Persistent Currents in Mesoscopic Rings: Ensemble Average and Half-Flux-Quantum Periodicity

Abstract

The transport properties of micron size metallic samples at low temperature have been shown to exhibit features characteristic of the quantum coherence of the electronic wave function along the whole sample. One of the most striking has been the detection of Aharonov-Bohm oscillations in the resistance of a loop pierced by a magnetic field perpendicular to its plane [1] . In such resistivity measurements, the electric probes induce a coupling of the sample with reservoirs of electrons in which dissipation occurs. Isolated metallic rings are predicted to present an even more spectacular behavior: Büttiker, Imry and Landauer [2] suggested the existence of persistent currents in the presence of a magnetic field. These currents are a consequence of the sensitivity of the eigenstates to the boundary conditions along the ring. In the presence of a magnetic field, the periodic boundary conditions are indeed modified into

$$\psi \left( {x + L} \right) = \psi \left( x \right){e^{i\varphi }}$$

((1))

where φ = 2πΦ/Φ ₀ . Φ is the magnetic field through the loop. Φ ₀ is the flux quantum h/e. The persistent current is related to the flux derivative of the eigenenergies by

$$I\left( \phi \right) = \sum\limits_n {\partial {E_n}/\partial \phi } $$

((2))

where n labels the N filled energy levels. The current has thus the period Φ ₀. There has been recently a growing experimental interest in the detection of this persistent current. On the theoretical side, it is of course of essential interest to estimate the amplitude of this current in a model as close as possible to the experimental situation.