Brillouin-zone unfolding of perfect supercells having nonequivalent primitive cells illustrated with a Si/Ge tight-binding parameterization

Numerical calculations of nanostructure electronic properties are often based on a nonprimitive rectangular unit cell, because the rectangular geometry allows for both highly efficient algorithms and ease of debugging while having no drawback in calculating quantum dot energy levels or the one-dimensional energy bands of nanowires. Since general nanostructure programs can also handle superlattices, it is natural to apply them to these structures as well, but here problems arise due to the fact that the rectangular unit cell is generally not the primitive cell of the superlattice, so that the resulting E(k) relations must be unfolded to obtain the primitive-cell E(k) curves. If all of the primitive cells in the rectangular unit cell are identical, then the unfolding is reasonably straightforward; if not, the problem becomes more difficult. Here, we provide a method for zone unfolding when the primitive cells in a rectangular cell are not all identical. The method is applied to a Si(4)Ge(4) superlattice using a set of optimized Si and Ge tight-binding strain parameters.