Correlating Josephson supercurrents and Shiba states in quantum spins unconventionally coupled to superconductors
Felix Küster, Ana M. Montero, Filipe S. M. Guimarares, Sascha Brinker, Samir Lounis, Stuart S. P. Parkin, Paolo Sessi
CCorrelating Josephson supercurrents and Shiba statesin quantum spins unconventionally coupled to supercon-ductors
Felix K ¨uster ‡ , Ana M. Montero ‡ , Filipe S. M. Guimar˜aes , Sascha Brinker , Samir Lounis , ∗ ,Stuart S. P. Parkin ∗ , Paolo Sessi ∗ Max Planck Institute of Microstructure Physics, Halle 06120, Germany Peter Gr¨unberg Institut and Institute for Advanced Simulation, Forschungszentrum J¨ulich & JARA,J¨ulich D-52425, Germany Faculty of Physics, University of Duisburg-Essen, 47053 Duisburg, Germany ‡ These authors contributed equally to this work. ∗ Emails: [email protected], [email protected], [email protected]
Local spins coupled to superconductors give rise to several emerging phenomena di-rectly linked to the competition between Cooper pair formation and magnetic exchange.These effects are generally scrutinized using a spectroscopic approach which relies on de-tecting the in-gap bound modes arising from Cooper pair breaking, the so-called Yu-Shiba-Rusinov (YSR) states. However, the impact of local magnetic impurities on the supercon-ducting order parameter remains largely unexplored. Here, we use scanning Josephson spec-troscopy to directly visualize the effect of magnetic perturbations on Cooper pair tunnelingbetween superconducting electrodes at the atomic scale. By increasing the magnetic impu-rity orbital occupation by adding one electron at a time, we reveal the existence of a direct a r X i v : . [ c ond - m a t . s up r- c on ] F e b orrelation between Josephson supercurrent suppression and YSR states. Moreover, in themetallic regime, we detect zero bias anomalies which break the existing framework based oncompeting Kondo and Cooper pair singlet formation mechanisms. Based on first-principlecalculations, these results are rationalized in terms of unconventional spin-excitations in-duced by the finite magnetic anisotropy energy. Our findings have far reaching implicationsfor phenomena that rely on the interplay between quantum spins and superconductivity.Introduction The competition between magnetism and superconductivity is one of the most fascinating, highlydebated, and intriguing topics in condensed matter physics. After the formulation of the BCStheory , it became clear that superconductivity in the spin singlet state is destroyed by a mag-netic exchange mechanism which tends to align the opposite spins of Cooper pairs in the samedirection, thus preventing their formation, i.e. the so-called paramagnetic effect
2, 3 . Consistentlywith theoretical expectations, early experimental works using heat-capacity, transport, and tunnel-ing junctions measurements evidenced a reduction of the superconducting transition temperaturewhen magnetic impurities were introduced into the system . However, by averaging over theentire sample’s area, these techniques rely on the assumption of equivalent impurities, inevitablyincluding spurious effects related to sample inhomogeneity or contaminants. Overall, this severelycomplicated the task of disentangling the role of spin from that of the local environment. Thisshortcoming has been overcome by the invention of experimental methods capable of capturingthe rich physics taken place at the nanoscale by atomic resolution imaging . In a seminal scanning2unneling microscopy (STM) work, Eigler and colleagues visualized the effect of single magneticimpurities coupled to an elemental superconductor, demonstrating the presence of an enhanceddensity of states residing inside the superconducting energy gap . By using a classical spin model,these results were explained in terms of magnetic exchange-induced quasi particle resonances, i.e.the so-called Yu-Shiba-Rusinov (YSR) states . In recent years, a tremendous progress has beenmade in understanding YSR excitations . These efforts were mainly driven by the identifica-tion of superconducting-magnetic interfaces as viable routes towards the creation of topologicalsuperconductors supporting Majorana modes
21, 22 , which are essential ingredients for topologicalquantum computation schemes
23, 24 . This progress was made possible by the development of rou-tinely available low-temperature STM-based spectroscopic techniques with an energy resolutionwell below the meV range which allowed one to precisely identify YSR resonances and directlylink them to the single impurity ground state .However, previous studies suffer from two main limitations, namely: the inability to directlyaccess the effect of magnetic perturbations on the superconducting order parameter and the focuson single specific perturbations, an approach that impedes the discovery of well-defined trends andcorrelations. Here, we overcome these limitations by (i) systematically spanning the 3d orbitaloccupation adding one electron at a time and (ii) scrutinizing the impact of each impurity in threedifferent spectroscopic regimes: Shiba, Josephson and metallic. Scanning Josephson spectroscopymeasurements are used to directly map the effect of magnetic impurities by visualizing the sup-pression they induce on Cooper pairs tunneling between superconducting electrodes . Thisallows to discover the existence a direct correlation between Cooper pairs tunneling and Shiba3tates, revealing a stronger suppression of the Josephson supercurrent for impurities hosting mul-tiple YSR within the energy gap, an effect directly linked to their higher spin state. In agree-ment with ab-initio calculations, this correlation is directly linked to the existence of an orbitaloccupation-dependent oscillatory behaviour, with vanishing magnetic interactions for elements atthe opposite extremes of the 3d element series. Moreover, by driving the system in the normalmetallic regime, we reveal the emergence of zero-bias anomalies which, in sharp contrast to ex-pectations, become progressively stronger by approaching the quantum phase transition from theKondo to the free spin regime in the well-known phase diagram of magnetic impurities coupled tosuperconductors . Supported by ab-initio calculations based on density functional theory (DFT),relativistic time-dependent DFT (TD-DFT) and many-body perturbation theory (MBPT)
33, 34 ,these low-energy spectroscopic features are identified as unconventional spin-excitations emergingfrom a finite magnetic anisotropy energy.Overall, our results shed new light on how local spins interact with superconducting conden-sates. They provide a self-consistent experimental picture allowing the discovery of new effectsand the visualization of new trends that always escaped experimental detection so far and with farreaching implications especially within the realm of engineered topological superconductivity.
ResultsExperimental lineup.
The experimental lineup used to scrutinize the aforementioned aspects isschematically illustrated in Figure 1. Local spins coupled to an electron bath are characterized4y a magnetic exchange term
J S with J being the s–d exchange coupling of the localized spinof the impurity S , carried here by d-electrons, and the conduction electrons of the substrate. Itseffects are expected to manifest in three distinct ways, schematically illustrated in panels a-c. Inthe superconducting regime, it represents a scattering potential breaking Cooper pairs and givingrise to in-gap YSR states (a). Additionally, it is expected to directly affect the superconductingorder parameter by suppressing the strength of the pairing interaction, resulting in a reduction ofthe Josephson current flowing between superconducting electrodes (b). Finally, a strong couplingbetween magnetic impurities and the electron bath can open additional tunneling channels. Theseresult from inelastic spin-excitations induced by the magnetic anisotropy, which opens a gap in thespectra and are experimentally signaled by a step-like increase in the experimentally detected localdensity of states (LDOS), as sketched in (c) . As described in the following, instead of the usualtwo steps expected at positive bias and negative bias voltage, the inelastic spectra can display anunconventional shape, in accordance to recent predictions
33, 34 .Panel d illustrates the portion of the periodic table of the 3d elements investigated in thepresent study. By scrutinizing the 3d occupation scenario adding one electron at a time, it is possi-ble to analyze the role of orbital-occupation in determining the magnetic impurity-superconductorinteraction strength. As superconducting material, we choose niobium single crystals which havebeen prepared according to the procedure described in Ref. 36. Niobium represents an optimalchoice compared to other superconductors such as Pb
14, 15 , Re
37, 38 , and Ta used in previousstudies. Indeed, by having the highest transition temperature ( T = 9 . ) among all elementalsuperconductors, it allows to clearly disentangle in-gap states from superconducting gap thermal5 ipSample S Je - e - Δ YSR V e - TipSample
S Je - e - Δ YSR spectroscopy Josephson spectroscopy TipSample
S Je - e - Spin-excitation spectroscopy V a b cd e V Vanadium 24 Cr Chromium 25 Mn Manganese 26 Fe Iron Co Cobalt
Fe Cr O L D O S L D O S H e i g h t highlow Figure 1: Experimental lineup . Schematic illustration of ( a ) Yu-Shiba-Rusinov (YSR), ( b )Josephson , and ( c ) spin-excitation spectroscopy. These three different spectroscopic modes can beused to provide a complete spectroscopic characterization of spin-related effects in the supercon-ducting as well as in the metallic regimes. V corresponds to the bias applied across the tunnelingjunction, J indicates the exchange coupling of the localized spin of the impurity S , while ∆ is thesuperconducting energy gap of electrons e − condensing into Cooper pairs, ( d ) 3d elements scruti-nized in the present study, ( e ) topographic image showing Cr and Fe single atoms coupled to theNb(110) surface. Scanning set-point: V = -300 meV, I = 300 pA.6roadening effects. Panel e shows a topographic image where different magnetic impurities (Feand Cr) have been deposited onto the clean Nb(110) surface prepared according to the proceduredescribed in the Methods section and Supplementary Figure 1. The very same approach has beenused for all atomic species, i.e. V, Cr, Mn, Fe, and Co (see Supplementary Figure 2 for the deter-mination of the adsorption sites). To investigate their impact onto the superconducting condensate,full spectroscopic maps have been acquired at temperature T = 1 . using superconducting Nbtips. Compared to conventional metallic tips, their use brings two crucial advantages: (i) they al-low to enhance the energy resolution while simultaneously (ii) opening the fascinating possibilityto measure the Josephson effect at the atomic scale. YSR spectroscopy.
Figure 2 reports the spectroscopic characterization of the superconductinggap obtained by positioning the tip directly on top of the different magnetic perturbations. As de-scribed in the Supplementary Figure 3, the use of superconducting tips shifts the “zero energy” by ± ∆ tip with respect to the Fermi level, ∆ being the superconducting energy gap. Hence, the singleparticle coherence peak appears at energies ± (∆ tip + ∆ sample ) . In the present case, this correspondsto approximately ± , with slight variations resulting from tips characterized by different Nbclusters at their apex (see Supplementary Figure 4). An inspection overview of Figure 2 allowsus to immediately identify the existence of an oscillatory behaviour which directly correlates withthe filling of the 3d orbitals. In particular, V and Co do not show any impact onto the supercon-ducting properties, with scanning tunneling spectroscopy spectra taken by positioning the tip overthe adatoms (orange line) perfectly overlapping, within our experimental resolution, the spectrataken over the bare Nb substrate (black dashed line). Despite no difference in the d I /d U curves,7 Å5Å
CoFeMn cde d z2 d z2 d z2 d z2 d yz d yz d xz d xz d xz* * d xzE-E F (meV) d I/ d U ( a r b . un i t s ) d I/ d U ( a r b . un i t s ) d I/ d U ( a r b . un i t s ) Cr b d z2 d z2 d xy d xy d yz d yz d I/ d U ( a r b . un i t s ) a d I/ d U ( a r b . un i t s ) V -Δ tip Δ tip -(Δ tip +Δ sample ) (Δ tip +Δ sample ) d I / d U highlow Figure 2: Yu-Shiba-Rusinov (YSR) spectroscopy . ( a-e ) Scanning tunneling spectroscopy for V,Cr, Mn, Fe, and Co. While no YSR state is detected for V and Co, a rich set of in-gap states emergefor Cr, Mn, and Fe. The insets show the spatial distribution of the YSR states, which reflect theirorbital character and allows one to quantify their spatial extension. Stabilization parameters: V =5 meV, I = 500 pA. 8 very weak d z -derived YSR state can be detected for V, which is energetically overlapping withthe single particle coherence peak at the edge of superconducting gap. These results suggest a verysmall and vanishing magnetic moment for both V and Co, respectively which are located at theopposite extremes of the 3d orbital scenario analyzed in the present study. Both elements beingcharacterized by a partially filled 3d shell, this behaviour might appear surprising and it highlightshow the hybridization with the substrate can dramatically impact the magnetic properties. Simi-larly to our finding, Co adatoms can be non-magnetic on Re surface as revealed by a YSR studylimited to Mn, Fe and Co impurities . As described in the following, the trend unveiled by ourexperiments is confirmed by ab-initio calculations (see subsection Ab-initio Simulations).In contrast, well-defined YSR states emerging within the superconducting gap are visible forCr, Mn, and Fe. As expected, all YSR states appear in pairs symmetrically located around theFermi level. Their energy position (cid:15) within the superconducting gap is generally described con-sidering pure magnetic scattering mechanisms, being determined by the strength of the exchangecoupling terms J through the following expression: (cid:15) = ± ∆ 1 − α α with α = πρJ S , S being the impurity’s spin, and ρ the sample density of states at the Fermi levelin the normal state . For each pair, the different intensities between occupied and unoccupiedresonances can be used to identify whether the YSR state is in a screened-spin (higher intensity forhole injection, i.e. E < E F ) or a free-spin configuration (higher intensity for electron injection, i.e.E > E F )
16, 20 . 9n the case of Fe, a single pair of YSR states is detected. It energetically overlaps with thesingle particle coherence peaks visible at the edge of the superconducting energy gap. Spatiallymapping its intensity allows one to assign it to a d z scattering orbital (see colormaps in Figure 2d).Cr and Mn show a more complicated spectrum supporting multiple YSR pairs. As for Fe, a d z scattering orbital is clearly visible, which moves towards smaller binding energies by progressivelydecreasing the atomic number. The additional YSR pairs are located at different energies within thesuperconducting gap. Their spatial distribution is far from being isotropic, resembling well-defined d level symmetries. These observations prove that the magnetic exchange scattering potentialsare strongly orbital-dependent . Interestingly, in Figure 2c, the Mn d xz -derived Shiba pair showdistinct spectral maps at positive and negative energies, signalling a strong particle-hole asymmetryin the wavefunctions, similarly to d z YSR bound states. An additional pair is visible at energies ± (∆ tip − (cid:15) ) . These states correspond to thermal replica of the d xz -derived Shiba pair: they becomepopulated by particles and holes due to their proximity to the Fermi level. This assignment isfurther confirmed by their shape, which energetically mirrors that of the original states .Interestingly, these results allow to systematically follow the evolution of the d z -derivedYSR state, visualizing how it progressively moves toward higher binding energies by increasingthe 3 d orbital occupation. Within the generally assumed framework of competing singlet formationmechanism, i.e. Kondo vs. Cooper pairs, this is expected to result in Kondo resonances becom-ing progressively stronger by moving from Cr to Mn, and, Fe. However, as demonstrated in thefollowing, this is far from being the case (see section Spin excitations and related discussion).10 osephson spectroscopy. Although YSR measurements can be effectively used to infer importantinformation on the magnetic coupling strength, they are characterized by a strong fundamentallimitation: they can not visualize the effect of magnetic impurities on the superconducting orderparameter. Indeed, the local pairing suppression which is expected to take place in presence ofmagnetic perturbation can not be directly reflected in the YSR spectra. As illustrated in Figure 2,these show a suppression in the intensity of the coherence peaks at the edge of the superconductinggap, their spectral weight being redistributed to the in-gap bound states, but without any energyshift of their position as compared to the substrate. This distinction between detecting the effectsof magnetic impurities on the local density of states and on the superconducting order parameteris well-known and consistent with theoretical expectations .To overcome this limitation, we perform scanning Josephson spectroscopy measurementswhich allow, by measuring the tunneling between Cooper pairs in superconducting electrodes,to directly extract information on local variation of the superconducting pairing amplitude at theatomic scale. Results for all investigated impurities are summarized in Figure 3, with left panelsshowing the I-V characteristics and the right ones reporting their respective d I /d U signal. The or-ange and dashed black lines have been acquired by positioning the tip atop the magnetic impuritiesand on the bare substrate, respectively. While no difference with respect to the bare Nb substrate isdetected for Co, a small reduction of the Josephson current is observed for V. This effect becomesstronger for all other impurities according to the following order: Fe, Mn, and Cr. These resultsconfirm the oscillatory magnetic behavior visualized in Figure 2: the magnetic moment becomestrongly suppressed at the opposite extreme of the 3d filling, while Fe, Mn, and Cr preserve a sub-11 oFeMnCrV abcdef g h Figure 3: Scanning Josephson spectroscopy . ( a-e ) I − V characteristics (left panels) and re-spective d I /d U signals (right panels) for all elements. ( f ) Apparent heights of the adatoms. ( g,h ) Reduction induced by the different magnetic perturbations in the maximum Cooper pairs cur-rent I max and intensity of the d I /d U peak, respectively, both showing the same trend. Stabilizationparameters: V = 5 meV, I = 15 nA. 12tantial magnetic character. The effect of the different magnetic impurities on the superconductingorder parameter can be quantitatively analyzed based on the fact that, both the maximum Cooperpairs current I max as well as the intensity of the d I /d U peak are proportional to the the square ofthe intrinsic Josephson current I c . A direct comparison among the different adatoms evidencesa suppression of the Cooper pairs tunneling, an effect directly linked to the reduction of the su-perconducting order parameter, which becomes progressively stronger by moving from Fe to Mnand, finally, Cr. In agreement with theoretical expectations, the same trend is consistently foundfor both I max (panel g) and the d I /d U peak intensity (panel h) . Although different magneticimpurities have a different apparent height, as shown in panel e, we can safely exclude that ourobservations are related to tip-height artifacts. Indeed, despite having Co an apparent height ofapproximately . ˚A, Josephson currents are not altered when compared to the case when the tip ispositioned over the bare substrate. Furthermore, Cr, which shows the strongest perturbation on thesuperconducting order parameter, has an apparent height which is smaller than Mn. Additional ex-perimental evidence ruling out tip-height effects is provided in the Supplementary Figure 5 where,by using atomic manipulation techniques, we create a Cr dimer. Although being apparently higherthan a single Cr adatom, the dimer does not have any impact on the superconducting order pa-rameter, an observation consistent with its antiferromagnetic ground state resulting in a total spinS=0. Consequently, our measurements directly fingerprint effects induced by a finite spin onto thesuperconducting order parameter, suggesting a progressively increasing magnetic moment whilemoving from Fe to Mn and finally Cr. As discussed in the following, these results follow the sametrend of the magnetic moments obtained by our theoretical calculations, and highlight the very13igh sensitivity of our measurement protocol. Ab-initio Simulations.
The theoretical interpretation of the trends observed in both YSR andJosephson spectra requires a detailed knowledge of the spin-resolved orbital structure of the adatomsand their coupling to the substrate. This is analyzed in the following on the basis of ab-initio sim-ulations of the 3d series of adatoms deposited on Nb(110) surface (see Supplementary Notes 1–3for more details). Figure 4a reports the spin-resolved local density of states (LDOS) for V, Cr, Mn,Fe and Co with upper and lower panels corresponding to minority- and majority-spin channels,respectively. The LDOS broadening is a direct consequence of the crystal field, which splits thedegeneracy of the different 3d orbitals. A detailed discussion is provided in Supplementary Notes1–3. Its inspection immediately reveals the appearance of a well-defined trend: a substantial im-balance between majority- and minority-spin resonances is found for Cr, Mn, and Fe, while thedifference between majority- and minority-spins is found negligible for V and totally absent forCo. These results follow the usual inverse parabolic behavior across the 3d series, with spin mag-netic moments reaching a maximum in the middle followed by a decrease toward the end of theseries. In agreement with our experimental observations, only four adatoms remain magnetic, withelements at half filling of the d-states carrying the largest moments (V: ∼ . µ B ; Cr: ∼ . µ B ;Mn: ∼ . µ B ; Fe: ∼ . µ B ) while Co is non-magnetic. Note that a non-negligible magneticmoment is induced in the bare Nb substrate at the vicinity of the adatoms, to which it generallycouples antiferromagnetically, except for V. This effect modifies the total adatoms–substrate com-plex spin moments, resulting in V: ∼ . µ B , Cr: ∼ . µ B , Mn: ∼ . µ B , and Fe: ∼ . µ B . Thesevalues correlate well with the trend visualized by Josephson–spectroscopy measurements reported14n Figure 2, allowing to establish a direct link between the magnitude of the magnetic moment andthe induced suppression of Cooper pairs supercurrents.The strength of the orbital-average impurity-substrate hybridization, Γ , between adatoms andsubstrate is rather large for all the adatoms, and it decreases by increasing the 3d orbital occupation,i.e. by moving from left to right across the 3d series (V: .
11 eV ; Cr: .
98 eV ; Mn: .
88 eV ; Fe: .
72 eV ; Co: .
57 eV ). This trend is related to the contraction of the 3d-states of the atoms whenincreasing their atomic number, which disfavors hybridization with neighboring atoms. While thehybridization strength is paramount for the description of YSR-bound states, it is worth stressingthat its effect can be counteracted by the exchange splitting, 2 U , and the energy of orbital m , E m . A full ab-initio description of the YSR states is currently challenging. Here, we follow asimplified model where the aforementioned quantities encode the magnitude of the orbital- andspin-dependent impurity-substrate s–d interaction I σm , where σ = ± depending on the spin ofconducting electrons. By virtue of the Schrieffer-Wolff transformation , I σm = ( V m + σJ m S ) , with V m and J m corresponding to non-magnetic and magnetic scattering contributions, respectively. Theenergies of the YSR states can then elegantly be cast into
13, 41 : (cid:15) m ∆ = ± cos ( δ + m − δ − m ) , (1)where the phase shifts are given by tan δ σm = πρ I σm .This approach is capable of mapping the scattering phase-shifts and the YSR energies di-rectly from our ab-initio results (see Supplementary Notes 2–3). The complexity of the problem isdirectly related to the very different energies scales coming at play: the interactions J and V de-15end on quantities of the eV range, while the energies of the YSR states are of the order of meV andsub-meV. This impedes a perfect one-to-one comparison between all the theoretically calculatedand experimentally measured spectra. However, our appraoch is effectively capable of capturingthe observed experimental trends, as discussed in the following. The theoretically predicted energyposition for Cr and Mn YSR states are summarized in Figure 4 b.The Cr d z Shiba state is predicted to be located at a lower energy than that of Mn, in agree-ment with the experimental data (see Supplementary Note 3 for a detailed discussion on the roleof non-magnetic and magnetic impurity-substrate interactions in determining the energies of theYSR states). Similarly to what is observed in Fig.2, the d yz state of Cr is theoretically expected ata higher energy than the YSR state of d z -symmetry, while for Mn the two states are found aroundthe same energy. The calculated d xz state is located at a lower energy than the d z state for both Crand Mn. While this is confirmed experimentally for the d xz state of Mn, it was not detected for Cr,the corresponding peak being either too weak or difficult to disentangle from the adjacent domi-nant resonances. Note that all YSR states are characterized by a finite broadening which is relatedto both the experimental energy resolution and their intrinsic lifetime. This explains why, althoughab-initio simulations predict that each of the orbitals of Cr and Mn adatoms carry a spin moment,resulting in five distinct YSR states, not all of them are detectable experimentally, as shown in Ref.14. Fe and V, on the other hand, are found to have a colossal adatom-substrate interactions, whichis favoured by the LDOS resonances located at the Fermi energy. In both cases, because of thevery strong interaction for all orbitals, all YSR features are expected to appear at the edge of theSC gap, with the d z orbital dominating the scene because of its larger extension into the vacuum,16hich facilitates its experimental detection, in agreement with our tunneling spectra. Spin excitations.
The interaction of magnetic impurities with superconducting condensates isgenerally described within the framework of competing singlet formation mechanisms, i.e. Kondoscreening vs. Cooper pairs. This competition is captured within a phase diagram where the mag-netic impurities can be either in a Kondo-screened or free-spin state depending on the impurity-superconductor coupling strength. In the strong coupling regime, k B T K (cid:29) ∆ , with k B be-ing the Boltzmann constant and T K the Kondo temperature, while in the weak coupling regime k B T K (cid:28) ∆ . A quantum phase transition between these two regimes takes place for k B T K ≈ ∆ ,i.e. when Kondo screening and the superconducting gap are characterized by similar energies .To scrutinize these aspects, a magnetic field has been applied perpendicular to the sample surfacein order to quench the superconducting state. Note that all elements are characterized by a well-defined d z -state, which allows to precisely map its evolution. This is found to progressively movetowards the single particle coherence peak located at the edge of the superconducting gap by in-creasing the orbital occupation, which should result in a progressively stronger Kondo resonancewhile moving from Cr to Mn and Fe. However, our measurements clearly reveal that this is farfrom being the case. As illustrated in Figure 5, our data reveal a strong zero-bias anomaly (ZBA)with a step-like feature for Cr adatom, also observable in the superconducting phase as shownin Supplementary Figure 6. A similar behaviour is observed for Mn and Fe although the signalis much weaker than for Cr (see Supplementary Figure 7 for a direct overlap of Cr, Mn, and Fespectra) and becomes progressively broader. Finally, V and Co spectra appear totally flat.17 b Figure 4: Local density of states and energies of YSR states . ( a ) Spin-resolved electronicstructure for V, Cr, Mn, Fe and Co – upper (lower) panel for minority- (majority-) spin channel.( b ) Theoretically obtained table of orbital-dependent energies of YSR states normalized by thesuperconducting gap. 18 oFeMnCr V Figure 5: Spin excitation spectroscopy . For all adatoms, the experimentally obtained spectra arereported as solid lines following, for each element, the same color coding scheme used across themanuscript. To exclude spurious effects related to the use of different microtips, all spectra arenormalized to the substrate. To drive the system into a metallic state, a magnetic field has beenapplied perpendicular to the sample surface. All spectra are overlapped with the theoretically cal-culated spin-excitation spectra (solid black line). The inset shows the theoretical spectra with anartificial Gaussian broadening of .
20 meV , .
98 meV and .
78 meV for Cr, Mn and Fe, respec-tively (dashed lines). Stabilization parameters: Cr, Fe, Mn: V = 10 meV, I = 500 pA; V, Co V =10 meV, I = 1 nA. 19t has recently been predicted that inelastic spin-excitations can also lead to unconventionalspectral shapes centered around the Fermi level . To verify if this is the case, the experimentaldata are compared to relativistic first-principles simulations, combining TD-DFT with MBPT (seeMethod section and Supplementary Notes 4–5), reported as solid black lines in Figure 5. Thetheoretical inelastic spectra qualitatively reproduce the experimental features (more details on theorigin of the step-shapes is provided in the Supplementary Notes Notes 4–5) Cr has a weak MAEleading to small excitation energies. The amount of electron-hole excitations responsible for thedamping of the ZBA are therefore weak, which favors the observation of the inelastic features.Electron-hole excitations are proportional to the MAE and to the product of density of states ofopposite spin-character at the Fermi energy
31, 32 . Therefore, although V has a weak MAE, itssmall exchange splitting leads to a large LDOS at the Fermi energy and a consequent number ofelectron-hole excitations, heavily decreasing the lifetime of the spin-excitations. The interplayof these two mechanisms, MAE and LDOS, broadens the features obtained for Mn and Fe aswell. The experimental ZBA of the latter adatoms seem broader than those calculated, whichcan be resulting from a slight theoretical underestimation of the spin-excitation energy or of theelectron-hole excitation energies as shown in Supplementary Figure 12. Here we account forthis underestimation by broadening the theoretical spectra using a Gaussian broadening, which isshown in the inset of Figure 5. For the three shown cases of Cr, Mn, and Fe we used a broadeningof .
20 meV , .
98 meV and .
78 meV , respectively, to match the theoretically predicted spectrawith the experimental spectra. 20 iscussion
Overall, our data allow to establish a unified picture of different spin-related phenomena emergingfrom magnetic impurities coupled to superconductors. By systematically mapping the impact ofsingle magnetic perturbations onto the Josephson effect, we unveil the existence of a direct linkbetween superconducting order parameter suppression and YSR states. This correlation followsa well-defined orbital occupation-dependent trend. Moreover, by comparing YSR and metallicregimes, our data challenge existing theoretical models that explain the interaction between mag-netic impurities and superconductors in terms of competing singlet-formation mechanisms, i.e.Kondo vs. Cooper pairs. Indeed, according to this picture, the asymmetry in the YSR inten-sity can be used to identify whenever the magnetic impurity is in a Kondo-screened ( S = 0 )or a free spin ( S > ) state, with the peak intensity being stronger below and above the Fermilevel, respectively. Mn and Cr are both characterized by a strong spectral weight below the Fermilevel, and they are thus supposed to be in a Kondo-screened ground state ( S = 0 ). In particu-lar, we detect zero-bias anomalies which become stronger by progressively approaching the freespin regime, indicating their unlikeliness to be Kondo resonances. Our ab-initio simulations sup-port this analysis reproducing the zero-bias anomalies by considering inelastic spin-excitations.The latter hinges on the magnitude of the magnetic anisotropy energy of the adatoms. Becauseof the relevance of magnetic-superconducting interactions in different topological qubit concepts,which lay at the foundation of advanced quantum computation architectures, the significance ofour findings goes beyond the single-impurity level, evidencing that new and unexpected phasescan emerge, subjected to the interplay of orbital-dependent spin-substrate interactions, magnetic21oments and magnetic anisotropy energies. This can only be explored through the systematic useof a rich workbench of spectroscopy techniques for magnet-superconducting interfaces. MethodsSample and tip preparation.
Nb(110) single crystals (Surface Preparation Laboratory) have beenprepared in ultra high vacuum conditions and measured using a Tribus STM head (Scienta Omi-cron) operated at T = 1.9 K. The samples have been flashed hundreds of times at a temperature T = 2300 K for 12 s using an home-built electron beam heater. As illustrated in the Supplemen-tary Information, this procedure is necessary to progressively reduce the oxygen contamination,resulting in clean surfaces. The high quality of the surface is further confirmed by scanning tun-neling spectroscopy measurements showing, in agreement with theoretical calculations, a sharppeak energetically located at E = -0.45 eV below the Fermi level which originates from a surfaceresonance of d z character. Single magnetic adatoms have deposited onto the Nb(110) surfaceusing an electron-beam evaporator while keeping the sample at T = 10 K. Superconducting Nbtips have been prepared by indenting electrochemically etched W tips inside the Nb(110) for sev-eral nanometers. d I /d U spectra were measured using a lock-in technique, modulating the samplebias with 50 µ V (r.m.s.) ac bias at a frequency 733 Hz. More experimental details are given inSupplementary Figures 1–7.
Ab-initio.
The ground state properties of the adatoms deposited on Nb(110) were calculated in atwo-pronged approach based on density functional theory (DFT). First, the Quantum Espresso
42, 43 package was utilized for geometrical optimization of the adatom-substrate complexes. A × su-22ercell is considered with 3 Nb layers and a k -mesh of × × is used. Exchange and correlationseffects are treated in the generalized gradient approximation using the PBEsol functional , andwe used ultrasoft pseudopotentials from the pslibrary with an energy cutoff of 500Ry. Second,the calculated positions were then used in the simulations based on the the full-electron scalar-relativistic Korringa-Kohn-Rostoker (KKR) Green function including the spin-orbit interactionself-consistently
46, 47 . KKR permits the embedding of single adatoms in an otherwise perfectsubstrate. We assume the local spin density approximation (LSDA) and obtain the full chargedensity within the atomic sphere approximation. The angular momentum cutoff of the orbital ex-pansion of the Green function is set to (cid:96) max = 3 and a k -mesh of × is considered. Thetrend of the atomic relaxations obtained with Quantum espresso agree with the simulations (Cr:17%; Mn: 18%; Fe: 29%; Co: 29% of the Nb bulk interlayer distance), except for V where thetheory predicts a relaxation of 22%, while from the corrugation shown in Figure 3(f), we expect apossible extra relaxation of 10%.The energies of the YSR-states of the adatoms are modeled by a realistic tight-binding modelwith parameters from DFT. The model considers the d orbitals of the adatoms and accounts for theNb substrate via an effective Hamiltonian construction. Further details can be found in Ref. 49 andSupplementary Notes 2.The spin-excitations were investigated utilizing a framework based on time-dependent den-sity functional theory (TD-DFT) including spin-orbit interaction. Many-body effects trig-gered by the presence of spin-excitations are approached via many-body perturbation theory extended to account for relativistic effects . The single-particle Green functions pertaining to23he ground state are employed for the calculation of the the bare Kohn-Sham dynamical magneticsusceptibility, χ KS ( ω ) . The latter is renormalized to χ ( ω ) via the Dyson-like equation to accountfor many-body effects χ ( ω ) = χ KS ( ω ) + χ KS ( ω ) K χ ( ω ) . (2) K represents the exchange-correlation kernel, taken in adiabatic LSDA (such that this quantityis local in space and frequency-independent ). A magnetization sum rule permits an accurateevaluation of the energy gap in the spin excitation spectra . The theory was successful todescribe spin-excitations measured by STM (see e.g. ).The self-energy describing the interactions of the electrons and the spin-excitations is calcu-lated from a convolution of the Green function, G , and susceptibility, Σ ∝ K χG K in Refs.
33, 34, 54, 55 .The impact of spin-orbit coupling is incorporated as described in Ref. . The self-energy is thenused to renormalize the electronic structure to account for the presence of spin-excitations by solv-ing the Dyson equation g = G + G Σ g .The theoretical spectra shown in Figure 5 are local densities of states calculated the vac-uum above the adatoms, which on the basis of the Tersoff-Hamann approach correspond tothe differential conductance measured by STM. More details on the simulations are provided inSupplementary Notes 1–5. Acknowledgments
A.M.M. and F.S.M.G. thank Juba Bouaziz for fruitful initial discussions on the method-ology related to the description of spin-excitations. S.L. acknowledge discussions with Nicolas Lorent´e.A.M.M, F.S.M.G., S.B. and S.L. are supported by the European Research Council (ERC) under the Euro- ean Union’s Horizon 2020 research and innovation programme (ERC-consolidator grant 681405 — DY-NASORE). We acknowledge the computing time granted by the JARA-HPC Vergabegremium and VSRcommission on the supercomputer JURECA at Forschungszentrum J¨ulich and at the supercomputing centreof RWTH Aachen University. Authors contributions
F.K. performed the STM measurements and analyzed the data. A.M.M. and S.B.performed the ab-initio simulations. S.B. and S.L. conceived the theoretical framework describing the YSRstates. A.M.M., F.S.M.G., S.B. and S.L. analysed the theoretical data. S.L. and P.S. wrote the initial versionof the manuscript to which all authors contributed. S.L., S.S.P.P. and P.S. supervised the project.
Competing Interests
The authors declare no competing interests.
Data and materials availability
All data needed to evaluate the conclusions in the paper are present inthe paper and/or the supplementary materials. Additional data related to this paper may be requested fromthe authors. The KKR Green function code that supports the findings of this study is available from thecorresponding author on reasonable request. eferences
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