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High-resolution tunnelling spectroscopy of a graphene quartet

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Electrons in a single sheet of graphene behave quite differently from those in traditional two-dimensional electron systems. Like massless relativistic particles, they have linear dispersion and chiral eigenstates. Furthermore, two sets of electrons
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  LETTERS High-resolution tunnelling spectroscopy of agraphene quartet YoungJaeSong 1,2 ,AlexanderF.Otte 1,2 ,YoungKuk 3 ,YikeHu 4 ,DavidB.Torrance 4 ,PhillipN.First 4 ,WaltA.deHeer 4 ,Hongki Min 1,2 , Shaffique Adam 1 , Mark D. Stiles 1 , Allan H. MacDonald 5 & Joseph A. Stroscio 1 Electrons in a single sheet of graphene behave quite differently from those in traditional two-dimensional electron systems. Likemasslessrelativisticparticles,theyhavelineardispersionandchiraleigenstates. Furthermore, two sets of electrons centred at differentpointsinreciprocalspace(‘valleys’)havethisdispersion,givingriseto valley degeneracy. The symmetry between valleys, together withspin symmetry, leads to a fourfold quartet degeneracy of theLandau levels, observed as peaks in the density of states producedby an applied magnetic field. Recent electron transport measure-ments have observed the lifting of the fourfold degeneracy in very large applied magnetic fields, separating the quartet into integer 1–4 and, more recently, fractional 5,6 levels. The exact nature of thebroken-symmetrystatesthatformwithintheLandaulevelsandliftthese degeneracies is unclear at present and is a topic of intensetheoretical debate 7–11 . Here we study the detailed features of thefour quantum states that make up a degenerate graphene Landaulevel. We use high-resolution scanning tunnelling spectroscopy attemperatures as low as 10mKin an applied magnetic field tostudy thetoplayerofmultilayerepitaxialgraphene.WhentheFermilevellies inside the fourfold Landau manifold, significant electron cor-relationeffectsresultinanenhancedvalleysplittingforevenfilling factors, and an enhanced electron spin splitting for odd filling factors. Most unexpectedly, we observe states with Landau levelfilling factors of 7/2, 9/2 and 11/2, suggestive of new many-body states in graphene. Whenmatterisputunderextremeconditions,itoftenexhibitsnew quantum phases; examples include superconductivity  12 , the integerquantum Hall effect 13 and the fractional quantum Hall effect 14 . Thehistory of fractional quantum Hall states in semiconductor hetero- junctions suggests that studying graphene at lower temperatures andhigher magnetic fields will also reveal new quantum phases of matter.The phases of two-dimensional electron systems (2DESs) in semi-conductor quantum wells are most commonly probed by electrontransport measurements. However, tunnelling spectroscopy, whichhas long been recognized as a powerful probe of 2DESs, has beendifficult to apply to the buried 2DESs in semiconductor quantumwells. Fortunately, graphene 2DESs are located on the surface of thematerial, and quantum Hall physics can be studied now that instru-mentsareavailablethatworkatthenecessarytemperaturesandfields.Recentscanningtunnellingspectroscopymeasurementsinanexternalmagnetic field,  B  , have verified the expected magnetic quantizationrelation in epitaxial graphene layers grown on SiC (ref. 15), withLandau level energies given by   E  N  ~ sgn ( N  ) c  1  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 e  B B  j N  j p   , where N  5 0, 6 1, 6 2, … is the orbital quantum index,  c  *  is the graphenedispersion velocity,  e   is the elementary charge and  B  is Planck’s con-stant divided by 2 p . In the current work, we achieve much higherenergyresolutionwithascanningprobemicroscopesystemoperatingat temperatures as low as  T  5 10mK in magnetic fields up to 15T(Methods). These temperatures yield a spectroscopic resolution( , 3 k  B T  , where  k  B  is the Boltzmann constant) that is orders of mag-nitude higher than previous measurements of graphene by scanningtunnelling spectroscopy at 4K (refs 15, 16). Recent spectroscopicmeasurements of the density of states in 2DESs 17 , which have similarresolution,haveobservedelectroncorrelationeffectsinquantumwellsystems. The present measurements resolve spectroscopic featureswhich suggest that new many-body quantum states also occur underthese extreme conditions in epitaxial graphene.The growth of epitaxial graphene (Fig. 1, left inset) on the carbonface of SiC results in a multilayer film with rotational misalignment 1 Center for Nanoscale Science and Technology, NIST, Gaithersburg, Maryland 20899, USA.  2 Maryland NanoCenter, University of Maryland, College Park, Maryland 20742, USA. 3 Department of Physics and Astronomy, Seoul National University, Seoul, 151-7474, South Korea.  4 School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA. 5 Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA. 28    d    I    /   d    V    (  n   S   ) Sample bias (mV)24207654 Linear fit B  = 2 T B  = 3 T B  = 4 T B  = 0 T B  = 2 T B  = 3 T B  = 4 T N   = 0 N   = 13(  NB  ) 1/2 210200150100500250    E    N   –    E    0    (  m  e   V   ) 1612843201000–100–200200 1 nm Figure 1  |  Landau level spectroscopy of epitaxial graphene on SiC. Tunnelling spectroscopy of the Landau level states in a plot of differentialconductivity versus sample bias. Tunnelling parameters are as follows: set-point current, 400pA; sample bias, 2 300mV; modulation voltage, 1mV.Left inset, high-resolution scanning tunnelling microscopy image of thegraphene honeycomb lattice. Tunnelling parameters: set-point current,100pA; sample bias, 2 250mV;  T  5 13mK. Right inset, Landau level peak energy position (relative to the  N  5 0 level) versus the square root of   NB .Excellent scaling is observed in the linear relationship, yielding a carrier velocity of (1.08 6 0.03) 3 10 6 ms 2 1 (1 s ). Vol 467 | 9 September 2010 | doi:10.1038/nature09330 185 Macmillan Publishers Limited. All rights reserved  ©2010  betweenthelayers 18,19 ,leadingtomoire´ interference patternsinscan-ning tunnelling microscopy images (Supplementary Fig. 1) 15,20 . Thevisible moire´ unit-cell dimension determines the rotational anglebetween the top two layers 20 . We measure a unit-cell length of 5.7nm (Supplementary Fig. 1) and, hence, a rotation angle of 2.3 u .The small-angle rotation reduces the interlayer coupling, resulting inlayers that have electrical properties nearly identical to those of asingle graphene layer 15,18 . This single-layer-like behaviour is con-firmed by the orbital Landau quantization measured by scanningtunnelling spectroscopy (Fig. 1), which shows excellent single-layer-graphene scaling with carrier velocities of  , 10 6 ms 2 1 (Fig. 1,right inset). In Fig. 1, the  N  5 0 Landau level ( E  0 ) is located at 2 134mV, which is close to the zero-field Dirac point ( E  D ) of  2 125 mV corresponding to an  n  -type doping of 1 3 10 12 cm 2 2 (SupplementaryFig.2).Wenotethatthemagneticoscillationintensity drops smoothly with energy difference from the Fermi energy. Thisenvelopemayberelatedtoquasiparticlelifetimes,whichdecreasewiththe energy difference from  E  F . We do not observe any Landau levelswith negative orbital index, presumably as a result of this envelope.We first concentrate on the field dependence of the Landau levels,using measurements made over a large energy range. Figure 2 showsthe evolution of the series of Landau levels as a function of magneticfield.Eachpaneloftheindicatedfieldconsistsof21d I/  d V  (differentialconductance) spectra acquired at different spatial points along a line(thesameforeachpanel)40nmlong.Thehorizontalaxisforallpanelsis the sample bias voltage, and the vertical axis in each panel indicatesthe position along the line where the spectrum was acquired; theresulting d I/  d V   intensity is shown in a colour scale. In this way, theLandau levels appear as a spectral ‘fingerprint’, showing the repro-ducibility of the spectra and their evolution with magnetic field. Inparticular,thelower- N  Landaulevels(forexamplethosewith N  5 0,1and2,respectivelyLL 0 ,LL 1 andLL 2 )areeasilyseenandcanbetrackedvisually as the magnetic field is varied (Fig. 2). In addition, there are anumberofnon-Landau-levelrelatedpeakswhenLL 1 crosses E  F atzerosample bias, between 11 and 14T. These additional states (whichappearingroupsoffour)varywithspatialpositionandmagneticfieldmuch more than the Landau levels themselves. Other sets of thesepeaks are seen weakly when LL 2  crosses  E  F , at 5T. Analysis of thesenon-Landau-level peaks is beyond the scope of this work.We can make several observations from these spectral prints. ThepositionofLL 0 inFig.2showsreproduciblestepsinitsenergypositionas a function of magnetic field, in particular between 7 and 8T andbetween 11 and 12T. For an isolated graphene sheet, the position of the  N  5 0 Landau level should be fixed at the Dirac charge-neutrality point,  E  D , and should not depend on the magnetic field. However, asseen in Fig. 2, the position of the LL 0  peak shifts to lower energy withincreasing magnetic field.This shift coincides with a shift of the otherLandaulevelsaswell,suchthatthespectrapreservethescalingrelationwith magnetic field as indicated in Fig. 1. The density of electrons inthe top layer is easily calculable when all Landau levels are filled orempty:  n  5 n B  / W 0 , where  n  is the filling factor (where we count twolevelsfor N  5 0 plus four eachfor N  . 0) and W 0 is theflux quantum.Figure3ashowsthatthisdensityvariesbyafactorofalmosttwoasthefield is swept. This variation indicates that charge is transferredbetween the top layer and layers below. Such transfers arise becausethe magnetic field changes the densities of states in the carbon sheetsbelowthetopmostlayer,inturnchangingtheenergybalancebetweenthe single-particle energies and the electrostatic potential. This com-petition apparently drives the charge flow between the layers to keepthe Landau levels in the top layer either filled or empty. Even thoughthe charge density in the top layer is not fixed, there is still a tendency for the filling factor to decrease with field.A close examination of the  N  5 0 to  N  5 2 Landau levels reveals aremarkablepatternofsplittings(Fig.2).ThesplittingsofLL 1 andLL 2 areclearatlowerfields,whereastheLL 0 splittingismorevisibleabove12T because of the broadening of this level at the higher energy. Weattributethelargestsplittingsweobservetobrokenvalleydegeneracy,by examining LL 1  in high-resolution measurements. Each Landaulevel splits into a quartet at higher applied fields (Fig. 4). Figure 4ashows a series of d I/  d V   line scans, obtained in the same spatial loca-tions as in Fig. 2 but with ten times the energy resolution, whichrevealreorganizationsoftheLL 1 densityofstatesasitcrossesthroughthe Fermi level, at magnetic fields between 11 and 12T. Figure 4bshows representative line spectra for each magnetic field extractedfrom Fig. 4a. Starting with the lowest panel in Fig. 4a (11T), the fourpeaks (bright lines) correspond to the fourfold manifold of LL 1 . Thefour separate peaks in Fig. 4b at 11T reveal a LL 1  quartet in whichboth the spin and the valley degeneracies are lifted. The schematicinset in Figure 5a shows the energy level structure of the  N  5 1Landau level. The electron spin-up and spin-down levels are indi-cated by the arrows. Their assignment is confirmed by more detailedmeasurements as a function of field, described below.Thespectrumat11T(thebottompanelinFig.4a,b)showsallfourpeaks clearly below   E  F , indicating a filling factor of   n 5 6 (two filledlevels for  N  5 0 plus four for  N  5 1). From the third panel from thebottom, at 11.25T, we see that the Fermi level is located between therightmost two spin-split levels of LL 1,  leading to a polarized state at afilling factor of   n 5 5 and a greatly enhanced spin splitting due to astrong exchange interaction. At the lower magnetic field of 11.125T(secondpanelfromthebottom),anewpeakappearsinLL 1 ,makingathreefold Landau level spin submanifold. An examination of thespectrashowsthattheintensitiesoftherightmosttwopeaksoneitherside of   E  F  are half the intensity of the third spin-up peak of thatsubmanifold. Thus, the srcinal spin-down feature splits into twospin-down peaks with half the intensity on either side of   E  F . Thisstate corresponds to a new stable quantum state with a top-layerfilling factor of   n 5 11/2. Progressing to higher fields in the seriesshown in Fig. 4a, b, we see another stable half-filled state at  n 5 9/2 BN   = 0100040  D    (  nm )   500Sample bias (mV)–50–100–150–200 N   = 1 N   = 21 T2 T3 T4 T5 T6 T10 T9 T8 T7 T11 T12 T13 T14 T–1 nS12 nS Figure 2  |  Landau levels of epitaxial graphene on SiC as a function ofmagnetic field.  A series of d I  /d V   line scans, taken vertically through themoire´ region in Supplementary Fig. 1, as a function of magnetic field. Eachpanel shows the d I  /d V   intensity in a colour scale (from 2 1 to 12nS). The vertical axis within each panel is distance, D , from 0 to 40nm. A splitting of the  N  5 0, 1 and 2 Landau levels can been seen in different field ranges.Tunnelling parameters: set-point current, 200pA; sample bias, 2 250mV;modulation voltage, 250 m V;  T  5 13mK. LETTERS  NATURE | Vol 467 | 9 September 2010 186 Macmillan Publishers Limited. All rights reserved  ©2010   just before both spin-split states empty and move above  E  F  at11.625T and  n 5 4. Finally, a third half-filled state is observed athigher fields (14T), when  E  F  lies inside the  K   valley (leftmost peaks),at a filling factor of   n 5 7/2 (Fig. 4c). An examination of fine fieldadjustments shows that these half-filled quantum states are the only stable states that are observed between the  n 5 4 and  n 5 6 states(Fig. 3b, c). Figure 3b shows the integrated intensity of the d I/  d V  spectra forthefilledstatesofthe K  9 valley(rightmost peaks)fromthemidpoint of the  N  5 1 Landau level to the Fermi level (zero bias). Aseach Landau level empties, the integrated intensity jumps down to anew plateau that is stable over a finite field range. The size of each jump in Fig. 3b corresponds to half the intensity of a single Landaulevel, indicating that these are stable half-filled states. Normalizingthe integrated intensity to a single Landau level (that is, the value at n 5 5) yields filling factors of 5.46 6 0.06 and 4.54 6 0.04 (error, 1 s )between  n 5 6 and 5 and, respectively,  n 5 5 and 4. The energy sepa-ration between the half-filled Landau levels ranges from 3meV at n 5 11/2 to about 5.5meV at  n 5 7/2, at 14T.Earlier transport measurements 1,2 at filling factors of 0 and  6 1showed the lifting of the level degeneracies and were initially inter-pretedasdemonstratingtheliftingofthespin degeneracy followedby the lifting of valley degeneracy for LL 0 . States at a filling factor of  6 4were also observed 1 and tentatively ascribed to the lifting of the spindegeneracy for LL 1 . In Fig. 5, we plot the energy separations betweenspectralpeakpositionsintheLL 1 manifoldinFig.4todeterminetheirsrcin.Weidentifythelargerenergysplittingasarisingfromthelifting a 3.5    D  e  n  s   i   t  y ,   n    (   1   0    1   2    c  m   –   2    ) 3.02.52.01.5 121086420 1 4    υ   = 7  υ   = 6  υ   = 5  υ   = 4  υ   = 3 b    I  n   t  e  g  r  a   l  o   f   fi   l   l  e   d  s   t  a   t  e  s   (  n   S  m   V   ) 20151050  υ   = 4  υ   = 9/2  υ   = 5  υ   = 11/2  υ   = 6Magnetic field, B  (T)11.7511.5011.2511.0012.00 c    D  e  n  s   i   t  y ,   n    (   1   0    1   2    c  m   –   2    ) 3.02.53.52.0  υ   = 4  υ   = 9/2  υ   = 5  υ   = 11/2  υ   = 611.7511.5011.2511.0012.00 Figure 3  |  Electron density and filling-factor variation as a function ofmagnetic field in epitaxial graphene. a , The electron density,  n 5 n B / W 0 ,determinedfromthefilling ofthe Landaulevels (redsymbols),asa functionof  B ,where n isthefillingfactorand W 0 isthefluxquantum.Thedashedbluelinescorrespondtodensitiesat constantfillingfactors rangingfrom n 5 3 to14. b ,Theintegralofthed I  /d V  spectraofthefilled K  9  valley(rightmostpeaksin Fig. 4), from the middle of the  N  5 1 Landau level to zero sample bias, E  F ,as a function of field (Fig. 4). The plateaux correspond to stable filling factors. c ,Electrondensityversus B intheregionaround11.5T,showingthetransitions between the half-filled states at  n 5 9/2 and  n 5 11/2 between n 5 4 and 5 and,respectively, n 5 5 and 6. Fillingfactors weredeterminedby takingtheratiooftheintegratedareasin b anddividingbytheareaofasingleLandaulevel(areaat n 5 5)andaddingfour(twofor  N  5 0andtwoforthe K   valley (leftmost peaks in Fig. 4)). The calculated filling factors are5.46 6 0.06, 4.54 6 0.04 and 3.52 6 0.05 (1 s ). Error bars, 1 s . Sample bias (mV) E  F 1050040    d    I    /   d    V    (  n   S   )    D     (  n  m   ) 20–5–10–151511.875 T  ν  = 411.75 T  ν  = 411.625 T  ν  = 411.5 T  ν  = 9/211.375 T  ν  = 511.25 T  ν  = 511.125 T  ν  = 11/211 T  ν  = 612 T  ν  = 4Half-filled LLHalf-filled LL E  F E  F Δ E  SF Δ E  SF Δ E   V B  = 14 T B  = 11.75 T B  = 11.5 T B  = 11.375 T B  = 11.25 T B  = 11.125 T B  = 11 T abc –1 nS10 nS159630–6–303691210501510501050105096309630–3–6–9–12–15530 Half-filled LLHalf-filled LLHalf-filled LL  ν  = 7/2  ν  = 4  ν  = 9/2  ν  = 5  ν  = 5  ν  = 11/2  ν  = 6 Δ E  SE Δ E  SL  Δ E  SR Figure 4  |  High-resolution Landau levelspectroscopyofthefourfoldstatesthatmakeupthe N 5 1 Landau level. a , A series of d I  /d V   linescans focusing on the Fermi level region of the  N  5 1Landaulevel(LL),madeinthesamespatiallocation as in Fig. 2. At 11.125 and 11.5T, new stable half-filled Landau levels appear at filling factors of 11/2 and 9/2.  b , Single d I  /d V   spectrafrom the middle regions of the panels in  a  for theindicated magnetic fields. The level separationenergies are defined in the various panels.  D E  V ,the lifting of the valley degeneracy (blue and redlines), is measured from the centres of the twospin-split states (see spectrum with B 5 11.75T).The yellow lines indicate the position of theFermi level at zero sample bias. We define threeenergy separations for the spin split peaks:  D E  SL and  D E  SR   for the left and right spin-split levels,respectively ( B 5 11T);  D E  SE  measures theenhancedspin splittingwhen the Fermi levelfallsbetween the spin-split levels ( B 5 11.25T); and D E  SF  measures the separation between the twohalf-filled Landau levels.  c , Tunnelling d I  /d V  spectrum at a filling factor of 7/2 showing asimilar 1/2-fractional state when E  F  is positionedbetweenthelefttwospinsplitstates( K   valley)atahigher magnetic field of 14 T. Note change inhorizontal scale and position of   E  F .Tunnelling parameters: set-point current, 200pA; samplebias, 2 250mV, modulation voltage, 50 m V; T  5 13mK. NATURE | Vol 467 | 9 September 2010  LETTERS 187 Macmillan Publishers Limited. All rights reserved  ©2010  of thevalleydegeneracies because the g  -factors of thesmallersplittingarecloseto2,asexpectedforelectronspins.Thelargerenergysplittingobserved in LL 1  is also observed in LL 0  and LL 2  (Fig. 2) for a limitedrange of magnetic fields. For LL 1 , the energy splitting is about tentimes the value of the Zeeman energy for electron spin ( g  m B B  , where m B  is the Bohr magneton). A fit of the linear portion of the valley energy splitting yields an effective  g  -factor of   g  V (LL 1 ) 5 18.4 6 0.4(1 s ). The valley splitting is enhanced in LL 1  as  E  F  becomes centredthere(Fig.4)atafillingfactorof  n 5 4,whichgivesrisetoapeakinthesplitting energy as a function of field (Fig. 5a). Figure 5b shows thevarious energy separations between the spin-up and spin-downLandaulevelsofthesamevalleypolarization.Theseparationsbetweenthe polarized levels of the left and right valleys are shown in red andblue symbols, respectively, and differ slightly, by  , 5%. A linear fit yields  g  -factors of   g  SL 5 2.36 6 0.01 and  g  SR  5 2.23 6 0.01 (1 s ). Anenhancement is observed in the spin-split states when they becomepolarized at odd filling ( n 5 5 in Fig. 4 and orange symbols Fig. 5b).This enhancement is a factor of three greater than normal Zeemansplittingandisdrivenbytheexchangeinteractionofthefullypolarizedspin level, similar to exchange enhancements observed in silicon2DESs 21 and later investigated in GaAs two-dimensional systems 22,23 .Forgraphene,weexpectadifferentenhancementbecausethescreeningand dielectric propertiesare significantly differentfrom thosein semi-conductor 2DESsIn graphene systems with weak disorder, interaction-driven gapsare expected 7 to lift Landau level degeneracies. Interaction physics isexpected to dominate in this epitaxial graphene, where, on the basisof the observed Landau level linewidths, which are of the order of 0.5meV, the sample is in the weak-disorder limit. Although we donot have a complete explanation for all the effects we observe, par-ticularly those at fractional top-layer filling factors, we highlightsome considerations that are likely to be important.The LL 1  valley splitting varies nonlinearly with field above 7T. Webelieve that the valley splitting is due in part to single-electron inter-layer coupling effects that are sensitive to the energies of submergedlevels, and in part to enhancement by interactions. The valley split-ting has a sharp peak at 12T, when  E  F  is exactly centred in LL 1 ,corresponding to  n 5 4. This enhancement suggests that electroncorrelations enhance the valley splitting at  n 5 4.Overthefieldrangebetween10and14T,the N  5 1Landaulevelof the top layer progressively empties. The filling factor is not inversely proportional to field, but decreases in a sequence of jumps corres-ponding to first-order phase transitions between especially stablestates (Fig. 3b, c). In particular, we observe unexpected new states inwhich the top-layer, spin- and valley-split Landau level at the Fermienergy splitsfurtherinto separate occupied and empty contributions.Viewing the tunnelling spectra as a correlated-state fingerprint, wedetermine that the state with these spectral features is stable over afinite range of applied field, giving way to states with even and oddfilling factors at the extremes of its stability range (Figs 3c and 4).Because the integrated intensity of the two smaller spectral featuresishalftheintensityofthemajorpeak(Fig.3b),thenewquantumstatesoccur with top-layer filling factors  n 5 11/2, 9/2 and 7/2. The abruptfield dependence of the density of states signals a first-order phasetransition between electronic states with qualitatively different elec-tronic correlations. Interaction-induced gaps within a Landau level,such as the ones we have observed, frequently signal interesting new physics. An example is the fractional quantum Hall effect, which hasrecently been observedin high-mobilitysuspended graphenesamplesat  n  5 1/3 (refs 5, 6). Our experimental observations suggest thatelectron–electron interactions are responsible for especially stablestates of   N  5 1 graphene electrons with top-layer filling factors closeto odd integers and half-odd integers. Although further experimentsmaybe necessary to clarify thephysics of the N  5 1 half-filledLandaulevel states, the following considerations provide some guidance.Thetoplayerthatweprobeisincloseproximitytofiveburiedlayersthatinfluencetop-layerphysicsnotonlybyservingasachargereservoirbut also by screening interactions between top-layer electrons. Thescreening by the buried layers can be described crudely by modellingthem as an effective metallic screening layer that is separated from thetop layer by a distance  d  eff  , which should be smaller than the totalthickness of the carbon layer, that is, less than , 2nm. Because  d  eff   issmaller than the magnetic length,  l  B  < 10nm, the Coulomb interac-tions of electrons in the top layer are screened at distances larger than d  eff  ,givinganeffectiveshort-rangeinteractionpotential V  0 5 4 p e  2 d  eff  / e ,where  e  accounts crudely for top-layer screening effects due to bothinter-Landau-leveland s -bondpolarizationanddielectricscreeningby  5 11/29/26 6543210 7/24 ab 1412 N   = 1 K K   Δ E  S Δ E   V Δ E  SL Δ E  SR Δ E  SE       Δ    E    S    (  m  e   V   )       Δ    E    V    (  m  e   V   ) Δ E  SF 108642161810Filling factor,  ν Filling factor,  ν 467/201412108 Magnetic field, B  (T)Magnetic field, B  (T)642016 14.013.513.012.512.011.511.0 Figure 5  |  Energies in the epitaxial graphene N 5 1 Landau level. a , Valley splitting, D E  V , measured for the  N  5 1 Landau level as a function of magneticfield. The solid blue line is a linear fit of the  N  5 1 data for fields less than 7T.Thelinearfityieldstheeffective  g  -factor  g  V (LL 1 ) 5 18.4 6 0.4(1 s ).Thesmoothcyanlineisaguidetotheeyeandshowsthe  N  5 1splittingpeakingwhen E  F iscentredinthe  N  5 1Landaumanifoldatafillingfactorof4.Inset,schematicof theenergylevelstructureofthe  N  5 1Landaulevel,showingthebreakingofthe valley symmetry,  D E  V , followed by the breaking of the spin symmetry,  D E  S . b , Spin level energy separations as functions of magnetic field. The solid linesare linear fits yielding the  g  -factors  g  SL 5 2.36 6 0.01 and  g  SR  5 2.23 6 0.01(1 s ). The level separation energies are defined in Fig. 4. Error bars, 1 s . LETTERS  NATURE | Vol 467 | 9 September 2010 188 Macmillan Publishers Limited. All rights reserved  ©2010  the SiC substrate. Short-range repulsive interactions favour spin-polarized states at odd-integer filling factors and contribute anexchange-correlation contribution  D XC 5 n  LL V  0 5 2 e  2 d  eff  / e l  B  2 to thegap between majority and minority spin levels 21,22 . This approximateexpression yields values consistent with the experimentally observedgap between majority and minority spin levels at filling factor  n 5 5(Fig. 5b).A number of possibilities come to mind for the gaps at fractionaltop-layer fillings. They could, for example, be associated with theformation of charge density-wave states 24,25 , such as those that occurin semiconductor quantum wells when  N  . 0 Landau levels arepartlyfilled 26,27 ,orwiththeexoticgroundstatessuspectedtodescribethe n 5 5/2 quantum Hall effects in semiconductors 28 . An alternativeexplanation, however, is suggested by the surprisingly large value of the fractional gap relative to the enhanced spin splitting. In multi-layer systems, stable states can form at fractional filling factor perlayer by forming counterflow superfluid states with spontaneousinterlayer coherence 29 . The fractional-state gap size,  D f  , would thenbe proportional to the interlayer interaction strength, which in oursystemshouldbenearlyaslargeas D XC becauseofthecloseproximity between different layers. Experimentally, we find that  D f  < D XC , inagreement with this hypothesis.Our observations suggest a number of future experiments. Theability to gate the graphene–SiC system and change the filling factorat fixed field would allow more magnetic quantum levels to be placedat the Fermi level, in particular the  N  5 0 Landau level, where 1/3-fractionalquantumstatesshouldbeobservedinthetunnellingdensity of states 17 . We expect that, as in the semiconductor case, differentcorrelated states will be revealed when Landau levels with differentvalues of   N   are pushed to the Fermi energy. In addition, the spatialvariation oftheLandaulevelsislikelytoberich becausethezero-fieldenergy scale of the disorder potential is of the same order as thesplitting energies of the Landau levels. In the future, we expect thatultralow-temperature, high-field scanning probe microscopy willcontinue to reveal new physics related to the electronic structure of graphene. METHODS SUMMARY We performed the experiments using a newly commissioned ultrahigh-vacuumscanning probe microscopy facility at NIST that operates with a base temper-ature of 10mK and a 15-T magnetic field capability. The epitaxial graphenesample was grown on C-face SiC at the Georgia Institute of Technology usingthe induction furnace method 30 . The graphene thickness was estimated to be six layers as determined by elliposometry measurements. The sample was trans-ported to NIST in a vacuum suitcase. 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Rev.Lett.  82,  394 – 397 (1999).28. Nayak, C., Simon, S. H., Stern, A., Freedman, M. & Das Sarma, S. Non-Abeliananyons and topological quantum computation.  Rev. Mod. Phys.  80,  1083 – 1159(2008).29. Eisenstein, J. P. & MacDonald, A. H. Bose – Einstein condensation of excitons inbilayer electron systems.  Nature  432,  691 – 694 (2004).30. Berger,C. etal. Ultrathinepitaxialgraphite:2Delectrongaspropertiesandaroutetoward graphene-based nanoelectronics.  J. Phys. Chem. B  108,  19912 – 19916(2004). Supplementary Information  is linked to the online version of the paper atwww.nature.com/nature. Acknowledgements WethankN.Zhitenevfordiscussions,S.Blankenship,A.Bandand F. Hess for their technical contributions to the construction of the millikelvinscanning probe microscopy system, V. Shvarts for his advice on cryogenics andU.D.Hamfor instructions onmakingsilver probe tips. This work was supported inpart by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD) (KRF- 2006-214-C00022), the NSF (DMR-0820382 [MRSEC],DMR-0804908, DMR-0606489), the Welch Foundation and the SemiconductorResearch Corporation (NRI-INDEX programme). Author Contributions  Y.J.S, A.F.O, Y.K. and J.A.S designed and constructed themillikelvin scanning probe microscopy system. The graphene scanning tunnellingmicroscopy/scanning tunnelling spectroscopy measurements were performed byY.J.S, A.F.O and J.A.S. The graphene sample was grown by Y.H and W.A.d.H., andthe surface was prepared and characterized by D.B.T and P.N.F. A theoreticalanalysis of the epitaxial graphene multilayer system was performed by H.M., S.A.,M.D.S and A.H.M. Author Information  Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial interests.Readers are welcome to comment on the online version of this article atwww.nature.com/nature. Correspondence and requests for materials should beaddressed to J.A.S. (joseph.stroscio@nist.gov). NATURE | Vol 467 | 9 September 2010  LETTERS 189 Macmillan Publishers Limited. All rights reserved  ©2010
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