Solid State Physics - Ashcroft/Mermin PDF

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15 ~Harcourt ~College Publishers

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A Harcourt Higher Learning Compony

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Now you will find Saunders College Publishing's distinguished innovation, leadership, and support under a different name .•• a new brand that continues our unsurpassed quality, service, and commitment to education.

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We are combining the strengths of our college imprints into one worldwide brand: ~Harcourt Our mission is to make learning accessible to anyone, anywhere, anytime-reinforcing our commitment to lifelong learning. We are now Harcourt College Publishers. Ask for us by name. -

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""VIhere learning

Com s to Life." \vvvvv.harcourtcollege.com )

tate Neil W. Ashcroft N. David Mermin Cornell University

Saunders College Publishing Harcourt College Publishers Fort Worth Philadelphia San Diego New York Orlando Austin San Antonio Toronto Montreal London Sydney Tokyo

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Copyright© 1976 by Harcourt. Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any fonn or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Requests for permission to make copies of any part of the work should be mailed to: Permissions Department, Harcourt, lnc., 6277 Sea Harbor Drive, Orlando, FL 32887-6777.

This book was set in Times Roman Designer. Scott Olelius Editor: Dorothy Garbose Crane Dnn~ings: Eric G. Hieber Associates. Inc.

Library of Congress Cataloging in Publication Data Ashcroft, Neil W. Solid state physics. I. Solids. II. Title.

I. Mermin, N . David. joint author.

QC176.A83 530.4'1 74-9772 ISBN 0-03-083993-9 (College Edition)

Printed in the United States of America SOLJD STATE PHYSICS ISBN# 0-03-083993-9 (College Edition)

0 I 2 3 4 5 6 7 8 9 076 35 34 33 32 31 30 29 28 27

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for Elizabeth, Jonathan, Robert, and Jan

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P1e ace We began this project in 1968 to fill a gap we each felt acutely after several years of teaching introductory solid stare physics to Cornell students of physics, chemistry, engineering, and materials science. In both undergraduate and graduate courses we had to resort to a patchwork array of reading assignments, assembled from some half dozen texts and treatises. This was only partly because of the great diversity of the subject ; the main problem lay in i1s dual nature. On the one hand an introduction to solid state physics must describe in some detail the vast range of real solids, with an emphasis on representative data and illustrative examples. On the other hand there is now a weB-established basic theory of solids, with which any seriousl y interested student must become familiar. Rather to our surprise, it has taken us seven years to produce what we needed: a single introductory text presenting both aspects of the subject, descriptive and analytical. Our aim has been to explore the variety of phenomena associated with the major forms of crystalline matter. while laying the foundation for a working understanding of solids through clear, detailed, and elementary treatments of fundamental theoretical concepts. Our book is designed for introductory courses at either the undergraduate or graduate level. 1 Statistical mechanics and the quantum theory lie at the heart of solid state physics. Although these subjects are used as needed, we have tried, especially in the more elementary chapters, to recognize that many readers. particularly undergraduates, will not yet have acquired expertise. When it is natural to do so, we have clearly separated topics based entirely on classical methods from those demanding a quantum treatmenL In the latter case, and in applications of statistical mechanics, we have proceeded carefully from explicitly stated first principles. The book is therefore suitable for an introductory course taken concurrently with first courses in quantu m theory and statistical mechanics. Only in the more advanced chapters and appendices do we assume a more experienced readership. The problems that foil ow each chapter are tied rather closely to the text. and are of three general kinds : (a) routine steps in analytical development are sometimes relegated to problems, partly to avoid burdening the text with formulas of no intrinsic interest, but, more importantly, because such steps are better understood if completed by the reader with the aid of hints and suggestions; (b) extensions of the chapter (which the spectre of a two volume work prevented us from including) are presented as problems when they lend themselves to this type of exposition ; (c) further numerical and analytical applications are given as problems, either to communicate additional

1

Sugf!CStion~

for how to use the text in

I

cou~

of varying length and level ore gh·~n on pp. xviii -xxt.

vii

,·iii

Preface

information or to exercise newly acquired skills. Readers should therefore examine the problems. even if they do not intend to attempt their solution. Although we have respected the adage that one picture i~ wonh a tho usand \\Ords. we are also aware that an uninformative illustration, though decorative, takes up the space that could usefully be filled by several hundred . The reader will rhus cncowuer stretches of expository prose unrelieved by figures, when none are necessary, as well as sections that can profitably be perused entirely by looking at the figures and their captions. We anticipate use of the book at different levels with different areas of major emphasis. A particular course is unlikely to follow the chapters (or even selected chapters) in the order in which they are presented here, and we have written them in a way that permits easy selection and rearrangement. 2 Our particular choice ofsequence follows certain major strands of the subject from their first elementary exposition to their more advanced aspects, with a minimum of digression. We begin the book3 with the elementary classical [I] and quantum [2] aspects of the free electron theory of metals because this requires a minimum of background and immediately introduces, through a particular class of examples, almost all of the phenomena with which thcorie~ of insulators. semiconductors. and metals mu~t come to grips. The reader is thereby spared t he impression that nothing can be understood tlntil a host of arcane definitions (relating to periodic strucll!res) and elaborate quantum mechanical explorations (of periodic systems) have been mastered. Periodic structures are introduced only after a survey (3) of those metallic properties that can and cannot be understood without investigating the consequences of periodicity. We have tried to alleviate the tedium induced by a first exposure to the language of periodic systems by (a) separating the very importan~ consequences of purely translational symmetry [4. 5] from the remaining but rather less essential rotational aspects [7), (b) separating the description in ordinary space [4] from that in the less familiar reciprocal space [5), and (c) separating the abstract a nd descriptive treatment of periodicity from its elementary application to X-ray diffraction [6]. Armed with the terminology of periodic systems, readers can pursue to whatever point seems appropriate the resolution of the difficulties in the free e lectron model of metals or. alternatively, can embark directly upon the investigation of lattice vibrations. The book follows the first line. Bloch ·s theorem is described and its implications examined [8] in general tenns, to emphasize that its consequences transcend the illustrative and very impo rtant practical cases of nearly free electrons (9] and tigh t binding [10]. Much of the content of these two chapters is suitable for a more advanced c:ourse, as is the following survey of methods used to compute real band structures [II]. The remarkable subject of semiclassical mechanics is introduced and given elementary applications [I 2] before being incorporated into the more elaborate semiclassical theory of transport (I 3]. The description of methods by which Fcm1i surfaces arc measured [14] may be more stlitable for advanced readt:rs, but m uch of the survey : The Tabh: o n pp. xi" x>ilh a mmimum uf dig.-.,.sion it i~ n ccc..' be asl..cd to read the chapters beano~ on the lc:cturc::. a~ well.

One-Semester Introduction

Chapter

Prerequisites

I

Two-Semester Introduction

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I I

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LECTURES

READING

READING

LECTURFS

I. Drude

None

All

All

2. Sommerfeld

I

All

All

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3. Failures of free-electron model 4. Crystallattices

5. Reciprocal lattice

- -

6. X-ray diffraction 7. Crystal symmetries 1-

8. Bloch's theorem 9. Nearly free electrons

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132-143

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8 (6)

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11 . Computing band structure

8 {9)

12. Semiclassical dynamics

2, 8 12 '

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152- 166 --

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·- .- -

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All

--

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All 176- 184

192- 193 214- 233

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All

176

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All

All

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5

I0. Tight binding

-··-·--

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96- 104 -

13. Semiclassical transport

All

All

4

-

All

Summarize

5

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All

1-

4

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None

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----

All

2

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--

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184- 189

All

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214- 233 -

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244- 246

Chuptl!r

Prerequisite.'/

l..ECJ'URI!S

14. Moasunng the Fermi surface I 5.

~md

Bcynnc. relnxation-tin"le approximation

17. Beyond tndepcndcnl clcclron a:ppruxi:matiun

I

-

19. Classificn1ion o f solids

f--

Fajlurcs of $la tic lauJcc model

2C>4- 27.S All

2

337 - 342

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2, 4 (6, 8)

354 3(i4

354 3(>4

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23. Quunlum harm(mic crystal

22

-

All AU

396-410

All

.

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-

All

422 - 4 37

All

452- 464

All

-

l

-

470- 481

All

2, 23

25. Anhurmortie drccl$

23

26. Phooons in metals

17. 23 (16)

27. o .elcctnc properties

19. 22

28. Homc.lgencous semiconductors

2, 8 , ( 12)

562- 580

All

28

590-600

All

-I

499 505

534- 542

31.

DUmlaJPICiism. Paratn.bgnctism

4 (8, 12. 19 . 22. 28. 29)

L

.All 512 519 523 - 526

523 - 526

29. Inhomogeneous

!!.

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24. Measuring pbonons

30. DefectS

All

/\II

628 - 63(> All

(2, 4. 14)

661 - 665

32. Magnetic inlcrt\t tions

31 (2.8.10. 16, 17)

672

33. Magnetic ordering

4,5.J2

694-700

All

1. 2(26)

726- 736

All

1 34

Superconducllvily

345 - 351

All

19 (17)

22. C la.,;c:imple picture:. and rough estimates of properties "hose more prec•~c compn:hcnston may require analysis of considerable complex H) . The failures of 1he Drude model to account f,>r some experiments. a nd t he conceptual puzz.les it rat~ed. defined the problems \\llh which 1he theory of metals was to grapple over the ne'\1 q uancr century. These found their resolution only in the rich and ~ubtle structure o f the quantum theory of solid:..

BASIC ASSUMPTIONS OF THE ORL"DE !\lODEL

J. J . Thomson's discovery of the e lectron in 1897 bad a vast :111d unmcdiare impact on theories of the structure of maller. a nd suggested an ob,•ious mechanism fo r conduction in metals. Three years after Thomson's disCO\'ery Urudc constructed hts theory of electrical and therma l conduction by applymg the h1ghly successful kinetic theory of gases to a metal. considered as a gas of electrons. In its simplest form kmettc theory treats the molecules of a gas as identical solid spheres. which move in stra tght lines until they collide \nth one anot her. 2 The time taken up by a sing le colliston IS assumed to be negligtblc. and. e>.cept for the forces commg momentaril} mto pia} during each collision. no o t her forces are assumed to act between the panicles. Altho ugh the re IS only one kind of panicle presem in the stmplest gases. in a metal there must be a t least two. for the electrons are negaurely charged. yet the metal is e lectrically neutral. Drude assumed that the compensating positive charge was at· '

lmiCih•n dlh lh~ ".11'- f the ""'~d cnl.unon!l th~m. af''ing mct.tl< unk ' ), ''-ne i~ 1nh:~tcd 1n \Cr~ f1 nc \\ I h."> thtn "h""t:ts.. c r- c:ff..~ts ut thc 'urr.-.x

Basic .\ s.'m (not to scale). !b) In a metal the nucleu~ and iun core rct;un their conligurauon 111 the free :11om. but the valence electrons leave the a tom to form the electron l!ill>.

-

tached to much heavier particles, wh ich he considered to be immobilt:. At his time, however, there was no precise notion of the origin of the light. mobile electrons a nd the heavier. immobile, positively charged panicles. The solution to this problem is o ne of the fundamental achievements of the modern quantum theory o f solids. ln this discussion of the Drudc model however, we shall simply assume (and in many metals this assumption can be justified) that when atoms o f a metallic element are brought together to form a metal. the valence electro ns become detached and wander freely through the metal. while tbe metallic ions remain intact and play the role of the immobile positive pan ides in Drudc·s theory. This model is indicated schematically in Figure L l. A single isolated atom of the metallic element has a nucleus of charge eZ0 • where Zo is the atomic nu mber and e is the magnitude of the electronic charge 3 : e = 4.80 x 10- 10 electrostatic units (csu) = 1.60 x 10- ' 9 coulombs. Surrounding the nucleus are Za electrons of total charge - eZa. A few of these, Z, are the relatively weakly bound valence electrons. The remainingZa - Z electrons art: relatively tightly bound to the nucleus, play much less of a role in chemical reactions. and are known as the core electrons. When these isolated atoms condense to form a metal, the core electrons remain bound to the nucleus to form the metallic ion, but the valence electrons are allowed to wander far away from their parent atoms. In the meta llic context they are calh:d conduction elecuons.4 • We shall al"'a}'S take(' to be a pos11JVC number. • When. as in the Drude model. the core elect rons play a pa:.-,•ve role and the iun ucts as an indivisible inert entity. o ne often refers to the conduct ion electrons simply as ·· thc electrons." s;.l\ing the fullterm for Lim~'S when the distincllon bel ween conduction and core electrons is to be emphasized.

4

Chapcer 1 The Drude Theory of Metals

Drude applied kinetic theory to this ~gas" of conduction electrons of mass m, which (in contrast to the molecules of an ordinary gas) move against a background of heavy immobile ions. The density of the electron gas can be calculated as follows : A metatlic clement contains 0.6022 x 1024 atoms per mole (Avogadro's number) and p,.,j A moles per cm 3 • where p,., is the mass density (in grams per cubic centimeter} and A is the atomic mass of the clement. Since each atom contributes Z electrons, the number of electrons per cubic centimeter. n = N / V. is , = 0.6022 x 1024

z;,.,.

(1.1)

Table 1.1 shows the conduction electron densities for some selected metals. They are typically o f order IOn conduction electrons per cubic centimeter, varying from 0.91 x 1022 for cesium up to 24.7...