Impurity scattering and Mott’s formula in graphene

We present calculations of the thermal and electric linear response in graphene, including disorder in the self-consistent $t$-matrix approximation. For strong impurity scattering, near the unitary limit, the formation of a band of impurity states near the Fermi level leads to that Mott’s relation holds at low temperature. For higher temperatures, there are strong deviations due to the linear density of states. The low-temperature thermopower is proportional to the inverse of the impurity potential and the inverse of the impurity density. Information about impurity scattering in graphene can be extracted from the thermopower, either measured directly or extracted via Mott’s relation from the electron-density dependence of the electric conductivity.

Figure 1

(Color online) (a) The impurity self-energy and (b) the density of states for large impurity potential $V_{\mathrm{imp}}$. An impurity band of width $\gamma \sim {ϵ}_c\sqrt{n_{\mathrm{imp}}}$ is formed near the Fermi level at an energy ${ϵ}_r\sim {ϵ}_c^2∕V_{\mathrm{imp}}$. For $ϵ⪢\gamma$, the spectrum is linear.

Figure 2

(Color online) Temperature dependence of (a) the charge conductance, (b) the heat conductance, and (c) the thermopower for three values of the impurity density $n_{\mathrm{imp}}$. The solid and broken lines are for a large impurity potential $V_{\mathrm{imp}}=20{ϵ}_c$, while the dotted lines are for the strict unitary limit $V_{\mathrm{imp}}\to \infty$. In (a) and (b), we have normalized the conductivities by the low-temperature asymptotics ${\sigma }_0=4e^2∕\left(\pi h\right)$ and ${\kappa }_0\left(T\right)=4\pi k_B^{2}T∕\left(3h\right)$. In (d), we show the details at low temperatures. The dotted lines are the thermopower computed through the Mott relation. In (e), we show the Lorenz ratio $L=\kappa ∕\left(\sigma T\right)$ in units of the value $L_0=({\pi }^2∕3)(k_B^2∕e^2)$ appearing in Wiedemann-Franz law. We used ${ϵ}_c=1\mathrm{eV}\to 11605\mathrm{K}$ to convert the temperature scale to Kelvin.

Figure 3

(Color online) The charge conductance (a) and the thermopower (b) as function of chemical potential at $T=50\mathrm{K}$. In (c), we show how thermal smearing leads to an approximate Mott’s relation based on the charge conductance as function of chemical potential. For $T⪡\gamma$, the Mott relation is exact to leading order (dash-dotted line).