Integer Quantum Hall Effect in Trilayer Graphene
A. Kumar, W. Escoffier, J.M. Poumirol, C. Faugeras, D. P. Arovas, M. M. Fogler, F. Guinea, S. Roche, M. Goiran, B. Raquet
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Integer Quantum Hall Effect in Trilayer Graphene
A. Kumar , W. Escoffier ∗ , J.M. Poumirol , C. Faugeras , D. P. Arovas ,M. M. Fogler , F. Guinea , S. Roche , , M. Goiran and B. Raquet Laboratoire National des Champs Magn´etiques Intenses (LNCMI), CNRS-UPR3228,INSA, UJF, UPS, Universit´e de Toulouse, 143 av. de rangueil, 31400 Toulouse, France Laboratoire National des Champs Magn´etiques Intenses (LNCMI),CNRS-UPR3228, INSA, UJF, UPS, 25 rue des Martyrs, 38042 Grenoble, France Department of Physics, University of California at San Diego,9500 Gilman Drive, La Jolla, California 92093, USA Instituto de Cienca de Materiales de Madrid, CSIC, Cantoblanco E28049 Madrid, Spain CIN2 (ICN-CSIC) and Universidad Autonoma de Barcelona,Catalan Institute of Nanotechnology, Campus UAB, 08193 Bellaterra (Barcelona), Spain and ICREA, Instituci ´ o Catalana de Recerca i Estudis Avan¸cats, Barcelona, Spain (Dated: October 2, 2018)The Integer Quantum Hall Effect (IQHE) is a distinctive phase of two-dimensional electronicsystems subjected to a perpendicular magnetic field. Thus far, the IQHE has been observed insemiconductor heterostructures and in mono- and bi-layer graphene. Here we report on the IQHEin a new system: trilayer graphene. Experimental data are compared with self-consistent Hartreecalculations of the Landau levels for the gated trilayer. The plateau structure in the Hall resistivitydetermines the stacking order (ABA versus
ABC). We find that the IQHE in ABC trilayer grapheneis similar to that in the monolayer, except for the absence of a plateau at filling factor ν = 2. Atvery low filling factor, the Hall resistance vanishes due to the presence of mixed electron and holecarriers induced by disorder. PACS numbers: 61.72.Bb, 71.55.Cn
More than 30 years after its initial discovery in semi-conductor two dimensional electron gases (2DEG), theInteger Quantum Hall Effect (IQHE) remains one ofthe most fascinating phenomena in condensed matterphysics. The recent discovery of graphene [1] boostedthis research field by providing a new 2D system whereDirac-like electronic excitations with Berry’s phase π leads to a new form of IQHE [2, 3], with plateaus at σ xy = ( n + ) ge /h , where g is the Landau level degen-eracy due to spin and valley degrees of freedom. Soonafterward, a third type of IQHE was reported in bilayergraphene, where the 2 π Berry’s phase of charge carriersresults in a conventional quantization sequence, exceptthat the last Hall plateau is missing [4]. As the dynamicsof charged carriers change every time an extra graphenelayer in added, it was theoretically anticipated that theLandau Level (LL) spectrum of N -layer graphene sys-tems would result in distinctive IQHE features arisingfrom an N π
Berry’s phase. In trilayer graphene, thezero-energy LL is expected to be 12-fold degenerate sothat the Hall resistance plateau sequence follows the sameladder as in graphene, but the plateau at ν = ± ∗ Author to whom correspondance should be addressed of both high field magneto-transport and Raman spec-troscopy, we clearly identified a trilayer graphene sam-ple and report on a fourth type of IQHE in this sys-tem. Self-consistent Hartree calculations of the gatedtrilayer Landau levels based on the Slonczewski-Weiss-McClure (SWMC) tight binding model [10–12] have beenperformed and favorably compared to the experimentaldata. Thus, a comprehensive knowledge of the under-lying electronic properties of trilayer graphene is pre-sented, together with an unambiguous determination ofthe stacking order.Many few layer graphene flakes were deposited onto a d = 280 nm thick thermally grown silicon oxide on sili-con substrate using the standard micro-mechanical exfo-liation of natural graphite. The flake measured in thisstudy was first roughly identified as few-layer grapheneusing optical microscopy. The Raman scattering spec-trum was measured at room temperature using a con-focal micro Raman scattering set-up using He-Ne laserexcitation ( λ = 632 . ∼ µ m diameter spot. The 2D band feature(also called G ′ feature) of this sample is shown in figure1-c and appears in the form of a multicomponent featurecharacteristic of multi-layer graphene specimens [13], dif-ferent from the one observed for a mono-layer graphenespecimen processed in the same way (also shown in fig-ure 1-c). The experimental IQHE of the sample understudy is displayed in figure 1-d, together with bi-layerand mono-layer graphene fingerprints of other samples.These samples have an equivalent carrier density andsimilar mobility (see legend of figure 1 for details). Forthe sample under consideration, the sequence of the Hall Bi-layer b) Bi-layerTri-layer Mono-layer d) Mono-layerSample understudy (tri-layer) R XY ( k ) B (T)
Mono-layerSample understudy (tri-layer) a) c) S c a tt e r i ng i n t. ( a r b . un i t s ) Raman shift (cm -1 ) =(e.B)/(n.h) R XY ( un i t o f h / e ) / / / / / / FIG. 1: Identifying the sample under study as a trilayer graphene. a) and b) schematic representations of the IQHE in mono-,bi- and tri-layer graphene. The degeneracy of the zeroth Landau Level, equally shared by electrons and holes, is f = g -folddegenerate in graphene, f = 2 g -fold degenerate in bilayer graphene and f = 3 g -fold degenerate in trilayer graphene where g = 4, leading to a sequence of the Hall resistance plateaus successively shifted by 2 h/e Ω. Possible LL degeneracy lifting athigh field is not represented in these drawings. c) Raman spectrum of trilayer graphene and mono-mayer graphene measured onthe same substrate. d) Experimental IQHE in tri, bi and mono-layer graphene for equivalent carrier density n = 3 . × cm − and quasi-equivalent mobility. The optical image of the trilayer graphene sample is shown in insert. As contact 1 was provendefective, a constant current of i = 1 µ A is injected through contacts 2 and 4. The Hall resistance is measured between contacts3 and 5. resistance plateaus is described by R xy = h/νe where ν = 6, 10, 14 ... . Taking into account the results of bothRaman spectroscopy and IQHE, we thus unambiguouslyidentify the sample as trilayer graphene. This assertionis further reinforced when comparing the optical contrastof the flake with other mono-layer graphene flakes undermicroscope (see the supplementary information). As ex-pected, the IQHE in trilayer graphene is indistinguish-able from its monolayer counterparts except at very highfield where the ν = 2 quantized Hall resistance plateauis absent.Before further proceeding with a detailed presenta-tion of IQHE in graphene trilayer, let us review someimportant experimental considerations. Once identified,the sample was connected to multi-terminal electrodesmade by successive thermal evaporation of Ti [5 nm] and Au [40 nm] through a PMMA mask (see figure1-d, inset). The flake was further patterned in theHall bar geometry using electron beam lithographyand oxygen plasma etching. The contact resistanceswere estimated to a few hundred ohms each. A gatevoltage V g between the sample and the substrate wasused to electrically induce electrons or holes up todensities of 6 × cm − . The sample was annealedalternatively in vacuum and in helium gas at a tem-perature T=110 ◦ C. The induced carrier density as afunction of the gate voltage follows the simple planecapacitor model n = κ ǫ ( V g − V CNP ) /ed where ǫ is thevacuum dielectric permittivity, κ = 3 . and V CNP = − .
75 V is the gatevoltage required to reach the Charge Neutrality Point(CNP), indicating the presence of a n -type residual n=5.90x10 cm -2 n=5.10x10 cm -2 n=4.94x10 cm -2 n=4.82x10 cm -2 n=4.30x10 cm -2 R XY ( k ) B (T) =6=10=14-4 -3 -2 -1 0 1 2 3 4 5 61234567
Vg (V) (b)
T=4.2 K R - ( k ) n (x10 cm -2 )T=4.2 K (a) CN P n ~ 0.85 x 10 cm -2 ~ 1200 cm / V.s FIG. 2: a) Electrical resistance of the sample as a functionof carrier concentration at T = 4 . [14] together with thededuced field-effect mobilities at residual carrier density of0 . × cm − b) Quantized Hall resistance profile in tri-layer graphene for selected charge carrier density far fromCNP. doping estimated to n = 0 . × cm − . Figure 2-ashows the typical two-probe longitudinal resistance asa function of the carrier density at T = 4 . µ e ∼ ±
100 cm . V − · s − for electrons and µ h ∼ ±
300 cm . V − · s − for holes.The Hall resistance displays well defined plateaus at R xy = h/ ( νe ) with ν = 6, 10, and 14. We noticethat the Hall resistance slightly overshoots the ν = 6resistance plateau at very high magnetic field. Thissurprising feature triggered the need for a detailedtheoretical analysis of the LL spectrum [15, 16] whichgoes beyond the simple model presented earlier in theintroduction.To model the gated trilayer, we employ an SWMCtight-binding parametrization of the local hopping am-plitudes [10, 11] and treat the Coulomb interactions via aself-consistent Hartree approach. There are two possiblestacking orders to consider: Bernal (ABA) and rhom-bohedral (ABC), both illustrated in figure 3. For the Bernal case, we take γ = 3000 meV, γ = 400 meV, γ = −
20 meV, γ = 300 meV, γ = 150 meV, and γ = 38 meV. In addition, there is an on-site energyshift of ∆ = 18 meV for each c -axis neighbor. For therhombohedral case [12], the parameter γ does not enter.When a gate voltage is applied to the device, the chargecarriers will be distributed among the three layers. In-tegrating Gauss’ law across the layers provides a set ofthree equations which have to be solved self-consistentlyin order to obtain the trilayer potentials (see supplemen-tary informations for details). We assume that the sys-tem contains stray charges which are located on the toplayer of the device. These charges are responsible forthe offset voltage V CNP necessary to bring the system toneutrality. The zero temperature Hall conductivity shiftsby e /h when the electrochemical potential crosses thecenter of a Landau level. We include Zeeman splitting,assuming g = 2. Finally, some disorder is introduced inthe model through a convolution of the density of stateswith a square distribution of half-width W = 10 meV,estimated from the sample’s mobility.Figure 3 shows the LL energies and theoretical Hallresistance for both ABC and ABA stacking for gate volt-age V g = +50 V, along with the experimental resultsfor R xy (the full set of IQHE data is reported in thesupplementary information). For fields up to 40 T, themeasurements agree fairly well with the theoretical pre-dictions for the ABC trilayer, and fail to reproduce thetheoretical Hall plateau sequence for the ABA trilayer.The contrasting plateau sequences for ABC and ABAtrilayers arise due to the significant differences in theirrespective Landau level structures, as is evident in figure3. Indeed, the rhombohedral stacking order accounts forthe absence of Hall plateaus at some filling factors, like ν = 8 and ν = 12. A similar effect occurs in monolayergraphene, where the plateaus at ν = 4, 8, 12, . . . aremissing due to valley degeneracy arising from the inver-sion symmetry of the honeycomb lattice. With no biasvoltage, the ABC trilayer is inversion symmetric, whilethe ABA trilayer is not [17]. The presence of an electricfield across the graphene layers, due to the gate voltageinduced charge redistributions [9, 18], breaks the latticeinversion symmetry. However, this field-induced splittingbetween LLs arising from different valleys is small in theABC case, except for the six levels closest to the Diracenergy. Neglecting Zeeman splitting, quantum Hall stepsof amplitude ∆ σ xy = 2 .e /h should be observed for eachplateau-to-plateau transition. This holds in particular forthe Bernal type of stacking as the LLs originating fromvalleys K and K ′ are quite distinct from each other dueto the absence of inversion symmetry. On the other hand,the ABC-stacked LL band-structure is much less affectedby electrostatic effects. In high enough magnetic field,the LLs evolve roughly by bunches of four and, whendisorder effects are taken into account, lead to quantumHall steps of ∆ σ xy = 4 .e /h , as experimentally observed.It has been demonstrated that the stacking in graphenetrilayers can be analyzed making use of micro- Raman FIG. 3: a) Illustration of ABC-stacked trilayergraphene (top panel), theoretical Landau levelstructure (middle panel) and quantized Hall resis-tance (bottom panel) using the following parame-ters : V g = 50 V, V CNP = − .
75 V, T = 4 . W = 10 meV and g = 2. In the middle panel, thesolid and dashed curves indicate the Landau levelsoriginating from valleys K and K’, respectively. Inthe bottom panel, the experimental Hall resistanceis displayed as the red curve, superimposed withtheoretical results as the black line. b) For com-parison, the same information for the ABA stackedtrilayer graphene spectroscopy [19]. However, the Raman spectral signa-ture of stacking is quite subtle and undoubtedly incon-clusive for the present experiment (figure 1-c). Althoughit has been found that ABC-stacked graphene trilayerflakes are much rarer than their ABA-stacked counter-part using the micro-mechanical cleavage method, thetheoretical analysis of IQHE allows an unambiguous de-termination of the rhombohedral staking order for thesample considered here.Interestingly, the IQHE fails to be reproduced atvery low filling factor (high magnetic field and lowcarrier density). To further investigate this issue, weanalyze the Hall resistance for various charge carrierconcentrations close to CNP. Focusing at figure 4-a,we begin the analysis with the Hall resistance for V g = +40 V ( n = 4 . × cm − ), which displays welldefined quantized Hall plateaus. As the gate voltageis decreased, the corresponding set of curves is shiftedso that the quantized plateaus occur at lower magneticfield, as expected from IQHE theory. On the other hand, as the Fermi energy is driven closer to the CNP, the lowfield Hall effect is no longer linear reflecting the presenceof electrons and holes that both contribute to transport.In the range −
50 V < V g < +20 V, the initial ratiobetween electron/hole density evolves as the magneticfield is increased to accommodate the field-inducedredistribution of quantum states available in the lowestLL [21]. Actually, the electron and hole densities tendto equilibrate and consequently the Hall resistancevanishes at high field. This effect is a hallmark of thedisordered 2DEG [22], where the presence of electronand hole puddles allows both types of carriers for agiven Fermi energy close to CNP. Indeed, the inevitablepresence of charged impurities or lattice imperfectionsintroduces long-range disorder and leads to a spatiallyinhomogeneous and fluctuating potential landscape,resulting in some local accumulation of charge carriers[23–27]. Alternatively, its consequences on charge trans-port can be monitored through temperature-dependentmeasurements. The main frame of figure 4-b shows the -60-40-200204060 R - ( k ) V g ( V ) T ( K ) a)b) Vg = 40V Vg = 30V Vg = 10V Vg = 0V Vg = -5V Vg = -10V Vg = -15V Vg = -20V Vg = -25V Vg = -30V R XY ( k ) B (T) R m a x ( k ) a t CN P T (K)
0 50 100 150 200 250 300 = 1150 cm / V.s= 58 meV FIG. 4: a) Hall resistance for various back gate voltage inthe close vicinity of the CNP. Notice that the Hall resistancetends to vanish at very high magnetic field as the ratio be-tween electrons and holes tends to equilibrate. b) Longitudi-nal resistance R − as a function of gate voltage and temper-ature. The maximum resistance as a function of temperatureis shown in the insert. A theoretical fit according to the modeldeveloped in [20] is also shown. The theoretical adjustmenttakes into account the increase of carriers density with tem-perature and the resulting activated conductance through aninhomogeneous system made of electron/hole puddles. two-probe longitudinal resistance R − with varyinggate voltage and temperature. The resistance increaseswith decreasing temperature in the vicinity of the CNPwhile it remains constant for higher doping. This trendis consistent with what has been previously observed in disordered-like samples, for which the highly inho-mogeneous carrier density can lead to both metallicand activated transport. The temperature evolution ofthe resistance maximum at CNP is given in the insetof figure 4-b. A good agreement with the predictionsof reference [20] is achieved with fitting parameters µ = 1150 cm / V · s and ∆Φ = 58 meV, where ∆Φstands for the rms amplitude of the fluctuating potentiallandscape. This value is slightly larger than thosereported in the literature for bilayer graphene [28, 29](∆Φ = 21 . µ ≈ / V · s), as predicted forFLG [30].To summarize, we report for the first time the observa-tion of a new form of IQHE in a gated graphene trilayer.The filling factor sequence associated with the quantizedHall resistance plateaus is identical to that for graphene,but the plateau at ν = 2 is missing. The experimentaldata are supported by a theoretical analysis whereboth Bernal and rhombohedral stacking have beenconsidered. The main experimental IQHE features arereproduced only for the rhombohedral case, emphasizingthe importance of stacking order in the electronic prop-erties of graphene trilayers. Usually, graphene trilayersdeposited on SiO show a poor mobility of the order of1000 cm / V · s [29–31], justifying the need for very highmagnetic field for IQHE studies. Important progressesare expected in samples with higher mobility, e.g. insuspended or boron nitride-deposited graphene trilayers. This research was supported by EuroMagNETII program under EU Contract No. 228043,by the French National Agency for Research(ANR) under Contract No. ANR-08-JCJC-0034-01, and by the technological platform of LAAS-CNRS, member of the RTB network. One ofthe authors (D.P.A.) was supported by NSFgrant DMR-1007028. 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