Imaging phonon-mediated hydrodynamic flow in WTe2 with cryogenic quantum magnetometry
Uri Vool, Assaf Hamo, Georgios Varnavides, Yaxian Wang, Tony X. Zhou, Nitesh Kumar, Yuliya Dovzhenko, Ziwei Qiu, Christina A. C. Garcia, Andrew T. Pierce, Johannes Gooth, Polina Anikeeva, Claudia Felser, Prineha Narang, Amir Yacoby
IImaging phonon-mediated hydrodynamic flow inWTe with cryogenic quantum magnetometry Uri Vool, , ∗ Assaf Hamo, ∗ Georgios Varnavides, , , ∗† Yaxian Wang, ∗ Tony X. Zhou, , Nitesh Kumar, Yuliya Dovzhenko, Ziwei Qiu, , Christina A. C. Garcia, Andrew T. Pierce, Johannes Gooth, , , Polina Anikeeva, , Claudia Felser, , Prineha Narang, † Amir Yacoby, , † John Harvard Distinguished Science Fellows Program,Harvard University, Cambridge, MA 02138, USA Department of Physics, Harvard University, Cambridge, MA 02138, USA John A. Paulson School of Engineering and Applied Sciences,Harvard University, Cambridge, MA 02138, USA Department of Materials Science and Engineering,Massachusetts Institute of Technology, Cambridge, MA 02139, USA Research Laboratory of Electronics,Massachusetts Institute of Technology, Cambridge, MA 02139, USA Max-Planck-Institut f¨ur Chemische Physik Fester Stoffe, Dresden, Germany Institut f¨ur Festk¨orper und Materialphysik,Technische Universit¨at Dresden, 01062 Dresden, Germany ∗ These authors contributed equally. † To whom correspondence should be addressed; E-mail: [email protected] (G.V.),[email protected] (P.N.), [email protected] (A.Y.).
In the presence of strong interactions, electrons in condensed matter systemscan behave hydrodynamically thereby exhibiting classical fluid phenomenasuch as vortices and Poiseuille flow. While in most conductors large screen-ing effects minimize electron-electron interactions, hindering the search for a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p ossible hydrodynamic candidate materials, a new class of semimetals has re-cently been reported to exhibit strong interactions. In this work, we study thecurrent flow in the layered semimetal tungsten ditelluride (WTe ) by imagingthe local magnetic field above it using a nitrogen-vacancy (NV) defect in dia-mond. Our cryogenic scanning magnetometry system allows for temperature-resolved measurement with high sensitivity enabled by the long defect spin co-herence. We directly measure the spatial current profile within WTe and findit differs substantially from the uniform profile of a Fermi liquid, indicatinghydrodynamic flow. Furthermore, our temperature-resolved current profilemeasurements reveal an unexpected non-monotonic temperature dependence,with hydrodynamic effects strongest at ∼
20 K . We further elucidate thisbehavior via ab initio calculations of electron scattering mechanisms, whichare used to extract a current profile using the electronic Boltzmann transportequation. These calculations show quantitative agreement with our measure-ments, capturing the non-monotonic temperature dependence. The combina-tion of experimental and theoretical observations allows us to quantitativelyinfer the strength of electron-electron interactions in WTe . We show thesestrong electron interactions cannot be explained by Coulomb repulsion aloneand are predominantly phonon-mediated. This provides a promising avenuein the search for hydrodynamic flow and strong interactions in high carrierdensity materials. Main text
When microscopic scattering processes in condensed matter systems conserve momentum, elec-trons flow collectively, akin to a fluid, deviating significantly from the expected diffusive flow2f a Fermi liquid. This behavior, termed hydrodynamic electron flow ( ), has been reportedin transport measurements in (Al,Ga)As ( ), graphene ( ), PdCoO ( ), and WP ( ). Evenwhen momentum is conserved, the presence of boundaries does not need to respect this con-servation, leading to a distinct spatial signature when current flows along a channel, known asPoiseuille flow. The resulting current profile is characterized by enhanced flow in the centerof the channel, and reduced flow along the edges, as has recently been shown by spatially-resolved measurements in graphene ( ). In other hydrodynamically-reported materials,however, the presence of significant carrier density presents unsolved mysteries regarding themicroscopic origins of hydrodynamics.Tungsten ditelluride (WTe ) is a layered semimetal which has stirred significant interest inrecent years as part of a new class of quantum materials. The bulk crystal exhibits large mag-netoresistance ( ) and pressure-driven superconductivity (
14, 15 ), and has been predicted ( )and observed ( ) to be a Type-II Weyl semimetal. In the monolayer, WTe can be elec-trostatically gated into a quantum spin Hall insulator ( ) or a superconductor (
22, 23 ). Theseeffects are due to the rich band structure of the material, its high conductance (carrier mean-free-paths as long as µ m ( )), and strong electron-electron interactions. Since momentumis conserved during electron-electron interactions, spatially resolved measurements of hydro-dynamic flow in WTe are uniquely positioned to probe interactions in the material.In this Report , we present the spatial profile of current flowing in a WTe flake and show it isconsistent with hydrodynamic flow. The spatial profile depends strongly and non-monotonicallyon temperature. Through a combination of experiment and theory we extract the characteristiclength scale of electron-electron interactions as a function of temperature. In contrast withother hydrodynamically-reported materials, where direct Coulomb interactions dominate, ourfindings indicate that electron-electron interactions in WTe are mediated by a virtual phonon. T d -WTe crystallizes in the orthorhombic lattice (space group P mn ), with two layers3igure 1: ( A ) Crystal structure highlighting the layered-structure of WTe . ( B ) CalculatedFermi surface for the WTe lattice, with compensated electron (blue) and hole (red) pock-ets ( ). ( C ) Experimental setup. Our scanning magnetometry microscope is mounted in aflow cryostat. The outer layer is filled with liquid He and houses the vector magnet. A needlevalve connects this bath to the central chamber, allowing for the flow of He vapor into it to con-trol temperature by balancing flow rate and heating. The microscope consists of a diamond tipwith a NV defect in contact with the sample, which can be moved by piezo-electric controllers.The diamond tip is attached to goniometers which allow for angle-control ( ). The bottomof the cryostat has a window for optical access, allowing us to measure the NV defect’s spinstate. The objective, which focuses the light on the defect, can also be moved. ( D ) A closeup ofthe diamond tip with the NV defect, and the WTe flake. Current is flowing through the flake,generating magnetic field which is measured by the NV.4n the unit cell bonded by weak van der Waals interactions along the crystallographic ˆ c axis,shown in Fig. 1A. The semimetal is charge-compensated ( ), as illustrated in Fig. 1B by theWTe Fermi surface with the electron (hole) pockets displayed in blue (red) ( ). We exfoliatea WTe flake of ∼
60 nm thickness, cleaved along its ˆ a crystallographic axis. To image thecurrent flow in WTe , we use a scanning probe based on a Nitrogen-Vacancy (NV) defect indiamond ( ), an atomic-size quantum magnetometer. The NV is sensitive to the magneticfield parallel to its crystal axis through the Zeeman effect, and our experiments take advantageof the long coherence time of the NV to perform echo magnetometry ( ) and achieve ∼
10 nT magnetic sensitivity. The unique custom-built cryostat and scanning system used in this study,allowing imaging at variable temperature, is shown schematically in Fig. 1C. Fig. 1D shows acloseup of the diamond tip with the NV defect above the WTe sample. At each tip location,the NV sensor detects the magnetic field generated by the current flowing along the sample. Inour notation, the current is flowing along the ˆ y -axis, generating non-zero magnetic field in the ˆ x - ˆ z plane. The NV defect axis is oriented along the ˆ y - ˆ z plane, and thus we are only sensitive tothe ˆ z component of the magnetic field.In Fig. 2A, we present the ˆ z -component of the magnetic field ( B z ) measured by our NVtip scanning along two 1D line scans, one taken above a gold contact (blue markers) and theother taken above our WTe sample (orange markers). Both measurements correspond to achannel width of W = 1 . µ m and total current I tot = 20 µ A , and were taken at a tip heightof h = 140 nm . We observe noticeable differences between the two scans, with the WTe measurement showing a sharper slope in the center of the channel and an inward shift of theextrema positions. Such differences are discernible due to the high signal-to-noise ratio ofour measurement ( ). To visualize the difference in the underlying current profile, we obtainthe ˆ x -component of the magnetic field ( B x ) by Fourier reconstruction (
25, 30 ) of the data inFig. 2A. This is shown by the blue and orange markers in Fig. 2B respectively, with the WTe - . . C u rr en t d i s t r i bu t i on J y ( A / m ) po s i t i on ( µ m ) M ea s u r ed f i e l d B z ( µ T ) -2 -1 0 1 2-6-4-20246 position ( µ m) AuWTe B x B z R e c on s t r u c t ed f i e l d B x ( µ T ) -1 0 10246 position ( µ m) A B C
Figure 2: ( A ) B z magnetic field, measured by the NV sensor in a 1D scan across a channel ofwidth W = 1 . µ m . The blue and orange markers show measurements along the gold contactand the WTe device respectively, normalized to have a total current I tot = 20 µ A . The blueand orange lines correspond to the expected B z field generated by the currents in (C) at a heightof h = 140 nm and neglecting the thickness of the sample. ( B ) Reconstruction of the B x fieldalong the scan, from the data shown in (A) for the corresponding blue and orange markers. Theblue and orange lines correspond to the expected B x field generated by the currents in (C). Notethat there is an asymmetry at the edges of the sample, with both curves appearing above theexpected values on the left and below them on the right. This is likely due to a ∼ ◦ anglemismatch of our NV sensor. ( C ) Theoretical current distributions along a channel of width W = 1 . µ m with total current I tot = 20 µ A . The blue line shows a uniform profile, expectedfor diffusive flow. The orange line shows a curved current profile with flow enhanced in thecenter of the channel and reduced along the edges (see discussion of Fig. 4 for more details onthe profile). 6easurement indicating enhanced current density in the center and reduced density along theedges, suggestive of hydrodynamic flow. These observations can be made more quantitative byexamining two theoretical examples of current distributions, as shown in Fig. 2C. The blue linecorresponds to uniform flow, where electron-electron interactions play a negligible role - as ex-pected for diffusive behavior. The orange line shows a non-diffusive current distribution whosequantitative details will be discussed below. The blue and orange lines shown in Fig. 2A,B arethe calculated B z and B x for the corresponding profiles in Fig. 2C. The good agreement withexperiment confirms that current flow in gold is indeed uniform, but notice that the finite heightoffset of the NV measurement results in a non-flat B x profile even for a fully diffusive currentdistribution. By contrast, the flow in WTe deviates significantly from diffusive flow.We then apply our apparatus to probe the temperature dependence of electron interactionsin WTe and their influence on hydrodynamic behavior. Fig. 3A shows a contour plot of the B x magnetic field profile across our WTe sample, taken at different temperatures. The magneticfield at the center of the channel becomes higher with lower temperature, until it peaks at around −
20 K , and then becomes lower again. Similarly the width of the profile, highlighted bythe gray contour at B x = 4 µ T , is also narrowest around −
20 K . This non-monotonicbehavior ( ) is evident in three field profiles taken at
90 K ,
20 K , and (Fig. 3B). The profileat
20 K is both the narrowest and has the highest peak value, indicating hydrodynamic effectson current flow are maximal at this temperature.To understand the underlying microscopic origin of hydrodynamic behavior in WTe andits non-monotonic temperature dependence, we investigate the competition of electron-electroninteractions with boundary scattering and enhanced momentum-relaxing scattering against im-purities in the sample. From an independent ab initio theory we extract the temperature de-pendence of the different scattering mechanisms in WTe . These are used to solve the Boltz-mann transport equation (BTE) under the relaxation time approximation (
2, 8, 25 ), to obtain7 position ( µ m) position ( µ m) position ( µ m) T e m pe r a t u r e ( K ) -0.5 0 0.56.57 WTe A B x ( µ T) B x ( µ T ) B x ( µ T ) B
90 K20 K5 K-1 0 1 1.55102040 T e m pe r a t u r e ( K ) -1 0 1-0.5 0 0.56.57B x ( µ T)
90 K15 K5 K DC e- e-e- e-e-e- e-e- phphimp ab initio experiment ab initio experiment Figure 3: ( A ) Contour plot of the magnetic field profile B x across the WTe device ( ˆ x -axis) atdifferent temperatures ( ˆ y -axis). The scans were taken along the same y -position to isolate theeffects of temperature on current flow ( ). The contour at B x = 4 µ T is highlighted in gray toshow the narrowest profile appears between −
20 K , where the peak height is also maximal.( B ) 3 line-cuts of (A) taken at
90 K ,
20 K , and as can be seen by the respective arrows.The inset zooms in on the peak of the profiles. The profile at
20 K , also shown in orange inFig. 2C, is both the narrowest and has the highest peak value. ( C ) Contour plot of the magneticfield profiles obtained from numerical transport calculations using ab initio scattering rate in-puts for electron-electron and electron-phonon interactions, and assuming an electron-impuritymean free path of . µ m . The theory captures the non-monotonic temperature dependence,peaking around
15 K . ( D ) 3 line-cuts of (C) taken at
90 K ,
15 K , and , showing quantitativeagreement with the experimental data in (B). 8 ean f r ee pa t h ( µ m ) Temperature (K)
A B C -2 -1 l m r / W -2 -2 -1 -1 Temperature (K) ∞
40 8010 l mc / W B pea k l e-eC l e-eph l e-ph l mr l e-imp l e-imp ( µ m) B peak J y B x W ph e - e - e- e - ph e - e - e - e - e - e - e - e - imp Figure 4: ( A ) Temperature dependent ab initio electron mean free paths for WTe , obtainedusing density functional theory ( ). We consider electron-phonon ( l e − ph ), electron-impurity( l e − imp ), electron-electron scattering mediated by a screened Coulomb interaction ( l Ce − e ), andby a virtual phonon ( l phe − e ). The corresponding Feynman diagrams are schematically shown inthe inset. The overall momentum-relaxing, l mr , electron mean free path is calculated usingMatthiessen’s rule. ( B ) Normalized B x magnetic field phase diagram, extracted from currentprofiles computed using the Boltzmann transport equation allowing for momentum-conserving( ˆ x -axis) and momentum-relaxing ( ˆ y -axis) interactions (
2, 8, 25 ). The color corresponds to themagnetic field B x in the center of the channel, normalized to be for uniform flow and for perfect parabolic Poiseuille flow as highlighted by the inset. Overlaid arrows indicate thedecreasing-temperature trajectories predicted ab initio in (A) for various values of the impuritymean free path, l e − imp , following the legend in (C). Points correspond to fits of the magneticfield profile to the experimental data at different temperatures ( ), following the colors in (C).( C ) 1D line cuts along the arrow trajectories in (B), indicating the ab initio theory captures thenon-monotonic temperature dependence of the observed hydrodynamic phenomena for l e − imp larger than ∼ µ m , after which the position of the peak at ∼ K stays unchanged. The pointscorrespond to experimental data at different temperatures shown in Fig. 3.the steady-state current profiles, used to extract magnetic field profiles, which are shown asa function of temperature in Fig. 3C. The theoretical predictions capture the non-monotonictemperature dependence, with the peak ( ∼
15 K ) showing quantitative agreement with exper-iment. At high temperature, theory predicts a more diffusive profile than the one measured inexperiment (Fig. 3D).The theoretical current profiles can be described generally as a function of two non-dimensionalparameters, the ratio of the momentum-relaxing ( l mr /W ) and momentum-conserving ( l mc /W )9rocesses’ mean free paths to the channel width W . Microscopically, we consider two momentum-relaxing processes, electron-phonon scattering, l e − ph , and electron-impurity scattering, l e − imp ,and two momentum conserving processes, electron-electron scattering mediated by a screenedCoulomb interaction, l Ce − e , and by a virtual phonon, l phe − e . Fig. 4A shows the full temperaturedependence of these mean free paths for WTe , given by our ab initio calculation. The mo-mentum conserving mean free path is dominated by the phonon-mediated interaction, as l Ce − e is more than two orders of magnitude higher than l phe − e at all temperatures. In contrast, boththe electron-phonon and electron-impurity scattering processes contribute to the momentum-relaxing mean free path. We represent the resulting current profiles, as a function of these twonon-dimensional parameters, by the normalized B x magnetic field peak in the middle of thechannel in Fig. 4B. It is instructive to identify four limits in our phase-diagram: i) in the ab-sence of momentum-conserving events ( l mc (cid:29) W ) and numerous momentum-relaxing events( l mr (cid:28) W ), i.e. bottom-right corner of the phase diagram in Fig. 4B, electron flow is diffusive;ii) in the absence of both momentum-conserving and momentum-relaxing events ( l mc (cid:29) W and l mr (cid:29) W ) (top-right), boundary effects dominate leading to ballistic flow; iii) in thepresence of numerous momentum-conserving events ( l mc (cid:28) W ) and absence of momentum-relaxing events ( l mr (cid:29) W ) (top-left), electrons flow collectively and we observe hydrodynamicflow; iv) finally, in the presence of both momentum-relaxing and momentum-conserving events( l mc (cid:28) W and l mr (cid:28) W ) (bottom-left), the flow is momentum-‘porous’, and the regime isreferred to as ‘Ziman’ hydrodynamics ( ).While Fig. 4A highlights that both l mr and l mc increase monotonically with decreasing tem-perature, Fig. 4B confirms there exist 1D temperature trajectories that support non-monotonichydrodynamic behavior. Overlaid arrows in Fig. 4B mark the decreasing temperature trajec-tories following the ab initio calculations in Fig. 4A, for empirical values of l e − imp between . µ m and . µ m . The infinite impurity arrow corresponds to momentum relaxation only10ue to electron-phonon interactions. Similarly, we can fit the experimental profiles at differ-ent temperatures to one such trajectory, shown by the overlaid points in Fig. 4B ( ). The abinitio predictions agree well with experimental profiles, albeit the experimental data suggests aweaker temperature-dependence. This deviation may be attributed to the finite thickness of thesample, position-dependent mean-free-path due to sample imperfections, or modification of thephonon spectrum by the presence of the quartz substrate ( ). The non-monotonic behavior isshown more clearly in Fig. 4C, which plots these 1D trajectories for empirical values of l e − imp between . µ m and . µ m (Fig. 4C). While a minimum impurity mean free path is necessaryto observe non-monotonic behavior, the location of the peak is largely independent of l e − imp above this threshold. This allows us to extract a robust estimate for the electron-electron inter-action length, of l phe − e ∼
200 nm at −
20 K . It is important to note that the ab initio calculationwas performed on a fully relaxed WTe lattice. When instead the lattice is constrained to ac-commodate Weyl nodes, electron-electron interactions increase, resulting in the non-monotonicbehavior peaking at ∼ ( ). This is in contrast with our observations, suggesting that oursample was not in the Weyl phase.In this work, we image the current profile of electron flow in a WTe channel. Our lownoise quantum magnetometry measurements allow us to differentiate the WTe current profilefrom uniform diffusive flow. Utilizing our setup’s tunable cryogenic capabilities, we study thetemperature dependence of the current profile. Curiously, these show non-monotonic temper-ature dependence, with hydrodynamic effects peaking between −
20 K . We compare ourresults with independent ab initio solutions to the BTE and obtain good quantitative agreement.This allows us to extract an estimate of the electron-electron interaction length in WTe . Im-portantly, our ab initio calculations, quantitatively consistent with experimental observations,indicate that the strong electron-electron interactions are likely phonon-mediated (
32, 33 ). Thisopens the possibility of observing hydrodynamic effects and strong interactions even in high11arrier density materials where direct electron-electron interactions are screened. Furthermore,the combination of spatially-resolved current imaging and ab initio theory could reveal themicroscopic mechanisms underlying strongly interacting systems, such as high-temperature su-perconductors and strange metals (
34, 35 ). In this work we focus on one WTe crystallographicorientation. Our setup, however, allows exploration and imaging of anisotropic hydrodynamicflows in 3D crystals. By studying different geometries of such anisotropic materials, future workwill permit observations of non-classical fluid behavior such as steady-state vortices, couplingof the fluid stress to vorticity and 3D Hall viscosity ( ). References and Notes
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The authors thank Mark Ku and Adam S. Jermyn for fruitful discus-sions. The authors also thank Pat Gumann and Rainer St¨ohr for the initial design and con-struction of the microscope. T.X.Z. thanks Ronald Walsworth and Matthew Turner for dia-mond processing assistance.
Funding:
This work was primarily supported by ARO GrantNo. W911NF-17-1-0023 and the Gordon and Betty Moore Foundations EPiQS Initiativethrough Grant No. GBMF4531. Fabrication of samples was supported by the U.S. Depart-ment of Energy, Basic Energy Sciences Office, Division of Materials Sciences and Engi-neering under award DE-SC0019300. A.Y. also acknowledges support from ARO grantsW911NF-18-1-0316, and W911NF-1-81-0206, the NSF grants DMR-1708688 and the STCCenter for Integrated Quantum Materials, NSF Grant No. DMR-1231319, and the AspenCenter of Physics supported by NSF grant PHY-1607611. G.V., Y.W., and P.N. acknowledgesupport from the Army Research Office MURI (Ab-Initio Solid-State Quantum Materials)14rant No. W911NF-18-1-0431 that supported development of computational methods todescribe microscopic, temperature-dependent dynamics in low-dimensional materials. G.V.acknowledges support from the Office of Naval Research grant on High-Tc Superconduc-tivity at Oxide-Chalcogenide Interfaces (Grant Number N00014-18-1-2691) that supportedtheoretical and computational methods for phonon-mediated interactions. Y.W. is partiallysupported by the STC Center for Integrated Quantum Materials, NSF Grant No. DMR-1231319 for development of computational methods for topological materials. This researchused resources of the National Energy Research Scientific Computing Center, a DOE Officeof Science User Facility supported by the Office of Science of the U.S. Department of Energyunder Contract No. DE-AC02-05CH11231 as well as resources at the Research ComputingGroup at Harvard University. This work was performed, in part, at the Center for NanoscaleSystems (CNS), a member of the National Nanotechnology Infrastructure Network, whichis supported by the NSF under award no. ECS-0335765. CNS is part of Harvard University.P.N. is a Moore Inventor Fellow and gratefully acknowledges support through Grant No.GBMF8048 from the Gordon and Betty Moore Foundation. C.A.C.G. acknowledges sup-port from the NSF Graduate Research Fellowship Program under Grant No. DGE-1745303.A. T. P. acknowledges support from the Department of Defense (DoD) through the NationalDefense Science & Engineering Graduate Fellowship (NDSEG) Program.
Author contri-butions:
J.G., P.N., and A.Y. conceived the project. U.V., A.H., T.X.Z., and Y.D. designedand developed the experimental setup under the guidance of A.Y.; N.K. and J.G. grew thebulk WTe crystals under the guidance of C.F.; U.V, A.H., Z.Q., and A.T.P fabricated andcharacterized the WTe flakes; G.V. implemented the numerical BTE methods under theguidance of P.A. and P.N.; G.V., Y.W., and C.A.C.G. developed and implemented the ab-initio theoretical methods under the guidance of P.N.; U.V., A.H., G.V., Y.W., P.N., and A.Y.analyzed the data and discussed the results. All authors contributed to the writing of the15anuscript. Competing interests:
The authors declare no conflict of interest.
Data andmaterials availability:
All data presented in the main text and the supplementary materialsare available upon request. 16 upplementary material for
Imaging phonon-mediated hydrodynamic flow in WTe with cryogenic quantum magnetometry Our magnetic sensing experiment is based on a diamond scanning probe with a Nitrogen va-cancy (NV) defect close to its surface. The design and fabrication of our probes is presented indetail in Ref. ( ). However, our cryogenic setup requires a variation of this design which wedescribe in this section.Our scanning tip is glued to a standard quartz tuning fork, and monitoring of the tuning forkfrequency allows us to maintain constant height above our sample during the scan. Supplemen-tary Fig. 1a shows the two legs of our tuning fork, and a quartz rod glued to one of the legs.The quartz rod is itself glued to a diamond cube µ m thick, µ m wide, and µ m long. Acloseup of the diamond is shown in Supplementary Fig. 1b, and the µ m etched diamond pillarcan be seen. Our NV defect is located
20 nm away from the pillar edge. Notice that our tuningfork is positioned horizontally, as opposed to the vertical position which is common in atomicforce microscopy (AFM) systems. This is due to size restrictions in our setup, imposed by theshort ( µ m ) focal length of our objective.In the configuration described above, our diamond tip is in contact with the measured sur-face. This method was problematic for certain materials (including the WTe we measured),as the tip was damaging the surface of the material during the scan. In addition, the material1 a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p a) (b) Tuning fork legsQuartz rod 50 µ mDiamond Diamond pillar Figure 1: a An optical image of our scanning tip. A quartz rod is glued to the edge of one ofthe prongs of a quartz tuning fork. A diamond cube of × × µ m is glued to the otherside of the rod. b A closeup image of the diamond cube, with a µ m etched pillar visible. Thispillar is used as our scanning probe, and an NV defect is located
20 nm away from the pillaredge. µ mSample Scanningpillar DiamondContactpillarSubstrate θ (a) (b) Figure 2: a A schematic for multiple pillar sensing. The diamond is tilted at an angle θ (usinggoniometric motors) so that one pillar is in contact with the substrate, while the scanning pillarmaintains a constant height above the sample without touching it. The angle θ control the heightof the NV probe above the sample. b An optical image of a multiple pillar scanning tip with7 pillars. The extreme pillars are made to be thicker so they are optimized for durability incontact, while the central pillars are optimized for NV sensing.2ccumulated on the tip itself, which reduced the optical contrast of the NV and hurt our mea-surement. To overcome this problem, we fabricated diamond tips with several pillars so thatone pillar is used for AFM contact, while another is used for magnetic sensing. This techniqueuses the angle control available in our setup due to goniometric motors (notice the motors label φ and θ in Main text Fig. 1a), as by tuning the angle we can control which tip will be in contactand the height of the scanning tip above the sample. Supplementary Fig. 2a shows a sketch ofour scanning method, where one pillar is in contact with the substrate (so it does not damagethe device), while another is used for scanning the sample. The angle θ of the diamond tip isused to tune the height of the NV above the sample. Supplementary Fig. 2b shows an opticalimage of such a multiple pillar diamond with 7 pillars in a row. The outer contact pillars aremade to be thicker ( ∼
800 nm diameter on the edge) so they are optimized for durability oncontact. The central pillars are thin ( −
400 nm diameter in the middle) so they contain oneNV defect per pillar on average, and for optimal optical waveguiding.
Single crystals of WTe were grown by excess Te-flux. In the typical synthesis, pieces of W(Chempur, . ) and Te (Alfa Aesar, . ) were weighed according to the stoichio-metric ratio of W . T e . ( ) and kept in a quartz crucible. This crucible was vacuum sealedinside a quartz tube. The reaction content was heated to ◦ C at a rate of ◦ C / h in aprogrammable muffle furnace. This temperature was maintained for
10 h to get a homogenousmolten phase, after which the temperature was lowered slowly to ◦ C at a rate of ◦ C / h . Atthis temperature, excess molten flux was removed by centrifugation to obtain shiny plate-likesingle crystals of WTe . The total carrier density of the bulk sample was measured (at ) tobe . × cm − . The average mobility is . × cm / Vs .We exfoliated WTe flakes and searched for elongated samples which maintain constant3 Te contact (gold)RF delivery line (gold) 10 µ m Figure 3: Optical image of the WTe sample used for the majority of the data in the manuscript,including Main text Fig. 2, 3, and 4. The µ m long flake is contacted at its ends with goldcontacts. The path along which our magnetic field profiles were taken is shown in white. Anadditional gold line nearby is used for delivery of RF power to manipulate the NV.width along a segment, allowing us to assume uniform current flow in this segment. The chosenflake was then transferred onto our quartz substrate. The flake was placed close to a gold lineused for delivery of a radio frequency (RF) signal, necessary for manipulating the states ofthe NV. This RF signal required the use of an insulating substrate, as the conducting plane ofa typical Silicon substrate would screen the RF signal and prevent NV manipulation. Afterplacing the flake, we etched the edges using an argon ion mill and evaporated gold contacts(without breaking vacuum).Supplementary Fig. 3 shows an optical image of the WTe sample used for the experimentsshown in Main text Fig. 2-4. It is a
60 nm thick and µ m long flake. We scanned along a . µ m wide profile in a uniform segment of the flake (see dashed white line). The gold scanshown as a blue line in Main text Fig. 2b was taken on the gold contact shown to the right ofthe sample in the image. The evaporated gold was also
60 nm thick to match the WTe flakethickness. There are several ways to extract the magnetic field from the NV defect, and here we willdescribe the method used in the paper, which utilizes the long coherence of the NV as a quantumsensor. 4he NV electron spin is a spin-1 system, and its low energy state are the triplet states |− i = |↓↓i , | i = √ ( |↑↓i + |↓↑i ) , and | i = |↑↑i . The | i is split in energy from |− i and | i by thezero field splitting and |− i and | i are split by applying a bias magnetic field due to Zeemansplitting. We apply an external bias field of parallel to the NV axis, where the energysplitting between the |− i and | i states is . . This is the only transition used in ourexperiment, and so the system can now be described as a qubit made of these two states.An external magnetic field parallel to the NV axis shifts the frequency of our qubit by gµ B B || due to the Zeeman effect, where g ∼ is the NV g-factor, µ B is the Bohr magneton, and B || is the applied parallel magnetic field. This shift is approximately .
87 MHz / G . Measuring theresonance frequency of the qubit allows us to infer the local parallel magnetic field.This frequency is often measured by a continuous spectroscopy method (ESR) in whichthe NV is probed with RF radiation at different frequencies to find the frequency at which theNV is excited, leading to reduced red counts in the optical measurement. While this method issimple, it has several drawbacks. The continuous drive can heat up the system being measured,and lead to broadening of the resonance peak. Also, the contrast in this measurement is halvedas the states compared are | i and an equal mixture of ( | i , |− i ) as opposed to full contrastbetween | i and |− i . Another method is Ramsey interferometry, in which the qubit is placedin a superposition state √ ( | i + i |− i ) . This superposition evolves to acquire a phase √ ( | i + ie i ∆ t |− i ) due to the change ∆ in the qubit resonance frequency. A measurement of this phaseprovides the shift in resonance frequency and therefore the magnetic field. This techniquerequires pulsed control, and thus overcomes many of drawbacks of continuous spectroscopy.The time over which we can acquire a phase is limited by the low frequency qubit coherencetime T ∗ .Pulsed manipulation of the qubit allows for more complicated sequences which cancelthe effect of low frequency noise, improving the qubit coherence time from T ∗ to T . In5 π/2 Y π X/Y ± π/2 Prepare 0Prepare Apply current I tot
Apply current -I tot
Measure Z0
X Z Y X Z Y X Z Y X Z Y X Z Y τ /2 τ /2 ϕ/2 ϕ X Z Y
Figure 4: A schematic of our echo magnetometry experiment. The pulses applied are shownbelow in sequence, and the Bloch sphere representation above shows the state of the qubit atthe corresponding place in the sequence. The red arrow indicates the current state of the qubit,while the green line indicates the path it took on the Bloch sphere since the last step. A detaileddescription of the sequence can be found in Section 3. For the last π/ , there are four possiblerotations, and the Bloch sphere representation shows the π/ around the X-axis.this experiment we used the simplest such sequence, the echo experiment, as shown in de-tail in Supplementary Fig. 4. The qubit is initially prepared in the | i state by illuminatinggreen light. Afterwards, an RF pulse is applied to perform a π/ pulse around the X-axis.This places the qubit in the superposition state √ ( | i + i |− i ) . Then a current I tot is ap-plied along the device for a duration τ / . This current leads to a parallel magnetic field B || and thus to a frequency shift ∆ = gµ B B || in the qubit resonance frequency shift. Duringa time τ / a phase ϕ/ τ / is acquired by the qubit, which is then left in the state √ ( | i + ie ϕ/ |− i ) . A π pulse around the Y-axis is then applied on the qubit, leading tothe state √ ( |− i − ie ϕ/ | i ) = √ ( | i + ie − ϕ/ |− i ) . Note that the phase accumulated hasbeen reversed and so applying the same current again will lead to a cancellation of the acquiredphase. Instead we apply the opposite current, leading to an acquisition of an additional − ϕ/ phase, leading to the state √ ( | i + ie − ϕ |− i ) . The phase has now been acquired and we needto extract it. For this, we repeat this experiment with four different pulse options: π/ pulse6round X-axis, π/ pulse around the Y-axis, − π/ pulse around the X-axis, and − π/ pulsearound the Y-axis. For each option we then measure the Z component of the qubit by applyinggreen laser light and collecting red photons. From the 4 measurements of probabilities to be inthe |− i state collected after their respective pulses we can extract ϕ as: ϕ = arctan (cid:18) M Xπ/ − M X − π/ M Y − π/ − M Y π/ (cid:19) , (1)where M Xπ/ is the measurement following the π/ pulse around X-axis and respectively forthe others. The different measurements allow us to cancel the effect of background counts andfocus only on the optical contrast due to the qubit state.Note that this sequence is only useful for sensing alternating magnetic fields, where the fieldfrequency is equal to the echo frequency. This is useful in current measurement where we canlock the application of current to the echo measurement, but is not useful for measuring staticmagnetic fields due to magnetization.As mentioned above, this measurement cancels low frequency noise and thus allows us toacquire a signal up to τ = T as opposed to T ∗ . The probe we used had long coherence time T = 150 µ s (and T ∗ = 3 µ s ). However, as will be discussed in Section 6, our measurementswere taken at shorter times τ = 21 µ s . This was chosen because we can afford to use relativelyhigh current, for which a longer integration time would lead to phase accumulation significantlyabove π - leading to ambiguity in the magnetic field as we can only determine the phase up to π . The measurements that are shown in the main text had to be measured in exceptional signal-to-noise ratio (SNR) in order to observe the slight changes over a small temperature range. Inorder to achieve that, we made use of a combination of relatively high current (15 − µA ) ≈ µs ). In this section we will try to estimate quantitativelythe SNR that we have in our measurements. The signal (in Counts / s ) is given by: Signal = ∆ φ · C · F · T · N Avg , (2)where ∆ φ is the accumulated phase in the echo experiment, C is the contrast, F is the fluo-rescence of the NV in Counts / s , T is the avalanche photo-diode (APD) acquisition time, and N Avg is the number of averages per second. The noise in the system is assumed to be only shotnoise:
N oise = p F · T · N Avg . (3)We want to calculate the sensitivity per second, assuming a SNR of 1: SignalN oise = 1 = ∆ φ · C ·√ F · T · p N Avg (4)From that we can extract the minimal ∆ φ : ∆ φ = 1 C ·√ F · T · p N Avg (5)In order to estimate the smallest magnetic field, we need to convert phase to field via: ∆ B = ∆ φ · πγ · τ , (6)where ∆ B is the magnetic field, τ is the spin-echo time, and γ = g · µ B = 2 . MHzGauss where g isthe g-factor and µ B is the permeability of the vacuum. From that we get: ∆ B = 12 πγ · C √ F · T · τ · p N Avg (7)To estimate the magnetic noise, we can approximate (by ignoring overheads and assuming longenough τ ) N Avg ≈ τ (where we assume the total time spent measuring is ). From that we get: B noise ≈ πγ · C √ F · T · √ τ (8)8n our NV center C = 0 . , F = 120 · Counts / s , T = 300 ns , and for our experiments inthe main text τ = 21 . µ s . From these values we can estimate the magnetic field noise level: B noise = 43 nT √ Hz (9)By comparing to theory, we empirically estimate of the noise in the magnetic field profile of thegold contact in Main text Fig. 2 and get: B noise = 104 nT √ Hz (10)The additional noise can be attributed to mechanical noise and other overheads. In our mea-surements, in order to reduce the noise even further we were averaging over
10 s which givesa noise of ∆ B ≈
13 nT . Together with a signal of ∼ µ T (peak to peak in Main text Fig. 2bwithout normalization) we get an estimate of the SNR of: SignalN oise ≈ (11)We also want to extract from this magnetic noise the minimal current density that is possibleto detect using our probe. Assuming a simple rectangular current profile, we can use the B z profile from Main text Fig. 2b, where we applied ∼ µ A when the device width is . µ m andgot ∼ µ T peak to peak. To achieve a SNR of for a
10 s measurement, we get the minimalcurrent density: ∆ J ≈
20 nA µ m (12)By taking advantage of our long T and measuring for µ s echo time, we are able to decreasethis number even more and get: ∆ J ≈ . µ m (13)9 uWTe R ab i F r equen cy [ M h z ] -5 -4 -3 -2 -1 0 127293133 µ m Y position ( µ m) XZ Y
Figure 5: Rabi frequency measured as a function of the position along y direction while keepingthe x position along the center of the device. The peak is indicated by dashed line and measured . µ m away another dashed line indicates the position in which we performed the x scan (thescan in figure 3 of the main text). In Main text Fig. 3 we have shown a dependence of the current profile on temperature. Dueto mechanical deformations (thermal expansion) of the scanning system at different tempera-tures, we had to determine the exact location along the device to perform the scan for everytemperature. We first used our optical microscope to determine the position roughly (up to afew microns). Following that, we performed a scan along the device (Inset, SupplementaryFig. 5) while keeping the tip at the center of the device. A point along the device happenedto have a strong distortion of the current which manifested as a peak in the Rabi frequency atthis point. In Supplementary Fig. 5 the measured Rabi frequency as a function of position isshown and a clear peak is visible at around Y = 0 µ m . The peak serves as a marker, and foreach temperature we performed our scan at the same distance ( . µ m ) from the peak (dashedvertical lines Supplementary Fig. 5). Using this technique, we were able to verify the locationalong the device for every current profile scan in Main text Fig. 3.10 position ( µ m) R e c on s t r u c t ed f i e l d B x ( µ T ) I tot =19 µ AI tot =4.6 µ A (norm.)
Figure 6: Reconstructed magnetic field B x as a function of the position along the device, takenfor two different current amplitudes. The red curve corresponds to a scan with a current of µ A and measured with an echo time of µ s . The blue curve corresponds to a scan withcurrent of . µ A and measured with an echo time of µ s . Both measurements were taken at
12 K . The bias applied on the WTe that was used throughout the paper was chosen to be smallenough that it is in the linear response regime yet large enough that the signal to noise ratiois maximized. In order to verify linear response, we performed two scans with two differentcurrents, one scan with relatively high current ( µ A ) and one scan with relatively low current( . µ A ). Importantly, to preserve the signal to noise ratio, we performed the two scans withdifferent corresponding echo times to accumulate the same final phase (see Section 3 for detailson AC magnetometry). For the high current scan, we used a µ s phase accumulation time andfor the low current scan we used µ s (which is the same ratio between the current amplitudes).As a result, the low current scan was times longer.The results can be seen in Supplementary Fig. 6, as two B x magnetic field profiles areshown. The blue one was taken with high current and the red with low current. As can beclearly seen, the difference between the two curves is very small which means that indeed we11re in the linear response regime. So, in order to maximize signal to noise ratio and to avoiddrifts of the mechanical scanning system over long scanning times we used the high current µ A and µ s echo time throughout the paper. The energy splitting of the NV electron spin states are sensitive to the component of the mag-netic field parallel to the NV axis. However, under certain conditions we can extract the fullmagnetic field vector from such a measurement ( ). This allowed us to reconstruct the B x component of the magnetic field, as shown in Main text Figs. 2c and 3.The NV used in our experiment was oriented in the y-z plane. As the current flows along they-axis, only the z-axis magnetic field component B z contributes field parallel to our NV. Thus,our NV directly measures B z as we scan above the sample, up to a factor of / √ due to theangle of the NV in the y-z plane.We measure the magnetic field in a plane above the sample. Assuming all sources of mag-netic field are below our plane, for the magnetic field in the plane we can assume source-freemagnetism: ∇ · ~B = 0 and ∇ × ~B = 0 . These additional equations allow us to extract the fullvector ~B from a single component. The full derivation is covered in the previous references, andhere we focus on the special case of extracting B x from B z which was used in our measurement.The equations then reduce to the simple form: b x ( k x , k y , z ) = − i k x p k x + k y b z ( k x , k y , z ) (14)where k x , k y are the x and y components of the planar wave vector, and b x/z ( k x , k y , z ) = R ∞−∞ R ∞−∞ B x/z ( x, y, z ) e − i ( k x x + k y y ) dxdy is the 2D Fourier transform of B x/z on the plane. Thus,in Fourier space B x and B z are related by a simple factor, allowing us to extract the perpendic-ular B x component of the magnetic field. 12or our analysis, we used the profile measured along the 1D path (x-axis) above the sample,and assumed the magnetic field pattern was constant along the y-axis. Note that in Main text Fig.2b, the magnetic field is still finite at the edge of our scanning window. As the Fourier integralrequired the field on the entire plane, this was problematic for the reconstruction. To improveour analysis, we extrapolated both edges of our scan with fits to /x tails before applying thereconstruction. At high temperatures, electron transport is usually resistive. This is in large part due to thenumerous momentum-relaxing scattering events, the microscopics of which are discussed be-low. Macroscopically, this leads to the well-known constitutive relation known as Ohm’s ‘law’,and can be described within the Drude theory of metals. At very low temperatures the elec-tron mean-free path, the average distance electrons travel in between scattering events, canexceed the geometry’s length scale leading to ballistic behavior. In certain materials, there ex-ists a collective-transport regime in between these two limits whereby mean free paths are low,suggesting numerous scattering events, but momentum is on-average conserved following scat-tering. This collective-transport behavior is commonly referred to as hydrodynamic flow due tothe resemblance of the resulting flow signatures to classical hydrodynamics. While experimen-tally these transport regimes are observed as a function of temperature, a more natural choice fornon-dimensional independent variables is the ratio of the momentum-relaxing and momentum-conserving mean free paths to the geometry’s length scale ( l mr ( T ) /W , l mc ( T ) /W ). Note thatthese depend implicitly on temperature. 13 .1 Electronic Boltzmann transport equation In the absence of momentum-conserving scattering, the steady-state evolution of non-equilibriumelectron distribution functions is given, at the relaxation time approximation (RTA) level, by: v s · ∇ r f ( s, r ) + e E · ∇ s f ( s, r ) = − f ( s, r ) − ¯ f ( s ) τ mrs , (15)where f ( s, r ) is the non-equilibrium distribution of electrons around position r with state label s (encapsulating the wavevector k and band index n ). Non-equilibrium electrons are allowed todrift in space with a particular group velocity, v s , as a result of an external forcing term, where e is the electron charge and E is an externally-applied electric field. These drifting electronsundergo momentum-relaxing events with an lifetime, τ mrs , which acts to returns them towardsan equilibrium distribution ¯ f ( s ) .We investigate the flow signatures eq. 15 permits in a two-dimensional channel, making thecommon assumption of a circular Fermi surface ( ). Due to translational invariance along thechannel, eq. 15 simplifies to read : v y ∂ y F ( y, θ ) − eE x v x = − F ( y, θ ) − h F ( y ) i θ τ mr (16) sin( θ ) ∂ y F ( y, θ ) + F ( y, θ ) l mr = eE x cos( θ ) , (17)where we have introduced the linearized electron deviation, F ( y, θ ) , via f ( y, θ ) ≈ f − (cid:0) ∂f ∂(cid:15) (cid:1) F ( y, θ ) ,and used v = v F (cid:16) cos( θ )sin( θ ) (cid:17) , h F ( y ) i θ = 0 , and l mr = τ mr v F , where v F is the Fermi velocity, insimplifying the last line. Finally, since eq. 17 is linear, we seek normalized solutions of theform: F ( y, θ ) = eE x cos( θ ) l eff ( y, θ ) by solving: sin( θ ) ∂ y l eff ( y, θ ) + l eff ( y, θ ) l mr = 1 , (18) Note that this section uses a different convention from the main text, with the channel current flowing alongthe ˆ x direction ˜ l eff ( y ) , which is directly pro-portional to current density, j x ( y ) : ˜ l eff ( y ) = Z π dθπ cos ( θ ) l eff ( y, θ ) (19) j x ( y ) = (cid:18) mπ ¯ h (cid:19) E F e E x mv F ˜ l eff ( y ) (20)Equation 18 can be readily solved using appropriate boundary conditions, which we take to becompletely diffuse, i.e. ∀ θ ∈ [0 , π ) l eff ( − W/ , θ ) = 0 (21) ∀ θ ∈ [ π, π ) l eff ( W/ , θ ) = 0 , (22)where W is the channel width. Supplementary Fig. 7a plots the solution to eq. 20 for various values of the first non-dimensionalparameter, namely l mr /W . It is instructive to identify the two limits. As lim l mr /W → , i.e. theelectrons undergo multiple momentum-relaxing scattering events before reaching the edge ofthe geometry, we recover a uniform ‘Ohmic’ profile, with zero current density at the boundaries.At the other extreme, as lim l mr /W →∞ , i.e. electrons reach the edge of the geometry withoutundergoing any scattering, we recover the (essentially) uniform ‘Knudsen’ profile, with nonzerocurrent density at the boundaries. In between the two limits, we observe a non-diffusive currentprofile, which peaks when l mr ∼ W . Note that, in contrast with the ‘Gurhzi’ flows describedbelow, the current density remains nonzero at the boundary. Before introducing momentum-conserving terms in our electronic BTE, it is instructive to ap-proach the problem from the opposite direction, i.e. introduce momentum-relaxing terms to a15ure hydrodynamic solution. To this end, our starting point is the electronic Stokes equation: − ν∂ y j x ( y ) + τ − mr j x ( y ) = ne E x m , (23)where ν = l mc v F is an effective electron viscosity, n is the carrier density, and m is the elec-tron effective mass. Using the (stricter) no-slip boundary conditions, i.e. j x ( ± W/
2) = 0 , thesolution to eq. 23 is given by: j x ( y ) = e l mr ¯ h r nπ E x (cid:18) − cosh( x/D )cosh( W/ D ) (cid:19) (24) = I total W − D tanh( W/ D ) (cid:18) − cosh( x/D )cosh( W/ D ) (cid:19) , (25)where D = √ l mc l mr is the geometric average of the momentum-conserving and momentum-relaxing mean free paths called the ‘Gurzhi’ length, and we have normalized all the physicalconstants with the total current, I total .Supplementary Fig. 7b plots the solution to eq. 23 for various values of the non-dimensionalparameter D/W . As in the ‘Knudsen’ case, as lim
D/W → we recover a uniform ‘Ohmic’ profile,with j x ( y ) = I total W . By contrast, as lim D/W →∞ we instead recover a fully parabolic ‘Poiseuille’profile, with j x ( y ) = I total (1 − y )2 W . Note that, by virtue of our stringent boundary conditions, allsolutions exhibit zero current density at the boundary and the peak current is monotonically in-creasing as D/W increases. Both these observations are in stark contrast with our experimentalobservations, and thus we turn back to the electronic BTE.
We seek to add additional scattering terms to eq. 18, which on-average conserve momentum: Z π dθv x ∂l eff ( y, θ ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) mc = 0 (26)16 a) (b) (c) Figure 7: a Normalized current density solutions to the momentum-relaxing electronic BTE as afunction of l mr /W , the value of which is denoted by color. The profiles transition from uniformdiffusive flow (at low l mr /W ), to non-diffusive flow (at l mr ∼ W ), to ballistic ‘Knudsen‘flow (at high l mr /W ). This is quantified by the curvature of the current (inset), peaking at0.25. b Normalized current density solutions to the electronic Stokes as a function of
D/W .The profiles transition monotonically from uniform diffusive flow (at low
D/W ), to parabolic‘Poiseuille’ flow (at high
D/W ). c Comparison of experimental profiles with the ab initio profiles afforded within the ‘Knudsen‘ picture, showing poorer agreement as compared to MainFig. 4c.The simplest possible such term is given by ( ): sin( θ ) ∂ y l eff ( y, θ ) + l eff ( y, θ ) l mr = 1 + − l eff l mc + ˜ l eff l mc ! (27) sin( θ ) ∂ y l eff ( y, θ ) + l eff ( y, θ ) l = 1 + ˜ l eff l mc , (28)where the first additional term ‘depopulates’ electrons according to an average mean free path l mc and the second term enforces these electrons are redistributed appropriately to conservemomentum. In the last line, we use Mathhiessen’s rule l − = l − mc + l − mr to combine the meanfree path terms. Note that the presence of ˜ l eff in eq. 28 makes our equation integro-differential.We solve this using both an iterative and an integral solver, to avoid numerical instabilities ofeach method. In particular, eq. 28 can be transformed to a Fredholm integral equation of the 2 nd ): ˜ l eff ( y ) − l mc Z W/ − W/ dy K ( y, y )˜ l eff ( y ) = ˜ l (0) eff ( y ) (29) ˜ l (0) eff ( y ) = l − lπ Z π/ dφ cos ( φ ) (cid:26) exp (cid:18) − W/ yl sin( φ ) (cid:19) + exp (cid:18) − W/ − yl sin( φ ) (cid:19)(cid:27) (30) K ( y, y ) := 2 π Z π/ dφ cos ( φ )sin( φ ) exp (cid:18) − | y − y | l sin( φ ) (cid:19) (31)To ensure numerical stability across a wide range of l mr ( mc ) /W , we solve this using the modifiedquadrature method ( ). Since the purely ballistic ‘Knudsen’ picture also permits non-diffusive flows which can exhibitnon-monotonic curvature behavior, we compare the experimental results using eq. 20. This isshown in Supplementary Fig. 7c, which should be compared with Main Fig. 4c. First, note thatthe temperature peak is slightly shifted from ∼ to ∼ . Second, the non-monotonictemperature dependence requires substantially purer samples. Finally, we note that the experi-mental profiles are more curved than the maximum curvature afforded by the ‘Knudsen‘ picture. The solutions to eqs. 20 and 28, as well as our spatial resolved measurements yield the currentdensity as a function of position. In Main Fig.4, we summarize our results on a two-dimensionalphase diagram, quantifying the profiles using a scalar metric. In this section, we discuss thevarious choices for scalar metrics and their slight differences. Supplementary Fig. 8a plots thecurrent density’s curvature as a function of l mc ( mr ) /W . While this is a natural choice, whichis by construction normalized, we do not have direct access to the current density from ourexperiments. Also note that the current density exhibits a sharp transition between the ‘porous’(bottom left) and hydrodynamic regimes (top left), making their differentiation challenging. By18 y B x B z (a) (b) (c) Figure 8: a Current density curvature phase diagram for various values of l mc ( mr ) /W . Insetshows the two curvature bounds for diffusive and parabolic flows respectively. b Normalized B x peak value phase diagram for various values of l mc ( mr ) /W . Inset shows the B x profiles fordiffusive and parabolic flows, which acts as our normalizing lower and upper bounds respec-tively. Note how the B x profile for diffusive flow is not flat, due to the finite height of the NVabove the sample. c Normalized B z extremum position phase diagram for various values of l mc ( mr ) /W . Inset shows the B z profiles for diffusive and parabolic flows, which acts as ournormalizing lower and upper bounds respectively. Note that for diffusive flow the B z extremapositions align with the sample edge, and move inwards for curved profiles.contrast, our measurements are sensitive to the magnetic field induced by this current densityabove the sample, and we plot the B x and B z components in Supplementary Fig. 8b-c respec-tively. The two metrics are very similar, but we prefer to use the normalized peak value of the B x profile since its connection to the current density is more intuitive. Finally, a physical scalar representation of the current density, available using transport mea-surements, is the conductivity in the channel given by: σ = ne m ∗ v F Z W/ − W/ dyW ˜ l eff ( y ) = ne m ∗ v F L eff , (32)where L eff is the overall effective mean free path ( ). Supplementary Fig. 9a plots L eff /W for various values of l mr ( mc ) /W , with the ab initio trajectories overlaid as arrows. This indeedallows for non-monotonic behavior, highlighted by the 1D-cuts at fixed values of l mr /W in Sup-plementary Fig. 9b. While transport measurements of conductivity can in principle extract the19 a) (b) (c) Figure 9: a Overall effective mean free path for various values of l mr ( mc ) /W . Note the densityplot is clipped at L eff /W = 5 , to emphasize the regions of interest. Overlaid arrows represent ab initio trajectories. b One-dimensional cuts of (a) for fixed values of l mr /W , illustratingthe possibility of non-monotonicity. c ab initio extracted conductivity, showing monotonictemperature dependence.non-monotonicity we observe in our spatially-resolved measurements, their dynamical range isdifferent and in-fact our particular sample the non-monotonicity would not be observed. Thisis highlighted in Supplementary Fig. 9c, which further illustrates the importance of spatially-resolved measurements and theory. Ab initio calculations
WTe is a semimetal with considerable density of states at the Fermi level, where the electronsmainly scatter against other electrons and phonons. Here we consider four microscopic elec-tron scattering events in WTe : the momentum conserving electron-electron (e-e) scatteringmediated by Coulomb screening ( τ Wee ) and by a phonon ( τ PHee ), plus the momentum relaxingelectron-phonon (e-ph) scattering ( τ eph ) and electron-impurity (e-imp) scattering ( τ imp ) (Fig. 4,main text). Details of the methodology can be found in earlier work ( ) and a similar studyon type-II Weyl semimetal WP has successfully revealed the electron scattering microscop-20cs ( ). Electron-phonon scattering.
For an electron with energy ε n k (with n band index and k wavevector) scattered by a phonon of energy ω q ν (with q the momentum and ν the branchindex), the electron final state has momentum of k+q and energy of ε m k + q (with m the newband index). The electron-phonon scattering time ( τ eph ) can be obtained from the electron selfenergy by Fermi’s golden rule following τ − ( n k ) = 2 π ¯ h X mν Z BZ d q Ω BZ | g mn,ν ( k,q ) | × [( n q ν + 12 ∓
12 ) δ ( ε n k ∓ ω q ν − ε m k + q )] , (33)where Ω BZ is the Brillouin zone volume, f n k and n q ν are the Fermi-Dirac and Bose-Einsteindistribution functions, respectively. The e-ph matrix element for a scattering vertex is given by g mn,ν ( k,q ) = (cid:18) ¯ h m ω q ν (cid:19) / h ψ m k + q | ∂ q ν V | ψ n k i . (34)Here h ψ m k + q | and | ψ n k i are Bloch eigenstates and ∂ q ν V is the perturbation of the self-consistentpotential with respect to ion displacement associated with a phonon branch with frequency ω q ν .The momentum-relaxing electron scattering rates are evaluated by accounting for the change inmomentum between final and initial states based on their relative scattering angle following (cid:0) τ mreph ( n k ) (cid:1) − = 2 π ¯ h X mν Z BZ d q Ω BZ | g mn,ν ( k,q ) | × (cid:18) − v n k · v n k | v n k || v n k | (cid:19) (35) × [( n q ν + 12 ∓
12 ) δ ( ε n k ∓ ω q ν − ε m k + q )] , where v n k is the group velocity.We calculate the temperature dependent momentum relaxing τ mreph by taking a Fermi-surfaceaverage weighted by | v n k | and the energy derivative of the Fermi occupation for transportproperties following τ mreph = R BZ d k (2 π ) P n ∂f n k ∂ε n k | v n k | τ mreph ( n k ) R BZ d k (2 π ) P n ∂f n k ∂ε n k | v n k | . (36)21 honon mediated electron-electron scattering. The electron-electron scattering rate medi-ated by a virtual phonon can be estimated within the random phase approximation by ( τ PH ee ) − = π ¯ h k B T g ( ε F ) X ν Z Ω BZ d q (2 π ) G q ν × Z + ∞−∞ ω d ω | ¯ ω q ν − ω | sinh hω k B T . (37)Here, g ( ε F ) is the density of states at the Fermi level, and ¯ ω q ν = ω q ν (1 + iπG q ν ) is the complexphonon frequency corrected by the phonon-electron scattering linewidth. Each phonon mode isweighed within the Eliashberg spectral function by G q ν = X mn Z Ω BZ d k (2 π ) | g mn,ν ( k,q ) | δ ( ε n k − ε F ) δ ( ε m k+q − ε F ) . (38) Coulomb screening mediated electron-electron scattering.
The Coulomb mediated electron-electron scattering rate is obtained by the imaginary part of the quasiparticle self energy at eachmomentum and state (
ImΣ n k ( n k ) ) as τ − ( n k ) = 2 π ¯ h Z BZ d k (2 π ) X n X GG ˜ ρ n k ,n k ( G ) ˜ ρ ∗ n k ,n k ( G ) × πe | k − k + G | Im[ (cid:15) − GG ( k − k , ε n, k − ε n k )] , (39)where ˜ ρ n k ,n k ( G ) is the plane wave expansion of the product density P σ u σ ∗ n k ( r ) u σn k ( r ) of theBloch functions with reciprocal lattice vectors G , and (cid:15) − GG ( k − k , ε n, k − ε n k ) is the microscopicdielectric function in a plane wave basis calculated within the random phase approximation.We then utilize the analytical relation of τ ee with dependence on temperature according tothe conventional Fermi-liquid theory since WTe has a considerable density of states at ε F .There, the electron-electron scattering rate grows quadratically away from the Fermi energyand with temperature as τ − ( ε, T ) ≈ D e ¯ h [( ε − ε F ) + ( πk B T ) ] . (40)We obtain τ − by fitting all the self energies in the entire Brillouin zone for all energy bands at298 K, extracting D e and then adding the temperature dependence ( ).22inally, taking into account the impurity scattering, the overall momentum relaxing meanfree path ( l mr ) and momentum conserving mean free path ( l mc ) are estimated by Matthiessensrule, l mr = v F ( τ mreph ) − + ( τ imp ) − (41) l mc = v F ( τ PHee ) − + ( τ Wee ) − , with v F being the Fermi surface averaged velocity and τ imp the impurity scattering time thatdoes not have temperature dependence but varies in different samples. The ab initio calculations were performed with the open source density functional theory (DFT)code JDFTx ( ). First we fully relaxed the T d -WTe (Fig. 1c, main text) using fully relativis-tic Perdew-Burke-Ernzerhof pseudopotentials ( ) and Grimme’s D-2 van der Waals ap-proach ( ). A kinetic cutoff energy of 40 Ha was used along with a × × Gamma-centered k -mesh, and a Fermi-Dirac smearing of 0.01 Ha for the Brillouin zone integration. Both the lat-tice constants and the ion positions were relaxed until the forces on all atoms were less than10 − Ha/Bohr. The relaxed lattice constants were found to be a = 3 . ˚A, b = 6 . ˚A, and c = 13 . ˚A, respectively. To compute the e-ph scattering time, we performed frozen phononcalculations in a × × supercell, and obtained 88 maximally localized Wannier functions(MLWFs) by projecting the plane-wave bandstructure to W d and Te p orbitals, which allowedus to converge the electron scattering calculation on a much finer × × ( × × ) k and q grid for T > ( < )20 K. τ PHee was collected on 56 irreducible q points in the Brillouinzone. For e-e scattering, we reduced the k -mesh to × × due to the computational cost,for which we verified the electronic structure. A dielectric matrix cutoff of 120 eV was used toinclude enough empty states, with a energy resolution of 0.01 eV.23urther, since the electronic structure of T d -WTe is known to be sensitive to lattice strain ( ),the Weyl semimetal phase (WSM) was realized by confining the lattice constants to a = 3 . ˚A, b = 6 . ˚A, and c = 14 . ˚A. Here we employed fully relativistic Perdew-Wang ( ) pseu-dopotentials with the same settings as described earlier to obtain stable phonons. An extensivecomparison between the Relaxed phase and
WSM phase is shown in Sec. 10.
10 Relaxed phase vs. Weyl semimetal phase
As mentioned in the previous section, the WSM phase has ∼ % tensile strain along thestacking ˆ z axis. The phonon modes only show slight softening along the Γ − Z directioncompared to the relaxed structure, without significant difference overall. However, the relaxedphase has much more dispersive electronic structure along Γ − Z direction due to a shorterdistance between the van der Waals layers (Supplementary Fig. 10(a) and (c)). Consequently,the Fermi energy cuts through these bands, leading to a slightly more ‘metallic’ behavior inthe relaxed phase. One may expect leaving a larger hole pocket (Supplementary Fig. 10(b) and(d)). However, when we examine the spatially resolved momentum relaxing electron-phononlifetimes on the Fermi surface in these two structures, the WSM phase features longer-livedelectrons than the relaxed phase.To better understand their hydrodynamic behavior, we performed the same numerical calcu-lation with ab initio τ ee , τ PHee , and τ eph . We found that while τ mr does not show predominant dis-tinction, τ ee is greatly decreased in the WSM phase. As a result, the non-monotonic temperaturedependence of the resulting current density profiles peaks at ∼ , as shown in SupplementaryFig. 11. 24 b)(d)(a)(c) 𝜏 eph (fs) 𝜏 eph (fs) xy Figure 10: Electronic bandstructures and Fermi surfaces for relaxed (a-b) and WSM (c-d) WTe phases. In (b) and (d) we show the electron-phonon lifetimes at 28 K projected onto the Fermisurfaces to highlight the anisotropic feature. 25 mpurityMean Free Path, [ nm ] ∞ [ K ] j y ( x ) C u r v a t u r e Figure 11:
Ab initio current density curvature solutions as a function of temperature for theWeyl semimetal phase. Note, the peak occurs at a lower temperature, and the curvature valuesare significantly higher than those observed experimentally.
11 Discussion on high temperature deviations between pre-dicted behavior and experimental observations
As shown in Main Figs. 3,4, the experimental observations above ∼ deviate significantlyfrom the predicted behavior. While the theoretical predictions admit almost entirely diffusiveprofiles, experimentally we observe curved profiles. In this section, we expand on possiblecauses for this discrepancy, which can be attributed to approximations made by the theory andexperimental uncertainties. First, oxidation effects in the WTe flake result in a non-uniformdistribution of impurities, which in turn suggests a position-dependent mean free path. It isinstructive to consider the extreme case in which such impurities permit no current densityat the edges, effectively reducing the sample width, artificially suggesting enhanced flow inthe center. Allowing for ∼ change in the effective width, for example, accounts for thediscrepancy at
90 K . Similarly, while the theoretical model assumes an infinite two-dimensional26ibbon, the sample has both finite length ( ∼ µ m ) and finite thickness ( ∼
50 nm ). Since theelectron mean free paths (Main Fig. 4a) are of the same order as the sample thickness, this willresult in a transition between ‘Knudsen’ and ‘Poiseuille’ flow along the ˆ z direction, modifyingthe profiles along the ˆ x direction. Finally, since our sample is a thin WTe flake exfoliated on aquartz substrate, the phonon spectrum will likely be modified away from that of the bulk crystal.Furthermore, considering the entire WTe phonon spectrum ( < − ) is contained withinthe quartz phonon spectrum, bulk phonons from the WTe can scatter into the substrate withlittle interface resistance, thus providing another competing scattering mechanism. References and Notes
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