Springer Series in Solid-State Sciences. Edited by Manuel Cardona and Peter Fulde

Transcription

1 41 Springer Series in Solid-State Sciences Edited by Manuel Cardona and Peter Fulde

2 Springer Series in Solid-State Sciences Editors: M. Cardona P. Fulde H.-J. Queisser Volume 40 Semiconductor Physics - An Introduction By K Seeger Volume 41 The LMTO Method By H.L. Skriver Muffin-Tin Orbitals and Electronic Structure 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 Microstructure II Three-Dimensional Models By I.A. Kunin Volume 45 Electronic Propetties of Doped Semiconductors By B. I. Shklovskii and A. L. Efros Volume 46 Topological Disorder in Condensed Matter Editors: F. Yonezawa and T. Ninomiya Volume 47 Statics and Dynamics of Nonlinear Systems Editors: G. Benedek, H. Bilz, and R Zeyher Volume 48 Magnetic Phase Thansitions Editors: M. Ausloos and R J. Elliott Volume 49 Organic Molecular Aggregates, Electronic Excitation and Interaction Processes Editors: P. Reineker, H. Haken, and H. C. Wolf Volume 50 Multiple Diffraction of X-Rays in Crystals By Shih-Lin Chang Volume 51 Phonon Scattering in Condensed Matter Editor: W. Eisenmenger Volumes 1-39 are listed on the back inside cover

3 Hans L. Skriver The LMTO Method Muffm-Tin Orbitals and Electronic Structure With 34 Figures Springer - erlag Berlin Heidelberg New York Tokyo 1984

4 Dr. Hans L. Skrroer Fysikafdelingen, Forsogsanlag RiS0 DK-4000 Roskilde, Denmark Series Editors: Professor Dr. Manuel Cardona Professor Dr. Peter Fulde Professor Dr. Hans-Joachim Queisser Max-Planck-Institut fur Festkorperforschung, Heisenbergstrasse 1 D-7000 Stuttgart 80, Fed. Rep. of Germany ISBN-13: DOl: / e-isbn-13: Library of Congress Cataloging in Publication Data. Skriver, Hans L., The LMTO method. (Springer series in solid-state sciences; 41) Bibliography: p. Includes index. 1. Energy band theory of solids. 2. Linear free energy relationship. I. Title. II. Series. QCI76.8.E4S ' 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 of the 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 1984 Softcover reprint of the hardcover 15t edition 1984 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. 2153/

5 Preface The simplifications of band-structure calculations which are now referred to as linear methods were introduced by Ole K. Andersen almost ten years ago. Since then these ideas have been taken up by several workers in the field and translated into computer programmes that generate the band structure of almost any material. As a result, running times on computers have been cut by orders of magnitude. One of the strong motivations behind the original proposal was a desire to give the conventional methods' a physically meaningful content which could be understood even by the non-specialist. Unfortunately, this aspect of linear methods seems to have been less well appreciated, and most workers are content to use the latter as efficient computational schemes. The present book is intended to give a reasonably complete description of one particular linear method, the Linear Muffin-Tin Orbital (LMTO) method, without losing sight of the physical content of the technique. It is also meant as a guide to the non-specialist who wants to perform band-structure calculations of his own, for example, to interpret experimental results. For this purpose the book contains a set of computer programmes which allow the user to perform full-scale self-consistent band-structure calculations by means of the LMTO method. In addition, it contains a listing of self-consistent potential parameters which, for instance, may be used to generate the energy bands of metallic elements. A major part of this listing is by Dieter Glotzel, and I wish to thank him for permission to include his results. Although I have been involved with linear methods for a long time, I was by no means the first to make working LMTO programmes. I have in this respect benefitted greatly from early access to the computer codes generated by my predecessors Ove Jepsen and Uffe Kim Poulsen. In this context, I should also mention that Dve Jepsen performed the first calculations of the canonical bands for the fcc, bcc, and hcp crystal structures which are included in this book. v

6 Part of my work has been performed at the laboratory of the National Research Council (NRC) of Canada in Ottawa and at the Kamerlingh Onnes Laboratory in Leiden, and I owe my sincere thanks to my colleagues at these institutions as well as to those at my home base Ris. I am especially grateful to J.-P. Jan at NRC who, from a desire to interpret his pioneering de Haas-van Alphen measurements on intermetallic compounds, helped perform the first LMTO calculations on alloys in 1975, and who continued to be a true collaborator until his untimely death in The present book would not have existed without the work of my colleague and friend, Ole Krogh Andersen. In fact, large parts of the present monograph are based directly on his notes, and I am extremely grateful to him for allowing me to use these unpublished results. An important role in the genesis of this book has also been played by Allan Mackintosh, who was largely responsible for the initiation of bandstructure calculations in Denmark. He has furthermore kindly read the manuscript at various stages of completion and made numerous useful suggestions. Thanks are also due to Niels, Egede Christensen who read the manuscript in its penultimate version, and who spotted several weak points in the text. The remaining flaws are my responsibility. It is a pleasure to thank the Danish Natural Science Foundation for financial support during the writing of the book, and Ris National Laboratory for providing office and writing facilities. In this context, I wish to thank Agnete Gjerl v for her expert typing of the manuscript and her everlasting patience in the preparation of the half-dozen versions in which each chapter has appeared over the writing period. Finally, to quote a fellow countryman, I should like to thank my parents for making the book possible, and my wife and children for making it necessary. Roskilde, July 1983 Hans L. Skriver VI

7 Contents 1. Introduction 1.1 The One-Electron Approximation 1.2 The Energy-Band Problem Energy-Band Methods Brief History of Linear Methods i. 5 Organi sat i on of the Book Canonical Band Theory Muffin-Tin Orbitals and Tail Cancellation Structure Constants and Canonical Bands Potential Function and the Wigner-Seitz Rule Potential Parameters, Unhybridised, and Hybridised Bands 2.5 Hybridised Canonical Theory State Densities and Energy Scaling One-Electron States in a Single Sphere Radial Basis Functions Partial Waves and Their Energy Derivatives Logarithmic Derivative and Laurent Expansion Potential Function and Bandwidth Matrix Elements and Variational Estimate of Energies Physically Significant Parameters 4.1 The Four Potential Parameters 4.2 How to Choose Ev!!'...,......, Chromium 3d Bands: An Example Free-Electron Potential Parameters 4.5 Volume Derivatives of Potential Parameters 4.6 Potential Parameter Relations VII

8 5. The Linear Method Partial Waves for a Single MUffin-Tin 5.2 Muffin-Tin Orbitals 5.3 Expansion Theorem for MTO Tails Energy-Independent Muffin-Tjn Orbitals One-Centre Expansion and Structure Constants The LCMTO Secular Matrix The LMTO Method The Atomic-Sphere Approximation (ASAl The Kinetic Energy K An Error Estimate The Atomic Sphere and the ASA The Canonical Structure Constants Muffin-Tin Orbitals ih the ASA Relation Between the LMTO and KKR Matrices, 6.7 Wave Functions and I Character Projected State Density and Density of Electrons The Combined Correction Term Ground-State Properties Cohesive Properties Density-Functional Theory Spin-Density-Functional Theory Self-Consistent Band-Structure Problem Electronic Pressure Relation First-Order Pressure Relation Chromium: An Example Many Atoms per Cell Molecules and Clusters Tail Cancellation The Two-Centre Approximation The LMTO Formal ism Muffin-Tin Orbitals and One-Centre Expansion Structure Constants and the LMTO Method Total Energy and Self-Consistent Energy Bands VIII

9 9. Computer Programmes , The Self-Consistency Loop Structure Constant Programme STR Lattice Summations in STR Listing of STR... '" Execution of STR Sample Output from STR Correction Structure Constant Programme COR Listing of COR Execution of COR Sample Output from COR Linear Muffin-Tin Orbital Programme LMTO LMTO and Cholesky Decomposition Listing of LMTO Execution of LMTO." Sample Output from LMTO Energy-Band File BND/YY/B Projected State-Density Programme DONS The Tetrahedron Method Mesh and Tetrahedra Listing of DONS Execut i on of DONS Sample Output from DONS The Self-Consistency Programme SCFC The Dirac Equation Without Spin-Orbit Coupling Listing of SCFC Execution of SCFC Sample Output from SCFC Self-Consistent Potential Parameters for 61 Metals List of Symbols References..., Subject Index IX