Accurate tight-binding Hamiltonians for two-dimensional and layered materials

We present a scheme to controllably improve the accuracy of tight-binding Hamiltonian matrices derived by projecting the solutions of plane-wave ab initio calculations on atomic-orbital basis sets. By systematically increasing the completeness of the basis set of atomic orbitals, we are able to optimize the quality of the band-structure interpolation over wide energy ranges including unoccupied states. This methodology is applied to the case of interlayer and image states, which appear several eV above the Fermi level in materials with large interstitial regions or surfaces such as graphite and graphene. Due to their spatial localization in the empty regions inside or outside of the system, these states have been inaccessible to traditional tight-binding models and even to ab initio calculations with atom-centered basis functions.

Figure 1

All-electron and pseudoatomic functions for the $\{s,s^{\prime }\}$ components of the ${\mathrm{PAO}}_3$ set. The parameters used to compute the pseudo-atomic-orbitals are defined by the same PAW dataset used in the DFT calculation.

Figure 2

Band structure of graphite using atomic-orbital (at the single-zeta, double-zeta polarized, and triple-zeta doubly polarized level) and well-converged plane-wave basis sets.

Figure 3

Band structure of graphene using atomic-orbital (at the single-zeta, double-zeta polarized, and triple-zeta doubly polarized level) and well-converged plane-wave basis sets. The first three panels use atomic-orbital-like sets: SZ, TZDP, and DZP with empty atoms from Ref. [32].

Figure 4

Projectability $p_{n\mathbf{k}}$, in color scale, of Bloch states ${\psi }_{n\mathbf{k}}$ of graphite on the ${\mathrm{PAO}}_1, {\mathrm{PAO}}_2$, and ${\mathrm{PAO}}_3$ sets. The fourth panel shows the interpolated band structure in green obtained from the TB Hamiltonians built on the ${\mathrm{PAO}}_3$, superimposed to the plane-wave band structure. The color bar is shown in Fig. 6.

Figure 5

Projectability of the Bloch states ${\psi }_{n\mathbf{k}}$ of graphite onto the four PAO sets defined in Table 1. The plot shows the minimum value of projectability for each bin of a discretized energy grid. The dotted line indicates the projectability threshold of 0.95. All $\mathbf{k}$ points in the reciprocal unit cell are included in the calculation.

Figure 6

Projectabilities of the Bloch states of graphene on the ${\mathrm{PAO}}_1$ and ${\mathrm{PAO}}_3$ sets. The interpolated TB band structure, computed using the ${\mathrm{PAO}}_3$ basis set, is shown in green on the third panel superimposed to the PW bands.