Abstract
Impurities in superconductors and their induced bound states are important both for engineering novel states such as Majorana zeroenergy modes and for probing bulk properties of the superconducting state. The hightemperature cuprates offer a clear advantage in a much larger superconducting order parameter, but the nodal energy spectrum of a pure dwave superconductor only allows virtual bound states. Fully gapped dwave superconducting states have, however, been proposed in several cuprate systems thanks to subdominant order parameters producing d + is or d + id′wave superconducting states. Here we study both magnetic and potential impurities in these fully gapped dwave superconductors. Using analytical Tmatrix and complementary numerical tightbinding lattice calculations, we show that magnetic and potential impurities behave fundamentally different in d + is and d + id′wave superconductors. In a d + iswave superconductor, there are no bound states for potential impurities, while a magnetic impurity produces one pair of bound states, with a zeroenergy level crossing at a finite scattering strength. On the other hand, a d + id′wave symmetry always gives rise to two pairs of bound states and only produce a reachable zeroenergy level crossing if the normal state has a strong particlehole asymmetry.
Introduction
It is wellestablished that a single magnetic impurity induces socalled YuShibaRusinov (YSR) bound states inside the energy gap of conventional swave superconductors^{1,2,3}. In systems with spinorbit coupling, such intragap bound states have recently been proposed to give rise to emergent Majorana zeroenergy modes at the endpoints of chains of magnetic impurities^{4,5,6}. This has lead to a surge of interest in impurityinduced bound states in superconductors, with both experimental and theoretical work focusing on properties ranging from high angular momentum scattering and complex internal structure of the impurities to quantum phase transitions and spontaneous current generation, as well as many other aspects^{7,8,9,10,11,12,13,14,15,16}. In addition, the physical properties of an impurity give valuable information about the bulk itself and can thus be a decisive probe for establishing the properties of the bulk superconducting state^{17,18,19}.
One severely limiting factor in all these studies is the low superconducting transition temperature accompanied by the small energy gap associated with conventional swave superconductors. The cuprate superconductors with their much high transition temperatures would here be a tantalizing option, if it were not for their dwave order parameter symmetry which enforces a nodal energy spectrum^{20}. The lowenergy nodal quasiparticles prevent impurity bound states and thus a pure dwave superconductor can only host virtual bound states^{7,21}. However, in small islands of YBa_{2}Cu_{3}O_{7−δ} a fully gapped spectrum has recently been discovered and attributed to the existence of subdominant order parameters, with the superconducting symmetry likely being either wave (d + is) or chiral wave (d + id′)^{22,23}. Both of these subdominant orders produce a hard gap and spontaneously break timereversal symmetry, since the free energy very generally is minimized for subdominant parameters with an overall π/2phase shift relative to the dominant order. The d + id′wave state is also a chiral state with its nontrivial topology classified by a Chern number N = 2^{24}. Evidence also exists that surfaces, especially the (11) surface^{25}, as well as certain impurities^{26} also spontaneously generate a timereversal symmetry breaking superconducting state with either d + is or d + id′wave symmetry.
In this work we establish the properties of both potential and magnetic impurities in these two fully gapped dwave superconductors. More specifically, we investigate the intragap bound states due to potential and magnetic impurities using both an analytic continuum Tmatrix formulation and numerical tightbinding lattice calculations. We show that impurities create entirely different bound states in d + iswave and chiral d + id′wave superconductors, despite both being fully gapped and with a dominant parent dwave state. For a d + iswave superconductor we find that a potential impurity does not induce any bound states, while a magnetic impurity gives rise to a pair of bound states, which behaves very similar to the YSR bound states in conventional swave superconductors. This includes the behaviour of the energy spectrum when tuning the scattering strength U_{mag} of the magnetic impurity. Since the d + iswave state is topologically trivial and with a lowenergy swave gap, this resemblance with a conventional swave superconductor is very plausible. More specifically, we find that the magnetic impurity bound states have a zeroenergy level crossing at a finite critical scattering . We are able to extract an analytical expression for the critical coupling which depends only on the ratio of the dominant dwave to the subdominant swave order parameter. Moreover, through selfconsistent tightbinding calculations we find a firstorder quantum phase transition at , which also induces a local πphase shift for the swave component of the order parameter.
For the chiral d + id′wave state we find a very different behaviour. Here both potential and magnetic impurities induce two pairs of bound states. For superconductors with a particlehole symmetric normal state, the bound states are twofold degenerate and there is no level crossings for any finite coupling. Instead, it is only in the unitary scattering limit (U^{c} → ∞) that the bound states approach the middle of the gap. Doping the normal state away from particlehole symmetry, the degeneracy is lifted for a magnetic impurity but not for a potential impurity. Moreover, for finite doping there is now a zeroenergy level crossing, but for low doping it occurs only at very large scattering strengths. Selfconsistent calculations for a single impurity in a d + id′wave superconductor finds a firstorder phase transition at the level crossing, but no local phase shifts in either the dominant or subdominant order parameters. Considering that recent experiments have demonstrated access to adjustable magnetic scattering strengths U_{mag}^{9}, magnetic impurities offer a very intriguing way to clearly distinguish between the chiral d + id′wave state and the likewise timereversal symmetry breaking but topologically trivial d + iswave state.
These results have several important consequences. In spite of impurities only generating localized imperfections to the lattice structure, we find that they are very suitable for probing and differentiating the symmetry of the superconducting state. Our results show that the symmetry of the order parameter can even be probed by simply counting the number of induced subgap states. This is in sharp contrast to the virtual bound states in the pure dwave state, which persist even above the superconducting transition temperature and consequently, can not be considered to be a good probe of the symmetry of the superconducting state^{27}. Moreover, understanding the behaviour of single impurities is the inevitable first step for studying impurity wires or even larger impurity domains, with the aim of constructing nontrivial topological phases with Majorana fermion boundary modes. Here fully gapped dwave superconductors offer a tantalizing alternative due to potentially much higher superconducting transition temperatures. Our results show that magnetic impurities in both d + is and d + id′wave superconductors are interesting systems in this regard, as they generate zeroenergy impurity states with no band degeneracies.
Results
Analytic Tmatrix calculations
Impurityinduced bound states only exist in dwave superconductors with a fully gapped energy spectrum. Introducing a subdominant superconducting order parameter will achieve this, since it very generally align with a complex π/2 phase relative to the dominant dwave state. Here we consider a twodimensional (2D) wave superconducting state with a complex subdominant order parameter such that the order parameter takes the form of . More specifically, we treat the two most likely candidates:  and wave symmetries. In order to achieve a good analytical understanding of the effect of impurities we here first perform Tmatrix calculations. Later, we confirm and extend these results by also performing selfconsistent tightbinding lattice calculations.
The Hamiltonian in the presence of a single impurity (magnetic and/or potential) reads (using ħ = 1)
where we use the Nambu space spinor . Here U_{pot} and U_{mag} are the potential and magnetic scattering matrix elements induced by the impurity, while the kinetic energy is ξ(k). The exact form of ξ(k) is unimportant as we can linearise the spectrum around the Fermi level, setting . Most contributions to the superconducting state come from quasiparticles very near the Fermi level. This allows us to linearise the spectrum, sum only over states close to the Fermi level, and assume no notable dependence on k for the order parameter. The dominant order parameter is Δ_{1}(k), while Δ_{2}(k) represents the subdominant order parameter. We assume that the magnetic moment is large enough to ignore quantum fluctuations and thus we treat the impurity as a classical spin. The local moment of the impurity is directed along the z easy axis, but it is straightforward to show that the results are not affected by this assumption, since the electrons pair in the spinsinglet channel. This is different from magnetic impurities in pwave superconductors, where the direction of the impurity generally affects the bound state energy^{14}. The matrices τ_{i} and σ_{i} are the Pauli matrices acting in particlehole and spin spaces, respectively, while τ_{0} and σ_{0} are unit matrices. The bare Green’s function for the superconductor is
with the energy spectrum . The Green’s function in the presence of a single impurity then reads , with the Tmatrix ^{28}. Therefore, finding the roots of the denominator of the Tmatrix gives the energy of the impurityinduced bound states. For these analytical calculations we assume that the order parameter does not notably depend on the magnitude of the wave vector k, but only on its direction , but note that this assumption is not needed in the numerical lattice calculations.
Let us first consider a fully particlehole symmetric spectrum for the normal state, which imposes . To access the Tmatrix denominator the summation over the bare Green’s function is needed, , where we have defined F_{i}(ω) as
Here ρ = k_{F}/(2πν_{F}) is the density of states of the 2D free electron gas at the Fermi level. The above result is for a particlehole symmetric normal state, but this symmetry is often broken in reality. We use the chemical potential μ to measure the degree of particlehole symmetry breaking in the energy spectrum, thus leaving the case μ = 0 to represent full particlehole symmetry. Considering a small deviation from particlehole symmetry, such that where is the energy integration cutoff, the summation of the bare Green’s function also contains the term F_{3}τ_{3}σ_{0}. Up to first order in we find , while it is straightforward to show that in this limit F_{0}, F_{1}, and F_{2} remain unchanged from the particlehole symmetric case.
d + iswave state
First, we consider the state where the order parameter is of the form . In this case F_{1}(ω) = 0, due to the periodicity of the cosine function and the subdominant order parameter Δ_{s} not depending on ϕ. Then, in the limit of μ = 0, the bound states can be found as the solutions to
Since we are interested in real bound states, we only look for solutions . Further, to make sure that the bound states are isolated from the continuum spectrum of the superconducting quasiparticles, we limit ourselves to solutions that lie inside the gap, i.e. . For a purely potential impurity we find no bound states, while for a purely magnetic impurity there is one pair of solutions that do not depend on the sign of U_{mag}. In order to find the bound state energies for a magnetic impurity we rephrase F_{0}(ω) and F_{2}(ω) in terms of the complete elliptic integral of the first kind K^{29}, resulting in
where we have defined , , and . For , we naturally recover the YSR bound states found in a conventional swave superconductor: ^{1,2,3}. The bound state spectrum for a general d + iswave superconductor comes as the solution of Eq. (5) and is illustrated in Fig. 1(a). As the figure shows, there is one pair of intragap bound states appearing at the gap edges for a weak magnetic impurity and moving toward the middle of the gap, such that at a critical magnetic scattering a level crossing occurs. This behaviour is qualitatively similar to the YSR bound pair in a conventional swave superconductor, where the level crossing signals a quantum phase transition between two different ground states. For a magnetic impurity stronger than , the ground state will have one unpaired electron because it is energetically favoured by the system to break a Cooperpair to partially screen the impurity^{9,30}. The same quantum phase transition takes place also in the d + iswave superconductor. Interestingly, for a d + iswave superconductor, the critical coupling depends only on and not on Δ_{d} and Δ_{s} separately. By setting in Eq. (5), we find the analytically exact expression . Assuming , the critical coupling reads , which, as seen in Fig. 1(b), only deviates at small from the exact result. This clearly illustrates that, in order to find zero modes, a larger moment and/or coupling is needed when the subdominant swave parameter is small compared to the dwave order.
If we now break the electronhole symmetry of the normal state, i.e. assume , the bound states are instead found as the solution of
It is clear that also in this case, for a purely potential impurity there are no real roots and consequently, potential impurities never induce any bound states in a d + iswave superconductor. Moreover, the modifications of the bound state spectrum induced by a magnetic impurity is of the order and the change to the critical coupling is also of the same order of magnitude and thus negligible. In fact, critical coupling in this approximation reads .
d + id′wave state
Next, we turn to the impurity bound state formation inside the gap of a chiral dwave or wave superconductor. Here the order parameter is . Because of the periodicity of cos(2ϕ) and sin(2ϕ), all offdiagonal terms in the summation of the bare Green’s function, i.e. F_{i} with , vanish. Left for a particlehole symmetric normal state spectrum (μ = 0) is then only
In this case the bound states are found as the solutions to
Very interestingly, pure potential or pure magnetic impurities in a chiral dwave superconductor lead to exactly the same pairs of intragap bound states, these states are explicitly shown in Fig. 2. In earlier work it has been claimed that the number of bound states for a potential impurity is only two^{8}. However, according to our results, these bound states are doubly degenerate, and there are in total actually four bound states. This statement is valid for both potential and magnetic impurities. Staying at μ = 0 we find that for a potential impurity the negative energy branch (those occupied at zero temperature) consist of one spinup and one spindown state, and there is thus a Kramers degeneracy of the states. However, for a magnetic impurity the bound states with negative energy are both spindown quasiparticles. For an impurity with both potential and magnetic scattering effects on the charge carriers (U_{pot} and U_{mag} both nonzero), four nondegenerate bound states are generally present in a chiral dwave superconductor, such that the twofold degeneracy in Fig. 2 is lifted.
If the particlehole symmetry of the normal states is broken, here by setting , then the summation over the bare Green’s function also contains F_{3}τ_{3}σ_{0}, where, up to first order in , . The influence of this new term on the bound states is much more pronounced for the chiral dwave state compared to the d + iswave state. For a potential impurity the bound states energies are still twofold degenerate, but a nonzero μ shifts the energy of the bound states in the unitary limit, as is illustrated in Fig. 3(a). In fact, for the electron doped case, μ > 0, a level crossing now appears for repulsive impurity scattering (U_{pot} > 0), while for a hole doped system, μ < 0, the level crossing instead occurs for an attractive impurity (U_{pot} < 0). Thus doping can be used as a simple means to control the level crossing for potential impurities. A local chemical potential induced by a tunneling probe could even offer insitu tunability of the level crossing^{11}.
For a magnetic impurity the bound state degeneracy is lifted for finite μ, as illustrated in Fig. 3(b) for positive U_{mag}. In this case, one pair of bound states move away from the middle of the gap (thick red lines). For these states no zero modes are thus expected even in the unitary scattering limit for any positive scattering U_{mag} > 0. The other pair of bound states (thin blue lines) move toward the middle of the gap and consequently a level crossing appears at some . The bound state energy spectrum is symmetric with respect to , and thus there is another level crossing at the corresponding negative magnetic scattering. In addition to being symmetric with respect to the sign of U_{mag}, the bound states also do not depend on the sign of μ. Remarkably, the dimensionless critical coupling for reaching a zeroenergy state for both magnetic and potential scatterers is the same . As seen, this critical coupling can be decreased by increasing μ.
For the purpose of the forthcoming selfconsistent numerical calculation, we mention already here that even at halffilling, the energy degeneracy can still be lifted by adding a small amount of extended swave superconductivity to the chiral dwave. More precisely, if the order parameter takes the form , where Δ_{s′} is kindependent, the energy degeneracy for magnetic impurity bound states is lifted, while the bound states remain degenerate for a potential impurity. Therefore, the degeneracy of the bound states in the presence of a magnetic impurity in a chiral dwave superconductor is very fragile and can easily be lifted.
Numeric tightbinding lattice calculation
We now turn to discuss the results obtained from tightbinding lattice calculations. In all calculations we use a generic finitesize square lattice in which we consider a single impurity located at the middle site. Again we consider both a d + is and chiral d + id′wave superconductor. The effective Hamiltonian for the 2D superconducting host with an impurity with both potential and magnetic scattering elements located at R is refs 23,31
Here i = (i_{x}, i_{y}) represents a site in the square lattice, with the lattice spacing a set to be 1. The dominant dwave order exists on nearest neighbour bonds, while the subdominant swave order is an onsite parameter and the d′wave order reigns on next nearest neighbour bonds. For the selfconsistent calculations (see below) we do not a priori assume any symmetries or conditions for any of these three order parameters. However, in calculations with constant order parameters throughout the sample, i.e. nonselfconsistent calculations, we enforce the dwave order by setting for all sites, i.e. the order parameter on ydirected bonds is equal in magnitude but with opposite sign compared to the order parameter on xdirected bonds. Likewise, the d′wave state has opposite signs on bonds in the ±(x + y) direction compared to bonds in the ±(x − y) direction. We also enforce the subdominant order parameter (s or d′) to be purely imaginary in the nonselfconsistent calculations.
We solve Eq. (9) by performing a Chebyshev polynomial expansion of the corresponding Green’s function^{32,33,34}. This method allows us to investigate lattices with very large number of lattice points because the amount of necessary computational resources grow only linearly with the size of the system, far outperforming regular diagonalization. More specifically, we calculate the Green’s function for the impurity site and its closest neighbours. The imaginary part of Green’s function gives the local density of states (LDOS) and the bound states are easily identified as sharp peaks inside the energy gap at the impurity site and also its neighbouring sites.
We find that the tightbinding calculations fit exactly to the analytical Tmatrix results. For instance, in Fig. 4 we show the bound state spectrum generated by a magnetic impurity in a chiral dwave superconductor at finite doping, for both the numeric tightbinding method and an analytic Tmatrix calculation. In order to be able to do this comparison, we evaluate the summations appearing in the Tmatrix formalism for a discrete mesh over the first Brillouin zone of the square lattice using the same form of the kinetic energy. The main difference between the continuum and discrete calculation is that in the former we neglect the radial dependence of the order parameter in reciprocal space while in the latter, we keep both the radial and angular dependence. Therefore, the discrete summation over the full tightbinding model gives us a more relevant solution to compare with the finitesize numerical calculations. We also find an excellent agreement in the unitary scattering limit (U_{mag} → ∞), which in the tightbinding lattice calculation can be implemented by simply removing the impurity site, thus creating a vacancy.
Selfconsistent results
Above we simply assumed constant order parameters and enforced the correct condensate symmetries. Now we allow the condensate to appropriately respond to the impurity through a proper selfconsistent calculation. Since the superconductor symmetry is important for the properties of the bound states, this is the most accurate way to ensure a correct solution. In these selfconsistent calculations we only assume a finite and constant pair potential V in each pairing channel and calculate the order parameter(s) explicitly everywhere in the lattice. For a dwave state we use the selfconsistent condition , where i, j are nearest neighbour sites. In the selfconsistent calculation we start by guessing a value for Δ_{d} on each bond, solve Eq. (9), calculate a new Δ_{d} on each bond using the selfconsistent condition, and repeat until Δ_{d} does not change between two subsequent iterations. For the d + iswave state we also assume a finite V_{s} in addition to V_{d} and calculate separately selfconsistently. For the d + id′wave state is likewise finite, such that , where i, j are next nearest neighbour sites.
We take the initial guess for the subdominant order parameter to be purely imaginary but through the selfconsistency loop it is free to acquire a real component as well. Likewise, we emphasize that Δ_{d} on x and ydirected bonds are treated fully independent and the same applies to the d′wave state. Thus, we have not a priori assumed any symmetry for any of the pairing states. In the calculations we use V_{d}/t = 1.7, V_{s}/t = 1.7 for d + iswave state and V_{d}/t = 1.8, V_{d′}/t = 1.7 for chiral dwave state, but the results are not sensitive to these particular values. We also set μ/t = −1 for the most general case and to avoid the van Hove singularity at halffilling. We mainly use a 51 × 51 lattice, with similar results obtained with a 31 × 31 lattice, which guarantees that the results are not sensitive to the lattice size.
Using the selfconsistently calculated order parameters, we extract the LDOS at and close to the impurity site for both d + is and d + id′wave states in the presence of either magnetic or potential impurities. The selfconsistent tightbinding lattice calculations reveal that the results obtained for fixed order parameters are still largely valid. However, important effects appear around the critical scattering strength in the selfconsistent calculations. Starting with the d + iswave state, the selfconsistent results show that the intragap localized bound states from a magnetic impurity behave largely in a similar way to their nonselfconsistent counterpart as seen in Fig. 5(a). The main discrepancy is close to the critical scattering . As seen in the inset in Fig. 5(a), the energy of the bound states does not evolve smoothly at the transition point and there is instead a clear kink at . This is a finger print of a firstorder quantum phase transition. A similar effect has been found in a pure swave superconductor^{30}.
The order parameter at the impurity site shed more light on this transition as can be seen in Fig. 5(b), where sudden changes near the critical coupling are visible. Selfconsistently calculating the order parameters, there is in addition to the onsite swave and wave order parameters, also an extended swave order parameter. This extended swave order resides on the nearest neighbour bonds and appears only very close to the impurity. It is thus a direct consequence of the impurity weakening the dwave character in favour of the more disorderrobust swave symmetry. At the quantum critical point this extended swave state even becomes the dominant order parameter but notably only at the impurity site, farther away the dwave order parameter is still dominant. In addition, both the swave and extended swave order parameters develop a πphase shift on the impurity site across the critical coupling. This is in line with previous calculations for pure swave superconductors where the swave state undergoes a local πshift^{15,30,35}. However, note that we find that the phase of the dominant dwave state stays constant.
For the case of a magnetic impurity in a chiral dwave superconductor, there is only a level crossing for one pair of bound states as seen in Fig. 6(a), as also found in the nonselfconsistent tightbinding and Tmatrix calculations. The selfconsistent solution, however, shows a kink close to the critical scattering, signaling a firstorder phase transition as in the d + is case. Considering the order parameter, the selfconsistent calculation reveals that the dominant stays the dominant order parameter even beyond the critical scattering and the subdominant state is also always the d_{xy}wave state, as seen in in Fig. 6(b). In this case only very weak extended swave components appear on nearest and nextnearest neighbour bonds, defined here as Δ_{s} and Δ_{s′}, respectively. Interestingly, this means that for a chiral dwave superconductor, the impurity does not disturb the dominant dwave orders nearly as much as in the d + iswave case. Despite the smallness of the swave components generated in the selfconsistent calculations, we find that they are still responsible for lifting of the degeneracy of the impurity bands in the halffilled lattice case (μ = 0).
Conclusions
In this work we have investigated impurityinduced bound states in fully gapped dwave superconductors. The main results are summarized in Table 1. As illustrated by this table, we have shown that an impurity, whether magnetic or potential, induces two pairs of intragap bound states in a chiral d + id′wave superconductor, while for a d + iswave superconductor, there is only one (zero) pair of bound states for a magnetic (potential) impurity. As a result, the number of intragap bound states becomes a powerful means for establishing the symmetry of the superconducting state in a fully gapped dwave superconductor, such as that recently established in cuprate nanoislands or at certain cuprate surfaces^{22,25}. With potential impurities also tunable by localized potential scattering from a tunneling probe^{11}, there even exist possibilities to study the evolution of the bound states of a particular impurity for a range of effective impurity strengths.
Another important difference between d + iswave and chiral dwave superconductors is the behaviour of the zeroenergy level crossings for the impurityinduced states. For a d + iswave superconductor, increasing the magnetic scattering strength U_{mag} leads to a level crossing of the bound states, which means there always exists a critical coupling , separating two distinct ground states. However, for a chiral d + id′wave superconductor there is no level crossing for a particlehole symmetric normal band structure (here indicated by μ = 0) at any finite scattering strength, for either potential or magnetic impurities. Only at significant doping away from μ = 0 there is a zeroenergy level crossing at an experimentally achievable scattering strength. It is also important to notice that the impurity bound states in a chiral d + id′wave superconductor are often twofold degenerate for both potential and magnetic impurities. For a magnetic impurity the degeneracy is lifted by either a finite μ or by everpresent subdominant extended swave components, as we find in our selfconsistent calculations, nonetheless, there are often two nearly degenerate states.
The systems with zero energy states and no impurity band degeneracies, i.e. magnetic impurities in both d + is and d + id′wave superconductors, have the potential for hosting Majorana fermions. However, in the case of a single impurity, the two zero energy states at occupy the same point in space (the impurity site) and thus these Majorana fermions will simply combine to form a regular electron quasiparticle excitation. This is why 1D magnetic impurity wires or other higher dimensional objects are needed to spatially separate the Majorana fermions and harness their Majorana character. Moreover, at least a small spinorbit coupling has been shown to be essential in both swave^{4,5,6,36} and d + id′wave^{31,37} superconductors in order to generate a nontrivial topological phase. In the chiral d + id′wave superconductor the near degeneracy of the impurity bands however possess additional complications for Majorana fermions in an impurity wire. An even number of positive (or negative) near zeroenergy states in the single impurity limit will result in two putative Majorana end modes for a wire, which then hybridize and split off from zero energy, losing their Majorana character. Thus, in terms of the potential for generating Majorana modes, our results directly shows that only magnetic impurity wires in a d + iswave superconductor or in a heavily doped d + id′wave state are promising systems.
Finally, we emphasize that our selfconsistent calculations confirm our analytical results where we have assumed constant order parameters uninfluenced by the impurities. In addition, the selfconsistent calculations shed more light on the nature of the zeroenergy level crossings and show that for both d + is and d + id′wave superconductors, these are firstorder quantum phase transitions, with clear discontinuities in both energy levels and order parameters. For the d + iswave state we even find a local πshift at the phase transition for all subdominant order parameters, consistent with the behaviour in conventional swave superconductors^{30,35}. For the d + id′wave superconductor we, however, do not find any πshifts.
Additional Information
How to cite this article: Mashkoori, M. et al. Impurity bound states in fully gapped dwave superconductors with subdominant order parameters. Sci. Rep. 7, 44107; doi: 10.1038/srep44107 (2017).
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 1
Yu, L. Bound state in superconductors with paramagnetic impurities. Acta Phys. Sin. 21, 75–91 (1965).
 2
Shiba, H. Classical spins in superconductors. Prog. Theor. Phys. 40, 435–451 (1968).
 3
Rusinov, A. I. On the theory of gapless superconductivity in alloys containing paramagnetic impurities. Sov. Phys. JETP 29, 1101–1106 (1969).
 4
Mourik, V. et al. Signatures of Majorana fermions in hybrid superconductorsemiconductor nanowire devices. Science 336, 1003–1007 (2012).
 5
NadjPerge, S. et al. Observation of Majorana fermions in ferromagnetic atomic chains on a superconductor. Science 346, 602–607 (2014).
 6
Ruby, M. et al. End states and subgap structure in proximitycoupled chains of magnetic adatoms. Phys. Rev. Lett. 115, 197204 (2015).
 7
Balatsky, A. V., Vekhter, I. & Zhu, J.X. Impurityinduced states in conventional and unconventional superconductors. Rev. Mod. Phys. 78, 373–433 (2006).
 8
Wang, F., Liu, Q., Ma, T. & Jiang, X. Impurityinduced bound states in superconductors with topological order. J. Phys. Condens. Matter 24, 455701 (2012).
 9
Hatter, N., Heinrich, B. W., Ruby, M., Pascual, J. I. & Franke, K. J. Magnetic anisotropy in Shiba bound states across a quantum phase transition. Nat. Commun. 6, 8988 (2015).
 10
Ruby, M., Peng, Y., von Oppen, F., Heinrich, B. W. & Franke, K. J. Orbital picture of YuShibaRusinov multiplets. Phys. Rev. Lett. 117, 186801 (2016).
 11
Sau, J. D. & Demler, E. Bound states at impurities as a probe of topological superconductivity in nanowires. Phys. Rev. B 88, 205402 (2013).
 12
Pershoguba, S. S., Björnson, K., BlackSchaffer, A. M. & Balatsky, A. V. Currents induced by magnetic impurities in superconductors with spinorbit coupling. Phys. Rev. Lett. 115, 116602 (2015).
 13
Björnson, K., Pershoguba, S. S., Balatsky, A. V. & BlackSchaffer, A. M. Spinpolarized edge currents and majorana fermions in one and twodimensional topological superconductors. Phys. Rev. B 92, 214501 (2015).
 14
Kim, Y., Zhang, J., Rossi, E. & Lutchyn, R. M. Impurityinduced bound states in superconductors with spinorbit coupling. Phys. Rev. Lett. 114, 236804 (2015).
 15
Björnson, K., Balatsky, A. V. & BlackSchaffer, A. M. Superconducting order parameter πphase shift in magnetic impurity wires. arXiv 1609.07626 (2016).
 16
Vernier, E., Pekker, D., Zwierlein, M. W. & Demler, E. Bound states of a localized magnetic impurity in a superfluid of paired ultracold fermions. Phys. Rev. A 83, 033619 (2011).
 17
Slager, R.J., Rademaker, L., Zaanen, J. & Balents, L. Impurity bound states and Greens function zeroes as local signatures of topology. Phys. Rev. B 92, 085126 (2015).
 18
Kaladzhyan, V., Bena, C. & Simon, P. Characterizing pwave superconductivity using the spin structure of shiba states. Phys. Rev. B 93, 214514 (2016).
 19
Kaladzhyan, V., Bena, C. & Simon, P. Asymptotic behavior of impurityinduced bound states in lowdimensional topological superconductors. J. Phys. Condens. Matter 28, 485701 (2016).
 20
Tsuei, C. C. & Kirtley, J. R. Pairing symmetry in cuprate superconductors. Rev. Mod. Phys. 72, 969–1016 (2000).
 21
Balatsky, A. V., Salkola, M. I. & Rosengren, A. Impurityinduced virtual bound states in dwave superconductors. Phys. Rev. B 51, 15547–15551 (1995).
 22
Gustafsson, D. et al. Fully gapped superconductivity in a nanometresize YBa2Cu3O7−δ island enhanced by a magnetic field. Nat. Nanotechnol. 8, 25–30 (2013).
 23
BlackSchaffer, A. M., Golubev, D. S., Bauch, T., Lombardi, F. & Fogelström, M. Model evidence of a superconducting state with a full energy gap in small cuprate islands. Phys. Rev. Lett. 110, 197001 (2013).
 24
Volovik, G. E. On edge states in superconductors with time inversion symmetry breaking. JETP. Lett. 66, 522–527 (1997).
 25
Elhalel, G., Beck, R., Leibovitch, G. & Deutscher, G. Transition from a mixed to a pure dwave symmetry in superconducting optimally doped yba2cu3o7−x thin films under applied fields. Phys. Rev. Lett. 98, 137002 (2007).
 26
Balatsky, A. V. Spontaneous time reversal and parity breaking in a wave superconductor with magnetic impurities. Phys. Rev. Lett. 80, 1972–1975 (1998).
 27
Chatterjee, K. et al. Visualization of the interplay between hightemperature superconductivity, the pseudogap and impurity resonances. Nature Phys. 4, 108–111 (2008).
 28
Mahan, G. D. ManyParticle Physics (Springer Science & Business Media, 2000).
 29
Abramowitz, M. & Stegun, I. A. Handbook of Mathematical Functions (Courier Corporation, 1972).
 30
Salkola, M. I., Balatsky, A. V. & Schrieffer, J. R. Spectral properties of quasiparticle excitations induced by magnetic moments in superconductors. Phys. Rev. B 55, 12648–12661 (1997).
 31
Sato, M., Takahashi, Y. & Fujimoto, S. NonAbelian topological orders and Majorana fermions in spinsinglet superconductors. Phys. Rev. B 82, 134521 (2010).
 32
Weiße, A., Wellein, G., Alvermann, A. & Fehske, H. The kernel polynomial method. Rev. Mod. Phys. 78, 275–306 (2006).
 33
Covaci, L., Peeters, F. M. & Berciu, M. Efficient numerical approach to inhomogeneous superconductivity: the ChebyshevBogoliubov–de Gennes method. Phys. Rev. Lett. 105, 167006 (2010).
 34
Björnson, K. & BlackSchaffer, A. M. Majorana fermions at odd junctions in a wire network of ferromagnetic impurities. Phys. Rev. B 94, 100501(R) (2016).
 35
Flatté, M. E. & Byers, J. M. Local electronic structure of a single magnetic impurity in a superconductor. Phys. Rev. Lett. 78, 3761 (1997).
 36
NadjPerge, S., Drozdov, I. K., Bernevig, B. A. & Yazdani, A. Proposal for realizing majorana fermions in chains of magnetic atoms on a superconductor. Phys. Rev. B 88, 020407 (2013).
 37
BlackSchaffer, A. M. Edge properties and majorana fermions in the proposed chiral dwave superconducting state of doped graphene. Phys. Rev. Lett. 109, 197001 (2012).
Acknowledgements
This work was supported by the Swedish Research Council (Vetenskapsrådet), the Göran Gustafsson Foundation, the Swedish Foundation for Strategic Research (SSF), and the Wallenberg Academy Fellows program through the Knut and Alice Wallenberg Foundation. The computations were performed on resources provided by SNIC at LUNARC.
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M.M. performed all the analytical calculations and together with K.B. the numerical lattice calculations. A.B.S. conceived the idea. M.M. and A.B.S. analysed the results and wrote the manuscript with input from K.B.
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Correspondence to Mahdi Mashkoori.
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Mashkoori, M., Björnson, K. & BlackSchaffer, A. Impurity bound states in fully gapped dwave superconductors with subdominant order parameters. Sci Rep 7, 44107 (2017) doi:10.1038/srep44107
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