Theory

Empirical Tight-Binding Method

The empirical tight-binding method expands electronic states in a basis of atomic-like orbitals localized on each atom. For a zinc-blende crystal with two atoms per unit cell (anion and cation), the Bloch Hamiltonian at wave vector $\mathbf{k}$ takes the form:

\[H(\mathbf{k}) = \begin{pmatrix} E_a & V(\mathbf{k}) \\ V^\dagger(\mathbf{k}) & E_c \end{pmatrix}\]

where $E_a$ and $E_c$ are diagonal on-site energy matrices for the anion and cation, $V(\mathbf{k})$ contains the hopping (off-diagonal) terms modulated by structure factors, and $V^\dagger$ is the conjugate transpose of $V$.

Structure Factors

For the zinc-blende structure (FCC lattice with two-atom basis), the nearest-neighbor structure factors are:

\[g_0(\mathbf{k}) = \frac{1}{4}\left(e^{i\pi(k_1+k_2+k_3)/2} + e^{i\pi(-k_1-k_2+k_3)/2} + e^{i\pi(-k_1+k_2-k_3)/2} + e^{i\pi(k_1-k_2-k_3)/2}\right)\]

with $g_1, g_2, g_3$ formed by sign changes on the individual exponentials. These factors encode the crystal geometry and modulate the Slater-Koster hopping integrals.

Slater-Koster Parameters

The hopping matrix elements between orbitals are parameterized using the Slater-Koster two-center integrals ($V_{ss\sigma}$, $V_{sp\sigma}$, $V_{pp\sigma}$, $V_{pp\pi}$, etc.). These parameters are fitted to reproduce experimental or first-principles band structures at high-symmetry points.

References

• J.C. Slater, G.F. Koster, "Simplified LCAO Method for the Periodic Potential Problem," Phys. Rev. 94, 1498 (1954). DOI:10.1103/PhysRev.94.1498
• See also the References page for parameter source publications.