Ciprian Padurariu

Thibaut Jonckheere

Jérôme Rech

Régis Mélin

Denis Feinberg

Thierry Martin

Yuli V Nazarov

Closing the proximity gap in a metallic Josephson junction between three superconductors

Keywords:

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

The density of states in a normal metal is strongly modified by contact to one or multiple superconductors placed in its proximity. In junctions with two superconductors, the development of a minigap in the density of states has been in the focus of research for many years 1 . Recent attention has been given to the possibility of engineering states inside the minigap in the context of Majorana zero energy states 2,3 . To this end, it would be advantageous if the Andreev states in a Josephson junction could be brought to zero energy, i.e. at the Fermi level, by controlling only the superconducting phase of the junction. Zero energy Andreev states together with spin-orbit coupling provide the opportunity to manipulate single fermionic quasiparticles 4 . The motivation is to increase understanding of quasiparticle dynamics and facilitate designs of superconducting spin quantum memory bits 7 . Unfortunately, in two terminal Josephson junctions Andreev states do not cross the Fermi level except at a singular phase difference [START_REF] Beenakker | Single-Electron Tunneling and Mesoscopic Devices[END_REF][START_REF] Belzig | [END_REF] , ϕ = π, and only in systems with reflectionless transport channels.

Recent work [4][5][6] shows that the situation is different in a Josephson junction with multiple terminals (N ≥ 3). The Andreev states have been shown to cross the Fermi level if the superconducting phases wind by 2π around the junction 4 . The crossing points have been termed Weyl points, as they are analogous to Weyl singularities studied in 3D solids, with Andreev bound states corresponding to energy bands and the superconducting phase differences corresponding to quasi-momenta 5,6 . The presence of Weyl points has been illustrated theoretically by using scattering theory to model junctions consisting of a quantum dot connected to three 4 or four 5,6 su-FIG. 1. (Color online.) Left. Two superconducting rings can be used to control the two independent superconducting phase differences in a three terminal superconducting junction via fluxes Φ1,2. Right. Quantum circuit theory model of a three terminal superconduting device using a single node.

perconductors. In particular, in three-terminal junctions, Weyl points appear along a closed curve in the space spanned by two independent superconducting phases. The area enclosed by the curve is shown to be maximal for a fully transparent quantum dot 4 .

Weyl points have been found for some choices of the scattering matrix describing the junction, but not for all. The existence or absence of Weyl points in the spectrum is determined by the details of the junction scattering matrix. The requirements for presence of Weyl points are not yet well understood, but are very important in view of their experimental observation. From the viewpoint of the experimental realization, it is advantageous to study metallic junctions, that do not require the strict electrical confinement of quantum dots. Metallic junctions have a quasi-continuous spectrum of states, in contrast to the discrete states in a quantum dot. The larger number of states discourages use of the scattering theory approach. Local density of states at the Fermi level as a function of phase differences for a symmetric junction with transmission coefficient T = 0.9. The figure illustrates the gapped regime (dark colored regions) and gapless regime (light colored regions). We define the phase φ = ϕ1 = -ϕ3, with ϕ2 = 0, that parameterizes an axis along which the system undergoes transitions between gapped and gapless regimes. Right. Above: the density of states as a function of energy in the gapped regime, at φ = 0.44π. Below: the same in the gapless regime, at φ = 0.63π.

To model metallic junctions, quasiclassical methods are better suited.

In this paper, we study the local density of states in a disordered metallic junction between three superconductors (see Fig. 1). The dimensions of the junction are assumed short at the scale of the superconducting coherence length, but much larger than the mean free path of electrons in the metallic island, in the normal state. We show that the junction exhibits two regimes. One where the density of states has a finite minigap, similar to the minigap in two terminal junctions, and another where the minigap is closed and the density of states is finite at all energies. The transition between these regimes can be made in equilibrium, by varying the superconducting phase differences. The absence of the minigap is a manifestation of states crossing at the Fermi level. In metallic junctions the gapless regime corresponds to an area in the space spanned by the superconducting phase differences, as in Fig. 2, in contrast to quantum dot junctions where the regime corresponds only to a closed curve. The larger parameter space is an additional advantage of metallic junctions in view of the experimental observation of the effect.

The paper is organized as follows. In Sec. II we introduce the theoretical method used. Sec. III presents our analytical derivation of the local density of states in the junction, showing the presence of a gapped and a gapless regime. Sec. IV presents the comparison between analytical and numerical results, discussing the physical mechanism at the origin of the gapless regime. Sec. V discusses the effect of electron-hole decoherence in the junction. In Sec. VI we propose to observe the predictions using tunneling spectroscopy and provide a numerical simulation of the results of a typical measurement setup. Sec. VII presents our conclusions.

II. THEORETICAL METHOD

To determine the local density of states in the disordered metallic island, we use the method of quantum circuit theory 10 , whereby the junction is separated into circuit elements described by spatially independent quasiclassical Green's functions. If the size of the island is much smaller than the superconducting coherence length, the appropriate circuit contains a single node connected to three superconducting terminals (see Fig. 1). Transport in the circuit is described using the quasiclassical action that is extremized with respect to the unknown retarded Green's function in the island Ĝ. The retarded Green's function matrix is conveniently parametrized by Ĝ = ( n(θ, φ) • ˆ τ ), where ˆ τ is the vector of Pauli matrices in Nambu space. The spectral vector 11 , n(θ, φ), is generally parametrized by two complex numbers θ and φ, with n x = sin(iθ) sin(φ), n y = sin(iθ) cos(φ), and n z = cos(iθ). In the superconducting bulk with energy gap |∆|, assumed the same for all three superconductors, φ S takes the value of the superconducting phase ϕ i corresponding to superconductor i = {1, 2, 3}, and θ S has the following energy dependence,

θ S = -i π 2 + 1 2 ln 1+ω 1-ω , ω < 1, 1 2 ln ω+1 ω-1 , ω > 1,
where ω = E/|∆|. In the limit of low energy ω 1, the spectral vector in superconductor i has the following simple form, n i = (sin(ϕ i ), cos(ϕ i ), -iω). In terms of the spectral vector of the island n and of those of the superconductors n i , the quasiclassical action of the junction takes the form

S = 2Re i p ln 1 + T p,i 2 (( n • n i ) -1) , (1)
where p labels the open transport channels in each of the junction contacts and T p,i represents the corresponding transmission coefficient. The spectral vector in the metallic island n(θ, φ) is obtained by extremization of the action under the constraint n 2 = 1. We introduce a complex Lagrange multiplier λ such that the extremization problem reduces to a set of four complex equations

∇ n,λ Λ( n, λ) = 0, Λ( n, λ) = S( n) + λ n 2 . ( 2
)
In general, the equations are strongly non-linear in terms of the unknowns, n and λ, and are solved numerically.

The local density of states in the metallic island is obtained from n by ρ(ω)/ρ N = Re{n z } = Re{cos(iθ)}, where ρ N is the density of states in the normal state.

III. ANALYTICAL RESULTS

Before we discuss the full numerical solution, we present analytic results in the tunnel limit, i.e. the transmission coefficients of all channels are small, T p,i the lowest order, the action takes the form

S (1) = 2Re{( n • n s )}, (3)
where n s = i g i n i is a spectral vector that depends on the superconducting phase differences and g i = p (T p,i /2) is proportional to the normal state conductance of contact i in units of the conductance quantum. We find two types of solutions in terms of n s that differ in the structure of the local density of states. i. If n s is non-vanishing, the extrema of the action S (1) occur when the vectors n and n s are aligned. In this case, we find that the z-component of n is purely imaginary in the limit of low energy ω 1, n z = -iω, meaning that the local density of states in the island has a finite gap. ii. In the opposite case, when n s = 0, the action vanishes in the lowest order. To study this situation we include the 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 FIG. 5. (Color online.) Top left: Schematic of an asymmetric junction with two contacts described by transmission τ and the third fully transparent. Left to right: The same as in Fig. 3 for the asymmetric junction with a single fully transparent contact.

next order terms in the action

S (2) = 2Re i (g i + q i )( n • n i ) - q i 2 ( n • n i ) 2 , (4)
where

q i = p (T p,i /2)
2 is proportional to the conductance of coherent processes that transfer two pairs of quasiparticles between the leads. We solve Eq. 2 using the action S (2) , in the limit of low energy ω 1,

Re i [(g i + q i ) -q i ( n • n i )] n i + λ n = 0, ( 5
)
together with n z = 1 -n 2 x -n 2 y . The z-component of the equation is of order ω and can be eliminated if the Lagrange multiplier λ is also of order ω. The x-and ycomponents of the equation are of order 1 and can be solved for n x,y ,

n x = a 4 a 3 -a 1 a 5 2 (a 2 3 -a 2 a 5 )
;

n y = a 1 a 3 -a 4 a 2 2 (a 2 3 -a 2 a 5 )
;

n z = 1 -n 2 x -n 2 y . ( 6
)
where we define the coefficients a 1 = i (g i + q i ) sin(ϕ i ), a 2 = i q i sin 2 (ϕ i ), a 3 = i q i sin(ϕ i ) cos(ϕ i ), a 4 = i (g i + q i ) cos(ϕ i ), and a 5 = i q i cos 2 (ϕ i ). Components n x and n y are real, while component n z is real if n 2

x + n 2 y ≤ 1 and purely imaginary otherwise. The relation n 2

x + n 2 y ≤ 1 defines an area in the space of superconducting phases where the local density of states in the island is finite at small energies, i.e. the gap in the density of states is closed. Estimations of n x and n y yield n x,y ∝ T -1 , meaning that n 2

x + n 2 y ≤ 1 defines a smaller area for smaller transmission coefficients. We have compared the analytic results for the boundary of this area with our full numerical solution in Figs. 3,4, and 5 and have found excellent agreement up to large transmission coefficients.

IV. NUMERICAL RESULTS

The numerical method used to obtain the exact solution is presented in Appendix. To restrict the number of parameters, we assume in all calculations that the number of open channels is the same for all contacts and that the channels in each contact have equal transmission coefficients, i.e. they are independent of channel label p. We stress that our results are not restricted to this situation; the method can be applied for any type of contacts. We have checked that the conclusions we present are robust with respect to change of the details of the contacts, as well as to change of the relative size of the superconducting gap of the three superconducting leads.

The details of the contacts determine the shape of the area corresponding to the gapless regime. For a symmetric junction this area diminishes, but does not vanish, at low transparency (see Fig. 3). As the transparency is increased the area reaches a maximum at T = 1. It is notable that the maximal area resembles the full parameter space where the superconducting phases wind by 2π around the junction, i.e. the necessary condition for presence of Weyl points 4 , but does not fully cover it.

For the asymmetric junction with two fully transparent contacts, described in Fig. 4, the case of low transparency corresponds to a symmetric junction with only two, rather than three, superconductors. The symmetric two terminal junction exhibits reflectionless transport channels leading to a crossing at the Fermi level at the singular point ϕ 3 = π. As the transmission of the third contact increases, the line ϕ 3 = π is deformed gradually and the area corresponding to the gapless regime increases.

In the case of the asymmetric junction with a single fully transparent contact, described in Fig. 5, we find that the presence of a gapless regime is conditioned by the transmission coefficient exceeding a threshold value. In contrast, for the symmetric junction we find a nonvanishing area corresponding to the gapless regime at all low transparencies. This denotes an apparent significance of processes involving all three superconductors in the emergence of the gapless regime.

Let us discuss the physical mechanism that leads to closing of the minigap in the density of states. The gapless regime corresponds to the situation when the balance of currents is dominated by transport processes involving coherent transfer of multiple Cooper pairs between the leads. This follows from our analytic derivation where second order terms are needed to find extrema of the action in perturbation theory. Multi-pair processes are not special to the geometry with three superconductors, they also manifest in conventional Josephson junctions with two superconductors. However, in conventional junctions multi-pair processes do not give rise to a gapless regime 12 . It is possible that the gapless regime emerges as a result of non-local multi-pair processes that require multiple terminals (N ≥ 3). Non-local transport processes consist of exchanges of Cooper pairs that involve quasiparticles from at least three superconductors [13][14][15] . The superconducting current contributed by non-local transport processes depends simultaneously on all three superconducting phases involved.

The fundamental process at the origin of non-local transport is the non-local Andreev reflection, also termed crossed Andreev reflection. Crossed Andreev reflection has been studied for years in the context of junctions formed between a superconductor and two normal leads [16][17][18] . On the contrary, the physics of crossed Andreev reflection in junctions with three superconducting leads has received relatively little attention until now. Recent extensive studies have so far been restricted to transport through a single level dot 14,15 . The rigorous analysis of non-local processes in metallic junctions with three superconductors is beyond the scope of this paper and will be reported elsewhere.

V. DECOHERENCE

Let us turn our attention to the effect of electronhole decoherence that appears as a result of quasiparticles spending a finite dwell time, τ d , in the junction. Within quantum circuit theory, decoherence is modeled by adding the following term to the action that corresponds to a fictitious circuit element where electron-hole coherence can be dissipated 19 (see Fig. 1),

S f = 2Re i p ln 1 + T p,i 2 (( n • n f ) -1) , (7)
where n f = (|∆|τ d / )(0, 0, -iω). This additional term modifies only the z-component of the vector equation ∇ n Λ( n, λ) = 0 that can be eliminated at low energy ω 1, ( /|∆|τ d ). It does not modify the x-and ycomponents, leaving the low energy density of states unchanged. Thus, the boundary of the area in the space of superconducting phases where the gap is absent remains unaffected by decoherence.

On the contrary, away from the limit of low energy, ω min(1, ( /|∆|τ d )), decoherence plays an important role in the structure of the density of states, as illustrated in Fig. 6. In the gapless regime, the minigap is replaced by a dip in the density of states. As illustrated in Fig. 6b, the effect of decoherence is to decrease the width of this dip. The dip narrows more rapidly when /(|∆|τ d ) 1. In the gapped regime, we recover a similar effect of decoherence as in the two terminal junctions, where the size of the minigap depends strongly on /(|∆|τ d ) (see Fig. 6c).

Near φ = ϕ 1 = -ϕ 3 = π, the density of states exhibits a smile-shaped structure located just below the superconducting gap (see Fig. 6a), reminiscent of the secondary smile-shaped minigap predicted previously in two-terminal junctions [START_REF] Reutlinger | [END_REF] . We note that the structure is present at low decoherence, but vanishes for intermediate decoherence, when /(|∆|τ d ) 1 (see Fig. 6b). Further investigation of this structure will be presented elsewhere.

VI. MEASUREMENT

Our predictions can be observed experimentally by means of tunneling spectroscopy 12,21 . The minimal setup has an additional normal metal lead brought in tunnel contact to the junction, as in Fig. 7a. The differential conductance of the tunnel contact is proportional to the local density of states, ρ(eV ), measured at the applied voltage, provided the contact is sufficiently weak. In this case, the local density of states is not significantly perturbed by the presence of the tunnel probe.

To check this requirement for our three-terminal superconducting junction, we have used the quantum circuit model in Fig. 7b to calculate numerically the tunnel current as a function of bias voltage. The results obtained are presented such that the normalized quantities are independent of the contact transmission coefficient, chosen to t p = 0.01 in the numerics.

In Fig. 7c we show that the differential conductance of the tunnel contact plotted as a function of bias voltage reproduces the energy dependence of the density of states plotted in Fig. 2, for the same transport parameters. The signature of the gapless regime can also be observed by measuring the low bias current flowing through the tunnel contact as a function of superconducting phases. Fig. 7d shows that the low bias current matches the dependence depicted in Fig. 2, mapping out the local density of states at zero energy as a function of superconducting phase differences.

The plots in Fig. 7 have been obtained assuming the limit of vanishing temperature. Our theoretical approach is also valid for finite temperatures, assuming the temperature is sufficiently small to permit a well developed proximity effect, k B T min{|∆|, τ -1 d }. In this temperature window, the local density of states in the junction is independent of temperature.

However, from the viewpoint of the experimental realization, at finite temperature, the features presented in Fig. 7c are blurred by thermal broadening of electronic transport in the normal contact at the scale eV k B T . If min{|∆|, τ -1 d } k B T , the thermal broadening does not conceal the transition between the gapped and gapless regimes. The measurement simulated in Fig. 7c can be used as a signature of this transition. The measurement simulated in Fig. 7d does not depend on temperature if the tunnel current can be measured at a voltage bias chosen such that eV min{|∆|, τ -1 d } and eV k B T .

VII. CONCLUSION

In conclusion, we have shown that the minigap in the density of states in a three terminal metallic junction can be closed by varying the superconducting phases. The area in the space of phase differences where the minigap is closed increases with the junction transparency and its shape depends strongly on the asymmetry of the three contacts. These conclusions characterize the low energy physics and are unaffected by electron-hole decoherence in the junction. At large energies, the density of states depends strongly on decoherence, as in two terminal junctions. Our most important result is the possibility to efficiently switch between a regime where the spectrum of states in the junction shows a separation that can be as large as |∆|, to a regime where the neighboring states are separated by the smallest energy scale of the junction, the level spacing in the normal metal. This opens interesting opportunities for designs of single fermion manipulation schemes and for the realization of Majorana bound states in multi-terminal superconducting junctions. We propose a tunneling transport experiment that can observe the predicted results and calculate its measurement output.

Appendix: Exact numerical method

For the exact numerical solution we have used the matrix representation of Green's functions in the full Keldysh-Nambu space, rather than spectral vectors as in the main text. The Green's functions of the supercon- where τ denotes the vector of Pauli matrices in Nambu space. The choice of Ǧf is such that I f carries no particle or heat currents, but only the information about loss of electron-hole coherence.

The unknown Green's function Ǧ is determined by the balance of current matrices, equivalent to the extremization of the action 1 defined in the main text. The above equations can be solved numerically using the following iterative algorithm 10,22 .

1. Start with an initial guess for Ǧ. 2. Find the matrix P that brings both Ǧ and M ( Ǧ) in diagonal form, i.e. P contains the eigenvectors of M as its columns.

3. Diagonalize the matrix M , M = P -1 M P . 4. Update the Green's function of the node Ǧnew = P sgn Re M P -1 . 5. Repeat iteratively until convergence, i.e. matrices Ǧ and Ǧnew are within a predefined accuracy.

6. Repeat the iterative procedure for every point in energy space.

The solution at a given point in energy space provides a convenient initial guess for the iterative procedure at the next step in energy.

FIG. 2 .

 2 FIG.2. (Color online.) Left. Local density of states at the Fermi level as a function of phase differences for a symmetric junction with transmission coefficient T = 0.9. The figure illustrates the gapped regime (dark colored regions) and gapless regime (light colored regions). We define the phase φ = ϕ1 = -ϕ3, with ϕ2 = 0, that parameterizes an axis along which the system undergoes transitions between gapped and gapless regimes. Right. Above: the density of states as a function of energy in the gapped regime, at φ = 0.44π. Below: the same in the gapless regime, at φ = 0.63π.

FIG. 4 .

 4 FIG.4. (Color online.) Top left: Schematic of an asymmetric junction with one contact described by transmission τ and the other two fully transparent. Left to right: The same as in Fig.3for the asymmetric junction with two fully transparent contacts.

6 FIG. 6 .

 66 FIG. 6. (Color online.) a. Density of states as a function of energy plotted for different values of the phase φ defined in Fig. 2. b. Density of states plotted as a function of energy for different dwell times, τ d , and for φ chosen in the gapless regime, at φ = π. c. The same as for panel b, for φ chosen in the gapped regime, at φ = 0.1π.

6 FIG. 7 .

 67 FIG. 7. (Color online.) a. Schematic of the tunneling transport spectroscopy measurement setup. b. Quantum circuit theory model of the measurement setup. c. Differential conductance of the tunnel probe, G = dI/dV , plotted as a function of voltage for the same parameters as in Fig. 2. d.Current measured at low voltage, eV|∆|, τ -1 d , plotted as a function of phases for the same parameters as in Fig.2. The current is normalized by the maximum current at low voltage, Imax, obtained at (ϕ1, ϕ3) = (2π/3, 4π/3) and (4π/3, 2π/3), with ϕ2 = 0.

  i -ǦA i ) tanh(E/2k B T ).(A.4) where complex energies have been introduced = E+i0 + and ξ = + |∆| -|∆|. The transport properties are encoded into the current matrices flowing from reservoir i,I i = 2 s e 2 π p T p,i [ Ǧi , Ǧ] 4 + T p,i ({ Ǧi , Ǧ} -2) . (A.5)where Ǧ denotes the Keldysh-Nambu Green's function of the node.Decoherence is described by a matrix current flowing into a fictitious terminal,

8 )Ǧi 4 +

 84 It is convenient to express the current balance equation as a commutator between matrix Ǧ and a matrix M that depends on all Ǧi and on the unknown matrix Ǧ, T p,i ({ Ǧi , Ǧ} -2) + i={1,2,3} p T p,i Ǧf .(A.10)

  FIG.3. (Color online.) Top Schematic of a fully symmetric junction with transmission coefficient τ . Left to right: The local density of states is calculated at the Fermi level as a function of phases, for transmissions τ = 0.1, 0.3, 0.5, 0.7, 0.9, using the full numerical solution. The horizontal x-axis represents ϕ1, while the vertical y-axis represents ϕ3, with ϕ2 = 0. The boundary of the gapless region calculated using the result in Eq. 6 is shown in bright green.

	0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
	0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
	1. In

ACKNOWLEDGMENTS

We appreciate the useful discussions with B. Douçot, R.-P. Riwar, M. Houzet, J. S. Meyer, and T. Yokoyama.

We acknowledge support from the French National Research Agency (ANR) through the project ANR-Nano-Quartets (ANR-12-BS1000701). This work has been carried out in the framework of the Labex Archimède (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the "Investissements d'Avenir" French Government program managed by the French National Research Agency (ANR).

Yu. V. acknowledges the support of the Nanosciences Foundation in Grenoble, in the framework of its Chair of Excellence program.


Pierre Darancet

Valerio Olevano

Didier Mayou

Coherent electronic transport through graphene constrictions: sub-wavelength regime and optical analogies

Keywords: numbers: 72.10.-d, 72.10.Fk, 42.25.-p, 42.25.Fx

Graphene two-dimensional nature combined with today lithography allows to achieve nanoelectronics devices smaller than the Dirac electrons wavelength. Here we show that in these graphene subwavelength nanodevices the electronic quantum transport properties present deep analogies with classical phenomena of subwavelength optics. By introducing the concept of electronic diffraction barrier to represent the effect of constrictions, we can easily describe the rich transport physics in a wealth of nanodevices: from Bethe and Kirchhoff diffraction in graphene slits, to Fabry-Perot interference oscillations in nanoribbons. The same concept applies to graphene quantum dots and gives new insigth into recent experiments [1] on these systems.

Analogies play a preminent role in physics. They allow the transfer of notions and concepts from one field to another, thus leading to deeper insights and advances in both fields. In particular, attempts to explore an analogy between the fields of quantum transport and optics have been suggested in the past, both in the direction of optics, for instance in order to understand coherent multiple scattering of light [2], and also in the reverse direction [START_REF] Datta | Electronic Transport in Mesoscopic Systems[END_REF]. Optical concepts have also been exploited to explain different properties in graphene [START_REF] Cheianov | [END_REF]. In this work we develop the analogy between quantum transport and optics beyond those attempts and bridge coherent electronic transport with sub-wavelength optics. Indeed graphene [5,6,7], either in the exfoliated or in the epitaxial form, has extraordinary coherence properties. The electron mean free-path can be of the order of a micron at room temperature and the Dirac electrons wavelength can be up to 100 nm or even larger [8]. From a technological point of view, graphene can be manipulated by ordinary lithography to realize devices as small as few nanometers [9] thus achieving a sub-wavelength regime. This suggests that an analogy with sub-wavelength optics can reveal fruitful in order to have a better insight into graphene transport properties. We believe that this approach could ease the task to find a solution to the formidable quantum transport problem of calculating the conductance characteristics in nanodevices.

We will show that, by introducing the hybrid "electronics-optics" concept of electronic diffraction barrier to represent constrictions at contacts, we can describe in an easy way the rich transport physics in a wealth of sub-wavelength graphene nanodevices: from the simplest systems such as slits between two half graphene planes, to graphene nanoribbons or quantum dots sandwiched in between semi-infinite graphene sheets as leads [10,11,12,13,14,15,16]. Our methodology relies on an exact numerical calculation of the conductance within a tight-binding [17] and a Landauer quan-tum transport formalisms [START_REF] Datta | Electronic Transport in Mesoscopic Systems[END_REF]. The novelty here introduced is that the Landauer equation is solved via a new recursive numerical algorithm [18].

Our results give new insights on the conductance characteristics of experimentally synthetized graphene quantum dots [1,19] where a chaotic Dirac billiard behaviour has been recently observed [1]. The diffraction barrier concept explains electron localization and allows us to provide a new interpretation of experimentally observed features observed into this system as due to diffraction phenomena occurring at the divergent wavelength of electrons at the Dirac point. We believe that the analogy can be further pushed to play a fruitful and key role in the design of new graphene nanodevices exhibiting other original and interesting characteristics. We also point out that the present findings are not only specific of graphene Dirac electrons but could be extended to other sub-wavelength devices made by 2D electron gas such as GaAs/AlAs or in molecular electronics.

Conductance of graphene slits and diffraction by an aperture -The archetype experiment showing an optical diffraction behaviour is the transmission through a slit or hole. We considered slits made into a graphene sheet (Fig. 1) [24] and calculated the conductance. We took into account several slits differing by the width W , from the thinnest consisting only of a single motif, i.e. a single hexagon. We introduce the wavelength λ of the incident Dirac electrons, related to the channel energy

E via λ = 2π k = hvF E
where k is the Dirac electrons wavevector, E = hkv F is the energy, linearly dispersed in graphene with a Fermi velocity of v F 10 6 ms -1 [6].

In Fig. 1 we report the conductance of the slits as a function of W/λ. We first remark (Fig. 1 inset) an universal scaling behaviour independent of the slit width W and depending only on the ratio W/λ. This scaling breaks down when the wavelength is λ < ∼ 4 nm, that is the energy is E > ∼ 1 eV. Indeed, as shown below, this scaling is intimately related to the scaling properties of the Dirac equation which in graphene is valid only in the low energy or long wavelength limit.

We then notice that in the limit W/λ 1, the conductance is quadratic, with the conductance g(W/λ) 50(e 2 /h)(W/λ) 2 , while in the opposite limit W/λ 1, the conductance turns out to be linear with g(W/λ) 5(e 2 /h)(W/λ). The crossover between the two regimes occurs around W/λ 0.2 ∼ 0.5. This promptly reminds of an analogy with classical optics and hence offers an immediate interpretation: the conductance response of a graphene slit is a clear manifestation of a diffraction phenomenon. Indeed we can identify two different diffraction regimes: for wavelengths λ W , much larger than the aperture, we observe a Bethe-like diffraction regime [10] with a slit transmitted energy proportional to the square of the aperture. On the other hand, for λ W one has a Kirchhoff-like diffraction regime, the transmitted energy being proportional to the aperture. Note that weak-localization effects in quantum billards can also reveal diffraction effects in the Kirchoff regime at the lead mouth [20]. In the short wavelength regime the phenomenon of diffraction is a perturbation and a semi-classical description of the electrons as ballistic wavepackets is applicable, thus leading to a transmitted current proportional to the section of the slit W . By contrast, in the subwavelength regime the transmission is lower than given by the semi-classical picture and can be viewed as a tuneling process.

In the ordinary optical diffraction of the slit transmitted intensity, an universal scaling behaviour is observed due to the scaling invariance of Maxwell equations. Here it stems from the fundamental scaling invariance of the graphene 2D Dirac equation. Indeed if a spinor ψ(r) satisfies the Dirac equation for the energy E and wavelength λ then the spinor ψ(r) = ψ(xr) satisfies the Dirac equation for the energy Ẽ = E/x and wavelength xλ. This leads to the fact that the conductance undergoes a scale invariance g(xλ, xW ) = g(λ, W ). As a consequence the conductance only depends on a reduced argument W/λ, g(λ, W ) = g(W/λ). [25]. Let us note that a scaling relation can also exist for other geometries. In that case the conductance of the circuit will depend only on the ratio between the characteristic lengthes of the circuit and the wavelength λ.

Nanoribbons as Fabry-Perot interferometers and subwavelength waveguides -Here we study the quantum transport response of finite-length metallic graphene nanoribbons and show that they present an oscillating response, typical of optical interferometers, and behave as subwavelength waveguides. We consider two ribbons, a zigzag and an armchair both chosen with a metallic character and hence an available conductance channel at the Dirac energy [22]. The geometry is presented in Fig. 2 and we have taken a length of 6 nm for the armchair nanoribbon and 3 nm for the zigzag (see Fig. 3 for the detailed geometry of each junction). The calculated conductance for both ribbons is shown in Fig. 2 where we again observe a conductance going to 0 at the Dirac point. However in this case, the most evident features are large amplitude oscillations in the conductance, from maxima values of 1 to minima disposed along an envelope function. These systems behave as Fabry-Perot cavities, exactly as in subwavelength optical metallic waveguides [14,15,16]. To demonstrate this we show that the nanoribbons calculated oscillating conductance distributes like the Fabry-Perot standard transmittance which is the Airy function:

T FP (E) = 1 1 + F (E) sin 2 (φ(E)/2) ( 1
)
where φ(E) = 2k(E)L + 2 φ(E) is the phase difference after one loop in the nanoribbon. k(E) is the wavevector of the Bloch state in the infinite ribbon with energy E. φ(E) is the phase factor acquired at each reflection. L is the length of the Fabry-Perot interferometer.

F (E) = 4R(E)/(1 -R(E)) 2 is the finesse coefficient and R(E) = 1 -T (E)
is the reflection coefficient at each end of the nanoribbon, that is at the junction. The Airy function presents maxima T max FP = 1 equal to 1 when φ/2 is an integer m multiple of π. For sufficiently large L the phase φ(E) = 2k(E)L + 2 φ(E) varies rapidly with energy, as compared to F (E); and the minima occur essentially when sin 2(φ/2) is maximum, that is at φ/2 = mπ + π/2, and envelope along the function

T min FP (E) = 1 1 + F (E) = (1 -R(E)) 2 (1 + R(E)) 2 . ( 2
)
which is independent of the length of the ribbon. We indeed checked that the envelope functions of the conductance minima, as expected from Eq. ( 2), is independent of the ribbon length. In particular the minimum of 2) with R(E) taken from Fig. 3 the transmission tends to zero at the Dirac energy, which indicates that R(E) tends to 1 at zero energy.

Another feature of the Fabry-Perot resonnances concerns their width. The peaks full width at half maximum is δE FWHM = δφ dE dφ = δφ v(E) 2L where δφ = 2(1 -R)/ √ R and v(E) is the group velocity at energy E. The velocity is bounded, such as the peaks look thin when R is close to 1, and broaden when R → 0. In our case when E → 0 the peaks seem to have negligible width, indicating again that R → 1, and broaden for the higher energies, indicating lower values of R.

The relative variation of the conductance (i.e. the ratio between the maximum and minimum conductance between two peaks) is also maximum close to the Dirac energy, due to the nearly perfect confinement at this energy (R(E) 1). We consider in the following that resonances are well defined if the maximum to minimum ratio is greater than 2. In our calculations this corresponds to a criterion λ/W > ∼ 3 ∼ 5 where W is the width of the nanoribbon.

The concept of electronic diffraction barrier at a constriction -Fabry-Perot oscillations occur due to the reflection at the junctions/ends of the nanoribbons. We now analyse this reflection and the response of the elementary junction. We consider the same metallic zigzag and armchair ribbons as before, but take them as semiinfinite (Fig. 3). If the system were ballistic (infinite ribbon, no contact resistance and no reflexion), we would expect to observe a conductance equal to 1 quantum of conductance (2e2/h) (dot-dashed line in Fig. 3) in the range of the Dirac energy. Instead, the calculated transmittance T (E) through the junction (continous line in Fig. 3) is observed to drop to zero at the Dirac energy, no matter the chirality, exactly like in slits and the nanoribbons. The only effect of the ribbon electronic structure is to produce a little difference in the characteristics, with the zigzag conductance showing a more cusp-like feature at the Dirac energy.

Since the associated resistance is independent of the ribbon length, this behaviour is a genuine manifestation of the contact resistance at the junction [26]. The junction conductance we have here calculated represents the characteristics of the elementary electronic diffraction barrier which is at the basis of the universal behaviour observed in all the devices hereby studied, slits, nanoribbons or quantum dots. Indeed coming back to the nanoribbon case, if in Fig. 2 we plot (1 -R(E)) 2 /(1 + R(E)) 2 (dashed curve) with R(E) = 1 -T (E) as extracted from Fig. 3 for the elementary junction, we find that the curve turns out to be the envelope of the Fabry-Perot oscillations minima in the nanoribbon conductance, as expected from Eq. ( 2). This exemplifies the concept of electronic diffraction barrier at a constriction and gives a quantitative example of such a barrier.

Conductance of a quantum dot -We now turn to more complicated structures and consider a standard nanoelectronics quantum dot (Fig. 4) consisting of a purposely irregular shape graphene nanostructure contacted via small apertures to the two half-planes graphene leads. Applying again the concept of electronic diffraction barrier, we expect that the quantum dot is weakly coupled to the leads when the energy of the electrons in the graphene sheet is close to the Dirac energy, i.e. when their wavelength is sufficiently large. We note that the conductance of this system (Fig. 4) presents a maximum around 3 eV which is the energy of the Van-Hove singularity of bulk graphene where the density of states diverges. We also note many irregular peaks superimposed to the envelope, clearly showing characteristic resonances of the quantum dot to be associated to the particular (irregular) shape. The conductance decreases when going to the Dirac point, which is a signature of the lowest number of available states and of a diffraction barrier at the contact constrictions. The maximum to minimum transmission ratio between adjacent peaks is much larger in the vicinity of the Dirac energy than at other energies. This is also observed in the nanoribbons studied previously. Again this can be explained by the presence of two diffraction barriers at the contacts which are nearly completely reflecting (R = 1) at the Dirac energy. This leads to well defined states within the quantum dot, and thus well defined conduction channels through these states. If we take as criterion to identify a range with well defined resonances the same we already introduced (maximum to minimum ratio greater than 2), then we find that the critical wavelength is of order of λ 4nm for a width W = 1.2nm. This is the same range of values of λ/W as in nanoribbons, i.e. λ/W > ∼ 3 ∼ 5. Experimental implications -Our quantum dot computed characteristics (Fig. 4) looks very similar to the experimental result of Ponomarenko et al. (Ref. [1], Fig. 1).

In those experiments the relation between voltages V (here V is the voltage variation with respect to the Dirac point voltage expressed in Volts) and electronic wavelength λ is λ 130nm/ √ V [23]. From that one deduces that in their experiment the resonances are well defined for a wavelength λ to aperture W ratio λ/W > ∼ 3 ∼ 5 in reasonable agreement with the numerical results presented here. They reported also the low bias conductance of constrictions of 20 nm (Ref. [1], Fig. S2), similar to the slits studied here. First of all, after our study of slits the Kirchoff-like (linear) regime is reached at voltages V ≥ 10Volts and for higher voltages we predict for this 20nm slit a conductance g(W/λ) 5(e 2 /h)(W/λ) 0.75(e 2 /h) √ V . This is consistent with the experimental results and for instance at 100 Volts from the voltage corresponding to the Dirac point they measure G 5.5e 2 /h (Ref. [1], Fig. S2), while we predict G 7.5e 2 /h, in good agreement with the experiment. On the other hand, the comparison of our results with their smaller 10 nm constriction is not such good. Other mechanisms should influence the electronic transport in this case. This is in agreement with the evidence of thermally activated transport for the smaller experimental constriction.

To conclude, our work establishes an important link between nanoelectronics and subwavelength optics which could be tested in today's technology graphene nanodevices, thank to its remarkable properties. The concept of electronic diffraction barrier is central to the understanding of transport properties in the subwavelength regime. It allowed us to get insights into the properties of graphene junctions, slits, nanoribbons, and quantum dots and explain their conductance characteristics in terms of diffraction and interference phenomena. This analogy can be further pushed forward in the design of devices with new properties.

is perpendicular to the slit, passing by the center of the slit. The conductance for half graphene plane is g(λ, W ) = lim W →∞ g(λ, W , W ). We consider a continuous type model of graphene circuit [21]. In this model the mass term of the Dirac equation is zero in the graphene zone and infinite outside. This confines the electrons in the graphene zone. Then there is a one to one correspondence between the scattering states at wavelength λ rib-bon width W , slit width W and those at wavelength xλ, ribbon width x W and slit width xW . This means that the conductance satisfies g(λ, W , W ) = g(xλ, x W , xW ) and as a consequence g(xλ, xW ) = g(λ, W ) [26] We are considering metallic nanoribbons. For nonmetallic nanoribbons the resistance is exponentially increasing with length.

FIG. 1 :

 1 FIG. 1: Diffraction of electrons through a graphene slit. Left: schematic view of the geometry of a graphene slit. Right: universal scaling law of the graphene slit conductance as a function of W/λ. This shows the Bethe-like λ W quadratic regime and the Kirchhoff-like λ W linear regime (see text). The inset shows the conductance for slits of widths W 0.8p nm with p = 1, 2, 3...

FIG. 2 :

 2 FIG. 2: Graphene nanoribbons as Fabry-Perot interferometers. Left: schematic view of nanoribbons contacted to graphene half-planes. Right: Fabry-Perot oscillations inf the conductance of armchair (upper panel) and zigzag (lower panel) nanoribbons. The dashed line which almost perfectly envelops along the oscillations minima, is the plot of Eq. (2) with R(E) taken from Fig.3

FIG. 3 :

 3 FIG. 3: Electronic diffraction barrier at a contact. Left: the contact geometries of the armchair (upper panel) and zigzag (lower panel) semi-infinite ribbons coupled to graphene halfplanes. Right: Conductance g of the device represented on the corresponding left side. g = T 2e 2 /h with T the transmittance of the diffraction barrier.

FIG. 4 :

 4 FIG. 4: Conductance of the quantum dot, showing well defined resonances close to the Dirac energy. Inset: geometry of the irregular shaped quantum dot coupled to the graphene half-planes.

Acknowledgements -This work has benefited from exchanges with many colleagues. We thank in particular C. Berger, X. Blase, F. Balestro, D. Feinberg, L. Lévy, L. Magaud, P. Mallet, C. Naud, T. Lopez-Rios, P. Quémerais, F. Varchon, J. Y. Veuillen, W. Wernsdorfer, L. Wirtz. Computer time has been granted by IDRIS and Ciment/Phynum.