Fast algorithms for large scale quantum transport simulations with applications

Abstract

Time is limited. While sophisticated quantum formalism like non-equilibrium Green's function (NEGF) exists for solving complicated quantum transport problems, it remains almost useless if we could not compute it within a reasonable time scale. This is indeed the situation for a large portion of systems covered in the physical length scale that is small enough to crave for quantum description but not large enough to be described solely by the classical one. In extreme situations where macroscopic quantum phenomenon like quantum Hall takes place, quantum systems could go up to millimetre scale. In general, computational complexity grows with time T, system size N=N_x N_y N_z, basis N_b and k-points N_k, while a simple matrix inversion would require O(T N_k N_b N^3) so that required computational cost is easily beyond reach. In this thesis, fast algorithms for large scale quantum transport simulations are proposed. Here, the scales refers to two cases, temporal and spatial length scale. In the first part, fast algorithms for larger temporal scale simulation are proposed based on the NEGF-CAP method for transient current calculation which is suitable to combine with the first principles density functional theory calculation. That is made possible with the four key ingredients 1) exact solution based on NEGF that goes beyond wide band limit, 2) complex absorbing potential, 3) possibility of the separation of space and time domains and 4) the fast multipole method. The construction and benchmarking of the algorithm which is O(1) with respect to time T will be discussed and applied leading to the discovery of all-electrical generated spin polarized current. In the second part, fast and memory efficient algorithms for large spatial scale quantum transport simulation called Hierarchical self-energization (HSE) that based on nested dissection methods will be discussed. The computational scaling is first reduced by mean of hierarchical divide and conquer to O(N^{1.5}) for the two-dimensional (2D) case and O(N^2) for the three-dimensional (3D) case. Discussion for further reducing the order of complexity by the so-called skeletonization will be given. Finally, the fast 3D algorithm is applied to the study of the intriguing oscillation in the conductance and conductance fluctuation due to magnetic disorder in the 3D system with quantum spin Hall to quantum anomalous Hall phase transition.