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Higher-Than-Ballistic Conduction of Viscous Electron Flows

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Higher-Than-Ballistic Conduction of Viscous Electron Flows Higher-than-ballistic conduction of viscous electron flows Haoyu Guoa, Ekin Ilsevena, Gregory Falkovichb,c, and Leonid S. Levitova,1 aDepartment of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139; bDepartment of Physics, Weizmann Institute of Science, Rehovot 76100, Israel; and cInstitute for Information Transmission Problems, Moscow 127994, Russia Edited by Allan H. MacDonald, The University of Texas at Austin, Austin, TX, and approved January 24, 2017 (received for review July 25, 2016) Strongly interacting electrons can move in a neatly coordinated Conveniently, both regimes are accessible in a single constric- way, reminiscent of the movement of viscous fluids. Here, we tion, because transport is expected to be viscous at elevated tem- show that in viscous flows, interactions facilitate transport, allow- peratures and ballistic at T = 0. The crossover temperature can ing conductance to exceed the fundamental Landauer’s ballistic be estimated in terms of the ee scattering mean free path as limit Gball. The effect is particularly striking for the flow through l (T )=w = π2=16 ≈ 0:62: [2] a viscous point contact, a constriction exhibiting the quantum ee mechanical ballistic transport at T = 0 but governed by electron This relation is found by setting Rvis = Rball and expressing viscos- 1 hydrodynamics at elevated temperatures. We develop a theory ity as η = νnm = 4 vF leenm, where m is the carrier mass and the of the ballistic-to-viscous crossover using an approach based on kinetic viscosity ν is estimated in Eq. S21. The condition Eq. 2 quasi-hydrodynamic variables. Conductance is found to obey an can be readily met in micron-size graphene junctions. additive relation G = Gball + Gvis, where the viscous contribution Several effects of electron interactions on transport in con- Gvis dominates over Gball in the hydrodynamic limit. The super- strictions were discussed recently. Refs. 14 and 15 study junc- ballistic, low-dissipation transport is a generic feature of viscous tions with spatially varying electron density and, using the electronics. time-dependent current density functional theory, predict a sup- pression of conductance. A hydrodynamic picture of this effect electron hydrodynamics j graphene j strongly correlated systems was established in ref. 16. In contrast, here we study junctions in which, in the absence of applied current, the carrier density ree electron flow through constrictions in metals is often is approximately position-independent. This situation was ana- Fregarded as an ultimate high-conduction charge transfer lyzed in ref. 27 perturbatively in the ee scattering rate, finding a mechanism (1–5). Can conductance ever exceed the ballistic limit conductance enhancement that resembles our results. value? Here we show that superballistic conduction is possi- The relation Eq. 1 points to a simple way to measure viscos- ble for strongly interacting systems in which electron movement ity by the conventional transport techniques. Precision measure- resembles that of viscous fluids. Electron fluids are predicted ments of viscosity in fluids date as far back as the 19th century to occur in quantum-critical systems and in high-mobility con- (28). They relied, in particular, on measuring resistance of a vis- ductors, so long as momentum-conserving electron–electron (ee) cous fluid discharged through a narrow channel or an orifice, scattering dominates over other scattering processes (6–9). Vis- a direct analog of our constriction geometry. Further, viscosity- cous electron flows feature a host of novel transport behaviors induced electric conduction has a well-known counterpart in the (10–22). Signatures of such flows have been observed in ultra- kinetics of classical gases, where momentum exchange between clean GaAs, graphene, and ultrapure PdCoO2 (23–26). atoms results in a slower momentum loss and a lower resistance We will see that electrons in a viscous flow can achieve of gas flow. Viscous effects are responsible, in particular, for a through cooperation what they cannot accomplish individually. dramatic drop in the hydrodynamic resistance upon a transition As a result, resistance and dissipation of a viscous flow can be from Knudsen to Poiseuille regime. For a viscous flow through markedly smaller than that for the free-fermion transport. As a simplest realization, we discuss viscous point contact (VPC), Significance where correlations act as a ”lubricant” facilitating the flow. The reduction in resistance arises due to the streaming effect Free electron flows through constrictions in metals are often illustrated in Fig. 1, wherein electron currents bundle up to regarded as an ultimate high-conduction charge transfer. We form streams that bypass the boundaries, where momentum loss predict that electron fluids can flow with a resistance that occurs. This surprising behavior is in a clear departure from the is much smaller than the fundamental quantum mechani- common view that regards electron interactions as an impedi- cal ballistic limit for nanoscale electronics. The “superballis- ment for transport. tic” low-dissipation transport is particularly striking for the A simplest VPC is a 2D constriction pictured in Fig. 1A. The flow through a viscous point contact, a constriction exhibit- interaction effects dominate in constrictions of width w exceed- ing the quantum mechanical ballistic transport at zero tem- ing the carrier collision mean free path l (and much greater ee perature but governed by viscous electron hydrodynamics than the Fermi wavelength λF ). The VPC conductance, evalu- at a higher temperature. Unlike other mechanisms of low- ated in the absence of impurity scattering, scales as a square of dissipation transport, for example, superconductivity, the vis- the width w and inversely with the electron viscosity η, cous electron flows can be realized at elevated temperatures, πn2e2w 2 granting a new route for the low-power electronics research. G (w) = ; w l ; [1] vis 32η ee Author contributions: H.G., G.F., and L.S.L. designed research; H.G., E.I., G.F., and L.S.L. where n and e are the carrier density and charge. In the opposite performed research; and H.G., G.F., and L.S.L. wrote the paper. limit, lee w, the ballistic free-fermion model (1, 5) predicts the The authors declare no conflict of interest. 2 conductance Gball = 2e =h N , where N ≈ 2w/λF is the number This article is a PNAS Direct Submission. of Landauer’s open transmission channels. The conductance Gvis 1To whom correspondence should be addressed. Email: [email protected]. grows with width faster than Gball. Therefore, for large enough This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. w, viscous transport yields G values above the ballistic bound. 1073/pnas.1612181114/-/DCSupplemental. 3068–3073 j PNAS j March 21, 2017 j vol. 114 j no. 12 www.pnas.org/cgi/doi/10.1073/pnas.1612181114 Downloaded by guest on September 30, 2021 Understanding the behavior at the ballistic-to-viscous cross- over 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 harmon- ics of momentum distribution are the same. This model provides a closed-form solution for transport through VPC for any ratio of the length scales w and lee, predicting a simple additive relation GVPC = Gball + Gvis: [3] This dependence, derived from a microscopic model, interpo- lates between the ballistic and viscous limits, w lee and w lee, in which the terms Gball and Gvis, respectively, dominate. 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). (ηr2 − (ne)2ρ)v(r)= nerφ(r): [4] Here φ(r) is the electric potential, η is the viscosity, and the sec- ond term describes ohmic resistivity due to impurity or phonon scattering. Our analysis relies on a symmetry argument and invokes an auxiliary electrostatic problem. We model the con- w w striction in Fig. 1A as a slit − 2 < x < 2 , y = 0. The y ! −y sym- metry ensures that the current component jy is an even function of y whereas both the component jx and the potential φ are odd in y. As a result, the quantities jx and φ vanish within the slit at y = 0. This observation allows us to write the potential in the plane as a superposition of contributions due to different current elements in the slit, w Z 2 φ(x; y) = dx 0R(x − x 0; y)j (x 0); [5] Fig. 1. (A) Current streamlines (black) and potential color map for viscous w − 2 flow through a constriction. Velocity magnitude is proportional to the den- sity of streamlines. Current forms a narrow stream, avoiding the bound- 2 2 where the influence function R(x; y) = β(y −x ) describes aries where dissipation occurs and allowing the resistance, Eq. 1, to drop (x 2+y2)2 below the ballistic limit value. (B) Current distribution in the constriction potential in a half-plane due to a point-like current source at the for different carrier collision mean-free-path values. The distribution evolves edge, obtained from Eq. 4 with no-slip boundary conditions and from a constant in the ballistic regime to a semicircle in the viscous regime, 2η ρ = 0 (30). Here β = 2 , and, without loss of generality, we Eq. 10, illustrating the interaction-induced streaming effect. Parameters π(en) used are L = 3w and b = 105v. The distributions are normalized to unit focus on the y > 0 half-plane. total current. A Fourier space filter was used to smooth out the Gibbs phe- Crucially, rather than providing a solution to our problem, the nomenon. potential-current relation Eq. 5 merely helps to pose it.
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