Slow imbalance relaxation and thermoelectric transport in graphene

Abstract

We compute the electronic component $(\ensuremath{\kappa})$ of the thermal conductivity and the thermoelectric power $(\ensuremath{\alpha})$ of monolayer graphene within the hydrodynamic regime, taking into account the slow rate of carrier population imbalance relaxation. Interband electron-hole generation and recombination processes are inefficient due to the nondecaying nature of the relativistic energy spectrum. As a result, a population imbalance of the conduction and valence bands [i.e., a nonequilibrium state with ${\ensuremath{\mu}}_{e}+{\ensuremath{\mu}}_{h}\ensuremath{\ne}0$, where ${\ensuremath{\mu}}_{e}$ $({\ensuremath{\mu}}_{h})$ denotes the electron (hole) chemical potential] is generically induced upon the application of a thermal gradient. We show that the thermoelectric response of a graphene monolayer depends upon the ratio of the sample length to an intrinsic length scale ${l}_{Q}$ set by the imbalance relaxation rate. At the same time, we incorporate the crucial influence of the metallic contacts required for the thermopower measurement (under open circuit boundary conditions) since carrier exchange with the contacts also relaxes the imbalance. These effects are especially pronounced for clean graphene, where the thermoelectric transport is limited exclusively by intercarrier collisions. For specimens shorter than ${l}_{Q}$, the population imbalance extends throughout the sample; $\ensuremath{\kappa}$ and $\ensuremath{\alpha}$ asymptote toward their zero imbalance relaxation limits. In the opposite limit of a graphene slab longer than ${l}_{Q}$, at nonzero doping $\ensuremath{\kappa}$ and $\ensuremath{\alpha}$ approach intrinsic values characteristic of the infinite imbalance relaxation limit. Samples of intermediate (long) length in the doped (undoped) case are predicted to exhibit an inhomogeneous temperature profile, while $\ensuremath{\kappa}$ and $\ensuremath{\alpha}$ grow linearly with the system size. In all cases except for the shortest devices, we develop a picture of bulk electron and hole number currents that flow between thermally conductive leads, where steady-state recombination and generation processes relax the accumulating imbalance. Our analysis incorporates, in addition, the effects of (weak) quenched disorder.