Energy averaging and the flux-periodic phenomena in small normal-metal rings

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Abstract
The low-frequency ac conductivity, σ, is evaluated numerically with use of the Kubo formula with inelastic lifetimes for a small one-dimensional (1D) closed metallic ring with random potentials. The effects of thermal excitations at finite temperatures and of increasing inelastic scattering are considered. It is found that energy averaging over a range larger than the typical level spacing (in 1D) markedly reduces the fundamental (hc/e) and odd harmonic periodicities of σ as a function of the Aharonov-Bohm flux, φ, through the ring. Thus, energy averaging makes the lowest even harmonic (hc/2e) appear to be the fundamental period.