Graphene hosts a unique electron system in which electron-phonon scattering is extremely weak but electron-electron collisions are sufficiently frequent to provide local equilibrium above liquid nitrogen temperature. Under these conditions, electrons can behave as a viscous liquid and exhibit hydrodynamic phenomena similar to classical liquids. Here we report strong evidence for this long-sought transport regime. In particular, doped graphene exhibits an anomalous (negative) voltage drop near current injection contacts, which is attributed to the formation of submicrometer-size whirlpools in the electron flow. The viscosity of graphene's electron liquid is found to be ≈0.1 m 2 s -1, an order of magnitude larger than that of honey, in agreement with many-body theory. Our work shows a possibility to study electron hydrodynamics using high quality graphene.Collective behavior of many-particle systems that undergo frequent inter-particle collisions has been studied for more than two centuries and is routinely described by the theory of hydrodynamics (1,2). The theory relies only on the conservation of mass, momentum and energy and is highly successful in explaining the response of classical gases and liquids to external perturbations varying slowly in space and time. More recently, it has been shown that hydrodynamics can also be applied to strongly interacting quantum systems including ultra-hot nuclear matter and ultra-cold atomic Fermi gases in the unitarity limit (3-6). In principle, the hydrodynamic approach can also be employed to describe many-electron phenomena in condensed matter physics (7-13). The theory becomes applicable if electron-electron scattering provides the shortest spatial scale in the problem such that ℓ ee ≪ , ℓ where ℓ ee is the electron-electron scattering length, the characteristic sample size, ℓ ≡ F the mean free path, F the Fermi velocity, and the mean free time with respect to momentum-non-conserving collisions such as those involving impurities, phonons, etc. The above 2 inequalities are difficult to meet experimentally. Indeed, at low temperatures ( ) ℓ ee varies approximately as ∝ −2 reaching a micrometer scale at liquid-helium (14), which necessitates the use of ultra-clean systems to satisfy ℓ ee ≪ ℓ. At higher , electron-phonon scattering rapidly reduces ℓ. However, for two-dimensional (2D) systems with dominating acoustic phonon scattering, ℓ decays only as ∝ −1 , slower than ℓ ee , which should in principle allow the hydrodynamic description over a certain temperature range, until other phonon-mediated processes become important. So far, there has been little evidence for hydrodynamic electron transport. An exception is an early work on 2D electron gases in ballistic devices (ℓ ~ ) made from GaAlAs heterostructures (15). They exhibited nonmonotonic changes in differential resistance as a function of a large applied current that was used to increase the electron temperature (making ℓ ee short) while the lattice temperature remained low (allowing long ℓ).The nonmonotonic behavior ...

Motivated by recent experimental progress in preparing encapsulated graphene sheets with ultrahigh mobilities up to room temperature, we present a theoretical study of dc transport in doped graphene in the hydrodynamic regime. By using the continuity and Navier-Stokes equations, we demonstrate analytically that measurements of non-local resistances in multi-terminal Hall bar devices can be used to extract the hydrodynamic shear viscosity of the two-dimensional (2D) electron liquid in graphene. We also discuss how to probe the viscosity-dominated hydrodynamic transport regime by scanning probe potentiometry and magnetometry. Our approach enables measurements of the viscosity of any 2D electron liquid in the hydrodynamic transport regime.

Materials subjected to a magnetic field exhibit the Hall effect, a phenomenon studied and understood in fine detail. Here we report a qualitative breach of this classical behavior in electron systems with high viscosity. The viscous fluid in graphene is found to respond to non-quantizing magnetic fields by producing an electric field opposite to that generated by the classical Hall effect. The viscous contribution is large and identified by studying local voltages that arise in the vicinity of current-injecting contacts. We analyze the anomaly over a wide range of temperatures and carrier densities and extract the Hall viscosity, a dissipationless transport coefficient that was long identified theoretically but remained elusive in experiment. Good agreement with theory suggests further opportunities for studying electron magnetohydrodynamics.

In highly viscous electron systems such as high-quality graphene above liquid nitrogen temperature, a linear response to applied electric current becomes essentially nonlocal, which can give rise to a number of new and counterintuitive phenomena including negative nonlocal resistance and current whirlpools. It has also been shown that, although both effects originate from high electron viscosity, a negative voltage drop does not principally require current backflow. In this work, we study the role of geometry on viscous flow and show that confinement effects and relative positions of injector and collector contacts play a pivotal role in the occurrence of whirlpools. Certain geometries may exhibit backflow at arbitrarily small values of the electron viscosity, whereas others require a specific threshold value for whirlpools to emerge.