Magnetic impurities in spin-split superconductors
Hybrid semiconductor-superconductor quantum dot devices are tunable physical realizations of quantum impurity models for a magnetic impurity in a superconducting host. The binding energy of the localized sub-gap Shiba states is set by the gate voltages and external magnetic field. In this work we discuss the effects of the Zeeman spin splitting which is generically present both in the quantum dot and in the (thin-film) superconductor. The unequal -factors in semiconductor and superconductor materials result in respective Zeeman splittings of different magnitude. We consider both classical and quantum impurities. In the first case we analytically study the spectral function and the sub-gap states. The energy of bound states depends on the spin-splitting of the Bogoliubov quasiparticle bands as a simple rigid shift. For the case of collinear magnetization of impurity and host, the Shiba resonance of a given spin polarization remains unperturbed when it overlaps with the branch of the quasiparticle excitations of the opposite spin polarization. In the quantum case, we employ numerical renormalization group calculations to study the effect of the Zeeman field for different values of the -factors of the impurity and of the superconductor. We find that in general the critical magnetic field for the singlet-doublet transition changes non-monotonically as a function of the superconducting gap, demonstrating the existence of two different transition mechanisms: Zeeman splitting of Shiba states or gap closure due to Zeeman splitting of Bogoliubov states. We also study how in the presence of spin-orbit coupling, modeled as an additional non-collinear component of the magnetic field at the impurity site, the Shiba resonance overlapping with the quasiparticle continuum of the opposite spin gradually broadens and then merges with the continuum.
The interest in bound states induced by magnetic impurities in superconductors, predicted in the early works of Yu, Shiba, and Rusinov Yu (1965); Shiba (1968); Rusinov (1969), has been recently revived by the advances in the synthesis and characterization of semiconductor-superconductor nanostructures Goffman et al. (2000); Pillet et al. (2010); Deacon et al. (2010); Franceschi et al. (2010); Bretheau et al. (2013); Janvier et al. (2015) and in the tunneling spectroscopy of magnetic adsorbates on superconductor surfaces Yazdani et al. (1997); Ji et al. (2008); Franke et al. (2011); Ruby et al. (2015a); Randeria et al. (2016); Hatter et al. (2015). In particular, hybrid devices based on quantum dots can be used as fully controllable physical realizations of quantum impurity models with gapped conduction bands Sakurai (1970); Satori et al. (1992); Sakai et al. (1993); Salkola et al. (1997); Flatté and Byers (1997a, b); Yoshioka and Ohashi (2000); Morr and Stavropoulos (2003); Balatsky et al. (2006); Bauer et al. (2007); Martín-Rodero and Levy Yeyati (2011); Dirks et al. (2011). The ground state of the quantum dot can be tuned to be either a spin singlet or a spin doublet depending on the impurity level and the hybridization with the bulk superconductor Buitelaar et al. (2002); Pillet et al. (2010); Deacon et al. (2010); Maurand et al. (2012); Martín-Rodero and Yeyati (2012); Jarrell et al. (1990). The Coulomb interaction on the quantum dot favors the spin doublet ground state, while the spin singlet can be stabilized by the Kondo effect or by pairing due to the superconducting proximity effect Vecino et al. (2003); Oguri et al. (2004); Choi et al. (2004); Karrasch et al. (2008); Meng et al. (2009). The position of the in-gap (Shiba) resonances, as determined from the tunneling conductance, agrees even quantitatively with the calculations based on the simple single-orbital Anderson impurity model Pillet et al. (2013); Lee et al. (2016).
Very recently, research has focused on the effects of the magnetic field on the in-gap states Mourik et al. (2012); Lee et al. (2012); Das et al. (2012); Deng et al. (2012); Chang et al. (2013); Lee et al. (2014); Žitko et al. (2015); Wentzell et al. (2016); Jellinggaard et al. (2016); Bujnowski et al. (2016) because systems of this class have been proposed as possible building blocks for topologically ordered systems exhibiting Majorana edge states Kitaev (2001); Nadj-Perge et al. (2013, 2014); Ruby et al. (2015b). These are significant for fundamental reasons and might also find application in quantum computation Sau et al. (2010); Alicea et al. (2011); Sarma et al. (2015). When an external magnetic field is applied to a thin-film superconductor in the parallel (in-plane) direction, the superconducting state persists to relatively large fields. The quasiparticle states become, however, strongly spin polarized and the coherence peaks in the density of states become Zeeman split Meservey et al. (1970); Tedrow and Meservey (1971, 1973); Meservey and Tedrow (1994); Eltschka et al. (2014): systems in this regime are known as spin-split or Zeeman-split superconductors, and play a key role in the emerging field of superconducting spintronics Linder and Robinson (2015). The spectral function of a spin-split superconductor has two band edges with diverging coherence peaks separated by the bulk Zeeman energy, reflecting the fact that the Bogoliubov excitations have spin-dependent energies . Here is the energy level of electron with momentum measured with respect to the chemical potential , is the gap, is the -factor of the superconductor, is the Bohr magneton, is the magnetic field, and is the quasiparticle spin. Since the Shiba states can be considered as bound states of Bogoliubov quasiparticles, the spectral properties of magnetic impurities in spin-split superconductors are modified.
The theoretical work has, so far, mainly focused on the effect of a local magnetic field applied on the position of the impurity only Žitko et al. (2015); Wentzell et al. (2016). For bulk electrons in the normal state, this approximation is usually justified because the impurity magnetic susceptibility is typically much larger (, where is the Kondo temperature) than that of the bulk electrons (Pauli susceptibility, , where is the density of states at the Fermi level and is the bandwidth). In superconductors, however, the Zeeman splitting of the Bogoliubov quasiparticle bands and the Zeeman splitting of the doublet sub-gap states are of comparable magnitude: the splitting of the first is simply the Zeeman energy , while the splitting of the second is , where is the impurity -factor renormalized by the coupling with the bulk. Generically, both splittings are comparable with the possible exception of nanowire quantum dots made of materials with extremely strong spin-orbit (SO) coupling and hence very high bare . For this reason, it is important to include the Zeeman terms both in the impurity and in the bulk part of the Hamiltonian.
We introduce the ratio of the Landé -factors which describe the magnitude of the Zeeman splittings:
For many elemental superconductors the factor is close to the free electron value, . In semiconductors the factor usually differs strongly from this value due to SO coupling. The effective factors are quite variable Csonka et al. (2008): they can be very large positive, as well as very large negative, or can even be tuned close to 0. The control of can be achieved through strain engineering Nakaoka et al. (2005), nanostructuring van Bree et al. (2016), or by electrical tuning in quantum dots Csonka et al. (2008); Schroer et al. (2011); Deacon et al. (2011); Ares et al. (2013). In the limit, the Zeeman term is only present on the impurity site: this limit is appropriate for materials with very large positive or negative factor, where the Zeeman splitting in the superconductor is indeed negligible. Another special limit is , where all sites (bulk and impurity) have the same -factor. In general, however, the value of is essentially unconstrained.
Here we study, using analytical calculations for a classical impurity (with no internal dynamics) and with the numerical renormalization group (NRG) method Wilson (1975); Krishna-murthy et al. (1980); Yoshioka and Ohashi (2000); Hofstetter (2000); Žitko and Pruschke (2009); Žitko (2009); Hecht et al. (2008); Bulla et al. (2008) for a quantum impurity (which incorporates the effect of spin flips), the spectral properties of the Shiba states. In the classical case we perform a calculation along the lines of Refs. Yu, 1965; Shiba, 1968; Rusinov, 1969, but include the effect of the Zeeman term in the superconductor. In the quantum case we focus on the single-orbital Anderson impurity and discuss the changes in the singlet-doublet phase transition as the ratio of the -factors of the impurity and the bulk is varied. We study the fate of a sub-gap resonance when it approaches the continuum of the Bogoliubov quasiparticles with the opposite spin, with and without the additional transverse magnetic field that mimics non-collinearity in the presence of SO coupling.
Ii Classical impurity
Initially, the impurity is described using a quantum mechanical spin- operator, which is exchange coupled with the spin-density of the conduction band electrons at the position of the impurity at . The corresponding Hamiltonian is with
where is the magnetic field expressed in the energy units (i.e., the Zeeman splitting), , and . is the exchange coupling between the impurity and the host. All other quantities have already been defined in the previous section. The classical impurity limit consists of taking the limit while keeping . In this limit, the longitudinal component of the exchange interaction persists, while the transverse (spin-flip) components decrease as and hence drop out of the problem. The Hamiltonian then becomes non-interacting. We introduce the effective local field
and the dimensionless impurity coupling parameter
where is the density of states (DOS) at the Fermi level in the normal state. We will first assume that the bulk field and the effective local field are collinear and of the same sign. To be specific, we choose , .
The non-perturbed Green’s function of the Zeeman-split superconductor is
Here are the Pauli matrices, is the identity matrix, and is the frequency argument. To obtain the local Green’s function at the origin, , we sum over the momenta and switch over to an integral over energies assuming a flat DOS in the normal state. In the wide-band limit we find
The spin-up spectral function is , while the spin-down spectral function is .
The 11 (spin-up) matrix component of has two poles:
Only one pole has a finite residue. For (hence ) we find a sub-gap resonance in the spin-up spectral function at
Conversely, the spin-down spectral function has a resonance at :
We emphasize that the spin-projected spectral functions have a single sub-gap resonance, one for each spin. This is to be contrasted with the behavior of the quantum model discussed in the following section which has (in the spin-singlet regime for finite magnetic field) two resonances in each spin-projected spectral function. This is a clear indication of the different degeneracies of states in the classical and quantum impurity models.
Some representative spectra are plotted in Fig. 1. The case corresponds to the limit of a clean Zeeman-split superconductor. Each quasiparticle continuum branch has a characteristic inverse square root divergence at its edge.
For small , the Shiba bound states emerge out of the quasiparticle continuum, the spin-up resonance in the negative part of the spectrum, and the spin-down resonance in the positive part, in line with Eqs. (13) and (14) for small . The shift by is expected, since the spin-up Shiba state is generated by the Bogoliubov states with spin up, which are themselves shifted by the same amount. Conversely, the spin-down Shiba state is generated as a linear superposition of Bogoliubov states with spin down which are shifted by . We observe that all four branches of the quasiparticle band lose their inverse square-root singularity and contribute spectral weight to the nascent Shiba state, see also Ref. Bujnowski et al., 2016, not only the “inner” ones (spin-up occupied and spin-down unoccupied).
With increasing , the Shiba states move toward the gap center (chemical potential) and they cross when the condition
is met, i.e., at
For , as used here, this happens at . This signals the occurrence of the quantum phase transition in which the fermion parity of the (sub)system changes. We also note that alternatively, for constant , the transition can be driven by the external magnetic field.
For still larger , the spin-up Shiba resonance overlaps with the spin-down quasiparticle continuum (and vice versa for the spin-down Shiba resonance), but since the spin is assumed to be a good quantum number there is no broadening of the Shiba resonances. (See below, Sec. III.2, for a discussion of the SO effects in the case of a quantum impurity.)
For very large values of , the Shiba states eventually merge with the continuum again. This trend is accompanied by the reappearance of the inverse square-root resonances, an indication of which is visible for in Fig. 1.
We now discuss the case of anti-aligned fields, taking and . In this case, for small the spin-up Shiba state occurs at
and hence overlaps with the continuum of spin-down quasiparticles for . The quantum phase transition occurs for
For large the Shiba states again merge with the continuum at the inner edges of the Bogoliubov bands. The regimes that the system goes through for are thus in the opposite order to those for .
The main deficiency of the impurity model in the classical limit is the reduced multiplicity of the sub-gap states. Physically, this is due to the fact that in the classical limit the effective impurity potential for particle-like excitations is attractive for one spin orientation and repulsive for the other, hence a single bound state is generated for a given spin orientation. The spin-flip processes in the quantum model lead to a situation where the effective potential is attractive for both spin polarizations, hence twice the degeneracy. We discuss this more general situation in the following section.
Iii Quantum impurity
iii.1 Model and method
We consider a single spin- impurity level with on-site Coulomb interaction. The Hamiltonian is given by
is the creation operator on the impurity which is hybridized with the bulk by and has the energy level . , , , . The magnetic field couples with the quantum dot by the -factor equal to and with the superconductor by . The transverse magnetic field which can flip the spin is introduced through the parameter . We will consider a flat particle-hole symmetric band of half-width so that . The hybridization strength is characterized by .
We employ the NRG method to solve the problem. There are two ways to introduce a bulk Zeeman field in the NRG: as local Zeeman terms on all sites of the Wilson chain, or through a separate discretization of spin-up and spin-down densities of states shifted by the Zeeman term Höck and Schnack (2013). The former approach is suitable for models with a spectral gap, as discussed here, while the latter has to be used for spin-polarized metals with finite DOS at the Fermi level. We use a fine discretization mesh with twist averaging over grids so that high spectral resolution is possible inside the gap and in the vicinity of the gap edges, which are the regions of main interest in this work. The only conserved quantum number in the presence of an external field along the -axis is the projection of total spin , i.e., the problem has U(1) spin symmetry. Other parameters are , the NRG truncation cut-off energy is where is the energy scale at the -th step of the iteration and at least 200 states were used at late iterations when the gap is opened. The spectral functions are computed with the DMNRG algorithm Hofstetter (2000) with the scheme for patching the spectral functions. This approach allows maximal spectral resolution at zero temperature. The broadening is performed on a logarithmic mesh with a small ratio between two energies outside the gap, and on a linear mesh inside the gap. As can be seen in the figures further down, the use of these different broadening kernels leads to some artifacts at the continuum edges. All calculations are performed in the zero-temperature limit, .
Unless otherwise specified, the model parameters are , , .
The ground state of the Anderson impurity model, Eq. (III.1), in the absence of the magnetic field is either a singlet or a doublet depending on the ratio of the Kondo temperature Wilson (1975); Krishna-murthy et al. (1980) and the superconducting gap . The impurity spin is screened by the conduction electrons for forming a spin singlet, while for the local moment is unscreened and the ground state forms a spin doublet; here Satori et al. (1992); Sakai et al. (1993); Yoshioka and Ohashi (2000) in the limit . At the quantum phase transition the energy of the excited many-particle state goes to zero, and the energy levels cross. The transition is accompanied by a jump in the spectral weight of the in-gap resonances and a change of sign of the pairing amplitude Choi et al. (2004). The Zeeman field lifts the degeneracy of the doublet state Žitko et al. (2008); Lee et al. (2014); Žitko et al. (2015); Wentzell et al. (2016). For a spin singlet ground state, the in-gap resonances corresponding to the doublet state are split in the magnetic field . In the case of doublet ground state, the positions of the singlet Shiba resonances are shifted in the Zeeman field.
Fig. 2 shows a schematic phase diagram in zero magnetic field and the evolution of the energy levels of the ground and excited states with increasing Zeeman magnetic field. This evolution of the in-gap resonances with changes of the hybridization and the magnetic field has been recently observed in tunneling experiments and agrees with the theoretical predictions in the case when the field is coupled only with impurity Lee et al. (2014); Jellinggaard et al. (2016); Žitko et al. (2015); Wentzell et al. (2016). Here, we explore the fate of the subgap states when the magnetic field is also Zeeman coupled with the bulk superconductor.
We now discuss the spectral function of the impurity in different parameter regimes and identify the boundary of the singlet-doublet phase transition in the parameter plane for different values of the -factor ratio .
We first consider the case of singlet ground state. In the magnetic field the subgap resonance (which is a spin doublet) splits to its spin up and spin down components. The impurity spectral function for , is shown in Fig. 3(a) for (central panel), (top panel), and (bottom panel). The additional panels show how the position of the resonances shifts as the parameter is varied. For the expectation value of the spin projection at the impurity site is (see Fig. 4(c) and Appendix A). Such compensation holds also in the particle-hole asymmetric case as long as . If the -factors are different, there will be net magnetization at the impurity site even if the ground state is a spin singlet and there is a finite gap to excited states.
We next consider the case of smaller hybridization, , so that the impurity is in the doublet ground state. The spectral functions for , and , as well as the evolution between them, are shown in Fig. 3(b). A single resonance is now visible for , since the ground state has spin projection , and the only possible excitation is adding a spin-up particle to form a singlet state. We also observe notable differences in the appearance of the gap edges for both spin projections, related to the strong spin polarization of the impurity state in the doublet regime. We emphasize that this distinguishing feature is not present in the classical impurity model discussed above.
The phase diagram in the plane is shown in Fig. 4. This plot represents the main result of this work. In the absence of a magnetic field, the ground state changes from singlet to doublet for . Here, and for the chosen value of . For the transition can be also induced by changing the magnetic field. For the magnetic field is coupled only with the impurity. In this case, as shown in Ref. Žitko et al., 2015, the critical magnetic field for the singlet-doublet transition linearly depends on the gap, . For , however, has non-monotonic dependence on : it increases approximately linearly with as it gets reduced from , reaches a maximum and then decreases to zero as . For the singlet-doublet transition is a consequence of a competition of three characteristic energies: , and . For very small values of (for ) the singlet-doublet transition coincides with the closure of the superconducting gap for . The phase boundary for small value of is shown in Fig. 4(b). We note that for small the transition to the normal phase would actually occur for smaller value of , , known as the Clogston limit Clogston (1962); Chandrasekhar (1962). For the normal phase has lower free energy than the superconducting one. Our main focus is, however, on larger values of the superconducting gap when it is comparable to the Kondo temperature.
The average value of the projection of the local spin abruptly changes at the phase transition, Fig. 4(c). For , i.e., for , the average value in the singlet case (see also Appendix A). For , is nonzero but small for singlet ground state, and it jumps to large absolute value by increasing the magnetic field at the transition to doublet ground state. The pairing amplitude on the impurity, , shows a characteristic sign change at the transition, Fig. 4(d).
When the spin-up Shiba state begins to overlap with the spin-down branch of Bogoliubov excitations, it remains unperturbed, as in the classical impurity model. This is the case in spite of the spin-flip processes in the quantum model, and is a simple consequence of the conservation of the spin projection quantum number. In other words, the spin-up Shiba state is a bound state of spin-up Bogoliubov quasiparticles which are orthogonal to and do not mix with the spin-down Bogoliubov quasiparticles. This is illustrated in Fig. 5. Here and were kept constant, while the position of the up-spin resonance was changed by changing . A transverse magnetic field, however, flips the spin and the Shiba resonances broaden, as illustrated in Fig. 6. Such broadening effects are expected in realistic systems due to SO coupling.
We have analyzed the behavior of magnetic impurities coupled to superconductors subject to an applied magnetic field that does not fully suppress the superconducting order but strongly spin-splits the Bogoliubov quasiparticle continua because of the Zeeman coupling. This situation commonly occurs when the field is applied in the plane of a superconducting thin layer and leads to clearly observable effects.
For a classical impurity, approximated as a static local point-like magnetic field (and aligned with the external field), we find that the position of the Shiba state is shifted linearly with the external field as a simple consequence of the shifting edges of the quasiparticle bands. In fact, the only effect of the spin-splitting of the Bogoliubov states is that the frequency argument in the impurity Green’s function is shifted as for spin-up and for spin-down particles. The parity-changing quantum phase transition no longer occurs at , but rather when the condition is met. This occurs for . We observed that for large the Shiba state of a given spin may overlap with the quasiparticle continuum of the opposite spin and still remain a sharp resonance (a peak). This remains true as long as there is no matrix element linking the quasiparticles of both spins.
We then turned to the case of a quantum impurity with far more complex behavior. The Zeeman coupling is present both in the bulk and on the impurity site, and generically the corresponding -factors are different: this is typically indeed the case in the nanoscale hybrid superconductor-semiconductor devices. We find a very significant effect of the Zeeman splitting of the quasiparticle continua: the phase diagram of the possible many-particle ground states (singlet or doublet) in the plane actually has two very different regimes. In the limiting regime, the transition occurs because a strong enough field decreases the energy of spin-down doublet state below that of the singlet state. In this regime, the phase boundary in the plane has a negative slope: the closer is to , the smaller the separation between the singlet and doublet states in the absence of the field, hence a smaller Zeeman splitting is necessary to induce the transition. We have established that for finite the splitting between the doublet sub-gap states is larger than for , hence the separation between the singlet and the spin-down doublet is smaller, thus the transition occurs for a smaller value of the magnetic field. In the other limiting regime of small , the transition occurs because the gap between the spin-polarized Bogoliubov bands closes and the transition line is given asymptotically as , hence the transition line has a positive slope. In reality, such transition is of course preempted by a bulk transition to the normal state (Clogston limit). Nevertheless, even in the physically accessible regime we observe that the actual behavior is determined by a competition of both trends and that the slope of the transition line changes at some intermediate point where the system crosses over from one limiting behavior to another. The actual transition line is therefore bell-shaped and depends on the value of . The straight line found in the limit is, in fact, highly anomalous, and for realistic values of the ratio there will be a significant degree of curvature.
We have confirmed the possibility of a sharp Shiba resonance overlapping with the continuum of opposite-spin Bogoliubov quasiparticles. In addition, we have considered the gradual widening of the Shiba resonance if local spin-flip processes are allowed (generated, e.g., by SO coupling leading to non-collinear effective magnetic fields): such processes lead to the hybridization of the Shiba state and its gradual engulfing in the continuum.
In conclusion, we have established the importance of including the Zeeman splitting in the bulk of the superconductor when discussing the effect of the external magnetic field on the sub-gap states induced by magnetic impurities in superconductors.
Acknowledgements.W.V.vG. and D.T. were supported by the Serbian Ministry of Education, Science and Technological Development under Project ON171017, and by the European Commission under H2020 project VI-SEEM, Grant No. 675121. R. Ž. acknowledges the support of the Slovenian Research Agency (ARRS) under Program P1-0044 and J1-7259. The authors acknowledge support from the bilateral Slovenian-Serbian project “Strong electronic correlations and superconductivity“.
Appendix A Non-interacting model
For completeness, in this appendix we define the analytical expression for the non-interacting Anderson impurity model (), see also Ref. Machida and Shibata, 1972. We work in the Nambu space, , . The Hamiltonian can be written as
The Green’s function is given by ,
with being Pauli matrices and the identity matrix, so that
The impurity Green’s function is
In the wide-band limit
where . , on real axis, :
Finally, we have
Matrix inversion yields
Now assume . We consider two functions and . Taking into account the symmetry properties of , it is easily shown that both inside and outside the gap. This shows that as long as the system is in the singlet ground state, it is possible to shift the spectral functions of spin-up and spin-down sub-systems to make them overlap, thus their integrals over the negative energies (occupied states) are equal, hence . This is also the case in the interacting case. For , in the singlet regime will be non-zero but small. In the doublet regime, irrespective of the value of , is large.
- Yu (1965) L. Yu, “Bound state in superconductors with paramagnetic impurities,” Acta Phys. Sin. 21, 75 (1965).
- Shiba (1968) H. Shiba, “Classical spins in superconductors,” Prog. Theor. Phys. 40, 435 (1968).
- Rusinov (1969) A. I. Rusinov, “Superconductivity near a paramagnetic impurity,” JETP Lett. 9, 85 (1969), zh. Eksp. Teor. Fiz. Pisma Red. 9, 146 (1968).
- Goffman et al. (2000) M. F. Goffman, R. Cron, A. Levy Yeyati, P. Joyez, M. H. Devoret, D. Esteve, and C. Urbina, “ Supercurrent in Atomic Point Contacts and Andreev States,” Phys. Rev. Lett. 85, 170–173 (2000).
- Pillet et al. (2010) J.-D. Pillet, C. H. L. Quay, P. Morin, C. Bena, A. Levy Yeyati, and P. Joyez, “Andreev bound states in supercurrent-carrying carbon nanotubes revealed,” Nat. Physics 6, 965 (2010).
- Deacon et al. (2010) R S Deacon, Y Tanaka, A Oiwa, R Sakano, K Yoshida, K Shibata, K Hirakawa, and S Tarucha, “Interplay of Kondo and superconducting correlations in the nonequilibrium Andreev transport through a quantum dot,” Physical Review Letters 104, 076805 (2010).
- Franceschi et al. (2010) Silvano De Franceschi, Leo Kouwenhoven, Christian Schönenberger, and Wolfgang Wernsdorfer, “Hybrid superconductor-quantum dot devices,” Nat. Nanotechnology 5, 703 (2010).
- Bretheau et al. (2013) L. Bretheau, C. O. Girit, C. Urbina, D. Esteve, and H. Pothier, “Supercurrent Spectroscopy of Andreev States,” Phys. Rev. X 3, 041034 (2013).
- Janvier et al. (2015) C. Janvier, L. Tosi, L. Bretheau, C. O. Girit, M. Stern, P. Bertet, P. Joyez, D. Vion, D. Esteve, M. F. Goffman, H. Pothier, and C. Urbina, “Coherent manipulation of Andreev states in superconducting atomic contacts,” Science 349, 1199 (2015).
- Yazdani et al. (1997) A. Yazdani, B. A. Jones, C. P. Lutz, M. F. Crommie, and D. M. Eigler, “Probing the local effects of magnetic impurities on superconductivity,” Science 275, 1767 (1997).
- Ji et al. (2008) S. H. Ji, T. Zhang, Y. S. Fu, X. Chen, Xu-Cun Ma, J. Li, Wen-Hui Duan, Jin-Feng Jia, and Qi-Kun Xue, “High-resolution scanning tunneling spectroscopy of magnetic impurity induced bound states in the superconducting gap of Pb thin films,” Phys. Rev. Lett. 100, 226801 (2008).
- Franke et al. (2011) K. J. Franke, G. Schulze, and J. I. Pascual, “Competition of superconductivity phenomena and Kondo screening at the nanoscale,” Science 332, 940 (2011).
- Ruby et al. (2015a) Michael Ruby, Falko Pientka, Yang Peng, Felix von Oppen, Benjamin W Heinrich, and Katharina J Franke, ‘‘Tunneling Processes into Localized Subgap States in Superconductors,” Physical Review Letters 115, 087001–5 (2015a).
- Randeria et al. (2016) M. T. Randeria, B. E. Feldman, I. K. Drozdov, and A. Yazdani, “ Scanning Josephson spectroscopy on the atomic scale ,” Physical Review B 93, 161115 (2016).
- Hatter et al. (2015) Nino Hatter, Benjamin W Heinrich, Michael Ruby, Jose I Pascual, and Katharina J Franke, “Magnetic anisotropy in shiba bound states across a quantum phase transition,” Nature communications 6 (2015).
- Sakurai (1970) Akio Sakurai, “Comments on superconductors with magnetic impurities,” Prog. Theor. Phys. 44, 1472 (1970).
- Satori et al. (1992) Koji Satori, Hiroyuki Shiba, Osamu Sakai, and Yukihiro Shimizu, “Numerical renormalization group study of magnetic impurities in superconductors,” J. Phys. Soc. Japan 61, 3239 (1992).
- Sakai et al. (1993) Osamu Sakai, Yukihiro Shimizu, Hiroyuki Shiba, and Koji Satori, “Numerical renormalization group study of magnetic impurities in supercoductors. II. Dynamical excitations spectra and spatial variation of the order parameter,” J. Phys. Soc. Japan 62, 3181 (1993).
- Salkola et al. (1997) M. I. Salkola, A. V. Balatsky, and J. R. Schrieffer, ‘‘Spectral properties of quasiparticle excitations induced by magnetic moments in superconductors,” Phys. Rev. B 55, 12648 (1997).
- Flatté and Byers (1997a) Michael E. Flatté and Jeff M. Byers, “Local electronic structure of a single magnetic impurity in a superconductor,” Phys. Rev. Lett. 78, 3761 (1997a).
- Flatté and Byers (1997b) Michael E. Flatté and Jeff M. Byers, “Local electronic structure of defects in superconductors,” Phys. Rev. B 56, 11213 (1997b).
- Yoshioka and Ohashi (2000) Tomoki Yoshioka and Yoji Ohashi, “ Numerical renormalization group studies on single impurity Anderson model in superconductivity: a unified treatment of magnetic, nonmagnetic impurities, and resonance scattering ,” J. Phys. Soc. Japan 69, 1812 (2000).
- Morr and Stavropoulos (2003) Dirk K. Morr and Nikolaos A. Stavropoulos, “Quantum interference between impurities: Creating novel many-body states in s-wave superconductors,” Phys. Rev. B 67, 020502(R) (2003).
- Balatsky et al. (2006) A. V. Balatsky, I. Vekhter, and Jian-Xin Zhu, “Impurity-induced states in conventional and unconventional superconductors,” Rev. Mod. Phys. 78, 373 (2006).
- Bauer et al. (2007) J. Bauer, A. Oguri, and A. C. Hewson, “Spectral properties of locally correlated electrons in a Bardeen-Cooper-Schrieffer superconductor,” J. Phys.: Condens. Matter 19, 486211 (2007).
- Martín-Rodero and Levy Yeyati (2011) A. Martín-Rodero and A. Levy Yeyati, ‘‘Josephson and Andreev transport through quantum dots,” Advances in Physics 60, 899–958 (2011).
- Dirks et al. (2011) T. Dirks, T. L. Hughes, S. Lal, B. Uchoa, Y.-F. Chen, C. Chialvo, P. M. Goldbart, and N. Mason, “Transport through Andreev bound states in a graphene quantum dot,” Nature Physics 7, 386–390 (2011).
- Buitelaar et al. (2002) M. R. Buitelaar, T. Nussbaumer, and C. Schonenberger, “ Quantum dot in the Kondo regime coupled to superconductors ,” Phys. Rev. Lett. 89, 256801 (2002).
- Maurand et al. (2012) Romain Maurand, Tobias Meng, Edgar Bonet, Serge Florens, Laëtitia Marty, and Wolfgang Wernsdorfer, “First-order 0- quantum phase transition in the Kondo regime of a superconducting carbon-nanotube quantum dot,” Phys. Rev. X 2, 011009 (2012).
- Martín-Rodero and Yeyati (2012) A. Martín-Rodero and A. L. Yeyati, “The Andreev states of a superconducting quantum dot: mean field versus exact numerical results,” Journal of Physics: Condensed Matter 24, 385303 (2012).
- Jarrell et al. (1990) M. Jarrell, D. S. Sivia, and B. Patton, “Gap states in dilute-magnetic-alloy superconductors: A quantum Monte Carlo study,” Physical Review B 42, 4804 (1990).
- Vecino et al. (2003) E Vecino, A Martín-Rodero, and A Yeyati, “Josephson current through a correlated quantum level: Andreev states and junction behavior,” Phys. Rev. B 68, 035105 (2003).
- Oguri et al. (2004) Akira Oguri, Yoshihide Tanaka, and A. C. Hewson, “Quantum phase transition in a minimal model for the Kondo effect in a Josephson junction,” J. Phys. Soc. Japan 73, 2494 (2004).
- Choi et al. (2004) Mahn-Soo Choi, Minchul Lee, Kicheon Kang, and W. Belzig, “ Kondo effect and Josephson current through a quantum dot between two superconductors ,” Phys. Rev. B 70, 020502 (2004).
- Karrasch et al. (2008) C. Karrasch, A. Oguri, and V. Meden, “Josephson current through a single Anderson impurity coupled to BCS leads,” Phys. Rev. B 77, 024517 (2008).
- Meng et al. (2009) Tobias Meng, Serge Florens, and Pascal Simon, “Self-consistent description of Andreev bound states in Josephson quantum dot devices,” Phys. Rev. B 79, 224521 (2009).
- Pillet et al. (2013) J.-D. Pillet, P. Joyez, R. Žitko, and M. F. Goffman, “Tunneling spectroscopy of a single quantum dot coupled to a superconductor: From Kondo ridge to Andreev bound states,” Phys. Rev. B 88, 045101 (2013).
- Lee et al. (2016) E. J. H. Lee, X. Jiang, R. Žitko, C. M. Lieber, and S. De Franceschi, “Scaling of sub-gap excitations in a superconductor-semiconductor nanowire quantum dot,” arxiv:1609.07582 (2016).
- Mourik et al. (2012) V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. A. Bakkers, and L. P. Kouwenhoven, “ Signatures of Majorana Fermions in Hybrid Superconductor-Semiconductor Nanowire Devices ,” Science 336, 1003 (2012).
- Lee et al. (2012) E. J. H. Lee, Xiaocheng Jiang, Ramon Aguado, Georgios Katsaros, Charles M. Lieber, and Silvano De Franceschi, “Zero-bias anomaly in a nanowire quantum dot coupled to superconductors,” Phys. Rev. Lett. 109, 186802 (2012).
- Das et al. (2012) A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrikman, “ Zero-bias peaks and splitting in an Al-InAs nanowire topological superconductor as a signature of Majorana fermions ,” Nat. Physics 8, 887 (2012).
- Deng et al. (2012) M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caroff, and H. Q. Xu, “ Anomalous Zero-Bias Conductance Peak in a Nb-InSb Nanowire-Nb Hybrid Device ,” Nano Letters 12, 6414 (2012).
- Chang et al. (2013) W. Chang, V. E. Manucharyan, T. S. Jespersen, J. Nygard, and C. M. Marcus, ‘‘ Tunneling Spectroscopy of Quasiparticle Bound States in a Spinful Josephson Junction ,” Phys. Rev. Lett. 110, 217005 (2013).
- Lee et al. (2014) E. J. H Lee, X. Jiang, M. Houzet, R. Aguado, C. M. Lieber, and S. De Franceschi, “Spin-resolved Andreev levels and parity crossings in hybrid superconductor-semiconductor nanostructures,” Nature Nanotech. 9, 79 (2014).
- Žitko et al. (2015) R. Žitko, Jong Soo Lim, Rosa Lopez, and Ramon Aguado, “Shiba states and zero-bias anomalies in the hybrid normal-superconductor Anderson model,” Phys. Rev. B 91, 045441 (2015).
- Wentzell et al. (2016) N. Wentzell, S. Florens, T. Meng, V. Meden, and S. Andergassen, “ Magneto-electric spectroscopy of Andreev bound states in Josephson quantum dots ,” Phys. Rev. B 94, 085151 (2016).
- Jellinggaard et al. (2016) A. Jellinggaard, K. Grove-Rasmussen, M. H. Madsen, and J. Nygard, “ Tuning Yu-Shiba-Rusinov states in a quantum dot ,” Phys. Rev. B 94, 064520 (2016).
- Bujnowski et al. (2016) B. Bujnowski, D. Bercioux, F. Konschelle, J. Cayssol, and F. S. Bergeret, “Andreev spectrum of a Josephson junction with spin-split superconductors,” Europhysics Letters 115, 67001 (2016).
- Kitaev (2001) Alexei Kitaev, “Unpaired majorana fermions in quantum wires,” Usp. Fiz. Nauk (Suppl.) 171, 131 (2001).
- Nadj-Perge et al. (2013) S Nadj-Perge, I K Drozdov, B A Bernevig, and Ali Yazdani, “Proposal for realizing Majorana fermions in chains of magnetic atoms on a superconductor,” Physical Review B 88, 020407 (2013).
- Nadj-Perge et al. (2014) S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. jeon, J. Seo, A. H. MacDonald, B. A. Bernevig, and A. Yazdani, “ Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor,” Science 346, 602 (2014).
- Ruby et al. (2015b) Michael Ruby, Falko Pientka, Yang Peng, Felix von Oppen, Benjamin W Heinrich, and Katharina J Franke, “End states and subgap structure in proximity-coupled chains of magnetic adatoms,” Physical review letters 115, 197204 (2015b).
- Sau et al. (2010) Jay D. Sau, Roman M. Lutchyn, Sumanta Tewari, and S. Das Sarma, “Generic new platform for topological quantum computation using semiconductor heterostructures,” Phys. Rev. Lett. 104, 040502 (2010).
- Alicea et al. (2011) J. Alicea, Y. Oreg, G. Refael, F. van Oppen, and M. P. A. Fischer, ‘‘ Non-Abelian statistics and topological quantum information processing in 1D wire networks ,” Nat. Physics 7, 412 (2011).
- Sarma et al. (2015) S. Das Sarma, M. Freedman, and C. Nayak, “Majorana zero modes and topological quantum computation,” npj Quantum Information 1, 15001 (2015).
- Meservey et al. (1970) R. Meservey, P. M. Tedorow, and Peter Fulde, “Magnetic field splitting of the quasiparticle states in superconducting aluminum films,” Phys. Rev. Lett. 25, 1270 (1970).
- Tedrow and Meservey (1971) P M Tedrow and R Meservey, “Spin-Dependent Tunneling into Ferromagnetic Nickel,” Physical Review Letters 26, 192–195 (1971).
- Tedrow and Meservey (1973) P M Tedrow and R Meservey, “Spin Polarization of Electrons Tunneling from Films of Fe, Co, Ni, and Gd,” Physical Review B 7, 318–326 (1973).
- Meservey and Tedrow (1994) R Meservey and P M Tedrow, “Spin-polarized electron tunneling,” Physics Reports 238, 173–243 (1994).
- Eltschka et al. (2014) Matthias Eltschka, Berthold Jäck, Maximilian Assig, Oleg V Kondrashov, Mikhail A Skvortsov, Markus Etzkorn, Christian R Ast, and Klaus Kern, “Probing Absolute Spin Polarization at the Nanoscale,” Nano Letters 14, 7171–7174 (2014).
- Linder and Robinson (2015) J. Linder and J. W. A. Robinson, “Superconducting spintronics,” Nat. Phys. 11, 307 (2015).
- Csonka et al. (2008) C. Csonka, L. L. Hofstetter, F. Freitag, S. Oberholzer, C. Schönberger, T. S. Jespersen, M. Aegesen, and J. Nygard, “Giant fluctuations and gate control of the -factor in InAs nanowire quantum dots,” Nano Letters 8, 3932 (2008).
- Nakaoka et al. (2005) T Nakaoka, T Saito, J Tatebayashi, S Hirose, T Usuki, N Yokoyama, and Y Arakawa, “Tuning of g-factor in self-assembled In(Ga)As quantum dots through strain engineering,” Physical Review B 71, 205301–7 (2005).
- van Bree et al. (2016) J van Bree, A Yu Silov, M L van Maasakkers, C E Pryor, M E Flatté, and P M Koenraad, “Anisotropy of electron and hole gtensors of quantum dots: An intuitive picture based on spin-correlated orbital currents,” Physical Review B 93, 035311–10 (2016).
- Schroer et al. (2011) M D Schroer, K D Petersson, M Jung, and J R Petta, “Field Tuning the gFactor in InAs Nanowire Double Quantum Dots,” Physical Review Letters 107, 176811–5 (2011).
- Deacon et al. (2011) R S Deacon, Y Kanai, S Takahashi, A Oiwa, K Yoshida, K Shibata, K Hirakawa, Y Tokura, and S Tarucha, “Electrically tuned gtensor in an InAs self-assembled quantum dot,” Physical Review B 84, 041302–5 (2011).
- Ares et al. (2013) N Ares, V N Golovach, G Katsaros, M Stoffel, F Fournel, L I Glazman, O G Schmidt, and S De Franceschi, “Nature of Tunable Hole g factors in Quantum Dots,” Physical Review Letters 110, 046602–5 (2013).
- Wilson (1975) K. G. Wilson, “The renormalization group: Critical phenomena and the Kondo problem,” Rev. Mod. Phys. 47, 773 (1975).
- Krishna-murthy et al. (1980) H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wilson, “Renormalization-group approach to the Anderson model of dilute magnetic alloys. I. Static properties for the symmetric case,” Phys. Rev. B 21, 1003 (1980).
- Hofstetter (2000) Walter Hofstetter, “Generalized numerical renormalization group for dynamical quantities,” Phys. Rev. Lett. 85, 1508 (2000).
- Žitko and Pruschke (2009) Rok Žitko and Thomas Pruschke, “Energy resolution and discretization artefacts in the numerical renormalization group,” Phys. Rev. B 79, 085106 (2009).
- Žitko (2009) Rok Žitko, “Adaptive logarithmic discretization for numerical renormalization group methods,” Comp. Phys. Comm. 180, 1271 (2009).
- Hecht et al. (2008) T. Hecht, A. Weichselbaum, J. von Delft, and R. Bulla, “Numerical renormalization group calculation of near-gap peaks in spectral functions of the Anderson model with superconducting leads,” J. Phys. Condens. Mat. 20, 275213 (2008).
- Bulla et al. (2008) Ralf Bulla, Theo Costi, and Thomas Pruschke, “The numerical renormalization group method for quantum impurity systems,” Rev. Mod. Phys. 80, 395 (2008).
- Höck and Schnack (2013) Matrin Höck and Jürgen Schnack, “ Numerical renormalization group calculations of the magnetization of Kondo impurities with and without uniaxial anisotropy ,” Phys. Rev. B 87, 184408 (2013).
- Žitko et al. (2008) Rok Žitko, Robert Peters, and Thomas Pruschke, “Properties of anisotropic magnetic impurities on surfaces,” Phys. Rev. B 78, 224404 (2008).
- Clogston (1962) A. M. Clogston, “Upper limit for the critical field in hard superconductors,” Phys. Rev. Lett. 9, 266 (1962).
- Chandrasekhar (1962) B. S. Chandrasekhar, “A note on the maximum critical field of high-field superconductors,” Appl. Phys. Lett. 1, 7 (1962).
- Machida and Shibata (1972) K. Machida and F. Shibata, “Bound States Due to Resonance Scattering in Superconductor,” Progress of Theoretical Physics 47, 1817–1823 (1972).