Frequency and temperature dependence of the optical conductivity of granular metals: A path-integral approach

We study the finite-temperature optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega},T)$ of a granular metal using a simple model consisting of a array of spherical metallic grains. It is necessary to include quantum tunneling and Coulomb blockade effects to obtain the correct temperature dependence of $\ensuremath{\sigma}$, and to consider polarization oscillations to obtain the correct frequency dependence. We have therefore generalized the Ambegaokar-Eckern-Sch\"on (AES) model for granular metals to obtain an effective field theory incorporating the polarization fluctuations of the individual metallic grains. In the absence of intergrain tunneling, the classical optical conductivity is determined by polarization oscillations of the electrons in the grains, $\ensuremath{\sigma}(\ensuremath{\omega})=\ensuremath{-}(in{e}^{2}f\ensuremath{\omega}∕m)∕({\ensuremath{\omega}}^{2}\ensuremath{-}{\ensuremath{\omega}}_{r}^{2}\ensuremath{-}i\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}∕{\ensuremath{\tau}}_{\mathrm{grain}}),$ where ${\ensuremath{\omega}}_{r}=e\sqrt{(4\ensuremath{\pi}∕3m)n}$ is the resonance frequency, ${\ensuremath{\tau}}_{\mathrm{grain}}^{\ensuremath{-}1}$ is the relaxation rate for electron motion within the grain, and $f$ is the volume fraction occupied by the grains. At finite intergrain tunneling, we find that $\ensuremath{\sigma}(\ensuremath{\omega})=\ensuremath{-}(in{e}^{2}\ensuremath{\omega}f∕m)∕({\ensuremath{\omega}}^{2}\ensuremath{-}{\ensuremath{\omega}}_{r}^{2}\ensuremath{-}i\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}∕{\ensuremath{\tau}}_{\mathrm{rel}})+{\ensuremath{\sigma}}_{\mathrm{AES}}(\ensuremath{\omega},T),$ where ${\ensuremath{\tau}}_{\mathrm{rel}}^{\ensuremath{-}1}$ is the total relaxation rate that includes the intragrain relaxation rate ${\ensuremath{\tau}}_{\mathrm{grain}}^{\ensuremath{-}1}$ as well as intergrain tunneling effects, and ${\ensuremath{\sigma}}_{\mathrm{AES}}(\ensuremath{\omega},T)$ is the conductivity of the granular system from the AES model obtained by ignoring polarization modes. We calculate the temperature and frequency dependence of the intergrain relaxation time, $\ensuremath{\Gamma}(\ensuremath{\omega},T)={\ensuremath{\tau}}_{\mathrm{rel}}^{\ensuremath{-}1}\ensuremath{-}{\ensuremath{\tau}}_{\mathrm{grain}}^{\ensuremath{-}1},$ and find it is different from ${\ensuremath{\sigma}}_{\mathrm{AES}}(\ensuremath{\omega},T)$. For small values of dimensionless intergrain tunneling conductance, $g\ensuremath{\ll}1,$ the dc conductivity obeys an Arrhenius law, ${\ensuremath{\sigma}}_{\mathrm{AES}}(0,T)\ensuremath{\sim}g{e}^{\ensuremath{-}{E}_{c}∕T},$ whereas the polarization relaxation may even decrease algebraically, $\ensuremath{\Gamma}(\ensuremath{\omega},T)\ensuremath{\sim}(g∕{E}_{c}^{2})[{T}^{2}+(\ensuremath{\omega}∕2\ensuremath{\pi}{)}^{2}],$ when $\ensuremath{\omega},T\ensuremath{\ll}{E}_{c}.$