Abstract
Viscous electron fluids have emerged recently as a new paradigm of stronglycorrelated electron transport in solids. Here we report on a direct observation of the transition to this longsoughtfor state of matter in a highmobility electron system in graphene. Unexpectedly, the electron flow is found to be interactiondominated but nonhydrodynamic (quasiballistic) in a wide temperature range, showing signatures of viscous flows only at relatively high temperatures. The transition between the two regimes is characterized by a sharp maximum of negative resistance, probed in proximity to the current injector. The resistance decreases as the system goes deeper into the hydrodynamic regime. In a perfect darknessbeforedaybreak manner, the interactiondominated negative response is strongest at the transition to the quasiballistic regime. Our work provides the first demonstration of how the viscous fluid behavior emerges in an interacting electron system.
Introduction
Electron fluids, an exotic state of matter in which electron–electron (ee) interactions dominate transport, have been long anticipated theoretically^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15} but until recently they were far from experimental reality. This situation is currently changing owing to the discovery of new materials in which ee interactions are particularly strong or momentum relaxation due to disorder and phonons is weak. The inventory of experimental systems that can host viscous efluids, as we will call them for brevity, has been steadily growing in the last few years^{16,17,18,19}, stimulating wide interest in their properties. Efluids may exhibit new behaviors such as vortices^{20,21}, whirlpools^{16}, superballistic transport^{22,23}, Poiseuille flow^{10,11,13,14,18}, anomalous heat conduction^{17}, and viscous magnetotransport^{24,25}. The questions about the genesis of efluids, on the other hand, received relatively little attention. How does an electron system enter the fluid state? What happens when l_{ee} becomes comparable or larger than the system dimensions? What is the relation between electric current and potential at the transition? All these questions are at present poorly understood: neither there exists a detailed theory treating both ballistic and viscous electron regimes on equal footing, nor any systematic experimental study of the transition has been performed. Searching for the fluidity onset is the subject of this work.
So far, the behavior of efluids was mostly discussed deep in the hydrodynamic regime, where the mean free path l_{ee} was the shortest lengthscale of the system. However, the experimental conditions are usually such that l_{ee}, tunable by varying temperature T, is either comparable or at most a few times smaller than the system dimensions, putting the experimentally investigated efluids close to the onset of fluidity. As we will show below, this regime hosts an interactiondominated quasiballistic state, which exhibits a negative voltage response similar to that observed at nottoohigh T in the ref.^{16}. The negative response arises because ambient carriers, as a result of momentumconserving collisions with injected carriers, are blocked from reaching voltage probes. Furthermore, the negative response is enhanced by “memory effects”, so that it may exceed the negative response in the viscous state^{26}. Thus, the interactiondominated quasiballistic state, while quite distinct from the viscous fluid state, can in some cases serve as a proxy for the latter.
Graphene offers a convenient venue for this study. First, due to their exceptional cleanness and weak electron–phonon (el–ph) coupling, stateoftheart graphene devices support micrometerscale ballistic transport with respect to momentumnonconserving collisions over a wide range of temperatures^{27}, from liquidhelium to room T. Second, above the temperatures of liquid nitrogen, ee collisions become the dominant scattering mechanism, so that the behavior of the electron system resembles that of viscous fluids^{16,23}. Third, l_{ee} in graphene can be varied over a wide range^{23} by changing the carrier density n and T. This enables a smooth transition (or, more precisely, a crossover) between singleparticle ballistic and viscous transport regimes, allowing us to track how the electron system enters the collective fluid state.
Results
Experimental data
We explore the onset of the hydrodynamic state by studying graphene devices in the socalled vicinity geometry^{16}, illustrated in Fig. 1a: The current I is injected through a narrow contact into a wide graphene channel, and a local potential is probed at a small distance x from the injector. The main result of our study is that the vicinity resistance R_{v} = V/I reaches an extreme negative value at the onset of fluidity. In particular, this behavior manifests itself most clearly through the temperature dependence of R_{v} (Fig. 1b, c), with the quasiballistic and hydrodynamic regimes occurring at low and high T, respectively. We will show that the deep minimum at intermediate temperatures in the R_{v} (T) dependences is the hallmark of the transition. Furthermore, we will demonstrate that this transition can be conveniently quantified by the electron Knudsen number
taking values \({\mathrm{Kn}} \ll 1\) and Kn > 1 in the hydrodynamic and quasiballistic transport regimes, respectively, and approaching unity at the fluidity onset.
Importantly, the negative sign of R_{v}, observed across the entire transition, signals that ee interactions dominate transport in both the quasiballistic and hydrodynamic regimes. The hydrodynamic regime, where theory predicts dR_{v}/dT > 0^{20,28}, occurs only at high enough temperatures and low enough carrier densities. This regime is preceded by an extended quasiballistic regime with dR_{v}/dT < 0, discussed in detail below. The occurrence of two distinct interactiondominated regimes in a 2D electron system is a surprising finding, which is of interest from a fundamental perspective and important for possible applications.
To explore the onset of the fluid state experimentally, we fabricated highquality devices based on bilayer graphene (BLG) encapsulated between hexagonal boron nitride (for details, see Methods). The latter provides a clean environment for graphene’s electron system ensuring micrometerscale ballistic transport with respect to extrinsic momentumnonconserving scatterering. The devices were shaped in a form of dualgated multiterminal Hall bars (Fig. 1a), allowing us to study the distancedependent potential anticipated at the transition upon varying the carrier densities n. The dualgated design allowed us to maintain zero displacement between the graphene layers, so that one could tune the Fermi energy ε_{F} in BLG without altering its band structure (opening the band gap). We have strategically chosen the BLG system because it ε_{F} varies with n stronger than in monolayer graphene (MLG) (n vs. n^{1/2}). The standard dependence l_{ee} = ℏv_{F}ε_{F}/(k_{B}T)^{2} translates into the scaling l_{ee} ∼ n^{3/2}, which is much faster than the n^{1/2} dependence in MLG. This allowed us to explore a wider range of l_{ee} than in MLG by varying the carrier density for a given T (see below), providing a convenient knob to tune the Kn value and probe the quasiballistictohydrodynamic transition^{29}.
Notably, the signal measured in the vicinity configuration contains a nonnegligible offset due to momentumnonconserving scattering (by phonons and/or disorder) which we further refer to as an Ohmic contribution. To distill the viscous contribution, we employed the approach introduced in the ref. ^{16} in which the Ohmic term, expressed as bρ, was subtracted from the measured vicinity signal, assuming the additive behavior of these contributions^{28}. Here ρ = ρ(n, T) is the BLG sheet resistance measured in the conventional fourterminal geometry and b is the geometric factor that depends on sample dimensions and the distance between the injection point and the voltage probe^{16,28} (for example, b = 0.1 for the measurementent configuration shown in Fig. 1a). As discussed below, the procedure of subtracting the Ohmic contribution, while somewhat ad hoc, can be justified for the geometry of our experiment. Below we refer to this adjusted vicinity resistance using the same notation R_{v} unless stated otherwise.
Figure 1b shows R_{v} as a function of T measured in one of our BLG devices. Far away from the charge neutrality point (CNP) and at liquid helium T, R_{v} is positive for all experimentally accessible n. When the temperature is increased, R_{v} rapidly drops, reverses its sign, reaches a minimum and then starts to grow. Figure 2a details this observation by mapping R_{v} on the (n, T)plane. The nonmonotonic dependence R_{v} vs. T is observed for all n, whereby the temperature at which R_{v} dips, grows with increasing n (red dashed line).
To understand this nonmonotonic behavior, we first consider the limiting cases: the hydrodynamic regime \(l_{{\mathrm{ee}}} \ll x\), realized at large T, and the freeparticle regime \(l_{{\mathrm{ee}}} \gg W\), realized at the lowest T (here W is the device width). In the hydrodynamic regime, negative R_{v} arises as a result of viscous entrainment by the injected current of the fluid in adjacent regions^{16,20,28}. In the freeparticle regime, positive R_{v} is expected from singleparticle ballistic transport due to reflection of injected carriers from the opposite boundaries^{30,31}. Therefore, the sign of R_{v} must change from negative to positive upon lowering T, as indeed seen in the data shown in Fig. 1a. Furthermore, the hydrodynamic R_{v} is proportional to viscosity^{20,28}, giving the dependence \(R_{\mathrm{v}}\sim l_{{\mathrm{ee}}}(T)\). The quantity l_{ee}(T) increases as T decreases, leading to increasingly more negative R_{v}. The nonmonotonic temperature dependence R_{v}(T), implied by these observations, is indeed seen in our measurements (Figs. 1b, 2a).
Importantly, in between the freeparticle regime \(l_{{\mathrm{ee}}} \gg W\) and the hydrodynamic regime \(l_{{\mathrm{ee}}} \gg x\) lies an interesting regime x < l_{ee} < W that has hitherto been ignored in the literature. This intermediate regime, which for the lack of a better name will be called “quasiballistic”, features an interactiondominated response of a nonhydrodynamic nature, since the mean free path l_{ee} is greater than the distance from the injector to the probe. Conspicuously, R_{v} remains negative in this regime. However, since now \(R_{\mathrm{v}}\sim 1/l_{{\mathrm{ee}}}(T)\), the sign of dR_{v}/dT is reversed compared to the hydrodynamic regime. The negative sign of R_{v} can be understood by considering injected carriers that travel over a large distance of the order of l_{ee} > x and then scatter off ambient thermal carriers. After scattering, some of the injected carriers make it back into the probe, creating a positive contribution to R_{v}. Simultaneously, some of the ambient carriers, through scattering off the injected carriers, are blocked from reaching the probe. This process creates a negative contribution to R_{v}. Detailed analysis shows that the latter contribution dominates^{26}, giving rise to negative R_{v}. As T increases, R_{v} grows progressively more negative until the point l_{ee} = x, where the hydrodynamic behavior sets in and the sign of the T dependence is reversed. Interestingly, in the quasiballistic regime, the value R_{v} decreases with n and grows with T, in qualitative agreement with the behavior of a MLG R_{v} at nottoohigh T found in the ref. ^{16}. This suggests a possible resolution of the conundrum posed by the findings of ref. ^{16}, in which a hydrodynamiclike negative R_{v} was found to depend on n and T differently from what is expected in the hydrodynamic regime.
Theory and comparison with experiment
To capture all these different regimes in a single model, we employ the kinetic equation for quasiparticles in the graphene Fermi liquid. Transport in the geometry of Fig. 1 is described by solving the kinetic equation in an infinite strip of width W: −∞ < x < ∞, 0 < y < W, with diffuse boundary conditions at the strip edges y = 0, W. Current I is injected through a pointlike source at x = y = 0 and is drained on the far left, x = −∞. We find the potential at (x, 0) by evaluating the particle flux entering the probe (for details, see Methods). At low temperatures the ee rate γ_{ee} is small, and the ee collision term can be ignored^{12}. The model then describes ballistic particles bouncing between the strip edges, as illustrated in the upper inset of Fig. 3b. The net flux of particles into the probe then gives a positive value R_{v} = V_{p}(x)/I. At high T, on the other hand, the ee collision term dominates, and the distribution function approaches the local equilibrium. The resulting hydrodynamic behavior is then described by the Stokes equation that states the balance between the viscous friction and electric forces: eE/m = −ν▽^{2}v. (The latter follows directly from the Eq. (12) of Methods, multiplied by p, integrated over momenta and combined with an expression for the stress tensor obtained from 1/γ_{ee} expansion.) In this case, we obtain \(R_{\mathrm{v}}\sim \eta /(nex)^2\), where η is the dynamic viscosity given by \(\eta = \frac{1}{4}m^ \ast nv_{\mathrm{F}}l_{{\mathrm{ee}}}\) and m* is the carrier effective mass^{20,22}. The single parameter γ_{ee} allows us to explore both the ballistic and viscous regime through the dependence of R_{v} on T and n. Carrier dynamics in the quasiballistic regime is shown schematically in the lower inset of Fig. 3b.
In Fig. 1b, c we compare the experimental data for R_{v} vs. T with the results of our modeling, assuming the ee collision rate that depends on T and n as \(\hbar \gamma _{{\mathrm{ee}}}\sim T_e^2/\varepsilon _{\mathrm{F}}\) ^{12}. For bilayer graphene, the Fermi energy ε_{F} is related to the carrier density as n = m^{*}ε_{F}/(πℏ^{2}), where m^{*} = 0.033 m_{e}. The two panels flaunt good qualitative agreement; namely, our theory captures the main experimental features: positive R_{v} at small T that rapidly drops with increasing T and monotonically grows with n, so that the minima and sign changes in R_{v} occur at higher T for larger n.
Furthermore, our model reproduces some of the more subtle features of the data. For example, the nodes in R_{v} vs. T shift to higher T and the minima to lower T, as the distance to the probe x increases, see Fig. 3. An overall agreement is also found for the full R_{v}(n, T) maps shown in Fig. 2a, b that become nearidentical after rescaling the T axis.
Discussion
In our analysis, for simplicity, we disregarded the Ohmic effects due to the elph scattering. This is a reasonable starting point since the elph scattering mean free path l_{el−ph} is considerably larger than l_{ee} at the temperatures of interest (for details, see Methods). However, the flow can be distorted by the Ohmic effects at the lengthscales set by \(\xi = \sqrt {\eta /n^2e^2\rho } = \frac{1}{2}\sqrt {l_{{\mathrm{el}}  {\mathrm{ph}}}l_{{\mathrm{ee}}}}\), which lies between l_{ee} and l_{el−ph}^{20,21}. Thus caution must be exercised even when the elph scattering is weak. The procedure of extracting the viscous contribution by subtracting the Ohmic contribution is expected to work well so long as the Ohmic effects do not distort the current flow at the lengthscales which are being probed, i.e. when ξ exceeds the distance to the probe x ≈ 1 μm. Estimates show that the inequality \(\xi \gg x\) holds at nottoohigh temperatures, i.e. in the quasiballistic regime. At the fluidity onset, identified above as the turning point in the R_{v}(T) dependence, for the estimated typical values \(l_{{\mathrm{ee}}} \lesssim 0.2\) μm and \(l_{{\mathrm{el}}  {\mathrm{ph}}}\sim 3\,{\mathrm{\mu m}}\), the lengthscale ξ can become comparable to x. However, an analysis based on the Stokes equation indicates that, for the geometry of our experiment, the Ohmic and viscous contributions remain approximately additive even for ξ < x (for details, see Methods). We therefore believe that the subtraction procedure provides a reasonable approximation in the entire range of temperatures and dopings.
We also note that Figs. 2, 3 exhibit some discrepancy between the values of T at which theoretical and experimental R_{v} reach the minimum. This is not particularly surprising given the simplistic expression of \(\gamma _{{\mathrm{ee}}}\sim T^2\) used in the model. Since γ_{ee} is the only relevant temperaturedependent parameter in the model, the quantitative agreement can be improved through revising the dependence γ_{ee} vs. T. Indeed, there are various effects that can give rise to deviation from the standard Fermiliquid T^{2} dependence. One is the logarithmic enhancement of the quasiparticle decay rate due to collinear ee collisions^{32,33,34}. However, it is probably an unlikely culprit, since collinear collisions do not lead to angular relaxation. At the same time, recent analysis^{35} indicates that the effective γ_{ee} that determines electron viscosity depends on the lifetimes of the oddm angular harmonics, m = ±3, ±5,..., which relax considerably slower than the Fermiliquid T^{2} estimate would suggest. Accounting for this effect could, effectively, extend the quasiballistic behavior to higher temperatures, which would improve the agreement with the observed dependence R_{v}(T). Detailed analysis of these rates and of their impact on R_{v} is beyond the scope of this work.
The experimental and theoretical R_{v}(T) exhibit two prominent features: R_{v} first changes sign from positive to negative and then passes through a deep minimum. Should the sign change or the minimum be taken as the signature of the onset of fluidity? That question can be answered with the help of the data presented in Fig. 3, demonstrating that R_{v} is a nontrivial function of both l_{ee}/W and l_{ee}/x. We note in that regard that the sign reversal of theoretically computed R_{v} occurs at \({\mathrm{Kn}} \gg 1\), that is inside the quasiballistic regime, for all values of x (Fig. 3b). Indeed, R_{v} in Fig. 3b changes sign at T = 20 K which for a given n translates into l_{ee} ≈ 10 μm, a length scale significantly greater than the values \(x\sim 1  2\) μm for this device. On the other hand, the most negative R_{v} in Fig. 3b is found at Kn = 1–3, which corresponds to \(x\sim l_{{\mathrm{ee}}} < W\). Since in the hydrodynamic regime R_{v} is proportional to η and thus should drop with increasing T, we infer that it is the condition \({\mathrm{Kn}} \sim 1\) (where R_{v} is most negative) that describes the fluidity onset. Furthermore, R_{v} is expected to be negative in the quasiballistic regime^{26} when Kn > 1, so it is indeed the drop of R_{v} with temperature, rather than the sign reversal, that marks the onset of the viscous flow.
Experimental observation of this anomalous behavior at the onset of the fluid state enables a direct electrical measurement of the mean free path l_{ee} and electron viscosity. Good qualitative agreement of the experimental data and our theoretical model suggests further opportunities to study the physics of efluids, in particular the electron transport in the presence of magnetic field and/or confining potential, obstacles, funnels and electron pumps. Our work clearly shows that the initial deviation from the ballistic behavior observed experimentally in different systems^{13,14,16,18,23} may be due to an entry into the interactiondominated “quasiballistic” regime rather than the true onset of electron fluidity. It requires higher temperatures and the observation of the behavior consistent with viscosity gradually decreasing with increasing T to ascertain that the NavierStokes description can be applied.
Methods
Device fabrication
Our devices were made of bilayer graphene encapsulated between ≈50 nmthick crystals of hexagonal boron nitride (hBN). The hBNgraphenehBN heterostructures were assembled using the drypeel technique described elsewhere^{27,36} and deposited on top of an oxidized Si wafer (290 nm of SiO_{2}) which served as a back gate. After this, a PMMA mask was fabricated on top of the hBNgraphenehBN stack by electronbeam lithography. This mask was used to define contact areas to graphene, which was done by dry etching with fast selective removal of hBN^{37}. Metallic contacts (usually, 5 nm of chromium followed by 50 nm gold) were then deposited onto exposed graphene edges that were a few nm wide. As the next step, another round of electronbeam lithography was used to prepare a thin metallic mask (40 nm Al) which defined a multiterminal Hall bar. After this, reactive ion plasma etching translated the shape of the metallic mask into encapsulated graphene. The Al mask also served as a top gate, in which case Al was wetetched near the contact leads to remove the electrical contact to graphene.
Distilling the hydrodynamic contribution in the presence of Ohmic effects
Here we assess the accuracy of the approach used in the main text to separate the viscous and Ohmic contributions to the R_{v} signal. In this approach, it was assumed that the contributions are approximately additive, and thus the viscous contribution can be distilled by subtracting the (suitably scaled) Ohmic resistivity measured in a fourprobe setup.
The validity of the additivity assumption can be verified using an exact solution of the hydrodynamic equations for current injected in a halfplane. The hydrodynamic approach applies when the ee mean free path is smaller than the elph scattering mean free path, \(l_{{\mathrm{ee}}} \ll l_{{\mathrm{el}}  {\mathrm{ph}}}\). At the scales larger than l_{ee} the electron flow satisfies the Stokes equation with an Ohmic term added to describe momentum relaxation:
Taking a curl and defining κ^{2} = ρ(en)^{2}/η = 1/ξ^{2}, we obtain the equation on the stream function:
where v = ∇ × (ψz). Following^{21}, we consider the flow in a halfplane y > 0 generated by the a point source on the boundary at x = 0: ψ_{x}(x, 0) = δ(x)I/ne. The stream function in this case has the form
where we defined \(q = \sqrt {k^2 + \kappa ^2} > 0\). The stream function can be used to evaluate the potential. Plugging Eq. (4) in Eq. (2), we see that only the first (harmonic) term in the stream function contributes the potential:
The yetundetermined quantity A(k) depends on the type of boundary condition. The nostress boundary condition at y = 0, which reads ψ_{yy}(x, 0) = 0, yields A(k) = q^{2}/κ^{2} = 1 + k^{2}/κ^{2}. Remarkably, the exact potential is a sum of the viscous and Ohmic contributions, with each contribution unaffected by the presence of the other contribution in this case:
where L is the system size. The subtraction procedure employed in analyzing the measurements is exact at all distances for the nostress boundary condition.
For the noslip boundary condition, on the other hand, the additivity is only an approximate property. In this case, ψ_{y}(x, 0) = 0 gives
The last term in this expression gives a contribution which depends both on viscosity and resistivity. As illustrated in Fig. 4, this contribution is nonnegligible at distances \(x \simeq \xi\), where R_{v} changes sign. Its magnitude, however, is small (under 10–15% of the total potential). Therefore, disregarding this contribution should provide a reasonably good approximation. Yet, this conclusion is almost certainly geometrysensitive, being valid for the point source at a halfplane edge but not necessarily for other geometries.
Estimates of the electronphonon scattering mean free path
Electronphonon scattering rate in graphene was discussed mostly for the singlelayer case^{38,39,40}. Here we modify this analysis for the bilayer case. The value of the mean free path l_{el−ph} is used in the main text to determine the lengthscales at which the the elph scattering does not distort the carrier flow.
We use the standard deformation potential Hamiltonian
where u(r, t) is the lattice displacement vector, D is the deformation potential coupling constant, ω_{k} = sk is the phonon frequency, and ρ is the surface mass density of graphene sheet. Plugging these quantities into the Golden Rule for the elph emission rate gives
where θ is the angle parameterizing the Fermi surface, and the deformation potential matrix element equals \(\left {V_{fi}} \right = \sqrt {\frac{\hbar }{{2\rho \omega _{\mathbf{k}}}}} D{\kern 1pt} {\mathbf{k}}{\kern 1pt} \left\langle {\psi _f\psi _i} \right\rangle\), with the overlap 〈ψ_{f}ψ_{i}〉 = cos(θ_{p′} − θ_{p}) accounting for the chirality of charge carriers. Here p and p′ are electron momenta, and k = p − p′. (Parenthetically, for monolayer graphene, the cos factor is to be replaced with cos((θ_{p′} − θ_{p})/2).) The density of final states equals ν = m^{*}/(2πℏ^{2}), where m^{*} = 0.033 m_{e} is the carrier effective mass; since electron–phonon scattering preserves carrier spin and valley index, the relevant degeneracies are not included in ν.
Phonon absorption is described by a similar expression with N_{ph}(k) + 1 replaced by N_{ph}(k). Since temperatures of interest are considerably larger than the BlochGruneisen temperature T_{BG} = ℏsk_{F}, we can approximate the Bose factors N_{ph}(k) and N_{ph}(k) + 1 as T/ℏω_{k}. Plugging N_{ph}(k) + N_{ph}(k) + 1 ≈ 2T/ℏω_{k} in the expression for dΓ and replacing k with p − p′, gives
Then the transport scattering rate equals
The electron–phonon mean free path is given by l_{el−ph} = v/Γ_{tr}, where v = ℏk_{F}/m^{*} is the carrier velocity. For bilayer graphene, we assume surface mass density ρ = 2 × 7.6 × 10^{−7} kg/m^{2}, the speed of sound s = 2 × 10^{4}m/s. In singlelayer graphene, transport measurements are consistent with deformation potential D of the order of 20 eV, see, e.g., ref. ^{41}. For bilayer graphene, abinitio calculations^{42} yield D = 15 eV. Assuming D in the range of 15–20 eV, we arrive at l_{el−ph} of the order of 3 μm for typical experimental conditions.
Table 1 provides a summary of the results for the singlelayer and bilayer graphene. These estimates are in agreement with the elph scattering rates extracted from the temperature dependence of the fourprobe resistance reported in the ref. ^{16}.
Details of the theoretical model
To describe the ballistic and viscous regimes on equal footing and provide a link between them, we use the kinetic Boltzmann equation for quasiparticles at the Fermi surface. Expanded to linear order in the deviation δf from the equilibrium FermiDirac distribution, Boltzmann equation reads
The collision operator I_{ee} in (12) describes scattering between singleparticle states via momentumconserving ee collisions. Near the Fermi surface, the distribution can be parameterized by the standard ansatz \(\delta f(p) =  \frac{{\partial f_0}}{{\partial \varepsilon }}\chi (p)\), where the energy dependence in χ can be ignored on the account of fast quasiparticle thermalization by collinear scattering at the 2D Fermi surface^{32,33}. We analyze the angular dependence χ(θ), where the angle θ parameterizes the Fermi surface and \(\hat p = ({\mathrm{cos}}\theta ,{\mathrm{sin}}\theta )\) is the unit vector along the carrier momentum. We assume that all nonconserved angular harmonics of χ(θ) relax with equal rates γ_{ee} = v_{F}/l_{ee},whereas the three angular harmonics corresponding to the conserved net momentum and particle number do not relax. The operator I_{ee}, linearized in χ, therefore takes the form^{13,22}:
The angular brackets denote angular averaging over θ′.
To model the current flow in the strip geometry shown in Fig. 5, the Eq. (12) is to be furnished with the boundary conditions describing momentum relaxation at the strip edges. We assume that particles are scattered diffusely, following Lambert’s law. Hence the edges y = 0 and y = W effectively become isotropic current sources. At y = 0, we write
The choice of the coefficient 1/2 in the second term is dictated by current conservation. Indeed, for an isotropic distribution of outgoing particles, χ(θ > 0) = χ_{0}, the outgoing particle flux, \(\nu \mathop {\int}\limits_0^\pi v_{\mathrm{F}}{\mathrm{sin}}\theta \chi (\theta ){\mathrm{d}}\theta /2\pi\), is given by νv_{F}χ_{0}/π. Here ν is the density of particle states, and v_{F} is the Fermi velocity. In the absence of current injection, this quantity must be equal to the incoming flux which is given by the integral in the second term. Similarly, an isotropic current source attached to the boundary, I(x, 0), is described via J(θ, x) = πI(x)/(eνv_{F}) in the Eq. (14). For the opposite orientation of the boundary, y = W, positive and negative angle values in the Eq. (14) are to be interchanged.
In general, distribution of particles in the Knudsen regime is not represented by a local equilibrium Fermi function, and a local chemical potential cannot be introduced. This poses a difficulty in relating the signal on a probe contact to the distribution function. To resolve this, we adopt the model of a probe which is commonly used to describe leads in mesoscopic circuits, see e.g.,^{43}. A probe is a perfect absorber for nonequilibrium carriers, which are equilibrated inside the probe and subsequently reemitted into the fluid with an isotropic angular distribution. If the opencircuit condition is maintained in the probing circuit, the potential on the probe is proportional to the influx F of charge carriers into the probe,
Since outgoing charge carriers are in equilibrium with the probe potential V_{p}, they are characterized by the distribution function χ = eV_{p}, so that the outgoing flux is νeV_{p}/π. Balancing these fluxes, one finds the probe potential
In the hydrodynamic regime, the distribution function is given by an equilibrium expression, which can be related to the local electric potential, χ(θ, x) ≈ eϕ(x). In this limit, one finds V_{p}(x) = ϕ(x). For a generic nonequilibrium distribution, however, the relation between the local potential ϕ(x) and the probe signal V_{p}(x) is less straightforward. In particular, in the ballistic limit (l_{ee} = ∞) the probe attached to the edge of the sample does not register particles grazing along the edge. This suppresses the space charge effect, however this suppression is not a universal phenomenon, and should be viewed as an approximation.
Numerical modeling
To model the experimental geometry of Fig. 1a, we analyze the flow induced in a strip of width W, 0 < y < W by a point source on its edge at the point (0, 0) and a drain at x = −∞, see Fig. 4. Such a flow can be represented as a superposition of a symmetric flow emitted by the source with a uniform flow directed to the drain electrode. Both flows can be analysed numerically via the approach described below.
First, we pass to the Fourier representation with respect to the coordinate x along the strip, and discretize the transverse coordinate: y_{n} = nh, where h = W/N_{y} is the step size, n = 0,…, N_{y} − 1. We also discretize the momentum direction as θ_{i} = π(i + 1/2)/N_{θ}, i = 0,…2N_{θ} − 1. Hence the distribution function becomes a function of the wavevector k and two discrete coordinates,
We employ the following finitedifference representation of the kinetic equation (13):
For numerical stability, the scheme is made “upwind”: the form on the first line should be applied for upwardgoing particles (0 < θ_{i} < π), and the form on the second line describes particles propagating downwards. Due to the choice of the exponential factors, the exact solution of (12) in the collisionless limit (γ_{ee} = 0), \(\chi (k,y,\theta ) \propto {\mathrm{exp}}(  iky\,{\mathrm{cotan}}\theta )\), satisfies the discretized equation.
Thus, discretization of the advection term in Boltzmann equation, (v∇)δf, links the values of the distribution function at nearby sites. The discretized form of the collision integral (13) mixes propagation angles within the same site. Therefore, the above finitedifference system, together with the boundary conditions (14) can be recast into the wellknown threediagonal form in which only blocks on three adjacent sites n and n ± 1 are coupled:
Here A_{n,ij}(k), B_{n,ij}(k) and C_{n,ij}(k) are matrix operators acting on the angular index i describing propagation of particles and scattering between different momentum directions. The righthand side b_{n,i}(k) describes external sources of particles. Such a system can be efficiently solved via the standard threediagonal matrix algorithm^{44}.
The point source was represented as a source term in the Eq. (14), with Fourier image I(k) = 1. The uniform Poiseuillelike flow can be obtained by analyzing the k = 0 limit of the threediagonal system (19), in which the flow is dragged by an external bias field. The bias field is incorporated into the Eq. (12) via the term −eEcosθ. The value of the bias field E is then obtained by normalizing the solution to the total current of 1/2.
To make sure that the details of the boundary layer near the edges are simulated properly, we have chosen a rather fine grid, N_{y} = 5000. The propagation angles were discretized with N_{θ} = 50, which corresponds to 3.6° step in θ. The particle distribution χ_{n,i}(k) was calculated for kW < 50, which gives a satisfactory approximation to the distances of interest, 0.1W < x < W. The probe signal was then calculated as the particle flux (16), giving the results shown in Figs. 1c, 2b, 3b.
Data availability
The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
We gratefully acknowledge support from the Simons Center for Geometry and Physics where some of the research was performed. D.A.B. and A.K.G. acknowledge the financial support from Marie Curie program SPINOGRAPH, Leverhulme Trust, the Graphene Flagship and the European Research Council. R.K.K. research was supported by EPSRC Doctoral Prize fellowship. G.F. research was supported by the Minerva Foundation, ISF Grant 882 and the RSF Project 142200259. L.L. acknowledges support of the Center for Integrated Quantum Materials under NSF award DMR1231319; and Army Research Office Grant W911NF1810116.
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L.S.L., G.F., and A.K.G. designed and supervised the project. M.B.S. fabricated the devices. Transport measurements and data analysis were carried out by D.A.B., A.I.B. and R.K.K. Theory analysis was done by A.V.S., L.S.L., and G.F. The manuscript was written by A.V.S., D.A.B., L.S.L., A.K.G., and G.F. A.I.B. and I.V.G. provided experimental support. All authors contributed to discussions.
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Bandurin, D.A., Shytov, A.V., Levitov, L.S. et al. Fluidity onset in graphene. Nat Commun 9, 4533 (2018). https://doi.org/10.1038/s41467018070044
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