Closed Formulas for Tunneling Time in Superlattices

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 ${\ensuremath{\tau}}_{n}$, 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, ${\ensuremath{\tau}}_{n}$ remains larger than the free motion time ${\ensuremath{\tau}}_{f}$, while in the gap it can be less than ${\ensuremath{\tau}}_{f}$ (with strong substrate effects), but it is larger than the single-cell phase time ${\ensuremath{\tau}}_{1}$. 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.