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

the system. Fig. 1 shows also that the spin ordering is highly sensitive to pressure: the non-uniform spin structure is stabilized by lattice extention, and conversely the local spin polarization is destroyed by uniform contraction of the system. The energy scale of the frozen magnon grows deeper from -1.0 mRy at equilibrium to -1.6 mRy with a lattice parameter increase of 3 percent using the paramagnetic energy as a reference point, and the magnetic moment at the Ru sites is increasing

from 0.8 µBohr to 1.2 µBohr. As is obvious from Fig. 2, DOS at the Fermi level also reaches its minimum at the same spiral parameters, due to

local polarization which splits the electronic states and lowers the total DOS. In fact, this minimum is even more pronounced in case of a static SDW caused by antiferromagnetic ordering, which should be considered a strong indication that the system will undergo a metal-insulator phase transition.

To estimate the size of cluster that would be su cient to reproduce the nesting featured exchange interaction, the LSDA-derived energy of the helix θ, q depicted in Fig. 1 was mapped onto a general form bilinear in terms of spins. That is, a frozen spin wave energy for a Heisenberg system with a single spin variable per site

Mi = M {sin θ cos qri, sin θ sin qri, cos θ}

has to be

constant since the magnetization has considerable variation in the space of helices θ, q. A numerical fit in the form of the Fourier expansion over the modulation parameter q

M (θ, q) = sin θ (αi sin(qri + βi),

i

is able to quite accuratly reproduce the magnetization in the phase space, Fig. 1. A reasonable mapping for the helix energy surface in the same space has been made in accord with expression (1) for a helix in the Heisenberg form. An LSDA-derived energy (mRy) expansion at equilibrium lattice parameter in terms of modulation vector q = q[110]2π/a,

224 Spin Stability and Low-Lying Excitations in Sr2RuO4

where q is varying between 0 and 1/2 within the first Brillouin zone,

shows that the largest contribution comes from Ru sites located along the nesting vector. This is because the terms with even and odd numbers of πq correspond to the exchange interaction between nearest and nextnearest Ru sites along qn [110] chain of Ru, respectively. Obviously the long-range nesting-imposed RKKY mechanism is responsible for the

spiral ordering with J ≈ −0.25 mRy/µ2Bohr between next-nearest Ru moments, whereas the spin modulation along Ru-O-Ru chains, with much

smaller J ≈ 0.02 mRy/µ2Bohr between nearest Ru moments, is mediated by super-exchange [11]. Of course, separation of the exchange into these

two mechanisms is more conventional than can be strictly justified, par-

similar energy expansion for a stretched lattice, middle picture on the Fig. 1,

shows that super-exchange, with J ≈ −0.10 mRy/µ2Bohr, and RKKY mechanism, with J ≈ −0.33 mRy/µ2Bohr, have di erent behaviors with lattice parameter change. There are two types of spin-spin correlations

S√iSj : 1) the dominant RKKY type correlations on the ξRKKY qN−1 3 2a 15 ˚A distances; 2) essentially weaker ligand-type correlations on

the scale of lattice parameter ξligand a. With lattice expansion, the Ru moments grow, a trend which hints at the impurity model and will be the

subject of the next section. Therefore, the energy gain associated with RKKY interaction is increasing roughly as M 2N (EF ) with increasing a, whereas the correlations SiSj remain unchanged. On the other hand, the energy gain due to the ligand exchange is on the order of −t2/U , with t as the ligand hopping integral, much smaller than the former but also more sensitive to the lattice parameter. The super-exchange correlations follow the increase of M 2 but get weaker on account of diminishing t with lattice expansion, so that it becomes rather obscure to predict the resultant e ect in simple terms of the perturbation. In any case, the dependence on lattice parameter changes as well as thermodynamics are dominated by RKKY type of correlations.

Calculations for a supercell with three Ru ions reveal an SDW slightly favored over the spiral configuration. Fig. 3 illustrates the distribution of magnetization for both types of spin ordering. An SDW description can be facilitated by introduction of two spin momenta per Ru site, one for a transverse spiral with right helicity as shown on the Fig. 3, and the other with left helicity, assuming this feature is an intrinsic structure of spin density matrix. This approach implies a spin-density matrix consideration on the intra-atomic scale, that is, however, beyond the present numerical implementation. Therefore, an LSDA evaluation of SDW is much more challenging, and we could accomplish the problem only by an appropriate enlargement of the unit cell to fit the nesting vector. As a result, there are two inequivalent Ru sites, one with M = 0.8µBohr at the equilibrium lattice parameter which is also the amplitude of spiral moment, and the other two just half of that value and opposite in orientation, so that the net magnetization is vanishes. This means

that on-site exchange coupling ji,rl between spin moments of spirals with opposite helicites Mk,r and Mk,l, respectively, is essentially smaller than

inter-site exchange Jij, given that the following interaction model

is relevant. In fact, the last terms in eq. 4 which stand for the exchange interaction between opposite-helicity spirals, are numerically shown to be just 0.2 of the spiral energy. It is also in accordance with the calculations to claim that the on-site interaction between spirals is dominant, since a superposition of only undistorted spirals leads to the formation of the sinusoidal SDW, and therefore the middle term in Eq. 4 can be discarded.

15.3Formation of spin and orbital moments and pressure dependencies

Now, to better understand the trends under pressure, the mechanism of spin formation has been analyzed in terms of the Anderson impurity model [20] instead of the Stoner approach, more common for itinerant systems. The choice of the framework is motivated by the fact that the formation of a magnetic moment is mainly due to the exchange selfenergy arising as the Coulomb repulsion between opposite spins rather than the Stoner-type exchange between parallel spins. It was shown earlier [21] that the electronic structure of pure strontinum ruthenate can accurately be reproduced by a tight-binding Hamiltonian which includes three type of bands, since an almost ideal octahedral crystal field

Figure 4. Symmetry-resolved partial density of states on Ru site: tetragonal point symmetry group D4h implies partially (2/3) occupied t2g states split into doubledegenerated eg xz and yz orbitals, and b2g in-plane xy orbital. Shown are also other irreducible states a1g and b1g with z2 and x2 − y2 symmetries, clustered however far away from the Fermi level.

splits the d-states of Ru into two groups of states with irreducible representation eg for two unoccupied and t2g for three partially occupied xz, yz and xy orbitals. With a more accurate treatment, the tetragonal point symmetry group D4h, relevant for tetragonally distorted octahedra (Jahn-Teller distortion), lifts the degeneracy and implies the following irreducible symmetry split of the Ru d-states: as shown on Fig. 4, partially occupied t2g states cast upon double-degenerated eg-states, with 1.4 occupancy per orbital, and single b2g-orbital, with 1.3 electrons. There are also single orbitals of a1g and b1g symmetry, with 0.95 and 0.8 occupancy respectively, which are, however far from the Fermi level.

The first two degenerate orbitals constitute quasi-one-dimensional bands with a width of about 1 eV and nested around the Fermi level, whereas the third band is two-dimensional and as almost 2 eV broad, with its xy orbital located 0.5 eV below the Fermi level. Therefore, a Hartree-Fock Hamiltonian with two degenerate t2g orbitals xz and yz seems quite relevant for a qualitative description of the moment formation. In this approximation, the condition for moment formation is formulated in terms of four spin occupation numbers niσ, two σ = +, − per each orbital i = xz, yz, Coulomb repulsion between opposite-spin same orbitals U , and “ordinary” exchange interaction between same-

Figure 5. Sr2RuO4, electronic structure and projected density of states for spinrestricted and simple spiral configurations, at equilibrium lattice parameter a = 7.29 Bohr. Orbitals mostly contributing to the states shaping the nesting of the Fermi surface and causing the local magnetization are mostly of xz and yz angular character, with total occupation of nearly 2.8 electrons per Ru site. The broader states built out of in-plane Ru xy orbitals are occupied by 1.3 electrons.

contrast, U becomes smaller under positive pressure due to an increase in Coulomb screening, which leads to a magnetization collapse with only a 3% reduction of the lattice constant. The spin moment formation condition given by Eq. (7) at the same time reads (U + J)ηiσ(EF ) 1.65 since of ηiσ(EF ) ≈ 0.65 st/eV/spin, which is large enough to satisfy the condition given by Eq. (7) and the exchange splitting if not the e ects of quantum fluctuations.

Formation of a local orbital moment requires, along with the spin moment formation, a certain local symmetry imposed by electrostatic crystal fields and relativistic coupling between spin and orbital moments. The latter is always present in relativistic considerations, whereas the crystal-field e ect has to be strong enough in order for the Hartree fields Ei+ for the two orbitally degenerate states to di er.

The origin of the orbital polarization can easily be traced out in terms of double-group symmetry classification for irreducible representation. Notice that by non-relativistic consideration the basis orbitals have the following spin-angular part

| φ1s √i (Y2,+1 + Y2,−1)χˆs,

2

1

| φ2s √ (Y2,+1 − Y2,−1)χˆs,

2

with χˆ as the Pauli spinors, which leads to vanishing orbital momentum

ˆ

because of LzYlm = mYlm and orbital degeneracy 1s = 2s . Eigenvaluesn=1,2s(k) in the periodic crystal will have similar degeneracy. In the

ˆ

presence of spin-orbit coupling hso,

the degeneracy in general will be lifted everywhere except perhaps at some high-symmetry points in the Brillouin zone. By construction of single-particle eigenvectors from Bloch combinations of the orbitals, the respective eigenvalues can in lowest perturbation order, easily be shown to be split at every k-point

¯ns(k) = (+) + s 2(−) + ξ2, (9)

where the spin-orbit coupling parameter ξ2 =| ψ1slzψ2s |2+ | ψ1+l+ψ2− |2, and (+), (−) stand for half-sum and half-di erence of the non-relativistic bands ns, respectively. Thus, electrostatic crystal fields and the spin-

orbit coupling split the energy levels

at general k-point, with s = ± (not a spin index anymore) and ξ =

ˆ ˆ

φ1slzφ2s since l± is vanishing on the Ru sites. The corresponding

is

which for the angular part implies

| k1s Y2,+1χˆs, | k2s Y2,−1χˆs,

since αis is either one or zero if no magnetic field or exchange splitting is present. More formally this result could be obtained within doublegroup classification when the wave function is to be expanded in terms of

direction eˆζ will be di erent from zero only if the spin degeneracy of each state is lifted. Then, in the lowest order of the perturbation theory in spin-orbit coupling [23–25]

is determined through the density of states nii ,σ = n aiσ ai σδ( −nσ(k)) with non-relativistic eigenenergies nσ(k) and expansion coe - cients ai σ, and remains unchanged for any orientation of the spin quan-

ˆ

tization axis. The orbital moment L ζ depends on the di erence of the Hartree field for the same-spin states (crystal-field splitting), the amount of spin-orbit splitting | ξ |, which is about 0.15 eV for the Ru 4d-orbitals, as well as the orbital-projected spin and particle DOS. At the same time, the magneto-crystalline energy

is related to the di erence between same orbital moment spincomponents, given that the spin-flip e ects are small. The angular dependence is given by iσ | lσ | i σ only, which in case of uniaxial symmetry is simply K0 + K1 sin2 θ. This specifically means that the same