Bloch-Wannier theory of persistent current in a ring made of the band insulator: Exact result for one-dimensional lattice and estimates for realistic lattices

TL;DR

This paper develops an exact theoretical framework for calculating persistent currents in 1D band insulator rings, deriving explicit formulas and estimating their magnitude in real materials, revealing measurable effects despite exponential decay.

Contribution

It introduces a novel recipe linking Bloch states of infinite crystals to persistent currents in finite insulator rings, providing exact results and practical estimates for real materials.

Findings

Persistent current depends on the flux-dependent on-site energy.

The current decays exponentially with ring length.

Measurable persistent currents can exist in real insulator rings.

Abstract

In this work, persistent currents in rings made of band insulators are analyzed theoretically. We first formulate a recipe which determines the Bloch states of a one-dimensional (1D) ring from the Bloch states of an infinite 1D crystal created by the periodic repetition of the ring. Using the recipe, we derive an expression for the persistent current in a 1D ring made of an insulator with an arbitrary valence band E(k). To find an exact result for a specific insulator, we consider a 1D ring represented by a periodic lattice of N identical sites with a single value of the on-site energy. If the Bloch states in the ring are expanded over a complete set of N on-site Wannier functions, the discrete on-site energy splits into the energy band. At full filling, the band emulates the valence band of the band insulator and the ring is insulating. It carries a persistent current equal to the product of N and the derivative of the on-site energy with respect to the magnetic flux. This current is not zero if one takes into account that the on-site Wannier function and consequently the on-site energy of each ring site depend on magnetic flux. To derive the current analytically, we expand all N Wannier functions of the ring over the infinite basis of Wannier functions of the constituting infinite 1D crystal and eventually determine the crystal Wannier functions by a method of localized atomic orbitals. Finally, we estimate the persistent current at full filling in rings made of real band insulators (GaAs, Ge, InAs). The current decays with the ring length exponentially due to the exponential decay of the Wannier functions. In spite of that, it can be of measurable size.