Two-dimensional hydrodynamic electron flow through periodic and random potentials

We study the hydrodynamic flow of electrons through a smooth potential energy landscape in two dimensions, for which the electrical current is concentrated along thin channels that follow percolating equipotential contours. The width of these channels, and hence the electrical resistance, is determined by a competition between viscous and thermoelectric forces. For the case of periodic (moiré) potentials, we find that hydrodynamic flow provides a route to linear-in-$T$ resistivity. We calculate the associated prefactors for potentials with $C_3$ and $C_4$ symmetry. On the other hand, for a random potential the resistivity has qualitatively different behavior because equipotential paths become increasingly tortuous as their width is reduced. This effect leads to a resistivity that grows with temperature as $T^{10/3}$.

Figure 1

A device sketch with the imposed external potential $U$ (blue and red contour lines), where the shaded dark-blue lines correspond to current flow under the applied electric field $E$. The thin current channels of width $h$ are concentrated about equipotential contours of $U$. (a) The case of square periodic potential $U=U_{0}cos(2\pi x/\xi )cos(2\pi y/\xi )$. (b) The case of a random potential, for which the equipotential contours and the current channels meander and become tortuous in nature, controlled by the hull correlation length ${\xi }_h$ and the hull perimeter ${\ell }_h$.

Figure 2

Numerical solutions for the current density $\mathbf{j}$ in various periodic potentials. Each row corresponds to a particular type of potential, illustrated in the first column (top to bottom: square potential, triangular potential with $\psi =\pi /6$, triangular potential with $\psi =0$). Columns 2–4 show the simulated current density (arrows show direction and color darkness shows magnitude) for each potential. Columns 2–4 correspond to $\alpha \sim {10}^3,{10}^5,$ and ${10}^7$, respectively. As $\alpha$ increases, the current-carrying channels become increasingly narrow.

Figure 3

Numerical results for the averaged current density profile ${\langle j_x\rangle }_x\equiv \int dx{j}_x/L_x$ in a square-periodic potential, normalized by total current $I$. The three values of $\alpha$ are the same as those in Fig. 2. The solid lines correspond to the results of direct numerical simulation, while dashed lines correspond to our approximate variational solution.