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Solution of the Poisson Equation for Two Dimensional Periodic Structures in an Overlapping Localized Site Density Scheme

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Bertaut's equivalent electric density idea (E. F. Bertaut, Journal de Physique 39, 1331 (1978)) is applied to the case of two dimensional periodic continuous charge density distributions. The following derivation differs from what was introduced by Bertaut. The presented method solves the Poisson equation for the scheme of overlapping localized site densities with periodic boundary conditions in the (x,y) plane and with the general finite voltage boundary condition in the perpendicular z-direction. As usual the long-range potential is calculated in the Fourier space. For the K ≠ 0 case a Fourier transformation helps to calculate the solution in a three dimensional periodic sense, while for K = 0 the required charge neutrality is the starting point. For both cases suitable representations of the spherical harmonics are needed to arrive at expressions that are convenient for numerical implementation. In this localized density scheme an explicit relation can be derived between the finite voltage in z-direction and the z-component of the dipole density. The application to band structure calculation is tested.
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Document Type: Research Article

Publication date: September 1, 2007

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  • Journal of Computational and Theoretical Nanoscience is an international peer-reviewed journal with a wide-ranging coverage, consolidates research activities in all aspects of computational and theoretical nanoscience into a single reference source. This journal offers scientists and engineers peer-reviewed research papers in all aspects of computational and theoretical nanoscience and nanotechnology in chemistry, physics, materials science, engineering and biology to publish original full papers and timely state-of-the-art reviews and short communications encompassing the fundamental and applied research.
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