Coulomb Drag of Massless Fermions in Graphene
Seyoung Kim, Insun Jo, Junghyo Nah, Z. Yao, S. K. Banerjee, E. Tutuc
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Coulomb Drag of Massless Fermions in Graphene
Seyoung Kim, Insun Jo, Junghyo Nah, Z. Yao, S. K. Banerjee, and E. Tutuc ∗ Microelectronics Research Center, The University of Texas at Austin, Austin, TX 78758 Department of Physics, The University of Texas at Austin, Austin, TX 78712 (Dated: July 16, 2018)Using a novel structure, consisting of two, independently contacted graphene single layers sepa-rated by an ultra-thin dielectric, we experimentally measure the Coulomb drag of massless fermionsin graphene. At temperatures higher than 50 K, the Coulomb drag follows a temperature and car-rier density dependence consistent with the Fermi liquid regime. As the temperature is reduced,the Coulomb drag exhibits giant fluctuations with an increasing amplitude, thanks to the interplaybetween coherent transport in the graphene layer and interaction between the two layers.
PACS numbers: 73.43.-f, 71.35.-y, 73.22.Gk
Bilayer systems formed by two layers of carriers inclose proximity are a fascinating testground for electronphysics. In particular, the prospect of electron-hole pair(indirect exciton) formation, and dipolar superfluidity has fueled the research of electron-hole bilayers inGaAs/AlGaAs heterostructures . Graphene is a par-ticularly interesting material to explore interacting bilay-ers. The symmetric conduction and valence bands, andthe large Fermi energy favor correlated electron states atelevated temperatures . The zero energy band-gap al-lows a seamless transition between electrons and holes ineach layer, and obviates the large inter-layer electric fieldrequired to simultaneously induce electrons and holes inGaAs bilayers . Coulomb drag, a direct measurement ofinter-layer electron-electron scattering can provide in-sight into the ground state of two- and one- electronsystems, as well as correlated bilayer states . Here wedemonstrate a novel, independently contacted graphenebilayer, and investigate the Coulomb drag in this system.Two main ingredients render the realization of inde-pendently contacted graphene bilayers challenging. First,an ultra-thin yet highly insulating dielectric is requiredto separate the two layers. Second, a method to positionanother graphene layer on a pre-existing device with min-imum or no degradation is needed to create the secondlayer of the structure investigated here. The fabricationof our independently contacted graphene bilayers is de-scribed in Fig. 1. First, the bottom graphene layer ismechanically exfoliated onto a 280 nm thick SiO dielec-tric, thermally grown on a highly doped Si substrate.E-beam lithography, metal lift-off, and etching are usedto define a Hall bar on the bottom layer [Fig. 1(a)]. A7 nm thick Al O is then deposited on the bottom layerusing a 2 nm oxidized Al interfacial layer, followed by 5nm of Al O atomic layer deposition . The second, topgraphene layer is also mechanically exfoliated on a similarSiO /Si substrate. A poly methyl metacrylate (PMMA)film is applied on the top layer and cured. Using anNaOH etch , the PMMA film along with the graphenelayer, and the alignment marks are detached from thehost substrate, forming a free standing membrane. Themembrane is placed face down on the substrate contain-ing the bottom graphene layer [Fig. 1(b)], and aligned with it. A Hall bar is subsequently defined on the toplayer [Fig. 1(c)]. Ten back-gated, independently con-tacted graphene bilayers have been fabricated and inves-tigated in this study, all with similar results. We focushere on data collected from three samples, labeled 1, 2,and 3, with mobilities between 4,200-12,000 cm /Vs forthe bottom layer, 4,500-22,000 cm /Vs for the top layer,and with inter-layer resistances of 1-20 GΩ. These struc-tures are markedly different from graphene bilayers exfo-liated from natural graphite, consisting of two graphenemonolayers in Bernal stacking .We now turn to the individual layer characterization.The layer resistivities ( ρ ) and Hall densities measuredfor sample 1 at a temperature T=4.2 K, as a functionof back-gate bias ( V BG ) are shown in Fig. 2(a) and2(b), respectively. The potential of the both layers isheld at zero (ground) for all measurements presented inthis study. The bottom layer dependence on the applied V BG shows ambipolar conduction and a finite resistanceat the charge neutrality (Dirac) point, consistent withthe expected response of gated monolayer graphene .More interestingly, the top layer resistivity also changesas a result of the applied V BG . This observation indicatesan incomplete screening of the gate-induced electric fieldby the bottom layer , which is most pronounced in thevicinity of the charge neutrality point, consequence of thereduced density of states in graphene. As we show below,we can quantitatively explain the layer resistivities anddensities dependence on V BG .Figure 2(c) shows the band diagram of the graphenebilayer at V BG =0 V; for simplicity the gate Fermi en-ergy and the charge neutrality point in the two layersare assumed to be at the same energy. Once a finite V BG is applied, finite charge densities are induced in bothtop ( n T ) and bottom ( n B ) layers [Fig. 2(d)]. The dif-ference between the gate and bottom layer Fermi lev-els is distributed partly across the SiO dielectric, andpartly on the Fermi energy of the bottom graphene layer: eV BG = e ( n B + n T ) /C SiO + E F ( n B ) (1); E F ( n ) =¯ hv F √ πn is the graphene Fermi energy measured withrespect to the charge neutrality point at a carrier den-sity n , e is the electron charge, v F =1.1 × m / s is thegraphene Fermi velocity, and C SiO denotes the SiO di- FIG. 1: (color online) Optical micrographs (top) and schematic representation (bottom) of the fabrication process of anindependently contacted graphene bilayer. (a) Hall bar device fabrication on the bottom graphene layer, followed by the Al O inter-layer dielectric deposition. (b) Top graphene layer isolation on a separate substrate, followed by transfer onto the bottomlayer. (c) Top layer Hall bar realization by etching, lithography, metal deposition, and lift-off. The yellow (inner) and red(outer) dashed contours in the optical micrograph represent the top and bottom layers, respectively. The scale bars in all panelsare 10 µ m.FIG. 2: (color online) (a) Layer resistivities, and (b) densities vs. V BG measured at T =4.2 K in sample 1. Depending on V BG , both electrons or holes can be induced in the bottom layer; the top layer contains electrons in the available V BG window,owing to unintentional doping. The symbols in panels (a) and (b) represent experimental data, while the lines represent thecalculated values according to Eqs. (1) and (2). (c) Band diagram across the graphene bilayer heterostructure at V BG =0 V,and (d) at a positive V BG . Both layers are assumed to be at the charge neutrality point, and aligned with the back-gate Fermilevel at V BG = 0. The layers are held at the ground potential, and their thicknesses exaggerated to show the Dirac cones. Theapplied V BG induces voltage drops V SiO , and V Al O across the bottom, and inter-layer dielectrics respectively. electric capacitance per unit area. Similarly, the Fermienergy difference between the two layers is responsiblefor the potential drop across the Al O inter-layer di-electric: E F ( n B ) = e n T /C Al O + E F ( n T ) (2); C Al O is the Al O dielectric capacitance per unit area. Thefinite Fermi energy of the bottom layer, E F ( n B ) in Eq.(2), plays the same role with respect to the top layer, asthe applied V BG in Eq. (1) for the bottom layer. Equa-tions (1) and (2) allow us to determine n B and n T as afunction of V BG . This model can be adjusted to includefinite layer densities at V BG =0 V. The layer resistivitydependence on V BG can be understood using a Drudemodel ρ T,B = ( n ∗ T,B eµ T,B ) − , where µ T , and µ B arethe top and bottom layer mobilities, and the layer den- sities n T,B = q n T,B + n T, B are adjusted to allow forfinite carrier densities ( n T , n B ) at the charge neutralitypoint. The data of Figs. 2(a) and 2(b) show a good agree-ment between the measured layer resistivities and den-sities (symbols), and the calculations (solid lines). Thelayer mobilities, determined from Hall measurements, are µ B =5,400 cm /Vs, and µ T =4,500 cm /Vs at T =4.2 K.Key insight into the physics of the graphene bilayer sys-tem can be gained from Coulomb drag measurements .A current ( I Drive ) flown in one (drive) layer leads to amomentum transfer between the two layers, thanks tothe inter-layer electron-electron interaction. To counterthis momentum transfer, a longitudinal voltage ( V Drag )builds up in the opposite (drag) layer. The polarity of
FIG. 3: (color online) Coulomb drag in graphene. (a) Layerresistivities ( ρ T,B ) and ρ Drag vs. layer densities ( n T,B ) forsample 2, measured by sweeping V BG at T =250 K. The bilayerprobes three different regimes: hole-hole, electron-hole, andelectron-electron. (b) ρ Drag vs. V BG at different T values,from 250 K to 77 K (solid lines). Inset: maximum ρ Drag vs. T in the electron-hole and electron-electron regimes. Thedifferent x -axis, i.e. n B and n T of panel (a) and V BG of panel(b), apply to both panels. V Drag depends on the carrier type in the two layers, andis opposite (same) polarity as the voltage drop in thedrive layer when the both layers have the same (oppo-site) type of carriers. The drag resistivity is defined as ρ Drag = (
W/L ) V Drag /I Drive , where L and W are thelength and width of the region where drag occurs. ρ Drag vs. V BG measured at T =250 K in sample 2 is shownin Fig. 3(a), along with the layer resistivities, ρ T and ρ B . Unlike sample 1 data (Fig. 2), the charge neutral-ity (Dirac) points of both layers can be captured in theexperimentally accessible V BG window. Consequently,depending on the V BG value, sample 2 can probe threedifferent regimes: a hole-hole bilayer, for V BG < -15 V,an electron-hole bilayer for -15 V < V BG < -2 V, andan electron-electron bilayer for V BG > -2 V. The depen-dence of ρ B and ρ T on V BG of Fig. 3 is also in goodagreement with the model presented in Fig. 2. Consis-tent with the above argument, ρ Drag is positive in theelectron-hole bilayer regime, negative in the hole-hole orelectron-electron regime, and changes sign when either -20 -10 0 10 201 Ω ρ D r ag ( Ω ) V BG ( V ) FIG. 4: (color online) ρ Drag vs. V BG measured in sample 2for T ≤
77 K. As T is reduced, ρ Drag exhibits mesoscopicfluctuations with increasing amplitude, which fully obscurethe average drag at the lowest T . The traces are shifted forclarity; the horizontal dashed lines indicate 0 Ω for each trace. the top or the bottom layer are at the charge neutralitypoint. Standard consistency checks ensured the mea-sured drag is not affected by inter-layer leakage current.For two closely spaced two-dimensional systems, whenthe ground state of each layer is a Fermi liquid, andthe inter-layer interaction is a treated as a perturbation,the ρ Drag depends on layer density ( n ) as ∝ /n / ,on temperature as ∝ T , and inter-layer distance ( d )as ∝ /d . Likewise, the Coulomb drag resistivity ingraphene, calculated in the Fermi liquid regime usingBoltzmann transport formalism and the random phaseapproximation for the dynamic screening is : ρ Drag = − he ζ (3)32 ( k B T ) d ǫ n / B n / T (3); k B is the Boltzmann con-stant, ζ (3) ∼ = 1 . ǫ and is the dielectric permittivity. Aseparate effect which has been theoretically advanced asthe representative Coulomb drag mechanism in grapheneis trigonal warping . Figure 3(b) data shows ρ Drag vs. V BG measured for sample 2 for T values between 77 Kand 250 K. Away from the bottom layer charge neutral-ity point, the ρ Drag magnitude decreases with increasing n B and n T . A power law, ρ Drag ∝ / ( n αB n αT ) fitting toFig. 3 data for V BG > α = 1.25 ± .
25, which depends little on temperature. We notethat the magnitude of ρ Drag is a factor of ∼ lowerthan the values expected according to Eq. (3). Whilefurther theoretical work is needed to explain this dis-crepancy, a possible explanation is that Eq. (3) is validfor high densities and/or large inter-layer spacing such FIG. 5: ρ Drag vs. B measured at T =0.3 K in sample 3,showing mesoscopic fluctuations similar to Fig. 4 data. Inset: ρ Drag vs. B data autocorrelation. that k F · d ≫ k F denotes the Fermi wave-vector) ;for Fig. 3 data k F · d ≤ ρ Drag ∝ ( k B T ) dependence, which stems from the al-lowed phase space where electron-electron scattering oc-curs, is followed closely for temperatures between 70 Kand 200 K (Fig. 3(b) inset), and softens for T > ρ Drag through 0 Ω, from the electron-hole to the electron-electron regime [blue (shaded) corridor of Fig. 3(b)]. Thecrossover can be explained by the co-existence of electronand hole puddles near the charge neutrality point of thebottom layer, which generate drag electric fields of oppo-site sign, and cancel the ρ Drag .A remarkable transition in the drag resistance is ob-served for T lower than 50 K (Fig. 4). As T is re-duced, the ρ Drag data starts to develop fluctuations su-perposed on the average ρ Drag vs. V BG dependence ofFig. 3 and Eq.(3), valid for diffusive transport. The ρ Drag fluctuations, which are reproducible in differentmeasurements, grow in amplitude as T is reduced, andfully obscure the average diffusive drag below 20 K. Thismanifestation of mesoscopic physics at elevated temper-atures is a consequence of the phase coherence length( L ϕ ) increasing with reducing T , and represents thecounterpart of universal conductance fluctuations in Coulomb drag . Figure 4 data reveal that ρ Drag fluc-tuation amplitude reaches a maximum near the chargeneutrality point of the bottom layer ( V BG = − T is decreased. Theo-retical arguments indicate that the drag conductivity( σ Drag = ρ Drag /ρ T ρ B ) fluctuation amplitude ( δσ Drag )depends on relevant length scales and temperature as δσ Drag ∝ T · ( L ϕ l ) /L ; l is the electron mean free path.For the temperature range examined in Fig. 4, l canbe considered constant, as the mobility is weakly depen-dent on T . Assuming the electron-electron interactionis the main phase-breaking mechanism in graphene ,hence L ϕ = l p E F / k B T , the T dependence of δσ Drag and δρ Drag should follow a ∝ T − / dependence, in goodagreement with Fig. 4 data.To probe the signature of weak localization in Coulombdrag, in Fig. 5 we show an example of ρ Drag vs. per-pendicular magnetic field ( B ) data, measured in sam-ple 3 at T =0.3 K and V BG =0 V; both layers containelectrons with layer densities n T = 1 . × cm − ,and n B = 1 . × cm − . Similar to the V BG de-pendence of Fig. 4, the ρ Drag vs. B data shows re-producible mesoscopic fluctuations. The auto-correlationfunction ( C (∆ B )) of Fig. 5 data reveals a correlationfield B c = 47 mT, which corresponds to a phase co-herence length L ϕ = p h/eB c = 300 nm. Similar L ϕ values have been extracted from ensemble average mea-surements using scanning gate microscopy .In summary, we demonstrate independently contactedgraphene bilayers, and probe the Coulomb drag in thissystem. At elevated temperatures the drag resistance de-pendence on density and temperature are consistent withthe Fermi liquid theory. At reduced temperatures, thedrag exhibits mesoscopic fluctuations which obscure theaverage drag, a result of the interplay between electron-electron interaction and phase coherent transport.We thank A. H. MacDonald, W. K. Tse, and B.Narozhny for discussions, and NRI-SWAN for support.Part of our work was performed at the National HighMagnetic Field Laboratory, which is supported by NSF(DMR–0654118), the State of Florida, and DOE. ∗ Electronic address: [email protected] Y. E. Lozovik, V. I. Yudson, JETP Lett. , 274 (1975). M. Pohlt et al. , Appl. Phys. Lett. , 2105 (2002). A. F. Croxall et al. , Phys. Rev. Lett. , 246801 (2008);J. A. Seamons, C. P. Morath, J. L. Reno, M. P. Lilly, Phys.Rev. Lett. , 026804 (2009). K. S. Novoselov et al. , Science , 666 (2004). K. S. Novoselov et al. , Nature , 197 (2005); Y. Zhang et al. , Nature , 201 (2005). H. Min, R. Bistritzer, J.-J. Su, A. H. MacDonald, Phys.Rev. B , 121401 (2008). C.-H. Zhang, Y. N. Joglekar, Phys. Rev. B P. M. Solomon, P. J. Price, D. J. Frank, D. C. La Tulipe, Phys. Rev. Lett. , 2508 (1989). T. J. Gramila, J. P. Eisenstein, A. H. MacDonald, L. N.Pfeiffer, K. W. West, Phys. Rev. Lett. , 1216 (1991). M. Yamamoto, M. Stopa, Y. Tokura, Y. Hirayama, S.Tarucha, Science , 204 (2006). G. Vignale, A. H. MacDonald, Phys. Rev. Lett. , 2786(1996). E. Tutuc, R. Pillarisetty, M. Shayegan, Phys. Rev. B. ,041303(R) (2009). S. Kim et al. , Appl. Phys. Lett. , 062107 (2009). A. Reina et al. , J. Phys. Chem. C , 17741 (2008). E. McCann, V. I. Fal’ko, Phys. Rev. Lett. , 086805(2006). S. Adam, E. H. Hwang, V. M. Galitski, S. Das Sarma,
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