Determination of integrals at surfaces using the bulk crystal symmetry

The symmetry arguments of Appelbaum, Baraff, and Hamann for integrals of physical quantities for the (110) and (100) surface orientations of zinc-blende-structure crystals are generalized for other crystals and surface orientations of lower symmetry. We define the surface region by identifying boundaries in the interior of the crystal consisting of segments of bulk symmetry planes. As an example, we consider (111) and (bar 1 bar 1 bar 1) surfaces and interfaces of zinc-blende crystals; we give an explicit construction of the boundaries and a (111) symmetry-adapted cell that enables the determination of integrals at these surfaces