Violation of the Wiedemann-Franz Law in Hydrodynamic Electron Liquids

The Wiedemann-Franz law, connecting the electronic thermal conductivity to the electrical conductivity of a disordered metal, is generally found to be well satisfied even when electron-electron ($e\text{−}e$) interactions are strong. In ultraclean conductors in the hydrodynamic regime, however, large deviations from the standard form of the law are expected, due to the fact that $e\text{−}e$ interactions affect the two conductivities in radically different ways. Thus, the standard Wiedemann-Franz ratio between the thermal and the electric conductivity is reduced by a factor $1+\tau /{\tau }_{\text{th}}^{ee}$, where $1/\tau$ is the momentum relaxation rate and ${\tau }_{\text{th}}^{ee}$ is the relaxation time of the thermal current due to $e\text{−}e$ collisions. Here we study the density and temperature dependence of $1/{\tau }_{\text{th}}^{ee}$ of two-dimensional electron liquids. We show that at low temperature $1/{\tau }_{\text{th}}^{ee}$ is $8/5$ of the quasiparticle decay rate; remarkably, the same result is found in doped graphene and in conventional electron liquids in parabolic bands.

Figure 1

Panel (a) represents the thermal-current linear response function. The solid dot is the bare vertex ${\mathbf{\Lambda }}^{(0)}$, $\mathbf{\Lambda }$ is the dressed vertex, and double solid lines stand for Green’s functions dressed by the self-energy of panel (b). The choice of the $GW$ self-energy and the requirement of fulfilling the Ward identities uniquely determine the irreducible interaction $I$ and the self-consistent Bethe-Salpeter equation [panel (c)] [10].

Figure 2

Panel (a) shows the renormalized WF ratio $L$, as defined in Eq. (3), in units of $L_0$ [Eq. (1)], plotted as a function of the density $n$ (in units of $10^{12}{\mathrm{cm}}^{-2}$). We fixed the temperature $T=300\mathrm{K}$, the ratio ${\mathcal{D}}_{\text{th}}/{\mathcal{D}}_c=1$, and the momentum relaxation time $\tau$ to reproduce the electron mobility ${\mu }_e=10000{\mathrm{cm}}^2/(\mathrm{V}\mathrm{s})$. We show three curves for different values of the dimensionless coupling constant of $e\text{−}e$ interactions, i.e., ${\alpha }_{ee}=0.5$ (solid line), ${\alpha }_{ee}=0.9$ (short-dashed line), and ${\alpha }_{ee}=2.2$ (long-dashed line). Panel (b) is the same as in panel (a) but shown as a function of temperature (in units of K) and for a fixed excess carrier density $n=10^{12}{\mathrm{cm}}^{-2}$ (corresponding to a Fermi temperature $T_F\sim 1300\mathrm{K}$). The weak ${\alpha }_{ee}$ dependence of the curves is due to subleading corrections to ${\tau }_{\mathrm{qp}}^{ee}$ beyond the leading order of Eq. (8).

Figure 3

The thermal conductivity of graphene as a function of the temperature for three different values of the carrier density ($n=1,4$, and $7\times 10^{11}{\mathrm{cm}}^{-2}$). Here ${\alpha }_{ee}=0.5$, while $\tau$ is chosen to reproduce an electron mobility of ${\mu }_e=250000{\mathrm{cm}}^2/(\mathrm{V}\mathrm{s})$. A broad maximum (tracked by the dotted line) occurs when ${\tau }_{\text{th}}^{ee}$, with increasing temperature, becomes smaller than $\tau$. This maximum marks the onset of the hydrodynamic regime; its observation will provide unequivocal confirmation of our theory. A further rise of ${\sigma }_{\text{th}}$ at higher temperature due to electron-phonon interactions [31] is not included here.