Magnetic-field driven ambipolar quantum Hall effect in epitaxial graphene close to the charge neutrality point
A. Nachawaty, M. Yang, W. Desrat, S. Nanot, B. Jabakhanji, D. Kazazis, R. Yakimova, A. Cresti, W. Escoffier, B. Jouault
MMagnetic-field driven ambipolar quantum Hall effect in epitaxial graphene close to thecharge neutrality point
A. Nachawaty,
1, 2
M. Yang, W. Desrat, S. Nanot, B. Jabakhanji, D.Kazazis,
5, 6
R. Yakimova, A. Cresti, W. Escoffier, and B. Jouault Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Universit´e de Montpellier, Montpellier, F-France. Laboratoire de Physique et Mod´elisation (LPM), EDST, Lebanese University, Tripoli, Lebanon Laboratoire National des Champs Magn´etiques Intenses (LNCMI-EMFL), UPR 3228,CNRS-UJF-UPS-INSA, 143 Avenue de Rangueil, 31400 Toulouse, France College of Engineering and Technology, American University of the Middle East, Egaila, Kuwait. Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud,Universit´e Paris-Saclay, C2N Marcoussis, 91460 Marcoussis, France Laboratory for Micro and Nanotechnology, Paul Scherrer Institute, 5232 Villigen-PSI, Switzerland. Department of Physics, Chemistry and Biology,Link¨oping University, SE-58183 Link¨oping, Sweden Univ. Grenoble Alpes, CNRS, Grenoble INP, IMEP-LaHC, F-38000 Grenoble, France Laboratoire National des Champs Magn´etiques Intenses,INSA UPS, CNRS UPR 3228, Universit´e de Toulouse,143 avenue de Rangueil, 31400 Toulouse, France
We have investigated the disorder of epitaxial graphene close to the charge neutrality point (CNP)by various methods: i) at room temperature, by analyzing the dependence of the resistivity on theHall coefficient ; ii) by fitting the temperature dependence of the Hall coefficient down to liquidhelium temperature; iii) by fitting the magnetoresistances at low temperature. All methods convergeto give a disorder amplitude of (20 ±
10) meV. Because of this relatively low disorder, close to theCNP, at low temperature, the sample resistivity does not exhibit the standard value (cid:39) h/ e butdiverges. Moreover, the magnetoresistance curves have a unique ambipolar behavior, which hasbeen systematically observed for all studied samples. This is a signature of both asymmetry inthe density of states and in-plane charge transfer. The microscopic origin of this behavior cannotbe unambiguously determined. However, we propose a model in which the SiC substrate stepsqualitatively explain the ambipolar behavior. I. INTRODUCTION
Undoubtedly, the best known exotic two-dimensionalelectron system is graphene. Among various excitingproperties, this material has demonstrated a half-integerquantum Hall effect (QHE) which is very robust intemperature, because of the extremely large energy sep-aration between the first Landau levels (LL) lying closeto the bottom of conduction and valence bands.The QHE in graphene is strongly influenced by disor-der and hence by the choice of the substrate. The quan-tum Hall plateaus observed in graphene on SiC (G/SiC)have a high breakdown current and appear at lowermagnetic fields with respect to graphene deposited onSiO . The quantum plateaus in G/SiC devices are muchlarger in magnetic field than those obtained in grapheneencapsulated in hBN, because they are stabilized bycharge transfer and disorder. Thanks to these proper-ties, it was recently demonstrated that G/SiC can act asa quantum electrical resistance standard, even in ex-perimental conditions relaxed with respect to the state-of-the-art in GaAs-based quantum wells. To date, fundamental and practical questions remainopen. In particular, the fate of the G/SiC quantumplateaus close to the charge neutrality point (CNP) hasstill to be elucidated. Achieving and controlling low dop-ing is of primary interest to use graphene as a resistance
FIG. 1: Magnetoresistance observed in sample G31 at lowtemperature T = 1 . I = 10 nA. A quan-tum Hall plateau at ρ xy = R K / B increases up to 0.5 T. At higher B , the Hall resistance changessign and a long plateau appears at ρ xy = R K /
2. The longitu-dinal resistance ρ xx shows a pronounced maximum when theHall resistance changes sign. ρ xy < >
0) corresponds toholes (electrons). standard at even lower magnetic fields, or in cryogen-freesystems. G/SiC could be also a material of choice fortesting theoretical models predicting additional quantumplateaus depending on the type of disorder.
However, there exist only a few experimental analy-ses of the transport properties close to the CNP, mainlybecause G/SiC is intrinsically strongly n -doped by theSiC substrate, thus requiring the system to be compen- a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug sated by a top gate. For graphene on SiO , close to theCNP, QHE reveals that electrons and holes coexist evenwhen the energy spectrum is quantized and the carrierspartially localized, but for G/SiC even this ratherintuitive picture has not been thoroughly tested. Theamplitude of the disorder potential fluctuation has beenevaluated by various methods, but the type of disor-der and its spatial and energy distribution close to CNPremain mostly unknown.In this paper, we present magnetotransport experi-ments in G/SiC to evaluate the disorder and to testthe stability of the QHE close the CNP. The results arelargely unexpected and reveal new physics in compari-son to what is observed on other substrates. We showthat not only electrons and holes coexist in high mag-netic fields, but the carriers also redistribute unexpect-edly as a function of magnetic field B , as illustrated inFig. 1. In this figure, the graphene is tuned very closeto the CNP. The magnetoresistance reveals a magneticfield driven ambipolar QHE. At low B , a quantized p -likeplateau starts to develop. However, when B is increasedfurther, this plateau collapses and is replaced by a quan-tized n -like plateau of opposite sign. We show that thisbehavior is robust and reproducible for the range of dop-ing where both types of carriers coexist and we proposea model based on disorder and in-plane charge transfer,after a detailed analysis of the sample disorder close tothe CNP. II. METHODS AND METHODOLOGYA. Sample fabrication
The SiC/G samples have been grown epitaxially onthe Si-face of a semi-insulating 4H-SiC substrate at ahigh temperature T = 2000 ° C. The as-grown sampleshave large uniform monolayer areas. Atomic force micro-scope analysis revealed the presence of SiC steps, approx-imately 500 nm wide and 2 nm high, uniformly covered bythe graphene layer. Additional Raman analysis revealedthe presence of elongated bilayer graphene patches, ap-proximately 10 µ m long and 2 µ m wide, covering around5% of the total surface.Hall bars of various size and geometry were then fab-ricated by standard electron-beam lithography. Thegraphene was covered by two layers of resist, as describedin Ref. 18. The resist acts as a chemical gate and stronglyreduces the intrinsic carrier density of the graphene layer.Magnetotransport measurements were performed onfour SiC/G Hall bars, named G14, G31, G21 and G34.The bars have a length of 420 µ m and a width of 100 µ m,except for G34, which has a width of 20 µ m. B. Corona preparation
After the lithography process, the graphene layer wassystematically n -doped with n (cid:39) × cm − at roomtemperature. Carrier density control was performed withion projections onto the resist bilayer covering graphene.Negative ions were produced by repeated corona dis-charges with a time interval of 17 s, following the methoddescribed in Ref. 19. The distance between the sampleand the corona source was 12 mm. Changes in the elec-tronic properties of graphene upon exposure to coronaions were detected by continuous measurements of theresistance and Hall coefficient K H at room temperature,using low magnetic field ( B = 0.05 T) and dc current I = ± µ A. The evolution of the resistivity ρ xx as a func-tion of the Hall coefficient K H during exposure to ions ispresented in Fig. 2. There is some point dispersion dueto the absence of correlation between the discharges andthe electrical measurements. However, the carrier den-sity clearly changes from its initial n -doping of 6 × cm − to a p -doping of 5 × cm − after a few hundredscycles of corona discharge. After this point, additionalion projections are inefficient to increase further the p -doping. When the discharges are stopped, the carrierdensity drifts slowly towards n -doping and K H stabilizesaround +1 k Ω /T within a few hours. The initial car-rier density is not recovered even when the sample is leftseveral months under ambient atmosphere. III. ESTIMATION OF DISORDERA. Disorder estimated from K H ( ρ xx ) at roomtemperature To describe the sample evolution seen in Fig. 2, weuse the usual equations that give the conduction of ahomogeneous sample, in which both electrons and holesparticipate in the conduction in parallel because of ther-mal activation. The model is detailed in Annex. Themodeled ( ρ xx , K H ) curve is plotted for µ e = µ h = 4 , /Vs and T = 300 K as a blue solid line in Fig. 2a when µ ch spans the energy window around the CNP. Here, µ e , µ h and µ ch are the electron mobility, hole mobility andchemical potential respectively. The model fits fairly wellthe data but with an obvious deviation on the hole side,where the K H coefficient is overestimated. The asymme-try of the ( ρ xx , K H ) curve indicates that µ e and µ h differ.The data can be modeled more precisely in two ways:i) the mobility ratio µ e /µ h increases when the numberof deposited ions increases because negative ions have alarger cross section for p -type charge carriers and theirpresence can decrease significantly the hole mobility ;ii) µ e /µ h > s as an additional fit-ting parameter, it is possible to fit the asymmetry of the ρ xx ( K H ) curve, as shown by the black line in Fig. 2. The FIG. 2: ρ xx vs K H during the corona preparation for sampleG14. The initial state before the ion deposition is indicatedby the blue filled circle. The colored squares are various statesfrom which the G14 sample has been cooled down at low tem-perature, see Fig. 4a. The blue line is a fit without disorder( µ e = µ h = 4 ,
300 cm /Vs), the black line a fit where the dis-order is taken into account ( s = 25 meV, µ h = 2 ,
750 cm /Vs, µ e /µ h = 1 . fitting parameters are s =25 meV, µ e /µ h = 1.5 and µ h =2,750 cm /Vs. All these parameters are in agreementwith the literature. B. Disorder estimated from n Hall ( T ) The disorder can also be estimated from the tempera-ture dependence of the Hall coefficient. Fig. 3a shows theevolution of the Hall density n Hall = 1 / ( K H e ) of sampleG14 when T is lowered from room temperature down to1.7 K. The initial corona preparation has been chosen toillustrate that one should not expect a T dependenceof the Hall density as a universal trend, as assumed inRef. 5. In Fig. 3a, the Hall density diverges at a criticaltemperature T c (cid:39)
140 K, is negative below T c and posi-tive above T c . This divergence is a clear signature thatboth electrons and holes participate in the conductionand from Eq. 5, T c can be identified as the temperaturefor which n h µ h = n e µ e . We stress that this change ofsign of n Hall does not imply that the net carrier densitychanges with T . Let us assume, as in Refs. 20 and 5,that the net carrier density n = n e − n h does not dependon T . It is possible to fit the data taking into account 3parameters: the disorder amplitude s , the Fermi energyat zero temperature E F = lim T → µ ch , and the mobilityratio µ e /µ h . From the fit of the temperature dependenceshown in Fig. 3a, we extract s = 14 meV, E F = -4 meVand µ e /µ h = 1 .
13. The fit matches very well the data,except in the vicinity of T c , where the experimental erroris larger as the measured voltage cancels. Fig. 3b showsthat n e and n h increase quadratically with T . FIG. 3: (a) Temperature dependence of the Hall carrier den-sity n Hall = 1 / ( ρ xy e ) for sample G14 (open circles). A goodfit (red solid curve) is obtained by including disorder ( s = 14meV) with two other fitting parameters: the Fermi energy E F =-4 meV at T = 0 K and the mobility ratio µ e /µ h = 1.13.By taking s = 0, the experimental data cannot be fitted prop-erly. (b) electron and hole density vs T given by the best fitin (a) are indicated by black and red solid lines. From various n Hall ( T ) fits corresponding to 11 differentinitial corona preparations, we found s = (21 ±
12) meVand µ e /µ h = 1 . ± .
15. These parameters are roughlyin agreement with those extracted from the K H ( ρ xx ) fit.However, the precision is not good enough to evidencea possible evolution of µ e /µ h with the progressive iondeposition. IV. AMBIPOLAR QUANTUM HALL EFFECTA. Experimental results
Fig. 4a,b shows the longitudinal and transverse magne-toresistances at T = 1 . B = 0 T, see Fig. 4a. The peak valueis close to h/e at T = 1.7 K, and largely exceeds thepseudo-universal value of ρ xx (cid:39) h/ e usually observedin graphene on SiO close to the CNP. We checked that ρ xx increases even further, up to (cid:39)
70 kΩ, when the tem-perature is decreased down to 280 mK. This insulatingbehavior, see Fig. 4c, shares strong similarities with aprevious experiment, where an insulating behavior ob-served in graphene encapsulated in hBN was attributed FIG. 4: (a) Longitudinal and (b) transverse magnetoresistances observed in sample G14 at T = 1 . I = 10 nA. Thedifferent curves correspond to five different sample preparations, indicated by the colored squares in Fig. 2. (c,d) Temperaturedependence of the longitudinal and Hall magnetoresistances ( ρ xx and ρ xy ) for an additional sample preparation, close to thered square in Fig. 2. to Anderson localization. Following Ref. 23, such a local-ization can only be observed if the carrier density in thepuddles is small enough (around 10 cm − in Ref. 23), ingood agreement with our own estimation of the disorderin G/SiC ( s (cid:39)
25 meV yields n (cid:39) . × cm − .)We now focus on the Hall magnetoresistance. First,the usual half-integer QHE is observed for high p -doping( n h (cid:39) cm − , panel b, cyan curve) or low n -doping( n e (cid:39) cm − , panel b, black curve). Between thesetwo dopings, the plateaus are not well defined. How-ever, the Hall magnetoresistance systematically followsa remarkable behavior. Increasing B , there is first adecrease of ρ xy , followed by a saturation at or before ρ xy = − R K / p -like plateau), where R K = h/e . Then, ρ xy collapses, changes its sign and finally stabilizes ata positive value close to R K / n -like plateau). Usingpulsed magnetic fields, we checked that there is no morechange of sign of ρ xy at least up to 30 T. We neverobserved the opposite transition, from n -like to p -likeplateau.In some other measurements, as shown in Fig. 1, awell defined plateau has been observed at R K /
2. Themagnetic field at which ρ xy cancels is also related withthe appearance of a bump on ρ xx , see Fig. 1 and Fig. 4a.The position of this bump seems to be also related with the initial doping obtained at room temperature. Whengraphene is slightly p -doped at room temperature, this ρ xx bump appears at low B (red curve in panel a). Whengraphene becomes progressively more p -doped, the bumpshifts to higher B (green and blue curves in panel a) andfinally disappears (cyan curve).Finally both ρ xx and ρ xy show a clear temperature de-pendence, as shown in panels c,d. The additional ρ xx bump goes to higher B when T increases and then dis-appears. The ρ xy behavior is even more striking, as theambipolar behavior disappears above T (cid:39)
20 K and isreplaced by an almost quantized plateau correspondingto an apparent p -doping. B. Inadequacy of the standard two fluid model
The two fluid Drude model (Eqs. 5 and 6) can explainin some cases a change of sign of ρ xy . This is because atlow B , ρ xy (cid:39) ( n e µ e − n h µ h ) / ( n h µ h + n e µ e ) whereas athigh B , ρ xy (cid:39) / ( n e − n h ). However, Fig. 4d shows thatin the high field limit B (cid:39)
10 T, ρ xy is positive at low T . This is in contradiction with the previous tempera-ture analysis, which indicates that n e − n h < T .Moreover ρ xy is negative at low B , which, in the frame- FIG. 5: (a) Top-view sketch of the Hall bar, embedding re-gions (HD, orange color) whose CNP is at higher energy thanthe surrounding graphene region (LD, blue color). The edgestates of the devices are also presented as dark blue lines. (b)Profile of the LL near the CNP. The LD and HD regions areindicated by blue and orange lines respectively. The LL splitsat the sample edges. When B increases, the electrochemicalpotential tries to maintain a constant total charge and maycross the CNP of the LD region at a finite B . (c) Estimatedcarrier density n LD in the LD region vs B , for the model givenin the main text, µ LDCNP = 14 meV, Γ = 15 meV, α = 30 %. work of two fluid Drude model, indicates that holes aremore mobile than electrons, in contradiction with all ourprevious analyses. Additionally, the Drude model is notvalid in the quantum Hall regime studied here and pre-dicts only monotonous positive longitudinal magnetore-sistance, in contradiction with the observation of a bumpin ρ xx at finite B . Obviously, the two fluid model fails toexplain the magnetoresistances and another explanationis needed to account for all of these observations. C. Disorder-induced charge transfer model
A rather similar ambipolar behavior was reported ingraphene on SiO substrate and interpreted as being dueto important disorder. Here, ρ xy changes sign at muchsmaller B and the QHE is rather well preserved, suggest-ing that the disorder amplitude is much smaller.Besides, charge transfer has already been identified asbeing a major source of charge redistribution in G/SiCwhen the magnetic field evolves. The SiC interface statesand graphene are coupled by a quantum capacitance, which depends on the graphene density of states (DOS)and hence on B . Additional in-plane charge transfers,induced by disorder, have also been identified in G/SiCwhen the overall top gate is degraded. Here, the photo-chemical top gates have kept their integrity (as routinelyobserved optically). Nevertheless, we show in the fol-lowing that in-plane charge transfer explain the observedambipolar QHE.Let us assume that the main part of the Hall bar ishomogeneous, with a low and uniform p -doping. Thisregion is referred to as the low doped (LD) region. How-ever, the device also contains a few highly doped (HD)regions where the p -doping is higher (Fig. 5a). The keyingredient of the model is that, in the quantum regime,the conductivity is governed by the edge states of the LDregion. The HD regions do not percolate, do not partici-pate in the conduction and only act as additional chargereservoirs.The possible microscopic origins of these regions aremultiple. For instance, the role of ionized acceptors ina quantizing magnetic field is well documented in theliterature. They trap electrons, create an impurity bandin the DOS of the LLs at high energies and shift the posi-tion of the quantum plateaus in two-dimensional electrongases. In our case, the ions trapped in the resist couldact as magneto-acceptors, but we have no definitive prooffor this scenario.In what follows, we propose another microscopic ori-gin for the HD regions. It is well known that the SiCsurface is not flat but has a step-like structure. Thegraphene layer is lifted from the SiC surface close to theSiC step edges. There, the quantum capacitance betweengraphene and the SiC interface is reduced, leading to alarger p -doping. At low magnetic field, graphene can be prepared insuch a way that µ ch is below the delocalized states of theLD region and the conductivity is governed by edge stateswhich have a hole character, as illustrated in Fig. 5b(green line). When the magnetic field is increased, theLL degeneracy increases. As the total net carrier density(including LD and HD regions) tends to remain constant, µ ch moves to higher energy (red line). When the mag-netic field is high enough, µ ch has shifted between theLL energies of the LD and HD regions. There, the LDregion has become n -doped whereas the HD puddles arestill p -doped. The conductivity is then governed by theLD edge states which have now an electron character.Experimentally, the magnetic field at which ρ xy reversescan be as low as B m = 0 . (cid:112) (cid:126) /eB m (cid:39)
40 nm.Below we derive numerical estimates for this modelassuming that the inhomogeneity comes from the step-like structure. First, because of the quantum capacitancetaking place between states at the G/SiC interface andgraphene , any modification of the DOS structure dueto the magnetic field induces charge displacement. Foreach region LD and HD, the balance equation gives: n i ( µ ch ) = − n g + β i ( A + µ i CNP − µ ch ) (1)where i labels the LD and HD regions, n g is the gatecharge density, A is the workfunction difference betweenundoped graphene and the interface states, β i is an effec-tive density of states, µ i CNP is the potential of the CNPin region i . We choose µ LDCNP = 0 as a reference. β i isgiven by β i = (cid:15) / ( (cid:15) + e d i γ ) where (cid:15) is the vacuumdielectric constant, γ is the density of interface states, d i is the distance between graphene and the interface.At equilibrium, the electro-chemical potential is thesame in the whole sample. Following Ref. 25, the equa-tion to solve is then obtained by summing the contribu-tions of n HD and n LD :(1 − α ) n LD ( µ ch ) + αn HD ( µ ch ) = − n g +(1 − α ) β LD ( A − µ ch )+ αβ HD (cid:0) A + µ HDCNP − µ ch (cid:1) , (2)where α is the proportion of HD region. On the leftside of this equation, both n LD and n HD can be numeri-cally estimated at a given magnetic field, where the DOSis given by a sum of Landau levels, each with a fixedGaussian broadening Γ. The chosen parameters are n g = 1 . × cm − , A = 0.4 eV, γ =5 × cm − eV − ,Γ= 15 meV. They have been chosen in accordance withthe literature. We assume that the LD and HD regionsonly differ by a very small difference between d LD and d HD : d LD = 0.3 nm and d HD = 0 . µ HDCNP = 14 meV (cid:39) s .Fig. 5c shows n LD ( B ) obtained from solving Eq. 2, with α = 30%. As the model assumes that the conduction isgoverned by the LD region alone, the magnetic field de-pendence of n LD ( B ) reproduces the ambipolar behaviorof the transverse magnetoresistance. Note that a qualita-tive agreement of the T -dependence is also obtained. Atlow B , n LD remains essentially T -independent, in agree-ment with the previous n Hall ( T ) analyses. By contrast,the critical magnetic field at which the sign of ρ xy ( i.e. the sign of n LD ) reverses is shifted to higher B when T increases. Experimentally this trend is indeed observed,with a shift which is even more pronounced, suggestingthat some other parameters ( e.g. A or γ ) could also be T -dependent.Finally, this model relates unambiguously the ρ xx bump to the conduction through the delocalized states ofthe LD region. The model also predicts that this bumpshifts to higher B when the initial p -doping increases, asindeed experimentally observed. V. CONCLUSION
To conclude, we have investigated the disorder of epi-taxial graphene close to the charge neutrality point byvarious analyses of the transport properties. All theseanalyses converge to give a disorder amplitude of the order of a few tens of meV. Remarkably , the magne-toresistance curves have an ambipolar behavior drivenby the magnetic field. We interpret this as the signatureof a very specific disorder combined with in-plane chargetransfer between different regions in the graphene layer.The origin of disorder cannot be unambiguously deter-mined but numerical estimations show that it could berelated to the stepped SiC substrate.
Acknowledgment
We thank F. Schopfer and W. Poirier (Laboratoire na-tional de m´etrologie et d’essais, France) for fruitful dis-cussion. This work has been supported in part by theFrench Agence Nationale pour la Recherche (ANR-16-CE09-0016) and by Programme Investissements d’Avenirunder the program ANR-11-IDEX-0002-02, referenceANR-10-LABX-0037-NEXT. Part of this work was per-formed at LNCMI under EMFL proposal TSC06-116.
Annex: Model of disorder
To describe the sample evolution seen in Fig. 2, wereproduce below the usual equations that give the con-duction of a homogeneous sample, in which both elec-trons and holes participate in the conduction in parallelbecause of thermal activation. The total electron densityis given by: n e = (cid:90) ∞−∞ D e ( E ) f ( E − µ ch ) dE, (3) D e ( E ) = D Eθ ( E ) is the density of states (DOS)for electrons, D = g s g v / (2 π (cid:126) v F ), g s = 2 and g v = 2 are the spin and valley degeneracies, f =1 / [1 + exp(( E − µ ch ) /k B T )] is the Fermi distributionfunction, θ is the Heavyside function and µ ch is the chem-ical potential. The total hole density is given by: n h = (cid:90) ∞−∞ D h ( E )(1 − f ( E − µ ch )) dE, (4)where D h ( E ) = D e ( − E ) is the DOS for holes. At lowmagnetic fields µ e B, µ h B (cid:28)
1, the Hall and longitudinalresistivities are given by: ρ xy = − ρ yx = − e ( n h µ h − n e µ e ) + µ h µ e B ( n h − n e )( n h µ h + n e µ e ) + µ h µ e ( n h − n e ) B B (5)and ρ xx = 1 e n h µ h + n e µ e + ( n e µ e µ h + n h µ h µ e ) B ( n h µ h + n e µ e ) + µ h µ e ( n h − n e ) B (6)where µ e and µ h are the electron and hole mobility re-spectively, − e is the electron charge.In the limit B →
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