Low energy electron diffraction has been a major technique in the determination of surface atomic structures. While the experimental techniques have been improved steadily, the theoretical calculation has been practically stayed with the method developed decades ago: the multiple-scattering (MS) method with a major approximation that the crystal potential around a nucleus is treated isotropic. While the MS theory succeeded on metal surfaces, it's much less accurate on semiconductor surfaces due to anisotropic potentials.
To make more accurate calculation, a few new approaches have been emerged that allow the use of full crystal potential. One impressive approach among them is the finite difference (FD) method [1]. In this method 3-D uniform grids are put to a unit cell with an unknown wave function value at each grid point so that the Schrodinger equation at a grid point is replaced by a linear combination of wave function values at several neighboring grid points. In the close vicinity of a nucleus wave functions are expanded in terms of radial function and spherical harmonics. All unknowns can be solved from a system of linear equations. This method is accurate but is too computationally demanding.
We have developed a method to significantly reduce the computation cost of the accurate FD scheme. In this method, a unit cell is divided into many thin slabs parallel to the surface and the amplitude reflection and transmission coefficients of distinguishable slabs are calculated. The actual reflectivity of each LEED spot of the system is obtained by combining together the contributions of all thin slabs with a simple iteration formula. By also the use of full symmetry of a system, this method can save storage memory and computation time by two orders of magnitude. The calculation on a Si (111)1x1 surface yielded more accurate LEED spectra than by MS method. For the first time, large unit cells were able to be calculated on a single CPU with reasonable calculation time, for example a fake Si(111)2x2 surface and a fake C(111)3x3 surface. With parallel computing this method should be able to handle calculations on realistic nanostructures with full potentials.
[1] Y. Joly, Physica Review B63, 125120 (2001).
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