4.1.3 Comparison with Experiment

In order to compare our results with experimental data, we have calculated the nuclear spin–lattice relaxation rate for an 17O ion in an RuO2 plane for di erent external magnetic field orientations (i = a, b, and c),

q

where Apq is the q–dependent hyperfine coupling constant and χp is the imaginary part of the corresponding spin susceptibility, perpendicular to the direction i. Similary to the experiment [11], we have used an isotropic hyperfine coupling constant (17Aq 22 kOe/µB ).

First, we discuss the spin anisotropy due to inclusion of spin–orbit coupling. In Fig. 4.7 we show the calculated temperature dependence of the spin–lattice relaxation for an external magnetic field parallel and perpendicular to the RuO2 plane, together with experimental data. At T = 250 K, the spin–lattice relaxation rate is almost isotropic. Owing to the anisotropy in the spin susceptibilities arising from spin–orbit coupling, the relaxation rates become di erent with decreasing temperature. The largest anisotropy occurs close to the superconducting transition temperature, in good agreement with experimental data [11]. Thus, our results clearly demonstrate the essential significance of spin–orbit coupling for the spin–dynamics even in the normal state of the triplet superconductor Sr2RuO4. We find that the magnetic response becomes strongly anisotropic even within an RuO2 plane: while the in–plane response is mainly ferromagnetic, the out–of–plane response is mainly antiferromagnetic–like.

In order to discuss both the long–wavelength and the short–wavelength limit of χ(q, ω) we compare in Fig. 4.8a our calculation of the temperature dependence of the uniform spin susceptibility χ(0, 0) with experiment, where this quantity is measured by the 17O Knight shift [3]. For the calculation of χ(0, 0), we have approximated U (q) by U (0) = 0.177 eV [31] which gives agreement with Knight shift measurements and was also used in previous calculations. In agreement with experiment, we obtain a tendency towards ferromagnetism2. Note that we also take into account the fact that there are four electrons for three t2g bands, which gives every χi0 an additional weight of 4/3. The maximum in χRP A(0, 0) at about 25 K results from thermally activated changes in the populations of the bands near EF . Because the agreement with experiment is remarkably good (although the results were obtained without consideration of spin–orbit coupling), these comparisons shed light on the validity of our results for χ(q, ω).

In Fig. 4.8b, we compare our results for Imχ(Qi, ω), again without spin– orbit coupling, with INS data [2]. In this case we must take UQi = 0.345 eV

2In the original paper [3], the e ect of cross-susceptibilities was not considered. Therefore, we compare our results with the total susceptibility χxy + χyz + χxz .

186 4 Results for Sr2RuO4

Fig. 4.7. Calculated normal–state temperature dependence of the nuclear spin– lattice relaxation rate T1−1 of 17O in an RuO2 plane for an external magnetic field applied parallel to the c axis (dashed curve) and to the ab plane (solid curve). The corresponding susceptibilities include spin–orbit coupling. The triangles pointing up and down are experimental points taken from [11] for the corresponding magnetic– field directions.

in order to fit χ(q, ω) to the position and height of the peak at ω = 6 meV observed in INS. While an uncertainty in the INS data (shown in Fig. 4.8b) is present, our results for χ(q, ω) should be a useful basis for further calculations. In general, the normal-state properties of χ(q, ω) control also the symmetry of the superconducting order parameter. Physically speaking, the antiferromagnetic spin excitations result in incommensurate antiferromagnetic alignment Ru spin at distances larger than the nearest–neighbor spacing. Hence, if Cooper pairing involves nearest neighbor Ru spins, incommensurate antiferromagnetic fluctuations will also cause triplet pairing because neighboring Ru spins see a partly ferromagnetic environment.

In order to briefly summarize the comparison with experimental data, we can say the following: using hybridized bands and taking into account all cross–susceptibilities, we are able to explain successfully the 17O Knight shift and INS data from our calculated dynamical spin susceptibility χ(q, ω) based on the Fermi surface topology. However, in order to explain the anisotropy of the spin–lattice relaxation rate of 17O in the RuO2 planes with respect to the crystallographic direction of the external applied field, one needs to include spin–orbit coupling. As a result of this, the anisotropy of the orbital subspace is reflected in the spin response of the system. Note that the γ band is approximately a circle, whereas the α and β bands are quasi–one–dimensional and have strong nesting properties. Using λ = 100 meV as a parameter for the strength of the spin–orbit coupling and taking into account the large Stoner enhancement factor of the spin susceptibility (within the RPA), we are able to explain the unexpected by large anisotropy of the spin response of Sr2RuO4, namely that the in-plane response is mainly ferromagnetic–like