Viscous electron fluids

In 1963 Soviet theorist Radii Gurzhi asked how a viscous electron flow could reveal itself in an experiment.¹ He assumed the existence of a metallic system in which $lee$ was the shortest length scale that electrons would travel, much shorter than both the sample size $W$ and the mean free path $l$ of electrons whose collisions—for instance, with phonons and crystal defects—did not conserve momentum. Given that assumption, frequent collisions between electrons should be able to establish a collective flow, illustrated in figure 1b, because their total momentum and energy is not lost to the outside world.

Gurzhi found that the resistance $R$ of such an imaginary metal would have to decrease with increasing $T$⁠. That’s a shocking result because the standard definition of a metal is that its $R$ increases with $T$⁠. Nonetheless, the theoretical prediction was unambiguous and could be traced to the fact that the electron viscosity $ν$ in metals decreases with $T$⁠. Intuitively, it also makes sense: As a system warms, it becomes less viscous, which allows easier passage of a fluid. The anomaly is usually referred to as the Gurzhi effect, and it explains that if a metal enters the hydrodynamic regime—where $lee≪W$ and $l$—it should exhibit a $T$ dependence that is the opposite of metallic and is more like that of semiconductors, whose $R$ decreases with $T$⁠.

Unfortunately, finding a system that satisfies those conditions turned out to be nearly impossible. One usually thinks of large, clean crystals at cryogenic $T$⁠, which would mitigate the effect of phonons and thus increase $l$⁠. Indeed, clean, three-dimensional metals at low $T$ exhibit values of $l$ that are nearly a centimeter. However, $lee$ also rapidly increases with decreasing $T$ because of what’s known as Pauli blocking. (See box 1 for details on the fundamental properties of electron systems.)

Fermi statistics greatly limits the available phase space for e–e collisions when $T$ is well below the Fermi temperature $TF$⁠. As a result, $lee$ diverges as $(TF/T)2$ with decreasing $T$⁠. That low-$T$ regime is precisely where Landau quasiparticles are long lived and the single-particle model of electrical conductivity is justified.

The only way to reach the hydrodynamic regime is to work at elevated $T$⁠, such that the Fermi sphere becomes “softer” and Pauli blocking less obstructive to e–e scattering. At those higher $T$⁠, phonons become the main hindrance and limit $l$ to the electron–phonon scattering length, $lep$⁠. The resulting condition, $lee≪lep$⁠, required to observe viscous behavior is extremely difficult to satisfy because $lep$ often decreases faster with increasing $T$ than does $lee$⁠. (For 3D metals, $lep$ usually varies as $T−3$⁠, whereas $lee$ varies as $T−2$⁠.) That scaling narrows the materials systems one could use and the $T$ interval in which electron hydrodynamics could possibly be observed.

An elegant attempt to break the impasse² was undertaken in the 1990s. Researchers applied a high electrical current that increased the electron $T$ of a 2D electron semiconductor (2DES) system and shortened $lee$⁠. Even so, the crystal lattice remained close to liquid-helium $T$⁠, which kept the electron–phonon scattering low as well. Measuring the differential resistance revealed a small but distinct bump as a function of applied current, a feature the researchers interpreted as plausible evidence for the Gurzhi effect.

Gurzhi and coworkers immediately disagreed with that interpretation,³ and they pointed out that peculiarities of e–e scattering in 2D materials demand an even more stringent condition than that in 3D metals—namely, $lee≪W(T/TF)$⁠, which had not been achieved in the experiment. Their rejection left the research status in limbo: For a half century after the Gurzhi theory was postulated, no electronic system had been found to exhibit unambiguous signs of hydrodynamic behavior.

Despite having a Nobel Prize behind it, graphene did not initially look like a promising candidate for studies of electron hydrodynamics. It was filled with impurities, with a mean free path barely exceeding 100 nm (see the article by Andre Geim and Allan MacDonald, Physics Today, August 2007, page 35, and Physics Today, December 2010, page 14). But that changed around 2011, when researchers found that encapsulating graphene in hexagonal boron nitride dramatically improved its electronic quality. The encapsulation shielded graphene from outside impurities and flattened the crystal by suppressing scattering at microscopic corrugations.

Today, graphene is one of the highest quality electronic materials ever produced: Its low-T mean free path is currently limited only by the device size $W$⁠, at least up to 10 μm, and exceeds a micron even at room $T$⁠. More importantly, graphene is extremely stiff, a feature that suppresses phonon scattering and increases $lep$⁠. And unlike what happens in 3D metals, electron–phonon scattering in 2D graphene increases slowly with $T$⁠; $lep∝T−1$⁠, with a small proportionality coefficient that accounts for stiffness. As noted earlier, e–e scattering rises much faster, with $lee∝T−2$⁠.

Therefore, above a certain $T$⁠, $lee$ is expected to become the shortest scattering length in graphene. Moreover, graphene’s $TF$ is typically greater than 1000 K. That’s neither too small, as it would be in semiconductor 2DESs, where the Fermi surface is largely destroyed at room $T$⁠, nor too high for the required 2D condition $lee≪W(T/TF)$ to be consequential. In short, it is hardly possible to imagine a better material than graphene for studying viscous electron flows.

Despite the promise of that expectation, in 2015 when the first contemporary experiments began probing the phenomenon, graphene’s resistance showed no sign of the Gurzhi effect at any $T$⁠. In hindsight, one can understand why the viscous effects did not show up straightforwardly. The kinematic viscosity $ν$ enters the Navier–Stokes equation as a coefficient in front of the second spatial derivative of velocity $v(x,y)$ (see box 2). In the standard resistance measurements that use a long strip of a uniform width, only $vx(y)$—the $y$ dependence of flow velocity in the x direction—is nonzero. Unless significant momentum losses occur at the strip boundaries, the $y$ dependence tends to be weak. The result is a fairly uniform flow profile. And without a significant velocity gradient, the viscosity term contributes little to the solution of the Navier–Stokes equation and, hence, to the resistance $R$⁠.

That insight offered a tip for how to proceed: To maximize the hydrodynamics effects in experiment, it is essential to create a current flow as inhomogeneous as possible.⁴ 

One geometry that provides large velocity gradients is a narrow current injector, shown schematically in figure 2. According to the Navier–Stokes equation, the electric potential changes its sign at a characteristic distance of order of $Dv=leel/2$ from the injector.^4–6 One can measure that local potential in the so-called vicinity geometry—that is, by placing a voltage probe sufficiently close to the injector. The corresponding resistance $RV$—the local voltage divided by the injected current—has the normal, positive sign for noninteracting electrons in both diffusive and ballistic transport regimes. Negative $RV$⁠, by contrast, is a smoking gun for viscous flow.⁴ 

However, one must be careful. As $T$ increases, the initial sign change indicates that ballistic transport is strongly affected by e–e interactions, and the hydrodynamic regime develops only later, at higher $T$ when collisions among electrons become more frequent.⁷ The observation of negative $RV$ in graphene and its comparison with behavior expected by Navier–Stokes theory allowed the first measurements of an electron fluid’s viscosity. At liquid-nitrogen $T$⁠, $ν$ turns out to be 100 times as great as honey. Reassuringly, that result agrees with many-body theory.⁴ 

Navier–Stokes theory also predicts another spectacular effect in the conductivity of metals because of viscosity.^4–6 The negative region of electric potential near the injector is predicted to develop into a whirlpool of electrical current. Whirlpools are familiar phenomena in the laminar flow of ordinary fluids, but in the vicinity geometry^4,6 of figure 2a, they are theoretically expected to exist near a narrow injector. Only the size of $Dν$ depends on the actual value of $ν$⁠.

For other geometries that create a nonuniform flow,⁵ current whirlpools generally disappear if $Dν$ gets smaller than the characteristic device size $W$⁠, even though the negative potential anomaly doesn’t change.

In 1908 Martin Knudsen observed that the speed of gas flowing through a small aperture suddenly increased when he increased the gas’s density. The experiment implies that a higher viscosity boosts the gas flow, which is a counterintuitive result. The effect is well understood today as the transition from Knudsen flow to Poiseuille flow, or in the language of metal physics, from ballistic-electron transport to viscous-electron transport. The phenomenon observed by Knudsen can be viewed as the analogue of the Gurzhi effect for gases rather than electrons.

An experiment similar to Knudsen’s was recently performed on graphene.⁸ As shown in figures 3 and 4, a narrow aperture of width $w$ connects two wider regions, a geometry known as point contact (PC). In the ballistic regime at low $T$⁠, such PCs were first made and studied by Yuri Sharvin in the 1960s. He found that even in the ideal case—without any disorder and scattering—a PC exhibited finite electrical conductance. Its value is given by the number of electron-wave modes that can fit inside the aperture.

Until recently, researchers have tacitly accepted that Sharvin’s conductance was the highest possible value. The absence of disorder seemed to imply the best-case scenario for unimpeded electron transport. But that turned out to be wrong. Figure 4b shows that when $T$ is increased and a system enters the hydrodynamic regime, the resistance measured in a graphene PC drops below the ideal ballistic limit. For the experiment in the figure, the drop was caused by the transition from ballistic to viscous electron transport. It was also accompanied by a semiconductor-like $T$ dependence—the first unambiguous manifestation of the Gurzhi effect.

How is it possible for viscosity to lower the electrical conductivity? After all, basic physics tells us that greater electron scattering should increase the resistance—a trend known as Matthiessen’s rule. Making the transition from the low-$T$ regime, where Sharvin’s description applies, to the higher-$T$ hydrodynamic regime, electron viscosity sets up a funnel-like current pattern through the aperture, akin to what happened in Knudsen’s experiment.

Imagine an electron moving toward the PC, as in figure 3. In the ballistic regime, it hits the wall and stops contributing to the conductance. But in the hydrodynamic regime, the same electron is dragged by electron collisions toward the opening and forced to funnel through it. That funneling is what raises the conductance above Sharvin’s ballistic limit. Mathematically, the superballistic flow happens because conductivities are added—the so-called anti-Matthiessen’s rule⁹ described in box 3. By comparing experimental results and theory, the two of us and our colleagues were able to accurately measure graphene’s viscosity as a function of electron concentration and $T$⁠.

Another knob that can be turned to explore viscous flow is the magnetic field $B$⁠. In traditional metallic systems, $B$ causes the Hall effect, a potential drop perpendicular to the direction of both current flow and the magnetic field. How is the Hall effect influenced by electron viscosity? The presence of a magnetic field breaks down time-reversal symmetry and produces a new kinematic coefficient $νH$ in the Navier–Stokes equation. The coefficient, known as the Hall viscosity, is odd under reversal of $B$ and is dissipationless. The Hall viscosity gives rise to an extra term in the Navier–Stokes equation that is proportional to $νH$⁠, acts against the Lorentz force, and suppresses the resulting potential drop.

The suppression of the Hall effect is local and extends only over distances of $Dv$⁠, typically about 0.5–0.6 µm. By placing voltage probes close to a narrow current injector, we measured a local Hall effect.¹⁰ For graphene in the hydrodynamic regime, it was found to be notably smaller than the standard Hall effect, measured simultaneously at some distance from the current contact.

Now that we know how to force hydrodynamics to show up in experiments, we expect to soon observe viscous phenomena in many systems, including 2DESs in semiconductors, graphite, bismuth, topological insulators, and Weyl metals. Evidence already exists for viscous flow in delafossites,¹¹ and local (vicinity and PC) geometries should help make those observations. Materials in which electrons and holes coexist and interact strongly present another interesting challenge.^12,13 

Let’s also not forget about materials that defy the Fermi-liquid paradigm. They are called strange metals^14,15 and have Planckian transport scattering times on the order of $ℏ/(kBT)$ down to the lowest $T$⁠. Those metals are also expected to exhibit viscous electron motion, albeit with a tiny viscosity conjectured to be close to a universal lower bound predicted by string-theory methods. Experimental evidence of the lower bound has been reported in ultrahot nuclear matter, such as quark–gluon plasmas, and in ultracold atomic Fermi gases, but not in condensed-matter physics.

Yet another enticing project would be to extend existing hydrodynamic studies into the regime where nonlinear terms in the Navier–Stokes equation can no longer be ignored. In classical fluids, those terms are responsible for nonlinear phenomena such as turbulence. Similar physics is expected to occur in electron fluids, but studying such fluids would require materials with smaller $ν$ and longer $τ$ compared with the 2DESs studied so far.

For all those new ventures, one should use not only electrical probes but also the visualization tools that are now available. Scanning probe microscopes that can sense voltages or magnetic fields are one example. They can image local distributions of electrical current at submicron scales and reveal electron hydrodynamics at an entirely new, more spectacular level. Watch out for beautiful images of electron whirlpools and viscous flows coming soon.

The European Union’s Horizon 2020 research and innovation program (Graphene Flagship) supported this work. We are grateful to everyone who contributed to the research—particularly Denis Bandurin, Alexey Berdyugin, Roshan Kumar, Leonid Levitov, Francesco Pellegrino, Leonid Ponomarenko, Alessandro Principi, Andrea Tomadin, and Iacopo Torre. Author Marco Polini dedicates this article to Rachele.

Marco Polini is a professor at the University of Pisa in Italy, a professor at the University of Manchester in the UK, and an external collaborator of the Italian Institute of Technology in Genoa. Andre Geim is the Regius Professor at the University of Manchester.