Wannier functions in crystals with surfaces

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Abstract
Generalized Wannier functions (GWF) are defined for one-dimensional crystals with surfaces, using an approach similar to that developed by Kohn and Onffroy for crystals with impurities. We show how an appropriate set of exponentially localized orthonormal GWF may be constructed for each band of the crystal; both the surface states and bulk states associated with a given band can be expressed as linear combinations of these GWF. The GWF have the same exponential decay into the crystal as the Wannier functions (WF) of the infinite lattice, even though the surface states themselves may have a longer range. In addition, successive GWF, starting from the surface, reduce to the WF in an exponential manner. These localization properties give the GWF several advantages over the energy eigenstates as a set of basis functions; and, using variational methods, they may actually be easier to compute.