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Compressibility Instability of Interacting Electrons in Bilayer Graphene Xin-Zhong Yan1 and C. S. Ting2 1Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100190, China 2Texas Center for Superconductivity, University of Houston, Houston, Texas 77204, USA (Dated: January 11, 2011) Using the self-consistent Hartree-Fock approximation, we studythe compressibility instability of theinteractingelectrons inbilayergraphene. Thechemicalpotentialand thecompressibility of the electrons can be significantly altered byan energy gap (tunableby externalgate voltages) between the valence and conduction bands. For zero gap case, we show that the homogeneous system is stable. When the gap is finite, the compressibility of the electron system becomes negative at low 1 carrierdopingconcentrationsandlowtemperature. Thephasediagramdistinguishingthestableand 1 0 unstableregionsofatypicallygappedsystemintermsoftemperatureanddopingisalsopresented. 2 PACSnumbers: 73.22.Pr,81.05.ue,51.35.+a n a J Bilayer graphene has attracted considerable atten- 0 tion because of its promised application in electronic 1 devices.1–19 In contrast to the Dirac fermions in monolayer graphene, the energy bands of the free electrons ] el itnhebivlaayleenrcgeraapnhdencoenadruecthiyopnerbbaonldics.a5nMdogsatpliemsspobrettawnetelyn, t1 - r an energy gap opening between the valence and conduc- t s tion bands can be generated and controlled by external . t gatevoltages. Atlowcarrierdoping,becausetheenergy- a t momentum dispersion relation around the Fermi level m is relatively flat, the Coulomb effect is expected to be - d significant in the interacting electron system of bilayer n graphene. The Coulomb effect has been studied by a o numberoftheoreticalworksusingtheHartree-Fock(HF) FIG. 1: Structure of Bernal stacking bilayer graphene. The c andrandom-phaseapproximations.11–19 Oneofthether- energiesofintra-andinterlayerelectronhoppingbetweenthe [ modynamicquantitiesdirectlyreflectingtheCoulombef- A(white) and B(black) atoms aregiven by tand t1, respec- 3 fectistheelectroniccompressibility. Recently,thisquan- tively. v tity has been measured by experiments.2–4 For timely 8 coordinating with the experimental measurements, it is 5 necessary to theoretically study the combined effect of interlayer distance is c = 3.34 ˚A 1.4a where a 2.4 3 4 Coulomb interactions and the gap opening in the com- ˚A is the lattice constant of mono≈layer graphene.≈The pressibility. . energyofelectronhopping betweenthe nearest-neighbor 0 In this work, using the self-consistent HF (SCHF) (nn) carbon atoms in each layer is t 2.82 eV,21 while 1 0 approximation,19 we investigate the energy bands, the the interlayer nn hopping is t1 0.39≈eV.22 ≈ 1 chemical potential and its derivative with respect to the The Hamiltonian describing the electrons is given by : carrier density for the interacting electrons in bilayer v graphene. With these results, the compressibility insta- i X bilitiesforgappedandungappedelectronsystemsareex- H = ψ† h0ψ + 1 n v n (1) r amined. We showthatthe compressibilityis alwayspos- X pσ pσ 2X i ij j a p ij itive for zero gap case, while for the gapped system the compressibility becomes negative at low doping and low temperaturethatimpliesthesystemwouldbecomeinho- where ψ† = (c† ,c† ,c† ,c† ) with p = (k,v,σ) pσ p,a,1 p,b,1 p,a,2 p,b,2 mogeneous. Thephasediagramdistinguishingthestable standing for the momentum k, the valley v and spin and unstable regions in the temperature-density plane is σ, and c† creating an electron in p state at a lat- p,a,ℓ presented for a typically gapped homogeneous electron tice of the ℓ layer (with the top layer labeled as 1), system in bilayer graphene. SCHF is a conserving ap- n is the electron density at the site i, and v is the i ij proximation satisfying microscopic conservation laws.20 Coulomb interaction between electrons at sites i and j. The conserving conditions are crucial in the many parti- Here, for the low energy electrons in bilayer graphene cle systems. with interlayerhopping, the matrix h0 is an extension5,8 The atomic structure of bilayer graphene is shown in of the Dirac fermions in monolayer graphene.23,24 By Fig. 1. The two sublattices in each layer are denoted denoting k + ik = kexp(iφ), h0 can be written as x y by A (white) and B (black) atoms, respectively. The h0 = T(φ)h(k)T†(φ) with T(φ) = Diag(1,eiφ,eiφ,ei2φ) 2 1.0 (a) (a) (b) 0.5 (b) 0.0 /DE = 0 D/E = 0.02 )k 0 0 ( E -0.5 (c) = + -1.0 SCHF at T = 0 Free electrons -1.5 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 FIG. 2: (color online) Diagrammatic expressions for (a) the ka ka self-energy Σ(k), (b) the irreducible density-density response function, and (c) the vertex correction. The solid line with anarrowistheGreen’sfunctionandthewavylinerepresents FIG.3: (coloronline)Energybandstructureoffreeelectrons theCoulomb interaction. (dashed lines) and the interacting electrons under the self- consistent Hartree-Fock approximation at T =0 (solid lines) for (a) ∆/E0 = 0 and δ = 0 and (b) ∆/E0 = 0.02 and and δ=10−5. ∆ k 0 0 k ∆ t 0  h(k)= 1 (2) whereU(k)is anunitary matrixandV isthe eigenop- 0 t ∆ k pν  1 −  eratorofaquasiparticlewithenergyE (k). Thechemical 0 0 k ∆ ν − potential µ is determined by the doped electron density in units of E √3t/2 = a = 1. Here ∆ is the energy 0 ≡ 4 gap parameter, which is a consequence of the potential n= f[E (k)] 2 (7) V X{X ν − } difference between the top and back gates (attached to k ν the top and lower layers, respectively). The form of h0 wherethefactor4comesfromthespinandvalleydegen- reminds us to take the transform ψ = T(φ)Φ and pσ pσ eracy, and f(E) = 1/ exp[(E µ)/T]+1 is the Fermi then work with the scalar momentum k. We here take { − } distribution function. The diagonalization of H and the cutoff of k as k = 1. Under the SCHF approxima- eff c determination of Σ and µ are carried out by iteration tiontotheCoulombinteractions,oneobtainsaneffective µν until the self-consistency is achieved. Hamiltonian In Fig. 3, we show the energy band structure of the Heff =XΦ†pheff(k)Φp (3) electrons at temperature T = 0 for (a) ∆ = 0 and elec- p trondopingconcentrationδ √3a2n/8(dopedelectrons ≡ per carbon atom) = 0 and (b) ∆/E = 0.02 (the typi- with h (k)=h(k)+Σ(k) and 0 eff cal magnitude in experiments2,3) and δ = 10−5. The 1 Σµν(k) = −V Xvµν(|~k−~k′|)cosθµνhΦ†p′νΦp′µi,(4) sboylitdhleinCesoualroemtbheinetneerargciteiosnosf.tTheheelfeocutrroennserregnyobrmanadliszeodf k′ free electrons denoted by the dashed lines are given by 0 α α 2α E0(k) = [∆2 +t2/2+k2 t4/4+(t2+4∆2)k2]1/2. α 0 0 α ± 1 ±p 1 1 (θ ) = (5) Theupperandlowerbandsofthefreeelectronsaresym- µν α 0 0 α   metric about zero energy. For the interacting electrons, 2α α α 0  the energies shift downwards because of the self-energy where v (~k ~k′ ) is the Coulomb interaction, α is the contributions, and the upper and lower bands are not µν angle betw|een−~k a|nd ~k′, and Φp′µ [with p′ = (k′,v,σ)] sisymnomtetcrhiacnagneydmboyre.CFoourlotmhebzienrtoe-rgaacptiocansse., tNheotziceerotghaapt is the µth component of Φp′. For electrons in the same the contribution from the two lower bands to the self- layer, v (q) = 2πe2/ǫq with ǫ = 4. For interlayer elec- µν energyisincludedhere. Ifitis aconstantforanydoping tron interactions, v (q) = 2πe2exp( cq)/ǫq. The self- µν − and temperature, it can be subtracted from the begin- energy Σ(k) is diagrammatically shown in Fig. 2 (a) ning. However,inthe SCHF, the valence band itselfand in terms of the Green’s function. In the numerical cal- therebyitscontributiontotheself-energyvarywithdop- culations, special care needs to be paid to a logarith- ing and temperature. Especially, in the zero gap case, mic singularity stemming from the azimuthally integral this variation is significant. The contribution from all of v (~k ~k′ )cosθ . µν | − | µν the two lower bands cannot be considered as a constant H can be diagonalized by using the transformation eff and subtracted from the beginning. Φ = U (k)V (6) Figure 4 shows the chemical potential µ as a function pµ X µν pν of δ for the system with ∆/E =0.02 at severaltemper- ν 0 3 600 -0.25 /DE = 0.02 0 T/T = 0 0 T/T = 0.5 0 400 Free electrons at T= 0 T/T = 1 E0 -0.30 0 / m T/T = 1, D = 0 0 200 T= 0 D= 0 T/T = 0.5 -0.35 0 T/T = 1 0 0 -0.8 -0.4 0.0 0.4 0.8 -0.50 -0.25 0.00 0.25 0.50 d (10-3) d(10-3) FIG. 4: (color online) Chemical potential µ as function of FIG. 5: (color online) (∂µ/∂n)T as function of doping con- doping concentration δ for ∆/E0 = 0 at T/T0 = 1 (black centrationδ atT =0(bluedash-dotline),T/T0=0.5(green dashed line) and ∆/E0 = 0.02 at T/T0 = 0 (blue dotted dashed line), and T/T0 = 1 (red solid line) for ∆ = 0. The line), T/T0 = 0.5 (green dash-dot line) and T/T0 = 1 (red black dotted line is theresult of thefree electrons at T =0. solid line). aturesandthat ofthe zero-gapsystematT/T0 =1 with (∂n/∂µ)T can be calculated by performing the deriva- T 0.01E 283 K (close to the room temperature). tive with the obtained result as shown in Fig. 4. On the 0 0 For≡the gappe≈d system, the low doping behavior of µ is otherhand,thisfactorisactuallytheirreducibledensity- delicate. At T = 0, µ varies discontinuously at δ = 0. densityresponsefunctionχnndiagrammaticallyshownin Thisdiscontinuitystemsfromtheenergygapbetweenthe Fig. 2 (b) with the vertex correction given in Fig. 2 (c). valence and conductions bands. From both sides of the Here, both results are the same because the SCHF ap- carrier doping close to δ =0, µ is not a monotonic func- proximation for Σ(k) and χnn satisfies the microscopic tionofδ. Incontrastto this zeroT behavior,atfinite T, conservation laws.20 µincreasesdramaticallyatsmalldoping. Itiswellknown In Fig. 5, we show (∂µ/∂n) as a function of δ for T that µ at finite T is lower than that at T =0 for a one- ∆ = 0 at various temperatures. At low doping close band electron system with parabolic energy-momentum to δ = 0, there is a sharp decrease in (∂µ/∂n) at T dispersion relation because the particles occupy energy T = 0 with decreasing δ. But even close to δ = 0, levels above the Fermi energy with a lowered chemical (∂µ/∂n) is positive. This result is different from the T potential. In the present gapped case, the conduction existing calculation based on the perturbative HF ap- band is the effective one band system. At smaller elec- proach on a bilayer graphene model.25 With increasing tron doping, the Fermi energy is lower, the temperature the temperature, (∂µ/∂n) at zero doping increasesand T effect is much more pronounced. The situation for the the function at low doping tends to show a flattened be- zero-gapsystemis different. The temperatureeffect inµ havior. Since (∂µ/∂n) > 0, the homogeneous electron T ofthezero-gapsystemisnotsonotableasinthe gapped system of ∆ = 0 is mechanically stable at any doping system. For ∆ = 0, the electrons in the valence band and any temperature. The result for the free electrons canbe thermallyexcitedtohighlevelsinthe conduction at T = 0 is also plotted in Fig. 5, which is a monoton- bandwithoutsignificantchangeinthechemicalpotential ically decreasing function of δ . Recent experiments on because the density of states is nearly symmetric about the compressibility2,3 seem q|ua|litatively consistent with thetouchpointofthevalenceandconductionbands. The this free electron behavior. Why the Coulomb effect is chemical potential for ∆ = 0 at T/T0 = 1 is almost the not observed is still an open question. same as at T =0. Shown in Fig. 6 is (∂µ/∂n) as a function of δ for T The behavior of the chemical potential is closely re- ∆/E = 0.02 at various temperatures. At T = 0, 0 lated to the compressibility κ and the spin susceptibility (∂µ/∂n) is negative at small δ consistent with the neg- T χ of the doped carrier system. They are defined as ative slope of µ as shown in Fig. 4. This means that 1 ∂n the homogeneous electron system with (∂µ/∂n)T < 0 is κ= n2(∂µ)T (8) unstable and favors to become an inhomogeneous state. At finite temperature, in principle, (∂µ/∂n) is positive T and at δ = 0 because of the behavior of µ as shown in Fig. 4. At T > 0, it is seen from Fig. 6 that (∂µ/∂n) at ¯hµ ∂n T B χ= ( )T (9) δ = 0 is likely infinitive. At finite but low temperature, 2 ∂µ (∂µ/∂n) decreaseswithin a small regionclose to δ =0, T with µ as the Bohr magneton. The common factor thenincreases,andfinallydecreasesslowlyatlargeδ. As B 4 600 using the self-consistent Hartree-Fock approximation. WehavestudiedthecombinedeffectsduetotheCoulomb /DE = 0.02 T/T = 1 interactionsandthe energygapbetweenthe valenceand 0 0 400 conductionbandsinthechemicalpotentialandthecom- T/T = 0.5 0 pressibility of the electrons. We find that the homoge- T= 0 neoussystemwithzerogapisalwaysstable. Foragapped 200 system,the compressibilitybecomes negative atlow carrier doping concentrations and low temperature, leading tothe instability ofthe homogeneoussystem. The phase 0 diagram distinguishing the stable and unstable regions of a typical gapped homogeneous system is given by the present calculation. -200 -1.0 -0.5 0.0 0.5 1.0 This work was supported by the Robert A. Welch d (10-3) Foundation under Grant No. E-1146, the TCSUH, the National Basic Research 973 Program of China under FIG.6: (coloronline)(∂µ/∂n)T asfunctionofdopingconcen- Grant No. 2011CB932700, NSFC under Grants No. tration δ for gapped system of ∆/E0 = 0.02 at T = 0 (blue 10774171and No. 10834011,and financial support from dash-dot line), T/T0 =0.5 (green dashed line with squares), 1.0 and T/T0 =1 (red line with circles). 0.8 k> 0 showninFig. 6,atT/T =1,(∂µ/∂n) ispositiveatall 0 T the doping concentrations. We then deduce that the ho- 0.6 mogeneoussystemisstableatanydopingconcentrations 0 for T/T0 >1. T/T 0.4 To know whether the systemis stable ornot is helpful k < 0 k < 0 for both the theoretical understanding of the thermody- 0.2 namic propertiesand the technologicalapplicationofbi- layer graphene. We have determined the phase diagram of the electron system with ∆/E = 0.02 as shown in 0.0 0 -0.4 -0.2 0.0 0.2 0.4 Fig. 7. The red curve in Fig. 7 divides the T δ plane − d (10-3) into homogeneous stable (κ > 0) and unstable (κ < 0) regions. Note that at the border, κ and χ go to infinity. κ means the system undergoes a phase separation FIG. 7: (color online) T −δ phase diagram of the typical → ∞ or Wigner crystallization. χ implies there is fer- gapped electron system of ∆/E0 = 0.02 in bilayer graphene. → ∞ romagnetization in the system. Therefore, in the region Thered curveis theborderbetween thehomogeneous stable with (∂µ/∂n) < 0, the case of an inhomogeneous sys- (κ>0) and the unstable(κ<0) states. T tem, a Wigner crystal,26 or a ferromagnetic state9 may be possible. In summary, we have investigated the compressibility instabilityoftheinteractingelectronsinbilayergraphene the Chinese Academy of Sciences for advanced research. 1 T. Ohta, A. Bostwick, T. Seyller, K. Horn, E. Rotenberg, 8 M.KoshinoandT.Ando,Phys.Rev.B76,085425(2007). Science 313, 951 (2006). 9 J. Nilsson and A. H. Castro Neto, Phys. Rev. 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