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# Higher-than-ballistic conduction of viscous electron flows

Edited by Allan H. MacDonald, The University of Texas at Austin, Austin, TX, and approved January 24, 2017 (received for review July 25, 2016)

## Significance

Free electron flows through constrictions in metals are often regarded as an ultimate high-conduction charge transfer. We predict that electron fluids can flow with a resistance that is much smaller than the fundamental quantum mechanical ballistic limit for nanoscale electronics. The “superballistic” low-dissipation transport is particularly striking for the flow through a viscous point contact, a constriction exhibiting the quantum mechanical ballistic transport at zero temperature but governed by viscous electron hydrodynamics at a higher temperature. Unlike other mechanisms of low-dissipation transport, for example, superconductivity, the viscous electron flows can be realized at elevated temperatures, granting a new route for the low-power electronics research.

## Abstract

Strongly interacting electrons can move in a neatly coordinated way, reminiscent of the movement of viscous fluids. Here, we show that in viscous flows, interactions facilitate transport, allowing conductance to exceed the fundamental Landauer’s ballistic limit

Free electron flow through constrictions in metals is often regarded as an ultimate high-conduction charge transfer mechanism (1⇓⇓⇓–5). Can conductance ever exceed the ballistic limit value? Here we show that superballistic conduction is possible for strongly interacting systems in which electron movement resembles that of viscous fluids. Electron fluids are predicted to occur in quantum-critical systems and in high-mobility conductors, so long as momentum-conserving electron–electron (ee) scattering dominates over other scattering processes (6⇓⇓–9). Viscous electron flows feature a host of novel transport behaviors (10⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–22). Signatures of such flows have been observed in ultraclean GaAs, graphene, and ultrapure PdCoO_{2} (23⇓⇓–26).

We will see that electrons in a viscous flow can achieve through cooperation what they cannot accomplish individually. As a result, resistance and dissipation of a viscous flow can be markedly smaller than that for the free-fermion transport. As a simplest realization, we discuss viscous point contact (VPC), where correlations act as a ”lubricant” facilitating the flow. The reduction in resistance arises due to the streaming effect illustrated in Fig. 1, wherein electron currents bundle up to form streams that bypass the boundaries, where momentum loss occurs. This surprising behavior is in a clear departure from the common view that regards electron interactions as an impediment for transport.

A simplest VPC is a 2D constriction pictured in Fig. 1*A*. The interaction effects dominate in constrictions of width

Conveniently, both regimes are accessible in a single constriction, because transport is expected to be viscous at elevated temperatures and ballistic at **S21**. The condition Eq. **2** can be readily met in micron-size graphene junctions.

Several effects of electron interactions on transport in constrictions were discussed recently. Refs. 14 and 15 study junctions with spatially varying electron density and, using the time-dependent current density functional theory, predict a suppression of conductance. A hydrodynamic picture of this effect was established in ref. 16. In contrast, here we study junctions in which, in the absence of applied current, the carrier density is approximately position-independent. This situation was analyzed in ref. 27 perturbatively in the ee scattering rate, finding a conductance enhancement that resembles our results.

The relation Eq. **1** points to a simple way to measure viscosity by the conventional transport techniques. Precision measurements of viscosity in fluids date as far back as the 19th century (28). They relied, in particular, on measuring resistance of a viscous fluid discharged through a narrow channel or an orifice, a direct analog of our constriction geometry. Further, viscosity-induced electric conduction has a well-known counterpart in the kinetics of classical gases, where momentum exchange between atoms results in a slower momentum loss and a lower resistance of gas flow. Viscous effects are responsible, in particular, for a dramatic drop in the hydrodynamic resistance upon a transition from Knudsen to Poiseuille regime. For a viscous flow through scatterers spaced by a distance

The peculiar correlations originating from fast particle collisions in proximity to scatterers can be elucidated by a spatial argument: Particle collisions near a scatterer reduce the average velocity component normal to the scatterer surface,

The viscosity-induced drop in resistance can be used as a vehicle to overcome the quantum ballistic limit for electron conduction. Indeed, we can compare the values **1** can be modeled in this way using the time of momentum diffusion across the constriction **1** predicts resistance below the ballistic limit values so long as

Understanding the behavior at the ballistic-to-viscous crossover is a nontrivial task. Here, to tackle the crossover, we use a kinetic equation with a simplified ee collision operator chosen in such a way that the relaxation rates for all nonconserved harmonics of momentum distribution are the same. This model provides a closed-form solution for transport through VPC for any ratio of the length scales

## The Hydrodynamic Regime

We start with a simple derivation of the VPC resistance in Eq. **1** using the model of a low-Reynolds electron flow that obeys the Stokes equation (29).*A* as a slit **4** with no-slip boundary conditions and

Crucially, rather than providing a solution to our problem, the potential-current relation Eq. **5** merely helps to pose it. Indeed, a generic current distribution would yield a potential that is not constant inside the slit. We must therefore determine the functions **5** must be treated as an integral equation for an unknown function **5** as

A solution of this integral equation such that **6**. Potential **9** describes the net contribution of the external field *A*).

Charge density on the strip, found from **9** with the help of Gauss’s law, **8**, gives a semicircle current distribution,*A* is then obtained by plugging this result into Eq. **5**. The flow streamlines are obtained from a similar relation for the stream function (see ref. 30). Evaluating the current **1**. The inverse-square scaling

Potential, inferred from the 2D/3D correspondence, is

## Crossover to the Ballistic Regime

Our next goal is to develop a theory of the ballistic-to-viscous crossover for a constriction. Because we are interested in the linear response, we use the kinetic equation linearized in deviations of particle distribution from the equilibrium Fermi step (assuming

In the presence of momentum-conserving collisions, transport is succinctly described by “quasi-hydrodynamic variables” defined as deviations in the average particle density and momentum from local equilibrium (31). These quantities can be expressed as angular harmonics of the distribution

To facilitate the analysis, we model

To simplify our analysis, we replace the constriction geometry by that of a full plane, with a part of the line

We will analyze the flow induced by a current applied along the **12**, reads**17** as **15** as an operator,

Next, we derive a closed-form integral equation for quasi-hydrodynamic variables by projecting the quantities in Eq. **18** on the **13**. Acting on Eq. **18** with

The quantity **20**. However, rather than attempting to invert **20** in 1D, on the line

We start with finding

The matrix **21**. Plugging Eq. **22** into

In what follows, it will be convenient to transform **k**, translationally invariant integral operators in position representation and diagonal operators in momentum representation.

Next, we evaluate the matrix that represents the operator **25**; as a result we obtain a block diagonal matrix,

Evaluating the integral over **25**, we obtain**20**, giving

The origin of the **29**, and its relation with the properties of the operator **27** and **28**.

We obtain current distribution by solving, numerically, Eq. **29**, discretized as in Eq. **30**, and subsequently Fourier-transforming *Supporting Information*). A large value *B*, features interesting evolution under varying **10**. Current suppression near the constriction edges is in agreement with the streaming picture discussed in the Introduction.

The solution on the line **20** on **x** being a 2D coordinate and **29**, as

This relation provides a route to evaluate resistance. Namely, because of charge neutrality, the density *A* shows

As a quick sanity check on Eq. **33**, we consider the near-collisionless limit **29** turns into an algebraic equation, which is solved by a step-like distribution,**32**. Integrating and taking the limit

The dependence *b*, cf. Fig. S1). Second, quite remarkably, inverting this quantity and plotting *B*. The straight line, which is identical for all *B*.

The additive behavior of conductance at the ballistic-to-viscous crossover comes as a surprise and, to the best of our knowledge, is not anticipated on simple grounds. It is also a stark departure from the Matthiessen’s rule that mandates an additive behavior for resistivity in the presence of different scattering mechanisms, as observed in many solids (32). This rule is, of course, not valid if the factors affecting transport depend on each other, because individual scattering probabilities cannot be summed unless they are mutually independent. The independence is certainly out of question for momentum-conserving ee collisions that do not, by themselves, result in momentum loss but can only impact momentum relaxation due to other scattering mechanisms. Furthermore, the addition rule for conductance, Eq. **3**, describes a striking “anti-Matthiessen” behavior: Rather than being suppressed by collisions, conductance exceeds the collisionless value.

## SI Integral Equation on a Circle

The integral Eq. **29**, which describes current distribution in the constriction, is defined on a line **S1** in the collisionless limit **28**.

In the first case, **S1** turns into an algebraic equation. This equation is solved by

In the second case, **6**. We will now show that the integral Eq. **S1**, in the limit **S3**, we can carry out the integral in Eq. **S1** by the method of residues, closing the integration path through the upper half-plane for **S1**, we determine the normalization factor **33**. The resulting resistance value is**S21**), we find**1**.

Next, to facilitate numerical analysis, we put our 2D problem on a cylinder, choosing a large enough cylinder circumference **S5** is identical to that in Eq. **S1**, because any function **5**, after being continued periodically outside the domain **S1**. We note, however, that such a prescription generates functions that are nonzero not only in the constriction interval **1**, must vanish in the limit

To handle the **S5** in momentum space, with momentum taking discrete values**S5** by inserting a resolution of identity **S6** are limited by

We solve Eq. **S8** numerically to obtain current distributions pictured in Fig. 1*B*: the calculation was done by first finding the distribution

In the plots, the value

Using the solution **33**, giving the conductance

## SI The Hydrodynamic Regime

As an illustration, here we use the approach developed in the main text to solve for the 2D potential distribution, current flow, and conductance in the hydrodynamic regime **S1** is a semicircle,**31**, with **24**, and approximating **S11**, the **24**, vanish at **32** we can compute the density distribution,**S1** (see discussion following Eq. **S4**). Evaluating conductance as in the main text, we find **1**) and the numerical results in Fig. 3. The latter yield the best-fit slope

## SI Hydrodynamic Modes

Here we derive hydrodynamic modes using the method of quasi-hydrodynamic variables, developed in the main text; this will allow us to relate the collision rate **12** in the absence of boundary scattering, **12** takes the form**S16** as **S18** is performed by writing **v** and wavevector **k**, and integrating over

As we now show, the equations

This transformation brings the **S21** agrees with that obtained from the linearized Navier–Stokes equation

The acoustic mode can be obtained from the even-mode **S21**.

## Acknowledgments

We thank M. Reznikov for useful discussions and acknowledge support of the Center for Integrated Quantum Materials (CIQM) under NSF Award 1231319 (to L.S.L.); partial support by the US Army Research Laboratory and the US Army Research Office through the Institute for Soldier Nanotechnologies, under Contract W911NF-13-D-0001 (to L.S.L.); The US-Israel Binational Science Foundation (L.S.L.); MISTI MIT-Israel Seed Fund (L.S.L. and G.F.); the Israeli Science Foundation Grant 882 (to G.F.); and the Russian Science Foundation Project 14-22-00259 (to G.F.).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: levitov{at}mit.edu.

Author contributions: H.G., G.F., and L.S.L. designed research; H.G., E.I., G.F., and L.S.L. performed research; and H.G., G.F., and L.S.L. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1612181114/-/DCSupplemental.

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