Speeding up the solution of the Bethe-Salpeter equation by a double-grid method and Wannier interpolation

The Bethe-Salpeter equation is a widely used approach to describe optical excitations in bulk semiconductors. It leads to spectra that are in very good agreement with experiment, but the price to pay for such accuracy is a very high computational burden. One of the main bottlenecks is the large number of $k$ points required to obtain converged spectra. In order to circumvent this problem we propose a strategy to solve the Bethe-Salpeter equation based on a double-grid technique coupled to a Wannier interpolation of the Kohn-Sham band structure. This strategy is then benchmarked for a particularly difficult case, the calculation of the absorption spectrum of GaAs, and for the well-studied case of Si. The considerable gains observed in these cases fully validate our approach, and open the way for the application of the Bethe-Salpeter equation to large and complex systems.

Figure 1

Calculated RPA absorption spectra starting from $GW$ corrected bands for bulk GaAs with $N_v=N_c=2$ for various $k$ grids, without (a) and with (b) double-grid method. The shaded area in (b) shows the spectrum corresponding to the $40\times 40\times 40$ $k$ grid of panel (a) for better comparison between the double-grid method and the standard calculations.

Figure 2

Calculated BSE absorption spectra for bulk GaAs with $N_v=2$ and $N_c=3$ using several $k$ grids without (a) and with (b) double-grid method. In (a) one spectrum is calculated on a symmetric $10\times 10\times 10$ $k$ grid, while the others are on shifted grids to accelerate convergence. The shaded area in both panels indicate the experimental spectrum at 22 K.[20]

Figure 3

Calculated RPA (a) and BSE (b) absorption spectra with LFE starting from $GW$ corrected bands for bulk Si with and without double-grid method employing a broadening of $0.05$ eV. For (a) $N_v=N_c=2$ were used, while for (b) $N_v=2$ and $N_c=3$. With the double-grid method both spectra (solid lines) converge faster. The shaded area in (b) shows the experimental spectrum.[32]