Efficient Chebyshev polynomial approach to quantum conductance calculations: Application to twisted bilayer graphene

Abstract

In recent years, Chebyshev polynomial expansions of tight-binding Green's functions have been successfully applied to the study of a wide range of spectral and transport properties of materials. However, the application of the Chebyshev approach to the study of quantum transport properties of noninteracting mesoscopic systems with leads has been hampered by the lack of a suitable Chebyshev expansion of Landaeur's formula or one of its equivalent formulations in terms of Green's functions in Keldysh's perturbation theory. Here, we tackle this issue by means of a hybrid approach that combines the efficiency of Chebyshev expansions with the convenience of complex absorbing potentials to calculate the conductance of two-terminal devices in a computationally expedient and accurate fashion. The versatility of the approach is demonstrated for mesoscopic twisted bilayer graphene (TBG) devices with up to 2.3 × 10 6 atomic sites. Our results highlight the importance of moiré effects, interlayer scattering events, and twist-angle disorder in determining the conductance curves in devices with a small twist angle near the TBG magic angle θ m ≈ 1.1 • .

Fast Fourier-Chebyshev Approach to Real-Space Simulations of the Kubo Formula

“…Owing to the full-spectral capability of the algorithm, it is possible to zoom in on small features, such as the conductance peak at the charge neutrality point caused by residual dispersion of the conduction bands. The absence of ballistic conductance steps is due to the channel mixing caused by elastic scattering between the layers, and is thus a direct result of moiré supercell effects [32]. Based on our extensive tests, we estimate that the largest TBG simulation (i.e., M ¼ 56 000, corresponding to a spectral resolution of 0.5 meV, and taking 40 core hours with FastCheb), would require around 2400 core hours using the standard kernel polynomial approach.…”

“…( 2) [σ aa ðE F Þ ≃ ð4e 2 =hΩÞ⟪ σr a ðε F Þ⟫ R ] using the framework for two-terminal devices recently developed in Ref. [32]. Note that in the zero-temperature limit, this quantity is a pure Fermi surface property and the conductivity kernel directly yields the dc response.…”