Tuning the coupling of an individual magnetic impurity to a superconductor:quantum phase transition and transport

Laëtitia Farinacci,1 Gelavizh Ahmadi,1 Gaël Reecht,1 Michael Ruby,1 Nils Bogdanoff,1

Olof Peters,1 Benjamin W. Heinrich,1 Felix von Oppen,2 and Katharina J. Franke1

1Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany2Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany

(Dated: November 1, 2018)

The exchange scattering at magnetic adsorbates on superconductors gives rise to Yu-Shiba-Rusinov (YSR) bound states. Depending on the strength of the exchange coupling, the magneticmoment perturbs the Cooper pair condensate only weakly, resulting in a free-spin ground state,or binds a quasiparticle in its vicinity, leading to a (partially) screened spin state. Here, we usethe flexibility of Fe-porphin (FeP) molecules adsorbed on a Pb(111) surface to reversibly and con-tinuously tune between these distinct ground states. We find that the FeP moment is screened inthe pristine adsorption state. Approaching the tip of a scanning tunneling microscope, we exerta sufficiently strong attractive force to tune the molecule through the quantum phase transitioninto the free-spin state. We ascertain and characterize the transition by investigating the transportprocesses as function of tip-molecule distance, exciting the YSR states by single-electron tunnelingas well as (multiple) Andreev reflections.

The exchange coupling of magnetic impurities to a su-perconductor induces localized Yu-Shiba-Rusinov (YSR)bound states [1–3]. Even for a single impurity, the lo-cal nature of the superconducting ground state dependsqualitatively on the strength of the exchange coupling J(Fig. 1a). For weak coupling, the bound subgap stateremains unoccupied, the superconducting ground statefully paired, and the impurity spin free. At strong cou-pling, the bound state becomes occupied and the super-conducting ground state involves an unpaired electronwhich (partially) screens the impurity spin [4–8]. Thelevel crossing between these states at a critical couplingJc, commonly referred to as a quantum phase transition,is protected by fermion-parity conservation. As statesof different fermion parity exchange roles at the tran-sition, the level crossing is most immediately reflectedin the single-particle addition spectrum. A discontinu-ity in the corresponding spectral weights as well as anabrupt change in the screened impurity spin make thisa first-order transition. When including effects of self-consistency, additional discontinuities are predicted inthe bound state energy and the local order parameter[9–11].

Experimental probes of magnetic-impurity physics usequantum dots or magnetic adatoms. For quantum dots,the exchange coupling J can be controlled electrically.Superconductor (SC)–quantum dot (QD)–SC junctionsthen provide indirect evidence for the quantum phasetransition through a 0–π transition of the Josephson cur-rent [12–14]. More immediate spectroscopic evidenceemerges from measurements on asymmetric SC–QD–normal metal (N) junctions [15–17]. Typically obtainedin the Coulomb-blockade regime at mK temperatures,the spectra are dominated by Andreev reflections at thejunction and are thus insensitive to the spectral weightsof the (single-particle) addition spectrum.

In contrast, probing magnetic adatoms with a scan-ning tunneling microscope (STM) tip readily providesaccess to both, single-electron and Cooper-pair (An-dreev) tunneling by controlling the tunnel gap and vary-ing the junction conductance over several orders of mag-nitude [18]. In the regime of single-electron tunneling,STM experiments measure the single-particle additionspectrum and thus provide crucial information about thequantum phase transition. In particular, tunneling spec-troscopy (STS) provides access to the asymmetry be-tween electron- and hole-like YSR excitations [18–26]which changes abruptly at the transition.

STM experiments are thus particularly well suited toprobe the first-order nature of the transition. However,earlier experiments [21, 27, 28] could only access a dis-crete set of exchange couplings J determined by theadatom’s adsorption site. Here, we use the flexibility ofa single molecule adsorbed on superconducting Pb(111)to modify the molecule’s interaction with the surface andtune the system continuously through the quantum phasetransition. The superconducting tip exerts a mechanicalforce on the molecule which can be approximated by aLennard-Jones potential (Fig. 1b). At large tip–moleculedistances, the force is attractive and the molecule is liftedfrom the surface. This modifies characteristic parametersin the junction and in particular reduces the exchangecoupling [29–31]. (For a discussion of the influence ofother parameters, see supplementary material [32].) Asthe tip approaches, repulsive forces eventually dominate,push the molecule back towards the surface, and increasethe exchange coupling again. When combined with de-tailed conductance measurements to resolve the differenttunneling processes, this technique allows for an in-depthanalysis of the quantum phase transition.

We use Fe(II)-porphin (FeP) molecules deposited on aclean Pb(111) surface from a powder of Fe(III)-porphin-

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Figure 1. a) Schematic dependence of ground and excitedstates of a magnetic adatom on a superconducting substrateon exchange coupling strength J . At weak coupling, theground state is a free-spin state, while the excited state has aquasiparticle bound to the adatom. At a critical coupling Jc,the roles of ground and excited states reverse. b) Sketch of theforces acting between tip and molecule vs. tip–molecule dis-tance. At large distances, attractive forces pull the moleculeaway from the surface. At closer distance, repulsive forcespush the molecule back towards the surface. c) STM topog-raphy image of a FeP island (V = −45 mV, I = 50 pA) witha molecular model (inset). d) dI/dV spectra acquired with aPb tip above bare Pb(111) (black), center (blue) and ligand(red, offset for clarity) of a molecule.

chloride (FeP-Cl). These molecules lose their Cl ligandby deposition below room temperature followed by an-nealing to 370 K (see supplementary material [32]). STMexperiments at a temperature of 1.6 K reveal the forma-tion of well-ordered islands, as shown in Fig. 1c. Theindividual molecules are identified by their clover shapewith a bright protrusion at the Fe center.

The superconducting Pb tips which we use to probethe YSR states of the molecule provide an effective en-ergy resolution beyond the Fermi-Dirac limit. The mea-sured signal is a convolution of the tip’s Bardeen-Cooper-Schrieffer (BCS) density of states with the substrate den-sity of states. Apart from peak intensity changes, theconvolution shifts all spectral features of the substrateby the tip’s superconducting energy gap ∆t. The BCSpeaks of the substrate are thus observed at bias voltagesof ±(∆ + ∆t)/e = ±2.65 mV [33], where ∆ denotes thegap of the substrate. Inside the gap, we find one pair ofYSR states at ±2.2 mV. These are resolved both on theFe center and the ligand (Fig. 1d) so that both positionscan be employed to investigate the tip-induced forces andthe quantum phase transition.

We first characterize the junction conductance as the

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2/h)vs. tip offset ∆z. b) 2D false-color plot of dI/dV spectrarecorded at various tip-sample distances, normalized to theirconductances at 5 mV. The spectra were acquired after open-ing the feedback at V = 5 mV, I = 200 pA and subsequentlyvarying the tip height by an offset ∆z (lock-in modulationVrms = 15 µV ). c) Spectra before (∆z = −140 pm, browntrace), at (∆z = −150 pm, claret), and after (∆z = −160 pm,blue) the quantum phase transition (offset for clarity). d) Ex-tracted YSR energies and intensity asymmetries vs. junctionconductance. Around G = 0.02 × G0 (∆z = −150 pm), theYSR state energy crosses zero and the asymmetry changessign, indicating the quantum phase transition. e) Single-electron tunneling processes for YSR resonances at positiveand negative bias voltages.

tip approaches the Fe center. We measure the tip ap-proach ∆z from a set point at V = 5 mV and I =200 pA. As shown in Fig. 2a, the conductance first in-creases exponentially with ∆z as expected for a tunneljunction. (Small deviations are due to nonlinearities inthe I-V converter at small current densities.) A super-exponential increase beyond ∆z = −150 pm and subse-quent leveling off indicate the transition to the contactregime between tip and molecule. Contact formation isfully reversible with identical approach and retractioncurves, enabling precise tuning of the junction conduc-tance.

As described above, the strength of the exchange cou-pling and hence the binding energy of the YSR statesdepend sensitively on the tip approach ∆z. As the tipapproaches the Fe center, we observe that the YSR statesshift deeper into the superconducting gap (Fig. 2b). At∆z = −150 pm, the YSR resonance occurs at a bias volt-

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age equal to the tip‘s gap parameter, i.e., at zero energy(see zoom in Fig. 2c). Approaching the tip further, theYSR states shift back towards the superconducting gapedge before eventually reversing at ∆z = −190 pm (redarrow in Fig. 2b) and reaching zero energy for a secondtime (blue arrow in Fig. 2b). Beyond this point, at evencloser tip–molecule distances, the spectra are dominatedby peaks at zero bias and ∆t.

Individual dI/dV spectra do not allow for a completeidentification of the binding energy of the YSR state sincethe same excitation energy may signify a screened or afree-spin ground state (see Fig. 1a). However, the ob-served shift of the YSR states deeper into the supercon-ducting energy gap upon approach allows for an unam-biguous assignment. As attractive tip–molecule forcesinitially lift the Fe center from the surface and weakenthe exchange coupling J , the pristine system must be inthe strong-coupling regime with a screened-spin groundstate and a negative YSR energy.

We observe asymmetric intensities of the electron- andhole-like YSR resonances at positive and negative volt-ages, respectively. This asymmetry is a consequence ofthe potential scattering by the magnetic impurity. Atlarge tip-molecule distances, the electron-like excitationhas more spectral weight than the hole-like excitation.The relative strength changes with tip approach and ac-companying YSR energy shift. In Fig. 2d, we plot thebinding energy � of the YSR state and the asymmetry(I+ − I−)/(I+ + I−) as a function of junction conduc-tance, with I+/− being the YSR intensities at positiveand negative bias voltages, respectively. While the bind-ing energy passes smoothly through zero, the zero-energycrossing is associated with an abrupt change in sign of theasymmetry.

These observations provide detailed evidence for afirst-order quantum phase transition from a screened-spinto a free-spin ground state as the tip–molecule distanceis reduced. For sufficiently weak tunneling, the currentis dominated by single-electron tunneling (sketches inFig. 2e) [18]. Hence, we can relate the observed intensi-ties of the electron- and hole-like resonances to the localweights of the electron and hole wave functions of theYSR bound state (see [32] for additional details). Excit-ing the system out of the screened-spin YSR state anni-hilates a bound quasiparticle. For single-electron tunnel-ing, this process involves γ0 = ucα − vc†β , where γ0 (cα)is the annihilation operator of the bound quasiparticle(electron with spin α). Correspondingly, the intensityof the electron-like excitation at positive bias voltages isgiven by |v|2, as the bound electron combines with thetunneling electron to form a Cooper pair. Similarly, thehole-like excitation at negative bias is given by |u|2, re-flecting a bound electron tunneling out into the tip. Theroles of u and v are reversed when exciting the system outof the free-spin ground state and creating a bound quasi-particle, as described by γ†0 = u

∗c†α−v∗cβ . Now, electron

tunneling at positive bias occupies the bound state andinvolves |u|2, while tunneling at negative bias breaks aCooper pair and involves |v|2. The abrupt change in theasymmetry at a tip approach of ∆z = −150 pm, wherethe YSR state crosses zero energy, is thus a hallmark ofthe first-order quantum phase transition. We also notethat the gradual increase of the asymmetry before thequantum phase transition is in agreement with a decreasein the exchange coupling [9, 34].

Self-consistency causes additional discontinuities in thebound-state energy and the local order parameter nearthe impurity [9–11]. We do not find indications of thesediscontinuities. In the supplementary material [32], wederive analytical estimates for their magnitude. The dis-continuity of the local gap parameter is substantial, oforder δ∆ ∼ −∆/ ln(ωD/∆) within a few Fermi wave-lengths of the impurity (ωD is the Debye frequency), butcannot be directly probed by single-electron tunneling.The latter probes the bound state energy, whose jump isof order δ� ∼ ∆2/[EF ln(ωD/∆)] ∼ 10−2 µeV. This isseveral orders of magnitude below our experimental reso-lution of order ∼ 100 µeV. Instead of a discontinuity, weobserve that the asymmetry vanishes right at the tran-sition. This suggests that both |u|2 and |v|2 are probedsimultaneously as is natural for a bound state whose en-ergy vanishes within the resolution.

Next, we place the tip above the ligand. The G −∆zcurves exhibit the same stability and reversibility as onthe center and allow for probing both the tunneling andthe contact regime (Fig. 3a). At large tip-molecule dis-tances, we find YSR states at the same bias voltagesas above the Fe center, but with reversed asymmetry(Fig. 1d). The dI/dV spectra at different junction con-ductances (Fig. 3b) also show the initial shift of theYSR states deeper into the superconducting energy gap.However, the YSR states do not reach or cross zero en-ergy. The YSR states come closest to zero energy for∆z = −150 pm before shifting back to higher energies asa result of the molecule being pushed back to the surface(Fig.1b). An analysis of the asymmetry confirms thatthe system does not pass through the quantum phasetransition (Fig. 3c). Indeed, the asymmetry does notchange sign at ∆z = −150 pm and the hole-like exci-tation remains stronger than the electron-like excitationthroughout (see detailed spectra in Fig. 3d). A change inasymmetry occurs only upon further approach (Fig. 3c,d)when resonant Andreev reflections become the dominanttunneling process [18].

Importantly, the overall trend of the YSR shift resem-bles the case with the tip above the Fe center, reflect-ing an analogous dependence on the tip–molecule force.Hence, despite the inverse asymmetry, the YSR state re-flects the same ground state. Given that we observe thesame energy on ligand and center, we suggest that theYSR state arises from a delocalized spin state associatedwith Fe dπ states hybridized with π states of the lig-

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Figure 3. Tip approach above Fe ligand of FeP molecule (seeFig. 1b) (for more data see [32]). a) Evolution of conductancewith tip offset ∆z. b) 2D false-color plot of dI/dV spectrarecorded at various tip-sample distances, normalized to theirconductances at 5 mV. c) Extracted YSR energies and inten-sity asymmetries vs. junction conductance. d) Four spectraof this approach set (offset for clarity). The YSR state doesnot reach zero energy and the asymmetry remains positiveduring most of the approach, see the brown, claret, and bluespectra in panel d). The asymmetry reverses only aroundG = 0.1 × G0 where Andreev reflections become dominant,see blue and purple spectra in panel d.

and [35–37]. The absence of a quantum phase transitionon the ligand reflects differences in the elastic responseof the molecule to the applied force.

So far, we could understand all spectral features interms of single-electron excitations of the YSR states.We already noted that on both Fe center and ligand,the asymmetry is reduced at tip approaches of ∆z ≈−190 pm. The continuous change of spectral intensitycan be explained by resonant Andreev reflections gainingstrength as the tunnel conductance increases [18]. Fur-thermore, we observe the onset of a zero-bias resonanceas a fingerprint of Cooper pair (Josephson) tunnelingupon further increase of the junction conductance. (Refs.[31, 38] also observed Josephson peaks in STM spectraof single adatoms and molecules at close tip–sample dis-tance.) In addition, we find peaks at eV = ±∆t. We in-terpret these as the threshold for the lowest-order (n = 2)multiple Andreev reflections (MAR), generally expectedat eVn = ±(∆ + ∆t)/n for an (n − 1)-fold Andreev re-flection (n = 2, 3, ...) [39]. We observe additional subgappeaks, which shift in energy with tip approach. The set

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Figure 4. a) Closer view of the approach set in Fig. 3b.b) Four spectra (offset for clarity) of this set showing twopeaks due to the Andreev processes depicted in c). An elec-tron (hole) is reflected through the junction until its energymatches the YSR state. d) Extracted positions of the peaksshown in a and b (rectangles) as well as traces showing theexpected positions of a thermal excitation of the YSR state(black trace) and of the processes shown in c (yellow and pink)given the position of the YSR state.

of peaks following eV = ∆t − � arises from thermal ex-citations [18]. Interestingly, the contrast-enhanced plotof dI/dV spectra (Fig. 4a,b) shows two other sets ofpeaks, which follow the relation eV = (∆t + �)/n withn = 2, 3 [38] (Fig. 4d). They originate from the exci-tation of the YSR state by electron/holes that are An-dreev reflected from the superconductor (see sketches inFig. 4c). Unlike conventional MARs, these processes re-flect the asymmetry of the YSR states and might also beusable as fingerprints of the quantum phase transition.

Finally, when the tip is in contact with the molecule(as deduced from the flat G −∆z curves in Fig. 2a andFig. 3a), the Andreev reflections through the YSR statesmerge with the conventional MARs. In this contactregime, the coupling of the impurity to the tip and thesurface start to compete which each another, leading toan effective coupling of the two electrodes. As a result,an applied bias drives the junction out of equilibrium sothat signatures of local excitations can be no longer de-tected in spectroscopic measurements [40].

In conclusion, we have realized a magnetic-impurityjunction whose exchange coupling can be precisely,continuously, and reversibly tuned through mechanicalforces exerted by an STM tip. Moreover, the junction

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conductance can be varied by almost three orders ofmagnitude which enables us to access various transportregimes. We have employed this setup for a detailed in-vestigation of the quantum phase transition between thescreened-spin and free-spin ground states. We confirmthe first-order nature of the transition in that the spectralweights of the single-particle addition spectrum changeabruptly at the transition. We do not observe the pre-dicted discontinuous jumps in the YSR energy and thelocal order parameter. Our analytical estimates showthat the expected jumps in YSR energies are by ordersof magnitude smaller than the energy resolution even atmK temperatures. Finally, we emphasize that our re-sults highlight a method for unambiguously determiningthe nature of the ground state through variation of theexchange coupling.

We thank S. Loth for discussions. Financial sup-port by Deutsche Forschungsgemeinschaft through grantsHE7368/2 and CRC 183, as well as by the Euro-pean Research Council through the consolidator grant“NanoSpin” is gratefully acknowledged. FvO performedpart of this work at the Aspen Center for Physics sup-ported by NSF grant PHY-1607611.

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6

[42] B. W. Heinrich, L. Braun, J. I. Pascual, and K. J. Franke,Nat. Phys. 9, 765 (2013).

[43] F. Pientka, L. I. Glazman, and F. von Oppen, Phys.Rev. B 88, 155420 (2013).

http://dx.doi.org/10.1038/nphys2794http://dx.doi.org/10.1103/PhysRevB.88.155420http://dx.doi.org/10.1103/PhysRevB.88.155420

7

SUPPLEMENTAL MATERIAL

THEORETICAL CONSIDERATIONS

In this section, we summarize relevant theoretical considerations to understand the experimental results, includingconsiderations of relevant parameters for driving the quantum phase transition, new analytical results on the effectof self-consistency on the local gap parameter ∆(r), and the energy � of the YSR state.

Relation between experimental and model parameters

A minimal model for the scattering of substrate electrons by a magnetic impurity is the s−d model (H = (V +JS·σ))which includes the hybridization between the impurity orbitals and the substrate through the exchange coupling Jand the potential scattering V (as obtained by a Schrieffer-Wolff transformation from the Anderson model). Treatingthe impurity spin S as classical, the energies of the YSR states are given by

� = ±∆

1− α2 + β2√(1− α2 + β2)2 + 4α2

(S1)with α = πρ0SJ and β = πρ0V . Here, ρ0 is the normal-state density of states of the superconductor. As shown inthe main manuscript, the approach of the STM tip to the FeP molecule shifts the YSR states substantially within thesuperconducting gap. Specifically, it shifts the YSR energies across the Fermi level, corresponding to the quantumphase transition. While according to Eq. S1, the YSR energies depend on several parameters, we argue that theexchange coupling strength J is the most relevant parameter for driving the quantum phase transition. In thefollowing we explain why the other parameters are expected to affect the YSR energies only slightly, and cannot drivethe quantum phase transition by themselves.

Exchange coupling J : The tip approach towards the molecule locally modifies the molecule-substrate distance asa result of tip-molecule forces. This changes the molecule-substrate hybridization and hence the exchange couplingJ . As can be seen from Eq. (S1), J is essential for changing the sign of the YSR energy and, hence, the quantumphase transition. Moreover, all qualitiative features of our experimental data can be captured by the variation of thisparameter in response to the force field of the STM tip, which induces conformational changes of the molecule.

Potential scattering V : The molecule-substrate hybridization also induces the potential scattering amplitude V .As V (and hence β) depends on the relative position of the singly and doubly occupied orbitals with respect to theFermi level, V also accounts for changes in the charge transfer between impurity and substrate. However unlike J ,the potential scattering V does not grow under renormalization group transformations of the s− d model, so that Vis generically expected to be weaker than J .Normal density of states ρ0 and gap parameter ∆: One can safely assume that ρ0 cannot drive the quantum phase

transition as it is a bulk parameter. Self-consistent calculations predict a J-dependent change in ∆ at the impuritysite. However, we show explicitly in the next section that this modification of ∆ causes only minimal shifts of theYSR state energy which are substantially below the energy resolution of the experiment.

Impurity spin S: The tip approach may also modify the magnetic moment and thus the effective spin. However,we expect that the spin state is only changed by a small fraction of the total spin moment. If the spin magneticmoment changed by more than ∆S = 1/2, we would expect more substantial changes in the subgap spectrum suchas a variation in the number of YSR states. A small change is insufficient to explain the substantial shift in the YSRstate energy.

While some of these additional parameters may thus be relevant for explaining quantitative details of the experiment,it suffices to focus on the exchange coupling J to understand the qualitiative features of the experimental data. Thisis the approach which we take in the main manuscript.

Gap parameter and YSR state energy near the quantum phase transition

In the absence of the magnetic impurity, the superconducting ground state involves an even number of electrons(even fermion number parity). Excited states with odd numbers of electrons (odd fermion parity) have a continuous

8

spectrum with an excitation energy larger than the superconducting gap ∆. A magnetic impurity induces a localizedquasiparticle state at subgap energies [1].

For weak exchange coupling between impurity and electrons in the superconductor, this YSR state is unoccupiedin the ground state so that the ground state remains fully paired (even fermion parity). But there is now a discreteexcited state with odd fermion parity at an excitation energy � below ∆ in which the YSR state is occupied by aquasiparticle.

The excitation energy of this odd fermion parity state reduces with increasing exchange coupling until it reacheszero at a critical coupling Jc. Beyond the critical coupling, the bound quasiparticle state becomes occupied in theground state and empty in the excited state, i.e., the ground state is now an odd fermion parity state while the excitedstate has even parity. This level crossing between states with even and odd fermion parity at the critical coupling isa true level crossing as it is protected by fermion parity conservation [1].

At zero temperature, tunneling into superconductors at subgap energies proceeds via Andreev processes whichtransfer Cooper pairs into the superconductor. At finite temperatures, inelastic processes open an additional tun-neling channel in which single electrons are transferred. An electron tunnels into the superconductor occupying asubgap quasiparticle state and is subsequently excited inelastically into the quasiparticle continuum (or recombinesinelastically into a Cooper pair with another subgap excitation). This single-particle tunneling dominates over An-dreev processes when the tunneling rate is slow compared to the inelastic relaxation rate. This is the case in STMexperiments at sufficiently low tunneling conductance [2].

We start from a 4× 4 Bogoliubov-deGennes (BdG) Hamiltonian

H = ξpτz + (V τz + JSi · σ)δ(r) + ∆τx. (S2)

in the basis in which the four-component Nambu operator takes the form Ψ = [ψ↑, ψ↓, ψ†↓,−ψ

†↑] in terms of the

electronic field operator ψσ(r) and, for definiteness, consider a classical magnetic impurity. Here, ξp = p2/2m − µ

with the chemical potential µ, V the strength of the potential scattering by the impurity, and J denotes the strength ofthe exchange coupling between the magnetic impurity with spin S and the electrons in the superconductor. The Paulimatrices σi (τi) operate in spin (particle-hole) space. Choosing the impurity spin S to point along the z direction,the 4× 4 Hamiltonian separates into independent 2× 2 Hamiltonians

H± = ξpτz(V τz ± JS)δ(r) + ∆τx. (S3)

A standard calculation [3] shows that H+ (H−) has one subgap solution [u�(r), v�(r] ([u−�(r), v−�(r)]), whose energywe denote by � (−�). Since the BdG formalism doubles the degrees of freedom, it is sufficient to consider only thesolutions of one of these Hamiltonians, say H−. Its subgap (YSR) state starts out at positive energies at smallexchange couplings and crosses to negative energies for strong coupling. Keeping only the contribution of the subgapstate with Bogoliubov operator

γ� =

∫dr[u∗� (r)ψ↓(r)− v∗� (r)ψ

†↑(r)], (S4)

we can write the electron operators as

ψ↓(r) = u�(r)γ� + . . . (S5)

and

ψ↑(r) = −v∗� (r)γ†� + . . . , (S6)

where the ellipses indicate the contributions of above-gap quasiparticles. The even parity state satisfies γ�|even〉 = 0and the odd-parity state is |odd〉 = γ†� |even〉. Note that γ†� removes an electron spin of 1/2 from the electron system,so that the spin of the odd parity state is lower by 1/2 compared to the (spinless) even parity state.

In the regime of single-electron tunneling, the differential conductance at eV = ±(∆t + �) (for a superconductingtip with gap ∆t) provides access to the Bogoliubov-deGennes wave function of the YSR bound state (energy �) at thetip position r [2]. Consider first a positive bias voltage with electrons tunneling into the superconducting substrate.

• For J < Jc, the tunneling electrons excite the system from the even parity ground state to the odd parityexcited state. In view of the spin polarization of the YSR state, the differential conductance is proportional to|〈odd|ψ†↓(r)|even〉|2 (Fermi’s Golden Rule). We therefore deduce that the differential conductance is proportionalto |u(r)|2 in terms of the electron component of the YSR state.

9

• For J > Jc, the differential conductance involves the matrix element |〈even|ψ†↑(r)|odd〉|2, since |odd〉 (|even〉) isnow the ground (excited) state. The differential conductance is thus proportional to |v(r)|2 involving the holewave function of the bound state.

As the electron and hole wave functions are generically different, one expects a discontinuous change in the differentialconductance at the critical coupling Jc.

Now consider negative bias voltages where electrons are tunneling out of the superconducting substrate. Thedifferential conductance is then proportional to |〈odd|ψ↑(r)|even〉|2 for J < Jc and |〈even|ψ↓(r)|odd〉|2 for J > Jc.Consequently, the roles of u and v invert relative to positive bias voltages.

This picture of the quantum phase transition must be amended when effects of self-consistency are taken intoaccount. Both the local gap parameter and the energy of the YSR state have discontinuities at the critical coupling.This has been investigated repeatedly by numerical simulations [4, 10, 11], but to the best of our knowledge, thereare no analytical estimates for the magnitudes of these discontinuities in the literature. Such estimates are providedin the following.

The jump in the local gap parameter can be understood when writing the (zero-temperature) gap equation in termsof the Bogoliubov-deGennes wave functions un(r) and vn(r) of, say, H−,

∆(r) = g∑n

un(r)v∗n(r), (S7)

where the sum is over positive-energy eigenstates within an energy band about the Fermi energy whose width is givenby the Debye frequency ωD. The parameter g > 0 denotes the coupling constant of the attractive interaction. As thesystem passes through the quantum phase transition, the positive energy subgap state crosses to negative energiesand no longer contributes to the sum on the right hand side of the gap equation. Thus, we find

δ∆(r) = −gu�(r)v∗� (r). (S8)

for the jump in the gap parameter. Here, we assume that the effect of the discontinuity on the other Bogoliubov-deGennes eigenstates is weak. We will confirm below that this is indeed the case. For a classical magnetic impurityand uniform gap parameter, one finds that u� and v� can be chosen real and have the same sign at the impurityposition [3]. Thus, this indeed describes a local suppression of the order parameter.

The discontinuity in the gap parameter is limited to the region in which the YSR bound state is localized. Atits center, say r = 0, and at the critical exchange coupling Jc, the YSR bound state has an amplitude of |u(0)|2 ∼|v(0)|2 ∼ ν0∆ [3], where ν0 is the (normal-state) density of states of the superconductor at the Fermi energy. Insertingthis into the expression for the jump in the gap parameter, we have δ∆(0) ∼ −gν0∆. Using the relation

∆ ∼ ωDe−1/gν0 , (S9)

we therefore obtain

δ∆(0) ∼ − ∆ln(ωD/∆)

(S10)

for the local and discontinuous reduction in the gap parameter at the critical coupling.As the gap is locally suppressed at the quantum phase transition, it can even change sign. This happens when

the discontinuity is larger than the value of the local gap parameter just before the quantum phase transition. ForJ < Jc, the magnetic impurity suppresses the gap parameter continuously and this suppression of the gap parameterhas been previously estimated within perturbation theory [6], yielding a suppression by α2∆ for small exchangecoupling α = πν0JS. As α ∼ 1 at the quantum phase transition, the value of ∆(0) just before the quantum phasetransition may indeed be smaller than the magnitude δ∆(0) of the jump, so that the gap parameter changes signacross the transition.

Since the jump in ∆(r) is associated with the bound state contribution to the gap equation, it is localized inthe vicinity of the impurity. To provide an estimate for the associated jump in the energy of the YSR state, weapproximate the jump in ∆(r) by a δ-function [6],

δ∆(r) = a3δ∆(0)δ(r). (S11)

Here, a is the linear dimension of the region over which the gap parameter is suppressed, which can be estimated as[5] a ≈ 2/kF in terms of the Fermi wave vector of the superconductor. The associated jump in the energy of the YSR

10

state can then be obtained from first-order perturbation theory which yields

δ� ∼ ∆2

EF ln(ωD/∆). (S12)

In deriving this expression, we have again used the magnitude of the YSR wave function at the position of the impurity.This expression shows that the effect of the gap reduction on the Shiba states is small in the parameter ∆/EF whichis of order 10−5 for conventional superconductors.

Clearly, this predicts a discontinuity in the energy of the Shiba state which is below experimental resolution byorders of magnitude. Moreover, in the regime of single electron tunneling, STM experiments are sensitive to thechange in the local gap parameter only via the associated change in the bound state wave function which is negligible.

ADDITIONAL EXPERIMENTAL DATA

Fe(III)-porphine-chloride molecules

As mentioned in the main text, Fe(III)-porphine-chloride molecules are evaporated on a Pb(111) sample heldbelow room temperature, followed by annealing the preparation to 370 K. We show in Fig. S1a a topography imageobtained on a sample before annealing. There, we observe the formation of ordered islands in which a few molecules(≤ 1%) exhibit a pronounced protrusion above their center. As shown in Fig. S1b, the apparent height of sucha molecule is approximately 0.7 Å larger than the one of molecules with a clover shape. A similar elevation hasbeen observed for Cl ligands on other Fe porphin derivatives, e.g., Fe-octaethylporphyrin-chloride (Fe-OEP-Cl) onAu(111) [7] and Pb(111) [8]. The dI/dV spectra on these bright protrusions reveal the presence of sharp peaks outsidethe superconducting energy gap at bias voltages of ±4 mV and ±5.4 mV (see Fig. S1c). These peaks reflect inelasticexcitations of the molecule on a superconductor [8], which can be assigned to spin excitations of the Fe center. In thepresence of the Cl ligand the Fe center lies in +3 oxidation state with S = 5/2. The anisotropic environment lifts thedegeneracy of the d levels. Together with spin-orbit coupling, this introduces magnetocrystalline anisotropy to the Festates. The inelastic peaks in the dI/dV spectrum arise from transitions between these anisotropy-split levels. Theabsence/presence of these excitations thus indicates the absence/presence of the Cl ligand. All molecules discussed inthe main text have lost their Cl ligand after annealing the deposited molecules to 370 K

1nm543210

2.5

2.0

1.5

1.0

0.5

0.0

X (nm)

Z (Å

)

a)

1nm

150

100

50

0

dI/d

V (n

A/V

)

-10 -505 10Sample Bias (mV)

c)b)

Figure S1. a) STM topography image (V = −45 mV, I = 100 pA) after deposition of FeP-Cl on a sample held below roomtemperature. A few molecules kept their Cl ligand and show a protrusion above their center. b) Line profile along the red lineshown in a) showing the larger apparent height of the molecule indicated by an arrow. c) dI/dV spectrum acquired above thecenter of such a molecule (feedback opened at V = −45 mV, I = 100 pA, Vrms = 50 µV) showing the presence of inelasticspin excitations in accordance with [8].

Quantum phase transition in molecules in different surroundings

The results presented in the main text are acquired above a molecule at the border of an island. Here, we showthat the same observations can be made for molecules in different surroundings.

The evolution of the junction conductance above the center and ligand of such a molecule within an island showsa reversible contact formation similar to the molecule in the main text [Fig. S2(ii)]. A set of spectra recorded at

11

Asymm

etry

0.2

0.1

0.0

-0.1-0.2

Ԑ (m

eV)

1.00.50.0

-0.5-1.0

Conductance (G0)10-1 10-2 10-3 10-4

1nm

4-4 2-2 0Sample bias (mV)

dI/d

V (a

rb. u

nits

)

Lo

Hi

2Δt

Δz (pm)

-160

-80

0

80

-240

Asymm

etry

0.20.10.0-0.1-0.2

Ԑ (m

eV)

1.00.50.0

-0.5-1.0

Conductance (G0)10-1 10-2 10-3

0.3

-0.3

Con

duct

ance

(G0)

0.001

0.01

0.1

-250 -200 -150 -100 -50 0Δz (pm)

-250 -200 -150 -100 -50 0Δz (pm)

Con

duct

ance

(G0)

0.01

0.1

2Δt

Sample bias (mV)4-4 2-2 0

dI/d

V (a

rb. u

nits

)

0

1

2

3

4

-190 pm

-200 pm

-210 pm

-170 pm

-180 pm

-190 pm

-220 pm

Sample bias (mV)4-4 2-2 0

dI/d

V (a

rb. u

nits

)

0

1

2

3

4

1nm

4-4 2-2 0Sample bias (mV)

dI/d

V (a

rb. u

nits

)

Lo

Hi

Δz (pm)

-160

-240

-80

0

802Δt

a) i.

iii.

ii.

iv.

v.

b) i.

iii.

ii.

iv.

v.

Figure S2. Evolution of the spectroscopic features above the center a) and ligand b) of a molecule within an island. i. STMtopography image showing the tip location (black dot). ii. Dependence of the junction conductance upon tip approach. iii.Set of dI/dV spectra normalized to their conductance at 5 mV recorded at different tip height. iv. Extracted YSR energyand asymmetry as a function of the junction conductance. v. Selected spectra (offset for clarity) showing that the system goesthrough the quantum phase transition on the center of the molecule (a), and does not go through the quantum phase transitionby approaching on the ligand (b).

dI/d

V (a

rb. u

nits

)

Lo

Hi

a) 2Δt

0 3-3Sample bias (mV)

-280

-240

-200

-160

Δz (pm)

b) c)

-A2 A2 A1

0 1-1Sample bias (mV)

dI/d

V (a

rb. u

nits

)

0

1

2-A1

1.8

2.0

2.2

2.4

2.6

-180-200-220-240-260Δz (pm)

Peak

pos

ition

(mV)

VYSR/2

Figure S3. a) Close-up view of the approach set of Fig. S2b highlighting the presence of resonances due to multiple Andreevreflections. b) Four spectra of this approach set (offset for clarity) showing a peak shifting upon tip approach. c) Extractedpeak position as well as the expected position of the Andreev process according to the position of the YSR state.

different tip–sample distances is shown as a false 2D color plot in Fig. S2(iii). At far tip distance, a pair of YSR statesis observed at ±2.5 mV at both the center and the ligand of the molecule. At both locations the YSR states first shifttoward zero energy upon tip approach, indicating a screened spin ground state and negative YSR energy. Above thecenter, the YSR state crosses zero energy and reverses its asymmetry around ∆z = −200 pm [see Fig. S2a (iv and v)]indicating that the system undergoes the quantum phase transition. In contrast, above the ligand, the YSR energyremains negative and a reversal of the YSR asymmetry only occurs at high junction conductance (G ≥ 0.1 × G0)when Andreev reflections dominate the transport through the junction [see Fig.S2b (iv and v)].

As shown in Fig. S3, resonances within V = ±∆t build up upon further tip approach. In addition to a zero-biaspeak due to Cooper pair tunneling, resonances are observed at V = ±∆+∆t2 and V = ±

∆+�2 (see Fig.S3c) arising from

multiple Andreev reflections, as discussed in the main text.

12

[1] A. V. Balatsky, I. Vekhter, and J. X. Zhu, Rev. Mod. Phys. 78, 373 (2006).[2] M. Ruby, F. Pientka, Y. Peng, F. von Oppen, B. W. Heinrich, and K. J. Franke, Phys. Rev. Lett. 115, 087001 (2015).[3] F. Pientka, L. I. Glazman, and F. von Oppen, Phys. Rev. B 88, 155420 (2013).[4] M. I. Salkola, a. V. Balatsky, and J. R. Schrieffer, Phys. Rev. B 55, 12648 (1997).[5] M. E. Flatté and J. M. Byers, Phys. Rev. B 56, 11213 (1997).[6] T. Meng, J. Klinovaja, S. Hoffman, P. Simon, and D. Loss, Phys. Rev. B 92, 064503 (2015).[7] B. W. Heinrich, G. Ahmadi, V. L. Müller, L. Braun, J. I. Pascual, and K. J. Franke, Nano Lett. 13, 4840 (2013).[8] B. W. Heinrich, L. Braun, J. I. Pascual, and K. J. Franke, Nat. Phys. 9, 765 (2013).

http://dx.doi.org/10.1103/RevModPhys.78.373http://dx.doi.org/ 10.1103/PhysRevLett.115.087001http://dx.doi.org/10.1103/PhysRevB.88.155420http://dx.doi.org/10.1103/PhysRevB.55.12648http://dx.doi.org/https://doi.org/10.1103/PhysRevB.56.11213http://dx.doi.org/ 10.1103/PhysRevB.92.064503http://dx.doi.org/ 10.1021/nl402575chttp://dx.doi.org/10.1038/nphys2794

Tuning the coupling of an individual magnetic impurity to a superconductor: quantum phase transition and transportAbstract Acknowledgments References Supplemental Material Theoretical considerations Relation between experimental and model parameters Gap parameter and YSR state energy near the quantum phase transition

Additional experimental data Fe(III)-porphine-chloride molecules Quantum phase transition in molecules in different surroundings

References