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 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.