Wave packet tunneling and imaginary wave vector dispersion

Dispersion associated with the complex band structure of a periodic potential can be used to explain single-electron wave packet tunneling through a finite-sized tunnel barrier. The imaginary $k$-space dispersion in a tunnel barrier may be viewed as a filter that controls the arrival time, shape, and group velocity of a transmitted wave packet. This $k$-space filter can be designed to increase or decrease the group velocity of the transmitted pulse and a transmitted peak-pulse probability arrival time can be engineered to occur before or after that of an electron wave packet propagating in the absence of the tunnel barrier.