Bose-Einstein condensate-mediated superconductivity in graphene
M. Sun, A. V. Parafilo, K. H. A. Villegas, V. M. Kovalev, I. G. Savenko
BBose-Einstein condensate-mediated BCS-like superconductivity in graphene
Meng Sun,
1, 2
A. Parafilo, K. H. A. Villegas,
1, 3
V. M. Kovalev,
4, 5 and I. G. Savenko
1, 4 Center for Theoretical Physics of Complex Systems,Institute for Basic Science (IBS), Daejeon 34126, Korea Basic Science Program, Korea University of Science and Technology (UST), Daejeon 34113, Korea Division of Physics and Applied Physics, Nanyang Technological University, 637371 Singapore, Singapore A. V. Rzhanov Institute of Semiconductor Physics,Siberian Branch of Russian Academy of Sciences, Novosibirsk 630090, Russia Novosibirsk State Technical University, Novosibirsk 630073, Russia (Dated: September 3, 2020)We propose a mechanism of robust BCS-like superconductivity in graphene placed in the vicinityof a Bose-Einstein condensate. Thus electrons in graphene interact with the excitations above thecondensate, called the Bogoliubov quasiparticles (or bogolons). It turns out that bogolon-pair-mediated interaction allows us to surpass the long-standing problem of vanishing density of statesof particles with a linear spectrum in the vicinity of the Dirac points. It results in a dramaticenhancement of the superconducting properties of graphene while keeping its relativistic dispersion.We study the behavior of the superconducting gap and calculate critical temperatures in the casesof single-bogolon and bogolon-pair-mediated pairing processes, accounting for the complex bandstructure of graphene. Surprisingly, both intravalley and intervalley bogolon-mediated pairings turnout equally possible.
Graphene is conventionally accepted as a two-dimensional (2D) material [1] with extremely high con-ductivity [2, 3]. Its chemical potential can be controlledby an external electric field, which makes it possibleto choose the type of the carriers of charge. Electronsand holes in graphene represent massless relativistic par-ticles [4] described by the 2D Dirac equation, whichopens perspectives for outstanding transport character-istics and allows for the study of the interplay betweenrelativity and superconductivity [5]. One interesting con-sequence of this interplay is the specular Andreev reflec-tion in graphene-superconductor junctions [6], which isnot typical for normal metal-superconductor junctions,where instead retroreflection takes place. When grapheneis deposited on a substrate, it can adopt the propertiesof the latter, such as ferromagnetism and superconduc-tivity, which can be induced in graphene due to the prox-imity effect [7, 8], even though neither superconductivitynor ferromagnetism belong to the set of intrinsic proper-ties of graphene. All this makes hybrid graphene-basedstructures, such as graphene-superconductor interfaces,an intense topic of research, which can be broadly called mesoscopic transport in graphene . In particular, it bringsin the term mesoscopic superconductivity [9].Why is bare graphene not intrinsically superconduct-ing? The primary reasons are the absence of electron-electron screening at small electron densities and thesmallness of the electron density of states [10], whichis linear in energy and thus it vanishes at the Diracpoint. As a consequence, the electron-phonon interactionin graphene is strongly suppressed [11]. And since theBardeen-Cooper-Schrieffer (BCS) electron pairing belowthe transition temperature T c involves basically the samematrix elements of electron-phonon interaction as thescattering matrix elements above T c [12–14], there might be no BCS superconductivity in graphene but for theinduced one. The bigger are the matrix elements ofelectron-phonon interaction, the larger the gap opens,and the larger is T c [15], which are the benchmarks ofrobust superconductivity. In other words, “ good ” con-ductors such as gold, copper or graphene are “ bad ” su-perconductors.Recently, there have been reported ways to turn amulti-layer graphene into a superconductor by twistingthe bilayer graphene [16, 17] or depositing it on a SiCsubstrate [11]. A trilayer graphene under a vertical dis-placement field also exhibits superconductivity [18]. Allthese approaches are targeted at increasing the electrondensity of states at the Fermi energy by building a flatband [19–21], including the recent progress in photonicgraphene [22]. Unfortunately, it usually destroys the rel-ativistic aspect of the problem since the dispersion ofgraphene remains no longer linear [23].In this Letter, we propose a non-conventional mech-anism of (i) strong electron scattering (above T c ) and(ii) electron-electron pairing interaction (below T c ) ingraphene, beyond the acoustic phonons and impuritychannels [24]. We consider a hybrid system consisting ofgraphene and a 2D Bose-Einstein condensate (BEC) [25–28] and show, that resistivity of graphene in its normal(non-superconducting) state becomes dominantly con-trolled by the condensate. The excitations above theBEC, called Bogoliubov quasiparticles or bogolons, pos-sess properties of sound. We show, that bogolon-pair-mediated resistivity turns out to be orders of magnitudelarger than that of any other scattering processes in thesystem. It makes us guess that graphene might acquirestrong superconducting (SC) properties below T c and inthe vicinity of Dirac points due to the bogolon-mediated(as opposed to acoustic phonon-mediated) pairing of elec- a r X i v : . [ c ond - m a t . s up r- c on ] S e p FIG. 1. System schematic: A hybrid system consisting of anelectron gas in graphene (top layer) separated by the distance l from a two-dimensional Bose-Einstein condensate, repre-sented by indirect excitons, where n-doped (red) and p-doped(blue) layers are separated by a spacer with the thickness d .The electrons in graphene and the excitons are coupled by theCoulomb force. trons. We check this assumption and prove it valid. Thisway one state of matter (Bose condensate) can induceanother state of matter (SC condensate) in graphene,avoiding a twist and keeping its relativistic dispersion.Let us consider the system consisting of a 2D electrongas in graphene and a layer of the exciton gas (Fig. 1).The layers are spatially separated and the particles arecoupled by the Coulomb interaction [29, 30], describedby the Hamiltonian H = (cid:90) d r (cid:90) d R Ψ † r Ψ r g ( r − R ) Φ † R Φ R , (1)where Ψ r and Φ R are the field operators of electrons andexcitons, respectively, g ( r − R ) is the Coulomb interac-tion strength, r and R are the in-plane coordinates of theelectron and exciton center-of-mass motion, respectively.Furthermore, let us assume the excitons to be in theBEC phase. Then we can use the model of a weaklyinteracting Bose gas and split Φ R = √ n c + ϕ R , where n c is the condensate density and ϕ R is the field operator ofthe excitations above the BEC. Then Eq. (1) breaks intothree terms, H = √ n c (cid:90) d r Ψ † r Ψ r (cid:90) d R g ( r − R ) (cid:104) ϕ † R + ϕ R (cid:105) , H = (cid:90) d r Ψ † r Ψ r (cid:90) d R g ( r − R ) ϕ † R ϕ R , (2)and H = gn c (cid:82) d r Ψ † r Ψ r , which gives a small correctionto the Fermi energy µ and can usually be disregarded [31].We now can express the field operators of bosonic exci-tations as the Fourier series of the linear combinations ofbogolon annihilation (creation) operators b q ( b † q ), ϕ R = 1 L (cid:88) p e i p · R ( u p b p + v p b †− p ) , (3) where L is a characteristic size, and the Bogoliubov co-efficients read [32] u p = 1 + v p = 12 (cid:34) (cid:18) M s ω p (cid:19) (cid:35) / ,u p v p = − M s ω p , with M the exciton mass, s = (cid:112) κn c /M the soundvelocity of Bogoliubov quasiparticles, κ = e d/ (cid:15) d theexciton–exciton interaction strength in the reciprocalspace, e the electron charge, (cid:15) d the dielectric constant, ω p = (cid:126) sp (1 + p ξ h ) / the spectrum of bogolons with p = | p | , and ξ h = (cid:126) / M s the healing length.Since the electron spectrum in graphene consists of twononequivalent cones with the minima at the Dirac points K and K (cid:48) , we can define the electron field operator asΨ r = 1 L (cid:88) k ,σ (cid:16) e i ( K + k ) · r c , k ,σ + e i ( K (cid:48) + k ) · r c , k ,σ (cid:17) , (4)where c α, k ,σ are electron annihilation operators with α = 1 , σ = ↑ , ↓ the electron spinprojection. Using Eqs. (2)-(4), we find H = √ n c (cid:88) k , p (cid:88) α,β,σ g αβ p L (cid:104) ( v − p + u p ) b p (5)+ ( v p + u − p ) b †− p (cid:105) c † α, k + p ,σ c β, k ,σ , H = (cid:88) k , p , q (cid:88) α,β,σ g αβ p L (cid:104) u q − p u q b † q − p b q + u q − p v q b † q − p b †− q + v q − p u q b − q + p b q + v q − p v q b − q + p b †− q (cid:105) c † α, k + p ,σ c β, k ,σ , where g αβ p is the Fourier image of the electron-exciton in-teraction. The terms H and H describe single-bogolon(1b) and bogolon-pair (2b)-mediated processes, corre-spondingly. Before considering the SC properties ofgraphene, let us, first, study its transport properties inthe normal state of the electron gas. Bogolon-pair-mediated resistivity.
Let us assume thatthe ambient temperature is above the critical temper-ature of SC transition but not too high (smaller than ∼
200 K), thus we can neglect the intervalley scatter-ing [33] and thus put g αβ p = g p δ αβ in Eq. (5). To studythe resistivity of graphene in the normal state, we will fol-low the Bloch-Gr¨uneisen approach [34, 35]. The deriva-tion starts with the Boltzmann equation (see Supplemen-tal Material for the detailed derivation [36]) e E · ∂f p (cid:126) ∂ p = I { f p } , (6)where f p is the nonequilibrium electron distribution func-tion, E is a probe electric field, and I { f p } is the collisionintegral describing 2b-mediated scattering processes (the1b scattering has been studied elsewhere [31]). (a) (b) FIG. 2. (a) 1b-mediated (dashed) and 2b-mediated (solid)resistivity of graphene as a function of temperature for theexciton condensate densities n c = 10 (black) and 10 cm − (red). Vertical lines show the Bloch-Gr¨uneisen temperature(for the curves of the same color). Other parameters are: d = 3 . l = 10 nm and n e = 1 . × cm − . (b)2b-mediated resistivity of graphene with (solid) and without(dashed) the temperature correction to the condensate den-sity, ˜ n c = n c (cid:2) − ( T /T
BEC ) (cid:3) with T c = 100 K for MoS .Other parameters are the same as in (a). The resistivity reads the general definition [24], ρ − = e D ( µ ) v / τ , where τ is an effective elec-tron scattering time, v is the Fermi velocity, D ( µ ) =( g s g v / π (cid:126) ) µ/v is the density of states in graphene atthe energy µ = (cid:126) v p F , p F = (cid:112) πn e /g s g v is the Fermiwave vector with n e the concentration of electrons ingraphene, g s = 2 and g v = 2 the spin and valley g -factors.Finally, we find the bogolon-pair-mediated resistivity ofgraphene, ρ ∨ = s M π e v ( (cid:126) p F ) ∞ (cid:90) L − dp | g p | p sinh (cid:16) (cid:126) sp k B T (cid:17) ln ( pL ) , (7)where the symbol ∨ reflects the linear spectrum (thusthe doping µ is assumed to be smaller than 1 eV); g p = e (1 − e − pd ) e − pl / (cid:15) d p ; and we accounted for theinfrared divergence [37]. Formula (7) contains the firstimportant result of this Letter. Figure 2 shows the com-parison between the contributions of 2b and 1b [31] scat-tering processes to the resistivity of graphene. Buildingthe curves, we account for the decay of the condensatedensity with temperature [38], ˜ n c = n c (cid:104) − ( T /T
BEC ) (cid:105) .We conclude, that 2b processes dominate over 1b scat-tering in the whole temperature range. Bogolon-pair-mediated superconductivity.
Now, let usdecrease the temperature below some T c and find the SCgap, considering a singlet pairing. Using Schriffer-Wollftransformation we find an effective Hamiltonian for 1b or 2b pairings ( λ = 1b, 2b), H ( λ )eff = H + (8)+ (cid:88) k , k (cid:48) , p (cid:88) σ,σ (cid:48) (cid:88) α,β V αβλ ( p )2 L c † α, k + p ,σ c β, k ,σ c † α, k (cid:48) − p ,σ (cid:48) c β, k (cid:48) ,σ (cid:48) + (cid:88) k , k (cid:48) , p (cid:88) σ,σ (cid:48) (cid:88) α (cid:54) = β V λ ( p )2 L c † α, k + p ,σ c α, k ,σ c † β, k (cid:48) − p ,σ (cid:48) c β, k (cid:48) ,σ (cid:48) , where H is a kinetic energy of electrons in graphene, V λ ( p ) , V λ ( p ) = V λ ( p ) are the matrix elements re-sponsible for the intravalley electron scattering, and V λ ( p ) = V λ ( p ∓ Q ) are the intervalley scatteringswith Q = | K − K (cid:48) | the momentum difference betweenthe Dirac points. The corresponding Feynman diagramsfor two-electron scattering are plotted in Fig. 3.After algebraic calculations (see Supplemental Mate-rial [36]), we find V b ( p ) = − n c M s g p ≡ − χ , (9) V b ( p ) = − χ p π p/ (cid:90) L − N q dq (cid:112) p − q , (10)where in (9) we introduced χ , and χ = M sg p / (cid:126) .The term V b ( p ) corresponds to the electron-electronpairing with an exchange of single Bogoliubov excitationof the BEC, whereas V b ( p ) describes the electron pair-ing mediated by an exchange of a pair of bogolons. Toderive these formulas, we followed the BCS approach and ε k , ↑ ε k , ↓ ε k + q , ↑ ε k − q , ↓ V λ (a) ε k , ↑ ε k , ↓ ε k + q , ↑ ε k − q , ↓ V λ (b) ε k , ↑ ε k , ↓ ε k + q , ↑ ε k − q , ↓ V λ (c) ε k , ↑ ε k , ↓ ε k + q , ↑ ε k − q , ↓ V λ (d) ε k , ↑ ε k , ↓ ε k + q , ↑ ε k − q , ↓ V λ (e) ε k , ↑ ε k , ↓ ε k + q , ↑ ε k − q , ↓ V λ (f ) FIG. 3. Feynman diagrams, illustrating intravalley (a-b) andintervalley (c-f) pairings, in accordance with Hamiltonian (8).Red and black lines describe electrons in K and K’ valleys,respectively. considered constant attractive interaction between elec-trons with energies smaller than a cut-off ω b = (cid:126) s/ξ h ,which appears by analogy with the Debye energy in theBCS theory. Since ξ h ∼ /s and thus ω b ∼ s ∼ n c ,the typical energy scale of the attractive interaction canbe controlled by the density of the condensate, n c . Inorder to satisfy the applicability of the BCS theory, wealso assume that ω b (cid:28) (cid:126) v /a , where a is the interatomicdistance in graphene [39].From the comparison of Eqs. (9) and (10) we see, thatin contrast with the 1b-mediated superconductivity, thestrength of the 2b-mediated pairing potential contains anadditional temperature-dependent term, proportional tothe Bose distribution of bogolons, N q ≡ [exp( (cid:126) ω q /k B T ) − − . Note, in (10) we introduced a cut-off L − , which isstandard for 2D systems, to avoid logarithmic divergenceat small momenta [37].Now, we are armed to proceed with the SC order pa-rameter, which structure is nontrivial due to the presenceof two valleys [40]. Indeed, we can distinguish betweentwo SC gaps, diagonal and nondiagonal in valley indices:∆ ααλ and ∆ αβλ with α (cid:54) = β . They characterize the forma-tion of Cooper pairs by electrons residing the same anddifferent valleys, correspondingly.To figure out which type of pairing (intra- or interval-ley) is more favorable in our system, we can solve thesystem of Gor’kov’s equations, (cid:18) − ∂∂τ − ξ p (cid:19) ˆ G ( p, τ ) + ˆ∆ λ ( p ) ˆ F † ( p, τ ) = 1 , (11) (cid:18) − ∂∂τ + ξ p (cid:19) ˆ F † ( p, τ ) + ˆ∆ † λ ( p ) ˆ G ( p, τ ) = 0 , where ξ p = ε p − µ with ε p = (cid:126) v p is the dispersion of elec-trons in graphene; G αα ( p, τ ) = −(cid:104) T τ c α,p, ↑ ( τ ) c † α,p, ↑ (cid:105) and F αβ ( p, τ ) = (cid:104) T τ c α,p, ↓ ( τ ) c β,p, ↑ (cid:105) are normal and anoma-lous Green’s functions in imaginary time ( τ = it ) rep-resentation, together with the equation for the SC gapmatrix in the valley space,ˆ∆ λ ( p ) = − (cid:88) p V λ ( p ) (cid:18) F F F F (cid:19) ( p,
0) (12) − (cid:88) p,j = ± V λ ( p + jQ ) (cid:18) F F (cid:19) ( p, . In order to make analytical estimations of the criticaltemperatures of the intra- and intervalley pairings, let usdisregard the temperature-dependent term in Eq. (10)and solve Eqs. (11), (12). We can consider two limitingcases of low ( | µ | < ω b ) and high ( | µ | > ω b ) doping. If wetake | µ | (cid:29) ω b , we find T intra c = 1 . ω b exp (cid:18) − V λ ( p F ) D ( µ ) (cid:19) , (13) T inter c = 1 . ω b exp (cid:18) − V λ ( p F ) + V λ ( p F )] D ( µ ) (cid:19) . (14) Obviously, both intra- and intervalley order parameters(and the corresponding critical temperatures) are with agood accuracy equal to each other due to the smallnessof V λ ( p F ) = V λ ( p F ) ∝ g Q . Indeed, g Q is exponentiallysuppressed due to large intervalley momentum Q . Thus,we can put ∆ ααλ = ∆ αβλ ≡ ∆ λ /
2. In the low-dopingregime, we come to the same relations.Then, the equation for the BCS-like SC gap reads1 = 12 (cid:90) d p (2 π ) | V λ ( p ) | ζ p [1 − n F ( ζ p )] , (15)where ζ p = (cid:112) ξ + ∆ λ is the energy of quasiparticles inthe superconductor, and ξ = ± (cid:126) v p − µ is the spectrumof electrons in doped graphene (where ± correspond tothe conduction and valence bands).The 1b-mediated gap at zero temperature reads∆ (0)1 b ( | µ | > ω b ) ≈ ω b exp (cid:18) − χ D ( µ ) (cid:19) , (16)∆ (0)1 b ( | µ | < ω b ) ≈ | µ | exp (cid:18) − χ D ( µ ) + ω b | µ | − (cid:19) . (17)In equations above, we assumed p F d, p F l (cid:28) g p . This assumptionimposes a restriction on the maximal allowed value of n e for considered distances d and l . Note, the SC gap (16)has a standard form of the BCS gap, while the expres-sion (17) is mostly determined by the doping µ ratherthan ω b [39].The 2b-mediated gap at zero temperature is the samefor both the limits of high and low dopings,∆ (0)2 b ≈ ω b exp (cid:18) − π (cid:126) v χ (cid:19) . (18)The order parameter (18) is orders of magnitude largerthan the SC gap mediated by the 1 b processes. First andforemost, Eq. (18) does not contain the density of statesof the Dirac electrons in graphene D ∝ | µ | . Second,2b-mediated SC gap is not determined by the chemicalpotential µ in the µ < ω b limit. Both of these featuresappear due to the nature of the 2b-mediated electron in-teraction, which matrix elements u p v p ∝ ( pξ h ) − . Asa result, the emerging 1 /p -term in the pairing poten-tial (10) compensates the smallness due to D ( µ ) ∼ p .Figure 4 illustrates the full numerical solution of Eq. (15)for 2b processes and shows that both the SC gap and thecritical temperature grow with n c .We should note, that our BCS-like theory is only appli-cable as long as we are in the weak coupling regime [41],which imposes a restriction χ / π (cid:126) v <
1, thus we use χ in the range (0 . − . π (cid:126) v . Otherwise, the Eliashbergequations treating the strong coupling regime should beused [42–44], which is beyond the scope of this Letter. Conclusions.
We have studied bogolon-mediated inter-action of electrons in graphene in the vicinity of a two-dimensional Bose-condensed dipolar exciton gas. In the T [ K ] Δ [ m e V ] n c [× cm - ] T c [ K ] (a) (b) FIG. 4. (a) Temperature dependence of the SC gapfor bogolon-pair-mediated processes with account of the N q -containing term (dashed) and without N q contribution(solid). We used the condensate density n c = 2 . × (red) and n c = 2 . × cm (black); the density of freeelectrons n e = 1 . × cm − (thus µ > ω b ); the separation d = 1 nm. We also accounted for the temperature dependenceof n c : ˜ n c = n c (cid:2) − ( T /T
BEC ) (cid:3) with T BEC = 100 K. (b) Crit-ical temperature of SC transition as a function of conden-sate density for bogolon-pair-mediated interaction with (reddashed) and without (black solid) the N q -containing contri-bution. normal state of graphene, we calculated the temperaturedependence of bogolon-mediated resistivity and foundthat in such a hybrid system, the dominant scatteringmechanism is provided by the interaction with pairs ofBogoliubov quasiparticles. Furthermore, we have studiedthe bogolon-mediated electron pairing in graphene andcalculated the critical temperature of the superconduct-ing transition. We have shown, that bogolon-mediatedelectron interaction allows one to solve the problem ofsmallness of the density of states in two-dimensionalDirac materials at small momenta. Acknowledgements.
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