Bound on the Group Velocity of an Electron in a 1D Periodic Potential

TL;DR

This paper proves that the group velocity of an electron in a 1D periodic potential cannot exceed that in free space, and shows energy bands are always below free-electron bands when aligned.

Contribution

It introduces a new upper bound on the group velocity of Bloch electrons in 1D periodic potentials, linking it to free-electron velocities and band positioning.

Findings

Group velocity in a 1D periodic potential is always ≤ free-electron velocity.

Energy bands in 1D crystals lie below corresponding free-electron bands when aligned.

The bound is derived from a recently established limit on equilibrium current in a ring.

Abstract

By using a recently derived upper bound on the allowed equilibrium current in a ring, it is proved that the magnitude of the group velocity of a Bloch electron in a one-dimensional periodic potential is always less than or equal to the group velocity of the same Bloch state in an empty lattice. Our inequality also implies that each energy band in a one-dimensional crystal always lies below the corresponding free-electron band, when the minima of those bands are aligned.