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Scattering states for realistic surfaces with multi-grid acceleration |
 Fig. 1: Image of the charge density in the yz plane at x=0 with the z interval shown from -24 Å to 38 Å. In the surface parallel y direction two unit cells are shown. The atoms lying in the plane are drawn black, and in the right cell the projected positions of the remaining atoms are grey. The different scales for the y and z directions are indicated by the Å scale. The normal incident electron beam has an energy of 18 eV, which in vacuum corresponds to a kinetic energy of 12.75 eV. The z interval used in the calculation (-85 Å to 85 Å) encloses 45 layers with 90 atoms per surface cell in the crystal and includes a vacuum part of almost the same extension. |
Electron spectroscopies at low energies are
sensitive to the shape of the surface near potential.
Sophisticated bandstructure calculations involve such potentials,
but almost exclusively consider bound states.
To obtain the same accuracy for current carrying states is still a
challenge. Here we have calculated
scattering solutions of the Schrödinger equation
with a realistic potential of the
GaAs (110) surface. We consider final states of photoemission,
which are time-reversed states of low energy electron diffraction
(LEED). As a first application a charge-density distribution of such a scattering state is shown in Fig. 1. The interface crystal vacuum, the decay of the wave function into the crystal, and the interference pattern of superimposed waves in the vacuum are clearly visible. The importance of the correct treatment of the potential barrier is illustrated by the strong charge fluctuations in the surface region. It is interesting to see that there are well localized regions of high charge density even at this nonbonding energy. The wave function is described with a reciprocal lattice representation parallel to the surface and a discretization of the real space perpendicular to the surface. The Schrödinger equation leads to a system of linear one-dimensional equations. The correct asymptotic behavior was obtained by the quantum transmitting boundary method. Several multigrid methods and direct solvers have been tested. The fastest were a routine in band storage mode with LU factorization and, when stopping at a slightly lower accuracy, a two-grid method. The multigrid method is competitive with direct solvers due to its higher flexibility. See also:
Acknowledgment: This work was supported by the Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie.
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