Low-Energy Electron-Diffraction Dispersion Surfaces and Band Structure in Three-Dimensional Mixed Laue and Bragg Reflections

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Abstract
Although the theory of the propagation of electron waves in periodic solids and the theory of the elastic diffraction of electron waves by periodic solids have developed independently into band theory and dynamical electron-diffraction theory, respectively, they are in fact formally identical. The electron wave functions which can exist in the crystal are determined by a seven-dimensional hypersurface in energy-complex K space which defines the totality of solutions to the wave equation in the infinite crystal. In the diffraction problem, the introduction of the crystal surface together with the magnitude and direction of the external electron wave vector selects the particular set of eigenfunctions which are excited during a given experiment and which correspond to the allowed electron states in the crystal. This set constitutes the wave field in the self-consistent multiple-scattering approach. It is demonstrated that the energy band diagram of band theory and the constant energy dispersion surface of dynamical theory are in fact sections of the same hypersurface. The complex nature of the dispersion surface leads to the excitation of evanescent waves both in the crystal and in the vacuum. The diffraction boundary conditions, notably conservation of total energy and of momentum parallel to the crystal surface, can easily be introduced, geometrically, by means of a constraint surface which contains the crystal normal. In a given experiment the excited wave functions are determined by the intersection of the hypersurface with the appropriate constraint surface. It is shown that the most useful constraint surfaces are those at constant energy and furthermore that the dispersion hypersurface is an ameniable method for the discussion of low-energy electron-diffraction (LEED) cases of high order and high symmetry, several of which are outlined in detail. The variation of the reflected intensities observed in electron-diffraction experiments is discussed in terms of the changes in the allowed electron wave functions as calculated by three-dimensional band-structure and/or dynamical diffraction theory. In particular, we predict zeros in the reflected Bragg intensities for certain special geometries in the case of two simultaneous reflections (mixed Bragg-Laue case).