Double k-Grid Method for Solving the Bethe-Salpeter Equation via Lanczos Approaches

Abstract

Convergence with respect to the size of the k-points sampling grid of the Brillouin zone is the main bottleneck in the calculation of optical spectra of periodic crystals via the Bethe-Salpeter equation (BSE). We tackle this challenge by proposing a double grid approach to k-sampling compatible with the effective Lanczos-based Haydock iterative solution. Our method relies on a coarse k-grid that drives the computational cost, while a dense k-grid is responsible for capturing excitonic effects, albeit in an approximated way. Importantly, the fine k-grid requires minimal extra computation due to the simplicity of our approach, which also makes the latter straightforward to implement. We performed tests on bulk Si, bulk GaAs and monolayer MoS2, all of which produced spectra in good agreement with data reported elsewhere. This framework has the potential of enabling the calculation of optical spectra in semiconducting systems where the efficiency of the Haydock scheme alone is not enough to achieve a computationally tractable solution of the BSE, e.g., large-scale systems with very stringent k-sampling requirements for achieving convergence.

Optical properties enhancement via WSSe/silicene solar cell junctions

“…50. In order to better resolve the optical spectrum, we take advantage of the double-grid approach, 51 where the electron-hole kernel matrix elements are calculated in a course grid and then interpolated on a fine mesh of 180 Â 180 Â 1 for the monolayer systems and 90 Â 90 Â 1 for the interface. Five valence bands and five conduction bands were included for WSSe, while for silicene three valence bands and four conduction bands were included.…”

Linear scaling approach for optical excitations using maximally localized Wannier functions

“…Such operations can be performed very efficiently in linear scaling as shown in the figure. For comparison, the time-evolution approach in a Bloch representation, where the Hamiltonian is dense, would scale with O(N 2 ) [42], which is similar to implementations that use a Lanczos-Haydock approach as implemented in the Yambo code [58]. Note that a direct diagonalization of the Hamiltonian scales with O(N 2 ) in the case of a sparse matrix or with O(N 3 ) in the case of a dense matrix.…”