Electron screening and excitonic condensation in double-layer graphene systems
aa r X i v : . [ c ond - m a t . s t r- e l ] A ug Electron screening and excitonic condensation in double-layer graphene systems
Maxim Yu. Kharitonov and Konstantin B. Efetov , Theoretische Physik III, Ruhr-Universit¨at Bochum, Germany L.D. Landau Institute for Theoretical Physics, Moscow, Russia (Dated: October 26, 2018)We theoretically investigate the possibility of excitonic condensation in a system of two graphenemonolayers separated by an insulator, in which electrons and holes in the layers are induced byexternal gates. In contrast to the recent studies of this system, we take into account the screeningof the interlayer Coulomb interaction by the carriers in the layers, and this drastically changes theresult. Due to a large number of electron species in the system (two projections of spin, two valleys,and two layers) and to the suppression of backscattering in graphene, the maximum possible strengthof the screened Coulomb interaction appears to be quite small making the weak-coupling treatmentapplicable. We calculate the mean-field transition temperature for a clean system and demonstratethat its highest possible value T max c ∼ − ǫ F . ǫ F is the Fermi energy). Inaddition, any sufficiently short-range disorder with the scattering time τ . ~ /T max c would suppressthe condensate completely. Our findings renders experimental observation of excitonic condensationin the above setup improbable even at very low temperatures. PACS numbers: 73.63.-b, 72.15.Rn, 81.05.Uw
INTRODUCTION AND MAIN RESULT
The possibility of excitonic condensation (EC) inmetallic systems was originally proposed by Keldysh andKopaev [1] for semimetals with overlapping conductionand valence bands. They have shown that the attractiveCoulomb interaction between electrons and holes leads toan instability towards formation of bound electron-holepairs analogous to the Cooper instability in supercon-ductors. Somewhat later, it was suggested [2] that ECcould be realized in a double-layer system of spatiallyseparated electrons and holes. Experimental efforts to-wards the observation of EC were mainly concentrated onsemiconductor double quantum well systems [3, 4, 5, 6]and experimental data speak in favor of the existence ofEC in electron-hole bilayers [3, 4, 5] and electron-electronbilayers in the quantum Hall regime [6].Since the carrier density in graphene, including its po-larity, can effectively be controlled by various means,graphene-based systems may also seem attractive for therealization of EC. Indeed, several ideas on how one couldobtain EC in graphene have been suggested recently. Onepossible way to create interacting electrons and holes isto apply a strong in-plane magnetic field to a single layerof graphene [7]. Such a magnetic field acts on the spins ofthe carriers only, and the Zeeman splitting creates elec-trons with one spin polarization and holes with the op-posite polarization in an initially neutral sample. A de-tailed theory of EC in such a setup has been developedin Ref. [7].A double-layer graphene system (Fig. 1) as a candi-date for the observation of EC was proposed recently inRefs. [8, 9, 10]. If two graphene layers are separated byan insulator, electrons in one layer and holes in the othercan be obtained by applying external gate voltage. Rel-atively high values of the Fermi energy ǫ F ∼ . FIG. 1: Excitonic condensate in a system of two spatiallyseparated graphene layers. Electrons and holes in the layersare induced by applying the external gate voltage. can be achieved in graphene by using gates [11] are anobvious advantage, since ǫ F serves as a high-energy scaleof the effect in such a setup. Solving the gap equationsnumerically, the authors of Refs. [9, 10] provided an es-timate T BKT ∼ . ǫ F for the critical temperature of theBerezinski-Kosterlitz-Thouless (BKT) transition and ar-gued that T BKT could thus reach room temperatures.However, in the analysis of Refs. [9, 10], the screen-ing of the Coulomb interactions by the carriers in thegraphene layers was not taken into account. Clearly, thetwo-dimensional screening cloud formed around a probecharge in a graphene sheet screens the field of the chargeboth in the off-plane and in-plane directions, althoughnot identically. Therefore, screening affects not only theintralayer but also the interlayer Coulomb interaction inthe double-layer setup.In this paper, we demonstrate that taking screeninginto account is essential as it drastically reduces the tran-sition temperature compared to the estimate obtainedin Refs [9, 10] neglecting screening. In fact, screeningsets the upper bound for the interaction strength, yield-ing for the maximum possible value of the dimensionlesscoupling constant λ max = 1 N = 18 . (1)Here N = N s N v N l = 2 = 8 is the total number ofelectron species in the system originating from two pro-jections of spin ( N s = 2), two valleys ( N v = 2), and twolayers ( N l = 2). Moreover, the chiral nature of quasipar-ticles in graphene leads to the suppression of backscat-tering. Consequently, the maximum interaction strength λ max c that determines the transition temperature appearsto be actually two times smaller than λ max , λ max c = λ max . (2)As follows from Eqs. (1) and (2), the large num-ber of electron species and suppression of backscatteringmake the maximum possible value λ max c of the interac-tion strength numerically quite small. This justifies theapplicability of the weak-coupling BCS approach to theproblem, since 1/16 can safely be considered as a smallparameter. As a result, for the highest possible value ofthe mean-field transition temperature we obtain T max c ≈ exp( − /λ max c ) ǫ F = exp( − ǫ F ≈ − ǫ F . (3)This is the highest possible value of the critical tempera-ture of the excitonic condensation that could be achievedin a perfectly clean double-layer graphene system. Enter-ing Eq. (3) as the exponent, the large value of the inverseinteraction strength 1 /λ max c = 16 results in a drastic re-duction of the transition temperature. In order to achievethe maximum value (3), the interlayer distance d mustbe much smaller than the Debye screening length κ − ,2 κ d ≪ . (4)The most optimistic estimate would thus be T max c ∼ ǫ F ∼ . d . . CALCULATIONS
We now present the details of derivation of Eq. (3).The bare strength of Coulomb interactions in grapheneis not that small. For SiO used as an insulator embed-ding graphene sheets, typical values of the dimensionlesscoupling constant are r s = e / ( εv ) ∼ ε is the dielectricconstant of the insulator and v is the velocity of the Diracspectrum, we set ~ = 1 throughout this section). Thisquestions the applicability of the weak-coupling approachto the problem of EC, suggesting, at the same time, thatthe transition temperature could be quite high [9, 10].It is known, however, that in a fermionic system with alarge number N ≫ N makes screening very effective, since all N species par-ticipate in the screening of interactions between fermionsof each particular species. Screening reduces the couplingconstant from r s to the value 1 /N ≪ r s → /N . Ef-fectively, the system becomes weakly interacting, despiteof the fact that the bare Coulomb interactions may benot weak ( r s & N approximation was already used for asingle-layer graphene [7, 12, 13] before. In a single layer,the number of species is equal to N = N s N v = 4 dueto two projections of spin ( N s = 2) and two valleys( N v = 2). This value is not exceptionally large, but doesgive hope that the large- N approach adequately describesgraphene physics. In a double-layer system the situationis better: since each electron can belong to either oneof the layers, one has an additional “which-layer” degreeof freedom ( N l = 2) making the total number of species N = N s N v N l = 8. It would already be quite reasonableto treat N = 8 as a large parameter. Therefore, large- N approximation seems to be particularly suitable for adouble-layer graphene system and is expected to providegood quantitative predictions.Below we employ the large- N approach to the double-layer graphene system (Fig. 1) treating N = 8 as a largeparameter and calculate the mean-field critical tempera-ture T c of EC. The calculations follow closely those ofRef. [7]. Of course, the mean-field treatment of two-dimensional systems is not necessarily a good one dueto strong thermal fluctuations. It is, however, sufficientfor our purposes, as our main goal is to demonstrate thatalready the mean-field critical temperature is extremelylow. The temperature T BKT of the actual BKT transi-tion can only be lower than the mean-field T c we calculatehere.Within the large- N approximation, the diagrammaticseries for the effective interaction between the electronsis identical to that of the random-phase approximation(RPA), which describes linear screening. For the prob-lem at hand, the relevant transfer momenta q are in therange 0 ≤ q ≤ p F , with p F = ǫ F /v the Fermi mo-mentum, and one has to use the exact expression for thepolarization operator (and not its limit form for q ≪ p F ).However, the static polarization operator in graphene [14]does not depend on momentum at all in this range andequals Π( ω = 0 , q ) = N s N v ν , where ν = ǫ F / (2 πv ) isthe density of states per one valley and one spin. As aresult, for the screened interlayer Coulomb interaction inthe momentum space we obtain V ( q ) = 2 πe ∗ exp( − qd ) q + 2 κ + κ [1 − exp( − qd )] /q , q ≤ p F . (5)In Eq. (5), q is the absolute value of the in-plane two-dimensional wave vector, d is the distance between thelayers, e ∗ is the effective electron charge screened by theinsulator embedding graphene sheets, e ∗ = e /ε , and κ = 2 πN s N v e ∗ ν is the inverse Debye screening length FIG. 2: The screened V ( q ) [Eq. (5), solid line] and unscreened V us ( q ) [Eq. (8), dashed line] interlayer Coulomb interaction,responsible for the excitonic instability. The values d = 0 and r s = e / ( εv ) = 1 were used, ν is the density of states. Atrelevant momenta q ∼ p F , the unscreened interaction poten-tial overestimates the actual screened one by about 10 times.The screened Coulomb interaction reaches its maximum at q = 0, the universal value (1) of which is achieved for 2 κ d ≪ in each layer. We assume the same Fermi momenta p F of electrons and holes (this can be achieved by tuning thegate voltage), since any difference between them wouldbe suppressing the condensate in a way Zeeman splittingsuppresses s-wave superconductivity.The screened Coulomb interaction (5) is a decreasingfunction of q (Fig. 2) and reaches its maximum at q = 0, V ( q = 0) = 2 πe ∗ κ + κ d . (6)The maximum of Eq. (6) is achieved, if the distance d between the layers is smaller than the Debye ra-dius [Eq. (4)], and equals V max ( q = 0) = 12 N s N v ν . (7)The factor 2 that enters the denominator in Eq. (7) isdue to “which layer” degree of freedom ( N l = 2), sinceeach carrier can belong to either one of the layers. Equa-tion (7) leads to Eq. (1) for the maximum value of thedimensionless coupling constant λ max = νV max ( q = 0).In contrast to the above calculation, in the analysis ofRefs. [9, 10] the unscreened form V us ( q ) = 2 πe ∗ exp( − qd ) /q (8)of the Coulomb interaction V ( q ) [Eq. (5)] was used, seeEq. (4) in Ref. [9] and inline formulas before Eq. (2) inRef. [10]. As seen from Eqs. (5) and (8) and Fig. 2, theunscreened form V us ( q ) is valid for q ≫ κ = N r s p F ,but significantly overestimates the actual screened inter-action V ( q ) for relevant momenta q ≤ p F . For the value r s = 1 used in Ref. [9] and typical for SiO as an insu-lator, one obtains V us ( p F ) ≈ V ( p F ). Using the un-screened form of the Coulomb interaction in Refs. [9, 10] resulted in the estimate T BKT ∼ . ǫ F and, as it ap-pears, led an overestimation of T BKT by a factor 10 , seeEq. (3).In order to obtain the mean-field transition temper-ature T c , we derive the linearized gap equation for theorder parameterˆ∆( r − r ′ ) = V ( r − r ′ ) h ˆ φ e ( r ) ˆ φ † h ( r ′ ) i . (9)Here, V ( r − r ′ ) is the interaction (5) in the coordinatespace and ˆ φ e,h are the Dirac spinor fields of electrons andholes in the graphene sheets. The matrix structure of theorder parameter in the sublattice space is predeterminedby chirality, but can be arbitrary in the valley and spinspaces. Using the standard BCS approach, we arrive atthe linearized gap equationˆ∆( n ) = ν ln ǫ F T Z d n ′ π V ( p F | n − n ′ | ) ˆ P ( n ′ ) ˆ∆( n ′ ) ˆ P ( − n ′ ) , (10)where the two-dimensional unit vectors n and n ′ repre-sent the direction of the electron momentum and ˆ P ( n ) =(1 + τ n ) /
2, with τ x and τ y the Pauli matrices in the sub-lattice space. The value of temperature T , at which anonzero solution ˆ∆( n ) to Eq. (10) appears, determines T c . Solving Eq. (10), we obtain T c ≈ exp( − /λ c ) ǫ F , (11)where λ c = ν Z π − π d θ π V (cid:18) p F sin θ (cid:19) θ . (12)The exact numerical value ∼ n ) inthe sublattice space are identical to those in Ref. [7] [seeEqs. (5.23)-(5.31) therein]. At the same time, the form(5) of the interaction V ( q ) is different here.The maximum possible value λ max c [Eq. (2)] of the in-teraction constant λ c [Eq. (12)] and, thus, the highestpossible transition temperature T max c , [Eq. (3)], are ob-tained by inserting Eq. (7) into Eq. (12). This corre-sponds to the limit p F ≪ κ ≪ /d [Eq. (4)], wherethe condition p F ≪ κ is automatically satisfied, since r s ∼ κ /p F = r s N ≫
1. The factor (1 + cos θ ) / λ max c by a factor 2compared to λ max [Eq. (1)], see Eq. (2).The obtained small value of λ max c = 1 /
16 justifiesthe very applicability of the weak-coupling BCS ap-proach to determining T c , within which the logarithmln( ǫ F /T ) ≈
16 has to be large. Therefore, the criti-cal temperature T c does exponentially depend on the in-verse coupling constant 1 /λ c , which leads to its extremelysmall value [Eq. (3)]. FIG. 3: Sensitivity of the excitonic condensate to the impu-rity scattering. Impurities with the size of potential smallerthan the interlayer distance scatter electrons and holes notidentically, thereby breaking electron-hole pairs and suppress-ing the condensate.
DISCUSSION AND CONCLUSION
Let us now discuss the obtained results. Remarkablyenough, as Eqs. (1)-(3) demonstrate, the specifics of thegraphene spectrum (chirality and valley degrees of free-dom) appears to be very unfavorable for the realizationof EC in graphene-based devices. At the same time,this is not so for double-layer systems based on materi-als with “conventional” metallic spectrum, such as, e.g.,GaAs/Al x Ga − x As heterostructures used so far experi-mentally [3, 4, 5, 6]. Indeed, in such systems the max-imum interaction strength λ max c = 1 / ( N s N l ) = 1 / λ max c = 1 / (2 N s N v ) = 1 / N l = 1) [see Eq. (5.29b) in Ref. [7]], and theexponential factor exp( − λ max c ) = exp( − ≈ · − inEq. (3) is not as small. However, the Zeeman splittingenergy enters Eq. (3) instead of ǫ F , which cannot be ex-tremely high even for experimentally very high magneticfields B . For B ≈
40 T one can estimate T max c ∼ p , any scattering process that changesthe momentum of electron and hole not identically, i.e., p → p e for electron and p → p h for hole, so that p e = p h , breaks the electron-hole pair (see Fig. 3). Thisis the case for any impurities with the range of the scat-tering potential less than the interlayer distance d , sincethe potential of such impurities differs in the two layers.The effect of the impurity scattering on the excitonic con-densate was studied analytically for conventional systemsin Refs. [2, 15] and the theory is analogous to Abrikosov-Gorkov’s theory for magnetic impurities in superconduc- tors. This approach has been very recently applied tographene in Ref. [16]. The main result of this study isthat sufficiently short-range impurities with the scatter-ing time τ destroy the excitonic condensate completelyas soon as ~ /τ & T c , (13)where T c is the transition temperature of the ideallyclean system. Equivalently, for the condensate to exist,electron momentum has to be conserved at the scale ofthe correlation length ~ v/T c . Since the mean free path vτ ∼ µ m of the order of the typical size of graphenesamples corresponds to ~ /τ ∼ T max c ∼ T ≤ T max c even inthe ballistic samples due to the boundary scattering.In conclusion, we have studied the possibility of theexcitonic condensation in double-layer graphene systems.We have demonstrated that in order to properly deter-mine the transition temperature, it is essential to takethe screening of the coupling interlayer Coulomb inter-action into account. The specifics of the graphene spec-trum (chirality and valley degrees of freedom) leads toa smaller interaction strength than in conventional semi-conductors and to an extremely small value . r s & r s ≪ p F d ≫ r s ∼
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