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 $E\left(\mathbf{k}\right)$ 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 $E\left(\mathbf{k}\right)$ 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.

Figure 1

Three of the bands along $\left[100\right]$ for the simple cubic $s{p}^3$ system studied here (see text). Bulk bands for this system (i.e., those of the primitive cell, with the lattice parameter $a$) are plotted in bold; the solid line is double degenerate. Bands for a $2\times 2\times 2$ supercell along $\left[100\right]$ are plotted with fine lines. Note that the bulk bands exactly overlap two of the supercell bands. The dashed line denotes the supercell Brillouin zone.

Figure 2

Bulk bands (bold lines) and supercell bands (fine lines) for the $2\times 2\times 2$ supercell of Table . The supercell state and one of the bulk states contributing to it are indicated with open circles. Note that the bulk bands are shown from the negative Brillouin zone face to zone center, while the supercell bands proceed from the zone center to their positive Brillouin zone face.

Figure 3

Bulk bands (bold lines) and supercell bands (fine lines) for the $1\times 2\times 3$ supercell of Tables . The supercell state at $\mathbf{K}=\mathbf{0}$ and one of the bulk states contributing to it are indicated with open circles.

Figure 4

Probability coefficients ${\mid a_{p,\mathbf{n}}\mid }^2$ for fixed ${\mathbf{G}}_{100}=\left(1,0,0\right)\pi ∕a$ from an imperfect $2\times 2\times 2$ supercell at $\mathbf{K}=(0.01,0.01,0.01)\left(\pi ∕a\right)$ (see text). There are 32 dots, one for each of the supercell states. Those energies at which the probability is greatest can be interpreted as being the band energies at this $\mathbf{k}=\mathbf{K}+{\mathbf{G}}_{100}$ for an averaged, translationally symmetric, Hamiltonian.