Critical currents in graphene Josephson junctions
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Critical currents in graphene Josephson junctions
J. Gonz´alez and E. Perfetto Instituto de Estructura de la Materia. Consejo Superior de Investigaciones Cient´ıficas. Serrano 123, 28006 Madrid. Spain. Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia,Universit`a di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy. (Dated: November 15, 2018)We study the superconducting correlations induced in graphene when it is placed between twosuperconductors, focusing in particular on the supercurrents supported by the 2D system. Forthis purpose we make use of a formalism placing the emphasis on the many-body aspects of theproblem, with the aim of investigating the dependence of the critical currents on relevant variableslike the distance L between the superconducting contacts, the temperature, and the doping level.Thus we show that, despite the vanishing density of states at the Fermi level in undoped graphene,supercurrents may exist at zero temperature with a natural 1 /L dependence at large L . Whentemperature effects are taken into account, the supercurrents are further suppressed beyond thethermal length L T ( ∼ v F /k B T , in terms of the Fermi velocity v F of graphene), entering a regimewhere the decay is given by a 1 /L dependence. On the other hand, the supercurrents can beenhanced upon doping, as the Fermi level is shifted by a chemical potential µ from the chargeneutrality point. This introduces a new crossover length L ∗ ∼ v F /µ , at which the effects of thefinite charge density start being felt, marking the transition from the short-distance 1 /L behaviorto a softer 1 /L decay of the supercurrents at large L . It turns out that the decay of the criticalcurrents is given in general by a power-law behavior, which can be seen as a consequence of theperfect scaling of the Dirac theory applied to the low-energy description of graphene. I. INTRODUCTION
Since the discovery of single atomic layers of carbon in 2004 , this new two-dimensional (2D) material (so-calledgraphene) has attracted a lot of attention . From the experimental point of view, the 2D carbon sheets have showna number of remarkable electronic properties. Thus, there has been evidence that graphene may have a finite lowerbound (4 e /h ) in the conductivity at the charge neutrality point . Furthermore, an anomalous integer quantumHall effect has been measured in the 2D system with plateaus at odd-integer values of the quantum of conductance .The absence of weak localization effects has also pointed at the unconventional effects that impurities and in generaldisorder may produce in the graphene sheet.Most of the remarkable transport properties of graphene have to do with its particular band structure at low energies.The undoped system has a finite number of Fermi points, placed at the corners of the hexagonal Brillouin zone. Onlytwo of such points can be taken as independent, with quasiparticle excitations which have conical dispersion above andbelow the Fermi level . This explains that the low-energy electronic states of graphene may be accommodated intotwo two-component spinor fields, governed by a Dirac hamiltonian which leads to a dispersion relation ε ( k ) = ± v F | k | .The electronic system displays hence a relativistic-like invariance at low energies, which is at the origin of the finitelower bound in the conductivity , the anomalous integer Hall effect , and the absence of backscattering inthe presence of long-range scatterers . Other exotic effects relying on the Dirac theory have been proposed, likethe selective transmission of electrons through a n - p junction or the specular Andreev reflection at a graphene-superconductor interface .Recently, the properties of graphene have been also investigated when the material is placed between superconduct-ing contacts. Thus, in the experiment reported in Ref. 16, it has been possible to measure supercurrents in grapheneby attaching wide superconducting electrodes with a spatial separation of ≈ . µ m. In another experiment, reportedin Ref. 17, a quite different geometry has been investigated by placing thin electrodes across a large 2D sample, witha minimum separation between the tips of ≈ . µ m. In this case, the evidence of the superconducting correlations ingraphene has been obtained in the form of Andreev reflection peaks in the I - V curves, as well as in the abrupt drop ofthe resistance at a temperature of ≈ ≈ ∼
10) between the width of the junction and the lead separation (ofthe order of a few hundreds of nanometers).It is therefore pertinent to study the way in which the superconducting correlations are induced in graphene whenit is placed between two superconductors, and how such correlations may depend on the geometry of the experimentalsetup. In this paper we are going to address this issue, focusing in particular on the supercurrents supported by thegraphene sheet. We will be using a formalism placing the emphasis on the many-body aspects of the problem. Thiswill allow us to clarify a number of questions, regarding the dependence of the critical currents on relevant variableslike the distance between the superconducting contacts, the temperature, and the doping level of the graphene sample.In this respect, our approach can be seen as complementary to that of Ref. 19, where the Josephson effect has beenstudied in terms of Andreev reflection at superconducting contacts, concentrating on junctions with relatively shortdistance between the electrodes. We will be dealing with a framework where the tunneling and propagation of theCooper pairs in graphene play the central role, placing in principle no restriction on the separation that may existbetween superconducting contacts.The content of this paper is distributed as follows. We will set up in section II the formalism needed to describe thetunneling and propagation of Cooper pairs in graphene. This will be applied to the computation of the critical currentsin section III, where we will also discuss the different regimes depending on the interplay between the temperatureand the distance between superconducting contacts. Section IV will be devoted to extend our analysis to the case offinite doping, showing the enhancement experienced then by the supercurrents. Finally, we will summarize our resultsand draw our conclusions in section V.
II. MODEL OF GRAPHENE JOSEPHSON JUNCTION
Our purpose is to build a model that incorporates the low-energy properties of electron quasiparticles in grapheneas well as the tunneling of electrons from graphene to the superconducting electrodes and vice versa. We take intoaccount in particular that, below an energy scale of ∼ ε on momentum k given by ε ( k ) ≈ ± v F | k | . We have to bear in mind that the 2D systemhas actually two independent Fermi points supporting such a conical dispersion, at opposite corners K, − K of thehexagonal Brillouin zone. The dynamics of the quasiparticles in graphene can be therefore described in terms of acouple of two-component Dirac spinors Ψ ( a ) , a = 1 ,
2, with a hamiltonian H = v F Z d r Ψ ( a ) † σ ( r ) σ ( a ) · ∂ Ψ ( a ) σ ( r ) (1)where { σ ( a ) } are two different suitable sets of Pauli matrices (we use units such that ¯ h = 1). In the above expression,the label of the spinor components is omitted for simplicity, and a sum is taken implicitly over the spin index σ aswell as over the index a running over the two different low-energy valleys of the dispersion.The above hamiltonian has to be then complemented with a term accounting for the tunneling of electrons fromthe graphene side to the superconducting electrodes and vice versa. In this respect, we are going to assume that thetunneling takes place with equal amplitude for the two sublattices of the graphene honeycomb lattice. This kind ofjunction may be realized in cases where the contacts between graphene and the superconductors preserve the structureof the graphene lattice. From a technical point of view, such condition implies that the different spinor componentsand the different low-energy valleys couple with equal amplitude to the superconductors. By denoting the electronfields in the respective superconducting electrodes by Ψ S and Ψ S , we may write the tunneling hamiltonian forcontacts along the coordinates x = 0 and x = L as H t = X j =1 , t Z W dy Ψ ( a ) † σ ( x j , y )Ψ Sj,σ ( x j , y ) + h . c . (2)where the parameter t represents the tunneling amplitude. We stress at this point that, while the contacts have awidth given by W in Eq. (2), the extension of the graphene layer along the transverse y direction is not constrained bythis parameter in our model. Thus, our description will apply in general to 2D graphene samples, with dimensions inboth the transverse and the longitudinal direction much larger than the contacts introduced by the superconductingelectrodes.The properties of the superconducting electrodes have to be also incorporated in the model of the Josephsonjunction. For the description of the supercurrents, it will be enough to specify the normal density of states ρ and theorder parameter ∆ in the superconducting state. We recall that a supercurrent arises in general from a gradient in thephase of the order parameter in a superconductor. In the case of a Josephson junction, the supercurrent is producedby a mismatch in the phases χ and χ of the respective order parameters in the superconducting electrodes. TheJosephson current I s is actually given by the derivative of the free energy with respect to the variable χ = χ − χ ,and it can be therefore expressed as I s = 2 e ∂∂χ k B T log (cid:16) Tr e − H/k B T (cid:17) (3)where T is the temperature and H stands for the full hamiltonian of the model.In order to compute the Josephson current from Eq. (3), we will resort to a perturbative expansion in the tunnelingamplitude t . The structure of the dominant contributions may be however very different depending on the actualgeometry of the Josephson junction . In cases where the distance L between the contacts is much smaller thanthe superconducting coherence length ξ , the supercurrents are built from processes with independent tunneling anduncorrelated propagation in graphene of the electrons of a Cooper pair. On the other hand, when L is much largerthan ξ , the behavior is governed by the fast tunneling and subsequent propagation of the Cooper pair in graphene,as shown schematically in Fig. 1. This situation corresponds to the case where the time of propagation between thecontacts is much larger than 1 / | ∆ | . Under the assumption of a large | ∆ | , the relevant properties of the superconductorsmay be encoded in the statistical average h Ψ Sj,σ ( x j , y ; − iτ )Ψ Sj, − σ ( x j , y ; − iτ ) i ≈ e iχ j ρ δ ( τ − τ ) (4)where the operators are ordered with respect to imaginary time τ . t t t L tt
SCgrapheneSC t FIG. 1: Schematic representation of the propagation of Cooper pairs in graphene between two superconductors (SC).
From inspection of the expansion of the r.h.s. in Eq. (3) in powers of the tunneling amplitude, we observe that thefirst nonvanishing contribution to I s appears to fourth order in t , from a statistical average of operators participatingof the condensates of the two superconductors. The expression of the maximum supercurrent I c (critical current) isworked out at that perturbative level in the Appendix, focusing on the regime corresponding to L ≫ ξ . After factoringout the relative tunnel conductances at the contacts (given in each case by the dimensionless quantity ρt W/v F ), weend up with an expression for the behavior of the critical current intrinsic to the 2D graphene layer: I (2 D ) c ( T ) ≈ ev F Z W dy Z W dy Z /k B T dτ h Ψ ( a ) †↑ (0 , y ; 0)Ψ ( − a ) †↓ (0 , y ; 0)Ψ ( b ) ↑ ( L, y ; − iτ )Ψ ( − b ) ↓ ( L, y ; − iτ ) i (5)We observe from (5) that the propagator of the Cooper pairs evaluated over a distance L plays the central role in thedetermination of the supercurrents. We will study in what follows the behavior of this propagator depending on thedistance L , the temperature, and the doping level. III. SUPERCURRENTS AT FINITE TEMPERATURE
We analyze first the behavior of the supercurrents in graphene when the system is undoped, but is placed at anonvanishing temperature T . The expectation values in the above formulas have to be understood then as statisticalaverages at that finite temperature. The building block for all the calculations is the electron propagator G ( a ) ( r , t ) = − i h T Ψ ( a ) σ (0 , ( a ) † σ ( r , t ) i (6)This is given in graphene by the propagator for Dirac fermions, in correspondence with the hamiltonian (1). In themany-body theory at temperature T = 0, the imaginary part of that object gets a specific term to account for thethermal effects. The full expression of the Dirac propagator becomes in momentum space G ( a ) ( p , ω p ) = ω p + σ ( a ) · p ω p − p + iǫ + i π ( ω p + σ ( a ) · p ) δ ( − ω p + p ) 11 + e | ω p | /k B T (7)The Cooper-pair propagator in (5) can be computed from the convolution of two Dirac propagators, bearing in mindthat they correspond to fields at opposite valleys of the graphene dispersion. In doing this operation, we will have tobe also consistent with our assumption that the tunneling at the superconducting contacts is the same for the twosublattices of the graphene lattice. This means that, when taking the average for the Cooper-pair propagator, we willalso take a trace in spinor space over the states of the Cooper pairs in sublattice A , given by Ψ ( a ) A, ↑ ( k + q )Ψ ( − a ) A, ↓ ( − q ),and in sublattice B , given by Ψ ( a ) B, ↑ ( k + q )Ψ ( − a ) B, ↓ ( − q ). The Cooper-pair propagator thus defined in momentum space, D ( k , ω ), can be expressed as D ( k , ω k ) = i Tr Z dω q π Z d q (2 π ) G ( a ) ( q + k , ω q + ω k ) G ( − a ) ( − q , − ω q ) (8)A nice feature of the diagrammatics of the many-body theory at T = 0 is that the terms carrying the dependenceon temperature do not need to be regularized by means of a high energy cutoff. The contributions at T = 0, however,remain finite only when the integrals over the momenta are suitably cut off. In the present model, it is convenientto choose a method of regularization of the integrals preserving the relativistic-like invariance of the theory. For thispurpose, we will adopt an analytic continuation in the number of space-time dimensions , that is, carrying out firstthe integrals at general dimension D , and then taking the limit D →
3. To implement this procedure, we first collectthe components of the momentum and the frequency to form 3D vectors, q ≡ ( v F q , ω q ), k ≡ ( v F k , ω k ). Next, wemay rotate all the 3D vectors to Euclidean space by introducing imaginary frequencies, ω q = − iω q . One can easilysee that the expression of the propagator (8) at general dimension D becomes D ( k , iω k ) | T =0 = Z dx Z d D q (2 π ) D q − k x (1 − x )( q + k x (1 − x )) = (cid:18) π / Γ (cid:18) − D (cid:19) − π / Γ (cid:18) − D (cid:19)(cid:19) Z dx p k x (1 − x ) (9)In the last passage we have made use of standard formulas in dimensional regularization. Quite remarkably, the resultturns out to be finite in the limit D →
3. After reverting the rotation back to real frequency, we finally get D ( k , ω ) | T =0 = − v F q v F k − ω (10)The part of the Cooper-pair propagator depending on temperature can be computed by using the second term in (7)to make the convolution (8). For our purposes, we can concentrate on the calculation of the Cooper-pair propagatorat zero frequency. By adding the result (10) to the temperature-dependent contribution, we get D ( k ,
0) = − v F | k | − log(2) πv F k B T + 12 πv F | k | Z dx √ − x
11 + e xv F | k | / k B T (11)From the results (10) and (11), we can already extract a number of conclusions regarding the behavior of thesupercurrents in long graphene Josephson junctions. From Eq. (5), we can express the critical current for L ≫ W as I (2 D ) c ( T ) ≈ ev F W Z ∞ dk π | k | J ( | k | L ) D ( k , e −| k | /k c (12)A short distance cutoff k c has been introduced to regularize the integral over the momentum. This is actually justifiedon physical grounds, since the description of graphene as a continuum in terms of the Dirac theory makes sense atdistances above the nanometer scale. A sensible choice corresponds to v F k c ∼ L ≫ k − c , the behavior of the critical current is in general not sensitive to the actual value of the cutoff.At T = 0, the dependence of the critical current on L can be obtained from the Cooper-pair propagator (10).Actually, we can derive an analytical expression for I (2 D ) c (0) by computing the integral in (12): I (2 D ) c (0) ∼ − ev F W Z ∞ dk | k | J ( | k | L ) e −| k | /k c (13)= ev F W k c ( k c L − p ( k c L + 1) (14)(15)From this result we check that, as expected, the behavior of the critical current is not affected by the cutoff k c in thelimit of large L . In this regime we find I (2 D ) c (0) ∼ ev F W L (16) H Μ m L ´ - ´ - c FIG. 2: Logarithmic plot of the critical current I (2 D ) c (in units of 10 − ev F k c ≈ . µ A) as a function of the distance L , taking W = 10 /k c (= 50 nm). The three curves correpond, from top to bottom, to different values of the temperature T = 2 K, 4 K,and 8 K. The strong power-law decay shown by (16) can be understood actually as a reflection of the linear dependence onmomentum of the quasiparticle energy, which dictates in turn the behavior of the Cooper-pair propagator (10) . Wereach anyhow the interesting conclusion that, while graphene has a vanishing density of states at the Dirac point, itmay still support a nonvanishing supercurrent when the Fermi level is at that charge neutrality point.The inspection of the full propagator (11) also reveals that the scaling is drastically modified when k B T ≫ v F | k | .Actually, we can distinguish between a high-temperature and a low-temperature regime of the Cooper-pair propagator,with quite different behaviors: D ( k , ≈ − v F | k | if k B T ≪ v F | k | (17) ≈ − log(2) πv F k B T − π | k | k B T if k B T ≫ v F | k | (18)The existence of this crossover in the momentum gives rise to an abrupt decay of the supercurrent beyond the thermallength L T = v F /k B T . This is illustrated in Fig. 2, where the critical current I (2 D ) c ( T ) is represented as a function ofthe distance L at different temperatures. We observe for instance that, for a temperature of the order of T ∼ L is of the order of a few microns, in agreement with the expression of the thermal length.From a physical point of view, it becomes clear that the Cooper pairs do not feel the thermal effects during theirpropagation when L is shorter than the scale given by L T , while they are increasingly disrupted at distances largerthan the thermal length. At short distances such that L ≪ v F /k B T , the decay of the critical current represented inFig. 2 follows a 1 /L power-law, in agreement with the above analysis at T = 0. However, beyond the crossover clearlyidentified in the three curves, we see that a different power-law behavior opens up at long distance L ≫ v F /k B T .This regime can be analyzed by considering that, when T is very large, the second term in the approximation (18)dictates the long-distance decay of the critical current. In this case we can compute again analytically the integral in(12): I (2 D ) c ( T ) ∼ − ev F W k B T Z ∞ dk | k | J ( | k | L ) e −| k | /k c (19)= ev F W v F k B T k c (9 k c L − p ( k c L + 1) (20)The leading contribution to the critical current becomes then for L ≫ v F /k B TI (2 D ) c ( T ) ∼ ev F W v F k c k B T L (21)The existence of this stronger power-law decay is manifest in the results of the numerical computation of the criticalcurrent represented in Fig. 2, as it can be checked that the rightmost part of the lower curves in the plot correspondswith great accuracy to a power-law behavior with the exponent given by Eq. (21). H K L c H K L ´ - ´ - ´ - ´ - ´ - I c H K L ´ - ´ - ´ - ´ - ´ - ´ - I c (a) (b) (c)FIG. 3: Plot of the critical current I (2 D ) c ( T ) (in units of 10 − ev F k c ≈ . µ A) as a function of the temperature, for W = 10 /k c (= 50 nm) and a spatial separation between superconducting contacts L = 0 . µ m (a), 1 . µ m (b), and 2 . µ m (c). (cid:144) T * c (cid:144) I c H L FIG. 4: Combined plot of I (2 D ) c ( T ) /I (2 D ) c (0) represented as a function of the scaled variable T /T ∗ (with T ∗ = v F / L ), whereit is seen the collapse of the three curves corresponding to values of the distance L = 0 . µ m (full line), 1 . µ m (dotted line),and 2 . µ m (dashed line). In order to establish a comparison with experimental results, the relevant behavior is given by the critical currentrepresented as as function of the temperature at fixed length L . The existence of a thermal length has a reflectionhere in the form of a crossover temperature T ∗ , which marks the strong decay of the critical current for T > T ∗ . Wehave plotted in Fig. 3 the critical current I (2 D ) c ( T ), computed from Eq. (12), at different values of L between 0 . µ mand 2 . µ m. The shapes of the curves in the figure are quite similar, and it can be checked that they can be collapsedinto a single universal curve after rescaling the temperature by T ∗ ∝ v F /k B L , as shown in Fig. 4. This is consistentwith the expression of the critical current in Eq. (12), where it it seen that the effect of a variation of the length L on I (2 D ) c ( T ) /I (2 D ) c (0) can be compensated by a suitable change in the scale of T , in the regime where the critical currentis not sensitive to the precise value of k c .We observe that the behavior of the critical current is in all cases quite stable for T ≪ T ∗ and that there is even anupturn before the abrupt drop at the crossover temperature. These features have been also found in the theoreticalinvestigation of the supercurrents in one-dimensional (1D) electron systems and in carbon nanotubes . The shapeof the critical currents obtained there is qualitatively similar to that of the curves in Fig. 3. A major difference ishowever that the decay of the supercurrents in the carbon nanotubes is given by a 1 /L dependence in the ballisticregime, instead of the much stronger power-law decay (16) in graphene.It is worth mentioning at this point the experiment reported in Ref. 17, in which the properties of a grapheneJosephson junction have been measured in the regime of large distance between superconducting electrodes. In theexperimental setup described there, the minimum distance between superconducting contacts can be estimated as ≈ . µ m. While no supercurrent was observed below the critical temperature T c of the electrodes ( ≈ T ≈ T is in good correspondence with the crossovertemperature that we find in our model for a distance L = 2 . µ m, as can be seen from Fig. 3(c). It is thereforelikely that the sharp decrease measured in the resistance has its origin in the same suppression of the thermal effectsthat enhances the supercurrents at T < T ∗ . We also notice that the prediction from our model is that the criticalcurrents for such a large value of L should be well below the scale of 1 nA. This may explain the failure to establish asupercurrent in the experiment of Ref. 17, and it may also anticipate better perspectives in experiments with suitablyshort graphene junctions. IV. SUPERCURRENTS AT FINITE DOPING
We have seen that the origin of the relative smallness of the critical currents in undoped graphene lies in thevanishing density of states at the Dirac point. Therefore, a straightforward way to enhance the supercurrents maysimply consist in shifting the Fermi level away from the charge neutrality point, as shown in Fig. 5. In practice, thiscan be achieved by doping the graphene sheet. In our theoretical framework, we will assume that this effect can beaccounted for by means of a finite chemical potential µ . Thus, the hamiltonian for the graphene part of the junctionwill now read: H = Z d r Ψ ( a ) † σ ( r ) (cid:16) v F σ ( a ) · ∂ − µ (cid:17) Ψ ( a ) σ ( r ) (22) K’K
FIG. 5: Schematic representation of the two independent Dirac valleys at the corners of the hexagonal Brillouin zone, showingthe regions of occupied (dark) and unoccupied (white) energy levels in doped graphene.
Working at µ = 0 leads to significant modifications in the propagator of the Dirac fermions and in the Cooper-pairpropagator. The Dirac propagator corresponding to the hamiltonian (22) turns out to be (for µ > G ( a ) ( k , ω ) = ( ω + v F σ ( a ) · k )[ 1 ω − v F k + iǫ + iπ δ ( ω − v F | k | ) v F | k | θ ( µ − v F | k | )] (23)As shown in the Appendix, the representation (23) is nothing but a compact form of expressing the propagation ofquasiparticles with v F | k | > µ and quasiholes with ± v F | k | < µ , in the particular case of conical dispersion.The propagator (23) is very convenient to carry out calculations in the many-body theory and, in particular, itallows us to compute the dependence on µ of the Cooper-pair propagator as a correction to the expression (10) at µ = 0. In this procedure, we observe that the second term in the r.h.s. of (23) does not introduce any integralsrequiring regularization in the diagrammatics of the Dirac theory. By computing then the Cooper-pair propagatoraccording to Eq. (8), we obtain D ( k ,
0) = − πv F µ if v F | k | < µ = − v F | k | + 14 πv F | k | arcsin (cid:18) µv F | k | (cid:19) − πv F µ if v F | k | > µ (24)At large values of v F | k | ≫ µ , we recover from (24) the linear dependence on the momentum that is characteristic ofthe Cooper-pair propagator in the undoped system. However, the chemical potential introduces a clear deviation fromthat behavior at small | k | , which has significant consequences in the decay of the supercurrent at long distances. This
50 100 500 1000 5000L H nm L - - - - I c H Μ A L
100 500 1000L H nm L - - - - I c H Μ A L
100 500 1000L H nm L - - - - I c H Μ A L (a) (b) (c)FIG. 6: Plot of the zero-temperature critical current I (2 D ) c (0) as a function of the distance L , for W = 10 /k c (= 50 nm) andthree different values of the chemical potential µ = 1 meV (a), 5 meV (b), and 10 meV (c). The dashed straight lines in Fig.3(a) are drawn as a reference to the power-law dependences 1 /L and 1 /L . is illustrated in Fig. 6, where it can be appreciated the existence in general of a crossover length scale L ∗ mediatingthe transition towards a softer power-law decay.According to (24), we can express the critical current I (2 D ) c (0) at finite chemical potential in the form I (2 D ) c (0) = I (2 D ) c (0) + I (2 D ) c (0) (25)with I (2 D ) c (0) = − π eW µ Z ∞ dk π | k | J ( | k | L ) e −| k | /k c (26) I (2 D ) c (0) = − π ev F W Z ∞ µ/v F dk π | k | arccos (cid:18) µv F | k | (cid:19) J ( | k | L ) e −| k | /k c (27)The first contribution to (25) is not relevant, since we have I (2 D ) c (0) = − π eW µ k c p ( k c L + 1) (28)which is smaller than the estimate (16) at µ = 0 by a factor µ/v F k c . The second contribution may change howeverthe behavior of the critical current at large L , as the integrand is not analytic at | k | = 2 µ/v F . The expression for I (2 D ) c (0) is actually finite in the limit k c → ∞ , and we obtain I (2 D ) c (0) ∼ − ev F W µ v F Z ∞ dx x arccos (cid:18) x (cid:19) J ((2 µL/v F ) x ) (29) ∼ eW µ L for µL/v F ≫ I (2 D ) c (0) has oscillations has a function of L , arising from the own behavior of theBessel function J . The power-law decay (30) applies then to the envelope of the maxima of the critical current, as itis illustrated in Fig. 6(a). There it can be appreciated the crossover from the 1 /L behavior to the oscillatory regimewith softer power-law decay. From the numerical results represented in the figure, it can be checked that the 1 /L behavior is followed with great accuracy at larges values of L (compared to v F /µ ).From a practical point of view, the most important result that we obtain is the significant enhancement of thecritical currents at moderate values of the chemical potential. This is clearly observed in the plots of Fig. 6, wherethe crossover to the 1 /L decay is always found at a length scale consistent with the theoretical estimate L ∗ ∼ v F /µ .For a chemical potential µ ≈
10 meV, for instance, that scale is ≈
50 nm. The critical currents can be then enhancedto values above the nanoampere scale for spatial separation between superconducting contacts
L > ∼
500 nm (assumingthin electrodes as in our case with W ∼
50 nm). This should open good perspectives to establish supercurrents ingraphene Josephson junctions by suitable doping of the samples.
V. CONCLUSION
In this paper we have adopted a framework suited to address the many-body properties of graphene Josephsonjunctions. We have described the development of the supercurrents through the tunneling and propagation of Cooperpairs in the graphene part of the junction, with the aim of investigating the dependence of the critical currents onrelevant variables like the distance between the superconducting contacts, the temperature, and the doping level. Wehave been able then to characterize different regimes in the behavior of the supercurrents, depending on the relationbetween those variables.The supercurrents have a natural tendency to decay in the graphene part of the Josephson junction, following ingeneral a power-law behavior with respect to the distance L between the superconducting contacts. Such a power-law decay is particularly strong in undoped graphene, given the vanishing density of states at the charge neutralitypoint. We have shown that the critical currents display then at zero temperature a 1 /L dependence on the distance L . When temperature effects are taken into account, there is always a finite thermal length L T (of the order of ∼ v F /k B T ) beyond which the supercurrents are further suppressed, due to the disruption of the Cooper pairs bymany-body effects. When this takes place, the supercurrents enter a regime where the natural decay is given by a1 /L dependence.On the other hand, many-body effects can be also used in our benefit to enhance the critical currents, in this caseby shifting the Fermi level away from the charge neutrality point. This can be achieved in our framework by meansof a chemical potential µ = 0. Inducing in this way a finite density of states at the Fermi level, we have seen thatthe critical currents are enhanced beyond a new crossover length L ∗ ∼ v F /µ . This is actually the scale at which theeffects of the finite charge density start being felt, marking the transition from the previously discussed 1 /L behaviorto a softer 1 /L decay of the supercurrents at long distances.At this point, it is interesting to note that the 1 /L decay at zero temperature in undoped graphene is similar to thebehavior found in the investigation of mesoscopic junctions made of a diffusive metal . In this case, the product ofthe critical current times the normal resistance of the metal is proportional to the Thouless energy, which depends onlength L as 1 /L . This implies consequently a 1 /L decay of the critical current, which we have seen is characteristicof graphene under conditions of ballistic transport. The reminiscence of some of the properties of clean graphene withrespect to the behavior of a disordered normal metal has been remarked in several other instances . We have topoint out, however, that this resemblance does not go farther in our case, regarding other regimes of the grapheneJosephson junction. In particular, we have seen that the critical current does not follow an exponential decay atdistances larger than the thermal length. The decay of the critical current is always given in graphene by a power-law,which can be seen as a consequence of the perfect scaling of the low-energy Dirac theory.We have also to stress that our results refer to Josephson junctions with graphene layers which have large dimensionsin both the longitudinal direction along the junction and the transverse direction. This condition comes from ourconsideration of a system which is truly 2D, where in particular the size in the direction transverse to the junctionis not constrained by the width of the superconducting contacts. In this regard, the situation is quite different tothe case of long but narrow junctions, where the small transverse dimension may lead to the quantization of thetransverse component of the momentum. In such circumstances, the behavior of the system may be rather dictatedby a 1D propagation of the Cooper pairs, which is known to lead to a 1 /L decay of the supercurrents in the ballisticregime .Anyhow, the great advantage of the graphene Josephson junctions is that the interaction effects have little signifi-cance at the temperatures required to measure the supercurrents. In the long 1D junctions made of carbon nanotubes,for instance, it is known that the Coulomb interaction may induce a strong power-law suppression of the density ofstates, with the consequent reflection in the decay of the supercurrent . In 2D graphene, however, the electron systemhas the tendency to become less correlated at low energies, with a strong renormalization of the Coulomb interactionthat makes it practically irrelevant at the temperature scale of 1 K .In conclusion, our results highlight the role of the different parameters conforming the geometry of grapheneJosephson junctions in the determination of the critical currents. We have seen that the interplay with variables likethe temperature and the doping level is what establishes the different regimes of a junction. This information may beuseful in the design of experiments, for the purpose of enhancing the magnitude of the critical currents in real devices. Acknowledgments
We thank H. Bouchiat and F. Guinea for useful comments. The financial support of the Ministerio de Educaci´ony Ciencia (Spain) through grant FIS2005-05478-C02-02 is gratefully acknowledged. E.P. is also financially supportedby CNISM (Italy).0
AppendixA. Lowest-order contribution to the critical current
The Josephson current I s can be computed in a perturbative framework by expanding the free energy in Eq. (3) inpowers of the tunneling amplitude t . The first nonvanishing contribution is found to fourth order in this expansion,as the statistical average of operators leads then to the appearance of the condensates of the two superconductors inthe junction. We have actually I s ≈ e ∂∂χk B T t Y i =1 Z /k B T dτ i Z W dy i h Ψ S , ↑ (0 , y ; − iτ )Ψ S , ↓ (0 , y ; − iτ ) i×h Ψ ( a ) †↑ (0 , y ; − iτ )Ψ ( − a ) †↓ (0 , y ; − iτ )Ψ ( b ) ↑ ( L, y ; − iτ )Ψ ( − b ) ↓ ( L, y ; − iτ ) i×h Ψ † S , ↑ ( L, y ; − iτ )Ψ † S , ↓ ( L, y ; − iτ ) i (31)We can apply to Eq. (31) the approximations pertinent to the regime we want to study in the paper. Focusing onthe case of a large junction where the distance L is much larger than the superconducting coherence length ξ , the useof Eq. (4) and translational invariance in the variable τ leads to a maximum supercurrent I c (critical current) I c ( T ) ≈ eρ t W Y i =1 Z W dy i Z /k B T dτ h Ψ ( a ) †↑ (0 , y ; 0)Ψ ( − a ) †↓ (0 , y ; 0)Ψ ( b ) ↑ ( L, y ; − iτ )Ψ ( − b ) ↓ ( L, y ; − iτ ) i (32)At this point, it becomes convenient to factor out the relative tunnel conductances at the contacts, which are eachgiven by the dimensionless quantity ρt W/v F . We concentrate then on the behavior of the critical current intrinsicto the 2D graphene system, represented by the expression I (2 D ) c ( T ) ≈ ev F Z W dy Z W dy Z /k B T dτ h Ψ ( a ) †↑ (0 , y ; 0)Ψ ( − a ) †↓ (0 , y ; 0)Ψ ( b ) ↑ ( L, y ; − iτ )Ψ ( − b ) ↓ ( L, y ; − iτ ) i (33)This last equation highlights the connection between the critical current and the propagator of the Cooper pairs,which plays a central role in the discussion of Sections III and IV in the paper. B. Dirac propagator at µ = 0 In the many-body theory of Dirac fermions, it is usual to write the Dirac propagator at finite charge density in theform (assuming µ > G ( a ) ( k , ω ) = ( ω + v F σ ( a ) · k )[ 1 ω − v F k + iǫ + iπ δ ( ω − v F | k | ) v F | k | θ ( µ − v F | k | )] (34)If we specialize the expression (34) to modes such that the eigenvalue ε ( k ) of the matrix v F σ ( a ) · k is positive, weget G ( a ) ( k , ω ) (cid:12)(cid:12)(cid:12) ε ( k )= v F | k | = 1 ω − v F | k | − iπ ω + v F | k | v F | k | δ ( ω − v F | k | )= 1 ω − v F | k | + iǫ (35)for v F | k | > µ , and G ( a ) ( k , ω ) (cid:12)(cid:12)(cid:12) ε ( k )= v F | k | = 1 ω − v F | k | + iπ ω + v F | k | v F | k | δ ( ω − v F | k | )= 1 ω − v F | k | − iǫ (36)for v F | k | < µ . 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