FIRST-PRINCIPLES CALCULATIONS OF THE ELECTRONIC STRUCTURE OF GRAIN BOUNDARIES

We have developed a new, first-principles method for the calculation of the electronic structure of surfaces, grain boundaries, and other low-symmetry systems. Our method yields the electronic density-of-states (DOS) for unrelaxed as well as structurally-relaxed grain boundaries. I t is based on a real-space formulation of multiple-scattering theory (RSMST) and thus does not rely on perfect periodicity and lattice Fourier transforms. This method allows us to bridge the gap between atomistic calculations of relaxed grain-boundary configurations, as obtained within the Embedded Atom Method (EAM), and first-principles quantum-mechanical calculations. We present results for realistic twist and tilt grain boundaries in metals, and discuss our plans for future development. Electronic-structure calculations require two ingredients: Knowledge of the atomistic structure of a material, and the existence of methods, both formal and computational, for treating the structure at hand. In the case of materials with full translational symmetry, e.g. bulk'crystalline solids, both requirements can be satisfied. The atomistic structure may be determined by, e.g., X-ray crystallography. Once this is known, the electronic structure may be obtained with the aid of Bloch's theorem and the associated lattice Fourier transforms, which diagonalize the Hamiltonian in reciprocal space (k-space). In contrast, the determination of both the atomistic and electronic structure of materials in which translational symmetry is broken by surfaces or internal interfaces is much more problematic. Experimental determination of the atomistic structure of some surfaces is possible through, e.g., low-energy electron diffraction in reciprocal space or scanning tunneling microscopy in real space, but there are no techniques available for imaging atomic positions a t internal interfaces. In some cases, partial information may be obtained with electron microscopy, but often one must rely on rnmni i tor cimli lat innc c n r h ac thnco affnrrlorl hv tho omhorlrlorl a tnm mothnrl 11 9 '2 AI Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990153 Cl-336 COLLOQUE DE PHYSIQUE Because nearly all existing, fully first-principles formalisms for performing electronic-structure calculations are based on the properties of translational invariance and the use of Bloch's theorem, knowledge of atomic coordinates 0f.a surface or interface does not necessarily enable such calculations to be performed with conventional methods. Consequently, it has often been found necessary to invoke rather severe approximations with respect to the structure of a system, such as the use of slabs of finite thickness to treat surfaces, or of repeating slabs or supercells to study internal interfaces. Such approximations are highly undesirable for a number of reasons: First, they are "uncontrolled", yielding results which should be checked for dependence on slab thickness on a case-by-case basis. Second, they are conceptually unattractive, involving rather severe approximations to the underlying geometry of the system. Third, the extent to which they may yield accurate results may lull one into a false sense of accomplishment, obscuring the existence of a still unsolved problem. Stated briefly, this problem consists in finding the solution of the one-particle Schrodinger equation, within the Born-Oppenheimer approximation (fixed nuclei) and the local-density approximation to density-functional theory, that satisfies the proper boundary conditions imposed by the structure of a particular semi-infinite material. In a recent publication 151, a first-principles, multiple-scattering formalism, which allows the exact treatment of the problem just stated, was introduced. As the formal aspects of this real-space multiple-scattering theory (RSMST) method have been reviewed in previous work 15, 61, we shall forgo all but a brief description of this method here. The RSMST is based on the concept of semi-infinite periodicity (SIP), defined as the regular repetition along a given direction of a scattering unit (atom, planes of atoms, etc.), or a set of such units. Systems with SIP possess the property of removal invariance, which states that the scattering properties (scattering matrices) of any such system remain invariant when an integral number of scattering units is removed from, or added to, the free end of the system. Using this property in conjunction with multiple-scattering theory, one can determine the electronic Green function, and hence all one-particle quantities such as the density-of-states (DOS), directly in real space, bypassing often cumbersome reciprocalspace (k-space) integrations. This formalism provides a unified treatment of the electronic properties of a broad spectrum of systems that includes, but is not limited to, pure elemental solids, compounds, ordered alloys, surfaces and interfaces, and other low-symmetry systems. Only systems with no recognizable periodic structure, e.g. amorphous materials and liquids, fall outside the scope of the RSMST method. The essence of the method consists in a prescription for the proper renormalization of the scattering properties of the boundary sites of a cluster of atoms. Unrenormalized or "bare" sites in the interior of the cluster describe the region of interest, such as a grain boundary, while the renormalized boundary sites represent the infinite medium surrounding the cluster. This "dressing up" of the cluster is done independently for each part of a system that is characterized by its own SIP, so that grain boundaries between essentially arbitrary crystal structures can be treated. At its present stage of development, our codes can be applied to known atomistic configurations with known electronic oneparticle potentials. Work currently in progress is aimed at the incorporation of the often important effects of charge self-consistency, and of total-energy capability, into the program.