THEORY OF MOLECULAR EXCITONS

THEORY OF MOLECULAR EXCITONS

THEORY OF MOLECULAR EXCITONS A. S. Davydov Kiev State University Kiev, USSR Translated from Russian by Stephen B. Dresner g? SPRINGER SCIENCE+BUSINESS MEDIA, LLC 1971

Aleksandr Sergeevich Davydov was graduated from Moscow State University in 1939 and defended his doctoral dissertation in 1943. A member of the Academy of Sciences of the Ukrainian SSR, he is director of the Department of Nuclear Theory of the Institute of Theoretical Physics, director of the Department of Theoretical Physics of the Institute of Physics (both of the Academy), and Professor of Theoretical Physics at Kiev State University. Working in the area of the theory of excitons in molecular crystals and the theory of the atomic nucleus, in 1948 he predicted the phenomenon that is known as "Davydov splitting." Davydov has investigated theoretically the collective excited states of nonspherical nuclei (Davydov-Filippov model and Davydov-Chaban model). The original Russian text, published by Nauka Press in Moscow in 1968, has been corrected by the author for this edition. The English translation is published under an agreement with Mezhdunarodnaya Kniga, the Soviet book export agency. AMKcandp Cepdeeou^i ffaebtdoo TEOPHH MOJIEKyjIHPHHX 3KCHT0H0B TEORIYA MOLEKULYARNYKH EKSITONOV THEORY OF MOLECULAR EXCITONS Library of Congress Catalog Card Number 72-75767 SBN 306-30440-6 Springer Science+Business Media New York 1971 Originally published by Plenum Press, New York in 1971 Softcover reprint of the hardcover 1st edition 1971 All rights reserved No part of this publication may be reproduced in any form without written permission from the publisher ISBN 978-1-4899-5171-7 ISBN 978-1-4899-5169-4 (ebook) DOI 10.1007/978-1-4899-5169-4

Preface to American Edition My first book on the theory of excitons was published in Kiev in 1951 and in English translation (McGraw-Hill) in 1962. It reflected only the initial stage of development of the theory of excitons. The present monograph is entirely new. It is my hope that this American edition will aid Englishspeaking scientists in becoming familiar with the latest stage of development of the theory. This edition also includes three recent original papers. I take this opportunity to express my sincere thanks to the translator. Kiev September, 1969 A. S. Davydov v

Foreword In recent years, great advances have been made in the theoretical and experimental study of the optical properties of solids. These advances have to a considerable extent resulted from the wide use of the concept of the exciton, which was introduced as early as in 1931 by Ya. I. Frenkel. Interest in the theory of exciton states in solids has increased significantly with the improvement of experimental techniques with polarized light at low temperatures and of methods of growing single crystals, with the extensive use of luminescent crystals, and with the establishment of the great role of energy-migration processes in solids and biological systems. Theoretical investigations of exciton states have made particularly great progress recently, because of the use of methods of quantum electrodynamics in the theory of solids. Unfortunately, the progress in experimental and theoretical research on exciton states in solids has not yet been fully reflected in monographs. The author's monograph "Theory of Light Absorption in Molecular Crystals" was published in 1951, and an English translation, "Theory of Molecular Excitons," was published in 1962, but it represents only the initial stage of development of the theory. R. S. Knox, in his "Theory of Excitons," which was published in 1963 (the Russian edition appeared in 1966), succeeded in discussing in a small book the basic physical concepts underlying the theory of excitons, the limit models of excitons (Frenkel excitons and Wannier excitons), and a number of experimental studies whose interpretation required the concept of the exciton. vii

viii FOREWORD A recent book by Agranovich and Ginzburg, "Crystal Optics with Allowance for Spatial Dispersion, and the Theory of Excitons," presents a phenomenological investigation of the relationship between the theory of excitons and crystal optics. But this book does not discuss completely enough the methods for calculating the energy bands of excitons, nor does it examine the role of or methods for calculating the interactions of excitons with crystal-lattice vibrations. Such problems are discussed on the basis of microtheory in the present monograph. Exciton states (currentless collective electronic excitations) appear in solids of various kinds: in molecular, ionic, and semiconductor crystals, and in complicated organic compounds. Excitons have begun to be mentioned even in analyses of the properties of normal and superconducting metals. The excitons in different solids have many features in common, but the methods for their experimental and theoretical study differ substantially from one type of solid to another. Certain problems of the theory of excitons in molecular crystals will be examined in this monograph, which is not a survey of the literature on the theory of excitons. The primary attention will be given to three problems in whose development the author participated. The references, which are given at the end of the book, are not meant to be complete. The theory of excitons is progressing vigorously at the present time. This growth is occurring in a number of directions and through various methods. Some authors have expressed dissimilar and, in some cases, conflicting opinions about the nature of the observed phenomena. This monograph naturally reflects the point of view of its author. A great deal of attention has been given to the mathematical apparatus of the theory. Experimental research is discussed only to illustrate certain conclusions. The formulas in each chapter are numbered by two figures separated by a dot. The first figure indicates the chapter section and the second indicates the formula number. References to formulas from other chapters contain a Roman numeral indicating the chapter number. The author hopes that this monograph will further the development of the theory and the establishment of concepts that will

FOREWORD ix best reflect the real phenomena, and that it will be useful to scientists and students specializing in solid state physics. The author thanks V. M. Agranovich, A. F. Lubchenko, V. A. Onishchuk, E. N. Myasnikov, and B. M. Nitsovich for assistance in developing some of the topics in the monograph, and Yu. I. Sirotin, who read the manuscript and made some valuable comments. A. S. Davydov

Contents Chapter I................. 1 Fundamentals of the Phenomenological Theory of Electromagnetic Waves in Dielectric Media.. 1 1. The Macroscopic Body and the External Electromagnetic Field................ 1 2. The Dielectric Constant of a Macroscopic Body. 6 3. The "Transverse" Dielectric Constant.... 12 4. The Dielectric Constant that Determines the Response of the System to an External Influence.. 15 5. Phenomenological Theory of Excitons.... 17 Chapter n 23 Elementary Theory of Excitons in Coordinate Representation 23 1. Frenkel Excitons and Wannier Excitons... 23 2. Molecular Excitons in Crystals with One Molecule in a Unit Cell........ 31 3. Molecular Excitons in Crystals with Several Molecules in a Unit Cell........... 39 4. Exciton States and the Dielectric Constant..... 47 5. Calculation of the Resonance-Interaction Matrix... 55 6. Using Group Theory for Qualitative Interpretation of the Properties of Exciton States........ 68 7. Experimental Confirmations of the Presence of Exciton States in Crystals............ 82 8. Exciton Luminescence......... 97 xi

xii CONTENTS Chapter ITI................................ 113 Theory of Exciton States in the Second-Quantization Representation (Fixed Molecules).. 113 1. The Energy Operator of a Crystal with Fixed Molecules...,0 113 2. The Heitler-London Approximation in the Theory of Excitons. 119 3. The Theory of Excitons without the Heitler- London Approximation.. 127 4. Exciton States with Allowance for Several Adjacent Molecular Le'O'els 130 5. Photoexcitons... 136 Chapter IV.............................. 153 Interaction of Excitons with Phonons and Photons. 153 1. The Exciton - Phonon Interaction Operator. 153 2. The Green Function Method in the Theory of Excitons... 0 169 3. Relationship Between the Dielectric Constant and the Retarded Green Function for Photons. 188 4. Green Functions for Excitons at Absolute Zero 191 5. Temperature Matsubara Green Functions for Interacting Excitons and Phonons. 206 6. Retarded Two-Time Green Functions for Excitons at Nonzero Temperatures 213 7. The Dielectric Constant of Simple Molecular Crystals with Allowance for Interaction Retardation.. 0 " 221 8. The Dielectric Constant of Complicated Molecular Crystals with Allowance for Retardation. 236 9. Elementary Excitations in a Crystal with Complete Allowance for Retardation. 241 Chapter V..... 0 245 The Dielectric Constant of Molecular Crystals with Allowance for Lattice Vibrations 245 1. Theory of the Width of Exciton Absorption Bands in One-Dimensional Molecular Crystals. 245 2. Dispersion and Absorption of Light by Three- Dimensional Molecular Crystals. 254

CONTENTS xiii 3. Dispersion and Absorption of Light in strong Interaction of Electronic Excitations with Phonons... g Ii 265 4. Theory of Strong Coupling of Electronic Excitations with Phonons in the Second-Quantization Representation.......... 276 5. Excitons in Thin Crystals..... 281 6. Elementary Theory of the Urbach Rule. 289 Append ix........ 0 0 297 1. Unitary Transformation of the Operators 297 2. The Weyl Identity Operator... 298 3. Calculation of the Mean Values of the Bose Operators...... 300 4. The Statistical Averages of the Phonon Operators III 301 References............... 305 Index.. 0 01 0., 311