We analyze the response of bilayer graphene to an external transverse
electric field using a variational method. A previous attempt to do so in a
recent paper by Falkovsky [Phys. Rev. B 80, 113413 (2009)] is shown to be
flawed. Our calculation reaffirms the original results obtained by one of us
[E. McCann, Phys. Rev. B 74, 161403(R) (2006)] by a different method. Finally,
we generalize these original results to describe a dual-gated bilayer graphene
device.Comment: 4 pages, 1 figur

Fogler, M. M.

McCann, E.

English

arXiv

M. M. Fogler

E. McCann

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Physical Review B

Comment on “Screening in gated bilayer graphene”

We analyze the response of bilayer graphene to an external transverse electric field using a variational method. A previous attempt to do so in a recent paper by Falkovsky [Phys. Rev. B 80, 113413 (2009)] is shown to be flawed. Our calculation reaffirms the original results obtained by one of us [E. McCann, Phys. Rev. B 74, 161403R (2006)] by a different method. Finally, we generalize these original results to describe a dual-gated bilayer graphene device

Fogler, M M

McCann, Edward

Lancaster E-Prints

Comment on “Screening in gated bilayer graphene”M. M. Fogler1 and E. McCann21Department of Physics, University of California–San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA2Department of Physics, Lancaster University, Lancaster LA1 4YB, United KingdomReceived 29 March 2010; revised manuscript received 21 September 2010; published 22 November 2010We analyze the response of bilayer graphene to an external transverse electric field using a variationalmethod. A previous attempt to do so in a recent paper by Falkovsky Phys. Rev. B 80, 113413 2009 is shownto be flawed. Our calculation reaffirms the original results obtained by one of us E. McCann, Phys. Rev. B 74,161403R 2006 by a different method. Finally, we generalize these original results to describe a dual-gatedbilayer graphene device.DOI: 10.1103/PhysRevB.82.197401 PACS numbers: 73.22.Pr, 73.20.AtThe physics of monolayer and bilayer graphenes BLGhas been a subject of much interest recently.1 A unique fea-ture of BLG is its tunable band structure: its band gap de-pends on the external electric field, which can be controlledby doping or gating. This effect was first analyzedtheoretically2–4 and recently studied experimentally.3,5–18The subject of this Comment is the value of the gap at theBrillouin zone BZ corners, which we denote by 2U.Within the conventional mean-field Hartree theory,2,3 2U co-incides with the electrostatic energy difference per electronin the two layers,2U = eEmdm, dm = 0.33 nm, 1where Em is the component of the electric field inside theBLG directed from the bottom to the top layer and dm is theinterlayer spacing. In experimental practice, this field de-pends not only on the external gate voltages but also on theinduced electron densities nt and nb of the layers, see Fig. 1.For a given chemical potential , these densities, and thusthe total electron density of BLG n=nt+nb are nonlinearfunctions of U. In general, they can be calculated onlynumerically.3,4,19–23 However, in a range U 1 anasymptotic analytical formula for the interlayer bias was de-rived by one of us42U  1N − 2NdtMc−1 + N −12lnN, N nn. 2The derivation was done within the commonly used approxi-mation that neglects certain small electronic structure param-eters 3, 4, and  for their physical meaning and a dis-cussion of their numerical values, see Ref. 23. Theparameters that are retained include the interlayer hoppingenergy 1=0.41 eV and the nearest-neighbor in-plane hop-ping 0=3.0 eV, which define the characteristic densityscale in the problemn =43a21202 = 1.2 1013 cm−2, 3where a=2.44 Å is the lattice constant. The remaining no-tations in Eq. 2 areNdt = ndt/n, 4which is the scaled background density of positive charge ndton the top layer or above it, see below andMc =2e2dmnm1 1, 5which is the dimensionless strength of the interlayer screen-ing, with m	1 being the effective dielectric constant of themedium between the layers. The screening is particularlysignificant in the narrow gap regime. If Ndt=0, it is realizedin the limit n→0. According to Eq. 2, the screened field Emis suppressed compared to the external field Eb by a diver-gent logarithmic factor see Fig. 1 and also Eq. 30 below.As a result, 2U has a superlinear dependence on n.Formula 2 was recently challenged by Falkovsky.24 Un-der the same conditions and assumptions, he obtained thesimple linear law for Ndt=0,2U = c1N, c =26.27842 − 1 = 0.3227. 6The goal of this Comment is to show that Ref. 24 containsegregious mistakes. Once they are corrected, Eq. 2 is re-covered.These mistakes are apparent on physical grounds. Onekey formula, Eq. 8 of Ref. 24, states in our notations thatthe BLG polarizationdbdmEbEmEtndtntnb+U−UFIG. 1. Color online Device schematic. The BLG is shown astwo horizontal lines in the middle. The thicker line at the bottomrepresents the gate. The arrows show the electric field in the systemand the crosses depict positive charge density ndt on or on andabove the top layer of the BLG.PHYSICAL REVIEW B 82, 197401 20101098-0121/2010/8219/1974014 ©2010 The American Physical Society197401-1n  nt − nb 7is positive for U ,	0, i.e., the BLG is polarized against,not along the electric field. In principle, such a phenomenoncan arise if a system has a negative compressibility25–27 dueto exchange and correlations see the discussion at the end.However, neither exchange nor correlation effects are in-cluded in the theory of Ref. 24. To expose the mistakes inRef. 24, we carry out below our own derivation using themethod proposed by Falkovsky in Ref. 24. Its idea is to treatU and  as variational parameters and get their actual valuesby minimizing the total energy density  of the system at afixed n.Let us proceed. In the mean-field approximation  is thesum of the kinetic and the Hartree interaction terms, =kin+H. The Hartree term is straightforward,H =e22cbn − ndt − ndb0 2 +2e2dmmnt − ndt2, 8which can be understood as the energy of two parallel-platecapacitors in series. Here cb=b / 4db is the capacitanceper unit area between the BLG and the bottom gate, b is thedielectric constant below the BLG, and ndb0 is the density ofadditional positive background charge on the bottom layernot shown in Fig. 1. For ndb0=ndt, Eq. 8 agrees with Eq.10 in Ref. 24.The kinetic term kin requires a little care. The author ofRef. 24 seems to assume that it coincides with the sum of theoccupied single-particle energiessum =1A  − EE, 9where  is a short-hand notation for all quantum numbers,x is the unit step function, and A is the area of the system.In fact, each energy E, which is an eigenvalue of the BLGHamiltonian H=Hkin+ also contains a potential term:E= Hkin+ . Here  stands for the electrostaticenergy that is equal to U in the top bottom layer. Lump-ing together sum and H is incorrect as it leads to doublecounting of the interaction energy. Instead, the proper for-mula iskin =1A  − EHkin = sum − Un . 10Next, using the Hellmann-Feynman theorem, we getn =  − EAHU =sum − nU,which entailsnt,b =nn2=12n U sum − n . 11The correct variational functional to minimize is = H + sum − UUsum − n − n , 12where  is a Lagrange multiplier. The sought minimum isdetermined by the equations  /= /U=0, whichleads to2U = 4e2dm/mnt − ndt . 13This equation is obviously correct because it follows fromGauss’s law. Indeed, it is the starting equation Eq. 2 ofRef. 4. The closest to Gauss’s law in Ref. 24 is Eq. 11 ofthat paper but the electric field is defined in the wrong direc-tion in that equation, as can be seen by comparison with Fig.3 in Ref. 24.Let us now compute nt by the present method to demon-strate where Ref. 24 contains another mistake. The total po-larization n=s1,s2ns1s2 is the sum over all four bands ofBLG, where, in keeping with notations of Ref. 28, we labelthe bands as follows: s1= distinguishes the conduction andvalence bands while s2= denotes the outer inner bands.The band dispersions are given byEs1s2 = s1122 + 2q + U2 + s22q , 142 144+ 212 + 4U2 , 15where q is the quasimomentum and q is the energy dis-persion in monolayer graphene. Near a BZ corner K, theDirac approximation is valid q= 3 /20ak, wherek=q−K.For simplicity, we consider the case where the Fermi sur-face is singly connected and includes only the +− states sothat n=kF2 /, where kF is the Fermi momentum measuredfrom the nearest BZ corner and = E+−kF U2 + 4kF12  U2 + 12N2. 16As explained above, if U is positive, then n should benegative. In Ref. 24 however only the conduction bandterm,4,24n+− =nU1lnU+2U2− 1 , 17is included while the much larger negative contributionn−n−++n−− of the completely filled valence bands isneglected. This important term is given byn−=UBZ 2d2q22E−+q + E−−qU=0U	0. 18Within the Dirac approximation, the above integral can bedone exactly by extending the integration domain to infinity,with the resultCOMMENTS PHYSICAL REVIEW B 82, 197401 2010197401-2n−= −nu1 + 4u22uu + 1 + u22+1 + 2u21 + 4u2sinh−11 + 2u2u+ sinh−1 2u , 19where uU /1. The leading term for U1 isn− −nU1ln21U, 20in agreement with Ref. 4. The nonvanishing and, in fact,dominant contribution of the filled valence bands to the totalpolarization is not an artifact of the Dirac approximation. Toprove that we repeated the calculation of n−within thesimplified tight-binding model, usingq = 0eiq·1 + eiq·2 + eiq·3 21in Eq. 14, with  j being the vectors connecting a lattice siteto its three nearest neighbors. The integration in Eq. 18 wasdone numerically and the results are shown in Fig. 2. Asexpected, the Dirac approximation remains accurate up toU0=3.0 eV. At 1U0, the polarization varies qua-dratically with U because it is dominated by the properties ofa single layer, where the density of states is linear inenergy. Finally, when 2U exceeds the bandwidth60=max q−min q of the monolayer dispersion, n−quickly approaches its minimum possible value, which isequal to the total atomic density of BLG na=43 /3a2, withthe minus sign.Most relevant for current experiments is the case U1,where Eq. 20 applies. Combined with Eqs. 16 and 17, ityields the result of the correct sign for the total polarization,n  −nU1ln2N + N2 + U/12. 22Substituting it into nt= n+n /2 and using Eqs. 4, 5, and13, we obtain the equation for the function UN,2UN  1N − 2NdtMc−1 +12ln2N + N2 + U/12. 23This formula is actually valid for arbitrary signs of U and N.In the region2U0/1 N 1 24it coincides with Eq. 2 within the accuracy of this calcula-tion. Here2U0  −21NdtMc−1 +12ln1Ndt25is the interlayer bias at N=0. Note that the gap 2U vanishesonly at N=2Ndt, i.e., n=2ndt, wherent = nb = ndt. 26The gap remains nonzero at all other densities includingn=0, in disagreement with the claim made in Ref. 24 but inagreement with the numerical results of Ref. 4. A finite gapat zero density has indeed been observed in quantum Halleffect measurements3 of doped BLG.As mentioned in the beginning of this Comment, thebackground positive charge density ndt does not have to re-side directly on the top layer. Equation 23 remains un-changed if some of this charge is located above the BLG, assketched in Fig. 1. For example, for a dual-gated BLGndt = ndt0 +cte2eVt −  + U , 27where the first term is the fixed donor density on the toplayer and the second term is the tunable charge density onthe top gate. Parameter ct is the capacitance between the topgate and the BLG per unit area. Similarly, the total density isgiven byn = ndt0 + ndb0 +ct − cbe2U + a=t,bcae2eVa −  , 28where Vb is the voltage on the bottom gate. The chemicalpotential  enters these equations because the measured“voltages” Vb and Vt of the two gates are not simply theelectrostatic but instead the electrochemical potential differ-ences between these gates and the BLG, cf. Refs. 17 and 27.However for distant gates,da dma2Mcm, a = t,b , 29one can in the first approximation neglect U and  comparedto eVb and eVt, leading to the simplified equations2U =eE0dm1 +Mc2ln2N + N2 + U/12, 300 2 4 6 8 10−1−0.8−0.6−0.4−0.20U (eV)∆n−/naFIG. 2. Color online BLG polarization n−due to the com-pletely filled valence bands in the Dirac approximation Eq. 19thin red line and the tight-binding model thick black line. Pa-rameter na=43 /3a2 is the atomic density.COMMENTS PHYSICAL REVIEW B 82, 197401 2010197401-3E0 2emndb0 − ndt0 + cbVbe−ctVte , 31N =nn1nndb0 + ndb0 + ctVte+cbVbe , 32which may be useful in experimental practice. Note howeverthat unavoidable disorder creates additional corrections tothese expressions,23 which can be as large as 30%.In closing, we comment on exchange and correlations ef-fects. Our Eq. 2 is in qualitative agreement with density-functional theory DFT calculations,20,29 which includesome of these effects. On the other hand, spontaneous polar-ization n0 and the corresponding gap generation havebeen predicted to occur19,30–34 even in the absence of theexternal electric field.35 Such a symmetry breaking can beincorporated into either mean-field theory or the DFT byintroducing a suitable self-energy difference between the lay-ers, which is a function of N and U. In fact, we already havea similar parameter in our formalism—it is our Ndt. Thisinteresting problem warrants further study.This work is supported by the NSF under Grant No.DMR-0706654 M.M.F. and by EPSRC First Grant No. EP/E063519/1 E.M.. We thank J. Stanley for comments on themanuscript and L. M. Zhang for help with producing Fig. 2.1 A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov,and A. K. Geim, Rev. Mod. Phys. 81, 109 2009.2 E. McCann and V. I. Fal’ko, Phys. Rev. Lett. 96, 086805 2006.3 E. V. Castro, K. S. Novoselov, S. V. Morozov, N. M. R. Peres, J.M. B. Lopes dos Santos, J. Nilsson, F. Guinea, A. K. Geim, andA. H. Castro Neto, Phys. Rev. Lett. 99, 216802 2007.4 E. McCann, Phys. Rev. B 74, 161403R 2006.5 T. Ohta, A. Bostwick, T. Seyller, K. Horn, and E. Rotenberg,Science 313, 951 2006.6 J. B. Oostinga, H. B. Heersche, X. Liu, A. F. Morpurgo, and L.M. K. Vandersypen, Nature Mater. 7, 151 2008.7 E. A. Henriksen, Z. Jiang, L.-C. Tung, M. E. Schwartz, M. Ta-kita, Y.-J. Wang, P. Kim, and H. L. Stormer, Phys. Rev. Lett.100, 087403 2008.8 Z. Q. Li, E. A. Henriksen, Z. Jiang, Z. Hao, M. C. Martin, P.Kim, H. L. Stormer, and D. N. Basov, Phys. Rev. Lett. 102,037403 2009.9 Y. Zhang, T.-T. Tang, C. Girit, Z. Hao, M. C. Martin, A. Zettl, M.F. Crommie, Y. R. Shen, and F. Wang, Nature London 459,820 2009.10 K. F. Mak, C. H. Lui, J. Shan, and T. F. Heinz, Phys. Rev. Lett.102, 256405 2009.11 A. B. Kuzmenko, E. van Heumen, D. van der Marel, P. Lerch, P.Blake, K. S. Novoselov, and A. K. Geim, Phys. Rev. B 79,115441 2009.12 A. B. Kuzmenko, I. Crassee, D. van der Marel, P. Blake, and K.S. Novoselov, Phys. Rev. B 80, 165406 2009.13 A. B. Kuzmenko, L. Benfatto, E. Cappelluti, I. Crassee, D. vander Marel, P. Blake, K. S. Novoselov, and A. K. Geim, Phys.Rev. Lett. 103, 116804 2009.14 B. E. Feldman, J. Martin, and A. Yacoby, Nat. Phys. 5, 8892009.15 S. Kim and E. Tutuc, arXiv:0909.2288 unpublished.16 Y. Zhao, P. Cadden-Zimansky, Z. Jiang, and P. Kim, Phys. Rev.Lett. 104, 066801 2010.17 E. A. Henriksen and J. P. Eisenstein, Phys. Rev. B 82, 0414122010.18 A. Young, C. Dean, I. Meric, S. Sorgenfrei, H. Ren, K. Wa-tanabe, T. Taniguchi, J. Hone, K. Shepard, and P. Kim,arXiv:1004.5556 unpublished.19 J. Nilsson, A. H. Castro Neto, N. M. R. Peres, and F. Guinea,Phys. Rev. B 73, 214418 2006.20 H. Min, B. Sahu, S. K. Banerjee, and A. H. MacDonald, Phys.Rev. B 75, 155115 2007.21 E. McCann, D. Abergel, and V. Fal’ko, Solid State Commun.143, 110 2007.22 J. Nilsson, A. H. Castro Neto, F. Guinea, and N. M. R. Peres,Phys. Rev. B 78, 045405 2008.23 L. M. Zhang, Z. Q. Li, D. N. Basov, M. M. Fogler, Z. Hao, andM. C. Martin, Phys. Rev. B 78, 235408 2008.24 L. A. Falkovsky, Phys. Rev. B 80, 113413 2009.25 M. S. Bello, E. I. Levin, B. I. Shklovskii, and A. L. Efros, Sov.Phys. JETP 53, 822 1981.26 O. V. Dolgov, D. A. Kirzhnits, and E. G. Maksimov, Rev. Mod.Phys. 53, 81 1981.27 J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 50,1760 1994.28 L. Zhang, M. Fogler, and D. Arovas, arXiv:1008.1418 unpub-lished.29 P. Gava, M. Lazzeri, A. M. Saitta, and F. Mauri, Phys. Rev. B79, 165431 2009.30 H. Min, G. Borghi, M. Polini, and A. H. MacDonald, Phys. Rev.B 77, 041407R 2008.31 F. Zhang, H. Min, M. Polini, and A. H. MacDonald, Phys. Rev.B 81, 041402 2010.32 R. Nandkishore and L. Levitov, Phys. Rev. Lett. 104, 1568032010.33 O. Vafek and K. Yang, Phys. Rev. B 81, 041401 2010.34 Y. Lemonik, I. Aleiner, C. Toke, and V. Falko, arXiv:1006.1399unpublished.35 Some evidence for this phenomenon has been reported in Ref.14. However, certain minor band-structure effects may have alsoplayed a role in these experiments, see Ref. 28.COMMENTS PHYSICAL REVIEW B 82, 197401 2010197401-4

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Screening in gated bilayer graphene via variational calculus

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