Coulomb drag of viscous electron fluids: Drag viscosity and negative drag conductivity

We show that Coulomb drag in bilayer systems in the regime of electron hydrodynamics leads to additional viscosity terms in the hydrodynamic equations, the drag, and drag-Hall viscosities, besides the well-known kinematic and Hall viscosities. These additional viscosity terms arise from a change of the stress tensor due to the interlayer Coulomb interactions. All four viscosity terms are tunable by varying the applied magnetic field and the electron densities in the two layers. At certain ratios between the electron densities in the two layers, the drag viscosity dramatically changes the longitudinal transport resulting in a negative drag conductivity.

Figure 1

Coulomb drag setup: An electric field $E_x^{\mathrm{a}}\widehat{\mathbf{x}}$ is applied to the active layer causing a Poiseuille current profile $J^{\mathrm{a}}\left(y\right)\widehat{\mathbf{x}}$. This current induces electron motion in the passive layer controlled by the drag coefficient ${\gamma }_d^{\mathrm{p}}$ and the drag viscosity ${\nu }_d^{\mathrm{p}}$. When the electron density of the passive layer is much higher than that of the active layer, ${\gamma }_d^{\mathrm{p}}$ becomes very small and $J^{\mathrm{p}}$ changes signs. The magnetic field $B_z\widehat{\mathbf{z}}$ and horizontal flow $J^{\lambda }\left(y\right)\widehat{\mathbf{x}}$ cause a charge buildup and a transversal electric field $E_y^{\lambda }\widehat{\mathbf{y}}$ that can be used to probe the drag-Hall viscosity ${\nu }_{dH}^{\lambda }$.

Figure 2

The viscosities in both layers. (a), (e) The kinematic viscosity ${\nu }^{\lambda }$; (b), (f) the drag viscosity ${\nu }_d^{\lambda }$; (c), (g) the Hall viscosity ${\nu }_H^{\lambda }$; and (d), (h) the drag-Hall viscosity ${\nu }_{dH}^{\lambda }$ in the active and passive layers, respectively, for different magnetic fields, drag coefficients ${\mathrm{\Gamma }}_d={\gamma }_d/{\gamma }_{ee}$, and density ratios $r={\overline{n}}_{\mathrm{a}}/{\overline{n}}_{\mathrm{p}}$. In (b), (c), (e), and (g), we have multiplied the red lines (${\mathrm{\Gamma }}_d=0.5$ and $r=0.1$) by the factors written in the plot. Here ${\nu }_0={\left(v_F^{\mathrm{a}}\right)}^2/\left(4{\gamma }_{ee}^{\mathrm{a}}\right)$.

Figure 3

(a) Conductivity ${\sigma }^{\mathrm{a}}$ and (b) drag conductivity ${\sigma }^{\mathrm{p}}$ as a function of $r={\overline{n}}_{\mathrm{a}}/{\overline{n}}_{\mathrm{p}}$ for several values of ${\tilde{\gamma }}_d$. In this plot, we have used ${\nu }_d/\nu =1/3$ and ${\sigma }_0=n_{\mathrm{a}}{e}^2{w}_h^2/\left(3m\nu \right)$. The dashed lines correspond to ${\tilde{\gamma }}_d=0.1$ and neglect the drag viscosity.

Figure 4

(a) Hall angles at the first layer $tan{\theta }_{\mathrm{a}}/{\tilde{\omega }}_c$ and (b) at the the second layer $tan{\theta }_{\mathrm{p}}/{\tilde{\omega }}_c$ as a function of $r={\overline{n}}_{\mathrm{a}}/{\overline{n}}_{\mathrm{p}}$ for several values of ${\tilde{\gamma }}_d={\mathrm{\Gamma }}_d\mathcal{R}$. Here ${\tilde{\omega }}_c={\omega }_c/{\gamma }_{ee}$ and ${\mathrm{\Gamma }}_d=0.5$. The dashed lines are for ${\tilde{\gamma }}_d=0.1$ and neglecting the drag-Hall viscosity ${\nu }_{dH}=0$ and the dotted lines are for ${\tilde{\gamma }}_d=0.1$ and ${\nu }_{dH}=0$ and ${\nu }_d=0$.