Interplay between Yu-Shiba-Rusinov states and spin-flip excitations on magnetic impurities on superconducting NbSe substrate
Exchange coupling between a magnetic impurity and a superconducting substrate results in the formation of Yu-Shiba-Rusinov (YSR) bound states, which have been recently used in artificial designer structures to realize exotic quasiparticles known as Majorana fermions. At strong coupling, the energies of YSR states are deep in the superconducting gap, and the impurity can even undergo a transition from high-spin to low-spin magnetic state by trapping one or several Bogoliubov quasiparticles from the bulk. At weak coupling, the YSR states migrate towards the superconducting gap edge. Additional spectral features can appear in the presence of magnetic anisotropy with spin , in particular spin-flip excitations outside the superconducting gap. Despite extensive experiments on magnetic impurities that exhibit separately either spin-flip excitations or YSR states, these phenomena have not been observed simultaneously. Here, we investigate the spectral evolution in different metal phthalocyanine molecules on NbSe surface as a function of the coupling with the substrate. Using scanning tunneling microscopy (STM), we tune the exchange coupling strength and for manganese phthalocyanine (MnPc) we demonstrate a smooth spectral crossover from the YSR states to intrinsic quantum spin states. These results provide crucial insight into the fundamental behaviour of magnetic/superconducting hybrid systems and could have significant ramifications for the design and control of atomic scale magnetic devices.
Precise control of the properties of magnetic impurities on surfaces, such as the spin state and magnetic anisotropy, is one of the ultimate goals in fabricating atomic or molecular scale devices for data storage or computing purposes. However, the properties of magnetic impurities are strongly influenced by the atomic environment. In the extreme case, the interactions with the environment (substrate) can create entirely new electronic states, such as the Kondo effect on metallic substrates Li et al. (1998); Madhavan et al. (1998); Franke et al. (2011), or the formation of Yu-Shiba-Rusinov (YSR) bound states on superconductors Yu (1965); Shiba (1968); Rusinov (1969); Yazdani et al. (1997); Heinrich et al. (2018). Recently, YSR states have received intense interest as it has become possible to create artificial designer structures, where the interaction between the YSR states gives rise to Majorana modes Nadj-Perge et al. (2014); Ruby et al. (2015a); Pawlak et al. (2016); Kezilebieke et al. (2018); Ruby et al. (2017); Ménard et al. (2017); Ruby et al. (2018); Kim et al. (2018). The YSR states are very sensitive to the immediate environment of the impurity spin; therefore, YSR states can also be used to understand the role of the local environment on the exchange interaction of an impurity spin with a superconductor Yazdani et al. (1997); Ji et al. (2008); Franke et al. (2011); Heinrich et al. (2013); Ménard et al. (2015); Ruby et al. (2015b, 2016); Cornils et al. (2017); Ruby et al. (2018); Kezilebieke et al. (2018); Kim et al. (2018).
The bulk of recent experimental work on YSR states on superconducting (SC) substrates has demonstrated that the strength of the exchange interaction can be significantly influenced by a small change in the adsorption site of the impurity or by spacers between the impurity and substrate Franke et al. (2011); Hatter et al. (2015, 2017); Heinrich et al. (2013, 2015). While increasing eventually induces a quantum phase transition to a reduced-spin ground state Franke et al. (2011); Hatter et al. (2017); Farinacci et al. (2018); Malavolti et al. (2018), reducing causes YSR states to migrate towards the SC gap edge. New symmetric features with respect to , arising from inelastic spin-flip excitations, can appear outside the gap for Heinrich et al. (2004); Hirjibehedin et al. (2006, 2007); Heinrich et al. (2013, 2015); Wiesendanger (2009); Ternes (2015, 2017). These two distinct phenomena - YSR states and spin-flip excitations - are illustrated in Fig. 1a and b.
High-spin systems are subject to the magnetocrystalline anisotropy, i.e., the dependence of the energy of the magnetic moment on its orientation relative to the atomic environment. It has been experimental observed that exchange coupling strongly affects the anisotropy of an impurity coupled to an electronic bath Oberg et al. (2013). However, despite extensive experimental and theoretical studies on both spin-flip excitations or YSR states in similar systems, the interplay and cross-over between spin-flip excitations and YSR states has not yet been demonstrated. Here we experimentally and theoretically investigate the spectral evolution arising from the competition between the YSR states and inelastic spin excitations in different spin systems of Mphthalocyanine molecules (MPc, where M: Co,Mn,Cu,Fe, Fig. 1c) on NbSe surface. By using the scanning tunneling microscope (STM) tip, we can tune the exchange coupling strength , and follow the spectral evolution from the YSR states to intrinsic quantum spin states characterized by well-developed spin-flip excitations. Our experiments, combined with calculations, show that this is a smooth cross-over. The excitation energies furthermore reveal a significant renormalization of the anisotropy by the exchange coupling with the superconducting electrons and, in addition, the coupling has a strong effect on the excited spin lifetime. Our results provide detailed understanding of the low-energy quantum states in magnetic/superconducting hybrid systems and could have significant ramifications for the design and control of atomic-scale magnetic devices.
Figure 1a illustrates how the exchange coupling with the substrate competes with the superconducting pairing energy . The interaction of the local spin with the Cooper pairs gives rise to a low-lying excited state within the gap of the quasiparticle excitation spectrum Yu (1965); Shiba (1968); Rusinov (1969); Yazdani et al. (1997); Heinrich et al. (2018). When is decreased, the direct interaction of the local spin with the Cooper pairs is reduced and the YSR states merge with the SC coherence peaks. In the simplified theory of Yu, Shiba and Rusinov, the position of the YSR state is given by with proportional to , , where is the normal-state density of states of the substrate at the Fermi level and is the impurity spin. The bound state results from the spin-dependent scattering of Bogoliubov quasiparticles on the impurity and is thus associated with the longitudinal part of the exchange interaction, , where the lower-case represents the spin-density of the substrate electrons at the position of the impurity. Furthermore, internal spin transitions in combination with magnetic anisotropy can give rise to symmetric features with respect to outside the superconducting gap as shown in Fig. 1b. These are associated with the spin-flip events, whereby the spin projection changes by Heinrich et al. (2004); Ternes (2015, 2017). The renormalization of the magnetic anisotropy, associated with the transverse part of the exchange interaction , approximately follows a dependence on the exchange coupling (here is the bare anisotropy, the renormalized value).
In the simple picture, the relative magnitudes of these two channels (YSR and spin-flip) are not constrained and, in principle, both of these effects should be observed simultaneously Berggren and Fransson (2014, 2015). Nevertheless, so far no experiment has shown the presence of both at the same time. This suggests that the complete picture of the interplay between these two effects is more complicated, requiring a full many-body treatment of the quantum mechanical spin degree of freedom interacting with a (gapped) continuum of electrons.
Individual MPc molecules on NbSe. Figure 1c shows topographic STM images of isolated MPc molecules on NbSe (see Methods for experimental details). Their topographic appearance already reflects differences that allow us to classify them into two groups: The metal ion appears as a protrusion in FePc, CoPc and MnPc and as a depression in CuPc. The different contrast is due to the large respectively small coupling of the out of plane - orbitals near the Fermi level with the tip states Lu et al. (1996).
To characterize the spin states of different MPcs, we recorded the differential conductance spectra (d/d curves) with a superconducting tip placed over the center of the molecules, see Fig. 2a. For CoPc and MnPc, there are two peaks that are located at symmetric bias voltages within the superconducting gap. These sub-gap peaks are due to the formation of YSR states and indicate a sizable magnetic interaction with the superconducting substrate caused by an unpaired spin in the -orbital (see Fig. 2b for the spin states of the molecules). This orbital is subject to strong coupling with the electronic states of the substrate due to its symmetry, while the spins on the - and -orbitals are instead expected to be only weakly coupled Kügel et al. (2014). The d/d curve taken on the CuPc molecules shows the unperturbed superconducting gap of NbSe due to the absence of unpaired spin in the orbital (Fig. 2b). As a system, CuPc is also not expected to show any spin-flip excitations.
While the d/d curves taken on FePc molecules do not show any features within the gap, there are remarkable features outside it. Their symmetric appearance indicates that these features result from inelastic excitations Jaklevic and Lambe (1966); Stipe et al. (1998); Heinrich et al. (2004). This identification is further confirmed by the fact that their shape is similar in form as the edges of the superconducting band Heinrich et al. (2013). Interestingly, unlike in the previous studies that required decoupling the magnetic molecules using an extra organic ligand as a spacer Heinrich et al. (2013, 2015), we observe these signals already when the molecule is directly adsorbed on the superconducting substrate. Since FePc is also expected to have a dominant -character, the molecule-substrate interaction should be similar to CoPc and MnPc. The surprising absence of YSR states on FePc suggests that the magnetic interaction with the superconducting substrate is actually weak, which would be at odds with an unpaired spin occupying a -orbital. FePc has spin triplet electronic ground state and DFT calculations suggest that FePc on NbSe has the same spin as in the gas phase (Supplementary Information). Although the exact configuration of the ground state is still under debate Fernández-Rodríguez and Toby (2015); Tsukahara et al. (2016), two configurations separated by 80 meV have been proposed, see Fig. 2b. The configuration with two electrons on the -orbital is predicted to have lower energy Fernández-Rodríguez and Toby (2015), and the absence of a YSR state in our experiments also suggests that the ground state does not have an unpaired -electron.
We have verified the spin triplet state of FePc and that the observed transitions do indeed correspond to inelastic spin excitations by acquiring the d/d spectra at different strengths of the magnetic field () that is applied perpendicular to the sample surface (see Supplementary Information for details). The energies of the first and second feature change with in a way consistent with a spin-state with transverse anisotropy. Fitting the data with a phenomenological spin Hamiltonian , where is the Landé factor, the Bohr magneton and the spin component along the field direction, we obtain meV and meV indicating easy-plane magnetic anisotropy. The positive value of is comparable to the bulk value (in contrast to the measurements on oxidized Cu(110) surface giving )Tsukahara et al. (2009, 2016).
Tuning the exchange coupling with the substrate. The exchange coupling between the magnetic impurity and the substrate can be modulated by changing the adsorption site of the molecule Franke et al. (2011); Bauer et al. (2013). We successfully positioned CoPc and FePc molecules on different adsorption sites through STM manipulation (see Supplementary Information Fig. S3). Interestingly, the energy positions of both the YSR states and the inelastic features change as the adsorption site is altered. On CoPC, the YSR states can even change between particle- or hole-like character depending on the adsorption site, similarly to the reported results on MnPc adsorbed on Pb(111) Hatter et al. (2015). On FePc, adsorption site causes variations of both and with typical values in the range of meV and meV, respectively.
The sensitivity of the YSR states and spin-excitations to small variations in the adsorption configuration suggests that they could be tuned continuously by the force exerted by the STM tip. Figure 3 shows a series of tunneling spectra measured at different tip-sample distances above the central ion of the MPc molecule. We start all the experiments at a distance determined by the set-point conditions and approach the tip by a distance of before recording the spectrum. The d/d spectra on a CoPc molecule, Fig. 3a, show a shift of the YSR resonances towards higher bias. At pm, the YSR resonances have merged with the superconducting coherence peaks at the gap edge. The d/d spectra on MnPc measured under similar conditions, Fig. 3b, also show a clear shift of the YSR resonance towards the superconducting gap edge. Moreover, at around pm, as the YSR resonance is merging with the superconducting coherence peak, a new symmetric pair of peaks is emerging outside the gap (marked with ). Finally, in FePc, Fig. 3c, the two spin excitation energies monotonously increase with decreasing tip-sample distance.
These variations in the YSR and spin excitations states of the MPc molecules on NbSe surface can be understood considering the force induced by the proximity of the STM tip to the molecules Ternes et al. (2011); Farinacci et al. (2018); Malavolti et al. (2018). Under this force, the metal ion is expected to be pulled towards the STM tip. This has the largest effect on the out-of-plane -orbitals ( and ), where the overlap with the substrate wave function will be strongly affected.
In the case of the molecule CoPc, the force affects the exchange coupling strength between the local spin and the Cooper pairs in NbSe by pulling the impurity towards the tip. This reduces the coupling between the -orbital and the NbSe and the YSR states move to the SC coherence peaks ( decreases) (Fig. 3a). In the case of FePc (Fig. 3c), the effect of the tip-sample force results in a monotonous increase of both spin excitation energies. In addition to modifying the exchange coupling with the substrate by reducing the hybrisation, the presence of the tip might also affect the energies of the -orbitals slightly. This causes the bare magnetic anisotropy , which is an admixture of low-lying excited states of the molecules Wang et al. (1993), to be directly affected by the tip-sample distance. In this system, it is not possible to distinguish whether the effective anisotropy changes dominantly through the variation of or .
In MnPc (), the coupling strength decreases as the tip-sample distance decreases, which leads to the migration of the YSR states towards the superconducting gap edge and the recovery of the superconducting coherence peaks. When YSR are close to the gap edges, the spectra also show new features outside the gap that are symmetric with respect to zero bias, which are due to spin-flip excitations. In MnPc, the only possible spin-excitation is between and levels, which are separated in energy by in the presence of axial magnetic anisotropy. Subtracting the value of the tip superconducting gap yields an axial anisotropy parameter of meV, which is close to the bulk value Barraclough et al. (1974) (see below for more detailed analysis).
Within the conventional picture of the formation of YSR states and the spin-flip excitations, where the quantum fluctuations are neglected, there is no reason to expect a cross-over behaviour from the YSR states to spin-excitations. These conduction channels should be independent and readily observable at the same time. To resolve this disagreement with our experimental results, we need to go beyond the simple theory and consider the full effect of the substrate on the impurity spin.
Full microscopic model and comparison with experiment. The magnetic impurity is described by a multi-orbital Anderson model for the relevant -shell orbitals Georges et al. (2013), which can be reduced in a given charge state to an effective model that takes the form of a Kondo Hamiltonian with the spin degree of freedom, , only Horvat et al. (2016, 2017) (see Methods and Supplementary Information for details). Since the total spin operator has contributions from all -orbitals, it is in general exchange coupled to different symmetry-adapted combinations of states from the substrate with different values of Kondo coupling strength, . These differences are due to the unequal orbital energies and hybridization strengths. Only those that are sufficiently large need to be retained; in the problems considered here, a single orbital is always strongly dominant, as evidenced by the presence of a single pair of sub-gap YSR peaks. In the model we also take into account the spin-orbit coupling that leads through orbital excitations to residual magnetic-anisotropy terms. We solve the resulting single-channel anisotropic Kondo model with high spin using the numerical renormalization group (NRG) method. This method provides a numerically exact solution to this problem.
Figure 4a shows the calculated spectral functions as a function of the exchange coupling between the magnetic impurity and the superconductor for an impurity with , as relevant for MnPc. In the absence of anisotropy (), the binding of a Bogoliubov quasiparticle would lead to an emergence of a YSR bound state with the spin reduced by (YSR screening from to ), giving rise to a single pair of YSR peaks. The phenomenology in the isotropic case is similar to that in the classical model, where the impurity is described as a static local magnetic field that binds Bogoliubov quasiparticles of the opposite spin direction. The many-body character of the sub-gap states is, however, revealed in the presence of magnetic anisotropy, which leads to the splitting of the sub-gap multiplet into the low-energy state and the high-energy states Žitko et al. (2011). For , such splitting is directly observable Hatter et al. (2015). When the anisotropy is large, as is the case here, the high-energy states are pushed instead into the continuum and only the sub-gap state is observable. Furthermore, one finds additional features outside the gap that are related to the transitions within the unscreened multiplet Heinrich et al. (2013). These spectral steps correspond to the spin-flip excitations, similar to the those in systems with normal-state substrates Hirjibehedin et al. (2007); Otte et al. (). The exchange coupling to the substrate determines the energy shift as well as the lifetime broadening of the excited states Heinrich et al. (2013); Oberg et al. (2013). NbSe is a soft-gap superconductor with a small but finite concentration of sub-gap states, hence the excited spin states have even for relatively short lifetimes compared to hard-gap superconductors, because the dominant decay channel (emission of particle-hole excitations) is here open. For this reason, a finite value of leads to significant broadening of the spin-flip excitations. Considering the required magnitude of for the YSR states to be visibly separated from the gap edges, it is unlikely to be able to experimentally well resolve both spectral features at a given value of the exchange coupling.
The theory curves in Fig. 4a can be compared with the experimental results shown in Fig. 4b. At small values of pm, only the YSR resonances are visible. As the tip is approached further, the YSR peaks merge with superconducting gap edges and spin-flip features emerge at bias voltage between 3 and 4 mV. The energies of the YSR peaks and spin-flip excitations are shown in Fig. 4c (the superconducting gap of the tip has been subtracted). We can estimate the exchange coupling with the substrate by comparing the theoretical and experimental YSR energies (see Supplementary Figure S7). This allows plotting the renormalized (effective) magnetic anisotropy and the intensity of the spin-flip transitions as a function of the estimated exchange coupling as shown in Figs. 4d,e. Such a calibration is possible because the energy of the YSR state is found to depend only very weakly on the precise value of in the experimentally relevant interval of parameter values (Supplementary Fig. S7). The comparison with theoretical results (solid lines in Fig. 4d) yields a value of for the non-renormalized (bare) value of the magnetic anisotropy. The experimental values actually deviate from the theoretical trend expected for a fixed value of . This is the case for all couplings , but more particularly for small , which corresponds to the largest tip-molecule interaction in the experiment. It is likely that the interaction with the tip distorts the molecular geometry or induced charge transfer with the metal ion resulting in a change in the bare value of itself. The presence of the YSR resonances thus allows us to disentangle the two contributions to the variation of . The extracted value as a function of the tip-sample distance is plotted in Supplementary Fig. S8.
The intensity of the spin-flip transition is plotted in Fig. 4e, where we observe a monotonous decrease with increasing (except the first two points, where the reason is again likely to be the tip-molecule interaction). The value at low is close to the theoretically expected value of 0.4 for pure spin-slip excitations on a normal metal substrate (see Fig. S9 for details) Ternes (2015, 2017). The experimental values can be compared to the theoretical predictions (solid line) again highlighting the strong correspondence between theory and experiments. Details on extracting the visibility of the spin-slip transitions from computed spectral functions are given in the Supplementary Information.
Conclusions. We have experimentally and theoretically investigated the spectral evolution in different metal phthalocyanine molecules as a function of the exchange coupling between the magnetic impurity and the superconducting substrate. In the case of MnPc, we have demonstrated for the first time the evolution from the YSR states to spin-flip transitions as the coupling is reduced. The spin-flip excitations are broadened and washed out at higher values of the exchange coupling making them difficult to detect experimentally simultaneously with the YSR states. Our results provide fundamental new insight on the behaviour of atomic scale magnetic/superconducting hybrid systems. The detailed understanding of these systems is invaluable for the future design and control of atomic scale magnetic systems, including e.g. artificial topological superconductors and spin logic devices.
Sample preparation. Sample preparation and subsequent STM experiments were carried out in an ultrahigh vacuum system with a base pressure of 10 mbar. The –NbSe single crystal (HQ Graphene, the Netherlands) was cleaved in situ by attaching a tape to the crystal surface and pulling the tape in vacuum in the load-lock chamber using the sample manipulator. MPc molecules (Sigma-Aldrich) were deposited from an effusion cell held at 390C onto a freshly cleaved NbSe at room temperature.
STM measurements. After the MPc deposition, the sample was inserted into the low-temperature STM (Unisoku USM-1300) and all subsequent experiments were performed at K. STM images were taken in the constant-current mode. d/d spectra were recorded by standard lock-in detection while sweeping the sample bias in an open feedback loop configuration, with a peak-to-peak bias modulation of V at a frequency of 709 Hz. The procedure for acquiring a spectrum was as follows: the tip was moved over the molecule at the imaging parameters (e.g. V and pA), the tip-sample distance was reduced by changing the setpoint to e.g. 200 pA at 100 mV. Finally, after disconnecting the feedback at the beginning the d/d spectrum, the tip-sample distance was decreased by a further pm () to increase the signal to noise ratio. The detailed numbers are given in the figure captions.
The NbSe tip was prepared by indenting the tip into the NbSe surface by a few nanometers while applying a voltage of 10 V. Manipulation of the MPc was carried out by placing the tip above the centre of the molecule with a bias voltage of 0.1 V and the current was increased to 1 nA with the feedback engaged. The tip was then dragged towards the desired location.
DFT calculations. Density functional theory calculations were performed with the FHI-AIMS computational package Blum et al. (2009); Havu et al. (2009) and the PBE generalized gradient approximation for the exchange-correlation functional Perdew et al. (1996). We used the standard ”light” numerical settings and basis sets of numeric atomic-centered orbitals tested and recommended by FHI-AIMS. Periodic NbSe supercells were sampled with a -point grid centred on the point. Van der Waals interactions were included by the post-SCF Tkatchenko-Scheffler correction Tkatchenko and Scheffler (2009). Before computing the electronic structure, all atomic forces were relaxed to eV/Å.
NRG calculations. The quantum impurity problems were solved with the numerical renormalization group method Wilson (1975); Bulla et al. (2008) using the ”NRG Ljubljana” implementation. We used the discretization parameter with interleaved discretization grids, keeping up to 12000 multiples at each step of the iteration. The spectra were computed using the density-matrix NRG algorithm Hofstetter (2000). The energies of the sub-gap states were extracted directly from the renormalization-group flow diagrams, while the energies of the spin-flip excitations were extracted from the peak position in the transverse component of the dynamical spin susceptibility of the impurity. Further details are provided in the Supplementary Information.
This research made use of the Aalto Nanomicroscopy Center (Aalto NMC) facilities and was supported by the European Research Council (ERC-2017-AdG no. 788185 “Artificial Designer Materials”), Academy of Finland (Academy Research Fellow no. 256818 and Postdoctoral Researcher nos. 309975 and 316347), and the Aalto University Centre for Quantum Engineering (Aalto CQE). Our DFT calculations were performed using computer resources within the Aalto University School of Science âScience-ITâ project and the Finnish CSC-IT Center for Science. R.Ž. acknowledges the support of the Slovenian Research Agency (ARRS) under P1-0044 and J1-7259.
S.K., T.O. and P.L. conceived and planned the experiment. S.K. performed the measurements. S.K. and P.L. analysed the STM data. R.Ž. and T.O. carried out the theoretical analysis of the system. M.D. carried out the DFT calculations. All authors jointly authored, commented, and corrected the manuscript.
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Geometries and spin densities from DFT calculations
Experiments on FePc in external magnetic field
We can verify that the observed transitions in FePc do indeed correspond to inelastic spin excitations by acquiring d/d spectra at different magnetic () field strengths aligned perpendicular to the sample surface. The energies of the first and second feature change with the external field, demonstrating that these are spin excitations (see Fig. S2). The zero-field splittings (ZFS) are well described by the following spin Hamiltonian:
where is the axial ZFS constant to determine the magnetic anisotropy, is the transverse anisotropy, and are the spin operators. The selection rule for the spin excitations implies that they can only occur between the states differing by =. For spin-state, we would expect only one step in the absence of transverse anisotropy () and at T. Transverse anisotropy () mixes the states and therefore a second step can occur. Hence, the d/d spectra on FePc on NbSe suggest spin-state with transverse anisotropy. We fitted our data using a phenomenological spin Hamiltonian Heinrich et al. (2004); Hirjibehedin et al. (2007); Ternes (2015, 2017)
where is the Landgé factor, the Bohr magneton, and is the component of spin along the direction of the magnetic field. Using the numerical code by M. Ternes Ternes (2015), we obtain meV, and meV as the best fit with the experimental data. The positive value of indicates easy plane magnetic anisotropy.
Effect of the adsorption site on the YSR states and the spin-flip excitations
Numerical renormalization group calculations
The impurity model is solved using the numerical renormalization group (NRG) method Wilson (1975). The NRG is a numerical procedure for solving quantum impurity models that is based on a logarithmic discretization of the continuum of electrons (here itinerant electrons from the substrate hybridized with the molecular states). The discretization is controlled by the parameter that determines the coarseness of the frequency grid, . The discretized Hamiltonian is transformed to a linear tight-binding representation that is diagonalized iteratively. One can compute static properties (expectation values of various operators), thermodynamics, as well as dynamic properties (spectral functions). The results, in particular spectra, can be significantly improved by performing several calculations for interleaved discretizations grids and averaging the results. This tends to remove the discretization artifacts which are periodic functions in with period ; using grids leads to a good cancellation of the fundamental frequency and the first harmonics.
In this work, the calculations have been performed with the “NRG Ljubljana” implementation of the technique. We used , , and kept up to 12000 states/multiplets (or states with energies up to , with the characteristic energy scale at the -th step of the iteration). The conduction band was assumed to have a constant density of states ranging in energy from to ; also serves as the energy unit. The gap is fixed at unless otherwise specified. The spectra were computed using the density-matrix NRG algorithm Hofstetter (2000).
The energies of the sub-gap states can be directly extracted from the renormalization-group flow diagrams in the NRG. This approach is significantly more accurate than determining the peak positions in the spectra, and does not suffer from any broadening artifacts. Example results are shown in Fig. S4
The positions of the spin-flip excitation energies can be determined from the spectral function, for instance by locating the peak position or the inflection point that approximately corresponds to the center of the step. Due to strongly changing line-shape of the equilibrium spectral function, this procedure is somewhat ill-defined, and furthermore suffers from possible broadening artifacts. In theoretical calculation using the NRG one can, however, use a more robust and direct approach for very accurately extracting the energies of spin excitations. It consists of computing the transverse part of the dynamical spin susceptibility function
The spin excitation energies can then be directly read off from the peak position, which is unique and very well defined. This is illustrated in Fig. S5
Properties of the anisotropic Kondo model
The spectral functions for a range of values of the exchange coupling for are shown in Fig. S6 (with parameter values corresponding to the experiments on MnPc).
The insensitivity of the YSR state energy on is illustrated in Fig. S7, which enables us to calibrate the exchange coupling based on the experimental YSR energies.
Bare magnetic anisotropy as a function of the tip-sample distance
Comparison of the spectral function with normal metal and superconducting substrate
Extraction of inelastic-excitation step heights
Extracting the amplitude of the inelastic excitations in the case of a normal-state substrate is simple: extract the step amplitude and reference it to the asymptotic value. For a superconducting substrate, the inelastic excitation corresponds to a spectral feature with a complicated shape that also changes with the increasing value of the exchange coupling.
In order to compare these results with the experiments, we have used two different approaches to extract the amplitude from computed spectral functions. For low (up to ), one can define the height as the difference between the minimum value of LDOS (this point occurs between the edge of the gap and the spin-flip resonance peak) and the asymptotic value. For larger , this minimum no longer exists. Instead, a plateau-like structure exists for up to . In this range, one can define the amplitude as the difference between the value at the center of this plateau and the asymptotic value. Finally, for large , no meaningful definition exists.
The maximum of the spectral function at the position of the spin excitation does not provide information about the amplitude of the spin-flip excitation, but rather reflects the properties of the bottom edge of the continuum (i.e., for small , information about the superconducting coherence peak in the impurity spectral function).
The procedure above yields the results shown in Fig. 4e of the main manuscript and corresponds physically to the visibility of the inelastic features. It is in spirit similar to how the spin-flip amplitude can be (and was) estimated experimentally. On the other hand, the true inelastic contribution to the total spectral function can be assessed by comparing the calculated response to that obtained with the same and large value of . Example of such curves (at ) is shown in Fig. S10a. This plot shows the calculated response with meV (black line) and with meV (red line). The asymptotic value of their difference (blue line) gives direct access to the inelastic contribution to the total spectral function. This is plotted as a function of in Fig. S10b showing values very close to the expected magnitude of 0.4 for pure spin-flip excitations on a normal metal substrate.