Low-Scaling GW Algorithm Applied to Twisted Transition-Metal Dichalcogenide Heterobilayers

Abstract

The GW method is widely used for calculating the electronic band structure of materials. The high computational cost of GW algorithms prohibits their application to many systems of interest. We present a periodic, low-scaling, and highly efficient GW algorithm that benefits from the locality of the Gaussian basis and the polarizability. The algorithm enables G₀W₀ calculations on a MoSe₂/WS₂ bilayer with 984 atoms per unit cell, in 42 h using 1536 cores. This is 4 orders of magnitude faster than a plane-wave G₀W₀ algorithm, allowing for unprecedented computational studies of electronic excitations at the nanoscale.

Figure 1

G₀W₀ band gap of monolayer WS₂, MoS₂, WSe₂, and MoSe₂ calculated from eq 23 as a function of the supercell size (TZVP-MOLOPT basis set, without spin–orbit coupling (SOC)).

Figure 2

Number of floating point operations (real double precision) needed for executing G₀W₀ algorithms. Green: low-scaling G₀W₀ algorithm from this work using a TZVP-MOLOPT basis set. Black: computing the irreducible polarizability χ in a plane-wave basis. Gray: inverting the dielectric matrix ϵ in a plane-wave basis. Underlying computational parameters are typical for monolayers MoS₂, MoSe₂, WS₂, and WSe₂; see the detailed raw data and discussion available in the SI.

Figure 3

Execution time of a G₀W₀ calculation for MoSe₂ 9 × 9–14 × 14 supercells (TZVP-MOLOPT basis set) on Supermuc-NG (Intel Skylake Xeon Platinum 8174). Magenta points show the computational cost to diagonalize the polarizability χ(k), which allows us to remove all spurious negative eigenvalues of χ(k) to ensure numerical stability. Dashed lines show a fit of αN_at^β to the execution time, where α and β are fit parameters. Raw data are available in the SI.

Figure 4

Band gap of a MoSe₂/WS₂ heterostructure as a function of the twist angle. Inset: The unit cell (black rhomboid) for a 19.4° twist angle contains 984 atoms.