Proximity induced interface bound states in superconductor-graphene junctions
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Proximity induced interface bound states in superconductor-graphene junctions
P. Burset , W. Herrera and A. Levy Yeyati Departamento de F´ısica Te´orica de la Materia Condensada C-V,Facultad de Ciencias, Universidad Aut´onoma de Madrid, E-28049 Madrid, Spain Departamento de F´ısica, Universidad Nacional de Colombia, Bogot´a, Colombia (Dated: December 4, 2018)We show that interface bound states are formed at isolated graphene-superconductor junctions.These states arise due to the interplay of virtual Andreev and normal reflections taking place atthese interfaces. Simple analytical expressions for their dispersion are obtained considering interfacesformed along armchair or zig-zag edges. It is shown that the states are sensitive to a supercurrentflowing on the superconducting electrode. The states provide long range superconducting correla-tions on the graphene layer which may be exploited for the detection of crossed Andreev processes.
PACS numbers: 73.23.-b, 74.45.+c, 74.78.Na, 73.20.-r
Introduction:
Several striking transport propertieshave been predicted to emerge from the peculiar elec-tronic band structure of single atom graphite layers,known as graphene [1]. Of particular interest is the caseof a graphene layer in contact with a superconductingelectrode, a situation which has been explored in severalrecent experiments [2]. Here, as in the case of normalmetals, the mechanism which dominates the electronictransport at subgap energies is the Andreev reflection,i.e. the conversion of incident electrons into reflectedholes with the creation of Cooper pairs in the supercon-ductor. However, while in the case of normal metals thereflected hole has typically the opposite mean velocityto the incident electron ( retroreflection ) in the case ofgraphene it is possible to have Andreev reflections witha specular character, as first shown in [3].Transport and electronic properties at graphene-superconductor junctions have been analyzed in severalworks [4]. It has been shown that the special character ofAndreev reflection in graphene leads to modifications inthe differential conductance compared to that of conven-tional N-S junctions [3, 5]. The effect on the local densityof states (LDOS) has been studied in Refs. [6, 7]. Theproperties of bound states arising from multiple Andreevreflections in S-graphene-S junctions have been analyzedin Refs. [8]. However, as we show in this work, the spe-cial electronic properties of graphene are such that boundstates can be formed even at isolated single junctions.The mechanism for the emergence of these states canbe understood from the scheme depicted in the left panelof Fig. 1. As is usually assumed the junction can bemodeled as an abrupt discontinuity between two regionsdescribed by the Bogoliubov-de Gennes-Dirac equation,taking a finite superconducting order parameter ∆ andlarge doping E SF ≫ ∆ on the superconducting side andzero order parameter and small doping E F ∼ ∆ on the normal side. For the analysis it is instructive to includean artificial intermediate normal region with ∆ = 0 and E IF = E SF , whose width, d , can be taken to zero at theend of the calculation. This intermediate region allows FIG. 1: (Color online) Simple model for the emergence ofIBSs (left panel). It illustrates the scattering processes tak-ing place at a graphene-superconductor interface with an in-termediate heavily doped normal graphene region of width d .Cases (i) and (ii) correspond to the case ~ vq < | E ± E F | and ~ vq > | E ± E F | respectively with ∆ > E > E F . (Right panel)Graphene-superconductor junctions along different edges. Onthe superconducting side (shaded areas) the on-site order pa-rameter ∆ is finite and the doping level is high ( E SF ≫ ∆). to spatially separate normal reflection due to the Fermienergy mismatch from the Andreev reflection associatedto the jump in ∆. As shown in Fig. 1 (case i), an incidentelectron from the normal side with energy E and parallelmomentum ~ q such that ~ vq < | E − E F | is partially trans-mitted into the intermediate region and after a sequenceof normal and Andreev reflections would be reflected as ahole. This process can either correspond to retro or spec-ular Andreev reflection depending on whether E < E F or E > E F [3]. For ~ vq ≥ | E ± E F | neither electronor holes can propagate within the graphene normal re-gion. However, virtual processes like the one depicted inFig. 1 (case ii) would be present. These correspond tosequences of Andreev and normal reflections within theintermediate region. A bound state emerges when thetotal phase φ accumulated in such processes reach theresonance condition φ = 2 nπ .The aim of the present work is to demonstrate theexistence of these interface bound states (IBS) and toanalyze their properties for different types of graphene-superconductor junctions. After completing the anal-ysis for the case of the simple model sketched above,which implicitly assumes a decoupling of the two val-leys in the graphene band structure, we consider moremicroscopic models for junctions formed along armchairor zig-zag edges. We study the effect of an additionalpotential barrier at the interface and the possibility tomodify the states by a supercurrent flowing through thesuperconductor. We finally discuss the potential use ofthese states for the generation of non-locally entangledAndreev pairs.It is quite straightforward to determine the dispersionrelation for the IBS from the model represented in the leftpanel of Fig. 1. The phase accumulated by a sequenceof normal and Andreev reflections in the intermediate re-gion can be obtained from the corresponding coefficients r e , r h and r A . Following Ref. [3] one obtains r e,h = e iα Ie,h e − iα Ie,h − e − iα e,h e iα Ie,h + e − iα e,h , (1)where α ( I ) e,h = arcsin ~ vq/ ( E ± E ( I ) F ). The condition E IF ≫ ∆ , E, ~ vq allows to take α Ie,h ≃
0. On the otherhand, in the region of evanescent electron and hole statesfor graphene ( | ~ vq | > | E ± E F | ) r e,h become a pure phasefactor e iϕ e,h , with ϕ e,h = − q/ ( E ± E F )) arctan e λ e,h and λ e,h = sign( q )arcosh( ~ vq/ | E ± E F | ). For the An-dreev reflection coefficient between regions I and S onehas r A = e iϕ A where ϕ A = arccos E/ ∆, as it correspondsto the Andreev reflection at an ideal N-S interface with E SF ≫ ∆ [9]. In the limit d → φ = 2 ϕ A + ϕ e + ϕ h , from which one obtainsthe following dispersion relation E ∆ = ± e ( λ e + λ h ) / − sign( E − E F ) e − ( λ e + λ h ) / √ cosh λ e cosh λ h . (2)This dispersion simplifies to E/ ∆ = ± ~ vq/ p ( ~ vq ) + ∆ at the charge neutrality point(i.e. for E F = 0). In this case the IBS approacheszero energy for q → q . Notice also thatthe decay of the states into the graphene bulk region( x < e x/ξ e,h ,where ξ e,h = ~ v/ ( | E ± E F | sinh( λ e,h )) for the electronand hole components respectively, which can be clearlymuch larger than the superconducting coherence length ξ = ~ v/ ∆ when E F ≪ ∆. It is also interesting to noticethat the IBSs survive when E F > ∆, i.e. in the regimecorresponding to the usual Andreev retroreflection, butwith a much smaller spatial extension. In order to analyze the existence and the charac-teristics of the IBSs for different types of graphene-superconductor junctions we make use of the Green func-tion formalism based on tight-binding models for thesejunctions which was introduced in Ref. [7]. Within thisformalism the retarded green functions at the interfaceˇˆ G ( E, q ) are given by h ˇˆ g − − ˇˆΣ i − , where ˇˆ g correspondsto the surface of the uncoupled semi-infinite graphenelayer and ˇˆΣ is the self-energy associated to the couplingwith the superconductor. In general all these quantitieshave a 2 × h ˇˆ g − − ˇˆΣ i = 0. Interface along an armchair edge:
We first consideran interface constructed along an armchair edge, asschematically depicted in the right panel of Fig. 1. Ina rather generic way one can write ˇˆ g = ˆ g e (ˇ τ + ˇ τ z ) / g h (ˇ τ − ˇ τ z ) / /t g = β ˇ τ z ˇ g BCS ˇ τ z + γ ˇ τ z ˆ σ x , whereˆ g e,h describe the propagation of e and h components inthe uncoupled graphene layer, ˇ g BCS = g ˇ τ + f ˇ τ x with g = − Ef / ∆ = − E/ √ ∆ − E being the BCS dimen-sionless Green functions, and β and γ are parameterswhich allow to control the transparency and the type ofinterface. As discussed in Ref. [7] γ = 0 correspondsto a model in which the coherence between the sublat-tices of graphene is broken on the superconducting side(bulk-BCS model), whereas for β = √ / γ = 1 / E, ∆ , ~ vq ≪ t g in ˆ g e,h of Ref. [7], where t g denotesthe hopping element between neighboring sites in thegraphene layer. In this case and for ~ vq > | E ± E F | , ˆ g e,h adopt the form t g ˆ g e,h = − (cid:2) √ µ e,h ˆ σ + ν e,h ˆ σ y ) ± ˆ σ x (cid:3) ,where µ e,h = sign( q ) / sinh λ e,h and ν e,h = sign( E ± E F ) / tanh λ e,h . The Green functions matrix has theproperty ˆ g − e,h = − t g ˆ g Te,h ∓ t g σ x . Using this property andthe definition for the self-energy the equation for the IBSsin this case becomesdet (cid:2) t g ˆ g h ˆ g e + βgt g (ˆ g e + ˆ g h ) + γt g (ˆ σ x ˆ g e − ˆ g h ˆ σ x ) − ( β + γ ) (cid:3) = 0 . (3)For the HDSC model (i.e. β = √ / γ = 1 /
2) theequation for the IBSs reduce to the one already foundwithin the simple analytical model (Eq. (2)). This leadsto a single root for arbitrary doping which is four-fold de-generate due to valley and spin symmetry. Fig. 2 showsa color-scale plot of the spectral density at a distance s pe c t r a l den s i t y ( a r b . un i t s ) -4 -2 0 2 4 PSfragreplacements E / ∆ ¯ hvq/ ∆¯ hvq/ ∆ E F FIG. 2: Gray-scale plot of the spectral density at a distance ∼ ξ from the interface defined along an armchair edge. Theresults were obtained using the HDSC model of Ref. [7] for E F = 0 (left panel) and E F = ∆ / ∼ ξ from the interface on the graphene layer with twodifferent doping conditions. The full lines correspond tothe IBS dispersion obtained by solving Eq. (2). As canbe observed, the minimal energy for the IBSs, E min , de-pends on E F . Further analysis of Eq. (2) reveals that itsatisfies the cubic equation E min + E min E F − ∆ E F = 0,thus evolving between 0 and ∆ as E F increases. Thetransition between E min > E F and E min < E F occursat E F = ∆ / √
2. The presence of the IBSs manifests alsoin the appearance of singularities in the LDOS around E = ± ∆ (see Ref. [7]). The behavior of the LDOS isanalyzed in more detail below.On the other hand, for the bulk-BCS model (i.e. γ = 0and β ∈ (0 , β g ( µ e µ h + ν e ν h −
1) + √ βg ( µ e + µ h )(1 + β )+ β (1 + 2 β )2 + 34 (1 + 2 β )( ν e ν h − µ e µ h ) + 14 = 0 . (4)In this case the degeneracy associated to the two val-leys in the band-structure of graphene is generally broken(except for E F = 0). The roots gradually evolve towardsthe linear dispersion | E + E F | = ~ vq as β →
0, whichcorresponds to the armchair edge state of the isolatedgraphene layer [10].
Interface along a zig-zag edge:
We now consider aninterface along a zig-zag edge as illustrated in the rightpanel of Fig. 1. The Green functions for the semi-infinitezig-zag edge can be obtained following the same formal-ism as in Ref. [7]. In the continuous limit ˆ g e,h becomes t g ˆ g e,h = (cid:18) ie − iα e,h ∓ e iπ/ ∓ e − iπ/ (cid:19) , (5)where as in Eq. (1) sin α e,h = ( ~ vq ) / ( E ± E F ) but with q measured with respect to the point K = 2 π/ a , where a is the lattice constant indicated in the right panel of Fig.1. There exists an additional branch where q is measuredfrom the opposite Dirac point at − K . The self-energy β = 1 β = 0.75 β = 0.5 β = 0.25 β = 0.1 PSfragreplacements E / ∆ K − K FIG. 3: (Color online) Dispersion relation for the IBSs on azig-zag interface for decreasing parameter β controlling thecoupling with the superconductor. The parallel momentum q in Eq. (6) is measured from the Dirac points at K = ± π/ a . due to the coupling with the superconductor is in thiscase ˇˆΣ = βt g (ˆ σ + ˆ σ z ) (ˇ τ z ˇ g BCS ˇ τ z ) /
2. The equation forthe IBSs then becomes E ∆ = ± e ( λ e + λ h ) / − sign( E − E F ) β e − ( λ e + λ h ) / p ( e λ e + β e − λ e )( e λ h + β e − λ h ) , (6)which looks very similar to Eq.(2) except for the presenceof the parameter β controlling the coupling and the al-ready mentioned redefinition of the parallel momentum q .An interesting property of zig-zag edges is the presence ofzero energy states for total parallel momentum between( − K , K ) and E F = 0 [11]. When the coupling to the su-perconductor is turned on by increasing the parameter β ,one observes that the zero energy states evolve acquiringa finite slope. These states can thus be identified with theIBS for this type of interface. This is illustrated in Fig.3. When the coupling parameter β reaches 1 the usualdispersion of the simplest analytical model is recovered. Effect of a supercurrent: a supercurrent flowing on thesuperconducting side of the junction modifies the spa-tial variation of the phase of the order parameter whichproduces a Doppler shift in the energy of the quasi-particles. This shift, obtained from the Bogoliubov-deGennes-Dirac equations within the Andreev approxima-tion, is given by η = ( ~ v ) q s q/E SF , where ~ q s is the mo-mentum of the Cooper pairs assumed to be parallel tothe interface. This result is equivalent to the one foundin Refs. [12] for conventional and two-band superconduc-tors. Notice that for this analysis we go beyond the limit E SF → ∞ taken in the initial simple model. The expres-sion for the reflection coefficients of Eq. (1) still holdsbut α Ie ≃ − α Ih ≡ α I = arcsin ~ vq/E SF is kept finite. Onthe other hand, the phase of the Andreev reflection coef-ficient between the intermediate region and the current-carrying superconductor becomes ϕ A = arccos E ′ / ∆( q s ),where E ′ = E + η . At zero temperature, due to Lan-dau criterion, the order parameter is unaffected by thesupercurrent while ~ vq s . ∆ (0) [12]. Therefore, in thiscondition, ∆( q s ) ≃ ∆(0) ≡ ∆. For the case E > E F we thus get the following modified equation for the IBSs PSfragreplacements E / ∆ ¯ hvq/ ∆ ρ ( E ) / ρ E/ ∆ FIG. 4: (Color online) Effect of a supercurrent flowing onthe superconducting electrode on the dispersion relation (leftpanel) and on the local density of states at a distance ∼ ξ / ρ = ∆( a/ ~ v ) / π (rightpanel). The results correspond to ~ vq s / ∆ = 0 . , . , . .
75 with E SF = 100∆. within the simple model sketched in Fig. 1 E ′ ∆ = ± sinh ( λ e + λ h ) / α I sinh ( λ e − λ h ) / p (cosh λ e − sin α I )(cosh λ h + sin α I ) . (7)Figure 4 illustrates the effect of a supercurrent both inthe dispersion relation of the IBS (left panel) and in thelocal density of states close to the interface (right panel).For q s = 0, the IBS manifest in a finite LDOS for E < ∆and a sharp peak at E = ∆. Qualitatively, the presenceof a supercurrent breaks the symmetry with respect toinversion of the parallel momentum ~ q and leads to asplitting of the singularity at E ≃ ∆ in the LDOS. Notethat this implies the appearance of an induced net cur-rent on the graphene side (for | x | . ξ ). For E F = 0 thedistortion of the dispersion relation for finite and small q s is given by E ( q s , q ) = E (0 , q ) + ( ~ vq ) η/ (( ~ vq ) + ∆ ).A more quantitative analysis of the effect of a super-current requires the estimation of the parameter E SF .This parameter is very much dependent on the fabri-cation methods and material properties of the metallicelectrodes deposited on top of the graphene layer. Ac-cording to the ab-initio calculations of Ref. [13] for Pd ongraphene a typical estimate would be E SF ∼ . eV , whichfor a superconductor like Nb gives a ratio E SF / ∆ ∼ Conclusions:
We have shown that interface boundstates appear at graphene-superconductor junctions.The properties of these states are sensitive to the typeof edge forming the interface, its transparency and thedoping conditions of the graphene layer. We have demon-strated that the interface states evolve towards the edgestates of the isolated graphene layer when the trans-parency of the interface is reduced. We have also shownthat they can be modulated by a supercurrent flowingthrough the superconductor in the direction parallel tothe interface. Even when our analysis has been restrictedto interfaces along armchair or zig-zag edges we expectthe appearance of IBSs to be a general property of anyedge orientation. We also notice that inclusion of weak disorder along the interface introducing a small uncer-tainty in the parallel momentum δq would not preventthe emergence of IBSs provided that δq ≪ ∆ / ~ v .As a final remark we would like to comment that theexistence of IBSs induce long range superconducting cor-relations between distant points on the graphene layerthat are close to the interface. This property could beexploited to detect crossed Andreev processes and there-fore entangled electron pairs using weakly coupled STMprobes on a graphene-superconductor junction, in a con-figuration like the one proposed in Ref. [14]. The analysisof non-local transport in this system will be the objectof a separate work.The authors would like to thank correspondence withC.W.J. Beenakker and useful discussions with J.C.Cuevas and A. Mart´ın-Rodero. Financial support fromSpanish MICINN under contracts FIS2005-06255 andFIS2008-04209, by DIB from Universidad Nacional deColombia, and by EULA-Nanoforum is acknowledged. [1] For a review see A. Geim and K.S. Novoselov, NatureMat. , 183 (2007).[2] H. B. Heersche et al. , Nature (London) , 56 (2007); A.Shailos et al. , Europhysics Letters , 57008 (2007); F.Miao et al. , Science , 1530 (2007); X. Du, I. Skachkoand E. Andrei, Phys. Rev. B , 184507 (2008).[3] C.W.J. Beenakker, Phys. Rev. Lett. , 067007 (2006).[4] J. C. Cuevas and A. Levy Yeyati, Phys. Rev. B , 241403 (2006); A. Ossipov, M. Titov and C. W.J. Beenakker, Phys. Rev. B , 241401 (2007); J. Linderand A. Sudbo, Phys. Rev. Lett. , 147001 (2007); J.Cayssol, Phys. Rev. Lett. , 147001 (2008); D. Rainis et al. , Phys. Rev. B , 115131 (2009); C. Benjamin andJ.K. Pachos, Phys. Rev. B , 155431 (2009).[5] S. Bhattacharjee and K. Sengupta, Phys. Rev. Lett. ,217001 (2006).[6] G. Tkachov, Phys. Rev. B , 235409 (2007); A.M.Black-Schaffer and S. Doniach, Phys. Rev. B , 024504(2008).[7] P. Burset, A. Levy Yeyati and A. Mart´ın-Rodero, Phys.Rev. B 77, 205425 (2008).[8] M. Titov, A. Ossipov and C.W.J. Beenakker, Phys. Rev.B , 045417 (2007); D.L. Bergman and K.Le Hur,arXiv:0806.0379.[9] G.E. Blonder, M. Tinkham and T.M. Klapwijk, Phys.Rev. B , 4515 (1982).[10] K. Sengupta, R. Roy and M. Maiti, Phys. Rev. B ,094505 (2006).[11] L. Brey and H.A. Fertig, Phys. Rev. B , 235411 (2006).[12] D. Zhang, C.S. Ting and C.-R. Hu, Phys. Rev. B ,172508 (2004); V. Lukic and E.J. Nicol, Phys. Rev. B , 144508 (2007).[13] N. Nemec, D. Tom´anek and G. Cuniberti, Phys. Rev. B , 125420 (2008).[14] J.M. Byers and M.E. Flatt´e, Phys. Rev. Lett.74