Spin excitations in the heavily overdoped monolayer graphene superconductor: an analog to the cuprates
aa r X i v : . [ c ond - m a t . s up r- c on ] F e b Spin excitations in the heavily overdoped monolayer graphene superconductor: ananalog to the cuprates
Wei-Jie Lin , ∗ W. LiMing , ∗ and Tao Zhou , † Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,Department of Physics, School of Physics and Telecommunication Engineering,South China Normal University, Guangzhou 510006, China Guangdong-Hong Kong Joint Laboratory of Quantum Matter,Frontier Research Institute for Physics, South China Normal University, Guangzhou 510006, China (Dated: February 23, 2021)Recently it was reported experimentally that the monolayer graphene can be doped to beyond theVan Hove singularity. We study theoretically the possible superconductivity and the correspondingspin excitations of the monolayer graphene in this doping region. A static spin-density-wave stateis favorable due to the nested Fermi surface as the Fermi level is doped to the Van Hove singu-larity point. Superconductivity may be realized upon further doping. The spin excitations in thesuperconducting state are studied theoretically based on the random phase approximation. Theoverall features are qualitatively the same with those in cuprate superconductors. Thus we haveproposed an exciting possibility, namely, the heavily overdoped monolayer graphene can become anovel platform to study the unconventional superconductivity.
High-T c superconductivity in the family of cuprate ma-terials has been studied intensively for more than thirtyyears, while so far its mechanism remains puzzling [1].This has motivated the great effort to seek for cuprateanalogs. Previously, several possible candidate super-conducting families have been proposed, including theiron-based superconductors [2] and the recently discov-ered nickelate superconductor [3].The realization of superconductivity in graphene-basedmaterials has been paid considerable attention sincethe first production of graphene in 2004 [4]. Previ-ously, evidence of superconductivity has indeed been re-ported in several graphene-based materials with differ-ent methods [5–11]. Especially, it was reported that thetwisted bilayer graphene will exhibit flat bands near theFermi energy, leading to the correlated parent insulatingstates [12]. The superconductivity emerges upon dop-ing electrons into this parent state. As a result, thetwisted bilayer graphene has become a novel platformto study unconventional superconductivity and has at-tracted tremendous interest in the past several years [10].The band structure of the monolayer graphene has sad-dle points at the M point of the Brillouin zone. Thequasiparticle dispersion near this point is flat leading tothe divergent density of states according the Van Hovesingularity (VHS) scenario. Of particular interest is totune the Fermi energy to the VHS point upon doping(the VHS filling) [13, 14]. Then the Fermi surface isperfectly nested leading to a spin-density-wave (SDW)instability [15–18]. On the other hand, the VHS near theFermi level may provide an effective attractive poten-tial then superconducting pairing is also favored [17–24].Previously the competition between the SDW order and ∗ These two authors contributed equally to this work. † Corresponding author: [email protected] the superconducting pairing has been studied theoreti-cally [17, 18, 20]. Such competition is of interest and isindeed similar to that in the Cuprate compound.Very recently, it was reported that the graphene dopingtechnique reaches a new level, namely, the electron fill-ing beyond the VHS filling was realized experimentallyfor the first time [25]. The electronic structure of themonolayer graphene at this doping region is similar tothat of twisted graphene material [10–12]. At the VHSfilling, the magnetic order is induced by the flat bandat the Fermi energy. The static magnetic order is sup-pressed and the superconducting state may emerge uponfurther doping. Therefore, we expect that the monolayermay become another platform to study the unconven-tional superconductivity.For cuprate superconducting materials, it is generallybelieved that spin excitations may play a fundamentalrole and mediate the superconducting pairing. Experi-mentally, the momentum and energy dependence of spinexcitations are obtained directly through the inelasticneutron scattering (INS) experiments [26–32]. Theoret-ically, the INS experimental results can be comparedthrough exploring the imaginary part of the dynamicalspin susceptibility [33–37]. One of the most importantresults is the resonant spin excitation in the supercon-ducting state [26–28, 33–37]. The spin resonance occursat an energy proportional to the superconducting transi-tion temperature [28, 35], so that it is believed to be in-timately related to superconductivity. For the graphene-based material, previously it has been proposed theoret-ically that spin fluctuations may account for the super-conductivity [23, 38]. Therefore, now it is timely and ofimportance to study the spin fluctuations in the super-conducting monolayer graphene and compare the resultswith those in the high-T c superconducting materials.In this paper, we study theoretically the dynamic spinsusceptibility of the mono-layer graphene material at thedoping level beyond the VHS filling based on the randomphase approximation (RPA). We consider a typical d + id pairing state, consistent with previous theoretical predic-tions [17–24]. The resonant spin excitation is indeed re-vealed. The overall features of spin excitations are quali-tatively the same with those in high-T c superconductors.Our results indicate that the mono-layer graphene mate-rials are indeed analogous to the cuprate materials andmay become a novel platform to study the mechanism ofunconventional superconductivity.We start from the Hamiltonian including the bare su-perconducting Hamiltonian and an onsite repulsive inter-action, expressed as, H = H SC + H int . (1) H SC includes the hopping term and the superconduct-ing pairing term. Considering the superconducting pair-ing between two nearest-neighbor sites, this term can bewritten as: H = − t X h ij i ,σ ( c † i σ c j σ + h . c . ) + X h ij i (∆ ij c † i ↑ c † j ↓ + h . c . ) − µ X i ,σ c † i σ c i σ , (2)where j is the nearest-neighbor site of the site i with j = i + e α . Here each site i has three nearest-neighbor siteswith e = ( √ , e = ( −√ ,
1) and e = (0 , − ij represents the superconducting pairing magnitude. H int is the on-site interaction term, expressed as H int = U X i n i ↑ n i ↓ . (3)In the momentum space, the bare Hamiltonian in thesuperconducting state can be expressed as a 4 × H ( k ) = − µ f ( k ) 0 ∆( k ) f ∗ ( k ) − µ ∆( − k ) 00 ∆ ∗ ( − k ) µ − f ∗ ( − k )∆ ∗ ( k ) 0 − f ( − k ) µ , (4)with f ( k ) = − t P j e i k · e j and ∆( k ) = P j ∆ j e i k · e j . Forthe d + id pairing symmetry, we have ∆ , , = ∆ e i π ,∆ e i π , and ∆ .The bare spin susceptibility from Eq.(4) can be writtenas a 4 × χ l ,l l ,l (0) ( q , ω ) = 1 N X k X α,β =1 [ ξ αl ( k ) ξ α, ∗ l ( k ) ξ βl ( k + q ) ξ β, ∗ l ( k + q )+ ξ αl ( k ) ξ α, ∗ l +2 ( k ) ξ β, ∗ l ( k + q ) ξ βl +2 ( k + q )] f ( E β k + q ) − f ( E α k ) ω + E α k − E β k + q + iη , (5) where l i = 1 , E α k and ξ α ( k ) are the eigenvalue and the eigen-vector of the bare Hamiltonian H ( k ). f ( x ) is the Fermidistribution function. -3-2-101230.0 1.4-1.0-0.50.00.51.0 0.00.51.01.52.02.53.0 e ( k ) k G: (0,0) M: (0, 2p/3) K:(2p/3 3, 2p/3) G (a) m=0.8 m=1.0 m=1.2(b) m=0.8 m=1.0 m=1.2 M i n q A ( U ) U(c) I m c ( q , w = . ) kG M K G(d) FIG. 1. (Color online) (a) The normal state energy bandsalong the highly symmetrical lines in the Brillouin zone. (b)The normal state Fermi surfaces with different chemical po-tentials. (c) The minimum value of the RPA factor as a func-tion of the interaction U with ω = 0. (d) The imaginarypart of the renormalized normal state spin susceptibility as afunction of the momentum with ω = 0 . µ = 1 . The renormailized spin susceptibility χ ( q , ω ) can beobtained through the RPA, given byˆ χ ( q , ω ) = [ ˆ I − ˆ χ ( q , ω ) ˆ U ] − ˆ χ ( q , ω ) , (6)where ˆ I is the 4 × U matrix include U l ,l l ,l = U for l = l = l = l . Thephysical spin susceptibility can be obtained through thesum of the elements of ˆ χ with l = l and l = l .The normal state energy bands and Fermi surfaces [ob-tained by setting ∆ = 0 in Eq.(4)] are displayed inFigs. 1(a) and 1(b), respectively. As is seen, at the M point [ Q M = (0 , π/ E = 1 .
0, leading to the VHS at thisenergy. When the chemical potential is below the VHSfilling ( | µ | < µ = 1, the Fermi surface is perfectly nested. Be-yond the VHS filling ( µ > ,
0) point.As discussed previously [15–18], the nested Fermi sur-face will lead to the SDW instability. With the RPAframework, the SDW instability can be explored throughthe RPA factor A ( q , ω ), with A ( q , ω ) = det | ˆ I − ˆ χ ( q , ω ) ˆ U | . (7)Generally at the zero energy the imaginary part of A ( q , ω ) is zero. If its real part Re A ( q , ω = 0) at a certainwave-vector q is negative, then the magnetic instabilityoccurs. In this case the RPA method cannot be used di-rectly and a static SDW order should be induced to de-scribe the system. The minimum value of the RPA factorwith ω = 0 as a function of the onsite interaction U isplotted in Fig. 1(c). At the VHS filling ( µ = 1 . A ( U ) is -0.4 -0.2 0.0 0.2 0.40123456 0.0 0.2 0.4 0.6 0.804812 0.00 0.04 0.080.00.10.2-0.2 0.0 0.20.00.30.6 q x /p w (a) q y =2p/3w r w c SC state Normal state I m c ( q , w ) q x /p(c) M a x q I m c w(d) w r D (b) Q ’1 Q FIG. 2. (Color online) (a)The imaginary part of the spin sus-ceptibility as functions of the energy and the momentum alongthe line q y = 2 π/ = 0 .
04. (b) The imaginary partof the spin susceptibility as a function of the momentum with ω = ω r = 0 .
12. (c) The imaginary parts of the spin suscepti-bility as a function of the momentum in the superconductingstate and normal state with ω = ω r . (d) The maximum valueof Im χ ( q , ω ) in the Brillouin zone as a function of the energywith different gap magnitudes. Inset: The resonant energy ω r as a function of the gap magnitude ∆ . less than zero as the interaction U is larger than a criticalvalue U c ( U c = 0 . U c increases signifi-cantly. In the following presented results, we study thespin excitations beyond the VHS filling with µ = 1 . U = 1 .
6. With these parameters the static SDW orderdisappears and the RPA technique is effective to studythe spin fluctuation. The imaginary part of the renor-malized normal state spin susceptibility as a function ofthe momentum with ω = 0 . δ, π/
3) near the M point.We now study the spin excitations in the supercon-ducting state. The intensity plot of the imaginary partof the spin susceptibility (Im χ ) as functions of the en-ergy and the momentum along the line q y = 2 π/ = 0 .
04 is presented in Fig. 2(a). Here two typicalenergies, i.e., ω r ≈ .
12 and ω c ≈ .
4, are revealed andindicated in Fig. 2(a). At the energy ω r , Im χ reaches itsmaximum value, at an incommensurate momentum with q = ( ± δ, π/ δ ≈ . π is the incommensurability. Asthe energy increases, the incommensurability decreases.At the energy ω c , the spin excitation is commensuratewith the maximum value appearing at the momentum Q M = (0 , π/ ω c , the spin excitation becomes incommensurateagain. The dispersion of the maximum spin excitationsin the whole momentum and energy space has a hourglassshape.Let us study in more detail the spin excitation at the energy ω r . The intensity plot of the imaginary part of thespin susceptibility in the whole Brillouin zone at the en-ergy ω = ω r is displayed in Fig. 2(b). As is seen, the spinexcitation has six-fold symmetry with the maximum ex-citation appearing at the incommensurate momentums Q and Q ′ (or their symmetrical momentums). Thetwo dimensional cut of Im χ along the line q y = 2 π/ q x is replotted in Fig. 2(c). The imag-inary part of the spin susceptibility in the normal statewith ω = 0 .
12 is also plotted in Fig. 2(c). As is seen,the intensity of the maximum Im χ at the energy ω r inthe superconducting state is significantly stronger thanthat in the normal state. The enhanced spin excitationin the superconducting state indicates the signal of thespin resonance. Previously, the resonant spin excitationhas attracted broad interest in various unconventionalsuperconductors [33–37]. It has been verified that theresonant energy is proportional to the gap magnitude orthe superconducting transition temperature T c [35]. Themaximum spin excitations in the whole momentum spaceas a function of the energy with different gap magnitudes(from ∆ = 0 .
02 to ∆ = 0 .
08) are plotted in Fig. 2(d).The intensity of the maximum value of Im χ increases sig-nificantly as ∆ increases. The possible resonant energy ω r can be obtained from Fig. 2(d) through the positionof maximum Im χ . The energy ω r as a function of the gapmagnitude is presented in the inset of Fig. 2(d). As isseen, here ω r is proportional to ∆ . These features verifythat the resonant spin excitation indeed exists in heav-ily overdoped superconducting graphene materials. Thisresonant spin excitation intimately related to the super-conducting pairing. Such resonant behavior may be usedto identify the unconventional superconductivity in thisfamily.We would like to compare the above numerical resultsof the spin excitation with those in high-T c cuprate su-perconductors. In cuprate superconductors, the spin ex-citations are material dependent. For the La-based mate-rial (e.g., La − x Sr x CuO ), two energy scales are revealedexperimentally [32]. The maximum spin excitation ap-pears at a low energy about 18 meV and an incommensu-rate momentum. At a higher energy (about 50 meV), thespin excitation is commensurate. For Y-based high-T c superconducting material (e.g., YBa Cu O − x ), a reso-nant spin excitation at the energy about 40 meV is re-vealed [26, 27]. The spin excitation at this energy is com-mensurate. For both two materials, the dispersions havehourglass shape [29–32]. Here obviously, the spin exci-tations are analogous to those in La-based high-T c su-perconducting materials, including the two energy scalesand the hourglass shape dispersion. Moreover, we havechecked numerically that if a rather strong gap magni-tude is considered (∆ ≥ . t ), the spin excitation at theresonant energy will become very strong and appears atthe commensurate momentum Q M (the numerical resultsare not presented here). In this case the results are sim-ilar to those of Y-based superconductors. Therefore, ourresults indicate that the spin excitations in the doped su- Rec Imc c ( Q M , w ) w w r w c FIG. 3. (Color online) The real and imaginary parts of thebare spin susceptibility with ∆ = 0 . perconducting graphene materials are indeed analogousto those in high-T c cuprate superconductors.The above features of spin excitations can be under-stood well based on the RPA framework and the topol-ogy of the Fermi surface. The general origin of the spinresonance in an unconventional superconductor has beenstudied intensively [33–37]. The renormalized spin sus-ceptibility includes two parts of contributions, namely,the bare spin susceptibility χ ( q , ω ) and the RPA fac-tor A ( q , ω ). The real and imaginary parts of the barespin susceptibility are displayed in Fig. 3. As is seen, theimaginary part of the bare spin susceptibility is nearlyzero due to the presence of the spin gap. At the edgeof the spin gap Im χ increases rapidly, then Re χ has apeak structure due to the Kramers-Kronig relation. Asa result, the real part of the RPA factor reaches the min-imum value at this energy. Thus the imaginary part ofthe renormalized spin susceptibility in the superconduct-ing state will be enhanced at this energy, leading to theresonant spin excitation at this energy. At higher ener-gies, the RPA factor plays minor role. The spin excita-tion is mainly determined by the bare spin susceptibilityand the topology of the Fermi surface. The bare spinexcitation has the maximum value at the energy ω c , asindicated in Fig. 3. This can be understood further fromthe nesting of the energy contour.The bare spin susceptibility is mainly contributed bya particle-hole excitation. Then we have Im χ ( q , ω ) ∝ P k δ [ ω − Ω( q , k )] with Ω( q , k ) = E ( k ) + E ( k + q ). Gen-erally, to explain the spin excitation at a certain energy ω , one needs to study the scattering between the energycontour ω = ω / E = ω r / .
06 and E = ω c / . E = 0 .
06 contains a flatpiece, indicating the nesting feature. The best nestingvectors are indicated in Fig. 4, namely, Q and Q ′ , wellconsistent with the numerical results shown in Fig. 2. Asthe energy increases, the energy contours become larger.Naturally the incommensurability will decrease. As the E=0.06 E=0.2 Fermi Surface Q Q1’
FIG. 4. (Color online) The constant energy contours with E ( k ) = 0 .
06 and E ( k ) = 0 . E ( k ) = 0 . energy is much greater than ∆ , the superconductingpairing term plays a minor role. As a result, the en-ergy contours E = 0 . µ = 1 . µ = 1 .
4. Sincethe normal state Fermi surface with µ = 1 . q = Q M . As a result, thespin excitation at the energy ω = 0 . Q M .As the energy increases further, the nesting condition ofthe energy contour is broken and the spin excitation willbecome incommensurate again.At last, we summarize the similarity between the possi-ble superconductivity in heavily overdoped graphene ma-terial with that in cuprates. Firstly, the general phasediagram may be similar. For the graphene material, atthe VHS filling, the Fermi surface is nested, leading tothe SDW instability. Beyond the VHS filling, the staticSDW order is suppressed and the superconductivity mayemerge. For the cuprates, at the half-filling, the system isin the antiferromagnetic state. Upon doping the antifer-romagnetic order is suppressed and the superconductivityis realized. Secondly, although so far no consensus hasbeen reached about the mechanism of superconductiv-ity in cuprates, while many believe that spin fluctuationshould play essential role. Here for the heavily overdopedgraphene material, strong spin fluctuation should existdue to the nesting of the Fermi surface. Such strong fluc-tuation could account for possible unconventional super-conductivity graphene materials. Thirdly, as presented inour present work, the overall features of spin excitationsin superconducting graphene materials are qualitativelysimilar with those of cuprate superconductors, includingthe resonant spin excitation at a certain energy and thehourglass dispersions. Therefore, here we propose thatthe highly overdoped monolayer graphene material maybe a potential candidate to be the cuprate analog.In conclusion, we have studied theoretically spin ex-citations in the superconducting state of the monolayergraphene material beyond the VHS filling. Two typicalenergies are revealed. At a low energy, a resonant spinexcitation is revealed at an incommensurate momentum.At a higher energy, the spin excitation is commensurate.The dispersion of the spin excitation has a hourglassshape. The overall features are qualitatively the same with those in superconducting cuprate materials. Wepropose that the mono-layer graphene beyond the VHSfilling are candidate to become cuprate analogs. All ofthe results can be used to identify the unconventionalsuperconductivity in this compound.This work was supported by the NSFC (Grant No.12074130) and Science and Technology Program ofGuangzhou (Grant No. 2019050001). [1] P. A. Lee, N. Nagaosa, and X.-G. Wen, Doping a mott in-sulator: Physics of high-temperature superconductivity,Rev. Mod. Phys. , 17 (2006).[2] G. R. Stewart, Superconductivity in iron compounds,Rev. Mod. Phys. , 1589 (2011).[3] D. Li, K. Lee, B. Y. Wang, M. Osada, S. Cross-ley, H. R. Lee, Y. Cui, Y. Hikita, and H. 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