Hydrodynamics of the electronic Fermi liquid: a pedagogical overview

TL;DR

Addresses how to describe charge transport in ultraclean electronic materials using hydrodynamics. The authors develop a long-wavelength hydrodynamic framework for a Fermi-liquid, deriving continuity equations for charge and momentum and constitutive relations that yield Stokes–Ohm dynamics with a Gurzhi length $\lambda=\sqrt{\nu/\gamma_\text{mr}}$. They discuss a range of hydrodynamic phenomena, including viscous flow profiles, Corbino geometry, superballistic transport, thermal transport, current noise, and nonlinear effects, and extend the discussion to symmetry-broken, electron-hole/Dirac fluids and electron-phonon hydrodynamics. The results show characteristic departures from Ohm’s law and the Wiedemann–Franz law, nonlocal dissipation, and geometry-dependent signatures that provide a robust experimental toolkit. The work thus offers a unified, universal framework for understanding and exploiting hydrodynamic electron transport in materials like graphene and beyond.

Abstract

For over a hundred years, electron transport in conductive materials has been primarily described by the Drude model, which assumes that current flow is impeded primarily by momentum-relaxing collisions between electrons and extrinsic objects such as impurities or phonons. In the past decade, however, experiments have increasingly realized ultra-high quality electronic materials that demonstrate a qualitatively distinct method of charge transport called hydrodynamic flow. Hydrodynamic flow occurs when electrons collide much more frequently with each other than with anything else, and in this limit the electric current has long-wavelength collective behavior analogous to that of a classical fluid. While electron hydrodynamics has long been postulated theoretically for solid-state systems, the plethora of recent experimental realizations has reinvigorated the field. Here, we review recent theoretical and experimental progress in understanding hydrodynamic electrons using the (hydrodynamic) Fermi liquid as our prototypical example.

Figures (7)

• Figure 1: A plot of the normalized current density $j$ that arises from a solution to the Stokes-Ohm equation in a rectangular channel with no-slip boundary conditions. $h$ is the width of the channel and $I$ is the total current. The thin line corresponds to $\lambda/h \rightarrow \infty$, the dotted line to $\lambda/h = 0.1$, the dashed line $\lambda/h = 0.01$, and the thick solid line (uniform flow) $\lambda/h = 0$. Adapted from Ref. Torre2015.
• Figure 2: A cartoon of the Gurzhi effect. At low temperatures, impurity scattering dominates and the resistance is temperature-independent. As the temperature increases, eventually $\gamma_\text{ee} \gg \gamma_\text{imp}$ and the material becomes hydrodynamic. Since the viscosity decreases with temperature, so too does the resistance. This decrease is known as the Gurzhi effect. As the temperature increases past the Bloch-Grünesein temperature, electron-phonon scattering becomes dominant, the material leaves the hydrodynamic window, and resistance increases again. Diagram courtesy of Felicia Setiono.
• Figure 3: A plot of electron-electron scattering rate $\tau_\text{ee}^{-1}$ (or inverse viscosity $\nu^{-1} \propto \tau_\text{ee}^{-1}$) against temperature for monolayer graphene across a number of experiments. Adapted from a figure courtesy of Yihang Zeng and Cory Dean.
• Figure 4: A sketch of viscous flow through a constriction geometry. Viscosity couples adjacent layers of flow, pulling them through the constriction. This effect enables a superballistic conductivity. Adapted from Ref. KrishnaKumar2017.
• Figure 5: A plot of current flow and electric potential in the Corbino geometry in the hydrodynamic limit. The expanding flow implies a non-zero viscous stress and therefore non-trivial viscous heating. Despite this production of heat, the electric potential is constant in the bulk of the device (i.e., there is no driving electric field). This apparent inconsistency is known as the Corbino paradox. It is resolved by the sharp drops in voltage at the inner and outer contacts. Adapted from Ref. Hui2022 and Ref. Shavit2019.