Collision-dominated conductance in clean two-dimensional metals

We study the temperature-dependent corrections to the conductance due to electron-electron ($e-e$) interactions in clean two-dimensional conductors, such as lightly doped graphene or other Dirac matter. We use semiclassical Boltzmann kinetic theory to solve the problem of collision-dominated transport between reflection-free contacts. Time-reversal symmetry and the kinematic constraints of scattering in two dimensions (2D) ensure that inversion-odd and inversion-even distortions of the quasiparticle distribution relax with parametrically different rates at low temperature. This entails the surprising result that at lowest temperatures the conductance of very long samples tends to the noninteracting, ballistic conductance, despite the relaxation of the quasiparticle distribution to a drifting equilibrium. The relative correction to the conductance depends on the ratio of relaxation rates of even and odd modes and scales as $\delta G/G_{\mathrm{ballistic}}\sim \left(T/{\varepsilon }_F\right)\sqrt{log\left(\frac{{\varepsilon }_F}{T}\right)}$, in stark contrast to the behavior in other dimensionalities. This holds generally in 2D systems with simply connected and convex but otherwise arbitrary Fermi surfaces, as long as $e-e$ scattering processes are dominant and umklapp scattering is negligible. These results are especially relevant to the bulk of wide and long suspended high-mobility graphene sheets.

Figure 1

Free-hanging graphene sheet in the $x-y$ plane. The leads are gated so as to have a finite Fermi surface, with quasiparticles described by Fermi-liquid theory. A potential difference is applied along the $x$ axis via weakly coupled leads at $x=\pm \frac{L}{2}$. The bias results in a longitudinal current density $J_x$.

Figure 2

The upper row illustrates the distribution of quasiparticles flowing in from two leads. Close to the leads, the incoming particles retain the distribution of the leads, and the full distribution resembles the one observed in noninteracting ballistic transport, as shown in the bottom row for comparison. However, within a distance ${\ell }_{\mathrm{inel}}$ from the leads, inelastic $e-e$ scattering relaxes the distribution to a drifting equilibrium, i.e., a displaced Fermi sphere. Despite the substantial difference in the quasiparticle distribution, the conductances in the two cases only differ by a term scaling as $T/{\varepsilon }_F{[log({\varepsilon }_F/T)]}^{1/2}$ at low temperature.