Influence of the electron-phonon interaction on the Aharonov-Bohm effect

The effect of the elastic and inelastic scattering of electrons by acoustical phonons on the dc electrical conductivity as a function of the magnetic flux, which penetrates the sample, is considered. Within linear response theory and the first Born approximation it is shown that energy conservation leads, for elastic collisions, to oscillations of the relaxation time with period ${\phi }_0$/2=hc/2e if the mean free path is small compared to the size of the sample. For inelastic collisions the period of the oscillations remains the same but the amplitude is reduced by a factor which depends on the effective mass, the radius of the sample, and the speed of sound. These oscillations persist in the pure-metal limit. For very low temperatures the period obtained is ${\phi }_0$ despite the presence of the harmonic ${\phi }_0$/2. As the temperature increases, the oscillations ${\phi }_0$ die out and the harmonic ${\phi }_0$/2 dominates. The results are independent of the geometry for one- and two-dimensional systems (rings or cylinders).