Fig. 4.8. (a) Calculated temperature dependence of the uniform spin susceptibility with U0 = 0.177 eV, compared with 17O Knight shift measurements. The peak is due to thermal activation involving the γ, α and β bands. (b) Calculated frequency dependence of Imχ(Qi, ω) using UQi = 0.345 eV, compared with INS data. Note that in order to obtain good agreement with both quantities (Knight shift and INS data), only hybridized bands, and no spin–orbit coupling, are needed.

and that the out–of–plane response is mainly antiferromagnetic-like. The implications of these findings for triplet Cooper pairing, following the equations derived in Chap. 2, will be discussed in the following section.

4.2 Symmetry Analysis

of the Superconducting Order Parameter

For the analysis of superconductivity in Sr2RuO4, we take into account the fact that experiment indicates non–s–wave symmetry of the order parameter, which strongly suggests spin–fluctuation–mediated Cooper pairing. Then, assuming spin–fluctuation–induced pairing as derived in Chap. 2, it is possible to analyze in general the symmetry of the superconducting state on the basis of the gap equation and our calculated results for the spin excitation spectrum χ(q, ω), which consists of pronounced peaks at wave vectors Qi and

188 4 Results for Sr2RuO4

qi. Thus we demonstrate how triplet pairing arising from ferromagnetic and strong antiferromagnetic excitations is possible.

4.2.1 Triplet Pairing Arising from Spin Excitations

Let us first discuss our results without spin–orbit coupling. Using appropriate symmetry representations [26], we discuss the solutions of the self–consistency equation for the superconducting order parameter (i.e. the gap equation, (2.49)) for the p–, d–, and f –wave symmetries of the order parameter:

The largest eigenvalue λl of (2.49) will yield the superconducting symmetry of ∆l in Sr2RuO4. When we solve (2.49) for the γ band numerically in the first Brillouin zone down to 5 K, we find f –wave symmetry to be slightly favored. As expected, p– and f –wave symmetry Cooper pairing are close in energy (λf = 0.76 > λp = 0.51). However, these calculations were restricted to a single RuO2 plane. A more complete analysis taking into account the coupling between RuO2 planes and an interband U might yield a more definite answer. For example, if the energy di erence between p– and f –wave symmetry in the RuO2 planes is larger than the energy gain for superconductivity resulting from interplane coupling, one may determine the pairing symmetry from the in–plane electronic structure. Note that to obtain a combined energy gain from the antiferromagnetism and Cooper pairing, one would require an order parameter with nodes in the RuO2 plane and possibly also with respect to the c direction. However, this question cannot be answered if 2.49 is solved for the two–dimensional case.

Let us now come back to the question of why triplet pairing is possible as a result of antiferromagnetic spin excitations. To be more precise, the solutions of (2.49) can be characterized by Fig. 4.9, where we present our results for the Fermi surface, the wave vectors Qi and qi defined in Fig. 4.6, and symmetry of the order parameter in Sr2RuO4. The areas with ∆f > 0 and ∆f < 0 are denoted by + and −, respectively. The summation over k in the first BZ is dominated by the contributions due to Qi and a contribution due to the background and qi. Thus, we obtain approximately the following for the γ–band contribution (l = f or p)

where the sum is over all contributions due to Qi and qi. As can be seen from Fig. 4.9, in the case of f –wave symmetry the wave vector qi in (1.14) bridges the same number of portions of the FS with opposite and equal signs.

Fig. 4.9. Illustration of possible triplet pairing due to antiferromagnetic spin excitations in the first BZ. α, β, and γ denote the Fermi surfaces of the corresponding hybridized bands. For simplicity, no spin–orbit coupling has been taken into account. The wave vectors Qi and qi are the wave vectors of the pronounced features in the susceptibility shown in Fig. 4.6. In a weak–coupling approach, these wave vectors of the γ band dominate the pairing potential entering the gap equation (2.49) yielding to (4.5). Thus these wave vectors would then determine the symmetry of the order parameter in a simple way. Also, for f –wave symmetry, the nodes of the real part of the order parameter are shown (dashed lines), together with the regions where the f –wave superconducting gap is positive and negative (+ and −, respectively). Note that for the real part of the p–wave order parameter the node occurs along kx = 0. It is clear from Fig. 4.6 that increasing nesting favors triplet f -wave symmetry. Our illustration shows also that singlet d–wave Cooper pairing is not possible.

Therefore, the second term in (1.14) is approximately zero for triplet pairing. We see from Fig. 4.9 that Qi bridges portions of the FS with equal signs of the superconducting order parameter. Thus, a solution of (1.14) for ∆f is indeed possible.

In the case of p–wave pairing, the real part of the order parameter has a node only along kx = 0 in the kx, ky plane. Then, the wave vectors Qi bridge portions of the FS where Re∆p has either the same or the opposite sign. Regarding the qi contributions, the situation is similar to that in the case of f –wave symmetry. Hence, we expect λp ≤ λf for the eigenvalues, as in the result of the algebraic solution of (2.49). Note that for increasing nesting, Fig. 4.9 also suggests that f –wave symmetry is favored more than p–wave. An eigenvalue analysis of the possible solutions ∆f and ∆p + i∆f should increasingly rule out the latter for stronger nesting.

Also, using similar arguments, we can rule out singlet pairing on the basis of (2.47). In particular, assuming dx2−y2 symmetry for Sr2RuO4, we obtain a change of sign of the order parameter upon crossing the diagonals of the BZ. According to (2.49), wave vectors around Qi connecting areas marked + and

190 4 Results for Sr2RuO4

− contribute constructively to the pairing. Contributions due to qi and the background connecting areas with the same sign subtract from the pairing (see Fig. 4.9, with nodes at the diagonals, for an illustration). Therefore, we find that the four contributions due to qi and the background cancel the pairbuilding contribution due to Qi. As a consequence, we obtain no dx2−y2 -wave symmetry.

Thus, without spin–orbit coupling, as a result only of the topology of the FS and the spin susceptibility, we obtain the strongest pairing for p and f waves and can definitely exclude d–wave pairing. This is in agreement with the numerical solution of (2.49). We have used general arguments which become exact for a given frequency (e.g. ω = 0) if the main pairing occurs at the Fermi level. However, strong feedback e ects of the self–energy on the one–particle spectrum, and mode–mode coupling e ects have been neglected. Nevertheless, our general arguments seem still to be valid in our approximation. Within this approximation, we also find that f –wave symmetry pairing is slightly favored over p–wave symmetry in Sr2RuO4 owing to strong nesting of the bands.

Consideration of Spin–Orbit Coupling

What, now, is the role of spin–orbit coupling in determining the symmetry of the superconducting order parameter? First, the spin–orbit coupling affects the spin dynamics and, as we have shown, induces an anisotropy in the spin subspace. In particular, the two–dimensional IAF fluctuations at Qi = (2π/3, 2π/3) are polarized along the z direction. This simply means that the antiferromagnetic moments associated with these fluctuations are aligned parallel to the z direction, and roughly no nested spectral weight is transferred to the γ band. At the same time, the ferromagnetic fluctuations are in the ab plane. This is in striking contrast to the case discussed above, where when we neglect spin–orbit coupling but include the hybridization between the xy, xz and yz bands, both the ferromagnetic and the IAF fluctuations are located within the ab plane. This would lead to nodes within an RuO2 plane. However, owing to the magnetic anisotropy induced by spin– orbit coupling, a nodeless p–wave pairing is possible in an RuO2 plane, as experimentally observed, and a node would lie between the RuO2 planes. The corresponding situation is illustrated in Figs. 4.10 and 4.11. The important IAF wave vector Qi now connects the nested portions of the β band, while the small qi bridges the γ band. Therefore, since the interaction in the RuO2 planes (kz = 0) is mainly ferromagnetic, nodeless p–wave pairing is indeed possible, and the superconducting order parameters in all three bands will not have nodes. On the other hand, the situation between the RuO2 planes will be determined mainly by the IAF excitations, which are mainly located in the β band and polarized along the z direction. This implies that the magnetic interaction between the planes has to be antiferromagnetic rather than

Fig. 4.10. Symmetry analysis of the superconducting order parameter for triplet pairing due mainly to qi in the first BZ for kz = 0 (solid curves) and kz = π/2c (dotted curves), after inclusion of spin-orbit coupling. α, β, and γ denote the corresponding Fermi surfaces and the wave vectors Qi and qi are the corresponding spin excitations shown in Fig. 4.2. For clarity, again the nodes of the real part of an order parameter with fx2−y2 –wave symmetry are shown, and also the regions + and − where the gap has positive and negative sign, respectively. Note that although a p–wave order parameter has no nodes, for the symmetry analysis we consider the real part of it, which has a node for kx = 0. The γ band plays the dominant role in Cooper pairing (Nγ (0) > Nα,β (0)).

ferromagnetic. In the superconducting state, this antiferromagnetic interaction would disturb triplet Cooper pairing and induce a line of nodes between neighboring RuO2 planes. This was shown by two groups in a phenomenological approach [8, 9].

The second result of the spin–orbit coupling is the orientation of the orbital moment of the Cooper pair. As is known from experimental data, the superconducting state is characterized by violation of time–reversal symmetry [32] and equal spin pairing within the basal plane of the tetragonal crystal lattice [33]. The single symmetry representation consistent with these experiments is d(k) = ˜z(sin kx ± i sin ky ) in an RuO2 plane in the standard vector notation. The important consequence of this order parameter is the orbital angular momentum of the Cooper pair (the “chiral” state). Taking into account the fact that there are other symmetry representations that possess p–wave symmetry, the important question is why the “chiral” state has a lower energy than the other representations. A theoretical study of this question, assuming magnetically mediated Cooper pairing, was done by Ng and Sigrist [30] and partly by Kuwabara and Ogata [27]. As we indicated previ-