Contour integral method for obtaining the self-energy matrices of electrodes in electron transport calculations

We propose an efficient computational method for evaluating the self-energy matrices of electrodes to study ballistic electron transport properties in nanoscale systems. To reduce the high computational cost incurred for large systems, a contour integral eigensolver based on the Sakurai-Sugiura method combined with the shifted biconjugate method is developed to solve exponential-type eigenvalue problem for complex wave vector. A remarkable feature of the proposed algorithm is that the numerical procedure is very similar to that of e conventional band structure calculations, thereby enabling the use of fast and sophisticated numerical techniques and allowing us to perform large scale electron transport calculations. We implement the developed method in the framework of the real-space higher-order finite difference scheme with nonlocal pseudopotentials. Numerical tests for a wide variety of materials validate the robustness, accuracy, and efficiency of the proposed method; in all cases, this method achieves speed-up of up to three orders of magnitude compared with the conventional method using the standard eigensolver.