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Visualizing Poiseuille flow of hydrodynamic electrons

Joseph Sulpizio, Lior Ella, Asaf Rozen, John Birkbeck, David Perello, Debarghya Dutta, M. Ben Shalom, Takashi Taniguchi, Kenji Watanabe, Tobias Holder, Raquel Queiroz, Alessandro Principi, Ady Stern, Thomas Scaffidi, Andre K. Geim, Shahal Ilani

Published Jan 1, 2019• 100 citations
OpenAlex

Abstract

Abstract Hydrodynamics, which generally describes the flow of a fluid, is expected to hold even for fundamental particles such as electrons when inter-particle interactions dominate1. Although various aspects of electron hydrodynamics have been revealed in recent experiments2–11, the fundamental spatial structure of hydrodynamic electrons—the Poiseuille flow profile—has remained elusive. Here we provide direct imaging of the Poiseuille flow of an electronic fluid, as well as a visualization of its evolution from ballistic flow. Using a scanning carbon nanotube single-electron transistor12, we image the Hall voltage of electronic flow through channels of high-mobility graphene. We find that the profile of the Hall field across the channel is a key physical quantity for distinguishing ballistic from hydrodynamic flow. We image the transition from flat, ballistic field profiles at low temperatures into parabolic field profiles at elevated temperatures, which is the hallmark of Poiseuille flow. The curvature of the imaged profiles is qualitatively reproduced by Boltzmann calculations, which allow us to create a ‘phase diagram’ that characterizes the electron flow regimes. Our results provide direct confirmation of Poiseuille flow in the solid state, and enable exploration of the rich physics of interacting electrons in real space.

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Abstract Hydrodynamics, which generally describes the flow of a fluid, is expected to hold even for fundamental particles such as electrons when inter-particle interactions dominate1. Although various aspects of electron hydrodynamics have been revealed in recent experiments2–11, the fundamental spatial structure of hydrodynamic electrons—the Poiseuille flow profile—has remained elusive. Here we provide direct imaging of the Poiseuille flow of an electronic fluid, as well as a visualization of its evolution from ballistic flow. Using a scanning carbon nanotube single-electron transistor12, we image the Hall voltage of electronic flow through channels of high-mobility graphene. We find that the profile of the Hall field across the channel is a key physical quantity for distinguishing ballistic from hydrodynamic flow. We image the transition from flat, ballistic field profiles at low temperatures into parabolic field profiles at elevated temperatures, which is the hallmark of Poiseuille flow. The curvature of the imaged profiles is qualitatively reproduced by Boltzmann calculations, which allow us to create a ‘phase diagram’ that characterizes the electron flow regimes. Our results provide direct confirmation of Poiseuille flow in the solid state, and enable exploration of the rich physics of interacting electrons in real space.
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 [Formula: see text] The effect is particularly striking for the flow through a viscous point contact, a constriction exhibiting the quantum mechanical ballistic transport at [Formula: see text] but governed by electron hydrodynamics at elevated temperatures. We develop a theory of the ballistic-to-viscous crossover using an approach based on quasi-hydrodynamic variables. Conductance is found to obey an additive relation [Formula: see text], where the viscous contribution [Formula: see text] dominates over [Formula: see text] in the hydrodynamic limit. The superballistic, low-dissipation transport is a generic feature of viscous electronics.
Hydrodynamic phenomena seem to be ubiquitous in the universe. Hydrodynamics provides an effective, long-wavelength description of both classical and quantum systems. We study the universal spatial features of certain non-equilibrium steady states corresponding to flows of strongly correlated fluids over obstacles. We identify spatial collective modes of the strongly coupled many body system. These modes can be both hydrodynamical or non-hydrodynamical. Analytic examples can be found in the large-dimension limit of the bulk theory, but in three dimensions as well. We construct the general first-order hydrodynamic theory invariant under time translations, the Euclidean group of spatial transformations and preserving particle number. Such theories are important in many situations, from hydrodynamics of graphene to flocking behaviour of birds and motion of self-propelled organisms. We apply our non-boost invariant model of hydrodynamics to concrete flow configurations - the Poiseuille, Couette and Taylor-Couette flows. We also test the instability of the Poiseuille flow.
Generic interacting many-body quantum systems are believed to behave as classical fluids on long time and length scales. Due to rapid progress in growing exceptionally pure crystals, we are now able to experimentally observe this collective motion of electrons in solid-state systems, including graphene. We present a review of recent progress in understanding the hydrodynamic limit of electronic motion in graphene, written for physicists from diverse communities. We begin by discussing the 'phase diagram' of graphene, and the inevitable presence of impurities and phonons in experimental systems. We derive hydrodynamics, both from a phenomenological perspective and using kinetic theory. We then describe how hydrodynamic electron flow is visible in electronic transport measurements. Although we focus on graphene in this review, the broader framework naturally generalizes to other materials. We assume only basic knowledge of condensed matter physics, and no prior knowledge of hydrodynamics.
The emergence of hydrodynamic behavior in electronic flow within clean, particle-hole-symmetric systems at half filling is a nontrivial problem. Navier-Stokes (NS) equations describe the momentum flow, while experimental measurements typically capture the current flow profiles. However, in particle-hole-symmetric systems, electric current and momentum flow are entirely decoupled, because electrons and holes move in opposite directions with equal distribution functions. This makes it challenging to link NS equations to observed flow patterns. In this Letter, we demonstrate that the hydrodynamic behavior of the charge current at half filling can emerge despite the absence of momentum flow. By combining Boltzmann transport theory with numerically exact quantum Monte Carlo simulations of clean graphene samples, we show that NS-type equations can be derived directly for the charge current, eliminating the need for any additional mechanism coupling the velocity field and charge current in explaining the experimentally observed hydrodynamic flow profiles in graphene at half filling. We show that a new transport quantity-the current diffusion coefficient-replaces viscosity and expect this description to be valid for any particle-hole-symmetric system. Our results provide new insights into the interpretation of experimental data and demonstrate how quantum Monte Carlo calculations can serve as an alternative to experiments in transport measurements to verify the kinetic theory results.
The quantum Hall effect realizes a quantized Hall resistance Rxy=h/(νe2)R_{xy} = h/(νe^2)Rxy​=h/(νe2) whereas the longitudinal resistance vanishes. The quantized value consists of the fundamental physical quantities, the elementary charge eee and the Planck constant hhh, along with an integer or fractional constant ννν. High precision measurements of RxyR_{xy}Rxy​ allude to a linear relation between the applied current III and the Hall voltage VHV_\mathrm{H}VH​. Here, we argue that a nonlinear relation between III and VHV_\mathrm{H}VH​ could arise when the electric field is spatially inhomogeneous. We first discuss that the linear III-VHV_\mathrm{H}VH​ relation holds with Galilean invariance. Then we consider a hydrodynamic description of a quantum Hall liquid to deal with an axially symmetric electric field. It reveals a nonlinear electronic response arising from the centrifugal force exerted on a curved flow and the density gradient invoked by vorticity.
Electronic transport in Fermi liquids is usually Ohmic, because of momentum-relaxing scattering due to defects and phonons. These processes can become sufficiently weak in two-dimensional materials, giving rise to either ballistic or hydrodynamic transport, depending on the strength of electron-electron scattering. We show that the ballistic regime is a quantum critical point (QCP) on the regime boundary separating Ohmic and hydrodynamic transport. The QCP corresponds to a \emph{free} conformal field theory (CFT) with a dynamical scaling exponent z=1z = 1z=1. Its nontrivial aspects emerge in device geometries with shear, wherein the regime has an intrinsic universal dissipation, a nonlocal current-voltage relation, and exhibits the critical scaling of the underlying CFT. The Fermi surface has electron-hole pockets across all angular scales and the current flow has \emph{vortices} at all spatial scales. We image the fluctuations in high-definition and animate their emergence as experimental parameters are tuned to the QCP (movie links in comments). The vortices clearly demonstrate that Pauli exclusion alone can produce collective effects, with low-frequency AC transport mediated by vortex dynamics. The scale-invariant spatial structure is much richer than that of an interaction-dominated hydrodynamic regime, which only has a single vortex at the device scale. Our findings provide a theoretical framework for both interaction-free and interaction-dominated non-Ohmic transport in two-dimensional materials, as seen in several contemporary experiments.