Efficient linear-scaling quantum transport calculations on graphics processing units and applications on electron transport in graphene

We implement, optimize, and validate the linear-scaling Kubo–Greenwood quantum transport simulation on graphics processing units by examining resonant scattering in graphene. We consider two practical representations of the Kubo–Greenwood formula: a Green–Kubo formula based on the velocity auto-correlation and an Einstein formula based on the mean square displacement. The code is fully implemented on graphics processing units with a speedup factor of up to 16 (using double-precision) relative to our CPU implementation. We compare the kernel polynomial method and the Fourier transform method for the approximation of the Dirac delta function and conclude that the former is more efficient. In the ballistic regime, the Einstein formula can produce the correct quantized conductance of one-dimensional graphene nanoribbons except for an overshoot near the band edges. In the diffusive regime, the Green–Kubo and the Einstein formalisms are demonstrated to be equivalent. A comparison of the length-dependence of the conductance in the localization regime obtained by the Einstein formula with that obtained by the non-equilibrium Green’s function method reveals the challenges in defining the length in the Kubo–Greenwood formalism at the strongly localized regime.