THERMOELECTRIC POWER OF METALS

Transcription

1 THERMOELECTRIC POWER OF METALS

2 THERMOELECTRIC POWER OF METALS Frank J. Blatt, Peter A. Schroeder, and Carl L. Foiles Michigan State University East Lansing, Michigan and Denis Greig University of Leeds Leeds, England PLENUM PRESS. NEW YORK AND LONDON

3 Library of Congress Cataloging in Publication Data Main entry under title: Thermoelectric power of metals. Includes bibliographical references and index. 1. Thermoelectricity. 2. Metals-Thermal properties. 3. Metals-Electric properties. I. Blatt, Frank J. QC621.T ' ISBN-13: e-isbn-13: : / Plenum Press, New York Softcover reprint of the hardcover 1st edition 1976 A Division of Plenum Publishing Corporation 227 West 17th Street, New York, N. Y All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming. recording, or otherwise, without written permission from the Publisher

4 Preface Thermoelectric and related transport properties of metals have been a source of information and, also, exasperation to physicists for over a century. Perhaps the principal reasons for interest in these phenomena are their sensitivity to composition, structure and external fields and, until fairly recently, the distressing fact that often even gross experimental features such as the sign of the thermopower eluded theoretical understanding. During the past two decades many of the previously perplexing aspects of thermoelectricity have yielded to more sophisticated theoretical treatment. As a result of this effort and concomitant experimental work using advanced measurement techniques, there is now good reason to believe that thermoelectric phenomena can shed much light on the interactions between electrons and phonons, impurities, and other defects. The last few years have witnessed new and fascinating developments that promise to stimulate new activity in this field. In contrast to the more conventional transport properties, second- and high-order contributions in electron scattering theory appear to play a profound role in thermoelectricity-the controversy surrounding ordinary and "phony" phonon drag is far from resolved; the startlingly large effect of magnetic fields on the thermopower of metals appears to be linked intimately to scattering anisotropy; quantum oscillations of thermopower are orders of magnitude larger than corresponding oscillations of the magnetoresistance; a new approach to thermoelectric studies allows extension of thermopower measurements into the millikelvin region of temperature; finally, the advent of superconducting detection devices permits the precise measurement of extremely small voltages, an essential requirement in this field. Since several review articles and a book on thermoelectricity have appeared in the last few years, we have placed special emphasis on those v

5 vi Preface facets of the field that appear to be of current interest and promise to hold the attention of experimenters and theorists for at least a few years. Following a brief introductory chapter and a cursory survey of the electronic theory of metallic conduction is a chapter on experimental techniques which stresses Josephson junction devices as used in thermopower studies. This is followed by a detailed discussion of phonon drag. For comparison purposes, the provocative theoretical work of Nielson and Taylor on the effect of second-order scattering on diffusion thermopower is included in the same chapter. The next two chapters are closely related; one is concerned with the thermopower of transition metals, with special attention to the relationship between this and their magnetic properties, and the other on dilute magnetic alloys wherein the still not fully resolved Kondo effect is highlighted. The final chapter is devoted to effects of pressure and of magnetic fields on thermoelectricity, areas where there have been a number of fascinating developments in the last years. Although we have not compiled a complete bibliography, we have made an attempt to include a large and representative list of references. We apologize, in advance, for the many omissions. Most of the illustrations in this book are reproduced from articles that have appeared in the periodical literature or from other books; the source of each such illustration appears in the figure caption. We are grateful to the many authors and publishers who gave their permission to reproduce these illustrations. In the preparation of the manuscript we have had the benefit of discussions with many individuals. One deserves particular mention; Dr. Jon Opsal not only gave it a thorough critical reading, but he also made numerous valuable suggestions and is largely responsible for Section 4.7. We also express our thanks to Mrs. Jean Strachan for her patience and skill in typing the manuscript. June 1976 East Lansing, Michigan Leeds, England F.J.B. P.A.S. CL.F. D.G.

6 Contents 1. Introduction Seebeck, Peltier, and Thomson Effects 1.2 Transport Coefficients and Onsager Relations Survey of the Theory of Electronic Conduction in Metals Electrons in Metals a Free Electron Gas b EnergyBands ' Transport Properties Relaxation Time Anisotropy for Spherical Fermi Surfaces Thermopower: Isotropic Relaxation Time Approximation S Thermopower: Real Metals 24 2.Sa Alkali and Noble Metals 24 2.Sb Polyvalent Metals Phonon Drag Thermopower of Alloys a Diffusion Thermopower b Phonon Drag Techniques in Thermoelectric Measurements 3.1 Introduction 3.2 Seebeck Effect SO vii

7 viii Contents 3.3 Peltier Effect Thomson Effect: The Absolute Thermopower of Lead Measurement of Unconventional ThermoelectricCoefficients Measurement of Temperature and of Small Voltages a Temperature Measurement b Voltage Measurement SuperconductingDevices a Superconducting Modulators b Weak Link and Josephson Junction Devices- SQUIDS and Slugs bl SQUIDS b2 Slugs Phonon Drag Introduction and General Relations Sg at High Temperatures Sg at Low Temperatures a Low-Temperature Phonon Drag in the Alkali Metals b Low-Temperature Phonon Drag in Other Metals Anisotropy of Relaxation Times and Phonon-Drag Thermopower Sg at Intermediate Temperatures Sgof Alloys Phonon Drag or Phony Phonon Drag? a Pure Metals b Dilute Alloys c Evidence for "Phony Phonon Drag" d Effects of Higher-Order Scattering Processes The Thermoelectric Power of Transition Metals Special Problems in Transition Metals 5.1a s- and d-conduction 5.1 b Electron-Electron Collisions

8 Contents ix 5.1c Magnetic Effects: Collective Electrons and Isolated Spins d Magnetic Effects: Magnons and Paramagnons le Magnetic Effects: Spin Mixing H Magnetic Effects: The Curie and Neel Temperatures The Diffusion Thermopower of Transition Metals a Phonons and Impurities b Electron-Electron Scattering c Magnons and Paramagnons d Two-Current Conduction and Spin Mixing e Curie Point Anomalies The Phonon-Drag Thermopower of Transition Metals Magnon Drag Transition Elements: Summary of Experimental Results a The Magnetic Elements: Cr, Mn,Fe, Co, andni b The "Thermocouple" Elements: Fe, Pt, Re, and W Commercial Thermocouples a Copper vs. Constantan (Type T) b Iron vs. Constantan (Type J) c Chromel vs. Constantan (Type E) d Chromel vs. Alumel (Type K) e Tungsten vs. Tungsten-Rhenium (Types G* and c*) f Platinum vs. Platinum-Rhodium (Types R, S, and B) Dilute Magnetic Alloys Introduction The Virtual Bound State a Electronic Properties for VBS Systems b Survey of Experimental Results for VBS Systerns 6.3 Kondo Alloys a Theory of the Kondo Effect b Thermoelectric Power of Kondo Alloys

9 x Contents 6.4 Spin-Fluctuation Models 6.5 Closing Comment Effects of Pressure and Magnetic Field on the Thermoelectric Power Pressure Dependence 7.1a Introduction. 7.1 b Practical Thermocouples 7.1c Fundamental Studies. 7.1c 1 Diffusion Thermopower 7.lc2 Phonon-Drag Thermopower 7.2 Magnetic Field Dependence. 7.2a Introduction b Practical Thermocouples 7.2c Fundamental Studies. 7.2d Landau Quantization Effects References. Author Index Subject Index

10 Notation First used Symbol Definition on page Equation an Direct lattice vector c Mole fraction of point defect Ce Electronic specific heat a Cg Lattice specific heat Cm Magnon specific heat e Electronic charge 7 E Thermal emf E Electric field vector EL Energy of virtual bound state 194 I Non equilibrium distribution function for electrons 10 Equilibrium Fermi-Dirac distribution It F External driving force 17 G Thermoelectric ratio H Magnetic field j Polarization index 14 I Exchange integral 141.J Electrical current density Ii Generalized flux of type i k Boltzmann constant 15 k Electron wave vector 14 xi

11 xii Notation First used Symbol Definition on page Equation k f Fermi wave vector 124 KnoKn Transport integrals Kn Reciprocal lattice vector l(e ), Electron mean free path I(k) Electron mean free path Phonon mean free path L Wiedemann-Franz ratio Lo Lorenz number 11 rt/ Transport tensor m Electron mass M Atomic mass Ms(T) Saturation magnetization 162 no Electron density N(e) Density of states for electrons Nd(e) Density of states for d electrons 136 Ns(e) Density of states for s electrons 146 No(q),N(q) Equilibrium and perturbed ,4.3 phonon distributions p Electron momentum 15 q Phonon wave vector 14 Clmin Minimum q for Umklapp process 35 Q Heat current density Q p Peltier heat r Position vector 17 S Absolute thermopower tensor SA,SB Absolute thermopowers for isotropic material SAB Thermopower of thermocouple Sd Diffusion thermopower S~ of d electrons S~e for electron-electron scattering 152 S~ for phonon scattering S~ for jth scattering mechanism S~ of s electrons S~, SdO for residual impurity scattering 99, Sg Phonon drag thermopower

12 Notation xiii First used Symbol Definition on page Equation SO g for electron phonon scattering only SN g for normal scattering processes 35 SU g for Umklapp scattering 35 processes Sg, for ith region of Fermi surface Sm Magnon drag thermopower So 1T 2 etb/3e7] T Temperature Tc Curie temperature 160 TK Kondo temperature 205 TN ~eel temperature 146 v Velocity of sound v Particle velocity 7 V Potential V Volume V(jq) Velocity associated with 88 phonon (jq) W Electronic thermal resistivity Wee for electron-electron scattering 152 W; for electron-phonon scattering "'i for scattering from impurity j Wr for residual impurity scattering x S/So XbAxl x=xl+4xl Xj,Xj Friedel parameters x hw/kt Zj Valence of host 40 ~ Valence of impurity 40 Z Zs-Zj 150 4Z Charge on ion-charge on 194 host ion a(jq; kl, k'l') Phonon scattering probability aj d lnpj/d In V 'Y Griineisen constant 'Y (." -ed)/kt 147

13 xiv Notation First used Symbol Definition on page Equation [ Half width of virtual bound state 195 8/ Phase shift of lth partial wave 195 d [/2 203 do Atomic volume Electron energy 7 y Surface area in k space T/ Fermi energy 7 ()v Debye temperature 23 ()L Debye temperature for longitudinal waves ()T Debye temperature for transverse waves ()* hvqrnin/ k K Electronic thermal conductivity (j = 0) Ke Electronic thermal 8 conductivity (E = 0) Kg Lattice thermal conductivity Ai Phonon mean free path for impurity scattering Ae Phonon mean free path for electron scattering J.t Thomson coefficient J.te Effective magnetic moment IlA' IlAB Peltier coefficients Ile (= 0-1 ) 8 Ilt (= IlA) 8 P Resistivity Pee electron-electron scattering 138 Pi phonon scattering Pj scattering by impurity j p, residual impurity scattering 196 (J Electrical conductivity 11 (Ja adiabatic 11 (Jd of d electrons (Ji of ith region of Fermi surface

14 Notation xv First used Symbol Definition on page Equation (Ts of s electrons (T, Isothermal electrical a conductivity T Electron relaxation time Tdd for d-d scattering 146 Tsd for s-d scattering 146 TO(t), Tom for impurity scattering of 154 electrons with spins (t) and m Tg Mean phonon relaxation time Tp Phonon relaxation time Tpe for electron scattering Tpi for impurity scattering Tpp for phonon scattering I Tp for all scattering processes other than phonon-electron l/j Magnetic flux 74 l/j Scattering temperature l/jo Flux quantum Entropy