PHYSICAL RE VIE% B VOLUME 21, NUMBER 4 15 FEBRUARY 1980 Tight-binding Green's functions for surfaces, thin films, and solid interfaces. A random-walk theory approach Alberto Robledo and Carmen Varea~ Division de Fstudios Superiores, Facultad de Quimica, Universidad ¹cional Autonoma de Mexico, Mexico 20, D. F. (Received 27 November 1978) We present a formalism, derived with the aid of random-walk theory, which yields the Green s function for a system with planar defects in terms of the associated bulk Green's functions. Exact analytical expressions are obtained for semi-infinite, thin-film, and solid-solid-interface systems. From these, we calculated the local density of states for (a) the (100), (110), and (111) surfaces of a model of rocksalt (two interwoven fcc sublattices with nearest-neighbor interactions); (b) a transition-metal film modeled by a two- band bcc structure; and (c) a solid interface formed by joining two semi-infinite sc one-band regions. Our results are compared with the corresponding bulk properties. I. INTRODUCTION 'The knowledge of the l. ocal density of states (LDOS) is of importance in many surface prob- lems because it provides a way of computing electronic properties of the system without need- ing to know its eigenstates. Consequently, con- siderable effort has been put into developing re- liable methods for its direct calculation. Among these, those that have in common the employment of tight-binding Hamiltonians have proved to be very productive in the study of the electronic structure of surfaces. The various techniques for calculating the LDOS in the tight-binding approximation (and which do not require the evaluation of the state functions) can be classified into two general categories. In the first we include those methods which take ad- vantage of the two-dimensional. periodicity of the crystal along directions parallel to the surface. " By means of Bloch's theorem the three-dimen- sional problem is seen to take on the same form as a one-dimensional problem. The associated Green's function is obtained in terms of a wave vector paral. lel to the surface, and an integration on a surface Brillouin zone (SBZ), corresponding to the arrangement of atoms in the surface planes, becomes necessary. The second category includes those procedures based on path- or walk-counting techniques on a cluster of atoms near the surface. The methods of Cyrot-Lackmann' and of Haydock et al. 'belong to this category. In both methods, the diagonal elements of the Green's function are given. by a continued-fraction expression. The moment method of Cyrot-Lackmann constructs the continued fraction from the first moments of the density of states. These in turn are calculated by computing all the possible closed walks on the cluster of atoms. In the method of Haydock et al. the counting of walks is performed through a re- cursion procedure defining a unitary transforma- tion which takes the Hamiltonian matrix into a tr idiagonal matrix. 4 The main disadvantage of the walk-counting methods is that, when accuracy in the results (e. g. , near the singularities of the density of states) requires large sizes of clusters, the nec- essary computations may become impractical. However, their range of applicability is quite ample, for it is always possible to calculate the first few moments or iterations in complicated situations which may include the consideration of surface dilation, ~' adatoms or layers of adatoms, " degenerate bands, ' stepped surfaces, ' etc. On the other hand, for some simple crystals with only one limiting surface (clean and unre- laxed), the methods that make use of Bloch's theorem yield closed expressions for the LDOS in terms of the bulk Green's function. '" More realistic surfaces need more attention and their analysis has-been complemented with some spec- ial techniques such as the transfer matrix ap- proach. " However, when this approach is ex- tended to treat crystal films one is led to iterative procedures. '2 8 loch's-theorem method has been applied recently to the study of chemisorption by calculating the change on the LDOS due to the interaction of the adatom with the clean sur- face. " Also, the effect of direct adatom-adatom interactions was obtained from a Dyson equation which couples the Green's function of a clean sur- face with that of a monolayer of adatoms. " In spite of these developments, al. l of the work to date' '4 has been confined to calculations on specific crystal models [often for (100) surfaces on sc crystals with nearest-neighbor one-electron Hamiltonians] and thus the general validity of their results (particularly those which appear in closed analytical form) has not been fully ex- 21 1469 1980 The American Physical Society