Halilov (ed), Physics of spin in solids.2004

Table 1. SDW configuration: magnetization, spin Sζ and orbital Lζ contributions at di erent orientation of the spin quantization axis eˆζ , in units of µBohr. Oei and Oai refer to oxygens equatorial and apical in respect to Ru1 and Ru2, respectively.

matrix elements are present at any orientation of spin quantization axis. From the formulas given by Eqs. 12 and 13, it is easy to project a similar scaling of orbital momentum and energy if the shell is more than half-filled [24], since the orbital polarization of spin-majority states has to vanish. Our LSDA-DFT calculations find an opposite orientational trend, which is obviously related to the e ects of hybridization between opposite-spin states. Notwithstanding a remarkable spin anisotropy with data collected in the Table 1, the energy shows practically no dependence on the orientation of the quantization axis. The in and out of plane di erence is less than 0.01mRy, whereas the spin moment shows considerable out of plane enhancement. We will use this fact to illustrate the destructive role of quantum spin fluctuations on long-range ordering. Note also that it follows from Eq. 12, considering only two orbitals would give K0 = K1. This would imply a factor of two di erence polarization of a factor of two between in-plane and out-of-plane orientation of the spin quantization axis, which only qualitatively correlates with the numbers from the Table 1, which includes hybridization e ects between all orbitals. Therefore, there is only limited legitimacy for the two-orbital model by interpretation of the orbital moment formation.

The sign of the projection is obviously a matter of the local occupation (3-rd Hund’s rule) and in accord with the Eq. (12) the exchange-caused

ˆz will be pointed in the direction of spin moment, which is common l

for orbitals with greater than half-filling. On the other hand, for the numbers for orbital occupation and spin magnetization caused by an

¯

external magnetic field, lz is pointed against the spin vector, since all orbitals will now be involved in the magnetic response, with an e ective occupation less than half-filling for Ru4+. ion.

Variation of the volume causes a certain trend in the amplitude and stability of the spiral configuration. As illustrated in Fig. 6, there is

obviously a higher paramagnetic density of states ησ(EF ) = 1.8 st/eV with lattice expansion of 3%, which in accord with the Eq. (7) facilitates

spin formation with an increase of the moment M = 1.0µBohr. With lattice compression of the same amount, the increased hopping between

Ru and O ions leads to a diminishment of the DOS, and the condition of Eq. (7) is no longer fulfilled.

That the system is close to a metal-insulator transition caused by the formation of AFM spin ordering, is clearly seen from Fig. 7, where the electronic structure of an SDW configuration is plotted along with the spinand atom-projected DOS at equilibrium and expanded lattice parameter.

The AFM gap itself is driven by the aforementioned local magnetic moment formation mechanism and the long-range nesting-imposed oscillating exchange interaction. As the spin configuration is stabilized due to covalency reduction between Ru orbitals and oxygen orbitals through expansion of the crystal, all three Ru xz, yz and xy orbitals become more involved in hybridization with each other as a result of the weakening of the octahedral crystal field e ect and therefore less distinguishible from each other. Therefore, by lattice expansion the occupation of the three Ru orbitals approaches half-filling (4 electrons per entire t2g-subshell), causing the Ru orbitals to split symmetrically around the Fermi level, due to the exchange interaction.

Of course, the amplitude of orbital polarization is still to be verified by more accurate numerical calculations, which indeed confirm the situation. The calculations of magneto-crystalline Ising-type anisotropy have been probed for the pure system with an SDW in the ground state at several lattice parameters. The magnitude of the uni-axial anisotropy amounts to less than 0.01 mRy per Ru unit at equilibrium, which is rather small on account of the tetragonal symmetry of the crystal. Note that, in the case of tetragonal symmetry, the e ect must be of second order in spin-orbit coupling, i.e. of the order of 0.1 mRy provided the second Hund’s rule is fulfilled. But this is not the case, as we already deduced on the basis of the impurity model. The situation changes only slightly with variation of the lattice parameter around the equilibrium: while remaining in the area of tetragonal elongation of the octahedra, the spin alignment starts to stabilize by increased anisotropy energy only with considerable lattice expansion of 3%. This behavior shows that the quenching of the orbital moment is a characteristic feature of ideal strontium ruthenate within a range of tetragonal-symmetry-conserving lattice distortion around the equilibrium. Upon octahedral rotation, orthorhombic crystal fields will likely enhance the spin and orbital polarization. Although considered speculation, this dependence of orbital

Figure 7. Sr2RuO4, electronic structure and projected density of states for spindensity wave configuration (ground state), for equilibrium lattice with a = 7.29 Bohr and extended lattice with a = 7.49 Bohr. The most prominent feature is the opening of the gap on the density of states at the Fermi level, what gives evidence for an AFM-caused metal-insulator trend in the pure compound.

polarization on tetragonal strain can at least partially be used to explain why the moments induced by Ti4+ on the Ru sites exhibit strong Ising anisotropy in the experiment [28]. Of course, by partial replacement of Ru by Ti the octahedral distortion is actually a local e ect and can’t properly be simulated by a simple uniform lattice expansion or compression as used in our attempt to describe the respective trend in the formation of the orbital moment.

The vanishing value for the anisotropy at equilibrium is certainly crucial for the formation of the magnetic moment since zero-point spin fluctuations will now destabilize the ground state SDW in the absence of a mechanism for pinning the orientation of local moments. This e ect of renormalization of the local spin momentum by quantum fluctuations was considered in detail for 2D antiferromagnetic Heisenberg systems on the basis of a 1/S spin expansion, see e.g. [26, 27], and it was asserted, though speculatively, that if the “trial” spin is small enough and there is no anisotropy then the staggered magnetization can be destroyed completely. There are rigorous results [29] which show that at T = 0 there is long-range order for spin moments S ≥ 1. There is growing evidence that in case of the 2D Heisenberg antiferromagnet the critical value for the spin momentum below which the staggered magnetization vanishes is around physically accessible Sc 1/2 within the model. In our case with an SDW in the ground state, and the hydrodynamic description as a more appropriate model, this value shoud be even less. The largest spin moment on the Ru site 0.4 is less than the Heisenberg critical value and most likely will collapse in presence of quantum fluctuations.

15.4Anisotropy of static magnetic susceptibility

We also checked the magnitude of the paramagnetic static spin susceptibility, which is related to the DOS at the Fermi level, and found reasonable agreement with the available experimental data, obtained from neutron scattering. For an applied magnetic field of H = 1 T, the calculated induced total spin moment per unit cell turns out to be

Mind = 1.15 × 10−3µBohr, which corresponds to χisotr(q = 0, ω = 0) ≈ 1.15×10−3[µBohr/T ] = 6.64×10−4emu/mol for the isotropic susceptibility. This theoretical value for the non-relativistic and therefore spatially

isotropic χ correlates well with experimental values which are in the range of 5.29 × 10−4 − 9.0 × 10−4 emu/mol [12, 13]. Given that the susceptibility is proportional to the DOS at the Fermi level, this seems to have a reasonable magnitude. Therefore, the observed e ect of local spin polarization is obviously imposed not by the high value of the DOS, which otherwise would be an artifact had the DOS been overestimated in

Table 2. Static paramagnetic susceptibility χ(ω = 0, q = 0) in units of 10−4µBohr/T, determined as a response to an external magnetic field applied along [001], [010], and [110] axis. The shown are partial spin χS and orbital χL contributions from Ru and oxygens. Oe1,e2 and Oap stand for oxygens equatorial and apical in respect to Ru, respectively. For symmetry reason, the equatorial oxygens are di erent only at H [010].

calculations, but rather by the Ru on-site exchange enhancement, which is responsible for the first Hund’s rule.

Inclusion of spin-orbit coupling causes a double anisotropy e ect. First, there is a very modest magneto-crystalline anisotropy of about less than 0.01 mRy as mentioned above. Second, the static paramagnetic longitudinal susceptibility also gains a small anisotropy χab(q 0)/χc(q 0) 1.09 at vanishing q vectors. This means the amplitude of spin fluctuations is slightly enhanced within the plane. The partial spin and orbital contributions, to the total magnetic susceptibility, resolved also in consituents of octahedron, is given in Table 2 (numbers are rounded). For symmetry reasons, equatorial oxygens along [100]- and [010]-bonds with Ru labeled as Oe1 and Oe2 respectively, are not equivalent at H [010].

Note from Table 2 that there is considerable orbital polarization of the equatorial oxygen ions in the presence of an external field, which makes the Ru ions orbitally polarized against the local spin moment, no matter what the orientation of the field. On the other hand, apical oxygens and Sr have very low magnetic polarizability, which again justifies the 2Dcharacter of the Fermi system and particularly makes the inter-planar exchange coupling immaterial. This fact also indicates that incorporation of orbitals on both cation and anion sites in any analytical model of spin susceptibility is mandatory in the perovskite. This could explain the unrealistically large anisotropy [14] of the magnetic susceptibility obtained within an RPA-designed two-band Hubbard model similar to the one used above. The two-band model seems, however, quite adequate for a qualitative discussion of the orientational e ect, since in the case of spontaneous polarization, oxygens are practically uninvolved in the low-energy angular momentum dynamics.

To reflect the aforementioned e ects of anisotropy, the following formula

can be used to adequately describe the basal-plane anisotropy of magnetic response with local C4v symmetry and uniaxial anisotropy along c-axis. The notations used are: , stand for components parallel/perpendicular to c-axis, e.g. external field H = H + H , and inplane unit-vector h ≡ H /H . Thus

(15)

the latter set is the most favorite for polarization, apparently because of the spin-orbit interaction causing most of the charge transfer between cations and anions when the field is along [110]. The induced orbital and spin moments on Ru sites are antiparallel, an expected result in view of the fact that the entire Ru+4 d-shell is less than half-filled. This observation is in contrast to the prediction of the two-orbital model discussed above in accord with which the orbital L and spin S moments would be expected to be collinear since the 2/3 occupation of the orbitals is clearly above the half. In fact, L and S are collinear by spontaneous spin polarization of Ru ions in the incommensurate SDW phase, which shows that the on-site exchange field responsible for the respective spontaneous splitting is contained mainly within the two considered Ru orbitals as having the highest Coulomb repulsion U between the same spins. When a uniform external field is applied, the Zeeman interaction acts on all sites and all orbitals, thus involving the entire d-shell on the Ru+4 sites with e ective occupation of about 4 electrons (less than half-filling), as mentioned above. In this case the orbital polarization has to adopt the minimum total moment J =| S − L | within the manifold of the Lande levels of the given L, S multiplet. By selectively switching on spin-orbit coupling parameter on various ions it is established that the dominant relativistic e ect comes from the d-orbitals of Ru, also the polarizing the neighbour equatorial oxygens. The polarizability on the sites of apical oxygens and adjacent ions of Sr+2 is vanishingly small, presumably as a result of complete ‘undressing’ of the Sr ions.

The following ratios for the susceptibility components

χ ≈ 1.09χ ,

complete the anisotropy mapping of the magnetic susceptibility at zero wave vector. So far we have discussed the anisotropy found for static χ at zero wave vector, particularly emphasizing stronger ferromagnetic fluctuations within the ab-plane. Since the anisotropy of χ is closely related to anisotropy of lz ζ the same magnitude of anisotropy would be expected for finite q as well. From the above discussion of the orientational dependence in Sec. 15.4, it is clear that the e ect of the energy denominator which doesn’t depend on the orientation of spin quantization axis (external magnetic field), will a ect the susceptibility at the nesting vector leading to an enhancement of both longitudinal χzz and transversal χ+− components in rather similar way. This conclusion is in contradiction to the earlier prediction [14] that χ+− has no features related to the Fermi surface nesting whereas χzz does. Apparently only

ˆ

matrix elements of lz are contributing to magnetic susceptibility within the two-orbital model, and they do that at any orientation of eˆζ. For wave vectors around the nesting qN , both paramagnetic longitudinal and transverse components are expected to be enhanced, with a rather modest level of anisotropy, since this is solely an e ect of the energy denominator.

15.5Summary: possible magnetic low-lying excitations and impact upon superconductivity

The presence of a SDW ground state, although suppressed by zeropoint fluctuations as mentioned above, raises questions about specific magnetic low-lying excitations, a somewhat general concern that might have some interesting consequences for the observables. Establishment of magnetic type low-lying excitations becomes even more intriguing with reference to potential mediators of superconducting pairing since earlier lattice dynamics investigation [4] shows no sign of phonon mode softening.

We have demonstrated that there is a certain trend toward the formation of a spin-density wave already in pure stronthium ruthenate. On the other hand, in view of the immaterial magneto-crystalline anisotropy the quantum spin fluctuations are most likely to destroy the staggered magnetization density. Yet, the magnetic susceptibility is expected to be structured by a type of low-lying magnetic excitations, which in spinwave approximation, can be considered as phasons with limited life-time. The latter type of excitations is characteristic of a simple helix in the

ground state. The problem was first considered some time ago [17, 18] within a Heisenberg model and recently within LSDA [19] in conjunction with the heavy rare earth metals, where the ground and meta-stable states are vary from ordinary FM through the AFM helix and FM cone. The dispersion of phasons, like antiferromagnetic magnons, is linear in wave vector and has no gap, since in presence of incommensurate spin modulation the azimuthal phase is averaged out from the total energy. The phasons exemplify the Goldstone mode as a result of spontaneous symmetry breaking in the system.

There is the question of whether the phasons, perhaps better termed para-phasons, might be involved in the superconductivity. If they are, then the classification in terms of singlet or triplet would be unnecessary, since on atomic scale the spinor symmetry is triplet, but on the scale of the helix the symmetry is a singlet. Although, as argued above, a vanishingly small magnetocrystalline anisotropy on the Ru sites cannot stabilize the long range order in the presence of zero-point spin motion, the latter itself has to reflect the symmetry of incommensurate helical superstructure or spin-density wave.

Since the observation of a temperature-independent spin susceptibility deep into the superconducting state has been used to justify a timereversal symmetry breaking triplet order parameter within Cooper pairing formalism, a detailed analysis shows that the “chiral” vector order developed for 3He is also relevant for the crystal in question (see Ref. [30] for a thorough discussion of this and related problems). Here we restrict ourselves to a brief discussion of possible complications of our results for the triplet pairing models.

Similar to the B phase of 3He, broken symmetry in the unconventional superconductor Sr2RuO4 governs the relative orientation of spin and orbital momentum. The on-site exchange on Ru sites seems to be strong enough to produce local ferromagnetic fluctuations of spin density, correlated in helical or spin-density wave structure by the RKKY-type interaction, which is a clear manifestation of the crystalline structure. On the other hand, the relativistic interaction is large enough to keep spin and orbital moments together up to 1000 K, but the vanishingly small anisotropy decouples the entire set of spin-orbit pairs from certain directions in the crystal. Therefore the superconducting order parameter seems similar to the one in the B phase of 3He, but modulated spatially as a helical or spin-density wave order parameter. Here we want to emphasize once again, that the magnitude of spin-orbit coupling alone cannot determine the anisotropy of the order parameter. Instead, the SO e ect has to be combined with the subtle e ects of crystal field and state occupation, which in our particular case leads to the vanishingly

small anisotropy of magnetic fluctuations. Note also, as a support for our finding, that there is so far no experimental proof that the vector order parameter is pinned along a certain direction in the crystal.

Clearly, this observation based on the exchange structure of the spin interaction, has to be complemented by the anisotropy mapping of the magnetic susceptibility which is particularly suited to the liquid 3He B phase model of superfluidity for the tetragonal crystalline structure of strontium ruthenate. By giving a credit to LSDA calculations, we have to admit that the zero wave-vector spin fluctuations are at maximum within the plane of the essentially 2D Fermi system along the [110 ] Ru-chain resembling the A phase of 3He, and strong enhancement of fluctuations at the Fermi surface nesting vector with rotationally broken symmetry as in the 3He B phase due to the vanishingly small orientational anisotropy in that case.

Acknowledgments

The authors are grateful to M. Johannes, A. Shabaev, I. Mazin, M. Pederson and B. Gyor y for useful discussions.

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