Preturbulent regimes in graphene flow

Para-hydrodynamics from weak surface scattering in ultraclean thin flakes

Electron hydrodynamics typically emerges in electron fluids with a high electron-electron collision rate. However, new experiments with thin flakes of WTe₂ have revealed that other momentum-conserving scattering processes can replace the role of the electron-electron interaction, thereby leading to a novel, so-called para-hydrodynamic regime. Here, we develop the kinetic theory for para-hydrodynamic transport. To this end, we consider a ballistic electron gas in a thin three-dimensional sheet where the momentum-relaxing (lmr) and momentum-conserving (lmc) mean free paths are decreased due to boundary scattering from a rough surface. The resulting effective mean free path of the in-plane components of the electronic flow is then expressed in terms of microscopic parameters of the sheet boundaries, predicting that a para-hydrodynamic regime with lmr ≫ lmc emerges generically in ultraclean three-dimensional materials. Using our approach, we recover the transport properties of WTe₂ in the para-hydrodynamic regime in good agreement with existing experiments.

Direct observation of vortices in an electron fluid

Vortices are the hallmarks of hydrodynamic flow. Strongly interacting electrons in ultrapure conductors can display signatures of hydrodynamic behaviour, including negative non-local resistance^1-4, higher-than-ballistic conduction^5-7, Poiseuille flow in narrow channels^8-10 and violation of the Wiedemann-Franz law¹¹. Here we provide a visualization of whirlpools in an electron fluid. By using a nanoscale scanning superconducting quantum interference device on a tip¹², we image the current distribution in a circular chamber connected through a small aperture to a current-carrying strip in the high-purity type II Weyl semimetal WTe₂. In this geometry, the Gurzhi momentum diffusion length and the size of the aperture determine the vortex stability phase diagram. We find that vortices are present for only small apertures, whereas the flow is laminar (non-vortical) for larger apertures. Near the vortical-to-laminar transition, we observe the single vortex in the chamber splitting into two vortices; this behaviour is expected only in the hydrodynamic regime and is not anticipated for ballistic transport. These findings suggest a new mechanism of hydrodynamic flow in thin pure crystals such that the spatial diffusion of electron momenta is enabled by small-angle scattering at the surfaces instead of the routinely invoked electron-electron scattering, which becomes extremely weak at low temperatures. This surface-induced para-hydrodynamics, which mimics many aspects of conventional hydrodynamics including vortices, opens new possibilities for exploring and using electron fluidics in high-mobility electron systems.

Evidence for Local Spots of Viscous Electron Flow in Graphene at Moderate Mobility

Dominating electron-electron scattering enables viscous electron flow exhibiting hydrodynamic current density patterns, such as Poiseuille profiles or vortices. The viscous regime has recently been observed in graphene by nonlocal transport experiments and mapping of the Poiseuille profile. Herein, we probe the current-induced surface potential maps of graphene field-effect transistors with moderate mobility using scanning probe microscopy at room temperature. We discover micrometer-sized large areas appearing close to charge neutrality that show current-induced electric fields opposing the externally applied field. By estimating the local scattering lengths from the gate dependence of local in-plane electric fields, we find that electron-electron scattering dominates in these areas as expected for viscous flow. Moreover, we suppress the inverted fields by artificially decreasing the electron-disorder scattering length via mild ion bombardment. These results imply that viscous electron flow is omnipresent in graphene devices, even at moderate mobility.

Transport coefficients for hard-sphere relativistic gas

Transport coefficients are of crucial importance in theoretical as well as experimental studies. Despite substantial research on classical hard sphere or disk gases in low- and high-density regimes, a thorough investigation of transport coefficients for massive relativistic systems is missing in the literature. In this work a fully relativistic molecular dynamics simulation is employed to numerically obtain the transport coefficients of a hard sphere relativistic gas based on Helfand-Einstein expressions. The numerical data are then used to check the accuracy of Chapmann-Enskog (CE) predictions in a wide range of temperature. The results indicate that while simulation data in low-temperature regime agrees very well with theoretical predictions, it begins to show deviations as temperature rises, except for the thermal conductivity which fits very well to CE theory in the whole range of temperature. Since our simulations are done in low density regimes, where CE approximation is expected to be valid, the observed deviations can be attributed to the inaccuracy of linear CE theory in extremely relativistic cases.

Electron-phonon instability in graphene revealed by global and local noise probes

Understanding and controlling nonequilibrium electronic phenomena is an outstanding challenge in science and engineering. By electrically driving ultraclean graphene devices out of equilibrium, we observe an instability that is manifested as substantially enhanced current fluctuations and suppressed conductivity at microwave frequencies. Spatial mapping of the nonequilibrium current fluctuations using nanoscale magnetic field sensors reveals that the fluctuations grow exponentially along the direction of carrier flow. Our observations, including the dependence on density and temperature, are consistently explained by the emergence of an electron-phonon Cerenkov instability at supersonic drift velocities. These results offer the opportunity for tunable terahertz generation and active phononic devices based on two-dimensional materials.

Prospects for the Detection of Electronic Preturbulence in Graphene

Based on extensive numerical simulations, accounting for electrostatic interactions and dissipative electron-phonon scattering, we propose experimentally realizable geometries capable of sustaining electronic preturbulence in graphene samples. In particular, preturbulence is predicted to occur at experimentally attainable values of the Reynolds number between 10 and 50, over a broad spectrum of frequencies between 10 and 100 GHz.

Modelling electron-phonon interactions in graphene with curved space hydrodynamics

We introduce a different perspective describing electron-phonon interactions in graphene based on curved space hydrodynamics. Interactions of phonons with charge carriers increase the electrical resistivity of the material. Our approach captures the lattice vibrations as curvature changes in the space through which electrons move following hydrodynamic equations. In this picture, inertial corrections to the electronic flow arise naturally effectively producing electron-phonon interactions. The strength of the interaction is controlled by a coupling constant, which is temperature independent. We apply this model to graphene and recover satisfactorily the linear scaling law for the resistivity that is expected at high temperatures. Our findings open up a new perspective of treating electron-phonon interactions in graphene, and also in other materials where electrons can be described by the Fermi liquid theory.

Hydrodynamics of electrons in graphene

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.

Linking Spatial Distributions of Potential and Current in Viscous Electronics

Viscous electronics is an emerging field dealing with systems in which strongly interacting electrons behave as a fluid. Electron viscous flows are governed by a nonlocal current-field relation which renders the spatial patterns of the current and electric field strikingly dissimilar. Notably, driven by the viscous friction force from adjacent layers, current can flow against the electric field, generating negative resistance, vorticity, and vortices. Moreover, different current flows can result in identical potential distributions. This sets a new situation where inferring the electron flow pattern from the measured potentials presents a nontrivial problem. Using the inherent relation between these patterns through complex analysis, here we propose a method for extracting the current flows from potential distributions measured in the presence of a magnetic field.

Kinetic approach to relativistic dissipation

Despite a long record of intense effort, the basic mechanisms by which dissipation emerges from the microscopic dynamics of a relativistic fluid still elude complete understanding. In particular, several details must still be finalized in the pathway from kinetic theory to hydrodynamics mainly in the derivation of the values of the transport coefficients. In this paper, we approach the problem by matching data from lattice-kinetic simulations with analytical predictions. Our numerical results provide neat evidence in favor of the Chapman-Enskog [The Mathematical Theory of Non-Uniform Gases, 3rd ed. (Cambridge University Press, Cambridge, U.K., 1970)] procedure as suggested by recent theoretical analyses along with qualitative hints at the basic reasons why the Chapman-Enskog expansion might be better suited than Grad's method [Commun. Pure Appl. Math. 2, 331 (1949)0010-364010.1002/cpa.3160020403] to capture the emergence of dissipative effects in relativistic fluids.

Higher-than-ballistic conduction of viscous electron flows

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.

Holography of the Dirac Fluid in Graphene with Two Currents

Recent experiments have uncovered evidence of the strongly coupled nature of graphene: the Wiedemann-Franz law is violated by up to a factor of 20 near the charge neutral point. We describe this strongly coupled plasma by a holographic model in which there are two distinct conserved U(1) currents. We find that our analytic results for the transport coefficients for the two current model have a significantly improved match to the density dependence of the experimental data than the models with only one current. The additive structure in the transport coefficients plays an important role. We also suggest the origin of the two currents.

Poiseuille flow in curved spaces

We investigate Poiseuille channel flow through intrinsically curved media, equipped with localized metric perturbations. To this end, we study the flux of a fluid driven through the curved channel in dependence of the spatial deformation, characterized by the parameters of the metric perturbations (amplitude, range, and density). We find that the flux depends only on a specific combination of parameters, which we identify as the average metric perturbation, and derive a universal flux law for the Poiseuille flow. For the purpose of this study, we have improved and validated our recently developed lattice Boltzmann model in curved space by considerably reducing discrete lattice effects.

Flow through randomly curved manifolds

We present a computational study of the transport properties of campylotic (intrinsically curved) media. It is found that the relation between the flow through a campylotic media, consisting of randomly located curvature perturbations, and the average Ricci scalar of the system, exhibits two distinct functional expressions, depending on whether the typical spatial extent of the curvature perturbation lies above or below the critical value maximizing the overall scalar of curvature. Furthermore, the flow through such systems as a function of the number of curvature perturbations is found to present a sublinear behavior for large concentrations, due to the interference between curvature perturbations leading to an overall less curved space. We have also characterized the flux through such media as a function of the local Reynolds number and the scale of interaction between impurities. For the purpose of this study, we have also developed and validated a new lattice Boltzmann model.

Exact lattice Boltzmann equation

The lattice Boltzmann equation is derived from the Bhatnagar-Gross-Krook kinetic equation using the Euler-Maclaurin integration formula. Unlike previous attempts to connect the lattice Boltzmann method with the kinetic theory, the result is free of any relaxation-type approximation. It shows that the conventional lattice Bhatnagar-Gross-Krook simulations of hydrodynamics belong to a parameter domain which is disconnected from the kinetic theory domain.

Gaussian quadrature and lattice discretization of the Fermi-Dirac distribution for graphene

We construct a lattice kinetic scheme to study electronic flow in graphene. For this purpose, we first derive a basis of orthogonal polynomials, using as the weight function the ultrarelativistic Fermi-Dirac distribution at rest. Later, we use these polynomials to expand the respective distribution in a moving frame, for both cases, undoped and doped graphene. In order to discretize the Boltzmann equation and make feasible the numerical implementation, we reduce the number of discrete points in momentum space to 18 by applying a Gaussian quadrature, finding that the family of representative wave (2+1)-vectors, which satisfies the quadrature, reconstructs a honeycomb lattice. The procedure and discrete model are validated by solving the Riemann problem, finding excellent agreement with other numerical models. In addition, we have extended the Riemann problem to the case of different dopings, finding that by increasing the chemical potential the electronic fluid behaves as if it increases its effective viscosity.

Hydrodynamic model for conductivity in graphene

Based on the recently developed picture of an electronic ideal relativistic fluid at the Dirac point, we present an analytical model for the conductivity in graphene that is able to describe the linear dependence on the carrier density and the existence of a minimum conductivity. The model treats impurities as submerged rigid obstacles, forming a disordered medium through which graphene electrons flow, in close analogy with classical fluid dynamics. To describe the minimum conductivity, we take into account the additional carrier density induced by the impurities in the sample. The model, which predicts the conductivity as a function of the impurity fraction of the sample, is supported by extensive simulations for different values of ε, the dimensionless strength of the electric field, and provides excellent agreement with experimental data.

Transition in the equilibrium distribution function of relativistic particles

We analyze a transition from single peaked to bimodal velocity distribution in a relativistic fluid under increasing temperature, in contrast with a non-relativistic gas, where only a monotonic broadening of the bell-shaped distribution is observed. Such transition results from the interplay between the raise in thermal energy and the constraint of maximum velocity imposed by the speed of light. We study the Bose-Einstein, the Fermi-Dirac, and the Maxwell-Jüttner distributions, and show that they all exhibit the same qualitative behavior. We characterize the nature of the transition in the framework of critical phenomena and show that it is either continuous or discontinuous, depending on the group velocity. We analyze the transition in one, two, and three dimensions, with special emphasis on twodimensions, for which a possible experiment in graphene, based on the measurement of the Johnson-Nyquist noise, is proposed.

Clustering of inelastic soft spheres in homogeneous turbulence

In this paper we numerically investigate the influence of dissipation during particle collisions in an homogeneous turbulent velocity field by coupling a discrete element method to a lattice-Boltzmann simulation with spectral forcing. We show that even at moderate particle volume fractions the influence of dissipative collisions is important. We also investigate the transition from a regime where the turbulent velocity field significantly influences the spatial distribution of particles to a regime where the distribution is mainly influenced by particle collisions.