RKKY interaction in Layered Superconductors with Anisotropic Pairing

The RKKY interaction between rare-earth (RE) ions in high-Tc superconductors is considered at T ≪ Tc. It is shown that this interaction consists of two terms: conventional oscillating one and the positive term, which is proportional to the gap function and decreases in the 2D case inversely proportional to the distance. In the antiferromagnetic state of the RE subsystem this positive interaction gives rise for frustrations which diminishes the Neel temperature. In the case of strongly anisotropic gap function this frustration produces two different values of the effective nearest neighbor exchange coupling between RE ions along the a and b. This anisotropy has been established experimentally in Ref. [6–8]. Typeset using REVTEX 1 In many High-Tc superconductors substitution of some ions by rare-earth elements (RE) leads to the low-temperature antiferromagnetism in the RE subsystem with TN of order of 1 K and without sizeable change of the superconducting transition temperature Tc. The mostly investigated compounds are YBa2Cu3O7−x (1 : 2 : 3) and YBa2Cu4O8 (1 : 2 : 4) with yttrium being substituted by different RE ions (see [1–5] and references therein). The magnetic dipolar, superexchange and RKKY interactions has been attracted for explanation of this antiferromagnetism in the RE subsystem. The detailed calculations, however, have been carried out only for dipolar interaction. Corresponding results have been presented in [4] and [5] for (1 : 2 : 3) and (1 : 2 : 4) compounds, respectively. In particular, it was pointed out in [4] that there is no correlation between the ground state energies calculated in the dipolar approximation and observed Neel temperatures. Hence the dipolar interaction alone cannot explain the magnetic properties of the considered systems. Moreover, a striking feature of the magnetic interaction between the RE ions is the very large difference between effective nearest neighbor exchange interactions along crystallographic a and b directions. Indeed, the specific heat data near TN for 1 : 2 : 3 system with Sm and Nd ions was very well fitted [6,7] using 2D Ising model with strongly different exchange constants J1 and J2. The ratio J1/J2 was found to be ≃ 11 and 50 for Sm and Nd samples respectively [6]. Recent measurements for NdBa2Cu3O7−δ [7] confirmed this result and revealed that the ratio J1/J2 depends on doping and increases rapidly with δ. As it was emphasized in Ref. [7] such a behaviour indicates that chains cannot be the direct reason of this anisotropy, since the structural anisotropy decreases with depleting of the chains. More direct confirmation of this anisotropy has been provided [8] by the inelastic neutron scattering in HoBa2Cu3O7δ. The values of Ja = (0.0005± 0.0002)meV and Jb = −(0.0028± 0.0008)meV were reported there. The orthorombic distortion in the considered compounds is very small [(a − b)/a ∼ 10 − 10] and weakly depends on δ. Therefore, it cannot explain the anisotropy of superexchange and dipolar interactions. At the same time it is well known that the hole system responsible for the superconductivity possesses pronounced anisotropy in the a-b 2 plane which strongly depends on the oxygen content δ. This anisotropy was established for the Fermi surface [9,10] and is apparently presented in the superconducting gap function ∆k [10,11]. Moreover, the anisotropy of ∆k should persist in any system with anisotropic Fermi surface. As a result we may suggest that the above exchange anisotropy of the RE subsystem stems from the interaction between the RE ions and holes in CuO2 planes. To date we have no direct information about interaction between the RE ions and CuO2 planes in metal state. Meanwhile for antiferromagnetic compound HoBa2Cu3O6.13 this interaction has been evaluated on the base of experimentally observed splitting of the Holmium Γ5 doublet. This splitting appears due to the interaction of Ho ion with spinwaves in CuO2 planes [12]. It was found that exchange integral between Ho total angular momentum and Cu spin is equal to 2.4 meV. Therefore we see that this interaction is not so weak and the same should be valid for the metal state, as well. The RE-hole interaction gives rise to the RKKY interaction between the RE ions, which should be anisotropic due to the anisotropies of the Fermi surface and of the gap function, mentioned above. In this paper we consider the problem of anisotropy of the RKKY interaction in superconducting state which stems from the anisotropy of the gap function. The effects arising from the non-circular shape of Fermi surface will be considered elsewhere. We obtain the following principal result. At T ≪ Tc the 2D RKKY interaction is a sum of two terms. The first one is the conventional oscillating contribution and the second term is positive and decreases as R. It is proportional to |∆R|/EF where ∆R is the gap function along R direction. Both terms are screened at distances of order of VF/|∆R|. The second term should strongly frustrate the AF ground state due to its long-range behavior. If the gap ∆R has not square symmetry this frustration is different in the a and b directions. As a result the effective exchange parameters Ja and Jb should be different, too. This paper is organized as follows. In Sec.II we evaluate the 2D RKKY interaction for kFR ≫ 1 in the case of anisotropic gap function and in Sec.III we discuss the physical consequences of these results.