PHYSICAL REVIEW B VOLUME 28, NUMBER 8 15 OCTOBER 1983 Electronic structure of line defects by means of the scattering theoretical method. Application to lines of vacancies in the simple cubic lattice E. Louis Centro de InUestigacion y Desarrollo, Empresa Xacional del Aluminio Sociedad Anonima, Alicante, Spain J. A. Verges Departamento de Fisica del Estado Solido, UniUersidad Autonoma, Cantoblanco, Madrid 34, Spain (Received 22 April 1982; revised manuscript received 2 August 1982) The scattering theoretical method recently used to describe the electronic structure of point defects in solids is extended to study line defects. The method is applied to the case of lines of vacancies in the simple cubic lattice within a tight-binding model. Lines of vacancies in the [100], [110], and [111]directions are considered, and their formation energies are estimated within the independent- electron approximation. The interaction energy between parallel lines of vacancies is also studied and the results are compared with those obtained for the interaction energy between single vacancies; both interaction energies are shown to be remarkably similar. I. INTRODUCTION In the last few years there has been a strong revival of interest in the electronic properties of defects in solids. ' Point defects, such as vacancies, have been studied in the bulk, surfaces, and interfaces of both metals and semicon- ductors. ' ' On the other hand, line defects, such as dislocations, have received little attention. ' ' This is mainly due to the following: (i) the most dramatic effect of dislocations is to be found in the mechanical properties of solids; therefore these properties have attracted most in- terest, and (ii) the difficulties in treating the severe break- ing of symmetry introduced by line defects. Nowadays it is generally recognized that, as do point defects, line de- fects should dramatically affect the electronic properties of solids and may play a fundamental role in determining such an important property as the catalytic activity of metal surfaces. Recently, there have been several attempts to extend the methods of calculation used to study sur- faces and vacancies to the case of line defects. ' ' Those analyses have always treated line defects by means of su- perlattice methods. The purpose of the present work is to show how the scattering theoretical method recentIy used to describe the electronic structure of surfaces, interfaces, and point de- fects in solids can be extended to treat simple line defects. In particular, we shall discuss its application to the case of lines of vacancies in the three main directions of the simple-cubic lattice. The formation energies of lines of vacancies (the difference between the energy of a crystal containing a line of N vacancies and the energy of a crys- tal with N isolated vacancies) will be estimated in the in- dependent electron approximation, and the interaction be- tween parallel lines of vacancies will be studied in the same approximation. Our results will be compared with those obtained by Yaniv' ' for the case of single vacan- cies. Of course, and due to the approximate nature of the model Hamiltonian (simple cubic lattice, one orbital per site, interactions up to first neighbors, neglect of the electron-electron and ion-ion interactions, and absence of ionic relaxations), our emphasis will be on the techniques required by the extension of the scattering method to han- die line defects, but not on the study of a realistic situa- tion. II. METHOD OF CALCULATION The formalism we shall use is based upon the Green's- function method. The single-particle Green's function as- sociated with a given Hamiltonian Hp is given by (E + irl Hp )Gp — 1, — E being the energy and g a positive infinitesimal. The va- cancy or line of vacancies is introduced through a local- ized perturbation V. The perturbed Schrodinger equation is now written as H ~ 1/J ) = (H p+ V) ~ 1( ) =E ~ q) . The perturbed Green's function is then given by Dyson's equation, namely, G =Gp+GpVG . The phase shift associated with this scattering problem is given by 5(E)=arg[D(E)] . where D (E) is obtained through D (E) =- det( 1 — Gp V) . Finally, the change in the density of states caused by the localized perturbation V is written as In the present work we shall consider a simple single- state Hamiltonian, which in the site representation takes the form Hp — — gh~~ i)(j ~ i and j run over the lattice sites and h, & is restricted to nearest-neighbor interactions, namely h;J. = h p if i and j are nearest neighbors and zero otherwise. 1983 The American Physical Society