# Fileset

[PhysRevResearch.5.043088.pdf](https://mdr.nims.go.jp/filesets/fbcef85a-599c-4364-a1e0-28f6802f9a5f/download)

## Creator

Lev V. Ginzburg, Yuze Wu, Marc P. Röösli, Pedro Rosso Gomez, Rebekka Garreis, Chuyao Tong, Veronika Stará, Carolin Gold, Khachatur Nazaryan, Serhii Kryhin, Hiske Overweg, Christian Reichl, Matthias Berl, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Werner Wegscheider, Thomas Ihn, Klaus Ensslin

## Rights

[Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/)

## Other metadata

[Long distance electron-electron scattering detected with point contacts](https://mdr.nims.go.jp/datasets/0200fa91-6158-4534-ae27-a91cd24855eb)

## Fulltext

Long distance electron-electron scattering detected with point contactsPHYSICAL REVIEW RESEARCH 5, 043088 (2023)Long distance electron-electron scattering detected with point contactsLev V. Ginzburg ,1,* Yuze Wu ,1 Marc P. Röösli,1 Pedro Rosso Gomez ,1 Rebekka Garreis ,1 Chuyao Tong ,1Veronika Stará ,2 Carolin Gold ,1,3 Khachatur Nazaryan ,4 Serhii Kryhin ,5 Hiske Overweg ,1 Christian Reichl,1Matthias Berl,1 Takashi Taniguchi,6 Kenji Watanabe ,7 Werner Wegscheider ,1 Thomas Ihn ,1 and Klaus Ensslin 11Solid State Physics Laboratory, ETH Zürich, 8093 Zürich, Switzerland2Central European Institute of Technology, Brno University of Technology, Purkyňova 123, 612 00 Brno, Czech Republic3Department of Physics, Columbia University, New York, New York 10027, USA4Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA5Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA6International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan7Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan(Received 31 March 2023; revised 12 August 2023; accepted 18 August 2023; published 26 October 2023)We measure electron transport through point contacts in an electron gas in AlGaAs/GaAs heterostructures andgraphene for a range of temperatures, magnetic fields, and electron densities. We find a magnetoconductancepeak around B = 0. With increasing temperature, the width of the peak increases monotonically, while itsamplitude first increases and then decreases. For GaAs point contacts the peak is particularly sharp at relativelylow temperatures T ≈ 1.5 K: the curve rounds on a scale of a few tens of microteslas, hinting at length scalesof several millimeters for the corresponding scattering processes. We propose a model based on the transitionbetween different transport regimes with increasing temperature: from ballistic transport to few electron-electronscatterings to hydrodynamic superballistic flow to hydrodynamic Poiseuille-like flow. The model is in qualitativeand, in many cases, quantitative agreement with the experimental observations.DOI: 10.1103/PhysRevResearch.5.043088I. INTRODUCTIONElectron transport can often be described by a semiclas-sical picture of charged particles moving through a materialand interacting with impurities, phonons, and sample bound-aries. The two widely used models of electron flow—ballisticand diffusive (Ohmic)—correspond to two opposite limitswithin this picture. Ballistic transport usually describes thesituation with few impurities and phonons, so that electronsmostly scatter with the sample boundaries, while the diffusiveflow represents the case where momentum relaxation occursmostly in the bulk of the system.This picture changes considerably if electron-electron scat-tering becomes significant. In clean systems, where theelectron-electron mean free path lee is much shorter than boththe characteristic sample size and the transport mean free pathlτ , electron transport is similar to viscous flow of a classicalfluid. This is known as the viscous (or hydrodynamic) electrontransport regime [1].Viscous electron flow was observed in different materials,including GaAs [2–5], graphene [6–8], PdCoO2 [9], WP2*glev@phys.ethz.chPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.[10], and WTe2 [11]. Experimental evidence for hydrody-namic behavior comes from superballistic flow through pointcontacts [4,7], negative nonlocal resistance [3,6], the Gurzhieffect [2], Stokes flow [12], and scanning probe experimentsinvestigating Poiseuille flow [11,13–15]. Most experimentswere performed at zero magnetic field. Magnetic fields highenough that the cyclotron radius becomes the shortest relevantlength scale in the system will eventually eliminate hydrody-namic effects [16]. However, the intermediate regime of smallmagnetic fields offers an interesting playground where severallength scales compete. A magnetic field introduces a Lorentzforce acting on the electron system. Furthermore, it modifiesthe viscosity and adds a second viscosity coefficient, usuallycalled the Hall viscosity [16–18]. The interplay of viscousflow and magnetic field was experimentally investigated in avicinity geometry [8] as well as in channels wider than theelectron-electron mean free path [19,20].In this paper we focus on one of the simplest possi-ble structures in two-dimensional electron gases (2DEGs):point contacts (PCs). Electron transport through the PCs atlow temperatures is well understood and, in general, canbe explained well within ballistic approximations. It wasshown before that at higher temperatures hydrodynamic ef-fects begin to play a significant role: at zero magneticfield the conductance exceeds the fundamental ballistic(Sharvin) limit due to the collective movement of electronsreducing momentum loss. This effect was predicted theoret-ically [21] and observed experimentally in graphene [7] andGaAs [4].2643-1564/2023/5(4)/043088(13) 043088-1 Published by the American Physical Societyhttps://orcid.org/0000-0001-8929-1040https://orcid.org/0000-0002-1060-5826https://orcid.org/0000-0002-4320-6936https://orcid.org/0000-0002-1233-998Xhttps://orcid.org/0000-0003-4947-6002https://orcid.org/0000-0003-4818-4366https://orcid.org/0000-0003-1700-3462https://orcid.org/0000-0002-9234-1835https://orcid.org/0000-0002-3159-5284https://orcid.org/0000-0002-9107-4763https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-2138-5558https://orcid.org/0000-0002-5587-6953https://orcid.org/0000-0001-7007-6949http://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevResearch.5.043088&domain=pdf&date_stamp=2023-10-26https://doi.org/10.1103/PhysRevResearch.5.043088https://creativecommons.org/licenses/by/4.0/LEV V. GINZBURG et al. PHYSICAL REVIEW RESEARCH 5, 043088 (2023)Nonzero magnetic fields add further complexity to the sys-tem: a peak in the magnetoconductance is observed aroundzero magnetic field at elevated temperatures (≈10 K in GaAsPCs). This peak was observed previously in GaAs PCs [22,23]and was associated with electron-electron interactions. Herewe expand on these early observations for PCs in GaAs2DEGs and investigate the nonmonotonic behavior of themagnetoconductance peak as a function of temperature, car-rier density, and split-gate voltages. Peculiar properties ofthe conductance peak that were previously unnoticed includeextreme sharpness of the peak (rounding on a scale of afew tens of microteslas) at relatively low temperatures (T <2 K) and slow disappearance of the peak at high tempera-tures (T > 10 K). Furthermore, we demonstrate that some ofthese effects can be observed in graphene PCs at ≈100 K,which is the temperature range where viscous flow occurs ingraphene [7].We argue that the observations can be explained by acontinuous transition between different transport regimes withincreasing temperature. At very low temperatures (T < 1 Kfor GaAs PCs) transport is mostly ballistic. At 1 < T <2 K electron-electron interactions become more important,and the system is not ballistic but is also not yet fullyhydrodynamic. This transitional regime results in a small,but very sharp, peak. At higher temperatures lτ � lee, andelectron transport becomes hydrodynamic. For T < 10 K thePC width d remains small compared to the other relevantlength scales. Electron transport is superballistic; the peakin the magnetoconductance is present. At even higher tem-peratures d becomes comparable to or smaller than theelectron-electron mean free path, and electron transport startsto resemble Poiseuille flow. As a consequence, the peak inconductance becomes less pronounced and eventually almostdisappears. For graphene PCs the relevant temperatures aregenerally higher, and we observe only ballistic and super-ballistic regimes. The model that we present agrees with ourobservations qualitatively and, in most cases, quantitatively.II. METHODSIn this paper we use both GaAs and graphene devices.The first device is based on an AlGaAs/GaAs heterostruc-ture with a 2DEG 200 nm below the surface. The globalpatterned back gate allows us to change the electron densitybetween 1.5 × 1011 and 2.7 × 1011 cm−2 [24]. The low-temperature (below 1 K) mobility is up to 7 × 106 cm2/Vs,corresponding to a transport mean free path of more than60 µm. The device has the shape of a large multiterminalHall bar (1800 × 400 µm2) with several top-gate-defined PCs(lithographic width d = 250 nm) in the central part of the Hallbar.The second device comprises monolayer graphene encap-sulated between hexagonal boron nitride (hBN) crystals witha graphite back gate. The stack is made with the standarddry-transfer technique and is placed on top of a silicon chipwith SiO2 surface. The Hall bar shape of the device and thePCs are etched through the top hBN crystal with reactiveion etching. The widths of the PCs are dnarrow = 150 nm anddwide = 350 nm.All linear conductance measurements were performed in4He and 4He / 3He systems at temperatures between 0.25 and25 K for the GaAs device and between 4.2 and 120 K for thegraphene device. Standard lock-in techniques at 31 Hz wereused. The carrier densities n were measured using the clas-sical Hall effect (GaAs) and Shubnikov–de Haas oscillations(graphene). The magnetic field B is always perpendicular tothe surface of the sample. All measurements are four-terminal.For the GaAs device, only one pair of top gates was used at atime, with all other top-gates grounded.III. MEASUREMENTSFirst, measurements in the GaAs device were performed.Below the data for one PC are presented; the data for otherPCs were consistent with these observations and displayedsimilar features (see Appendix B).Figure 1(a) shows an example curve for PC conductanceG as a function of B at T = 1.3 K and n = 2.7 × 1011 cm−2.The overall behavior is well known: the conductance increaseslinearly with |B| [25] until the onset of Shubnikov–de Haasoscillations, which for this sample is visible above |B| >200 mT. A peak in conductance is present around B = 0.First, we focus on a relatively narrow temperature rangebetween 1.3 and 4.3 K [Fig. 1(b)]. The magnetoconductancepeak becomes more pronounced with increasing temperatureand less pronounced but sharper with decreasing tempera-ture. Figure 1(c) shows �G(B, T ) = G(B, T ) − G(0, T ) asa function of B in a narrow range of B. Note that at lowtemperature T = 1.3 K the peak is rounded on a scale of afew tens of microteslas. Features of this size in magnetic fieldare, in general, highly unusual in electron transport and, to thebest of our knowledge, were not reported for AlGaAs/GaAsheterostructures before.Second, we explore the properties of the peak in a widertemperature range. Figures 2(a) and 2(b) show the conduc-tance G of the PC as a function of B and T for two differentelectron densities, n = 2.7 × 1011 and 1.5 × 1011 cm−2, re-spectively. For the higher electron density a small step inconductance occurred at T = 13.6 K, which we attribute toa random impurity being charged or discharged close to thePC; in order to compensate for it, the conductance above thistemperature is multiplied by 0.997.The background increase in conductance with |B|, men-tioned above, is clearly present at low temperatures andbecomes less pronounced with increasing temperature. Thepeak in conductance around B = 0 is present at all T > 0.5 Kand is most visible around T ≈ 12 K.The amplitude of this peak first increases and then de-creases with temperature, with the maximum being around≈10 K for all measured electron densities. The maximumis placed at slightly higher temperatures for higher electrondensities. The width of the peak along the magnetic field axisincreases monotonically with temperature for all densities.The details of G(B) can be seen as line cuts at constanttemperatures in Figs. 2(c) and 2(d).Similar behavior was observed for all available electrondensities in several PCs in this device as well as in a PC ina different GaAs device (no back gate, fixed electron den-sity of 1.8 × 1011 cm−2, mobility up to 4.1 × 106 cm2/Vs,043088-2LONG DISTANCE ELECTRON-ELECTRON SCATTERING … PHYSICAL REVIEW RESEARCH 5, 043088 (2023)FIG. 1. (a) shows the conductance of a GaAs PC as a function of magnetic field G(B) at electron density n = 2.7 × 1011 cm−2 andtemperature T = 1.3 K. The inset shows an AFM image of the PC. The range close to B = 0 (gray area) is shown in more detail and fortemperatures between T = 1.3 K and T = 4.3 K with a step of �T = 0.3 K in (b). In (c) we further zoom into a very small range of B.Conductance with respect to the peak value for each temperature �G(B, T ) = G(B, T ) − G(0, T ) is shown. The curves in (c) are verticallyoffset for clarity.PC width d = 500 nm; see Appendix B). For as long as thePC was defined, the top-gate voltage VTG and therefore theeffective width of the PC affected the amplitude of the peakbut not its width along the magnetic field axis. Some indi-cations of the conductance peak are present even in a verywide PC (lithographic width d = 4 µm; see Appendix B fordetails).Similar, but quantitatively different, effects were observedin graphene PCs (Fig. 3) at temperatures ≈100 K. Fig-ures 3(a)–3(e) show the normalized conductance G(B, T )FIG. 2. Conductance of a GaAs PC G as a function of magnetic field B and temperature T . (a) and (c) correspond to the electron densityn = 2.7 × 1011 cm−2; (b) and (d) correspond to electron density n = 1.5 × 1011 cm−2. The full data set is presented in 2D maps in (a) and(b). The solid green lines represent Dν = RC (above the lines Dν < RC, below Dν > RC). The upper green dashed line represents lτ = 10lee;the lower one represents lτ = lee. (c) and (d) show line cuts at constant temperatures of (a) and (b), respectively. The step in temperaturebetween two lines is 1.5 K for (c) and 1 K for (d). The lines are shifted artificially by 0.075e2/h for (c) and 0.1e2/h for (d). Large dots onthe lines correspond to the magnetic field value where Dν = RC (shown only for positive B). The dash-dotted black lines show the ballistic fit(suppressed backscattering) for the two lowest temperatures. Calculations of Dν and RC use no fitting parameters; the ballistic fit calculationsuse G(B = B0) as the only fitting parameter (here B0 = 2.5 mT is the effective zero of the magnetic field).043088-3LEV V. GINZBURG et al. PHYSICAL REVIEW RESEARCH 5, 043088 (2023)FIG. 3. Conductance of graphene PC G as a function of magnetic field B and temperature T at different carrier densities. Negative carrierdensities correspond to holes; positive ones correspond to electrons. [(a)–(e)] Two-dimensional maps, with green dashed lines representingthe magnetic field values for which Dν = RC. Conductance is normalized by G0 = G(B = 0, T = 4.2 K) for each plot. [(f)–(j)] Line cuts atconstant temperatures (starting at 4.2 K, then going from 10 to 120 K with a step of 10 K). Magnetic field B such that Dν = RC is shown forevery temperature by an arrow of corresponding color. Red dashed lines show the ballistic fit (suppressed backscattering). Calculations of Dνand RC use no fitting parameters; ballistic calculations use G(B = 0) as the only fitting parameter.for different carrier densities n (positive n correspondsto electrons) with corresponding line cuts presented inFigs. 3(f)–3(j). At low temperatures we observe an increase inconductance with increasing |B| due to suppressed backscat-tering [25]. Unlike for GaAs PCs, there are oscillationssuperimposed on the V-shaped background. These oscilla-tions are not symmetric in B and decay with temperature. FastFourier transformation of the curves provides a leading periodin magnetic field proportional to√|n| and the correspondingcyclotron radius RC ≈ 2.5 µm; these oscillations likely resultfrom magnetic focusing between the PCs and the narrowvoltage probes.Like for GaAs PCs, we observe an increase of conductancearound B = 0 with increasing temperature. This behavior ismore pronounced and visible as a peak in G(B) for high holeor electron densities [e.g., −2.9 × 1012 cm−2 for holes and2.5 × 1012 cm−2 for electrons; Figs. 3(f) and 3(j)]. At lowercarrier densities the peak is not present, but there is still anoticeable increase in G(B) around zero magnetic field athigher temperatures [Figs. 3(h) and 3(i)].In contrast to GaAs PCs, the conductance peak forgraphene PCs does not become very sharp at any measuredtemperature or carrier density; that is, the rounding of the peakalways happens on a scale of tens of milliteslas.IV. DISCUSSIONA. Hydrodynamic modelIn this section we discuss possible explanations for theconductance peak behavior as shown in Fig. 2 and proposea quantitative model.To begin with, we exclude possible explanations basedon a single-particle picture. The conductance peak is not aclassical size effect since the width of the peak depends sig-nificantly on temperature, while the cyclotron radius does not.Neither is it caused by weak antilocalization, as here the effectbecomes stronger with increasing temperature (at least up to acertain temperature) and is not present in bulk measurements.The observed effect is also not caused by the filling of asecond subband of the 2DEG: while this could produce apeak of conductance around B = 0 [26] and the amplitudeof this peak could theoretically increase with temperature ifthe occupation of the second subband increases, this effectshould strongly depend on electron density, which is not thecase here. No other single-particle effects seem to be able toproduce the observed behavior.Both the temperature range in which the effect is observedand the nonmonotonic temperature dependence suggest anexplanation related to electron-electron interactions. Belowwe show that the viscous electron transport model can, indeed,explain our observations.First, let us consider B = 0. In this case the PC conduc-tance can be written as the sum of two terms (the so-calledsuperballistic electron flow model [21]):G = Gball + Gvis. (1)The first part, Gball, is the same as expected for purely bal-listic electron transport (Sharvin conductance [27]), Gball =Ngvgse2h , where N is the number of modes in the PC, gs is thespin degeneracy, and gv is the valley degeneracy. The secondpart, Gvis, is the viscous contribution, originating from theStokes equation (low Reynolds number, no magnetic field,steady-state approximations):�J − D2ν∇2 �J = −σ0∇φ, (2)where σ0 is the Ohmic “Drude-like” conductance due to im-purities and phonons, φ is the electrostatic potential, �J is thecurrent density, andDν =√leelτ2(3)043088-4LONG DISTANCE ELECTRON-ELECTRON SCATTERING … PHYSICAL REVIEW RESEARCH 5, 043088 (2023)is the length scale responsible for viscous flow (also describedas the vorticity diffusion length) [28].At high enough magnetic field, such that the cyclotronradius RC = h̄kFeB is much shorter than the length scale forthe viscous transport, hydrodynamic effects are not present,and we recover the ballistic result. The conductance increaseswith increasing |B| due to the suppression of backscatteringas Gball = 1R0−|B|/en ≈ G0 + G20|B|en [25], with possible oscilla-tions on top due to the Shubnikov–de Haas effect [here G0 =Gball(B = 0)]. Higher temperatures remove the oscillationsand reduce the dependence of conductance on the magneticfield due to electron-phonon scattering. If we use this formulafor Gball and extend the definition of Gvis to nonzero magneticfield so that it is zero in high B, Eq. (1) becomes applicable inboth zero and high B.The applicability of Eq. (1) in intermediate fields is notknown; however, if we assume a smooth transition betweenzero B and high B regions, we arrive at a result that fits ourobservations. Indeed, we experimentally observed a V-shapedbackground in G(B) that becomes less pronounced at highertemperatures, which corresponds to the results of ballisticcalculations [shown as dash-dotted black lines in Figs. 2(c)and 2(d)]. On top of this, the viscous contribution Gvis createsa conductance peak centered around B = 0 [see Fig. 3(a)].This peak grows with increasing temperature as lee and lτbecome shorter (see Appendix A for the temperature depen-dence of the length scales). Increasing the temperature further(such that optical phonons become relevant) would eventuallydestroy both ballistic and hydrodynamic effects and recoverpurely diffusive behavior.Next, we discuss the width of the conductance peak alongthe magnetic field axis. A magnetic field introduces an addi-tional Lorentz-like term to Eq. (2) and modifies the viscousterm; both additions have a corresponding length scale RC.To the best of our knowledge, the analytical solution to theresulting equation is not known, so we attempted to extractsome estimates from a comparison of the length scales. Thereare three different length scales in this problem: RC from themagnetic field, Dν from the original Stokes equation, and thewidth of the PC d from the boundary conditions. d does notdepend on magnetic field or temperature and is not likely togive us the observed nontrivial behavior of the conductancepeak. Therefore, dimensionality dictates RC ∼ Dν . Here wehypothesize that RC = Dν is the transition point for electronbehavior: in this case, for RC > Dν some increase in conduc-tance above the V-shaped background due to viscous effectswould still be present; for RC < Dν the superballistic behaviorwould mostly be gone.In order to compare this hypothesis with the experiment,we numerically calculate lee [29] (see Appendix A for thedetails) and extract lτ from the direct measurements of thebulk resistance (the large size of the Hall bar allows the useof the Drude model). Together these two mean free pathsgive us the viscous length Dν (T ) [Eq. (3)]. From the as-sumption RC = Dν we solve RC(B) = Dν (T ) and get BC(T )for the transition in electron behavior. The solid green linesin Figs. 2(a) and 2(b) and large dots in Figs. 2(c) and 2(d)correspond to B(T ) = ±BC(T ) + B0, where experimental off-set B0 = 2.5 mT. The hydrodynamic model is applicable onlywhen lτ � lee; in order to show the region of model validitywe add two green dashed lines to Figs 2(a) and 2(b): thelower ones correspond to temperatures where lτ = lee, andthe upper ones correspond to lτ = 10lee. Indeed, we see thatthe observed peak of conductance lies between the two solidgreen lines and above the dashed ones, i.e., in the region whereDν (T ) < RC(B) and the hydrodynamic model is applicable.The positions of B(T ) = ±BC(T ) + B0 closely correspondto the minima of G(B, T ). Consequently, the measurementssupport the hypothesis that RC = Dν is the relevant condition.The expected change in BC(T ) with experimental param-eters is also consistent with observations. A lower electrondensity gives higher values of BC at a given temperature,which correspond to a wider peak in Fig. 2(b) compared toFig. 2(a).Following the same approach as for GaAs, we compare thecyclotron radius RC with the viscous length Dν for graphene.We estimate lτ from the bulk resistance measurements. Unfor-tunately, unlike in the GaAs device, the width of the Hall barand the distance between the contacts in the graphene deviceare not large compared to lτ ; therefore, the Drude modelbased calculations of lτ are not precise. The real transportmean free path is likely longer than the calculated one. Forthe estimate of lee we use the experimental and numericalresults from [7], where a similar device was used. We scalelee to the electron densities in our measurements accordingto lee ∼√nln(n) . The resulting BC(T ) is shown in Fig. 3. Theobserved peak of conductance is somewhat narrower alongthe magnetic field axis than the calculated BC; this can beexplained by the underestimated transport mean free path lτ .Aside from this inconsistency, the experimental results for thegraphene device correspond to the suggested model.B. Sharpness of the conductance peakThe extreme sharpness of the conductance peak does notfollow from the hydrodynamic model described above. Inaddition, not only the size but also the shape of the peakchanges significantly with temperature: at low temperatures,the peak is wide at the bottom and sharp at the top, while athigh temperatures it has a more rounded shape. This change inshape hints at the presence of a second relevant length scale,in addition to Dν , with a temperature dependence differentfrom Dν . This new length scale should be particularly large atlow temperatures (up to a few millimeters) in order to explainthe observations of the sharp conductance peak at T = 1.3 K[Fig. 1(c)].The hydrodynamic model is based on the assumptionof many electron-electron interactions before momentum isdissipated, i.e., lτ � lee. At 1 < T < 2 K, however, this as-sumption is not valid anymore for GaAs: lτ > lee, but lτ � leedoes not hold, and the number of e-e scattering events beforemomentum is dissipated is not large. This transitional regimebetween ballistic and hydrodynamic transport potentially of-fers the additional length scale described above.Let us consider a single electron e added above an equilib-rium distribution, moving through a PC from left to right atB = 0, represented by the blue arrow in Fig. 4(a). After trav-eling for ∼lee (several microns at T ≈ 1.5 K; see Appendix Afor details), this electron scatters with another electron e′from the equilibrium distribution [yellow star in Fig. 4(a)].043088-5LEV V. GINZBURG et al. PHYSICAL REVIEW RESEARCH 5, 043088 (2023)FIG. 4. Schematic of the electron backscattering as a hole rele-vant at low temperatures (1 < T < 2 K) in (a) zero and (b) nonzeromagnetic field. Black areas represent depleted parts of the 2DEGforming the point contact. The electron moving through the PC isshown with blue arrows, and the backscattered hole is shown with redarrows. The electron-electron scattering event happens at the positionmarked with a yellow star.Electron-electron scattering tends to be a head-on collisiondue to phase-space arguments [30–32]; that is, e′ was movingtoward the PC before the scattering event. In addition, d   leemeans that it is unlikely for either of the two electrons to movetoward the PC after the scattering event. Consequently, fromthe point of view of the total current through the PC, the mostprobable scattering event can be rephrased as an electron ebackscattering as a hole h toward the PC [red arrow in theFigure 4(a)], where h represents the absence of e′ that hadbeen moving toward the PC. In this case holes are defined notin a band structure meaning (as quasiparticles in the valenceband), but as quasiparticles missing from the equilibrium dis-tribution in the conductance band.This backscattered hole can go back through the PC andprovide additional current, leading to a small increase in con-ductance. Multiple scattering events are also possible; that is,an electron can backscatter as a hole, which can backscatteragain before reaching the PC as an electron, and so on. Herewe consider only a single scattering event.Let us add a small magnetic field B perpendicular to the2DEG. Now both the electron (before the scattering) and thebackscattered hole move in circular arcs, and their trajectoriesdo not coincide anymore. Figure 4(b) shows an example casewhere the electron originates from one edge of the PC and theresulting hole goes symmetrically through the PC at the otheredge. In this case it can be geometrically derived thatRC ≈ l2eed. (4)Here we assumed RC � lee � d . Shorter RC (higher magneticfield) would result in larger deviations between the trajectoriesof the electron and the hole, preventing the hole from goingbackward through the PC and eliminating this additional con-tribution to conductance.In a given temperature window Eq. (4) results in longRC (up to several millimeters) and, consequently, small B(≈ 10 μT). Below this value of B, the increase in conduc-tance due to electron backscattering is mostly unaffected byB; above this value, an increase in B would cause some ofthe backscattered holes to miss the PC and would lead to adecrease in conductance. This can explain the sharpness ofthe conductance peak observed in the experiments.The model is also consistent with a change in the peakshape with temperature. In leading order, lee ∼ T −2. There-fore, the magnetic field describing the sharpness of the peakscales as B ∼ T 4. This temperature dependence is muchstronger than the one following from the hydrodynamic model(Sec. IV A): B ∼ D−1ν ∼ T 3/2 or slower (here the typical tem-perature dependence for electron-phonon scattering is used,lτ ∼ T −1, and impurities will lead to saturation at low T ; seealso Appendix A). Consequently, the sharpness of the peakchanges much faster with T (at low temperatures) than theoverall width of the peak [compare to Figs. 1(b) and 1(c)].The simplified argument above gives us an estimate of thelength scale important for the problem. A complete modelwould have to account for multiple factors, including theangular distribution for the electrons arriving through the PC,different possible distances between the PC and the first scat-tering, multiple e-e scattering events, the finite possibility ofmomentum-relaxing scattering, and the details of the electro-static potential at the PC. Creating this detailed model goesbeyond the scope of this paper.Interestingly, we observe the sharp conductance peak onlyin GaAs devices and not in the graphene device. This can beexplained by the smaller size of the graphene device, which isa technical limitation compared to GaAs: at low temperatures,where the described effect is important, an electron movingthrough the PC is more likely to scatter at a graphene flakeboundary and not with another electron, suppressing the rele-vant contribution to the PC conductance at low magnetic field.C. Peak amplitudeIn the previous section we examined the transition regimebetween hydrodynamic and ballistic electron transport at lowtemperatures. Here we show that at higher temperatures thereis another change in the system’s behavior not predicted bythe hydrodynamic superballistic flow model. This can be seenby analyzing the temperature dependence of the amplitude ofthe conductance peak.Following Eq. (1), which is applicable for the superbal-listic flow model, the measured conductance is split into thebackground Gball and viscous Gvis contributions. For eachtemperature the V-shaped background is approximated as a043088-6LONG DISTANCE ELECTRON-ELECTRON SCATTERING … PHYSICAL REVIEW RESEARCH 5, 043088 (2023)FIG. 5. Conductance peak height �G for GaAs PC. (a) Example of the �G calculation for one temperature (7 K) and electron density(2.7 × 1011 cm−2): the blue curve is the conductance G as a function of B at temperature 7 K and electron density 2.7 × 1011 cm−2. The reddashed lines are linear fits at high magnetic field; �G is the excess conductance above the linear fits. (b) and (c) �G calculated this way isshown as a function of temperature for two densities (blue dots). Hydrodynamic calculations of superballistic conductance contributions atzero magnetic field are shown by yellow dashed lines for no-slip boundary conditions and purple dash-dotted lines for no-stress boundaryconditions.linear fit from the high magnetic field data [see red dashedlines in Fig. 5(a) for an example]. This fit is subtracted fromG(B) to acquire the zero B viscous contribution �G as theheight of the conductance peak. �G is shown as a function oftemperature for two different electron densities in Figs. 5(b)and 5(c). �G is not presented for the entire temperature rangesince at higher temperatures the linear parts of G(B) are notvisible and the linear fit is not well defined. Next, the theoret-ically predicted viscous conductance contribution at B = 0 iscalculated (superballistic conductance) [21]:Gthvis = e2d2eff8h̄√πn21lee, (5)where n is the bulk electron density and deff is the effectivewidth of the PC. In general, deff is smaller than the litho-graphic width d due to the side depletion below top gates.Since the dependence of deff on temperature is weak, it canbe calculated in the ballistic case from the lowest temperaturezero B value of G:deff = h2e2√π2nPCG∣∣∣T =Tmin,B=B0. (6)The electron density in the PC nPC present in the equa-tion above is significantly lower than the bulk density n. Weestimate it by comparing the Hall voltage and the diagonalvoltages in the quantum Hall effect measurement. The resultof the calculations is shown in Figs. 5(b) and 5(c) as yellowdashed lines. However, Eq. (5) is derived under no-slip bound-ary conditions, which is likely not the case for the gate-definedGaAs PCs. A paper by Li et al. [33] predicted that for theno-stress boundary condition Gvis should be two times higher[purple dash-dotted lines in Figs. 5(b) and 5(c)].The theoretical curves Gthvis obtained in this way do notmatch the experimental data �G. At low temperatures theexperimental and theoretical values of conductance are close,but the experimental temperature dependence is weaker thanthe theoretically predicted one. Indeed, the hydrodynamictheory predicts Gthvis ∼ 1/lee ∼ T 2 [neglecting the corrections∼ log(T )], but the observed curves are much closer to lineardependence (see Appendix B for the details).At high temperatures the experimental curve �G decreaseswith increasing T above a certain temperature. This decreasein conductance does not follow from the superballistic flowmodel, and it cannot be explained by Ohmic resistance ofthe 2DEG in series with the PC either (the Ohmic resistanceneeded for that is more than order of magnitude above the totalmeasured resistance of the system). Notably, for both elec-tron densities at the maximum of Gvis the electron-electronmean free path approaches lee ≈ 1.0 µm. If this experimen-tal dependence continued to higher temperatures, the peakwould disappear at T ≈ 20 K for high electron density andat T ≈ 12 K for low electron density. Both cases correspondto lee ≈ 200 nm ≈ d (see Appendix A).This observation suggests the following explanation. Atrelatively low temperatures [i.e., 2 < T < 7 K for highelectron density; Fig. 5(b)] electron transport is mostly hy-drodynamic, and the electron-electron mean free path is muchlonger than the width of the PC (lτ � lee � d). Conse-quently, the PC is the injection point for electrons, and all theinteractions happen in the 2DEG outside of the PC. This cor-responds to the superballistic flow model, with the predictionsdescribed in Sec. IV A. The conductance peak amplitude in-creases with T , but the functional dependence does not agreewith the hydrodynamic prediction. Interestingly, a similar ap-proximately linear dependence was observed previously forthe superballistic contribution of graphene PC conductance[7]. While the two results cannot be compared directly due tothe different methods used to extract the viscous contributionto the PC conductance, these observations hint that a morecomprehensive model might be needed.At high temperatures (T > 18 K) electron transport isstill hydrodynamic, but the electron-electron mean free pathbecomes comparable to or smaller than the width of thePC (lτ � d � lee). Electron-electron interactions now hap-pen in the PC itself, and the electron transport tends towardPoiseuille flow through the channel rather than superballistictransport through the small PC. No conductance peak G(B)is predicted for the hydrodynamic flow through the channel,and even the conductance peak from the classical size effectwould be suppressed by hydrodynamic effects [34]. The PC inour experiment is not a long channel, but the top-gate-definedpotential provides a finite length of the PC which is more thanits width d .Consequently, even within the hydrodynamic model sev-eral regimes can be seen: superballistic flow is observed forlower temperatures, and Poiseuille-like flow can be seen at043088-7LEV V. GINZBURG et al. PHYSICAL REVIEW RESEARCH 5, 043088 (2023)FIG. 6. (a) Conductance of GaAs PC G as a function of magnetic field B for different top-gate voltages VTG. The temperature is fixed at4.2 K, the electron density is 2.7 × 1011 cm−2. (b) Conductance peak as a function of B for different VTG, with the background V-shaped curvesubtracted as in Fig. 5. The inset shows the background conductance at effective zero magnetic field (for example, see the crossing point ofthe two red dashed lines in Fig. 5). (c) Same data, but scaled along the vertical axis. The inset shows the scaling factor a as a function of thebackground conductance at B = 0. The red dashed line is a linear fit. The scaling factor a is independent of B.higher temperatures. At intermediate temperatures a gradualtransition between these two modes would be expected.D. ScalingThe superballistic flow observed above (Sec. IV A) can beadditionally verified by examining G(B) at different top-gatevoltages VTG at a fixed temperature and electron density. Itcan be shown that the curves can be scaled to collapse onto asingle curve and that the scaling parameter is close to the pre-diction of the hydrodynamic model. For these measurementswe chose the temperature T = 4.2 K such that lτ � lee � dand the model described in Sec. IV A is applicable [Fig. 6(a)].Similar to the procedure used in the previous section, weseparate the measured conductance into the V-shaped ballisticbackground and the hydrodynamic peak. Figure 6(b) showsthe peak of conductance Gvis with the ballistic backgroundsubtracted for different values of VTG, while the inset depictsthe ballistic background conductance Gball at B = 0 as a func-tion of VTG. The peak amplitude changes with VTG, whileits shape and width along the magnetic field axis are almostconstant. Below we analyze this observation and compare itto the predictions of our model.The amplitude of the peak explicitly depends on the effec-tive width of the PC deff [Eq. (5)], which explains the changein the amplitude with VTG. The constant shape of the observedpeak can be demonstrated by scaling the curves at differentVTG according toa(VTG)Gvis(B,VTG) = Gscaledvis (B,VTG), (7)where the scaling parameter a depends only on VTG and is cho-sen such that the mean square difference between the curvesis minimal. Indeed, it can be seen that the scaled curves coin-cide [Fig. 6(c)]. The previously discussed model (Sec. IV A)explains the constant width of the peak. The scaling behaviorfor the shape of the peak does not directly follow from themodel; however, the model can be used to study the behaviorof the peak amplitude.Below we consider the simple approximation of the PC as arectangular potential well, where VTG affects only the effectivewidth deff of the PC. In this case the ballistic conductance atzero magnetic field would be proportional to the PC width(Gball ∼ deff , which follows from the Sharvin formula), whilethe viscous contribution Gvis ∼ d2eff [Eq. (5)]. Therefore, onecan expect the relation a ∼ G−γball between the scaling factora and the ballistic conductance Gball, where γ = 2. The insetin Fig. 6(c) shows the corresponding double logarithmic plotof the scaling parameter a as a function of Gball. The curve isclose to linear, and the extracted γ is between 1.5 and 1.8.This shows that the amplitude of the peak increases fasterwith VTG than the ballistic background. In the argument abovewe ignored the dependence of the electron density in thePC on VTG (which is considerable for the narrow PCs), sosome discrepancy between the measured and predicted γ isexpected.V. CONCLUSIONSWe performed measurements of conductance through pointcontacts in a GaAs 2DEG and graphene at different tempera-tures, bulk carrier densities, and magnetic fields. At elevatedtemperatures we observed a peak of conductance around zeromagnetic field. The width of the peak along the magneticfield axis increases monotonically with temperature. The peakamplitude first increases and then (at least for GaAs de-vices) decreases with temperature. The shape of the peakalso depends on temperature. For GaAs devices, the peak isparticularly sharp (rounding on a scale of tens of microteslas)at lower temperatures, T ≈ 1.5 K.We proposed a model based on a transition betweendifferent transport regimes and a comparison of relevantlength scales, which explained the observations qualitativelyand, in many cases, quantitatively. For GaAs point con-tacts, the transition is from ballistic electron transport at verylow temperatures (no magnetoconductance peak) to the fewelectron-electron interaction regime (small sharp peak) to hy-drodynamic superballistic flow (the amplitude of the peakincreases) to hydrodynamic Poiseuille-like transport (the peakslowly disappears). For graphene point contacts, only the firstand third transport regimes were observed directly, althoughthe last one should be possible at temperatures even higherthan available in our experiment.043088-8LONG DISTANCE ELECTRON-ELECTRON SCATTERING … PHYSICAL REVIEW RESEARCH 5, 043088 (2023)FIG. 7. Relevant length scales for GaAs as a function of temperature for electron densities (a) n = 2.7 × 1011 cm−2 and (b) n =1.5 × 1011 cm−2. The solid blue line shows calculated the electron-electron mean free path lee, the dot-dashed red line shows the measuredtransport (momentum-relaxing) mean free path lτ , and the dashed yellow line shows the viscous length Dν .ACKNOWLEDGMENTSThe experimental data were discussed at an early stagewith L. Levitov. His insights led to further experiments and toan intuitive understanding of the narrow magnetoconductancepeak. We gratefully acknowledge these discussions. We thankO. Zilberberg, A. Leuch, F. K. de Vries, and H. Duprez forvaluable discussions and comments and P. Märki, T. Bähler,and the FIRST staff for technical support. We acknowledgefinancial support from the European Graphene Flagship Core3Project, H2020 European Research Council (ERC) SynergyGrant under Grant Agreement No. 951541, the EuropeanUnion’s Horizon 2020 research and innovation program un-der Marie Sklodowska-Curie Grant Agreement No. 766025,Eidgenössische Technische Hochschule Zürich (ETH Zurich),and the Swiss National Science Foundation via the Na-tional Center of Competence in Research Quantum Scienceand Technology (NCCR QSIT). K.W. and T.T. acknowledgesupport from the JSPS KAKENHI (Grants No. 19H05790 andNo. 20H00354).APPENDIX A: RELEVANT LENGTH SCALES FOR GaAsThe transport mean free path lτ , also known as themomentum-relaxing mean free path, was extracted from stan-dard mobility measurements in the wide Hall bar geometry.Viscous length Dν was calculated according to (3).The electron-electron mean free path was calculated nu-merically according to [29], assuming a spin-independentscattering function in the random-phase approximation. Theformulas for the electron-electron scattering length lee of aparticle at energy ξ̃k = 1kBT ( h̄2k22m∗ − EF ) (dimensionless, rela-tive to the Fermi energy EF ) interacting with the equilibriumFermi sea at temperature T were acquired by combiningEqs. (7), (17), (18), (32), and (33) of [29]:lee(ξ̃k, T ) = π2h̄4√2πn3m∗2w f b(kBT )2[1 − nF (ξ̃k )]{12(π2 + ξ̃ 2k)[ln 82 + ln( EFkBT)] − F (ξ̃k )}[1 + exp(−ξ̃k )], (A1)whereF (ξ̃k ) = 12[1 − nF (ξ̃k )]−1∫ +∞−∞d ξ̃k′∫ +∞−∞d ξ̃p ln |(ξ̃k′ − ξ̃p)(ξ̃k − ξ̃k′ )|nF (ξ̃p)[1 − nF (ξ̃k′ )][1 − nF (ξ̃k + ξ̃p − ξ̃k′ )], (A2)w f b =1 + (1 + 1√2rs)−22, rs = m∗e28π h̄2εε0√πn, (A3)and nF (ξ̃ ) = 11+exp(ξ̃ )is a Fermi-Dirac distribution. The final result for the electron-electron mean free path (Fig. 7) follows fromthe averaging of lee(ξ̃k, T ) weighed by the derivative of the Fermi-Dirac distribution:lee(T ) =∫ +∞−∞lee(ξ̃k, T )(−∂nF∂ξ̃k)d ξ̃k . (A4)APPENDIX B: ADDITIONAL MEASUREMENTSAdditional measurements were performed to confirm thereproducibility of the results and provide further evidence forthe suggested explanations.1. GaAs point contactsFigure 8(a) shows conductance G of the GaAs PC de-scribed in Secs. II and III as a function of top gate voltageVTG for T = 1.3 K and T = 4.3 K. At low temperature, the043088-9LEV V. GINZBURG et al. PHYSICAL REVIEW RESEARCH 5, 043088 (2023)FIG. 8. (a) PC conductance G(VTG ) for n = 2.7 × 1011 cm−2 and two temperatures, T = 1.3 K and T = 4.3 K. Two arrows denote valuesof VTG where the magnetoconductance measurements were performed, which are shown in (b) and (c) as �G(B, T ) = G(B, T ) − G(0, T ). In(b) and (c) the curves are offset for clarity; the step in temperature between the curves is 0.3 K.standard conductance steps of 2e2/h are observed. At hightemperature, the conductance steps are much less pronounced,and the overall conductance is increased more for highervalues of VTG. Measurements are performed at n = 2.7 ×1011 cm−2.Examples of �G(B, T ) = G(B, T ) − G(0, T ) in a narrowrange of B are shown in Figs. 8(b) and 8(c) for two differentvalues of VTG (one between the conductance plateaus, onealmost on the plateau). The observed behavior is the same forthese two cases.In Sec. IV C the dependence of the amplitude of the mag-netoconductance peak on T is discussed. The same data as inFigs. 5(b) and 5(c) are shown in double logarithmic scale inFigs. 9(a) and 9(b). The observed curves at low temperaturesare much better described by �G ∼ T than the �G ∼ T 2expected from hydrodynamic theory. �G is defined as inSec. IV C.Figure 10(a) shows the conductance of a different PCwith dimensions similar to the one described in Secs. II andIII, fabricated on the same GaAs wafer. Measurements wereperformed at electron density n = 2.7 × 1011 cm−2. The ob-served G(B, T ) is qualitatively and quantitatively similar tothe one presented in Fig. 2(a).Figure 10(b) shows the conductance of a very wide con-striction d = 4 µm (same GaAs wafer, density n = 2.7 ×1011 cm−2). A weak magnetoconductance peak is presentaround B = 0, although it is much less pronounced than inthe case of narrow PCs.A similar conductance peak was observed in a differentGaAs wafer [Fig. 10(c)]. This wafer has no back gate, electrondensity is n = 1.8 × 1011 cm−2, low-temperature mobility is4 × 106 cm2/Vs, PC width d = 500 nm, and the 2DEG is130 nm below the surface.Instead of increasing the temperature of the sample, higherbias voltage can be applied to achieve a qualitatively similareffect of broadening of the Fermi distribution. Figure 10(d)shows the conductance of the GaAs PC (wafer and geome-try as described in Sec. II) as a function of B and appliedDC bias eVsd,DC/2kB. The measurements were performed atT = 4.2 K, and applied bias is a sum of a small AC signalfor the conductance measurement and a large DC signal forincreasing the effective temperature of the 2DEG. The mag-netoconductance peak around B = 0 is present for all appliedvalues of the DC bias. The peak width is almost constantfor eVsd,DC/2kB < 5 K, where the effective temperature ofthe 2DEG is mostly defined by the lattice temperature andtherefore almost constant. The peak width increases with theDC bias for eVsd,DC/2kB > 5 K, where the effective 2DEGtemperature is mostly proportional to the DC bias. This resultis in agreement with the temperature dependences G(B, T )described above.2. Graphene point contactsUnlike monolayer graphene, bilayer graphene (BLG) de-vices allow gate-defined structures and therefore better controlFIG. 9. Magnetoconductance peak amplitude �G (as defined in Sec. IV C) as a function of temperature shown in a double logarithmic scalefor two electron densities, n = 2.7 × 1011 cm−2 and n = 1.5 × 1011 cm−2. Dashed lines show linear and parabolic functions for comparison.043088-10LONG DISTANCE ELECTRON-ELECTRON SCATTERING … PHYSICAL REVIEW RESEARCH 5, 043088 (2023)FIG. 10. (a) Conductance G(B, T ) of a second GaAs PC on the same wafer as described in Sec. II, with electron density n = 2.7 ×1011 cm−2 and lithographic width d = 250 nm. (b) Conductance G(B, T ) of a wide d = 4 µm GaAs PC on the same wafer. (c) ConductanceG(B, T ) of a GaAs PC on a different wafer (electron density of 1.8 × 1011 cm−2, mobility up to 4 × 106 cm2/Vs, PC width of 500 nm).(d) Conductance G as a function of magnetic field B and DC bias voltage Vsd,DC for the original PC (as described in Sec. II), measured atT = 4.2 K.of geometry. Bilayer graphene is also known to show hy-drodynamic behavior [8]. Unfortunately, the gap in the bandstructure induced by the vertical electrostatic displacementfield in BLG is small compared to the band gap of conven-tional semiconductors, such as GaAs. At temperatures highenough for hydrodynamic effects (≈100 K) the current leakthrough the depleted region below the top gates is significantand prevents proper confinement in the PCs [35]. Regardless,we attempted to measure the magnetoconductance peak inthe gate-defined PCs in the BLG device. The data shownin Fig. 11 are measured at T = 95 K and electron densityn ≈ 2 × 1012 cm−2. The sample is sample I described inAppendix E of [35]. A slight increase in conductance wasobserved around B = 0. The effect is much weaker than inmonolayer graphene devices and was observed only in anarrow window of parameters. In principle, higher displace-ment fields can produce higher band gap and correspondinglysmaller leakage currents at high temperatures, potentially im-proving the data quality. However, for our sample this was notpossible due to a leak between gates and graphene at highergate voltages. Further studies are needed in order to draw aclear conclusion regarding the magnetoconductance peak in agate-defined BLG PCs.FIG. 11. Conductance G(B) of a PC in a bilayer graphene sam-ple, with T = 95 K. The measurement was repeated nine times;each one is shown by a narrow line of a different color. Black dotsrepresent the average of the nine measurements.043088-11LEV V. GINZBURG et al. PHYSICAL REVIEW RESEARCH 5, 043088 (2023)[1] R. N. Gurzhi, Minimum of resistance in impurity-free con-ductors, J. Exptl. Theoret. Phys. (U.S.S.R.) 44, 771 (1963)[JETP Lett. 17, 521 (1963)].[2] M. J. M. de Jong and L. W. Molenkamp, Hydrodynamic elec-tron flow in high-mobility wires, Phys. Rev. B 51, 13389 (1995).[3] B. A. Braem, F. M. D. Pellegrino, A. Principi, M. Röösli,C. Gold, S. Hennel, J. V. Koski, M. Berl, W. Dietsche, W.Wegscheider, M. Polini, T. Ihn, and K. Ensslin, Scanninggate microscopy in a viscous electron fluid, Phys. Rev. B 98,241304(R) (2018).[4] L. V. Ginzburg, C. Gold, M. P. Röösli, C. Reichl, M. Berl,W. Wegscheider, T. Ihn, and K. Ensslin, Superballistic electronflow through a point contact in a Ga[Al]As heterostructure,Phys. Rev. Res. 3, 023033 (2021).[5] A. C. Keser, D. Q. Wang, O. Klochan, D. Y. H. Ho, O. A.Tkachenko, V. A. Tkachenko, D. Culcer, S. Adam, I. Farrer,D. A. Ritchie, O. P. Sushkov, and A. R. Hamilton, Geometriccontrol of universal hydrodynamic flow in a two-dimensionalelectron fluid, Phys. Rev. X 11, 031030 (2021).[6] D. A. Bandurin, I. Torre, R. K. Kumar, M. Ben Shalom, A.Tomadin, A. Principi, G. H. Auton, E. Khestanova, K. S.Novoselov, I. V. Grigorieva, L. A. Ponomarenko, A. K. Geim,and M. Polini, Negative local resistance caused by viscouselectron backflow in graphene, Science 351, 1055 (2016).[7] R. Krishna Kumar, D. A. Bandurin, F. M. D. Pellegrino, Y.Cao, A. Principi, H. Guo, G. H. Auton, M. Ben Shalom, L. A.Ponomarenko, G. Falkovich, K. Watanabe, T. Taniguchi, I. V.Grigorieva, L. S. Levitov, M. Polini, and A. K. Geim, Su-perballistic flow of viscous electron fluid through grapheneconstrictions, Nat. Phys. 13, 1182 (2017).[8] A. I. Berdyugin, S. G. Xu, F. M. D. Pellegrino, R. KrishnaKumar, A. Principi, I. Torre, M. Ben Shalom, T. Taniguchi,K. Watanabe, I. V. Grigorieva, M. Polini, A. K. Geim, andD. A. Bandurin, Measuring Hall viscosity of graphene’s elec-tron fluid, Science 364, 162 (2019).[9] P. J. W. Moll, P. Kushwaha, N. Nandi, B. Schmidt, andA. P. Mackenzie, Evidence for hydrodynamic electron flow inPdCoO2, Science 351, 1061 (2016).[10] J. Gooth, F. Menges, N. Kumar, V. Süß, C. Shekhar, Y. Sun,U. Drechsler, R. Zierold, C. Felser, and B. Gotsmann, Thermaland electrical signatures of a hydrodynamic electron fluid intungsten diphosphide, Nat. Commun. 9, 4093 (2018).[11] U. Vool, A. Hamo, G. Varnavides, Y. Wang, T. X. Zhou, N.Kumar, Y. Dovzhenko, Z. Qiu, C. A. C. Garcia, A. T. Pierce,J. Gooth, P. Anikeeva, C. Felser, P. Narang, and A. Yacoby,Imaging phonon-mediated hydrodynamic flow in WTe2, Nat.Phys. 17, 1216 (2021).[12] G. M. Gusev, A. S. Jaroshevich, A. D. Levin, Z. D. Kvon, andA. K. Bakarov, Stokes flow around an obstacle in viscous two-dimensional electron liquid, Sci. Rep. 10, 7860 (2020).[13] J. A. Sulpizio, L. Ella, A. Rozen, J. Birkbeck, D. J. Perello, D.Dutta, M. Ben-Shalom, T. Taniguchi, K. Watanabe, T. Holder,R. Queiroz, A. Principi, A. Stern, T. Scaffidi, A. K. Geim, andS. Ilani, Visualizing Poiseuille flow of hydrodynamic electrons,Nature (London) 576, 75 (2019).[14] M. J. H. Ku, T. X. Zhou, Q. Li, Y. J. Shin, J. K. Shi, C. Burch,L. E. Anderson, A. T. Pierce, Y. Xie, A. Hamo, U. Vool, H.Zhang, F. Casola, T. Taniguchi, K. Watanabe, M. M. Fogler,P. Kim, A. Yacoby, and R. L. Walsworth, Imaging viscousflow of the Dirac fluid in graphene, Nature (London) 583, 537(2020).[15] Z. J. Krebs, W. A. Behn, S. Li, K. J. Smith, K. Watanabe, T.Taniguchi, A. Levchenko, and V. W. Brar, Imaging the breakingof electrostatic dams in graphene for ballistic and viscous fluids,Science 379, 671 (2023).[16] P. S. Alekseev, Negative magnetoresistance in viscous flowof two-dimensional electrons, Phys. Rev. Lett. 117, 166601(2016).[17] J. E. Avron, Odd viscosity, J. Stat. Phys. 92, 543 (1998).[18] F. M. D. Pellegrino, I. Torre, and M. Polini, Nonlocal transportand the Hall viscosity of two-dimensional hydrodynamic elec-tron liquids, Phys. Rev. B 96, 195401 (2017).[19] G. M. Gusev, A. D. Levin, E. V. Levinson, and A. K. Bakarov,Viscous transport and Hall viscosity in a two-dimensional elec-tron system, Phys. Rev. B 98, 161303(R) (2018).[20] G. M. Gusev, A. D. Levin, E. V. Levinson, and A. K. Bakarov,Viscous electron flow in mesoscopic two-dimensional electrongas, AIP Adv. 8, 025318 (2018).[21] H. Guo, E. Ilseven, G. Falkovich, and L. S. Levitov, Higher-than-ballistic conduction of viscous electron flows, Proc. Natl.Acad. Sci. USA 114, 3068 (2017).[22] V. T. Renard, O. A. Tkachenko, V. A. Tkachenko, T. Ota, N.Kumada, J.-C. Portal, and Y. Hirayama, Boundary-mediatedelectron-electron interactions in quantum point contacts, Phys.Rev. Lett. 100, 186801 (2008).[23] M. Y. Melnikov, J. P. Kotthaus, V. Pellegrini, L. Sorba, G.Biasiol, and V. S. Khrapai, Influence of e-e scattering on thetemperature dependence of the resistance of a classical ballisticpoint contact in a two-dimensional electron system, Phys. Rev.B 86, 075425 (2012).[24] M. Berl, L. Tiemann, W. Dietsche, H. Karl, and W.Wegscheider, Structured back gates for high-mobility two-dimensional electron systems using oxygen ion implantation,Appl. Phys. Lett. 108, 132102 (2016).[25] H. van Houten, C. W. J. Beenakker, P. H. M. van Loosdrecht,T. J. Thornton, H. Ahmed, M. Pepper, C. T. Foxon, and J. J.Harris, Four-terminal magnetoresistance of a two-dimensionalelectron-gas constriction in the ballistic regime, Phys. Rev. B37, 8534 (1988).[26] H. van Houten, J. G. Williamson, M. E. I. Broekaart, C. T.Foxon, and J. J. Harris, Magnetoresistance in a GaAs-AlxGa1−xAs heterostructure with double subband occupancy,Phys. Rev. B 37, 2756 (1988).[27] Yu. V. Sharvin, A possible method for studying Fermi surfaces,J. Exptl. Theoret. Phys. (U.S.S.R.) 48, 984 (1965) [J. Exp.Theor. Phys. 21, 655 (1965)].[28] I. Torre, A. Tomadin, A. K. Geim, and M. Polini, Nonlocaltransport and the hydrodynamic shear viscosity in graphene,Phys. Rev. B 92, 165433 (2015).[29] T. Jungwirth and A. H. MacDonald, Electron-electron inter-actions and two-dimensional–two-dimensional tunneling, Phys.Rev. B 53, 7403 (1996).[30] R. N. Gurzhi, A. N. Kalinenko, and A. I. Kopeliovich,Electron-electron collisions and a new hydrodynamic effect intwo-dimensional electron gas, Phys. Rev. Lett. 74, 3872 (1995).[31] P. Ledwith, H. Guo, and L. Levitov, Angular superdiffusionand directional memory in two-dimensional electron fluids,arXiv:1708.01915v2.043088-12http://www.jetp.ras.ru/cgi-bin/e/index/e/17/2/p521?a=listhttps://doi.org/10.1103/PhysRevB.51.13389https://doi.org/10.1103/PhysRevB.98.241304https://doi.org/10.1103/PhysRevResearch.3.023033https://doi.org/10.1103/PhysRevX.11.031030https://doi.org/10.1126/science.aad0201https://doi.org/10.1038/nphys4240https://doi.org/10.1126/science.aau0685https://doi.org/10.1126/science.aac8385https://doi.org/10.1038/s41467-018-06688-yhttps://doi.org/10.1038/s41567-021-01341-whttps://doi.org/10.1038/s41598-020-64807-6https://doi.org/10.1038/s41586-019-1788-9https://doi.org/10.1038/s41586-020-2507-2https://doi.org/10.1126/science.abm6073https://doi.org/10.1103/PhysRevLett.117.166601https://doi.org/10.1023/A:1023084404080https://doi.org/10.1103/PhysRevB.96.195401https://doi.org/10.1103/PhysRevB.98.161303https://doi.org/10.1063/1.5020763https://doi.org/10.1073/pnas.1612181114https://doi.org/10.1103/PhysRevLett.100.186801https://doi.org/10.1103/PhysRevB.86.075425https://doi.org/10.1063/1.4945090https://doi.org/10.1103/PhysRevB.37.8534https://doi.org/10.1103/PhysRevB.37.2756http://www.jetp.ras.ru/cgi-bin/e/index/e/21/3/p655?a=listhttps://doi.org/10.1103/PhysRevB.92.165433https://doi.org/10.1103/PhysRevB.53.7403https://doi.org/10.1103/PhysRevLett.74.3872http://arxiv.org/abs/arXiv:1708.01915v2LONG DISTANCE ELECTRON-ELECTRON SCATTERING … PHYSICAL REVIEW RESEARCH 5, 043088 (2023)[32] S. Kryhin and L. Levitov, Collinear scattering and long-livedexcitations in two-dimensional electron fluids, Phys. Rev. B107, L201404 (2023).[33] S. Li, M. Khodas, and A. Levchenko, Conformal maps of vis-cous electron flow in the Gurzhi crossover, Phys. Rev. B 104,155305 (2021).[34] T. Scaffidi, N. Nandi, B. Schmidt, A. P. Mackenzie, and J. E.Moore, Hydrodynamic electron flow and Hall viscosity, Phys.Rev. Lett. 118, 226601 (2017).[35] H. Overweg, Electrostatically induced nanostructuresin bilayer graphene, Ph.D. thesis, ETH Zurich,2018.043088-13https://doi.org/10.1103/PhysRevB.107.L201404https://doi.org/10.1103/PhysRevB.104.155305https://doi.org/10.1103/PhysRevLett.118.226601