Incoherence, metal-to-insulator transition, and magnetic quantum oscillations of interlayer resistance in an organic conductor

Abstract

An analytic theory is developed for the conductivity across the layers ${\ensuremath{\sigma}}_{zz}$ in a layered conductor in perpendicular magnetic field under the conditions of interlayer incoherence. The latter assumes a small hopping integral between the layers $t⪡\ensuremath{\hbar}∕\ensuremath{\tau}$ and the presence of localized states in the tails of broadened Landau levels (LLs) ($\ensuremath{\tau}$ is the electron scattering time within the layers). In the incoherent regime, ${\ensuremath{\sigma}}_{zz}$ strongly depends on the in-plane conductivity mechanisms because electrons spend most of their time within the weakly coupled layers. At high fields $\ensuremath{\Omega}\ensuremath{\tau}⪢1$, an integer quantum Hall effect (IQHE) within the layers develops which changes dramatically magnetic quantum oscillations in the ${\ensuremath{\sigma}}_{zz}$ compared to the standard Lifshitz-Kosevich theory ($\ensuremath{\Omega}$ is the cyclotron frequency). At low fields, ${\ensuremath{\sigma}}_{zz}$ displays Shubnikov--de Haas (SdH) oscillations which in the limit $\ensuremath{\Omega}\ensuremath{\tau}⪢1$ transforms into sharp peaks. The peaks reach their maximum values ${\ensuremath{\sigma}}_{zz}\ensuremath{\propto}\frac{\ensuremath{\hbar}\ensuremath{\Omega}}{T}$ when LLs cross the chemical potential $\ensuremath{\mu}$. When $\ensuremath{\mu}$ falls into the tails between the LLs, the ${\ensuremath{\sigma}}_{zz}$ displays first a thermal activation behavior ${\ensuremath{\sigma}}_{zz}\ensuremath{\propto}\mathrm{exp}[\ensuremath{-}(\ensuremath{\hbar}\ensuremath{\Omega}\ensuremath{-}\ensuremath{\mu})∕T]$ and, then at lower temperatures $T$, crosses over into a variable-range-hopping regime with ${\ensuremath{\sigma}}_{zz}\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}\sqrt{{T}_{0}∕T})$, where ${T}_{0}\ensuremath{\propto}{\ensuremath{\mid}B\ensuremath{-}{B}_{0}\ensuremath{\mid}}^{\ensuremath{\gamma}}$. Above ${B}_{0}$, the in-plane electrons are in the quantum-Hall-insulator regime and the background interlayer magnetoresistance ${R}_{b}$ has an insulatorlike temperature dependence. Below ${B}_{0}$, the in-plane electrons are in the conventional SdH oscillation regime and ${R}_{b}$ has a metal-like temperature dependence. On the insulating side, ${R}_{b}$ displays a universal dependence on the scaling variable $(B\ensuremath{-}{B}_{0})∕{T}^{\ensuremath{\kappa}}$. Scaling is destroyed in tilted magnetic fields at angles corresponding to the spin zeros. All the above features in the ${\ensuremath{\sigma}}_{zz}$ have been observed in the ${\ensuremath{\beta}}^{\ensuremath{''}}\text{\ensuremath{-}}{(\mathrm{BEDT}\text{\ensuremath{-}}\mathrm{TTF})}_{2}\mathrm{S}{\mathrm{F}}_{5}\mathrm{C}{\mathrm{H}}_{2}\mathrm{C}{\mathrm{F}}_{2}\mathrm{S}{\mathrm{O}}_{3}$, in which the critical exponent is equal to $\ensuremath{\kappa}=1∕\ensuremath{\gamma}=0.65$. The IQHE regime at high fields in this quasi-two-dimensional organic conductor is favored by the fixed value of the chemical potential. It is shown that at low temperatures $(T⪡\ensuremath{\hbar}∕\ensuremath{\tau})$, oscillations of the conductivity and magnetization are related by the condition ${\ensuremath{\sigma}}_{zz}\ensuremath{\propto}{B}^{2}\ensuremath{\partial}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{M}∕\ensuremath{\partial}B$, in agreement with observations in ${\ensuremath{\beta}}^{\ensuremath{''}}\text{\ensuremath{-}}{(\mathrm{BEDT}\text{\ensuremath{-}}\mathrm{TTF})}_{2}\mathrm{S}{\mathrm{F}}_{5}\mathrm{C}{\mathrm{H}}_{2}\mathrm{C}{\mathrm{F}}_{2}\mathrm{S}{\mathrm{O}}_{3}$. The analysis shows that the above features in the conductivity cannot be explained within the model with a narrow-band dispersive electron transport across the layers because the model is incompatible with the incoherence condition $t⪡\ensuremath{\hbar}∕\ensuremath{\tau}$. Moreover, in the self-consistent Born approximation, this model yields a nonphysical negative conductivity ${\ensuremath{\sigma}}_{zz}<0$.