Excitonic condensation of massless fermions in graphene bilayers
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Excitonic condensation of massless fermions in graphene bilayers
C.-H. Zhang and Yogesh N. Joglekar
Department of Physics, Indiana University-Purdue University Indianapolis (IUPUI), Indianapolis, Indiana 46202, USA (Dated: October 29, 2018)Graphene, a single sheet of graphite with honeycomb lattice structure, has massless carriers withtunable density and polarity. We investigate the ground state phase diagram of two graphene sheets(embedded in a dielectric) separated by distance d where the top layer has electrons and the bottomlayer has holes, using mean-field theory. We find that a uniform excitonic condensate occurs over alarge range of carrier densities and is weakly dependent on the relative orientation of the two sheets.We obtain the excitonic gap, quasiparticle energy and the density of states. We show that both, thecondensate phase stiffness and the mass of the excitons, with massless particles as constituents, varyas the square-root of the carrier density, and predict that the condensate will not undergo Wignercrystallization. PACS numbers:
Introduction:
Over the past three years, graphene hasemerged as the unique candidate that provides a real-ization of two-dimensional massless fermions whose car-rier density and polarity are tunable by an external gatevoltage [1]. Subsequent experimental and theoretical in-vestigations have led to a thorough re-examination ofsome of the properties of linearly dispersing masslessfermions [2, 3]. The truly two dimensional (2D) natureof graphene permits control and observation of local car-rier density and properties [4, 5, 6]. In graphene bilayers,the ability to change the carrier polarity of an individ-ual layer implies that the interlayer Coulomb interactioncan be tuned from repulsive to attractive. This raisesthe possibility of formation of electron-hole bound statesor indirect excitons, albeit with massless fermions as itsconstituents. Properties of such bound states of masslessparticles are an open question; the only other example,to our knowledge, is the proposed color superconductiv-ity in dense quark matter [7]. Graphene bilayers providean ideal and unique candidate for straightforward exper-imental investigations of such phenomena.A uniform Bose-Einstein condensate of excitons inelectron-hole bilayers occurs when the interlayer dis-tance is comparable to the distance between the parti-cles within each layer [8, 9]. These excitons have mass m ex = m e + m h where m e ( m h ) is the band massof the electron (hole). At high densities, dipolar re-pulsion between the excitons can lead to a condensateground state with broken translational symmetry: a su-persolid [10]. Biased bilayer quantum Hall systems neartotal filling factor one have shown uniform excitonic con-densation [11]. In this case, the exciton mass is deter-mined solely by interlayer Coulomb interaction and is in-dependent of the bias voltage [12, 13]. These observationsraise the questions: What is the mass of an exciton withmassless constituents? Will such an excitonic condensatelead to a supersolid if the dipolar repulsion between suchexcitons (with a nonzero mass) is increased?In this paper, we investigate the excitonic condensa- tion in two graphene sheets embedded in a dielectric andseparated by a distance d ≫ a ( a =1.4˚A is the honey-comb lattice size) so that the tunneling between the lay-ers is negligible, but interlayer Coulomb interaction isnot. The layers have opposite polarity and equal densityof carriers n D . We remind the Reader that in graphene,in the continuum limit, the length-scale 1 /k F and theenergy-scale E F are both set by the density of carriers n D ( k F = √ πn D is the Fermi momentum, E F = ~ v G k F is the Fermi energy, and v G ∼ c/
300 is the speed ofmassless carriers). Therefore, the ground state phasediagram depends only on one dimensionless parameter k F d . This is markedly different from conventional bi-layer systems parameterized by ( d/a B , r s ) where a B isthe band Bohr radius and r s = 1 / p πa B n D [9], as wellas biased bilayer quantum Hall systems, parameterizedby ( d/l B , ∆ ν ) where l B is the magnetic length and ∆ ν isthe filling factor imbalance [13, 14, 15, 16].We use the mean-field theory to obtain the ground-state phase diagram as a function of k F d . We find thata) excitonic condensation occurs at all densities as longas k F d ∼
1. b) the condensate properties are weakly sen-sitive to the relative orientation of the two sheets (stack-ing). c) the superfluid phase stiffness ρ s and the excitonmass have a √ n D dependence. d) the excitonic conden-sate does not undergo Wigner crystallization in spite ofdipolar repulsion between excitons with a nonzero mass.The plan of the paper is as follows. In the next section,we present the mean-field Hamiltonian [17] and brieflysketch the outline of our calculations. In the subsequentsection, we show the results for the excitonic gap ∆ k , thequasiparticle energy E k , and the quasiparticle density ofstates D ( E ). We discuss the density dependence of thesuperfluid stiffness ρ s and the mass of the excitons. Inthe last section, we show that these results are equivalentto absence of Wigner crystallization, and mention theimplications of our results to experiments. Mean-field Model:
We consider two graphene sheets em-bedded in a dielectric separated by distance d with chem-ical potentials in the two layers adjusted so that the toplayer (denoted by pseudospin τ = +1) has electrons andthe bottom layer (denoted by pseudospin τ = −
1) hasholes with the same density. We consider two stackings:the Bernal stacking that occurs naturally in graphite, andthe hexagonal stacking in which each sublattice ( A and B ) in one layer is on top of the corresponding sublatticein the other layer. Since the Hamiltonian in the con-tinuum description is SU(4) symmetric in the spin andvalley indices, we ignore those indices for simplicity. Inthe continuum limit, the single-particle Hamiltonian forcarriers in layer τ is [18]ˆ H = Σ k α ( α ~ v G k ) c † k ατ c k ατ (1)where k is the momentum measured from the K -pointand α = ± denote the conduction and valance bandsthat result from diagonalizing the Hamiltonian in thesublattice-basis. c † k ατ ( c k ατ ) is creation (annihilation)operator for an electron in band α in layer τ with mo-mentum k . We point out that for the hexagonal stack-ing, c † k ατ = [ c † k Aτ + αe − iθ k c † k Bτ ] / √ τ . For the Bernal stacking, the creationoperators in the two layers are related by complex con-jugation, c † k ατ = [ c † k Aτ + αe − iτθ k c † k Bτ ] / √ θ k =tan − ( k y /k x ). The interaction Hamiltonian consists ofintralayer Coulomb repulsion V A ( q ) = 2 πe /ǫq and in-terlayer Coulomb attraction V E ( q ) = − V A ( q ) exp( − qd ).( ǫ is the dielectric constant). Using standard mean-field techniques [17], we obtain the following mean-fieldHamiltonianˆ H = X k (cid:2) e † k h − k (cid:3) (cid:20) ǫ k − µ ∆ k ∆ ∗ k − ǫ k + µ (cid:21) (cid:20) e k h †− k (cid:21) . (2)Here e † k = c † k ++ creates an electron in the conductionband ( α = +) in the top layer ( τ = +1) and h †− k = c k −− creates a hole in the valance band ( α = − ) in the bot-tom layer ( τ = − ). The term ǫ k contains single-particleenergy, capacitive Hartree self-energy and the intralayerexchange self-energy. The off-diagonal term ∆ k is pro-portional to the excitonic condensate order parameter h h − k e k i . The eigenvalues of the mean-field Hamiltonianare given by ± E k = ± p ( ǫ k − µ ) + ∆ k . We considermean-field states with a real ∆ k = ∆ ∗ k , and spatiallyuniform density. It is straightforward to diagonalize theHamiltonian and obtain the mean-field equations [9] ǫ k = ~ v G k + e n D C − Z k ′ V A ( k − k ′ ) (cid:20) − ξ ′ k E ′ k (cid:21) (3)∆ k = − Z k ′ V E ( k − k ′ ) f ( θ k , k ′ ) ∆ k ′ E ′ k (4)where ξ k = ǫ k − µ , C = ǫ/ πd is the capacitance per unitarea, and θ k , k ′ = θ k − θ k ′ . The form factor for the twostackings are f ( θ k , k ′ ) = (cid:26) (1 + cos θ k , k ′ ) Hexagonalcos θ k , k ′ (1 + cos θ k , k ′ ) Bernal . (5) We point out that the self-energy in Eq.(3) takes intoaccount both intrinsic and extrinsic contributions thatcancel the cos θ k , k ′ -dependent terms in the form factorand make the results independent of the ultra-violet cut-off [19, 20]. Therefore the intra-layer self-energy in Eq.(3)is the same as that for a conventional system [19, 20]. Thechemical potential µ is determined by the carrier densitythat takes into account the four-fold spin and valley de-generacy n D = 4 Z k (cid:20) − ξ k E k (cid:21) . (6)It is straightforward to derive similar equations for a con-ventional electron-hole system [9]. They are obtained bychanging the single-particle dispersion to a quadratic andreplacing the form factor f ( θ k , k ′ ) by a constant f = 2.We solve Eqs. (3), (4) and (6) iteratively to obtain self-consistent results. ∆ k / E F and E k / E F k/k F E k ∆ k BernalHexagonalusual 2DEG
FIG. 1: (Color online) Excitonic gap ∆ k and the quasiparticleenergy E k in graphene bilayer for Bernal (green dashed) andhexagonal (red solid) stacking with k F d = 1. The quasiparti-cle spectrum E k becomes linear with a renormalized velocity˜ v G > v G for large k ≫ k F . The dotted blue curves showcorresponding results for an electron-hole system at r s = 2 . k F d = 1 when plotted using relevant (atomic) unit forenergy [9]. Results:
Figure 1 shows the excitonic gap ∆ k and thequasiparticle energy E k for the Bernal (green dashed)and the hexagonal (red solid) stacking. The excitonic gap∆ k is maximum at the Fermi momentum k F where thequasiparticle energy E k is minimum. Since the electron-hole Coulomb interaction is always attractive, the exci-tonic condensate order parameter is nonzero down to thebottom of the Fermi sea, ∆ k =0 = 0. Our results predictthat the hexagonal-stacked system will have a larger exci-tonic gap than the Bernal-stacked system.The quasipar-ticle energy E k becomes linear at large k ≫ k F , since theconstituent particles of the exciton have a linear disper-sion. The speed of these quasiparticles is increased dueto intralayer exchange self-energy [20, 21, 22] althoughthe increase is modest, ∼ ∆ m a x / E F k F dBernalHexagonalusual 2DEG FIG. 2: (Color online) Dependence of the graphene bilayer ex-citonic gap ∆ m = Max(∆ k ) on interlayer distance d for Bernal(green dashed) and hexagonal (red solid) stacking. This gapcan be tuned by changing n D for a given sample. Corre-sponding result for a conventional system at r s = 2 . Figure 2 shows the dependence of the maximum exci-tonic gap ∆ m on k F d . We find that ∆ m is weakly depen-dent on the stacking and decays rapidly when k F d ≫ This result implies that the excitonic condensation is arobust phenomenon that will not require precise align-ment of the two graphene sheets when they are being em-bedded in a dielectric . With typical graphene carrier den-sities n D ∼ /cm and d ∼ k F d ∼
1, theexcitonic gap is appreciable, ∆ m ∼
30 meV.A direct probe of the excitonic gap is the quasiparti-cle density of states. For graphene with no interactions,the density of states is linear, D ( E ) = 2 E/π ~ v G . Inthe excitonic condensate phase, for intermediate energies∆ m ≤ E ≤ E k =0 there are two rings in the phase-spaceconsistent with that energy: one with k < k F and theother with k > k F . Therefore the quasiparticle densityof states is given by D ( E ) = D < ( E ) + D > ( E ) where D < ( D > ) denotes the density of states from respective rings.Figure 3 shows D < ( E ) (green dashed) and D > ( E ) (redsolid); they are both zero for E < ∆ m and diverge at∆ m as is expected. Note that D < ( E ) = 0 for E > E k =0 ,since there are no states for k < k F with energies higherthan E k =0 . The asymmetry in D < and D > for E ≫ ∆ m is due to the linear dispersion of carriers and the nonzeroelectron-hole pairing that extends to the bottom of theFermi sea, ∆ k =0 = 0 [23]. The inset shows correspondingresults a conventional system, where the density of stateswithout interactions is constant, D ( E ) = m/π ~ .Superfluidity of a uniform Bose-Einstein condensate ischaracterized by a non-zero phase stiffness ρ s that quan-tifies the energy of a condensate with a linearly wind- D ( E ) E/E F D > (E)D < (E)k F d = 1, Hexagonal ∆ max FIG. 3: (Color online)Quasiparticle density of states contri-butions D < ( E ) (green dashed) associated with states with k ≤ k F , and D > ( E ) (red solid) associated with states k ≥ k F .These results are for hexagonal stacked graphene bilayers with k F d = 1. Both diverge at E = ∆ m , as expected. The totaldensity of states D = D < + D > can be probed by differentialconductance for tunneling from a metal into the condensate.The inset shows corresponding results for an electron-hole sys-tem at r s = 2 . k F d = 1. All results are expressed in theirrespective units. ing phase, E ( Q ) = ρ s Q A/ A is the area ofthe sample and the phase of the condensate varies asΦ( x ) = Qx . For graphene, since E F is the sole energyscale (at zero temperature), it follows from dimensionalanalysis that phase stiffness must scale linearly with theFermi energy, ρ s = g ( k F d ) E F where g ( x ) is a dimension-less function that satisfies g ∼ O (1) [24] when 0 ≤ x . g → x ≫
1. Hence, the phase stiffness isgiven by ρ s = g ( k F d ) ~ v G √ πn D . The condensate en-ergy E ( Q ) can also be expressed, in the particle-picture,as the kinetic energy of excitons that have condensedin a state with center-of-mass momentum ~ Q . Thus, E ( Q ) = N ~ Q / m ex where m ex ( N ) is the mass (num-ber) of condensed excitons [25]. Equating the two ex-pressions for energy implies m ex = n D ~ /ρ s ∝ √ n D .Thus we predict that the phase stiffness ρ s and the exci-ton mass will both vary as the square root of the carrierdensity . We emphasize that these results are unique tographene and, as we will show in the next section, are equivalent to the absence of excitonic Wigner crystalliza-tion in graphene bilayers [10, 26]. Discussion:
In this paper, we have investigated the prop-erties of excitonic condensates in graphene bilayers. Ourcalculations predict that excitonic condensation will oc-cur at all carrier densities as long as k F d ∼
1, and thatthe strength of the condensate, as measured by the exci-tonic gap ∆ m is relatively insensitive to the stacking.The mean-field results presented in this paper are ob-tained at zero temperature T =0. (Finite temperatureanalysis gives a critical temperature T MF /E F ∼ . T MF ∼
20 meV. This is an artifact of the mean-fieldapproximation.) In two dimensions, the critical temper-ature T c for Bose-Einstein condensation is zero, but thesuperfluid properties survive for T ≤ T KT where T KT isthe Kosterlitz-Thouless transition temperature. There-fore, our results will be valid at nonzero temperature T ≪ T KT [27]. A weak disorder will suppress the ex-citonic condensate order parameter and reduce the exci-tonic gap, an effect equivalent to increasing the value of k F d . Therefore, we have ignored the effects of a weakdisorder potential.In our analysis, we have only considered excitonic con-densation with uniform density. In conventional (quan-tum Hall electron-hole) bilayers, varying d and r s ( ν )leads to excitonic condensates with lattice structure [10,26, 28]. The origin of the lattice structure is Wignercrystallization of carriers in an isolated layer at large r s (small ν ). Graphene does not undergo Wigner crystal-lization as its carrier density is changed [29]. Therefore,we expect that the excitonic condensate in graphene bi-layers remains uniform. Now we show that this result isequivalent to our predictions for density dependence of ρ s and m ex . The quantum kinetic energy of an exciton,associated with localizing it within a distance 1 /k F , is K = ~ k F / m ex . The potential energy due to the dipo-lar repulsion between them is P = e d k F /ǫ . Hence theirratio is given by P/K = e d k F m ex /ǫ ~ . Wigner crys-tallization occurs when the ratio P/K ≫
1. This ratiowill solely be a function of k F d - no matter what thevalue of d is - if and only if m ex ∝ k F = √ πn D . There-fore, results in the last section show that the excitoniccondensate in graphene will not undergo Wigner crystal-lization in spite of the dipolar repulsion between excitonswith a quadratic dispersion. This result, too, is unique tographene and is markedly different from the behavior ofdipolar excitonic condensates in conventional bilayers. Itis interesting that the mass of these effective bosons hasthe same density dependence and order of magnitude asthe cyclotron mass of fermionic carriers in graphene [1].The onset of excitonic condensation can be detectedby a divergent interlayer drag [30]. A uniform in-planemagnetic field B || between the two graphene sheets isexpected to induce a (counterflow) supercurrent J d insuch a condensate [31], J d = 2 ρ s e dB || / ~ . The phasestiffness ρ s and its density dependence can be directlyobtained from experimental measurements of the coun-terflow supercurrent. The verification (or falsification)of our predictions, including the density dependence of ρ s and m ex , will deepen our understanding of propertiesand condensation of excitons with massless fermions asconstituent particles. Acknowledgments:
This work was supported by the Re-search Support Funds Grant at IUPUI. After this work was completed, we became aware of a recent relatedwork [27]. [1] K. S. Novoselov et al. , Nature , 197(2005); Y. Zhang et al. , Nature , 201(2005).[2] A.K. Geim and K.S. Novoselov, Nature Materials , 183(2007) and references therein.[3] K. Yang, Solid State. Comm. , 27 (2007).[4] J.R. Williams, L. DiCarlo, and C.M. Marcus, Science , 638 (2007).[5] D.A. Abanin and L.S. Levitov, Science , 641 (2007).[6] J. Martin et al. , Nature Physics , 414 (2008).[7] For a review, see M.G. Alford et al. , cond-mat/0709.4635and references therein.[8] S.A. Moskalenko and D.W. Snoke, Bose-Einstein Con-densation of Excitons and Biexcitons (Cambridge Uni-versity Press, 2000).[9] X. Zhu et al. , Phys. Rev. Lett. , 1633 (1995).[10] Y. N. Joglekar, A. V. Balatsky, and S. Das Sarma, Phys.Rev. B , 233302 (2006).[11] J.P. Eisenstein and A.H. MacDonald, Nature (London) , 691 (2004); J.P. Eisenstein, Science , 950 (2004);and references therein.[12] K. Yang, Phys. Rev. Lett. , 056802 (2001).[13] Y.N. Joglekar and A.H. MacDonald, Phys. Rev. B ,235319 (2002).[14] C. Hanna, Bull. Am. Phys. Soc. , 533 (1997).[15] E. Tutuc et al. , Phys. Rev. Lett. , 076802 (2003).[16] I.B. Spielman et al. , Phys. Rev. B , 081303(R) (2004).[17] J.W. Negele and H. Orland, Quantum Many Particle Sys-tems (Addison Wesley, New York, 1988).[18] G. W. Semenoff, Phys. Rev. Lett. , 2449 (1984).[19] E.H. Hwang, B.Y.-K. Hu, and S. Das Sarma, Phys. Rev.Lett. , 226801 (2007).[20] Y. Barlas et al. , Phys. Rev. Lett. , 236601 (2007).[21] S. Das Sarma, E.H. Hwang, and W.-K. Tse, Phys. Rev.B , 121406(R) (2007).[22] R. Roldan, M.P. Lopez-Sancho, and F. Guinea, Phys.Rev. B , 115410 (2008).[23] In N -dimensions, fermions with dispersion ǫ k ∼ k α and excitonic gap ∆ will have D < ( > ) ( E ) = D [ µ ∓√ E − ∆ ] N/α − for ∆ ≤ E ≤ p µ + ∆ , as k reducesfrom k F to 0. Thus, for N = α , D < ( E ) = D > ( E ) = D where D = E/ √ E − ∆ .[24] T = 0 phase stiffness, calculated in Ref. [27] via mean-field analysis reproduces the same functional form andimplies that g = 1 / π .[25] In our model, exciton formation and condensation occurat the same k F d . Hence, the T = 0 superfluid excitondensity is the same as the carrier density n D .[26] X. M. Chen and J. J. Quinn, Phys. Rev. Lett. , 895(1991).[27] H. Min et al. , arXiv:0802.3462.[28] C.H. Zhang and Y.N. Joglekar, cond-mat/0711.4847.[29] H. P. Dahal, Y. N. Joglekar, K. S. Bedell, and A. V.Balatsky, Phys. Rev. B , 233405 (2006).[30] S. Vignale and A.H. MacDonald, Phys. Rev. Lett. ,2786 (1996).[31] A.V. Balatsky et al. , Phys. Rev. Lett.93