The numerical operator method to the real time dynamics of currents through the nanostructures with different topologies

We present the numerical operator method designed for the real time dynamics of currents through nanostructures beyond the linear response regime. We apply this method to the transient and stationary currents through nanostructures with different topologies, e.g., the flakes of square and honeycomb lattices. We find a quasi-stationary stage with a life proportional to the flake size in the transient currents through the square flakes, but this quasi-stationary stage is destroyed in the presence of disorder. However, there is no quasi-stationary stage in the transient currents through the honeycomb flakes, showing that the transient current depends strongly upon the topologies of the nanostructures. We also study the stationary current by taking the limit of the current at long times. We find that the stationary current through a square flake increases smoothly as the voltage bias increasing. In contrast, we find a threshold voltage in the current-voltage curve through a honeycomb flake, indicating a gap at the Fermi energy of a honeycomb flake.Comment: 13 pages, 4 figure