Spin and orbital effects of Cooper pairs coupled to a single magnetic impurity

Abstract

The Kondo effect strongly depends on spin and orbital degrees of freedom of unconventional superconductivity. We focus on the Kondo effect in uniformly gapped superconducting systems ͑the two-dimensional p x ϩip y -wave and d x 2 Ϫy 2ϩ id xy -wave superconductors here͒ to compare the magnetic properties of the spintriplet and spin-singlet Cooper pairs. The difference appears when both of the paired electrons couple to a local spin directly. For the p x ϩip y wave, the ground state is always a spin doublet for a S imp ϭ1/2 local spin, and it is always a spin singlet for S imp ϭ1. The latter is due to uniaxial spin anisotropy of the triplet Cooper pair. For the d x 2 Ϫy 2ϩ id xy wave, the interchange of ground states occurs, which resembles a competition between the Kondo effect and the superconducting energy gap in s-wave superconductors. Thus the internal degrees of freedom of Cooper pairs give a variety to the Kondo effect. 65 094434-1 impurity ͑2.26͒where S Ϯ ϭS x ϮiS y , and the orbitally dependent exchange couplings are given by ͑2.27͒We can find that uniaxial anisotropy is introduced to the spin of the total system through the couplings J Ј z J Ј Ќ for finite values of J 0 and J 1 . On the other hand, for the d x 2 Ϫy 2ϩ d xy wave, the transformation corresponding to Eq. ͑2.25͒ is given byfor the lϭ2 electrons. The H 0 for the lϭ0 and lϭ2 electrons is given by the same form with Eq. ͑2.26͒ except for the exchange couplingswhere J 2 is a coupling constant between the impurity and l ϭ2 electrons. In this case, the total spin keeps its isotropy: J Ј z ϭJ Ј Ќ is satisfied for all values of J 0 and J 2 . Thus, by introducing the l 0 scattering into the exchange interaction, we can see the difference between the triplet and singlet pairings which comes from the phases of the wave functions of the l 0 electrons. III. RESULTS A. Short-range scattering impurityFirst we summarize main results for a case of short-range scattering impurity with S imp ϭ1/2, the details of which were discussed in our previous papers. 17,18 Both J 0 and ⌬ are SPIN AND ORBITAL EFFECTS OF COOPER PAIRS . . . PHYSICAL REVIEW B 65 094434 094434-3

Single Impurity Effects in Multiband Superconductors with Different Sign Order Parameters

“…We examine the magnetic impurity problem as in the d x 2 −y 2 + id xy -wave case. 26) In Fig. On the other hand, for a small T K /∆, the doubly degenerate bound (spin-singlet) state appears in the superconducting energy gap, like a nonmagnetic impurity in §2.…”

“…It was found that low energy states appear with four-fold symmetry near the impurity. [18][19][20][21][22] About a single magnetic impurity, we previously investigated a quantum spin in unconventional superconductors using the NRG method and focused on a fully gapped chiral superconductor expressed by p x ± ip y -wave or d x 2 −y 2 ± id xy -wave type order parameters, [23][24][25][26][27] where orbital effect of the Cooper pair plays an important role.…”

Effects of magnetic anisotropy on the subgap excitations induced by quantum impurities in a superconducting host

“…Lee et al have studied the isotropic high-spin impurities in a side-coupled configuration 48 and the two-level impurity model 49 . The twochannel models have been studied in the context of unconventional superconductors 50 and iron pnictide superconductors 51 . Moca et al have demonstrated the presence of multiple sub-gap states in a multi-orbital model for Mn impurity in MgB 2 52 .…”

Anomaly in the heat capacity of Kondo superconductors

“…A reliable theoretical tool is the numerical renormalisation group (NRG) 11,12,13 . The NRG has been applied, for example, to study the sub-gap bound states in isotropic 14 and anisotropic 15 Kondo model and in the Ander-son model 16 in both conventional and unconventional 17,18 superconductors. Further NRG calculations focused on dynamical properties such as the spectral function 19,20,21,22 and more recently on the transport properties of quantum dots attached to superconducting leads 23,24,25,26,27 .…”