The boundary conditions of viscous electron flow

TL;DR

This paper develops a kinetic theory to determine the temperature-dependent slip length at rough interfaces in viscous electron flows, revealing how boundary conditions influence electron transport in graphene and Fermi liquids.

Contribution

It introduces a model for calculating slip lengths at rough boundaries in quantum fluids, connecting microscopic scattering to macroscopic flow boundary conditions.

Findings

Slip length is of the order of the electron mean free path in strongly disordered edges.

Nearly specular boundaries have large slip lengths, approaching no-stress conditions.

At low temperatures, the flow is always in the no-stress regime due to diverging mean free path.

Abstract

The sensitivity of charge, heat, or momentum transport to the sample geometry is a hallmark of viscous electron flow. Therefore, hydrodynamic electronics requires the detailed understanding of electron flow in finite geometries. The solution of the corresponding generalized Navier-Stokes equations depends sensitively on the nature of boundary conditions. The latter are generally characterized by a slip length $\zeta$ with extreme cases being no-slip $\left(\zeta\rightarrow0\right)$ and no-stress $\left(\zeta\rightarrow\infty\right)$ conditions. We develop a kinetic theory that determines the temperature dependent slip length at a rough interface for Dirac liquids, e.g. graphene, and for Fermi liquids. For strongly disordered edges that scatter electrons in a fully diffuse way, we find that the slip length is of the order of the momentum conserving mean free path $l_{ee}$ that determines the electron viscosity. For boundaries with nearly specular scattering $\zeta$ is parametrically large compared to $l_{ee}$. Since for all quantum fluids $l_{ee}$ diverges as $T\rightarrow0$, the ultimate low-temperature flow is always in the no-stress regime. Only at intermediate $T$ and for sufficiently large sample sizes can the slip lengths be short enough such that no-slip conditions are appropriate. We discuss numerical examples for several experimentally investigated systems.