eters in bulk alloys, namely, the so-called Connolly-Williams inversion scheme ...... 24 J. Kudrnovský, I. Turek, V. Drchal, P. Weinberger, N. E. Chris- tensen, and ...
PHYSICAL REVIEW B
VOLUME 54, NUMBER 11
15 SEPTEMBER 1996-I
Ab initio theory of surface segregation: Self-consistent determination of the concentration profile V. Drchal and J. Kudrnovsky´ Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-180 40 Praha 8, Czech Republic and Institute for Technical Electrochemistry, Technical University of Vienna, Getreidemarkt 9, A-1060 Vienna, Austria
A. Pasturel Experimentation Nume´rique, Maison des Magiste`res, CNRS, Boˆ ite Postale 166, 38042 Grenoble Cedex, France
I. Turek
Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zˇizˇkova 22, CZ-616 62 Brno, Czech Republic
P. Weinberger Institute for Technical Electrochemistry, Technical University of Vienna, Getreidemarkt 9, A-1060 Vienna, Austria ~Received 26 February 1996; revised manuscript received 3 May 1996! The parameters of the effective Ising Hamiltonian governing segregation and ordering phenomena in the surface region of an alloy are determined from first principles. Employing the force theorem, the total energy of semi-infinite disordered alloys is mapped onto an Ising Hamiltonian. The band term is treated within the generalized perturbation method, and, in addition, the contributions to the on-site terms of the Ising Hamiltonian from core states, and the double-counting and Madelung terms are included. The concentration profile is determined by using Monte Carlo simulations. The electronic structure and the Ising Hamiltonian parameters are then recalculated for the profile, and the whole procedure is repeated until self-consistency between the electronic and atomic structures of the alloy surface is achieved. As an illustration, the results for an fcc~001! surface of the Cu-Ni alloys as calculated within the all-electron fully relativistic tight-binding linear-muffintin-orbital coherent potential approximation method are discussed. @S0163-1829~96!02835-4#
I. INTRODUCTION
A detailed understanding of observed metallurgical and magnetic properties of transition-metal alloys requires a complete knowledge of their phase diagrams. The description and prediction of the phase diagrams from the underlying electronic structure is of considerable practical and theoretical importance. Any theoretical study of phase stability has to rely upon an accurate description of the internal energy and the entropy of the system as a function of alloy composition and temperature. The free energies for competing alloy configurations can then be calculated, and the stability of phases studied. Very often at alloy surfaces the concentrations in the first few surface layers differ from those in the bulk. This is the so-called surface segregation phenomenon which has been the subject of extensive experimental1–3 and theoretical4–7 studies. Segregation plays an important role in such diverse phenomena as catalysis, chemisorption, and crystal growth, since all of them depend sensitively on surface properties. An understanding of these surface-related phenomena requires a knowledge not only of the atomic structure of the surface, but also of the surface composition, which, in turn, depends on the thermodynamics of surface segregation. The usual modeling of the phase stability and surface segregation at a microscopic level is based on expanding the energy of the system in occupation numbers of the individual atomic sites. Order-disorder transformations and/or the equilibrium composition of the surface layers then follows from classical statistical mechanics. In practice, the energy of the 0163-1829/96/54~11!/8202~11!/$10.00
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alloy and its surface is approximated by an appropriate Hamiltonian of the Ising type, with coefficients usually derived phenomenologically. The statistical part of the problem now seems to be well under control, and usually the cluster variation method ~CVM! ~Ref. 8! in the tetrahedron approximation, and its two-dimensional generalization,9 is employed. The well-known molecular field, or Bragg-Williams approximation, is the first term of the CVM expansion, and gives reasonably good results for high temperatures. Recent advances in Monte Carlo simulations6 extended these methods to systems with macroscopic inhomogeneities, thus making them applicable to the study of segregation phenomena. We remark that an alternative formulation is employed in the concentration-wave method10 for homogeneous bulk alloys which is based on the grand-canonical potential V configurationally averaged within the coherent potential approximation ~CPA!,11 and on a mean-field solution of the statistical part of the problem also including the Onsager cavity field. The success and reliability of statistical simulations, however, rely heavily upon the correct determination of the parameters of the Ising Hamiltonian. The primary task of an ab initio approach to problems of phase stability and segregation is the determination of the Ising Hamiltonian parameters, often called effective cluster interactions ~ECI’s!. Two approaches have been developed for calculating such parameters in bulk alloys, namely, the so-called Connolly-Williams inversion scheme ~CWI!,12 and the generalized perturbation method ~GPM! ~Refs. 13 and 14! and related approaches.15 The CWI method extracts ECI’s from total-energy calculations for a set of ordered structures. The GPM is based on the 8202
© 1996 The American Physical Society
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AB INITIO THEORY OF SURFACE SEGREGATION: . . .
expansion of the band-energy term of the total energy of a completely disordered alloy by making use of the so-called force or structural energy theorem.16 Recently an attempt was made to remove the limitation of the GPM to the band term of the total energy in bulk alloys17 by including the double-counting and electrostatic terms. The CWI method yields concentration-independent ECI’s, while those determined by the GPM depend on the alloy concentration. Both approaches, the CWI and GPM, were recently generalized to the case of alloy surfaces. The CWI method applied to surfaces extracts the ECI’s from total energies either of a set of ordered structures ~special configurations!,18 or of disordered semi-infinite alloys with varying concentration profiles.19 Within the GPM, ECI’s for the surface case are calculated both in model20 and ab initio21,22 studies by using the parameters of the electronic Hamiltonian corresponding to the homogeneous bulk alloy, and by using the CPA effective medium of a homogeneous bulk alloy as a reference medium. The parameters of such an Ising model, however, cannot be determined self-consistently together with the segregation profile. The aim of this paper is to develop a fully self-consistent method for a determination of concentration profiles in the surface region of segregating alloys. The main features of the method can be summarized as follows: ~i! The surface electronic structure is calculated from first principles for an arbitrary ~inhomogeneous! concentration profile, taking properly into account the charge redistribution and related electrostatic fields in the surface region in a self-consistent manner, and subsequently evaluating the corresponding Ising Hamiltonian parameters. ~ii! A concentration profile is determined from the Ising Hamiltonian parameters by methods of statistical mechanics. ~iii! Steps ~i! and ~ii! are repeated until a consistency between the electronic structure and concentration profile is achieved. The Ising Hamiltonian parameters are determined using the force theorem, which leads to a neglect of local environment effects. It is assumed that the potential on each site is independent of the occupation of the neighboring sites. It then follows from the force theorem that double-counting, core, and electrostatic terms have to be included in calculation of ECI’s. The paper is organized as follows. The central part, Sec. II, describes the mapping of the total energy of a semiinfinite alloy within the single-site approximation onto the effective Ising Hamiltonian. The segregation profiles are calculated using Monte Carlo simulations ~Sec. III!, the results of which are presented in Sec. IV. The electronic structure is calculated by the all-electron relativistic version of the tightbinding linear-muffin-tin-orbital coherent potential approximation ~TB-LMTO-CPA! method within the atomic sphere approximation ~ASA!. As an example, the segregation profiles for the Cu x Ni 12x alloy system (0.05
