Frequency dependence of the Nyquist noise contribution to the dephasing rate in disordered conductors

Abstract

We calculate the Nyquist noise contribution to the dephasing rate $1/{\ensuremath{\tau}}_{\mathrm{NN}}(\ensuremath{\omega},T)$ of disordered conductors in d dimensions in the regime where the frequency \ensuremath{\omega} is larger than the temperature T. For systems with a continuous spectrum we find at zero temperature $1/{\ensuremath{\tau}}_{\mathrm{NN}}(\ensuremath{\omega},0)\ensuremath{\propto}{\ensuremath{\nu}}_{d}^{\ensuremath{-}1}(\ensuremath{\omega}{/D)}^{d/2},$ which agrees qualitatively with the inelastic quasiparticle scattering rate. Here ${\ensuremath{\nu}}_{d}$ is d-dimensional density of states, and D is the diffusion coefficient. Because at zero frequency and finite temperatures $1/{\ensuremath{\tau}}_{\mathrm{NN}}(0,T)\ensuremath{\propto}[T/({\ensuremath{\nu}}_{d}{D}^{d/2}){]}^{2/(4\ensuremath{-}d)}$ for $d<2,$ the frequency dependence of $1/{\ensuremath{\tau}}_{\mathrm{NN}}(\ensuremath{\omega},0)$ in reduced dimensions cannot be obtained by simply replacing $\stackrel{\ensuremath{\rightarrow}}{T}\ensuremath{\omega}$ in the corresponding finite-temperature expression for $1/{\ensuremath{\tau}}_{\mathrm{NN}}(0,T).$ We also discuss the dephasing rate in mesoscopic systems with length L and show that for $\ensuremath{\omega}\ensuremath{\gtrsim}{D/L}^{2}$ the spectrum is effectively continuous as far as transport is concerned. We propose weak localization measurements of the ac conductivity in the GHz range to clarify the origin of the experimentally observed zero-temperature saturation of the dephasing rate.