Magnetophonon resonances in quantum wires

A theory of magnetophonon resonances in quantum wires is presented. The magnetoconductivity ${\mathrm{\sigma }}_{\mathit{x}\mathit{x}}$ calculated using the Kubo formula is found to consist of two types of contribution; one is related to the current carried by one electron hopping motion between the localized cyclotron orbits through the electron-phonon interaction, and the other is caused by the current carried by electron motion affected by the confinement potential. The former, ${\mathrm{\sigma }}_{\mathit{e}\mathrm{-}\mathrm{ph}}$, is directly proportional to the coupling constant α of the electron–optical-phonon interaction, whereas the latter, ${\mathrm{\sigma }}_{\mathrm{po}}$ is found to be inversely proportional to α. At resonances, ${\mathrm{\sigma }}_{\mathrm{e}\mathrm{-}\mathrm{ph}}$ exhibits maxima and ${\mathrm{\sigma }}_{\mathrm{po}}$ minima. For weak confinement potential, i.e., for wide quantum wires, ${\mathrm{\sigma }}_{\mathit{e}\mathrm{-}\mathrm{ph}}$ is dominant and total magnetoconductivity ${\mathrm{\sigma }}_{\mathit{x}\mathit{x}}$ shows maxima at resonances as pointed out by Vasilopoulos et al. [Phys. Rev. B 40, 1810 (1989)]. On the other hand, in the case of strong confinement potential, or in narrow quantum wires, ${\mathrm{\sigma }}_{\mathit{x}\mathit{x}}$ is dominated by ${\mathrm{\sigma }}_{\mathrm{po}}$, resulting in minima in an extremely strong confinement potential.