An alternating-direction hybrid implicit-explicit finite-difference time-domain method for the Schrödinger equation

This paper proposes a novel hybrid FDTD method for solving the time-dependent Schrödinger equation, which is fundamental for modeling materials and designing nanoscale devices. The wave function is propagated on nonuniform grids by applying explicit updates in part of the grid and implicit updates elsewhere. The latter are based on the Alternating-Direction Implicit (ADI) scheme while the former are constructed with a central difference for the time derivative. A rigorous stability analysis proves that spatial steps can be selectively removed from the stability criterion thus combining the unconditional stability of the ADI scheme with fast explicit calculations. The scheme excels in its flexibility by efficiently discretizing and balancing explicit with implicit updates, as such expediting the computations. Moreover, it retains the linear complexity of explicit schemes with respect to space and time, making it especially scalable to numerically large problems. Several numerical experiments, including a laterally tunnel coupled quantum wire and a nanowire double-barrier resonant-tunneling diode, show the validity of the scheme by demonstrating its high accuracy and decreased CPU time compared to traditional methods.