Beyond Ohm's law: Bernoulli effect and streaming in electron hydrodynamics

Recent observations of nonlocal transport in ultraclean two-dimensional materials raised the tantalizing possibility of accessing hydrodynamic correlated transport of a many-electron state. However, it has been pointed out that nonlocal transport can also arise from impurity scattering rather than interaction. At the crux of the ambiguity is the focus on linear effects, i.e., Ohm's law, which cannot easily differentiate among different modes of transport. Here we propose experiments that can reveal rich hydrodynamic features in the system by tapping into the nonlinearity of the Navier-Stokes equation. Three experiments we propose will each manifest a unique phenomenon that is well known in classical fluids: the Bernoulli effect, Eckart streaming, and Rayleigh streaming. Analysis of known parameters confirms that the proposed experiments are feasible and the hydrodynamic signatures are within reach of graphene-based devices. Experimental realization of any one of the three phenomena will provide a stepping stone to formulating and exploring the notions of nonlinear electron fluid dynamics with an eye to celebrated examples from classical nonlaminar flows, e.g., pattern formation and turbulence.

Figure 1

Proposed experimental setups and sketches of their observed effects. (a) The Venturi geometry, comprised of a circular wedge of the hydrodynamic material in yellow. A nonlinear $I\text{−}V$ characteristic with $I\sim \sqrt{V_0}$ behavior is expected, marked in blue. The gray dashed line represents an unstable solution branch, while the gray region represents a possible instability toward turbulent and/or intermittent flow. (b) Eckart streaming. A voltage oscillation of zero mean is driven on one side of a back-gated device, leading to a rectified dc current $\overline{I}$. For large $l$, the dc current scales as $l^{-1}$. For small $l$, oscillations due to interference with the reflected wave become visible. (c) Rayleigh streaming. In a similar back-gated geometry of (b), a standing wave of current oscillations of amplitude $u_0$ and of period $\lambda$ along $x$ is imposed, leading to an oscillating magnetic field pattern of period $\lambda /2$ along $x$. These magnetic fields arise due to the formation of vortical current cells of size $\lambda /4$ along $x$ and $h/2$ along $y$, shown in the lower panel.

Figure 2

A top view of the Venturi geometry, with inner radius $r_0$, outer radius $r_1$, and total wedge angle ${\theta }_0$.

Figure 3

Main: a parametric plot of the voltage-symmetrized current $I_{\text{sym}}\left(V_0\right)\equiv \frac{1}{2}[I\left(V_0\right)+I(-V_0)]$ against total current $I\left(V_0\right)$. Inset: the $I\text{−}V$ characteristic. The solid lines are obtained analytically from Eq. (9) in the $\nu \to 0$ limit, and the points are obtained numerically with finite $\nu$. Fixed-voltage boundary conditions are taken. The inner and outer radii are 5 and $10\mu \mathrm{m}$, respectively, with wedge angle ${\theta }_0=\pi /2$, with graphene parameters $\nu =0.1{\mathrm{m}}^2/\mathrm{s}$ and $\gamma =650$ GHz. Since $r_d\sim 0.4\mu \mathrm{m}$ and lengths are $\sim 10\mu \mathrm{m}$, viscous corrections to the analytic $\nu \to 0$ solution should be $\sim 5\%$.

Figure 4

Main: A plot of $\overline{I^{\left(2\right)}}$ at fixed input current amplitude $I_0$ for device length $l=30\mu \mathrm{m}$ and graphene-hBN parameters stated in Sec. 2b, in units of $A_0=\frac{I_0^2}{{\rho }_e^{\left(0\right)}h}\frac{1}{2l\gamma }(1-\frac{2r_d}{h}tanh\frac{h}{2r_d})$. We remark that this is also a scaled plot of the Reynolds number ${Re}_{\gamma }$. Inset: a blowup of the yellow highlighted portion. At high frequencies, ${Re}_{\gamma }$ saturates to a constant $A_0=\frac{I_0}{{\rho }_e^{\left(0\right)}h}\frac{1}{2l\gamma }$, while at sufficiently low frequencies the interference oscillations become more visible. The gray box demarcates the low-frequency region $\omega \ll \gamma$, where perturbation theory in $V_0$ breaks down for a fixed $I_0$.

Figure 5

A plot of the bulk vorticity distribution $\overline{{\mathrm{\Omega }}_{\text{bulk}}^{\left(2\right)}}\equiv \mathbf{\nabla }\times \overline{{\mathbf{J}}_{\text{bulk}}^{\left(2\right)}}$ induced by Rayleigh streaming for $h=5\mu \mathrm{m}$ and $\omega =2$ THz with graphene-hBN parameters as in Sec. 2b. The local bulk vorticity corresponds to a Coulomb-like point source of magnetic field due to Ampere's law.