Wigner wave packets: Transmission, reflection, and tunneling

A numerical solution to the time evolution equation of the Wigner distribution function (WDF) with an accuracy necessary to simulate the passage of a wave packet past a barrier is developed, where quantum effects require high accuracy and fine discretization. A wave packet incident on a barrier, a portion of which tunnels through, demonstrates behavior that can define various characteristic transmission and reflection delay (TARD) times useful in the simulation of electron emission. A model for the TARD times is proposed that relies only on the asymptotic maxima of the position $\ensuremath{\rho}(x,t)$ and wave number $\ensuremath{\rho}(k,t)$ densities given by the WDF and applied to a ballistic trajectory model for the (faster) transmitted and (slower) reflected parts. The dependence of the TARD times on barrier width, symmetry, and abruptness is analyzed. For symmetrical barriers with characteristics similar to field emission barrier heights and widths, TARD times are on the order of a fraction of a femtosecond. The TARD times for when tunneling predominates are contrasted to tunneling times in the literature. Use of the TARD times in simulations of field emission in nanogap devices or to model ultrashort pulses generated under rapidly changing conditions for electron sources are proposed.