbor, MI, June 1999b. Heremans, J., D. T. Morelli, D. L. Partin, C. H. Olk, C. M. Thrush, and T. A. Perry, Phys. Rev. B 38, 10280 (1988). Heremans, J., C. M. Thrush, Z. Zhang, X. Sun, M. S. Dresselhaus, J. Y. Ying, and D. T. Morelli, Phys. Rev. B 58, R10091-R10095 (1998). Heremans, J. P., C. M. Thrush, Y.-M. Lin, S. Cronin, and M. S. Dresselhaus, Bull. Am. Phys. Soc., 2000a. APS March Meeting: abstract M28.10. Heremans, J., C. M. Thrush, Yu-Ming Lin, S. Cronin, Z. Zhang, M. S. Dresselhaus, and J. F. Mansfield, Phys. Rev. B 61, 2921-2930 (2000b). Hicks, L. D. The Effect of Quantum-Well Superlattices on the Thermoelectric Figure of Merit. Ph.D. thesis, Massachusetts Institute of Technology, Department of Physics, 1996. Hicks, L. D., and M. S. Dresselhaus, Phys. Rev. B 47, 12727-12731 (1993a). Hicks, L. D., and M. S. Dresselhaus, Phys. Rev. B 47, 16631 (1993b). Hicks, L. D., and M. S. Dresselhus, in Semiconductor Heterostructuresfor Photonic and Electronic Applications: M R S Symposia Proceedings, Boston, Vol. 281, edited by C. W. Tu, D. C. Houghton, and R. T. Tung, p. 821, Materials Research Society Press, Pittsburgh, PA, 1993c. Hicks, L. D., T. C. Harman, and M. S. Dresselhaus, Appl. Phys. Lett. 63, 3230 (1993). Hicks, L. D., T. C. Harman, X. Sun, and M. S. Dresselhaus, Phys. Rev. B 53, R10493-R10496 (1996). Holy, V., G. Springholz, M. Pinczolits, and G. Bauer, Phys. Rev. Lett. 83, 356 (1994). Hong, K., F. Y. Yang, K. Liu, D. H. Reich, P. C. Searson, C. L. Chien, F. F. Balakirev, and G. S. Boebinger, J. Appl. Phys. 85, 6184 (1999). Houston, B., R. E. Stranka, and H. S. Belson, J. Appl. Phys. 39, 3913 (1968). Huber, T. E., and R. Calcao, in Sixteenth International Conference on Thermoelectrics: Proceedings, ICT "97, Dresden, Germany, edited by Armin Heinrich and Joachim Schumann, p. 404, Institute of Electrical and Electronics Engineers, Inc., Piscataway, NJ, 1997. IEEE Catalog Number 97TH8291; ISSN 1094-2734. Huber, C., M. Sadoqi, and T. E. Huber, Adv. Mater. 7, 316 (1995). Huber, T. E., F. Bocuzzi, and L. Silver, Phys. Rev. B 59, 7446 (1999a). Huber, T. E., M. J. Graf, and C. A. Foss, in Thermoelectric Materials--The Next Generation Materials for Small-Scale Refrigeration and Power Generation Applications: M R S Symposium Proceedings, Boston, Vol. 545, edited by T. M. Tritt, H. B. Lyon, Jr., G. Mahan, and M. G. Kanatzidis, pp. 227-231, Materials Research Society Press, Pittsburgh, PA, 1999b. Ichiguchi, T., S. Nishikawa, and K. Murase, Solid State Commun. 34, 309 (1980). Ioffe, A. V., and A. F. Ioffe, Dokl. Akad. Nauk SSSR 98(5), 757 (1954).

118

M . S . DRESSELHAUS ET AL.

Isaacson, R. T., and G. A. Williams, Phys. Rev. 185, 682 (1969). Ismail, K., in Low-Dimensional Electronic Systems, edited by G. Bauer, F. Kuchar, and H. Heinrich, p. 333 (Springer-Verlag, Berlin, 1992). Ivanov, G. A., V. A. Kulikov, V. L. Naletov, A. F. Panarin, and A. R. Regel, Soviet Phys. Semicond. 6, 1134 (1973). Jain, A. L., Phys. Rev. 114, 1518 (1959). Kai Liu, Chien, C. L., Searson, P. C., and Kui Yu-Zhang, Appl. Phys. Lett. 73, 1436 (1998). Kasper, E., H.-J. Herzog, H. Jorke, and G. Abstreiter, Superlattices and Microstructures 3, 141 (1987). Ketterson, J. (private communication). Kittel, C. Introduction to Solid State Physics, 6th ed., p. 426 (John Wiley & Sons Inc., New York, 1986). Koga, T., S. B. Cronin, T. C. Harman, X. Sun, and M. S. Dresselhaus, in Semiconductor Process and Device Performance Modeling: M R S Symposia Proceedings, Boston, Vol. 490, edited by S. T. Dunham and J. S. Nelson, pp. 263-268, Materials Research Society Press, Pittsburgh, PA, 1998a. Koga, T., X. Sun, S. B. Cronin, and M. S. Dresselhaus, Appl. Phys. Lett. 73, 2950 (1998b). Koga, T., X. Sun, S. Cronin, and M. S. Dresselhaus, in Thermoelectric Materials--The Next Generation Materials for Small-Scale Refrigeration and Power Generation Applications: M R S Symposium Proceedings, Boston, Vol. 545, edited by T. M. Tritt, H. B. Lyon, Jr., G. Mahan, and M. G. Kanatzidis, pp. 375-380, Materials Research Society Press, Pittsburgh, PA, 1999a. Koga, T., X. Sun, S. B. Cronin, and M. S. Dresselhaus, Appl. Phys. Lett. 75, 2438 (1999b). Koga, T., X. Sun, S. Cronin, and M. S. Dresselhaus, in The 18th International Conference on Thermoelectrics: ICT Symposium Proceedings, Baltimore, p. 378, Institute of Electrical and Electronics Engineers, Inc., Piscataway, NJ, 1999c. Koga, T., T. C. Harman, S. B. Cronin, and M. S. Dresselhaus, Phys. Rev. B 60, 14286-14293 (1999d). Koga, Takaaki. Concept and Applications of Carrier Pocket Engineering to Design Useful Thermoelectric Materials Using Superlattice Structures. Ph.D. thesis, Harvard University, Division of Engineering and Applied Sciences, June 2000. Krishnamurthy, S., A. Sher, and A.-B. Chen, Phys. Rev. B 33, 1026 (1986). Lax, B., and J. G. Mavroides, in Advances in Solid State Physics (Academic Press, New York, 1960), Vol. 11. Lee, S.-M., D. G. Cahill, and R. Venkatasubramanian. Appl. Phys. Lett. 70, 2957-2959 (1997). Lee, S. T., N. Wang, Y. F. Zhang, and Y. H. Tang, MRS Bull. 24, 36 (1999). Lenoir, B., M. Cassart, J. P. Michenaud, H. Scherrer, and S. Scherrer, J. Phys. Chem. Solids 57, 89-99 (1996). Li, S. S. NBS Special Publication 400-33, 13 (1977). Li, F., L. Zhang, and R. M. Metzger, Chem. Mater. 10(9), 2470 (1998). Lieber, C. (private communication). Lin, Yu-Ming, Doping Bi nanowires. Master's thesis, Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, January 2000. Lin, Y. M., X. Sun, and M. S. Dresselhaus, Phys. Rev. B 62, 4610 (2000). Lin, Y. M., S. B. Cronin, J. Y. Ying, and M. S. Dresselhaus, Appl. Phys. Lett. 76, 3944 (2000a). Lippmann, G., P. Kfistner, and W. Wanninger, Phys. Status Solidi A 6, K159 (1971). Madelung, O., in Landolt-B6rnstein Numerical Data and Functional Relationships in Science and Technology, New Series, edited by K.-H. Hellwege (Springer-Verlag, Berlin, 1987a), Vol. 21a. Madelung, O., in Landolt-B6rnstein Numerical Data and Functional Relationships in Science and Technology, New Series, Group III." Crystal and Solid State Physics, edited by O. Madelung and M. Schultz (Springer-Verlag, Berlin, 1987b), Vol. 22a.

1

QUANTUM WELLS AND QUANTUM WIRES

119

Mahan, G. D., in Solid State Physics, edited by H. Ehrenreich and F. Spaepen. Academic Press, 1997a. Mahan, G. D., Physics Today March (1997b). Masuda, H., H. Yamada, M. Satoh, H. Asoh, M. Nakao, and T. Tamamura, Appl. Phys. Lett. 71(19), 2772 (1997). McSkimin, H. J., and P. Andreatch, Jr., J. Appl. Phys. 34, 651 (1963). McSkimin, H. J., and P. Andreatch, Jr., J. Appl. Phys. 35, 2181 (1964). Miller, A. J., G. A. Saunders, and Y. K. Yogurtcu, J. Phys. C 14, 1569 (1981). Mitchell, D. L., E. D. Palik, and J. N. Zemel, Proc. 7th Int. Conf. Phys. Semicond., p. 325 (1964). Murayama, Y., in Encyclopedia of Applie Physics, edited by G. L. Trigg (VCH Publishers, New York, 1994). Nimtz, G., in Landolt-B6rnstein Numerical Data and Functional Relationships in Science and Technology, New Series, Group III." Crystal and Solid State Physics, p. 170, edited by K.-H. Hellwege and O. Madelung (Springer-Verlag, Berlin, 1983), Vol. 17. Noothoven Van Goor, J. M., Philips Res. Rep. 4, 1 (1971). Novikova, S. I., and N. Kh. Abrikosov, Soy. Phys. Solid State 5, 1397 (1964). Pimenta, M. (private communication). Piraux, L., S. Dubois, J. L. Duvail, A. Radulescu, S. Demoustier-Champagne, E. Ferain, and R. Legras, J. Mater. Res. 14, 3042 (1999). Preier, H., Appl. Phys. 20, 189 (1979). Rabin, Oded (unpublished). Ravich, Y. I., B. A. Efimova, and I. A. Simirnov, Semiconducting Lead Chalcogenides (Plenum Press, New York, 1970), Chapters 3, 4, and 6. Ravich, Yu. I., B. A. Efimova, and V. I. Tamarchenko, Phys. Stat. Sol. (b) 43, 11 (1971a). Ravich, Yu. I., B. A. Efimova, and V. I. Tamarchenko, Phys. Stat. Sol. (b) 43, 453 (1971b). Rieger, M. M., and P. Vogl, Phys. Rev. B 48, 14276 (1993). Sakurai, M., N. Satoh, S. Tanuma, and I. Yoshida, in Sixteenth International Conference on Thermoelectrics: Proceedings, ICT '97, Dresden, Germany, edited by Armin Heinrich and Joachim Schumann, p. 180, Institute of Electrical and Electronics Engeineers, Inc., Piscataway, NJ, 1997. IEEE Catalog Number 97TH8291; ISSN 1094-2734. Samsonov, G. V., ed., Handbook of the Physicochemical Properties of the Elements (IFI/Plenum, New York, 1968). Sander, M. S., and R. Gronsky, Bull. American Physical Society, 2000. APS March Meeting. Sander, M. S., Y.-M. Lin, M. S. Dresselhaus, and R. Gronsky, in Nanophase and Nanocomposite Materials III." M R S Symposium Proceedings, Boston, December 1999, edited by S. Komareni, J. C. Parker, and H. Hahn, Materials Research Society Press, Pittsburgh, PA, 2000. Saunders, G. A., and Z. Siimengen, Proc. R. Soc. London A 329, 453 (1972). Scherrer, H., Z. Dashevsky, V. Kanster, A. Casian, I. Sur, and A. Sandu, in Thermoelectric Materials 1998--The Next Generation Materials for Small-Scale Refrigeration and Power Generation Applications: M R S Symposium Proceedings, Boston, Vol 545, edited by T. M. Tritt, M. G. Kanatzidis, G. D. Mahan, and H. B. Lyon, Jr., pp. 99-104, Materials Research Society Press, Pittsburgh, PA, 1999. Schiferl, D., and C. S. Barret, J. Appl. Crystalloyr. 2, 30 (1969). Schlichting, U., Dissertation Technische Universit~it Berlin (1970). Shin, S. C., J. E. Hilliard, and J. B. Ketterson, Thin Solid Films 111, 323 (1984a). Shin, S. C., J. E. Hilliard, and J. B. Ketterson, J. Vac. Sci. Technol. A 2, 296 (1984b). Shin, S. C., J. B. Ketterson, and J. E. Hilliard, Phys. Rev. B 30, 4099 (1984c). Sipatov, A., V. Volobuev, A. Fedorov, E. Rogacheva, and I. Krivulkin, in The 18th International Conference on Thermoelectrics." ICT Symposium Proceedings, Baltimore, Institute of Electrical and Electronics Engineers, Inc., Piscataway, NJ, 1999. Sitter, H., K. Lischka, and H. Heinrich, Phys. Rev. B 16, 680 (1977).

120

M . S . DRESSELHAUS ET AL.

Slack, G. A., and M. A. Hussain, J. Appl. Phys. 70(5), 2694 (1991). Smith, R. A., Semiconductors, 2nd ed. (Cambridge University Press, Cambridge, 1978). Smith, G. E., and R. Wolfe, J. Appl. Phys. 33, 841 (1962). Springholz, G., and G. Bauer, Appl. Phys. Lett. 62, 2399 (1993). Springholz, G. Ihninger, G. Bauer, M. M. Olver, J. Z. Pastalan, S. Romaine, and B. B. Goldberg. Appl. Phys. Lett. 63, 2908 (1993). Springholz, G., V. Holy, M. Pinczolits, and G. Bauer, Science 282, 734 (1998). Stacy, A. (private communication). Suhrmann, R., and G. Wedler, Z. Angew. Phys. 14, 70 (1962). Sun, Xiangzhong. The Effect of Quantum Confinement of the Thermoelectric Figure of Merit, Ph.D. thesis, Massachusetts Institute of Technology, Department of Physics, June 1999. Sun, L. Z., and J. R. Bolton, J. Phys. Chem. 100, 4127 (1996). Sun, X., G. Chen, K. Wang, and M. S. Dresselhaus, in Seventeenth International Conference on Thermoelectrics: Proceedings, ICT '98, Nagoya, Japan, edited by Kunihito Koumoto, Institute of Electrical and Electronics Engineers, Inc., Piscataway, NJ, 1998. Sun, X., S. B. Cronin, J. Lin, K. L. Wang, T. Koga, M. S. Dresselhaus, and G. Chen, in The 18th International Conference on Thermoelectrics." ICT Symposium Proceedings, p. 652, Baltimore, Institute of Electrical and Electronics Engineers, Inc., Piscataway, NJ, 1999a. Sun, X., Y. M. Lin, S. B. Cronin, M. S. Dresselhaus, J. Y. Ying, and G. Chen, in The 18th International Conference on Thermoelectrics." ICT Symposium Proceedings, p. 394, Baltimore, Institute of Electrical and Electronics Engineers, Inc., Piscataway, NJ, 1999b. Sun, X., J. Liu, S. B. Cronin, K. L. Wang, G. Chen, T. Koga, and M. S. Dresselhaus, in Thermoelectric Materials--The Next Generation Materials for Small-Scale Refrigeration and Power Generation Applications." M R S Symposium Proceedings, Boston, Vol. 545, edited by T. M. Tritt, H. B. Lyon, Jr., G. Mahan, and M. G. Kanatzidis, pp. 369-374, Materials Research Society Press, Pittsburgh, PA, 1999c. Sun, X., Z. Zhang, and M. S. Dresselhaus, Appl. Phys. Lett. 74, 4005-4007 (1999d). Sun, X., Z. Zhang, G. Dresselhaus, M. S. Dresselhaus, J. Y. Ying, and G. Chen, in Thermoelectric Materials--The Next Generation Materials for Small-Scale Refrigeration and Power Generation Applications: M R S Symposium Proceedings, Boston, Vol. 545, edited by T. M. Tritt, H. B. Lyon, Jr., G. Mahan, and M. G. Kanatzidis, pp. 87-92, Materials Research Society Press, Pittsburgh, PA, 1999e. Ueta, A. Y. MBE Growth and Characterization of PbTe/Pbl _xEuxTe Epitaxial Layers, Ph.D. thesis, the Johannes Kepler Universit~it Linz, Department of Physics, January 1997. Van de Walle, C. G., Phys. Rev. B 39, 1871 (1989). Vecchi, M. P., and M. S. Dresselhaus, Phys. Rev. B 10, 771 (1974). Verdun, H. R., and H. D. Drew, Phys. Rev. Lett. 33, 1608 (1974). Vining, C. B., in CRC Handbook of Thermoelectrics, edited by D. M. Rowe, p. 329 (CRC Press, New York, 1995). Walther, K., Phys. Rev. 174, 782 (1968). White, G. K., J. Phys. C 5, 2731 (1972). Windmiller, L. R., Phys. Rev. 149, 472-484 (1966). Wood, C., Rep. Pro9. Phys. 51, 459 (1988). Wyckoff, R. W. G., in Crystal Structures (Interscience: New York, 1964), Vol. 1. Yasaki, T., J. Phys. Soc. Jpn. 25, 1054 (1968). Yim, W. M., and A. Amith, Solid State Electron. 15, 1141-1165 (1972). Yoshida, I., S. Tanuma, and J. Takaahashi, J. Mag. Magnetic Mater. 126, 608-611 (1993). Yuan, Shu, H. Krenn, G. Springholz, and G. Bauer, Phys. Rev. B 47, 7213 (1993). Yuan, Shu, G. Springholz, G. Bauer, and M. Kriechbaum, Phys. Rev. B 49, 5476 (1994). Yuan, S., H. Krenn, G. Springholz, Y. Ueta, G. Bauer, and P. J. McCann, Phys. Rev. B 55, 4607 (1997). Zemel, J. N., J. D. Jensen, and R. B. Schoolar, Phys. Rev. A 140, 330 (1965).

1

QUANTUM WELLS AND QUANTUM WIRES

121

Zhang, Zhibo, B. Fabrication Characterization and Transport Properties of Bismuth Nanowire Systems, Ph.D. thesis, Massachusetts Institute of Technology, Department of Physics, February 1999. Zhang, Z., X. Sun, M. S. Dresselhaus, J. Y. Ying, and J. Heremans, Appl. Phys. Lett. 73, 1589-1591 (1998a). Zhang, Z., J. Y. Ying, and M. S. Dresselhaus, J. Mater. Res. 13, 1745-1748 (1998b). Zhang, M. S. Dresselhaus, and J. Y. Ying, in Thermoelectric Materials--The Next Generation Materials for Small-Scale Refrigeration and Power Generation Applications: M R S Symposium Proceedings, Boston, Vol. 545, edited by T. M. Tritt, H. B. Lyon, Jr., G. Mahan, and M. G. Kanatzidis, pp. 351-356, Materials Research Society Press, Pittsburgh, PA, 1999a. Zhang, Z., D. Gekhtman, M. S. Dresselhaus, and J. Y. Ying, Chem. Mater. 11, 1659 (1999b). Zhang, Z., X. Sun, M. S. Dresselhaus, J. Y. Ying, and J. Heremans, Phys. Rev. B 61, 4850-4861 (2000).

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SEMICONDUCTORS AND SEMIMETALS, VOL. 71

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Thermoelectric Transport in Quantum Well and Quantum Wire Superlattices D. A. Broido DEPARTMENTOF PHYSICS BOSTONCOLLEGE CHESTNUT HILL, MASSACHUSETTS

T L. Reinecke NAVAL RESEARCHLABORATORY WASHINGTON, D.C.

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . II. SEMIQUANTITATIVE THEORY OF THE POWER

FACTOR . . . . . . . . . .

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III. QUANTITATIW THEORY OF Tim POWER FACTOR . . . . . . . . . . . . 1. Q u a n t u m Well Superlattices . . . . . . . . . . . . . . . . . . 2. Q u a n t u m Wire Superlattices . . . . . . . . . . . . . . . . . . 3. Results and Discussion . . . . . . . . . . . . . . . . . . . .

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V. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES

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1.

135 139 140 152 153

Introduction

In the past half dozen years there has been renewed interest in finding novel materials and systems for use in cleaner, more efficient cooling and power generation applications (Mahan et al., 1998). The thermoelectric figure of merit Z (Goldsmid, 1964) provides a useful measure of the desirability of a material for cooling applications that is independent of device configuration. It is given by P Z = -

(1)

K

Here P is the power factor: P oS 2, where a is the electrical conductivity, S is the thermoelectric power (Seebeck coefficient), and K is the thermal conductivity. The thermal conductivity x is composed of electrical and =

123 Copyright 9 2001 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-752180-1 ISSN 0080-8784/01 $35.00

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lattice parts, x = x e + KL. Thermoelectric transport is controlled by the Peltier effect, and the power factor P gives a measure of the electrical contribution to Z*. On the other hand, the thermal conductivity describes a heat current that opposes the Peltier effect, and it is described by K in the denominator of Z. The figure of merit has units of inverse temperature and is typically given in the dimensionless form Z T, where T is the temperature. Since the 1950s, a wide range of materials have been examined to assess their potential for providing high values of ZT. Semiconductors and related materials have been the main focus of interest because they generally have a desirable combination of the parameters a, S, and ~:. Bi2Te 3 and related alloys have been found to have the highest room temperature values of Z T ~ 1, but over the years little increase in Z T has been found above these values. Even modest increases in Z T would provide important opportunities for applications (Mahan et al., 1998). Much of the recent renewed interest in thermoelectrics has been stimulated by the prospect that superlattices and related composites could provide increased Z T s . In recent years, novel materials growth and synthesis capabilities, such as molecular beam epitaxy, have made it possible to tailor materials with electronic or thermal properties not previously available in nature. For example, semiconductor quantum wells and quantum wires exhibit quasi-two- and quasi-one-dimensional electrical properties and can have high electrical conductivities. In addition, thermal transport in these systems can be modified by interface scattering and by changes in phonon dispersion. These systems offer the prospect of new materials for thermoelectric applications, and they also exhibit novel transport phenomena of interest in a wider range of research. Quantum wells and superlattices made from them have been grown of materials of interest in thermoelectrics, and quantum wires are now being fabricated using several innovative techniques. This work is described in the accompanying chapters in this volume. Theoretical work on thermoelectric transport in quantum wells was begun some years ago (Friedman, 1984; Tao and Friedman, 1985). In the past half dozen years, this work has expanded and become more detailed and realistic. A sketch of the potential profile of a superlattice is shown in Fig. 1, where there is a potential barrier Vo between the bulk conduction band edge in quantum wells of width a composed of one material and that of the barriers of width b of another material. This example represents a type I semiconductor system with conduction band minima and valence band maxima in the same material. *In the followingwhen discussing the electronic contribution to thermoelectric transport, we give results for the power factor in order to make clear which effects arise from electronic properties and which from lattice effects. This is different from the practice in earlier papers on this topic that gave results for the figure of merit of superlattices using the bulk lattice thermal conductivity.

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THERMOELECTRICTRANSPORT

P-I-el

I

125

V0

FIG. 1. Sketch of the potential profile of a superlattice with well width a, barrier width b, and potential offset Vo.

In 1993 Hicks and coworkers (Hicks and Dresselhaus, 1993a, 1993b; Hicks et al., 1993) argued that Z T in superlattices would be greatly e n h a n c e d over bulk values at small well sizes. They used a model in which the potential barriers were taken to be infinite and barriers had zero width to discuss Z T in Bi2Te 3. In effect, this model described a series of single q u a n t u m wells with infinite potentials. Results of calculations of Z T similar to theirs are shown in Fig. 2. The increase in Z T for decreasing width arises mainly from the increasing density of states per unit volume of the effectively two-dimensional q u a n t u m wells with infinite potentials. The density of states per unit volume is given by m/nh2a, where m is the carrier mass and a is the q u a n t u m well width. In the case of q u a n t u m wires with infinite

5

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,

,

,

,

,

,

I

,

,

,

,

,

,

,

,

,

,

,

i

,

,

,

,

,

,

,

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,

i

20

,

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,

40 Well

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80 ( A

,

,

,

100

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FIG. 2. Thermoelectric figure of merit, ZT, for model Bi2Te 3 superlattice as a function of well width with infinite potential offsets Vo and barrier widths b = 0. The calculations are made using the constant relaxation time approximation for the carrier scattering and a model of the band structure in which the valley degeneracy is not lifted, and they are described in Section II.

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D.A. BROIDOAND T. L. REINECKE

potentials the density of states per unit volume goes as 1/a 2, and thus P and Z T in this case would be still greater than the quantum well case. This work attracted considerable interest in the thermoelectrics community. Subsequent theoretical work undertook to describe reliably the qualitative features of P and Z T in fully three-dimensional superlattice systems and to develop a quantitative theory of thermoelectric transport in them. Mahan and Lyon (1994), Sofo and Mahan (1994b), Lin-Chung and Reinecke (1995), and Broido and Reinecke (1995a, 1995b, 1995c) described the effects of heat transport in the barriers in superlattices using different approaches. They showed that in realistic superlattices, the charge carriers flow preferentially in the quantum well regions but that heat flows through the whole structure. This effectively increases • of the whole superlattice as compared to that of the quantum well alone. In addition, Sofo and Mahan (1994) and Broido and Reinecke (1995a, 1995b, 1995c) studied thermoelectric transport by carriers in superlattices with finite barrier heights including carrier tunneling through the barriers of the band structures and using the constant relaxation time approximation for carrier scattering. They found that P and Z T remained finite for all widths and that they reached maxima for widths of a few tens of angstroms. The magnitudes of the maxima were found to increase with increasing potential offset. This work gave a good qualitative description of P in superlattices. Broido and Reinecke subsequently developed (Broido and Reinecke, 1997, 1999, forthcoming a; Reinecke and Broido, 1997, 1998) a more quantitative theory of the power factor in semiconductor superlattices, and they used it to describe important qualitative features of P in these systems. They gave detailed results for PbTe quantum well and quantum wire superlattices with different crystallographic directions. More recently, they used three-dimensional band calculations for the electronic states of superlattices in an inelastic Boltzmann equation approach for carrier scattering (Broido and Reinecke forthcoming a). They showed that even in the case of infinite potentials and zero barrier width, P did not diverge for small well and wire sizes. In quantum well superlattices with finite potential barriers, they found that P is nearly independent of potential offset and that in quantum wire systems P has a strong dependence on potential offset. They also gave quantitative results for the lifting of valley degeneracy in systems grown in different crystallographic directions. This work has provided a reliable and reasonably complete theoretical treatment of the P in superlattices. An inelastic Boltzmann treatment of P has also been given for PbTe single quantum wells with finite barriers (Koga et al., 1999), and was used to discuss experiment. For most materials of interest, the total thermal conductivity K is dominated by the lattice contribution. Thus K plays a key role in determining ZT. This quantity is not yet as well understood as is P. Measurements of ~ both along the planes (Yao, 1987) in GaAs-A1As superlattices and in

2 THERMOELECTRICTRANSPORT

127

the growth direction (Capinski and Maris, 1996) indicate that these are reduced as compared to the bulk case. In addition, there is evidence for a decrease of K in the growth direction in Si-Ge superlattice systems (Lee et al., 1997). Models have been introduced by several groups (Hyldgaard and Mahan, 1996, 1997; Chen, 1996; Walkauskas et al., 1999; Simkin and Mahan, 2000) in order to try to understand these effects, particularly the role of interface scattering and of superlattice phonon dispersion. Experimental work has been directed at growing and fabricating good quality quantum well systems and at obtaining reliable measurements of their transport parameters (Harman et al., 1996; Hicks et al., 1996), and innovative techniques for growing quantum wire systems (Zhang et al., 1999; Huber et al., 1999) from materials of interest in thermoelectrics is being done. This work is described in the accompanying chapters in this volume. We also note that there has been experimental and theoretical work done on transport along the superlattice growth direction, which is also described in the accompanying chapters. In the following, we review theoretical work on the power factor for transport along the quantum well planes and along quantum wires. This will be done mainly from the point of view of our contributions. Other work and other topics will be covered in separate chapters in this volume. First, we describe a semiquantitative theoretical approach based on a constant relaxation time approximation for the Boltzmann equation. It gives a good qualitative picture of P in superlattices including the effects of carrier tunneling through barriers. Then, we give a quantitative theory of P in superlattices using 3D electronic band structure calculations in an inelastic Boltzmann equation treatment. Finally, a phonon Boltzmann approach is used to describe the lattice thermal conductivity due to interface scattering in free-standing quantum wells and wires. Our objective in the work described here has been to develop reliable, quantitative theoretical tools and a physical understanding of the essential ingredients needed to describe thermoelectric transport in superlattices. These tools enable meaningful comparisons with experiment and predictions to be made.

II.

Semiquantitative Theory of the Power Factor

Any theoretical treatment of the electrical contribution to thermoelectric transport involves (1) a description of the electronic band structure of the system and (2) a treatment of the carrier scattering. Many, but not all, of the important qualitative and semiquantitative features of the power factor P in thermoelectric transport can be described by a relatively straightforward approach (Sofo and Mahan, 1994b; Broido and Reinecke, 1995a,

128

D.A. BROIDOAND T. L. REINECKE

1995b, 1995c). It involves an envelope function description of the superlattice band structure and the constant relaxation time approximation (CRTA) for carrier scattering. We outline this treatment briefly now, and we give typical results for the power factor P as a function of superlattice period and of potential offset from this treatment for a model of BizTe a superlattices. In general, the electronic transport coefficients a, S, and ~ce are given in terms of the electric and heat currents Je and Je that result from carrier motion when a weak electric field and a small temperature gradient are applied to a system

( -Je/e'] e2 -JQIk,J = aST ekB

(Vec~'] \Vk, T ] -~BB/

(2)

Here ~b is the electrostatic potential, and 7 is the thermal conductivity at zero electric field, with ~ = ~ce + ToS 2, where ~ce is the usual thermal conductivity at zero electric current. In general, a, S, and 7 are tensors. Here we will treat transport in the direction parallel to the quantum well planes and take these quantities to be scalars. Here we will consider transport by electrons in the conduction band. The superlattice potential is illustrated in the sketch in Fig. 1. It is used to describe the superlattice where a periodic potential is imposed on the effective mass electron band of a bulk semiconductor. That is, the barrier and well materials will be taken to be the same except for the potential offset. Within the envelope function approximation, the electron energy in a quantum well superlattice is the sum of a contribution along the confinement direction, e,(kz), taken to be the z direction, and a parabolic but anisotropic dispersion in the plane of the quantum wells. For simplicity, in this section, we take e,(kz) in a tight-binding approximation in which the wells are weakly coupled. This approximation gives a superlattice dispersion of the form

e,,(k) =

e,o + A,(1 -T- cos

kzd ) + -~ \mx + my/

(3)

where e,o is the energy at the bottom of the nth subband, n = 1, 2 , . . . , and the - ( + ) sign goes with the odd (even) values of n. The overlap integrals A, are fit to the bandwidth of the exact dispersion obtained by solving the Schr/Sdinger equation for the superlattice potential shown in Fig. 1. The A, depend on the potential Vo and on the well and barrier widths, a and b.

2 THERMOELECTRICTRANSPORT

129

#J

9

#energy FIG. 3. Sketch of carrier density of states (DOS) for quantum well superlattices.

Within this approximation, the electronic density of states of the superlattice is obtained analytically by summing over all wave vectors at a given energy (Friedman, 1984; Tao and Friedman, 1985). The density of states for each subband is

x//mxmr gn(~) =

7~h2

_1{dn-lkzn ,

~;no < e < e.o +

d 1,

e > e,o + 2A,

2A, (4)

where k2n = d- 1 cos- 1[1 - (e - e,o)/A,). A sketch of this density of states is given in Fig. 3. The flat portions correspond to quasi-two-dimensional subbands and are separated by dispersive regions that arise from tunneling between the subbands. For decreasing Vo when the tunneling becomes large, 9,(0 goes over into the form for free particles in the bulk. Carrier scattering is described by a Boltzmann equation. In Section III, we give a full solution of the inelastic Boltzmann equation for phonon scattering. In this section, we use the simpler CRTA in which the total carrier scattering is approximated by a single relaxation time. Then the solution of the Boltzmann equation for steady state transport gives the linearized nonequilibrium distribution function, f,,kx = fo + 6f, kx, where fo is the Fermi function and

(C3fo~ hkx (eEx ~f~k~ = r \ c3eJ m---~

_~

-

T

(5)

is the deviation from the equilibrium distribution function for the nth carrier subband. Here the electric field and temperature gradient are taken to be in the x direction, # is the chemical potential, and ~ is the isotropic constant relaxation time, which is taken to be independent of energy and of subband index and is chosen to fit the bulk mobility. The electric and heat currents

130

D . A . BROIDO AND T. L. REINECKE

are given by

dk3 V. kx C~ Je = - e ~ f -~ f . kx

(6) (7)

The transport coefficients are given straightforwardly by Ashcroft and Mermin (1976)

a = L o,

1

aS=--~L1, er

and

7=

1

L2

e~

(8)

where

f ( Jo) a(e) = e2r ~

f ~"* 6(e -

(9)

e.(k))v2x

(10)

With the carrier dispersion in Eq. (3), the transport coefficients a, S, and ?, are given by

a = elZx ~ h .

(11)

n

aS = el4 ~ (2g. - (A. + (.)h. + 2.)

(12)

n

7 = ektx ~ (392 -- 2(An + 2(.)g. + (.(2A. + (.)h. + 2~. - 2{.~..)

(13)

n

where

r,.= ff d~fo(~)o.(~),~.= fod~fo(~)o.(~)~, e,,-2 = fo 2,, = A,, f l a . de fo(e)g,,(e ) (sinkz,,d) \ kz,,d , _

dp. = A.

de fo(e)g .(e)e 2

def~

\

(14)

sin z. ) k=,,d

(15) Here the energy e and chemical potential of the carriers ~. = (/t - eno) are measured from the bottom of the nth subband, and /~x = (e'c)/mx is the carrier mobility in the direction of transport x.

2 THERMOELECTRICTRANSPORT

131

1.1

0.9 IxI

a. 0.8

0.7 0.6

0.5 F 0.4

1018

a= 00;"

//

~/

%%%i

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

_

''~176 .~

,

J

i

J

,

,

, i I

10 ~9 Density

( cm 3 )

FIG. 4. Calculationsof the power factor P as a function of carrier density for model Bi2Te 3 superlattices for Vo = 200 meV for indicated well widths a and for barrier width b = a.

To illustrate the results of this treatment of P, we give calculations for a model band structure of BizTe 3 superlattices. The transport properties of bulk BizTe 3 are highly anisotropic. We take the x direction to be in the quantum well plane and along the ao axis of the hexagonal unit cell. The superlattice growth direction z is taken to be along the bo axis of the unit cell, which gives m x = 0.021, my = 0.32, and mz = 0.081 and a mobility #x = 1200 cmZ/V s [parameter values for BizTe 3 are taken from Hicks and Dresselhaus (1993a)]. In general, the sixfold degeneracy of the bulk electron valleys will be lifted by confinement in superlattices. A treatment of these band structure effects, including anisotropy and the multiband character, will be given in the next section. For the moment, we take account of this degeneracy by multiplying a by 6, both in the bulk and in superlattices, which will give an overestimate of cr in the superlattice case. The following results are all for temperature T = 300 K. Calculations of ~r and S are made from Eqs. (11) and (12) and are used to give P = crS2. In general, P depends on the carrier density through the chemical potential #. Figure 4 gives P as a function of the carrier density for a quantum well superlattice with varying well widths. For each width, P has a maximum as a function of density. In the following, we evaluate P at its maximum with respect to density. In practice, such changes in density can be obtained with doping. Figure 5 gives P for BizTe 3 superlattices as a function of well width a for several values of b/a and for fixed Vo. For wide wells, P lies below the bulk

132

D.A. BROIDOAND T. L. REINECKE 1.2

,

,

,

,

,

,

,

1.1

~.

,

,

,

,

,

,

,

,

,

,

,

,

b=a/2

~~~ " ...... a'"

--'-=0.9 Q.. n

0.8

0.7

"'.

0.6

0.5

0.4 20

40 Well

Thickness

60

80

100

( A )

FIG. 5. P o w e r factor of model Bi2Te 3 q u a n t u m well superlattices scaled by the calculated bulk P as a function of well width a for several values of the ratio of barrier to well widths, b/a. Here Vo = 200 meV.

values, and this effect is more pronounced for the larger barrier thicknesses. This arises from the factor of 1/d = 1/(a + b) in the density of states in Eq. (4) which enters a and aS in Eqs. (11) and (12). Physically, the charge carriers are constrained to flow preferentially through the quantum wells, resulting in a decreased current per total unit area as compared to bulk and in a decreased P. This effect in turn has implications for the figure of merit Z T because the heat current flows both through barriers and through wells, and thus Z T is decreased by what can be thought of as a "parasitic" heat current flowing through the barriers. This point has been made by several groups (Mahan and Lyon, 1994; Sofo and Mahan, 1994b; Lin-Chung and Reinecke, 1995; Broido and Reinecke, 1995a, 1995b, 1995c) using different approaches. For decreasing a in Fig. 5, P increases and then reaches a maximum, and for still smaller a it decreases again. The value of the well width at which the maximum P occurs decreases for increasing barrier widths b. The peak values of P for these superlattices are found to be somewhat larger than the bulk P and to be nearly independent of the barrier width b for this range a and b. The enhancement of P over the bulk value arises from the changes in the carrier density of states upon electron confinement in relatively narrow wells. For small widths, P again approaches the bulk value, as expected, which occurs because of carrier tunneling for finite Vo and for small a and b. Figure 6 gives the dependence of P on the potential offset Vo. For large periods, d, P increases for decreasing Vo and approaches the bulk value

2 THERMOELECTRICTRANSPORT 3

,

,

,

,

9

,

,

,

,

,

,

,

,

,

,

,

,

133 ,

,

' V0=o o

2.5

',b=O

2 Vo=1000me V ".

J~- 1.5

b-"

"'~1769

1

"~176176 ~

0

~

b=a

0.5

0

,

,

A

I

20

,

,

,

1

40

,

,

,

i

60

,

,

,

i

80

. . . .

100

Well Thickness ( A ) FIc. 6. Power factor of a model of Bi2Te 3 quantum well superlattices scaled by the calculated bulk P as a function of well width for several values of the potential offset Vo.

because in this case the electrons spread out approaching bulk behavior for small Vo. On the other hand, for d near the value that gives the maximum P, the power factor increases with increasing Vo. This behavior arises from increasing quantum confinement for large Vo. The case of Vo = ~ and barrier width b = 0 also is shown. For it, P increases monotonically for decreasing well widths giving the behavior shown earlier in Fig. 2, which is similar to that in Hicks and Dresselhaus (1993a) for this case. Within the CRTA, the existence of a maximum in P for finite Vo as a function of size arises from carrier tunneling through the barriers. Figure 7 gives P for quantum wire superlattices using the same tightbinding treatment of the band structure as for quantum wells (Broido and Reinecke, 1995b). These quantum wires have square cross sections with side a, and they are arranged on a square lattice with barrier widths b = a. It is noteworthy that in this superlattice there is three times as much barrier material as wire material. The qualitative features of P are similar to those in Fig. 6 for quantum well superlattices. Once again for large a, P lies below the bulk value for nonzero Vo. For decreasing a, P reaches a maximum, and for smaller a, it again decreases due to tunneling. The maximum value of P as a function of a is higher for these quantum wire superlattices than it is for the quantum well case even though the ratio of barrier material to well or wire material is larger for the wires. This is because in wires, the density of electronic states scales is 1/d 2 as compared to lid in wells. The relative magnitude of these maxima for wells and wires is discussed in more detail in Broido and Reinecke (1995b).

134

D.A. BROIDOAND T. L. REINECKE 1.6

' ' '

~,V'o=3000m'eV

1.4 /'

= 1.2 Q.

-'-.~ 9 ..... 0.8

......

b=a

Vo=1000m b=a

v_0=_2~176 "'-..~.. . . . . . . . . . . . . .

~

0.6 0.4

0

20

40

60

80

100

Wire Thickness ( ,~, )

FIG. 7. Power factor of a model of Bi2Te 3 quantum wire superlattices as a function of wire width for several potential offsets Vo. In all cases, the barrier width b = a.

The theoretical approach discussed in this section also has been used to discuss the thermoelectric efficiency in Bi superlattices (Broido and Reinecke, 1995c). Bi is a semimetal that would have desirable thermoelectric properties in the bulk except that the effects of electron and hole carriers tend to cancel one another in the thermopower. The electron and hole degeneracy is lifted by quantum confinement in superlattices giving rise to higher values of P than in the bulk (Hicks et al., 1993). More recently, CRTA calculations of Z T for Bi nanowires (Sun et al., 1999) have given Z T values much larger than 1 for narrow Bi wires, and experimental work on these structures is ongoing (Zhang et al., 1999; Huber et al., 1999). The theory in this section is relatively straightforward to implement, and it provides a good qualitative picture of P in superlattice systems. In particular, it includes the effects of the three-dimensional superlattice and the effects of carrier tunneling through the barriers.

III.

Quantitative Theory of the Power Factor

The need to go beyond the CRTA treatment of carrier scattering described in the preceding section is motivated by two factors: (1) scattering of carriers is dependent on their energy, well-wire superlattice dimensions, the barrier height, and a variety of other quantities and (2) for some scattering mechanisms, carrier scattering is inelastic. For inelastic scattering a relaxation time cannot be defined (see, for example, Nag, 1980). In

2 THERMOELECTRICTRANSPORT

135

addition, we note that many important thermoelectric materials have multiple carrier valleys. Confinement lifts the bulk degeneracy of these valleys leading to energetically separate subband ladders deriving from different bulk valleys. This affects how many subbands contribute to conduction. Here we describe a theoretical treatment of the power factor that addresses these issues for quantum well and quantum wire superlattice systems. In thermoelectric materials, room temperature scattering of carriers is dominated by phonons. Impurity scattering is typically less important and it depends on doping configurations. Here we focus on phonon scattering and later comment on impurity effects. We consider carrier scattering by acoustic phonons via the deformation potential (DP) interaction and by polar optical phonons (POPs) via the Fr6hlich interaction. In the following, we do not include carrier scattering between valleys, and we treat each valley independently. The contribution to the thermoelectric transport coefficients is calculated from the subbands deriving from each conducting valley and summed to give the total transport coefficients. For notational simplicity, we suppress a valley index in the following expressions.

1.

QUANTUMWELL SUPERLATTICES

Thermoelectric transport in a model quantum well superlattice shown in Fig. 1 with the growth axis in the z direction is considered here. For each valley, an electron state is specified by a subband index n and wave vector k = (kll, k=), where kll = (k x, ky) is the in-plane component. The Boltzmann equation for the superlattice for steady state transport in the presence of electric field E and temperature gradient VT is (see, for example, Nag, 1980):

eE. Vkf, +

Ot

_vf 8~ 3

h

-h

9V T - - = - t~T c~t

(16)

dk'[W,,,(k', k)f,,(k')(1 - f,(k)) - W,,,(k, k')f,(k)(1 - f,,(k'))] (17)

Here, f,(k, r) is the distribution function for electrons in superlattice state

(n, k), W,,,(k, k') is the scattering probability for electrons from state (n, k) to state (n', k'), and e,(k) is the electron energy. The collision operator c~fc/c~t accounts for intra- and intersubband inelastic scattering into and out of the state (n, k).

136

D.A. BROIDOAND T. L. REINECKE

We focus on transport in the plane of the quantum well layers, which is the direction of interest in most of the experimental and theoretical work to date. Materials of interest in thermoelectrics typically have multiple ellipsoidal conduction band valleys. In the effective mass approximation, the quantum well superlattice subband structure deriving from each bulk valley has the form

e,,(k) = e,,(kz) + -~ kmx + my/

(18)

Here, e,,(kx) is the superlattice dispersion, which is obtained along with the superlattice Bloch functions ~9,kz(Z) by numerical solution of the Schr/Sdinger equation for the periodic potential in Fig. 1. Note that here we do not make the tight-binding approximation to the band structure used in Section II. For each valley, the x and y directions are oriented along the principal axes of the elliptic constant energy surfaces. We will consider weak electric fields E and weak thermal gradients V T in the plane of the quantum wells. These fields in general will not lie along the principal axes of the elliptic energy surfaces for each valley. Thus, the currents will in general not be along these directions. The transport coefficients are, however, independent of the magnitudes of E and V T for small values of these quantities. We first evaluate the transport coefficients for the directions of the principal axes and then obtain them for arbitrary directions by adding the contributions from the components of the field and temperature gradient in the directions of the principal axes (Nag, 1980). Thus, for E and VT along one of the principal axes, taken to be the x direction, the distribution function for the nth superlattice subband can be expressed in terms of its deviation, 6f,, from the equilibrium Fermi distribution: fnk = fo + C~fnk, with

6f"k

= (afo'] hkx eEx,r,ln(k ) -k- 1 dT Z2n(k) ) \ a e J m---f

-T -d-~x

(19)

In quantum well systems, the scattering functions Vl,(k) and Z2n(k ) a r e anisotropic because of the superlattice band structure e,(kz) and because of any anisotropy of the bulk band structure. Substituting Eq. (19) into the Boltzmann equation leads to two sets of coupled integral equations for the scattering functions:

Lc(z in(k))

--

1

Lc(zz,(k))

= e- p

(20)

where the collision operator is

L~((z,,,(k)}) = Z,,,f

1 -- f o ( k ) )

-- ~ Tin'(kt)

(21)

2 THERMOELECTRICTRANSPORT

137

These equations are analogous to those obtained for isotropic bulk materials (Nag, 1980; Sofo and Mahan, 1994a). Here, they are extended to include multiple superlattice subbands and anisotropy. Scattering of electrons by acoustic phonons and polar optical phonons are the dominant scattering mechanisms at room temperature. We take the phonons to be unaffected by the superlattice structure. The scattering rates by optic phonons are obtained to a good approximation by bulk plane waves (Riicker et al., 1992; Knipp and Reinecke, 1993), and the superlattice periodicity has only small effects on the acoustic phonons. We take the optic phonon branch to be dispersionless and given by hcoo, and the acoustic branch to be linear with the wave vector with averaged isotropic velocity given in terms of the elastic constants Vo = (3Cl1 + 2C12 + 4C44)/5 (Zak, 1964). The electron-phonon scattering probabilities for quantum well superlattices are:

WP~162 nn' •.., k') = ~_~ (N

l~p2POPl*'a ~I/tPOPL/,. 0 Jr- 1 ---~- -21~-~ nn' t.a'~, k,)(~( s n' (k,)

]F,k.,.k~(K.,)l 2 .]~/fPOP(~.. nn' ~..., k') = 2mAk2 + Ak2 + (Akz + Kin)2'

__ ~n (k) -.+.- hO)o) (22)

2he 2 C2~ = lc* V coo (23)

Here, the + ( - ) sign is for phonon emission (absorption), Ak i = k ' i - k i, i = x , y , z , K m = 27rm/d is the reciprocal superlattice vector; l/K*= 1/~Co~ --1/~Co,~Co(~C~) is the static (high-frequency) dielectric constant; and N o = 1/(exp(hcoo/ksT) - 1). The function F,kz,,k~(Km) = | Jo

ei(k~-kz-Km)z,h,

~,.,~ (z)~ , ~ ( z ) dz

(24)

is given from the superlattice Bloch functions, I P n k z ( Z ) = e i k z Z U n k z ( Z ) with U,kz(Z + d) = U,kz(Z). These wave functions can be expressed in closed form (Kittel, 1986), which enables F to be evaluated analytically. At room temperature, the average electron energy is considerably larger than that of acoustic phonons involved in the deformation potential scattering. The scattering probability for this mechanism is, to good approximation, elastic and has the form:

~.,De (k,

k') =

C ~ . M . ."" , ( k ~ , k~)6(e.,(k') ' - s.(k))

M,,DP(k~, k'~) = ~ [Fnk~n,k~(Km)[2 m

E2~ks T C2p -- 2 Vpv 2

(25)

(26)

138

D . A . BROIDO AND T. L. REINECKE

Using the preceding expressions for the transition probabilities to calculate the collision operator in Eq. (21), we obtain Eq. (20) as

~, = S~

- ~ S+,,(e)z,,,(e + hcoo) - ~ S+,(e)r,,,(e - he)o) Of'

(27)

Og'

where o~ = (n, k r, kz) and ~i ~- 1 for i = 1 and ~ i - - - e - - ~ / / for i = 2. These equations are solved using an extension of the Ritz iterative method (Nag, 1980; Sofo and Mahan, 1994a), which will be described elsewhere (Broido and Reinecke, forthcoming b). From Eq. (27), it is evident that the collision operator connects energy surfaces differing by multiples of the LO phonon energy hcoo. Neglect of this energy in the scattering functions gives a relaxation time approximation. We will consider the significance of this approximation in Subsection 3 of Section III. For transport along one of the two principal axes of the in-plane ellipsoidal energy surface (x), we obtain the contributions from each valley

as e2knT (mxmy) 1/2 mx rch2d ~ Io'(l~, T)

(28)

aS = - e k 2 T (mxmy)l/2 m~-f-- rch2d Z I1,(~, T)

(29)

k~T 2 (mxmr) 1/2 7 = mx uh2d 2 I2"([a, T)

(30)

n

where

Io'(it, T) = -~u

dxf~ dx (xz l'(x, y))

(31)

dxf~ dx (x(x - ~'(y)))~ l'(x, y))

(32)

dy --TO

if; f:

Ii'(lt , T) = 2n

dy

Tr

12..(,. T) = ~l f ]

dy f o dxf~ dx d (x(x -- ('(Y))Z 2"(x, Y))

(33)

and

Zin(X, y) = _1~02~dO COS20rin(X, y, 0)

(34)

with y = k=d, x = (~ - e'(k=))fl, fo = fo( x, ~'(kx)), and ~'(k=) -(/~ - e'(kz))fl

2 THERMOELECTRICTRANSPORT

139

being the scaled chemical potential. The full transport coefficients for a field along x are obtained by adding the contributions for all valleys. From these results and those for the other principal axis directions we obtain the total transport coefficients and the power factor, P = aS 2. It should be noted that Eqs. (28) through (30) reduce to those in the CRTA (Sofo and Mahan, 1994b; Broido and Reinecke, 1995a) if we take z l, = z, ZZn = (e -- fl)T.

2.

QUANTUMWIRE SUPERLATTICES

For quantum wire superlattices, the Boltzmann equation can again be cast in the form of Eqs. (20) and (21). The wire superlattice dispersion is

h 2 kx2

(35)

where x is the transport direction. Here, e,(ki) is the quantum wire superlattice dispersion, with k• = (kr, kz). The quantum wire superlattice potential, which is just the extension of Fig. 1 to two dimensions, is nonseparable, and the two-dimensional superlattice SchriSdinger equation in (y,z) does not have a closed form solution. Instead, we expand the superlattice Bloch functions,

~,k~ (p) = e 'k~p u,k~ (p)

(36)

u,k~ (p) = ~ c,k~ (K)e -ix.o,

(37)

in a plane wave basis:

K

where K = (2n/d)(m 1)V + m22) and solve the resulting matrix equation for the c.ki, to obtain e.(k• and ff,k~. The electron-phonon scattering probabilities for quantum wire superlattices have a form similar to that for quantum well superlattices, but have a 2D reciprocal lattice:

IF.k~.'k~(Kin,, Km2)l2 "]k/J'Pnn' OP/~'x,.", ~" kt) -- ml,m2E A k 2 _.]_ ( A k y -]- Kin,) 2 .ql_ ( A k z %- Kin2 )2

MD.P~(k• k'• = ~ m l ,m2

IF,,k+,,'kl(Km,, Km~)l2

(38) (39)

140

D.A.

BROIDO AND T. L. REINECKE

with the overlap factor,

F,,k~,,,g(K) = f c~,, ei(kl-k'-x)PtP*k;(P)~"k~ (p) dp

(40)

Here Kin1 --(2rc/d)ml~ and similarly for Kin2. The solution of the inelastic multisubband Boltzmann equation for quantum wire superlattices is similar to that for wells in Subsection 1 of this section and expressions analogous to Eqs. (28) to (30) are obtained for the transport coefficients (Broido and Reinecke, forthcoming b).

3.

RESULTS AND DISCUSSION

In the following, we illustrate results of this treatment mainly for PbTe systems, which are of particular interest in studies of superlattices. It will be helpful to illustrate some ideas with results for GaAs, which has a particularly simple, isotropic, single-valley conduction band. All results are for T = 300 K. It is also helpful to discuss the limit of infinite potential barriers, Vo = oo. For if the superlattice subbands become dispersionless, and the scattering probabilities for wells and wires can be reduced to simpler form (Price, 1981; LeBurton, 1984). Figure 8 illustrates the calculated total POP scattering rate for the lowest

_ ~,,~=~ oA ~'\ ,, ~

a=50A

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

."..--..................................

c-

_

rO

.~

0

2

4

6

8

10

/E POP FIG. 8. Carrier scattering rates for GaAs quantum wells with Vo = oo and b = 0, and well widths indicated. The dotted line corresponds to bulk.

2 THERMOELECTRICTRANSPORT

141

2.5

2 r~

E 0

I

1.5

I

I

I

I

I

I

I

I

I|

I t t t t t

t I

t

I

I~

I

,-%

I

:E

0.5

. . . .

i

. . . .

1O0

1

. . . .

200

1

. . . .

300

i

. . . .

400

500

Well Thickness ( A ) Fx6. 9. Carrier mobility due to polar optical phonon scattering for GaAs quantum wells with Vo = ~ as a function of well width in a relaxation time model (dashed line) and from the inelastic solution of the Boltzmann equation (solid line).

subband, n = 1, for GaAs single quantum wells with infinite barriers. This is obtained by integrating Eq. (22) for phonon emission and absorption over k' and kz. Two well widths, a = 50 A and a = 10 ]k, are shown along with the bulk result (a = ~). The carrier energy is scaled by the phonon energy h~o. Notice the energy dependence of the scattering, with a peak occurring at ho~o where the onset of phonon emission occurs. With decreasing a, this peak increases to values above the bulk scattering rate. The energy and well width dependences of the P O P scattering rate in Fig. 8 show that for narrow wells with large Vo, the scattering cannot be represented accurately by a single constant relaxation time. The importance of inelastic scattering in the Boltzmann equation is illustrated in Fig. 9, which gives calculations of the room temperature POP-limited carrier mobility in GaAs single quantum wells with Vo = ~ as a function of well thickness. The density has been taken to be 1018 cm-3. The dashed line is from a relaxation time approximation in which ho~o in the scattering functions from Eq. (27) is neglected (Tsuchiya and Ando, 1993). The solid line is from the inelastic Boltzmann equation. The decrease in mobility for decreasing a below 150,/k arises from the increase in the scattering rates with decreasing widths in Fig. 8. For increasing widths the

142

D.A. BROIDOAND T. L. REINECKE

peak and decrease results from the onset of intersubband scattering, which becomes significant when the subband confinement energies decrease. The relaxation time approximation in Fig. 9 gives a mobility higher than the bulk value for wells with a ~ 150 A. In the inelastic treatment the added intra- and intersubband inelastic scattering reduce the mobility. For large well widths, the contributions from many subbands converge, and the bulk value is recovered. The lower mobilities obtained for the inelastic treatment for large well widths are in agreement with the bulk mobilities in GaAs (Landhold and B6rstein, 1982a) and point to the importance of an inelastic treatment to describe transport accurately in lower dimensional systems. PbTe is a good room temperature thermoelectric material in bulk with a Z T ~ 0.4. Its thermoelectric properties have also been studied in PbTe/ PbEuTe quantum well systems (Harman et al., 1996; Hicks et al., 1996), and enhanced power factors were observed in them. Bulk PbTe has four highly anisotropic ellipsoidal valleys along the [111] crystallographic directions. Here, we consider PbTe quantum well superlattices with growth axes along both the [111] and [001] directions. For the [111] direction, the confinement lifts the fourfold valley degeneracy resolving one set of minibands from the longitudinal valley lying below another threefold degenerate set of subbands from the oblique valleys. In the [001] orientation, the masses for all four valleys along the confinement direction are the same. Superlattices in this orientation retain the fourfold valley degeneracy of the bulk. We will also give results for quantum wire superlattices with two orientations. For the "[001]" quantum wires, the confinement directions are [001] and [010], and the transport direction is along [100]. For "[111]" quantum wires, the confinement directions are [111] and [112], and the transport direction is [110]. In the latter case, the valley degeneracy is lifted, while in the former it is not. The quantum wires have square cross section and are arranged on a square lattice as described in Section II. Material parameters for PbTe are taken to be: m g = 0.35, mt = 0.034 (Nimitz and Schlicht, 1982), and E~ = 25meV, ho~o = 14meV, x o =414, ~ = 33, Cxx = 1.072 • 107 N/cm 2, C12 = 7.68 • 105 N/cm 2, C44 = 1.322 • 106 N/ cm 2 (Landholt and B6rstein, 1982b). As noted in Section II, P is always a function of carrier density. In the following we evaluate P at the densities for which P is a maximum. Figure 10 shows P for [001] PbTe quantum well superlattices with Vo =- ~ and b = 0 under several treatments of the carrier scattering; P is scaled to the bulk value for the same treatment of carrier scattering. As discussed in Section II, previous calculations within the CRTA for these strongly confined systems have suggested that P should increase monotonically with decreasing well and wire widths attaining values well above the bulk value. This behavior is given in Fig. 2 and by the dash-dotted line in Fig. 10. The dotted line gives P with carrier scattering by only optic phonons. The dashed curve shows P with only scattering of carriers by acoustic phonons,

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which is independent of well width. In the high-temperature approximation to DP scattering (Nag, 1980), which is valid for PbTe at room temperature, the carrier relaxation time for this fully confined case, ~, is inversely proportional to the density of electronic states per unit volume, D, which itself goes as 1/a. Since the ~ ~ D~, it becomes independent of a for acoustic phonon scattering. The solid line shows P including both P O P and DP scattering. The relatively weak increase of P with decreasing well thickness reflects the fact that DP scattering is dominant in PbTe in strongly confined well geometries. These results are dramatically different from those in the CRTA and will be seen to have implications for PbTe superlattices with finite Vo, particularly for the dependence on Vo. Figure 11 shows P for PbTe [001] quantum wells with Vo = ~ and corresponding results for GaAs. Here, the plot extends over a wider range of well widths. For both cases, at large well widths, P approaches the bulk value. To obtain this limit correctly, many 2D subbands had to be included. For narrow wells, P for each material approaches a constant value, reflecting the dominance of the DP scattering as a ~ 0 for infinite potentials, as discussed above. For bulk GaAs, the DP scattering is much weaker than P O P scattering, and thus, the limiting value of P for small widths is larger. We now consider superlattices with finite potential barriers. Figure 12 gives the scattering rates for PbTe [001] quantum wells with several barrier heights and b - a - 50 A. With increasing barrier height, the scattering rate

144

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increases from the bulk value to that for Vo = oc. Figure 13 gives the corresponding mobility as a function of well thickness. For Vo = 200 meV, the effects of confinement are small and the mobility remains near the bulk value. For Vo = 1000 meV, the mobility is closer to that for Vo = oo for large well widths, but it approaches the bulk value for narrow wells where carrier tunneling becomes important. Figure 14 gives P for [001] q u a n t u m wells with b = a and increasing Vo. Note that qualitatively the dependence of P on a for finite Vo is similar to that for the C R T A in Figs. 5 and 6. However, an important difference is that the m a x i m u m P is not much affected by Vo. This point is shown in Fig. 15, where the m a x i m a in the curves of P vs a are given as functions of Vo. In the CRTA, these m a x i m a of P increase with a, whereas in the full calculations they are nearly independent of a. Physically we find that for increasing Vo, the increased scattering rates due to stronger carrier confinement (see Fig. 12) offset the density of states enhancement resulting in a weak dependence of P on Vo. This weak dependence of P on Vo can be traced to features of the c a r r i e r - p h o n o n scattering. The limiting value of P for a ~ 0 in the case Vo = oo and b = 0 in Fig. 11 is ~ 2, and it is traced to the dominant acoustic p h o n o n scattering in PbTe. This limit gives an effective upper bound on P in PbTe superlattices with finite Vo and nonzero b. F r o m Fig. 11, the small size limit of P in GaAs for Vo = oo and b -- 0 is greater than that for PbTe. This difference results from the greater importance of P O P scattering in

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GaAs. In general, we find that the dependence of P on Vo varies from system to system but that in all quantum well superlattices we have examined it is weaker than that predicted by less complete treatments. Figure 16 gives P for [111] quantum wells with b = a and increasing Vo. In contrast to the case for [001] wells, here P decreases with increasing confinement because of the lifting of the valley degeneracy giving fewer conducting subbands. The results for PbTe [001] and [111] superlattices in Figs. 14 and 16 show how important the bulk valley degeneracy is in determining P in semiconductor superlattices. Figure 17 gives P for PbTe [001] quantum wire superlattices. Here the two confinement directions are [001] and [010], and the transport is along [100]. For them, P increases with Vo, which is different from that for the [100] quantum well superlattices in Fig. 14. For the quantum wire case, substantial portions of the carrier wave functions extend into the barriers even for relatively strong confinement, thereby leading to lower scattering rates and to an increase of P with Vo. The dependence of maxima in P versus a is shown in Fig. 18. In contrast to the quantum well case, these P increase with increasing Vo in a similar way as those in the CRTA. For high Vo, P decreases approaching the result for Vo = ~ and b = a, as seen in Fig. 17. For quantum wire superlattices with Vo = ~ and b = 0, P is higher than that for finite barriers, but it remains finite for all a. This suggests that "free-standing" quantum wire systems should give high P and figures of merit ZT.

148

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BROIDO AND T. L. REINECKE

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2 THERMOELECTRICTRANSPORT

149

The [001] quantum wire superlattices above retain the bulk valley degeneracy on confinement. We have also considered PbTe wire superlattices where the valley degeneracy is lifted. They have confinement directions along [111] and a perpendicular direction: [112]. Results for them are qualitatively similar to those for [ 111] quantum well superlattices" For weak confinement, P remains close to the bulk value, while for strong confinement and b = a, the lifting of the valley degeneracy causes P to lie below the bulk value. Phonon scattering typically is the dominant carrier scattering at room temperature in semiconductor systems. Impurities are often introduced when doping carriers into the quantum well superlattices for thermoelectrics. Impurity scattering can be included straightforwardly in the treatment described earlier and we have done so for a variety of systems. We take the free carrier screening to be described by a 3D Thomas-Fermi dielectric function, and we calculate the quasi-2D scattering rate from the singly charged Coulomb centers. The free carrier screening is also included for polar optic phonons and acoustic phonons. In these calculations, we typically take the carrier density equal to the impurity density. Then, with increasing density, the increased impurity scattering reduces a and P, but the free carrier screening moderates this reduction. When impurity scattering and free carrier screening are included for PbTe quantum well superlattices, P remains essentially unchanged. This is because the dielectric constant in PbTe is large (~c0 ~ 400) and greatly reduces the impurity scattering rate because of efficient screening. Thus, results for PbTe when impurity scattering and free carrier screening are included are essentially the same as those in the preceding. We note that for materials with high static dielectric constants such as PbTe, modifaction in the impurity configuration (e.g., modulation doping as compared to uniform doping) should not enhance the room temperature transport coefficients for quantum wells because the effects of impurity scattering is small. The effects of impurities in quantum wire superlattices are greater than for quantum well systems. In strictly 1D systems, carrier screening is not possible, and the interaction of carriers with ionized impurities can lead to carrier localization. This should occur in very narrow wires with strong confinement such that only a single wire subband is occupied. Thus, methods for removing impurities from the narrow wires, such as modulation doping, are important to achieve high values for P.

IV.

Lattice Thermal Conductivity and the Figure of Merit

The room temperature lattice thermal conductivity ~:s of bulk semiconductors plays an important role in determining the figure of merit Z T [Eq. (1)]. Even in the bulk, this is a difficult quantity to calculate microscopically

150

D.A. BROIDOAND T. L. REINECKE

because the dominant scattering mechanism, anharmonic phonon-phonon scattering (Ziman, 1960), has a complicated form and the coupling coefficients are not known well. In quantum well and wire superlattices, further complications arise from interface scattering and from changes in the dispersion of acoustic phonons due to the superlattice structure. Studies of the thermal conductivity of quantum well systems have been made both experimentally (Yao, 1987; Capinski and Maris, 1996; Lee et al., 1997) and theoretically (Hyldgaard and Mahan, 1996, 1997; Chen, 1996; Walkauslas et al., 1999; Simkin and Mahan, 2000), and substantial decreases in ~cL have been found compared to the bulk value. Models have been introduced by several groups to try to understand the ~cL in these superlattice structures (Hyldgaard and Mahan, 1996, 1997; Chen, 1996; Walkauslas et al., 1999; Simkin and Mahan, 2000). We (Walkauslas et al., 1999) have used a simple model (Hyldgaard and Mahan, 1996; Fuchs, 1938; Dingle, 1950) to calculate the thermal conductivity free-standing quantum wells and wires. In this model, ~cL is obtained from the phonon Boltzmann equation in the relaxation time approximation for bulk phonon-phonon scattering and employing physically motivated boundary conditions for scattering at the interfaces. The interface scattering is taken to be either diffusive or specular (Ziman, 1960). A phonon undergoing diffusive scattering will emerge in a random direction and thus be in local equilibrium with the surface and will not contribute to transport. On the other hand, a phonon scattered specularly will emerge with the same velocity component along the well or wire and thus be unaffected by the interface. This approach is similar to that used to describe electrical conduction in thin films (Fuchs, 1938) and wires (Dingle, 1950). We have used two descriptions for the bulk phonon relaxation time z. In the first, z has a constant bulk mean free path, z = l/v, where v is the acoustic phonon velocity, and l is determined from the bulk thermal conductivity (Ziman, 1960). This form for z is often used in thermal conductivity calculations (Hyldgaard and Mahan, 1996, 1997; Chen, 1996; Walkauslas et al., 1999; Simkin and Mahan, 2000; Tien and Chen, 1994). In the second description, a frequency dependent relaxation time is used, z - A/co 2, where A is chosen from the bulk ~cL. This description has been used for both bulk materials (Callaway, 1959) and quantum wells (Ren and Dow, 1982) to model the third-order anharmonic phonon-phonon scattering, which is the scattering mechanism that is expected to dominate at room temperature. Figure 19 shows the calculated ~cL for free standing PbTe quantum wells and quantum wires as a function of the well-wire size using purely diffusive interface scattering. The dashed curves are for the constant mean free path model, while the solid lines are for the frequency-dependent relaxation time model. It is evident that ~cL for PbTe wells and wires decreases substantially below the bulk value for decreasing well and wire thickness. This results

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from the increased boundary scattering for decreasing size. The reduction is larger in wires because of their larger surface to volume ratio. Within this model, KL goes to zero as d ~ 0. For small enough sizes, a more microscopic treatment is required and would include such effects as disorder-induced localization of the phonons. Further, changes in the phonon dispersion might have to be included. The effects of these changes have been found to be quite important for transport along the growth direction in superlattices (Simkin and Mahan, 2000). Such calculations have not yet been done for transport in the plane of a quantum well or along quantum wires, nor for the free standing structures considered here. To illustrate the potential effects of these large changes in ~cL, in Fig. 20 we plot Z T at room temperature for free standing PbTe quantum wires as a function of wire size. The power factor is chosen from the earlier [001] PbTe wire results for Vo = ~ and b = 0. The dashed curve is obtained using the bulk ~cL. The solid curve is for free-standing wires obtained using the calculated ~L shown in Fig. 19 for the frequency dependent relaxation time model. The large values of Z T for small well and wire widths are mostly a consequence of both the large reduction in KL due to the interface scattering and the enhancement of the power factor for decreasing well and wire

152

D.A. BROIDOAND T. L. REINECKE '

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thickness. The large increase in Z T for the quantum wire systems shown here compared to bulk suggests that such systems may show promise for achieving high ZT.

V.

Summary

In this chapter, we described a theoretical treatment of the electronic contribution to thermoelectric transport in superlattices that has been developed over 5 years. This work clarified the physical ingredients needed to understand the power factor. It gives a reliable description of this quantity, of its essential qualitative dependences on both structure and potential, and it also provides fully quantitative results. This understanding will guide choices of systems that have the greatest potential for high power factors, and it also makes possible quantitative predictions for specific systems. We noted that many of the essential qualitative featfures of the power factor can be understood on the basis of a relatively straightforward description based on the constant relaxation time approximation for carrier scattering but that fully quantitative results require a theory based on an inelastic carrier scattering treatment. We have also noted that an understanding of the lattice thermal conductivity of these systems is needed in order to describe their figures of merit and that progress is now being made in this direction.

2

THERMOELECTRICTRANSPORT

153

ACKNOWLEDGMENTS

This work was supported in part by the U.S. Office of Naval Research.

REFERENCES Ashcroft, N. W., and N. D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976), Chap. 13. Broido, D. A., and T. L. Reinecke, Effects of superlattice structure on the thermoelectric figure of merit, Phys. Rev. B 51, 13797 (1995a). Broido, D. A., and T. L. Reinecke, Thermoelectric figure of merit of quantum wire superlattices, Appl. Phys. Lett. 67, 100 (1995b). Broido, D. A., and T. L. Reinecke, Comment on "Use of quantum well superlattices to obtain high figure of merit from non-conventional thermoelectric materials," Appl. Phys. Lett. 67, 1170 (1995c). Broido, D. A., and T. L. Reinecke, Thermoelectric transport in semiconductor superlattices, Appl. Phys. Lett. 70, 2834 (1997). Broido, D. A., and T. L. Reinecke, in Thermoelectric transport in superlattices, in Thermoelectric Materials 1998, The Next Generation of Materials for Small-Scale Refrigeration and Power Generation Applications, ed. T. M. Tritt, M. G. Kanatzidis, G. D. Mahan, and H. B. Lyons, 1999, p. 485. Broido, D. A., and T. L. Reinecke, Thermoelectric power factor of superlattices (forthcoming a). Broido, D. A., and T. L. Reinecke (forthcoming b). Callaway, J., Model of lattice thermal conductivity at low temperatures, Phys. Rev. 113, 1046 (1959). Capinski, W. S., and H. Maris, Thermal conductivity of GaAs/A1As superlattices, Physica B 219 and 220, 699 (1996). Chen, G., Size and interface effects on thermal conductivity of superlattices and periodic thin-film structures, in Proc. National Heat Transfer Conference, HTD-Vol. 323, p. 121, 1996. Dingle, R. B., The electrical conductivity of thin wires, Proc. R. Soc. London, Ser. A 201, 545 (1950). Friedman, L., Thermopower of superlattices as a probe of the density of states distributions, J. Phys. C 17, 3999 (1984). Fuchs, K., The conductivity of thin metallic films according to electron theory of metals, in Proc. Cambridge Philos. Soc. 34, 100 (1938). Goldsmid, H. J. Thermoelectric Refrigeration (Plenum, New York, 1964). Harman, T. C., D. L. Spears, and M. J. Manfra, High thermoelectric figures of merit in PbTe quantum wells, J. Electron. Mater. 25, 1121 (1996). Hicks, L. D., and M. S. Dresselhaus, Effect of quantum well structure on the thermoelectric figure of merit, Phys. Rev. B 47, 12727 (1993a). Hicks, L. D., and M. S. Dresselhaus, Thermoelectric figure of merit of a one-dimensional conductor, Phys. Rev. B 47, 16631 (1993b). Hicks, L. D., T. C. Harman and M. S. Dresselhaus, Use of quantum well superlattices to obtain a high figure of merit from nonconventional thermoelectric materials, Appl. Phys. Lett. 63, 3230 (1993). Hicks, L. D., T. C. Harman, X. Sun and M. S. Dresselhaus, Experimental study of the effect of quantum-weU structures on the thermoelectric figure of merit, Phys. Rev. B 53, 10493 (1996).

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D . A . BROIDO AND T. L. REINECKE

Huber, T. E., M. J. Graf and C. A. Foss, in Thermoelectric Materials 1998--The Next Generation Materials for Small-Scale Refrigeration and Power Generation Applications, eds. T. M. Tritt, M. G. Kanatzidis, G. D. Mahan, and H. B. Lyons, 1999, p. 227. Hyldgaard, P., and G. D. Mahan, Phonon Knudson flow in GaAs/AIAs superlattices, in Thermal Conductivity 23 (Technomic Publishing Co. Inc., Lancaster, PA, 1996), 172. Hyldgaard, P., and G. D. Mahan, Phonon superlattice transport, Phys. Rev. B 56, 10754 (1997). Kittel, C., Introduction to Solid State Physics, 6th ed. (Wiley, 1986), Chap. 7. Knipp, P. A., and T. L. Reinecke, Effects of boundary conditions on confined optical phonons in semiconductor nanostructures, Phys. Rev. B 48, 18037 (1993). Koga, T., T. C. Harman, S. B. Cronin, and M. S. Dresselhaus, Mechanism of the enhanced thermoelectric power in (111)-oriented n-type PbTe/Pb 1_xEuxTe multiple quantum wells, Phys. Rev. B 60, 14286 (1999). Landholt and B6rstein, Numerical Data and Functional Relationships in Science and Technology, New Series, Vol. 17e (Springer-Verlag, Berlin, 1982a), pp. 218-258. Landholt and B6rstein, Numerical Data and Functional Relationships in Science and Technology, New Series, Vol. 17f (Springer-Verlag, Berlin, 1982b), pp. 170-180. LeBurton, J. P., Size effects on polar optical phonon scattering of 1-D and 2-D electron gas in synthetic semiconductors, J. Appl. Phys. 56, 2850 (1984). Lee, S. M., D. G. Cahill, and R. Venkatasubramanian, Thermal conductivity of Si-Ge superlattices, Appl. Phys. Lett. 70, 2957 (1997). Lin-Chung, P. J., and T. L. Reinecke, Thermoelectric figure of merit of composite superlattice systems, Phys. Rev. B 51, 13244 (1995). Mahan, G. D., and H. B. Lyon, Thermoelectric devices using semiconductor quantum wells, J. Appl. Phys. 76, 1899 (1994). Mahan, G., B. Sales and J. Sharp, Thermoelectric materials: new approaches to an old problem, Physics Today 50, 42 (1998). Nag, B. R., Electron Transport in Compound Semiconductors (Springer-Verlag, Berlin, 1980). Nimitz, G., and B. Schlicht, Narrow-Gap Semiconductors (Springer-Verlag, Berlin, 1982), pp. 1-106. Price, P. J., Two-dimensional electron transport in semiconductor layers, Ann. Phys. (NY) 133, 217 (1981). Reinecke, T. L., and D. A. Broido, Thermoelectric transport in superlattices, Proc. MRS Res. Soc. 478, 161 (1997). Reinecke, T. L., and D. A. Broido, Thermoelectric transport in superlattices, in Proc. of the International Conference on Thermoelectrics, Dresden, 1997 (IEEE, Piscataway, NJ, 1998), p. 424. Ren, S. Y., and J. D. Dow, Thermal conductivity of superlattices, Phys. Rev. B 25, 3750 (1982). Riicker, H., E. Molinari and P. Lugli, Microscopic calculation of the electron-phonon interaction in quantum wells, Phys. Rev. B 45, 6747 (1992). Simkin, M. V., and G. D. Mahan, Minimum thermal conductivity of superlattices, Phys. Rev. Lett. 84, 927 (2000). Sofo, J. O., and G. D. Mahan, Optimum band gap of a thermoelectric material, Phys. Rev. B 49, 4565 (1994a). Sofo, J. O., and G. D. Mahan, Thermoelectric figure of merit of superlattices, Appl. Phys. Lett. 65, 2690 (1994b). Sun, X., Z. Zhang, and M. S. Dresselhaus, Theoretical modeling of thermoelectricity in Bi nanowires, Appl. Phys. Lett. 74, 4005 (1999). Tao, T., and L. Friedman, Thermoelectric power factor of superlattices II, J. Phys. C 18, L455 (1985). Tien, C. L., and G. Chen, Challenges in microscale conductive and radiative heat transfer, J. Heat Transfer 116, 799 (1994), and references therein.

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Tsuchiya, T., and T. Ando, Mobility enhancement in quantum wells by electronic-state modulation, Phys. Rev. B 48, 4599 (1993). Walkauskas, S. G., D. A. Broido, K. Kempa, and T. L. Reinecke, Lattice thermal conductivity in quantum wires, J. Appl. Phys. 85, 2579 (1999). Yao, T., Thermal properties of A1As/GaAs superlattices, Appl. Phys. Lett. 51, 1798 (1987). Zhang, Z., M. S. Dresselhaus, and J. Y. Ying, in Thermoelectric Materials 1998--The Next Generation Materials for Small-Scale Refrigeration and Power Generation Applications, ed. T. M. Tritt, M. G. Kanatzidis, G. D. Mahan, and H. B. Lyons, 1999, p. 351. Ziman, J. M., Electrons and Phonons (Oxford University Press, Oxford, 1960). Zook, J. D., Piezoelectric scattering in semiconductors, Phys. Rev. 136, A869 (1964).

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SEMICONDUCTORS AND SEMIMETALS, VOL. 71

CHAPTER

3

Thermionic Refrigeration G. D. Mahan SOLID STATEDIVISION OAK RIDGE NATIONALLABORATORY OAK RIDGE, TENNESSEE

I. I N T R O D U C T I O N

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

II. VACUUM DEVICE

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

III.

ONE-BARRIER SOLID-STATE DEVICE

IV.

MULTILAYER DEVICES

g.

W H Y BALLISTIC 9.

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

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

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

VI. DISCUSSION

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

REFERENCES

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

I.

157 160 163 166 170 172 172

Introduction

Thermionic refrigeration is a solid-state refrigerator based on a different principle than thermoelectric refrigeration. The name arises from thermionic emission, which is the thermal excitation of hot electrons from a metal surface. Metals have an energy difference between the chemical potential and the vacuum energy, which is called the work function edp. Richardson's equation describes the current from such a surface with a temperature T

JR(T) = ATZexp I - ek-~T]

(1)

where kB is Boltzmann's constant. The constant A is given by fundamental constants and has a numerical value close to A = 120A/K 2 cm 2. Oldfashioned electronic amplifiers used radio tubes, where the electrons in the gaseous parts of the tube were generated by thermionic emission. In the 1950s, thermionic emission was proposed as a power generator (Hatsopoulos and Kaye, 1958; Houston, 1959). Two metal plates, separated by a vacuum, could be at different temperatures. More electrons flow from hot to cold than from cold to hot. The cold cathode charges up, which creates a voltage difference between the two electrodes. This voltage difference can 157 Copyright 9 2001 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-752180-1 ISSN 0080-8784/01 $35.00

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G.D. MAHAN

be made to do work through an external circuit. The operation is similar in practice to a solar cell, with an open circuit voltage, and so on. They were used as power supplies in some Soviet space stations. The efficiency of the power generation is increased as the work function is lowered. In practice, the metal surfaces were cesiated to lower their work function. The space between the electrodes is then filled with cesium vapor. The cesium molecules tended to be ionized by the electron currents, so that the space became a plasma. This detrimental feature reduced the efficiency of the power generator. The situation could be avoided by reducing the voltages, which made the device less efficient (Scott, 1981). The concept of thermionic refrigeration was proposed by Mahan in 1994. Given two metal plates, at different temperatures, applying a voltage to encourage electrons to flow from cold to hot through the vacuum gap causes the cold electrode to become colder. The two plates are connected by a battery, which drives the current between the electrodes and returns electrons to the cold side at the energy of the chemical potential. The returned electrons carry negligible net heat. One has created a device with a legitimate Maxwell demon: two metal plates separated by an air or vacuum gap serves as an energy filter so that only the hottest electrons carry heat over the work function and between the metal plates. With a suitable value of the work function eq5 the efficiency of the thermionic refrigerator is quite high: higher than any other known refrigerator. The relevent value of the work function can be derived from Eq. (1). A refrigerator needs to have a cooling power of about 10 W / c m 2 in order to overcome thermal losses. Since voltages tend to be of the order of 1 V, the current densities need to be of the order of J ~ 10A/cm 2. Setting T = 300 K, in Eq. (1) the only unknown is the work function. Solving gives a value of etk = 0.3 eV. Our modeling, which is described in the following, shows that a pair of electrodes with this work function value would have an efficiency at room temperature that is much higher than a freon-based compressor. One advantage of a vacuum gap refrigerator is that thermal losses are mainly due to radiation, which is a small but not negligible effect. Unfortunately, there is no known material with this low work function value. There has been a hundred year search for a material with a low work function value, and none has been found. The lowest value reliably reported at room temperature is 0.9 eV, which is silicon with cesium oxide on the surface (Levine, 1973). Langmuir (1923) showed that a low value can not be attained except if the space gap is very narrow. As the work function becomes smaller, the electrons are thermally excited from the electrodes into the vacuum region. They form a plasma, which creates its own potential barrier. The barrier depends on the distance between the electrodes (Mahan, 1994). At room temperature, for a gap of 1 mm, the plasma barrier is 0.7 eV even if the work function is zero. The plasma barrier becomes smaller as the plate separation is reduced. In order to get down to 0.3 eV, the gap must be

3 THERMIONICREFRIGERATION

159

in the range of a micrometer. This plasma barrier has been discussed by Mahan for thermionic refrigeration (1994). Huang and Dye (1990) reported a work function of 0.4 eV in an organic material at T = - 8 0 ~ The reported currents were only picoamps per centimeter squared, and the current lasted only a few seconds from a fresh surface. A later remeasurement showed the work function is actually 0.7 eV, and the observed current was from impurities on the surface; it was not actually current from the bulk of the material. The organic material also degrades within an hour of being made. It does not seem to be a suitable material for the electrodes of a thermionic refrigerator. Nevertheless, papers continue to be written on the rosy prospects of this electrode, and its use in refrigeration has been patented (Edelson, 1997). Thermionic refrigeration is an example of evaporative cooling. Any system will cool if its most energetic particles are removed regularly. Examples are wind flowing by a wet blanket, and pumping away the vapor of liquid 4He. In both cases, the most energetic atoms (He) or molecules (H20) have enough energy to escape from the fluid to become a gas atom or molelcule. Pumping the gas particles away does not allow them to return to the fluid, and the fluid cools. Another example of evaporative cooling are quantum dot refrigerators, which cool below 1 K (Edwards et al., 1993, 1995; Nahum et al., 1994; Leivo et al., 1996). Thermionic refrigeration is an evaporative cooler for room temperature applications. The present development of the thermionic refrigerator is as an all-solidstate device. At the interface between two different materials there is often an energy barrier to the flow of electrons. These "non-ohmic" contacts are usually a nuisance, but here they are potentially useful. Typical examples are the band offsets between different semiconductors, or the Schottky barrier between a metal and a semiconductor. Barriers of the order of 0.3 eV are quite typical, so that internal thermionic emission might make a practical device. In fact, following Mahan's original suggestion of a vacuum device (Mahan, 1994), there were numerous suggestions of periodic barriers in a multilayer solid as a means of filtering all but the energetic electrons (Rowe and Min, 1994; Whitlow and Hirano, 1995; Bogomolov et al., 1995; Moyzhes, 1996; Shakouri and Bowers, 1997; Mahan and Woods, 1998; Mahan et al., 1998; Shakouri et al., 1998, 1999; Zhou et al., 1999; Nolas and Goldsmid, 1999). These qualitative proposals are similar to the devices that are presently being constructed and tested. The great liability of using solid-state barriers between the conducting electrodes is that the thermal conductivity of the barrier allows vibrational heat to flow from the hot to the cold side of the refrigerator. These thermal losses are much larger than those for radiative heat transfer and they completely change the modeling of the thermionic refrigerator. Very large current densities are required to overcome these heat losses due to phonons. From Richardson's equation, large currents are attained only by small work

160

G.D. MAHAN

functions. The modeling described in the following suggests that internal work functions are needed that are only two to four times the thermal energy, or 50-100 meV. Such small barriers are still attainable at solid-state interfaces. The thermal losses are reduced by having the thermal conductivity of the barrier be as small as possible. There has been much recent work on the thermal conductivity of semiconductor superlattices for conductivity along the layers, and also perpendicular to the layers (Chen et al., 1994; Capinski and Maris, 1996; Lee et al., 1997; Venkatasubramanian et al., 1999; Hyldgaard and Mahan, 1997; Chen and Neagu, 1997; Tamura et al., 1999; Simkin and Mahan, 2000). Since band offsets would form periodic barriers to the flows of electrical current perpendicular to the barriers, this geometry is now investigated for the relevance to thermionic refrigeration. Measurements of thermal conductivity of superlattices, perpendicular to the layers, show that the value can be less than the random alloy of the materials in the superlattice. Multiple interfaces act as a phonon filter, similar to the Fabry-P6rot in optics. Theory and experiment also show that there is a minimum in the thermal conductivity when it is plotted vs superlattice period (Venkatasubramanian et al., 1999; Simkin and Mahan, 2000). Very small values of the thermal conductivity are required to make the thermionic refrigerator relatively efficient. The minimum in the thermal conductivity, as a function of superlattice period, occurs at barrier thicknesses of the order of 3-5 nm. These are too narrow for use in a thermionic refrigeration. The low values of thermal conductivity of superlattices may not benefit the thermionic refrigerator, which requires thicker barriers.

II.

Vacuum Device

The original conception of the device was for two parallel metal plates separated by a vacuum or air gap (Mahan, 1994). The plates are at different temperatures, which are called the hot (temperature Th) and the cold (temperature T~) electrodes. An energy level diagram for refrigeration is shown in Fig. 1. Each metal electrode has a horizontal line that represents the chemical potential /~h,Pc. There is an applied voltage e V a = l a c - Ph, which raises the chemical potential of the cold side above that of the hot size. The work function e~b is assumed to be the same for the two metal electrodes. The refrigerator behaves similarly even when the work functions are different. The applied potential Va has the polarity to draw electrons from the cold electrode to the hot electrode. Any electron with enough kinetic energy to surmount the barrier gets drawn away from the cold surface to the hot electrode. In contrast, for an electron to go from the hot to the cold surface, it must surmount the barrier of e(4~ + Va). In this

3 THERMIONICREFRIGERATION

Th

t

t

161 Te

e~

e~

FIG. 1. Energylevel diagram for vacuum thermionic refrigerator.

geometry, the net current density is J(Va) -- JR(Tc) - Jg(Th)e-eV,,/k, Th

(2)

Note J R ( T ) is an increasing function of T. Positive current is defined as going from cold to hot. At zero applied voltage, then J ( V a = 0) is negative since J n ( T h ) > JR(Tc) if Th > Tc. However, the application of a suitable voltage reduces the second term, and one can make a net current go from the cold electrode to the hot electrode. The power required is P = J(Va)Va. Electrons carry heat. Denote the heat current as J e , which has the units of watts per centimeter squared. M a h a n (1994) showed that eJo. = [ed? + 2kBT~]JR(Tc) -- [edp + 2kaTh]JR(Th)e -eva/knT" + a[Tc 4 -- Th4]

(3) Here tr is the S t e p h a n - B o l t z m a n n constant, and the last term is the radiative transfer of energy from the hot to the cold electrode. The first two terms are the net heat current due to thermionic emission. This term must be larger than the radiative losses to get any net cooling. One way to understand this expression is to examine how it behaves in the limit that the temperature difference AT = Th - Tc is small. Define T = ( T h + Tc)/2 as the average temperature. Then a small a m o u n t of algebra enables one to write the preceding expression as (k e - ka/e ) Jo. = [~ + 2 k e T - ' ] J - A T [ k B J R ( T c ) + kBJg(Th) e-eva/k"T" + 4a7"3]

(4)

The thermoelectric Boltzmann constant k e = kn/e = 86.17/W/K. The expression is very analogous to the heat current in a thermoelectric device

162

G.D. MAHAN

(Goldsmid, 1986; Mahan, 1989, 1998; Mahan et al., 1996) of length L K

(5)

Jo. = S T J - ~ A T

Comparing these two expressions, the thermionic refrigerator has an effective Seebeck coefficient S and thermal conductivity K of (Mahan et al., 1998) S = ke[2 + b],

b-

ec~ kaT

K = L[kBJR(Tc) + kBJR(Th)e -ev~

(6) + 4 a 7 "3]

(7)

These concepts are useful and are used in the following. For the modeling of the vacuum device, however, we employ the full expressions in Eq. (3). The Carnot efficiency of a refrigerator is defined as the heat leaving the cold electrode JQ divided by the input power

.(vo) = Jo(V.) J(Va)Vo

(8)

Figure 2 shows the efficiency as a function of the applied voltage Va. Selected values are Th = 300 K, T c = 260 K, and e~b = 0.30 eV. These values give that AT = 40 K and T = 280 K. The maximum values are above 5.0. The maximum efficiency allowed by a perfect Carnot refrigerator is qc = T ~ / A T = 6.5 for these values. Any value less than 6.5 does not violate the second law of thermodynamics. The household refrigerator based on a freon compressor has a maximum efficiency of 1.5, while commerical thermoelectric devices have an efficiency of 0.7. These values depend on the preceding choice of (Th, T~). The thermionic refrigerator has a maximum efficiency of about 5, which is thrice that of a freon compressor. The curve is not shown at the smallest values of Va since it goes through gyrations as J and Je change sign at different values of V,. The calculated curve is unrealistic since no material has a work function as small as 0.3 eV. If we use the smallest known value, e#) - 0.9 eV, then there is no cooling since the radiative losses are larger than the thermionic energy currents. The graph of efficiency vs voltage rises sharply to its maximum value, and then declines slowly with increasing voltage. Note that the current J and the heat current JQ saturate at high voltage, when e V~/kaTh > 1, and the current from the hot side becomes negligible. In this limit, the efficiency rI ~ JQ(OO)/[VaJ(oo)], which is the hyperbolic shape apparent in Fig. 2. It is assumed that electrons traverse the barrier region without scattering. The only scattering in a vacuum device is due to electron-electron interactions with the electrons in the electrodes. These interactions cause plasmon m

3 THERMIONICREFRIGERATION

0.0

i

|

0.1

0.2

163

0.3

V(eV) FIG. 2. Efficiency rl(Va) of a v a c u u m - g a p thermionic refrigerator for the operating conditions that Th = 300 K, T~ = 260 K, and e~b = 0.30 eV. This curve is unrealistic since no material has a work function as small as 0.3 eV.

modes in the bulk of the electrodes and surface plasmons outside of the surface of the electrode. Two parallel electrodes have coupled surface plasmons, of which one mode extends to small values of frequency. These low-frequency modes serve as bosons, which could inelastically scatter electrons while they are in the vacuum region between the two electrodes. Our calculations show that this process has a very small probability. The result is small because the electrons have a small kinetic energy that is within kBT of the vacuum. There is not much phase space accessible for scattering. Ballistic transport between the electrodes is a good model for the vacuum device.

III.

One-Barrier Solid-State Device

The lack of electrodes with small work functions makes the vacuum device unworkable. Small energy barriers are available in devices that are all solid state. Next, consider a solid-state thermionic refrigerator composed of three solid layers: metal-semiconductor-metal. The Schottky barrier at the metal-semiconductor interface provides the "internal work function," which is a barrier to the flow of electrons. We assume the electron can

164

G.D. MAHAN

traverse the semiconductor barrier region without scattering, and the ballistic transport model is valid. In the following, we discuss whether this assumption is achievable. For ballistic transport, the same equations can be used as for the vacuum device. The only difference is that now we must include the heat flow by phonons from the hot side to the cold side. This heat flow is large and alters the design of the device. The radiative and phonon heat flows, from hot to cold, are given by

KAT

0 = o[Th 4 -- T~4] - . I . - ~ L

(9)

where L is the thickness of the semiconductor barrier. A typical value of L is 1 #m. The thermal conductivity K is not just that of the barrier material, but also includes the thermal boundary resistance at the interfaces between the metal and the barrier. Using a small value of thermal conductivity K = 1 W / ( m K ) and A T - 4 0 K , the heat conduction by phonons is 4000 W/cm 2, which is quite large! In contrast, the radiative heat flow is only 0.02 W/cm 2, which is negligible by comparison. The radiative heat flow will be neglected compared to the phonon heat flow. The energy current becomes K

eJo = [edp + 2k~T~]Jg(Tc) - [edp + 2kBTh]JR(Th)e -ev"/k"Th- - - A T L

(10)

The large value of the phonon heat flow makes the solid-state device less efficient compared to the vacuum device. It also completely changes the optimal value of the internal work function e~b. The preceding equations can be used to determine the efficiency of a single-barrier solid-state device. The efficiency in Eq. (8) is evaluated using the energy current in Eq. (10). Figure 3 shows the result as a function of the applied voltage V, for several different values of barrier height e~b. Other values used in this figure are Th - 300 K, T~ = 260 K, K = 1 W/(m K), and L = 1.0/tm. The efficiencies are less than the vacuum device, but greater than that of a freon compressor. One result that is apparent from this figure is that the useful values of the internal work function (b = ec/)/kBT ) are around 2-4. So the best internal work functions at room temperature are in the range of 50-100 meV. This small value produces a very large current density and a very large energy current density, which is required to overcome the large backflow of heat due to phonons. The phonon heat flow is controlled by the ratio K/L. In the present modeling the phonon heat flow reduces the maximum efficiency (COP) by about 20% compared to the result obtained if K = 0. Raising the value of K / L a factor of 10 means that the thermionic refrigerator could not work.

3 THERMIONICREFRIGERATION

165

4.0 3.5 3.0 2.5 ~"

2.0 1.5 1.0 "

0.5 0.0 0.0

// I/ y // / I/ ,,Ill

!

0.1

0.2

Va(Volts) FIG. 3. Efficiency of thermionic refrigerator with a single barrier. Results for different barrier heights ~b = 0.050, 0.075, and 0.100 V as a function of applied voltage Va. Highest efficiency is from smallest ~b. Increasing the work function reduces the maximum efficiency.

The thermionic cooling is less than the backflow of heat from the phonons. The requirement of ballistic transport controls all of the device characteristics. This requirement keeps L small, so that the phonon heat flow (KA/L) is large. To counter this large backflow, very large thermionic currents are required, which demands small barrier heights. The reason for ballistic transport is discussed in the next section. The value we selected for the thermal conductivity K is typical of a good thermoelectric material such as alloys of Bi2Te 3. The layer thickness we selected (L = 1.0/~m) is typical of the electron mean free path in a narrowband semiconductor. These values reflect our requirements to have barrier materials with a small value of thermal conductivity and a large value of the electron mean free path. The latter enables large L. These two requirements are needed to have small values of K/L. The barrier material must be a poor conductor of phonon heat and a good conductor of electrical current. The same two requirements are needed in good thermoelectric materials. Glen Slack has called these twin requirements: "phonon glass and electron crystal." A list of the best possible barrier materials is identical to the list of good thermoelectric materials. An additional requirement for the thermionic refrigerator is that the metal-semiconductor Schottky barrier must have a value of the order of ~b = 0 . 0 5 0 - 0.100eV. A literature review shows that almost no Schottky barrier measurements have been reported on these

166

G.D. MAHAN

classes of materials. Values of Schottky barriers are usually about one-third to one-half of the energy gap of the semiconductor barrier. Small values of the Schottky barrier will be found in materials with small energy gaps. This criteria is again similar to the selection of thermoelectric materials, which also have small energy gaps. Here is another reason that the choice of barrier material is confined to the list of good thermoelectrics.

IV.

Multilayer Devices

There are several reasons to consider whether the thermionic refrigerator is more efficient as a multilayer device. The first is that a single-barrier device is relatively more efficient for small temperature differences of 6T ~ 1 K per barrier. By "relative efficiency" we mean compared to the Carnot limit of qc = Tc/AT. A large temperature difference A T ~ 40 K can be attained by having N layers, where N is large enough to get macroscopic cooling. The efficiency of a multilayer device can not be calculated by just scaling up the result for a single barrier. It is important to include the Joule heating of the electrons in the middle of the device, and to allow for the flow of this heat out of the ends. The second reason for using multilayers is that the thermal conductivity is very small in these nanostructured devices. This feature is discussed in the following. The device consists of alternate layers of a conductor and a barrier. The barrier layers are semiconductors. The conducting layer could be either a metal or a semiconductor. In the latter case, the multilayer device is a semiconductor superlattice. It is assumed the electrons traverse the barrier layers ballistically, and then thermalize in the next conducting layer. If the applied voltage acrossed barrier j is Vj, then the electron gains an energy e Vj traversing the barrier. This energy is converted to phonon energy by thermalization, which is the source of the Joule heating. This heat flows out the ends of the device by phonon transport. The voltage drop Vj across a single barrier is small if 6Tj is small. The transport equations can be expanded in powers of the smallness parameters 6TjT, Vjke T, where T is the average temperature in the barrier. The currents are assumed to be one dimensional. This expansion gives (Mahan et al., 1998)

j _ e JR(T) kB T [ V j - ke6Tj(b + 2)] JQ = keT(b + 2)J - 6 T j

2keJ a +

(111 (12)

These equations have the same mathematical form as the thermoelectric

3 THERMIONICREFRIGERATION

167

equations. The various coefficients provide a thermoelectric analogy to thermionic multilayers (Mahan et al., 1998) JRLi

= ~

(13)

S = ke(b + 2)

(14)

K = Kp + K e

(15)

K e = 2JRkeL J = 2 a T k 2

(16)

keT

The parameters are electrical conductivity a, Seebeck coetficient S, and thermal conductivity K e. The thermionic thermal conductivity K e obeys a Wiedemann-Franz law that relates it to the electrical conductivity. The advantage of the analogy to thermoelectrics is that one can now use the vast knowledge of thermoelectrics to do modeling of the thermionic multilayer refrigerator or generator. In thermoelectrics, it is known that the efficiency of the solid-state refrigerator or generator depends on the dimensionless figure of merit called z = ZT. The goal of material science is to make z be as large as possible, since the efficiency of the solid-state refrigerator and generator increase with increasing z. The figure of merit for thermionics is

2TI

=

Bri =

aTS 2 ke(b + 2)2JRLj Kp + K e = Kp h- 2 J g k e L j

(b + 2) 2

(b + 2) 2

2 + Kp/(KeJRLj)

2 + eb/Bri

k eA T 2 Lj Kp

(17) (18) (19)

In deriving this expression, we have used JR = A T 2 exp(-b). The parameter Brt is dimensionless. At room temperature T = 300 K, and Richardson's constant is A = 1.2 MA/m 2. Taking a typical barrier thickness as L = 1/tm, and phonon thermal conductivity as Kp = 1.0 W/(m K), then we get that B T I = 9.3 (1/Lm) (\ w / (Kp m K))

(20)

Convert this value into a figure of merit. We optimize ZTI in Eq. (18) by differentiating with respect to b and setting the derivative equal to zero. This gives the nonlinear equation for the maximum figure of merit as bo eb~ = 4BT,

(21)

168

G.D. MAHAN TABLE I DIMENSIONLESS FIGURE OF MERIT FOR A THERMIONIC MULTILAYER REFRIGERATORAS A FUNCTION OF BTI

BTI

bo

ZTI

0.1 0.3 0.6 1.0 2.0 3.0

0.30 0.64 0.94 1.20 1.61 1.86

0.34 0.84 1.38 1.92 2.90 3.60

Then we can calculate b o and ZTi(bo) for various values of BTI as shown in Table I. There are several important results in this table. Present thermoelectric devices have a dimensionless figure of merit about 1 at room temperature. Values as high as 2 have been attained at higher temperatures. A value of zrt = 4 would revolutionize refrigeration, while a value of zTt = 2 at room temperature would enable many new uses for solid-state refrigeration and power generation. For example, solid-state power generators using waste heat from power plants become feasible as a means of adding 5% more power to a plant with small capital investment. All of this requires a Brx value of about 1-3. Is that attainable? According to Eq. (20), such a value can be attained if L = 100 nm and Kp= 1 W/(m K), or else L = 25 nm and K p - 0 . 2 5 W/(m K). The other important result in Table I is that the factors of bo are small. Recall that b = eck/k~T. So if bo ~ 1 then e~ ~ kBT. The barrier heights are the same size as the thermal energy for the optimal operation of the multilayer thermionic refrigerator. The B-factor was originally introduced into the thermoelectic field by Chasmar and Stratton (1959) (they called it fl). Vining and Mahan (1999) noted that the figure of merit for thermoelectrics has an equation similar to that for thermionics (2 - r/)2 ZTE ~-

2+

mkB(ksT)22T BTe= 2n3/2hZKv '~T = "r2VT -- ~.~

BTZ =

(22)

e-'I/BTe

/2k~T m

mkB(kBT)2L

2nzhaK p

(23)

(24)

(25)

3 THERMIONICREFRIGERATION

169

We have rewritten the expression for BTI. The different appearance from our earlier expression is that we have written out the expression for Richardson's constant A = emk2/(2~z2h3). The factor of t / = la/kBT is the chemical potential of the semiconductor divided by the thermal energy. In writing the above equation, we have assumed that the chemical potential is below the conduction band edge so that r / < 0. In that case the replacement r/---, - b makes the two expressions identical. They are further identical in that r/is a parameter that can be adjusted by the experimentalist by doping the semiconductor. It can also be varied to attain the highest figure of merit. Note that both - r / a n d b are the energies we need to get from the chemical potential into the conducting region. Therefore it is reasonable to ask whether thermionics or thermoelectrics has the largest value of B? The highest efficiency is obtained by the largest value of B. Note that the two expressions have many of the same parameters. Their ratio is

BTI = ~ L

(26)

The requirement of ballistic transport means that L < 2 r so that BTI < BTE. A thermoelectric device is more efficient than a thermionic device if both have the same value of lattice thermal conductivity Kp. Again the requirement of ballistic transport seems to be the limiting factor on the efficiency of the thermionic device. A big advantage of using superlattices is they may have a very low value of phonon thermal conductivity for transport perpendicular to the layers. A low value of Kp directly increases the B-factor. A superlattice is also a possible design of a multilayer thermionic device that can take advantage of the low value of thermal conductivity. However, the preceding argument shows that there is even a bigger advantage of making a thermoelectric device using a superlattice. One might think that the band offsets impede electrical transport perpendicular to the layers. In most cases, that seems to be the case. There is one important exception. Venkatasubramanian et al. (1999) have used molzcular beam epitaxy to grow multilayers of Bi2Te 3 and Sb2Te 3 and their alloys. They have prepared samples with many different superlattice widths, and find that the thermal conductivity has a minimum for a layer width around 2-3 nm. A theory of Simkin and Mahan (2000) predicts a minimum at exactly the same value. At this minimum the thermal conductivity has a value of Kp = 0.25 W/(m K) and the value of ZTE ~ 3. These are the largest room temperature values ever reported. An interesting feature of this particular superlattice is that it is electrically isotropic. The host materials, BizTe 3 and Sb2Te3, are naturally layered and are anisotropic. The superlattice is more isotropic than the pure materials or their alloys. No band offsets have been detected, and they seem to be thermoelec-

170

G.D. MAHAN

tric devices. Note that a thermionic device is unable to take advantage of this low value of thermal conductivity, since the conductivity minimum occurs at a barrier thickness L ~ 2 - 3 nm, which is small and causes Br~ to be small. Our conclusion at this time is that the BizTea-SbzTe 3 superlattice has an effective value of zre that is larger than is going to be attained by any thermionic device. However, one should only believe the theory so far, and the experimentalist should keep exploring thermionic devices.

V.

Why Ballistic?

The preceding discussion always assumed that the thermionic device worked by having electrons ballistically traverse the barrier region. If L is the barrier thickness and 2 r is the mean free path of the electron, then the ballistic transport is defined as the case where L < 2 r. The other limit is where L >> 2 r. In this case, the barrier is long compared to the mean free path of the carriers. The material acts just like an ordinary thermoelectric material. The thermionic phenomena only influences the boundary conditions at the ends of the thermoelectric material. Mahan (2000a) showed that the maximum efficiency of this device was not affected by the boundary conditions. They basically only affect the current density J. However, the current is an experimental variable that is controlled to produce the maximum efficiency. The fact that the ends of the material have density variations has no bearing on the overall efficiency. The length scale, which is poorly understood, is the case that the barrier width L is two or three times as long as the mean free path. Shakouri has discussed this case and argued that the thermionic device works well in this limit also (Shakouri and Bowers, 1997; Shakouri et al., 1998, 1999). That conclusion seems reasonable if the scattering, which produces the mean free path, is mostly in the forward direction. Then particles that are scattered only once or twice still traverse the barrier quasiballistically. The same phenomena is well known in optics, where the photons that are scattered a few times in the forward directions are called "snakes" (Alfano et al., 1995). The word refers to the geometry of their path. If large angle scattering is important, however, then scattering only once or twice takes the carriers into the diffusive regime where we showed there was no benefit from thermionic behavior at the ends. It is useful to have an estimate of electron mean free paths in semiconductors. The electron can scatter from either acoustical or optical phonons, and also from defects. At room temperature, the scattering from acoustical phonons is usually negligible in semiconductors, and the mean free path is from scattering by either optical phonons or impurities. In polar semicon-

3 THERMIONICREFRIGERATION

171

TABLE II ELECTRON M F P DATA AT T = 300K Units m*

me

eo ~ hWLo

meV

zo vT

ps km/s

1)TT 0

#m

GaAs

InP

InAs

InSb

0.064 12.8 10.9 35.4 0.067 0.40 461 0.18

0.078 12.5 9.5 42.8 0.125 0.26 418 0.11

0.027 15.15 12.25 29.6 0.055 0.43 712 0.31

0.013 17.7 15.7 23.6 0.020 1.05 1025 1.08

The top lines have material constants. The lower group of lines have estimated mean free paths from optical phonon scattering.

ductors, the polar coupling between phonons and optical phonons provides the largest scattering for intrinsic semiconductors (Mahan, 2000b). Table II shows typical mean-free-path (MFP) data for some common semiconductors. The dimensionless polar coupling constant ~ is defined in terms of the effective mass m*, and the low (eo) and high ( ~ ) frequency dielectric constants: e2( m* ~1/2 ~X-- -~- \ 2 h W L o , J

1

Joe

lo)

(27)

ehWLo/kBT~ ] (28) -co =

20r

/ 3kBT

2 0 = TOUT ~ TO~/ m---~

(29)

The calculation of ~ is done using the material parameters at the top of Table II. The mean free path 2 o from polar scattering is found by multiplying the lifetime by the thermal velocity. The values range between 110 nm and 1.08 #m. These semiconductors were chosen for this calculation since their properties are well known. They all have relatively small bandgaps and high dielectric functions, which are also typical of thermoelectric materials. The same calculation on thermoelectric materials is harder because of the multiple valleys in the conduction band. The mean free path depends on direction. The semiconductors in Table II are cubic, and the mean free path does not depend on direction. These results show that a barrier can have a thickness of between 100 nm and 1/~m, depending on material. These large values are suitable for large values of the parameter BTI, as explained in the preceding.

172

G.D. MAHAN VI.

Discussion

The preceding discussion concluded that the thermionic refrigerator and generator would be relatively efficient, compared to typical thermoelectric devices, if a barrier material could be found with the properties: (1) Schottky barrier of O(kBT), and (2) 2/Kp ratio of O(1), where 2 is the mean free path in micrometers, and Kp is the lattice thermal conductivity in units of W/(m K). Whether or not this combination of factors is achievable is not yet determined. There are several experimental programs trying to construct multilayer thermionic refrigeration. Shakouri et al. (1999) and Shakouri and Bowers (1999) have reported several devices made from narrow-gap III-V semiconductors. They have achieved cooling from 1 to 4~ on several devices. Kim and Weitering (unpublished) have constructed several devices with semiconductor barriers and metal electrodes, but have not yet measured cooling.

ACKNOWLEDGMENTS

I wish to acknowledge the many discussions with my colleagues. The theorists are Lilia Woods, Jorge Sofo, Mirek Bartkowiak, and Mikhail Simkin. My experimental collaborators are Georg Kim, Hanno Weitering, Dave Zehner, and Frank Modine. Research has been supported by a grant from ONR N00014-98-1-0742. Research support is also acknowledged from the University of Tennessee and from Oak Ridge National Laboratory, which is managed by Lockheed Martin Energy Research Corp. for the U.S. Department of Energy under contract DE-AC05-96OR22464.

REFERENCES Alfano, R. R., X. Liang, L. Wang, and P. P. Ho, Time-resolved Imaging of Oil Droplets in Highly Scattering Soil Solution, Proc. SPIE 2389, 2 (1995). Bogomolov, V. N., D. A. Kurduykov, A. V. Prokofiev, Yu. I. Ravich, L. A. Samoilovich, and S. M. Samoilovich, Cluster Superlattice as 3D-array of Thermionic Energy Converters, Proc. 14th Int. Conf. Thermoelectrics (1995). Capinski, W. S., and H. J. Maris, Thermal Conductivity of GaAs/A1As Superlattices, Physica B 219 & 220, 699 (1996). Chasmar, R. P., and R. Stratton, The Thermoelectric Figure of Merit and its Relation to Thermoelectric Generators, J. Electron. Control 7, 52 (1959). Chen, G., and M. Neagu, Thermal Conductivity and Heat Transfer in Superlattices, Appl. Phys. Lett. 71, 2761 (1997). Chen, G., C. L. Tien, X. Wu, and J. S. Smith, Thermal Diffusivity Measurement of GaAs/ A1GaAs Thin-film Structures, J. Heat Transfer 116, 325 (1994).

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173

Edelson, J. S., Method and Apparatus for Vacuum Diode-Based Devices with Electride-Coated Electrodes, U.S. Patent 5,675,972 (1997). Edwards, H. L., Q. Niu, and A. L. deLozanne, A Quantum Dot Refrigerator, Appl. Phys. Lett. 63, 1815 (1993). Edwards, H. L., Q. Niu, and A. L. deLozanne, Cryogenic Cooling Using Tunneling Structures, Phys. Rev. B 52, 5714 (1995). Goldsmid, H. J., Electronic Refrigeration (Pion Limited, London, 1986). Hatsopoulos, G. N., and J. Kaye, Measured Thermal Efficiencies of a Thermo Electron Engine, J. Appl. Phys. 29, 1124 (1958). Houston, J. M. Theoretical Efficiency of a Thermionic Energy Converter, J. Appl. Phys. 30, 481 (1959). Huang, R. H., and J. L. Dye, Low Temperature Thermionic Electron Emission from Alkalides and Electrides, Chem. Phys. Lett. 166, 133 (1990). Hyldgaard, P., and G. D. Mahan, Phonon Superlattice Transport, Phys. Rev. B 56(10), 754 (1997). Kim, G., and H. Weitering (unpublished). Langmuir, I., The Effect of Space Charge on the Potential Distribution and Thermionic Current Between Parallel Plate Electrodes, Phys. Rev. 21,419 (1923). Lee, S. M., D. G. Cahill, and R. Venkatasubramanian, Thermal Conductivity of Si-Ge Superlattices, Appl. Phys. Lett. 70, 2957 (1997). Leivo, M. M., J. P. Pekola, and D. V. Averin, Efficient Peltier Refrigeration by a Pair of NIS Junctions, Appl. Phys. Lett. 68, 1996 (1996). Levine, J. D. Structural and Electronic Model of Si/Cs/O Surface, Surf. Sci. 34, 90 (1973). Mahan, G. D. Figure of Merit for Thermoelectrics, J. Appl. Phys. 65, 1578 (1989). Mahan, G. D. Thermionic Refrigeration, J. Appl. Phys. 76, 4362 (1994). Mahan, G. D., Good Thermoelectrics, in Solid State Physics, Vol. 51, Ed. H. Ehrenreich and F. Spaefen (Academic Press, 1998), p. 81. Mahan, G. D., Density Variations in Thermoelectrics, J. Appl. Phys. 87, 7326 (2000a). Mahan, G. D., Many-Particle Physics, 3rd ed. (Plenum, 2000b), chapter 8. Mahan, G. D., B. C. Sales, and J. Sharp, Thermoelectric Materials: New Approaches to an Old Problem, Physics Today (March, 1996), p. 42. Mahan, G. D., J. O. Sofo, and M. Bartkowiak, Multilayer Thermionic Refrigerator and Generator, J. Appl. Phys. 83, 4683 (1998). Mahan, G. D., and L. M. Woods, Multilayer Thermionic Refrigeration, Phys. Rev. Lett. 80, 4016 (1998). Moyzhes, B., Possible Ways for Efficiency Improvement of Thermoelectric Materials, Proc. 15th Int. Conf. Thermoelectrics (1996), p. 183. Nahum, M., T. M. Eiles, and J. M. Martinis, Electronic Microrefrigerator Based on a NIS Tunnel Junction, Appl. Phys. Lett. 65 3123 (1994). Nolas, G. S., and H. J. Goldsmid, A Comparison of Projected Thermoelectric and Thermionic Refrigerators, J. Appl. Phys. 85, 4066 (1999). Rowe, D. M., and G. Min, Multiple Potential Barriers as a Possible Mechanism to Increse the Seebeck Coefficient and Electrical Popwer Factor, Proc. 13th International Conference on Thermoelectrics, ed. B. Mathiprakasam and P. Heenan (ALP, 4/:316, 1994), p. 339-342. Shakouri, A., and J. E. Bowers, Heterostructure Integrated Thermionic Coolers, Appl. Phys. Lett. 71, 1234 (1997). Shakouri, A., and J. E. Bowers, Heterostructure Thermionic Coolers, U.S. Patent 5,955,772 (1999). Shakouri, A., C. LaBounty, J. Piprek, P. Abraham, and J. E. Bowers, Thermionic Emission Cooling in Single Barrier Heterostructures, Appl. Phys. Lett. 74, 88 (1999). Shakouri, A., E. Y. Lee, D. L. Smith, V. Narayanamurti, and J. E. Bowers, Thermoelectric Effects in Submicron Heterostructure Barriers, Microscale Thermophys. Eng. 2, 37 (1998).

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Scott, J. B., Extension of Langmuir's Space-charge Theory, J. Appl. Phys. 52, 4406 (1981). Simkin, M. V., and G. D. Mahan, Minimum Thermal Conductivity of Superlattices, Phys. Rev. Lett. 84, 927 (2000). Tamura, S., Y. Tanaka, and H. J. Maris, Phonon Group Velocity and Thermal Conduction in Superlattices, Phys. Rev. B 60, 2627 (1999). Venkatasubramanian, R., E. Siivola, T. Colpitts, and B. O'Quinn, Phonon-Blocking ElectronTransmitting Structures, Proc. Int. Conf. Thermoelectrics, Baltimore (1999, in press). Vining, C. B., and G. D. Mahan, The B Factor in Multilayer Thermionic Refrigeration, J. Appl. Phys. 86, 6852 (1999). Whitlow, L. W., and T. Hirano, Superlattice Applications to Thermoelectricity, J. Appl. Phys. 78, 5460 (1995). Zhou, R., D. Dagel, and Y. H. Lo, Multilayer Thermionic Cooler with a Varying Current Density, Appl. Phys. Lett. 74, 1767 (1999).

SEMICONDUCTORS AND SEMIMETALS, VOL. 71

CHAPTER

4

Phonon Blocking Electron Transmitting Superlattice Structures as Advanced Thin Film Thermoelectric Materials R a m a Venkatasubramanian RESEARCH TRIANGLEINSTITUTE RESEARCHTRIANGLEPARK, NORTH CAROLINA

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Low-TEMPERATURE HETEROEPITAXY OF B i 2 T e 3 - S b 2 T e 3 SUPERLATTICES . . . .

III. IN-PLANECARRIERTRANSPORTIN Bi2Te3-Sb2Te 3 SUPERLATTICES . . . . . . IV. PHONONTRANSPORTIN Bi2Te3-Sb2Te 3 SUPERLATTICES ........ V. MEASUREMENTS OF CROSS-PLANE THERMAL CONDUCTIVITY

. . . . . . . .

V I . LATTICE THERMAL CONDUCTIVITY IN SUPERLATTICES . . . . . . . . . . . V I I . MEAN FREE PATH REDUCTION IN SUPERLATTICES

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

VIII. DIFFUSIVETRANSPORTANALYSIS . . . . . . . . . . . . . . . . . . . IX. PHONON REFLECTION AT SUPERLATTICE INTERFACES

. . . . . . . . . . .

X. EQUIVALENCE BETWEEN DIFFUSIVE TRANSPORT AND LOCALIZATION

. . . . . .

XI. K L AND /MFP OF ULTRA-SHORT-PERIOD SUPERLATTICES . . . . . . . . . . . X I I . LOCALIZATION-LIKE BEHAVIOR IN S i - G e

SUPERLATTICES

. . . . . . . . . .

X I I I . CROSS-PLANE CARRIER TRANSPORT IN B i E T e 3 - S b E T e 3 SUPERLATTICES

. . . .

X I V . ADIABATIC PELTIER EFFECT IN THIN FILM THERMOELEMENTS . . . . . . . . . X V . DIFFERENTIAL COOLING IN BULK AND SUPERLATTICE THERMOELEMENTS

. . . .

X V I . SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . REFERENCES

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

I.

175 176 179 181 182 184 186 187 189 190 192 193 194 196 197 198 200

Introduction

H i g h figure-of-merit ( Z T ) thermoelectric materials can enable efficient solid-state refrigeration and p o w e r conversion. The d o m i n a n t state-of-theart thermoelectric materials are based on solid solution alloys in the p-type (BixSbl_x)2Te 3 system and n-type Bi2(SevTel_v) 3 system for near 3 0 0 K applications or in the SiGe alloy system for h i g h - t e m p e r a t u r e applications. The rationale for solid solution alloying (Wright, 1958) is that the lattice thermal conductivity is reduced m u c h m o r e strongly than the electrical conductivity so that an overall e n h a n c e m e n t in Z T can be achieved for certain alloy compositions. Thin film superlattices offer the possibility of a 175 Copyright 9 2001 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-752180-1 ISSN 0080-8784/01 $35.00

176

RAMA VENKATASUBRAMANIAN

significant reduction in lattice thermal conductivity (Venkatasubramanian et al., 1993) even compared to solid-solution alloys. Many of the thin film approaches (Hicks and Dresselhaus, 1993; Venkatasubramanian and Colpitts, 1997; Mahan and Woods, 1998) being investigated to enhance the Z T could benefit from the reduced lattice thermal conductivity in low-dimensional structures. Utilizing this advantage, we have recently observed a factorial enhancement in Z T with BizTe3-SbzTe 3 superlattices at 300K, relative to state-of-the-art bulk alloys (Venkatasubramanian et al., in press).

II. Low-Temperature Heteroepitaxy of Bi2Te3-Sb2Te 3 Superlattices In our endeavor to demonstrate the potential for high figure of merit in thin film BizTe3-SbzTe 3 superlattice structures, we first had to develop a novel low-temperature approach to metal-organic chemical vapor deposition (MOCVD) (Venkatasubramanian et al., 1999). BizTe 3 and SbzTe 3 are layered chalcogenides with a weak van der Waals-like bonding along their c axis. Invariably, the c axis is the natural growth axis in thin films of these materials. Low substrate temperatures are ideal for establishing the rather weak interlayer van der Waals bonding in these materials. In fact, the epitaxy of such materials may be categorized as van der Waals epitaxy (Koma, 1992). Although MOCVD is a major technique for the deposition of thin films (Stringfellow, 1989), the growth temperatures required for the pyrolysis of the organometallic and hydride precursors are typically high, which can cause interlayer diffusion in superlattices. The high growth temperatures also exacerbate the "inevitable" lattice mismatch-thermal mismatch problems in heteroepitaxy. To mitigate these concerns, it is desirable to maintain the substrate at lower temperatures during growth. We have described a low-temperature growth approach (Venkatasubramanian et al., 1999), where the substrate is not in direct contact with the hot susceptor, but instead, it is on a quartz separator with its surface parallel to the gas flow. The poor thermal conductivity of quartz and its limited contact area with the heated susceptor result in negligible conduction heating of the substrate. Instead, the substrate is heated by the radiation from the hot susceptor, controllably transmitted through the quartz separator by varying its thickness and the like. This enables the substrate temperature to be significantly lower than the temperature of pyrolysis occurring at the hotter front end of the precursors. For example, using infrared measurements (with emissivity corrections) we have observed substrate temperatures that are 100 to 200~ lower than the respective susceptor temperatures for various geometries of the separator. The lower substrate temperature with the quartz separator reduces the mobility of the ad-atoms on the substrate surface during growth. This apparently aids the layer-by-layer (Frank-van der Merwe) growth instead of a three-dimen-

4 PHONON BLOCKING ELECTRON TRANSMITTING SUPERLATTICE STRUCTURES

177

FIG. 1. A high-resolution TEM image of the GaAs-Bi2Te 3 interface; the circled area indicates the kink site introduced by the intentional misorientation of the substrate.

sional (Volmer-Weber) growth, the preferred equilibrium morphology in the presence of strain associated with lattice misfit (Council on Materials Science, 1989). Note the in-plane lattice mismatch between GaAs and the BizTe 3 layer is about 22%. A high-resolution transmission electron microscopy (TEM) image of the GaAs-BizTe 3 interface is shown in Fig. 1. Note the atomic step (highlighted by the circle) at the heterointerface, anticipated from the intentional 2-3 ~ misorientation of the [100] GaAs substrate. Although the substrate indicates some roughness for a thickness of ~ 20 ,/k from the heterointerface, the BizTe 3 film shows uniform layering along the growth axis. Few defects propagate into the heteroepitaxial BizTe 3 film from this interface in spite of the 22% lattice mismatch. We do observe that high crystalline perfection has been predicted in such materials, due to the van der Waals bond along the growth axis, even for pronounced lattice mismatch like 50% (Koma, 1992; Lang et al., 1994). Of course, the establishment of the rather weak van der Waals bond, in the first place, is facilitated with lower substrate temperatures. The layered chalcogenides, due to the van der Waals gap along the growth axis, also enable ultrasharp interfaces in heterostructures (Lang et al., 1994). The TEM micrograph of a typical 1 0 A - 5 0 A BizTe3-SbzTe 3 superlattice is shown in Fig. 2(a). The structure is indeed periodic; this is remarkable, given that 10 ~ represents a minimum repeat distance in this material system (Landolt-Bornstein, 1983).

178

RAMA VENKATASUBRAMANIAN

FIG. 2. (a) TEM image of a 10,/k-50,/k Bi2Te3-Sb2Te 3 superlattice, (b) intensity oscillations seen in the 10/k-50/k structure and within the 10-,/k subunits of 50-fk SbETe3, and (c) Fourier transform of the TEM image showing the satellites due to the superlattice and the (0003) reflection.

It is also worth commenting on the quality of the TEM image, given some inherent difficulties to obtaining strong material contrast (due to Bi vs Sb). The difficulty partly stems from that the Bi and Sb atomic layers are physically separated by the intervening van der Waals-like Te-Te bonding. Also, the van der Waals-like bond produces its own periodic contrast with

4

PHONONBLOCKING ELECTRON TRANSMITTINGSUPERLATTICE STRUCTURES

179

respect to the covalent structure. Thus, to enhance the atomic (Z) contrast in the images, we utilized thicker TEM specimens. This inevitably leads to some loss of resolution and artifacts related to the variation of sample thickness. Also, the Ar ion milling used for the preparation of TEM specimens has been found to easily damage these "soft" materials. Even so, intensity contrasts corresponding to the TEM image of the 1 0 A - 5 0 A BizTe3-SbzTe 3 superlattice were readily observable with an image algorithm (courtesy of the U.S. National Institutes of Health at http:// rsb.info.nih.gov/nih-image//) as indicated in Fig. 2(b). This scan clearly reveals not only the ~60-A superlattice period but also the expected 10-A contrast oscillations within the 50-A SbzTe 3 layer, due to the covalent bond-van der Waals bond-related fluctuations in electron transmission. The Fourier transform of the image in Fig. 2(a), equivalent to an electron diffraction spectrum, is shown in Fig. 2(c). We observe the first and the second satellite at (0000) along with the (0003) resonance (d spacing ,~ 10 A).

III.

In-Plane Carrier Transport in Bi2Te3-Sb2Te 3 Superlattices

The structural quality of superlattices grown by the low-temperature approach was evident in their in-plane carrier transport characteristics. In Fig. 3, we compare the in-plane Hall mobilities of p-type BizTe3-SbzTe 3 superlattice structures as a function of their period. Note that the ultrashort-period ( ~ 20-A)structures exhibit carrier mobilities as good as those seen in larger-period structures. This suggests weak interface scattering of carriers as their in-plane mobilities in superlattices are sensitive to interfacial structural imperfections (Yao, 1987; Fukul and Saito, 1986). We also observe that the hole mobilities in the p-type BizTe3-SbzTe3 superlattices are higher than those in corresponding BiSbTe 3 alloys of comparable carrier concentration. The lower mobilities in alloys stemming from microscopic inhomogenieties and/or as a result of carrier scattering between the respective valence bands of the unalloyed components (Tietjen and Weisberg, 1965) is well known. The enhancement in carrier mobilities in the monolayerrange superlattices, relative to solid solution alloys, perhaps due to the ideality possible with the van der Waals type bonding along the growth axis, is noteworthy. We believe interface carrier scattering has hampered prior demonstrations of similar results in other material systems. The weak interface scattering was further evident in the temperature dependence of carrier mobilities. A temperature range of 300-130 K was considered so that impurity scattering does not dominate the transport process. Figure 4 indicates the Hall mobility carrier concentration product as a function of temperature for the superlattice structures and comparable alloys; the product was used to normalize any carrier freeze-out effects.

180

RAMA VENKATASUBRAMANIAN

9 p - 1 x 1019

400

<> p~3.5 x 1019 & p~l x 1019, alloy

3501

/k p~3.5 x 1019, alloy

300 -

m p-1 x 1019, Sb2Te 3 9 p~l x 1019, Bi2Te 3

O

= 250 r

E

200 " =. ,., =,, ,,.

,,, . . . . . . .

u .. ,. - .. -- O

o 1 0 0 ~ ~r

500

0

I

I

I

50

100

150

Superlattice

Period

200

(,~)

FIG. 3. The 300K hole mobility in Bi2Te3-Sb2Te3 (dBi2Te3--dsb2Te3)superlattice structures as a function of their period along with those of reference alloys and unalloyed thin films.

2 x 1022 E 1.6x 1022 x

9SL, p(300K) - 9.7 x 1018 A Alloy, p(300K)~ 7.8 x 1018 9SL, p(300K) 4.2 x lO 19

I~

.4x o,

0 W

~ 1.2x 1022 E

~4

0

Qx 8 . 0 x l 021

~..1

y_T -1.o4

~ ,I0

O 4.0 x 1021

5

0 100

I

150

I

I

200

250

Temperature (K)

y-T

-1.42

y-T

-1.34

y-T

-1.76

I

300

I

350

FIG. 4. Temperature dependence of the mobility carrier concentration product in superlattice structures and comparable alloys at different carrier levels (in cm-3).

4

PHONON BLOCKING ELECTRON TRANSMITTING SUPERLATTICE STRUCTURES

181

Given the potential complexity of multiprocess scattering of carriers, the following discussion is intended to be only qualitative. For the lower carrier samples, we observe that ~alloy P ~ T-1"76 and PSL P ~ T-1"34. The temperature dependence of alloy-scattering-limited mobility is given by #a~oy ~ T -~ due to either random inhomogeneiety or band-edge-difference effects (Tietjen and Weisberg, 1965). The larger negative exponent for the alloy film is expected when two mechanisms, lattice phonon scattering and alloy scattering, are at work simultaneously (Seeger, 1982). In the superlattice structures, the interface-scattering-limited mobility can be modeled as proportional to Tn (with n ~ 1) as this scattering relaxation time is expected to increase with carrier thermal velocity, decrease with carrier mean free path, and be inversely proportional to Debye screening length (Seeger, 1982). The interface carrier scattering, if strong, is expected to considerably reduce the exponent in the temperature dependence of the overall mobility. In the data of Fig. 4, however, we see only a small reduction in the temperature exponent from -~ (associated with lattice phonon scattering). Consider the behavior when the impurity level is increased; here, we compare intentionally chosen samples of nearly the same ratio of impurity levels for both the alloy and superlattices. From this data, we observe that the temperature variation of the ( T " , m > 0) impurityscattering-limited mobility has as much a role in the superlattices as in the alloys. This suggests, again, weak interface scattering of carriers in the superlattices. In the next section, we consider, in marked contrast, the strong scattering of phonons from a reflection process at the superlattice interfaces (Venkatasubramanian, 2000).

IV. Phonon Transport in Bi2Te3-Sb2Te3 Superlattices Reduced thermal conductivity (Yao, 1987) and controllable transmission characteristics of phonon waves (Narayanamurti et al., 1979) in superlattices had been reported prior to the first observation (Capinski and Maris, 1996a, 1996b; Venkatasubramanian, 1996a, 1996b) of a significant reduction in the thermal conductivities below those of solid solution alloys with superlattice structures. Also, the weaker carrier-phonon and phonon-phonon interactions, leading to slow decay of optically excited systems and slower dissipation of local Joule heating, respectively, have been documented in A1GaAs-GaAs superlattices relative to bulk GaAs (Kelly, 1985). Wave transport models predicting Bragg reflections at acoustic phonon frequencies have also been discussed (Narayanamurti et al., 1979; Tamura et al., 1988). It is not clear, however, if the transport behavior of long-wavelength heat-conducting phonons, diffusing under a local temperature gradient in a material, would be different from that observed for "externally generated"

182

RAMA VENKATASUBRAMANIAN

waves in the acoustic phonon frequency regime (Narayanamurti et al., 1979; Tamura et al., 1988). Reliable data on thermal conductivities in superlattice structures, owing significantly to the thin film 3-03 method developed by Cahill et al. (1994), are just becoming available. Work by Lee et al. (1997) on the thermal transport properties of Si-Ge superlattices have confirmed the significant reduction in thermal conductivities. The benefits of such superlattice structures for advanced thermoelectrics have prompted detailed theoretical understanding of the thermal conductivities in superlattices using a Boltzmann transport model (Hyldgaard and Mahan, 1997; Chen, 1998). Several factors, including phonon dispersion relations and scattering at the superlattice interfaces, have been considered in the thermal transport models (Chen, 1998). The various fitting parameters to account for the different mechanisms have yielded apparent agreement with experimental data. We have carried out an extensive study based on over 50 high-quality superlattice samples of the variation of thermal conductivity with the superlattice period and free-carrier levels. Based on this study a model for the reduction of lattice thermal conductivity in superlattice structures has been developed (Venkatasubramanian, 2000). A phonon diffusive transport analysis indicates a low-frequency cutoff (03cutoff) in the spectrum of phonons that transport heat. A physical model for the reduction of lattice thermal conductivity based on the coherent backscattering of phonon waves at the superlattice interfaces has been outlined. Essentially, a localization-like behavior depending on the frequency of phonons has been proposed. It is similar to the photon localization in highly scattering media, such as superlattices (John, 1987) and semiconductor powders (Wiersma et al., 1997); this model is consistent with the experimentally observed selective transmission of high-frequency phonons in superlattices (Narayanamurti et al., 1979).

V.

Measurements of Cross-Plane Thermal Conductivity

The thin film 3-03 method (Cahill et al., 1994) has been utilized for the measurement of thermal conductivities normal to the BizTe3-SbzTe 3 superlattice interfaces. Note that superlattice structures, by the very nature of their interfaces, are expected to exhibit anisotropic thermal conductivities. This is true even in materials that do not exhibit anisotropy in bulk form (Chen et al., 1994). When heater widths are significantly larger than the film thickness, however, the thin film 3-03 method (Cahill et al., 1994) is expected to measure the cross-plane thermal conductivities in superlattices (Lee et al., 1997). The thermal conductivities measured by the 3-03 method are comparable to those from the thermoreflectance technique, another inde-

4

PHONON BLOCKING ELECTRON TRANSMITTING SUPERLATTICE STRUCTURES 1,000

850

_

/

183

GaAs+Si3N4+SL,

10 ~m wide heater

700 550 400

_

j

. / G a A s + S i 3 N 4, 10 I~m

m

S-

250 100

GaAs+Si~N4+SL, 20 prn

7.5

II ......

O -NN. . . . . .

-0

. . . . .".

O- ......

-n .... ~;--n ......

.\

I

I

8

8.5

O- ...... -II

0 -II

GaAs+Si3 IN4, 20 l~m

9

In(2~) FIG. 5. Graph of AT vs ln2co for two heater widths used to elucidate the cross-plane thermal conductivity measurement by the 3<0 method in BizTe3-Sb2Te 3 superlattices.

pendent cross-plane thermal conductivity measurement method, in Si-Ge superlattices. The requirements for accurate 3-co measurements of thin films with anisotropic thermal conductivities are described in Venkatasubramanian (2000). Examples of the raw 3-co data (300K) from a BizTe3-SbzTe 3 superlattice structure with the Si3N 4 dielectric isolation for two heater widths are shown in Fig. 5. Here, the superlattice film thickness is ~0.44 #m. We observe very well behaved 3
184

RAMAVENKATASUBRAMANIAN

further checked using a mesa geometry (Ju e t al., 1999). The mesa structure ensures that the heat transport is essentially one-dimensional through the superlattice interfaces. The thermal conductivities from the conventional and the mesa-etched thin film structures were within 5 % of each other, again consistent with the one-dimensional heat transport through the thickness of the film. Most importantly, we also note that the cross-plane thermal conductivities measured by the 3-o~ method in the Bi2Te3-Sb2Te3 superlattices were found to be in agreement with those obtained from throughthickness Peltier effect measurements in thin film thermoelements processed with these superlattice films (Venkatasubramanian et al., in press). 3-o~ measurements between 300 and 150K on a limited number of Bi2Te 3Sb2Te 3 superlattice samples indicate that the general behavior is comparable to that reported for Si-Ge superlattices (Lee et al., 1997).

VI. Lattice Thermal Conductivity in Superlattices The 3-co method measures the total cross-plane thermal conductivity (KT), consisting of the lattice component (KL) and the electronic component (Ke), as shown in Eq. (1); Ke is given by Eq. (2), where L o is the Wiedemann-Franz constant, q is the charge of the carrier, #• is the carrier mobility in the cross-plane direction, p is the carrier concentration, and T is the absolute temperature. K T -- K L + K e

Ke

-

-

L o T(qpp •

(1) (2)

We can obtain K L from the variation of K T as a function of p, if we assume that/z• is nearly independent of carrier level. This would be valid for a narrow range of flee-carrier levels. For carrier transport perpendicular to the superlattice interfaces, any small reduction in carrier mobility from increased impurity scattering at higher carrier level is expected to be offset by a higher transport function across the potential barrier (even if only shallow in these superlattices); this can be understood from the movement of the Fermi level toward the valence band edge with higher hole concentration. Thus, we expect a near constant #• across the superlattice interfaces for the conditions of our experimental set. This is likely to be the case even if the carrier transport across the superlattice interfaces is through a miniband conduction process. Figure 6 shows the K T vs p variation for some sets of Bi2Te3-Sb2Te 3 superlattice structures. The behavior is nearly linear in all the cases, supporting the preceding remarks. The lattice thermal conductivities of the 10 ]k-50/k and the 30 A-30 A Bi2Te3-Sb2Te 3 superlattices with the same period are the same, ~0.25 W/m-K (Venkatasubramanian, 2000). A similar

4

PHONON BLOCKING ELECTRON TRANSMITTING SUPERLATTICE STRUCTURES

185

O 30/60 SL; y = 2.25 x 10"2ox + 0.417 0.9 -

920/30 SL; y = 2.43 x 10-2ox § 0.224 910/50 SL; y = 2.93 x 10-2oX + 0.247 9 30/30 SL; y = 1

0.8

m

0.7

A

E

,

0.6

0.5

0.4

0.3

1

1

I

I

4 x 10 TM

8 x 10 TM

1.2 x 1019

1.6x 1019

i 2x

1019

Hole Concentration (cm'3) FIG. 6. Total thermal conductivity as a function of free-carrier concentration in some of the Bi2Te3-Sb2Te 3 superlattice structures.

study of K T v s carrier conductivity gives a K L of 0.49 W/m-K for BixSbl_xTe 3 (x ~ 0.9 +_ 0.15) alloy films along the c axis. This measured K L is in agreement with that of Rosi and Ransberg (1960) for an alloy of similar composition as well as with that deduced from the data of Scherrer and Scherrer (1995) for comparable alloys. Figure 7 shows the variation of the lattice thermal conductivity ( K L ) a s a function of the superlattice period. Three points are worth noting: 1. The K L of the superlattice structures shows a minimum for a period of 50 A. The minimum K L i s -'~ 0.22 W/m-K; this value is nearly a factor of 2.2 smaller than that obtained for a solid-solution alloy. 2. The K L of larger-period superlattices exceed that of the solid-solution alloy but begins to approach that of a weighted average of Bi2Te 3 ( K L , ~ 1.05 W/m-K) and Sb2Te 3 (KL~0.96 W/m-K). 3. For superlattice periods < 50 ,~, K L begins to increase from the above minimum and starts approaching that of the solid-solution alloy (Venkatasubramanian, 2000). In addition to this first report of lattice thermal conductivities approaching that of the alloy for ultra-shortperiod superlattices, preliminary observations of increase in total thermal conductivity with ultrashort superlattice periods have been reported (Venkatasubramanian, 1996a, 1996b) and in another study (Yamasaki et al., 1998).

186

RAMA VENKATASUBRAMANIAN

10

0"89~'~'---- Sb2Te3

A 0.7~ [2 0.6

8


BiSbTe 3 Alloy

0

m" 6 4 v

4 o~ &~

0.2

0.1 0

A

.

I

40

2 I 80

,

I 120

1 160

200

0

Superlattice Period (A)

FIG. 7. Experimentallattice thermal conductivityand calculated average phonon mean free path as a function of the period in Bi2Te3-SbzTe3 superlattices.

VII.

Mean Free Path Reduction in Superlattices

Kinetic theory (Kittel, 1976) gives the average phonon mean free path (/MFP) from Eq. (3) K L = (1/3)cp(v,)IMF P

(3)

where c is the specific heat, p is the density, and ( v t ) is the average phonon velocity. Even in the presence of localization-like effects, as discussed later, this velocity can be thought of as the speed at which the energy is transported locally by wave diffusion in a "viscous" medium (Sheng, 1995). The variation of IMFP as a function of the superlattice period is shown in Fig. 7; the behavior essentially resembles the variation of lattice thermal conductivity, accounting for the (cp) product in the various superlattices (Venkatasubramanian, 2000). For the calculations of/MFP, C and p in a superlattice were assumed to be weighted averages of the individual layer components Bi2Te 3 and SbETe 3. Note that the Debye temperatures of both these materials are ~ 160K (Landolt-Bornstein, 1983), much lower than the temperature of discussion (300 K). In addition, these materials have a rather large number of atoms per unit cell. Thus, the specific heat at 300 K would be dominated by the high-frequency acoustic and optical phonons due to the frequency dependence of density of states. This seems apparent even including the effect of singularities in the phonon density of states vs frequency using a B o r n - y o n Karman model (Landolt-Bornstein, 1983). Thus the specific heats of the constituents of the superlattice should be comparable to their bulk values,

4

PHONON BLOCKING ELECTRON TRANSMITTINGSUPERLATTICESTRUCTURES

187

independent of the propagation (or lack of it) of the low-frequency heatconducting phonons. This is especially so for (T/ODebye) > 1. We have used a (vt) of ~2.4 • 105 cm/s in the estimation of lMva in the various superlattices, the alloy, and SbzTe 3. This (vt) is about the average phase velocity of sound in these materials along the trigonal axis. Preliminary estimation of the average group velocity indicates a (v,) in the range of 2.0 x 105 cm/s. This value is obtained from the slope of the experimental longitudinal and transverse o)-k curves along the trigonal axis in the low to mid frequencies of the their respective acoustic phonon spectra (Landolt-Bornstein, 1983), using a weighted average for the varying BizTe 3 and SbzTe 3 components in the superlattice or an alloy. The closeness of the estimated (vt) to the phase velocity (Vp) is due to the near-linear experimental dispersion curves in these materials (for most part of the longitudinal and transverse acoustic phonon spectra) along the trigonal axis. Certainly, more complicated, but, rigorous calculations of (vt) are possible when localization effects are at work, perhaps as discussed in Sheng (1995). Let us consider the estimated lMva for the SbzTe 3, the BiSbTe 3 alloy, and the superlattices based on the preceding approach. For example, in SbzTe 3 we obtain a /MFP of ~9.6 A from its K L. This /MFP is apparently consistent with the anharmonicity built into the crystal by comparison with the spacing of inter-van der Waals bonding along the c axis. This weak bonding by its very nature could potentially vary from layer to layer, thereby, creating a potential anharmonicity along the axis of heat flow. Similarly, for the BiSbTe 3 alloy film, we estimate a /MFP of ~ 4.9 ]k from its measured K L. Remarkably, this is about the average separation between a Bi and a Sb atom in a lattice of the random alloy, potentially its region of anharmonicity. For superlattices with the lowest K L, we obtain a lMva of ~ 2.2 A. We believe, in these superlattices, the region of anharmonicity may be in the vicinity of the van der Waals gap, perhaps due to the different covalent components (Bi2Te 3 or Sb2Te3) on its either side.

VIII. Diffusive Transport Analysis From/MFP and the effective diffusivity (D) of phonons, estimated from Eq. (4) below, we can obtain the spectrum of frequencies that potentially conduct heat.

D = (1/3)(vt)lMr a

(4)

We start from the one-dimensional continuity relation (Carslaw and Jaeger, 1959) for the transport of heat to get the familiar diffusion Eq. (5): 82T c?T (o) c~x~ = ,~t

(5)

188

RAMA VENKATASUBRAMANIAN

The frequency domain solution to this problem is well known and provides an estimate for the range of the temperature wave (Seeger, 1982) for a given angular frequency (co) via Eq. (6)" l =

(6)

As the linear heat conduction by phonons would occur for a length scale /UFP the phonons would have been scattered in another direction, we have proposed that (Venkatasubramanian, 2000), a given D and/MFP would set a low-frequency cutoff, COcutoff. From Eqs. (4) and (6), we obtain Eq. (7): (/)cutoff

=

(2/3)(vt)(lmvP)

(7)

- ~

Thus the reduced /MFPin superlattice structures (Fig. 7), compared to the solid-solution alloy, can lead us to conclude that the low-frequency phonons would be inhibited from transport while the high-frequency phonons would be allowed to propagate in superlattices. This is in agreement with the experimental observation of a loss in transmission of low-frequency phonons and the near complete transmission of the high-frequency phonons in GaAs-A1GaAs superlattices (Narayanamurti et al., 1979). The calculated low-frequency cut-off as a function of the superlattice period for the BizTe3-SbzTe 3 superlattices is shown in Fig. 8. Thus we believe the origin of a reduction in lattice thermal conductivity in superlattices starts with the reduction of the phonon mean free path from 1.25 x 1012 A N

=: >,l U C:

9

:::3 0" 0 I=,

u. :1= ? =

/k

/k/k

1 x 1012 7 . 5 x 1011

A 5 x 1011 I

~

A

BiSbTe3Alloy

A

A

2.5 x lO 11 I Sb2Te 3

0

0

I

I

I

40

80

120

Superlattice

Period

....

I 160

200

(,~)

FIG. 8. Calculated low-frequency cutoff as a function of period in Bi2Te3-Sb2Te 3 superlattices compared with that of an alloy and Sb2Te 3.

4 PHONONBLOCKINGELECTRONTRANSMITTINGSUPERLATTICESTRUCTURES

189

the anharmonicity. The reduction of mean free path in turn leads to the inhibition of the propagation of the heat-conducting, low-frequency phonons.

IX.

Phonon Reflection at Superlattice Interfaces

Although we can relate the reduction in the average /MFP to the blocking of low-frequency phonons, we need a physical model for the reduction of /MFP from the anharmonicity in the superlattice. In other words, how does the anharmonicity, as indicated earlier, reduces the /MVP? Here, we invoke the coherent backscattering of phonon waves at the superlattice interfaces, the regions of anharmonicity. This is similar to photon localization in highly scattering media (John, 1987; Wiersma et al., 1997). Let us consider a simple picture of a uniform alloy made of two components M~ and M 2 and that of a superlattice consisting of two individual layers of M x and M 2 as shown in Fig. 9. The alloy would exhibit an average "acoustic impedance" Zave that is related to the masses M x and M 2 , as described by Brillouin (1946). In the superlattice, each of the two layers Z 1 and Z 2, respectively, has a characteristic impedance. At first glance, it would appear that the specific arrangements of atoms among the available lattice sites are irrelevant to the propagation of long-wavelength acoustic phonons. However, note that the acoustic mismatch would lead to reflection of the phonon waves at the interface (Brillouin, 1946). Thus, in a superlattice, the acoustic long-wavelenyth phonons not only see the varyin9

composition (as much in the random alloy) but also experience reflection at the

a)

Zav e

M2

b)

M1

Z1 M1

M1

M2 ! !

,, MI',+M2 I I i

M1

M2

Z2 M2

M2

Interface FIG. 9. Schematicof an alloy and that of a superlattice showing the potential for reflection of waves at the interface of a superlattice.

190

RAMA VENKATASUBRAMANIAN

periodic interfaces. The reflection aspect in conjunction with a Bragg reflection model has been invoked by Narayanamurti et al. (1979) in explaining their observation of loss of transmission in certain low-frequency phonons. However, they used the acoustic impedance mismatch model to indicate that the transmission should be minimized when the superlattice period is ~ 2/2, where 2 is the phonon wavelength, in analogy with optical filters. However, in a subsequent paper, Kelly (1985) indicated that the transmission should be a maximum when the phonon wavelength matches the superlattice period. Kelly (1985) pointed out that the difference might be that inelastic processes are involved in the observations of Narayanamurti et al. (1979). However, we propose that the reflection of phonon waves leads to an enhanced probability of a phonon wave coming back to its point of origin (in a local sense), thereby providing a scenario for a localization phenomena. This coherent backscattering is frequently referred to as weak localization or as a precursor to wave localization, provided randomness is available to initiate a tendency for localization. We believe a possibility for randomness may be the varying amount of van der Waals bonding along the various superlattice interfaces, exemplified by the fluctuation in the electron diffraction intensity that we observe within the bilayer of the superlattice in Fig. 2. The rather striking nonlinear behavior of the K L versus the superlattice period (Fig. 7) and the observation that the larger period superlattices are more opaque to phonon transport than smaller period samples, for below 50 ]k, support this hypothesis. The localization model may be appropriate in dealing with potential reflection of diffusing waves (John, 1987; Sheng, 1995). The backscattering should lead to a "viscous" medium (Sheng, 1995) for phonon waves in the superlattice structures, over and above that attainable in solid-solution alloys. For one to observe the effects of such backscattering, of course, the superlattice interfaces probably must be of high quality. The transmission electron microscopy studies and in-plane electrical transport data of the BizTe3-SbzTe 3 superlattices (Venkatasubramanian et al., 1999) do indicate that the superlattices under consideration are abrupt.

X.

Equivalence between Diffusive Transport and Localization

In this section, we discuss the equivalence of the results from the diffusive transport analysis and the possible localization phenomena that stem from back scattering at the superlattice interfaces. The theme of the equivalence arguments is summarized in Fig. 10. The Anderson localization phenomena can be inferred from a calculation of the klMFP product (Sheng, 1995), where k is the wave vector. From the knowledge of /MFP and the phase velocity (Vp), we can estimate the k/MFP

4

PHONON BLOCKING ELECTRON TRANSMITTING SUPERLATTICE STRUCTURES

[Thermal Conductivity I

I

Anderson Localization (k.lmfp) ~; 1

191

Kinetic Theory

Diffusive Transport Model

1

1

Cut-off Frequency

Cut-off Frequency

1

2 ~ ~ " Vp

kcut'~

FIG. 10.

Comparison between the diffusive transport and the localization models.

product over the entire range of the acoustic phonon spectrum. The calculated kIMFp products for some of the structures under discussion are shown in Fig. 11. We observe that in the short-period superlattices, the condition of klMFP < 1 is strongly met for a larger ranye of low frequencies. This should predict localization-like behavior for the low-frequency phonons in a superlattice. The cutoff frequency estimated for the various structures (in Fig. 11), from the application of the Anderson localization

4.0 9Sb2Te3

3.5

9BiSbTe 3

O 10/10 SL 0 20/20 SL

3.0

9

930130 SL z~20/30 SL

2.5

9

E '~" 2.0 1.5

B

1.0

9

9

O OO O O

O O

0.5 0

4.0 • 1011

8.0 x 1011

1.2 x 1012

1.6 x 1012

Frequency (Hz) FIG. 11. Calculated klMFa product for some of the superlattices, alloys, and Sb2Te 3 over the range of acoustic phonon spectrum.

192

RAMAVENKATASUBRAMANIAN

criterion were comparable to those estimated from the diffusive transport model (shown in Fig. 8). This is not surprising since from Eqs. (7) and (8), we obtain Eq. (9). This relation can be compared with the Anderson localization criterion of klMva ~ 1 and the Ioffe-Riegel criterion of 0.5 < klMFP < 0.985 (Sheng, 1995): kcut~

27~ = '~'cutoff

(8)

2 (V,)

kcutofflMFP~3. Vp

XI.

(9)

K L and IMFPof Ultra-Short-Period Superlattices

In light of the preceding discussion, we can consider the rise in K L and the IMva (Fig. 7) in ultra-short-period superlattices (period <50 A). This behavior is not attributable to phonon-tunneling effects even though we observe a near-exponential rise in thermal conductivity (Fig. 7) with (period)- a. We discount this possibility, as the 10 A-50 A, 20 A-40 A, and 30 ,/t-30 A superlattices (all with a period ~ 60 A) show nearly the same KL and about similar mean free paths. If phonon tunneling through a 10-]k region were dominant in order to explain the difference between the 10 A-10 ,/k and the 30 ,~-30 A superlattice, then one would expect the K L of the 10 ,/t-50 A superlattice to be markedly different from that of a 30 A-30 superlattice. The behavior of ultra-short-period superlattices can be understood if we compute the wavelengths using the phase velocity for the cutoff frequencies shown in Fig. 8. The cutoff wavelength (2cutoff) is plotted as a function of the superlattice period in Fig. 12 (upper). We observe that near the point where the minimum lattice thermal conductivity (equivalent to minimum average phonon transmission) is approached, the superlattice period is ~22cutoff. Under this condition, we can consider a schematic of the phonon waves, for the critical frequency, in the two individual layers of the superlattice as shown in Fig. 12 (lower) for dBizTe3 ~ dsb2Te3- When the cutoff wavelengths calculated by the diffusive transport analysis starts approaching and exceeding the individual layer thickness in both the layers, the two layers probably become coupled and so the effect of acoustic mismatch starts to disappear. From this viewpoint, the cutoff wavelength would not decrease any further with the reduction of the superlattice period and the long-wavelength phonons would begin to be transported across the interfaces. It is interesting to observe that the condition for maximum transmission as per Kelley (1985) is at a period ~ 2 and we indicate a minimum at a period ~ 2,~cutoff. Thus, based on a model of phase coherence for minimum transmission, one would obtain a Bragg condition leading to period ~ 2/2 (Narayana-

4

PHONON BLOCKING ELECTRON TRANSMITTING SUPERLATTICE STRUCTURES

193

10oj. IP ~ -

0<'~,~ = ~>

Sb2Te 3

80t~,,

60

t: '~ 2o~

BiSbTe 3 Alloy

AA

A

A

A

-!

AA~

0/

I

0

40

I 80

120

Superlattice

,

I 160

200

Period (/~)

SLperiod 1

SL interface

FI6. 12. (Upper)Low-frequencycutoffwavelength as a function of superlattice period and (lower) the approximate situation when the superlattice period -~ 2Acutoff.

murti et al., 1979). However, we arrive at period ~22cutoff using a model of diffusing phonon waves being able to couple from one individual layer to adjacent layer of a superlattice.

XII.

Localization-Like Behavior in S i - G e Superlattices

The general behavior of lattice thermal conductivity in superlattices, with a minimum at certain intermediate period, is not unique to the BizTe 3Sb2Te a superlattices. The in-plane lattice thermal conductivities at 300 K in Si-Ge superlattices as a function of their period is shown in Fig. 13. A first order estimate of the lattice thermal conductivity (KL) was made from the total thermal conductivity (Venkatasubramanian et al., 1998), accounting

194

RAMA VENKATASUBRAMANIAN 14.0

Sio.6sGeo.35All, ~y "T'..,,.

lo.o i. ~,

9

8.0

~ 6.0 4.0 2.0

9

0.0 0

20

40

60

80

100

Superlattice Period (,&) FIG. 13. Variation of in-plane lattice thermal conductivity as a function of superlattice period in the Si-Ge system.

for the rather small in-plane electronic portion from electrical conductivity using the Weidemann-Franz law. We observe a rather sharp minimum for superlattice periods of 65 to 75 A. Note, however, the dip is not introduced because of the electronic correction but is seen in the total thermal conductivity. A realistic modeling of this data is complicated by the fact that the 0Deby e of these materials (450 to 600 K) is significantly larger than the temperature of measurement. Also, the in-plane heat transport, especially in such periodic structures, may have to be effectively treated as a threedimensional problem. Even so, our preliminary calculations suggest that phonon-reflection effects leading to a localization-like behavior may be at work in this system as well for superlattice periods of 65 to 75 A. We also observe that the ultra-short-period structures in this system show additional features, probably related to disorder-induced thermal conductivity reduction, for superlattice periods below 40A. When such disorder reduces the lattice thermal conductivity, however, it also reduces the electrical transport carrier mobility. Thus such superlattices (with potential disorder) do not appear advantageous, as is the case when phonon reflection effects are at work in reducing thermal conductivities, in terms of obtaining enhanced thermoelectric figure of merit. For Si-Ge superlattices with a period of 65 to 75 A, we have seen a factorial enhancement in the figure of merit compared to SiGe bulk alloys (Venkatasubramanian et al., 1998).

XIII.

Cross-Plane Carrier Transport in Bi2Te3-Sb2Te3 Superlattices

Let us now consider the carrier transport perpendicular to the BizTe 3SbzTe 3 superlattice interfaces; this is a key, in concert with the reduced

4

PHONON BLOCKING ELECTRON TRANSMITTING SUPERLATTICE STRUCTURES

(a)

195

(c)

]~ Lt00i

Lt00i I

Ltl

L.5

Metal Contact

Ltl I .L.5

I i i iii.!~i!!!!

Substrate

i i

, ..................... i,hl SL Film

Substrate

(b)

(d) 9 E

I-

J

-2Lt0

~,ope = ~es/stiPvlityne x

Ri(0)

rv "(1

Spacing

FIG. 14.

( 1)

,

Mesa Height (h)

Schematic of TLM cross-plane resistivity measurement.

lattice thermal conductivities in this superlattice system, to achieving high ZT. The electrical resistivity perpendicular to the plane of the film was obtained using a modification to the transmission line model (TLM) technique [Figs. 14(a) and 14(b)] (Berger, 1972). This adaptation was feasible due to the low specific contact resistivities ( ~ 1 • 10-7~-cm 2) achievable in these materials. With low specific contact resistivities, the transfer lengths (Lt) are small and so a "squeezing" of the current occurs effectively through a small region of the contact, as shown in Fig. 14(a). Then, with an etch step as shown in Fig. 14(c), we can sensitively measure the increase in the intercept resistance associated with the current flow perpendicular to the plane of the film. At each etch step, the additional incremental intercept resistance is measured; the slope of the incremental specific resistance versus mesa height (noting two vertical sections are involved in each measurement) gives the electrical resistivity perpendicular to the plane of the film [Fig. 14(d)]. This method has been evaluated on thin films of n-type BizTe 3 of extrinsic doping (to ensure one-band conduction but of only moderate degeneracy) and known theoretical (Drabble et al., 1957) and measured experimental bulk electrical anisotropy (Hicks and Dresselhaus, 1993). This data is summarized in Table I. The validity of method has also been evaluated in several films of p-type BixSbl_xTe 3 ( ~ 0 . 6 3 _ 0.12) alloys with varying carrier level (Sherrer and Sherrer, 1995). Note that the cross-plane TLM

196

RAMA VENKATASUBRAMANIAN TABLE

I

ELECTRICAL ANISOTROPY MEASURED IN n-Bi2Te 3 THIN FILMS BY THE CROSS-PLANE T L M TECnYIQt~

Carrier conc. Theory (Drabble e t al., 1957) Bulk experiment (Scherrer and Scherrer,

(cm-3)

1995)

3.5 • 1019 5.2 • 1019

Thin film

1.5 • 1020 1.4 • 1020

Anisotropy

(Pc/Pa-b)

4.1 5.3 5.3 6.67 3.8 + 0.4

method presented here estimates the electrical resistivity under isothermal conditions. This is because of the large (parallel) thermal shunting across the region of current flow, the small cross section with Lt. Examples of measured cross-plane resistivities in some of the superlattices are compared with their in-plane resistivities measured by the conventional van der Pauw method in Table II. The measured variation of electrical anisotropies with the superlattice period is apparently consistent with an equilibrium-diffusive-miniband model for cross-plane carrier transport in such superlattices (Capasso et al., 1986).

XIV.

Adiabatic Peltier Effect in Thin Film Thermoelements

The lower K L in superlattice materials have been corroborated by processing prototypal unipolar thermoelements, albeit of nonoptimal aspect ratio, and measuring the Peltier voltage (Vo) developed across them under the application of a quasi-steady-state current. This is a part of the familiar

T A B L E II TYPICAL IN-PLANE AND CROSS-PLANE ELECTRICAL RESISTIVITIES IN SOME SUPERLATTICE STRUCTURES B i 2 T e 3 - S b 2 T e 3 superlattice structure (xA/yA)

10/10 10/50 20/20 20/40 60/60 120/120

In-plane electrical resistivity (fl cm)

Cross-plane electrical resistivity (fl cm)

3.1 x 10 -3 8.8 • 10 - 4 4.7 • 10 -3

3.5 • 10 -3 7.2 • 10 - 4 6.5 x 10 -3

1.5 • 10 -3

1.5 • 10 -3

3.6 • 10 -3 2.4 • 10 -3

3.2 • 10 -2 3.3 • 10 -3

4

197

PHONON BLOCKING ELECTRON TRANSMITTING SUPERLATTICE STRUCTURES

Harman method (Harman, 1959) for Z T determination. Note that the

(002/K product that enters into the Z T value of a thermoelectric material can be readily obtained from Vo, the voltage across the thermoelement just after the flow of current (I) is terminated. The (e)2/K product is given by Vo = ( ~ 2 / K r ) T ( l / a ) I

(10)

Here T is average absolute temperature, and l/a is the aspect ratio. To minimize heat loss from the thin film thermoelement, a fine-tip ( ~ 12-#m) electrical probe is placed on a typical 100-#m-square thin-film mesa thermoelement of several #m height. The fine tip is adequate to measure Vo as it is measured just after the current is turned off. However, this precludes the direct measurement of the resistive voltage (VR) across the mesa device during current flow. So VR was estimated for each current from p• (obtained from cross-plane TLM technique described above) and l/a, using Eq. (11); VR and Vo were used to determine Z T value from Eq. (12): VR = p • (l/a)I

(11)

ZT=

(12)

(Vo/VR)

Thermoelements have been made in reference bulk materials and p-type BixSbl_xTe 3 (x ~ 0.63 _+ 0.12) alloys (nonsuperlattice structures) with aspect ratios similar to those of superlattice thin films and their Z T values have been determined by the preceding method. The Z T values of state-ofthe-art bulk single-crystal thermoelements were routinely determined to be ~ 1 along the a - b axis. The estimated Z T values of the alloy films along the c axis (normal to the growth-plane) were in the range of 0.4 _+ 0.13 for carrier levels of ~ 3 x 1019 cm-3; these are in agreement with those reported for comparable bulk materials along the same c axis (Scherrer and Scherrer, 1995). The estimated Z T values in some of the superlattice thermoelements are in the range o f ~ 1 . 7 to ~ 3 . 5 at 30OK. Generally high Z T values were observed in short-period superlattices, resulting from a combination of their low lattice thermal conductivities and ideal-near-ideal cross-plane electrical conductivities. More detailed Z T measurements of such superlattice thermoelements are currently underway. Larger-period superlattices that show little reduction in lattice thermal conductivity compared to alloys also exhibit poor carrier transport perpendicular to the interfaces; thus they offer Z T lower values than those of alloys.

XV.

Differential Cooling in Bulk and Superlattice Thermoelements

The measured Peltier voltage (Vo) can be related to net cooling AT across the thermoelement using the experimentally measured in-plane Seebeck

198

RAMA VENKATASUBRAMANIAN

]L~..

9 6

~ N O O

i

3

4 ak

9

_%

~

O Peltier Voltage -v

Method 9

13 -.i

12 -150

-100

-50

i

0

50

IR-camera measurement ,, i

I00

150

Current (mA) FIG. 15. Net AT measured in bulk thermoelements using an IR-imaging camera and the Peltier voltage measurement.

coefficient. This assumes that the Seebeck coefficient is relatively isotropic; this is known in bulk materials and we observe that it is the case with the superlattice structures as well. The AT values estimated by this approach and those measured with an IR-imaging camera have shown good agreement in large-area thermoelements as shown in Fig. 15. By estimating net AT using measured Peltier voltages, we have been able to compare the cooling for some of the high Z T superlattice thin film thermoelements with those of bulk thermoelements of similar l/a ratios. This is shown in Fig. 16. A higher Z T apparently translates to a larger A T across the superlattice thermoelements for a comparable current.

XVI.

Summary and Conclusions

In conclusion, a low-substrate-temperature variant to the conventional MOCVD has been developed (Venkatasubramanian, 2000b) for the epitaxy of BizTe3-SbzTe 3 superlattices on GaAs substrates. High-resolution transmission electron microscopy studies indicate that the interface between GaAs substrate and BizTe 3 film is qualitatively defect-free and that periodic structures are formed in BizTe3-SbzTe 3 superlattices, with one of the individual layers as small as 10A. These structures apparently have good interfaces, enabling the demonstration of enhanced carrier mobilities in monolayer-range superlattices (relative to alloys) through the elimination of alloy scattering and the absence of random interface scattering of carriers. These results are perhaps due to the ideality achievable in heterointerfaces

4

PHONON

BLOCKING

ELECTRON

TRANSMITTING

SUPERLATTICE

STRUCTURES

199

9 B u l k , Z T - 1, l/a = l / c m 9 T h i n - f i l m , Z T - 3.2, l/a = l . 4 / c m tx T h i n - f i l m , Z T - 2.3, l/a = l . 1 6 / c m . .

4.5 4 ~

3.5

["

3

~

2.5

"'~ o o

1.5

" Z

1 0.5

2

0

0

70

140 210 Current (mA)

280

350

FIG. 16. Net cooling estimated with bulk and superlattice thermoelements from measured Peltier voltages.

involving van der Waals-like bonding, the establishment of which is aided by the low-temperature growth process along the growth axis. These superlattices denoted as phonon-blocking electron-transmitting structures could also enable the optimization of electron and phonon transport properties. The heat transport in superlattice structures may offer a scope for understanding the potential effects related to phonon localization phenomena. Most importantly, however, the lattice thermal conductivity reduction offered by the superlattices may be very useful to obtaining highperformance thermoelectric materials (Venkatasubramanian et al., 1998, in press). We have provided preliminary evidence that engineered superlattices can attain the desirable characteristics of phonon-blocking electron-transmitting properties for high-performance thermoelectric devices. Initial data suggests intrinsic Z T values in the range of 1.7 to 3 at 300 K with p-type BizTe 3SbzTe 3 superlattice materials from adiabatic Peltier effect and isothermal electrical resistivity measurements. Further work is in progress to ascertain the significant Z T enhancement. We have also obtained detailed evidence for enhanced performance of superlattices based on differential Z T measurements as well as using the novel idea of relating Z T values to the ratio of thermal conductivity at zero electric field to that at zero current (Vining, 1992). It appears that intrinsic Z T values in the range of 2.5 are achievable with such phonon blocking electron transmitting structures. In addition ambient cooling of about 12 K have been obtained with such thermoelements. This work (Venkatasubramanian et al., to be published) should provide the first critical results needed for a major push toward the implementation of thin film superlattice materials for a variety of electronics cooling applications.

200

RAMA VENKATASUBRAMANIAN ACKNOWLEDGMENTS

Dr. Stuart Wolf and Dr. Valerie Browning at the Defense Advanced Research Projects Agency (DARPA) and Dr. John Pazik at Office of Naval Research (ONR) are acknowledged for the support of this work. Mr. Terry Stark of Materials Analytical Services, Raleigh, North Carolina, and Prof. Nadia E1-Masry of North Carolina State University are thanked for the TEM results. Dr. Kevin Stokes is thanked for the optimization of the 3-co measurement facility. Mr. Edward Siivola, Mr. Thomas Colpitts, and Mr. Brooks O'Quinn of the Research Triangle Institute are gratefully acknowledged for their technical assistance during the course of the work leading to some of the results described in this chapter.

REFERENCES Berger, H. H., Solid State Electron. 15, 145 (1972). Brillouin, L., Wave Propagation in Periodic Structures, McGraw-Hill, New York (1946). Cahill, D. G., M. Katiyar, and J. R. Abelson, Phys. Rev. B 50, 1464 (1994). Capasso, F., K. Mohammed, and A. Y. Cho, IEEE J. Quantum Electron. QE-22, 1853 (1986). Capinski, W. S., and H. J. Maris, Bull. Am. Phys. Soc. 41, 692 (1996a). Capinski, W. S., and H. J. Maris, Phys. B 219&220, 699 (1996b). Carslaw, H. S., and J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, Oxford (1959). Chen, G., Phys. Rev. B 57, 14958 (1998). Chen, G., C. L. Tien, X. Wu, and J. S. Smith, J. Heat Trans. 116, 325 (1994). Council on Materials Science, Div. of Materials Science, Fundamental Issues in Heteroepitaxy, U.S. DOE Panel Report, Monterey, CA, 1989. Drabble, J. R., R. D. Groves, and R. Wolfe, Proc. Phys. Soc. 71, 430 (1957). Fukui, T., and H. Saito, Inst. Phys. Conf. Ser. 79, 397 (1986). Harman, T. C., J. Appl. Phys. 30, 1373 (1959). Hicks, L. D., and M. S. Dresselhaus, Phys. Rev. B 47, 12727 (1993). Hyldgaard, P., and G. D. Mahan, Phys. Rev. B 56, 10754 (1997). John, S., Phys. Rev. Lett. 58, 2486 (1987). Ju, Y. S., K. Kurabayashi, and K. Goodson, Thin Solid Films 339, 160 (1999). Kelly, M. J., J. Phys. C: Solid State Phys. 18, 5963 (1985). Kittel, C., Introduction to Solid State Physics, John Wiley, New York (1976). Koma, A., Thin Solid Films 216, 72 (1992). Landolt-Bornstein, Numerical Data and Functional Relationships in Science and Technology, Vol. 17, Springer-Verlag, Berlin (1983). Lang, O., R. Schlaf, Y. Tomm, C. Pettenkofer, and W. Jagermann, J. Appl. Phys. 75, 7805 (1994). Lee, S.-M., D. G. Cahill, and R. Venkatasubramanian, Appl. Phys. Lett. 70, 2957 (1997). Mahan, G. D., and L. M. Woods, Phys. Rev. Lett. 80, 4016 (1998). Narayanamurti, V., H. L. St6rmer, M. A. Chin, A. C. Gossard, W. Wiegmann, Phys. Rev. Lett. 43, 2012 (1979). Rosi, F. D., and E. G. Ransberg, in Thermoelectricity, ed. by P. H. Egli, John Wiley, New York (1960).

4

PHONON BLOCKING ELECTRON TRANSMITTING SUPERLATTICE STRUCTURES

201

Scherrer, H., and S. Scherrer, CRC Handbook of Thermoelectrics, ed. by D. M. Rowe, CRC Press, New York (1995), p. 211. Seeger, K., Semiconductor Physics, Springer-Verlag, Heidelberg (1982). Sheng, P., Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, Academic Press, London (1995). Stringfellow, G. B., Organometallic Vapor Phase Epitaxy." Theory and Practice, Academic Press, San Diego (1989). Tamura, S., D. C. Hurley, and J. P. Wolfe, Phys. Rev. B 38, 1427 (1988). Tietjen, J. J., and L. R. Weisberg, Appl. Phys. Lett. 7, 261 (1965). Venkatasubramanian, R., Bull. Am. Phys. Soc. 41, 693 (1996a). Venkatasubramanian, R., Nay. Res. Rev. 48(4), 31 (1996b). Venkatasubramanian, R., Phys. Rev. B 61, 3091 (2000). Venkatasubramanian, R., Low Temperature Chemical Vapor Deposition and Etching Apparatus and Method, United States Patent No. 6,071,351 (2000b). Venkatasubramanian, R., and T. Colpitts, Mater. Res. Soc. Syrup. Proc., Vol. 478 (Materials Research Society, Warrendale, PA, 1997), p. 73. Venkatasubramanian, R., M. L. Timmons, and J. A. Hutchby, Proc. 12th International Conference on Thermoelectrics, Yokohama, ed. by K. Matsuura, 322 (1993). Venkatasubramanlan, R., E. Siivola, and T. Colpitts, Proc. 17th Inter. Conf. on Thermoelectrics, IEEE Press, Cat. No. 98TH8365, 191 (1998). Venkatasubramanian, R., T. Colpitts, B. C. O'Quinn, S. Liu, N. El-Masry, and M. Lamvik, Appl. Phys. Lett. 75, 1104 (1999). Venkatasubramanian, R., E. Siivola, T. Colpitts, and B. C. O'Quinn, in Proc. 18th International Conference on Thermoelectrics, Baltimore (IEEE, New York, in press). Venkatasubramanlan, R., E. Siivola, T. Colpitts, and B. C. O'Quinn, to be published. Vining, C. B., Proc. 11 th International Conference on Thermoelectrics, Arlington, Texas (1992). Wiersma, D. S., P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997). Wright, D. A., Nature 181, 834 (1958). Yamasaki, I., R. Yamanaka, M. Mikami, S. Sonobe, Y. Mori, and T. Sasaki, Proc. 17th Inter. Conf. on Thermoelectrics, IEEE Press, Cat. No. 98TH8365, 211 (1998). Yao, T., Appl. Phys. Lett. 51, 1798 (1987).

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SEMICONDUCTORS AND SEMIMETALS, VOL. 71

CHAPTER

5

Phonon Transport in Low-Dimensional Structures G. C h e n MECHANICAL AND AEROSPACE ENGINEERING DEPARTMENT UNIVERSITY OF CALIFORNIA, LOS ANGELES

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . II. PHONONS IN BULK AND Low-DIMENSIONAL MATERIALS . . . . . . . .

1. Phonon Thermal Conductivity in B u l k Materials . . . . . . . . . 2. Phonon Dispersion in Nanostructures . . . . . . . . . . . . . . III. THIN FILM THERMAL CONDUCTIVITY MEASUREMENT TECHNIQUES . . . .

1. Microsensor Methods . . . . . . . . . . . . . . . . . . . . . 2. Optical Pump-and-Probe Methods . . . . . . . . . . . . . . . 3. Optical-Electrical Hybrid Methods . . . . . . . . . . . IV. ANALYTICAL TOOLS . . . . . . . . . . . . . . . . . . .

1. 2. 3. 4. 5.

Lattice Dynamics and Phonon Dispersion Analysis . . . . . Boltzmann Transport Equation . . . . . . . . . . . . . Boundary Conditions f o r B T E . . . . . . . . . . . . . Monte Carlo Simulation . . . . . . . . . . . . . . . . Molecular Dynamics Simulation . . . . . . . . . . . .

V. THERMAL CONDUCTIVITY OF NANOSTRUCTURES

1. 2. 3. 4.

. . . . . . .

Thermal Conductivity o f Single-Layer Thin Films . . Thermal Conductivity o f Superlattices . . . . . . . . Thermal Conductivity o f One-Dimensional Structures Heat Conduction in Nanoporous and Mesostructures .

. . . .

. . . . . . . .

. . . . . . . .

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

VI. PHONONENGINEERINGIN NANOSTRUCTURES . . . . . . . . . . . . . . . VII. CONCURRENTELECTRON-PHONONMODELING . . . . . . . . . . . . . . VIII. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I.

203 206 206 210 214 214 219 221 222 222 224 225 228 228 230 230 234 243 244

246 250 250 253

Introduction

Solid-state energy conversion technologies such as thermoelectric and thermionic refrigeration and power generation require materials with a low thermal conductivity (k) but g o o d electrical conductivity (a) and Seebeck coefficient (S), as determined by the dimensionless thermoelectric figure of merit Z T ( = a S 2 T / k ) (Goldsmid, 1964) for thermoelectric devices. It is well k n o w n that the best thermoelectric materials are found in semiconductors, which can have a large Seebeck coefficient and electrical conductivity. The 203 Copyright 9 2001 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-752180-1 ISSN 0080-8784/01 $35.00

204

G. CHEN

thermal conductivity of semiconductors is composed of contributions of electrons and phonons, with the latter dominating heat conduction in most semiconductor materials. Reducing the phonon thermal conductivity without causing much degradation of the power factor (S2a) thus becomes a central issue in the search of highly efficient thermoelectric materials. The largest increment in Z T occurred in the 1950s to 1960s (Ioffe et al., 1957), when the alloying method was introduced to scatter phonons that leads to a significant reduction on the lattice thermal conductivity as compared to those of their constituent materials. Examples of these materials are alloys of Si-Ge, Bi2Te3-Sb2Te3, and Bi2Te3-Bi2Se 3, which are used in today's thermoelectric coolers and power generators. Other methods to scatter phonons such as particle inclusions, grain boundaries, and the like have also been introduced, but none of them have reached a level comparable to the alloying method (Rowe, 1995). The increment in Z T from 1960s to the 1990s was thus very small. The next significant progress occurred in the 1990s when the phonon rattler concept was introduced and was used to reduce the lattice thermal conductivity in materials with high power factors (Nolas et al., 1996), particularly in cage-type materials such as skutterudites, which are reviewed in this series (Uher, 2001). While the bulk materials research has mainly focused on discovering materials with good power factors and ways to reduce the lattice thermal conductivity, new approaches emerged based on quantum effects in lowdimensional structures to increase the power factor of existing materials (Hicks and Dresselhaus, 1993a, 1993b; Harman et al., 1996). Reviews of this approach are given in Chapters 1 and 2 of this volume (Dresselhaus et al., and Broido and Reinecke). By using low-dimensionality effects, such as quantum confinement, tunneling, and stress, new structures with novel electronic transport properties can be designed with known materials (Koga et al., 1998, 1999). High Z T values have been reported in some of the quantum structures in literature (Harman et al., 1999). In addition to the use of low-dimensionality effects, thermionic emission in low-dimensional structures is also being considered for cooling and power generation applications (Shakouri and Bower, 1997; Mahan et al., 1998). More details on this topic can be found in Chapter 3 of this volume (Mahan). The low-dimensionality also offers another advantage. Starting from Casimir's work (Casimir, 1938), it has long been known that boundary scattering imposes another limit on the phonon transport. In bulk materials, the boundary scattering appears only at very low temperatures when the phonon mean free path (MFP) becomes comparable to the sample size. In thin films, boundary scattering can dominate heat transfer even at room temperature if the film thickness is comparable or smaller than the phonon MFP. Since the early investigation of the thermal conductivity of thin films (Tien et al., 1969; Nath and Chopra, 1973; Decker et al., 1984), many studies on the thermal conductivity of thin films have been carried out (Tien et al.,

5

PHONONTRANSPORT IN Low-DIMENSIONAL STRUCTURES

205

1998). A moderate thermal conductivity reduction in superlattices was predicted (Ren and Dow, 1982). The first few experimental data (Yao, 1987; Chen et al., 1994) provided on superlattice structures, however, showed a much larger reduction in thermal conductivity compared to the predictions. Modeling based on the Boltzmann transport equation indicates a large thermal conductivity reduction in quantum wells (Chen and Tien, 1993). Tien and Chen (1992) suggested that quantum structures such as superlattices can be made into super thermal insulators. Early interest in the thermal conductivity of quantum structures was primarily due to their applications in microelectronics and optoelectronic devices (Chen, 1996; Goodson and Ju, 1999), for which the low thermal conductivity of these structures is a concern for the device design and reliability. Venkatasubramanian (1992, 2000a) proposed the use of cross-plane transport in superlattices for thermoelectric applications and recently reported high Z T values in BizTe 3SbzTe3 superlattices, as summarized in this series. Although there have been increasing studies on the heat conduction mechanisms in low-dimensional structures, predicative methods are still lacking. This situation is understandable if one examines past studies on the thermal conductivity of bulk materials, most of which were based on the linearized Boltzmann transport equation (BTE) (Klemens, 1958). This phenomenological approach requires fitting to experimental data to determine the unknown coefficients in the relaxation time (Klemens, 1958; Callaway, 1959). On the other hand, it is possible to use the relaxation time derived from bulk materials to study the boundary and interface effects in thin films and superlattices (Tellier and Tosser, 1982). Other methods, such as molecular dynamics, lattice dynamics, and Monte Carlo simulation, are also being introduced to understand heat conduction mechanisms in nanostructures. In this chapter, we summarize recent research and understanding on heat conduction in superlattices and nanostructured materials that have been considered for thermoelectric applications. We start from a brief review of heat conduction in bulk materials and the phonon dispersion characteristics in low-dimensional structures to illustrate the fundamental differences between bulk materials and their nanostructures (Section II). Since measurements of the thermal conductivity of low-dimensional structures are extremely difficult, Section III provides a brief review of the experimental techniques for measuring thin film thermal conductivity. Section IV discusses the theoretical tools for modeling the thermal conduction in low-dimensional systems. Section V summarizes experimental data and modeling results on the thermal conductivity in single-layer thin films, superlattices, 1D structures, and nanoporous and mesostructures. In Section VI, we discuss the minimum thermal conductivity and strategies for engineering phonon transport in low-dimensional structures. Concurrent electron and phonon modeling is briefly touched on in Section VII, followed by a summary of this chapter.

206

G. CHEN II.

Phonons in Bulk and Low-Dimensional Materials

The thermal conductivity of a material is typically composed of two parts: contributions from electrons and phonons, (1)

k = k e -k- kp

In nondegenerate semiconductors, the electronic contribution to the total thermal conductivity is typically very small compared to that of phonons, because of the small number of carrier density. Good thermoelectric materials are usually heavily doped semiconductors with a nonnegligible electronic contribution, which is often estimated from the WiedemannFranz law (Kittel, 1996), o"

=L

keT

where L is the Lorentz number. For metals, the Lorentz number is a constant. For semiconductors, the Lorentz number depends on the Fermi level (Goldsmid, 1964). Our emphasis in this chapter is on the phonon contribution to the thermal conductivity. Phonons are the quantized normal modes of lattice vibrations. An example is the sound wave, which consists of low-frequency phonons. Heat conduction processes involve phonons of all allowable energies, as determined by the phonon dispersion relation of each material. There are typically three acoustic branches (two transverse and one longitudinal phonons) and 3 ( N - 1) optical phonons branches, where N is the number of atoms at each lattice point. In Si crystals, for example, N - 2, so there are three optical branches.

1.

PHONON THERMAL CONDUCTIVITY IN BULK MATERIALS

In a bulk material, the superposition of phonon waves leads to wave pockets carrying energy at the group velocity. These wave pockets can be treated as particles and phonon transport in a crystal is similar to that of gas molecules inside a container. Under the single-mode relaxation time approximation, the following expression for the lattice thermal conductivity in the x direction can be derived from the BTE (Ziman, 1960; Berman, 1976)

lfimx

k x = -3

1;omax

v 2 r h ~ -dT D ( co ) do9 = -~

Vx2"cC ,o d o9

(2)

where Co is the specific heat at each frequency ~, D is the phonon density of states, f is the Bose-Einstein distribution, v is the phonon group velocity,

5

PHONONTRANSPORT IN Low-DIMENSIONAL STRUCTURES

207

z is the phonon relaxation time, and e~max is the maximum phonon frequency. The preceeding expression is similar to the kinetic theory for the thermal conductivity of gases CvA

k= ~

(3)

3

where A is the M F P of gas molecules, except that Eq. (2) takes into consideration the spread of the phonon energy and different polarization of phonons in crystals. In Eq. (2), the density of states and the group velocity are derived from the phonon dispersion relation between the phonon frequency and the wave vector. These dispersion relations can be computed by considering only the harmonic force interaction among atoms and are thus relatively straightforward. On the other hand, it is more difficult to evaluate the phonon relaxation time, which is determined by the anharmonic force interaction among atoms. Different scattering processes, such as phonon-phonon scattering, phonon-impurity scattering, and phonon-boundary scattering, may dominate at different temperatures or coexist at other temperatures. The phonon-phonon scattering can be further divided into the normal and umklapp scattering. Resistance to the heat flow is caused by the umklapp process (Ziman, 1960). Although the normal scattering process does not create resistance to the heat flow, it participates in the redistribution of phonons. Callaway (1959) devised a method to take the normal process into consideration from the BTE. Studies have also been carried out to investigate the effect of defects. For example, the point defect scattering follows the familiar Rayleigh law z" ,'~ o9-4

(4)

The total relaxation time is often obtained from the Mathiessen rule (Ziman, 1960) 1 .

T,

1 .

.

.

T,p_p

1 ~ ~

"~p-i

1 +

+

...

(5)

7Jp-b

where T,p_p, T,p_i, and T,p_ b are the phonon-phonon, phonon-impurity, and phonon-boundary scattering relaxation times, respectively. Although the boundary scattering is often grouped together with other scattering mechanisms according to the Mathiessen rule, this must be done with extreme care because unlike the phonon-phonon scattering and phonon-impurity scattering, which are volumetric processes, the phonon-boundary scattering is a surface process.

208

G. CHEN lo 2

.

.

.

.

.

.

.

!

,

Holland (l 9;4) ..... ]

A

Amith et al. (1965) ]

[-, > 101 [-, L) N

Z

oL~ ,d i0 ~ <

&

00

102 101 TEMPERATURE (K)

103

FIG. 1. Thermal conductivity of bulk GaAs as a function of temperature based on Holland's model (1964). The total contributions to thermal conductivity are divided into longitudinal phonons (kla) and low-frequency (k,o) and high-frequency transverse phonons (kt,) (from Chen and Tien, 1993).

As mentioned, the direct computation of the phonon relaxation time for the thermal conductivity modeling is very difficult. The majority of past modeling work on the thermal conductivity of bulk materials is based on approximate expressions for the relaxation time, which are usually derived from Fermi's golden rule on scattering (Klemens, 1958). These expressions usually contain unknown coefficients that are determined through fitting the thermal conductivity model with experimental data. More vigorous methods that do not involve the relaxation time approximation, such as the variational method (Hamilton and Parrot, 1969) and direct calculation of scattering term in the BTE (Omini and Sparavigna, 1997) have also been developed, but they are not easy to implement. Figure 1 shows an example of the relaxation time approach used to fit the experimental thermal conductivity of GaAs bulk crystals (Chen and Tien, 1993) based on the approach developed by Holland (1964). By fitting the experimental data, one could gain information on the scattering mechanisms. At room temperature, the phonon-phonon scattering typically dominates the thermal resistance. As the temperature decreases, the impurity scattering becomes important as the strength of the umklapp scattering decreases. The decrease of the thermal conductivity at low temperatures is caused by the boundary scattering, which could be approximated as temperature independent. Thus,

5

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES

209

the temperature dependence of the thermal conductivity follows the trend of the specific heat; that is, T a at low temperatures. Equation (2) points out that the key to reduce the thermal conductivity of a given material is to shorten the relaxation time. The first major development in reducing the thermal conductivity was the alloying method developed by Ioffe (Ioffe, 1957). While it is well known that the point defects scatter phonons, the density of point defects in a crystal is not large enough to cause significant reduction in thermal conductivity at room or higher temperatures. Ioffe introduced the alloying method to increase phonon scattering. A theoretical model was established by Abeles (1963) based on point defect scattering inside a virtual crystal to explain the experimental results. In the 1990s, the phonon rattler concept was developed by Slack (Slack and Tsoukala, 1994) and successfully applied to a variety of caged structures such as skutterudites, as reviewed in this series. A convenient parameter to gauge whether a material is a good thermal conductor or thermal insulator is the phonon MFP. Often this parameter is estimated from the thermal conductivity, the specific heat, and the speed of sound, according to Eq. (2). This way of estimating the phonon MFP, however, usually leads to an underestimation of the M F P for those phonons that are actually carrying the heat because of the following reasons. 1. Phonons are dispersive and their group velocity varies from the speed of sound at the Brillouin zone center to zero at the zone edge. The average phonon group velocity is much smaller than the speed of sound. 2. Optical phonons contribute to the specific heat but typically contribute little to heat flux due to their low group velocity and their high scattering rates. 3. Phonon scattering is highly frequency dependent. High-frequency phonons are usually scattered more strongly than low frequency phonons. We use Figs. 2(a) through 2(c) to strengthen these points (Chen, 1997a). These figures compare the modeling results with experimental data of the in-plane thermal conductivity for GaAs-A1As superlattices based on different ways of estimating the phonon MFP. In Fig. 2(a), the M F P is estimated based on the simple kinetic theory, the bulk specific heat and the speed of sound, which leads to a phonon M F P about 209 A in GaAs and 369 A in AlAs. The underestimation of phonon MFP is clear from Fig. 2(a), since results for even the worst scenario (i.e., totally diffuse interfaces) lead to theoretical values much larger than the experimental results. In Fig. 2(b), we estimate the average phonon group velocity and M F P based on a sinefunction approximation to the phonon dispersion relation and totally neglecting optical phonon contributions to thermal transport. The phonon

210

G. Cm~N

It

.',':

a

,I""

js

'

1

W~. '

......

L.AYF.RT H I ~ I ~ 1~

(a)

s

0

w'

~

WTHIClCNlU(A)

1el

(b)

~e"

~ "- ~ ~" - ' - :

1104

~

s A

o.v.d.,.cn,~ I

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

W

" .-:

S

~

!'2o-.-.--~_:~.- _ 9

.,""

"----'--:;-';""'!

(A) 10 '

10 9

(e)

FIG. 2. Comparison of experimental and calculated thermal conductivity of GaAs-AIAs superlattices based on different ways of estimating the phonon M F P in bulk materials: (a) kinetic theory, (b) average phonon M F P over the acoustic branches and neglect contributions from optical phonons, and (c) frequency-dependent phonon relaxation time (from Chen, 1997a).

MFP values thus obtained are 1058 ]k for GaAs and 2248 ,/k for AlAs. The agreement with experimental data is much better. In Fig. 2(c), we show the calculated thermal conductivity based on the frequency-dependent phonon relaxation time, and again good agreement with experimental data is obtained, albeit with a very different value for the interface specularity parameter that represents the fraction of specularly scattered phonons. Naturally, one would think that the frequency-dependent treatment is more accurate. Note, however, that the frequency-dependent relaxation time is obtained by fitting with the bulk thermal conductivity data of those materials. Those bulk thermal conductivitiy models are subject to uncertainty that will propagate into the thin film models.

2.

PHONON DISPERSION IN NANOSTRUCTURES

Phonons in nanostructures are subject to additional constraints imposed by boundaries and interfaces, which may fundamentally change the vibrational characteristics. Many studies have been published on phonon dispersion in superlattices (Colvard et al., 1985; Tamura et al., 1989), quantum wells (Bannov et al., 1995), quantum wires (Nishiguchi et al., 1997), and quantum dots (Fu et al., 1999). In a superlattice, for example, the transfer matrix method (Naranyamamurti et al., 1979; Tamura et al., 1989), the elasticity equations (Colvard et al., 1985), and the lattice dynamics (Tamura et al., 1999) method have been used to obtain the phonon dispersion relations. Figures 3(a) to 3(d) show the lattice dynamics calculation of the phonon dispersion of a 2 • 2 Si-Ge-like superlattice (two monoatomic layers of Si and two monoatomic layers of Ge) in the direction perpendicular and parallel to the superlattice plane (Bao and Chen, 2000a). Compared to bulk materials, more phonon

5 PHONONTRANSPORTIN Low-DIMENSIONALSTRUCTURES 80

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t (a)

.

iBulkSir t

,

=

(b)

211

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~"

20

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0.5 ka/~

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Fxc. 3. Phonon dispersion in (a) bulk Si, (b) bulk Ge, (c) the cross-plane direction, and (d) the plane direction (100) of a 2 x 2 Si-Ge superlattice, based on a fcc lattice dynamics model and neglecting bulk optical phonons (from Bao and Chen, 2000a).

branches show up in the superlattice. This is because of the phonon folding in the reduced zone representation. One way to understand this reduced zone representation is to imagine a new unit cell for the superlattice structure that contains four atoms as a basis. The folded phonons are the optical branches of the new unit cell. Minigaps are also created at the zone boundaries and inside the Brillouin zone (Tamura et al., 1989) as a result of the phonon interference and mode conversion. The effect of phonon interference is clearly seen in Fig. 4(a), which plots the phonon transmissivity of a 5-x 5-,/k Si-Ge-like superlattice, obtained from the transfer matrix method (Chen, 1999). The stop bands correspond to the minigaps in the superlattice phonon dispersion. Despite the similarity, however, there is a subtle difference between the transfer matrix method and the lattice dynamics simulation. In a lattice dynamics simulation, one period is included as a unit cell. Each wave vector used in Fig. 3 represents the same spatial direction in both layers. In the transfer matrix method, however, a wave vector in one layer is conjugated to the adjacent layer through the Snell law that governs the refraction of acoustic waves at the interface (Auld, 1990). The latter distinction facilitates the expression of the total internal reflection and tunneling, as shown in Fig. 4(a), while the phonon spectra of superlattices only implicitly include the same processes. For quantum wells and quantum wires with rigid or free boundaries, the phonon propagating wave vector can be only in the direction parallel to the film plane or the wire axis. Transverse and longitudinal phonons in bulk materials are coupled inside quantum wells (or wires) and the normal modes are classified into dilatational, flexural, torsional, and shear modes (Bannov et al., 1995). The phonon dispersion is usually plotted as a function of the wavevector along the film plane or wire axis, as shown in Fig. 5(a) for that

212

G. CHEN

FIG. 4. Transfer matrix method calculation of (a) phonon transmissivity as a function of wavelength and incidence angle for a 5-x 5-/k Si-Ge-like superlattice and (b) thermal conductance of superlattices as a function of layer thickness (from Chen, 1999).

of a quantum well (Bao and Chen, 2000b). Clearly, the phonon dispersion in quantum wires/wells deviates significantly from that in bulk materials. One would expect that thick films or large wire diameters should have dispersion similar to the bulk materials, which is difficult to imagine from Fig. 5(a). Answers to this dilemma are as follows. First, the phonon dispersion for nanostructures is only valid if the phase coherence at each wavelength can be preserved over a length scale larger or at least comparable to the film thickness or wire diameter. The coherence length at each frequency will be determined by the strength of scattering, or roughly equal to the phonon M F P at each frequency. Second, although no traveling wave can be supported other than in the direction parallel to the film plane, each phonon branch can be thought as a result of the superposition of the

5

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES

213

FIG. 5. (a) Phonon dispersion in a free-standing Si quantum well obtained from a fcc lattice dynamics model and (b) illustration of the superposition of incident and reflected waves at the boundary that leads to a propagating wave vector parallel to the film plane (from Bao and Chen, 2000b).

incident and reflected waves at different angles coming towards the boundary, as shown in Fig. 5(b). The net effect of such a superposition is the generation of the required traveling wave. Each phonon branch corresponds to a specific incident angle. Unlike phonon dispersion in bulk materials and in superlattices, however, the minimum phonon energy in quantum wells can no longer be zero due to the requirement of standing waves in the lateral direction. From this picture, it is easy to understand that the group velocity, obtained from the phonon dispersion relations such as from Fig. 5(a), is only the group velocity projection along the in-plane direction or the wire axis direction. The modification of the phonon dispersion in nanostructures compared to those of bulk materials has several implications on the lattice thermal conductivity, as can be inferred from Eq. (2). First, the phonon group velocity of nanostructures can be different from that in macrostructures. Normally, one would associate the optical phonons with a slower group velocity and thus anticipate a smaller heat flux carried by these phonons. In superlattices, the number of optical branches increases with increasing thickness of the constituent layers. Clearly, one should not expect that all

214

G. CHEN

the optical-like phonons have a small group velocity because many branches are just folded acoustic phonons. Second, the density of states of nanostructures may be different from that of bulk materials, which will be reflected in the specific heat (Prasher and Phelan, 1998). Third, the relaxation time in nanostructures can be different from that in bulk materials. And fourth, the integration limit for each branch will also change. The impacts of these effects on thermal conductivity are discussed in more detail in later sections. Before such discussions, we turn attention to experimental techniques for measuring the thermal conductivity of low-dimensional structures.

III.

Thin Film Thermal Conductivity Measurement Techniques

Thermal conductivity characterization of nanostructures is challenging because it is difficult to establish and to measure the temperature difference over a small distance. Fortunately, a large amount of work has been done in the 1990s to develop experimental techniques for characterizing the thermal conductivity of thin films for various applications. Here, we briefly review some of the techniques used in the characterization of thermoelectric thin films, particularly superlattices. Interested readers should consult other reviews appeared in literature (Cahill, et al., 1989; Hatta, 1990; Tien et al., 1998; Volklein and Starz, 1997; Goodson and Ju, 1999).

1.

MICROSENSORMETHODS

Direct thermal conductivity measurements require the knowledge of the amount of heat input into the sample. For thin films, this is best done by lying down microheaters directly onto the sample through standard photolithography techniques established in the semiconductor industry. Since the thermal conductivity of nanostructures is generally anisotropic, one must be very careful in choosing the models, preparing the samples, and depositing microsensors and heaters. We choose two techniques to illustrate those points. a.

3co M e t h o d

An excellent method for measuring the thermal conductivity of thin films in the cross-plane direction is the 3co method (Cahill, 1990; Lee and Cahill, 1993). In this method, a heater is deposited on the thin film. An ac heating current at frequency co is applied to the heater. This generates a 2co temperature variation inside the sample and a corresponding 2co resistance change through its temperature dependence. The voltage drop across the heater thus contains a third harmonic (3co) signal that depends on the

5

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES

215

temperature rise. By measuring the third harmonics and calibrating the resistance of the heater as a function of temperature, the actual ac temperature rise of the sample at different modulation frequency can be determined. The 3o9 method can be used to determine the thermal conductivity of substrate materials as well as thin films on a substrate. The substrate thermal conductivity is often determined from the following approximate relation between the slope of the in-phase temperature rise and the modulation frequency (Cahill, 1990):

k= =

P/(2~zl) d[ln(rac,r)]/df

(6)

where P is the power input, and l is the heater length. For a thin film deposited on the substrate, only the surface temperature at the heater location can be measured. To determine the temperature drop across the film, the temperature at the interface between the film and the substrate must be determined. Under adequate modulation frequency, Eq. (6) can be applied to determine the thermal conductivity of the substrate beneath the film. After determining the substrate thermal conductivity, the temperature rise inside the substrate can be calculated. The temperature drop across the film is inferred from the measured total temperature rise and the calculated temperature rise of the susbtrate (as shown in Fig. 6), using a onedimensional steady-state heat conduction model (Lee and Cahill, 1993) P

/. 'Lr -

21b(Tt

_

7"=)

(7)

FIG. 6. Measurements of the thermal conductivity of thin films by the 3~ method for a SiN x thin film deposited on the substrate. The substrate thermal conductivity is determined from the slope and used to calculate the temperature rise of the substrate. The difference between the measured temperature and the calculated temperature is used to calculate the thin-film thermal conductivity.

216

G. CHEN

FIG. 7. Comparative3~omethod (a) illustration of samples and (b) experimental results on a Si-Ge superlattice and its buffer (from Borca-Tasciuc et al., 1999).

where b is the half width of the heater, Tr is the temperature rise at the film surface, and T~ is the calculated temperature rise in the substrate. Compared to the steady-state method, the 3co method has the following major advantages: (1) The effect of the finite substrate thickness can be avoided by increasing the modulation frequency, (2) the surface radiation loss is minimized, and (3) the lock-in detection offers high sensitivity. Measurements of the thermal conductivity of thermoelectric thin films by the 3o~ method are complicated by the fact that the films are conducting. An insulating layer must be deposited to isolate the thermoelectric film from the heater. The thermal conductivity of this insulating layer is usually unknown. In superlattices with a buffer layer, the thermal conductivity of the buffer layer is also unknown. For such situations, we found that the best way to implement the 3co technique is to use a differential method, as shown in Fig. 7(a). The reference has everything as the sample except the film to be measured. The temperature difference between the real sample and this reference sample obtained under identical heating condition is used as the temperature drop across the film. Figure 7(b) shows an example of such a differential method applied to a Si-Ge superlattice and the buffer (BorcaTasciuc et al., 1999). This differential method requires that the films in the reference sample have same properties as in the test sample, which is realized by ensuring that all samples go through identical processing steps. Special attention must be paid to the starting surface since insulating films deposited on different starting surface may have different microstructures.

5

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES 11.5

. . . .

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FIG. 8. Fitting of 309 experimental results based on one-dimensional steady-state model [-Eq. (7)] and an exact model that compensates for heat spreading in the film as well as the thermal conductivity mismatch between the film and the substrate (from Borca-Tasciuc et al., 2000a).

There are several factors that one should keep in mind when using the 309 method (Borca-Tasciuc et al., 2000a). First, the approximate formula to determine the substrate thermal conductivity is conditional and depends on the substrate thickness and the heater width, as well as the modulation frequency. If the modulation frequency is too low, the other surface of the substrate reflects the thermal wave and affects the signal. On the other hand, if the modulation frequency is too high, the heat capacity of the heater becomes important. Second, the approximate formula for determining the film thermal conductivity is also conditional and depends on the ratio of the film thermal conductivity to the substrate thermal conductivity. This point becomes clear if one considers the limit case when the film and the substrate have the same thermal conductivity. In this case, no temperature drop should be calculated or measured. This will lead to an infinite thermal conductivity of the film, according to Eq. (7). Finally but not lastly, the lateral spreading due to heat conduction inside the film will also introduce an error to the thermal conductivity. Figure 8 shows the experimental results for an Alo.9Gao.1Aso.ovSbo.93 alloy film on a GaSb substrate (BorcaTasciuc et al., 2000a) based on the one-dimensional steady-state model given by Eq. (7) and an exact formulation considering the heat-spreading effect inside the film as well as the film-substrate thermal conductivity mismatch.

218 b.

G. CHEN Membrane and Bridge Methods

The 3o9 method is most suitable for thermal conductivity measurements across the thin film plane, although there are attempts to extend the measurements to both the in-plane and cross-plane directions (Chen et al., 1998a; Kurabayashi et al., 1999). One way to measure the in-plane thermal conductivity is to remove the substrate to form free-standing membranes or bridges. We can further categorize the methods according to whether heat conduction is perpendicular to the heating wire or along the heating wire, as shown in Figs. 9(a) and 9(b). In Fig. 9(a), heat conduction along the heater creates a nonuniform temperature distribution along its axial direction. The average temperature rise of the heater, and thus the resistance variation, depends on the thermal conductivity of the heater-film composites, as well as the convection or radiation heat loss from the heater to the ambient. By measuring the average temperature rise under different geometrical or working conditions, the thermal conductivity of the film can be determined. Volklein and Kessler (1984) used the method to measure thermal conductivity of bismuth thin films. Tai et al. (1987) and Mastrangelo and Muller (1988) used microfabricated bridges to measure the in-plane thermal conductivity and thermal diffusivity of polysilicon thin films. This method is best suited if the film itself is conducting so that it also serves as the heater. Figure 9(b) is a more favorable arrangement to measure the in-plane thermal conductivity of thin films. In this method, energy dissipated in the metallic or semiconducting heater is conducted through the membrane surrounding the heater. If radiation or convection is neglected, the thermal conductivity of the film can be easily determined from the measured temperature rise of the heater (Volklein, 1990). The radiation effect can be further reduced through transient measurement or through periodic heating of the heater. Alternatively, arrays of heaters can also be

FIG. 9. Illustration of membrane and bridge methods based on heat conduction (a) along the heater bridge and (b) perpendicular to the axis of the heating wire.

5

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES

219

used to measure temperature distribution (Zhang and Grigoropoulos, 1995; Venkatasubramanian et al., 1998), from which thermal conductivity and dittusivity of the film can be determined. Since thermoelectric materials have low thermal conductivity values, it is important to design the structure and experimental conditions such the heat loss through the heater into the substrate is minimized.

2.

OPTICAL PUMP-AND-PROBE METHODS

The optical heating methods (optical pumping) use radiation energy as the heat source. Because it is usually difficult to determine exactly the amount of heat absorbed in the sample, optical-heating-based methods typically use the dynamic response of the sample under time-varying heating conditions. From the temporal response caused by the sample temperature rise, the thermal diffusivity or thermal effusivity (the product of the thermal conductivity and the volumetric specific heat) of studied samples are often obtained, rather than direct measurements of the thermal conductivity of the sample. The variation of the sample temperature under time-varying heating conditions generates different signatures that can be detected (probed) and used for obtaining the thermal diffusivity/effusivity of the samples. Examples of these signatures (as shown in Fig. 10) are the thermal emission, refractive index change in both the sample and the surrounding media, and the thermal expansion in both the solids and the surrounding media. All these signatures have been explored in the 1980s and 1990s to determine thin film thermal diffusivity or effusivity. Conference proceedings and monographs on photothermal and photoacoustic phenomena are excellent resources on these methods (Mandelis, 1992; Scudieri and Berlotti, 1998). We briefly discuss some of these methods according to the heating mode.

FIG. 10. Illustration of selected photothermal signals that have been used in pump-andprobe techniques for thin film thermal diffusivity or effusivity determination.

220 a.

G. CHEN Pulsed Heating Methods

For thin film thermophysical properties measurements, it is usually desirable to have the pulse length shorter than the thermal penetration depth of the film. Order of magnitude speaking, this can be put as d2

t. <

(8)

where d is the film thickness, and ~ is the thermal diffusivity of the film. For a film thickness of 1 #m and thermal diffusivity 10 -6 mZ/s, the pulse range should be less than 10 -6 s to avoid the influence of the substrate. Commercial available laser flash apparatus typically cannot reach such a temporal resolution. In the past, femtosecond, picosecond (Paddock and Eesley, 1986; Capinski et al., 1999), and nanosecond lasers (Goodson et al., 1995) were used in the measurement of the thermal diffusivity of thin films. The temperature response is typically probed with another laser beam through the change of the reflectivity of the samples caused by the temperature dependence of the refractive index. For nanoscale laser heating, the temporal temperature response can be directly measured with fast radiation detectors. The temperature variation generated by a single pulse is captured, but averaging of many pulses is required to reduce the noise. Because femtosecond and picosecond lasers do not have matching detectors, the temporal response is often measured by a delayed probe beam that measures only one point for each heating pulse. Thermal response to a single pulse is inferred from repetitive measurements of the probe response to many identical heating pulses. Uncertainty has been estimated around 10% in the reported experimental data for GaAs-A1As superlattices with picosecond pump-andprobe experiments (Capinski et al., 1999). b.

Periodic Optical Heating Methods

By periodically heating the samples, the signal detection can be accomplished through the use of lock-in amplifiers. Depending on the signature detected, various pump-and-probe methods have been developed and named differently (Hess, 1989; Scudieri and Berlotti, 1998). In the photothermal method, the reflectance change or the thermal emission from the samples are measured (Wu et al., 1993). In the photoacoustic method, the acoustic waves generated by the heating of the sample, either in the surrounding ambient or through the samples, can be measured and used to fit the thermophysical properties of the sample (Hu et al., 1999). The photothermal displacement method measures the thermal expansion of the sample (Bertolotti et al., 1994), while the mirage method uses the deflection

5

PHONONTRANSPORT IN Low-DIMENSIONAL STRUCTURES

221

FIG. 11. Temperature distribution (a) amplitude and (b) phase surrounding a modulated focused laser beam (15 kHz) for a SiO2 film deposited on Si substrate (from Borca-Tasciuc and Chen, 1998).

of laser beams when passing through a temperature gradient (Kuo et al., 1986). Signals are detected in the frequency domain, that is, as a function of frequency. The thermal penetration depth, determined by the modulation frequency ~(o~/yr,f) 1/2, is typically longer than the film thickness so that the substrate effect must be taken into account in most of the experiments. If the properties of the substrate are not accurately known, a large uncertainty will be incurred in the film properties. One possibility to overcome this difficulty is to measure the temperature distribution (phase and amplitude) surrounding a tightly focused laser beam at different frequencies (Hartman et al., 1997; Borca-Tasciuc and Chen, 1998). It is possible to fit properties of thin films as well as the substrate. Figure 11 shows an example of the phase and amplitude distribution surrounding a modulated laser beam.

3.

OPTICAL-ELECTRICAL HYBRID METHODS

Hybrid methods that combine the electrical heating and optical detection or vice versa are also developed. One example is the ac calorimetry method, in which a laser beam is used to heat the sample and the detection is done by small thermocouples or sensors directly patterned onto the sample (Hatta, 1990; Yu et al., 1996), as shown in the insert of Fig. 12. The distance between the laser and the sensor is varied. Under appropriate conditions, the thermal diffusivity along the film plane can be calculated from either the phase or the amplitude data,

% = (d lnlOI/dx) 2

% = (dq)/dx) 2

(9)

Figure 12 gives an example of the amplitude data for GaAs-A1As superlat-

222

G. CHEN 0.20

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.

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POINT SOURCE, [Inl Ol0ol+0.51n(rlro )]lftr2

UNIFORM ILLUMINATION, Inl 0/0o]/P ~ /

r~ 0.15

LINE SOURCE, In[ 010oJ/f lr*

_w .J o. =i < a 0.10 uJ N .J

,.......

7.

"

a::: 0 0.05 z

0.0(]

0

50

100

150

200

250

300

350

DISPLACEMENT FROM TEMPERATURE SENSOR(pm)

FIG. 12. Amplitude signal as a function of the relative displacement of a laser beam away from the thermocouple in the ac calometry measurements of the in-plane thermal diffusivity of thin films (from Yu et al., 1996).

tices (Yu et al., 1996). Cross-plane thermal diffusivity can also be measured, as demonstrated by Chen et al. (1994). Alternatively, the sample can be heated with an electrical signal and the thermal signal can be detected by the reflectance change (Goodson et al., 1995) or thermal emission from the sample (Indermuehle and Peterson, 1999).

IV.

Analytical Tools

After the discussion in Section II on the difficulties involved in the modeling of the bulk thermal conductivity and the phonon dispersion in nanostructures, it is easy to appreciate that modeling and quantitative prediction of the thermal conductivity in nanostructures will encounter similar problems inherent in bulk materials as well as new issues related to the surfaces and interfaces. In this section, we discuss some of the tools that have been used to understand and to explain experimental data.

1.

LATTICE DYNAMICS AND PHONON DISPERSION ANALYSIS

The phonon dispersion in nanostructures is very different from that in macrostructures, as discussed in Section II. The new dispersion relation modifies the phonon density of states (DOS), group velocity, scattering

5

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES 8

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FIG. 13. Comparison of phonon (a) D O S and (b) thermal energy propagating factor (v 2) of a 5 x 5 S i - G e superlattice with those of bulk Si and Ge (from Bao and Chen, 2000a). The (v 2) is the square of the group velocity in a specific direction (x is parallel to the film plane and z is perpendicular to the film plane) weighed against the density of states.

mechanisms, and the like in nanostructures. All the dispersion analysis methods, such as the lattice dynamics, the continuum mechanics, and the transfer matrix method, however, are based on the assumption of harmonic force interaction between atoms, which does not lead to the calculation of phonon relaxation time. Several publications consider the effect of phonon group velocity change on the thermal conductivity of superlattices (Hyldgaard and Mahan, 1997; Tamura et al., 1999; Bao and Chen, 2000a), quantum wells (Balandin and Wang, 1998), and quantum wires (Khitun et al., 1999). Figure 13(a) shows an example of the DOS of Si-Ge superlattices, which is roughly the arithmetic average of those of bulk Si and Ge. This is consistent with the specific heat calculation of superlattices, which shows very little size dependence (Grille et al., 1996). Although the DOS does not change much with size, the projection of the group velocity in certain specific directions may change significantly in nanostructures, leading to a large reduction of the lattice thermal conductivity in these directions. Figure 13(b) shows the thermal energy propagation factor, defined as the product of the square of the phonon group velocity and the density of states, vp2 x DOS in Eq. (2), as a function of the frequency for Si-Ge superlattices in directions parallel and perpendicular to the superlattice film plane. The cross-plane direction shows a drop of about a factor of 5 compared to the average of the corresponding bulk values for Si and Ge, while that in the in-plane direction it only drops to 90% of the bulk values. The relatively large reduction in the cross-plane direction, although still smaller than that observed in certain samples, puts the group velocity reduction as a potential cause for explaining the experimental data. In the

224

G. CHEN

in-plane direction, however, the relatively small reduction of this factor clearly cannot explain the experimental data. Although the phonon spectrum analysis gives physical insights on the potential effects of group velocity reduction on the lattice thermal conductivity, a comparison of results from such an analysis with experimental data is at most partially successful. The thermal conductance predicted from the consideration of the thermal energy propagation factor decreases with increasing period thickness (Tamura et al., 1999; Bao and Chen, 2000a) and approaches a constant. This is due to the phonon tunneling effect, as is seen in Figs. 4(a) and 4(b). Experimentally, it is found that the thermal conductivity of superlattices decreases with decreasing period thickness if each layer is thicker than a few monolayers. In BizTe3-SbzTe3 superlattices, the thermal conductivity recovers to a higher value after passing through a minimum at ,-~60A (Venkatasubramanian, 2000a). The lattice dynamics calculation and elastic wave calculation (Chen, 1999) can explain the recovery of thermal conductivity at very low layer thickness but cannot explain the experimental results on thicker films. Possible reasons for this discrepancy are (1) the relaxation time change in nanostrucutures, (2) the loss of phonon coherence, and (3) diffuse and inelastic interface scattering. While it is possible to include the first effect within the thermal conductivity calculation, such a consideration is unlikely to reverse the trend since the phonon DOS becomes closer to those of their bulk materials as the layers become thicker. The last two factors can be better taken into consideration by treating phonons as particles that can be described by the BTE. Simkin and Mahan (2000) introduced an imaginary wave vector in the lattice dynamics model. Results from such a treatment could lead to qualitative agreement with experimental data of BizTe3-SbzTe 3. This method is equivalent to the introduction of an absorption coefficient in optical waves. It has the effect of reducing the coherence and tunneling effect, but suffers from the drawback that an imaginary wave vector cannot impose the energy conservation for phonon propagation. One can appreciate this point by considering a damped optical wave due to absorption, which converts photons into heat so that photon numbers are not conserved. In reality, the phonon scattering reduces their coherence but the scattered phonons continue to propagate. A more vigorous treatment can be conceived based on the fluctuation-dissipation theory that are developed for photons (Rytov, 1959), and extend it to phonon propagation that includes both absorption and scattering (Chen, 1997b).

2.

BOLTZMANNTRANSPORT EQUATION

The BTE is by far the most popular way to explain thermal conductivity of nanostructures, despite various drawbacks mentioned in Section II.

5

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES

225

Similar to the thermal conductivity of bulk materials, the single-mode relaxation time approximation is often invoked, which leads to the steadystate Boltzmann transport equation as

v'Vff =

f-fo

(10)

where fo is the Bose-Einstein distribution. For bulk materials, the firstorder perturbation is often adopted, V~f ~ V~fo = VT"

dfo dT

(11)

The size effect persisting in bulk materials at low temperatures is often lumped into the relaxation time according to the Mathessien rule, as indicated in Eq. (3). The application of the Mathessien rule to the boundary scattering, however, is highly questionable when both the internal scattering and the boundary scattering coexist (Ziman, 1960). This is because the internal scattering processes, such as phonon-phonon and phonon-impurity scattering, are volumetric while the boundary scattering is clearly not. By solving the BTE with the boundary scattering as an interface condition, Chen and Tien (1993) showed that for heat conduction along the in-plane direction of a thin film, the boundary scattering term must be combined with the internal scattering to make the Mathessien rule applicable. For heat conduction in the direction perpendicular to the film plane, their study suggests that the rule is more appropriate. For phonon transport in nanostructures, the approximation made in Eq. (11) is no longer true in the directions that limit the size of the structure, such as the thickness direction for thin films, because phonon distribution deviates significantly from equilibrium in those directions. Solving the BTE under appropriate boundary conditions thus seems to be a natural approach. A typical BTE approach would involve determining the phonon relaxation time in bulk materials, applying the BTE to the considered nanostructures and the heat flow direction, establishing the boundary conditions, and solving the BTE for the specified geometry. For many nanostructures and heat transfer configurations, past solutions of the BTE for electron transport, neutron transport, and photon transport can be applied (Tellier and Tosser, 1982; Majumdar, 1993).

3. BOUNDARY CONDITIONS FOR B T E

Establishing the correct boundary conditions for BTE is the crucial step for analyzing size effects in nanostructures. This requires a clear understand-

226

G. CHEN

MEDIUM

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"~ I(':.,'-.

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REFLECTiVITY

0.20

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[",. 9

TRAI~ISMI.SSIVITY 9g "00 30 sO INCIDENT ANGLE (DEGREE) Co)

FIG. 14. (a) Schematic illustration of phonon reflection and transmission at an interface and (b) calculated reflectivity and transmissivity at an interface similar to Si-Ge for a transverse phonon polarized in the plane of incidence (SV phonon) coming from the Ge side, showing the mode conversion [from SV wave into a longitudinal wave (L)] and the total internal reflection phenomena (from Chen, 1999).

ing of phonon-scattering processes at boundaries and interfaces. Most studies of phonon-scattering mechanisms at interfaces come from the past investigation of the thermal boundary resistance, or Kapitza resistance, that exists at the interface of two materials when heat flows across the interface (Little, 1959; Swartz and Pohl, 1989). This thermal boundary resistance originates from phonon reflection at an interface. For the long-wavelength acoustic waves, the phonon reflection is due to the difference between the acoustic impedance across the interface of two materials. Here the acoustic impedance is defined as the product of the mass density and the speed of sound. In the most general case [as illustrated in Fig. 14(a)] an incident acoustic wave may excite three reflected and three transmitted phonon waves of different polarization (Auld, 1990). The phonon reflectivity and transmissivity can be derived from the continuity requirements for the atom displacement and the traction force at the interface. Figure 14(b) shows the calculated phonon reflectivity and transmissivity at an interface similar to that between Si and Ge, albeit with isotropic properties such that the two transverse phonon modes are degenerate (Chen, 1999). The incident phonon is from the Ge side with a transverse wave polarized in the plane of incidence. This figure shows (1) the mode conversion and (2) the total reflection phenomena. Some of the transverse phonons are converted into longitudinal phonons. For incident angle higher than ~ 33 ~ the critical angle, total internal reflection occurs. Beyond the critical angle, an evanescent wave exists in the Si side. Such an evanescent wave does not carry net energy flow into the second medium, but it can become propagating in multilayer structures due to tunneling, as shown in Figs. 4(a) and 4(b).

5 PHONONTRANSPORTIN Low-DIMENSIONALSTRUCTURES

227

Once the phonon reflectivity and transmissivity at an interface are known, the thermal boundary resistance can be calculated (Little, 1959). Predictions for the thermal boundary resistance, based on the perfect acoustic-mismatch model, are in reasonably good agreements with experimental data at low temperatures, but often fail at high temperatures (Swartz and Pohl, 1989). One possible reason is that at high temperatures, the dominant phonon wavelength is shorter and the interface roughness becomes more important, causing diffuse scattering of phonons. Another possible source of discrepancy lies in the mismatch of the phonon spectra between the two materials. Due to the mismatch in the phonon spectrum, the high-frequency phonons must split into lower-frequency phonons to transmit into the adjacent materials, or they will be confined inside the original medium, causing total reflection. Past work indicates that inelastic scattering should be taken into account in explaining the experimentally measured thermal boundary resistance at high temperatures (Stoner and Maris, 1993). Modeling of the diffuse and inelastic interface scattering is difficult. A highly idealized model was proposed by Swartz and Pohl (1989) as the diffuse limit at which interface scattering makes the origin of reflected and transmitted phonons indistinguishable. This leads to simple expressions for the phonon reflectivity and transmissvity at interfaces. Chen (1998) extended these models to include the effects of high temperatures and inelastic scattering. The acoustic mismatch and the diffuse scattering models are the specular and diffuse limits of an interface. Clearly, the phonon-scattering processes will depend on the wavelength and interface conditions, as well as the interatomic potentials between atoms. A compromise between the two limits is the partially specular and partially diffuse interfaces. A highly idealized model was established by Berman et al. (1955) based on pure surface roughness consideration, leading to the following expression for the surface specularity parameter, defined as the ratio of the specularly reflected phonons to the total incident phonons:

p = 1 - exp

87~2~2) ,~2

(12)

where r/is the surface roughness, and 2 is the incident phonons wavelength. Clearly, whether the interface is specular or diffuse depends on the phonon wavelength. By assuming partially specular and partially diffuse interfaces, boundary conditions can be established for the BTE. Chen (1997a, 1998) used this approach for superlattices in the in-plane and cross-plane directions, assuming a frequency-independent specularity parameter. Asheghi et al. (1998) considered the frequency dependence in the modeling of the thermal conductivity of single-layer Si crystalline thin films.

228 4.

G. CHEN MONTE CARLO SIMULATION

Several papers published in the past used the Monte Carlo method to simulate heat conduction in a confined space (Klitsner et al., 1988; Peterson, 1994). The Monte Carlo simulation needs the relaxation time as in the BTE approach. The advantage of this method is that the simulation code can be easily adapted to complex geometries. So far, the few reported Monte Carlo simulations are for simple geometry, for which solution of the BTE leads to identical results (Majumdar, 1993).

5.

MOLECULARDYNAMICS SIMULATION

The molecular dynamics (MD) simulation provides one potential route for the direct computation of the thermal conductivity. The MD techniques trace the trajectory of individual atoms and calculate the transport properties from the atomic trajectories. Unlike modeling based on BTE and the relaxation time approximation, the MD-based modeling does not require additional parameters other than a good interatomic potential. Despite the quite large volume of existing literature on the MD simulation of thermal conductivity, few studies have been reported on crystalline materials, presumably due to the long phonon mean free path in these materials and, consequently, the perceived large simulation domain required for performing such a calculation. The MD approach, however, is ideal for nanostructures that have significantly smaller number of atoms. Figures 15(a) and 15(b) show examples of the thermal conductivity simulation based on the Green-Kubo formalism for bulk silicon crystals and Si nanowires, respectively (Volz and Chen, 1999, 2000). The simulated thermal conductivity of bulk Si crystals increases with the simulation size, demonstrating the insufficiency of the number of atoms in the simulation domain. The simulated thermal conductivity of Si nanowires, however, approaches a constant as the wire length increases, suggesting that direct simulation of the thermal conductivity is feasible. Even for bulk crystals such as Si, our recent studies (Volz and Chen, 2000) show that by combining modeling with the MD simulation, good agreements with experiment can be obtained, as shown in Fig. 16. Unlike the BTE-based approach, the modeling-based MD method requires only the interatomic potential. The MD simulation of thermal conductivity of superlattices has also been explored (Volz et al., 1999; Liang and Shi, 1999). Although the classical MD simulation could include phonon wave effects within the accuracy of the harmonic oscillator approximation, it may not correctly account for the scattering mechanisms due to the differences between the Bose-Einstein and the Boltzmann distribution functions. We

5 25

80

Section:

500K

60

A 2.14nm x 2.14nm

A

229

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES

~" 20 E

,

40 20

t t

"0 C

+ § +

0

+

0

9

i

i

3

6

9 12

Transv. Dim. (nm]

8 10 0 ,C

~.

5

0 400

.

.

.

.

,

.

.

.

.

.

450

.

9

.

.

.

,

.

.

.

.

,

500 550 Temperature (K)

.

.

.

.

600

650

(a)

~" 3.5

1.61nmx1.61nm A 2.14nmx2.14nm 2.68nmx2.68nm (~ Temperature : 200K 5.37nmx5.37nm [7

E

3 3

t t

Sections: Temperature = 300K

2.5

U

= "o e~

o

u

2

1.5

m

ID ,1=

~"

0.5 9

.

,

,

|

.

.

.

.

|

,

|,

J

9

|

.

.

.

.

|

5 10 15 20 N a n o w i r e Length (Lattice C o n s t a n t s )

,

,

,

,

25

(b)

FIG. 15. Molecular dynamics simulation of the thermal conductivity of (a) bulk Si crystals and (b) Si nanowires based on the Green-Kubo formalism. Thermal conductivity of bulk Si increases with increasing simulation domain size due to limitations of the number of atoms in the simulation domain, while the thermal conductivity of nanowires approaches a constant as the length increases, indicating simulation size does not affect the final results for nanowires (from Volz and Chert, 1999, 2000).

anticipate that the M D technique will be valid at high temperatures, where the difference between these two statistical distribution functions is small. The quantum M D simulation, although it may overcome these problems, is currently too limited in terms of the number of atoms it can deal with.

230

G. CHEN 300 o

o

Section: 21.4 A x 21.4 A

A

E 250 Natural Si

>

0,.

200 o "1o o o 150

.tt

t~

E o ,,c

~- 100

0

280

9

-

-

|

!

380

|

9

9

9

|

480 Temperature (K)

.

.

.

'-t .

i

580

FIG. 16. Molecular dynamics simulation of the thermal conductivity of bulk silicon crystals by combining the spectral Green-Kubo formalism with modeling, suggesting the MD method as a viable tool to directly compute the thermal conductivity of nanostructures as well as bulk crystalline materials (from Volz and Chen, 2000).

V.

Thermal Conductivity of Nanostructures

In this section, we discuss experimental, theoretical, and modeling studies on the thermal conductivity of low-dimensional structures related to thermoelectric applications, including single-layer thin films, superlattices, onedimensional nanowires, and mesoscopic and nanoporous structures.

1.

THERMAL CONDUCTIVITY OF SINGLE-LAYER THIN FILMS

In the 1970s, a series of experiments on the thermal conductivity of metal thin films and metal-semiconductor multilayers were reported (Chopra and Nath, 1974; Nath and Chopra, 1973). A number of studies have since been reported on the thermal conductivity of semiconductors and dielectric thin films. Here, we focus on the study of thermoelectric thin films. Other excellent resources for this topic can be found in several review articles (Tien et al., 1998; Goodson and Ju, 1999). Among thermoelectric thin films, Bi thin films have drawn most attention. Abrosimov et al. (1974) and Volklein and Kesseler (1984) measured the in-plane thermal conductivity of polycrystalline Bi thin films deposited through thermal evaporation, as well as other thermoelectric properties.

5

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES

231

Although bulk single-crystal Bi has a fairly small thermal conductivity, say, 5-10 W/mK, depending on the crystallographic direction (Gallo et al., 1963), the measured thermal conductivity shows size dependence for films as thick as ~ 1000 nm. Abrosimov et al. (1974) observed a peak in the thermoelectric figure of merit for Bi films with a thickness ~ 100 nm and a systematic shift of the peak toward large thickness as temperature decreases. Such a peak behavior, however, was not seen in Volklein and Kessler's data (1984). Volklein and his coworkers also studied the in-plane thermal conductivity and other thermoelectric thin films including Sb thin films (Volklein and Kessler, 1990), BixSbl_ x thin films (Volklein and Kessler, 1987), and (Bil_xSbx)2Te 3 thin films (Volklein et al., 1990). For both the Bi and the Sb thin films, after substracting the electronic contribution to the thermal conductivity according to the Wiedemann-Franz law, the phonon contribution to thermal conductivity increases with increasing temperature. In the cross-plane direction, Baier and Volklein (1990) measured the thermal conductivity of Bio.sSb~.2Te 3 films between 50 and 1000 nm, but no thickness dependence was reported. The thermal conductivity at room temperature is ~0.37 W/mK, lower than bulk alloys (Goldsmid, 1964). Song et al. (1999a) measured thermal conductivity of polycrystalline C o S b 3 and IrSb 3 thin films and their alloy films. Figure 17 shows the temperature dependence of the thermal conductivity of these films and a comparison with those of their bulk counterparts. A significant reduction in the thermal conductivity is observed in comparison to their bulk counterparts, even for

10 2

....

, ....

, ....

, ....

, ....

, ....

A E ~101 >

I::) r~

(.1 z O o10 o

~7

..J < 5 n,

Ir.sCOo.sSb3 (150 nm) Iro.sCo0.sSba (Bulk, esL) 23 Ir0.12C00 Sbs (Bulk, exp.) 23

IrSb 3 (Bulk) 6 C o S b (Bulk): IrSb (175 nm) CoSb. (110 nm)

uJ 2: I10-1

. . . . . . . .

50

100

. . . . . . .

,

. . . .

150 200 TEMPERATURE

,

250 (K)

. . . . . . . . . 300

350

FIG. 17. Cross-plane thermal conductivity of CoSb 3 and IrSb 3 thin films and their alloys as a function of temperature and comparison to that of their bulk counterparts. The thermal conductivity can be lower than filled skutterudites (from Song et al., 1999a).

232

G. CHEN

the alloy films. More interestingly, the thermal conductivity of the alloy film is comparable to filled skutterudites. Volklein and Kessler (1986) developed a model to explain the size dependence of the in-plane phonon thermal conductivity in Bi thin films. Their model is based on the well-established Fuch theory (Tellier and Tosser, 1982) for the electrical conductivity of thin films but takes into consideration the grain boundary scattering. Quantum perturbation theory is used to evaluate the strength of grain boundary scattering, and the final expression for the scattering rate obtained depends on the angle and the grain size. The grain boundary scattering is combined with background scattering according to the Mathiessen rule to obtain the internal relaxation time. The surface effects are included through the Fuch's solution for the BTE in a thin film. The grain boundary scattering was also considered by Goodson (1996) in the study of diamond thin films, through the consideration of phonon trajectories. Although Volklein and Kessler's model leads to good fitting with experimental results, the fitting depends on the estimation of the relaxation time in bulk materials. In their work, Volklein and Kessler (1986) used the Debye model to fit the relaxation time in bulk Bi, which will overestimate the group velocity of optical phonons and acoustic phonons close to the boundary of the first Brillouin zone. As discussed in Subsection 2 of Section II, this leads to the underestimation of the phonon MFP in bulk materials. More evidence of such an underestimation is provided in the experimental and theoretical study of the thermal conductivity of single crystalline silicon thin films (Ju and Goodson, 1999), which shows that the phonon MFP can be as long as ~ 300 nm, close to that estimated by Chen (1998). Chen and Tien (1993) developed a model for the thermal conductivity of quantum wells based on the BTE for both the in-plane and the cross-plane directions. The in-plane thermal conductivity is based on the Fuch solution, while the cross-plane direction is based on the approximate solution for the photon transfer equation. Although the original paper stated that the thermal boundary resistance between the quantum well and the cladding media is not included, the approximate solution of the BTE employed actually included this resistance. Figure 18 shows the modeled cross-plane thermal conductivity of A1As-GaAs-A1As quantum wells. The significant reduction of the thermal conductivity is due to the thermal boundary resistance. The BTE approach, however, does not consider the phonon spectrum change in the thin film. It is natural to ask what the effect is of phonon interference, tunneling, and confinement on the thermal conductivity. The impact of phonon interference and tunneling, however, seems to be quite small in single-layer structures, as shown by Chen (1999) through considering phonon propagation through a G e - S i - G e sandwich structure in the cross-plane direction. Figure 19 illustrates the calculated phonon heat

5 10 2

i

233

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES

.

.

.

.

.

.

.

.

.

!

.

.

.

.

.

.

.

.

i~

.

0.729cm or-1 0

~'

'

"

....,~'-'"'O,~t~.~ /" ~ ~ / / " O '~O ~

e 10 1

.

.

.

.

.

.

.

0.729 cm, (x=0.41 0.I lam, or=0.41 -] 20 nm, or=0.41 20 nm, ot=l.0 | Holland(1964) ":[

~ ~ O

[,,., G~ 10 0 Z~ @ ,< 10 -1

"~,.,.,.,

10 -2

10 -3 10 0

.

.

.

.

.

. 10 1

.

.

.

.

~ 10 2

. . . . . . . .

lO 3

TEMPERATURE (K) FIG. 18. Modeled thermal conductivity of GaAs quantum well in the cross-plane direction (from Chen and Tien, 1993).

0.035 '7

'

'

'

I

'

'

,

~

I

t

'

'

'

'

I

'

'

'

'

TOTAL AT 300 K

,?,

[i

~: 0.030

TOTAL

M

Z .<

0.025

[-

Z~ @

0.020

< ~1 [.,,

0.015

RAY TRACING m . a i m ~ m

0.01e 0

i

....

m

i

m .... 5

T m

m

i

O ==

u

T ~

I .... 10

m

A a

m

L --

,~,,m

~ ~,

== ,=, ,m

,=

m .... 15

20

A) FIG. 19. Thermal conductance of a G e - S i - G e heterostructure obtained from the transfer matrix method that includes the phonon coherence effect and from the ray tracing that does not include the coherence effect. The increase of conductance is due to the phonon tunneling above the critical angle (from Chen, 1999).

234

G. CI-IEN

conductance based on acoustic wave propagation and based on ray tracing. The two approaches lead to same value after a few monoatomic layers. For films within a few monolayers, phonon tunneling increases the conductance.

2.

THERMAL CONDUCTIVITY OF SUPERLATTICES

The first measurement on the thermal conductivity of superlattices was reported in 1987 (Yao, 1987). The in-plane thermal conductivity of GaAs-A1As superlattices with equal thickness of GaAs and AlAs from 40 A to 500 A was measured at room temperature, and it was found that the thermal conductivity generally decreases with decreasing layer thickness (Fig. 2). Chen et al. (1994) reported the thermal diffusivity of a GaAsAlo.67Gao.33As vertical-cavity surface-emitting laser structure. Figure 20 shows a transmission electron microscopy (TEM) viewgraph of the structure (Walker et al., 1991). The Alo.67Gao.33As layer is approximated by 6 x 3 short period superlattices. Interfaces are also digitally graded for A1 content from 0 to 0.67. The measured thermal diffusivity for such a structure in the in-plane direction is 0.062 cmZ/s while in the cross-plane direction is 0.026 cmZ/s. The in-plane thermal conductivity is larger than that of the corresponding bulk alloys for the studied structures. The cross-plane direc-

FIG. 20. TEM of the laser structure studied by Chen et al. (1994) (from Walker, 1993, courtesy of Dr. Walker).

5

235

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES

tion, when converted into thermal conductivity, is ~4.5 W/mK, smaller than their corresponding alloy ( ~ 10 W/mK). Yu et al. (1995) reported the temperature dependence of the in-plane thermal conductivity of a 700 ]k700~ GaAs/A1As superlattice. All these data were obtained using modulated radiation heating. Capinski and coworkers reported the thermal conductivity of GaAs-A1As in the cross-plane direction measured with a picosecond pump-and-probe method (Capinski and Maris, 1997; Capinski et al., 1999). Their data clearly show that the thermal conductivity of superlattices may be smaller than that of their alloys. Recent studies on the thermal conductivity and heat conduction mechanisms in superlattices were summarized by Chen and coworkers (Chen et al., 1998b, 1999). Several papers reported the thermal conductivity of Si-Ge materials. Lee et al. (1997) measured the cross-plane thermal conductivity of Si-Ge superlattices grown by metal-organic chemical vapor deposition (MOCVD) on GaAs substrate with a Ge buffer layer. They found that the thermal conductivity increases with increasing period thickness until it reaches a peak and then drops sharply after 200 A. They attributed the drop to the relaxation of the superlattice and the associated high dislocation density. The cross-plane thermal conductivity is found to be much smaller than that of Sio.sGeo. 5. The in-plane thermal conductivity of similar Si-Ge structures was reported by Venkatasubramanian et al. (1998). A pronounced dip was observed in their thermal diffusivity data at a period of 66 A. The thermal conductivity is also found to be smaller than that of Sio.sGeo. 5 for certain thickness range. Because these samples were grown on a Ge buffer, the Si and Ge layers are not of equal thickness. Borca-Tasciuc et al. (1999) studied symmetrized Si-Ge superlattices grown on Sio.sGeo.5 buffers by molecular beam epitaxy (MBE). The cross-plane thermal conductivity behavior is similar to those of MOCVD superlattices, as shown in Fig. 21.

....

, ....

j ....

, ....

, ....

| ....

Tempar'd~e

E

A

~ 8 7 K

9 9 9

9

9

,

~3 8

$

|2 50

o

o

9

$ 9

9

9 $ 9 O

-~K

9

$ 9 O

, O

~as

$ 9

9 JL158 (4Anm) 9JL187 (9nm) 9 JL155 (14nm) 9 lpm SiGe Alloy Film (Lee et al 1997) 100 150 200 250 300 350 400 450 Temperature (K)

(a)

;~k

"0 co u

I

"~-

Lm st al. 1987

'~~.~

3

2.8 (D e-

I-

~,~

2

0

SO

. . . . . . . . . .

100 180 200 280 ~ c e period (J~)

300

Co)

FIG. 21. Cross-plane thermal conductivity of MBE-grown Si-Ge superlattices: (a) temperature dependence and (b) thickness dependence (from Borca-Tasciuc et al., 1999).

236

G. CHEN

The cross-plane thermal conductivity of the thermoelectric superlattice BizTe3-SbzTe3 is also reported (Venkatasubramanian, 1996, 2000; Yamasaki et al., 1998). The thermal conductivity is found to be smaller than that of their corresponding alloy. It decreases with decreasing layer thickness until when the period thickness ~60./t, below which the thermal conductivity actually recovers to higher values. Phonon transfer matrix calculation (Chen et al., 1998b and Chen, 1999) suggest that this recovery is due to phonon tunneling above the critical angle of incidence, as indicated in Fig. 4(b). This is also consistent with the lattice dynamics calculation (Tamara et al., 1999; Bao and Chen, 2000a; Simkin and Mahan, 2000) although the latter cannot clearly show the tunneling phenomena, as explained in Subsection 1 of Section II. More recently, the thermal conductivities of several other materials systems are also measured, including pulsed-laser deposited CoSb3-IrSb3 (Song et al., 1999a) and MBE-grown InSb-A1Sb superlattices (BorcaTasciuc et al., 2000). The CoSb3-IrSb 3 superlattice thermal conductivity is shown in Fig. 22. The thermal conductivity of the skutterudite superlattices is comparable to that of an alloy sample with comparable thickness but it should be remembered that these values all seem to be lower than those of their corresponding bulk materials. Compared to Si-Ge and CoSb3-IrSb3 superlattices, the thermal conductivity of InSb-A1Sb superlattices show a more complex temperature dependence that resembles those of single

4.0

Thickness [Film (nm)/Period (nm)]

~'3.5 3.0 > F_.2.5

---c>- IrSb31CoSb3 (175/6) IrSb31CoSb3 (205/14) --C-- IrSbJCoSb 3 (160/16) IrSb31CoSb3 (140/25)

~ ~

IrSb31CoSb3 (225/75) Ir.sCo0.Sb3 (150) , IrLaGe3Sb I (Bulk) 6

a z2.0 0 o ..i < 1.5 =i iv, uJ zl.0 i-. 0.5

50

100

150 200 T E M P E R A T U R E (K)

250

300

FIG. 22. Cross-planethermal conductivity of skutterudite superlattices (from Song et al., 1999a).

5

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES

237

3.6 .

.. ~,3.4 E ~3.2

~ 3.0 o

-,-vtv NOT ANN. o o~

oooOOooooo_ ~ "~176 o

-:TGRoWTH----4250C~ ~ ' x mm 9 .AN:. u 9 1 4 9

~z 2 . 8

0 u

~ 2.6 <

99

w 2.4

50

100

--

----v

"

i

J~

000"/

-Un uu nun

.vvvVVVVvvvv~,v_ 9

Ix:

2,2

*** NOT ANN. 4~ t t O 4~4, 4~4~4~O O 4~4~4~4'

\

ANN. 9

\ \ 9

InAslAISb-33.4A/31.8A -~ \ 9 2800 92790 T GROWT=390~ " 9 802 9 80 150 200 250

TEMPERATURE(K)

300

. 350

FIG. 23. Cross-plane thermal conductivity of InAs-A1Sb superlattices showing the effect of the growth temperature and annealing (from Borca-Tasciuc et al., 2000).

crystals (Fig. 23). The temperature dependence, however, is very weak while that of single crystals follows k/T" with n - 1-1.5. The thermal conductivity of GaAs-A1As also shows a similar crystalline behavior (Capinski et al., 1999). Borca-Tasciuc and coworkers studied the effects of the growth temperature and annealing on the thermal conductivity of Si-Ge (Borca-Tasciuc et al., 1999) and InSb-A1Sb (Borca-Tasciuc et al., 2000b) superlattices. It was anticipated that samples grown at higher temperatures would have more diffuse interfaces. Similarly, annealing will also increase the mixing of atoms at the interface. Figure 23 shows the effect of the growth temperature annealing on the thermal conductivity of InSb-A1Sb superlattices. Since we expect that interfaces are rougher after annealing and for samples grown at higher temperatures, the experimental results indicate that the thermal conductivity decreases as the interface becomes rougher. For Si-Ge systems, an opposite trend was observed when two identical superlattices were grown at different temperatures. While the effect of the buffer change may be a source of uncertainty for the observation, the difference in dislocation density in samples grown at different temperatures is a plausible explanation. For the in-plane thermal conductivity, two similar models have been developed. Chen (1997a) solved the BTE by assuming partially specular and partially diffuse interfaces. The original work (Chen, 1996) underestimated the phonon MFP. Hyldgaard and Mahan (1996) used Fuch's solution for the diffuse interfaces and a simple proportionality model to account for partially specular and partially diffuse interfaces. From Chen's work (1997a),

238

G. CHEN A

80

" B'UdK . . . . . . . . . . . .

]

To

T=300 K GaAslAIAs EQUAL THICKNESS

~" 60 50

~,.#~ ~*

40

.--

...........

20 ...--

)

II,]

J JI=

LAYER THICKNES 1000 A..-'"" / ~1 --'" 9 S i ............ .s" /'-I ............. 300A ,,,/ /.

30

~

t ~j . s l~l

.--

..--7.....--_ . . . . . . . . . . . .

.

.

INTERFACE SPECULARITY PARAMETER p

FIG. 24. Dependence of the in-plane thermal conductivity of GaAs-A1As superlattices on the interface specularity parameter.

the Fuch's solution is indeed valid at the totally diffuse interface limit but no longer holds true for partially specular and partially diffuse interfaces. After taking into consideration the frequency dependence of the phonon relaxation time, a much better agreement with experimental data was obtained, as shown in Fig. 2(c). In these models, the interface specularity parameter was left as a fitting parameter. All other properties are based on the bulk properties and the best estimate of the phonon relaxation time in bulk materials. The model was also used to estimate the thermal conductivity of Si-Ge superlattices in the in-plane direction (Chen and Neagu, 1997). From those studies, it appears that the phonon diffuse scattering is the major reason for the observed thermal conductivity reduction. The exact cause of the diffuse scattering is not clear, although interface mixing and roughness are likely to be responsible. These studies indicate that the thermal conductivity is very sensitive to the interface specularity parameter, as shown in Fig. 24. By manipulating interfaces, it may be possible to avoid the scattering of electrons while maintain high scattering rates for phonons. In the cross-plane direction, several approaches have been taken to explain the observed thermal conductivity reduction, including the solution of the BTE with partially specular and partially diffuse interface conditions (Chen and Neagu, 1997, Chen, 1998), phonon dispersion and group velocity calculation (Hyldgaard and Mahan, 1997; Tamura et al., 1999; Bao and Chen, 2000a; Simkin and Mahan, 2000), and transfer matrix calculation of phonon transmission through superlattices (Chen, 1999). The phonon particle model developed by Chen and coworkers (Chen and Neagu, 1997; Chen, 1998) included the effect of partially diffuse interface reflection and transmission, inelastic scattering, as well as internal reflection. Figure 25 shows an example of the model results in comparison with experimental

5 PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES

~o~

239

4.5

~4.0 ~3.5

i ~

3.0

..0 o

[]

0

2.0 1.5 V 1.0 50

LINES CURRENTMODEL-

100 150 200 250 300 350 400 TEMPERTURE (K) (a)

'~ ........ BULK

,

= ~

I"

~

-

....... 1

,-'"" . . . . . . ~=

..,"

,,fl

|,,,P='

,,"

~"

/

," Si osGeosALLOY,""

S0|I/'"

t

///

,,* P~,I:/~

~

~ 10 o

/

,~ /I

/

/1

]

//

I.,";

| /

10 -i10 t

Si/Ge

10 2 PERIOD

1 |

10 3

THICKNESS (A) (b)

FIG. 25. Modeled cross-plane thermal conductivity of Si-Ge superlattices as a function of (a) temperature and (b) thickness (from Chen, 1998).

G. CHEN

240 z

2.5 "Q

O IIl ne

'

"...

I

'

"'-..

I

'

I

'

INELASTIC dl=d z=50 A

p'-~-*...~ |

|

2.0

..= |

!

w nr I-

1.5

GaAs@AIAs

uJ

-

|

! |

LU I-1.0 .J < Z

om r

z

LU

=E

o.s-

"''~.

- ....p:.o 2

-

.I

~

i

i

-

;

;

,==,=

r-t Z

o z

0.0 , I , i , 0.0 0.1 0.2 0.3 0.4 NONDIMENSIONAL COORDINATE

FIG. 26. Temperature distribution inside one period of a GaAs-A1As superlattice, demonstrating that most of the temperature drop occurs at the interface (from Chen, 1998).

data. The major reason for the thermal conductivity reduction is due to the interface reflection (total internal reflection) and the associated thermal boundary resistance. Figure 26 shows the temperature distribution inside one period of the superlattice, and it is clear that most of the temperature drop occurs at the interface. In fact, phonons inside each layer are highly nonequilibrium. The temperature as defined is only a measure of the local phonon energy rather than the conventional definition for the system at local thermal equilibrium. It is argued (Chen, 1998) that phonons will probably not all be confined, otherwise the model will lead to a too small thermal conductivity compared to experimental data. Hyldgaard and Mahan (1997) used a simple lattice dynamics model for a 2 x 2 Si-Ge superlattice and found that the group velocity and total internal reflection will cause an order of magnitude reduction in thermal conductivity. Similar results were obtained by Tamura et al. (1999) based on a refined model. Although no direct comparison with experimental data was made, the calculated results are generally higher than experimental data, particularly for GaAs-A1As. Chen's calculation (Chen et al., 1998; Chen, 1999) based on transfer matrix method also lead to similar conclusion that total internal reflection and phonon confinement can create a large reduction of phonon

5

PHONONTRANSPORT IN Low-DIMENSIONAL STRUCTURES

241

transmissivity. For the very thin layer limit, however, their calculation shows that phonon tunneling can actually cause partial recovery of the phonon conductance. This may explain the experimental results on Bi2Te 3Sb2Te 3 (Venkatasubramanian, 1996; Yamasaki et al., 1998). Lattice dynamics modeling by Tamara et al. (1999) shows a similar trend. However, the lattice dynamics or the transfer matrix method cannot capture the increasing trend of thermal conductivity at the thicker film limit. In this regime, the phonon particle picture dominates the transport, and the Boltzmann transport equation is more appropriate for describing the transport. Although the use of an imaginary wave vector (Simkin and Mahan, 2000) in the lattice dynamics model shows the correct trend, such an treatment replaces scattering by absorption and cannot impose energy balance for phonons. While Figs. 2 and 25 show that the diffuse interface scattering reduces the thermal conductivity in both the in-plane and the cross-plane thermal conductivity, we should be more careful to generalize the modeling results for special-material systems. For the in-plane direction, it is conceivable that the diffuse interface scattering is better than the specular interface scattering. For the cross-plane direction, however, this is not generally true. Whether diffuse or specular interfaces are more favorable to the thermal conductivity reduction depends on which type of process creates more phonon reflection. For GaAs-A1As and Si-Ge interfaces, our modeling indicates that diffuse interface is better if inelastic scattering occurs (Chen, 1998). For interface between SiO2 and diamond, however, modeling shows that specular interface causes more phonon reflection and thus leads to a smaller thermal conductivity (Zeng and Chen, 1999). Clearly, the BTE and the lattice dynamics approaches have their own merits. The BTE approach actually captures the most significant factors contributing to the thermal conductivity reduction: interface reflection, particularly total internal reflection. The phonon confinement effect can also be partially taken into account in the BTE approach, if not exactly. However, the BTE approach does not consider tunneling and propagating of long-wavelength phonons. The latter maybe responsible for the slight increasing in the thermal conductivity in InSb-A1Sb and GaAs-A1As superlattices with decreasing temperature. An ideal model would take into consideration of the phonon dispersion in superlattices, and exact internal and interface scattering mechanisms. We believe that an extension of the fluctuation-dissipation theorem developed for electromagnetic field to lattice waves should provide a unified approach covering both the wave and the particle regimes, as is demonstrated for radiation transport (Chen, 1997b). Although the phonon group velocity reduction and total internal reflection can partially explain the cross-plane thermal conductivity, it cannot explain the thermal conductivity reduction in the in-plane direction. Bao and Chen (2000a) calculated the in-plane thermal conductivity using the

242

G. CHEN 600

.

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.

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......... A b ~

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Number of monolayers n

FIG. 27. Anisotropic of the thermal conductivity of (a) Si-Ge and (b) GaAs-A1As superlattices from a lattice dynamics model that considers the change of the thermal energy propagation factor (from Bao and Chen, 2000a). Subscripts x and z represent directions parallel and perpendicular to film planes, respectively. The in-plane thermal conductivity reduction is very small, suggesting that the experimentally observed reduction in the in-plane direction is due not to phonon confinement but to interface diffuse scattering.

same lattice dynamics model developed by Tamura et al. (1999) and found that the thermal conductivity reductions in the in-plane direction for both S i - G e and GaAs-A1As superlattices are too small to explain the experimental data, as shown in Fig. 27. This establishes the importance of interface diffuse scattering on thermal conductivity reduction, as predicted by the BTE models (Chen, 1997a; Chen and Neagu, 1997). To summarize, it is our belief that the thermal conductivity reduction of superlattices in the in-plane direction is dominated by diffuse phonon scattering. Increasing the diffuse scattering components should decrease the thermal conductivity. In the cross-plane direction, the major cause for the thermal conductivity reduction is the interface reflection, particularly the total internal reflection. Although phonon confinements can create a large cross-plane thermal conductivity and have been observed in different superlattices, it is unclear what fraction of phonons is actually confined to create the desired effect. Tunneling can cause partial recovery of thermal conductance if the layers are a few monolayers thick. Whether the diffuse or

5

PHONONTRANSPORT IN Low-DIMENSIONAL STRUCTURES

243

specular interfaces are more beneficial for the thermal conductivity reduction in the cross-plane direction depends on which type of interface creates more reflection of phonons. Generally, we can anticipate that if the mismatches between density, specific heat, and group velocity of adjacent layers are large, the specular interface will create a higher reflectivity and thus a larger thermal conductivity reduction. In the opposite case, diffuse interfaces may be better. Other effects, including dislocations, stresses and strains, may also impact the superlattice thermal conductivity, but their rules are not clear yet.

3.

THERMAL CONDUCTIVITY OF ONE-DIMENSIONAL STRUCTURES

The preceding discussion focused on phonon transport in two-dimensional structures. It can readily appreciated that similar but stronger size effects occur inside one-dimensional (quantum wires) and zero-dimensional (quantum dots) structures. So far, there is no experimental study on the thermal conductivity of thermoelectric wires. Few studies reported on the thermal conductivity of nanowires include GaAs nanowires (Potts et al., 1991), metallic nanowires (Seyler and Wybourne, 1992), and SiN x nanowires (Tighe et al., 1997), all at very low temperatures. Experimental results were also reported for carbon nanotube bundles (Hone et al., 1999), but extraction of the thermal conductivity of single nanowires involved many assumptions. More recently, Heremans and Thrush (1999) measured the thermal conductance of bismuth nanowire arrays embedded in A1203 templates. One concern in the template-based nanowire fabrication approach is the degrading effect of the thermal conductivity of the template on the device performance. We have measured the thermal conductivity of commercially available anodized alumina template in the direction perpendicular to the nanochannel axis using the ac calorimetry method. The average thermal conductivity is ~ 1.7 W/mK. This is much lower than typical ceramic materials but is not surprising considering that the anodized alumina is essentially amorphous. This thermal conductivity, however, cannot be ignored in the design of thermoelectric devices. Theoretically, there was a debate on the phonon localization and ballistic transport possibilities in fine wires (Jackle, 1981; Kelly, 1982). Thermal conductivity quantization is predicted (Angelescu et al., 1998), which is observable only at very low temperatures. At temperatures that are more relevant for thermoelectric applications, the BTE approach is probably more practical as has been used in thin films and superlattices. Ziman (1960) gave general solutions that can be extended to nanowires (Volz and Chen, 1999). Walkauskas et al. (1999) solved the BTE for square nanowires and

244

G. CHEN 10

~" E

(J

p=0.50

T=200K

BTE 9 Solution Data points MD 9 Results

8

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Temperature (K) FIG. 28. Thermal conductivity of Si nanowires as a function of temperature and wire cross-section area obtained from direct MD simulation (dots) and solution of BTE (from Volz and Chen, 1999).

compared the thermal conductivity of nanowires to thin films. As expected, additional lateral surfaces cause a more pronounced reduction in the thermal conductivity of wires compared to films. In addition to the BTE, the small number of atoms within a nanostructure suggests that the MD method may be a viable approach to directly compute the lattice thermal conductivity. Volz and Chen (1999) used the MD method to directly compute the thermal conductivity of Si nanowires. Figure 28 shows the MD simulation and the BTE solution of the thermal conductivity of Si nanowires, demonstrating a significant reduction of the thermal conductivity of such wires. The limitation of the MD method, however, is the computational power and the accuracy of the interatomic potential. Embedded in the MD results are the physics of the phonon heat conduction processes. It is not a trivial task to understand the physics behind the MD results.

4.

HEAT CONDUCTION IN NANOPOROUS AND MESOSTRUCTURES

Since the 2D and 1D structures reduce thermal conductivity, 0D structures such as dots may be even better. It is, however, difficult to establish temperature difference across a single dot, so interconnected dots have been proposed for thermoelectric applications. The interconnected dots can form

5

PHONONTRANSPORT IN Low-DIMENSIONAL STRUCTURES

245

regular structures such as opals, or random porous structures. In the past, porous media have been suggested as a vehicle to degrade thermal conductivity more than electrical conductivity. Lidorenko et al. (1970) reported up to 30% increase in the ratio of the electrical conductivity to the thermal conductivity for porous bulk Si-Ge alloys in certain range of porosity. The increment in Z T through porosity is possible under several conditions. 1. When the phonon M F P is larger than the electron MFP, size effects on phonons will be more profound than on electrons. 2. If the electron can go through the pores while phonons cannot. This is possible if electron tunneling and thermionic emission occurs. Such a scenario would occur more easily in nanoprous structures or nanocomposite structures than in microporous structures. Thermal conductivity data of regular opal structures have been reported. Arutyunyan et al. (1997) measured the effective thermal conductivity of SiO2 opals filled with PbSe. Based on an effective thermal conductivity model, they backed up the thermal conductivity of the PbSe crystals. The temperature dependence of the PbSe crystals inside the opal pores shows crystalline behavior, albeit is strongly influenced by the boundary scattering. Baughman (1998) reported the thermal conductivity of Bi and S i O 2 opal composites as well as the thermal conductivity of inverse Bi opals. The thermal conductivity values of the Bi/opal composites is in the range of 1.5-2 W/mK from 220 to 330 K. When the SiO2 is removed, the inverse Bi opal thermal conductivity reduces to 0.7 W/mK. Thermal conductivity of nanoporous Bi thin films was investigated experimentally (Song et al., 1999b). Nanoporous Bi films with a thickness in the range from 18 to 400 nm are deposited on Si substrate through sol-gel processing (Shen et al., 1998). The thermal conductivity of these films is measured using a 3o~ method. Figure 29(a) shows the thermal conductivity of porous Bi films in the cross-plane direction with different thickness and porosity, and Fig. 29(b) shows their corresponding electrical conductivity, albeit in the direction parallel to the film plane. It seems that in the low-porosity limit, the porosity has a much smaller effect on the thermal conductivity than on the electrical conductivity. Electron tunneling through the pores is one plausible explanation. Compared to the opal structures (Baughman, 1998), the nanoporous Bi films have smaller thermal conductivity. There have been many modeling studies in the past on the thermal conductivity of porous materials due to their applications in thermal insulation (Chan and Tien, 1973). These models, however, cannot be applied to nanoporous materials because they did not include size effects, as is clearly required from experimental data (Song et al., 1999b). Several attempts to include the size effects have been made in the modeling of the

246 1.0

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(a)

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TEMPERATURE

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(K)

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FIG. 29. (a) Cross-plane thermal conductivity and (b) in-plane electrical conductivity of nanoporous bismuth thin films (from Song et al., 1999b).

thermal conductivity of nanoporous Si (Gesele et al., 1997; Lysenko et al., 1998; Chung and Kaviany, 2000). Modeling of phonon transport in opal structures was also reported (Mahan, 1998).

VI.

Phonon Engineering in Nanostructures

An intriguing question is whether the nanostructures share the same minimum thermal conductivity limits as established for bulk materials (Slack, 1979; Cahill et al., 1992). In this section, we provide a qualitative discussion of the minimum thermal conductivity of low-dimensional structures and suggest that the thermal conductivity of low-dimensional structures can be lower than the limits for their corresponding bulk materials. The discussion also leads to directions for phonon engineering in lowdimensional materials to further reduce the thermal conductivity of those structures. We start with a brief review of the key ideas behind Slack's minimum thermal conductivity theory (Slack, 1979). Slack argued that the minimum phonon MFP must be of the order of its wavelength, and later Cahill et al. (1992) further limited it to half the wavelength. With such an MFP, Eq. (2) is used to calculate the minimum thermal conductivity. For acoustic phonons, the speed of sound is used for estimating the minimum thermal conductivity and for optical phonons, the velocity is replaced with the interatomic spacing multiplied by the phonon frequency. While the minimum thermal conductivity is very intuitive for bulk

5

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES

247

materials, low-dimensional structures may have a lower minimum thermal conductivity value from their corresponding bulk materials. Take superlattices as an example. The unit cell of a superlattice is much larger than those of its parent bulk materials and the number of atoms at each basis is proportional to the thickness of the period. Accordingly, there are more optical phonons, as is clearly shown in Fig. 3. These optical phonons are folded acoustic phonons in the bulk material but may have smaller group velocity than their bulk materials. Simply treating these phonons as optical phonons and estimating their contribution to the lattice thermal conductivity, however, will probably underestimate their contribution due to their acoustic phonon parenthood. Some experimental data on the thermal conductivity of superlattices have lead to thermal conductivity values comparable to the theoretical minimum of bulk materials, as shown in Fig. 30 for a highly dislocated Si-Ge superlattice (Borca-Tasciuc et al., 1999). More experimental data are needed, as well theoretical studies, to conclusively demonstrate that low-dimensional structures have lower minimum values compared to their parent materials. The minimum thermal conductivity theory starts from Eq. (2) and assumes isotropic scattering. This leads to the conclusion that the best approach to reduce the thermal conductivity is to reduce the phonon relaxation time. The alloy method and the phonon rattler concept that have been successfully implemented for bulk thermoelectric materials manifest the effectiveness of this approach. In nanostructures, the preceding discussion clearly shows that phonon transport is highly anisotropic, and so are the group velocity and the relaxation time. By considering the formulation of

3.0 A

E

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2.5

2.0 f Ge

a 1.5

0000'"

Z

;" 9 9 ' , . . . . . . . -&~ . .

0 o 1.0 -.I < =! ~" LU O.5

.

-

SiGe . .

......................... .(~e,-..................... DOTS FOR JL92 SilGe (20A/20A) SUPERLATTICE LINES FROM MINIMUM CONDUCTIVITY THEORY

'I" I--

0.0

Si ----&~itwe-eeeeee4

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

0

50

100 150 200 TEMPERATURE (K)

250

300

FIG. 30. Comparison of thermal conductivity of a highly dislocated Si-Ge superlattice with the predictions from the minimum thermal conductivity theory prediction (from Borca-Tasciuc et al., 1999). Solid lines are based on Slack's (1979) theory and dashed lines are based on Cahill et al.'s theory (1992).

248

G. CHEN

FIG. 31. Thermalconductivity formulation for anisotropic systems.

thermal conductivity in anisotropic systems, it would become clear that there are new alternative ways to reduce thermal conductivity based on various low-dimensional effects that we have discussed throughout this chapter. Considering an anisotropic structure (as illustrated in Fig. 31), we could rewrite the thermal conductivity expression to account for the directional dependence of the relaxation time and group velocity as (Chen et al., 1998b)

1//max[

K = ~--~

sin 2 q~dq~

<;o

C(o~)v (o9, O, q~)A(~o,0, q~) COS20sin OdO

)]

d~o

(13) where 0 and q~ are the polar and azimuthal angles formed with the heat flux direction. The task of reducing thermal conductivity is to reduce the values of the above integral. From the preceding equation, we can see the following alternative approaches to decrease the thermal conductivity value, in addition to the traditional method of reducing the phonon mean free path, which is possible through engineered nanostructures. First, the group velocity can be altered in nanostructures. The formation of standing waves in nanostructures means that the group velocity becomes smaller, thus reducing the thermal conductivity. In superlattices, the bulk acoustic phonons can be changed into optical phonons, drastically reducing their group velocity. Second, it is possible to induce anisotropic scattering in low-dimensional structures. For example, interface reflection and transmission are highly angular-dependent. As another example, the optical phonons in two materials have totally different energies. It is likely that the scattering of optical phonons at the interface will be highly directional, that is, the optical phonons will be scattered backward. In the phonon dispersion curve, this is called the phonon confinement. Third, the specific heat of nanostructures can be changed by changing (1) the density of states and (2) the degrees of freedom in the atomic vibrations.

5

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES

249

Theoretical studies on superlattices, however, suggest that these changes are not strong except at low temperatures. This is clear from the DOS of superlattices [as shown in Fig. 13(a)], which is approximately the arithmetic average of their parent materials. Fourth, it is possible to change the limits of the integrals in Eq. (13) for both the angular integration and the frequency integration. For example, in the angular integration, the phonon transmission above the critical angle is zero, as shown in Fig. 4(a). Also, for the frequency integration, the phonon confinement effect may lead to a lower integration limit, as shown in Fig. 13(b). These possible alternative approaches, combined with preceding discussion on the heat conduction mechanisms, lead to the following strategies that can be pursued to engineer the phonon transport for reducing the lattice thermal conductivity: 1. For transport along the interfaces (i.e., along the film plane and wire axis), the thermal conductivity can be reduced by creating diffuse interface scattering and reducing the interface separation distance. In addition to naturally existing interface roughness due to the mixing of atoms, other possibilities are artificially corrugated interfaces, such as thin films grown on step-covered substrates, and quantum dots interface. 2. For transport perpendicular to the interfaces, increasing the phonon reflectivity is the key to reduce the thermal conductivity. This could be realized by increasing the mismatch of the properties such as density, group velocity, specific heat, and phonon spectrum between adjacent layers. The effects of interface roughness can be positive or negative, depending on whether the diffuse phonon scattering actually decreases or increases the phonon reflectivity at the interfaces. Experiments and modeling so far seem to indicate that diffuse scattering is more effective when the mismatch in material properties is not large. Phonon confinement occurs due to the mismatch of the bulk phonon dispersion, and a large difference in the dispersion favors more phonon confinement. How much phonons can be confined, however, is an open question. 3. Some long-wavelength phonons may not see the interfaces in structures such as superlattices. The localization of these phonons can further decrease the lattice thermal conductivity. Using aperiodic rather than periodic superlattices is one possibility to localize some, if not all, of these long-wavelength phonons. 4. Defects, particularly dislocations, can be another vehicle to reduce the lattice thermal conductivity in low-dimensional systems. Superlattices with multiple periodicity are another possibility. Clearly, whether all these strategies or some will work for the improvement of the energy conversion efficiency depend on the impacts of these

250

G. CHEN

measures on the electron-hole energy conversion capabilities. Models that include the low-dimensional effects on both electrons and phonons must be developed. In the next section, we discuss some of the effort along this direction.

VII.

ConcurrentElectron-Phonon Modeling

It is now generally accepted that low-dimensional systems may benefit the electron energy conversion due to quantum size effects, stress effects, and interface effects, as discussed in several chapters of this volume. The preceding discussion also illustrates the fundamental differences of phonon transport in nanostructures from that in macrostructures. Combining the low-dimensional effects on electrons and phonons thus seems to hold great promise to increase the energy conversion efficiency. Due to the difficulties in quantitative modeling of phonon thermal conductivity, however, not many studies have been directed at concurrent electron-phonon modeling. In the study of the electron quantization effects on Z T, Hicks and Dresselhaus (1993a) included the phonon boundary scattering effect by estimating the lattice thermal conductivity from the simple kinetic relation k = 1 / 3 C v A and replacing the phonon MFP by the film thickness of wire diameter. On one hand, such a primitive treatment overestimates the size effect because the actual MFP may be longer. On the other hand, the specific heat and group velocity of those phonons that carry the heat may be smaller than the bulk specific heat and the speed of sound. Sun et al. (1998) and Koga et al. (1998, 1999) reported modeling of the in-plane thermoelectric figure of merit of Si-Ge and GaAs-A1As systems considering both the electron confinement effects and phonon interface scattering. An example is given in Fig. 32 for Si-Ge superlattices. In the cross-plane direction of thin films and superlattices, Mahan et al. (1998) and Radtke et al. (1999) considered thermal conductivity reduction on the thermoelectric and thermionic effect in the cross-plane direction by assuming reduced thermal conductivity values. Consistent models that include both electron and phonon transport have yet to be developed. A recent attempt carried out by Zeng and Chen (2000) included both thermoelectric and thermionic effects in heterostructures through solving the BTE for electrons. Concurrent modeling of the nonequilibrium electron and phonon transport based BTEs is in progress.

VIII.

Summary

Through this chapter, we have demonstrated the fundamental differences between phonon heat transport in nanostructures and that in macrostructures. The major points of this chapter are summarized as follows:

5 PHONONTRANSPORTIN Low-DIMENSIONALSTRUCTURES

251

2.5 .... a = IOOA ----- a= 75A

2.0

~ a -

5oA

a

2

1.5

1.0

0.5

o. ~

1" . . . . . . . .

lO~2 n=o (cm "2)

'1 o'13

. . . . .

1o "

FIG. 32. Resultsof concurrent electron-phonon modeling for the Z T of Si quantum wells (from Sun et al., 1998).

1. Interfaces and surfaces play a pivotal role for the thermal conductivity reduction. 2. For heat flow parallel to interfaces, diffuse interface scattering is the key to reduce thermal conductivity along the film-wire plane. 3. For heat conduction perpendicular to interfaces, phonon reflection, particularly total internal reflection, can greatly reduce the heat transfer and thermal conductivity. Phonon confinements can theoretically create a large thermal conductivity reduction but how many phonons are actually confined is an open question. Whether diffuse or specular interfaces are more effective in reducing thermal conductivity depends on which type of interfaces cause more phonon reflection. Tunneling, which occurs when each layer is within a few monolayers, causes partial recovery of thermal conductance of superlattices. 4. When the superlattice period is very small (within a few monolayers), the phonon wave model should be used and refined. For thicker superlattices, the phonon particle model based on BTE is adequate. The latter can capture most of the significant factors that dominate the

252

G. CHEN

,

,

thermal conductivity reduction (i.e., interface reflection and the associated group velocity reduction) and phonon confinement effects (through modeling). It could handle better interface diffuse scattering. It cannot, however, deal with long-wavelength phonons spreading over the superlattices. The crystalline behavior in the temperature dependence of the thermal conductivity of InAs-A1Sb and GaAs-A1As superlattices may be due to these long-wavelength phonons. Defects, particularly dislocations, can create a large thermal conductivity reduction. The mismatch in the lattice constants among different materials in nanocomposite structures may create high dislocation density that further decreases the lattice thermal conductivity. It is likely that the minimum thermal conductivity of the nanostructured materials is lower than that of their corresponding bulk materials. The anisotropy intrinsic in some low-dimensional structures offers alternative ways to reduce the phonon thermal conductivity for thermoelectric applications through engineering the phonon transport.

Compared to the large amount work that has been done on the electrical transport in nanostructures, research of thermal transport in nanostructures is still relatively scarce. Some of the many challenges that should be addressed in future studies are listed here: 1. Measurement techniques. Accurate techniques for the anisotropy of thin film thermal conductivity with minimum disturbance to samples should be developed and improved. Concurrent thermal and thermoelectric transport property measurements of thin films have yet to be demonstrated. Thermal properties of single wires and wire arrays await to be explored. 2. Phonon interface scattering mechanisms. What causes the diffuse phonon scattering and how to predict it? How many phonons are confined? 3. Thermal conductivity modeling. Integration of wave characteristics as well as the diffuse and inelastic interface scattering mechanisms into a consistent model. 4. Modeling and experiments on 1D and mesostructures. 5. Concurrent modeling of electron and phonon transport in low-dimensional structures. It should be clear from the discussion in this chapter and related chapters of this series that low-dimensionality offers the possibility to increase Z T through enhancing the power factor and reducing the lattice thermal conductivity. Opportunity windows may exist to simultaneously increasing

5

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES

253

the power factor while decreasing the thermal conductivity. Clearly identifying such windows requires continued understanding of electron and phonon thermoelectric transport in low-dimensional structures.

ACKNOWLEDGMENTS

I would like to thank Professors M. S. Dresselhaus, K. L. Wang, B. Dunn, M. S. Goorsky, T. Sands, and R. Gronsky for stimulating discussion, providing samples, and carrying out structural characterization for several low-dimensional systems, including Si-Ge, skutterudites, and porous Bi, and Professors S. Pei and T. Lin for InAs-A1Sb systems. I am also indebted to post-docs and students in my laboratory for contributing to the experimental and theoretical studies of phonon transport in low-dimensional systems, including Dr. S. Volz, Dr. T. Zeng, T. Borca-Tasciuc, B. Yang, W. L. Liu, D. Song, and D. Achimov. Funding for the work summarized here comes from the DOD/ONR MURI program on thermoelectrics, the DARPA HERETIC program, and an NSF Young Investigator Award.

REFERENCES Abeles, B.,"Lattice Thermal Conductivity of Disordered Semiconductor Alloys at High Temperatures," Phys. Rev. 131, 1906 (1963). Abrosimov, V. M., B. N. Egrorov, N. S. Liderenko, and V. A. Karandashev, "The Dimensional Effect of the Transfer Coefficients in Oriented Films of Bismuth Grown on Polymer Susbtrates," High Temperatures 12, 456 (1974). Amith, A., I. Kudman, and E. F. Steigmeier, "Electron and Phonon Scattering in GaAs at High Temperatures," Phys. Rev. 138, A1270 (1965). Angelescu, D. E., M. C. Cross, M. L. Roukes, "Heat Transport in Mesoscopic Systems," Superlattices Microstruct. 23, 673 (1998). Arutyunyan, L. I., V. N. Bogomolov, N. F. Kartenko, D. A. Kurdyukov, V. V. Popov, A. V. Prokof'ev, I. A. Smirnov, and N. V. Sharenkova, "Thermal Conductivity of a New Type of Regular-Structure Nanocomposites: PbSe in Opal Pores," Phys. Solid State, 39, 510 (1997). Asheghi, M., M. N. Touzelbaev, K. E. Goodson, Y. K. Leung, and S. S. Wong, "TemperatureDependent Thermal Conductivity of Single-Crystal Silicon Layers in SO1 Substrates," J. Heat Transf 120, 30 (1998). Auld, B. A., Acoustic Fields and Waves in Solids, 2nd ed., Krieger, Florida (1990). Baier, V., and F. Volklein, "Thermal Conductivity of Thin Films," Phys. Stat. Sol. (a) 118, K69 (1990). Balandin, A., and K. L. Wang, "Significant Decrease of the Lattice Thermal Conductivity due to Phonon Confinement in a Free-Standing Semiconductor Quantum Well," Phys. Rev. B 58, 1544 (1998). Bannov, N., V. Aristov, V. Mitin, and M. A. Stroscio, "Electron Relaxation Times due to the Deformation-Potential Interaction of Electrons with Confined Acoustic Phonons in a Free-Standing Quantum Well," Phys. Rev. B 51, 9930 (1995).

254

G. CHEN

Bao, Y., and G. Chen, "Lattice Dynamics Study of Anisotropy of Heat Conduction in Superlattices," Proc. MRS Spring Meeting Symposium Z, in press (2000a). Bao, Y., and G. Chen, "Lattice Dynamics Study of Phonon Heat Conduction in Quantum Wells," Phys. Low-Dim. Struct. 516, 37 (2000b). Baughman, R. H., "Advanced Nanostructured Thermoelectrics for Aerospace Applications," Proc. DARPA Thermoelectric Materials Workshop, Washington Dulles Airport Hilton, Herndon, Virginia, Oct. 14-16 (1998). Berman, R., Thermal Conduction in Solids, Clarendon, Oxford (1976). Berman, R., E. L. Foster, and J. M. Ziman, "Thermal Conduction in Artificial Sapphire Crystals at Low Temperatures," Proc. R. Soc. London 231, 130 (1955). Bertolotti, M., G. L. Liakhou, A. Ferrari, V. G. F. Ralchenko, A. A. Smolin, E. Obraztsova, K.G. Korotoushenko, S. M. Pimenov, and V. I. Konov, "Measurements of Thermal Conductivity of Diamond Films by Photothermal Deflection Technique," J. Appl. Phys. 75, 7795 (1994). Borca-Tasciuc, T., and G. Chen, "Thin-Film Thermophysical Property Characterization by Scanning Laser Thermoelectric Microscope," Int. J. Thermophys. 19, 557 (1998). Borca-Tasciuc, T., W. L. Liu, J. L. Liu, T. Zeng, D. W. Song, C. D. Moore, G. Chen, K. L. Wang, M. S. Goorsky, T. Radetic, R. Gronsky, X. Sun, and M. S. Dresselhaus, "Thermal Conductivity of Si/Ge Superlattices," Proc. 18th Int. Conf. Thermoelectric, ICT'99, IEEE Cat. No. 99TH8407 (1999); also to appear in Superlatt. and Microstruc. Borca-Tasciuc, T., R. Kumar, and G. Chen, "Improvements of 309 Method for Thin-Film Thermal Conductivity Determination," to be presented at 2000 National Heat Transfer Conference, Pittsburg, submitted for publication (2000a). Borca-Tasciuc, T., D. Achimov, W. L. Liu, G. Chen, C. -H. Lin, A. Delaney, and S. S. Pei, to be presented at Eng. Found. Conf. on Heat Transfer and Transport Phenomena in Microsystems, Oct. 15-20, 2000, Canada (2000b). Cahill, D. G., "Thermal Conductivity Measurement from 30-K to 750-K--The 3-Omega Method," Rev. Sci. Instrum. 61, 802 (1990). Cahill, D. G., H. E. Fischer, T. Klitsner, E. T. Swartz, and R. O. Pohl, "Thermal Conductivity of Thin Films: Measurement and Understanding," J. Vac. Sci. Technol., A7, 1259 (1989). Cahill, D. G., S. K. Watson, and R. O. Pohl, "Lower Limit to the Thermal Conductivity of Disordered Crystals," Phys. Rev. B, 46, 6131 (1992). Callaway, J., "Model for Lattice Thermal Conductivity at Low Temperatures," Phys. Rev. 113, 1046 (1959). Capinski, W. S., and H. J. Maris, 1996, "Thermal Conductivity of GaAs/A1As Superlattices," Physica B 220, 699 (1996). Capinski, W. S., H. J. Maris, T. Ruf, M. Cardona, K. Ploog, and D. S. Katzer, "ThermalConductivity Measurements of GaAs/A1As Superlattices Using a Picosecond Optical Pump-and-Probe Technique," Phys. Rev. B 59, 8105 (1999). Casimir, H. B. G., "Note on the Conduction of Heat in Crystals," Physica 5, 495 (1938). Chan, C. K., and C. L. Tien, "Conductance of Packed Spheres in Vacuum," J. Heat Transf 302 (1973). Chen, G., "Heat Transfer in Micro and Nanoscale Photonic Devices," Ann. J. Rev. Heat Transf 7, 1 (1996). Chen, G., "Size and Interface Effects on Thermal Conductivity of Superlattices and Periodic Thin-Film Structures," J. Heat Transf 119, 220 (1997a), see also Proc. 1996 Nat. Heat Transf Conf., ASME HTD, Vol. 323, 121 (1996). Chen, G., "Wave Effects on Radiative Transfer in Absorbing and Emitting Thin-Film Media," Microscale Thermophys. Eng. 1, 215 (1997b). Chen, G., "Thermal Conductivity and Ballistic Phonon Transport in Cross-Plane Direction of Superlattices," Phys. Rev. B 57, 14958 (1998), see also Proc. 1996 Int. Mech. Eng. Congr., DSC Vol. 59, 13 (1996).

5

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES

255

Chen, G., "Phonon Wave Heat Conduction in Thin Films and Superlattices," J. Heat Transl. 121, 945 (1999). Chen, G., and M. Neagu, "Thermal Conductivity and Heat Transfer in Superlattices," Appl. Phys. Lett. 71, 2761 (1997). Chen, G., and C. L. Tien, "Thermal Conductivity of Quantum Well Structures," AIAA J. Thermophys. Heat Transf., 7, 311 (1993). Chen, G., C. L. Tien, X. Wu, and J. S. Smith, "Measurement of Thermal Diffusivity of GaAs/A1GaAs Thin-Film Structures," J. Heat Transf. 116, 325 (1994). Chen, G., S. Q. Zhou, D.-J. Yao, C. J. Kim, X. Y. Zheng, Z. L. Liu, and K. L. Wang, "Heat Conduction in Alloy-Based Superlattices," Proc. 17th Int. Conf. Thermoelectrics, ICT'98, 202, IEEE Cat. No. 98TH8365 (1998a). Chen, G., S. G. Volz, T. Borca-Tasciuc, T. Zeng, D. W. Song, K. L. Wang, and M. S. Dresselhaus, "Thermal Conductivity and Phonon Engineering in Low-Dimensional Structures," MRS Proc. 545, 357 (1998b). Chen, G., T. Boca-Tasciuc, B. Yang, D. Song, W. L. Liu, T. Zeng, D.-A., Achimov, "Heat Conduction Mechanisms and Phonon Engineering in Superlattice Structures," Thermal Sci. Eng. 7, 43 (1999). Chopra, K. L., and P. Nath,, "Experimental Determination of the Thermal Conductivity of Thin Films," Thin Solid Films 18, 29 (1973). Chung, J. D., M. Kaviany, "Effects of Phonon Pore Scattering and Pore randomness on Effective Conductivity of Porous Silicon," Int. J. Heat Mass Transf. 43, 521 (2000). Colvard, C., T. A. Gant, M. V. Klein, R. Merlin, R., Fisher, H. Morkoc, and A. C. Gossard, "Folded Acoustic and Quantized Optic Phonons in (GaA1)As Superlattices," Phys. Rev. B 31, 2080 (1985). Decker, D. L., L. G. Koshigoe, and E. J. Ashley, "Thermal Properties of Optical Thin Film Materials," NBS Special Publication, 727, Laser Damage in Optical Materials, 291 (1984). Fu, H., V. Ozolins, and A. Zunger, "Phonons in GaP Quantum Dots," Phys. Rev. B 59, 2881 (1999). Gallo, C. F., B. S. Chandrasekhar, and P. H. Sutter, "Transport Properties of Bismuth Single Crystals," J. Appl. Phys. 34 144 (1963). Gesele, G., J. Linsmeier, V. Drach, J. Fricke, and R. Arens-Fischer, "Temperature-Dependent Thermal Conductivity of Porous Silicon," J. Phys. D: Appl. Phys. 30, 2911 (1997). Goldsmid, H. J., Thermoelectric Rrefrigeration, Plenum Press, New York (1964). Goodson, K. E., "Thermal Conduction in Nonhomogeneous CVD Diamond Layers in Electronic Microstructures," J. Heat Transf 118, 279 (1996). Goodson, K. E., and S. Ju, "Heat Conduction in Novel Electronic Films," Ann. Rev. Mat. 29, 261 (1999). Goodson, K. E., O. W. Kading, M. Rosier, and R. Zachai, "Experimental Investigation of Thermal Conduction Normal to Diamond-Silicon Boundaries," J. Appl. Phys. 77, 1385 (1995). Grille, H., K. Karch, and F. Bechstedt, "Thermal Properties of (GaAs),/(Ga~_xA1JAs), (001) Superlattices," Physica B 220, 690 (1996). Hamilton, R. A. H., and J. E. Parrot, "Variational Calculation of the Thermal Conductivity of Germanium," Phys. Rev. 178, 1284 (1969). Harman, T. C., D. L. Spears, and M. J. Manfra, "High Thermoelectric Figure of Merit in PbTe Quantum Wells," J. Electron. Mat. 25, 1121 (1996). Harman, T. C., D. L. Spears, and M. P. Walsh, "PbTe/Te Superlattice Structures with Enhanced Thermoelectric Figure of Merit," J. Electron. Mater. 28, L1 (1999). Hartmann, J., P. Voigt, and M. Reichling, 1997, "Measuring Local Thermal Conductivity in Polycrystalline Diamond with a High Resolution Photothermal Microscope," J. Appl. Phys. 81, 2966 (1997). Hatta, I., "Thermal Diffusivity Measurement of Thin Films and Multilayered Composites," Int. J. Thermophys. 11, 293 (1990).

256

G. CHEN

Heremans, J., and C. M. Thrush, "Thermal Transport in Bismuth Nanowires," Proc. in 25th International Conference, in press (1999). Hess, P., Ed., Photoacoustic, Photothermal, and Photochemical Processes at Surfaces and in Thin Films, Springer-Verlag, Berlin (1989). Hicks, L. D., and M. S. Dresselhaus, "Effect of Quantum-Well Structures on the Thermoelectric Figure of Merit," Phys. Rev. B 47, 12727 (1993a). Hicks, U D., and M. S. Dresselhaus, "Thermoelectric Figure of Merit of a One-Dimensional Conductor," Phys. Rev. B 47, 16631 (1993b). Holland, M. G., "Phonon Scattering in Semiconductors from Thermal Conductivity Studies," Phys. Rev. 134, A471 (1964). Hone, J., M. Whitney, C. Piskoti, and A. Zettl, "Thermal Conductivity of Single-Walled Carbon Nanotubes," Phys. Rev. B 59, R2514 (1999). Hu, H., X. Wang, and X. Xu, "Generalized Theory of Photo Acoustic Effect with a Multilayer Material," J. Appl. Phys. 86, 3953 (1999). Hyldgaard, P., and G. D. Mahan, "Phonon Knudson Flow in Superlattices," in Thermal Conductivity 23, 172, Technomic, Lancaster (1996). Hyldgaard, P., and G. D. Mahan, "Phonon Superlattice Transport," Phys. Rev. B 56, 10754 (1997). Indermuehle, S. W., and R. B. Peterson, "A Phase-Sensitive Technique for the Thermal Characterization of Dielectric Thin Films," J. Heat Transf 121, 528 (1999). Ioffe, A. F., Semiconductor Thermoelements and Thermoelectric Cooling, Infosearch Limited, London (1957). Jackle, J., "On Phonon Localization in Nearly One-Dimensional Solids," Solid State Commun. 39, 1261 (1981). Ju, Y. S., and K. E. Goodson, "Phonon Scattering in Silicon Films with Thickness of Order 100 nm," Appl. Phys. Lett. 74, 3005 (1999). Kelly, M. J., "Thermal Anomalies in Very Fine Structures," J. Phys. C: Solid State Phys. 15, L969 (1982). Khitun, A., A. Balandin, and K. U Wang, "Modification of the Lattice Thermal Conductivity in Silicon Quantum Wires due to Spatial Confinement of Acoustic Phonons," Superlatt. Microstruct. 26, 181 (1999). Klemens, P. G., "Thermal Conductivity and Lattice Vibration Modes," Solid-State Phys. 7, 1 (1958). Kittel, C., Introduction to Solid State Physics, 7th ed., Wiley, New York (1996). Klitsner, T., J. E. VanCleve, H. E. Fischer, and R. O. Pohl, "Phonon Radiative Heat Transfer and Surface Scattering," Phy. Rev. B 38, 7576 (1988). Koga, T., X. Sun, S. B. Cronin, and M. S. Dresselhaus, "Carrier Pocket Engineering to Design Superior Thermoelectric Materials Using GaAs/AIAs Superlattices," Appl. Phys. Lett. 73, 2950 (1998). Koga, T., X. Sun, S. B. Cronin, and M. S. Dresselhaus, "Carrier Pocket Engineering Applied to "Strained" Si/Ge Superlattices to Design Useful Thermoelectric Materials," Appl. Phys. Lett. 75, 2438 (1999). Kuo, P. K., M. J. L. Lin, C. B. Reyes, L. D. Favro, R. L. Thomas, D. S. Kim, S. Y. Zhang, J. L. Ingelhart, D. Fournier, A. C. Boccara, and N. Yacoubi, "Mirage-Effect Measurement of Thermal Diffusivity, Part I: Experiment," Can. J. Phys. 64, 1165 (1986). Kurabayashi, K., M. Asheghi, M. Touzelbaev, and K. E. Goodson, "Measurement of the Thermal Conductivity Anisotropy in Polyimide Films," J. Microelectromechan. Syst. 8, 180 (1999). Lee, S. M., and D. G. Cahill, "Heat Transport in Thin Dielectric Films," J. Appl. Phys. 81, 2590 (1993). Lee, S. M., D. G. Cahill, and R. Venkatasubramanian, "Thermal Conductivity of Si-Ge Superlattices," Appl. Phys. Lett. 70, 2957 (1997).

5

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES

257

Liang, X. G., and B. Shi, "2-Dimensional Molecular Dynamics Simulation of the Thermal Conductance of Superlattices with Lennard-Jones Potential," submitted for publication (1999). Lidorenko, N. S., O. M. Narva, L. D. Dudkin, and R. S. Erofeev, "Influence of Porosity and Intergrain-Boundary Quality on the Electrical and Thermal Conductivities of Semiconductive Thermoelectric Materials," Inorg. Mat. 6, 1853 (1970). Lysenko, V., V. Gliba, V. Strikha, A. Dittmar, and Others, "Nanoscale Nature and Low Thermal Conductivity of Porous Silicon Layers," Appl. Surface Sci. 123, 458 (1998). Little, W. A. "The Transport of Heat Between Dissimilar Solids at Low Temperatures," Can. J. Phys. 37, 334 (1959). Mahan, G. D., "Theory of Thermoelectrics and Thermionics," Proc. DARPA Thermoelectric Materials Workshop, Washington Dulles Airport Hilton, Herdon, Virginia, Oct. 14-16 (1998). Mahan, G. D., J. O. Sofo, and M. Bartkowiak, "Multilayer Thermionic Refrigerator and Generator," J. Appl. Phys. 83, 4683 (1998). A. Majumdar, "Microscale Heat Conduction in Dielectric Thin Films," J. Heat Transl. 115, 7 (1993). Mandelis, A., Principles and Perspectives of Photothermal and Photoacoustic Phenomena, Elsevier, New York (1992). Mastrangelo, C. H., and R. S. Muller, "Thermal Diffusivity of Heavily Doped Low Pressure Chemical Vapor Deposited Ploysiltalline Silicon Films," Sens. Mater. 3, 133 (1988). Narayanamurti, V., J. L. Stormer, M. A. Chin, A. C. Gossard, and W. Wiegmann, "Selective Transmission of High-Frequency Phonons by a Superlattice: the Dielectric Phonon Filter," Phys. Rev. Lett. 43, 2012 (1979). Nath, P., and K. L. Chopra, "Thermal Conductivity of Copper Films," Thin Solid Films 20, 53 (1974). Nishiguchi, N., Y. Ando, and M. N. Wybourne, "Acoustic Phonon Modes of Rectangular Quantum Wires," J. Phys. Cond. Matt. 9, 5751 (1997). Nolas, G. S., G. A. Slack, D. T. Morelli, T. M. Tritt, and A. C. Ehrlich, "The Effect of Rare-Earth Filling on the Lattice Thermal Conductivity of Skutterudites," J. Appl. Phys. 79, 4002 (1996). Omini, M., and A. Sparavigna, "Effect of Phonon Scattering by Isotope Impurities on the Thermal Conductivity of Dielectric Solids," Physica B 233, 230 (1997). Paddock, C. A., and G. Eesley, "Transient Thermoreflectance from Thin Metal Films," J. Appl. Phys. 60, 285 (1986). Peterson, R. B., "Direct Simulation of Phonon-Mediated Heat Transfer in a Debye Crystal," J. Heat Transf. 116, 815 (1994). Potts, A., M. J. Kelly, D. G. Hasko, C. G. Smith, D. B. Hasko, J. R. A., Cleaver, H. Ahmed, D. C. Peacock, D. A. Ritchie, J. E. F. Frost, and G. A. C. Jones, "Thermal Transport in Free-Standing Semiconductor Fine Wires," Superlatt. Microstruct. 9, 315 (1991). Prasher, R. S., and P. E. Phelan, "Size Effects on the Thermodynamic Properties of Thin Solid Films," J. Heat Transf. 120, 971 (1998). Radtke, R. J., H. Ehrenreich, and C. H. Grein, "Multilayer Thermoelectric Refrigeration in Hgl_xCdxTe Superlattices," J. Appl. Phys. 86, 3195 (1999). Ren, S. Y., and J. Dow, "Thermal Conductivity of Superlattices," Phys. Rev. B 25, 3750 (1982). Rowe, D. M., Ed., CRC Handbook of Thermoelectrics, CRC Press, Boca Raton, FL (1995). Rytov, S. M., Theory of Electric Fluctuations and Thermal Radiation, AFCRC-TR-59-162, Bedford, MA, 1959. Scudieri, F., and M. Bertolotti, Eds., Photoacoustic and Photothermal Phenomena, Proc. lOth Int. Conf., Rome, Italy, AIP (1998). Seyler, J., and M. N. Wybourne, "Acoustic Waveguide Modes Observed in Electrically Heated Metal Wires," Phys. Rev. Lett. 9 1427 (1992).

258

G. CHEN

Shakouri, A., and J. E. Bowers, "Heterostructure Integrated Thermionic Coolers," Appl. Phys. Lett. 71, 1234 (1997). Shen, W.-N., B. Dunn, F. Ragot, M. S. Goorsky, C. D. Moore, G. Chen, and R. Gronsky, "Preparation and Properties of Porous Bismuth Thin Films," MRS Proc. 545, 273 (1998). Simkin, M. V., and G. D. Mahan, "Minimum Thermal Conductivity of Superlattices," Phys. Rev. Lett. 84, 927 (2000). Slack, G. A. "The Thermal Conductivity of Nonmetallic Crystals," Solid State Phys. 34, 1 (1979). Slack, G. A., and V. G. Tsoukala, "Some Properties of Semiconducting IrSb3," J. Appl. Phys. 76, 1665 (1994). Song, D. W., W. L. Liu, T. Zeng, T. Borca-Tasciuc, G. Chen, C. Caylor, and T. D. Sands, "Thermal Conductivity of Skutterudite Thin Films and Superlattices," Proc. Int. Thermoelectrics Conf., ICT'99, 679, IEEE Cat. No. 99TH8407 (1999a). Song, D. W., W.-N. Shen, T. Zeng, W. L. Liu, G. Chen, B. Dunn, C. D. Moore, M. S. Goorsky, R. Radetic, and R. Gronsky, "Thermal Conductivity of Nano-Porous Bismuth Thin Films for Thermoelectric Applications," Proc. ASME, HTD-V.364-1, 339. (1999b). Stoner, R. J., and H. J. Maris, "Kapitza Conductance and Heat Flow Between Solids at Temperatures from 50 to 300 K," Phys. Rev. B 48, 16373 (1993). Sun, X., G. Chen, K. L. Wang, and M. S. Dresselhaus, "Theoretical Modeling of The Thermoelectric Figure of Merit in Si/Sil_xGe x Quantum Well Structures," Proc. 17th Int. Conf. Thermoelectrics, ICT'98, 47, 1998, IEEE Cat. No. 98TH8365. Swartz, E. T., and R. O. Pohl, "Thermal Boundary Resistance," Rev. Modern Phys. 61, 605 (1989). Tai, Y. C., C. H. Mastrangelo, and R. S. Muller, "Thermal Conductivity of Heavily Doped Low-Pressure Chemical Vapor Deposited Polycrystalline Solicon Films," J. Appl. Phys. 63, 1442 (1987). Tamura, S., D. C. Hurley, and J. P. Wolfe, "Acoustic-Phonon Propagation in Superlattices," Phys. Rev. B 38, 1427 (1989). Tamura, S., Y. Tanaka, and H. J. Maris, "Phonon Group Velocity and Thermal Conduction in Superlattices," Phys. Rev. B 60, 2627 (1999). Tellier, C. R., and A. J. Tosser, Size Effects in Thin Films, Elsevier, Amsterdam (1982). Tien, C. L., B. F. Armaly, and P. S. Jagannathan, "Thermal Conductivity of Thin Metallic Films and Wires at Cryogenic Temperatures," Proc. 8th Thermal Conductivity Conf., 13 (1969). Tien, C. L., A. Majumdar, and F. Gerner, Eds., Microscale Energy Transport, Taylor and Francis, Bristol, PA (1998). Tien, C. L., and G. Chen, "Challenges in Microscale Conductive and Radiative Heat Transfer," in Proc. Am. Soc. Mech. Eng., ASME HTD, Vol. 227, 1 (1992), also in J. Heat Transf. 116, 799 (1994). Tighe, T. S., J. M. Worlock, and M. L. Roukes, "Direct Thermal Conductance Measurements on Suspended Monocrystalline Nanostructures," Appl. Phys. Lett. 70, 2687 (1997). Uher, C., "Thermoelectric Properties of Skutterudites," in Semiconductors and Semimetals, Vol. 69, Academic Press, San Diego (2001). Venkatasubramanian, R., "Organometallic Epitaxy of Bi2Te 3 and Related Materials and the Development of Planar, Monolithically-Interconnected, Superlattice-Structured, HighEfficiency Thermoelectric Elements," Proc. 1st National Thermogenic Cooler Workshop (1992). Venkatasubramanian, R., "Thin-Film Superlattice and Quantum-Well Structures - - A New Approach to High-Performance Thermoelectric Materials," Nay. Res. Rev. 58, 44 (1996). Venkatasubramanian, R., "Lattice Thermal Conductivity Reduction and Phonon Localizationlike Behavior in Superlattice Structures," Phys. Rev. B 61, 3091 (2000). Venkatasubramian, R., E. Siivola, and T. S. Colpitts, "In-Plane Thermoelectric Properties of

5

PHONON TRANSPORT IN Low-DIMENSIONAL STRUCTURES

259

Freestanding Si/Ge Superlattice Structures," Proc. 17th Int. Conf. Thermoelectrics, ICT'98, 191 IEEE Cat. No. 98TH8365 (1998). Volklein, F., "Thermal Conductivity and Diffusivity of a Thin Film SiO2-Si3N4 Sandwich," Thin Solid Films 188, 27 (1990). Volklein, F., V. Baier, U. Dillner, E. Kessler, "Transport Properties of Flash-Evaporated (Bi 1- xSbx)zTe3 Films, I. Optimization of Film Properties," Thin Solid Films 187, 253 (1990). Volklein, F., and E. Kessler, "A Method for the Measurement of Thermal Conductivity, Thermal Diffusivity, and Other Transport Coefficients of Thin Films," Phys. Stat. Sol. (a) 81, 585 (1984). Volklein, F., and E. Kessler, "Analysis of the Lattice Thermal Conductivity of Thin Films by Means of a Modified Mayadas-Shatzkes Model: the Case of Bismuth Films," Thin Solid Films 142, 169 (1986). Volklein, F., and E. Kessler, "Thermoelectric Properties of Bil_xSb x Films with 0 < x ~< 0.3," Thin Solid Films 155, 197 (1987). Volklein, F., and E. Kessler, "Thermal Conductivity and Thermoelectric Figure of Merit of Thin Antimony Films," Phys. Stat. Sol. (b) 158, 521 (1990). Volklein, F., and T. Starz, "Thermal Conductivity of Thin Films--Experimental Methods and Theoretical Interpretation," Proc. 16th Int. Conf. Thermoelectrics, ICT'97, 711, IEEE Cat. No. 97TH8291 (1997). Volz, S. G., and G. Chen, "Molecular Dynamics Simulation of Thermal Conductivity of Si Nanowires," Appl. Phys. Lett. 75, 2056 (1999). Volz, S. G., and G. Chen, "Molecular Dynamics Simulation of the Thermal Conductivity of Silicon Crystals," Phys. Rev. B 61, 2651 (2000). Volz, S. G., J. B. Saulnier, G. Chen, and P. Beauchamp, "Molecular Dynamics Study of Heat Transfer in Si/Ge Superlattices," presented at 15th European Conference on Thermophysical Properties, September 5-9, Wuerzburg, Germany (1999). Walkauskas, S. G., D. A. Broido, K. Kempa, and T. L. Reinecke, "Lattice Thermal Conductivity of Wires," J. Appl. Phys. 85, 2579 (1999). Walker, J. D., "Vertical-Cavity Laser Diodes Fabricated by Phase-Locked Epitaxy," Ph.D. Thesis, UC Berkeley, 1993. Walker, J. D., D. M. Kutcha, and J. S. Smith, "Vertical-Cavity Surface-Emitting Laser Diodes Fabricated by Phase-Locked Epitaxy," Appl. Phys. Lett. 59, 2079 (1991). Wu, Z. L., M. Reichling, X. Q. Hu, K. Balasubramanian, and Others, "Absorption and Thermal Conductivity of Oxide Thin Films Measured by Photothermal Displacement and Reflectance Methods," Appl. Opt. 32, 5660 (1993). Yamasaki, I., R. Yamanaka, M. Mikami, H. Sonobe, Y. Mori, and T. Sasaki, "Thermoelectric Properties of BizTe3/Sb/Te 3 Superlattice Structures," Proc. 17th Int. Thermoelectrics Conf., ICT'98, 210, IEEE Cat. No. 98TH8365 (1998). Yao, T., "Thermal Properties of AIAs/GaAs Superlattices," Appl. Phys. Lett. 51, 1798 (1987). Yu, X. Y., G. Chen, A. Verma, and J. S. Smith, "Temperature Dependence of Thermophysical Properties of GaAs/A1As Periodic Structure," Appl. Phys. Lett. 67, 3554 (1995); 68, 1303 (1996). Yu, X. Y., Z. Liang, and G. Chen, "Thermal-Wave Measurement of Thin-Film Thermal Diffusivity with Different Laser Configurations," Rev. Sci. Instrum. 67, 2312 (1996). Zeng, T., and G. Chen, "Microscale Heat Transfer in Thin Films: Impacts of Thermal Boundary Resistance and Internal Heat Generation," CD-ROM Proc. 3rd ASME/JSME Joint Thermal Eng. Conf., San Diego, March 15-18 (1999). Zeng, T., and G. Chen, "Interplay between Thermoelectric and Thermionic Effects in Heterostructures," to be presented at IMECE 2000, ASME Proc., in press (2000). Zhang, X., and C. P. Grigoropoulos, "Thermal Conductivity and Diffusivity of Free-Standing Nitride Thin Films," Rev. Sci. Instrum. 66, 1115 (1995). Ziman, J. M., Electrons and Phonons, Clarendon Press, Oxford (1960).

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SEMICONDUCTORSAND SEMIMETALS,VOL.71

Index

Bloch functions, 136, 137, 139 Bohr hydrogen-like model, 68-69 Boltzmann constant, 157, 161 Boltzmann transport equation (BTE), 70-71, 127, 129, 135, 139, 140, 141-142, 182 boundary conditions for, 225-227 Fuch's solution, 232 low-dimensional materials and, 205-208, 224-227 Bose-Einstein distribution, 206, 225 Bragg reflection model, 190 Bulk materials, phonon thermal conductivity in, 206-210

Anderson localization, 190-192 Anharmonic phonon-phonon scattering, 150

B-factor, 168-169 Bi applications, 44 bulk properties of, 45-48 carrier pocket engineering, 49-51 Bi nanowires compared with Sb, 111-114 doping of, 68-70, 91-96 four-point resistivity measurement of, 89-91 magnetoresistance of, 96-103 mobility tensors at 77 K, 73-78 Seebeck coefficient, 103-107 semi-classical transport model for, 70-83 structure of, 54-57 summary, 114 temperature-dependent resistivity of, 83-96 T-point holes, effects of, 78-83 Bil_xSb x, 51-53 Bi2Te3-Sb2Te 3 superlattices cross-plane carrier transport, 194-196 cross-plane thermal conductivity, measurements of, 182-184 in-plane carrier transport, 179-181 lattice thermal conductivity, 184-185 low-temperature heteroepitaxy, 176-179 phonon transport, 181-182

Carnot efficiency, 162 Carrier pocket engineering, 11-17 for Bi, 49-51 for SiGe, 36-44 Carrier scattering, 128, 129, 135 Concurrent electron-phonon modeling, 250 Constant relaxation time approximation (CRTA), 128, 129, 133, 142-143, 145, 147 Cross-plane carrier transport, 194-196 Cross-plane thermal conductivity, measurements of, 182-184

Deformation potential (DP), 135, 143 Density of states (DOS), 4-5, 129, 222-224 Diffusive transport analysis, 187-189, 190192 261

262

Envelope function approximation, 128

Fabry-P6rot, 160 Fermi-Dirac distribution function, 71 Fermi-Dirac related integral, 5, 23 Fermi energy, 101-102 Fermi function, 129 Fermi's golden rule, 208 Figure of merit, 2, 5-6, 123, 124 lattice thermal conductivity and, 149-152 for SiGe, 30-33 in thermionic refrigeration, 167-168 for thin-film, 176 Fr6hlich interaction, 135 Fuch's solution, 232

Green-Kubo formalism, 228

Hall effect, 96 Hall mobilities, 179 Harman method, 197 Heteroepitaxy, low-temperature, 176-179 High-resolution electron microscopy (HREM), 57

INDEX

Landau levels, 101-103 Lattice dynamics and phonon dispersion analysis, 222-224 Lattice thermal conductivity and figure of merit, 149-152 thin-film materials and, 184-185 Localization, weak, 190 comparison between diffusive transport and, 190-192 -like behavior in SiGe, 193-194 Lorentz number, 206 Low-dimensional materials background information, 203-205 Boltzmann transport equation (BTE), 205-208, 224-227 concurrent electron-phonon modeling, 250 lattice dynamics and phonon dispersion analysis, 222-224 molecular dynamics (MD) simulation, 228-229 Monte Carlo simulation, 228 phonon dispersion in nanostructures, 210-214 phonon engineering in nanostructures, 246-250 phonon thermal conductivity in bulk materials, 206-210 summary, 250- 253 thermal conductivity of nanostructures, 230-246 thin film thermal conductivity measurement techniques, 214-222 Low-temperature heteroepitaxy, 176-179

M

In-plane carrier transport, 179-181 Ioffe-Riegel criterion, 192

Kronig-Penney model, 11-12, 13, 38-39

Magnetoresistance of Bi nanowires, 96-103 Matthiessen's rule, 74, 87, 207, 225, 232 Maxwell-Garnett approximation, 109, 110 Mean-free-path (MFP) data, 171, 186-187, 204, 207, 209-210, 232, 246 Metal-organic chemical vapor deposition (MOCVD), 176, 235 Minimum thermal conductivity theory, 246-250 Molecular beam epitaxy (MBE), 9, 124, 235

263

INDEX Molecular dynamics (MD) simulation, 228-229, 244 Monte Carlo simulation, 228 Multi-quantum-well superlattices of PbTe, 22-26

Nanostructures phonon dispersion in, 210-214 phonon engineering in, 246-250 thermal conductivity of, 230-246 Nanowires background information, 54 comparison between Sb and Bi, 111-114 electronic structure of, 58-68 Raman spectra and optical properties, 109-111 thermal conductivity, 107-109 Nanowires, bismuth compared with Sb, 111-114 doping of, 68-70, 91-96 four-point resistivity measurement of, 89-91 magnetoresistance of, 96-103 mobility tensors at 77 K, 73-78 Seebeck coefficient, 103-107 semi-classical transport model for, 70-83 structure of, 54-57 summary, 114 temperature-dependent resistivity of, 83-96 T-point holes, effects of, 78-83

Optical-electrical hybrid methods, 221- 222 Optical properties, nanowires and, 109-111 Optical pump-and-probe methods, 219-221

PbTe, 17 bulk properties of, 18-22 multi-quantum-well superlattices of, 2226 Peltier effect, 124, 196-197

Phonon dispersion in nanostructures, 210214, 222- 224 Phonon engineering in nanostructures, 246-250 Phonon reflection, 189-190 Phonon scattering, 135, 137, 139-149 Phonon thermal conductivity in bulk materials, 206- 210 Phonon transport, 181-182 Polar optical phonons (POPs), 135, 137, 140-141, 145, 147 Power factor quantitative theory of, 134-149 role of, 124 semiquantitative theory of, 127-134 Proof-of-principle studies, 8-11 for SiGe, 35-36

Quantitative theory of power factor, 134149 Quantum wells and wires, thermoelectric applications and background information, 1-3 carrier pocket engineering, 11-17 models, 3-8 proof-of-principle studies, 8-11 summary, 114 Quantum wells and wires, thermoelectric transport in background information, 123-127 lattice thermal conductivity and figure of merit, 149-152 quantitative theory of power factor, 134-149 semiquantitative theory of power factor, 127-134 summary, 152 Quantum wells and wires, two dimensional Bi, 44-53 PbTe, 17-26 SiGe, 26-44

Radioisotope thermoelectric generators (RTGs), 26

264 Raman spectra, nanowires and, 109-111 Rattling, 209 Rayleigh law, 207 Richardson's equation, 157, 159-160, 167, 169 Ritz iterative method, 138

Sb nanowires, Bi compared with, 111-114 Scanning electron microscopy (SEM), 55, 56 Schottky barrier, 159, 163, 165-166 Schrrdinger equation, 59, 60, 61, 136, 139 Seebeck coefficient, 2, 9, 10 in Bi nanowires, 103-107 in thermionic refrigeration, 162 in thin-film materials, 197-198 Selected-area electron diffraction (SAED), 56, 57 Semiquantitative theory of power factor, 127-134 Shubnikov-de Haas (SdH) oscillatory effect, 101-103 SiGe applications, 26 bulk properties of, 27-28 carrier pocket engineering, 36-44 figure of merit, 30-33 localization-like behavior in, 193-194 proof-of-principle study, 35- 36 superlattice formation, 29-30 temperature-dependent behavior, 34- 35 Stephan-Boltzmann constant, 161 Superlattices See also Bi2Te3-Sb2Te 3 superlattices quantum well, 135-139 quantum wire, 139-140 thermal conductivity of, 234-243 ultra-short-period, 192-193

Te-doped Bi nanowires, 91-96 Temperature-dependent resistivity of Bi nanowires, 83-96 Thermal conductivity, 107-109 cross-plane, measurements of, 182-184

INDEX

lattice, 149-152, 184-185 minimum thermal conductivity theory, 246-250 phonon, in bulk materials, 206-210 in thermionic refrigeration, 162 Thermal conductivity of nanostructures heat conduction in, 244-246 measurement techniques, 214-222 one-dimensional structures, 243-244 single-layer thin films, 230-234 of superlattices, 234-243 Thermal transport. See Quantum wells and wires, thermoelectric transport in Thermionic emission, 157 Thermionic refrigeration background information, 157-160 ballistic transport, 170-171 conclusions, 172 multilayer devices, 166-170 one-barrier solid-state device, 163-166 vacuum device, 160-163 Thin film thermal conductivity measurement techniques membrane and bridge methods, 218-219 microsensor methods, 214-219 optical-electrical hybrid methods, 221222 optical pump-and-probe methods, 219221 3o~ method, 214-217 Thin-film thermoelectric materials background information, 175-176 comparison between diffusive transport and localization, 190-192 cooling, 197-198 cross-plane carrier transport, 194-196 cross-plane thermal conductivity, measurements of, 182-184 diffusive transport analysis, 187-189, 190-192 in-plane carrier transport, 179-181 lattice thermal conductivity, 184-185 localization-like behavior, 193-194 low-temperature heteroepitaxy, 176-179 mean free path reduction, 186-187 Peltier effect, 196-197 phonon reflection, 189-190 phonon transport, 181-182 summary and conclusions, 198-199 ultra-short-period superlattices, 192-193

265

INDEX Thomas-Fermi dielectric function, 149 309 method, 214-217 T-point holes, effects of, 78-83 Transmission electron microscopy (TEM) studies, 10-11, 56, 177, 178-179

W Wiedemann-Franz law, 2, 33, 167, 194, 206 Work function, 157, 158, 159

X-ray diffraction (XRD), 56, 57, 68 Umklapp process, 207, 208

van der Waalls epitaxy, 176-179

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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. IV. Gobeli and I. 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

267

268

CONTENTSOF VOLUMESIN THIS SERIES Volume 3

O p t i c a l of P r o p e r t i e s I I I - 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. G. Mavroides, Interband Magnetooptical Effects H. Y 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. .4. 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 AnIBv 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

I n f r a r e d Detectors

H. Levinstein, Characterization of Infrared Detectors P. W. Kruse, Indium Antimonide Photoconductive and Photoelectromagnetic Detectors M. B. Pr&ce, 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. ,4. Lampert and R. B. Schilling, Current Injection in Solids: The Regional Approximation Method K Williams, Injection by Internal Photoemission ,4. M. Barnett, Current Filament Formation

CONTENTS OF VOLUMES IN THIS SERIES

269

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, MOS 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 I I I - V Compounds with Indirect Band Gaps R. W. Ure, Jr., Thermoelectric Effects in I I I - V Compounds H. Piller, Faraday Rotation H. Barry Bebb and E. IV. Williams, Photoluminescence I: Theory E. IV. 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

Volume 10

Transport Phenomena

R. L. Rhode, Low-Field Electron Transport J. D. Wiley, Mobility of Holes in I I I - V Compounds C. M. Wolfe and G. E. Stillman, Apparent Mobility Enhancement in Inhomogeneous Crystals R. L. Petersen, The Magnetophonon Effect

270

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 Teiluride

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

Hgl_xCdxSe alloys M. H. Weiler, Magnetooptical Properties of Hgl_ CdxTe Alloys P. W. Kruse and J. G. Ready, Nonlinear Optical Effects in Hgl_ CdxTe

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

271

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. IV. 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 W F H. Micklethwaite, The Crystal Growth of Cadmium Mercury Telluride P. E. Petersen, Auger Recombination in Mercury Cadmium Telluride R. M. Broudy and K J. Mazurczyck, (HgCd)Te Photoconductive Detectors M. B. Reine, A. K. Soad, and 7". J. Tredwell, Photovoltaic Infrared Detectors M. ,4. 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. K Pleskon, Photoelectrochemistry of Semiconductors

Volume 20 Semi-Insulating GaAs R. N. Thomas, H. M. Hobgood, G. IV. Eldridge, D. L. Barrett, T. 7'. 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

272

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 W. 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 T. 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 K 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

273

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, W. 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 W. T. Tsang, Molecular Beam Epitaxy for III-V Compound Semiconductors G. B. Stringfellow, 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. AraL and F. Koyama, Dynamic Single-Mode Semiconductor Lasers with a Distributed Reflector W. T. Tsang, The Cleaved-Coupled-Cavity (C 3) Laser

Part C R. J. Nelson and N. K. Dutta, Review of InGaAsP 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-pm Regions Y. Horikoshi, Semiconductor Lasers with Wavelengths Exceeding 2 pm 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

274

CONTENTS 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. MukaL 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 Muitiquantum 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

V o l u m e 25

275

Diluted Magnetic Semiconductors

w. Giriat and J. K. Furdyna, Crystal Structure, Composition, and Materials Preparation of

Diluted Magnetic Semiconductors W. M. Becket, Band Structure and Optical Properties of Wide-Gap A~x_xMnxBiv Alloys at

Zero Magnetic Field S. Oseroff and P. 11. 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

V o l u m e 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. E. C J. L.

P. Pouquet, Structural Instabilities M. Conwell, Transport Properties S. Jacobsen, Optical Properties C Scott, Magnetic Properties 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 A. M. Weiner and R. B. Marcus, Photoemissive Probing

276

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 A. 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

Volume 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 T. 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. Gerard, 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 3". 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

277

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. Gunshor, L. A. Kolodziejski, A. V. Nurmikko, and N. Otsuka, Molecular Beam Epitaxy of II-VI Semiconductor Microstructures

Volume 34

Hydrogen in Semiconductors

J. I. Pankove and N. M. Johnson, Introduction to Hydrogen in Semiconductors C. H. Seager, Hydrogenation Methods J. I. Pankove, Hydrogenation of Defects in Crystalline Silicon J. W. Corbett, P. De~k, 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. Haller, 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

Volume 35

Nanostructured Systems

M. Reed, Introduction 11. 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

Volume 36 D. .4. .4. O. D.

The Spectroscopy of Semiconductors

Heiman, Spectroscopy of Semiconductors at Low Temperatures and High Magnetic Fields K Nurmikko, Transient Spectroscopy by Ultrashort Laser Pulse Techniques K. Ramdas and S. Rodriguez, Piezospectroscopy of Semiconductors J. Glembocki and B. K 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

278

CONTENTS OF VOLUMES IN THIS SERIES

V o l u m e 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

V o l u m e 38

Imperfections in III/V Materials

u. Scherz and M. Sche~er, 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

V o l u m e 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 Ga 1_xlnxAs/InP Quantum Wells

CONTENTS OF VOLUMES IN THIS SERIES

Volume 41

279

High Speed Heterostructure Devices

F. Capasso, F. Beltram, S. Sen, A. PahlevL and A. 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. OhnishL 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 IV. 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, F. 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. Iwanczyk and B. E. Patt, 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

280

CONTENTS OF VOLUMES IN THIS SERIES

Volume 44 II-IV Blue/Green Light Emitters: Device Physics and Epitaxial Growth J. Han and R. L. Gunshot, 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. V. 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 A. 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 ,4. 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

281

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 Microbolometer 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 A1GalnP for High Efficiency Visible Light-Emitting Diodes F. A. Kish and R. M. Fletcher, A1GalnP 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 Isoclcctronic Impurities in Silicon and Silicon-Germanium Alloys and Supcrlattices J. Michel, L. V. C. Assali, M. T. Morse, and L. C. Kimerling, Erbium in Silicon E 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 Nonradiativc Processes in Silicon Nanocrystallites 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

282

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 III-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 IV. 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

Identification of 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. HautojiirvL 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. Kisielowski, 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

283

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. L 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_yCr and Sil_x_rGexCr Alloy Layers

284

CONTENTS OF VOLUMES IN THIS SERIES

Volume 57

Gallium Nitride (GaN) II

R. 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 W. G6tz and N. M. Johnson, Characterization of Dopants and Deep Level Defects in Gallium Nitride B. Gil, Stress Effects on Optical Properties C. Kisielowski, 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 A1GaN 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. Khurg&, 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 Mukai, Mitsuru Sugawara, Mitsuru Egawa, and Nobuyuki Ohtsuka, Metalorganic Vapor Phase Epitaxial Growth of Self-Assembled InGaAs/GaAs Quantum Dots Emitting at 1.3 #m

Kohki Mukai and Mitsuru Sugawara, Optical Characterization of Quantum Dots Kohki Mukai and Mitsuru Sugawara, The Photon Bottleneck Effect in Quantum Dots Hajime Shoji, Self-Assembled Quantum Dot Lasers Hiroshi Ishikawa, Applications of Quantum Dot to Optical Devices Mitsuru Sugawara, Kohki Mukai, Hiroshi Ishikawa, Koji Otsubo, and Yoshiaki Nakata, The Latest News

CONTENTS OF VOLUMES IN THIS SERIES

Volume 61

285

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. Haller, Hydrogen in III-V and II-VI Semiconductors S. J. Pearton and J. W. Lee, The Properties of Hydrogen in GaN and Related Alloys Jrrg 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. 1I. 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 LL Bruce Tredinnick, and Mel Hoffman, Consumables I: Slurry Lee M. Cook, CMP Consumables II: Pad Francois Tardif, Post-CMP Clean Shin Hwa LL Tara Chhatpar, and Frederic Robert, CMP Metrology Shin Hwa LL 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. W. Huang, M. R. Krames, and S. A. MaranowskL High-Etficiency A1GalnP Light-Emitting Diodes 1~ S. Kern, IV. Grtz, 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 1I. Bulovik, P. E. Burrows, and S. R. Forrest, Molecular Organic Light-Emitting Devices

286

CONTENTS OF VOLUMES IN THIS SERIES V o l u m e 65

E l e c t r o l u m i n e s c e n c e II

V. Bulovik and S. R. Forrest, Polymeric and Molecular Organic Light Emitting Devices: A Comparison Regina Mueller-Mach and Gerd O. Mueller, Thin Film Electroluminescence Markku Leskelgi, 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 H. Kurz, Optically Excited Bloch Oscillations-Fundamentals and Application Perspectives

V o l u m e 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 Durnitric(l, 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

287

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

Volume 70

Recent Trends in Thermoelectric Materials Research II

Brian C. Sales, David G. Mandrus, and Bryan C. Chakoumakos, Use of Atomic Displacement Parameters in Thermoelectric Materials Research S. Joseph Poon, Electronic and Thermoelectric Properties of Half-Heusler Alloys Terry M. Tritt, A. L. Pope, and J. liE. Kolis, Overview of the Thermoelectric Properties of Quasicrystalline Materials and Their Potential for Thermoelectric Applications Alexander C. Ehrlich and Stuart A. Wolf, Military Applications of Enhanced Thermoelectrics David J. Singh, Theoretical and Computational Approaches for Identifying and Optimizing Novel Thermoelectric Materials Terry M. Tritt and R. T. Littleton, IV, Thermoelectric Properties of the Transition Metal Pentatellurides: Potential Low-Temperature Thermoelectric Materials Franz Freibert, Timothy W. Darling, Albert MigliorL and Stuart A. Trugman, Thermomagnetic Effects and Measurements M. Bartkowiak and G. D. Mahan, Heat and Electricity Transport through Interfaces

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