Driving viscous hydrodynamics in bulk electron flow in graphene using micromagnets

We consider the hydrodynamic flow of an electron fluid in a channel formed in a two-dimensional electron gas (2DEG) with no-slip boundary conditions. To generate vorticity in the fluid, the flow is influenced by an array of micromagnets on the top of the 2DEG. We analyze the viscous boundary layer, and we demonstrate anti-Poiseuille behavior in this region. Furthermore, we predict a longitudinal voltage modulation, where a periodic magnetic field generates a voltage term periodic in the direction of transport. From an experimental point of view, we propose a method for a boundary-independent measurement of the viscosity of different electron fluids. The results are applicable to graphene away from the charge-neutrality point and to semiconductors.

Figure 1

Streamlines in a semi-infinite channel with a no-slip boundary at $y=0$. The parameters are chosen as $gD=0.88$, $\omega \tau =0.2,1.2,2.4$. Solid lines correspond to the numerically exact solution described in Sec. 3b, and dashed lines correspond to the solution given by Eq. (5) in the infinite device.

Figure 2

Ohmic, viscous, and total dissipation from solution of the NS equation (1) as a function of channel width and total dissipation in the bulk from solution of the ${\mathrm{NS}}^{+}$ equation (3). The dissipation is normalized to the ${\dot{E}}_0$, which is due to Drude resistance only, i.e., which is expected in the absence of B-field and hydrodynamics (${\dot{E}}_0/LW=n{m}^{*}{v}_0^2/\tau$). The value $\omega \tau =1.2$ corresponds to $B_0=75$ mT and $\omega \tau =2.4$ corresponds to $B_0=150$ mT. In these plots we assume that the viscosity is independent of the magnetic field, $\nu \left(B\right)=\nu \left(0\right)$. The plots correspond to the parameter set (11), where $gD=0.88$.

Figure 3

Longitudinal velocity profile at $x/D=0$ with $gD=0.88$ for various values of $\omega \tau$. At large magnetic fields anti-Poiseuille flow is observed and a small counterflow is observed at the opposite boundary due to the formation of a viscous vortex.

Figure 4

FEM results for the absolute value $\left|\mathbf{v}\right|$ of the velocity in a finite channel of length $L=100D$ for $gD=0.5$. There is an equilibration region near the inlet and outlet where uniform flow transitions to periodic flow, with periodic flow established from $-40<x/D<40$. Note that this color plot should not be confused with lines of flow, which are shown in Fig. 1 The velocity is measured in units convenient for numerical solution, and the system contains six magnetic strips. The region of high velocity, anti-Poiseuille flow, is shifted by half a period at opposite boundaries.

Figure 5

Left: top view of a periodic array of magnetic bars perpendicular to the flow direction shown by the red arrow. The micromagnets are magnetized parallel to the longitudinal axis of the channel. The direction of the magnetization is shown by the blue arrows. The period of the superlattice is $a$ and the width of each micromagnet is $d$. Right: Side view of the device. The vertical distance from the micromagnets to the 2DEG is $z$ and the thickness of each micromagnet is $\delta z$. First presented in Ref. [53].

Figure 6

Magnetic field in the 2DEG plane for NiFe bars. Left panel: $B_z$ vs $x$. Right panel: $B_x$ vs $x$. The parameters are $\delta z=0.20\mu \mathrm{m},z=0.40\mu \mathrm{m}$.

Figure 7

Longitudinal velocity profile at $x/D=0$ with $gD=0.88$ and $\omega \tau =4.8$ for a range of slip length $\beta$. Clearly the finite-slip cases interpolate between the no-slip and perfect-slip cases. As $\beta$ increases, the strength of the anti-Poiseuille flow peak increases until it saturates at the perfect-slip level, while counterflow at the other boundary quickly disappears with finite $\beta$.