Electronic transport of a large scale system studied by renormalized transfer matrix method: Application to armchair graphene nanoribbons between quantum wires

Study on the electronic transport of a large scale two dimensional system by the transfer matrix method (TMM) based on the Schrödinger equation suffers from numerical instability. To address this problem, we propose a renormalized transfer matrix method (RTMM) by setting up a set of linear equations from U times of multiplication of traditional transfer matrix (U=NS with N and S respectively being the atom number of length and the transfer steps), and smaller S is required for a wider system. Then we solve the above linear equations by Gaussian elimination method and further optimize to reduce the computational complexity from O(U3M3) to O(UM3), in which M is the atom number of the width. Applying the RTMM, we study transport properties of large scale pure and long-range correlated disordered armchair graphene nanoribbons (AGR) (carbon atoms up to 106 for pure cases) between quantum wire contacts. For a pure AGR, the conductance is superlinear with the Fermi energy, and linear with the width while independent of the length, showing characteristics of ballistic transport. For a disordered AGR with long-range correlation, there is a metal-insulator transition induced by the correlation strength of disorder. It is straightforward to extend the RTMM to investigate the electronic transports of large scale systems with various structures.