Mesoscopic persistent current in small rings

Small normal-metal rings threaded by a constant magnetic flux have been shown to carry a mesoscopic persistent current at low temperatures. The current is a few-electron effect and its sign and amplitude depend on the microscopic configuration of disorder. Assuming a Gaussian current distribution, we characterize the effect by three quantities, the rms or typical total current ${\mathit{I}}^{\mathrm{typ}}$=〈${\mathit{I}}^2$ ${\mathrm{〉}}_{\mathit{D}}^{1/2}$, the average current ${\mathit{I}}^{\mathrm{av}}$=〈I${\mathrm{〉}}_{\mathit{D}}$, and the typical single-level current ${\mathit{i}}^{\mathrm{typ}}$=〈${\mathit{i}}^2$ ${\mathrm{〉}}_{\mathit{D}}^{1/2}$. Specifically, we review and extend the analytical calculations for the typical total and single-level currents focusing on the case of noninteracting electrons in disordered rings in the regime of diffusive transport. We calculate and discuss those current-current correlation functions that describe the dependences of the persistent current on filling, flux, and disorder configuration. Only the single-electron contribution discussed in this paper is known to contribute to the first, ${\mathrm{\phi }}_0$-periodic harmonic of the total current in a single ring. The second harmonic also contains an interaction-induced contribution proposed by Ambegaokar and Eckern that survives the disorder average. The Thouless correlation energy ${\mathit{E}}_{\mathit{c}}$ is the characteristic energy scale for the amplitude of the total current and its dependences on filling, temperature, and inelastic scattering. The persistent current is sensitive to changing the position of a single impurity. We compare our results with the recent single-ring experiment by Chandrasekhar et al.