## Electrons that flow like a fluid

Electrons inside a conductor are often described as flowing in response to an electric field. This flow rarely resembles anything like the familiar flow of water through a pipe, but three groups describe counterexamples (see the Perspective by Zaanen). Moll *et al.* found that the viscosity of the electron fluid in thin wires of PdCoO_{2} had a major effect on the flow, much like what happens in regular fluids. Bandurin *et al.* found evidence in graphene of electron whirlpools similar to those formed by viscous fluid flowing through a small opening. Finally, Crossno *et al.* observed a huge increase of thermal transport in graphene, a signature of so-called Dirac fluids.

## Abstract

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 the temperature of liquid nitrogen. 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 transport regime. We found that 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 square meters per second, an order of magnitude higher than that of honey, in agreement with many-body theory. Our work demonstrates the possibility of studying electron hydrodynamics using high-quality graphene.

The collective behavior of many-particle systems that undergo frequent interparticle 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 that vary slowly in space and time. More recently, it has been shown that hydrodynamics can also be applied to strongly interacting quantum systems, including ultrahot nuclear matter and ultracold atomic Fermi gases in the unitarity limit (*3*–*6*). In principle, the hydrodynamic approach can also be used 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, so that ≪ *W*, , where is the electron-electron scattering length, *W *is the characteristic sample size, ≡ ν_{F}τ is the mean free path, ν_{F }is the Fermi velocity, and τ is the mean free time with respect to momentum-nonconserving collisions, such as those involving impurities and phonons. The above inequalities are difficult to meet experimentally. At low temperatures (*T*), varies approximately ∝*T*^{−2},^{} reaching a micrometer scale at liquid helium *T* (*14*), which necessitates the use of ultraclean systems to satisfy ≪ . At higher *T*, electron-phonon scattering rapidly reduces . However, for two-dimensional (2D) systems in which acoustic phonon scattering dominates, decays only ∝*T*^{–1}, slower than , which should in principle allow the hydrodynamic description to apply 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 ( ~ *W*) made from GaAlAs heterostructures (*15*). These devices exhibited nonmonotonic changes in differential resistance as a function of a large applied current *I*, which was used to increase the electron temperature (making short) while the lattice temperature remained low (allowing long ). The nonmonotonic behavior was attributed to the Gurzhi effect, a transition between Knudsen ( ≫ ) and viscous electron flows (*7*, *15*). Another possible hint about electron hydrodynamics comes from an explanation (*16*) of the Coulomb drag measured between two graphene sheets at the charge neutrality point (CNP) (*17*).

Here we address electron hydrodynamics by using a special measurement geometry (Fig. 1) that amplifies the effects of the shear viscosity ν and, at the same time, minimizes a contribution from ballistic effects that can occur not only in the Knudsen regime but also in viscous flows in graphene. A viscous flow can lead to vortices appearing in the spatial distribution of the steady-state current (Fig. 1, A and B). Such “electron whirlpools” have a spatial scale , which depends on electron-electron scattering through ν and on the electron system’s quality through τ (*18*). To detect the whirlpools, electrical probes should be placed at a distance comparable to *D*_{ν}. By using single- and bi-layer graphene (SLG and BLG, respectively) encapsulated between boron nitride crystals (*19*–*21*), we were able to reach a *D*_{ν} of 0.3 to 0.4 μm thanks to the high viscosity of graphene’s Fermi liquid and its high carrier mobility μ, even at high *T*. Such a large* D*_{ν}, which is unique to graphene, nevertheless necessitates submicron resolution to probe the electron backflow. To this end, we fabricated multiterminal Hall bars with narrow (~0.3 μm) and closely spaced (~1 μm) voltage probes (Fig. 1C and fig. S1). Details of the device fabrication are given in (*18*).

All our devices were first characterized in the standard geometry by applying *I* along the main channel and using side probes for voltage measurements. The typical behavior of longitudinal conductivity σ_{xx} at a few characteristic values of *T* is shown in Fig. 1D. At liquid helium* T*, the devices exhibited μ ~ 10 to 50 m^{2} V^{−1} s^{−1} for carrier concentrations *n* over a wide range of the order of 10^{12} cm^{−2}, and μ remained above 5 m^{2} V^{−1} s^{−1} up to room *T* (fig. S2). Such values of μ allow ballistic transport with > 1 μm at *T *< 300 K. At *T *≥ 150 K, decreases to 0.1 to 0.3 μm over the same range of *n* (figs. S3 and S4) (*22**, **23*). This allows the essential condition for electron hydrodynamics ( ≪ *W*, ) to be satisfied within this temperature range. If one uses the conventional longitudinal geometry of electrical measurements, viscosity has little effect on σ_{xx} (figs. S5 to S7), essentially because the flow in this geometry is uniform, whereas the total momentum of the moving Fermi liquid is conserved in electron-electron collisions (*18*). The only evidence for hydrodynamics that we could find in the longitudinal geometry was the Gurzhi effect that appeared as a function of the electron temperature, which is controlled by applying large *I*, similar to the observations in (*15*) (fig. S8).

To look for hydrodynamic effects, we used the geometry shown in Fig. 1C. In this setup, *I *is injected through a narrow constriction into the graphene bulk, and the voltage drop *V*_{v} is measured at the nearby side contacts located at the distance *L* ~ 1 μm away from the injection point. These can be considered as nonlocal measurements, although the stray currents are not exponentially small (dashed curves in Fig. 1E). To distinguish from the proper nonlocal geometry (*24*), we refer to the linear-response signal measured in our geometry as vicinity resistance, *R*_{v} = *V*_{v}/*I*. The idea is that, in the case of a viscous flow, whirlpools emerge as shown in Fig. 1B, and their appearance can then be detected as sign reversals of *V*_{v}, which is positive for the conventional current flow (Fig. 1A) and negative for viscous backflow (Fig. 1B). Figure 1E shows examples of *R*_{v} for the same SLG device as in Fig. 1D, and other SLG and BLG devices exhibited similar behavior (*18*). Away from the CNP, *R*_{v} is negative over a wide range of intermediate *T*, despite an expected substantial offset due to stray currents. Figure 2 details our observations further by showing maps of *R*_{v} (*n*,*T*) for SLG and BLG. The two Fermi liquids exhibited somewhat different behavior, reflecting their different electronic spectra, but *R*_{v} was negative over a large range of *n* and *T* for both. Two more *R*_{v} maps are provided in fig. S9. In total, seven multiterminal devices with *W* ranging from 1.5 to 4 μm were investigated, showing vicinity behavior that was highly reproducible both for different contacts on the same device and for different devices, independently of their *W*, although we note that the backflow was more pronounced for devices with the highest μ and lowest charge inhomogeneity.

The same anomalous vicinity response was also evident when we followed the method of (*15*) and used the current *I *to increase the electron temperature. In this case, *V*_{v} changed its sign as a function of *I* from positive to negative to positive again, reproducing the behavior of *R*_{v} with increasing *T* of the cryostat (fig. S10). Comparing figs. S8 and S10, it is clear that the vicinity geometry strongly favors the observation of hydrodynamic effects: The measured vicinity voltage changed its sign, whereas in the standard geometry, the same viscosity led only to relatively small changes in *dV*/*dI*. We also found that the magnitude of negative *R*_{v} decayed rapidly with *L* (fig. S11), in agreement with the finite size of electron whirlpools.

Negative resistances can in principle arise from other effects, such as single-electron ballistic transport ( ≫ ) or quantum interference (*18*, *20*, *24*). The latter contribution is easily ruled out, because quantum corrections rapidly wash out at *T *> 20 K and have a random sign that rapidly oscillates as a function of magnetic field. Also, our numerical simulations using the Landauer-Büttiker formalism and the realistic device geometry showed that no negative resistance could be expected for the vicinity configuration in zero magnetic field (*19*, *21*). Nonetheless, we carefully considered the possibility of any accidental spillover of single-electron ballistic effects into the vicinity geometry of our experiment. The dependences of the negative vicinity signal on *T*, *n*, *I*,* *and the device geometry allowed us to unambiguously rule out any such contribution (*18*). For example, the single-electron ballistic phenomena should become more pronounced for longer (that is, with decreasing *T* or electron temperature and with increasing *n*), in contrast to the nonmonotonic behavior of* V*_{v}.

Turning to theory, we can show that negative *R*_{v} arises naturally from whirlpools that appear in a viscous Fermi liquid near current-injecting contacts. As discussed in (*18*), electron transport for sufficiently short can be described by the hydrodynamic equations (1)and (2)where **J**(**r**) is the (linearized) particle current density, and ϕ(**r**) is the electric potential in the 2D plane. If *D*_{ν} → 0, Eq. 2 yields Ohm’s law –*e***J**(**r**) = σ_{0}**E**(**r**) with a Drude-like conductivity σ_{0 }≡ *ne ^{2}*τ/

*m*, where –

*e*and

*m*are the electron charge and the effective mass, respectively (

**E,**electric field). The three terms in Eq. 2 describe (i) the electric force generated by the steady-state charge distribution in response to the applied current

*I*, (ii) the viscous force (

*1*,

*2*), and (iii) friction caused by momentum-nonconserving processes that are parameterized by the scattering time τ(

*n*,

*T*).

Equations 1 and 2 can be solved numerically (*18*), and Fig. 3 shows examples of spatial distributions of ϕ(**r**) and** J**(**r**). For experimentally relevant values of* D*_{ν}, a vortex appears in the vicinity of the current-injecting contact. This is accompanied by the sign reversal of ϕ(**r**) at the vicinity contact on the right of the injector, which is positive in Fig. 3C (no viscosity) but becomes negative in Fig. 3, A and B. Our calculations for this geometry show that *R*_{v} is negative for* D*_{ν}0.4 μm (*18*). Because both τ and ν decrease with increasing *T*, *D*_{ν} also decreases, and stray currents start to dominate the vicinity response at high* T*. This explains why *R*_{v} in Figs. 1 and 2 becomes positive close to room *T*, even though our hydrodynamic description has no high-temperature cutoff (*18*). Despite positive *R*_{v} values, the viscous contribution remains considerable near room *T* (Fig. 1D and fig. S12). At low *T*, the electron system approaches the Knudsen regime, and our hydrodynamic description becomes inapplicable because ~ (*18*). In the latter regime, the whirlpools should disappear and *R*_{v} should become positive (fig. S13), in agreement with the experiment and our numerical simulations based on the Landauer-Büttiker formalism.

The numerical results in Fig. 3 can be understood if we rewrite Eqs. 1 and 2 as (3)where is the vorticity ( is the unit vector perpendicular to the graphene plane) (*2*). Taking the curl of Eq. 3, the vorticity satisfies the equation , where *D*_{ν} plays the role of a diffusion constant. The current *I* injects vorticity at the source contact, which then exponentially decays over the length scale* D*_{ν}. For ν = 0.1 m^{2} s^{−1} [as estimated in (*25*)] and τ = 1.5 ps (fig. S2), we find that *D*_{ν }≈ 0.4 μm, in qualitative agreement with the measurements in fig. S11.

Lastly, we can combine the measurements of *R*_{v} and resistivity ρ_{xx} with the solution of Eqs. 1 and 2 in Fig. 3 to extract the kinematic viscosity for SLG and BLG. Because the observed Gurzhi effect in ρ_{xx} is small at low currents (fig. S6), we can use ρ_{xx} = 1/σ_{0} = *m*/(*ne*^{2}τ) to determine τ(*n*,*T*) (*18*). Furthermore, for the experimentally relevant values of *D*_{ν}, we find that *R*_{v} is a quadratic function of *D*_{ν} (4)where *a* and *b* are numerical coefficients dependent only on the measurement geometry and boundary conditions, and *b *describes the contribution from stray currents (fig. S14). For the specific device in Fig. 3, we determined that *a *= –0.29 μm^{−2} and *b* = 0.056, and this allows us to estimate *D*_{ν}(*n*,*T*) from measurements of *R*_{v}. For the known τ and* D*_{ν}, we find that . The applicability limits of this analysis are discussed in (*18*), and the results are plotted in Fig. 4 for one of our devices. The figure shows that, at carrier concentrations of ~10^{12} cm^{−2}, the Fermi liquids in both SLG and BLG are highly viscous, with ν ≈ 0.1 m^{2} s^{−1}. In comparison, liquid honey has typical viscosities of ~0.002 to 0.005 m^{2} s^{−1}.

Figure 4 also shows the results of fully independent microscopic calculations of ν(*n*,*T*), which were carried out by extending the many-body theory of (*25*) to the case of 2D electron liquids hosted by encapsulated SLG and BLG. Within the range of applicability of our analysis in Fig. 4 (*n *~ 10^{12} cm^{−2}), the agreement in absolute values of the electron viscosity is good, especially taking into account that no fitting parameters were used in the calculations. Because the strong inequality ≫ required by the hydrodynamic theory cannot be reached even for graphene, it would be unreasonable to expect better agreement (*18*). In addition, our analysis does not apply near the CNP, because the theory neglects contributions from thermally excited carriers, spatial charge inhomogeneity, and coupling between charge and energy flows, which can play a substantial role at low doping (*16*, *18*). Further work is needed to understand electron hydrodynamics in the intermediate regime and, for example, to explain ballistic transport ( > *W*) in graphene at high *T* in terms of suitably modified hydrodynamic theory. The naive single-particle description that is routinely used for graphene’s ballistic phenomena even above 200 K (*19*, *21*) cannot be justified; a more complete theory is needed to describe the injection of a collimated electron beam into a strongly interacting 2D liquid. As for experimental approaches, the highly viscous Fermi liquids in graphene and their accessibility offer a promising opportunity to use various scanning probes for visualization and further understanding of electron hydrodynamics.

## Supplementary Materials

## References and Notes

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**Acknowledgments:**This work was supported by the European Research Council, the Royal Society, Lloyd’s Register Foundation, the Graphene Flagship, and the Italian Ministry of Education, University and Research through the program Progetti Premiali 2012 (project ABNANOTECH). D.A.B. and I.V.G. acknowledge support from the Marie Curie program SPINOGRAPH (Spintronics in Graphene). A.P. received support from the Nederlandse Wetenschappelijk Organisatie. R.K.K. received support from the Engineering and Physical Sciences Research Council.