Numerical solver for first-principles transport calculation based on real-space finite-difference method

We propose an efficient procedure to obtain Green\u27s functions by combining the shifted conjugate orthogonal conjugate gradient (shifted COCG) method with the nonequilibrium Green\u27s function (NEGF) method based on a real-space finite-difference (RSFD) approach. The bottleneck of the computation in the NEGF scheme is matrix inversion of the Hamiltonian including the self-energy terms of electrodes to obtain the perturbed Green\u27s function in the transition region. This procedure first computes unperturbed Green\u27s functions and calculates perturbed Green\u27s functions from the unperturbed ones using a mathematically strict relation. Since the matrices to be inverted to obtain the unperturbed Green\u27s functions are sparse, complex-symmetric, and shifted for a given set of sampling energy points, we can use the shifted COCG method, in which once the Green\u27s function for a reference energy point has been calculated the Green\u27s functions for the other energy points can be obtained with a moderate computational cost. We calculate the transport properties of a C60@(10,10) carbon nanotube (CNT) peapod suspended by (10,10)CNTs as an example of a large-scale transport calculation. The proposed scheme opens the possibility of performing large-scale RSFD-NEGF transport calculations using massively parallel computers without the loss of accuracy originating from the incompleteness of the localized basis set