7 Edited by Peter Fulde. Springer Series in Solid-State Sciences

Transcription

1 7 Edited by Peter Fulde Springer Series in Solid-State Sciences

2 Springer Series in Solid-State Sciences Editors: M. Cardona P. Fulde H.-J. Queisser Volume 40 Semiconductor Physics - An Introduction By K Seeget Volume 41 The LMTO Method By H.L. Skriver Volume 42 Crystal Optics with Spatial Dispersion and the Theory of Excitations By V.M. Agranovich and V.L. Ginzburg Volume 43 Resonant Nonlinear Interactions of Light with Matter By V.S. Butylkin, A.E. Kaplan, Yu.G. Khronopulo, and E.I. Yakubovich Volume 44 Elastic Media with Microstmcture II By I.A. Kunin Three-Dimensional Models Volume 45 Electronic Properties of Doped Semiconductors By B.1. Shklovsky and A. L. Efros Volume 46 Topological Disorder in Condensed Matter Editors: F. Yonezawa and T. Ninomiya Volumes 1-39 are listed on the back inside cover

3 E. N. Economou Green's Functions in Quantum Physics Second Corrected and Updated Edition With 52 Figures Springer-Verlag Berlin Heidelberg GmbH

4 Professor Ele/therios N. Economou, PhD Department of Physics, University of Crete, Heraklion, Crete, Greece Series Editors: Professor Dr. Manuel Cardona Professor Dr. Peter Fulde Professor Dr. Hans-Joachim Queisser Max-Planck-Institut fur Festkărperforschung, HeisenbergstraBe 1 D-7000 Stuttgart 80, Fed. Rep. of Germany ISBN ISBN (ebook) DOI / Library of Congress Cataloging in Publication Data. Economou, E. N., Green's functions in quantum physics. (Springer series in solid-state sciences ; v. 7). Bibliography: p. Inc1udes index. 1. Green's functions. 2. Quantum theory. 1. Title. II. Series. QC G68E ' This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, reuse of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 54 ofthe German Copyright Law, where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. by Springer-Verlag Berlin Heidelberg 1979 and 1983 Urspriinglich erschienen bei Springer-Verlag Berlin Heidelberg New York 1983 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

5 to Sophia

6 Preface to the Second Edition In this edition the second and main part of the book has been considerably expanded as to cover important applications of the formalism. In Chap.5 a section was added outlining the extensive role of the tightbinding (or equivalently the linear combination of atomic-like orbitals) approach to many branches of solid-state physics. Some additional information (including a table of numerical values) regarding square and cubic lattice Green's functions were incorporated. In Chap.6 the difficult subjects of superconductivity and the Kondo effect are examined by employing an appealingly simple connection to the question of the existence of a bound state in a very shallow potential well. The existence of such a bound state depends entirely on the form of the unperturbed density of states near the end of the spectrum: if the density of states blows up there is always at least one bound state. If the density of states approaches zero continuously, a critical depth (and/or width) of the well must be reached in order to have a bound state. The borderline case of a finite discontinuity (which is very important to superconductivity and the Kondo effect) always produces a bound state with an exponentially small binding energy. Chapter 7 has been expanded to cover details of the new and fast developing field of wave propagation in disordered media. The coherent potential approximation (a simple but powerful method) is presented with an extensive list of references to the current literature. Then the electrical conductivity is examined both because it is an interesting quantity in its own right and because it plays a central role in demonstrating how disorder can create a qualitatively different behavior. Since the first edition of this book, very significant advances in the field of random media have taken place. An effort has been made to present in a simple way the essential points (for the reader with a casual interest in this subject) and to review the current literature (for the sake of the reader whose research activities are or will be related to the field of disordered systems). VII

7 In this edition each chapter is preceded by a short outline of the material to be covered and it is concluded by a summary containing the most important equations numbered as in the main text. I would like to thank A. Andriotis and A. Fertis for pointing out to me several misprints in the first edition. I would also like to express my gratitude to Exxon Research and Engineering Company for its hospitality during the conclusion of this work. Heraklion, Crete January, 1983 E.N. Economou VIII

8 Preface to the First Edition This text grew out of a series of lectures addressed to solid-state experimentalists and graduate students beginning their research career in solidstate physics. The first part consisting of Chaps.1 and 2 is a rather extensive mathematical introduction which covers the material about Green's functions usually included in a graduate course on mathematical physics. Emphasis is given to those topics which are of significance in quantum physics. On the other hand, little attention is given to the important question of determining the Green's functions associated with boundary conditions on surfaces at finite distance from the source. The second and main part of the book is, in my opinion, a first effort to collect in a systematic but concise way various topics of quantum physics, where the Green's functions (as defined in Part I) can be successfully applied. Chapter 3 is a direct application of the formalism developed in Part I. In Chap.4 the perturbation theory for Green's functions is presented and applied to scattering and to the question of formation of bound states. Next, the Green's functions for the so-called tight-binding Hamiltonian (TBH) are calculated. The TBH is of central importance for solidstate physics because it is the simplest example of wave propagation in periodic structures. It is also important for quantum physics in general, because it is rich in physical phenomena (e.g., negative effective mass, creation of a bound state by a repulsive perturbation) and, at the same time, simple in its mathematical treatment. Thus one can derive simple, exact expressions for scattering cross sections and for bound and resonance levels. The multiple scattering formalism is presented within the framework of the TBH, and it is applied to questions related with the behavior of disordered systems (such as amorphous semiconductors). The material of Part II is of interest not only to solid-state physicists but to students of a graduate course in quantum mechanics (or scattering theory) as well. IX

9 In Part III, with the help of the second quantization formalism, the many-body Green's functions are introduced and utilized in extracting physical information about interacting many particle systems. Many excellent books have been devoted to the material of Part III (e.g., FETTER and WALECKA: Quantum Theory of Many-Particle Systems [4.1]). Thus the present treatment must be viewed as a brief introduction to the subject; this introduction may help the solid-state theorist approach the existing thorough treatments of the subject and the solid-state experimentalist to become acquainted with the formalism. I would like to thank Nuclear Research Center "Demokritos" and the Greek Atomic Energy Commission for their hospitality during the writing of the second half of this book. Athens, Greece, November 1978 E.N. Economou x

10 Contents Part I: Green's Functions in Mathematical Physics 1. Time-Independent Green's Functions Forma 1 ism Examples Three-Dimensional Case Two-Dimensional Case One-Dimensional Case Finite Domain Il Summary Time-Dependent Green's Functions First-Order Case Examples Second-Order Case Examp 1 es Summary Part 1/: Green's Functions in One-Body Quantum Problems 3. Physical Significance of G. Application to the Free-Particle Case General Relations The Free-Particle (Ho =p2/2m) Case The Free-Particle Klein-Gordon Case Summary Green's Functions and Perturbation Theory Formalism Time-Independent Case XI

11 4.1.2 Time-Dependent Case Application: Scattering Theory (E>O) Application: Bound State in Shallow Potential Wells (E<O) Summa ry Green's Functions for Tight-Binding Hamiltonians Introductory Remarks The Tight-Binding Hamiltonian (TBH) Green's Functions One-Dimensional Lattice Square Lattice Simple Cubic Lattice Green's Functions for Bethe Lattices (Cayley Trees) Brief Review of Applications in Solid-State Physics Summary Single Impurity Scattering Formal ism Explicit Results Three-Dimensional Case Two-Dimensional Case One-Dimensional Case Applications Leve 1 sin the Ga p The Cooper Pair and Superconductivity The Kondo Problem Lattice Vibrations in Crystals Containing "Isotope" Impurities... ' Summa ry Two or More Impurities; Disordered Systems Two Impurities Infinite Number of Impurities Virtual Crystal Approximation (VCA) Average t-matrix Approximation (ATA) Coherent Potential Approximation (CPA) Direct Extensions of the CPA Cluster Generalizations of the CPA Electrical Conductivity Definition and Some Basic Results XII

12 7.3.2 A General Formula for Conductivity The Conductivity in Terms of Green's Functions CPA for the Vertex Corrections Vertex Corrections Beyond the CPA Disorder and Localization One-Dimensional Case Sca 1 i ng Approach Numerical Methods Mapping the Problem to Other Problems Green's Functions Approach I: RPE Green's Functions Approach II: Vertex Corrections 'Summary Part III: Green's Functions in Many-Body Systems 8. Definitions Single Particle Green's Functions in Terms of Field Operators Green's Functions for Interacting Particles Green's Functions for Noninteracting Particles Summary Properties and Use of the Green's Functions Analytical Properties of the g's and 9'S Physical Significance and Use of the g's and 9'S Quasi-Particles Summary Calculational Methods for 9..., Equation of Motion Method Diagrammatic Method for Fermions at T = Diagrammatic Method for T I Partial Summations. Dyson's Equation Summary Applications Normal Fermi Systems. Landau Theory High-Density Electron Gas Dilute Fermi Gas Summary XIII

13 Appendix A: Analytic Behavior of G (z) Near a Band Edge Appendix 8: The Renormali zed Perturbation Expansion (RPE) Appendix C: Second Quantization References Subject Index XIV