Practical application of zone-folding concepts in tight-binding calculations

Modern supercell algorithms, such as those used in treating arrays of quantum dots or alloy calculations, are often founded upon local basis representations. Such local basis representations are numerically efficient, allow considerations of systems consisting of millions of atoms, and naturally map into carrier transport simulation algorithms. Even when treating a bulk material, algorithms formulated on a local basis generally cannot produce an $E(\mathbf{k})$ dispersion resembling that of a simple unit cell, due to zone folding. This paper provides an exact method for perfect supercells to unfold the zone folded $E(\mathbf{k})$ diagrams into a meaningful bulk dispersion relation. In addition, a modification to the algorithm for use with imperfect supercells is presented. With this method, questions such as algorithm verification, dispersions in nanowires, and dispersions in finite supercell heterostructures can be addressed.