0-Pi quantum transition in a carbon nanotube Josephson junction: Universal phase dependence and orbital degeneracy
R. Delagrange, R. Weil, A. Kasumov, M. Ferrier, H. Bouchiat, R. Deblock
00- π quantum transition in a carbon nanotube Josephson junction: Universal phasedependence and orbital degeneracy R. Delagrange, R. Weil, A. Kasumov, M. Ferrier, H. Bouchiat, and R. Deblock ∗ Laboratoire de Physique des Solides, CNRS, Univ. Paris-Sud,Université Paris Saclay, 91405 Orsay Cedex, France.
In a quantum dot hybrid superconducting junction, the behavior of the supercurrent is dominatedby Coulomb blockade physics, which determines the magnetic state of the dot. In particular, in asingle level quantum dot singly occupied, the sign of the supercurrent can be reversed, giving riseto a π -junction. This 0- π transition, corresponding to a singlet-doublet transition, is then drivenby the gate voltage or by the superconducting phase in the case of strong competition between thesuperconducting proximity effect and Kondo correlations. In a two-level quantum dot, such as aclean carbon nanotube, 0- π transitions exist as well but, because more cotunneling processes areallowed, are not necessarily associated to a magnetic state transition of the dot. In this proceeding,after a review of 0- π transitions in Josephson junctions, we present measurements of current-phaserelation in a clean carbon nanotube quantum dot, in the single and two-level regimes. In thesingle level regime, close to orbital degeneracy and in a regime of strong competition between localelectronic correlations and superconducting proximity effect, we find that the phase diagram of thephase-dependent transition is a universal characteristic of a discontinuous level-crossing quantumtransition at zero temperature. In the case where the two levels are involved, the nanotube Josephsoncurrent exhibits a continuous 0- π transition, independent of the superconducting phase, revealing adifferent physical mechanism of the transition. I. INTRODUCTION
Josephson junctions [1] refer to any non superconducting material sandwiched between two superconductors. Thesimplest Josephson junction (JJ), an insulator between two superconductors, is passed through by a non dissipativecurrent I = I C sin( ϕ ) , I C being the critical current and ϕ the phase difference of the superconducting order parameters.When a normal metal is inserted between the superconductors, the transmission of Cooper pairs takes place throughAndreev Bound States (ABS), that are confined in the normal region at an energy below the gap [2]. Due to theboundary conditions, the energy of these bound states depends on ϕ , leading to a phase dependence of the supercurrent:the current-phase relation (CPR).In a single level QD-JJ, the physics of the ABS is governed by four characteristic energies: the coupling Γ = Γ L +Γ R ( Γ L and Γ R are the coupling respectively to the left and right reservoirs, Γ L / Γ R is the asymmetry), the chargingenergy U , the level energy in the dot (cid:15) d and the superconducting gap of the contacts ∆ . We focus in this articleon the intermediate regime Γ ≈ U ≈ ∆ , where the Coulomb Blockade is strong enough to impose a well definedoccupancy and the coupling sufficient to observe a supercurrent [3]. The transfer of Cooper pairs then involvescotunneling processes, strongly dependent on the dot’s occupancy. When this occupancy is even, one has a 0-junctionwhose amplitude follows the transmission of the dot. For an odd occupancy, the first non-zero contribution to thesupercurrent involves fourth order processes, that imply reversing the spin-ordering of the Cooper pair. The sign ofthe supercurrent is thus reversed and its amplitude strongly reduced, this is called a π -junction. Experimentally, thesupercurrent can be precisely changed by tuning the parity of the dot with a gate voltage [4–6].In addition, an oddly occupied dot gives rise to Kondo effect. The interaction of the local magnetic moment withdelocalized conduction electrons through spin flip processes leads, in the normal state, to the formation of a stronglycorrelated state. This Kondo singlet state is characterized by the screening of the dot’s magnetic moment and by aresonance in the density of states at the Fermi energy for temperatures below the Kondo temperature T K [7–9]. Inthe superconducting state, when k B T K < ∆ , the Kondo screening is destroyed by superconducting correlations anddoes not affect the π -junction. But for k B T K (cid:29) ∆ , the unpaired electron’s spin is involved in a Kondo singlet thatopens a well transmitted channel in the system and facilitates the transfer of Cooper pairs. Therefore the 0-junctionis recovered and the supercurrent is enhanced due to the cooperation between superconductivity and Kondo effect[6, 10–12]. Since the Kondo temperature depends on U , Γ and (cid:15) [13], the junction can be tuned from 0 to π byvarying these parameters for a fixed parity and value of ∆ [14]. Measuring the CPR of the single-level QD Josephsonjunction directly gives insights into the magnetic state of the system: a doublet if one measures a π -junction, a ∗ author to whom correspondence should be addressed : [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec singlet state (purely BCS or Kondo) otherwise. Here, we are particularly interested in the specific regime of the 0- π transition, where the system undergoes a level-crossing quantum transition. The fundamental state of the system, 0or π , depends on the superconducting phase ϕ , meaning that the magnetic state of the system (singlet or doublet)can be controlled by this parameter [15–22]. The gate and/or phase induced 0- π transition measured on the CPRis directly related to the behaviour of the Andreev states [23–29], which coincide, for a certain range of parameters,with the Yu-Shiba-Ruzinov states when the quantum dot is oddly occupied [30, 31].In a multi-level quantum dot, the picture described previously is not valid anymore, as predicted theoretically [32–36]. The measurement of the current-phase relation is no longer a good indicator of the effective magnetic momentof the dot. Indeed, as soon as several energy levels participate in the transport, the available cotunelling processesare different and the properties of the wave-functions become determinant, making 0 and π -junction possible both foreven and odd occupations. This multi-level effects on the supercurrent in a quantum dot based Josephson junctionhave been experimentally observed by van Dam et al. [4] with an InAs nanowire. But no phase dependence ofthe supercurrent was shown and the analysis of the experiment was complicated by the fact that, unlike in carbonnanotube, the exact electronic configuration is not known.The aim of this work is to present the investigation of the CPR in a clean carbon nanotube (CNT) QD, where theorbital levels are nearly degenerate. Our results show that distinct behaviors emerge depending on the number of levelsinvolved in transport. This number is determined by the occupancy and the relative widths of the nearly degenerateorbital levels. In our sample, for most filling factors, the system is well understood in a single-level description. Inthis regime, for odd electronic occupation and intermediate transmission of the contacts, we probe experimentallythe existence of this phase-driven 0- π transition and demonstrate its universal character. On the other hand, in someodd diamonds with nearly degenerate orbital levels, we qualitatively confirm theoretical predictions about the gatedependence of the supercurrent in the two-level regime and its high sensitivity to the precise configuration of the twoorbital states involved in transport. In addition, the phase dependence of the supercurrent shows a continuous 0- π transition with a complete cancellation of the amplitude of the Josephson current, in contrast with the first ordersingle-level 0- π transition.The paper is organized as follows. In a first part, we remind the reader of the basics of the physics involved andreview the different kinds of existing 0- π transitions, and their relation with quantum phase transitions. Then, wepresent our experimental results. II. JOSEPHSON EFFECT IN A CARBON NANOTUBE QUANTUM DOT AND 0- π TRANSITIONS:STATE OF THE ART
It has been shown in 1999 that a carbon nanotube can sustain a supercurrent by proximity effect [37]. A contactedcarbon nanotube with barriers at the interface with reservoirs can be seen as a quantum dot, with quantized energylevels and in the Coulomb blockade regime if the temperature is low enough. The energy scales of the system areshown on fig. 1: the charging energy U needed to add an electron on the dot, the energy level spacing ∆ E and thewidth Γ of the levels (also the coupling between the dot and the reservoirs). The position of the energy levels can beshifted by (cid:15) d , controlled by a gate voltage. S S QD φ L φ R U+ΔE ε d Γ ? Figure 1: Schematic of a quantum dot sandwiched between two superconductors. The superconducting reservoirs are supposedidentical except that there is a difference of superconducting phase ϕ = ϕ R − ϕ L between them. The energy levels in thequantum dot are represented with their characteristic energy scales: the charging energy U, the energy level spacing ∆ E , thewidth Γ of the levels (also the coupling between the dot and the reservoirs) and (cid:15) d the shift of the energy levels, controlled bya gate voltage. Concerning the proximity effect, the situation is strongly dependent on the relative values of the parameters of thedot:If Γ (cid:29) U, ∆ , in the strong coupling regime, the energy levels in the dot are so broad that they overlap : the chargefluctuates in the dot. On the contrary, if Γ (cid:28) U, ∆ , the coupling is low such that the system is in a strong Coulombblockade regime: out of the charge degeneracy points, it is not possible to add or remove an electron on the dot.The most interesting case is the intermediate regime Γ ≈ U ≈ ∆ . In this limit, U is high enough to give rise toCoulomb blockade while Γ is sufficiently large to allow the cotunneling processes needed for the development of theKondo effect. In the normal state, these cotunneling processes allow an electron to enter the dot (and thus to changethe electronic occupancy) provided that another one goes out during a time shorter than (cid:126) /U (the typical tunnelingtime being (cid:126) / Γ ). In the superconducting state the situation is more complicated since, instead of a single electron, oneneeds to make the two electrons of a Cooper pair tunnel through the dot coherently in order to observe a supercurrent.Because of the exclusion principle, the situation will be strongly dependent on the parity of the number of electronsin the dot. In the next section, we give a qualitative image of what happens in this intermediate regime. A. Parity induced 0- π transition To transfer a Cooper pair through a quantum dot by cotunneling processes, one needs to break the pair (whichis possible during a time (cid:126) / ∆ ) and make the electron co-tunnel one by one (during typically the time (cid:126) / Γ ): this ispossible only if ∆ < Γ . Depending on the parity of the number of electrons in the dot, different processes are possible(see ref. [3, 38]).For an even number of electrons, the highest occupied energy levels is filled, as represented on fig. 2, the system isin a singlet state. To transfer a Cooper pair from the left to the right reservoir, one electron from the dot tunnels tothe right electrode (1) and is replaced by the electron of the Cooper pair with the same spin (2). In a second step, theelectron on the dot with the opposite spin tunnels to the right and is replaced by the second electron of the Cooperpair. At the end, the ordering of the Cooper pair is unchanged.For an odd number of electrons, the system is in a doublet state. Things are different since there is only oneelectron, spin up or down, in the highest occupied state. If, in the first step, we inter-exchange electrons of same spin,it will not be possible to reform a zero-spin Cooper pair in the right electrode. We thus have to invert the spins: inthe first step a spin down (for example) of the Cooper pair replaces a spin up in the dot. In the second step, spinup of the Cooper pair replaces a spin down in the dot (see fig. 2). At the end, the spin ordering of the Cooper pairformed in the right contact has been inverted. = - N even singletS=0
N odd doubletS=1/2 intermediate state
Figure 2: Qualitative explanation for the parity induced 0- π transition. Depending on the even or odd parity, the co-tunnelingprocesses are different (see text). These spin-flip processes strongly affect the supercurrent, as it has been first pointed out by Kulik in 1966 [39]. Theseprocesses are of fourth order and, in the doublet state, are the first one leading to a non-zero current. Consequently,the supercurrent is weakened [40] and the current-phase relation is dephased by π ( I = I c sin( ϕ + π ) = − I c sin( ϕ ) ):this is a π -junction [3, 11, 38]. This name is given by opposition to the singlet case, where the current-phase relationis a regular one: I = I c sin( ϕ ) .The transition from a 0 to a π junction is achieved by tuning the dot’s occupancy with a gate voltage, yielding agate-controlled 0- π transition. It has been first experimentally observed by Van Dam et al. [4] in an InAs nanowireQD and Cleuziou et al. [5] in a CNT. In both experiments, the QD is inserted in a superconducting loop so that thephase is controlled by a magnetic field, allowing the measurement of the current-phase relation. Jorgensen et al. [6]measured as well this gate-controlled 0- π transition, also in a CNT but without control of the phase. B. Competition between Kondo effect and superconductivity
We now focus on Coulomb diamonds corresponding to an odd number of electrons in the presence of the Kondoeffect. This effect, well known in dilute alloys [41–43], leads to the screening of the magnetic impurities embedded ina metal by the conduction electrons around the Fermi energy. A quantum dot connected by metallic electrodes andoccupied by an odd number of electrons constitutes an artificial realization of Kondo impurity [7, 9, 44].In this system, the Kondo effect manifests as enhanced cotunneling processes which overcome Coulomb blockade.This gives rise to a resonance in the density of states at the Fermi level, i.e. as a conductance resonance at zero bias,as well as a screening of the local magnetic moment of the dot. In other words, the Kondo effect transforms thedoublet state induced by Coulomb blockade in a Kondo singlet state. The characteristic energy associated to thiseffect defines the Kondo temperature, and can be expressed as [13]: k B T K = (cid:114) U Γ2 e π (cid:15) d ( (cid:15) d + U ) / ( U Γ) (1)When the electrodes of a Kondo quantum dot turn superconducting, different regimes can take place [3]:• If ∆ (cid:29) T K , there is no electron around the Fermi level in a range of energy T K that is able to participate tothe Kondo screening. The situation is not modified compared to the case described above: this is a π -junction.• In the ∆ (cid:28) T K case, the Kondo correlations coexist in the system with superconducting ones. The cotunnelingprocesses are enhanced, making the transfer of Cooper pairs easier, and favoring the formation of a Kondo/BCSsinglet state. Both Kondo effect and superconductivity cooperate so that a 0-junction is recovered, with possiblya very high supercurrent.• Intermediate regime ∆ ≈ T K : how does the system transit from 0 to π -junction in this regime of strongestcompetition? We will see that in this regime, the superconducting phase plays a role in the transition, aspredicted first by Rozhkov et al. [32] in 1999 and Clerk et al. in 2000 [10].This 0- π transition driven by the ratio ∆ /T K was predicted some time ago by Glazman and Matveev, in 1989 [11].The first occurrence of this interplay came in 2002 by Buitelaar et al. [45], who proved that the conductance at zerobias in the superconducting state is suppressed if ∆ > T K and enhanced otherwise. Then, in 2009, this result hasbeen supported by critical current measurements in a carbon nanotube by Eichler et al. [12] (only the critical currentwas measured, not the current-phase relation).Note that the problem has also be tacked through the spectroscopy of the Andreev bound states, as for examplein ref. [25, 27–29, 46]. These experiments enable in particular to visualize the crossing of the Andreev levels as afunction of the different parameters. C. Phase dependence of the ABS, current-phase relation and phase-driven 0- π transition. To understand the current-phase relation at the 0- π transition, we have to understand the phase dependence of theABS in the system. They can be calculated in the general case taking into account the Kondo correlations, as forexample in ref. [20]. On fig. 3 are represented qualitatively the phase-dependence of the Andreev Bound States ,inspired by ref. [15] in three different regimes, as well as the derivative as a function of the phase of the ones belowthe Fermi energy, giving the dominant contribution to the supercurrent at zero temperature.• The singlet state corresponds to the blue part on the phase diagram, where T K > ∆ such that the Kondo effectcooperates with the superconductivity to form the singlet-state. The most striking difference with a standardSNS junction is that the degeneracy of the ABS is broken : each ABS is split in two because of the Coulombinteraction [15]. Moreover the ABS are detached from the continuous spectrum above the gap ∆ . However, thesign of the current is the same as in a standard junction: this is a 0-junction. Doublet state: π -junction Singlet state: 0-junction φ -dependent state: 0/ π -transition Doublet π -junction Singlet: 0-junction π -transition φ / π ε /U E / Δ I ( ua ) Γ / U φ / π E / Δ φ / π E / Δ φ / π I ( ua ) φ / π I ( ua ) φ / π Figure 3: Andreev Bound States and corresponding current-phase relation in the three regions of the phase diagram (in thebottom right): the singlet state (in blue), doublet state (red), and at the transition (purple line). • The doublet state, where T K < ∆ such that the Kondo effect does not survive to superconductivity, correspondsto the red region. Then the bound states have been exchanged: the sign of the supercurrent is reversed, this isa π -junction.• The third case is the most intriguing, corresponding to the purple line on the phase diagram, where the Kondoeffect and superconductivity are of the same order of magnitude ( T K ≈ ∆ ). There, the inner ABS’s cross at theFermi energy for some values of superconducting phase (see fig. 3) such that the ground state around ϕ = π isa doublet and a singlet around ϕ = 0 . The consequence on the CPR is spectacular: it looks like a π -junctionaround ϕ = π and to a 0-junction around ϕ = 0 . Since the system transits from a doublet to a Kondo singletvarying the superconducting phase, we can see the situation as a phase-dependent Kondo screening.This doublet to singlet transition driven by the phase can also be shown using renormalization group theories[16, 19] and Quantum Monte Carlo [17, 21].However, before our work, this kind of current-phase relation had not been measured. The CPRs measured byMaurand et al. [14] at the 0- π transition exhibit some anharmonicities which may be related to this phenomenon,but the symmetry of these curves is problematic: they are not odd functions of the superconducting phase as theyshould be in absence of any breaking of the time reversal symmetry.The aim of this work is to measure accurately and systematically the current-phase relation at this 0- π transitionin order to demonstrate the driving of the 0- π transition by the superconducting phase. D. Josephson effect in a two-level quantum dot
Until now, we have considered a single-level (SL) quantum dot, where the measurement of the CPR (0 or π -junction)is related to the state of the system (singlet or doublet). But, in clean carbon nanotubes, each energy level is nearlyorbitally degenerated in addition to the spin degeneracy, such that they generally form two-level (2L) QD. This mayaffect dramatically the Josephson effect.The situation is summarized on fig. 4. We want to induce a supercurrent in a QD where the gate voltage is chosensuch that two levels may participate to the transport of Cooper pairs: these two levels are called A and B and areseparated by an energy δE (which represents the breaking of the orbital degeneracy). They may be differently coupledto the reservoirs and we call respectively Γ A and Γ B these couplings (they may be different for left and right reservoirs,but we will neglect this point in the following of this part).As stated above, the π -junction in a SL-QD with an odd number of electrons is possible since the Cooper paircannot be transfered without spin-flip during the cotunneling. But if other levels (empty or full) are available, thisis not true anymore since a new path become available for electrons. This has been experimentally pointed out in2006 by Van Dam et al. [4]. In this experiment, the Josephson effect has been measured in an InAs nanowire where δ EAB Γ A Γ B V sd V g U+ Δ E U -
1 2 3 0 α (U+ Δ E) α U E δ E Δ E δ E 2 δ E2 δ E' N=1 δ E' Figure 4: Left : Schematic of a two-level quantum dot such as the one we are interested in. The two energy levels, called Aand B, are separated by an energy δE . They may be differently coupled to the reservoirs. Right : Typical stability diagramexpected in a clean CNT, of charging energy U , where the energy levels are separated by ∆ E and the orbital degeneracy islifted by δE . The occupancy is indicated on the top of each Coulomb diamond. We call α the proportionality factor betweenthe energy level and the applied gate voltage Vg . The spacing between the inelastic cotunneling peaks in the double occupancy(N=2) is called δE (cid:48) . δE (cid:48) (cid:54) = δE because of exchange interactions [47]. Inset: corresponding electronic configuration for N=1. the Kondo effect was negligible and several levels participated to transport. They measured the supercurrent at afixed phase ϕ = π/ as a function of the gate voltage (this quantity can roughly be seen as the critical current forharmonic current-phase relations, with a positive sign for 0-junctions and negative for π -junctions). The supercurrentin some diamonds is quite surprising: it becomes negative for an even number of electrons. In addition to that, thesupercurrent is not symmetric compared to the center of the diamond. Such a behavior was actually predicted in1998 by Shimizu et al. [48] and is due to the participation of several levels to transport. In this work the authorshave shown that in a multi-level regime, using a random distribution of energy levels with different couplings, both 0and π -junctions are possible for both odd and even occupancies.To our knowledge, there is no other measurement of Josephson effect in a multi-level QD (except in ref. [49], butthis question is not raised explicitly). It seems that all the previous measurements of supercurrent realized in CNTwere in the SL regime ( δE (cid:29) U, Γ ). In the next section, some theoretical predictions in the two-level regime arepresented.
1. Two-level regime predictions without Kondo effect
The supercurrent in a two-level quantum dot has been calculated by Shimizu et al. using an Anderson-like Hamil-tonian and neglecting the Kondo correlations[48]. For each co-tunneling event, there are a priori
24 = 4! sequencespossible to transfer a Cooper pair through the QD. The sign of the contribution to the current of each sequenceis given by the number of electron permutations ( i.e. the number of processes with spin-flip) and the sign of thewave-function (see ref. [48] and supp. informations of ref. [4]).In the single level regime, when there is one electron in the dot, only six of them are available, and they contributenegatively to the current. When there are two electrons in the dot, only some processes giving positive contributionsto the currents are allowed. In a two-level QD, a second path is available, such that the twenty-four processes are apriori possible for both even and odd occupancies. That is why a π -junction is not necessary related to a doubletstate.Still without considering the Kondo effect, Yu et al. focused on the carbon nanotube case, with two spin-degeneratelevels [50]. Two parameters, specific to CNT, are taken into account: the small energy δE between the two levels, anda new parameter, T , which quantifies how much the orbital degree of freedom is conserved during tunneling: T = 0 if it is conserved and is non-zero otherwise. The main influence of δE/ ∆ is to increase the size of the N=2 diamond.The influence of orbital mixing ( T (cid:54) = 0 ) is more interesting: it suppresses the π behavior at N = 2 .Note that Droste et al. [36] and Karrasch et al. [35] have also calculated the supercurrent in double dots in series,a system a priori different from a CNT where both levels are connected to both reservoirs. The results are thus notdirectly transposable.Note as well that, in all these articles, no current-phase relations are presented in the ML-regime. Current-phaserelations are however presented in the article by Lee et al. , but it concerns only the 2L-QD occupied by two electrons. F φ0 π-π F φ0 π-π I φ0 π-π I c I c I c I φ0 π-π I c F φ π-π I φ0 π-π0-junction π-junctionπ'-junctionF φ π-π I φ0 π-π0'-junction (a) (b) (c) (d) Figure 5: Current-phase relation and free energy F (see text) for four different kinds of Josephson junctions.
2. What about the Kondo effect?
When the Kondo correlations are taken into account, the situation becomes much more complex. It has for examplebeen taken into account in an article by Lee et al. [34], but only in the case of a double occupancy of the 2L-QD.Zazunov et al. [33] have tackled more specifically the case of the SU(4) Kondo effect (for N=1 occupancy), Lim etal. [51] did it in presence of spin-orbit coupling. The calculations provide also a rich phase diagram.
E. 0- π transitions In the single-level regime, we have presented a transition from a 0 to a π -junction, originating from a crossing ofAndreev levels, induced by an interplay between Coulomb interactions and the superconducting correlations, helpedby the Kondo effect. However 0- π transitions can happen in other situations that are reviewed here.
1. Back to the π -junctions In order to generalize the notion of π -junction, we can consider the free energy F of the junction, from which theJosephson current is calculated with I = − e (cid:126) ∂F∂ϕ [52].If the current-phase relation is I = I c sin( ϕ ) (+ harmonics), the minimum of F is at ϕ = 0 : this is why wecall it a 0-junction (see fig. 5 (a)). On the other hand, for a CPR I = I c sin( ϕ + π ) (+ harmonics) the minimumenergy is at ϕ = π : this is a π - junction (fig. 5 (d)) [52]. What about the CPRs predicted at the single-level 0- π transition, as represented on fig. 3? We argued that the CPR has a 0 or a π behavior depending on the value of thesuperconducting phase, and is thus neither 0 nor π state. The situation is depicted on fig. 5 (b) and (c), taking intoaccount a temperature broadening : depending on the proportion of 0 and π behaviors, the global minimum of thefree energy is at ϕ = 0 or π . But there is as well a local minimum at respectively ϕ = π and : that is why they arecalled a (cid:48) or π (cid:48) junctions [17]. In the SL 0- π transition described above, the system thus transits from 0 to π through0’ and π (cid:48) states. Is it general to all 0- π transitions? Does it depend on their physical origin? What are the differentkinds of 0- π transitions? That’s the questions we try to address now.
2. Various 0- π transitions a. Single level Quantum Dot 0- π transition As stated above, this 0- π transition is the consequence of a transitionfrom a doublet to a Kondo/BCS singlet, driven by the ratio ∆ /T K . In this context, it is worth noting that the criticalcurrent is expected to be non-zero at the transition, since the system transits through 0’ and π (cid:48) states. b. Two-level Quantum Dot 0- π transition In a two-level QD, transitions from π to 0-junctions may happen whenthe contributions of the various processes (those leading to positive and negative current) vary with the gate voltage.How is the CPR at the transition? Lee et al. answer this question in the case of a double occupancy of the 2L QD.When the 0/ π transition corresponds to a transition between two different ground states (singlet/triplet transitions),it goes through intermediate (cid:48) and π (cid:48) states (with the corresponding anharmonic CPR). However, when the 0- π transition does not involve any change of magnetic state of the dot, the amplitude of the supercurrent simply goes tozero. To our knowledge, there is no generalization of this work for single occupancy of the 2L QD. c. SFS π -junctions/Zeeman π -junctions The most famous way of fabricating a π -junction is to make a ferromag-netic Josephson junction, a thin ferromagnetic layer sandwiched between two superconductors (SFS). In this context,this is equivalent to applying a Zeeman magnetic field. In a ferromagnetic material or in presence of a Zeeman field, itexists an exchange energy E ex between the spins up and down (see refs. [53, 54] for SFS junctions or [55] for Zeemanfield π -junctions). This exchange energy induces a phase shift between the electron and the hole of the Andreev pairafter propagation in the junction of length d f : ∆ φ = 2 E ex d f / ( (cid:126) v F ) . It follows an oscillation of the superconductingorder parameter, leading to a π -junction when ∆ φ ∈ [ π/ , π/ [56].Another way to see the phenomenon is to remark that the spin-degeneracy of the ABS is lifted by the ex-change/Zeeman field [55]. Depending on its amplitude, the position of the ABS is modified, potentially leadingto π -junctions or 0’/ π (cid:48) -junctions in the case of ABS crossings.To the best of our knowledge, there exists only very few measurement of Zeeman π -junctions, observed in a bismuthnanowire, benefiting of a large g-factor [57] or HgTe quantum wells [58]. However, in SFS junctions, the phenomenonhas been largely investigated and is still an intense subject of research, in relation with superconducting spintronics.The 0- π transition in SFS junctions, driven by the thickness of the F layer, has been observed by Kontos et al. in 2002[56] through critical current measurements, followed in 2003 by the phase dependence [59]. If the temperature T is ofthe order of the exchange energy, the 0- π transition can also be driven by the temperature [60]. The current-phaserelation has been measured at this T-driven 0- π transition by Frolov et al . [61], giving an interesting result: at thetransition, the amplitude of the CPR ( i.e the critical current) vanishes, or at least becomes too small to be measured.This result is in conflict with most theories predicting that, even though the first harmonic vanishes at the transition,it should remain a least a double harmonic contribution, so that the critical current is non-zero at the transition [62].This second harmonic has been indeed detected by Shapiro steps measurements done in 2004 [63]. If we considerthe transition as originating from a crossing of ABS, there should obviously be higher harmonics in the CPR at thetransition, but they may be difficult to measure at the temperatures at which the transition happens (of the order of3K). d. Controllable Josephson junctions If an SNS junction is thermally excited, the ABS above the Fermi energycan be populated. Then, they participate as well to the supercurrent giving a negative contribution: the amplitudeof the supercurrent can be controlled [64] or even reversed [65]. Because of the phase-dependence of the ABS, it iseasier to populate the excited levels for superconducting phases around ϕ = π . That is why 0’ and/or π (cid:48) phase arepredicted and measured [66]. It is interesting to note that the measured current-phase relations at the transition havethe same unexpected even symmetry as the one measured by Maurand et al. .
3. Quantum phase transitions?
We have presented a number of 0- π transitions, whose common characteristic is a sign reversal of the supercurrent.These sign reversals have a wide variety of origins: singlet-doublet transition, thermal excitation or a Zeeman/exchangesplitting of the ABS, two-level regime in QDs.Is it possible to classify these transitions among the (quantum) phase transitions?A quantum phase transition (QPT) is a phase transition happening at zero temperature, induced by the variationsof a parameter different from temperature [67]. As any phase transition, it results from the competition betweendifferent ground states of the system, leading to different macroscopic phases. In this respect, the 0- π transitioninduced by thermal excitation by Baselmans et al. [66] is not a QPT.Just like classical phase transitions, QPTs can be classified in first order and second (or higher) order transitions.First order ones result from a level crossing, such that the two states coexist during the transition. In a second orderQPT, the transition is continuous and driven by quantum fluctuations instead of thermal ones [68]: there exists aquantum critical region which is described by critical exponents, providing a very rich physics.Speaking of quantum phase transition in a quantum dot is questionable. Is the reversing of the sign of the super-current a change of phase? However, if it is the manifestation of a transition from a doublet to a singlet ground state,then this terminology is clearly suitable provided that this ground state is a collective one.Some quantum phase transitions have been investigated in quantum dots. Mebrahtu et al. [69] studied resonanttunneling in a Luttinger liquid formed in a CNT between two barriers, where a second order QPT occurs varying thesymmetry of these barriers. Roch et al. [70] have measured an infinite order QPT (Kosterlitz-Thouless transition)in a single molecule QD, where the system transits from a singlet to a triplet ground state varying the gate voltage.In both cases (singlet and triplet), the Kondo effect is involved, inducing a correlated state between the dot and theleads: it is appropriate to speak of QPT. Another example of QPT of second order is given by the investigation ofthe crossover between one and two-channel Kondo effect by Iftikhar et al. [71]. I φπ-π 0 F φ π-π I φπ-π 0 F φ π-π (a) (b)First ordertransition Second ordertransition ? Figure 6: Qualitative representation of free energy and the current-phase relation expected for two kinds of 0- π transitions: (a)first order transition, as predicted for the 0- π transitions due to level-crossing and (b) an hypothetic second order transition.In the first order case, at the transition, both 0 and π states coexist at different ϕ , the critical current never vanishes and theCPR is anharmonic. In the second order case, the amplitude of the CPR decreases until 0, where the sign change occurs, andincreases again in the other state. In the case of the 0/ π transition in a single-level QD, the use of the term quantum phase transition is reasonable.It is clearly due to a crossing of Andreev levels, and is thus a first-order QPT [14]. Typical free energies F and thecorresponding CPRs are represented on fig. 6 (a): at the transition, F has two minima, one at 0 and the other at π (one of these minima can be local and the other global). That is why the CPR is strongly non-harmonic and itscritical current never vanishes. The transition is discontinuous since, between the 0 and π states, one can find mixstates (0’ and π (cid:48) ). Note that there is no quantum criticality associated to this transition.0- π transitions in SFS junctions and induced by a Zeeman field are also due to a level-crossing, the CPR at thetransition should be similar to fig. 6 (a).The situation is more complex in two-level quantum dots. According to Lee et al. who calculated the case of doubleoccupation, while 0- π transitions are the consequence of a transition from a magnetic state to the other (singlet,doublet, triplet), the system makes a transition through the intermediate states 0’ and π (cid:48) . But as soon as the 0- π transition does not involve a change of ground state of the system itself, there is no intermediate CPR and the criticalcurrent vanishes at the transition, as represented on fig. 6 (b). Then the amplitude of the CPR decreases untilvanishing at the transition, and increases again but with a sign reversal. In this case, there is no intermediate state0’ or π (cid:48) , it is the kind of transition that would be expected for a second order phase transition. Does it mean thatthese transitions without change of magnetic state are second order transitions or simply that they are not anymorequantum phase transitions? Is it possible to extend this result to single occupancies? According to us, these questionsremain open. III. MATERIALS AND METHODSA. Principle of the measurement
We present now our measurements of current-phase relation in a carbon nanotube quantum dot. The principle ofthe measurement, as well as the specific experimental setup used in this work is described in refs. [72–74]. The CNTQD Josephson junction is embedded in an asymmetric SQUID (Fig.7). We extract the current-phase relation of theCNT QD junction by measuring the switching current of the SQUID as a function of magnetic flux.0 C A B Φ B A C CNT I sin( φ ) φ I φ φ tunnelojunction carbononanotube (a) (b) I sin( φ ) I CNT f( φ ) V sd Figure 7: (a) Schematic of the measured asymmetric SQUID, containing two reference JJs in parallel with a CNT based Joseph-son junction. To phase bias the CNT junction, a magnetic flux Φ is applied with a magnetic field perpendicular to the SQUID. (b)Scanning electron microscopy image of the sample, with the layers constituting the tunnel junctions and the contacts of the CNT.Depending on the samples the first layer is made of Pd(7 nm)/Al(70 nm) (sample S-Al) or Pd(7 nm)/Nb(20 nm)/Al(40 nm)(sample S-NbAl).
1. Sample fabrication
The CNT are first grown by chemical vapor deposition on a silicon wafer covered by an oxide layer. We chose touse a doped silicon wafer so that it can play the role of a back-gate for the nanotube.The nanotube contacts and the tunnel junctions are made during one unique lithography step. A first aluminum-based multi-layers is deposited with an angle of 15°, is oxidized under oxygen and is covered by a second layer ofaluminum (120 nm) (fig. 7). The contacts of the nanotube are separated by a distance L = 400 nm . In this workwe have measured samples with contacts made of a niobium layer between palladium and aluminum ( ∆ P dNbAl =170 ± µeV , called S-NbAl) : Pd(7 nm)/Nb(20 nm)/Al(40 nm). The sample measured are similar to the onerepresented on fig. 7 (b).The sample is then cooled down in a dilution refrigerator of base temperature 50 mK. The phase difference acrossthe CNT-junction ϕ is controlled applying a magnetic field B perpendicularly to the sample. The magnetic fluxenclosed by the loop is Φ = B × S , with S ≈ µ m the loop area. B. Characterization of the sample in the normal state
The samples are first characterized in the normal state. To do so, we measure the differential conductance dI/dV sd of the sample as a function of the bias voltage V sd and the gate voltage V g in the normal state. This is done using astandard lock-in-amplifier technique and applying a magnetic field large enough to suppress the superconductivity inthe contacts. We explore different Coulomb diamonds, at different gate voltages, which correspond to different fillingfactor and parameters (Table I, [72, 73]). IV. CURRENT-PHASE RELATION MEASUREMENT IN A CNT QUANTUM DOT
We now restore superconductivity, by switching off the magnetic field, so that we can measure the current-phaserelation (CPR) of the QD JJ. In the following we focus on a sample with S-NbAl contact. A more detailed study canbe found in ref. [72, 73].
A. Current-phase relation in the single-level regime
In this part, we discuss the phase dependence of the supercurrent in the single-level regime, focusing on one samplewith occupation number N = 1 (called I to be consistent with [73]).The modulation of the switching current δI s of the SQUID versus magnetic field B, proportional to the CPR, ismeasured for various V g and is represented on Fig.8a together with CPRs extracted from the 0- π transition.1 (1)(3) (2) (4) (5) (6) π0 0 (a.2) Vg=6.238V (a.6)
Vg=6.2V (a.1)
Vg=6.249V (a.3)
Vg=6.234V (a.4)
Vg=6.232V (a.5)
Vg=6.23V (a) g (b)(c) Figure 8: (a) Modulation of the switching current of the SQUID δI s , proportional to the CPR, as a function of the magneticfield B and the gate voltage V g . Vertical cuts at the 0- π transition are represented for ridge I, showing the whole transition.The dashed lines are guides to the eyes and represent the contributions of the singlet (0-junction, in blue) and the doublet state( π -junction, in red). (b) Definition of the critical phase ϕ C such that the CPR has 0-behavior for ϕ ∈ [0 , ϕ C ] and π -behaviorfor ϕ ∈ [ ϕ C , π ] . (c) Critical phase ϕ c plotted as a function of (cid:15) d , yielding a phase diagram of the ϕ -controlled transition. Wecall δ(cid:15) the width of the transition. The different samples correspond to the ones described on table I. More details can be foundin ref [72–74]. On the edges of the diamonds, far from the transition (fig. 8 (a.1)), the junction behaves as a regular 0-junction,with a CPR proportional to sin( ϕ ) . In contrast, at the center of the diamond (Fig. 8.(a.6)), where T K is minimum,the CPR is π -shifted ( δI s ∝ sin( ϕ + π ) ) and has a smaller amplitude, characteristic of a π -junction. In between, theCPR is anharmonic: a distortion appears first around π and develops as T K decreases. The CPR is composite, with0-junction behavior around ϕ = 0 and a π -junction behavior around ϕ = π . The transition from one part to the otheris achieved by varying the superconducting phase. The global state of the system is called 0’ or π (cid:48) depending on whatis the dominant contribution (and where is the global minimum of the free energy, see section II E). a. Comparison with QMC calculations For a quantitative comparison between theory and experiment, in collab-oration with D. Luitz and V. Meden, we performed a CT-INT calculation in the superconducting state ( B = 0 ) indiamond I to obtain the CPRs in the transition regime [72]. Using the measured value of the superconducting gap ∆ = 0 .
17 meV and the previously determined parameters, the Josephson current has been computed as a function ofthe phase difference ϕ . The theoretical CPR are calculated at various (cid:15) d (related to V g by (cid:15) d = αV g ) and plotted asblack lines in comparison to our experiments in fig. 8 (a.2) to (a.5). Since our setup yields a switching current thatis necessarily smaller than the critical current, the experimental CPRs were multiplied by a unique correction factorchosen to obtain the best agreement with the QMC results. The agreement for the shape of the CPR is excellent;however a shift of the energy level δ(cid:15) d = 0 .
28 meV of the theoretical CPRs is needed to superimpose them with theexperimental ones: the QMC calculations predict a transition region centered around a smaller (cid:15) d than measuredexperimentally [72]. Note however that the width of this transition is very well reproduced. Interestingly, these datahave been fitted by Zonda et al. [75] with a perturbative theory: they argue that, with a charging energy of . instead of . , the theory works perfectly. It seems that this discrepancy between experiment and theory wouldoriginate from a bad estimation of U .These data are thus consistent with a phase controlled level-crossing quantum transition in a single-level QD. In2 Sample B C G I J rightN 1 3 ∆ Γ ∆ E δE k B T K ≈ ≈ δ(cid:15)/U (cid:15) t /U ∆ , Γ , ∆ E , δE , T K , δ(cid:15) and (cid:15) t , given in meV for the investigated diamonds. For diamondsB, C, G and I, there are two transitions (0 to π and π to 0), with different parameters. In diamond J, only the right side ofthe diamond exhibits a phase dependence of the transition. other words, one can control the magnetic state of the junction, doublet or singlet, with the superconducting phase.
1. Universal phase diagram of the first order transition
We present now a more quantitative study of the level-crossing quantum transition in the single-level regime. Wecall ϕ c the superconducting phase at which, at a fixed gate voltage, the system undergoes the transition from 0to π . Theoretically, this critical phase ϕ c is defined at T=0, where the transition is expected as a jump in thesupercurrent. At finite temperature the transition is rounded but, if T is small enough, ϕ c equals the phase at whichthe supercurrent is zero [19, 72]. On fig. 8 (c), we show ϕ c ( (cid:15) d ) from experiment for different 0- π transitions. In thisfigure each transition is characterized by two parameters: the value of (cid:15) d , called (cid:15) t , at which the junction changesfrom π to 0 at ϕ = π/ , and the width δ(cid:15) of the transition. These quantities are given in table I for the concerneddiamonds (B, C, G and I). δ(cid:15) is found to depend strongly on the parameters of the diamonds: large transition’s widthscorrespond to ratios T K ( (cid:15) = 0) / ∆ close to 1 (see left inset of fig. 8). To compare these eight transitions (left andright sides of four diamonds), we plot on fig. 8 the critical phase ϕ c as a function of ( (cid:15) d − (cid:15) t ) /δ(cid:15) . For diamonds B, Gand I, the scaled data fall on the same curve, with an arccosine dependence.All the transitions display the same characteristic shape. We try as well to compare these data to an analyt-ical formula obtained in the atomic limit of the Anderson impurity model with ∆ (cid:29) Γ and T = 0 : ϕ c ( (cid:15) ) =2 arccos (cid:112) g − ( (cid:15)/h ) : a fit of our data with this formula gives a very good agreement for g = 2 . and h = 0 .
72 meV .Note that since we are not at all in an atomic limit, these fit parameters do not correspond to the any physical oneof the experiment.
B. Effect of the two-level regime on the 0- π transition Now we focus on the two-level 0- π transitions, and more particularly on diamond J, corresponding to a N=3 fillingfactor. The modulation of I s versus the magnetic field, proportional to the CPR, is represented on Fig. 9 (a) as afunction of V g . CPRs are also shown for some particular values of gate voltage (fig. 9 (b)).On the right side of the diamond, close to the N = 3 to degeneracy point (see right part of fig. 4), the 0- π transition is achieved through 0’ and π (cid:48) states, similarly to the one investigated in the single-level regime (fig. 9 (b)(8)). In addition, ϕ c = f (( (cid:15) − (cid:15) t ) /δ(cid:15) ) collapses on the same arc-cosine shape as the single-level 0- π transitions (seefig. 8). But on the left side of the diamond, from the N = 3 to degeneracy point to the center of the diamond(fig. 9 (b) (1) to (3)) , the CPR behaves as a 0-junction. The supercurrent’s amplitude decreases with V g and evolvesfrom 0 to π continuously. Close to the transition, around the center of the diamond (fig. 9 (b) (2) to (4)), the CPR3 (1) (2) (3) (4) (5)(6) (7) (8) π-J two-level 0-J Single- level 0-J (1) (2)(3) (4) (5)(6) saturated colors (b) (7) (8) (a)
Figure 9: (a) Modulation of the switching current of the SQUID versus the magnetic field, proportional to the CPR, as afunction of the gate voltage V g , for the diamond J. The supercurrent at the transition being very low, the color scale issaturated. (b) Supercurrent versus the superconducting phase ϕ at some particular gate voltage, indicated by the numbers onpanel (a). Note the zero amplitude of the CPR (4) and the fact that the CPR (2), in the two-level 0-junction, has a strongeranharmonicity than (8), on the degeneracy point. Dashed line on (2): guide for the eyes representing a sine function, showingthat the continuous line is not perfectly harmonic.More details can be found in ref [73, 74] N=1
N=3 π (a) (b) d ε d =1.1 meV2 levels, ε d =0.1 meV φ / πφ / π I ( n A ) I ( n A ) (c) N=3
N=1 BA δ E Figure 10: (a) Critical current I c , defined as the maximum amplitude of the measured switching current, as a function of theenergy level (cid:15) . I c is defined as positive for a 0-junction, and negative for a π -junction. This quantity is plotted for the twodiamonds I (N=1, blue dots) and J (N=3, orange squares), respectively in the single-level and two-level regime. The dashedlines materialize discontinuities of I c , specific of first order transitions. Inset: focus on two 0- π transitions, centered around (cid:15) t : at (cid:15) d = − . in diamond I, and at (cid:15) d = 0 . in diamond J. (b) Current-phase relations around the 0- π transitionin diamond I, at (cid:15) d = 1 . , indicated by a green arrow. The CPR becomes anharmonic and the critical current nevervanishes. (c) Current-phase relations around the 0- π transition in diamond J, at (cid:15) d = 0 . , indicated by an orange arrow.The CPR is harmonic all over the transition, and the critical current vanishes at the transition. (c) Schematic explanation ofthe symmetry breaking observed in the supercurrent between N = 1 and N = 3 fillings. The lowest energy level A is bettercoupled to the reservoirs than the highest one B (see text). More details can be found in ref [73, 74]. becomes slightly anharmonic. But, unlike in the other transitions, no 0’ or π (cid:48) state is observed. Note as well that thecurrent-phase relation has a larger anharmonicity in the two-level 0-junction than on the charge degeneracy points,while its amplitude is smaller (fig. 9 (b) (2), to compare with fig. 9 (b) (1) or (8)).To go further, we consider now the critical current I c . This quantity is the maximum of the CPR and is extractedhere as the maximum amplitude of the modulation of the switching current, positive for a 0-junction, and negative fora π -junction. It is represented as a function of (cid:15) d on fig. 10 (a), for diamonds I and J, respectively in the single and4two-level regime. In diamond I (N=1), the phase-dependence of the single-level transitions gives rise to discontinuitiesof I c , which characterize a first order transition [14, 67]. For diamond J (N=3), the 0- π transition at (cid:15) d = 0 . does not exhibit this phase-dependence, yielding a vanishing critical current at the transition. This is not anymorea first order transition, contrary to the transition at the other side of the diamond J ( (cid:15) d = 1 . ). According toref. [34], this kind of 0- π transition, without intermediate state 0’ or π (cid:48) , is indeed possible when there is no changeof the magnetic state of the system. This emphasizes that we are facing a transition between a 0-doublet state and a π -doublet (instead of a 0-singlet and a π doublet as in the single level regime and on the right side of the diamond),specific to the two-level regime.This kind of gate dependence of the supercurrent is predicted in a carbon nanotube QD (see section II D), in absenceof Kondo effect. The comparison of our data with ref. [50] suggests that in our experiment, the channels associatedwith each orbital are mixed during the transfer of Cooper pairs. This is also why we do not observe two-level induced π -junctions for even occupancies of the dot.To explain why this two-level behavior is observed for N = 3 but not for N = 1 , we propose that the two orbitallevels A and B (see fig. 10) of the CNT are slightly differently coupled to the electrodes, as in ref. [76]. A detailedanalysis of the gate dependence of the inelastic cotunneling peaks in the even diamond between I and J in the normalstate shows indeed that Γ A ≥ Γ B . Following ref. [76], we roughly evaluate Γ A − Γ B ≈ .
07 meV .When two quasi-degenerated levels have different widths, the supercurrent is mostly carried by the broader one.We therefore expect different behaviors for N=1 and N=3, as pointed out theoretically by Droste et al. [36]. Here, thelowest level A is more coupled to the electrodes than the highest one (B). For N = 1 , the unpaired electron occupiesthe level A and the level B is too poorly coupled to participate to the transfer of Cooper pairs: we are in a single-levelsituation, the junction is π . For N = 3 , the unpaired electron is in the poorly-coupled-level B, which thus participatesto the transport: the system is in a two-level regime. According to this interpretation, in the opposite situation of alevel B better coupled than the level A, the N = 1 diamond would exhibit the two-level physics instead of the N=3diamond.There could be another reason for the electron-hole (N=1/3) symmetry to be broken in a carbon nanotube: spin-orbit coupling [36, 77, 78]. According to Brunetti et al. [79], the consequence on the supercurrent could be similarto what we observe. It is however difficult to be affirmative, since we are not able to measure the value of the SOcoupling in our system. It would indeed require to study the evolution of cotunneling peaks versus the magnetic fieldparallel to the nanotube.
1. Toward a ϕ -junction? On the CPR measured in the two-level regime, represented on fig. 11, it seems that, on the edges of the gate voltageregion, the CPR exhibits a small continuous phase shift.Until now, we have insisted on the fact that a current-phase relation is necessary an odd function of the phase: I ( − ϕ ) = − I ( ϕ ) such that I ( ϕ = 0) = 0 . But if the time reversal symmetry (TRS) is broken in the system, it ispossible to observe a non-zero current for a zero phase bias, called the anomalous current I a which can be defined as: I = I sin( ϕ ) + I a cos( ϕ ) (2)(with harmonics if the CPR is non-sinusoidal). Equivalently, we can write the supercurrent as I = I c sin( ϕ + ϕ ) , thisis why such a junction is called ϕ -junction.To quantify the measured effect, we represent on fig. 11 (c) the results of fit for both ϕ and I , I a as defined above.It seems indeed that a φ junction develops on the edges of the diamond.Is the feature on our measurements related to a breaking of the TRS? To assure this, one should prove thereproducibility of the effect, as it could be due for example to the trapping of a vortex in the coil. However, thesymmetry of the effect on both edges of the diamonds makes us think about something related to Coulomb blockade.Then the effect could be due to the presence of spin-orbit coupling in the nanotube, as well as to a small magneticfield associated to the participation of two-levels to the transport of Cooper pairs. Zazunov et al. [80] as well as Brunetti et al. [79] have shown that a substantial anomalous current could be observed in this kind of system, especially inoddly occupied diamonds, where Coulomb interactions enhance the effect. This requires further investigations, inparticular measuring other samples. V. CONCLUSION
In conclusion, we have measured in a carbon nanotube quantum dot junction the supercurrent as a function of thesuperconducting phase across it. We have measured this quantity in the regime where the Kondo and superconducting5 (a) (b)(c)
Figure 11: (a) Current-phase relations for various values of gate voltage on the left edge of the gate voltage region (square inblack on (b)). We can see that these CPRs look sinusoidal and that some of them are slightly dephased (in particular the redand orange). (c) Result of sinusoidal fits all over the diamond. In blue and pink are represented the critical current I and theanomalous current I a from I = I sin( ϕ ) + I a cos( ϕ ) , in green the phase shift ϕ from I = I c sin( ϕ + ϕ ) . Note that the errorbar diverges at the 0- π transitions: at the center where the amplitude almost vanishes, and on the right edge where the CPRsare strongly anharmonics. correlations are of the same order of magnitude and shown that the ground state of the system, singlet or doublet(corresponding respectively to 0 and π junctions), is then controlled by the superconducting phase, giving rise tostrongly anharmonic current-phase relations. We have also demonstrated that, if a second energy level participate inthe transport of Cooper pairs, the 0- π transition is not anymore a first order one, as it is the case when only one levelis involved. Acknowledgments
The authors acknowledge S. Guéron, J. Basset, P. Simon, A. Murani, F. Pistolesi, S. Florens and M. Filippone fordiscussions and technical help from S. Autier-Laurent. This work was supported by the French program ANR MASH(ANR-12-BS04-0016), DYMESYS (ANR 2011-IS04-001-01) and DIRACFORMAG (ANR-14-CE32-0003).
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