Geometry-Induced Skin Effect in Electron Hydrodynamics

TL;DR

The paper develops a minimal, continuum hydrodynamic description of electron flow in constrictions using Brinkman equations with a finite momentum-relaxation length $\lambda$, solving analytically via Fourier transform to obtain a singular integral equation for the aperture current and validating these insights with finite-element simulations. It identifies a geometry-induced skin effect where edge sharpness and finite dissipation concentrate current near constriction edges, producing double-hump profiles that resemble NV-imaging data in graphene even within a hydrodynamic regime. A crossover among viscous, Ohmic, and ballistic-like transport emerges, with a bifurcation between flow patterns occurring near $a/\lambda_c \approx 8.3$, highlighting the nontrivial role of geometry in shaping current distributions. Together, these results provide a transparent hydrodynamic baseline for interpreting current-flow images and clarify how geometry and momentum relaxation jointly sculpt observed patterns in ultra-clean 2D conductors.

Abstract

In ultra-clean 2d materials electron viscosity is as important as Ohmic dissipation and electron transport exhibits hydrodynamic features. Using a simple framework of Brinkman equations we find that hydrodynamic electron flows exhibit a geometric skin effect: sharp obstacles locally enhance the current suppressing it far from the edges where the flow is unobstructed. This effect arises within hydrodynamic transport with finite momentum relaxation and does not rely on ballistic dynamics. Our results provide a natural hydrodynamic interpretation of edge-enhanced and double-bump current profiles observed in constricted geometries. By comparing with recent scanning NV magnetometry experiments on gated graphene, we demonstrate that such flow patterns are consistent with viscous hydrodynamics shaped by geometry, clarifying the role of geometric effects in the interpretation of electronic flow experiments.

Figures (9)

• Figure 1: The flow geometry.
• Figure 2: (a) The flow profile at the aperture compared to the experimental data from jenkins2022imaging. The solid red line is the best fit with $\lambda=a/10$. The dashed lines correspond to $\lambda=a/20$ (blue) as well as $\lambda=a$ (orange) and $\lambda=\infty$ (green) virtually indistinguishable from one another. (b-g) The current density $|j|$ maps for momentum relaxation length $\lambda=\infty$, $a$, $a/3$, $a/6$, $a/10$, and $a/20$.
• Figure 3: FEM results for the flow through the channel with a finite length $l$: maps showing the exponent $\alpha$ fitted to the velocity profile $J$ across the aperture (the fitting line is marked by red dashed lines in the upper panels). Representative cases, with parameter $(\lambda,l)$ configurations -- showing four different flow regimes -- indicated by symbols on the map, are shown in the upper panels.
• Figure 4: The current density $|j|$ for various momentum relaxation length $\lambda$ with a convective term included in the FEM formulation.
• Figure S1: FEM computational box with marked various boundary conditions.