Spectral properties of excitons in the bilayer graphene
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec Spectral properties of excitons in the bilayer graphene
V. Apinyan ∗ , T. K. Kope´c Institute for Low Temperature and Structure Research,Polish Academy of SciencesPO. Box 1410, 50-950 Wroc law 2, Poland (Dated: July 18, 2018)In this paper, we consider the spectral properties of the bilayer graphene with the local excitonicpairing interaction between the electrons and holes. We consider the generalized Hubbard model,which includes both intralayer and interlayer Coulomb interaction parameters. The solution of theexcitonic gap parameter is used to calculate the electronic band structure, single-particle spectralfunctions, the hybridization gap, and the excitonic coherence length in the bilayer graphene. Weshow that the local interlayer Coulomb interaction is responsible for the semimetal-semiconductortransition in the double layer system, and we calculate the hybridization gap in the band structureabove the critical interaction value. The formation of the excitonic band gap is reported as thethreshold process and the momentum distribution functions have been calculated numerically. Weshow that in the weak coupling limit the system is governed by the Bardeen-Cooper-Schrieffer(BCS)-like pairing state. Contrary, in the strong coupling limit the excitonic condensate statesappear in the semiconducting phase, by forming the Dirac’s pockets in the reciprocal space.
PACS numbers: 68.65.Pq, 73.22.Pr, 73.22.Gk, 71.35.Lk, 71.35.-y, 71.10.Li, 78.67.Wj, 73.30.+y
1. INTRODUCTION
The electronic band gap of semiconductors and insu-lators largely determines their optical, transport proper-ties and governs the operation of semiconductor baseddevices such as p-n junctions, transistors, photodiodesand lasers [1]. Opening up a band gap in the bilayergraphene (BLG), by applying the external electric fieldand finding a suitable substrate are two challenges forconstituting the modern nano-electronic equipment [2, 3].The imposition of external electrical field can tune the bi-layer graphene from the semimetal to the semiconduct-ing state [2]. On the other hand, the possibility of for-mation of the excitonic insulator state and the excitoniccondensation in the bilayer graphene structures remainscontroversial in the modern solid state physics [4–14].In difference with the quasi two-dimensional (2D) semi-conducting systems, where those two states have beenobserved experimentally and well discussed theoretically[15–27], the formation of the excitonic condensate statesin the BLG system, from the original electron-hole pair-ing states, is much more obscure because of the compli-cated nature of the single-particle correlations in thesesystems [6, 8, 10, 13, 14]. The weak correlation dia-grammatic mechanism, discussed in the Refs.13, 14, isrestricted only to the closed loop expansion in the dia-grammatic series, and in this case, only the density fluc-tuation effects could affect the formation of the excitoniccondensate states. Meanwhile, it has been shown [28–30]that even the undoped graphene can provide a varietyof electron-hole type pairing chiral symmetry breaking ∗ Corresponding author. Tel.: +48 71 3954 284; E-mail address:[email protected]. orders especially for the strong Coulomb coupling case,which renders the treatments in Refs.13, 14 to be non-trivial. As the monolayer graphene, bilayer graphene hasa semimetallic band structure with the zero bandgap,which is unsuitable for many electronic device applica-tions, as routinely done with semiconducting devices. In2007, Allan H. MacDonald and his colleagues have pre-dicted that the electric displacement field, applied to thetwo layers of the BLG could introduce a tunable bandgapin the electronic band structure of the BLG [31], whichhas been proved later on, experimentally [32–34].In this paper, we show the band gap formation in thebilayer graphene’s band structure, mediated by the lo-cal interlayer Coulomb interaction parameter and with-out the external electric field. Particularly, we show howthe interlayer Coulomb interaction tunes the BLG fromthe semimetallic state into the semiconducting one (or apossible insulator state, for large interaction limit), fora fixed value of the interlayer hopping amplitude. Wewill show that the formation of the band gap in the BLGis a threshold process in our case. We calculate the ex-citonic hybridization gap for different values of the in-teraction parameter. Furthermore, by using the exactsolutions for the excitonic gap parameter and chemicalpotential, we calculate the single-particle spectral func-tions and the momentum distribution functions in theBLG for different values of the local interlayer interactionparameter. We will show that the system is governed bythe weak-coupling Bardeen-Cooper-Schrieffer (BCS)-likepairing states for the small and intermediate values of theinterlayer coupling parameter, and the behavior of the ex-citonic coherence length agrees well with the BCS typerelation ξ c ∼ ¯ µ/ ( k F ∆), with ∆, being the excitonic pair-ing gap parameter. Contrary, in strong coupling limit thecoherence length becomes proportional to ∆ ( ξ c ∼ ∆),and the system is in the excitonic condensate regime.In the Section 2, we introduce the bilayer Hubbardmodel for the BLG system. In the Section 3, we givethe action of the bilayer graphene system and we obtainthe coupled self-consistent equations for the excitonic gapparameter and chemical potential. The excitonic disper-sion relations are also given there. Next, in the Section4, we discuss the single-particle spectral properties andmomentum distribution functions for different interlayerinteraction limits and we calculate the interlayer exci-tonic coherence length in the bilayer graphene. In theSection 5, we give a conclusion to our paper.
2. THE METHOD
The BLG system, considered here, is composed of twocoupled honeycomb layers with sublattices A , B and ˜ A ,˜ B , in the bottom and top layers respectively (see inFig. 1). In the z -direction the layers are arranged ac-cording to Bernal Stacking (BS) order [35], i.e. the atomson the sites ˜ A in the top layer lie just above the atomson the sites B in the bottom layer graphene, and eachlayer is composed of two interpenetrating triangular lat-tices. For a simple treatment at equilibrium, we initiallysuppose the balanced BLG structure, i.e., with the equalchemical potentials in the both layers. When switch-ing the local Coulomb potential W between the layers,we keep the charge neutrality equilibrium through theBLG, by imposing the half-filling condition in each layerof the BLG. Next, we will pass to the Grassmann rep-resentation for the fermionic variables, and we write thepartition function of the system, in the imaginary timefermion path integral formalism [36]. For this, we in-troduce the imaginary-time variables τ , at each latticesite r . The time variables τ vary in the interval (0 , β ),where β = 1 /T with T being the temperature. Then,the Hamiltonian of the bilayer graphene, with the localinterlayer interaction, has the following form H = − γ X h rr ′ i X σ (cid:0) a † σ ( r ) b σ ( r ′ ) + h.c. (cid:1) − γ X h rr ′ i X σ (cid:16) ˜ b † σ ( r )˜ b σ ( r ′ ) + h.c. (cid:17) − γ X r σ (cid:0) b † σ ( r )˜ a σ ( r ) + h.c. (cid:1) − X r σ X ℓ =1 , µ ℓ n ℓσ ( r )+ U X r X ℓη [( n ℓη ↑ − /
2) ( n ℓη ↓ − / − / W X r σσ ′ [( n bσ ( r ) − /
2) ( n aσ ′ ( r ) − / − / . (1)Here, we have used the graphite nomenclature notations[35] for the hopping amplitudes γ and γ . Namely, theparameter γ is the intraplane hopping amplitude, and γ is the interlayer hopping amplitude in the BLG (seealso in Fig. 1). The summation h rr ′ i , in the first term, FIG. 1: (Color online) The bilayer graphene structure. Twodifferent sublattice site positions are shown ( A , B , and ˜ A , ˜ B )in two different layers of the bilayer graphene structure. in Eq.(1), denotes the sum over the nearest neighborslattice sites in the separated honeycomb layers in thebilayer graphene structure. We keep the small letters a, b and ˜ a, ˜ b for the electron operators on the lattice sites A, B and ˜ A, ˜ B respectively (see in Fig. 1). The index ℓ = 1 , ℓ = 1 for the bottom layer, and ℓ = 2for the top layer. Furthermore, we have η = a, b for ℓ = 1,and η = ˜ a, ˜ b , for ℓ = 2. The symbol σ denotes the spinvariables with two possible directions ( σ = ↑ , ↓ ). Next, n ℓσ ( r ) is the total electron density operator for the layer ℓ with a given spin direction σn ℓσ ( r ) = X η n ℓησ ( r ) , (2)and n ℓησ ( r ) = η † ℓσ ( r ) η ℓσ ( r ) is the electron density opera-tor for the η -type fermions with the spin σ . We considerthe BLG structure with pure electronic layers withoutinitial dopping in the system. The condition of half-fillingin each layer reads as h n ℓ i = 1, for ℓ = 1 ,
2, where n ℓ isthe total electron density operator for the layer ℓ summedover the spin index σ : n ℓ ( r ) = P σ n ℓσ ( r ). Further-more, U , in the Hubbard term in Eq.(1), parametrizesthe intralayer Coulomb interaction. The parameter W ,in the last term in Eq.(1), describes the local interlayerCoulomb repulsion between the electrons located on the B and ˜ A type of sites in the different layers of the BLG.We will put γ = 1, as the unit of energy in the system,and we set k B = 1, ~ = 1 throughout the paper.
3. THE GREEN’S FUNCTION MATRIX
The pairing between the electron and holes results ina gap in the excitation energy spectrum of the system.The pairing gap parameter between the particles withthe same spin orientations is ∆ σσ ′ = ∆ σσ δ σσ ′ , and we canalso assume that the pairing gap is real ∆ σσ = ∆ † σσ ≡ ∆.Then the excitonic pairing gap parameter will be definedas ∆ σσ = W (cid:10) b † σ ( r τ )˜ a σ ( r τ ) (cid:11) . (3)In order to find the momentum dependence of the exci-tonic gap parameter, we will pass to the Fourier spacerepresentation, given by the transformation η σ ( r , τ ) = 1 βN X k ν n η σ k ( ν n ) e i ( kr − ν n τ ) , (4)where N is the total number of sites on the η -type sub-lattice, in the layer ℓ , and we write the fermionic actionof the bilayer graphene system in the form S (cid:2) ψ † , ψ (cid:3) = 1 βN X k ν n X σ ψ † σ k ( ν n ) ˆ G − σ k ( ν n ) ψ σ k ( ν n ) . (5)Here, ν n = π (2 n + 1) /β with n = 0 , ± , ± , . . . , arethe fermionic Matsubara frequencies [37]. The four com-ponent Dirac spinors ψ σ k ( ν n ), in Eq.(5), have been in-troduced at each discrete state k in the reciprocal spaceand for each spin direction σ = ↑ , ↓ . Being the generalizedWeyl spinors, they are defined as ψ σ k ( ν n ) = h a σ k ( ν n ) , b σ k ( ν n ) , ˜ a σ k ( ν n ) , ˜ b σ k ( ν n ) i T . (6)The matrix ˆ G − σ k ( ν n ), in Eq.(5), is the inverse Green’sfunction matrix, of size 4 ×
4. It is defined asˆ G − σ k ( ν n ) = E ( ν n ) − ˜ γ k − ˜ γ ∗ k E ( ν n ) − γ − ∆ † σ − γ − ∆ σ E ( ν n ) − ˜ γ k − ˜ γ ∗ k E ( ν n ) . (7)Indeed, the structure of the matrix does not changeswhen inverting the spin direction, i.e., ˆ G − ↓ k ( ν n ) ≡ ˆ G − ↑ k ( ν n ). The diagonal elements of the matrix, in Eq.(7),are the single-particle quasienergies E ℓ ( ν n ) = − iν n − µ eff ℓ , (8)where the effectve chemical potentials µ eff ℓ with ℓ = 1 , µ eff1 = µ + U/ µ eff2 = µ + U/ W . The parameters ˜ γ ℓ k , in Eq.(7), are therenormalized (nearest neighbors) intralayer hopping am-plitudes and ˜ γ ℓ k = zγ ℓ k γ , where the k -dependent pa-rameters γ ℓ k are the usual tight-binding energy disper-sions in the BLG. Namely, we have γ ℓ k = 1 z X ~δ ℓ e − i k ~δ ℓ . (9)The parameter z , is the number of the nearest neigh-bors lattice sites on the honeycomb lattice. The com-ponents of the nearest-neighbors vectors ~δ ℓ , for the bot-tom layer 1, are given by ~δ (1)1 = (cid:0) d/ , d √ / (cid:1) , ~δ (2)1 = (cid:0) d/ , − d √ / (cid:1) and ~δ (3)1 = ( − d, ~δ (1)2 = ( d, ~δ (2)2 = (cid:0) − d/ , − d √ / (cid:1) , and ~δ (3)2 = (cid:0) − d/ , d √ / (cid:1) . Then, for the function γ k , we get γ k = 13 " e − ik x d + 2 e i kxd cos √ k y d ! , (10)where d is the carbon-carbon interatomic distance. It isnot difficult to realize that γ k = γ ∗ k ≡ γ ∗ k , and thereforewe have ˜ γ k = ˜ γ ∗ k ≡ ˜ γ ∗ k . The partition function of thebilayer graphene system will read as Z = Z (cid:2) D ψ † D ψ (cid:3) e −S [ ψ † ,ψ ] , (11)where the action of the system is given in Eq.(5) above.The form of the Green’s function matrix, given in Eq.(7),will be used in the next Section, in order to derive theself-consistent equations, which determine the excitonicgap parameter ∆ and the effective bare chemical poten-tial ¯ µ in the interacting BLG system. Particularly, thislast one plays an important role in the BLG theory andredefines the charge neutrality point (CNP) [38–40] inthe context of the exciton formation in the interactingbilayer graphene. Quite interesting experimental resultson that subject are given recently in Refs.38–40. A. The excitonic dispersion relations
We will perform the Hubbard-Stratanovich transfor-mation of the partition function in Eq.(11). In the Dirac’sspinor notations, the partition function will be trans-formed as follows Z = Z (cid:2) D ψ † D ψ (cid:3) e − βN P k νn P σ ψ † σ k ( ν n ) ˆ G − σ k ( ν n ) ψ σ k ( ν n ) ×× e βN P k νn P σ [ J † k σ ( ν n ) ψ k σ ( ν n )+ ψ † k σ ( ν n ) J k σ ( ν n ) ] ≈≈ e βN P k νn P σ J † k ( ν n ) ˆ G σ k ( ν n ) J k σ ( ν n ) , (12)where we have introduced the auxiliary fermionic sourcefield vectors J k σ ( ν n ), which are also Dirac’s spinors asthe ψ -fields, defined in the Section 3 J σ k ( ν n ) = (cid:2) j aσ k ( ν n ) , j bσ k ( ν n ) , j ˜ aσ k ( ν n ) , j ˜ bσ k ( ν n ) (cid:3) T . (13)Here, we use the condition of the half-filling in each layerof the graphene bilayer, in order to determine the exactchemical potential µ in the interacting system. For thelayer 1, this condition holds that h n a i + h n b i = 1 . (14)After evaluating the averages in the left-hand side inEq.(14), we get for the chemical potential the followingequation 4 N X k X i =1 α i k n F ( ξ i k ) = 1 . (15)The coefficients α i k , in Eq.(15) are defined in Eq.(21) inRef.46. The function n F ( ε ), in Eq.(15), is the Fermi-Dirac distribution function n F ( ε ) = 1 / (cid:2) e β ( ε − µ ) + 1 (cid:3) .For the parameters ξ i k , in the arguments of the Fermi-Dirac distribution function, in Eq.(15), we have ξ i k = µ − κ i k , (16)where the energy the parameters κ i k , define the elec-tronic band structure of the bilayer graphene with theexcitonic pairing interaction. We find that κ , k = − (cid:20) ∆ + γ ± q ( W − ∆ − γ ) + 4 | ˜ γ k | (cid:21) + ¯ µ,κ , k = − (cid:20) − ∆ − γ ± q ( W + ∆ + γ ) + 4 | ˜ γ k | (cid:21) + ¯ µ. (17)Here, we have introduced a new bare chemical poten-tial ¯ µ ¯ µ = µ eff1 + µ eff2 . (18)After evaluating the statistical average, given inEq.(3), we will have the self-consistent equation for theexcitonic gap parameter ∆∆ = W ( γ + ∆) N X k X i =1 β i k n F ( ξ i k ) . (19)Here, the coefficients β i k are given in Eq.(26) in Ref.46.As we will see later on, when evaluating numerically κ i k , the shift of the Dirac’s crossing energy level, dueto the non-zero condensate states in the noninteractingBLG (in the case W = 0 the condensate states in theBLG are due to the interlayer hopping mechanism) isdirectly related to the effective bare chemical potential¯ µ , which enters in Eq.(17). It is not difficult to verifythat for the case of the noninteracting BLG, i.e., when U = 0, W = 0 and ∆ = 0, the expressions in Eq.(17) arereducing to the usual tight-binding dispersion relations ε i = ± γ ± p ( kγ ) + ( γ / − µ, (20)with i = 1 , .. k = | γ k | ) discussed in Ref.41, in thecontext of the real-space Green’s function study of thenoninteracting bilayer graphene. The exact numerical solution for the excitonic pair-ing gap parameter ∆ in the BLG is given in Ref.46. Asthe numerical calculations show, in the mentioned paper,the pairing gap parameter decreases when augmentingthe temperature and is not destroyed at the very hightemperatures. It is also clear from the results, given inRef.46, that the weak coupling region corresponds to theBCS-like excitonic pairing states between the electronsand holes in different layers of the BLG, while the strongcoupling region corresponds to the excitonic condensatestates in the system. In the next Section, we will presentthe modifications of the BLG band structure due to theexcitonic effects and we will calculate the hybridizationgap ∆ H = | κ k − κ k | at the Dirac’s point K , betweenthe conduction band electrons and valence band holes inthe bilayer graphene. B. The excitonic band structure and thehybridization gap
Here, we will discuss in more details the numericalresults for the electronic band structure of the bilayergraphene in the vicinity of the Dirac’s point K and byconsidering the excitonic pairing mechanism in the BLG.The detailed band structure of the BLG with the exci-tonic pairing mechanism is discussed in Ref.46.In Figs. 2 and Fig. 3 the electronic band structures (seein Eq.(17)) are shown near the Dirac’s K point, wherethe hybridization gap starts to open at the critical valueof the interlayer interaction parameter W c = 0 . γ =0 .
396 eV (we have putted here the realistic value for theintralayer hopping parameter γ = 3 eV, according toRef.42). In the inset, in the first panel, in Fig. 3, wehave shown the exact solution for the chemical potential µ in the bilayer graphene as a function of the interlayerCoulomb interaction parameter W/γ . Different valuesof temperature have been considered there.For the noninteracting case ( W = 0), which is givenin the first of the left panels, in Fig. 3, we recover theusual BLG band structure modified by the displacementof the Dirac’s crossing energy ǫ D . This is due to the finitechemical potential solution in the interacting system. Forthe zero interaction limit, the bare chemical potential ¯ µ iscoinciding exactly with the Dirac’s crossing energy level:¯ µ = ǫ D . The differences appear at the non-zero values of W . Therefore, ¯ µ controls the position of the Fermi levelin the interacting BLG.It is calculated using the formula in Eq.(18), in theSection 3 A: ¯ µ = µ + κU + 0 . W (with κ = 0 . µ , after the self-consistent equations for the excitonic gapparameter ∆ and the chemical potential µ . It has beenshown in Ref.46, that the Fermi energy in the bilayergraphene at T = 0 has also a very large jump nearly atthe same value of W as the exact chemical potential µ : W ∼ . γ (see in the inset, in the first panel, in Fig. 3).In contrast, for higher temperatures T > . γ , this be- FIG. 2: (Color online) The Electronic band structure of the bilayer graphene in the vicinity of the Dirac’s point K , for variousvalues of the normalized interlayer Coulomb interaction parameter W/γ . The values of the hybridization gap ∆ H , and theFermi energy levels ¯ µ are shown in the panels. The interlayer hopping amplitude is fixed at the value γ = 0 . γ , and thezero temperature limit is considered.FIG. 3: (Color online) The electronic band structure of the bilayer graphene near the Dirac’s point K , for various values ofthe normalized interlayer Coulomb interaction parameter W/γ . The values of the hybridization gap ∆ H and the Fermi energylevels ¯ µ are shown in the panels. The interlayer hopping amplitude is fixed at the value γ = 0 . γ , and the zero temperaturelimit is considered. The inset, in the bottom of the first panel, shows the behavior of the chemical potential as a function ofthe interlayer interaction parameter W γ and for different values of temperature. havior is smoothed, and for the very high temperatures,we have practically the continuous variation of ¯ µ withrespect to W . The behavior of the Fermi energy, give inRef.46 explains, at least qualitatively, the similar behav-ior of the bilayer graphene chemical potential, observedexperimentally in [38], by a direct measurement of thechemical potential of the BLG as a function of its carrierdensity. For this purpose, a double-BLG heterostructurehas been built, and the bottom bilayer chemical potentialhas been mapped along the charge neutrality line of thetop bilayer (see in Ref.[38], for details).In the table I, we have presented the exact solutionsof the chemical potential µ , bare chemical potential ¯ µ ,and the Dirac’s crossing energies ε D for all consideredvalues of the parameter W/γ . The levels of the barechemical potential ¯ µ , which indeed play the role of theFermi energy for each given interaction value, are pre-sented in all panels, in Figs. 2 and Fig. 3. We seein the first panel in Fig. 2 that the hybridization gap∆ H opens in the electronic band structure of the bilayergraphene at the critical value W c = 0 . γ = 0 .
396 eVof the interlayer Coulomb interaction parameter. Never-theless, the system is formally semi-metallic in this case,because the Fermi level ¯ µ lies in the conduction band(¯ µ = − . γ ). This is the limit of the small gapsemiconductor when the system has the semi-metallicproperties yet and at the same time is in the opticallyactive regime. We observe also in Figs. 2 and 3 thatthe hybridization gap increases when augmenting the in-teraction parameter W . The true semiconducting limithappens for the higher values of W , which we see in themiddle panel in Fig. 2, where the system passes into thesemiconducting state at W = 0 . γ = 1 . µ = − . γ . We, intentionally, have not attributedto ¯ µ the Fermi energy notation ǫ F in the pictures, in or-der to distinguish it from the noninteracting case. For thelarger values of W (see in the panels, in Fig. 3), the Fermienergy lies practically in the middle of the hybridizationgap, reporting the well defined intrinsic semiconductingstate (or insulator state for the large enough values of W ) in the bilayer graphene. It is interesting to observethat the band structure of the BLG degenerates at theinterlayer interaction value W j = 1 . γ and T = 0.Indeed, we see in the first two panels in Fig. 3 that thereare two band structures for the same value of W = W j .This degeneracy is related to the chemical potential solu-tion in the system. We have two possible chemical poten-tial solutions at W = 1 . γ (see also in the table I).One is situated in the lower bound of the solution for thechemical potential µ (see in the inset, in the first panel,in Fig. 3) and the another solution is in the upper boundsolution of µ . Therefore, the chemical potential has alarge jump at W j = 1 . γ , from the lower bound tothe upper bound: ∆ µ = 1 . γ = 4 .
11 eV. For the highervalues of W : W > W j , we have only the upper boundsolutions for µ . Those mentioned values of µ at W = W j are given in the table I. It is worth to mention that the gap opening in thebilayer graphene’s band structure at the Dirac’s point K , discussed here, has not its analogs in the literatureyet. Particularly, it is not equivalent to the studies whenthe external electric field creates a tunable band gap inthe band structure of the bilayer graphene (see, for ex-ample in Refs.2, 3). In Fig. 4, we have presented thedependence of the hybridization gap on the interlayer in-teraction parameter W . Here, the hybridization gap isdue to the local interlayer Coulomb interaction effects,which is not equivalent to the electric field effects, dis-cussed in Refs.[2, 3]. Three different limits of tempera-ture have been considered in Fig. 4. Nearly linear depen-dence of ∆ H on W/γ , for the small and intermediatevalues of W , remains practically unchanged when vary-ing the temperature. This linear behavior correspondsto the decreasing values of the chemical potential solu-tion in the interval W ∈ (0 , γ ) (see the lower boundsolution for the chemical potential, in the inset, in thefirst panel, in Fig. 3. We observe, in Fig. 4, that the gapformation in the bilayer graphene is truly a threshold pro-cess, and for W < W c = 0 . γ we have ∆ H ≡
0. At W = 0 . γ = 0 .
396 eV the Fermi energy level ¯ µ is situ-ated in the conduction band, nevertheless the hybridiza-tion gap is not zero: ∆ H = 1 .
56 meV. The thresholdvalue of the interlayer interaction parameter at which thehybridization gap opens at T = 0 is W = 0 . γ = 0 . H = 0 . γ = 1 .
551 meV.For T = 0 . γ , we have ∆ H = 0 . γ = 1 .
32 meV,corresponding to W = 0 . γ = 0 . T = 0 . γ : ∆ H = 0 . γ = 1 .
11 meV, correspond-ing again to W = 0 . γ = 0 .
396 eV. We see that ∆ H is slightly decreasing when augmenting the temperature,while the critical value of the interlayer interaction pa-rameter W c remains the same: W c = 0 . γ = 0 . H in the inter-val W ∈ [1 . γ , . γ ) is related again to the behaviorof the chemical potential solution in this interval (see inthe inset, in the first panel, in Fig. 3, black line). In-deed, the non-linearity of ∆ H could not be explainedwith the behavior ¯ µ , nevertheless ¯ µ plays the role of theFermi level in the interacting bilayer graphene (see inthe Section 5, in Ref.46, the discussion on the parameter¯ µ ). The bare chemical potential ¯ µ , obtained in the pa-per is the effective chemical potential, which is directlyrelated to the measurements of the Fermi level in thebilayer graphene [38], while the original energy parame-ter µ is the prerequisite in order to explain the behaviorof the hybridization gap. The second linear part of thehybridization function curves (see the interval of the in-teraction parameter W ∈ [1 . γ , γ ]) corresponds tothe upper bound solutions of the chemical potential inthe bilayer graphene system. W γ γ γ γ γ (lower bound) 1.48999 γ (upper bound) 2 γ µ -1.863 γ -1.89 γ -1.894 γ -1.982 γ -2.103 γ -1.86956 γ -0.49935 γ -0.393 γ ¯ µ -1.363 γ -1.3306 γ -1.328 γ -1.2325 γ -1.103 γ -0.624 γ γ γ ε D -1.363 γ -1.3907 γ -1.39418 γ -1.4825 γ -1.603 γ -1.369 γ γ γ ∆ H γ γ γ γ γ γ TABLE I: The exact solutions of the chemical potential µ , Fermi energy ¯ µ , Dirac’s crossing energy ε D and the hybridizationgap ∆ H .FIG. 4: (Color online) The hybridization gap ∆ H /γ as afunction of the Coulomb interaction parameter W/γ and forvarious values of temperature. The interlayer hopping ampli-tude is fixed at the value γ = 0 . γ .
4. THE SINGLE-PARTICLE SPECTRALPROPERTIESA. The sublattice spectral functions
We will introduce here the normal Matsubara Green’sfunctions, according to the standard notations [36, 37],namely, for the sublattice- η , in the layer ℓ , we define thenormal local Green’s functions, as follows G ησ ( r τ, r τ ) = (cid:10) η σ ( r τ ) η † σ ( r τ ) (cid:11) . (21)In the Fourier space representation, and for thesublattice- A in the bottom layer 1, the Green’s function,in Eq.(21), takes the following form G aσ ( r τ, r τ ) = 1 βN X k ν n G aσ ( k , ν n ) , (22)where the Fourier transform G aσ ( k , ν n ) is given as G a ( k , ν n ) = 1 βN D a k ( ν n ) a † k ( ν n ) E . (23)The function G a ( k , ν n ), in Eq.(23), is the MatsubaraGreen’s function for the η = a -type fermions in the layer 1. Here, we have restricted only to the case σ = ↑ andwe have omitted the spin indexes in the Green’s functionsnotations. This is due to the spin-symmetry of the action,given in Eq.(5). The statistical average in the expressionof the Green’s function G a ( k , ν n ) could be evaluated withthe help of the partition function, given in Eq.(12), in theSection 3 A.The normal spectral functions for different sublattices(corresponding to the sites A and B ), in the layer 1, aredefined on the real frequency axis ν with the help of theretarded Green’s functions [37]. For the sublattice- A , wehave S a ( k , ν ) = − π ℑ G R a ( k , ν ) . (24)The retarded Green’s function, in turn, could be obtainedafter the analytical continuation of the correspondingMatsubara Green’s function G a ( k , ν n ), given in Eq.(23)above: G R a ( k , ν ) = G a ( k , ν n ) | iν n → ν + i + . (25)The explicit form of the Green’s function G a ( k , ν n ) couldbe obtained after the functional derivation with respectto the external source field variables and by using thepartition function, in Eq.(11). For the sublattice- A , inthe layer 1 of the BLG, we get G a ( k , ν n ) = X i =1 α i k iν n + κ i k . (26)Then, for the single-particle spectral function S a ( k , ν ),we get the following expression S a ( k , ν ) = 1 π X i =1 λα i k λ + ( ν + κ i k ) . (27)We have introduced the Lorentzian function represen-tation for the Dirac’s delta-function, and a broadeningparameter λ has been introduced in Eq.(27). Further-more, we calculate the normal spectral function for thesublattice- B , in the layer 1. Similarly, we obtain S b ( k , ν ) = 1 π X i =1 λγ i k λ + ( ν + κ i k ) (28)with γ i k = ( − i +1 Q j =3 , P ′ (3) ( κ i k )( κ k − κ k ) 1 ( κ i k − κ j k ) , if i = 1 , , Q j =1 , P ′ (3) ( κ i k )( κ k − κ k ) 1 ( κ i k − κ j k ) , if i = 3 , , (29)where P ′ (3) ( κ i k ) is the polynomial of third order in κ i k ,namely we have P ′ (3) ( κ i k ) = κ i k + ω ′ k κ i k + ω ′ k κ i k + ω ′ k (30)with the coefficients ω ′ i k , i = 1 , ...
3, given as ω ′ k = − µ eff1 − µ eff2 ,ω ′ k = µ eff1 (cid:0) µ eff1 + 2 µ eff2 (cid:1) − | ˜ γ k | ,ω ′ k = − µ eff2 (cid:0) µ eff1 (cid:1) + µ eff1 | ˜ γ k | . (31) B. The anomalous spectral function and theexcitonic coherence length
Another interesting function, to be considered here,is the anomalous spectral function S b ˜ a , defined betweenthe layers of the BLG, which (theoretically) gives a di-rect information about the excitonic pair formation andcondensation in the double layer graphene system. Thisfunction provides the simultaneous probability to find anelectron in the upper layer 2 and a hole in the lower layer1, for a given discrete quantum state ( k , ν ), thus, report-ing directly the excitonic pairs between the layers of thebilayer graphene. The frequency integrated expression ofthis function is especially relevant, due to its direct re-lation to the excitonic coherence length ξ c between thelayers. The spectral function S b ˜ a could be obtained withthe help of the imaginary part of the anomalous retardedGreen’s function G R b ˜ a ( k , ν ). We have G R b ˜ a ( k , ν ) = G b ˜ a ( k , ν n ) | iν n → ν + i + , (32)where G b ˜ a ( k , ν n ) is the Fourier transform of the anoma-lous Matsubara Green’s function. It is defined as follows: G b ˜ a ( k , ν n ) = 1 βN (cid:10) ˜ a ( k , ν n ) b † ( k , ν n ) (cid:11) . (33)Then, we have the anomalous spectral function S b ˜ a ( k , ν ) = − π ℑ G R b ˜ a ( k , ν ) . (34)Again, after using the functional derivation techniques,we get the following analytical form S b ˜ a ( k , ν ) = γ + ∆ π X i =1 λα i k λ + ( ν + κ i k ) . (35) It is not difficult to show that G ( k , ν n ) = G ( k , ν n )and G ( k , ν n ) = G ( k , ν n ), which follows from the sym-metry of the inverse Green’s function matrix, given inEq.(7). Therefore, we have for the spectral functions S ˜ a ( k , ν ) = S b ( k , ν ) , S ˜ b ( k , ν ) = S a ( k , ν ) . (36)The relations in Eq.(36) above explicitly imply that thespectral functions, for the layer 2 in the BLG, coincidewith that of the layer 1, just after interchanging thesublattice sites. After the summation over the Matsub-ara frequencies ν n in the expressions of the normal andanomalous Green’s functions, we will get the momentumdependent correlation functions of the system. Namely,the frequency dependent Matsubara Green’s functions inthe BLG have the following structures G a ( k , ν n ) = X i =1 α i k iν n + κ i k ,G b ˜ a ( k , ν n ) = ∆ ′ X i =1 β i k iν n + κ i k ,G b ( k , ν n ) = X i =1 γ i k iν n + κ i k , (37)where ∆ ′ in the last equation is ∆ ′ = ∆ + γ . Aftersumming over the fermionic Matsubara frequencies ν n g η ( k ) = β P ν n G η ( k , ν n ), we get the momentum distri-bution functions in the BLG g a ( k ) = X i =1 α i k n F ( − κ i k ) ,g b ˜ a ( k ) = ∆ ′ X i =1 β i k n F ( − κ i k ) ,g b ( k ) = X i =1 γ i k n F ( − κ i k ) . (38)In order to see the character of the exciton conden-sation in the momentum space, we have considered theanomalous momentum distribution function g b ˜ a ( k ) (seethe second equation in Eq.(38)) (see also in the Sections4 B 1 and 4 C below). This function is the exciton con-densation amplitude. Note that we use here the term“anomalous” to indicate that the number of electrons oneach of the “ a ” and “ b ” flavours is not conserved due tothe excitonic condensation, although the total number ofelectrons in a given layer ℓ is conserved. By using the ex-pression of the function g b ˜ a ( k ), we will evaluate the paircoherence length ξ c , which corresponds to the interlayerspatial size of the electron-hole pair and might be definedby the relation ξ c = sX k |∇ k g b ˜ a ( k ) | / X k | g b ˜ a ( k ) | . (39)In Fig. 5, we have presented the coherence length in thebilayer graphene as a function of the normalized inter-layer Coulomb interaction parameter W/γ . The specialnumerical package FADBAD++ [43] is used for the nu-merical differentiation, which implements the forward au-tomatic differentiation of the condensate amplitude func-tion g b ˜ a ( k ). Different values of temperature are consid-ered in the picture. The interlayer hopping amplitudeis fixed at the value γ = 0 . γ = 0 .
384 eV. We seein Fig. 5, that at the zero value of W , the interlayercoherence length is very large and it decreases when aug-menting the interaction parameter.The sadden jump of the coherence length at T = 0(and also for higher temperatures) in the interval W ∈ (1 . γ , . γ ) is related to the behaviour of the chemi-cal potential solution, presented in the inset, in the firstpanel, in Fig. 3 (see the drastic jump of µ in the case T = 0). In turn, the Fermi energy ¯ µ , also has a largejump in that interval of W and at T = 0 (see in Ref.46).For higher values of T , this behavior is smoothed. At thejump point of the chemical potential, which correspondsto the value W j = 1 . γ (see the band structure inthe first two panels in Fig. 3), the coherence length ξ c becomes comparable to the inter-atomic distances d inthe single-layer graphene in the BLG. It is remarkable toindicate that for the case T = 0 the coherence length isvery large for the values W > W j (for example, ξ c ∼ . d at the value W = 2 . γ = 8 . ξ c at the weak interaction limit is the manifestation ofthe excitonic BCS-like pairing, and the excitonic gap pa-rameter is very small in this case (see in Fig.4, in Ref.46).Indeed, in the BCS-like weak coupling region (this corre-sponds to the interval W ∈ (0 . , . γ )), the excitonic co-herence length ξ c and the excitonic gap parameter ∆ sat-isfy well the BCS-like relation ξ c ∼ ~ v F / ∆. On the otherhand, in the large momentum approximation, where thespectrum of the BLG is nearly linear [44, 46], we havefor the Fermi velocity: v F = | ¯ µ | / ( ~ k F ), where k F is theFermi wave vector at the Dirac’s points K and K ′ . Af-ter analysing the solutions for the excitonic gap param-eter ∆ and the bare chemical potential ¯ µ , for the case T = 0, we find that the relation between the coherencelength and excitonic gap parameter in our case reads as ξ c /d = α | ¯ µ | / ( k F ∆), where α is the coefficient showingthe deviations from the exact BCS relation above.As it was mentioned above, the bare average chemi-cal potential ¯ µ plays the role of the effective Fermi levelin the interacting bilayer graphene: ¯ µ = ǫ F . Thus, theBCS-like relation for the coherence length is well satis-fied in the case of the weak and intermediate couplings.Contrary, at the high interlayer coupling limit, i.e., when W > . γ , the excitonic coherence length becomes pro-portional to the excitonic gap parameter ξ ∼ ∆ andthis corresponds to the excitonic condensate regime inthe semiconducting phase. We see in Fig. 5 that in theBCS regime the excitonic coherence length could be verylarge, even at the high temperatures (see, for example, ξ c at T = 0 . γ in Fig. 5, the red curve). Contrary, in the FIG. 5: (Color online) The interlayer exciton coherence lengthas a function of the interlayer Coulomb interaction parameter
W/γ . Different values of temperature are considered in thepicture ( T = 0, T = 0 . γ , T = 0 . γ , T = 0 . γ and T = 0 . γ , from top to bottom). The interlayer hoppingamplitude is set at the value γ = 0 . γ . case of strong interaction regime, the excitonic coherencelength in the condensate regime is large (of the order offew inter-atomic distances) only in the case T = 0. Forthe non-zero temperatures, the coherence length becomesof the order of d in the semiconductor phase. This corre-sponds to the strong localization of the excitonic statesat the surfaces of the BLG layers.
1. Zero interlayer interaction
Here, we consider the case of the zero interlayerCoulomb interaction limit, i.e. when
W/γ = 0. Theexcitonic gap parameter vanishes, meanwhile, the bi-layer graphene is strongly correlated in that case (seethe discussion below). In Fig. 6, we have plotted thewave vector-resolved intensity plot of the normal spec-tral function S a ( k , ν ) for the zero interlayer Coulomb in-teraction regime. The chemical potential is finite andnegative in that case: µ = − . γ . The temperature,in Fig. 6, is set at zero, and the interlayer hopping is γ = 0 . γ . For the Lorentzian broadening small pa-rameter λ , we have chosen a very small value λ = 0 . K . Let’s men-tion here the role of the bare chemical potential ¯ µ inthe normal spectral function structure. We see in Fig. 6that the spectral lines are touching each other at the0 FIG. 6: (Color online) Two-dimensional intensity plot of the A -sublattice normal spectral function S a ( k , ν ). The interlayerCoulomb interaction parameter is set at zero and the zerotemperature limit is considered in the picture. The interlayerhopping amplitude is set at the value γ /γ = 0 . S b ˜ a ( k , ν ). The interlayerCoulomb interaction parameter is set at zero, and the zerotemperature limit is considered. The interlayer hopping am-plitude is set at the value γ = 0 . γ . Dirac’s crossing points K and K ′ , as it should be forthe case of the noninteracting BLG system [2, 3]. TheDirac’s crossing energy ε D is coinciding exactly with theabsolute value of the effective bare chemical potential ¯ µ : | ¯ µ | = 1 . γ = 4 .
089 eV. This value is much higherthan the known results for the undoped neutral BLG[45] (where ε F = ε D ∼ − γ ∼ − . FIG. 8: (Color online) The anomalous spectral function (thecondensation amplitude g b ˜ a ( k )). The interlayer Coulomb in-teraction parameter is set at zero, and the zero temperaturelimit is considered in the picture. The interlayer hopping am-plitude is set at the value γ = 0 . γ .FIG. 9: (Color online) Two-dimensional ( k x , k y ) map ofthe momentum distribution function g b ˜ a ( k ). The interlayerCoulomb interaction parameter is set at zero, and the zerotemperature limit is considered in the picture. The interlayerhopping amplitude is set at the value γ = 0 . γ . (with v F = 1 . × cm · s − ). This last effect is relatedto the electron-hole condensate states with a zero centerof mass momentum P c = 0, at the Dirac’s crossing points K and K ′ , and between the opposite layers in the BLG[46]. It is remarkable to note (this will be clear here-after when discussing the anomalous spectral functions)that the condensate states at zero interlayer interactionturn into the BCS-like weak coupling pairing states when W/γ = 0, with a finite excitonic pairing gap ∆ /γ = 0opening in the BLG.In Fig. 7, the anomalous spectral function S b ˜ a ( k , ν )is presented for the same values of parameters, as in thecase, given in Fig. 6. The spectrum is again gapless at theDirac’s points, and the principal change, compared with1 FIG. 10: (Color online) Two-dimensional intensity plot ofthe A -sublattice normal spectral function S a ( k , ν ). The in-terlayer Coulomb interaction parameter is set at the value W = 1 . γ , and the zero temperature limit is considered inthe picture. The interlayer hopping amplitude is set at thevalue γ = 0 . γ .FIG. 11: (Color online) Two-dimensional intensity plot ofthe B -sublattice normal spectral function S b ( k , ν ). The in-terlayer Coulomb interaction parameter is set at the value W = 1 . γ , and the zero temperature limit is considered inthe picture. The interlayer hopping amplitude is set at thevalue γ = 0 . γ . the result, presented in Fig. 6, is that we have only 2 spec-tral intensity lines at the place of 4, in the case of the nor-mal spectral function. The normal and anomalous single-particle spectral intensities, presented in Figs. 6 and 7,are of the same order and do not show any sign of the ex-citonic pair formation in the BLG system. Whether theexcitonic condensation is really present at this limit, willbe clear after analyzing the two-dimensional map of theanomalous momentum distribution function g b ˜ a ( k ) (orthe condensate amplitude) in the BLG. The condensateamplitude g b ˜ a ( k ), for the zero interlayer interaction limit,is shown in Fig. 8. The momentum distribution function g b ˜ a ( k ) shows a prominent “condensate”-type structure at FIG. 12: (Color online) Two-dimensional intensity plot ofthe anomalous spectral function S b ˜ a ( k , ν ). The interlayerCoulomb interaction parameter is set at the value W =1 . γ , and the zero temperature limit is considered in thepicture. The interlayer hopping amplitude is set at the value γ = 0 . γ . the corners of unconnected k -cell hexagons in the recip-rocal space. These states occur via the interlayer hoppingmechanism, as the Eq.(38) suggests. The interlayer hop-ping amplitude is fixed at γ /γ = 0 . g b ˜ a ( k ). Thus, the free excitonic pairing statesare absent, nevertheless, the function g b ˜ a ( k ) shows thatthe system is in the condensate regime. Therefore, thefundamental excitonic state in the BLG system, at thezero interlayer interaction regime, is governed principallyby the electron-hole pair condensed phase caused by theinterlayer hopping, and the individual free excitonic pairsare indeed absent in the BLG structure. C. Finite interlayer interactions
In Figs. 10, 11 and Fig. 12 we have presented thespectral functions S a ( k , ν ), S b ( k , ν ) and S b ˜ a ( k , ν ) for thesufficiently large Coulomb interaction parameter W =1 . γ = 3 .
75 eV. This is the value of W , at which theexcitonic gap parameter is maximal (see in Fig.4, in theRef.46 ). The interlayer hopping amplitude is fixed atthe value γ = 0 . γ and the zero temperature limitis considered in the pictures. The results, Figs. 10, 11and Fig. 12, are of great importance for the whole the-ory, presented here. We see that there is a large gap inthe intensity plots at the Dirac’s points K in the BZ ofgraphene. The double spectral lines corresponding to theelectron and hole channels in the normal functions plots,in Figs. 10, 11, are well separated now, and the intensitiescorresponding them are now different. Particularly, the A -sublattice electron intensities, (see the spectral func-tion S a ( k , ν )) in Fig. 10, are much higher than the holeintensities in the same picture. Contrary, the electron in-2 FIG. 13: (Color online) Two-dimensional map of the momen-tum distribution functions g a ( k ) (see the left panel) and g b ( k )(see the right panel). The interlayer Coulomb interaction pa-rameter is set at the value W = 1 . γ , and the zero tempera-ture limit is considered in the picture. The interlayer hoppingamplitude is set at the value γ = 0 . γ . tensities of the spectral function S b ( k , ν ), given in Fig. 11are much lower than the hole intensity lines. These re-sults show how the electron-hole pair correlations appearin the bilayer graphene at the finite value of the interlayerCoulomb interaction parameter W . Namely, the spectralintensities in Fig. 10 show that the sublattice sites A inthe lower layer of the BLG are mostly occupied by theelectrons, while the B -sublattice sites are occupied by the“hole” quasiparticles. Thus the electron-hole symmetryin the lower graphene sheet is well pronounced and showthat the “hole” quasiparticles, at the sites B , partici-pate to the formation of the electron-hole pairing statesdue to the local Coulomb interaction W induced betweenthe quasiparticles at the B -sublattice (in the lower layer)and ˜ A -sublattice (in the upper layer) site positions. Theintensity plots in Figs. 10 and 11 are also in well agree-ment with the intensity plot of the condensate spectralfunction S b ˜ a ( k , ν ), presented in Fig. 12, which shows theprobability of the local exciton formation between thequasiparticles at the sublattice sites B and ˜ A , for a givenquantum state ( k , ν ). The single particle “hole” line, inFig. 12, is much more intense than the electron spectralline, thus corresponds well to the discussion in Figs. 10and 11.In Fig. 13, we have shown the ( k x , k y ) map of thenormal momentum functions, given in Eq.(38). The in-terlayer interaction parameter is set again at the value W = 1 . γ , and the temperature is set at zero. Thefunction g a ( k ) is shown in the left side of the picture,and the function g b ( k ) is shown in the right panel, inFig. 13. We see, that the structure of the ( k x , k y ) mapshows again the possibilities for the excitonic excitationsin the system. Particularly, a remarkable particle-holesymmetry is observable in Fig. 13. The bright hexagonson the maps are the regions of the usual single-particleexcitations spectra in the system (see, for example, inRef.49, for more details), while the regions between them FIG. 14: (Color online) The condensate amplitude function g b ˜ a ( k ). The interlayer Coulomb interaction is set at the value W = 1 . γ , and the zero temperature limit is considered inthe picture. The interlayer hopping amplitude is set at thevalue γ = 0 . γ .FIG. 15: (Color online) Two-dimensional ( k x , k y ) map ofthe condensate amplitude function g b ˜ a ( k ). The interlayerCoulomb interaction parameter is set at the value W =1 . γ , and the zero temperature limit is considered in thepicture. The interlayer hopping amplitude is set at the value γ = 0 . γ . (see the 6-apex stars regions) show the particle-hole sym-metry in the bottom layer of the BLG. These are theregions, where, potentially, the excitonic condensationcould be isolated from the excitonic pairing region (aswe will see later on, in this paper). In Figs. 14 and 15we have shown the condensate amplitude function g b ˜ a ( k )in the bilayer graphene system at the finite interlayerCoulomb interaction parameter W = 1 . γ . We seethat the excitonic pair formation and condensation prin-cipally happens at the values of k x and k y , which corre-spond to the regions, where the particle-hole symmetryis apparent (see the 6-apex stars regions, in Fig. 13).It is very interesting to see also the behavior of themomentum distribution functions for the very high value3 FIG. 16: (Color online) Two-dimensional ( k x , k y ) map ofthe normal momentum distribution functions g a ( k ) (see theleft panel) and g b ( k ) (see the right panel). The interlayerCoulomb interaction parameter is set at the value W = 3 γ ,and the zero temperature limit is considered in the pic-ture. The interlayer hopping amplitude is set at the value γ = 0 . γ . of the interlayer interaction parameter W . Here, wewill consider the value W = 3 γ = 9 eV, (such avalue can be realized with the commonly used SiO sub-strate/dielectric with ǫ ∼ W = 3 γ . The left panel corresponds to the function g a ( k ) and the right panel shows the k -dependence ofthe function g b ( k ). First of all, the 6-apex stars re-gions, in the normal momentum distribution spectra, donot reflect anymore the particle-hole symmetry in thiscase. Thus, deep in these regions, the excitonic pairingcould not happen. Contrary, now, the whole quasiparti-cle excitation regions (see the k -cell hexagonal regions,in Fig. 16, marked as A and B ) represent the possible re-gions of the wave vector components k x and k y , at whichthe excitonic pair formation could appear, and these arethe regions now, which represent the particle-hole sym-metry in the problem. We see, in Fig. 17, that the exci-tonic pair formation regions (the corresponding momen-tum functions have very small amplitudes, as comparedto the weak interaction regime), in the BLG, cover nearlythe entire k -space, except the regions, corresponding tothe 6-apex stars pockets. In addition, we see, in Fig. 17,how the excitonic condensation peaks surround the freeexcitonic pair formation hexagonal k -cell regions, just atthe borders of the dark triangular Dirac’s pockets. The2D map of the function g b ˜ a ( k ) at W = 3 γ is shown inFig. 18). D. The k-space evolution of the condensate states
The principal conclusion that could be gained from theresults in Figs. 17 and 18 is the following: at the high val-ues of the interlayer Coulomb interaction parameter W ,we have the separation of the excitonic condensate statesfrom the free excitonic pair formation regions in the inter- FIG. 17: (Color online) The condensate amplitude function g b ˜ a ( k ) at T = 0. The interlayer Coulomb interaction is set atthe value W = 3 γ and the interlayer hopping amplitude is γ = 0 . γ .FIG. 18: (Color online) Two-dimensional ( k x , k y ) map ofthe condensate amplitude function g b ˜ a ( k ). The interlayerCoulomb interaction parameter is set at the value W = 3 γ ,and the zero temperature limit is considered in the pic-ture. The interlayer hopping amplitude is set at the value γ = 0 . γ . acting BLG, and the system is apparently in the strongcondensate regime, which coexists with the excitonic in-sulator state. Thus, at large W , we have a semiconduc-tor with the well defined interband and intraband bandgaps of the hole-decay spectrum, and the BLG is char-acterized by the mixed states, composed of the excitonicpair formation and condensate states. Another funda-mental observation from the obtained numerical resultsis related to the existence of the excitonic BCS-Bose-Einstein-Condensation (BEC)-like crossover mechanismin the BLG system, where the tunable parameter is theinterlayer Coulomb interaction W . The correspondingexperimental parameter, that could tune a similar tran-sition is the additional charge density on both layers of4BLG (see the works [38–40]).Here, we would like to demonstrate also, the k -spaceevolution of the condensate states appearing (in thesemiconducting phase) around the Dirac’s points K inthe reciprocal space by forming the condensate pockets(Dirac’s pockets) in the k -space. In Figs. 19 and 20,we have shown the evolution of the anomalous momen-tum distribution function as a function of the interlayerCoulomb interaction parameter W . A broad interval ofvalues of the interaction parameter W has been consid-ered: from very weak interaction limit: W = 0 . γ (see inthe first panel, in Fig. 19), up to very high value of W : W = 4 γ (see in the last panel, in Fig. 20).As it was discussed and explained previously, at thezero interaction limit, the BLG system is in the exci-tonic pair condensate regime, mediated by the local in-terlayer hopping parameter γ . For the finite interlayerinteraction value W = 0 . γ = 0 .
396 eV, a finite hy-bridization gap opens in the bilayer graphene. We haveshown that for a sufficiently large interval of W (when W ∈ (0 . , . γ )) the BCS-like weak coupling theoryis valid for the BLG. In this case, we converge to thelow-energy results, given in Refs.10–14, where the weakcoupling BCS theory is evaluated to obtain the exci-tonic pairing gap parameter. This has the similaritiesalso with the 2D square lattice excitonic systems [24–27],where the excitonic insulator state, at the low interbandCoulomb interaction limit, is governed by the BCS-likepair condensed state, which is transformed into the ex-citonic BEC state at the large interband U -interactionlimit. When augmenting the interlayer Coulomb inter-action parameter (see already in the second panel, inFig. 19, where W/γ = 0 . k -cell hexagons aremerging at the M symmetry points in the reciprocalspace (see also in the second panel, in Fig. 21, for the 2Dmap of the anomalous momentum distribution function),and the system passes into the mixed state, composed ofthe excitonic pairs. Particularly, the Dirac’s pockets startto develop, which surround the Dirac’s neutrality points K and K ′ . Furthermore, when increasing the parame-ter W (see in the rest of the panels in Fig. 19, where W = 1 .γ , W = 1 . γ , W = 1 . γ ), the excitonic pairformation states are replenishing the k -space pockets. Atthe value W = 1 . γ , (see in the last panel, in Fig. 19,and also in Fig. 21) the Dirac’s pockets in the k -spaceare completely replenished, and are isolated from eachother, along the directions K − M − K ′ , in the reciprocalspace. At this value of W , the bilayer graphene is in thesemiconducting phase, as the electronic band structureshows (see in the Section 3 B above). At the values of W , corresponding to the interval W > W j , where W j is the value at which the chemical potential passes intoits upper bound (with a sufficiently large jump), the ex-citonic condensates states appear and start to separatefrom the excitonic pair formation regions (see the concavelike structures, which appear in the middle of the Dirac’spockets, in the first three panels in Fig. 20 and also inFig. 22) at the cost of breaking of the mixed states. When continuing to increase the interlayer Coulomb interac-tion parameter, the excitonic condensation peaks appearin the reciprocal k -space at the borders of the pocketsregions, and the mixed state continues to break. Simul-taneously, we see that the free excitonic pair formationregions are formed (see the concave like bright hexagonalregions in Figs. 20 and 22) at the place of the emptydark hexagonal regions in k -space (see in the panels inFigs. 19 and 21).In the last panels, in Figs. 20 and 22, we have shownthe condensate amplitude function for the very high valueof the interaction parameter: W = 4 γ . We observe inFig. 20 that the amplitude of the anomalous momen-tum function is considerably reduced, and the concave-like excitonic pair formation regions become convex-like,improving the free excitonic pairing state in the system(like in the semiconducting systems). In addition, thereis a strong evidence for the excitonic condensation in theBLG, and the excitonic condensates states are completelyseparated from the excitonic pair formation regions (seethe condensates peaks at the borders of the Dirac’s pock-ets, in the reciprocal space, in Fig. 20). Thus, at thehigh values of the interlayer Coulomb interaction, theexcitonic condensate states are of the BEC-type, andwe observe a remarkable BCS-BEC-like crossover mech-anism in the bilayer graphene system, analogue to thatobserved in some intermediate valent semiconductor sys-tems [24–27], where the similar crossover is due to theinterband Coulomb interaction parameter U . Note, thatthe BCS-BEC type crossover in the extrinsic BLG struc-ture is observed also recently in Ref.8, in the presenceof the perpendicular magnetic field, when the localizedmagnetoexcitons form.We see particularly, in Fig. 22, how the excitonic con-densate formation develops in the reciprocal space, at thevalues W > W j by forming the Dirac’s pockets aroundthe symmetry points K and K ′ . In the panels in Figs. 21and 22 we see clearly how the dark hexagonal regions inFig. 21, are transforming into the bright hexagonal onesand the excitonic pairing states with small amplitudesare formed, and, at the same time, the condensate picksappear in the momentum space, which surround the holyDirac’s pockets in the k -space (see, for example, in thelast panel, in Fig. 22).
5. CONCLUSION
We have studied the influence of the excitonic effectson the spectral properties of the bilayer graphene. Thetheory, evaluated in the present paper, permits to con-struct a fully controllable theoretical model for the stud-ies of the excitonic condensation effects in the bilayergraphene system and to manipulate a number of physi-cal parameters, which are important from theoretical andexperimental points of views.We have reconstructed the electronic band structure ofthe bilayer graphene, taking into account the local exci-5
FIG. 19: (Color online) The condensate amplitude function g b ˜ a ( k ) for different values of the interlayer Coulomb interactionparameter W ( W = 0 . γ , W = 0 . γ , W = 1 . γ , W = 1 . γ , W = 1 . γ , from left to right). The zero temperature limit isconsidered in the picture. The interlayer hopping amplitude is set at the value γ = 0 . γ .FIG. 20: (Color online) The condensate amplitude function g b ˜ a ( k ) for different values of the interlayer Coulomb interactionparameter W ( W = 1 . γ , W = 1 . γ , W = 2 γ , W = 3 γ , W = 4 γ , from left to right). The zero temperature limit isconsidered in the picture. The interlayer hopping amplitude is set at the value γ = 0 . γ . tonic pairing interaction between the layers of the system.We have shown that a remarkable hybridization gap ap-pears in the electronic spectrum at sufficiently small val-ues of the interlayer coupling parameter. We estimatedthe values of the hybridization gap for a large interval ofthe interaction parameter. We have demonstrated thatthere is a semimetal-semiconductor transition in the BLGsystem, tuned by the local Coulomb interaction param-eter. We have shown that the chemical potential solu-tion has two bounds: lower and upper, which controlsthe borders of the excitonic pair formation and excitoniccondensate phases. Meanwhile, the additional effectivechemical potential ¯ µ appears in the energy spectrum ofthe bilayer graphene, which plays the role of the trueFermi energy in the bilayer graphene system. This resultexplains, at least qualitatively, the recent experimentaleffort to determine the chemical potential behavior inthe gated bilayer graphene systems [38–40]. The particle-hole symmetry, in the single layer of bilayer graphene, be-comes strongly apparent when analyzing the normal andanomalous momentum distribution functions properties.Namely, in the regions, where the particle-hole symme-try is conserved in the normal distribution function, theexcitonic pairing occurs at the non-zero values of the lo-cal interlayer Coulomb interaction parameter. In this limit of the very small and intermediate values of the in-terlayer Coulomb interaction, the double layer system isgoverned by the effective weak coupling BCS-like pairingstates [50, 51], and the excitonic coherence length satisfieswell the BCS-like relation ξ c = α ¯ µ/ ( k F ∆), with ∆, beingthe excitonic pairing gap parameter. Contrary, in strongcoupling limit the coherence length becomes proportionalto ∆ ( ξ c ∼ ∆), and the system is in the excitonic con-densate regime in that case. By analysing the anomalousmomentum distribution functions properties (directly re-lated to the excitonic pair formation or condensation)and by varying the interlayer interaction parameter (fromvery small up to very high values), we have found a re-markable BCS-BEC type crossover, which is different (atits origin) from the analogue transitions obtained in theintermediate-valent semiconductors, or rare-earth com-pounds [19–26]. We have shown the evolution of the ex-citonic pairing and condensate states as a function of theCoulomb interaction parameter and we have found theregions, where the system is in the excitonic pairing orcondensate regimes. It is remarkable to mention that, forthe large values of the coupling parameter, the excitoniccondensate states appears around the Dirac’s points K and K ′ , by creating the Dirac’s “hole” pockets in the re-ciprocal space. Namely, triangular k -pockets are formed,6 FIG. 21: (Color online) The 2D k -map of the condensate amplitude function g b ˜ a ( k ) for different values of the interlayerinteraction parameter W ( W = 0 . γ , W = 0 . γ , W = 1 . γ , W = 1 . γ , W = 1 . γ , from left to right). The zerotemperature limit is considered in the picture. The interlayer hopping amplitude is set at the value γ = 0 . γ .FIG. 22: (Color online) The 2D k -map of the condensate amplitude function g b ˜ a ( k ) for different values of the interlayerinteraction parameter W ( W = 1 . γ , W = 1 . γ , W = 2 γ , W = 3 γ , W = 4 γ , from left to right). The zero temperaturelimit is considered in the picture. The interlayer hopping amplitude is set at the value γ = 0 . γ . around which the new excitonic BEC states appear andcoexist with the excitonic pairing states, which cover thelarge regions in the k -space (creating by this the mixedregion: excitonic pairing+excitonic BEC). It is remark-able to note also that the surface forms, correspondingto the free excitonic pair formation regions (the excitonicinsulator state), transform from the concave-like into theconvex-like when augmenting the interlayer Coulomb in-teraction parameter.The results obtained in the present paper show thatthe spectral functions and the momentum distributionfunctions provide a direct proof of the excitonic effectsin the bilayer graphene system. On the other hand, theexperimental measurement of the anomalous functionsis extremely difficult, due to the very short lifetime ofexcitonic quasiparticles and fast electron-hole recombi-nation effects. We hope that these difficulties, relatedto the measurements of the anomalous momentum dis- tribution functions, will be overcome in the near futureby the improvement of the bilayer graphene fabricatingtechniques and by the setup architecture of the biaseddouble-layer heterostructures. The principal advantagefor such a success would be the simultaneous measure-ments of the photon’s absorption and emission spectra indifferent layers of the BLG. The results of the presentedpaper are particularly important in the context of therecent experimental results [52, 53] concerning the dis-covery of the laser-induced white light emission spectrain graphene ceramics. It was demonstrated in Ref.[52]that a large bandgap opening and light emission fromgraphene is possible by using the continuous-wave laserbeams with wavelengths from the visible to the near-infrared range. This hidden multistability of graphene isfundamental to create a semiconducting phase immersedin the semimetallic continuum one. [1] W. Ehrenberg, Electric conduction in semiconductorsand metals, Oxford University Press, 1958.[2] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S.Novoselov, A.K. Geim, Rev. Mod. Phys. 81 (2009) 109.[3] Eduardo V. Castro, K. S. Novoselov, S. V. Morozov, N.M.R. Peres, J.M.B. Lopes dos Santos, Johan Nilsson,F. Guinea, A.K. Geim, A.H. Castro Neto, Phys. Rev.Lett. 99 (2007) 216802.[4] J.P. Eisenstein, A.H. MacDonald, Nature 432 (2004) 691.[5] J.J. Su, A.H. MacDonald, Nat. Phys. 4 (2008) 799.7