Quantum Hall effect in exfoliated graphene affected by charged impurities: metrological measurements
J. Guignard, D. Leprat, D. C. Glattli, F. Schopfer, W. Poirier
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A ug Quantum Hall effect in exfoliated graphene affected by charged impurities:metrological measurements
J. Guignard, D. Leprat, D. C. Glattli, F. Schopfer, W. Poirier ∗ Laboratoire National de M´etrologie et d’Essais, 29 avenue Roger Hennequin, 78197 Trappes, France and Service de Physique de l’ ´Etat Condens´e, Commissariat `a l’ ´Energie Atomique, 91191 Gif-sur-Yvette, France
Metrological investigations of the quantum Hall effect (QHE) completed by transport measure-ments at low magnetic field are carried out in a-few- µ m-wide Hall bars made of monolayer (ML) orbilayer (BL) exfoliated graphene transferred on Si/SiO substrate. From the charge carrier densitydependence of the conductivity and from the measurement of the quantum corrections at low mag-netic field, we deduce that transport properties in these devices are mainly governed by the Coulombinteraction of carriers with a large concentration of charged impurities. In the QHE regime, at highmagnetic field and low temperature ( T < . (one standard deviation, 1 σ ) at theexpected rational fractions of the von Klitzing constant, respectively R K / R K / − accuracy usually achieved in GaAs, is the low value ofthe QHE breakdown current being no more than 1 µ A. The current dependence of the longitudinalconductivity investigated in the BL Hall bar shows that dissipation occurs through quasi-elasticinter-Landau level scattering, assisted by large local electric fields. We propose that charged impu-rities are responsible for an enhancement of such inter-Landau level transition rate and cause smallbreakdown currents.
PACS numbers: 73.43.-f, 72.80.Vp, 06.20.-f
I. INTRODUCTION
The discovery of the quantum Hall effect (QHE) in1980 has revolutionized resistance metrology by estab-lishing a universal quantum resistance standard at ra-tional fractions of the von Klitzing constant R K ≡ h/e where e is the electron charge and h Planck’s constant.Although the QHE was first observed in Si-MOSFETs,the cleaner two-dimensional electron gas (2DEG) madeby epitaxial growth of GaAs/AlGaAs heterostructureprovided more practical quantum resistance standards.They give accurate and reproducible representations of R K within an uncertainty below one part in 10 whenoperated at low temperature ( T = 1 . B = 10 T) . Following observation ofthe QHE in graphene with a sequence of Hall resis-tance plateaus at R H = ± R K / (4( n + 1 / n aninteger >
0) that survive even at room temperature ,an application to resistance metrology was considered .The peculiar QHE originates from the honeycomb lat-tice of carbon atoms in which charge carriers at lowenergy behave like chiral massless relativistic fermionswith Berry’s phase π . Under magnetic field, the den-sity of states becomes quantized in Landau levels (LLs)with a 4 eB/h degeneracy (valley and spin) that oc-curs at energies ± v F √ ~ neB . The robustness of theQHE on the first plateau comes from the energy spac-ing 36 p B [T] meV between the first two LLs being largerthan in GaAs (1 . B [T] meV). In bilayer graphene, whichconsists of two graphitic monolayers with Bernal stack-ing, the dispersion relation becomes parabolic and carri- ers behave like chiral massive ( m = 0 . × m e with m e the electron mass) Dirac fermions with Berry’s phase2 π . This leads to QHE with resistance plateaus at R H = ± R K / (4( n + 1)), with n an integer >
0. Theenergy gap between LLs occuring at ± ~ ω c p n ( n − ω c = eB/m is the cyclotron pulsation) is smaller thanin single graphene layer, especially at low magnetic field,but is larger than in GaAs systems. Larger energy gapsgive much hope that a more practical resistance stan-dard operating at a lower magnetic field or a highertemperature could be developed in both graphene sys-tems. In the short term, comparison of the Hall resis-tance in graphene systems and in GaAs would constitutea stringent test of the QHE universality. This wouldsupport ongoing efforts to make an historic evolution to-wards a Syst`eme International of units directly linked tofundamental constants of physics . More generally, themetrological approach can supplement the understand-ing of physics to the limits of instrumentation. Lastly,meeting the very demanding metrological requirementsfor the QHE application in graphene (quality of elec-trical contacts, control of electronic properties such asmobility and density over large mm-size scale) furtherenhances the severeness of the benchmark test offered bythe QHE for the quality of any two-dimensional material,and makes it very significant and useful for the develop-ment of industrial applications such as microelectronics.The metrological investigation has started shortly afterthe discovery of the QHE in graphene. Previously, theHall resistance R H was demonstrated to agree with R K /2on the plateau corresponding to Landau level filling fac-tor ν = n s h/ ( eB ) = 2 in exfoliated monolayer graphenewithin a relative uncertainty of 15 parts in 10 (one stan-dard deviation, 1 σ ), probably limited by the high resis-tance of contacts ( > . More recently, Tzalenchukand co-workers have reported an agreement within anuncertainty as low as 9 parts in 10 (1 σ ) in a large sam-ple (160 × µ m ) made of epitaxial monolayer graphenegrown on the Si-terminated face of silicon carbide (SiC),with a mobility of about 7500 cm V − s − when placedat B = 14 T and T = 0 . . Achieving the QHEquantization in graphene with similar uncertainty at afew teslas magnetic induction and higher temperature,which is required to develop a quantum resistance stan-dard challenging the GaAs ones, is still a critical issue.In this paper, we report on the accurate investigationof the QHE quantization in monolayer and bilayer exfoli-ated graphene lying on Si/SiO substrate. Measurementswere performed with a Cryogenic Current Comparator(CCC)-based resistance bridge. The objective was to de-termine limitations to the quantized Hall resistance ac-curacy that can be experienced in exfoliated graphene,which however turned out to be the reference techniqueenabling to unveil most of chiral Dirac fermions electronictransport properties. The understanding of these limita-tions could even be useful to overcome likely obstacles inthe development of quantum resistance standards withhigher performances in graphene grown either on SiC orby Chemical Vapor Deposition (CVD).The paper is organized as follows. In Section II, wereport on electronic transport properties of graphene in-vestigated by means of conductivity measurements at lowmagnetic field. In both the ML and BL samples, theanalysis of the conductivity dependence on charge car-rier density shows that carriers are mainly scattered by alarge concentration of charged impurities located about1 nm close to the graphene flakes. The major impact ofcharged impurities responsible for strong spatial fluctu-ations of the carrier density which survive at finite den-sity is also confirmed by measurements of quantum cor-rections to conductance (weak localization and univer-sal conductance fluctuations) in the BL sample. SectionIII reports on quantization tests performed by means ofcomparing the QHE in GaAs and in graphene systems.For monolayer and bilayer graphene, the Hall resistanceof the first plateau (Landau levels are spin and valleydegenerated) in the zero dissipation limit is found quan-tized within 5 parts in 10 (1 σ ) to R K /2 and R K /4 re-spectively. One main limitation to accuracy is the lowvalue of the QHE breakdown current limited to about ∼ µ A. In section IV, we show that the mechanism ofdissipation (or backscattering) in the BL sample, whichends up in the QHE breakdown is based on quasi-elasticinter-Landau level scattering (QUILLS) assisted by largelocal electric fields. This leads to discussing the role ofcharged impurities in enhancing inter-Landau level tran-sitions.
FIG. 1. (Color online) Optical images of the BL sample a) andof the ML sample b) with contacts resistance values indicatedbelow.
II. ELECTRONIC TRANSPORT PROPERTIESAT LOW MAGNETIC FIELD
Measurements were carried out on 15 × µ m and26 × . µ m Hall bars based on monolayer graphene(ML) and bilayer graphene (BL) respectively, which havebeen mechanically exfoliated from natural graphite (seeFig. 1). Flakes were transferred on top of highly dopedsilicon substrates covered by 90 nm (resp. 500 nm in BL)of thermally grown SiO used for backgating. Grapheneflakes are electrically contacted using Ti/Au (BL) andPd (ML) pads. Samples were then patterned with a Hallgeometry appropriate for QHE precision measurements.Graphene arms, at least 300 nm long, connect voltagemetallic contacts to the main channel. This geometryalso avoids electrode-induced doping of the main chan-nel. Samples were finally covered with a 300 nm-thickpolymethyl methacrylate (PMMA) resist layer. Trans-port properties were explored by four-terminal resistancemeasurements defined by R ij , kl = ( V k − V l ) /I i → j , where V i is the voltage potential at terminal i and I i → j is thecurrent flowing between terminals i and j. A. Influence of charged impurity scattering onconductivity
In both samples, the four-terminal conductivity σ =1 /ρ = 1 /R ij , kl × d kl W ( W is the sample channel width, d kl the distance between terminals k and l), deduced from R , and R , measurements in the ML and BL sam-ples respectively, was analyzed at zero magnetic field asa function of the gate voltage V G . It shows a typical min-imum that occurs at V Gmin (see Fig. 2a). At this valuethe carrier density defined as n s = C G ( V G − V Gmin ) /e (with C G /e = 2 . × cm − / V for ML and C G /e =4 . × cm − / V for BL) is zero on spatial aver-age. ¯ n = − C G V Gmin /e is the carrier density induced inthe graphene by surrounding charged impurities. Whileannealing the samples under vacuum at a temperatureof about 400 K, the conductivity dip becomes sharperand its position V Gmin shifts near zero indicating an in-crease in the carrier mobility µ and a decrease of | ¯ n | . At T = 1 . T = 0 .
35 K forthe BL sample) and at carrier density away from the re-gion of the minimum conductivity, σ ( V G ) is quite linearfor ML with no proof of sublinearity in the considereddensity range ( < . × cm − ), and is slightly super-linear for BL. These features indicate that the long-rangeCoulomb potential induced by charged impurities consti-tute the dominant source of scattering in the consideredsamples .Conductivity for the ML sample, except near the min-imum ( − ¯ n ± × cm − ), is well fitted by the the-oretical model based on Boltzmann transport theorywith charged scatterers σ ( C G V G /e ) = σ ( n s − ¯ n ) = G ( r s , d ) e h | n s | n i valid for electrons ( n s > n ∗ ) and for holes( n s < − n ∗ ) where n i is the density of charged impu-rities at an average distance d from the conductor (inthe silicon substrate or in the PMMA). n ∗ is a resid-ual density corresponding to the density of electron andhole puddles into which the system breaks at low den-sity because of the inhomogeneous density profile cre-ated by Coulomb impurities. r s describes the full di-electric environment of the sample that screens Coulombinteractions. Considering two semi-infinite media madeof SiO and PMMA on top of the device with dielec-tric constants ǫ SiO = 3 . ǫ PMMA = 4 . r s = 2 e / (4 πǫ ( ǫ PMMA + ǫ SiO ) ~ v F ) = 0 .
47 (with v F = 1 . × ms − ) and G ( r s = 0 . , d = 0) = 28 . G ( r s , d ) isonly weakly dependent on d , the approximated value G ( r s , d = 0) is valid while the electron/hole asymme-try in the conductivity curve remains weak, as observed,and thus is not considered. The mean impurity density(electron/hole average) deduced from the adjustment is n i ≈ . × cm − . At low density, assuming thisvalue of n i and a finite value of d in the range lowerthan 2 nm, the Boltzmann transport theory correctlyexplains (within a factor of 2, see ref. ) the experi-mental values of the conductivity minimum σ , of theplateau width minimum conductivity n ∗ and of the min-imum position − ¯ n . The size ξ and the density n ∗ ofelectron/hole puddles near the charge neutrality point(CNP) can be calculated from ξ = 1 / ( r √ n i ) = 32 nmand n ∗ = σ n i /G ( r s = 0 . , d = 0) he = 2 . × cm − (with σ = 4 e h ) respectively. One deduces that each pud-dle contains about 9 elementary charges in average. Thetheoretical model can also explain the conductivity curveasymmetry which corresponds to a constant mobility( µ = σ/ ( n s e )) higher for holes (4050 cm V − s − ) thanfor electrons (3400 cm V − s − ) by a typical < +5 ˚A-size shift of the distance d of charged impurities from thegraphene layer under the electric field effect producedby the back-gate voltage, assuming unequal numbers ofrandom positively and negatively charged impurities .A similar electron/hole asymmetry has already been ob- served in dirty samples . On the other hand thisasymmetry cannot be explained by the theory for attrac-tive vs. repulsive scattering of massless Dirac fermionsby Coulomb impurities predicting a higher mobility forelectrons for a negative value of ¯ n . Neither can a lo-cal doping due to the presence of metallic contacts ongraphene account for it since they are non-invasive in thestudied samples and would have induced sublinearity ofthe σ ( C G V G /e ) curve.In the BL sample, the conductivity can also be well fit-ted by a similar transport theory based on Coulomb in-teractions with charged impurities σ ( C G V G /e ) = σ ( n s − ¯ n ) ≈ π e h | n s | n i [1 + π p | n s | ( d + q − )] valid for elec-trons ( n s & n ∗ ) and holes ( n s . − n ∗ ) with q TF − =4 πǫ ( ǫ PMMA + ǫ SiO ) ~ / me = 0 . n i ≈ × cm − but also d ≈ thickness), this agreement strongly supports ourdescription of conductivity using the Boltzmann trans-port theory based on long-range Coulomb scatterers. Theextracted distance d is consistent with the position ofcharged impurities assessed in the ML sample and gen-erally measured in Si/SiO substrate . It appears that FIG. 2. (Color online) a) Conductivity as a function of car-rier density controlled by the back-gate voltage for the BL(green) and ML (blue) samples. Solid lines are fits given bya Boltzmann transport theory including charged impurities.b) Magneto-conductivity in the BL sample at T = 0 .
35 K atcarrier densities in the range n s = − × cm − ± ∆ n s / n s = 3 . × cm − . c) Magneto-conductivity afteraveraging on carrier density and adjustments by an appro-priate weak-localization theory (dotted lines) at T = 0 .
35 K(blue) and T = 1 . this model correctly predicts, except for the minimumposition ( − ¯ n ), the experimental values of σ and of theplateau width minimum conductivity n ∗ . From the val-ues of ξ = 11 nm calculated with the specific modeldeveloped for BL and the value of n ∗ = σ π he n i =2 . × cm − (with σ = 5 . e h ), one deduces that eachpuddle contains about 8 elementary charges. An elec-tron/hole asymmetry of the conductivity is also observed.But contrary to the ML sample, at n s = 2 × cm − the electron mobility (2300 cm V − s − ) is higher thanthe hole mobility (2000 cm V − s − ) by about 15%. Itcan again be explained by the shift of the mean distance d between the charged impurities and the graphene layerby a few ˚A under the electric field produced by the volt-age on the back-gate, but with impurities in excess witha sign opposite to the ML sample case . We note thatthe same amount of charged impurities leads to a lowercarrier mobility in the BL sample than in the ML sample,confirming that long-range Coulomb scattering is a veryefficient mechanism to spoil mobility in bilayer graphene.Beyond providing a very efficient source of scattering,charged impurities give rise to strong spatial fluctuationsof the charge carrier density with a correlation length ξ of 32 nm and 11 nm for ML and BL samples respec-tively. These fluctuations leading to electron/hole pud-dles landscape near the CNP are also known to persistat the higher carrier densities ( n s ≈ − × cm − )where the QHE has been investigated . Also, a moremacroscopic inhomogeneity of the carrier density at a µ m-size scale with a typical amplitude of a few 10 cm − has been observed. In particular, it manifests itself inthe BL sample through spatial variation of − ¯ n , σ andof d σ/ d V G slopes. These quantities depend for instanceon the conductor area probed in different configurations( e.g. R , and R , ). These carrier density fluctua-tions can be explained by spatial variations of n i by afew 10 cm − and of d by a few ˚A, which is also thetypical height of graphene flake ripples. Therefore thesamples are far from being homogeneous compared toGaAs based 2DEG commonly used for quantum resis-tance standards, where less than 10 cm − variation of n s can be achieved.In both samples, the diffusion coefficient D and trans-port mean free path l tr can be determined from conduc-tivity measurements at low temperature and carrier den-sities where the QHE was investigated. In the ML sam-ple, at T = 1 . n s = 6 . × cm − (electrons),corresponding to a Fermi energy of E F = ~ v F √ n s π =102 meV, D is calculated using the Einstein relation D = σ ( n s ) √ π ~ v F / (2 e √ n s ) = 2 . × − m s − and then l tr = 2 D/v F = 36 nm. In the BL sample, at T = 0 .
35 K,and n s = − × cm − (holes), i.e. E F = ~ √ n s π/m =72 meV, D = σ ( n s ) π ~ / (2 e m ) = 1 . × − m s − , and l tr = 2 D/v F ( n s ) = 34 nm, with v F ( n s ) = ~ √ n s π/m =8 . × ms − . These values confirm that electronictransport is diffusive with similar amount of disorderin both samples: k F l tr = 5 . . V − s − and density 5 . × cm − , l tr = 3 . µ m is 100 times higher and k F l tr ≈ B. Quantum corrections to conductivity
In the BL sample, where the QHE has beenmore extensively studied, quantum interference correc-tions to conductivity, both weak-localization correction(WL) and reproducible mesoscopic conductance fluctu-ations (CF), were investigated by performing magneto-conductivity measurements at low temperature. Actu-ally, both in monolayer and bilayer graphene the am-plitude of these corrections is not only ruled not by in-elastic scattering like in any other diffusive metal, butalso by elastic scattering mechanisms affecting the valleysymmetry (intravalley scattering and/or trigonal warp-ing of the conical band structure, intervalley scatter-ing). This is a consequence of the direct manifestationof chirality property in quantum interference effects. Forinstance, interferences between time-reversal symmetricdiffusive electron trajectories lead to weak-localization corrections to conductivity in bilayer graphene because ofthe charge carrier wave function 2 π Berry’s phase whileweak-antilocalization is expected in monolayer due to π Berry’s phase.Applying a magnetic field breaks the system time re-versal symmetry and suppresses the weak-localizationcorrections. This gives rise to a well-known magnetoresis-tance. Four-terminal magnetoresistance measurements R ( B ) were carried out at temperatures T = 0 .
35 K and T = 1 . I = 30 nA,thus the effective temperature of carriers assessed by T eff = eRI/k B = 0 .
52 K, where R is the resistanceper square, is slightly higher than the base temperature0 .
35 K. Fig. 2b reports a set of magnetoconductivitycurves recorded at densities around n s = − × cm − over a total range ∆ n s = 3 . × cm − . They all dis-play a characteristic dip at zero field, signature of the ex-pected weak-localization, the amplitude of which barelyexceeds reproducible fluctuations (CF) which are ana-lyzed below. To make the WL conductivity dip standout from fluctuations, magnetoconductivity curves wereaveraged over the full density range where the diffusioncoefficient D does not vary by more than 10%. The aver-aged curve (see Fig. 2c) is then adjusted by the appropri-ate weak-localization theory , ∆ σ ( B ) = σ ( B ) − σ (0) = e πh [ F ( τ − B τ − ) − F ( τ − B τ − +2 τ − )+2 F ( τ − B τ − + τ − + τ − ∗ )]. Here F ( z ) = ln ( z ) + ψ (1 / z − ), ψ ( x ) is the digamma func-tion, τ − B = 4 eDB/ ~ . τ − = D/L is the phasebreaking rate. τ − = D/L is the intervalley scatteringrate lifting the valley degeneracy of electronic states andwhich is caused by short-range defects with maximumsize of the order of the lattice spacing. τ − ∗ = D/L ∗ =2 τ − + τ − is an intravalley scattering rate. τ − is theintravalley chirality breaking rate caused by surface rip-ples, dislocations and atomically sharp defects, i.e. short-range defects. τ − is the intravalley p → − p symmetrybreaking rate (where p = ~ k F , k F is the carrier momen-tum at the Fermi level) caused by the anisotropy of theFermi surface in k space, i.e. the trigonal warping. Inbilayer graphene, assuming a quadratic Hamiltonian, itis expected that τ − = τ − where τ tr = l tr /v F is thetransport time .The adjustments of data at T = 0 .
35 K and T = 1 . L ΦWL ( T = 0 .
35 K) =0 . µ m and L ΦWL ( T = 1 . . µ m, the in-tervalley scattering length L i ( T = 0 .
35 K) = 0 . µ mand L i ( T = 1 . . µ m, the intravalley scat-tering length L ∗ ( T = 0 .
35 K) and L ∗ ( T = 1 . . . µ m. The extracted values are very similar tothose measured in bilayer graphene and reported in theliterature . L ΦWL below the sample size indicates thatelectronic transport is not fully quantum coherent. Itappears that L ΦWL ∼ L i and L i ≫ L ∗ . The WL ismade observable due to significant intervalley scattering,though much less than intravalley processes. It also ap-pears that L ∗ ∼ l tr . The fact that τ − ∗ = 2 τ − + τ − ∼ τ − means that τ − is small, since it is expected that τ − = τ − . Finally, the fact that L i ≫ L ∗ ∼ l tr , to-gether with τ − ≪ τ − demonstrate that short-rangescattering is not dominant. Moreover, L ΦWL appearsquasi constant between T = 0 .
35 K and T = 1 . L ΦWL at low temperature,well below the particle-particle interaction length ( L hh = q D [ σh πe ln [ σh/ (2 e )] 1 k B T ] = 1 . µ m at T = 0 .
35 K) , hasalready been observed in graphene samples near theCNP. It could be a feature of transport by percolationthrough electron/hole puddles persisting at finite den-sity (typically n s = − × cm − < n ∗ ) in the veryinhomogeneous BL sample. These results confirm theconclusion drawn from the analysis of the conductivitycurves σ ( C G V G /e ) that long-range Coulomb scatteringby charged impurities trapped in the silicon substrate orin the PMMA top-layer of the graphene-based sample isdominant.Conductance fluctuations were measured by varyingthe magnetic induction over a ± T = 0 .
35 K.The standard deviation is found to be δG B = 0 . e /h .In graphene, CF resulting from interference of phase-coherent chiral carrier diffusive paths are also expectedto depend on elastic scattering. In the BL sample,since L ΦWL ∼ L i , one expects the amplitude of CFto be properly described by the theory of well-knownuniversal conductance fluctuations (UCF) for diffusivemetals . Precisely, in the case of a two-dimensionalconductor, at a magnetic field larger than the typicalmagnetic field of WL magnetoresistance, it is given by δG = 0 . √ q WL min ( L Φ ,L T ) L e h where L and W are the length and the width of the conductor measured, L T = p ~ D/k B T is the thermal length. Assuming thevalues W ≈ . µ m, L ≈ µ m, T eff = eRI/k B = 0 .
87 K,one finds L T eff = 0 . µm < L ΦWL = 0 . µm whichresults in δG = 0 . e /h . The good agreement ofthe experimental magnitude of CF with the theoreti-cal value of the UCF in diffusive metals confirms that L i ∼ L Φ ≃ . µ m, and since l tr = 34 nm ≪ L i , thatlong-range scattering is dominant. On the other hand, itshows that conductance fluctuations as a function of themagnetic field are not sensitive to the observed carrierdensity inhomogeneity or presence of electron and holepuddles.The analysis of transport at low-magnetic field showsthat the dominant mechanism of scattering in our sam-ples is Coulomb interaction with a large concentration ofcharged impurities closely surrounding graphene flakes(in the silicon substrate and in the PMMA top layer cov-ering the devices). Beyond to drastically reducing thecarrier mobility, they are responsible for strong spatialfluctuations of the carrier density that might stay bipo-lar even at finite density (a few 10 cm − ). III. HALL RESISTANCE QUANTIZATIONTESTS OF THE QUANTUM HALL EFFECTREGIME
FIG. 3. a) Hall resistance ( R , ) (at magnetic inductions B = 2 , , , , , . R , )(at B = 18 . T = 0 .
35 K.b) R xx = R , × Wd (blue), R xx = R , × Wd (magenta)for currents I = 0 . , , µ A. c) R xx = R , × Wd as afunction of n s around ν = − T = 0 .
35 K and I = 0 . µ A(blue); T = 0 .
35 K and I = 1 µ A (deep blue); T = 1 . I = 0 . µ A (orange); T = 1 . I = 1 µ A (red). Ver-tical dashed lines underline reproducible fluctuations. Inset:ln( R xx ) as a function of the carrier density. d) Three-terminalresistance of contacts as a function of n s . In the QHE regime, all measurements were performedusing direct current (DC) measurements techniques.Each resistance value reported in the following is theaverage of values measured for both current directions.About notations, R xx is a longitudinal resistance valuenormalized to a square, for example R xx = R ij , kl × Wd kl ifthe longitudinal resistance is measured between terminalsk and l. Fig. 3a shows Hall and longitudinal resistance asa function of carrier density for the BL sample. Measure-ments clearly reveal ν = ± ν = ± n = 0 , n = 2) is 92 meV (1068 K equivalenttemperature) at B = 18 . T = 0 .
35 K. The longitudinal resistance R xx reported at B = 18 . n = 0 and n = 1 LLs andminima occuring simultaneously with Hall plateaus. Weonly investigated the physics of the ν = − R xx . Dissipation level in the 2DEG and qual-ity of contacts are essential quantization criteria of theQHE, as demonstrated by several experimental works as well as the Landauer-B¨uttiker theory . The quanti-zation is indeed directly related to the absence of dissi-pation ( i.e. of backscattering), the rate of which can bedetermined by the measurement of R xx . Fig. 3b showsthe behavior of R xx with hole density on the ν = − . µ A to 5 µ A. The R xx plateau shrinksand simultaneously the R xx minimum increases. Fig. 3balso shows that position and magnitude of R xx minimadepend on the sample region measured. Position varia-tion can be attributed to carrier density fluctuations witha magnitude of a few 10 cm − caused by charged im-purities, as already mentioned in Section II. In addition,the magnitude variation illustrates that the ignition ofQHE breakdown is a very spatially inhomogeneous phe-nomenon. Fig. 3c shows that the temperature effect on R xx between 0 .
35 K and 1 . . µ A and 1 µ A. It also shows that R xx has reproducible fluctuations as a function of n s witha similar pattern at the two different temperatures andcurrents. We will later discuss the origin of these fluc-tuations, particularly visible a bit away from the mini-mum because of a better signal to noise ratio. Averagingfluctuations (and noise) of R xx around specific densityvalues gives typical and relevant mean values of the lon-gitudinal resistance ¯ R xx . At T = 0 .
35 K and I = 0 . µ A,¯ R xx = ¯ R , × Wd is (2 ±
14) mΩ and (62 ±
9) mΩ at n s = − . × cm − and n s = − . × cm − respectively. These resistance values are to be comparedwith 100 µ Ω, the typical value of R xx which ensures a10 − R H accuracy in usual GaAs-based quantum stan-dards (LEP514 ). Contact quality was determined byperforming three-terminal measurements of resistance. R ( e. g. R ij , il ) gives the resistance value of the contact R c ( e. g. i) combined with a R xx ( e. g. R kj , il ) con- tribution. For a good contact, the drop to a negligiblevalue of R xx ( ≪ R giving an up-per bound of R c . As observed in Fig. 3d, for the goodTi/Au contacts of the BL sample, R c values deducedfrom R minima can be as low as 10 Ω (see Fig. 1a).Note that R minima occur at slightly different n s val-ues due to the carrier density spatial inhomogeneity. R for contact 3 does not exhibit such a flat minimum witha value higher than 428 Ω. The highest resistance valuewas found equal to 5 . k Ω for contact 8. These anoma-lous behaviors can be explained by a large fluctuation of n s in the voltage arm thin channel (2 µm ) or even by apartial breaking of the constriction probably caused bythe sample cooling down too fast. The complete breakingcan account for the infinite resistance observed for someother contacts. Contacts 3 and 8 were used as currentcontacts, rather than voltage, for the Hall resistance pre-cision measurements. It was indeed demonstrated thata very resistive detecting voltage contact can lead to a de-viation from quantization notably because being unableto restore the equilibrium of the edge state population .Although we used Pd instead of Ti/Au to make contactto graphene, similar observations are reported in the MLsample. The five contacts used to perform measurementshave low resistance values ranging from 15 Ω to 260 Ω(see Fig. 1b). FIG. 4. a) Relative Hall discrepancy ∆ R H / R H as afunctionof n s (and ν in upper-scale) at four currents. b) ∆ R H / R H asa function of I at n s = − . × cm − (blue) and at n s = − . × cm − (red). c) R xx = R , × Wd as a functionof I at n s = − . × cm − (filled blue square) and at n s = − . × cm − (filled red circle), R xx = R , × Wd as a function of I at n s = − . × cm − (unfilled greensquare). d) R xx = R , × Wd as a function of I for fillingfactors ν = − . ν = − . T = 1 . T = 0 .
35 K(unfilled symbols). Error bars correspond to uncertaintiesgiven within one standard deviation, 1 σ . We then performed accurate measurements of R H interms of R K using a resistance bridge equipped with aSQUID based cryogenic current comparator. In practice,the Hall resistance is compared to a well-known 100 Ωwire resistor calibrated in terms of a GaAs based quan-tum resistance standard (LEP514). In the BL sample,Fig. 4a reports the relative deviation of R H = R , from its nominal value ∆ R H / R H = R H /( R K /4)-1 as afunction of n s . All uncertainties are given within onestandard deviation (1 σ ). Let us note that the resis-tance measured not only includes a pure transverse re-sistance but also a longitudinal resistance contribution,because the line between voltage terminals is not per-pendicular to the one between current terminals. Mea-surements clearly show a flat resistance plateau within 3parts in 10 over a 2 × cm − carrier density rangewhen measured with a current below 1 µ A. At the low-est measurement current I = 0 . µ A, deviations fromquantization at highest carrier density agree with the ex-pected shape of the Hall plateau (decrease of resistanceon plateau edges). The shape evolution at higher cur-rents is attributed to a R xx contribution which adds tothe transverse resistance and increases with the current.This coupling between R H and R xx , which always ex-ists to some extend in GaAs based quantum resistancestandard , will be later discussed in more details. Theflatness appears worse at I = 2 µ A, as expected with re-gards to the large increase of R xx , fluctuating with carrierdensity as previously discussed. Fig. 4b confirms thatdeviations from quantization start to drastically increasefrom I = 2 µ A at n s = − . × cm − and from I = 3 µ A at n s = − . × cm − . As demonstratedin Fig. 4c, the increase of deviation due to current is ac-companied by a large increase of R xx at both densities.The weighted mean value of ∆ R H / R H values measuredat currents below these critical currents leads to smalldeviations of (0 . ± . × − and (3 . ± . × − at n s = − . × cm − and n s = − . × cm − re-spectively. Since R xx is the relevant parameter of quanti-zation, ∆ R H / R H as a function of R xx is then reported inFig. 5 from data of 4b and 4c for the two carrier densities.Although dissipation is inhomogeneous in the sample, asthe very different values of R xx measured using voltageterminal-pairs (2,4) and (3,4) at n s = − . × cm − express again, all deviations scale quite linearly with R xx ,indicating a common coupling mechanism between Halland longitudinal resistances. This linear relationship,which is usually observed in GaAs based quantum resis-tance standards, is generally explained in terms of an ef-fective misalignment of Hall probes, either due to a lack ofcarrier density homogeneity in the sample or to cur-rent flow chiral nature in finite width voltage terminals .In a good quantum Hall resistance standard, one usuallyfinds ∆ R H / R H = αR xx / R H with α ≃ . −
1. In our casevoltage terminals are really misaligned, which should leadto a unity coupling factor. But from slopes we deduce α values in the range 10 − to 10 − , depending where R xx is measured. This means that, due to inhomogene- ity, R xx values are not quantitative measurements of thedissipation level between Hall probes 2 and 4 when thecurrent flows between terminals 3 and 8. Nevertheless,the values as a whole give a qualitative representationof the dissipation current behavior in the sample. It istherefore justified to extrapolate ∆ R H / R H in the dissipa-tionless limit ( R xx = 0) at which the perfect quantizationis expected. In this limit, at n s = − . × cm − ,we find ∆ R H /R H ( R xx = 0) = ( − . ± . × − and ∆ R H /R H ( R xx = 0) = ( − . ± . × − us-ing R xx measurement with voltage terminal pairs (2,4)and (3,4) respectively. Agreement of these two valueswithin the measurement uncertainty corroborates ourextrapolation protocol. At n s = − . × cm − ,∆ R H /R H ( R xx = 0) = ( − . ± . × − , thusthe Hall resistance stays quantized within the mea-surement uncertainty. But the carrier density value − . × cm − seems to ensure a minimal sensi-tivity of the Hall resistance to dissipation. FIG. 5. ∆ R H / R H as a function of R xx = R , × Wd at n s = − . × cm − (filled blue square), ∆ R H / R H as a functionof R xx = R , × Wd at n s = − . × cm − (unfilledgreen square), ∆ R H / R H as a function of R xx = R , × Wd at n s = − . × cm − (filled red circle). Errors bars cor-respond to measurement uncertainties given within one stan-dard deviation, 1 σ . A similar study was carried out on the ML sample.Fig. 6a shows ν = ± ν = ± B = 11 . T = 1 . R H / R H = R H /( R K /2)-1 and R xx val-ues measured with two measurement currents 0 . µ Aand 1 µ A at n s = 6 . × cm − on the ν = 2plateau. Although the deviation strongly increases from I = 1 µ A, the extrapolation to zero dissipation gives∆ R H /R H ( R xx = 0) = (0 . ± . × − . The degree ofaccuracy achieved in the ML and BL samples is thereforesimilar. It is independent of the ratio of the energy gap tothermal energy since
92 meV0 .
35 K × k B in the BL sample is by 2 .
123 meV1 . × k B in the ML sample. The quantiza-tion accuracy is probably determined by the presence ofthe same high concentration of charged impurities in bothsamples leading to the carrier density inhomogeneity andlow carrier mobility. The impact of charged impuritieson the QHE breakdown will be discussed in the followingthrough detailed analysis of the current dependence of R xx in the BL sample. FIG. 6. a) Hall ( R H = R , ) and longitudinal ( R xx = R , × Wd ) resistances as a function of n s at B = 11 . I = 200 nA and T = 1 . R H / R H as a functionof R xx at n s = 6 . × cm − . Error bars correspond touncertainties given within one standard deviation, 1 σ . IV. DISSIPATION MECHANISM IN THE QHEREGIME IN BILAYER GRAPHENEA. Current dependence of the longitudinalresistance
Dissipation in GaAs/AlGaAs 2DEG was found to in-crease with temperature or current through several mech-anisms. At low temperature and low current, carriers canbackscatter from one edge to the opposite edge throughlocalized states by variable range hopping (VRH) withsoft Coulomb gap, characterized by a temperature behav-ior of the conductivity ( σ /T ) exp[ − ( T ( ξ loc ) /T ) / ]where k B T ( ξ loc ) = e / (4 πε ε r ξ loc ) and ξ loc is the local-ization length a lower bound of which is the mag-netic length l B = p ~ /eB . Current effect manifests it-self as an effective temperature k B T eff = eV H ξ loc /W . Ata higher temperature, conductivity is activated follow-ing the behavior σ exp[ − ( T Act /T )], where σ is close to e /h and weakly dependent on the electron-phonon cou-pling in case of a short-range potential but expected tobe universal and equal to 2 e /h in case of a long-rangepotential . T Act is typically related to the cyclotron gap.Experimentally, VRH mechanism was also observed inmonolayer graphene . In samples based on exfoliatedgraphene transferred on Si/SiO substrate, screening ofthe Coulomb interaction by the close metallic back-gateeven restores the usual two-dimensional VRH mechanismwith a temperature dependence exp[ − ( T ( ξ loc ) /T ) / ]where T ( ξ loc ) ∼ / ( g ( E F ) ξ ) and g ( E F ) is the den- sity of states . Conductivity activation by temperaturewas also observed in graphene systems .On the other hand, there are few reports dealingwith detailed investigation of the QHE breakdown byincreasing the current in exfoliated graphene. For semi-conductors 2DEGs, several electric field assisted mech-anisms have been considered to explain the large in-crease of longitudinal conductivity leading to the QHEbreakdown : quasi-elastic inter-Landau levels scatter-ing (QUILLS) possibly combined with intra Lan-dau levels scattering , increase of delocalized elec-tron states in Landau levels , ordinary electron heat-ing, bootstrap-type electron heating (particularly ef-ficient in large-size samples), and electron percolation be-tween sample edges by merging of compressible islands .In a sample made of exfoliated graphene on Si/SiO sub-strate, Singh and co-workers deduced from the mea-surement of breakdown current dependence on integerfilling factor that the QHE regime is broken by inter-Landau levels scattering in presence of large local electricfield.Fig. 4d reports on R xx dependence on current mea-sured at two filling factors ν (or n s values) near ν = − . .
35 K and 1 . R xx over three orders of magnitudeabove a critical current. More precisely, one can definea breakdown current I c by the value above which con-ductivity exceeds 2 . − S. I c linearly decreases for de-creasing ν values departing from the filling factor ν = − . µ A to 0 . µ A (seeFig. 8a). The breakdown current at T = 0 .
35 K is slightlyhigher than at T = 1 . . FIG. 7. Conductivity σ xx ( e /h ) as a function of I − at ν = − . ν = − . ν = − . T = 1 . T = 0 .
35 K).Solid lines and dots lines correspond to theoretical adjust-ments at T = 1 . T = 0 .
35 K respectively. Errorbars correspond to uncertainties given within one standarddeviation, 1 σ . Fig. 7 clearly displays the existence of four cur-rent regimes for the conductivity calculated by σ xx = R xx / ( R + R ). We will later discuss the first regimeI for very high currents. For currents down to 1 . µ A(second current regime II), conductivity σ xx decreasesby decreasing the current following a unique phenomeno-logical fitting function σ ,ν exp[ − ∆ E a ( ν ) /eR H I ] at bothtemperatures. σ ,ν is found quite universal around0 . e /h within 30% for all ν values. Fig. 8a shows that∆ E a ( ν ) scales linearly with ν , similarly to ∆ E th ( ν ) =( √ ~ ω c / ν + 4) /
2) which is the energy differencebetween the Fermi level and the center of the n = − E th ( ν ) and ∆ E a ( ν ) re-sults in a deviation of the ν value for which ∆ E a = 0from −
6, center of the n = − ν edge can be interpreted as the mobility edge whichseparates localized and extended states near the centerof the n = − σ xx at T = 0 .
35 K decreases more quicklywith I decreasing and departs from σ xx at T = 1 . . µ A leads tothe fourth regime IV where conductivity apparently satu-rates at values σ T,ν different for the two temperatures. At T = 0 .
35 K, the conductivity threshold cannot be deter-mined because of the increasing weight of some hystereticcharging effect altering measurements of σ xx for currentsbelow 0 . µ A. Consequently, the reasonable assumptionthat σ T,ν follows the typical temperature dependence ofVRH mechanism cannot be confirmed.We remark that the monotonous behavior ofconductivity following the current dependence σ ,ν exp[ − ∆ E a ( ν ) /eR H I ] dominates at T = 1 . I = 0 . µ A (and at T = 0 .
35 K inregime II), and cannot be explained by an activationeffect caused by a simple heating of electrons by currentsince it does not manifest itself at the lowest temper-ature T = 0 .
35 K in the low current regime III downto the same value I = 0 . µ A. More quantitatively, aconductivity increase due to heating by current shouldbe described by σ ,ν exp[ − ∆ E th ( ν ) /k B T el ] with T el theeffective electron temperature resulting from the heating.The correct adjustment of data at T = 1 . I = 0 . µ A by σ ,ν exp[ − ∆ E th ( ν ) /eR H I ]would result in k B T el = eV H = eR H I , leading to aneffective temperature T el of 52 K for I = 0 . µ A. Thisis not in agreement with the electronic temperature,which is obviously close to 1 . I = 0 . µ A, witha constant value expected to be determined by the bathtemperature. Finally, absence of strong asymmetry of R xx values (there are similar within 30%) with respectto current direction indicates that there is no stronglocal electron heating in current contact . This rulesout any strong role of current contacts 8 (5 . < . E th ( ν ) in current regime II rather directs towards a dis-sipation mechanism based on quasi-elastic inter-Landaulevels scattering (QUILLS) assisted by the electric field.In current regime III, the decrease of conductivity at T = 0 .
35 K suggests that the QUILLS mechanism iscombined with a blockade mechanism manifesting itselfapproximately below T = 1 . ∼ µ A), like a heating mechanism by current. It ap-pears that all conductivity curves under regimes II, IIIand IV, at both T = 1 . T = 0 .
35 K, can be ad-justed (see Fig. 7 and Fig. 4d) by a unique fitting function σ xx = σ T,ν + σ ,ν exp[ − ∆ E a ( ν ) /eR H I ] exp[ − E c / ( k B ( T + γσ xx V H2 ))]. σ ,ν ≃ . e /h for all ν values, E c =95 µ eV, γ = 0 .
48 K/pW and ∆ E a near ∆ E th as alreadyexplained. The contribution σ T,ν is chosen to adjust con-ductivity in the low current regime IV only at T = 1 . FIG. 8. a) ∆ E th (meV) as a function of ν (red solid line),∆ E a (meV) as a function of ν (red filled circle), breakdowncurrents I c as a function of ν at T = 0 .
35 K (blue filled square)and T = 1 . σ b)Schematics of inter-Landau levels transitions in case of disor-dered 2DEG. B. Phenomenological model based on Quasi-elasticInter-Landau Level Scattering (QUILLS)
In order to explain with the QUILLS mechanism themain current dependence of conductivity observed inregime II, let us first consider an homogenous electricfield and harmonic oscillator wave functions for the car-riers. The tilting of LLs by the electric field bringscloser localized states at Fermi energy and extendedstates in the nearest Landau levels, increases the wave-function overlap, and thus leads to an increased tran-sition probability P between LLs. Through scatter-ing processes by phonons and/or charged impurities, P is proportional to the wavefunction overlap givenby exp[ − Q l B ] where Q is the typical direct momen-tum between Landau levels Q = ω c B/E (see fig. 8b).This should lead to P ∝ exp[ − (∆ E th ( ν ) /eEl B ) ] =exp[ − (∆ E th ( ν ) /eV H ) ( W/l B ) ] , thus a transitionprobability different from the one observed. But, inpresence of disorder, at a length scale larger than l B ,0one expects a dependence exp[ − x/ξ loc ( ν )] of the local-ized state wavefunction tail where ξ loc ( ν ) is the local-ization length varying like ( ν − ν c ) − (2 . ± . with ν c thefilling factor of the Landau level center . This caseshould be particularly valid in bilayer graphene wherethe energy gap between LLs is very large. From thedistance x = ∆ E th ( ν ) /eE between an initial localizedstate at the Fermi energy and a final extended state inthe Landau level at the same energy, one therefore ex-pects P ∝ exp[ − ∆ E th ( ν ) / ( eEξ loc )]. This model canwell describe the main exponential current dependenceobserved in current regime II provided that large localelectric fields with a magnitude around V H /ξ loc are con-sidered. Assuming ξ loc ∼ l B = 6 nm at ν = − B = 18 .
5T actually leads to a high value of the electricfield V H /ξ loc ∼ V/m for I = 1 µ A. Besides, thecorrect adjustment of data by this model needs ∆ E th to be replaced by ∆ E a that can be interpreted as theenergy difference between the Fermi energy and the mo-bility edge of the nearest LL.We argue that the high concentration of charged impu-rities (2 . cm − ) in the substrate can lead to such largeelectric fields. In absence of magnetic field, it was demon-strated in section II that charged impurities create carrierdensity fluctuations with a magnitude of 10 cm − anda typical correlation length ξ = 11 nm in the consid-ered BL sample notably manifesting themselves as elec-tron and hole puddles near the CNP. These fluctuationsare combined with more macroscopic carrier density vari-ations extending over larger spatial scales. Their im-pact at high magnetic field in the QHE regime has beenaddressed. Scanning of graphene on Si/SiO substrateby tunneling spectroscopy or single electron transistortechnique has indeed shown that the potential land-scape drawn by charged impurities is partially screenedby the Coulomb interaction and leads to the existence ofcompressible islands surrounded by incompressible stripslike in AlGaAs/GaAs 2DEG . Jung and co-authors even show that electron or hole puddles at zero magneticfield turn into compressible islands surrounded by incom-pressible strips in the QHE regime. It turns out that thelocalization length ξ loc , or rather its lower bound l B , aswell as the characteristic length of incompressible stripsacross which Hall potential drops, could be similar to theelectron and hole puddle correlation length ξ . Thus, theexistence of large local electric field in the BL sample witha typical magnitude V H /ξ ∼ V/m should result fromthe strong carrier density fluctuations caused by largeconcentration of charged impurities. Similar explanationwas proposed by Sing and co-workers . Another way tounderstand the impact of the carrier density fluctuationsis to consider that they turn into spatial variations of thefilling factor in the QHE regime. Otherwise, the currentflows along a path minimizing the dissipation that is ex-pected to occur at ν = −
4. Given the correlation lengthof the filling factor (or similarly of carrier density) fluctu-ations and the small width of the sample, it is thereforelikely that the current flows along a narrow percolating incompressible path having a typical width ξ = 11 nm.The potential drop concentration across this path leadsto the existence of large local electric fields. Beyond theenhancement of the electric field, the role of charged im-purities in the QHE breakdown has been investigatedin conventional semiconductor heterostructure. Whilecharged impurities are kept away from the 2DEG by the10 nm to 40 nm thick spacer, acoustic electron-phononinteraction controls the QHE breakdown because elas-tic scattering by ionized impurities increases the inter-Landau level transition rate at higher electric field. Butnumerical work shows that the closer charged impu-rities are from the 2DEG the lower the electric field atwhich they are efficient. We therefore propose that ahigh concentration of charged impurities located at onlyabout 1 nm from graphene in the BL sample could itselfbe responsible for inter-Landau level transitions, whichare in addition enhanced by the strong electric fields in-troduced by the carrier density inhomogeneity these im-purities induce. This results in QHE breakdown cur-rents (typically 0 . I ≈ . µ A. On the other hand, in samplesmade from exfoliated monolayer graphene of higher mo-bility where short-range scatterers dominate transport atlow magnetic field , dissipation in the QHE regime wasobserved to occur through VRH when increasing cur-rent up to ≈ µ A.The term exp[ − E c / ( k B ( T + γσ xx V H2 ))] allows the de-scription of the temperature effect and the weak heat-ing effect by current, clearly visible in current regime III(below 1 µ A). It phenomenologically models a block-ade mechanism that can be activated by thermal energyabove a critical energy E c = 95 µ eV. The effective tem-perature of carriers given by T ∗ = T + γσ xx V H2 leadsto the best adjustment of data notably reproducing verywell the sharpness of the crossover between large and lowcurrent regimes at T = 0 .
35 K. Even at T = 1 . γ ∼ .
48 K / pW.At I = 1 µ A, T ∗ amounts to about 0 . . ν values -4.2, -4.5 and -4.8 respectively. Theelectronic temperature increases all the more so as the ν value departs from ν = −
4, because of the higher meanconductivity leading to more dissipation. The origin ofthis blockade mechanism manifesting itself at low tem-perature and typically clearly visibly below T = 1 . ~ c s /l B , where c s is the sound velocity, are 22 meV(25 . . . c s = 2 × ms − ) and of Si0 ( c s = 6 × ms − )respectively, thus well above 95 µ eV. On the other hand,1the energy value 1 . . c s = 1 . × ms − ) could be com-patible with our observations. An explanation based onCoulomb blockade effect in compressible islands is moreimprobable since the low value of E c would mean over-sized islands.Finally, the observed disappearance of the exponentialregime in current regime I (see Fig. 4d) can naturallybe explained by the QUILLS mechanism because of theoverlap integral saturation occurring when Landau levelsare very tilted. At higher currents, conductivity slowlyincreases with a polynomial dependence σ xx ∝ I β with β varying from 1/3 to 2/3 for ν values from -4.8 to -4.2. FIG. 9. a) ln R xx ( R , ) as a function of ν for four currentvalues at T = 0 .
35 K. b) ∆ E b = − ln[ R xx / ( σ ,ν R )] eR H I asa function of ν ; ∆ E a values are reported as black dot points. The linear dependence of ∆ E a ( ν ) on ν demonstratedin the range between ν = − . ν = − . ν from the R xx dependence on ν measuredat several currents. Fig. 9a reports ln( R xx ) as a func-tion of ν measured with current values 3 , , µ A forwhich the term exp[ − E c / ( k B ( T + γσ xx V H2 ))] ∼ E b = − ln[ R xx / ( σ ,ν R )] eR H I (with σ ,ν = 0 . e /h )approximatively merge into a unique curve, except farfrom ν = −
4. This nicely shows that conductivity wellfollows the current dependence σ ,ν exp[ − ∆ E b ( ν ) /eR H I ]and reinforces the meaning of ∆ E b ( ν ) as the energy dif-ference between the Fermi energy and the mobility edgeof the nearest Landau levels. ∆ E b ( ν ) draws the depen-dence of this energy difference on ν . This energy reachesa maximum value of 45 meV at exactly ν = −
4, whichis half the energy gap as expected. Fig. 9b first showsthat ∆ E a values deduced at ν = − . , − . , − . E b ( ν ) curve deduced from thefilling factor dependence of R xx at different currents. Sec-ond, it shows that ∆ E b ( ν ) linearity holds on both sideof ν = − ν = −
4, curves do not su-perimpose in a unique curve which means that QUILLSis no more the mechanism responsible for conductivity.The sub-linearity of ln( R xx ) as observed in Fig. 9a can rather be explained by a saturation of the wavefunctionoverlap at filling factors near mobility edges. Therefore,even if we expect an increase of density of states, theenergy determined near mobility edges in Fig. 9b is notrelevant. Extrapolating the linear behavior of ∆ E b ( ν )at zero energy should give a reasonable estimate of themobility edge filling factor of the n = − ν edge( n = − = − .
55. This value means that the mobilityedge energy should depart from the ( n = −
2) Landaulevel energy by 10 . ± . ~ / τ e .It matches the lower bound that is calculated equal to8 . τ e ∼ τ tr = 34 nm in bilayergraphene because of the 2 π Berry’s phase and ignoringthat τ tr could be larger than τ e because of the long-rangecharacter of the dominant scattering potential. On theother hand, the extrapolation of the linear behavior of∆ E b ( ν ) at zero energy between ν = − ν = 0 leadsto ν edge( n =0 , ≃ −
3, which corresponds to a mobilityedge shifted from the Landau level center by a largerenergy of 34 . n = − C. Longitudinal resistance reproduciblefluctuations
Fig. 3c shows that in the BL sample, at both T =0 .
35 K and T = 1 . R xx in the low current regime at I = 0 . µ Ais similar to that measured at I = 1 µ A, where con-ductivity mainly results from the QUILLS mechanism.Measurements at currents above the breakdown current( > µ A) have shown a strong decrease of the relativeamplitude of these fluctuations. From these observationswe deduce that the QUILLS mechanism adds a conduc-tivity contribution that does not itself fluctuate with car-rier density. Only the term σ T,ν manifesting itself in thelow current regime IV has fluctuations. The resistanceshift due to the QUILLS mechanism by increasing cur-rent from 0 . µ A up to 1 µ A is particularly visible inthe inset of Fig. 3c which reports ln( R xx ) as a functionof n s between − . × cm − and − . × cm − away from the R xx minimum at n s = − . × cm − .It also shows a quite linear relationship between ln( R xx )and n s for both currents. At I = 1 µ A such a behavioris expected since this is a feature of the QUILLS mech-anism, as also observed at higher current in Fig. 9a andFig. 9b. On the other hand, at I = 0 . µ A in currentregime IV, QUILLS cannot account for the linear behav-ior since it is no more the dominant dissipation mecha-nism, as observed in current dependence of conductivityin Fig. 7. But it turns out that VRH with soft Coulombgap predicts ln( R xx ) ∝ − ( T ( ξ loc ) /T ) / ∝ − ξ − / . As-suming ξ loc ∝ ( ν − ν c ) − . , VRH also leads to a near2linear behavior of logarithmic conductivity with ν , sinceln( R xx ) should be proportional to ( ν − ν c ) . / . At ν = − . n s = − . × − cm − ), VRH would leadto T = 5 K and ξ loc = 800 nm. VRH can also explainthat fluctuation amplitude decreases as ν increases (inabsolute value), and that it decreases slightly with in-creasing temperature and more strongly with increasingcurrent. This mechanism indeed predicts longitudinalresistance fluctuations resulting from gaussian fluctua-tions of the localization length ξ loc with an amplitude δ ln( R xx ) ∝ T ( − / ξ ( − / δξ loc that decreases as temper-ature and ξ loc increase. Decreasing of the amplitude withcurrent can be explained by the heating effect by currentbecoming very significant far from ν = − µ A, as observed in regime III for T = 0 .
35 K when ν decreases from − . − . T ∗ (1 µ A) = 0 . ν = − . T ∗ (1 µ A) = 5 . ν = − .
8. Thus, the reproducible fluctuations of R xx ob-served are compatible with the existence of VRH in theregime IV at low current and low temperature. Coulombblockade in compressible islands surrounded by incom-pressible strips could also be considered as a source ofconductivity fluctuations. In this hypothesis, peaks ofconductance would correspond to the addition of oneelectron into islands and π ∆ n s r = 1 with ∆ n s thecarrier density width of peaks. Considering the experi-mental value of ∆ n s leads to typical radius of islands r between 43 nm to 70 nm, thus in agreement with valuesfound by others groups but also not so far fromthe puddle correlation length at low field (11 nm). Al-though it is difficult to conclude about the mechanism atthe origin of fluctuations, they can explain fluctuationsof the Hall resistance R H observable in Fig. 4a due to theunavoidable residual coupling between R H and R xx . V. CONCLUSION
To conclude, we have performed quantization tests ofthe QHE in µ m wide Hall bars based on bilayer andmonolayer exfoliated graphene deposited on Si/SiO sub-strate where electronic transport properties at low mag-netic field are mainly governed by the Coulomb inter-action of carriers with a high concentration of chargedimpurities. On the Hall plateaus corresponding to Lan-dau level filling factor near ν = 2 in the ML sample and ν = − R H re-spectively agrees with R K / R K / , in the limit of zero dis-sipation or at low current below a few µ A. These experi-ments are therefore the most accurate QHE quantizationmeasurements to date in monolayer and bilayer exfoliated graphene. They contribute to generalize the universalityproperty of R K to the bilayer graphene material for whichthe QHE was not investigated metrologically so far. Atlow magnetic field, charged impurities probably locatedin the silicon substrate at about 1 nm below the surfaceand with density near 2 × cm − reduce mobility,more strongly in the BL sample ( µ < V − s − )than in the ML sample ( µ < V − s − ). Thesevery efficient long-range scatterers also induce large spa-tial fluctuations of carrier density that stays bipolar up tofinite density values (2 × cm − in BL). Such densityinhomogeneity can notably be responsible for the satura-tion of L Φ observed in the BL sample at low temperatureand at finite density. In the QHE regime, dissipationleading to the QHE breakdown mainly occurs throughquasi-elastic inter-Landau level scattering (QUILLS) inpresence of high local electric fields. We claim thata high concentration of charged impurities very closeto graphene efficiently assist elastic inter-Landau levelstransitions. In addition, charged impurities induce astrong filling factor spatial inhomogeneity which is fa-vorable to the existence of large local electric fields. Atlow temperature and low current, it is observed in theBL sample that dissipation also follows an activation lawwith a typical energy of 95 µ eV, the origin of which isnot understood. As a result, breakdown is very antici-pated at currents as low as 1 µ A by enhancement of theinter-Landau level transitions which prevent from mea-suring the Hall resistance quantization with better ac-curacy at higher currents. This is even more tragic inthe small graphene samples produced by exfoliation tech-nique. The role of charged impurities present in the MLsample is expected to be qualitatively the same in theanticipated breakdown, but possibly with quantitativedifferences resulting from particularities of the Coulombpotential screening. We then conclude that the develop-ment of a graphene based quantum resistance standardable to challenge GaAs would require large samples withhigher mobility and more homogeneous carrier density.To achieve this, the role of substrate on which grapheneis deposited or grown has to be carefully addressed what-ever the graphene fabrication technique considered. Thisis a consequence of the high sensitivity of graphene elec-tronic transport properties to its environment.
ACKNOWLEDGMENTS
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