The Propagation Matrix Method for the Band Problem with a Plane Boundary
Solution of electronic problems involving plane surfaces on crystals requires solution of the band problem for real energy E in complex k space, and superposition of the generalized Bloch functions at the surface. A compact and general formulation of the problem of finding these Bloch functions and matching them across a plane makes use of a numerical matrix, the propagation matrix P, obtained from the Schrodinger equation. The eigenvectors of P are just the desired Bloch functions, and the eigenvalues give all k⊥ values at given E, k// (component parallel to the surface). Thus once P is found, the band problem is reduced to an ordinary eigenvalue problem; the bands can be followed along any line in k space parallel to k⊥ by varying k//; the potential may be complex (to describe inelastic scattering). A procedure for generating P by integration of a matrix equation has the advantage that a general anistropic potential can be used, but the disadvantage of a Fourier expansion parallel to the surface plane which does not hold well near the nucleus; hence it applies best for potentials that are weak or have a small number of Fourier coefficients. By generating P for a single layer by a two-dimensional version of KKR, this limitation is avoided for muffin-tin potentials.
KeywordsWave Function Scattered Wave Evanescent Wave Schrodinger Equation Outgoing Wave
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