Stokes flow analogous to viscous electron current in graphene

Jonathan Mayzel,Victor Steinberg,Atul Varshney

doi:10.1038/s41467-019-08916-5

Abstract

Electron transport in two-dimensional conducting materials such as graphene, with dominant electron–electron interaction, exhibits unusual vortex flow that leads to a nonlocal current-field relation (negative resistance), distinct from the classical Ohm’s law. The transport behavior of these materials is best described by low Reynolds number hydrodynamics, where the constitutive pressure–speed relation is Stoke’s law. Here we report evidence of such vortices observed in a viscous flow of Newtonian fluid in a microfluidic device consisting of a rectangular cavity—analogous to the electronic system. We extend our experimental observations to elliptic cavities of different eccentricities, and validate them by numerically solving bi-harmonic equation obtained for the viscous flow with no-slip boundary conditions. We verify the existence of a predicted threshold at which vortices appear. Strikingly, we find that a two-dimensional theoretical model captures the essential features of three-dimensional Stokes flow in experiments.

Highlights

Electron transport in two-dimensional conducting materials such as graphene, with dominant electron–electron interaction, exhibits unusual vortex flow that leads to a nonlocal currentfield relation, distinct from the classical Ohm’s law
The dimensionless Reynolds number (Re) defines the ratio of inertial and viscous stresses in fluids[4]. They extended their model for viscous electron flow in a three-dimensional (3D) conducting slab of small, finite thickness, and showed that the extra dimension in the Stokes equation translates into an effective resistance term[2]
The numerical simulations were conducted for the 2D viscous flow without any wall friction and with zero size inlet-outlet channels, whereas the experimental flow is 3D with finite-size inlet-outlet channels (0.5 mm channel width compared to the cavity width of ~2 mm)

Summary

Introduction

Electron transport in two-dimensional conducting materials such as graphene, with dominant electron–electron interaction, exhibits unusual vortex flow that leads to a nonlocal currentfield relation (negative resistance), distinct from the classical Ohm’s law. Levitov and Falkovich (L&F) developed a theoretical model based on Stokes flow of strongly interacting electrons[2,16] They explored three transport regimes of electron fluids, namely: Ohmic, mixed ohmic-viscous and viscous, in an infinitely long 2D rectangular strip, with point source and drain contacts located at the center, on opposite sides of the strip. The dimensionless Reynolds number (Re) defines the ratio of inertial and viscous stresses in fluids[4] They extended their model for viscous electron flow in a three-dimensional (3D) conducting slab of small, finite thickness, and showed that the extra dimension in the Stokes equation translates into an effective resistance term[2]

Methods

Results

Conclusion