Electronic compressibility of layer polarized bilayer graphene
A. F. Young, C. R. Dean, I. Meric, S. Sorgenfrei, H. Ren, K. Watanabe, T. Taniguchi, J. Hone, K. L. Shepard, P. Kim
EElectronic compressibility of layer polarized bilayer graphene
A. F. Young, C. R. Dean,
2, 3
I. Meric, S. Sorgenfrei, H. Ren, K. Watanabe, T. Taniguchi, J. Hone, K. L. Shepard, and P. Kim Department of Physics, Columbia University, New York, New York 10027, USA Department of Electrical Engineering, Columbia University, New York, New York 10027, USA Department of Mechanical Engineering, Columbia University, New York, New York 10027, USA Advanced Materials Laboratory, National Institute for Materials Science, 1-1 Namiki, Tsukuba, 305-0044, Japan (Dated: October 27, 2018)We report on a capacitance study of dual gated bilayer graphene. The measured capacitanceallows us to probe the electronic compressibility as a function of carrier density, temperature, andapplied perpendicular electrical displacement D . As a band gap is induced with increasing D , thecompressibility minimum at charge neutrality becomes deeper but remains finite, suggesting thepresence of localized states within the energy gap. Temperature dependent capacitance measure-ments show that compressibility is sensitive to the intrinsic band gap. For large displacements,an additional peak appears in the compressibility as a function of density, corresponding to thepresence of a 1-dimensional van Hove singularity (vHs) at the band edge arising from the quarticbilayer graphene band structure. For D >
0, the additional peak is observed only for electrons,while
D <
The unique band structures of monolayer (MLG) andbilayer graphene (BLG) offer unprecedented tunabil-ity in a high quality two dimensional electron system(2DES). Within the independent electron approximation,both MLG and BLG are gapless, chiral systems. Thegapless spectra are related to the pseudospin degeneracy,which is tied to the symmetry between the two sublat-tices constituting the honeycomb. Whereas in the mono-layer a gap can be opened only by a potential modula-tion on the spatial scale of the lattice constant , in BLGthe relevant sublattices are located on different layers,allowing a gap to be induced by a modulation of the in-terlayer imbalance via the application of an electric fieldperpendicular to the BLG planes . Although the field-effect tunable gap in BLG has been observed optically , transport measurements show hopping conductivityat low temperatures. In a parallel plate capacitor made up of imperfect con-ductors, adding charge n to the plates costs the sum ofthe classical electrostatic energy, the kinetic energy dueto the resulting change in the chemical potential µ , andthe potential energy of Coulomb interactions between thecharge carriers. The measured differential capacitance, C = δn/δV , in such a system reflects this finite electroniccompressibility by manifesting a lowered effective capaci-tance, C − = C − + (cid:0) e Aν (cid:1) − , where C is the geometriccapacitance, A is the area of the device and ν ≡ ∂n/∂µ isthe electronic compressibility, which corresponds to thedensity of states in the noninteracting, zero temperaturelimit. In low dimensional systems the contribution ofthe compressibility to the capacitance—termed “quan-tum capacitance” —can be small even when the con-ductivity remains large, providing a powerful tool in thestudy of both one and two dimensional electronic sys-tems. Capacitance measurements are particularly power-ful in the study of disordered systems, as they are able todetect localized states whose contribution to transport is suppressed. Capacitance measurements are, as a result,crucial in understanding biased bilayer graphene, a sys-tem in which localization is known to play a role. More-over, the small but finite interlayer separation ( d ∼ .To produce dual gated graphene devices with high ge-ometric capacitance, we utilize single crystal hexagonalboron nitride (h-BN) flakes as the top gate dielectricfabricated by the process described in Ref. 19. Briefly,both graphene and single crystal h-BN, an insulating iso-morph of graphite, are exfoliated onto n-Si/SiO wafers(Fig. 1(a-b)). A thin (5-7 nm) h-BN flake is trans-ferred on top of the graphene using a wet etch processand micromechanical manipulation , followed by elec-tron beam lithography to form contact electrodes anda local top gate (Fig. 1c). The heavily doped siliconsubstrate, coated with 285 nm oxide, serves as the bot-tom gate. The double gated geometry allows independentcontrol of the electronic density, n , and the displacement, D = ε B d B (cid:16)(cid:0) V B − V B (cid:1) − C T C B (cid:0) V T − V T (cid:1)(cid:17) through the topand bottom gate voltages, V T and V B . Here, ε B and d B are the dielectric constant and thickness of the backgate dielectric layer, C T ( B ) is the geometric capacitanceof the top (bottom) gate, and V T ( B ) is the voltage offsetrequired to obtain minimal density and displacement inthe dual-gated region. We find that h-BN is an excel-lent gate dielectric, with ε ∼ ∼ .8 V/nm) to SiO thin films. In addition,we observe minimal degradation of graphene samples,with no additional doping contributed by the presenceof the top gate and typical post h-BN transfer mobilitiesof µ ∼ /V sec for graphene monolayersand µ ∼ /V sec for bilayers.Low temperature capacitance measurements were per- a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b SiO V B V T hBNCr/Augraphene d e C T C R δ V R δ V T V T ba c FIG. 1: (a-c) Optical microscope images. (a) a thin hBN sin-gle crystal is transferred onto a (b) mono- or bilayer grapheneflake and (c) contacts and gate electrodes deposited by elec-tron beam lithography. Scale bars are 10 µ m. (d) Cross sec-tion schematic of resulting dual gated device. (e) Schematiccircuit diagram of the capacitance bridge. A reference capac-itor (Johanson Technology R14S) is mounted on the probe,and a reference voltage is chosen to balance the capacitancebridge. A small AC excitation signal is added to the DC gatebias through a transformer (Triad Magnetics SP67). formed using a capacitance bridge circuit (Fig. 1(e))with a cold reference capacitor. All wires were shielded,and the sample package was encased in a Faraday cageto further reduce parasitic capacitances, which representan additive constant to the measured value of C T . Aceramic multilayer capacitor with minimal temperaturedependence was chosen for the reference capacitor ( C R )and connected near the sample at low temperature. Thenoise level of the bridge was ∼ e/ √ Hz , allowing sub-femtofarad resolution with averaging times of less than30 seconds for our typical top gate AC excitation volt-age δV T ∼ δV T =50 mV AC excitation volt-age on top of the DC gate bias and measured the cur-rent through the graphene device directly. Although thismethod results in poorer signal to noise than the bridgemeasurement, it eliminates calibration errors stemmingfrom the small temperature dependence of the referencecapacitor. The ability to measure the quantum contribu-tions to the capacitance relies on the use of a thin top gatedielectric layer. The 7 nm thick hBN dielectric results invalues of C T that are comparable to the quantum capac-itance C Q , so that variations in the measured C T withchanging electronic compressibility are easily detectable.We check the efficacy of this measurement scheme us- -3 0 3 0510 G ( e / h ) C (f F ) -3 0 3 V T (V) a b V T (V) FIG. 2: Measured conductance and capacitance of monolayergraphene at 2 K and B = 0 (a) and B = 9 T (b). In (a), ca-pacitance and conductance resemble each other closely due tothe fact that both contain a spurious contribution adding “inparallel”; G = (1 /R + 1 /R C ) − and C T = (cid:0) /C Q + 1 /C T (cid:1) − where R is the device resistance, C Q is the compressibility,or “quantum capacitance,” C T is the geometric capacitance,and R C is the contact resistance of our two-terminal devices.In (b), capacitance reflects the electron ing monolayer graphene capacitors. MLG is expected todisplay monotonically increasing ν as a function of ab-solute density | n | . In the presence of a strong magneticfield, ν is further modulated due to the formation of Lan-dau levels. Fig. 2 shows the measured capacitance C andconductance G of MLG as a function of top gate voltage V T at both zero and finite magnetic field. The loweredcompressibility stemming from the linear spectrum of ofMLG can be inferred from a depression in the capacitanceat zero density, while at high B the formation of the zeroLandau level leads to a peak at charge neutrality . Thehigh magnetic field capacitance traces show compressibil-ity oscillations due to the formation of higher LLs, whileconductance shows the electron-hole asymmetry that isthe signature of edge state transport in graphene hetero-junctions , which form due to the partial coverage ofthe graphene channel by the top gate. The peak inver-sion at zero density and electron-hole symmetry in highmagnetic field together confirm that the measured signalis indeed capacitive, and is sensitive to density of statesrather than lateral resistance effects.We now turn our attention to bilayer graphene (BLG)samples. Fig. 3 shows the capacitance of a BLG sampleat 1.5 K measured with the cold bridge. Tuning externalgates adjusts both n and D . For small values of D ≈ n = 0as expected for ungapped bilayer graphene, which has ahyperbolic band structure . As | D | increases, the n =0 minimum gets deeper, corresponding to the formationof a gap in the energy spectrum. The n=0 minimumdoes not go to zero at high values of | D | , and in factthe capacitance modulation is only 10% with respect tothe D = 0 value. In addition, a distinct local maximumdevelops next to the minimum. As we argue below, thepresence of both the dip and local peak in ν at high | D | b -4 0 4110115 V T (V) C T (f F ) V B =70 V B =-80 -4 0 4-80080 C (fF) V T (V) V B ( V ) aa -10 0 10110120 C T ( f F ) C B ∆ V B +C T ∆ V T (10 cm -2 ) D (V/nm) 1.2 .8 0 -.8 -1.2 -1.6
FIG. 3: Capacitance at B =0 and 1.5 K as a function of V T and V B . Colored traces in (b) are taken at 30 volt intervals in V B ,corresponding to the colored lines in (a). (c) Traces at constant D , extracted from the data set shown in (a). Data is plotted asa function of C T ∆ V T + C B ∆ V B , which would correspond to the density were the bilayer perfectly 2 dimensional and perfectlycompressible. Curves in (c) are offset for clarity by 2fF per V/nm in D . can be understood, at least qualitatively, from the bandstructure of gapped BLG, once the effects of disorder and the interlayer separation are taken into account.Within the nearest-neighbor tight binding approxima-tion, the energy spectrum of pristine, Bernal stackedbilayer graphene with finite interlayer asymmetry ∆ isgapped and has a “Mexican hat” structure . Evenin the presence of disorder, the absence of a positivequadratic term in the energy spectrum turns the problemof gapped, disordered bilayer graphene into one of a heav-ily doped semiconductor with quartic energy bands .We thus expect vestiges of a ν ∝ / √ E vHs-like featureto be present even in our low mobility bilayer samples,manifesting as a nonmonotonic-in-density feature at theband edge . In addition to smearing the band edgevHs, disorder has a dramatic effect on compressibility atcharge neutrality in the presence of a large gap. In con-trast to clean semiconductors, in which the depleted sys-tem is incompressible, our measured capacitance remainsfinite and large even for large D , a fact we attribute totails in the density of states representing localized in-tergap states . This explains the discrepancy betweenenergy scales that govern transport and the gap en-ergies observed optically . It also suggests that recentlypredicted topological edge conduction is not the domi-nant reason for incomplete “turn-off” in BLG devices oftypical quality.Quantitative analysis of the capacitance data requiresextracting the compressibility ν from the measured sig-nal, C T . For a perfectly two dimensional electron system( C BL → ∞ ), the measured top gate capacitance is C − T = (cid:18) C T + 1 Ae ν (cid:19) − + C para , (1)where C para are all parasitic capacitances between gateand contact electrodes and terms O (cid:16) C B C T (cid:17) ∼ .
034 have been neglected. Extracting ν thus requires subtract-ing both parallel (C para ) and series (C T ) capacitances,and dividing by A =31 µ m (determined by optical mi-croscopy). We determined C T /C B from the ratio of theback and top gate capacitances, measured by tracking thecharge neutrality point in the V T -V B plane (the darkdiagonal belt in Fig. 3(a)). For the BLG device pre-sented in this paper, we measured and C T /C B =29.5 ± . C B = 115 aF /µ m is the geometric capacitance ofthe bottom gate. As disordered BLG devices cannot beturned off completely, C para cannot be measured in situ as is common practice in depletable semiconductors and semiconducting carbon nanotubes . Instead, we de-termine C P =16 ± C para constitutes about 10% of the total capaci-tance signal. Due to the subsequent subtraction of the(series) geometric capacitance C T , the error in determi-nation of C para is least important when the capacitancediffers considerably from the geometric value. This is theregime in which we perform a quantitative analysis of thecompressibility of gapped BLG.Near overall charge neutrality at | D | (cid:29)
0, our samplesshow a hopping conductivity similar to that observed inRef. 10 from 1 K to ∼
150 K. Capacitance instead showsno significant temperature dependence up to 50 K, there-after slowly rising as might be expected for a spectrumwith a gap in 50-100 meV range (Fig. 4). This is con-sistent with the presence of disorder-induced tails in thedensity of states throughout the band gap: whereas tem-perature dependent transport is dominated by the hop-ping between these localized states, temperature depen-dent capacitance is dominated by thermal population ofthe much larger density of states near the band edge.The most interesting, and unexpected feature, of theexperimental data are the local maxima observed at the ( A e ν ) - ( p F - ) -4 -2 0 2 40.00.51.0 V BG =-80 VV BG =0 V V T (V) T (K) V BG =80 V1.001.25 0 100 200 T (K) A e ν ( p F ) G ( µ S ) V B =-60V FIG. 4: Bottom panel: Temperature dependence of the in-verse compressibility ( Ae ν ) − for V B = 80, 0 and -80 V.A single value of C para is chosen for all gate voltages at agiven temperature, but a different C para is chosen for eachtemperature so that the curves match at high density. Weattribute the variation in C para to thermal expansion of thebonding wires, the capacitance of which constitute the major-ity of C para . Top panel: comparison between the temperaturedependence of the minimal compressibility and minimal con-ductivity at V B =-60 V. The dashed line is a guide for the eye. band edge are associated with the 1D vHs inherent inthe BLG band structure . Interestingly, this fea-ture is only present on one edge of the band, appear-ing on the electron side for D >
D <
0. This inversion symmetry in the variables( D , n ) was observed in all devices measured, includingthose fabricated using a resist free shadow mask met-allization as well as seeded atomic layer deposition ofHfO . Understanding this asymmetry requires takinginto account the three dimensional structure of BLG ,which consists of two strongly coupled but spatially dis-tinct layers of carbon atoms. The charge distribution ona BLG flake is sharply concentrated on the two layers, n ( z ) (cid:39) n δ ( z ) + n δ ( z + d ), so that the system can bemodeled as a four plate capacitor. Solution of this elec-trostatic problem leads to the modified relation for themeasured capacitance in which C T depends on both the -10 0 10-6-303 ∆ C T ( f F ) C B ∆ V B +C T ∆ V T (10 cm -2 ) D (V/nm) 1.2 .8 -.8 -1.2 -1.6 -200 -100 0 100 20090100110 C T ( f F ) E (meV)
FIG. 5: Left Panel: Subtracted capacitance, ∆ C T = C T ( D ) − C T ( D = 0), as a function of approximate density for differentapplied displacements. Curves are offset by 1fF per V/nm in D . Right Panel: Calculated top gate capacitance for disor-dered bilayer graphene, following Ref. 17. The colors cor-respond to the displacements in the left panel; the disorderparameter is γ = 4 for all curves. Curves are offset by 3 fFper V/nm in D . interlayer capacitance, ν , and intralayer capacitances, ν and ν , where ν ij ≡ ∂n i /∂v j with n and v rep-resent layer indexed density and potential, and the in-dices i,j=1,2 denote the top or bottom layer. Crucially,while ν is symmetric with respect to layer interchange, ∂n ∂v is obviously asymmetric in the presence of interlayerasymmetry. Penetration field measurements of bilayergraphene depend only on layer-symmetric quantities, andthus probe fundamentally different physical quantities.As elaborated in Ref. 17, the 1 / √ E divergence associ-ated with the 1D vHs at the band edge principally on thelow energy layer within the BLG flake. The vHs man-ifests more strongly in the measured capacitance whenvHs-hosting layer is closest to the top gate; conversely,the gate sees a far-layer vHs only through the screenedfield penetrating the near layer. Counterintuitively, dis-order enhances this effect not only by smearing the to-tal density of states but by populating the normallydepleted non-vHs layer, thereby enhancing its ability toscreen. While an ideal experimental geometry would per-mit the simultaneous measurement of capacitance fromtwo sides of the BLG flake this is effectively accomplishedin our single local gate geometry by reversing the sign of D , thus reversing the order of the vHs bearing and non-bearing layers.In order to better compare our experimental data withtheory based on a parabolic two band model, it is conve-nient to subtract a background taken at D = 0. Becausethe high energy behavior depends only weakly on thedisplacement, this has the effect of isolating the low en-ergy part of the measured capacitance, and in additionremoves the effects of electron-hole asymmetric elementsof the band structure . The results of this subtractionresemble theoretical calculations which take into accountboth the interlayer separation as well as weak, short-range disorder (Fig. 5). In particular, the asymmetricappearance of the van Hove singularity can be under-stood as the effect of disorder enhanced interlayer screen-ing. However, quantitative understanding of the role ofdisorder in bilayer graphene will require experiments thatindependently control the disorder, as well as a more so-phisticated theory taking into account a wider varietyof effects including long range scattering and electron-electron interactions.In conclusion, we study the broken symmetry state ofbilayer graphene induced by an electrical displacementfield applied perpendicular to the bilayer. We observedthe formation of the displacement induced gap and its ac-companying 1D vHs, and estimate the gap size from tem-perature dependent measurements. We also discussedhow different measurement geometries make capacitancea probe of layer-pseudospin polarization in the bilayer.In BLG, layer symmetry breaking in BLG can also be expected to occur spontaneously, as a result of electronicinteractions at both finite and zero magneticfields. Similar measurement performed on high-qualitybilayers manifesting these effects will allow spontaneouslayer polarization to be probed directly through compar-ison of measurements in various capacitance geometries,as discussed at length in a companion paper . Acknowledgments
The authors acknowledge discussions with I.L. Aleiner,B.L. Altshuler, J.P. Eisenstein, E.A. Henriksen, L.S. Lev-itov, A.H. MacDonald, K.F. Mak, and E. McCann. Thiswork is supported by AFOSR MURI and INDEX (A.Y.Fand P. K.)). A. K. Geim and K. S. Novoselov, Nature Materials , 183(2007). A. H. Castro Neto et al. , Reviews of Modern Physics ,109 (2009). G. Giovannetti et al. , Phys. Rev. B , 073103 (2007). E. McCann and V. I. Fal’ko, Phys. Rev. Lett. , 086805(2006). E. McCann, Phys. Rev. B , 161403 (2006). E. V. Castro et al. , Phys. Rev. Lett. , 216802 (2007). T. Ohta et al. , Science , 951 (2006). Y. Zhang et al. , Nature , 820 (2009). K. F. Mak, C. H. Lui, J. Shan, and T. F. Heinz, Phys.Rev. Lett. , 256405 (2009). J. B. Oostinga et al. , Nature Materials , 151 (2008). K. Zou and J. Zhu, Phys. Rev. B , 081407 (2010). T. Taychatanapat and P. Jarillo-Herrero, Phys. Rev. Lett. , 166601 (2010). J. Yan and M. S. Fuhrer, Nano Lett. , 4521 (2010). S. Luryi, Applied Physics Letters , 501 (1988). S. Ilani, L. A. K. Donev, M. Kindermann, and P. L.McEuen, Nature Physics , 687 (2006). J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev.Lett. , 674 (1992). A. F. Young and L. S. Levitov, Phys. Rev. B , 085441(2011). Y. Kubota, K. Watanabe, O. Tsuda, and T. Taniguchi,Science , 932 (2007). C. R. Dean et al. , Nature Nanotechnology , 722 (2010). L. Jiao et al. , Nano Letters , 205 (2009). J. Martin et al. , Nature Physics , 144 (2008). D. A. Abanin and L. S. Levitov, Science , 641 (2007). J. R. Williams, L. DiCarlo, and C. M. Marcus, Science , 638 (2007). B. ¨Ozyilmaz et al. , Phys. Rev. Lett. , 166804 (2007). J. Xia, F. Chen, J. Li, and N. Tao, Nature Nanotechnology , 505 (2009). E. A. Henriksen and J. P. Eisenstein, Phys. Rev. B ,041412 (2010). J. Nilsson and A. H. Castro Neto, Phys. Rev. Lett. ,126801 (2007). V. V. Mkhitaryan and M. E. Raikh, Phys. Rev. B ,195409 (2008). P. Van Mieghem, Rev. Mod. Phys. , 755 (1992). J. Li, I. Martin, M. Buttiker, and A. F. Morpurgo, NatPhys , 38 (2011). J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer, and K. W.West, Phys. Rev. B , 7910 (1990). D. B. Farmer et al. , Nano Letters , 4474 (2009). Y. Barlas, R. Cˆot´e, K. Nomura, and A. H. MacDonald,Phys. Rev. Lett. , 097601 (2008). B. E. Feldman, J. Martin, and A. Yacoby, Nature Physics , 889 (2009). Y. Zhao, P. Cadden-Zimansky, Z. Jiang, and P. Kim, Phys.Rev. Lett. , 066801 (2010). H. Min, G. Borghi, M. Polini, and A. H. MacDonald, Phys.Rev. B , 041407 (2008). R. Nandkishore and L. Levitov, Phys. Rev. Lett.104