Fast algorithm for transient current through open quantum systems

Abstract

Transient current calculation is essential to study the response time and capture the peak transient current for preventing meltdown of nanochips in nanoelectronics. Its calculation is known to be extremely time consuming with the best scaling $T{N}^{3}$ where $N$ is the dimension of the device and $T$ is the number of time steps. The dynamical response of the system is usually probed by sending a steplike pulse and monitoring its transient behavior. Here, we provide a fast algorithm to study the transient behavior due to the steplike pulse. This algorithm consists of two parts: algorithm I reduces the computational complexity to ${T}^{0}{N}^{3}$ for large systems as long as $T<N$; algorithm II employs the fast multipole technique and achieves scaling ${T}^{0}{N}^{3}$ whenever $T<{N}^{2}$ beyond which it becomes $T{log}_{2}N$ for even longer time. Hence it is of order $O(1)$ if $T<{N}^{2}$. Benchmark calculation has been done on graphene nanoribbons with $N={10}^{4}$ and $T={10}^{8}$. This algorithm allows us to tackle many large scale transient problems including magnetic tunneling junctions and ferroelectric tunneling junctions.