Superconducting Quantum Interference at the Atomic Scale
S. Karan, H. Huang, C. Padurariu, B. Kubala, G. Morrás, A. Levy Yeyati, J. C. Cuevas, J. Ankerhold, K. Kern, C. R. Ast
SSuperconducting antum Interference at the Atomic Scale
Sujoy Karan, Haonan Huang, Ciprian Padurariu, Bj¨orn Kubala,
2, 3
Gonzalo Morr´as, AlfredoLevy Yeyati, Juan Carlos Cuevas, Joachim Ankerhold, Klaus Kern,
1, 5 and Christian R. Ast ∗ Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstraße 1, 70569 Stugart, Germany Institut f¨ur Komplexe antensysteme and IQST, Universit¨at Ulm, Albert-Einstein-Allee 11, 89069 Ulm, Germany Institute of antum Technologies, German Aerospace Center (DLR), S¨oinger Straße 100, 89077 Ulm, Germany Departamento de F´ısica Te´orica de la Materia Condensada and Condensed Maer Physics Center (IFIMAC),Universidad Aut´onoma de Madrid, 28049 Madrid, Spain Institut de Physique, Ecole Polytechnique F´ed´erale de Lausanne, 1015 Lausanne, Switzerland (Dated: February 26, 2021)A single spin in a Josephson junction can reverse the ow of the supercurrent. At mesoscopic length scales,such 𝜋 -junctions are employed in various instances from nding the pairing symmetry to quantum computing.In Yu-Shiba-Rusinov (YSR) states, the atomic scale counterpart of a single spin in a superconducting tunneljunction, the supercurrent reversal so far has remained elusive. Using scanning tunneling microscopy (STM),we demonstrate such a 0 to 𝜋 transition of a Josephson junction through a YSR state as we continuously changethe impurity-superconductor coupling. We detect the sign change in the critical current by exploiting a secondtransport channel as reference in analogy to a superconducting quantum interference device (SQUID), whichprovides the STM with the required phase sensitivity. e measured change in the Josephson current is asignature of the quantum phase transition and allows its characterization with unprecedented resolution. Two superconductors that are connected by a weak linkcan sustain a supercurrent, which is carried by Cooper pairs— the well-known Josephson eect [1]. Inserting a single spininto the junction may completely change its behavior by re-versing the direction of the supercurrent [2], which is theresult of a 𝜋 -shi in the phase across the junction. Such 𝜋 -junctions have been used in nding the pairing symmetry inunconventional superconductors [3–7] and they have beenproposed as building blocks for energy ecient quantumcomputing or high-speed memory [8–10]. At mesoscopiclength scales ( ≈
10 to 100 nm), 𝜋 -junctions may be realizedby singly occupied quantum dots or ferromagnetic interlay-ers [11–18]. At the atomic scale ( ≈ 𝜋 -shi[22–24].e hallmark of this quantum phase transition (QPT) inYSR states is a discontinuous change in the total spin ofthe respective ground states: a previously free impurity spinturns into a screened spin, when the magnetic exchange cou-pling increases beyond a critical value. Consequently, a re-versal in the ow of Cooper pairs through a YSR state hasbeen predicted [25]. Experimentally, the QPT can be identi-ed by a zero energy crossing of the YSR state in dieren-tial conductance spectra [26–30]. However, the actual conse-quences for the fundamental Josephson eect remain elusivein atomic scale junctions.e observation of a YSR state based 𝜋 -junction is exper-imentally challenging, because detecting such a phase shibetween superconducting ground states requires a referencechannel. At mesoscopic length scales, this is typically solvedby employing a superconducting quantum interference de- vice (SQUID) loop geometry [14–17]. To reach similar con-ditions at the atomic scale, a scanning tunneling microscope(STM) requires a rudimentary phase sensitivity through anadditional transport channel [15, 18].Here, we demonstrate a supercurrent reversal in an atomicscale Josephson junction through a YSR state as we moveacross the QPT. We produce a magnetic impurity at the apexof a superconducting vanadium tip (see Fig. 1(a)), whichis approached to a superconducting V(100) sample. As weapproach, the atomic forces pull on the impurity [26, 31–33], which reduces the impurity-superconductor coupling 𝛤 along with the magnetic exchange coupling. is concomi-tantly allows the YSR state to pass from the strong scaer-ing (screened spin) to the weak scaering (free spin) regimeas outlined in Fig. 1(b). e two scenarios are schematicallyillustrated in Fig. 1(c), where the total spin in the free spinregime is 𝑆 tot = / . In the screened spin regime, a Cooperpair is broken to screen the impurity spin changing the over-all parity of the system (indicating whether the total numberof particles is even or odd) as well as the total spin to 𝑆 tot = 𝐺 N is shown in Fig. 1(d). e YSR statemoves across the QPT when the YSR energies are closest toeach other. Because both tip and sample are superconduct-ing, in the spectrum the tip YSR states appear at voltages 𝑉 shied by the sample gap 𝛥 s , i.e. 𝑒𝑉 = 𝜀 + 𝛥 s with the YSR a r X i v : . [ c ond - m a t . s up r- c on ] F e b Figure 1:
Atomic YSR state. (a)Schematic of the tunnel junc-tion. e YSR impurity is atthe tip with two transport chan-nels (BCS and YSR) indicated asdashed lines. (b) Phase dia-gram of the YSR system as func-tion of impurity-superconductorcoupling and level (particle-hole)asymmetry. (c) Schematic of thefree spin and the screened spinregime. In the screened regime,a Cooper pair is broken chang-ing the overall parity of the sys-tem. (d) Dierential conductancespectra as function of bias voltage( 𝑥 -axis) and conductance ( 𝑦 -axis).e prominent peaks are the YSRstates, while the coherence peaksare only barely visible. (e) Hor-izontal line cut through (d) toshow the prominent YSR peaks.(f) Half-width at half maximum 𝛥 K of the Kondo peak in the samejunction at a magnetic eld of1.5 T, when superconductivity isquenched. e Kondo spectra areshown in the inset. state energy 𝜀 varying with the normal state conductance 𝐺 N .Interestingly, there are no distinct coherence peaks visible atthe sum of the tip gap and the sample gap ±( 𝛥 t + 𝛥 s ) , whichindicates that a second transport channel through an emptygap (i.e. without any YSR state and hence with coherencepeaks) has a much weaker, but still nite transmission com-pared to the YSR state. is can be more directly seen in asingle spectrum near the QPT, which is shown in Fig. 1(e).e coherence peaks at 𝑒𝑉 = 𝛥 t + 𝛥 s are greatly reduced andthe YSR peaks are prominently enhanced by almost a factorof 100.To conrm that the impurity-superconductor coupling (inour case the impurity-tip coupling) decreases with increas-ing conductance, we have measured the Kondo eect in thesame junction by quenching the superconductivity in a mag-netic eld of 1.5 T. e Kondo spectra are shown in the in-set of Fig. 1(f), from which we extract the half width at halfmaximum 𝛥 K . As 𝛥 K is directly related to the Kondo tem-perature and the magnetic exchange coupling, we concludethat the impurity-superconductor coupling decreases withdecreasing tip-sample distance (i.e. increasing junction con-ductance). Physically, the impurity is pulled away from thetip by the aractive atomic forces of the approaching samplesubstrate [33].e supercurrent, which is carried by tunneling Cooperpairs (Josephson eect), is visible throughout the range ofconductance values (see red arrow in Fig. 1(e)). In the dy-namical Coulomb blockade (DCB) regime, in which the STM operates [34], the typical voltage-biased measurement showsa negative current peak followed by a positive current peak ofequal size near zero bias voltage. e evolution of the Joseph-son eect as function of conductance is shown in Fig. 2(a).Each spectrum is shown in a bias voltage range of ± 𝜇 eVand oset horizontally. Assuming a harmonic current-phaserelation in the DCB (i.e. 𝐼 ( 𝜑 ) = 𝐼 C sin 𝜑 , where 𝐼 C is the criticalcurrent), the Josephson current is predicted to scale with thesquare of the critical current, i.e. 𝐼 ( 𝑉 ) ∝ 𝐼 ∝ 𝐺 [35–38]. Itcan be directly seen in the data that this square dependenceis not fullled in the data set in Fig. 2(a). In particular, theregion indicated by the horizontal bracket shows signicantdeviations, even a slight decrease in the Josephson currentwith increasing conductance. e conductance at which theQPT occurs is indicated by a vertical dashed line, which fallsdirectly into the region of the horizontal bracket.For a more quantitative analysis, we plot the current max-ima 𝐼 S (switching current) for each conductance as a blue linein a double logarithmic plot in Fig. 2(b). e expected squaredependence on the conductance ( 𝐼 S ∝ 𝐼 ∝ 𝐺 ) can be clearlyseen for very small and very large conductances. In the tran-sition region (indicated by the horizontal bracket), the behav-ior of the switching current 𝐼 S strongly changes. For com-parison, we plot the experimentally extracted energies of theYSR state (red line), which has a minimum at the QPT (ver-tical dashed line) [39]. is indicates a drastic change in thebehavior of the Josephson eect across the QPT.To put the evolution of the switching current in reference Figure 2:
Josephson Eect. (a) Josephsonspectra 𝐼 ( 𝑉 ) in a range of ± μ eV shied hor-izontally by the conductance at which theywere measured. e horizontal bracket indi-cates the region, where the evolution deviatesfrom the conventional Ambegaokar-Baratoformula. e quantum phase transition (QPT)is indicated by the vertical dashed line. (b)e switching current 𝐼 S , which is the currentmaximum indicated in (a) as function of thenormal state conductance 𝐺 N (blue line). esquare dependence at low and high conduc-tance is indicated by dashed lines (labeled by ∝ 𝐺 ). e YSR energy as function of conduc-tance is shown as a blue line. e minimum in-dicates the QPT (vertical dashed line). (c) isgraph shows √ 𝐼 S 𝑅 N (blue line) of the data in (b),which is proportional to 𝐼 C 𝑅 N for a harmonicenergy-phase relation ( 𝑅 N : renormalized resis-tance (see text), 𝐼 C : critical current). A referencejunction without any YSR states is shown asa yellow line indicating the expected evolutionaccording to the Ambegaokar-Barato formula. to other Josephson junctions, we calculate √ 𝐼 S 𝑅 N , which isshown in Fig. 2(c) ( 𝑅 N is the normal state tunneling resis-tance). is quantity is proportional to the product 𝐼 C 𝑅 N for aharmonic energy-phase relation. In this way, the overall con-ductance dependence is eliminated such that the measure-ment appears like a step in Fig. 2(c) with a sizeable reductionin height almost by a factor of two across the QPT. We willshow below that this is due to a supercurrent reversal in theYSR channel which leads to a crossover from a constructiveto a destructive interference between the two transport chan-nels.In order to compare the experimental data to the theory,we have to renormalize the normal state resistance 𝑅 N for theYSR spectra due to the enhanced density of states from Kondocorrelations (for details see the Supplementary Information[40]). e reference spectra (orange line in Fig. 2(c)) are mea-sured for a Josephson junction without any YSR states, wherethe √ 𝐼 S 𝑅 N is constant as expected from the Ambegaokar-Barato formula [41, 42].To understand the behavior of the Josephson eect in Fig.2, we rst have a look at the energy-phase relations far awayfrom the QPT at high and at low conductance. In Fig. 3(a),the energy-phase relations for the BCS channel and the YSRchannel, which is calculated from a mean eld Anderson im-purity model, are shown in red and blue, respectively (fordetails see the Supplementary Information [40]). To calcu-late the energy-phase relation, we apply a constant phase dif-ference 𝜑 across the tunnel junction, but no bias voltage. AFourier expansion of the energy-phase relation reveals thatthe most relevant contribution to the Josephson eect is theharmonic term proportional to cos ( 𝜑 ) . Zooming in to bothchannels (cf. Fig. 3(b)), we estimate that the ratio of the chan-nel transmissions is about 4:1 (YSR:BCS): is results in a sig-nicantly smaller amplitude for the energy-phase relation of the BCS channel (red) than in the YSR channel (blue) (Indi-vidual Channels in Fig. 3(b)). e coherent superposition ofthese two channels (Channel Sum in Fig. 3(b)) leads to anoverall sign change as well as dierent amplitudes, when thechannels are in phase (upper row in Fig. 3(b)) or out of phase(lower row in Fig. 3(b)). In the measurement, we are onlysensitive to the change in amplitude 𝐼 ( 𝑉 ) ∝ ( 𝐸 YSR + 𝐸 BCS ) though, which results in the obvious step in Fig. 2(c). We at-tribute the width of the step to the nite temperature in ourexperiment.Since the temperature in our experiment (10 mK) is stillnon-zero, we expect uctuations due to thermal excitationsclose to the QPT. e probability for the system to be inthe ground state (blue) or the excited state (orange) is indi-cated in Fig. 3(c) using an eective temperature of 75 mK. iswill broaden the expected sharp features associated with thequantum phase transition. Taking the excitation probabilitydue to the nite temperature into account, we can calculatethe expected Josephson current in the DCB regime (see Sup-plementary Information [40]). e t is shown in Fig. 3(d)with excellent agreement to the data. e only free parame-ters are the eective temperature 𝑇 e =
75 mK, which is de-termined by the width of the transition and the ratio of thetwo channel transmissions, which is determined by the stepheight. For a best t, we nd that the YSR channel contributes78.4% and the BCS reference channel contributes 21.6% to thetotal conductance relevant to the Josephson eect, which isconsistent with the prominent YSR states and the strongly re-duced coherence peaks in the quasiparticle spectra (see Fig.1(e)). All other parameters are given by the experimentallyextracted values. In this way, we demonstrate that the su-percurrent through an atomic scale YSR state reverses uponcrossing the QPT, which can be detected in the STM by meansof a BCS reference channel in analogy to a SQUID geometry
Figure 3:
Supercurrent Reversal. (a) Energy-phase relationfor the BCS channel (red) and the YSR channel (blue). For theJosephson current only the oscillation is relevant, but not theenergy oset. (b) shows a zoom-in to the oscillation for thetwo channels. e upper two panels in (b) show the in-phaseoscillation in the screened spin regime. e lower two pan-els in (b) show the out-of-phase oscillation in the free spinregime, which is indicative of the supercurrent reversal. eSTM is not sensitive of the sign of the supercurrent, but theconcomitant change in magnitude is clearly observable. (c)Probabilities for the system to be in the ground state (blue) orthe excited state (red) as function of YSR state energy. (d) e √ 𝐼 S 𝑅 N product comparing experimental data with a theoret-ical t. e only free parameters are the temperature, whichdenes the width of the QPT and the relative channel trans-mission, which denes the step height.Figure 4: Cooper Pair Tunneling. e tunneling process in thefree spin and in the screened spin regime is shown from the initialto the nal state via an intermediate (virtual) state in the excitationpicture. e order is set by the numbered red arrows. e totalspin 𝑆 tot describes the total spin of the YSR system including thespin of the impurity. (a) In the free spin regime, the order of thespins is exchanged compared to the intial state, which results in thesupercurrent reversal. (b) In the screened spin regime, the orderof the spins is retained, such that there is no sign change in thesupercurrent. (see Supplementary Information for more details [40]).To beer understand the origin of this supercurrent rever-sal and to illustrate the crucial role of the impurity spin, wediscuss the Cooper pair tunneling process in Fig. 4 using theexcitation picture. Zero energy denotes the ground state, 𝜀 is the energy of the excited YSR state, and 𝛥 marks the be- ginning of the quasiparticle continuum. e order is givenby the numbered red arrows. Figure 4(a) describes the freespin regime, where the total spin is 𝑆 tot = / [22, 25]. isindicates that the total parity (superconductor + impurity)must be odd. e Cooper pair transfer process involves aswap between two fermions, the one associated with the im-purity and one associated with the Cooper pair, as depictedby the arrows 3 and 4 in Fig. 4(a). Formally, this appears as anexchange of fermion operators inducing a negative sign ( 𝜋 -shi) [15, 43]. By contrast, Fig. 4(b) shows the screened spinregime, which has a ground state with total spin 𝑆 tot = 𝜋 junction) as the conductance increases.At the QPT, a system typically becomes very sensitive toexternal parameters, such as temperature. Here, we notethat the width of the QPT step in Fig. 3(d) depends onlyon temperature, but experiences no broadening from voltagenoise. is is in contrast to conventional scanning tunnel-ing spectroscopy, where temperature broadening is typicallyobscured by voltage noise as well as interactions with the en-vironment [34]. Hence, YSR-tip functionalization may opennew developments for low temperature thermomentry withhigh spatial resolution where measuring the slope of the QPTstep accesses the temperature.In summary, the experimental results directly reveal theconsequences of the discrete parity change across the QPTin YSR states as well as the role of the impurity spin, whichmanifests itself in the supercurrent reversal. Our results es-tablish an important connection to mesoscopic 𝜋 -junctionsproviding the perspective to transfer some of their concepts,for example as sensing tools, to the atomic scale. Having di-rect tunable access to the QPT could be exploited to enhancethe sensitivity in quantum sensing applications, such as a lo-cal temperature measurement. Also, demonstrating the co-herent superposition of dierent transport channels in theDCB regime introduces a rudimentary phase sensitivity inSTM measurements that can be exploited in other scenariosas well. Acknowledgments
We gratefully acknowledge stimulating discussionswith Annica Black-Schaer, Robert Drost, Berthold J¨ack,Francesco Tafuri, and Andreas eiler. We dedicate thismanuscript to the memory of Fabien Portier for the manyinspirational discussions that the authors had with him.is work was funded in part by the ERC ConsolidatorGrant AbsoluteSpin (Grant No. 681164) and by the Centerfor Integrated antum Science and Technology (IQ ST ). J.A.acknowledges funding from the DFG under grant numberAN336/11-1. C.P. acknowledges funding from the IQ ST andthe Zeiss Foundation. A.L.Y. and J.C.C. acknowledge fundingfrom the Spanish MINECO (Grant No. FIS2017-84057-P andFIS2017-84860-R), from the “Mar´ıa de Maeztu” Programmefor Units of Excellence in R&D (MDM-2014-0377). ∗ Corresponding author; electronic address: [email protected][1] B. D. Josephson,
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MATERIALS AND METHODS
Experiments were performed on Josephson nano-junctions built in a low temperature scanning tunnelingmicroscope (STM) operated at 10 mK. Approaching a super-conducting vanadium tip, tailored with a spin- ⁄ impurityat its apex, to a crystalline V(100) substrate, we drove theimpurity induced YSR states across the quantum phasetransition (QPT) and detected a reversal of the supercurrent.e V(100) substrate was cleaned by repeated cycles of Arion spuering, annealing to ∼
925 K, and cooling to ambi-ent temperature at a rate of 1-2 K/s. Oxygen diused fromthe bulk to the surface lead to typical surface reconstructions[1, 2], which did not inuence the characteristics of supercon-ducting vanadium. Surface defects mostly involve missingoxygen within the reconstruction, which appeared bright inSTM topographs [3]. Magnetic defects were found exhibitingYSR states at arbitrary energies within the gap as reported inRef. [3].e tip was spuered in ultra-high vacuum and treatedwith eld emission as well as subsequent indentation intothe vanadium substrate until the expected gap of bulk vana-dium appeared in the conductance spectrum. YSR tips weredesigned following the method of random dipping explainedin Ref. [3]. We purposefully chose to use YSR tips for our ex-periment as it gave beer stability of the junction at higherconductance. Moreover, it oered beer exibility in design-ing and dening the junction over magnetic surface defects,which were mostly found to have a spatial extent of around1 nm.
THE JOSEPHSON CURRENT
Calculating the Josephson current is typically based on theenergy-phase relation 𝐸 ( 𝜑 ) at zero applied bias voltage. Inthe DCB regime, the charging energy of the tunnel junctiondominates, such that tunneling is charge dominated and se-quential. e measurement is typically voltage-biased andthe Cooper pair tunneling relies on the exchange of energywith the environment. e corresponding Cooper pair trans-fer can be calculated by means of a Fourier transform of theenergy-phase relation 𝐸 ( 𝜑 ) = +∞ ∑︁ 𝑚 = −∞ 𝐸 𝑚 𝑒 𝑖𝑚𝜑 , (S1)where 𝑒 𝑖𝑚𝜑 corresponds to the charge transfer operator and 𝑚 is the number of Cooper pairs being transferred. eFourier components 𝐸 𝑚 from Eq. S1 are used to calculate the Josephson current 𝐼 ( 𝑉 ) = 𝜋 ℏ +∞ ∑︁ 𝑚 = | 𝐸 𝑚 | ( 𝑚𝑒 ) [ 𝑃 𝑚 ( 𝑚 𝑒𝑉 ) − 𝑃 𝑚 (− 𝑚 𝑒𝑉 )] , (S2)where 𝑃 𝑚 ( 𝐸 ) is the probability to exchange energy 𝐸 with theenvironment during the tunneling process, when 𝑚 Cooperpairs are tunneling. It is dened as a generalized 𝑃 ( 𝐸 ) -function [4] 𝑃 𝑚 ( 𝐸 ) = ∫ +∞−∞ d 𝑡 𝜋 ℏ 𝑒 𝑚 𝐽 ( 𝑡 )+ 𝑖𝐸𝑡 / ℏ (S3)with the phase correlation function 𝐽 ( 𝑡 ) = (cid:104)[ ˜ 𝜑 ( 𝑡 ) − ˜ 𝜑 ( )] ˜ 𝜑 ( )(cid:105) , (S4)accounting for Cooper-pair–phase uctuations ˜ 𝜑 = 𝜑 − 𝑒𝑉 𝑡 / ℏ around the mean value determined by the externalvoltage. For a more detailed discussion see Ref. [4]. DERIVATION OF THE ENERGY-PHASE RELATION
To describe the phase dependence of the energy of the YSRstates we follow Ref. [5, 6] and make use of a mean-eld An-derson model where a magnetic impurity coupled to super-conducting leads is described by the following Hamiltonian 𝐻 = 𝐻 t + 𝐻 s + 𝐻 i + 𝐻 hopping . (S5)Here, 𝐻 𝑗 , with 𝑗 = t , s (t stands for tip and s for substrate), isthe BCS Hamiltonian of the lead 𝑗 given by 𝐻 𝑗 = ∑︁ 𝒌 𝜎 𝜉 𝒌 𝑗 𝑐 † 𝒌 𝑗𝜎 𝑐 𝒌 𝑗𝜎 + ∑︁ 𝒌 (cid:16) 𝛥 𝑗 𝑒 𝑖𝜑 𝑗 𝑐 † 𝒌 𝑗 ↑ 𝑐 †− 𝒌 𝑗 ↓ + 𝛥 𝑗 𝑒 − 𝑖𝜑 𝑗 𝑐 − 𝒌 𝑗 ↓ 𝑐 𝒌 𝑗 ↑ (cid:17) , (S6)where 𝑐 † 𝒌 𝑗𝜎 and 𝑐 𝒌 𝑗𝜎 are the creation and annihilation opera-tors, respectively, of an electron of momentum 𝒌 , energy 𝜉 𝒌 𝑗 ,and spin 𝜎 = ↑ , ↓ in lead 𝑗 , 𝛥 𝑗 is the superconducting gap pa-rameter, and 𝜑 𝑗 is the corresponding superconducting phase.On the other hand, 𝐻 i is the Hamiltonian of the magnetic im-purity, which is given by 𝐻 i = 𝐸 U ( 𝑛 ↑ + 𝑛 ↓ ) + 𝐸 J ( 𝑛 ↑ − 𝑛 ↓ ) , (S7)where 𝑛 𝜎 = 𝑑 † 𝜎 𝑑 𝜎 is the occupation number operator on theimpurity, 𝐸 U is the on-site energy, and 𝐸 J is the exchange en-ergy that breaks the spin degeneracy on the impurity. Finally, 𝐻 hopping describes the coupling between the magnetic impu-rity and the leads that adopts the form 𝐻 hopping = ∑︁ 𝒌 ,𝑗,𝜎 𝑡 𝑗 (cid:16) 𝑑 † 𝜎 𝑐 𝒌 𝑗𝜎 + 𝑐 † 𝒌 𝑗𝜎 𝑑 𝜎 (cid:17) , (S8)where 𝑡 𝑗 describes the tunneling coupling between the impu-rity and the lead 𝑗 = t , s and it is chosen to be real.Now, it is convenient to rewrite the previous Hamiltonianin terms of four-dimensional spinors in a space resulting fromthe direct product of the spin space and the Nambu (electron-hole) space. In the case of the leads, the relevant spinor isdened as ˜ 𝑐 † 𝒌 𝑗 = (cid:16) 𝑐 † 𝒌 𝑗 ↑ , 𝑐 − 𝒌 𝑗 ↓ , 𝑐 † 𝒌 𝑗 ↓ , − 𝑐 − 𝒌 𝑗 ↑ (cid:17) , (S9)while for the impurity states we dene˜ 𝑑 † = (cid:16) 𝑑 †↑ , 𝑑 ↓ , 𝑑 †↓ , − 𝑑 ↑ (cid:17) . (S10)Using the notation 𝜏 𝑖 and 𝜎 𝑖 ( 𝑖 = , ,
3) for Pauli matricesin Nambu and spin space, respectively, and with 𝜏 and 𝜎 asthe unit matrices in those spaces, it is easy to show that theHamiltonian in Eq. (S5) can be cast into the form 𝐻 𝑗 = ∑︁ 𝒌 ˜ 𝑐 † 𝒌 𝑗 ˆ 𝐻 𝒌 𝑗 ˜ 𝑐 𝒌 𝑗 , (S11a) 𝐻 i =
12 ˜ 𝑑 † ˆ 𝐻 i ˜ 𝑑, (S11b) 𝐻 hopping = ∑︁ 𝒌 ,𝑗 (cid:110) ˜ 𝑐 † 𝒌 𝑗 ˆ 𝑉 𝑗 i ˜ 𝑑 + ˜ 𝑑 † ˆ 𝑉 i 𝑗 ˜ 𝑐 𝒌 𝑗 (cid:111) , (S11c)where ˆ 𝐻 𝒌 𝑗 = 𝜎 ⊗ ( 𝜉 𝒌 𝜏 + 𝛥 𝑗 𝑒 𝑖𝜑 𝑗 𝜏 𝜏 ) , (S12a)ˆ 𝐻 i = 𝐸 U ( 𝜎 ⊗ 𝜏 ) + 𝐸 J ( 𝜎 ⊗ 𝜏 ) , (S12b)ˆ 𝑉 𝑗 i = 𝑡 𝑗 ( 𝜎 ⊗ 𝜏 ) = ˆ 𝑉 † i 𝑗 . (S12c)e starting point for the description of the electronicstructure of our impurity system are the Green’s functions ofthe dierent subsystems which can be easily calculated from the previous matrix Hamiltonians as follows. First, the re-tarded and advanced Green’s functions of the leads resolvedin 𝒌 -space are dened as ˆ 𝑔 r , a 𝒌 ,𝑗 𝑗 ( 𝐸 ) = ( 𝐸 ± 𝑖𝜂 − ˆ 𝐻 𝒌 𝑗 ) − , where 𝑗 = t , s and 𝜂 = + is a positive innitesimal parameter, whichwe shall drop out to simplify things along with the super-script r , a, unless they are strictly necessary. Summing over 𝒌 , ˆ 𝑔 𝑗 𝑗 = (cid:205) 𝒌 ˆ 𝑔 𝒌 ,𝑗 𝑗 → 𝜌 𝑗 ( ) ∫ ∞−∞ 𝑑𝜉 𝒌 ˆ 𝑔 𝑗 𝑗 ( 𝜉 𝒌 ) , where 𝜌 𝑗 ( ) is thenormal density of states at the Fermi energy ( 𝐸 =
0) of lead 𝑗 , we arrive at the standard expression for the bulk Green’sfunction of a BCS superconductorˆ 𝑔 𝑗 𝑗 ( 𝐸 ) = − 𝜋 𝜌 𝑗 ( ) √︃ 𝛥 𝑗 − 𝐸 𝜎 ⊗ (cid:2) 𝐸𝜏 + 𝛥 𝑗 𝑒 − 𝑖𝜑 𝑗 𝜏 𝜏 (cid:3) . (S13)On the other hand, the impurity Green’s function is givenby ˆ 𝑔 ii ( 𝐸 ) = ( 𝐸 − ˆ 𝐻 i ) − (S14) = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) 𝐸 − 𝐸 U − 𝐸 J 𝐸 + 𝐸 U − 𝐸 J 𝐸 − 𝐸 U + 𝐸 J
00 0 0 𝐸 + 𝐸 U + 𝐸 J (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) . e dressed Green’s functions of the impurity system tak-ing into account the coupling to the superconducting leadsare calculated by solving the Dyson equationsˆ 𝐺 𝑗𝑘 = ˆ 𝑔 𝑗𝑘 + ∑︁ 𝛼𝛽 ˆ 𝑔 𝑗𝛼 ˆ 𝑉 𝛼𝛽 ˆ 𝐺 𝛽𝑘 , (S15)where the indices run over t, i and s and the couplings aregiven by ˆ 𝑉 𝑗 i = 𝑡 𝑗 ( 𝜎 ⊗ 𝜏 ) = ˆ 𝑉 † i 𝑗 . Assuming equal gaps inboth superconducting leads ( 𝛥 t = 𝛥 s = 𝛥 = μ eV), theseequations can be easily solved to obtain the following dressedGreen’s functions on the impurity siteˆ 𝐺 ii ( 𝐸 ) = (cid:18) ˆ 𝐺 ii , ↑↑ ( 𝐸 )
00 ˆ 𝐺 ii , ↓↓ ( 𝐸 ) (cid:19) , (S16)withˆ 𝐺 ii ,𝜎𝜎 ( 𝐸 ) = 𝐷 𝜎 ( 𝐸 ) (cid:32) 𝐸 ( 𝛤 s + 𝛤 t ) + ( 𝐸 + 𝐸 U − 𝐸 J 𝜎 )√ 𝛥 − 𝐸 𝛤 s 𝛥𝑒 𝑖𝜑 s + 𝛤 t 𝛥𝑒 𝑖𝜑 t 𝛤 s 𝛥𝑒 − 𝑖𝜑 s + 𝛤 t 𝛥𝑒 − 𝑖𝜑 t 𝐸 ( 𝛤 s + 𝛤 t ) + ( 𝐸 − 𝐸 U − 𝐸 J 𝜎 )√ 𝛥 − 𝐸 (cid:33) , (S17)where 𝐷 𝜎 ( 𝐸, 𝜑 ) = ( 𝛤 s + 𝛤 t ) 𝐸 ( 𝐸 − 𝐸 J 𝜎 ) + (cid:2) ( 𝐸 − 𝐸 J 𝜎 ) − 𝐸 − ( 𝛤 s + 𝛤 t ) (cid:3) √ 𝛥 − 𝐸 + 𝛥 √ 𝛥 − 𝐸 𝛤 s 𝛤 t sin ( 𝜑 / ) . (S18)Here, 𝜑 = 𝜑 t − 𝜑 S is the superconducting phase dierence, 𝐸 J ↑ = + 𝐸 J and 𝐸 J ↓ = − 𝐸 J , and we have dened the tunnelingrates 𝛤 𝑗 = 𝜋 𝜌 𝑗 ( ) 𝑡 𝑗 with 𝑗 = t , s.As usual, the local density of states (LDOS) projected ontothe impurity site can be obtained from the imaginary part ofdiagonal elements of the Green’s function above. e resultfor the total LDOS adopts the following form 𝜌 Total , imp ( 𝐸, 𝜑 ) = 𝜌 ↑ ( 𝐸, 𝜑 ) + 𝜌 ↓ ( 𝐸, 𝜑 ) , (S19)where 𝜌 𝜎 ( 𝐸, 𝜑 ) = 𝜋 Im (cid:40) 𝐸 ( 𝛤 s + 𝛤 t ) + ( 𝐸 + 𝐸 U − 𝐸 J 𝜎 )√ 𝛥 − 𝐸 𝐷 𝜎 ( 𝐸, 𝜑 ) (cid:41) , (S20)where 𝐷 𝜎 ( 𝐸, 𝜑 ) is given by Eq. (S18). e condition for theappearance of superconducting bound states is 𝐷 𝜎 ( 𝐸, 𝜑 ) = 𝐸 J (cid:29) 𝛥 (and they are inside the gap when also 𝛤 t (cid:29) 𝛥 ).Additionally, we focus on the tunnel regime where 𝛤 s (cid:28) 𝛤 t ,which is the relevant regime for our experiments. Note thatin our experiment the impurity is strongly coupled to the tip(t) and tunneling is between the impurity and the substrate(s). e phase-dependent energies of the YSR bound statesare approximately given by 𝐸 YSR ( 𝜑 ) = ± 𝛥 ( 𝛤 s + 𝛤 t ) + 𝐸 − 𝐸 − 𝛤 s 𝛤 t sin ( 𝜑 / ) √︃(cid:2) ( 𝐸 U − 𝐸 J ) + ( 𝛤 s + 𝛤 t ) (cid:3) (cid:2) ( 𝐸 U + 𝐸 J ) + ( 𝛤 s + 𝛤 t ) (cid:3) . (S21)e transmission 𝜏 YSR through the YSR channel is given by 𝜏 YSR = 𝛤 s 𝛤 t ( 𝐸 U − 𝐸 J ) + ( 𝛤 s + 𝛤 t ) + 𝛤 s 𝛤 t ( 𝐸 U + 𝐸 J ) + ( 𝛤 s + 𝛤 t ) . (S22)If the impurity is only coupled to one of the superconduc-tors, the energy of the YSR state is 𝜀 = ± 𝛥 𝛤 𝑗 + 𝐸 − 𝐸 √︂(cid:104) ( 𝐸 U − 𝐸 J ) + 𝛤 𝑗 (cid:105) (cid:104) ( 𝐸 U + 𝐸 J ) + 𝛤 𝑗 (cid:105) . (S23)At the QPT, the energy of the YSR state crosses zero ( 𝜀 = 𝐸 = 𝛤 𝑗 + 𝐸 = const . (S24) THE ENERGY-PHASE RELATION e energy-phase relation for a tunnel junction with twoequal conventional Bardeen-Cooper-Schrieer (BCS) gaps(i.e. 𝛥 t = 𝛥 s = 𝛥 ) in tip ( 𝑡 ) and sample ( 𝑠 ) is 𝐸 BCS ( 𝜑 ) = ± 𝛥 √︂ − 𝜏 BCS sin (cid:16) 𝜑 (cid:17) , (S25) Figure S1:
Atomic YSR State (Data Set (a) Dierential conduc-tance spectra as function of bias voltage ( 𝑥 -axis) and conductance( 𝑦 -axis). e YSR states and the coherence peaks are nearly equallyprominent. (b) Horizontal line cut through (d) to show the YSRpeaks in relation to the coherence peaks, which are more promi-nent in this data set compared to the data in the main text. isindicates that the transmission through the BCS reference channelis higher than for the data in the main text. where 𝜏 BCS is the BCS channel transmission. For small chan-nel transmission ( 𝜏 BCS (cid:28) 𝑚 =
1) following Eq. (S1) is 𝐸 BCS1 = 𝛥 𝜏 BCS , (S26)e energy-phase relation of a tunnel junction with oneYSR state is given in Eq. (S21). For the YSR junction, we ex-pect higher harmonic contributions only very close to theQPT. We can explain all experimental data based on therst harmonic contribution, so we will neglect higher ordercontributions here. e amplitude in the lowest harmonic( 𝑚 =
1) for Eq. (S21) is 𝐸 YSR1 = sgn [( 𝛤 s + 𝛤 t ) + 𝐸 − 𝐸 ] 𝛥𝛤 s 𝛤 t √︃(cid:2) ( 𝐸 U − 𝐸 J ) + ( 𝛤 s + 𝛤 t ) (cid:3) (cid:2) ( 𝐸 U + 𝐸 J ) + ( 𝛤 s + 𝛤 t ) (cid:3) , (S27)where sgn is the sign function. e only way that we canmodel the experimentally observed step is by coherentlyadding the two transport channels 𝐸 ( 𝜑 ) = 𝐸 BCS ( 𝜑 ) + 𝐸 YSR ( 𝜑 ) (S28)and by extension the two harmonic amplitudes, which enterinto the calculation of the Josephson current in Eq. S2 𝐸 = 𝐸 BCS1 + 𝐸 YSR1 . (S29)Equation (S29) is valid at zero temperature. At nite temper-ature, thermal excitations have to be taken into account. THERMAL EXCITATION OF THE YSR STATE
In the vicinity of the QPT, the energy of the YSR state be-come so small that it may be thermally excited. is probabil-0
Figure S2:
Josephson Eect (Data Set (a) Joseph-son spectra 𝐼 ( 𝑉 ) in a range of ± μ eV shied hori-zontally by the conductance at which they were mea-sured. e horizontal bracket indicates the region,where the evolution deviates from the conventionalAmbegaokar-Barato formula. e quantum phasetransition (QPT) is indicated by the vertical dashedline. (b) e switching current 𝐼 S , which is the cur-rent maximum indicated in (a) as function of the nor-mal state conductance 𝐺 N (blue line). e square de-pendence at low and high conductance is indicated bydashed lines (labeled by ∝ 𝐺 ). e YSR energy asfunction of conductance is shown as a blue line. eminimum indicates the QPT (vertical dashed line). (c)is graph shows √ 𝐼 S 𝑅 N (blue line) of the data in (b),which is proportional to 𝐼 C 𝑅 N for a harmonic energy-phase relation ( 𝑅 N : renormalized resistance (see text), 𝐼 C : critical current). A reference junction withoutany YSR states is shown as a yellow line indicatingthe expected evolution according to the Ambegaokar-Barato formula. ity has to be taken into account when calculating the Joseph-son current as it can signicantly broaden the region of theQPT even at mK temperatures. As the energy of the Andreevbound state in the BCS channel is 𝐸 BCS > 𝛥 √ − 𝜏 , it is un-likely for this state to be thermally excited. We, therefore,only consider the thermal excitation of the YSR state. Here,the energy of the excited state is − 𝐸 YSR ( 𝜑 ) . If we now dene 𝑝 as the probability for the YSR state to be thermally excited,we can write the lowest order harmonic coecient as 𝐸 = ( − 𝑝 ) (cid:12)(cid:12) 𝐸 BCS1 + 𝐸 YSR1 (cid:12)(cid:12) + 𝑝 (cid:12)(cid:12) 𝐸 BCS1 − 𝐸 YSR1 (cid:12)(cid:12) , (S30)where 𝑝 = /( + exp (| 𝜀 |/ 𝑘 B 𝑇 )) , 𝜀 is the energy of the YSRstate, and 𝑇 is the temperature. We use the coecient in Eq.S30 to calculate the Josephson current in Eq. S2, which is usedto calculate the t in the main text. THE 𝐼 C 𝑅 N PRODUCT FOR A YSR STATE
In the presence of a YSR state, the normal state resistance 𝑅 N , which is relevant for modeling the Josephson current maybe modied in the presence of Kondo correlations resultingfrom an enhanced density of states in the range of the super-conducting gap. erefore, the normal state conductance 𝐺 N that is typically measured outside the superconducting gaphas to be renormalized as the assumption that the density ofstates is a constant no longer holds. We have used referencemeasurements through a tunnel junction without any YSRstate [7], which has been shown previously to follow the ex-pected Ambegaokar-Barato formula [8], to nd the renor-malization coecient for the normal state resistance 𝑅 N inthe presence of a YSR state. For the data set presented in themain text, we nd 𝑅 N = /( . 𝐺 N ) , where 𝐺 N is the normalstate conductance outside the superconducting gap. We notethat this is a phenomenological result. e value of the renor-malization factor only holds for this particular data set as the contributions from Kondo correlations to the local density ofstates may vary between impurities. OTHER YSR TIPS
In Fig. S1, we show another data set with an STM tip thatis functionalized with a YSR state. As in the main text, theYSR state energy changes as a function of conductance (i.e.tip-sample distance) as shown in Fig. S1(a). e YSR stateundergoes a QPT at 0 . 𝐺 from a screened spin to a freespin regime (for comparison, the data in the main text has theQPT at 0 . 𝐺 ). A spectrum near the QPT is shown in Fig.S1(b) to show the transmission ratio between the transportchannel through the YSR state and the reference BCS gap.We can clearly see sizeable coherence peaks at ±( 𝛥 t + 𝛥 s ) indicating that the reference BCS channel is much strongerthan the YSR channel ( 𝛥 t = 𝛥 s = μ eV).e Josephson eect can be seen throughout the range ofconductance values shown in Fig. S1. Figure S2 shows theanalysis of the Josephson eect for the second data set inanalogy to Fig. 2 of the main text. In Fig. S2(a), we see the evo-lution of the Josephson spectra as function of conductance.Each spectrum is shown in a range of ± μ eV and horizon-tally oset by the corresponding conductance value. We canclearly see an increase in the signal, but which is again modu-lated in the region of the square bracket in Fig. S2(a). We plotthe switching current 𝐼 S (as indicated in Fig. S2(a)) as a func-tion of conductance in Fig. S2(b) (blue line). e correspond-ing YSR energies are ploed as a red line. We indicate theposition of the QPT by a vertical dashed line, where the YSRenergy has a minimum. e switching current also changesin the region of the QPT as in Fig. 2 of the main text.In Fig. S2(c), the √ 𝐼 S 𝑅 N product of the experimental datais ploed (blue line) in relation to a reference junction (yel-low line) and a t (orange line). For this data set, the renor-1malization of the normal state resistance 𝑅 YSRN = /( . 𝐺 N ) through the YSR state is not as pronoounced because thetransmission through the YSR channel is much reduced com-pared to the data set in the main text. For the t, we use thesame eective temperature of 𝑇 e =
75 mK as for the data inthe main text. For the relative channel transmission, we nd77.7% and 22.3% for the BCS and the YSR channel, respec-tively. e overall agreement between experimental data andt is excellent. ∗ Corresponding author; electronic address: [email protected][1] R. Koller, W. Bergermayer, G. Kresse, E. L. D. Hebenstreit,C. Konvicka, M. Schmid, R. Podloucky, and P. Varga, e struc-ture of the oxygen induced (1 ×
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