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Nicholas field without magnetic effect Hall geometric nonlinear, A www.pnas.org/cgi/doi/10.1073/pnas.1916406116 eateto hsc,TeUiest fCiao hcg,I 60637; IL Chicago, Chicago, of University The Physics, of Department per cosacretcryn iepae namagnetic potential a transverse in placed a wire that current-carrying discovered a Hall across appears Edwin 1879, n ntecnetoa alefc,we neeti current electric an when effect, Hall conventional the In distributions charge surface that realized Kirchhoff 1850, By is measure and predict we that voltage transverse The c h nioFriIsiue h nvriyo hcg,Ciao L60637 IL Chicago, Chicago, of University The Institute, Fermi Enrico The 2 (9). . mV. ≈0.5 q | asstruha ple antcfield magnetic applied an through passes aleffect Hall a,b,c,1 ai .Schuster I. David , | ufc charge surface | graphene a,b n inyR Nagel R. Sidney and , b h ae rnkIsiue h nvriyo hcg,Ciao L60637; IL Chicago, Chicago, of University The Institute, Franck James The i B, of iei eiiclranlswt ne radius the inner assume with annulus We semicircular radius straightforward. a is is wire potential this Calculating of Theory sign the reveals potential, potential, this Hall of a carriers. polarity of the polarity The the 1B. current to the Fig. of analogous in sign illustrated the on as depends carriers, differ- edges) potential wire the the thus the between (and arc, ence field the electric circular that radial mandates the a it this of centripetally; of direction that carriers shape the such cen- accelerate circle the current the to toward in ter radially the point wire must on force a electric force of For transverse magnetic a wire. component than exert the transverse and follows current—rather a field create the electric charges geometry— the of surface of path The result a forces. the are in that distributions curvature charge surface of to sensitive voltages verse n of direction the along Here doi:10.1073/pnas.1916406116/-/DCSupplemental at online information supporting contains article This 1 is ulse oebr1,2019. 18, November published First BY-NC-ND) (CC 4.0 NoDerivatives License distributed under is article access open This Submission.y Direct PNAS a is article performed This N.B.S. interest.y competing paper.y no research; the declare wrote authors designed S.R.N. The and S.R.N. N.B.S. and and data; analyzed D.I.S., N.B.S. research; N.B.S., contributions: Author cancels charges these by difference: 1A, potential caused Fig. transverse in field a illustrated create electric as edges, the force wire until the Lorentz at magnetic accumulates a Charge experience carriers the owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To sapoeo urn ah nconductors. in paths current of probe the a measurement within is transverse-potential paths nonlinear current this tortuous Thus, voltages to wires. transverse due exhibit origin wires geometric large experimentally of straight as We Even are that millivolts. voltage. wires as graphene Hall curved the in potentials measure observe mag- to a applying field charges, by netic current-carrying obtained traditionally the are potential of that This density properties present. and is sign field the magnetic geomet- reflects no a which is in This curve. effect the ric accel- follow to must radially The field current electric the wire. erate transverse the a across because wire potential arises curved potential electric a transverse through a flowing current produces electric that show We Significance n h ino h hrecarriers. charge the of sign the and oso htmgeimi o eesr o rdcn trans- producing for necessary not is magnetism that show To a,b,c n r out stecag-are est and density charge-carrier the is PNAS h iehsarcaglrcosscino width of cross-section rectangular a has wire the ; | eebr3 2019 3, December esrmn of measurement A B. ∆ n V n h inof sign the and . Hall y raieCmosAttribution-NonCommercial- Commons Creative = | . tnq y iB o.116 vol. https://www.pnas.org/lookup/suppl/ . | q t etk advantage take we , ∆ o 49 no. h iethickness wire the V Hall r in F | determines B 24475–24479 n outer and = q v F × B [1] B. to

PHYSICS    1 m 2 A ≈ 2 2 3 i in limit w  rin [4] rinwt n q    w m 2 ≈ 2 3 hj i in limit w  rin. [5] rin n q

The terms in square brackets depend only on the wire geome- B try. The expression in Eq. 3 is exact for a wire of any width; the approximations in Eqs. 4 and 5 are valid in the narrow- wire limit, w  rin, with hj i = i/wt being the average current density. As in the Hall effect, the potential ∆Vgeom is an odd power of the charge q, which means that the sign of q can be determined. We note that this result, essentially due to momentum con- servation, is equally valid in the diffusive limit where the charge carriers experience collisions that randomize their velocities. C D Momentum conservation still dictates that the electric field must include a radial component as in the Drude model (16),

d hp (t)i hp (t)i = qE − . [6] dt τ In steady state, with a constant azimuthal current, the left side of Eq. 6 must equal the transverse force necessary to keep the current traveling along a circular arc. This leads to Eq. 2. See SI Appendix for details. There are 2 significant differences between this transverse Fig. 1. Surface charge distributions that produce transverse potentials. (A) potential and the Hall effect, however. The first one is that In the classical Hall effect, the directions of current, i, and of the magnetic m field, B, determine the direction of the magnetic force. The surface charges the mass of the carriers, , enters into the expression for the produce a transverse electric field E. (B) In a curved wire without an applied potential difference. The second difference is that ∆Vgeom is magnetic field, the centripetal acceleration of the carriers is due to an elec- quadratic, rather than linear, in the current, i. tric force. Surface charges produce an electric field whose direction reveals the sign of the carriers. (C) Circuit for measurement of transverse poten- Experimental Validation tials due to wire geometry. Orange regions are metal and purple regions Circuit Design. To maximize the signal ∆Vgeom, we need high are exposed graphene. (D) Optical micrograph of the curved graphene wire current i (or current density j ) and low carrier density n. It is in a completed device. Graphene electrodes are visible in center at top and advantageous to use a conductor whose charge-carrier sign and bottom. (Scale bar, 20 µm.) density can be modulated to check whether the signal and carri- ers change sign concurrently. Monolayer graphene satisfies these conditions (17). w ≡ (r − r ) t out in and thickness . We force the radial edges of Due to graphene’s electronic band structure, as the Fermi the half annulus to be equipotentials, as illustrated in Fig. 1B, energy approaches the Dirac point, the effective mass m∗ of E = V − V with potential difference a b . Symmetry requires that the charge carriers vanishes and the carriers behave like rela- the current density vector field is purely azimuthal, such that tivistic Dirac particles (18). The effective mass is given by m∗ = the charge carriers on average follow semicircular trajecto- √ ( /vF ) πn2D, where vF is the Fermi velocity, is the reduced ries. We assume that the conductivity σ and carrier density n ~ ~ Planck constant, and n2D = nt is the 2D carrier density (19, 20). are uniform and that all charge carriers have the same mass By applying a gate voltage to place the Fermi energy far from m q and charge . the Dirac point, we ensure that the charge carriers in our cir- Unless the wire is a superconductor, the current-density mag- cuits have a nonzero effective mass. Additionally, we design our j (r) nitude, , is equal to the longitudinal (azimuthal) component wires to be orders of magnitude longer and wider than the elec- E = E/(πr) j (r) = of the electric field, θ , times the conductivity: tronic mean free path in room-temperature graphene (21), which σE/(πr) E . The transverse (i.e., radial) component, r , provides means that the nonrelativistic, diffusive transport model should the force necessary for the carriers to follow circular paths on be applicable to our experiments. average. Our circuit, illustrated in Fig. 1C, consists of a curved graphene In the curved wire that we consider, the magnitude of the local wire with measurement leads on either side and a straight hvi drift velocity of the carriers is proportional to the current graphene wire as a control. We use graphene grown by chem- density, hvi = j /nq, and the force for keeping these particles in 2 ical vapor deposition and transferred by the manufacturer a circular orbit of radius r is F = mhvi /r = Er q. The radial (Graphenea, grain size ≤20 µm) to a doped silicon wafer with electric field is thus a 300-nm oxide gap. We use photolithography, electron-beam evaporation, and plasma etching to pattern the graphene and dV (r) m [j (r)]2 to place Ti/Au contacts on it, as shown in Fig. 1D. (See Mate- Er (r) = − = 2 3 , [2] dr n q r rials and Methods for a detailed nanofabrication procedure.) We control the sign and density of the carriers in the graphene by where V (r) is the transverse electrostatic potential. This can be applying a back-gate voltage Vbg to the silicon. As initially fab- integrated across the width of the wire, w, to find ∆Vgeom ≡ ricated, the samples are highly doped; we current-anneal (22) Vout − Vin, them to cross the Dirac point at Vbg < 100 V. The fact that the signal is quadratic in the current may be  r −2 − r −2  m  ∆V = in out i 2 [3] exploited to remove several potential sources of measurement geom 2 2 2 3 2 2t (ln (rout/rin)) n q error. The i dependence means that an ac current at frequency

24476 | www.pnas.org/cgi/doi/10.1073/pnas.1916406116 Schade et al. Downloaded by guest on September 25, 2021 Downloaded by guest on September 25, 2021 caee al. et Schade o u sal mle hn hti h uvdscin efind annealed, does We current comparable section. been has curved be that the can in section that straight than, h|∆V smaller the usually in but signal to, the of nitude samples many h ie oee,w tl n inlwoepaei either is phase whose signal of a curvature the find by still out we picked φ not However, is wire. wire, straight the the of the sides for 2 account to small too is it the that in 2ω found harmonic and second the current voltage- measured driving 20 graphene also the We of least leads. length of the measuring at prediction varying by the S3) effect and Seebeck the Fig. observe we Appendix, signal Eq. ( SI the current-induced than be a smaller times should from contribution field Hall-effect magnetic the of nitude density current the though feeto n oepdlsrte hna nfr material uniform a as than rather assortment puddles random with hole a and as behave electron graphene inhomogeneities of the observations point, make Dirac prior defects the with near and consistent samples is graphene in This that than point. Dirac Rather the (23).] near singularity, graphene behav- of a the [This properties of showing geometry. electrical direction only of the in the with using typical shown determines hysteresis carriers as measurement shows charge minimum, this ior the conductance that of confirms the sign This of opposite 2C. that the Fig. has to carriers close the that the of age side find of other charge We the to the sign. level where Fermi point the move Dirac and sample the to h vrg antd ftesga is signal the Eq. a For of Using 2B. magnitude the Fig. average in in the scatter shown significant as voltage is samples driving There between 2A. magnitude Fig. signal in the shown with quadratically as rises current, potential measured the that confirms esrdtases oetast power a expected to is amplitude, potentials what transverse is a measured This on carriers. graphene any charge without in positive samples, the to at of sponding and all In annealing samples. difference, current our in potential wires the curved measured We results. Results our affect not do they that so contribution Seebeck the In 2) and effect. field magnetic current-induced a from voltage u oJuehaiggv iet ogtdnloclain nthe in oscillations longitudinal (See at to resistivity occur rise conductor’s that give the potential heating in Joule Fluctuations to signal. due the to tribute i wire, elec- the longitudinal within the to component due field at drop appear tric potential will The Earth, ignored. the of safely that at as such potential field, at the potentials out measure ing to amplifier lock-in ω ow edntwryaotipretaineto transverse of alignment imperfect that checking about By worry leads. measurement not need we so 2 m 0 = eesr htany that ensure we , h ino h oeta difference, potential the of sign The current the thin, are samples our Because fe urn neln,w a pl akgt voltage back-gate a apply can we annealing, current After w xrnossucso a of sources extraneous Two rdcsatases oeta tfrequency at potential transverse a produces /(n rnvreptnilta eobserve. we that potential transverse that Appendix) (SI checked We 3. n IAppendix SI ∆V straight or ≈ 2 t geom 0 2 φ IAppendix SI q 1,24–26). (15, = 3 |∆ i≈ |i 3 ) ,a hw nFg 2D. Fig. in shown as ),

π V h∆ n vrgn vrtesmls emeasure we samples, the over averaging and ≈ h vrg inlover signal average The 2B. Fig. in shown as , E geom 0.35 5 o details.) for V = × straight SiO ∆ .0V h urn is current the V, 1.00 ∝ | h|∆ 10 ω V edsrb o ehv iiie their minimized have we how describe we 2ω alptnildet cmagnetic dc a to due potential Hall A . 3ω 2 −6 ∆V i geom V β usrt 1,1,2) efitdthe fitted We 23). 17, (15, substrate ≈ i n efind we and , geom amnc ntecretd o con- not do current the in harmonics n ihrhrois u o at not but harmonics, higher and kg V geom j bg hne ina akgt volt- back-gate a at sign changes u nasnl ape h mag- the sample, single a in but 0, m svr ag.Teeoe h mag- the Therefore, large. very is sw sweep we As |i. 0 = 4 2ω ∆ ·C asssotl hog zero through smoothly passes V , −3 inlaedet )teHall the 1) to due are signal straight ∆V ∆ . E ∆ V ∆ θ geom hβ h∆V ieieocr at occurs likewise , geom V V i ∆ i straight ≈ hne in(utas (just sign changes straight 2 = V 370 geom spstv,corre- positive, is ω β spootoa to proportional is geom V V .00 2ω i n hscnbe can thus and bg ftecurrent the of bg 2ω ± ssaleven small is ewe the between , ≈ i sntdeto due not is costhe across , ± we,a is as sweep, hl filter- while 130 nasample a in euea use We . 0. .6mV. 0.46 This 11. and µA V 2ω ω bg . , es otg fesa diinltcnqefrsuyn such studying for technique trans- additional nonlinear con- The an heterogeneities. to path. offers necessary heterogeneous voltage are any verse distributions to Just charge not. currents are wires, fine paths curved current for internal as the straight, are themselves and signal mechanics observed elementary from The conductor prediction effect. the electrodynamics. a geometric bend not with purely instead edges consistent do a but is we wire create carriers analog, the the to a geometric of itself on inside the paths In accumulate carriers the current. bend charge charges the of that to paths transverse so the wire effect, curves Hall current- straight classical field a the In magnetic across alone. a geometry voltage to transverse due wire a carrying demonstrated have We Conclusion nooeete.Temaueeto transverse resistivity a as of ∆ themselves measurement manifest The polycrys- detailed would inhomogeneities. are a they features that for these enough (34) large talline, tensor are electrical wires conductivity our Graphene’s a Since orientation analysis. require 32). lattice they (31, the and on strains pud- (33), depending charge local anisotropic, to are and as uni- properties well 26) ascribed not as been (15, are (30), has boundaries graphene dles this grain as graphene at scattering such In to wires. conductors the 2D throughout in form and 29) (28, Eq. width, of in value wire width large smaller the observed the by for given account for can not value are smaller paths even w current an the give effective if would the However, Using carriers (17). the level of doping mass our for charge expected with value density electron the the carrier of a mass suggest bare the be larger to the of width, magnitude measured the smaller the the current, signal. den- of the current more total be the will As fill in fixed to quadratic (24). for is expands edges signal sity, it the the wire, Because along the width. wire’s and down flows contacts current metal the landscapes, the potential at electrostatic tran- on including irregular graphene impurities revealed of surface mapping has to Photocurrent sistors studies due 27). poor with (15, be graphene consistent doping may the is contact inhomogeneous that This introduce and wire. contacts (25) the metal down that fans moves Ti/Au showing then the current it the from as points; injected localized out is at current graphene the the into that leads path assume we meandering wire, a the taking current the to wire. straight the due the along in thus observed current section. is signal the curved The that sections is homogeneous. the behavior not in this to are for that paths place reason to, the magnitude from that comparable its conclude varies and but We samples voltage than, different this smaller in and is of wire straight sign the a along The across place signal wire. significant a of is section there that unexpected is It Discussion oeta rp ynal reso antd rmna the near from magnitude of orders 2F 2 where Fig. leads, in nearly shown by as drops drain and potential source metal of the over magnitude with average contact the with We that 2E. off find Fig. falls and in samples illustrated long-wire it, 4 across electrodes of u ahrb h eeoeet ftecnutvt,te we then conductivity, the of heterogeneity the by rather but , V eosresgasee nsrih ie.Atog h wires the Although wires. straight in even signals observe We hr scnieal vdneta urnsi hnmtlfilms metal thin in currents that evidence considerable is There hsitrrtto loofr nepaainfrtelarge the for explanation an offers also interpretation This of decrease the rationalize To emeasured We straight sa lgn rb ftetruu urn path. current tortuous the of probe elegant an is , d d ≈ h itnefo hr h rpeemakes graphene the where from distance the , PNAS ∆ 10 V ,t h etr where center, the to µm, straight 3. | eebr3 2019 3, December h∆ n 2D V rmaln tagtwr ih8pairs 8 with wire straight long a from geom ≈ 10 ftems nEq. in mass the If i. ∆ 12 V cm straight | −2 o.116 vol. h∆ hc sone-tenth is which , d rmteed of ends the from V ≈ geom | q 500 = o 49 no. i e ∆ µm. 2ω u data our , yuiga using by 3 V staken is | straight signal, The . 24477 n 2D .

PHYSICS A B

C D

E F

Fig. 2. Measured transverse potentials in graphene wires. (A) Signal across curved wire versus current. Colors correspond to different samples, each with rin= 10 µm, rout= 100 µm, and graphene measurement leads of length L ≥ 90 µm. Black line, average of power-law fits to individual data sets, has slope = 2.0. (B) Signals from curved (blue) and straight (red) wires in 34 samples at an average current hirmsi ≈ 370 µA. The radius and angle represent the magnitude and phase of the measurement. Error bars are smaller than data markers. (C) Signal across a curved wire (blue) and circuit’s conductance (black) versus Vbg after current annealing. (D) Measurements from curved wires when the Fermi level is below (Left) or above (Right) the Dirac point, controlled by changing Vbg after current annealing. Symbols correspond to different samples. For a curved wire phase φ = 0 indicates positive charge carriers. (E) Long-wire design with 8 pairs of measurement leads. (F) Signal versus distance from the metal edges at either end of the straight wire, averaged across back-gate voltage sweeps from −100 V to +100 V in 4 samples.

An important difference between the classical Hall effect and in those experiments and in turn those effects, if present, could the geometric effect that we have demonstrated is that the lat- masquerade as a geometric effect. ter depends on the mass of the charge carriers. If the relevant Low-temperature quantum Hall effects arise due to time- mass in Eq. 3 is the effective mass, then this transverse potential reversal symmetry breaking in a magnetic field. In the presence represents a simple way of measuring the effective mass in wires of quantum interactions, the magnetic Hall effect becomes par- where current path tortuosity is negligible. Alternative methods ticularly remarkable; it would be interesting to consider whether of measuring the effective mass of charge carriers in graphene any striking quantum effects could be observed at very low are nontrivial (19, 20). temperatures due to wire geometry alone. Recent work (9) has found a nonlinear Hall effect due to an induced Berry curvature (35) in bilayer conductors. Such an Materials and Methods effect is not expected to occur in a single-layer material such as Circuit Fabrication. We begin with CVD graphene that has already been graphene and is not predicted to depend on the wire curvature. transferred by the manufacturer (Graphenea) to a doped silicon wafer with Our purely geometric effect can contribute to the signals found a 300-nm SiO2 gap and diced to 10-mm × 10-mm chips in a class 1,000

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Kim, P. Stormer, L. H. Tan, Y.-W. Zhang, Y. 19. xeietlRsuc NFEC-520) oeo h S’ National NSF’s the of node Infrastructure. a Nanotechnology Coordinated Hybrid of ECCS-1542205), Nanotechnology and University (NSF Nanofabrica- Soft The Resource Pritzker the at from Experimental Engineering the support Molecular of receives for which use Institute Chicago, made the of work Facility This tion thank DMR-1404841. DMR-1420709 dis- We Program NSF helpful (NSF) Centers for assistance. and Foundation Koolstra, Engineering Son Science and T. G. technical National Science D. the Research as by for and Materials supported well Dean, Tang was R. as work F. C. This Poddar) impor- cussions. Panagopoulos, P. and C. gave and Littlewood, Chakram, who Lee, B. P. S. Huang J.-U. Lujie Earnest, Lee, to K.-H. N. Park’s Park, and Jiwoong thank (J. project We graphene. group this with nanofabrication of about advice stages tant early the on ACKNOWLEDGMENTS. in and text main the in present are Availability. Data Power Bipolar Instru- 6827A HP (National an using Accessory voltage Supply/Amplifier. back-gate Connector the Research Shielded control (Stanford We BNC-2110 ments). Preamplifier a Current Low-Noise and wires, SR570 Systems) graphene Model the frequencies of a (Stanford resistance at the use amplifier measure To we measurements lock-in Hz. exposed 250 potential SR830 and are Hz an transverse 10 samples between use for the We Systems) temperature. where Research room station at probe air a to using other circuits the V.graphene access +100 can than Measurements. less we voltage so Electrical back-gate V, a +100 using by to typi- point the 0 Dirac and done, of the minimum h is of range side conductance this 2.5 the Once graphene within least air. the falls at to typically to for exposed corresponding is device voltage sample the back-gate the of while drain overnight, by and cally samples source the applying the (22) by between anneal point Dirac current of the can current of a we side injecting However, other impracti- is the voltage. it to back-gate conditions level a these Fermi Under the point. move Dirac to the cal from far is level Fermi is fabrication once Annealing. Current vacuum low We under aligned. storage well and for intact chips complete. are the regions seal graphene and vacuum metal the that dry sure and alcohol isopropyl with sample the nitrogen. rinse with we it Afterward, h. 6 least at eisetec eieo h hpudra pia irsoet make to microscope optical an under chip the on device each inspect We alefc n er’ hs ngraphene. in phase Berry’s and effect Hall Phys. graphene. of measurements nanoribbons. engineering. strain by graphene in effect Lett. Rev. Phys. graphene. talline thickness. nm 7 gold to polycrystalline down nanobridges of conductivities thermal and electrical the nanofilms. on scattering ary semiconductors. graphene. in inhomogeneity point. neutrality (2009). the 1071 near graphene of properties devices. graphene in effects transistors. Lett. Science bias. current 653(2007). 163513 91, 9920 (2010). 1959–2007 82, hbio-eHa siltoso igelyrgahn ne dc under graphene layer single a of oscillations Haas Shubnikov-de al., et 1–1 (2013). 614–617 342, nuneo ea otcsadcag nooeet ntransport on inhomogeneity charge and contacts metal of Influence al., et n-iesoa lcrclcnatt w-iesoa material. two-dimensional a to contact electrical One-dimensional al., et hs e.B Rev. Phys. C Nano ACS alrn lcrcltasotars ri onaisi polycrys- in boundaries grain across transport electrical Tailoring al., et lcrcladtemlcnuto naoi ae deposition layer atomic in conduction thermal and Electrical al., et hs e.B Rev. Phys. .Ce.Phys. Chem. J. 481(2009). 046801 103, a.Mater. Nat. l aanee oeaut h ocuin nti paper this in conclusions the evaluate to needed data All Science PNAS siiilyfbiae,tesmlsaehgl oe;the doped; highly are samples the fabricated, initially As i 2172 (2010). 7221–7228 4, ≈ 319(2006). 134109 74, eaepriual rtflt Kh to grateful particularly are We 149(2011). 115429 84, 1314 (2012). 1143–1146 336, | A(hj mA 3 epromeetia esrmnso the on measurements electrical perform We a.Nanotechnol. Nat. 971(2008). 194701 128, 1510 (2015). 1195–1205 14, hs e.Lett. Rev. Phys. eebr3 2019 3, December a.Phys. Nat. ≈ i IAppendix SI aoLett. Nano 2–2 (2009). 722–726 5, 10 a.Phys. Nat. Nature 7 987(2009). 096807 102, A/cm 8–9 (2008). 486–490 3, 8–8 (2012). 683–686 12, 0–0 (2005). 201–204 438, | 2 oi tt Commun. State Solid .T oti,w pl 00V 10.0 apply we this, do To ). . 03 (2010). 30–33 6, o.116 vol. a- ´ | T I ˆ o 49 no. h worked who o ˆ pl Phys. Appl. 1068– 149, e.Mod. Rev. | 24479

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