Hall viscosity and hydrodynamic inverse Nernst effect in graphene

Motivated by Hall viscosity measurements in graphene sheets, we study hydrodynamic transport of electrons in a channel of finite width in external electric and magnetic fields. We consider electric charge densities varying from close to the Dirac point up to the Fermi-liquid regime. We find two competing contributions to the hydrodynamic Hall and inverse Nernst signals that originate from the Hall viscous and Lorentz forces. This competition leads to a nonlinear dependence of the full signals on the magnetic field and even a cancellation at different critical field values for both signals. In particular, the hydrodynamic inverse Nernst signal in the Fermi-liquid regime is dominated by the Hall viscous contribution. We further show that a finite channel width leads to a suppression of the Lorenz ratio, while the magnetic field enhances this ratio. All of these effects are predicted in parameter regimes accessible in experiments.

Figure 1

Velocity profile of an electron fluid in a graphene channel of width $W$, with applied electric field $\mathit{E}$ and magnetic field $\mathit{B}$. $l_s$ is the slip length and $l_G^{\mathrm{eff}}$ is the effective Gurzhi length. ${\mathit{f}}_B$ and ${\mathit{f}}_{{\eta }_H}$ are the Lorentz and Hall force densities, respectively.

Figure 2

Hall voltage $\mathrm{\Delta }V$ and temperature gradient $\mathrm{\Delta }T$ as functions of the magnetic field $B_z$. The other parameters are $T=120\mathrm{K}, E_x=-1000\mathrm{V}/\mathrm{m}, {\nabla }_{x}T=0, W=2\mu \mathrm{m}$. The dashed (dotted) lines show $\mathrm{\Delta }{V}_B$ ($\mathrm{\Delta }{V}_{{\eta }_H}$) for $\mu =2k_{B}T$ and $\mathrm{\Delta }{T}_B$ ($\mathrm{\Delta }{T}_{{\eta }_H}$) for $\mu =k_{B}T$.

Figure 3

The Lorenz ratio $L$ in units of $L_{\mathrm{WF}}$ as a function of the width $W$ and the magnetic field $B_z$ at $\mu =0.5k_{B}T$ and $T=120\text{K}$. When $W\to \infty$, it approaches $L_{\infty }/L_{\mathrm{WF}}\approx 4.9$.