ELECTRONIC STRUCTURE OF DISORDERED ALLOYS, SURFACES AND INTERFACES

Transcription

1 ELECTRONIC STRUCTURE OF DISORDERED ALLOYS, SURFACES AND INTERFACES

2 ELECTRONIC STRUCTURE OF DISORDERED ALLOYS, SURFACES AND INTERFACES IIja TUREK Institute of Physics of Materials, Bmo Academy of Sciences of the Czech Republic Vaclav DRCHAL, Josef KUDRNOVSKY Institute of Physics, prague Academy of Sciences of the Czech Republic and Institute for Technical Electrochemistry Technical University of Vienna, Austria Mojmî'r SOB Institute of Physics of Materials, Bmo Academy of Sciences of the Czech Republic Peter WEINBERGER Institute for Technical Electrochemistry Technical University of Vienna, Austria SPRINGER SCIENCE+BUSINESS MEDIA, LLC

3 Library of Congress Cataloging-in-Publication Data A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN ISBN (ebook) DOI / Springer Science+Business Media New Y ork Originally published by Kluwer Academic Publishers in 1997 Softcover reprint ofthe hardcover 1st edition 1997 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photo-copying, recording, or otherwise, without the prior written permission of the publisher, Springer Science+Business Media, LLC. Printed on acid1ree paper.

4 CONTENTS PREFACE xi 1 INTRODUCTION 1.1 Electronic Structure of Solids Born-Oppenheimer (Adiabatic) Approximation Selfconsistent Field Approximation Density Functional Theory Local Density Approximation to the Exchange-Correlation Functional Beyond the Local Density Approximation Methods of Electronic Structure Calculations for Perfect Solids 1.2 Systems with Reduced Symmetry 1.3 Tight-Binding Approximation Tight-Binding Hamiltonian Semiempirical and First-Principles Tight-Binding Calculations Resolvents and Green Functions 26 References LINEAR MUFFIN-TIN ORBITAL (LMTO) METHOD 2.1 Secular Equation Atomic Sphere Approximation Single-Site Problem M uffin-tin Orbitals and Tail Cancellation Lattice Fourier Transformations v

5 vi ELECTRONIC STRUCTURE 2.2 Variational Principle Energy Linearization Linear Muffin-Tin Orbitals Hamiltonian and Overlap Matrices Potential Parameters 54 References 58 3 GREEN FUNCTION METHOD Green Functions in Solids General Comments Green Functions within Atomic Sphere Approximation Green Functions in the LMTO Method Tight-Binding LMTO Method Basic Definitions and Properties Particular Representations Ab Initio Tight-Binding Hamiltonians Screening Transformations and Reference Potentials Relation to the KKR Method Green Functions in Layered Systems Principal Layers Partitioning Technique Surface Green Functions and Embedding Potentials Semi infinite Homogeneous Systems Green Function in the Intermediate Region Concluding Remarks Calculation of Observables Charge Densities Densities of States Bloch Spectral Functions 108 References COHERENT POTENTIAL APPROXIMATION (CPA) Configurational Average of Green Function Single-Site Approximation 120

6 Contents vii Transformation Properties of the CPA A Few Additional Remarks Calculation of Observables Properties and Limitations of the CPA 131 References SELFCONSISTENCY WITHIN ATOMIC SPHERE APPROXIMATION Charge Selfconsistency One-Electron Potentials Charge Densities Electrostatic (Madelung) Fields Crystalline Structures Layered Structures Total Energy 150 References RELATIVISTIC THEORY Relativistic TB-LMTO Method: Non-Magnetic Case Dirac Equation Solution for a Single Spherically Symmetric Potential Well Potential Parameters Relativistic Structure Constants Green Functions and the Coherent Potential Approximation Charge Selfconsistency Symmetry of Non-Magnetic Systems Summary Relativistic TB-LMTO Method: Spin-Polarized Case Dirac Equation in the Presence of a Magnetic Field Spin-Polarized Solution for a Single Spherically Symmetric Potential Well Potential Parameters Relativistic Structure Constants The LMTO Hamiltonian and Green Functions 183

7 viii ELECTRONIC STRUCTURE Selfconsistency for Charge and Spin Densities Projected Local Densities of States Magnetic Field and Symmetry Spin-Orbit and Exchange Splitting Parameters Summary 192 References BULK SYSTEMS, OVERLAYERS AND SURFACES Bulk Elemental Metals Bulk Transition-Metal Alloys Local Densities of States and Bloch Spectral Functions Bulk Semiconductor Alloys Charge-Transfer Effects in Random Alloys: Al-Li System Clean Surfaces and Overlayers Surface-Related Properties Layer-Resolved Densities of States and Bloch Spectral Functions Random Overlayers Surfaces of Random Alloys 219 References MAGNETIC PROPERTIES Ferromagnetic Bulk Alloys Compositional Dependence of Magnetization Magnetic Moments and Atomic Order Exchange and Spin-Orbit Splitting Magnetism of Surfaces and Interfaces Magnetism at Transition Metal Surfaces Magnetic Overlayers on Non-Magnetic Substrates Interfaces and Grain Boundaries Disordered Local Moments Paramagnetic State of Metals in the Presence of Local Moments 247

8 Contents ix Complex Magnetic Structures in Random Alloys 252 References EFFECTIVE INTERATOMIC INTERACTIONS IN ALLOYS Ising Model for Alloys Ising Hamiltonian Determination of Parameters Determination of Phase Diagrams Generalized Perturbation Method Single-Site Approximation for Charge Densities Effective One-Electron Potential Total Energy Charge-Transfer Effects Band Term Derivation of the Generalized Perturbation Method Core Contributions Double-Counting Terms and Non-Spherical Corrections Madelung Contributions Bulk Systems Case Study: CU50Ni5o Alloy Surfaces Case Study: fcc (001) Surface of CU50Ni50 Alloy Concluding Remarks 283 References NUMERICAL IMPLEMENTATION Tight-Binding Structure Constants Radial Schrodinger and Dirac Equations Complex Contour Energy Integration Analytic Continuation Brillouin Zone Integration Surface Green Functions Reciprocal Space Approach Real Space Approach 296

9 x ELECTRONIC STRUCTURE 10.7 Coherent Potential Approximation Local Spin Density Approximation One-Electron Potentials Selfconsistency Iterations 305 References 306 INDEX 311

10 PREFACE At present, there is an increasing interest in the prediction of properties of classical and new materials such as substitutional alloys, their surfaces, and metallic or semiconductor multilayers. A detailed understanding based on a microscopic, parameter-free approach is thus of the utmost importance for future developments in solid state physics and materials science. The interrelation between electronic and structural properties at surfaces plays a key role for a microscopic understanding of phenomena as diverse as catalysis, corrosion, chemisorption and crystal growth. Remarkable progress has been made in the past years in the understanding of behavior of ideal crystals and their surfaces by relating their properties to the underlying electronic structure as determined from the first principles. Similar studies of complex systems like imperfect surfaces, interfaces, and multilayered structures seem to be accessible by now. Conventional band-structure methods, however, are of limited use because they require an excessive number of atoms per elementary cell, and are not able to account fully for e.g. substitutional disorder and the true semiinfinite geometry of surfaces. Such problems can be solved more appropriately by Green function techniques and multiple scattering formalism. The linear muffin-tin orbital (LMTO) method, and in particular its tightbinding (TB) version (TB-LMTO), seems to be especially suitable for such studies due to its simplicity, flexibility, and applicability to various materials ranging from simple metals to high-temperature superconductors. The book by H.L. Skriver [1] on the LMTO method, published in 1984, described how to use the LMTO for calculating the properties of ideal crystals. The LMTO method is now widely used in many research laboratories. In the last two decades or so, there has been an increasing interest in more complex systems without three-dimensional translational symmetry, as e.g. substitutional alloys, surfaces, interfaces and other types of extended defects. Application of various Green function techniques to those problems is highly advantageous. There are textbooks dealing with employment of the Korringa-Kohn-Rostoker (KKR) Green function technique to disordered bulk alloys [2, 3], however, at present, xi

11 xii ELECTRONIC STRUCTURE there is neither a textbook nor a monograph available to cover the new development in application of this and other Green function techniques to surfaces, interfaces and extended defects. The purpose of the present book is to fill this gap. It is written as a comprehensive introduction to the study of basic electronic and magnetic properties of complex materials like alloys, their surfaces, interfaces, and extended defects. The book consists of two major parts. The first one (Chapters 1-6) reviews the theoretical background, while the second one (Chapters 7-10) is devoted to applications. Chapter 1 sets the scene by an overview of the density functional theory and the local density approximation, which constitute the basis of most modern electronic structure calculations. Empirical approaches as well as ab initio methods in the tight binding framework are briefly discussed, and their application to the ideal systems and to the systems with reduced symmetry is outlined. The TB-LMTO method is chosen as one of the most effective approaches to this problem, being computationally fast and sufficiently reliable and accurate. Chapter 2 summarizes the traditional aspects of the LMTO method in the atomic sphere approximation (ASA), while Chapter 3 presents a concise and comprehensible exposition of the Green function approach. Various LMTO representations and the TB-LMTO method are also introduced. The connection between the LMTO and KKR techniques is briefly explained within the Green function formalism. Further, definitions and properties of surface Green functions, the central quantities in a description of the electronic structure of layered systems, are discussed in detail. The coherent potential approximation (CPA) as an effective way of describing substitutional disorder in alloys is the subject of Chapter 4. Its derivation within a general LMTO representation valid for inhomogeneous systems like surfaces and interfaces is given. Special attention is paid to averages of site-diagonal (like density of states) and site non-diagonal (like Bloch spectral functions) observables. Chapter 5 deals with specific problems of application of the local density approximation to the systems with reduced symmetry treated within the ASA. An analysis of the corresponding Madelung fields is presented, an expression for the total energy is given, and limitations of the ASA are discussed. Chapter 6 deals with the fully relativistic version of the present formalism, both for non-magnetic and for spin-polarized systems. The applications start in Chapter 7 dealing with the electronic structure of disordered bulk alloys as well as with ground-state properties of solid surfaces. Calculated electronic structures of transition-metal and semiconductor bulk alloys are shown followed by properties of ideal surfaces, random overlayers on perfect substrates, and surfaces of disordered alloys. Magnetic ground-state

12 Preface xiii properties are then treated in Chapter 8. Here the CPA formalism is also applied to the description of magnetic properties at finite temperatures within the disordered local moment picture. The Ising Hamiltonian for random systems is a starting point for statistical studies of such complex phenomena like orderdisorder transitions, phase stability, and surface segregation. Its parameters are derived in Chapter 9 from the calculated electronic structure within the generalized perturbation method. Finally, Chapter 10 provides details of the numerical implementation of the tight-binding LMTO-CPA method. We aim at giving a clear explanation of the theory underlying the techniques used to calculate the electronic structure and related properties of complex materials (like random substitutional alloys, surfaces, interfaces, and multilayered structures), treating both bulk and layered systems on equal footing. Particular emphasis is put on computational aspects. The reader is assumed to be familiar with quantum mechanics and the theory of electronic structure of solids on an intermediate level. It should be stressed that the methods discussed require numerical effort increasing with the number N of layers as O(N). The Green function techniques described are mostly applicable to the closely related KKR method as well. We hope that the book will be useful to all those who wish to extend their applications of modern first-principles computational methods in these areas. The present book is a result of a common effort of all authors, LT. contributing mainly to Chapters 2, 3,5, 8, 10, V.D. to 6 and 9, J.K. to 4 and 7, and M.S. to l. V.D. and J.K. are greatly indebted to B. Velicky who guided them in the use and the applications of Green function techniques in alloy and surface theory. V.D., J.K. and M.S. would like to express their gratitude to O.K. Andersen for introducing them into the LMTO method and its tight-binding version. We would also like to acknowledge discussions with a number of our colleagues, especially with LA. Abrikosov, C. Blaas, S. Bliigel, S.K. Bose, N.E. Christensen, M.H. Cohen, P.H. Dederichs, F. Ducastelle, M.V. Ganduglia-Pirovano, A. Gonis, B.L. Gyorffy, J. Hafner, O. Jepsen, D.O. Johnson, J. Kollar, R. Monnier, V. Natoli, A. Pasturel, R. Podloucky, J. Redinger, A.V. Ruban, T. Schulthess, H.L. Skriver, J.B. Staunton, L. Szunyogh, P.E.A. Turchi, B. Ujfalussy, and V. Vitek. Obviously, they cannot be responsible for any errors or deficiencies that might remain in the book. Investigations connected with this book were supported by grants from the Grant Agency of the Czech Republic, the Grant Agency of the Academy of Sciences of the Czech Republic, the Center for Computational Materials Science

13 xiv ELECTRONIC STRUCTURE (CMS) in Vienna, the Austrian Science Foundation, the Austrian Ministry of Science, and the US-Czechoslovak Science and Technology Program. REFERENCES [1] H.L. Skriver, The LMTO Method (Springer-Verlag, Berlin, 1984). [2] P. Weinberger, Electron Scattering Theory for Ordered and Disordered Matter (Clarendon Press, Oxford, 1990). [3] A. Gonis, Green Functions for Ordered and Disordered Systems (North Holland, Amsterdam, 1992).

14 ELECTRONIC STRUCTURE OF DISORDERED ALLOYS, SURFACES AND INTERFACES