Closed Formulas for Tunneling Time in Superlattices

Abstract

New formulas for an exact and simple evaluation of the phase time associated with the passage of electrons or photons through a finite superlattice are derived. This time, named here the superlattice-tunneling time ${\tau }_n$, exhibits a resonant-band structure or a superluminal phase time behavior depending on whether the particle's energy lies in a band or a gap. In the band, ${\tau }_n$ remains larger than the free motion time ${\tau }_f$, while in the gap it can be less than ${\tau }_f$ (with strong substrate effects), but it is larger than the single-cell phase time ${\tau }_1$. Extremely good agreements with optical-pulse and superluminal delay times measured by Spielmann et al. and Steinberg et al. are found, including the superlattice-tunneling-time limit and the substrate effects. Conditions for measurements of earlier electron arrival are also analyzed.