Linking Spatial Distributions of Potential and Current in Viscous Electronics

Viscous electronics is an emerging field dealing with systems in which strongly interacting electrons behave as a fluid. Electron viscous flows are governed by a nonlocal current-field relation which renders the spatial patterns of the current and electric field strikingly dissimilar. Notably, driven by the viscous friction force from adjacent layers, current can flow against the electric field, generating negative resistance, vorticity, and vortices. Moreover, different current flows can result in identical potential distributions. This sets a new situation where inferring the electron flow pattern from the measured potentials presents a nontrivial problem. Using the inherent relation between these patterns through complex analysis, here we propose a method for extracting the current flows from potential distributions measured in the presence of a magnetic field.

Figure 1

Streamlines (black) and potential color map for current injected through a point in a half-plane, Eqs. (2) and (11). The velocity is shown by white arrows, its magnitude is proportional to the density of streamlines. Boundary conditions: (a) no stress (i.e., shear-stress free); (b) no slip.

Figure 2

Current streamlines (black) and potential color map for a flow across the strip. Arrows mark the streamlines nearest to separatrices. Stagnation points are labeled $s_{1,2}$, $s_{1,2}^{\prime }$. To elucidate the behavior near contacts, two regions are shown with a tenfold density of streamlines. Boundary conditions: (a) no stress, (b) no slip.