Determining the electronic properties of semi-infinite crystals

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A self-consistent method for the calculation of the electronic structure of crystalline surfaces is described. It is based on a semi-infinite geometry with individual surface atomic layers stacked onto an infinite number of bulk layers. Contrary to models based on slab or superlattice geometries there is no artificial distortion of the correct asymptotic behavior of the wave functions so that an exact distinction between surface and bulk effects is possible. Furthermore there are no principal restrictions on the shape of the self-consistent potential. A special form of wave-function matching is used to construct the discrete surface states as well as the continuum of bulk states from complete sets of solutions of the Schr\odinger equation in each single layer. The semi-infinite substrate is treated as a whole by means of the complex band structure which appears as an easily obtainable side-product of the theory. The main improvement at this step is the complete avoidance of the inherent numerical instability which prevented the application of similar matching techniques to other than very simple materials so far. The layer solutions of the Schr\odinger equation are obtained by means of the spline-augmented-plane-wave method providing very accurate wave functions. As a first application the (001) and (111) surfaces of aluminum were investigated. The results obtained include the self-consistent charge density, the work function, and the complete band structure of the surface states and resonances. All calculations are found to be in good quantitative agreement with experiment.