Quantum-critical relativistic magnetotransport in graphene

Abstract

We study the thermal and electric transport of a fluid of interacting Dirac fermions using a Boltzmann approach. We include Coulomb interactions, a dilute density of charged impurities and the presence of a magnetic field to describe both the static and the low frequency response as a function of temperature T and chemical potential mu. In the quantum-critical regime mu << T we find pronounced deviations from Fermi liquid behavior, such as a collective cyclotron resonance with an intrinsic, collision-broadened width, and significant enhancements of the Mott and Wiedemann-Franz ratio. Some of these results have been anticipated by a relativistic hydrodynamic theory, whose precise range of validity and failure at large fields and frequencies we determine. The Boltzmann approach allows us to go beyond the hydrodynamic regime, and to quantitatively describe the deviations from magnetohydrodynamics, the crossover to disorder dominated Fermi liquid behavior at large doping and low temperatures, as well as the crossover to the ballistic regime at high fields. Finally, we obtain the full frequency and doping dependence of the single universal conductivity sigma_Q which parametrizes the hydrodynamic response.

Figures (11)

• Figure 1: Dependence on the screening parameter $\eta$ of the conductivity of undoped graphene, $\alpha^2\sigma(\mu=0;\eta)$ in units of $e^2/\hbar$. At very small $\eta$, $\alpha^2\sigma(0;\eta)$ approaches its limiting value FSMS$0.121$ as $\alpha^2[\sigma(0;\eta)-\sigma(0;0)]\sim [\log(1/\eta)]^{-1}$, the solid line being a fit to $a+b[\log(1/\eta)]^{-1}$. The lower curve is a linear fit to the data obtained from the two mode approximation, which becomes asymptotically exact as $\eta\to 0$.
• Figure 2: The normalized transport coefficient $\sigma_Q(\mu,\omega=0)$ as a function of $\mu/T$.
• Figure 3: The d.c. conductivity, $\alpha^2\sigma_{xx}$ in units of $e^2/\hbar$, as a function of the ratio of doped carrier density to impurity density. The disorder strength and temperature have been chosen such that $\Delta/\alpha^2=0.25$. The plot shows the crossover from the quantum critical regime, where the conductivity is essentially limited by inelastic scattering between electrons, to the regime dominated by elastic scattering from Coulomb impurities at higher doping. In the latter regime the conductivity increases linearly with doped carrier density, see Eq. (\ref{['sigmaproptorho']}). The red and yellow curves correspond to the two terms in Eq. (\ref{['eq:fullsigma']}), respectively: the contribution from modes relaxing due to inelastic scattering, and the contribution from the momentum mode, which is only disorder limited and dominates at large density.
• Figure 4: The ratio $R\equiv -\frac{3e k_B^2 T}{\pi^2}\frac{\alpha_{xx}(\mu)}{d\sigma/d\mu(\mu)}$ as a function of $\mu/T$. In the Fermi liquid regime ($\mu\gg T$) $R$ tends to $1$ as predicted by Mott's law.
• Figure 5: The thermal conductivity at particle-hole symmetry scales as $\kappa_{xx}\sim B^{-2}$ both at small and large $B$. The plot shows the coefficient $C$ in the relation $\kappa_{xx}(\mu=0;B)= C\, \alpha^2/b^2 k_B^2T$ as a function of $b/\alpha^2 \sim \omega^{\rm typ}_c \tau_{\rm ee}$. It interpolates between the magnetohydrodynamic regime (\ref{['CMHD']}) and the large field limit of Eq. (\ref{['ClargeB']}).