A method for calculating surface electronic structures using semi-infinite boundary conditions

We have developed a new formalism for solving the Kohn-Sham equations in the layer geometry appropriate for studying equilibrium and transport properties of surfaces and interfaces. The formalism assumes that the electron-lattice interactions are modeled by pseudopotentials containing both local contributions and non-local terms represented by separable functions, and works especially well with the projector augmented wave ``PAW'' method.\footnote{ P.~E. Bl\"{o}chl, {\em Phys. Rev. B}, {\bf{50}},17953 (1994), A.~R. Tackett, N.~A.~W. Holzwarth, and G.~E. Matthews, {\em Comput. Phys. Comm.}, {\bf{135}}, 348 (2001); Website: \urllink{http://pwpaw.wfu.edu}{http://pwpaw.wfu.edu}} Based on the Numerov algorithm, a two-point recurrence relation is used to integrate the differential equations. The recurrence formalism is used to find the generalized Bloch waves in the bulk regions of the system as well as to find the propagating and surface states in the interface regions of the system. The wavefunction is matched at the boundary between the bulk and interface regions and at intermediate points to ensure stability. The formalism is demonstrated for a simple model of a semi-infinite system and compared with a boundary matching formalism developed by Choi and Ihm.\footnote{Y. J. Choi and J. Ihm, {\em{Phys. Rev. B}} {\bf{59}}, 2267 (1999).}