Wiedemann-Franz Relation and Thermal-transistor Effect in Suspended Graphene

We extract experimentally the electronic thermal conductivity, $K_{e}$, in suspended graphene which we dope using a back-gate electrode. We make use of two-point dc electron transport at low bias voltages and intermediate temperatures (50 - 160 K), where the electron and lattice temperatures are decoupled. The thermal conductivity is proportional to the charge conductivity times the temperature, confirming that the Wiedemann-Franz relation is obeyed in suspended graphene. We extract an estimate of the Lorenz coefficient as 1.1 to 1.7 $\times 10^{-8}$ W $\Omega$K$^{-2}$. $K_e$ shows a transistor effect and can be tuned with the back-gate by more than a factor of 2 as the charge carrier density ranges from $\approx$ 0.5 to 1.8 $\times 10^{11}$cm$^{-2}$.