Imaging Viscous Flow of the Dirac Fluid in Graphene

Article Imaging viscous flow of the Dirac fluid in graphene

https://doi.org/10.1038/s41586-020-2507-2 Mark J. H. Ku1,2,3,14,15, Tony X. Zhou1,4,15, Qing Li1, Young J. Shin1,5, Jing K. Shi1, Claire Burch6, Laurel E. Anderson1, Andrew T. Pierce1, Yonglong Xie1,7, Assaf Hamo1, Uri Vool1,8, Received: 26 May 2019 Huiliang Zhang1,3, Francesco Casola1,3, Takashi Taniguchi9, Kenji Watanabe9, Accepted: 14 April 2020 Michael M. Fogler10, Philip Kim1,4, Amir Yacoby1,4 ✉ & Ronald L. Walsworth1,2,3,11,12,13 ✉

Published online: 22 July 2020 Check for updates The electron–hole plasma in charge-neutral graphene is predicted to realize a quantum critical system in which electrical transport features a universal hydrodynamic description, even at room temperature1,2. This quantum critical ‘Dirac fuid’ is expected to have a shear viscosity close to a minimum bound3,4, with an interparticle scattering rate saturating1 at the Planckian time, the shortest possible timescale for particles to relax. Although electrical transport measurements at fnite carrier density are consistent with hydrodynamic electron fow in graphene5–8, a clear demonstration of viscous fow at the charge-neutrality point remains elusive. Here we directly image viscous Dirac fuid fow in graphene at room temperature by measuring the associated stray magnetic feld. Nanoscale magnetic imaging is performed using quantum spin magnetometers realized with nitrogen vacancy centres in diamond. Scanning single-spin and wide-feld magnetometry reveal a parabolic Poiseuille profle for electron fow in a high-mobility graphene channel near the charge-neutrality point, establishing the viscous transport of the Dirac fuid. This measurement is in contrast to the conventional uniform fow profle imaged in a metallic conductor and also in a low-mobility graphene channel. Via combined imaging and transport measurements, we obtain viscosity and scattering rates, and observe that these quantities are comparable to the universal values expected at quantum criticality. This fnding establishes a nearly ideal electron fuid in charge-neutral, high-mobility graphene at room temperature4. Our results will enable the study of hydrodynamic transport in quantum critical fuids relevant to strongly correlated electrons in high-temperature superconductors9. This work also highlights the capability of quantum spin magnetometers to probe correlated electronic phenomena at the nanoscale.

Understanding the electronic transport of strongly correlated quantum quantum-critical scattering rate is observed for electron–hole systems 11 12 matter is a challenging problem given the highly entangled nature of in monolayer and bilayer/trilayer graphene . With a fast τpp << τmr, interacting many-body systems. Nevertheless, many disparate cor- where τmr is the momentum-relaxation time, local equilibrium can be related systems share universal features owing to quantum criticality. established. As a result, a simple, universal description of collective A paradigmatic example is the electron–hole plasma in high-mobility particle motion emerges in which electrical transport resembles that of graphene at the charge-neutrality point (CNP), known as the Dirac hydrodynamic fluids. A quantum critical electron fluid involving a maxi- fluid4,10, which is expected to share universal features present in other mal ‘Planckian’ dissipative process is postulated to underlie the linear-T quantum critical systems, even at room temperature1. First, particle– resistivity observed in the strange metal phase of high-temperature 1 9 particle scattering occurs on the timescale τpp ≈ τħ, where the Planck- superconductors . Second, as a result of Planckian-bounded dissipa- ian time τħ = ħ/(kBT) is the fastest timescale for entropy generation tion, the Dirac fluid in graphene is expected to have a ratio of shear allowed by the uncertainty principle at temperature T (ħ is the reduced viscosity η to entropy density s that saturates at an ‘ideal fluid’ lower 2–4 Planck’s constant and kB is the Boltzmann constant). Such a universal bound η/s ≥ ħ/(4πkB).

1Department of Physics, Harvard University, Cambridge, MA, USA. 2Quantum Technology Center, University of Maryland, College Park, MD, USA. 3Harvard-Smithsonian Center for Astrophysics, Cambridge, MA, USA. 4John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA. 5Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY, USA. 6Harvard College, Harvard University, Cambridge, MA, USA. 7Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA. 8John Harvard Distinguished Science Fellows Program, Harvard University, Cambridge, MA, USA. 9National Institute for Materials Science, Tsukuba, Japan. 10Department of Physics, University of California, San Diego, La Jolla, CA, USA. 11Center for Brain Science, Harvard University, Cambridge, MA, USA. 12Department of Physics, University of Maryland, College Park, MD, USA. 13Department of Electrical and Computer Engineering, University of Maryland, College Park, MD, USA. 14Present address: Department of Physics and Astronomy, University of Delaware, Newark, DE, USA. 15These authors contributed equally: Mark J. H. Ku, Tony X. Zhou. ✉e-mail: [email protected]; [email protected]

Nature | Vol 583 | 23 July 2020 | 537 Article a cd Ohmic Hydrodynamic To camera Green Red Diamond probe

y x Objective

NV hBN–G–hBN b 1 Top gate

hBN Graphene 0 SiO B ( μ T) 2

Doped Si –1 Diamond with NV ensemble –0.5 0 0.5 –0.5 0 0.5 x/W x/W

Fig. 1 | Probing viscous electronic transport via magnetic field imaging. Local diamond probe containing a single near-surface NV spin, excited by green light. current density Jy in a 2D conductor oriented along the y direction for NV red photoluminescence is collected by a confocal microscope. conventional ohmic transport (left) and viscous Poiseuille flow (right). hBN-encapsulated graphene device (hBN–G–hBN) is fabricated on a SiO2/doped b, Magnetic field Bx(x) (solid line) and Bz(x) (dashed line) at z = d generated by an Si substrate. Typical stand-off distance ≱ 50 nm between NV and graphene. electronic flow Jy(x) in the y direction for ohmic (left) and viscous Poiseuille flow d, Wide-field magnetic imaging. An hBN-encapsulated graphene device with a (right). Here the magnetic field profile is shown for a channel width W = 1 μm, graphite top gate is fabricated on a macroscopic diamond chip containing a current I = 1 μA and stand-off distance d = 50 nm. Hence, a quantum spin high-density ensemble of near-surface NVs. Green excitation light is sent through magnetometer located at z = d from a 2D current distribution J probes the local an objective to illuminate the field of view. NV photoluminescence is collected by current via sensing the local magnetic field B generated by J. c, d, Magnetic field the same objective and imaged onto a camera with 420-nm 2D resolution. imaging modalities employed in this work. c, Scanning NV magnetometry with a

Compared with the non-interacting particle case, hydrodynamic low-mobility graphene. Below we describe our experimental findings electron flow is expected to be dramatically modified owing to viscosity, in detail and discuss them in light of existing theories. for example, leading to vortices near a constriction and a parabolic We begin with scanning single-NV magnetic microscopy22,25, which velocity profile in a channel (known as Poiseuille flow)2. Signatures has spatial resolution ≱ 50 nm (Methods). For the devices we interro- of such viscous electron flow have been observed in electrical trans- gate, the dimension along which the electrical current flows (the y port5–8,13,14, scanning gate measurements15 and scanning single-electron direction) is much longer than the device’s narrow dimension (along transistor measurement of the Hall field16. However, experiments in the x direction). Therefore, we perform a 1D scan of the single NV probe graphene have so far been unable to provide information on whether across the width of the device W to measure the projective stray field 5–8,10,16 viscous Dirac fluid flow occurs near the CNP . Hydrodynamic along the NV axis Bx||() (Fig. 2a–c), from which we obtain Jy(x) and sub- transport applies to the velocity field u, which is related to electrical sequently elucidate the presence of hydrodynamic flow. 17–19 current via J = e(ne − nh)u + δJ. Here e is the electron charge, ne (nh) As a benchmark, we first perform measurements on a thin palladium is the density of electrons (holes), and δJ is an ohmic correction. At channel (Fig. 2a) and a low-mobility graphene device (Fig. 2b) in which finite doping, |ne − nh| > 0 suppresses δJ and hence the electrical current disorder is intentionally introduced (Methods). Local current density can reveal hydrodynamic transport. At the CNP, ne − nh = 0, additional can be reconstructed via inverting the Biot–Savart law and assuming the coupling between u and J is needed for electrical current to manifest continuity equation ∇·J = 0. The resulting current profiles Jy(x) for the viscous flow. In theory, such coupling may arise from an applied mag- palladium channel and the low-mobility graphene device are shown in netic field or thermoelectric effect, but has not been experimentally Fig. 2e. The current profiles show a clear hallmark of ohmic transport, observed. Thus the question of whether there is viscous Dirac fluid with a near-uniform Jy that drops sharply at the edges x/W = ±0.5. flow near the CNP remains open. We next apply scanning NV magnetic microscopy to measure current Here we address this question by directly imaging the flow of profiles in high-quality graphene devices, with nanoscale resolution. the Dirac fluid at room temperature in a high-mobility graphene A device consists of a hexagonal boron nitride (hBN)-encapsulated channel via measurement of the flow-induced stray magnetic field B monolayer graphene on a standard SiO2/doped Si substrate. The encap- using nitrogen vacancy (NV) centres in diamond20. Inside a conductor sulation ensures orders-of-magnitude increase in mobility and reduc- where the dominant scattering is momentum relaxing, for example, tion in momentum-relaxing scattering, a necessary condition to enter owing to impurities, phonons or Umklapp processes, the conventional the hydrodynamic regime. Unless otherwise noted, all references to ohmic transport of electrons in a channel exhibits a uniform current graphene are for such high-quality, encapsulated samples. The meas- profile. In contrast, a hydrodynamic electron fluid can develop a para- ured B|| at the CNP is shown in Fig. 2c and the resulting current density bolic (Poiseuille) current profile (Fig. 1a). For a two-dimensional (2D) is shown in Fig. 2e. The CNP is identified by a transport measurement current distribution J, the inverse problem between J and B is unique21; of resistivity as a function of carrier density (Fig. 2d), as controlled by therefore, measurement of B(x, y, z = d), where d is the sensor–source the gate voltage. In contrast to the ohmic profile seen in the palladium stand-off distance, reveals the local current flow (Fig. 1b). For our experi- channel and the low-mobility graphene device, the current profile in ments, magnetic imaging of current flow is performed with two comple- the standard graphene device is distinctly parabolic and vanishes at mentary modalities: (1) scanning probe microscopy using a single NV the boundary within experimental uncertainty. Similar profiles are centre22,23 (Fig. 1c); and (2) wide-field magnetic imaging with an ensem- observed in four standard devices (Fig. 2f), three at small current (about ble of NVs24 (Fig. 1d). We find clear experimental evidence for non-slip 1 μA) and one at large current (20 μA, second profile from the left). carrier flow at channel boundaries; and strong deviation from a flat While the profile of non-interacting electrons also develops curvature current profile. These results are in stark contrast with otherwise similar when the momentum-relaxation mean free path lmr = νFτmr ~ W, where 6 −1 measurements we performed on a conventional ohmic conductor and the Fermi velocity νF = 10 m s , we rule out this ballistic transport

538 | Nature | Vol 583 | 23 July 2020 abe constitutes an observation of viscous Poiseuille flow of the Dirac fluid 0.5 1 in room-temperature graphene. In addition, the boundary condition for the electron fluid in (mT) ( μ T) || || 19 B 0 B 0 graphene was not previously known . Previous work that explored 5,7,8 1 viscous flow in graphene (at finite doping) involved vortex backflow 0 1 0 1 and superballistic conductance across point contacts6, but neither Vx (V) Vx (V) c d J /( I / W ) phenomenon requires a specific boundary condition. Our experimental finding that the current density vanishes at the channel edges identi- 0.5 1 fies a no-slip boundary condition at room temperature. Recent theory ( μ T) || U (k Ω ) 0 26 B 0 work enables an estimate of the slip length of about 80 nm << W at 0 –0.5 0 0.5 room temperature (Supplementary Information), consistent with the 0 1 –0.5 0 0.5 x/W experimental observation. Note that past experimental work at reduced V (V) 12 –12 x n (10 cm ) temperatures was not able to establish viscous current flow at the CNP and the boundary condition in graphene5–8. f 1.6 In all devices, we observe an isolated dip in the current profile, close 1.2 to the centre of the graphene channel. While the nature of this current 0.8 profile dip requires future investigation, we note that recombination

J /( I / W ) 27–29 0.4 is expected to play a role in the transport of electron–hole systems , leading to peaks in the current density associated with the edges of a 0 flow in a channel28,29. This scenario is compatible with the experimental –0.5 0 0.5 –0.5 0 0.5 –0.5 0 0.5 –0.5 0 0.5 conditions in our devices (Methods). x/W The measurement described so far assumes uniformity of the sys- Fig. 2 | One-dimensional scanning NV microscopy of ohmic flow in a metallic tem along the y direction as well as the continuity equation. As a com- channel and low-mobility graphene and viscous flow in hBN-encapsulated plementary measurement, we perform wide-field 2D imaging of the graphene at the CNP. a–c, Scanning magnetometry of the projective stray field B|| vector magnetic field resulting from current in a graphene device, measured with the palladium channel (a), the low-mobility graphene device which allows us to map the electronic flow with minimal assumptions. (b) and a standard (hBN-encapsulated) graphene device (c) at the CNP. The The measurement is performed on an hBN-encapsulated graphene horizontal axis is the piezovoltage that drives the scanning probe in the x direction. device with a graphite top gate (optical image in the inset of Fig. 3a) The lines are B|| from reconstructed current density Jy. d, Transport measurement fabricated on diamond with a high-density ensemble of near-surface of resistivity ρ versus carrier density n for hBN-encapsulated graphene device, NV spins (Fig. 1d). Figure 3a shows the measured Bx and By as a function same as in c. The standard graphene device is set to the CNP by tuning the gate of position (x, y) in the graphene device. The resulting 2D vector cur- voltage to achieve maximal resistivity. Data for the palladium channel are rent map is shown in Fig. 3b. The measured current density has a flow measured with ODMR and a sourced current of 1 mA. Data for the low-mobility and pattern with direction and magnitude that are consistent with how standard graphene devices are measured with spin-echo a.c. magnetometry (The the current is sourced and drained. More quantitatively, this vector details of the measurement technique are described in Methods). A source–drain current map allows us to verify that the continuity equation ·J = 0 voltage Vsd = 50 mV is applied for the low-mobility device and 5.8 mV for the ∇ standard device. The width is 1 μm for the low-mobility and standard graphene holds (Supplementary Information). Furthermore, the measured net devices and 800 nm for the palladium channel. The palladium channel is 30 nm incoming current flux at the top of the device and the outgoing flux thick and 100 μm long. All the graphene devices measured in this work have at the bottom-left side probe are both consistent with a source–drain lengths ≥5 μm. For all devices, scanning magnetometry is performed at a y position current of 100 μA. near the mid-point along the length of the channel. For graphene devices, we In Fig. 3c, we show a line-cut of the current density far away from ensure the y position of the scan is within 1 μm of the longitudinal mid-point via the device drain, Jy(x, y = 2.5 μm). We can then compare the measured atomic force microscopy with the diamond probe. e, Reconstructed current current density profile with calculated diffraction-limited profiles density Jy(x). Jy is normalized by the average charge carrier flux I/W, where of uniform and viscous electron flow, for the same total current, Ix=d∫ Jxy () is the total flux and W is the width of the channel. The spatial 100 μA. As seen in Fig. 3c, the experimentally measured Jy profile coordinate x is normalized by W and centred on the channel. Red points, graphene at the CNP clearly deviates from uniform flow and matches well a at the CNP; grey points, palladium channel; orange points, low-mobility graphene. parabolic Poiseuille profile. This result provides further confirmation Error bars correspond to the relative deviation of J that generates 2χ2, where χ2 is y of the observation of viscous Dirac fluid flow in room-temperature the cost function. Blue (green) lines, ideal viscous (uniform) flow with 5% error band; purple dashed curve, current profile of non-interacting electrons with graphene. diffusive boundary condition and momentum-relaxing mean free path We next investigate the carrier-density dependence of current profiles in graphene. Owing to photoinduced doping in back-gated lmr = 0.625W, which corresponds to the maximum possible curvature for a non-interacting flow (Supplementary Information). f, Current profile at the CNP encapsulated devices, a scanning measurement away from the CNP from four different standard hBN-encapsulated graphene devices. Second profile averages over a range of carrier densities (Supplementary Informa- from left is for current of 20 μA and is measured with ODMR, whereas other profiles tion). The top panel of Fig. 4a shows current profiles measured with have current ≱ 2 μA and are measured with spin-echo a.c. magnetometry. the scanning NV magnetic microscope averaged over several different ranges of carrier densities. In all cases, the current profile is consistent with viscous flow. Similar observations are made with wide-field vector magnetic imaging. The top-gated graphene-on-diamond device is scenario by noting the inconsistency of the current profile in this regime immune from photoinduced doping and allows NV measurements at with measured profiles and a magneto-transport experiment (Meth- a specified value of the carrier density30. The bottom panel of Fig. 4a ods). Instead, the measured profiles have a compelling match with a shows diffraction-limited current profiles at several carrier densities viscous flow profile. Furthermore, our data show that by turning on up to 1.5 × 1012 cm−2, all of which are consistent with a Poiseuille profile and off disorder, the current profile in graphene switches between and hence viscous flow. While previous imaging work at T = 75 K and uniform and parabolic. This result is consistent with a transition 150 K in a graphene channel with W = 4.7 μm has shown that electric between disorder-dominated ohmic and collision-dominated hydro- transport moves away from the Poiseuille regime as carrier density dynamic regimes. The consistency between experiment and theory increases16, the insensitivity of the current profile to carrier density

Nature | Vol 583 | 23 July 2020 | 539 Article a 2 100 100

50 B 50 B x y ( μ T) 1 0 ( μ T) 0

y ( μ m) –50 –50 0 –100 –100 –1 –4 –2 0 2 4 –4 –2 0 2 4 x (μm) x (μm) bc250 150

2 200

100 J (A m 150 ) –1 1 –1 (A m y ( μ m) ) 100 y

– J 50

50 0 0 0 –2 –1 0 1 –1.0 –0.5 0 0.5 1.0 x (μm) x (μm)

Fig. 3 | Wide-field magnetic imaging of viscous electron flow in graphene. extracted from vector field measurement. In a and b, the grey outline denotes a, The 2D spatial distribution of Bx (left) and By (right) measured by wide-field area covered by a metallic top-gate contact that obstructs light, and top/ NV vector magnetometry, for a source–drain current of 100 μA and NV bottom left black arrows indicate the direction of the bias from the source/ ensemble imager located in positive z direction. Inset: optical image of device drain. c, Horizontal line-cut of b far away from drain, Jy(x, y = 2.5 μm), measured channel W = 1 μm. Current is sourced from the top contact to the bottom-left at the CNP. Calculated diffraction-limited profiles of a uniform (black) and contact. b, Vector (arrow) and amplitude (colour) plots of current density J(x, y) viscous (blue) flow are also shown.

observed here may be due to the much narrower channel used in our with both our experimental result for the kinematic viscosity v in the experiment (Methods). Dirac fluid, as well as present and past measurements at finite carrier To determine the kinematic viscosity v of the Dirac fluid and as a density (Fig. 4b). Note that this good agreement with existing theory function of carrier density, we consider an electronic flow along the y does not extend to measurements on a wider channel (Methods), for direction in a 2D channel with |x| ≤ W/2, where W is the channel width. which lmr/W is smaller and hence deviations from hydrodynamic behav- 2 Evaluating the electronic Navier–Stokes equation , assuming a no-slip iour may be expected, including, for example, discrepancies in Dv and v. boundary condition Jy(x = ±W/2) = 0, gives the following current density For the narrow-channel (1 μm) results, we define the viscous scattering 28,31 2 and conductivity time as τνν ≡/vF and show the values of τv normalized by the Planckian

time τħ on the right vertical axis of Fig. 4b. τv is not necessarily the same ev2 τ n  cosh(/xD)  Fmr ν as the particle–particle scattering time τpp, but is expected to be the Jxy ()= E1−  (1) ħ π  cosh(/WD(2 ν))  same order of magnitude. The results for the viscous scattering time are consistent with the expectation for a quantum critical system, where 2 the Planckian time sets the scale for scattering. On the basis of our evFmτ r n  2Dν  W  σ = 1− tanh  (2) result and the quantum critical scattering time τpp = 5τħ measured for ħ π  W 2Dν  11 a Dirac fluid , we estimate (Supplementary Information) τv ≈ 1–3τpp. Here E is the bias electrical field, n is the carrier density and the Gur- Lastly, Fig. 4c shows the shear viscosity η = nmv determined from our measurements, where mħ=πnv/ F is the carrier effective mass. The zhi length Dν ≡ ντmr is a characteristic length that determines the scale shear viscosity is normalized by the entropy density s0 of neutral gra- of viscous effects. In the limit Dv >> W, Jy becomes the parabolic profile 2 2 4 evF n  W 2 phene at T = 300 K, which we determined from the calculation in ref. . of ideal viscous flow, J ()xE=− x  . As profiles with y 2ħν π ()2  For an ‘ideal fluid’, this quantity reaches the conjectured lower bound Dv/W > 0.3 are indistinguishable from an ideal viscous flow within η/s = ħ/(4πkB) = 0.08ħ/kB. Close to the CNP, we observe η/s0 ≈ 0.3–0.8ħ/kB, experimental uncertainty, we estimate the lower bound on the Gurzhi which lies between the theoretical value of 0.26ħ/kB for a Dirac fluid at 4 length to be 0.3W for the profiles presented in this paper for 1-μm-wide 300 K from ref. and an experimental estimate of about 10ħ/kB for a 18 channels. Accordingly, we then obtain bounds on v using equation (2) Dirac fluid at about 100 K from ref. . We note that our value for η/s0 and the measured conductivity. Figure 4b shows values for v as a func- near the CNP is comparable to the value of about 0.4ħ/kB obtained in 32 33 tion of carrier density corresponding to Dv/W = ∞ and 0.3, which set cold atoms and ≱ 0.5ħ/kB in quark gluon plasma . Thus, our result the upper and lower bounds respectively for the Gurzhi length; as well adds a new data point for η/s in strongly interacting quantum matter, as for Dv/W = 0.5, which provides the median value. Near the CNP, where from a condensed-matter system, and highlights the trend of strongly thermal energy kBT becomes comparable to or exceeds the Fermi interacting systems exhibiting η/s within an order of magnitude of the energy EħFF= vπn , the kinematic viscosity v for the Dirac fluid is ‘ideal fluid’ lower bound. extracted from our measurement using a carrier density n obtained The observation of viscous flow of the Dirac fluid in room-temperature, from EF(n) = kBT. Away from the CNP, at relatively large carrier density, high-mobility graphene indicates a coupling between the velocity field our experiment gives v ≈ 0.1 m2 s−1, which is consistent with past trans- and electric current at the CNP, which enables current measurements port measurements performed in graphene at finite doping5,6. We note to reveal the underlying hydrodynamic transport. Theoretically, it that a recent calculation based on kinetic theory17 compares very well is known that a number of potential mechanisms may lead to such

540 | Nature | Vol 583 | 23 July 2020 ab0.4 15 Online content 0.3 Any methods, additional references, Nature Research reporting sum-

) 10 maries, source data, extended data, supplementary information, –1 W Q s / W 2 0.2 acknowledgements, peer review information; details of author con- ħ tributions and competing interests; and statements of data and code v (m 5 J /( I / W ) availability are available at https://doi.org/10.1038/s41586-020-2507-2. 0.1

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Phys. Rev. B 98, 125111 these and other potential sources of velocity–current coupling. The (2018). role of recombination in the transport of the Dirac fluid is another topic 30. Cadore, A. R. et al. Thermally activated hysteresis in high quality graphene/h-BN devices. Appl. Phys. Lett. 108, 233101 (2016). that deserves further exploration. Beyond viscous flow, electronic 31. Holder, T. et al. Ballistic and hydrodynamic magnetotransport in narrow channels. Phys. turbulence may also be explored via magnetic imaging experiments, Rev. B 100, 245305 (2019). as one can tune the Reynolds number by increasing the current and 32. Cao, C. et al. Universal quantum viscosity in a unitary Fermi gas. Science 331, 58–61 (2011). 33. Luzum, M. & Romatschke, P. Conformal relativistic viscous hydrodynamics: applications observe the evolution of the time-averaged flow profile from laminar to RHIC results at sNN =200 GeV. Phys. Rev. C 78, 034915 (2008). to turbulent. Lastly, our magnetic imaging techniques can be deployed for nanoscale study of an increasing number of materials manifesting Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. viscous electronic flow, as well as other correlated electronic phenomena, such as topological transport in the quantum spin Hall effect. © The Author(s), under exclusive licence to Springer Nature Limited 2020

Nature | Vol 583 | 23 July 2020 | 541 Article Methods contacts, metallic electrodes are made directly on graphene without an RIE step before evaporating metals. NV spin physics An NV spin consists of a substitutional nitrogen atom and an adjacent Wide-field magnetic imaging vacancy in the diamond lattice. Its ground state is an S = 1 electronic The sample is positioned on a combined translational stage of Physik spin-triplet with ms = 0, ±1 eigenstates, each with magnetic moment Instrumente M686.D64 for coarse positioning with about 0.1-μm pre- −1 msγe, where γe = 2.8 MHz G is the gyromagnetic ratio of the NV elec- cision, and P-541.2DD for fine positioning with roughly nanometre tronic spin. An NV spin can be initialized into the ms = 0 state with opti- precision. NV centres are optically excited with a 532-nm laser (Coher- cal excitation at around 532 nm wavelength. Optical excitation also ent COMPASS 315M-100) switched on and off by an acousto-optic generates spin-dependent photoluminescence in the 640–800 nm modulator (Isomet 1250C-974). The light is directed onto the sample wavelength range, which allows optical spin readout. The ms = ±1 states with an Olympus 0.9NA ×100 objective. A Kohler-illumination sys- are split from the ms = 0 state by about 2.87 GHz at zero magnetic field, tem expands the laser beam to illuminate an area of about 5 μm in and are further split from each other at finite magnetic field due to the diameter on the sample. About 10 mW of excitation illuminates the Zeeman effect. sample. NV photoluminescence is collected by the same objective. Experiments employ electronic grade single crystal diamonds with a The collected light passes through a 552-nm edge dichroic (Semrock {100} front facet from Element Six, created using chemical vapour LM01-552-25), after which it is separated from the excitation, passes deposition. Atomic force microscopy ensures surface roughness is through another 570-nm long-pass filter to further reduce light not under 1 nm. Near-surface NV spins are created using 15N implantation in the photoluminescence wavelength range, and is imaged via a tube (Innovion) at 6 keV followed by annealing for both wide-field magnetic lens (focal length f = 200 mm) onto a camera (Basler acA1920-155um). imaging and scanning probe magnetometry. This implantation energy The output of a microwave synthesizer (SRS SG384) is controlled by a generates NV spins at a depth of about 10–20 nm beneath the diamond switch (Mini-Circuits, ZASWA-2-50DR+), then amplified (Mini-Circuits, surface. For wide-field magnetic imaging, a dosage [N] = 2 × 1012 cm−2 ZHL-16W-43-S+) and delivered onto the printed circuit board housing leads to an ensemble of [NV] ≈ 1011 cm−2. For scanning probe magnetom- the sample. A permanent magnet, whose position is controlled by three etry, a dosage is selected to ensure each probe contains on average one sets of motorized translational stages (Thorlabs MTS25-Z8), is used NV spin22. The diamonds are cleaned using a mixture of sulfuric, nitric to apply a uniform external magnetic field. A gate voltage is supplied and perchloric acids before deposition of van der Waals heterostructure by a Keithley 2420, and d.c. current for the measurement is supplied or fabrication of scanning probe. by a Keithley 6221. Optically detected magnetic resonance (ODMR) spectra are Device fabrication measured by capturing an NV photoluminescence image at different

To realize high-quality graphene devices, we generate a heterostructure microwave frequencies. The spectra contain both the ms = 0 ↔ −1 of graphene and hBN34. A polymer-free assembly method35 is used to (lower) and +1 (upper) resonances. To extract the resonance frequen- prepare a van der Waals heterostructure for the devices. hBN and mon- cies f±, Lorentzian fits of photoluminescence versus frequency olayer graphene (G) flakes are exfoliated on standard SiO2/p-doped Si are performed at each pixel; the fitting function consists of a single substrate. Polymer polycaprolactone36 is used to pick up the flakes. For Lorentzian or double Lorentzian if the 15N hyperfine splitting is scanning measurement, three-layer hBN–G–hBN stacks are prepared resolved. This procedure is applied with the external field (about on a standard substrate; the thickness ranges from 20 to 50 nm for 90 G) aligned along three of the four NV axes, labelled as i = 1, 2, 3. At the bottom hBN and from 10 to 20 nm for the top hBN. For each NV orientation, resonance frequencies for the upper transition wide-field magnetic imaging, a four-layer structure, consisting of an f+(±I) and lower transition f−(±I) are recorded for current through hBN-encapsulated graphene stack (top hBN 13 ± 1 nm thick and bottom the sample in the positive and negative polarity ±I. At a given NV hBN 27 ± 1 nm thick) together with a graphite top gate is deposited on orientation, the projected stray field generated by the current a diamond. can then be obtained as Bf|| =[((+++)If−(−)If)−((−−+)If−(−Iγ))]/(4)e . Electron-beam lithography is performed using Elionix 7000 with 495 Determination of three projective field components B||,i allow the PMMA C6 as the resist. To mitigate charge build-up during fabrication vector magnetic field in the NV layer to be determined for all three of the graphene-on-diamond device, an additional layer of Aquasave Cartesian directions. is spun onto the diamond and conductive carbon tape is attached to Four classes of orientations exist in the NV ensemble. Up to a rotation the diamond edges. To pattern the stack and to make edge contacts, in the xy plane, the orientation vectors are given by inductively coupled plasma reactive ion etch (RIE) with CHF3 chemistry 1 is employed (Surface Technology Systems). Electron-beam evapora- uˆ=1,NV (2,0,1) (3) tion deposits 10 nm of chromium as an adhesion layer followed by 3 gold of minimum 100-nm thickness for metallization and creation of other electrical structures (for example, microwave delivery for the 1 uˆ= (− 2,0, 1) (4) graphene-on-diamond device and bond pads). 2,NV 3 For the graphene-on-diamond device, it is necessary to create contact to the top gate graphite without shorting to the graphene. To accom- 1 uˆ= (0,2,1) (5) plish this, first electron-beam lithography is used to write a dielectric 3,NV 3 structure with hydrogen silsesquioxane, which serves to insulate the top gate contact from the graphene. The top gate contact is then cre- 1 ated. It is also necessary to ensure the top gate graphite is not shorted uˆ=4,NV (0,− 2,1) (6) 3 to graphene edge contacts. To accomplish this, graphite in the vicinity of the edge contacts is etched away using RIE with O2 chemistry, which Here the coordinates are defined by the diamond, which we call the does not affect hBN. NV frame, (xNV, yNV, z). This differs from the coordinate in the device Low-mobility graphene devices are prepared directly from exfoliated frame (x, y, z), which is such that the y direction is along the length of monolayer graphene flakes on SiO2/p-doped Si substrate without hBN the channel, by a rotation of angle θ in the x–y plane. For a vector meas- encapsulation. Furthermore, the samples do not undergo thermal urement, three projected fields are measured, Buii=⋅B ˆ ,NV. Then, the annealing, in contrast to hBN-encapsulated devices. Instead of edge vector field in the NV frame can be determined: average and half of the difference, respectively, of the photolumines- BBz =( 3/2)(+12B ) (7) cence signals of the ms = 0 and the ms = −1 states. Hence, PL0 + A is the

photoluminescence signal for the ms = 0 state and PL0 − A is the pho-

BBx,NV1=3/2 (−B2)/2(8) toluminescence signal for the ms = −1 state. Therefore, the NV phase can be determined as

BByz,NV3=3/2 (−B /3) (9) −1 PL−+yy−PL  ϕ =tan   (13) PL+−xx−PL  The vector field in the device frame can be obtained by a rotation angle θ The microwave pulses are tuned to the ms = 0 ↔ −1 transition as determined via ODMR. Before the start of a scan, a spin-echo sequence is Bx cos−θθsin Bx,NV run without V to determine the Hahn echo spin coherent time T , as   =    (10) a.c. 2 B     y   sincθθos By,NV well as locations of the NV spin coherence revival in the presence of the natural abundance 13C bath. τ ≈ 10 or 20 μs is used. During a scan, Current reconstruction is done using direct inversion of the Biot– the microwave drive Rabi frequency is measured at each spatial loca- Savart law in Fourier space tion to calibrate the microwave pulses. Spin-echo measurements are then performed with all four possible final microwave pulses ±(π/2) μ x,y 0 −d k to obtain a measurement of the NV phase as described above. bdx(,kk)= e(jy ) (11) 2 Differential measurements are also performed with the a.c. bias sequence +V , −V and −V , +V , leading to a total of eight μ a.c. a.c. a.c. a.c. 0 −d k pulse-sequence combinations. bdy(,kk)=− e(jx ) (12) 2 The current reconstruction for scanning NV microscopy is described in the Supplementary Information. where bx,y (jx,y) are the Fourier transforms of the magnetic field Bx,y

(current density Jx,y), k is the wave vector, μ0 is the vacuum permeabil- Transport measurement ity, and the stand-off distance (Supplementary Information) used is Transport measurement of device conductivity σ is performed with a d = 50 ± 10 nm. In this modality, uncertainty in the stand-off distance Stanford Research Systems lock-in amplifier (SRS SR830) at 17.77 Hz is the dominant source of error for the current density, which is about using a current-bias modality, with a large attached load of about 10 MΩ −1 9 A m . and source of about 100 nA. The gate voltage Vg is supplied by a Keithley

Optical diffraction limits the point spread function of the wide-field 2420. Carrier density n is determined using n = Cg(Vg − VD), where VD is magnetic imager, placing a Fourier low-pass filter on the current recon- the location of the CNP determined from location of the peak resistivity struction, with a cut-off spatial frequency much lower than the cut-off ρ, and Cg = ε0εr/(et) is the gate capacitance per area, with ε0 the permit- placed by the NV layer stand-off distance (that is, the imager spatial tivity of free space, εr the dielectric constant and t the thickness of the resolution 420 nm is large compared with the stand-off d ≈ 50 nm). Thus, dielectric. For the graphene-on-diamond device, εr = 4 and t = 13 nm are high-spatial-frequency noise is suppressed by the wide-field magnetic used for the hBN gate dielectric. For the devices made on the standard imager and direct inversion of the device current is appropriate. SiO2/Si substrates, εr = 3.9 and t = 285 nm are used.

Scanning NV magnetic microscopy Low-mobility graphene devices The setup is described in ref. 23. Scanning of the sample is performed with The low-mobility graphene devices serving as a control are distin- piezoelectric nanopositioners Attocube ANPxyz101 and ANSxyz100. guished from standard, high-quality graphene devices in two ways: The measurement pulse sequence is controlled by a Tektronic Arbi- the absence of hBN-encapsulation and thermal annealing. It is known trary Waveform Generator (AWG) 5014C. Microwaves are supplied by that such bare graphene devices without annealing are heavily doped, a Rhode Schwartz SMB100A. As the device for scanning measurement and are expected to have low mobility37,38, with the CNP shifted to high is fabricated on a substrate with a global back gate, microwaves can- gate voltage. The primary cause of strong disorder scattering in the not be delivered on-chip as capacitive effects lead to poor microwave low-mobility devices is twofold. First, the graphene is in direct contact transmission. Instead, we use a separate antenna made from a wire of with the SiO2 substrate, which leads to scattering with charged surface about 20-μm diameter placed about 50 μm above the device. The move- states, impurities, surface roughness and SiO2 surface optical phon- ment of the antenna is controlled by manual translational stages. The ons34. Second, the graphene is exposed to air and polymer used for diamond probe consists of a nanopillar containing a single near-surface lithography, leading to residue and contamination, which are known NV spin and is described in refs. 22,25. to act as external scattering centres and further limit the mobility38,39.

A scanning measurement consists of iterating the piezovoltage Vx Hence in these bare graphene devices, transport is dominated by dis- of the sample stage, and then performing OMDR or spin-echo mag- order even at room temperature. We prepared two such devices, with netometry at each value of Vx. For a scanning ODMR measurement, optical images shown in Extended Data Fig. 1a, b. An example of the current is supplied by a Keithley 2420. The ODMR measurement is gate-dependent transport curve is shown in Extended Data Fig. 1c. performed in the same manner as in wide-field magnetic imaging, From the slope of conductivity σ with respect to the gate voltage −1 except that only one projection of B|| is measured. For spin-echo a.c. Vg, we extract the mobility as μ = (eCg) (dσ/dVg), where Cg is the gate 2 −1 −1 magnetometry, a Hahn spin-echo (π/2)x − τ − (π)x − τ − ±(π/2)x,y is capacitance. We obtain μ ≈ 1,000 cm V s for both devices, which employed. The first (π/2)x pulse generates an equal NV superposition is typical for such unannealed graphene in the literature and is close state (|0⟩ −|i −1⟩)/2. A modulating square pulse of bias ±Va.c. is supplied to two orders of magnitude lower than a typical hBN-encapsulated to the graphene device from the AWG during the NV free precession graphene device. Extended Data Fig. 1d is the same figure as Fig. 2e, but time, generating a field of ±Ba.c on the NV. During the free precession, with (1) the palladium data removed and (2) the current profiles from the NV picks up a total phase of ϕ = 2γeτBa.c.. The NV is therefore in the both of the two low-mobility graphene devices displayed. We observe −iϕ superposition state (|0⟩ +ei |−1⟩)/ 2 . The last ±(π/2)x,y pulse leads that in both low-mobility graphene devices, the current profile is to a photoluminescence (PL) readout signal of PL±x = PL0 ± Acos(ϕ) for sharp at the edges and flat within the channel, consistent with ohmic

±(π/2)x and PL±y = PL0 − (± Asin(ϕ)) for ±(π/2)y. Here, PL0 and A are the transport. Article

experimental error bar; hence it is possible that variation in Dv due to Ruling out ballistic transport in encapsulated graphene device changing n leads to insignificant change in the current profile in the at room temperature standard 1-μm channel.

For a system of non-interacting (τpp = ∞) electrons moving in a chan- To investigate this scenario, we perform scanning NV microscopy in a nel −W/2 < x < W/2 with a completely diffusive boundary, the flow wider channel, W = 1.5 μm, across a few different carrier density ranges. profile is expected to develop a curvature in the regime where the In Extended Data Fig. 2a, we show the current profile at the CNP. The momentum-relaxing mean free path is on the order of the channel profile is clearly flatter compared with the ideal Poiseuille limit. We width, lmr ~ W. Therefore, it is important to rule out such ballistic trans- can compare this profile in real space to the profile measured in the port as a possible explanation of the measured current profiles that narrower 1-μm channel for the same device (Fig. 2e) by plotting the display curvature. We performed such a check using two different x axis in terms of micrometres (instead of the normalized unit x/W). approaches: (1) comparison of experimental profiles to calculated In Extended Data Fig. 2b, we see that at the CNP, by moving from non-interacting (ballistic) current profiles and (2) magneto-transport a narrower to a wider channel, a flatter profile is observed. measurements. Next, we show a measured current profile in a regime slightly away The derivation of non-interacting current profiles is described from the CNP in Extended Data Fig. 2c. Here the density range is esti- in the Supplementary Information. Comparison of a calculated mated to be n = 0.08 × 1012–0.30 × 1012 cm−2. This profile is closer to a non-interacting current profile with maximum curvature and an experi- parabola. In a third density regime, estimated to be n = 0.30 × 1012– mentally measured profile is shown in Fig. 2e. The spatial resolution 0.53 × 1012 cm−2, the measured profile (Extended Data Fig. 2d) becomes and signal-to-noise of our experiment allows us to identify directly the flatter again. In these measurements, the density range is assessed in the transport regime based on the measured current profile in the graphene same way as for the data shown in Fig. 4a (Supplementary Information). channel at the CNP. The observed profile smoothly decreases away This set of measurements shows that indeed, in a wider channel, the from the centre of the channel and vanishes at the boundaries, within current profile is more sensitive to the carrier density variation. We experimental uncertainty. A ballistic current profile has a discontinuity find that at finite carrier density, further increasing the carrier density at the boundaries, which is inconsistent with our observation and is leads to the system moving away from the Poiseuille regime, which is thus ruled out; whereas a viscous flow profile has a compelling match consistent with the observation of ref. 16. Near the CNP, we find that with the measured profiles. moving the system away from neutrality brings the system closer to the The magneto-transport measurement is described in the Supple- Poiseuille limit. In the regime of low carrier density and near the CNP, mentary Information. With this technique, we find that while the device there is neither existing theory nor experiment to which our observa- exhibits ballistic transport at 10 K, as expected given the device’s high tions can be compared. quality, ballistic transport is ruled out at room temperature. Here we comment on our observation in the context of the current understanding of the physics of graphene and carrier hydrodynam- Electron–hole recombination as possible source for ics. The shape of the current profile as a function of carrier density higher-order feature in current profiles is expected to be determined by the n dependence of both v and lmr With high-resolution scanning NV microscopy22,25, we observe an iso- As found in ref. 5, v is rather insensitive to n at finite doping and near lated dip in the current profiles of the high-mobility graphene devices, room temperature. However, it is known that lmr near room tempera- close to the centre of the graphene channel (Fig. 2e, f). We note that ture generally has a non-monotonic trend with n: increasing initially recombination is expected to play a role in the transport of electron– away from the CNP, and then decreasing as n continues to increase hole systems27–29 such as graphene at the CNP, leading to peaks in the (see, for example, ref. 35). This pattern has an intuitive explanation. current density associated with the edges of a flow in a channel28,29. At low carrier density, it is known that the lack of screening leads This scenario is compatible with the scale of the recombination length to contributions to momentum relaxation coming from disorder. in our devices, estimated to be about 1 μm in ref. 27, which is compa- Then, increasing density in this regime begins to screen disorder and rable to the channel width W. Furthermore, the dip is attenuated in hence reduces momentum relaxation. Once the system is at high car- 40 one graphene device (the last profile on the right of Fig. 2f), which has rier density, relaxation is limited by phonons and lmr decreases with higher carrier density inhomogeneity as indicated by a much broader increasing n. This non-monotonic behaviour of lmr as a function of car- density-dependent resistivity. We estimate a fluctuation potential of rier density n may be related to the trend in current profiles observed about 450–570 K for this device (Supplementary Information), which here. Further theoretical and experimental work is needed to address is substantial compared with the thermal energy and hence leads to an this possibility. effective doping. The appearance of an attenuated dip in this device Lastly, we note that while we observe current profiles in the wider is also compatible with recombination leading to additional features channel that deviate from ideal Poiseuille behaviour, it is not appropri- in the current density, as recombination is exponentially suppressed ate to extract quantitative values for Dv from these profiles by fitting upon doping. them to equation (1). A condition for hydrodynamics is that lmr is not 35 small compared with W. As lmr ≈ 1 μm near room temperature , equa- Current profiles in wider channel of encapsulated graphene tion (1) is less applicable for the wider channel. Consequentially, in the device wider channel, the smaller lmr/W ratio (that is, deviation from the hydro- The shape of a current profile in the hydrodynamic regime, given by dynamic regime) can also lead to a flatter current profile. Therefore, in equation (1), depends on the ratio of Dv/W. In principle, changing the the wider channel, fitting the current profile to equation (1) is likely to carrier density n may lead to changes in the kinematic viscosity v and extract a value of Dv that is smaller than the actual value. the momentum-relaxing mean free path lmr, and hence n may provide a means to tune the Gurzhi length Dν = νlmr and thus the shape of the current profile. For example, scanning single-electron transistor meas- Data availability urements16 in a 4.7-μm-wide graphene channel show that electron flow The data that support the plots and other analysis in this work are avail- deviates from the ideal Poiseuille regime as n increases at T = 75 and able upon request. 150 K. As shown in Fig. 4, in the standard 1-μm channel used in our encapsulated graphene devices, we observe qualitatively parabolic 34. Dean, C. R. et al. Boron nitride substrates for high-quality graphene electronics. Nat. 12 −2 Nanotechnol. 5, 722–726 (2010). current profiles up to a carrier density of about 10 cm . We note that 35. Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science current profiles with Dv/W > 0.3 are indistinguishable within 342, 614–617 (2013). 36. Son, S. et al. Strongly adhesive dry transfer technique for van der Waals heterostructure. grant number P300P2-158417. P.K. acknowledges support from ARO (W911NF-17-1-0574). Preprint at https://arxiv.org/abs/2006.15896 (2020). M.M.F. acknowledges support from the Office of Naval Research grant N00014-18-1-2722. A.T.P. 37. Cheng, Z. et al. Toward intrinsic graphene surfaces: a systematic study on thermal acknowledges support from the Department of Defense through the National Defense Science

annealing and wet-chemical treatment of SiO2-supported graphene devices. Nano and Engineering Graduate Fellowship (NDSEG) Program. Y.X. acknowledges partial support Lett. 11, 767–771 (2011). from the Harvard Quantum Initiative in Science and Engineering. This research used resources 38. Moser, J., Barreiro, A. & Bachtold, A. Current-induced cleaning of graphene. Appl. Phys. of the Center for Functional Nanomaterials, which is a US DOE Office of Science Facility, at Lett. 91, 163513 (2007). Brookhaven National Laboratory under contract number DE-SC0012704. This work was 39. Lindvall, N., Kalabukhov, A. & Yurgens, A. Cleaning graphene using atomic force performed, in part, at the Center for Nanoscale Systems (CNS), a member of the National microscope. J. Appl. Phys. 111, 064904 (2012). Nanotechnology Infrastructure Network, which is supported by the NSF under award no. 40. Hwang, E. H. & Das Sarma, S. Acoustic phonon scattering limited carrier mobility in ECS-0335765. CNS is part of Harvard University. two-dimensional extrinsic graphene. Phys. Rev. B 77, 115449 (2008). Author contributions M.J.H.K., A.Y. and R.L.W. conceived the project. M.J.H.K., T.X.Z., A.Y. and R.L.W. designed the experiments. M.J.H.K. and T.X.Z. performed scanning NV measurements Acknowledgements We thank B. Narozhny for helpful discussions, M. J. Turner for annealing and room-temperature electrical transport measurements and analysed the data. T.X.Z. diamond samples and S. Y. F. Zhao for assisting with the magneto-transport measurement. fabricated the scanning probes and built the scanning probe setup. M.J.H.K. and Q.L. built the This material is based on work supported by, or in part by, the United States Army Research NV wide-field imaging setup, performed the measurement, and analysed the data. M.J.H.K., Laboratory and the United States Army Research Office under contract/grant number T.X.Z., L.E.A., A.H. and U.V. performed the magneto-transport experiment and analysed the W911NF1510548 and number W911NF1110400, as well as the Quantum Technology Center data. M.J.H.K., Y.J.S., J.K.S., C.B., A.T.P., Y.X. and H.Z. fabricated the devices. M.M.F. provided the (QTC) at the University of Maryland. A.T.P. and Y.X. were primarily supported by the US theory on the current profile of non-interacting electrons. K.W. and T.T. grew the hBN single Department of Energy, Basic Energy Sciences Office, Division of Materials Sciences and crystals. M.J.H.K., T.X.Z., Q.L., F.C., P.K., A.Y. and R.L.W. contributed to the discussion. M.J.H.K., Engineering under award DE-SC0001819. T.X.Z., A.H. and U.V. were partly supported by ARO T.X.Z., A.Y. and R.L.W. wrote the manuscript with input from all authors. P.K., A.Y. and R.L.W. grant number W911NF-17-1-0023 and the Gordon and Betty Moore Foundations EPiQS Initiative supervised the project. through grant number GBMF4531. Fabrication of samples was supported by the US Department of Energy, Basic Energy Sciences Office, Division of Materials Sciences and Engineering under award DE-SC0019300. A.Y. also acknowledges support from ARO grants Competing interests The authors declare no competing interests. W911NF-18-1-0316 and W911NF-1-81-0206 and the STC Center for Integrated Quantum Materials, NSF grant number DMR-1231319. Part of this work was supported under the NSF Additional information grant number EFMA 1542807; and the Elemental Strategy Initiative conducted by MEXT, Japan, Supplementary information is available for this paper at https://doi.org/10.1038/s41586-020- and JSPS KAKENHI grant JP15K21722 (K.W. and T.T.). Finally, this work was partly carried out at 2507-2. the Aspen Center for Physics, which is supported by National Science Foundation grant Correspondence and requests for materials should be addressed to A.Y. or R.L.W. PHY-1607611. F.C. acknowledges partial support from the Swiss National Science Foundation Reprints and permissions information is available at http://www.nature.com/reprints. Article

Extended Data Fig. 1 | Low-mobility graphene devices. a, b, Optical image of d, Current profiles of two bare monolayer graphene devices. Grey is for the the two bare monolayer graphene devices (a and b), with experimental results device shown in a. Pink is for the device shown in b. The data are shown in reported in panels c and d. The width of each device is W = 1 μm. c, Conductivity comparison to the parabolic current profile measured for the encapsulated

σ as a function of the gate voltage Vg of the device shown in a. Data are shown in graphene device shown in Fig. 2e. Error bars correspond to the relative 2 2 red. The blue line is a linear fit, which allows for extraction of the mobility. deviation of Jy that generates 2χ , where χ is the cost function. a 1.6 b 1.6 1.4 1.4 1.2 1.2 1.0

) 1.0 ) W W / / I 0.8 I 0.8 J /( J /( 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0

-0.5 0.0 0.5 -1.0 -0.5 0.0 0.5 1.0 x/W x ( m)

c 1.6 d 1.6 1.4 1.4 1.2 1.2

) 1.0 ) 1.0 W W / / I 0.8 I 0.8 J /( 0.6 J /( 0.6 0.4 0.4 0.2 0.2 0.0 0.0

-0.4 0.0 0.4 -0.5 0.0 0.5 x/W x/W

Extended Data Fig. 2 | Scanning NV measurement of current profiles in a estimated carrier density range n = 0.08 × 1012–0.30 × 1012 cm−2 (c) and W = 1.5 μm channel. a, Measurement at the CNP. b, Data from a compared with n = 0.30 × 1012–0.53 × 1012 cm−2 (d). In a, c and d, the black dashed line is the ideal data from Fig. 2e. In this panel, the horizontal axis is x in micrometres, where uniform profile, the blue dashed curve is the ideal Poiseuille profile and the for all the other panels the horizontal axis is x/W. c, d, Measurement in the green curve is a fit to the data with equation (1). Article https://doi.org/10.1038/s41586-020-2507-2 Supplementary information

Imaging viscous flow of the Dirac fluid in graphene

In the format provided by the Mark J. H. Ku, Tony X. Zhou, Qing Li, Young J. Shin, Jing K. Shi, Claire Burch, Laurel E. authors and unedited Anderson, Andrew T. Pierce, Yonglong Xie, Assaf Hamo, Uri Vool, Huiliang Zhang, Francesco Casola, Takashi Taniguchi, Kenji Watanabe, Michael M. Fogler, Philip Kim, Amir Yacoby ✉ & Ronald L. Walsworth ✉

Nature | www.nature.com/nature Nature | www.nature.com | 1 Supplementary Information for Imaging Viscous Flow of the Dirac Fluid in Graphene

Mark J.H. Ku, Tony X. Zhou, Qing Li, Young J. Shin, Jing K. Shi, Claire Burch, Laurel E.

Anderson, Andrew T. Pierce, Yonglong Xie, Assaf Hamo, Uri Vool, Huiliang Zhang, Francesco

Casola, Takashi Taniguchi, Kenji Watanabe, Michael M. Fogler, Philip Kim, Amir Yacoby, and

Ronald L. Walsworth

1 1 Validity of mapping current density via magnetic ﬁeld imaging

In the case of a static 2D current distribution J(x, y) = (Jx(x, y),Jy(x, y)), there is a one-to-one mapping between J(x, y) and the associated stray magnetic ﬁeld

B(r) = (Bx(x, y, z),By(x, y, z),Bz(x, y, z)). Therefore, in principle one can equate the knowledge of the magnetic field distribution to the knowledge of the current distribution. Here, we evaluate several experimental factors that may invalidate this equivalence. We show that in our experimental setting, these factors are insignificant within experimental uncertainties, and hence it is valid to extract the local current density via magnetic field imaging.

First, we discuss the potential consequence of measuring B(r) over a ﬁnite ﬁeld-of-view.

According to the Biot-Savart law, the in-plane stray ﬁeld at (x, y, d) is

µ Z d B (x, y, d) = 0 dx0dy0 J (x0, y0) (S1) x 4π [(x − x0)2 + (y − y0)2 + d2]3/2 y µ Z d B (x, y, d) = − 0 dx0dy0 J (x0, y0) , (S2) y 4π [(x − x0)2 + (y − y0)2 + d2]3/2 x

where µ0 is the vacuum permeability. In other words, the in-plane stray ﬁeld can be considered as a convolutional ﬁlter (x2 + y2 + d2)−3/2 with a resolution length-scale d applied on the local current density. As a result, Bx(x, y, d) and By(x, y, d) are only sensitive to the current density in the neighborhood of (x, y) extending on the order of ∼ d. This is also apparent with the forms of the above equations in Fourier space1:

µ b (k, d) = 0 e−d|k|j (k) , (S3) x 2 y µ b (k, d) = − 0 e−d|k|j (k) , (S4) y 2 x

2 where k is the wave-vector, and bx,y (jx,y) is the Fourier transform of Bx,y (Jx,y). Therefore,

(Bx,By) is equivalent to (Jx,Jy) with a low-pass filter, and the current density can be obtained by inverting the low-pass filter on the stray-field (e.g., in the case of vectorial wide-field imaging).

Let us now consider the case where the measured stray ﬁeld involves the out-of-plane stray

ﬁeld Bz, for example in scanning magnetometry. From the Biot-Savart law, one has

µ Z (y − y0)J (x0, y0) − (x − x0)J (x0, y0) B (x, y, d) = 0 dx0dy0 x y . (S5) z 4π [(x − x0)2 + (y − y0)2 + d2]3/2

2 2 3/2 Here, the convolutional kernel has the form ∼ r/(r + d ) . Therefore, Bz is not simply a low-

pass ﬁlter applied on the current density. This is also apparent with the form of the above equation

in Fourier space1: µ k k b (k, d) = i 0 e−d|k| y j (k) − x j (k) . (S6) z 2 |k| x |k| y

The form of the kernel in Eq. S5 indicates that Bz(x, y, d) is insensitive to the current density in

the immediate neighborhood of (x, y), while contribution from further away has greater weight.

As a consequence, Bz has a slow ∼ 1/r decay far away from a current-carrying wire segment.

Therefore, when attempting to extract the local current density from a measurement that contains

Bz, one needs to consider the potential contribution from current outside of the ﬁeld of view. This

can happen when the current-carrying channel does not continue in a straight line indeﬁnitely, but

makes a turn. Let us consider how close the turning needs to occur from the point of measurement

so the contribution from far-away wire becomes important compared to the measurement

precision, typically ∼5% on the maximal value of the projective ﬁeld for our experiments. Let us

consider a channel initially along the y-direction, which changes direction to run along the

3 x-direction at a distance r away from the point of measurement. Using the expression of the ﬁeld away from a thin current-carrying wire, one can estimate the contribution to Bz as µ0I/(4πr).

There is an extra factor of 2 in the denominator because after the turn, we have a semi-inﬁnite wire. We estimate the ﬁeld magnitude at the point of measurement to be (µ0/2)(I/W ), with

W = 1 µm. Then, the contribution due to Bz from the far-away wire is comparable to 5% of the total measured field when r is closer than 3 µm. In the devices measured with NV scanning magnetometry, turning of current paths occurs on length-scales significantly greater than this distance. Therefore, the effect of slowly-decaying Bz from far-away current does not invalidate our approach to use magnetometry to probe the local current profile.

The discussion so far centers around the magnetostatic case. In AC magnetometry, the natural assumption is to replace J(x, y) and B(x, y, d) with J(x, y, t) and B(x, y, d, t). In general, this is not strictly true because of the delay in the propagation of the electromagnetic field; the generalized time-dependent formula for the stray magnetic field is given by Jefimenko’s equations which incorporate retarded time. Here, we show that our experiment is in the magnetostatic regime. Let us consider the current being modulated at a frequency ω. At a point of measurement, the stray field is sensitive to contributions from current over a length scale r given by the distance between the measurement point and the current source. Over this length scale, an electromagnetic

ﬁeld propagates to the point of measurement within a time τprop ∼ r/c, where c is the speed of light. We can assume the experiment is in the magnetostatic regime if this time is much faster than 1/ω. In the scanning magnetometry experiment, the scan range is no more than 5 µm, so r < 5 µm and τprop < 17 fs. Therefore, magnetostatics can be safely assumed for

4 ω/(2π) 60 THz, which is certainly the case in our experiment.

Lastly, in AC magnetometry, RC reactance can lead to a slow rise/fall in current modulation.

To assess the effect of RC reactance, we consider the effective circuit shown in Fig. S1 under a step voltage modulation V (t) = V Θ(t), where Θ(t) is the Heaviside step function. The resistor

R1 corresponds to the resistance of the graphene device and is at most R1 ≈ 20 kΩ as measured

with two-terminal transport measurement. The resistor R2 = 50 Ω corresponds to the terminal

load resistance. The capacitor C comes from a combination of bond pads and BNC cables, and is

estimated via geometry to be no more than 1 nF. Solution of the current through R1 is

V R (R + R ) I(t) = 1 + 2 exp − 1 2 t . (S7) R1 + R2 R1 CR1R2

The current initially overshoots before reaching the equilibrium (DC) value V/(R1 + R2).

The time scale CR1R2/(R1 + R2) ≈ 0.05 µs is much shorter than the typical modulation period of our experiments & 10 µs. Furthermore, the relative overshoot is R2/R1 .1%. Therefore, RC reactance has negligible effect in our AC measurements.

R1 V(t)

Figure S1: Effective circuit of the device.

5 2 Optimal sensitivity for optically detected magnetic resonance

The best sensitivity for optically detected magnetic resonance (ODMR) is achieved by minimizing √ the ratio Γ/(C N ), where Γ is the linewidth of the ODMR spectrum, C is the contrast, and N is the rate of photon collection. In general, both the linewidth and the contrast depend on the combination of the optical and microwave power. In both imaging and scanning experiments, we are limited by the laser intensity that can be delivered to the NV spins: we restrict the excitation power to be below saturation to avoid NV degradation (likely due to charge state interconversion).

In imaging, we are further restricted by the available laser power, which has to be distributed over a microns-scale area. Therefore, the only remaining knob is the microwave power. We obtain

ODMR spectra, measure Γ and C as a function of the microwave power, and use the microwave power that minimizes the ratio Γ/C for ODMR. In the case of imaging, this procedure is performed with the ODMR spectra integrated over the entire ﬁeld of view. In Fig. S2, we show an example spectrum at a single pixel of wide-ﬁeld ODMR imaging of the graphene device on diamond at the optimal magnetometry parameters.

6 Figure S2: Wide-ﬁeld imaging ODMR spectrum at a single pixel for the NV ms = 0 ↔ −1

(left panel) and ms = 0 ↔ +1 (right panel) transitions at +I (red circles) and −I (blue circles).

Photoluminescence (PL) is normalized by the PL value of ms = 0 state. The lines are ﬁts to double

Lorentzian line shapes split by 3 MHz hyperﬁne coupling due to 15N nuclear spin.

7 3 Determination of magniﬁcation and resolution for wide-ﬁeld imaging

For wide-ﬁeld magnetic imaging, we calibrate magniﬁcation with the following procedure. Using

the nano-positioner on which the graphene-on-diamond device is mounted, we move the device in

the xy plane over a range of about 7 µm, and record an image with each lateral movement of the

nano-positioner. By tracking features in the image and using the known pixel size of the camera

(5.86 µm × 5.86 µm), we obtain a magniﬁcation of about 117. Therefore, each pixel on the camera

corresponds to a 50 nm × 50 nm area on the sample. For data analysis and display, we perform

a 2 × 2 binning, so that each pixel in the ﬁgures corresponds to a 100 nm × 100 nm area. This

calibration procedure has an estimated 6% error due to measurement uncertainty.

The point spread function (PSF) is given by the Airy pattern with the form

2J (u)2 PSF (u) = 1 , (S8) Airy u

where J1 is the Bessel function of the ﬁrst kind of order one, and u is a dimensionless length scale.

The ﬁrst zero occurring at u = 3.8317 ≈ 4 corresponds to the resolution according to the Rayleigh criterion. For convenience, we use a Gaussian proﬁle to approximate the PSF. The Airy pattern to

2 4 −u2/4 leading order in u is PSFAiry = 1 − u /4 + O(u ). A Gaussian proﬁle of the form e has the

same expansion. Therefore, we use the following as our resolution function

1 2 2 2 −(x +y )/σr PSF(x, y) = 2 e (S9) πσr

and identify 2σr as the imaging resolution.

To experimentally determine the resolution, we make use of a sharp feature in the NV PL

8 generated by reactive ion etching (RIE) of the diamond surface. As the RIE employed for

graphene/hBN etching also etches diamond and hence near-surface NV spins, we can use RIE to

generate patterns in the NV PL. The pattern we use is a stripe (Fig. S3a). We image a line cut

(indicated by the blue line in Fig. S3a), normalize the slowly-varying background to unity, and

show the normalized PL in Fig. S3b. To model the PL distribution, we ﬁrst note the 1D PSF is

R −x2/σ2 √ PSF1D(x) = dy PSF(x, y) = e r /( πσr). The 1D line-cut of the stripe is then the convolution of the 1D PSF and the rectangular function Π(x/W ), which is unity for |x/W | < 0.5 and zero otherwise, with W the width of the stripe:

1 W − 2x W + 2x PSF1D ∗ Π(x/W ) = Erf + Erf , (S10) 2 2σr 2σr

where Erf(x) is the error function. From a ﬁt of this form to the normalized PL line-cut (Fig. S3b),

we obtain a resolution of 2σr = 420(40) nm. This is consistent with the estimated resolution

assuming a PL wavelength λ=650 nm and NA=0.9: 0.61λ/NA = 440 nm.

9 a) 3 Count s b) 1.0 (kC) 2 8 0.9

m) 1 6

µ 0.8

y ( 0 4 0.7 2 PL Normalized -1 0 0.6 -2 0.0 1.0 2.0 3.0 -4 -2 0 2 4 x (µm) x (µm)

Figure S3: (a) NV photoluminescence (PL) image of the graphene-on-diamond device. The red outline indicates the boundary of the graphene channel. Some of the dark regions are due to shadows cast by metallic contacts, which include electrical contacts to the graphene channel and side probes, as well as top gate contacts. These PL features are not sharp due to the thickness of the metallic structures. Therefore, we use RIE to generate sharp features in NV PL, such as the stripe marked by the blue line. The blue line indicates the 1D line-cut used to generate the proﬁle shown in (b), which we use to extract the resolution. We note that there is no PL feature corresponding to the outline of the graphene, because we did not etch all the way through the bottom hBN when patterning the hBN-encapsulated graphene heterostructure. However, for generating the stripe, we etched longer in order to etch through the hBN and hence remove the NV spins underneath.

(b) Normalized PL line-cut measurements (red circles). Here, we normalize the slowly-varying background to unity. Hence, the depletion of the stripe feature has a relative contrast of about

40% with respect to the background. We ﬁt the normalized PL to Eq. S10 (blue) to extract the resolution.

10 4 Pulse sequence for spin-echo AC magnetometry

In Fig. S4, we show the pulse sequence for spin-echo AC magnetometry. The NV spin is ﬁrst

optically polarized into the ms = 0 level. A (π/2)x microwave pulse resonant with the ms =

0 ↔ −1 transition creates a coherent superposition between these two states. The system evolves

for a period of time τ in the presence of a ﬁeld BAC generated by a bias VAC applied on the device, and after a (π)x refocusing pulse evolves for another τ in the presence of −BAC generated by −VAC. For read-out, the electronic spin state is projected onto the ms = 0, −1 basis by a

ﬁnal ±(π/2)x,y pulse, and the ground state population is detected optically via spin-dependent

ﬂuorescence. Here, (... )i refers to a rotation of the Bloch vector around Si axis, where (Sx,Sy,Sz) are the spin operators.

(π/2)x πx ±(π/2)x,y Microwave τ τ

Vsd(t) VAC

532 nm Excitation

Photon counting

Figure S4: Pulse sequence for spin-echo AC magnetometry.

11 5 Current reconstruction for scanning probe magnetometry applied on simulated current

proﬁles

To show the fidelity of the current reconstruction algorithm for scanning magnetometry, we test the algorithm on simulated current profiles. In Fig. S5, we show a simulation with three different types of current profiles: a uniform profile, a uniform profile with two bumps, and a parabola with two bumps. The total current, width, and noise level are chosen to be comparable to those in the experiment with graphene devices. The simulated data for B|| has a noise of 5% on the maximal

B||. As shown here, the reconstructed current profile is typically within 5% of the original current profile. For the simulation of a uniform current, while the reconstructed current occasionally deviates beyond 5% from the original profile, the standard error of the mean is 6%.

12 ed(e ons ln with along points) (red field have profiles All voltage. scanning piezo The the bumps. of two with parabola a (right) is and simulation bumps, two uniform with a right: distribution (middle) to distribution, uniform Left a (left) distributions: current profiles. different three simulated on performed to applied reconstruction Current S5: Figure adfrassigtegons ftereconstruction. the error of 5% goodness the is the band assessing blue for reconstructed light band The of curve). result blue (dark Bottom: original the field. to the compared (red) of profile value current maximal the on noise 5% has data Simulated J (A/ m ) B|| (µT) -0.2 0.0 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.0 0.2 0.2 0.4 0.4 V x V (V) x 0.6 0.6 0.8 0.8 B || B ( T) eeae rmtercntutdcretpol bu curves). (blue profile current reconstructed the from generated J || µ 1.0 1.0 0.0 0.0 0.2 0.2 13 I 0.4 0.4 V 1 = x V (V) x 0.6 0.6 µ A o:smltddt o h projected the for data simulated Top: . 0.8 0.8

J B|| (µT) 1.0 1.0 0.6 0.6 x ai ssoni units in shown is -axis 0.8 0.8 V x 1.0 1.0 V (V) x 1.2 1.2 1.4 1.4 6 Sensitivity of current reconstruction to uncertainty in the stand-off distance

NV centers used in this work are generated with nitrogen ion implantation at 6 keV. Stopping

and Range of Ions in Matter (SRIM) calculation gives an estimate of 10(3) nm for the depth of

the nitrogen, with the error bar denoting the ion straggle. However, experiments from Refs. 2, 3 indicate that the actual average NV depth for such shallow implants may be up to a factor of two larger compared to the estimate from SRIM. Therefore, for the NV spin ensemble in the wide-

ﬁeld imaging measurement, we estimate an average depth of 20 nm with uncertainty of 10 nm, which includes both the uncertainty of the average of the distribution as well as the straggle. The bottom hBN has a thickness close to 30 nm. Therefore, we use d =50(10) nm for the current reconstruction. The uncertainty in the stand-off distance leads to an 9 A/m (or 6%) error on the current density.

For the single NV scanning measurement, the stand-off distance d is one of the free parameters in the Biot-Savart functional and therefore is extracted. The extracted d has some distribution and can vary over a range up to 50 nm. On average, the extracted d is below 60 nm, which is consistent with the experimental setup: ≈ 20(10) nm from the 6 keV implant plus the thickness of the top hBN (. 20 nm) or the finite thickness of the Pd wire (≈ 30 nm). Given that a variation of d over ∼ 10 nm scale is not expected to change the generated stray field much, we do not expect to be able to extract d to high precision. This is reflected in the uncertainty of d which

is generally around 20 nm to 40 nm. We can evaluate the effect of the spread of d by extracting

the current profile when we fix d to a specific value. Fig. S6 displays variation in the

14 reconstructed current proﬁles of a Pd channel and an encapsulated graphene device shown in Fig.

2e of the main text, corresponding to d ﬁxed at several values. Here, the range of variation in d corresponds to the typical spread of the extracted d. For the Pd wire, with ±15% change in d, we

see a 2%-7% change in the current proﬁle; whereas with the same variation in d, even less change

in the current proﬁle is observed for graphene. The typical error on the current density estimated

here is consistent with ≈ 5% error assessed from simulated current proﬁles.

1.2 1.6 1.0 0.8 1.2

0.6 0.8

J/(I/W) J/(I/W) 0.4 0.4 0.2 0.0 0.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 x/W x/W

Figure S6: Variation in current proﬁles in Fig. 2e of the main text at various values of d. Red:

allowing d to be extracted from the current reconstruction algorithm. Extracted d is 60(10) nm for

the Pd channel (left panel) and 20(20) nm for the graphene device (right panel). Green (gray): d at

−15% (+15%) of the aforementioned value.

15 7 Transport characterization of the graphene-on-diamond device

In Fig. S7, we show resistance R as a function of the gate voltage Vg and carrier density n for the graphene-on-diamond device, obtained with two-terminal measurement. The location of the charge neutrality point (Dirac point) VD is determined from location of the peak resistance. Carrier density n is determined using n = Cg(Vg − VD), where Cg = ε0εr/(et) is the gate capacitance per area, with ε0 the permittivity of free space, εr the dielectric constant, and t the thickness of the dielectric. For the graphene-on-diamond device, εr = 4 and t = 13nm are used for the hBN gate dielectric.

12 -2 n (10 cm ) -2 -1.5 -1 -0.5 0 0.5 1 24

)

Ω 22

Resistance (k Resistance

16 -1.0 -0.5 0.0 0.5 1.0 Gate voltage (V)

Figure S7: Resistance R as a function of the gate voltage Vg and carrier density n obtained from two-terminal measurement.

16 8 Transport characterization of devices for scanning measurement

In Fig. S8, we show resistivity ρ as a function of carrier density n for three of the four devices shown in Fig. 2f of the main text. Resistivity was obtained with four-terminal measurements. The location of the charge neutrality point (Dirac point) VD is determined from location of the peak resistance. Carrier density n is determined using n = Cg(Vg − VD), where Cg = ε0εr/(et) is the

gate capacitance per area, with ε0 the permittivity of free space, εr the dielectric constant, and t the

thickness of the dielectric. For the devices made on the standard SiO2/Si substrates, εr = 3.9 and

t = 285 nm are used.

For the device corresponding to the second (from the left) data in Fig. 2f of the main text, the

geometry of the device precluded a four-terminal measurement of the resistivity. However, because

photo-doping brings the system to the Dirac point in steady state, the magnetometry measurements

performed on this device were at the Dirac point. The last two sets of data (from the right) of Fig.

2f of the main text correspond to the same physical device: initial transport characteristics are

shown in the middle panel of Fig. S8, and later, altered characteristics are shown in the right panel

of Fig. S8. The later state of the device has a signiﬁcantly broader ρ vs n and much lower peak

resistivity, indicating higher carrier density inhomogeneity. Therefore, we consider the two states

as two distinct devices.

17 1.5

) 1.0 ) )

Ω Ω Ω

(k (k (k ρ 0.5 ρ ρ

0.0 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 12 -2 12 -2 12 -2 n (10 cm ) n (10 cm ) n (10 cm )

Figure S8: Resistivity ρ as a function of carrier density n for three of four graphene devices shown in Fig. 2f of the main text. From the left to right, each panel corresponds to the (starting from left)

ﬁrst, third, and fourth data set shown in Fig. 2f of the main text. Resistivity was obtained using four-terminal measurements.

18 9 Variation of viscous proﬁle at different Gurzhi Lengths

The linearized electronic Navier Stokes equation4 for the current density J is

1 n −ν∇2J + J = e2 E , (S11) τmr m

where ν is the kinematic viscosity, e is the electronic charge, τmr is the momentum-relaxing time,

m is the effective mass, and E is the bias electric ﬁeld. For electronic carriers in graphene, m =

√ 6 ~ πn/vF, where vF = 10 m/s is the Fermi velocity. A ﬂow along the y-direction J = (0,Jy(x))

and under no-slip boundary condition, Jy(x = ±W/2) = 0 has following current density

2 r e vFτmr n cosh(x/Dν) Jy(x) = E 1 − , (S12) ~ π cosh(W/(2Dν)) √ where Dν ≡ ντmr. One can then obtain the conductivity deﬁned as σ ≡ I/(WE) where I = R dx Jy(x): e2v τ rn 2D W σ = F mr 1 − ν tanh . (S13) ~ π W 2Dν

In Fig. S9, we show proﬁles from the solution of the electronic Navier Stokes equation with different values of Dν/W . For Dν/W 1, the current density approaches a uniform proﬁle Jy =

2 σ0E and the conductivity σ = σ0 ≡ e τmrn/m becomes the Drude conductivity. For Dν/W 1,

2 2 e vF p n W 2 the current density approaches the ideal Poiseuille proﬁle Jy(x) = E − x , and 2~ν π 2

2 2 2 the conductivity σ = e nW /(12mν) has the ∼ W /ν scaling. For Dν/W = 0.3, the current

density approaches within 5% of the ideal Poiseuille proﬁle, while for Dν/W > 0.5, the current

density approaches within 2% of the ideal Poiseuille proﬁle.

19 1.6 Uniform Poiseuille 0.5 1.2 0.3 0.2 0.1 0.8 0.05 0.01

J/(I/W) 0.4

0.0 -0.5 0.0 0.5 x/W

Figure S9: Solution of the electronic Navier Stokes equation with different values of Dν/W . The band around the ideal Poiseuille proﬁle is 5%. The current density is normalized by the average

ﬂux I/W , and the spatial position is normalized by W .

20 10 Comparison of measured current proﬁle to the ﬂow of non-interacting electrons with

diffuse boundary

For a system of non-interacting (τpp = ∞) electrons moving in a channel −W/2 < x < W/2

with completely diffusive boundary, the ﬂow proﬁle develops a curvature in the regime when the

momentum-relaxing mean free path is on the order of the width, lmr ≈ W . In the main text, we

compared the NV measured proﬁle in graphene at the charge neutrality point (CNP) to the non-

interacting case, and concluded that the non-interacting case cannot explain our measurement. In

this section, we discuss the details of the calculation of non-interacting proﬁles.

The derivation of the current proﬁle of non-interacting electrons in a channel with diffuse

boundary can be done following previous work (see, e.g., ref. 5 and references therein). The local

current density is given by

Z d2p J(x) = eg vf(x, p), v = ∂ ε, (S14) (2π)2 p

where e = −|e| is the electron charge, g = 4 is the spin-valley degeneracy, ε = ε(p) is the

quasiparticle energy as a function of momentum p, and f(x, p) is the steady-state quasiparticle

distribution function. It is convenient to write the deviation of f from the equilibrium Fermi-Dirac

−1 (ε−εF )/T distribution f0(ε) = e + 1 in the form

df f(x, p) − f (ε) = − 0 F. (S15) 0 dε

If the temperature T is much smaller than the Fermi energy εF = ε(pF ), the dependence of F on

|p| proves to be weak, so that F = F (x, θ) where θ is the angle between vector p and the y-axis.

21 Accordingly, the current density is given by π dn Z dθ J(x) = e vF (x, θ), v = (−vF sin θ, vF cos θ) , (S16) dεF 2π −π R 2 2 where n = g f0(ε) d p/(2π) is the electron density and vF = dεF /dpF is the Fermi velocity.

Adopting the relaxation-time approximation for bulk scattering, we obtain the linearized

Boltzmann kinetic equation for F : π F − F Z dθ vx∂xF − evyE = − , F ≡ F. (S17) τmr 2π −π It is easy to see that F must be odd under reﬂection with respect to the x-axis: |θ| → π − |θ|. This implies that F = 0, so that the electron density and the electric ﬁeld E are uniform in the channel.

Equation (S17) becomes F sin θ ∂xF + = eE cos θ. (S18) lmr We need to solve this equation subject to the suitable boundary conditions (BC) at the edges. For

the case of purely diffuse scattering, the quasiparticle ﬂux reﬂected from the edges is isotropic.

Thus, the BC at x = W/2 edge is F (W/2, θ) = const at 0 < θ < π. Flux conservation ﬁxes this

constant to be 0 1 Z F (W/2, θ) = F (W/2, θ0) sin θ0dθ0, 0 < θ < π. (S19) 2 −π A similar boundary condition, for angles −π < θ < 0, holds at x = −W/2 edge. The desired

solution is therefore

W/(2l ) x/l F (x, θ) = eEl cos θ 1 − exp − mr + mr . (S20) mr | sin θ| sin θ

Substituting this expression into Eq. (S16), we obtain, after some elementary transformations, ∞ Z √ Jy(x) W + 2|x| W − 2|x| 2 du 2 −zu = 1 − G − G ,G(z) ≡ 3 u − 1 e , (S21) eσ0E 2lmr 2lmr π u 1

22 2 where σ0 = e nlmr/(vFm) is the standard Drude conductivity.

Function G(z) monotonically decreases at z > 0 and has the following asymptotic behavior:

1 2 2e−γ z2 1 2e−γ 7 G(z) ' − z ln − + z3 ln + , z 1, (S22) 2 π z 2 6π z 3 r 2 ' e−z, z 1, (S23) πz3 where γ = 0.577 is the Euler constant. The upper line is relevant if the scattering is dominated by the edges (i.e. ballistic limit), lmr W . The lower line is useful if the transport is limited by the bulk scattering (i.e. Ohmic limit), lmr W . In both such limits the current density enhancement

Jy(0)W/I at the center of the channel becomes small.

In general, this proﬁle has some curvature, which is maximal in the regime of lmr ≈ W .

In Fig. S10, we show the proﬁles at various values of lmr/W and compare to an ideal Poiseuille

flow. We see that the non-interacting profiles begin as a uniform profile for lmr/W 1, develop a curvature that is maximal at lmr/W ≈ 1, and then flattens again as lmr/W 1. In Fig. S11, we show the peak (central) value of Jy/(I/W ) as a function of lmr/W in order to quantify the curvature of the profile. The maximum occurs at lmr/W = 0.625, though there is little change in the range of lmr/W ∼0.5-1. From both Figs. S10 and S11, we see that even with the maximal curvature at lmr/W = 0.625, the non-interacting profile still significantly deviates from the ideal viscous case. In Fig. 2e of the main text, we compared an experimental profile at the CNP to both the ideal viscous profile and the non-interacting profile at lmr/W = 0.625. It is clear that the non-interacting profile cannot explain the measured profile in graphene, which instead matches well to the viscous profile.

23 1.6

1.4

1.2

1.0

0.8 Uniform Boltzmann

J/(I/W) 0.6 lmr/W=0.1 lmr/W=0.6 0.4 lmr/W=100 Viscous: 0.2 Dν/W=0.3 Dν/W>>1 0.0 0.0 0.2 0.4 0.6 x/W

Figure S10: Comparison of hydrodynamic proﬁles with Dν/W = 0.3, ∞ and non-interacting proﬁles with lmr/W =0.1, 0.6, and 100.

1.12

1.08

1.04

J(x=0)/(I/W)

1.00 0.001 0.01 0.1 1 10 100 lmr/W

Figure S11: Peak (central) value of J/(I/W ) of non-interacting proﬁles as a function of lmr/W .

24 11 Validity of hydrodynamics in graphene at room temperature

At elevated temperatures, the dominant mechanism for momentum-relaxation is electron-phonon

scattering. For hBN-encapsulated graphene, the associated momentum-relaxing mean free path

6–8 lmr ≈ 1 µm (refs. ) corresponds to τmr = lmr/vF ≈ 1 ps. From the main text, we have τν =

0.19 − 0.39 ps; τpp should be on the same order and hence is small compared to τmr. We also note

that the particle-particle collision mean free path is lpp ≡ τppvF ∼ 190 − 390 nm, corresponding to

a Knudsen number Kn≡ lpp/W ∼ 0.19 − 0.39, which is small compared to unity. Together, these

two conditions justify the hydrodynamic limit. As observed in this work and in ref. 8, for the usual

range of carrier density n < 1012 cm−2 accessed in a graphene experiment, ν ∼ 0.1 − 0.4 m2/s,

which corresponds to Dν ≈ 300 − 600 nm. For W = 1 µm, the resulting Jy calculated with Eq.

1 of the main text is within 4% to 1% of an ideal viscous proﬁle with the same total current. This is consistent with our observation that there is no discernible variation in the current proﬁle at different carrier densities.

25 12 Effect of photo-doping on scanning magnetometry experiment

Exposure to light is known to lead to photo-induced doping in an hBN-encapsulated graphene

9, 10 device fabricated on an SiO2/doped Si substrate . The mechanism is that defects residing in hBN are photo-dissociated, leading to free positive and negative charges that can migrate within the hBN layer. At any value of gate voltage Vg, the free charges experience the electric ﬁeld and migrate accordingly. The charges that migrate towards the graphene dope the graphene; however, the charges that migrate towards SiO2 cannot exit the hBN layer, remain trapped in hBN, and serve to screen the electric ﬁeld11, 12. Hence, upon light exposure, the system moves towards the CNP over time, and the steady state is the CNP. Nevertheless, the rate is slow enough that we can obtain scanning NV magnetic measurements as a function of carrier density range. In this section, we describe the relevant analysis to examine the effect of photo-doping in scanning measurement.

In scanning NV magnetometry, photo-induced doping occurs as the graphene device is exposed to the NV-excitation light scattered from the diamond probe. We observe photo-doping dynamics with the following experiment. With the probe retracted (about 10 µm above the device) and light focused on the probe, we set Vg to a ﬁnite value, and monitor the resistivity ρ as

a function of time. The density-dependent transport curve ρ vs n of this device is shown in the

middle panel of Fig. S8. In Fig. S12, we show ρ as a function of time for Vg = −10 V (left panel)

and Vg = +10 V (right panel). In both cases, the resistivity increases and saturates at the

resistivity associated with the CNP ≈ 1.5 kΩ (see the middle panel of Fig. S8). This indicates the

evolution of the system towards CNP and that the CNP is the steady state under light illumination.

26 1.5 1.5

) )

Ω 1.0 Ω 1.0

(k (k

ρ ρ 0.5 0.5

0.0 0.0 0 5 10 15 0 5 10 15 20 25 30 35 40 45 50 Time (hour) Time (hour)

Figure S12: Time-dependent evolution of resistivity ρ under exposure to NV-excitation light

scattered from the diamond probe. The measurement is done with the probe retracted at about

10 µm above the device. Time evolution is measured with Vg set to −10 V (left) and +10 V (right).

9 Note that while ref. observed photo-induced doping to occur only for Vg < 0 at 77 K, here we

observe photo-induced doping regardless of the sign of Vg at room temperature. Our observation

is consistent with the result of ref. 10.

Once the system has been photo-doped to the CNP, it is stable under light illumination. In the left panel of Fig. S13, we show ρ measured throughout the course of a scanning

magnetometry measurement when the device has been photo-doped to the CNP. Negligible

variation in ρ is observed.

Photo-induced doping presents complication for a scanning magnetometry measurement

away from the CNP. The time scale of the dynamics presented in Fig. S12 is on the order of

hours, but this is under the situation when the probe is retracted. When the probe is on the device,

27 the evolution becomes signiﬁcantly faster. Nevertheless, the dynamics is slow enough that during

a scanning magnetometry measurement, which lasts about 1.5 hours, the system evolves across a

range of carrier density, but does not completely reach the CNP. For the data presented in Fig. 4a

of the main text, we ﬁrst photo-dope the system to shift the CNP to Vg = −20 V. Then, we set

Vg = 0 V, and perform scanning magnetometry measurements. Before and after each set of scanning magnetometry measurements, we monitor the resistivity ρ to determine the carrier density range that the system has evolved across by matching the measured resistivity to the transport curve shown in the middle panel of Fig. S8. In the right panel of Fig. S13, we show the same transport curve as in the middle panel of Fig. S8, and shade the ranges of carrier density that the system has evolved across during the two measurements at ﬁnite carrier density shown in Fig.

4a of the main text. The carrier density range is n = 0.3 − 1.5 × 1012 cm−2 and

n = 0.1 − 0.3 × 1012 cm−2 respectively.

Unlike a device where gating is provided by a Si back gate separated from the device with an

SiO2 layer, a device where gating is provided by a graphite gate, separated from the graphene only

by an hBN layer, does not undergo photo-induced doping13. Therefore, the graphene-on-diamond

device, where gating is provided by a graphite top gate separated from the graphene by an hBN

layer, does not suffer from photo-induced doping.

28 1.6 1.5 1.2 ) ) 1.0

Ω Ω 0.8

(k (k ρ ρ 0.5 0.4 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 -0.4 -0.2 0.0 0.2 0.4 12 -2 Vx (V) n (10 cm )

Figure S13: Left panel: resistivity monitored as a function of piezo scanner voltage during a scanning NV magnetometry measurement. Right panel: transport curve ρ vs n from the middle panel of Fig. S8, and carrier density range (green checkers and blue vertical stripes) covered by the two scanning magnetometry measurements at ﬁnite carrier density shown in Fig. 4a of the main text. The carrier density ranges are n = 0.3 − 1.5 × 1012 cm−2 and n = 0.1 − 0.3 × 1012 cm−2.

29 13 Comparison of current reconstruction methods for scanning magnetometry

In this section, we compare the current reconstruction method for scanning magnetometry

described in the main text with an alternative approach where one directly inverts the Biot-Savart

law in Fourier space (Eqs. S3, S4, S6). We refer to the latter approach as Fourier inversion. We

show that the main text method gives the same bulk current proﬁle as Fourier inversion; however,

unlike Fourier inversion, the main text method preserves sharp edges in the current proﬁle (if

present) without amplifying noise.

In Fig. S14, we show the current proﬁle in graphene from Fig. 2e of the main text along with

current proﬁles obtained from the same scanning magnetometry data using the Fourier inversion

method for several values of standoff distance d. Regularization is achieved via placing a cutoff

kc in Fourier space, with kc chosen in each case such that the variation in the background (i.e., for

|x| > W/2) is similar to the peak error bar from the data of Fig. 2e. Over the range of d expected for the experiment, we see that the current proﬁle obtained with the main text method is consistent with that obtained from Fourier inversion.

In Fig. S15, we show the dependence on Fourier cutoff kc for d = 30 nm. The qualitative shape of the current density - a parabolic proﬁle - is insensitive to the choice of kc, which is

consistent with the expectation for a smooth proﬁle. The only difference kc makes is the amount

of noise and the prominence of the dip feature close to the center of the proﬁle. The optimal cutoff

−1 for this measurement is kc/(2π) = 3 µm .

30 In Fig. S16, we show the current proﬁle in Pd from Fig. 2e of the main text along with current

profiles obtained from the same data using the Fourier inversion method with d = 70 nm and for several values of kc. For low kc, the influence of high spatial frequency noise on the reconstruction is minimal, but sharp features in the current profile are washed out. As one increases kc, one begins to recover sharp edges, but when the sharp features are comparable to those of the reconstructed current profile from the main text (lower right panel), high spatial frequency noise is severely amplified. Lastly, we see that the width of the rise at the edges is given by 1/kc.

From this analysis, we see that the main text reconstruction method gives the same result as Fourier inversion when there are no sharp spatial features in the current proﬁle. Compared to

Fourier inversion, the main text method is able to preserve sharp features without amplifying noise.

31 1.5

1.0

J/(I/W) 0.5

0.0

-0.5 0.0 0.5 -0.5 0.0 0.5 -0.5 0.0 0.5 x/W x/W x/W

Figure S14: Comparison of current proﬁle in graphene from Fig. 2e of the main text (red circles) to current proﬁle extracted with Fourier inversion using several standoff distances d. Left panel: black stars are for d = 15 nm. Middle panel: blue open diamonds are for d = 30 nm. Right panel: green open squares are for d = 50 nm. In each case where Fourier inversion is applied, a cutoff in

Fourier space is chosen such that the resulting noise in the background is comparable to the peak error bar of the red data points.

32 1.5

1.0

J/(I/W) 0.5

0.0

-0.5 0.0 0.5 -0.5 0.0 0.5 x/W x/W

1.5

1.0

J/(I/W) 0.5

0.0

-0.5 0.0 0.5 -0.5 0.0 0.5 x/W x/W

Figure S15: Comparison of current proﬁle in graphene from Fig. 2e of the main text (red circles) to current proﬁle extracted with Fourier inversion (blue diamond) at d = 30 nm using several

−1 −1 Fourier cutoffs kc. Upper left: kc/(2π) = 1.7 µm . Upper right: kc/(2π) = 3 µm . Lower left:

−1 −1 kc/(2π) = 4.4 µm . Lower right: kc/(2π) = 5.7 µm .

33 1.5

1.0

0.5

J/(I/W)

0.0

-0.5 0.0 0.5 -0.5 0.0 0.5 x/W x/W

1.5

1.0

0.5

J/(I/W)

0.0

-0.5 0.0 0.5 -0.5 0.0 0.5 x/W x/W

Figure S16: Comparison of current proﬁle in Pd from Fig. 2e of the main text (red circles) to

current proﬁle extracted with Fourier inversion at d = 70 nm using several Fourier cutoffs kc.

−1 −1 Upper left (purple): kc/(2π) = 2.1 µm . Upper right (blue): kc/(2π) = 3.2 µm . Lower left

−1 −1 (black): kc/(2π) = 4.3 µm . Lower right (green): kc/(2π) = 4.6 µm . For each panel, the light

−1 blue points are located at ±(W − kc ), and the light blue bars extend from the light blue points

with a length 1/kc.

34 14 Heating due to laser and microwave delivery in wide-ﬁeld imaging

To estimate heating due to laser and microwave delivery, we use the Fourier law q = −k∇T , where q is the heat ﬂux and k is the thermal conductivity. Here, we use k = 2200 W/(mK) for diamond.

If the heat is being deposited at a power Q over an area ∆x2, where ∆x is the characteristic length scale of the heating, then based on the Fourier law we can write Q/∆x2 ≈ k∆T/∆x and hence

∆T ≈ Q/(k∆x).

For wide-ﬁeld imaging, about 10 mW of laser illuminates an area of about 5 µm in diameter.

If we assume the laser is completely absorbed on the diamond surface, then with Q ≈ 10 mW and

∆x ≈ 5 µm, we get an upper bound on the temperature change to be ∆T . 1 K.

Microwave delivery for wide-ﬁeld imaging consists of a -35 dBm signal ampliﬁed at 45 dB gain. Therefore, about 10 mW of power is sent through the microwave delivery. Once again, if we assume this power is entirely deposited on the diamond in the vicinity of the device, and using the dimension of the microwave delivery to estimate ∆x ≈ 10 µm, we get ∆T . 1 K.

In both cases, we see that heating due to laser and microwave delivery is negligible in wide-

ﬁeld imaging.

35 15 Estimate ﬂuctuation potential for high impurity device

One of the devices measured has higher charge impurity, as shown by the lower resistivity at the

Dirac peak. The current proﬁle for this device is shown as the right-most proﬁle in Fig. 2f and its

transport curve ρ vs. n is shown in the right-most panel of Fig. S8. In this section, we estimate the

ﬂuctuation potential δµ as a measure of the impurity level for this device.

It is insightful to plot the transport measurement data in the form of conductivity σ vs. Fermi √ wave vector kF = πn = EF/(~vF) (Fig. S17). For both the standard device and high impurity device, the data shows that initially for small kF, σ remains close to constant, and for large kF , there is a linear trend that extrapolates to an x-intercept corresponding to kF = kBT/(~vF). The difference between the standard device and the high impurity device is that the latter has a higher minimal conductivity.

In the viscous Poiseuille (Dν W ) regime, the conductivity can be written as σ = AEF/ν

2 2 2 where A = e W /(12π~ ). Note that the same form σ ∝ EF can be written in the Ohmic case,

except that the relevant quantity that determines transport is τmr instead of ν, and that W disappears

from the dependence. The data of σ vs kF data can be described by the following model:    A(kBT + δµ)/νDF EF < kBT + δµ σ = (S24)   A(EF − kBT )/νFL EF > kBT + δµ

In this model, ν is treated as taking on a constant value νDF near the CNP (the Dirac ﬂuid

regime), and another constant value νFL in the ﬁnite-density Fermi-liquid regime. Such

36 approximation is justiﬁed from the result shown in Fig. 4b.

In this model, the minimal conductivity σmin ∼ (kBT + δµ)/νDF. Assuming νDF is an intrinsic property of the system and hence is independent of inhomogeneity, what sets σmin at a given temperature is δµ. Therefore, when comparing two minimal conductivities, we have

σmin,1/σmin,2 = (kBT + δµ1)/(kBT + δµ2). We use two values of δµ for the standard device:

δµ = 0 and δµ = 50 K, which is typical for encapsulated device14. This allows us to estimate the

ﬂuctuation potential for the high impurity device: δµ = 450 − 570 K, which is large compared to the thermal energy.

37 6

(mS) 3

σ 2

0 0 20 40 60 80 100 120 -1 kF (µm )

√ Figure S17: Conductivity σ as a function of Fermi wave vector kF = πn. Red, black, and green circles correspond to transport measurements shown in the left, middle, and right panels of

Fig. S8. Red and black data are from standard devices, and green data is from the high-impurity

−1 device. The corresponding solid curves are linear ﬁts to the data for kF > 100 µm . Shaded blue region corresponds to kF < kBT/(~vF). Red (green) horizontal dashed lines correspond to the minimal conductivity for the standard devices (high-impurity device).

38 16 Transport regime of Pd

Here we substantiate that the transport regime of the Pd wire is Ohmic. Refs. 15, 16 have each obtained palladium lmr to be 9 nm and 25 nm respectively. The Fermi energy is EF = 8.16 eV,

∗ 6 and with effective mass m /m = 0.4, this gives us the Fermi velocity vF = 2.7 × 10 m/s.

We then have the elastic (momentum-relaxing) scattering time τmr = 3 − 9 fs. The mobility

∼ 10cm2/(Vs) is four magnitude lower compared to that of high quality graphene device such as in our experiment. For the Fermi liquid regime valid for Pd, the electron-electron scattering time can

2 be estimated τee ∼ ~EF/(kBT ) = 8 ps τmr. With such a short momentum-relaxing scattering time compared to the electron-electron scattering time, transport in Pd is diffusive (Ohmic).

39 17 Estimate of the slip-length

17 Poiseuille current ﬂow only occurs with a no-slip boundary condition uy(x = ±W/2) = 0,

where W is the channel width. In general, the boundary condition can be parameterized by the

slip-length ζ, deﬁned as ζ ≡ duy /u evaluated at the boundary17. A no-slip boundary condition dx y applies for ζ W , where the other limit is the no-stress boundary condition ζ W . In the

case of no-stress boundary condition, current ﬂow in a channel is uniform despite the presence of

viscosity.

The boundary condition for (lithographically patterned) graphene was not known prior to our

work. Viscous ﬂow of an electron ﬂuid in doped graphene has been explored either via a vortex

whirlpool detected using a negative vicinity voltage8, 18, 19 or superballistic conduction measured

with a point contact20. Formation of a vortex whirlpool is insensitive to the boundary condition8.

Superballistic flow through a point contact has a profile that is analogous to Poiseuille flow across a channel, but does not require a no-slip boundary condition; e.g., the analysis of ref. 20 was

performed with a no-stress boundary condition.

In our graphene experiments, we ﬁnd that the current continuously decreases away from the

center of the channel, vanishing at the boundary. This observation indicates a no-slip boundary

17 condition at the room temperature. Kiselev and Schmalian estimate ζ ≈ 0.6lpp for the diffuse

limit likely appropriate for lithographically patterned graphene. The authors also discuss a nearly

3 λT specular limit with ζ ≈ 0.008 h2h0 lpp. Here, λT = vF~/(kBT ) is the thermal wavelength and

the two length scales h and h0 are the roughness of the edge. The authors use h = h0 = 250 A˚ .

40 3 2 0 Thus in the nearly specular limit, the ratio λT /(h h ) ≈ 1 and ζ ≈ 0.008lpp. This is much smaller compared to the diffuse limit, and since we are interested to establish an upper bound on ζ from the authors’ estimate, it is appropriate to consider the diffuse limit.

In order to estimate ζ, we need lpp = vFτpp. The viscous scattering time τν determined in our

21 work is not the same as τpp. However, τpp is measured in ref. to be τpp = 5τ~ so we use that value here. At T = 300 K we have τpp = 127 fs and hence lpp = 127 nm. This gives ζ ≈ 76 nm W , consistent with the no-slip boundary condition observed in our experiment.

We note that a recent work ref. 22 estimates ζ ≈ 500 nm in their system at T = 75 K. With such a longer slip-length, Poiseuille ﬂow persists because a wider channel is employed in the work, with W = 4.7 µm. In addition, lpp ≈ 0.16W = 750 nm, which is much larger than in our system because of lower temperature. Hence one still has ζ ≈ 0.6lpp. In another word, the slip-length is large at lower temperature, and decreases as temperature increases.

As discussed in ref. 17, a large slip-length below 100 K is consistent with the observation by the Manchester group that the Gurzhi effect is not observed up to 100 K. However, at much higher temperature, the slip-length decreases signiﬁcantly; according to ref. 17, the slip-length would reach a sub-100 nm value at room temperature, consistent with our experimental observation of no-slip boundary condition.

41 18 Field-dependence of viscous proﬁles

To check if there is significant field dependence to the observed current profiles in graphene, we

performed scanning NV magnetometry at the CNP for several applied bias magnetic ﬁelds

(Fig. S18). The bias field was aligned with the orientation of the NV and the values were B0 =50, √ 200, 389 G. The out-of-plane field is B0,z = B0/ 3 = 29, 115, and 225 G. At all magnetic field

values, the observed current proﬁles maintain the overall shape for Poiseuille ﬂow and do not

have signiﬁcant, qualitative change.

The corresponding cyclotron radius is rC = EF/(evFB) = 8.9, 2.2, and 1.1 µm at the three bias ﬁeld values. Here, we use EF = kBT . In the regime where the cyclotron radius is larger than the width W , a hydrodynamic current proﬁle is not expected to change22, which is consistent with our experimental observation.

42 Ideal viscous 1.6 29 G 115 G 1.2 225 G

0.8

J/(I/W) 0.4

0.0 -0.8 -0.4 0.0 0.4 0.8 x/W

Figure S18: Current profiles at the CNP for three different out-of-plane bias magnetic fields. The current density J is normalized by average flux I/W . The x-axis is normalized by the width W .

The out-of-plane ﬁeld is 29 (green), 115 (red), and 225 G (blue), corresponding to cyclotron radius rC = 8.9, 2.2, and 1.1 µm. Solid black curve is for ideal viscous ﬂow.

43 19 Analysis of transport characterization of graphene device for scanning measurement

Here, we provide further analysis of the transport characterization of a graphene device studied

with scanning NV magnetometry. Analysis is performed for the data shown in the middle panel

of Fig. S8. The data shown in the left-most panel of Fig. S8 is almost identical, so here we only

show analysis of one set of data. The data shown in the right-most panel of Fig. S8 corresponds

to a device with signiﬁcantly larger charge impurity; an analysis of the impurity level is shown in

Section 15 of the Supplementary Information.

√ First, we extract the mobility µ = σ/(en) from the standard formula. For EF = ~vF πn <

kBT , we use n such that EF(n) = kBT , both here and in the subsequent analysis. µ as a function of

n is shown in Fig. S19a. The benchmark of device quality is seen in the high-density regime where

mobility is fairly constant. In that regime, we have ≈ 6 × 104 cm2/(Vs), which is on par with

the 5 × 104 cm2/(Vs) value typical for a high mobility encapsulated graphene device cited in the

literature (see for example the mobility of a typical device in the Manchester group experiment8).

Next, we extract the momentum-relaxing scattering length lmr and viscous scattering length

2 lν ≡ ν/vF. For lmr, one could apply the standard Drude formula lmr = σh/(2e kF) where the √ Fermi wave vector kF = πn. This formula is applicable where the current proﬁle is uniform.

The resulting lmr is shown as black dashed curve in Fig. S19b. Given that we observed a parabolic current proﬁle, the Drude formula will not accurately assess lmr. In order to assess the scattering

length scales in the Poiseuille regime, we make use of the results for three values of the kinematic

viscosity ν from the main text Fig. 4b, each corresponding to a value of Dν/W . lmr can be obtained

44 p from the deﬁnition of the Gurzhi length Dν ≡ lmrν/vF and the value of ν corresponding to each value of Dν/W . lmr corresponding to Dν/W = 0.3, 0.5 are shown in Fig. S19b. Note that lmr is not shown for Dν/W = ∞ because it corresponds to lmr = ∞. The viscous scattering length is then given by the deﬁnition lν ≡ ν/vF, as shown in Fig. S19c.

Lastly, we show the ratio of the scattering lengths lν/lmr, which is the same as τν/τmr, for

Dν/W = 0.3 (red) and 0.5 (green) in Fig. S19d. While τν is not necessarily the same as τpp, they are expected to be on the same order, and hence τν/τmr provides a measure to assess the validity of hydrodynamics. We found τν/τmr is bounded above by 0.4 and hence signiﬁcantly smaller than unity. Thus, we see that the transport measurement provides a self-consistent check that the system is in the hydrodynamic regime.

45 a) 1.0 b) /(Vs))

2 1.5 cm 5 0.5 1.0 ( µ m) mr l 0.5

Mobility (10 Mobility 0.0 0.0 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 12 -2 12 -2 n (10 cm ) n (10 cm ) c) 500 d) 0.5 400 0.4 300 0.3 mr / τ ν (nm) τ ν

l 200 0.2 100 0.1 0 0.0 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 12 -2 12 -2 n (10 cm ) n (10 cm )

Figure S19: (a) Mobility µ vs. carrier density n. (b) Momentum-relaxing mean free path lmr. Black dashed curve is the data from the Drude formula. Red and green curves are from a hydrodynamic analysis with Dν/W = 0.3 and 0.5. (c) Viscous scattering length lν extracted from a hydrodynamic analysis with Dν/W = 0.3 (red), 0.5 (green), and ∞ (blue). (d) Ratio of scattering times τν/τmr = lν/lmr for Dν/W = 0.3 (red) and 0.5 (green).

46 20 Range of channel width for Poiseuille ﬂow in graphene at room temperature

In this section, we discuss the range of channel width W in which one may observe Poiseuille current ﬂow in graphene at room temperature. As discussed in Supplementary Information

Section 11, one expects Dν ≈ 300 − 600 nm. Hence, W . 3Dν ≈ 1 − 2 µm is the necessary regime to observe a parabolic current proﬁle. On the other hand, W cannot be small such that the slip-length ζ starts to become signiﬁcant. Based on the work in ref. 17, Supplementary

Information Section 17 estimates ζ ≈ 80 nm. It is reasonable to place 10ζ . W as the criterion for Poiseuille ﬂow. Therefore, the range in which one can expect to observe Poiseuille ﬂow is

0.8 µm . W . 2 µm.

Previous works obtained a signature of viscous Poiseuille ﬂow in the scaling of the resistivity

−2 20 23 ρ ∝ W in a graphene point contact and WP2 (ref. ). The graphene point contact experiment covered a range of point contact width from 0.1 to 0.8 µm, a factor of 8, while the experiment with

WP2 ribbon covered a range of ribbon width from 0.4 to 9 µm, a factor of 22.5. In contrast, the range of graphene channel width at room temperature discussed above corresponds to a factor of

2.5. Hence it will be difﬁcult to observe the resistivity scaling in the case of a graphene channel at room temperature due to the narrow range in which one has to conduct such an experiment.

47 21 Veriﬁcation of continuity equation from the wide-ﬁeld imaging experiment

In this section, we show line-cuts of Jy at a few different y-values, and demonstrate that the

continuity equation ∇ · J = 0 is satisﬁed. First, we show line-cuts Jy at several y-values in

Fig S20b. Jy becomes ﬂatter as one approaches the lower part of the channel where the current

starts to bend around a corner towards the drain on the left.

In order to verify that this dynamics is real and not an imaging artifact, we probe the

divergence of the current to check that the continuity equation is satisﬁed. In Fig. S20c, we show

an example of the divergence of the current density ∇ · J from the wide-ﬁeld imaging experiment.

In the channel where we investigate Poiseuille ﬂow, the largest value of the divergence is around

δ(∆J/∆x) ≈ 170µA/µm2. As the pixel size of the image is ∆x = 0.1µm, we can provide

another independent assessment of the error on the current to be ∆xδ(∆J/∆x)/2 ≈ 9 A/m. The factor of 2 in the denominator comes from error propagation in calculating the divergence, where one needs to make use of values of the current density at four points. This error estimate is consistent with the assessment from uncertainty of the stand-off distance.

We note that ∇ · J at the bottom-left side-probe is signiﬁcantly larger compared to that in the main channel. This is due to the large magnetic gradient arising from the highly concentrated current, which leads to a distorted ODMR spectrum and inaccurate determination of the magnetic

ﬁeld. Due to such artifacts, one expects a much larger error in the extracted local current and hence divergence for images at this part of the device.

48 Next, we check that the continuity equation is satisﬁed in an alternative way. We can reliably R obtain vertical ﬂux dx Jy in the channel above the top gate metallic contact, and we observe the

ﬂux agrees with the source-drain current Isd = 100µA within 9µA, which is also consistent with R δJ = 9µA/µm. In addition, we checked that the horizontal ﬂux dy Jx at the bottom left side probe (at around x = −1µm) agrees with Isd.

Therefore, we conclude that the data is consistent with the continuity equation within the error bar. The check of the continuity equation benchmarks the reliability of our imaging technique.

Given that we verify the continuity equation, we conclude that the observed spatial evolution of the current density is real and not an artifact in the measurement. Presumably, this involves higher order dynamics of the electron ﬂuid as the current bends around the device corner, which leads the system to deviate from the equilibrium ideal Poiseuille ﬂow in the lower part of the channel.

49 a) J (A/m) b) 250 -150 2.72.7 OhmicOhmic 2.52.5 2 ViscousViscous 150 2.32.3 -100 22 1.7 150 1.7 m) 1.51.5 µ (A/m) (A/m) y 1 y -50 1.21.2 y ( -J 100 -J 11

0 50 0 0 -2 -1 0 1 2 -1.0 -0.5 0.0 0.5 1.0 x (µm) x/W c) d) 2.5 Div J ( Div J 500 ( Div J 2.0 -100 µ 1.5 µ A/ A/ A) 0 A) µ µ m) m) µ µ µ µ m 1.0 m -50 2 2 y ( y ( ) ) Flux ( 0.5 -500 Flux ( 0.0 0 -0.5 -2 -1 0 1 2 0.0 1.0 2.0 x (µm) y (µm)

Figure S20: (a) The same image from Fig. 3b for reference. (b) Line-cuts of Jy at a few different

y. The values of y (units in µm) corresponding to each set of data is shown in the legend. (c)

Example of measured divergence of current density ∇ · J in graphene from wide-ﬁeld imaging R experiment. (d) Vertical ﬂux dx Jy as a function of y. The gray band corresponds to the region

within 420 nm (imaging resolution) of the metallic top gate contact; in this region, imaging is

distorted due to the presence of the contact. Blue band corresponds to ±9 µA of the source-drain

current Isd = 100 µA.

50 22 Magneto-Transport Measurement

Magneto-transport measurement provides a way to determine whether transport is ballistic in

graphene as a function of temperature. The measurement is performed with the hBN-encapsulated

graphene device whose measured current proﬁle is shown in Fig. 2e of the main text. An optical

image of the device, together with the schematic of the transport measurement, is shown in

Fig. S21a. We measure the four-terminal resistivity ρ as a function of an out-of-plane magnetic

ﬁeld B and carrier density n. The measurement is carried out in a Janis SVT/MAGNET cryostat.

A 2D plot of the data is shown in Fig. S21b for temperature at 10 K and 270 K. At ﬁnite doping, the data for 10 K shows two prominent maxima in the resistivity, whereas the data for 270 K is largely featureless. To further highlight the difference at the two temperatures, we show line-cuts along B at several values of n. The horizontal axis is shown in units of W/RC , where W = 1 µm √ is the device width and RC = ~kF /(eB) is the cyclotron radius. Here kF = πn is the Fermi

wave vector. At 10 K, a peak in ρ occurs at W/RC ≈ ±1, and a kink occurs at W/RC ≈ ±2.

These are well-known signatures of ballistic transport in graphene24, 25. Hence, the device is demonstrated to exhibit ballistic transport at low temperatures, as expected for a standard hBN-encapsulated graphene device. As temperature increases, ballistic transport will disappear at some point. For our data at 270 K, the signatures of ballistic transport have disappeared. Hence we conclude that ballistic transport is no longer present at 270 K and above.

51 a) c) 10 K

V b)

10 K ρ (Ω) 400400 1

) 300 -2 300

cm 12 200200 0.5 -4 -3 -2 -1 0 1 2 3 4 n (10 100 270 K 100 W/RC

0 ρ (Ω) -0.5270 K 0.0 0.5 B (T) 800 1

) 600 -2

cm 12 400 0.5 n (10 200

0 -4-4 -3 -2-2 -1 0 1 2 3 44 W/R -0.5 0.0 0.5 W/RCC B (T)

Figure S21: Magneto-transport measurements of hBN-encapsulated graphene device at 10 K and

270 K. (a) Schematic of the magneto-transport experiment, performed via measurements of the

four-terminal resistivity ρ. (b) Resistivity ρ as a function of carrier density n and out-of-plane

magnetic ﬁeld B. The range of ρ displayed is chosen to make the relevant features visible in the

colormap. At low temperature (10K), we observe a double-peak feature in ρ as B is varied. This

feature is absent at 270 K. (c) 1D line-cuts of (b). The horizontal axis is shown in units of W/RC ,

52 √ where W = 1 µm is the device width and RC = ~kF /(eB) is the cyclotron radius. Here kF = πn

is the Fermi wave vector. Each linecut is offset vertically. Line-cuts are shown for n = 5.67 × 1011

12 −2 10 −2 to 1.36 × 10 cm at 3.78 × 10 cm intervals. At 10 K, a peak in ρ occurs at W/RC ≈ ±1, and a kink occurs at W/RC ≈ ±2. These features are absent at 270 K.

53 23 Current Reconstruction from Scanning NV Microscopy

Scanning NV microscopy measures only one projection of the magnetic ﬁeld generated by current

through the device. However, the Biot-Savart law together with continuity equation, which has the

form kxjx + kyjy = 0 in Fourier space, requires the map of one projection of the magnetic ﬁeld to contain all information of a 2D current distribution. In our experiment, the NV has an orientation vector uˆ = (p2/3 cos θ, p2/3 sin θ, p1/3), where θ is the angle between the NV orientation

projected on the xy plane and the x-axis. Here, the device being studied is used to deﬁne the

coordinate axis, with the long direction of the device taken as the y-direction. Therefore, the p p projected ﬁeld is B||(x) = 2/3 cos θBx(x) + 1/3Bz(x). The sample is placed on the setup in

such a way as to minimize θ.

There are two challenges with applying the Fourier inversion method described previously

to scanning measurements. First, B|| involves Bz, which has a long ∼ 1/x tail. Direct inversion

requires measurement over a large ﬁeld of view or otherwise leads to long-wavelength artifacts

(e.g., offsets or a slope), which in general can be corrected but nevertheless are inconvenient.

Second, current reconstruction involves inverting a low-pass ﬁlter which, in the absence of other

low-pass ﬁlter such as optical diffraction, leads to ampliﬁcation of noise at high frequencies.

Therefore, regularization is required26. However, regularization needs to be performed with care:

a naive application of a global regularization can smoothen out a sharp feature, such as at the

edges.

The goal is to have a current reconstruction procedure that is generally applicable to an

54 arbitrary current proﬁle Jy(x) ﬂowing in a channel oriented along the y-direction. First, the current density is parameterized as Jy = Jy({an}, x), with {an} being free parameters that determine Jy.

It is convenient to capture the expected overall shape of the current proﬁle with a minimal number of parameters in a function called J0. For this purpose, the functional form of Eq. 1 of the main text is chosen, which can vary from a parabola to a rectangular function. While J0 is the solution of the

electronic Navier Stoke equation, it is used here merely to set the functional form without assuming

hydrodynamics. The second part of the parameterization ∆J captures the remaining variation; it

is a linear interpolation with N equally spaced points inside the channel and parametrized by the

W W th values an = ∆J (xn), where xn ≡ − 2 + n N for n = 0, 1, ..., N is the n equally-spaced point along the width of the channel. Since the behavior at the edges is already captured by J0, it is allowed to set a0, aN = 0. The mathematical expression for the parameterization is

Jy(x, W, D, {an}) = A0J0(x, W, D) + ∆J(x, W, N, {an}) , (S25) cosh(x/D) J (x, W, D) ≡ 1 − Π(x/W ) , (S26) 0 cosh(W/(2D)) N X ∆J(x, W, N, {an}) ≡ Jlin(x − (xn−1 + xn)/2, W/N, an−1, an) , (S27) n=1 b − a a + b J (x, ∆W, a, b) ≡ x + Π(x/∆W ) . (S28) lin ∆W 2

Here, Π(x) is the rectangular function, which is unity for |x| < 1/2 and zero otherwise. Note that

no particular model is assumed for the current density; also the parameterization described above

allows Jy to take on any functional form. The only prior knowledge exploited in this procedure is

the fact that the current density is zero outside the channel.

55 With this parameterization, functional can be constructed that generates B|| for a given Jy:

r2 B (x, x , W, D, {a }, d, θ, B ) = B + cos(θ)α (x, d) ∗ J (x − x , W, D, {a }) || ctr n 0 0 3 x y ctr n 1 + √ αz(x, d) ∗ Jy(x − xctr, W, D, {an}) . (S29) 3

Here, xctr is the center of the channel, d is the stand-off distance, B0 is an offset that accounts for potential contribution in Bz from far away current ﬂowing not strictly in the y-direction, ∗ denotes a 1D convolution, and the kernels are

µ d α (x, d) ≡ 0 (S30) x 2π (d2 + x2) µ x α (x, d) ≡ − 0 . (S31) z 2π (d2 + x2)

To obtain the parameters {xctr, W, D, {an}, d, θ, B0}, the cost function is minimized:

2 X 2 χ = |B||(xi, xctr, W, D, {an}, d, θ, B0) − B||,i| , (S32) i where {xi,B||,i} is the experimental data.

Inverting the Biot-Savart law, which acts as a low-pass filter, without regularization leads to unphysical amplification of high-frequency noises on the magnetic field, and therefore regularization is routinely required for the inverse problem. Here, regularization is performed by

2 00 00 00 subjecting the χ minimization to a global constraint |∆J | < ∆Jmax, where ∆J is the

(numerical) second derivative of ∆J. This is equivalent to a low-pass ﬁlter of the form k−2 in

00 2 Fourier space. The constraint ∆Jmax is chosen such that the reduced χ is close to unity,

2 2 2 χν ≡ χ /[δB (Nd − Np)] ≈ 1. Here, δB is the typical error bar of the data, Nd is the number of data points and Np are the number of free parameters in the functional. We note that J0 itself is well-behaved, and therefore only regularization on ∆J is necessary.

56 24 Comparison of imaging techniques

In this section, we provide a summary of different imaging techniques for probing electronic transport in graphene22, 27–32 and discuss their utility in the context of imaging hydrodynamic current ﬂow. The table below provides a summary of the key speciﬁcations for each technique.

The most critical requirement is the ability to operate above 100 K, which is the temperature range of hydrodynamic current ﬂow in graphene. This requirement excludes Josephson junction,

SQUID, and scanning gate microscopy. Note that scanning gate microscopy has been applied to probe electron hydrodynamic in a 2D electron gas31, where hydrodynamics is observed at a much lower temperature compared to graphene.

Graphene Temperature Spatial Quantity References current imaging of operation resolution measured technique Single NV <600 K; this ~50 nm B-field This work scanning probe work 300K (projective) Josephson 10 mK ~200 nm Supercurrent Allen et al. Nat. Phys. 12, 128 junction (2015)

Ensemble NV <600 K; this 420 nm B-field (vector) This work, wide field work 300K Tetienne et al. Sci. Adv. 3, imaging e1602429 (2017).

Nano SQUID <4.2 K ~50 nm B-field (Bz) Uri et al. Nat. Phys. 16, 164 (2020). Scanning gate <4.2K ~50 nm*, Perturbation on *Aidala et al. Nat. Phys. 3, 464 microscopy ~200 nm# current from (2007), scanning gate #Braem et al. Phys. Rev. B 98, 241304(R) (2018) Scanning SET <300K 880 nm E-field Ella et al. Nat. Nanotechnol. 14, 480 (2019), Sulpizio et al. Nature 576, 75 (2019).

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