General(ized) Hartman effect

TL;DR

This paper proves that the Hartman effect, where tunneling time becomes independent of barrier width, extends from a single potential to periodic systems built from identical units, regardless of the gap size.

Contribution

It establishes a generalization of the Hartman effect from individual potentials to periodic systems with arbitrary gaps, showing the effect's universal applicability.

Findings

Hartman effect exists for arbitrary unit cell potentials

Tunneling time in periodic systems matches that of the unit cell for thick potentials

Hartman effect is independent of the gap size between unit cells

Abstract

In this letter we prove explicitly that if Hartman effect exists for an arbitrary `unit cell' potential, then it also exist for a periodic system constructed using the same `unit cell' potential repeatedly. We further show that if Hartman effect exists, the tunneling time in the limiting case of a sufficiently thick `unit cell' potential is same as that of its periodic system. This is true for any arbitrary value of the intervening gap between the consecutive `unit cell' of the periodic system. Thus generalized Hartman effect always occurs for a general potential constructed using multiple copies of single potential which shows Hartman effect.