Manipulating the magnetic state of a carbon nanotube Josephson junction using the superconducting phase R. Delagrange,1 D. J. Luitz,2 R. Weil,1 A. Kasumov,1 V. Meden,3 H. Bouchiat,1 and R. Deblock1 1Laboratoire de Physique des Solides, Univ. Paris-Sud, CNRS, UMR 8502, F-91405 Orsay Cedex, France. 2Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse and CNRS, 31062 Toulouse, France 3Institut für Theorie der Statistischen Physik, RWTH Aachen University and JARA—Fundamentals of Future Information Technology, 52056 Aachen, Germany Themagneticstateofaquantumdotattachedtosuperconductingleadsisexperimentallyshown to be controlled by the superconducting phase difference across the dot. This is done by probing the relation between the Josephson current and the superconducting phase difference of a carbon nanotubejunctionwhoseKondoenergyandsuperconductinggapareofcomparablesize. Itexhibits distinctively anharmonic behavior, revealing a phase mediated singlet to doublet transition. We obtain an excellent quantitative agreement with numerically exact quantum Monte Carlo calcula- 5 tions. Thisprovidesstrongsupportthatweindeedobservedthefinitetemperaturesignaturesofthe 1 phase controlled zero temperature level-crossing transition originating from strong local electronic 0 correlations. 2 PACS number(s): 74.50.+r, 72.15.Qm, 73.21.-b, 73.63.Fg n u J When a localized magnetic moment interacts with a implies reversing the order of particles within this pair. 0 Fermi sea of conduction electrons, the Kondo effect can This leads to a gate-controlled sign reversal of the CPR 1 develop: spin-flip processes lead to a many-body singlet whentheparityofthenumberofelectronsischanged,as state in which the delocalized electrons screen the mo- was observed experimentally [7–10]. In contrast, if the ] l ment. Quantum dots (QD) in the Coulomb blockade Kondo effect and thus local correlations prevail, the spin l a regime and particularly carbon nanotube (CNT) dots of the dot is screened by unpaired electrons leading to h constitute ideal systems for the investigation of Kondo a singlet ground state: the 0-junction is then recovered - s physics at the single spin level [1–3]. In these systems, it even though the parity of the dot is still odd. e m is possible to control the number of electrons on the dot The switching from 0- to π-junction behavior as a varying a gate voltage. For an odd occupation, the dot function of a variety of energy scales of the QD JJ in t. accommodatesamagneticmomentwhichisscreenedpro- the presence of local correlations was extensively stud- a m videdthatthisisnotprohibitedbyanenergyscalelarger ied theoretically [11–20]. The scales are the broadening - than the Kondo energy kBTK. Temperature is the most Γ of the energy levels in the dot due to the coupling to d obvious obstacle to the development of the Kondo effect the reservoirs, the superconducting gap ∆ of the con- n since TK can be smaller than 1K. However, if temper- tacts, the dot’s charging energy U and its level energy o ature is sufficiently low, the Kondo effect may compete (cid:15). When these parameters fall into the 0-π transition c with other quantum many-body phenomena such as su- regime, it was predicted that the ground state of the [ perconductivity, forwhichtheformationofCooperpairs system—singlet or doublet—depends on the phase dif- 3 of energy ∆ may prevent the screening of the dot’s spin. ference ϕ, undergoing a level-crossing transition. This v Thissituationcanbeinvestigatedusingsuperconducting leads to a characteristic anharmonicity of the CPR for 7 5 hybrid junctions, where a supercurrent is induced by the temperatureT >0andajumpatacriticalphaseϕ for C 3 proximity effect, for example in CNT-based QDs [4] or T = 0. In other words, in this particular regime of pa- 3 semiconductor-based ones [5]. rameters,themagneticstateofthedotispredictedtobe 0 governed by the superconducting phase difference across A setup of a high resistance tunnel barrier between . 1 the junction. twosuperconductors,alsocalledJosephsonjunction(JJ), 0 5 carries a supercurrent I =ICsinϕ, with the critical cur- ThisKondorelated0-π transitionwasearlierobserved 1 rent I . The superconducting phase difference across experimentallyasafunctionofthegatevoltage[9,21,22] C : thejunctionϕcontrolstheamplitudeandthesignofthe and the spectroscopy of Andreev bound states enabled v supercurrent. This is the Josephson relation, the most a better understanding of the involved physics [23–27]. i X famous example of a current-phase relation (CPR). In Measurements of the CPR of a QD JJ embedded in a r some peculiar systems such as ferromagnetic supercon- SQUID were also performed [22] and indeed showed an- a ductingjunctions,thetransmissionofCooperpairsgives harmonicities. However, in the region of 0-π transition, risetoaπ phaseshiftoftheCPR[6]. InQDJJs(tunnel the obtained CPRs are not odd functions of flux as they barrier replaced by QD) in the strong Coulomb blockade should be, indicating that the physics is spoiled by other regimewheretheKondoeffectisnegligible,suchaπshift effects [28, 29]. is observed as well since the tunneling of a Cooper pair Here we report on the successful measurement of the 2 The CNTs are grown by chemical vapor depo- b a 5.3 5.7 6.1 6.5 6.9 sition on an oxidized doped silicon wafer [33]. 1µm JJ A three-junctions SQUID is constructed around I A a selected nanotube with the following materials: Φ Pd(7 nm)/Nb(20 nm)/Al(40 nm), AlOx and Al(120nm) B C CNT 0e-123 01 23 0 123 01230 1[c3h20a]m3. bTerheofsaamdpilleutiisonthreerfmriaglelryataonrchoofrbedasetotetmhepemraixtuinrge 50 mK and measured through low-pass filtered lines. A c d magnetic field B is applied perpendicular to the loop to A modulate the phase difference across the CNT-junction by 2πBS/Φ , with the superconducting flux quantum 0 0e- 1 2 3 0 0e- 1 2 Φ0 =h/2e and S the loop area. We first characterize the sample in the normal state, B measuring the differential conductance dI/dV versus sd bias voltage V for various backgate voltages V , using sd g a lock-in-amplifier technique. The contacts are made of Figure1: aScanningelectronmicroscopyimageofthemea- Pd/Nb/Al with a gap of ∆ = 0.17 meV±10%, a value sured asymmetric SQUID (see text and [32]). b Differential veryclosetothegapofAlbutconsiderablysmallerthan conductance ( dI ) in the normal state of the CNT junction the Nb gap because of the Pd layer. A magnetic field of dVsd versus gate voltage Vg and bias voltage Vsd for a large range 1T is needed to suppress superconductivity in these con- of Vg. Figures c and d focus on two Kondo zones called re- tacts. Eventhoughsuchamagneticfieldsignificantlyaf- spectivelyAandB.A1Tmagneticfieldisappliedtodestroy fectstheKondoeffect,theresultsofFig. 1showCoulomb superconductivityinthecontacts. Thenumberofelectronsin diamondsandanincreaseoftheconductanceatzero-bias the last occupied energy levels is indicated in white. In light blue, dI (V ) at zero bias is plotted (axis on the right). In in some diamonds, a signature of the Kondo effect. The the nodrVmsdal sstdate, the reference JJ contribution is a constant fourfolddegeneracy,characteristicsofcleancarbonnan- whichwassubtractedtoobtaintheplots. Thewhitesymbols otubes with orbital degeneracy [34, 35], is clearly seen in c correspond to the theoretical fit of the conductance (see (Fig. 1 b). This allows us to determine the dot’s occu- text). pancyindicatedonthefigure. Wefocusedontworanges ofgatevoltages,correspondingtodiamondswithoddoc- cupancy,wherenon-zeroconductanceisobservedatzero CPR of a CNT-based hybrid junction over the entire bias, zone A (around V = 6.2 V) and zone B (around g 0-π transition. This constitutes the first experimental V =1.4 V) (see Figs. 1 c and d). Among all the Kondo g demonstration of the 0-π transition controlled by the su- ridges leading to a π-junction in the superconducting perconducting phase ϕ. A very important part of our state, those show the widest extent in gate voltage of analysis is the comparison between the measured CPRs the 0-π transition, which makes the measurements more and theoretical ones, computed for the Anderson model accurate. The height in V of the Coulomb diamonds sd with superconducting leads using a numerically exact gives the charging energy U =3.2 meV±10% in zone A quantum Monte Carlo (QMC) method. The excellent and2.5meV±10%inzoneB.Duetothecomplexinter- agreement provides strong support that we indeed ob- play of the Kondo scale k T and the Zeeman energy, B K servethetransitionresultingfromstronglocalelectronic that are of comparable size, Γ and the contact asymme- correlations. trycannotbedetermineddirectlyfromtheexperimental We fabricated a CNT-based QD, connected to super- results; theoretical modeling is required. conducting leads and embedded in an asymmetric mod- The junction is modeled by an Anderson impurity ified SQUID (Fig. 1 a). This device, a SQUID contain- model [11–20] with right (R) and left (L) BCS super- ing the QD JJ (here the CNT) and a reference JJ with conducting leads and superconducting order parameter critical current high compared to the one of the QD JJ, e iϕ/2∆. TheinteractionofelectronsontheQDisgiven ± allows us to determine the CPR of interest [31, 32]. The by a standard Hubbard term with charging energy U switching current I of the SQUID versus magnetic flux and the coupling of the leads to the QD is described by s is measured. The CPR of the QD JJ is then obtained by theenergyindependenthybridizationstrengthΓ . We L/R extracting the modulation of I around its mean value solveitusingthenumericallyexactCT-INTMonteCarlo s (cid:104)I (cid:105). Our device possesses a second reference JJ and a method [19] in the normal state (∆ = 0) in a magnetic s third connection as described in Ref. [32]. This allows field and calculate the finite temperature linear conduc- us to characterize each junction independently at room tance for different Γ as a function of the dot energy L/R temperature, and to measure both the CPR of the CNT (cid:15), defined relative to particle-hole symmetry. The ampli- and its differential conductance in the superconducting tudes of the magnetic field and the charging energy are state. fixed to the experimentally determined values B = 1T 3 and U =3.2meV of zone A [37]. A comparison with the a b Vg=1.447V (1) measured conductance at zero source drain voltage (i.e. in equilibrium) yields a set of parameters that fit the ex- perimentbest. WefindΓ +Γ =0.44meV,Γ /Γ =4. R L R L Vg=1.416V We also slightly varied T to estimate the electronic tem- (2) perature in the sample and obtain T = 150mK. Addi- tionally, this procedure gives a reliable way of extracting the conversion factor α = 39meV between the applied Vg=1.4V V (3) gate voltage and the dot on-site energy (cid:15) (as done in [20]). Our best fit is displayed in Fig. 1 c (white sym- (3) (2) (1) bols). Afterreliablyestimatingallparameters, usingthe formula TK = (cid:112)ΓU/2exp(−π|4(cid:15)82Γ−UU2|) [36], we can di- rectlyshowthatforzoneAthedotisindeedintheregime Figure3: aModulationoftheswitchingcurrentoftheSQUID ofstrongestcompetitionbetweenKondocorrelationsand for zone B, which exhibits a partial transition from 0- to π- superconductivity with k T ≈∆ (Fig. 4 a). junction. b CPR extracted from the previous data near the B K transition(greencontinuousline). Thedashedlinesareguides to the eyes and represent the contributions of the singlet (0- a b junction in blue) and the doublet state (π-junction in red). 6.16 6.2 6.24 Vg=6.242V This process is reproduced and averaged around 1000 times, the whole procedure being repeated at different 0 π 0 valuesofmagneticfieldbelowafewGauss,smallenough Vg=6.222V topreservesuperconductivity. Toobtainthemodulation (4)(2) π 1.0 1.2 1.4 1.6 of the switching current δIs versus magnetic field, the (6) (5)(3) (1) contribution of the reference junctions (around 90 nA) c is subtracted. As demonstrated in Ref. [32], δI is pro- Vg=6.249V Vg=6.238V Vg=6.234V s (1) (2) (3) portional to the CPR of the CNT junction. It should be noted that this kind of system, in particular near the 0-π transition, is very sensitive to the electromagnetic environment, whichthereforeneedstobeoptimized[28]. V(4g)=6.232V V(5g)=6.23V V(6g)=6.2V The main results of this work are presented in Figs. 2 and3,whereweshowtheextractedCPRsforgatevalues overtheentiretransitionregime(curvesc.1toc.6ofFig. 2 for zone A and curves b.1 to b.3 of Fig. 3 for zone B). We now analyze qualitatively the shape of these curves. On the edges of Kondo zone A, far from the transition Figure 2: a Modulation of the switching current of the (Fig. 2 c.1), the junction behaves as a regular JJ with SQUID, proportional to the CPR of the CNT junction, ver- a CPR proportional to sin(ϕ) (0-junction). In contrast, sus the magnetic field for various gate voltages in zone A. b at the center of the Kondo zone (Fig. 2 c.6) the CPR Modulation of the switching currents near the transition for is π-shifted (δI ∝ sin(ϕ+π)) and has a smaller ampli- severalgatevoltages. cCPRextractedfromthepreviousdata tudecharacteristicforaπ-junction. Inbetween,theCPR and rescaled by 1.33 (see the text) near the transition (green is composite with one part corresponding to 0- and an- continuous line). The theoretical predictions resulting from QMCcalculationsareshownasblacklines. Thedashedlines otherparttoπ-junctionbehavior. Thelatterfirstoccurs are guides to the eyes and represent the contributions of the around ϕ=π, giving rise to a very anharmonic CPR. In singlet(0-junction,inblue)andthedoubletstate(π-junction, themiddleofthetransitionregion,wefindperiodhalving in red). (Fig. 2c.4)[38]. ThisevolutionoftheCPRbetweena0- and π-junction is consistent with the finite temperature Next, superconductivityisrestoredbysuppressingthe transition of the dot’s magnetic state (between singlet 1T magnetic field and the CPR is measured in both and doublet) which is controlled by the superconducting Kondo zones, extracting the modulation of the switch- phase difference [12, 17, 19]. ing current δI versus magnetic field (Fig. 2) from the A more precise analysis of the transition allows to at- s critical current of the SQUID. To measure the switching tribute this 0-π transition to a competition between the current, the SQUID is biased with a linearly increasing Kondo effect and the superconductivity. Indeed, around current with a rate dI =37 µA/s and the time at which the center of Kondo zone A (Fig. 2 a), the π-junction dt the SQUID switches to a dissipative state is measured. extends over a range of 60 mV of gate voltage. Accord- 4 ing to the dI/dV in the normal state (Fig. 1) and the a b sd conversion factor α, the odd diamond has a width in π 0 gate voltage of about 82 mV, larger than the π-junction regime. Consequently,this0toπtransitionisnotsimply due to a change in the parity of the dot filling but to an increase in the ratio ∆/T (see Fig. 4 a). This is even K more obvious for Kondo zone B (Fig. 3 b) where the 0 U/2 to π transition is incomplete. c For a quantitative comparison between theory and ex- periment, we performed a second CT-INT calculation in the superconducting state (B =0) for zone A [37] to ob- taintheCPRsinthetransitionregime. Weusedthemea- sured value of the superconducting gap ∆ = 0.17meV andthepreviouslydeterminedparametersandcomputed the Josephson current as a function of the phase differ- ence ϕ. The theoretical CPR are calculated at various (cid:15) (related to V by (cid:15) = αV ) and plotted as black lines in g g comparison to our experiments in Fig. 2 c2 to c5. Since Figure 4: a Calculated TK (black line) is compared to ∆ our setup yields a switching current that is necessarily (dotted line) as a function of the level energy (cid:15) ((cid:15) = αδVg smaller than the supercurrent, the experimental CPRs with δVg the gate value measured from half-filling and α = 39 meV/V). In green dots, the critical phase ϕ is plotted, were multiplied by a unique correction factor chosen to C defined as the phase, different from 0 and π, for which the obtain the best agreement with the QMC results. The CPRequalszero(seetextandinsetof c). The0-π transition agreement for the shape of the CPR is excellent; how- occurringwhenϕ switchesfrom1to0,thesystemisindeed C ever a shift of the energy level δ(cid:15) = 0.28 meV of the in the regime k T ≈ ∆. b Fourier analysis of the CPRs B K theoretical CPRs is needed to superimpose them with of zone A near the transition for different level energies (cid:15). the experimental ones [30]. The QMC calculations pre- ThecirclescorrespondtotheexperimentalCPRswhereasthe dict a transition region centered around a smaller (cid:15) than continuouslinescorrespondtotheQMCcalculation(red,blue andblack: harmonics1,2and3). cMeasuredϕ (redcircles) measured experimentally (see supplementary materials); C versus(cid:15). ϕ extractedfromtheQMCcalculationandshifted adeviationbetweenexperimentandtheorywhichwecur- C is also shown (blue squares). The black line shows the result rently do not understand. Note however that the width ofatwo-parameterfitoftheanalyticalcurveobtainedinthe of this transition is very well reproduced. atomic limit (see text). The comparison of the measurements and calculations can even be refined employing Fourier decompositions I(ϕ) = a1sin(ϕ)+a2sin(2ϕ)+a3sin(3ϕ)+... of the gives ϕC((cid:15)) = 2arccos(cid:112)g−((cid:15)/h)2. For the experimen- 2π-periodic CPRs. The first three amplitudes suffice to tal ∆ (cid:47) Γ the dependence of g and h on the model pa- describe the experiment perfectly; see Fig. 4 b where rameters cannot be trusted; we rather fit both (g = 2.2, they are shown as functions of (cid:15). The theoretical model h = 0.72 meV) and obtain very good agreement. This thus exactly captures the nontrivial finite temperature showsthatthe(cid:15)-dependenceofϕ isastrongcharacter- C phase dependence of the measured Josephson current. istic of the 0-π transition. An important information that can also be extracted In conclusion we have experimentally shown that the from the experiment is the gate voltage dependence of magneticstateofaCNTquantumdotjunction,singletor the critical phase ϕ at which the system switches from doublet, can be controlled by the superconducting phase C 0 to π-junction behavior, i.e. the CPR has 0-behavior difference. This has to be contrasted to previously mea- for ϕ ∈ [0,ϕ ] and π-behavior for ϕ ∈ [ϕ ,2π − ϕ ] suredgatecontrolledtransition. Itisachievedbyprobing C C C (Fig. 4 c). At T = 0 this switching at ϕ is asso- withunprecedentedaccuracytheevolutionoftheJoseph- C ciated to a first order level crossing transition and ap- son current at the 0-π transition. We have shown that pears as a jump of the current from positive to negative. theCPRhasacompositebehavior,witha"0"anda"π" For T > 0 the transition is washed out and the CPR component and that the phase at which this transition is smoothed. However, at small enough T, ϕ depends occursisgatedependent. Themeasurementsaresuccess- C only weakly on T (cf. Refs.[17, 19, 30]). In Fig. 4 c, fully compared to exact finite temperature QMC calcu- we compare ϕ ((cid:15)) from experiment and theory. Both lations. The possibility to measure precisely the CPR of C display the same characteristic shape, that can be un- correlated systems motivates the study of systems with derstood based on the atomic limit of the Anderson im- different symmetry such as Kondo SU(4) [39] or with purity model with ∆ (cid:29) Γ. A straightforward extension strong spin-orbit coupling [40]. of the T =0 atomic limit calculation (such as presented Acknowledgments: Ra.D., H.B. and Ri.D. thank M. in [17]) to the case of asymmetric level-lead couplings Aprili, S. Autier-Laurent, J. Basset, M. Ferrier, S. 5 Guéron, C. Li, A. Murani, and P. Simon for fruitful dis- [22] R. Maurand, T. Meng, E. Bonet, S. Florens, L. Marty, cussions. V.M. is grateful to D. Kennes for discussions. and W. Wernsdorfer. Phys. Rev. X, 2, 019901 (2012). [23] J-D. Pillet, C. H. L. Quay, P. Morfin, C. Bena, and ThisworkwassupportedbytheFrenchprogramANR A. Levy Yeyati and P. Joyez. Nature Phys., 6, 965–969 MASH (ANR-12-BS04-0016) and DYMESYS (ANR (2010). 2011-IS04-001-01). D.J.L. also acknowledges funding by [24] J.-D. Pillet, P. Joyez, Rok Žitko, and M. F. Goffman. Phys. Rev. B, 88, 045101 (2013). the ANR program ANR-11-ISO4-005-01 and the alloca- [25] W. Chang, V. E. Manucharyan, T. S. Jespersen, tion of CPU time by GENCI (grant x2014050225). J. Nygård, and C. M. Marcus. Phys. Rev. Lett., 110, 217005 (2013). [26] B-K. Kim, Y-H. Ahn, J-J. Kim, M-S. Choi, M-H. Bae, K. Kang, J. Lim, R. López, N. Kim. Phys. Rev. Lett., 110, 076803 (2013). [1] M. Pustilnik and L. Glazman. Journal of Physics: Con- [27] R. S. Deacon, Y. Tanaka, A. Oiwa, R. Sakano, densed Matter 16(16), R513 (2004). K. Yoshida, K. Shibata, K. Hirakawa, and S. Tarucha. [2] D. Goldhaber-Gordon, J. Göres, M. A. Kastner, H. Phys. Rev. Lett., 104, 076805 (2010). Shtrikman, D. Mahalu and U. Meirav Phys. Rev. Lett. [28] In particular, we noticed that the CPR at the transition 81, 5225-5228 (1998). can be distorted by a bad environment, acquiring a spu- [3] S.M.Cronenwett,T.H.Oosterkamp,andL.P.Kouwen- rious even component as a function of the phase while hoven. Science 281, 540–544 (1998). thesupercurrentisexpectedtobeanoddfunctionofthe [4] A.Yu. Kasumov, R. Deblock, M. Kociak, B. Reulet, phase in absence of time-reversal symmetry breaking. H. Bouchiat, I.I. Khodos, Yu.B. Gorbatov, V.T. Volkov, [29] R.AvrillerandF.Pistolesi.Phys.Rev.Lett.,114,037003 C. Journet, and M. Burghard. Science, 284, 1508–1511 (2015). (1999). [30] Seesupplementarymaterialsforsupportinginformation. [5] Y.-J. Doh, J. A. van Dam, A. L. Roest, E. P. [31] M. L. Della Rocca, M. Chauvin, B. Huard, H. Pothier, A. M. Bakkers, Leo P. Kouwenhoven, and Silvano D. Esteve, and C. Urbina. Phys. Rev. Lett., 99, 127005 De Franceschi. Science 309, 272–275 (2005). (2007). [6] V.V.Ryazanov, V.A.Oboznov, A.Yu.Rusanov, A.V. [32] J. Basset, R. Delagrange, R. Weil, A. Kasumov, Veretennikov, A. A. Golubov, and J. Aarts. Phys. Rev. H.Bouchiat,andR.Deblock.JournalofAppliedPhysics, Lett. 86, 2427–2430 (2001). 116, 024311 (2014). [7] J. A. van Dam, Y. V. Nazarov, Erik P. A. M. Bakkers, [33] Y.A.Kasumov,A.Shailos,I.I.Khodos,V.T.Volkov,V.I. S.DeFranceschi,andL.P.Kouwenhoven.s. Nature 442, Levashov,V.N.Matveev,S.Gueron,M.Kobylko,M.Ko- 667 (2006). ciak,H.Bouchiat,V.Agache,A.S.Rollier,L.Buchaillot, [8] J.-P. Cleuziou, W. Wernsdorfer, V. Bouchiat, T. On- A.M. Bonnot, and A.Y. Kasumov. Applied Physics A, darçuhu, and M. Monthioux. Nature Nanotechnology 1, 88(4), 687–691 (2007). 53 (2006). [34] J.P.Cleuziou,N.V.N’Guyen,S.Florens,andW.Werns- [9] H. Ingerslev Jorgensen, T. Novotny, K. Grove- dorfer. Phys. Rev. Lett., 111, 136803 (2013). Rasmussen,K.Flensberg,andP.E.Lindelof. Nano Let- [35] D.R.Schmid,S.Smirnov,M.Marga,A.Dirnaichner,P. ters 7(8) 2441–2445 (2007). L. Stiller, M. Grifoni, K. H. Andreas, C. Strunk. Phys. [10] S. De Franceschi, L. Kouwenhoven, C. Schonenberger, Rev. B, 91, 155435 (2015). and W. Wernsdorfer. Nature Nanotechnology, 5, 703 [36] F. D. M. Haldane. Phys. Rev. Lett., 40, 6 (1978) (2010). [37] Computing the normal state linear conductance as well [11] A. Clerk and V. Ambegaokar, Phys. Rev. B 61, 9109- astheCPRswithCT-INTisnumericallycostly.Wethus 9112 (2000). focus on zone A, showing the complete transition. [12] L.I. Glazman and K.A. Matveev. JETP Lett. 49, 659 [38] It should be noted that this Φ0 periodicity may be acci- 2 (1989). dentalandnotfundamentalsinceitimpliesthatboththe [13] E.Vecino,A.Martin-Rodero,andA.LevyYeyati. Phys. 0andπcontributionshavethesameshapeandamplitude Rev. B 68, 035105 (2003). which, here, is made possible by the finite temperature. [14] M-S. Choi, M. Lee, K. Kang, and W. Belzig. Phys. Rev. [39] A.Zazunov,A.LevyYeyatiandR.Egger. Phys.Rev.B, B, 70, 020502 (2004). 81, 12502 (2010). [15] F. Siano and R. Egger. Phys. Rev. Lett. 93, 047002 [40] J.S.Lim,,R.Lopez,M.-S.Choi,andR.Aguado. Phys. (2004); 94, 039902(E) (2005). Rev. Lett., 107, 196801 (2011). [16] J.Bauer,A.Oguri,andA.C.Hewson.JournalofPhysics: [41] Ambegaokar, V. and Baratoff, A. Tunneling between Condensed Matter 19(48), 486211 (2007). superconductors. Phys. Rev. Lett. 10, 486-489 (1963). [17] C.Karrasch,A.Oguri,andV.Meden.Phys. Rev. B 77, 024517 (2008). [18] T. Meng, S. Florens, and P. Simon. Phys. Rev. B, 79, 224521 (2009). SUPPLEMENTARY MATERIALS [19] D.J.Luitz,F.F.AssaadPhys.Rev.B81,024509(2010). [20] D. J. Luitz, F. F. Assaad, T. Novotný, C. Karrasch, and Sample fabrication V. Meden. Phys. Rev. Lett., 108, 227001 (2012). [21] A.Eichler,R.Deblock,M.Weiss,C.Karrasch,V.Meden, The carbon nanotubes (CNT) are grown by Chemical C. Schönenberger, and H. Bouchiat. Phys. Rev. B, 79, 161407 (2009). Vapor Deposition (CVD) on an oxidized doped silicon 6 wafer [33]. The wafer is used as a backgate. Then the happens for a smaller (cid:15) in theoretical simulation than in three-junctions SQUID (with area S ≈ 40 µm2) is de- the measurement. signed on top of it by electron beam lithography, with Critical phase resin bridges for the tunnel junctions. A first trilayer of Pd(7 nm)/Nb(20 nm)/Al(40 nm) is deposited with an Thetransitionfromthe0-junctiontotheπ-junctionis angle 15° (Fig. 5). Its superconducting gap is found to afirstorderquantumtransitionandislinkedtoachange be inhomogeneous: the CNT, which is in contact with of the ground state from a doublet to a singlet state on the Pd, sees a gap ∆ = 0.17 meV while the gap for the the quantum dot at a critical value of the phase ϕ . At tunnel junction is ∆ = 0.63 meV . Then the Al is oxi- c finite but low temperatures, we only see a washed out dized and a layer of Al(120 nm) is deposited on top of it signal of this behavior but can nevertheless try to ex- with an angle -15 ° . This layer has a superconducting tract the phase boundary as a function of gate voltage. gap ∆=0.18 meV. We define the value of the critical phase at temperature The reference JJs obtained have tunnel resistances T by the conditions I(ϕ ,T) = 0 and ∂I(ϕ ,T) < 0. R = 1.6 kΩ and R = 1.9 kΩ, from which we esti- c ∂ϕ c n1 n2 At the particle-hole symmetric point (cid:15) = 0 and at suf- mate the expected critical currents [41]: I = 330 nA c1 ficiently low temperatures ϕ is known to only weakly and I =280 nA. c c2 dependent on temperature[17] and will therefore provide an accurate estimate of the phase boundary. In order to checkhowϕ dependsontemperatureawayfromthehigh c symmetrypoint(cid:15)=0,wehaveperformedaCT-INTcal- culation at different temperatures (expressed by inverse temperatures β = 1 ) and for the other parameters as in the experimentk.BTOur findings show that ϕ indeed c Γ=0.44meV,U=3.2meV,ΓL/ΓR=4,(cid:15)d=0.8meV Figure 5: Summary of the different layers constituting the 0.025 φc(T=0),extr. 232.1mK sample 0.020 580.2mK 193.4mK 386.8mK 165.8mK 0.015 290.1mK 145.1mK 0.010 /¯h] 0.005 ∆ Additional data for the 0-π transition J[e 0.000 2.4 0.005 OnFig. 6isplottedwithoutanyadjustmentthesuper- −−0.010 φc 12..60 currentatafixedsuperconductingphasedifferenceϕfor −0.015 0 200 400 600 T[mK] zoneA,asafunctionofthelevelenergyofthedot. While −0.0200 π4 π2 34π π the experimental data are represented with the red line, φ the dotted red line represents QMC data. Here, only the positivevaluesof(cid:15)havebeenplotted. The0-π transition Figure 7: Effect of temperature on the critical phase calcu- lated current phase relation at 0-π transition ((cid:15) = 0.8meV) fortheparametersoftheexperiment. TheCPRhasbeencal- culated for different temperatures from 145 to 580 mK. For temperatures low enough (lower than 200 mK), the critical phase ϕ does not depend anymore on the temperature. c depends on temperatures at high temperature but con- verges to the zero temperature value at low ones. The results displayed in Fig. 7 demonstrate clearly that at the temperature T =150mK relevant to our experiment (corresponding to β ≈ 77 1 ) ϕ essentially represents meV c thegroundstateresult. Thisshowsthattheresultshown inthemaintextisinfacttheaccurategroundstatephase Figure 6: Comparison of raw data for the 0-π transition diagram.