invited papers
Hartree–Fock study of orbital magnetic moments in 3d and 5f magnets and Xray magnetic circular dichroism
^{a}Department of Quantum Matter, ADSM, Hiroshima University, HigashiHiroshima 7398526, Japan
^{*}Correspondence email: jo@sci.hiroshimau.ac.jp
The spin (M_{s}) and orbital magnetic moments (M_{o}) of the uranium 5f state in the ferromagnetic compound uranium sulfide (US) and of the cobalt 3d state in various transitionmetal superlattices are calculated on the basis of a tightbinding model, in which the intraatomic f–f or d–d multipole interaction is taken into account using the Hartree–Fock (HF) approximation. The parameters in the model are determined on the basis of available firstprinciples calculations. For US, the calculated ratio M_{o}/M_{s} and spectrum for U M_{4,5} absorption are in good agreement with the experimental results. Inclusion of the expectation values of the spinoffdiagonal operators in addition to the number operators in the 5f state is found to be crucially important when describing the 5f magnetic state. A difference in enhancement of M_{o} of the Co atom between the Co/Pd and Co/Cu superlattices is discussed on the basis of a semiquantitative calculation, assuming ferromagnetism.
Keywords: orbital moment; uranium sulfide; superlattice; Hartree–Fock approximation; Xray magnetic circular dichroism.
1. Introduction
In magnets, the atomic spin (M_{s}) and orbital (M_{o}) magnetic moments are basic quantities and their separate determination is therefore important. In addition to the traditional method to measure M_{s} and M_{o} (Bonnenberg et al., 1986), the recently developed technique of Xray (XMCD) with coretovalence Xray absorption (Chen et al., 1995) combined with several sum rules (Thole et al., 1992; Carra et al., 1993) has attracted much attention as a method for site and symmetryselective determination of M_{s} and M_{o}. From the theoretical point of view, the firstprinciples localdensity approximation (LDA) of densityfunctional theory is a typical method for calculating magnetic quantities and has been successfully applied to various substances, especially 3d transitionmetal systems (Morruzi et al., 1978).
In the LDA theory, the Kohn–Sham equation is described by the local potential including the spindependent electron density. The M_{o}, is not, however, included. This means that there is no theoretical framework within which to determine M_{o} selfconsistently in the LDA, and M_{o} is calculated as a quantity induced by M_{s} through the spinorbit interaction (SOI). As a result, LDA theory is known to underestimate M_{o} in some cases. In bulk 3d transition metals, M_{o} is about one tenth of M_{s} (Bonnenberg et al., 1986) and M_{o} is sometimes neglected, i.e. the success of the firstprinciples LDA approach relies on the experimental fact that M_{o} ≪ M_{s}. In 4f rare earths, M_{o} and M_{s} are mostly atomlike, i.e. determined by the Hund rule, and a solidstate effect is in many cases irrelevant. In 5f actinide systems, especially in uranium systems, M_{o} and M_{s} of the 5f state are determined by solidstate effects, i.e. they are dependent on the atomic environment. LDA theory applied to ferromagnetic U compounds underestimates M_{o} (Kraft et al., 1995).
which describesMany theoretical attempts to improve the underestimation of M_{o} have been made. They may be roughly classified into two categories. An example of the first category is the socalled currentdensityfunctional theory, which was formulated to extend the densityfunctional theory to include the orbital current, which describes M_{o}, as an extra degree of freedom (Vignal & Rasolt, 1987; Skudlarski & Vignal, 1993; Higuchi & Hasegawa, 1997). An explicit form of the contribution of the is, however, at present unknown and a tentative application of the theory to Co still underestimates M_{o} of the Co atom (Ebert et al., 1997). The other category includes the socalled orbital polarization (OP) (Brooks, 1985; Eriksson et al., 1990) and LDA + U (U is an electron–electron interaction parameter) approaches (Solovyev et al., 1998), intended to calculate M_{o} practically. For problems with the OP approach, the reader is referred to the work of Solovyev et al. (1998) and Shishidou et al. (1999). In the LDA + U approach, the wavefunctions are prepared by the LDA calculation, the multipole intraatomic interaction between electrons is taken into account for the state of interest and the Hartree–Fock (HF) calculation is performed. The essential part of the LDA + U approach seems to be reproduced if an appropriate multiorbital tightbinding Hamiltonian (Hubbard model) is prepared. In fact, for the antiferromagnet CoO, the HF calculations, with rotational invariance, based on both LDA + U (Solovyev et al., 1998) and an extended Hubbard model (Shishidou & Jo, 1998) are found to give similar results for both M_{o} of Co (∼1 μ_{B}) and of the stable magnetic structure.
The purpose of the present work is to review recent HF calculations of M_{s} and M_{o} made on the basis of an extended Hubbard model including the intraatomic multipole interaction, for ferromagnetic US and superlattices of Co atoms in Pd or Cu matrixes. Parameters of the oneelectron part of the model are determined on the basis of data available from firstprinciples LDA calculations. Among the Slater integrals describing the multipole interaction, the HF value of F^{0} for an atom assuming a suitable is, in metals, reduced to a considerable extent by a screening effect, while that of F^{k} with k ≠ 0 is only reduced to 90∼70% (Norman, 1995). In the calculations described here, suitable reductions from the HF values for F^{k} with k ≠ 0 are assumed and values of F^{0} are determined so that the calculated total moment M_{s} + M_{o} agrees with the experimental result; the obtained ratio M_{o}/M_{s} is compared with the experimental value.
For US with NaCl structure, in addition to various magnetic measurements (Tillwick & de V. du Plessis, 1976; Lander et al., 1991), XMCD has recently been observed (Collins et al., 1995). Through an HF calculation with rotational invariance, the role of the multipole exchange interaction described by the Gaunt coefficient (Condon & Shortley, 1959) is clearly seen to be important in the estimation of M_{o}, which is the case for atoms obeying the Hund rule. Since the establishment of the XMCD sum rules relating M_{s} and M_{o}, the enhancement of M_{o} of Co or Fe compared with the bulk metal has been reported in many XMCD experiments on magnetic multilayers (Wu et al., 1992; Tischer et al., 1995; Nakajima et al., 1998). The reported enhancement factor of M_{o} is at most ∼2. For example, an enhancement of M_{o} of bulk hexagonal close packed (h.c.p.) Co from ∼0.15 μ_{B} to ∼0.25 μ_{B} has been reported, which is still much smaller than M_{s} of 1.6 μ_{B}. For Co or Fe impurities in Cs metals, on the other hand, a recent LDA + U calculation (Kwon & Min, 2000) has predicted a value for M_{o} of ∼3 μ_{B}, i.e. almost atomlike, as for rare earths. In the present work, M_{o} of Co atoms in a Pd or Cu matrix is calculated assuming ferromagnetism. The relation between M_{o} and the atomic environment is discussed and it is shown that M_{o} of Co can be comparable to M_{s} if Co is surrounded by Cu atoms as nearest neighbours.
2. Model
We assume the Hamiltonian
where H_{1} denotes the multipole d–d or f–f interaction, and the spinorbit interaction in d or f orbit is given by
with i and ν being the lattice point and the combined index of the magnetic and spin quantum numbers of the d or f orbit, m and σ, respectively; ζ denotes the The matrix element of the d–d or f–f interaction is expressed in terms of the Slater integral F^{k} and the Gaunt coefficient c^{k}(lm, lm′), with l = 2 and k = 0, 2 and 4 for the d orbit, and l = 3 and k = 0, 2, 4 and 6 for the f orbit, as follows (Condon & Shortley, 1959):
H_{2} denotes the interatomic and the atomic level,
with μ being the combined index of the magnetic and spin quantum numbers of all the orbits including d and f.
For the multipole d–d or f–f interaction in H_{1}, we apply the HF approximation: the first term of H_{1} is replaced by
with being the M_{s} and M_{o}:
of the operator including the number operator. By diagonalizing the obtained onebody Hamiltonian, we calculateand
The lattice point index i is omitted for simplicity.
3. 5f state and M_{4,5} XMCD in US
Taking into account the U 5f, 6p, 6d and 7s orbits, and the S 3s, 3p and 3d orbits, the parameters in H_{2} are determined by a firstprinciples LDA calculation with the fullpotential linear augmented plane wave (FLAPW) method in the paramagnetic state without the 5f spinorbit interaction. F^{0} = 0.76, F^{2} = 5.530, F^{4} = 4.669 and F^{6} = 2.881 eV are adopted in the Hamiltonian H_{1}. For details, the reader is referred to the work of Shishidou et al. (1999), including the F ^{0} dependence of M_{s} and M_{o}. As a result of the presence of a large M_{o}, magnetic anisotropy exits. Values of M_{5f} (= M_{s} + M_{o}) of 1.70 for the [111] direction, 1.51 for the [110] direction and 1.28 for the [001] direction are obtained, in units of μ_{B}. By comparing the total energies among the three directions, the [111] direction is found to be the easy axis in accordance with the experiment and hereinafter the magnetic moments are presented along the easy axis.
In Table 1, the calculated M_{5f}, M_{s} and M_{o} values obtained by several methods are compared. The results based on the conventional LDA with SOI reveal an absolute value of M_{5f} that is too small compared with the experimental value of 1.70 μ_{B} (Wedgwood, 1972) because of an underestimation of M_{o}. Brooks (1985) applied the OP method and obtained a larger magnitude of M_{o} and a considerable improvement in M_{}5f. According to Severin et al. (1993), the individual absolute magnitudes of M_{s} and M_{o} of Brooks are too large compared with those obtained from the analysis of the magnetic form factor. Severin et al. (1993) performed an HF calculation in which the expectation values of only the number operators specified by the spin (σ) and magnetic quantum numbers (m) are taken into account and the exchange integrals are scaled to the LDA (`scaled HF' method). Their result is, as far as M_{s} and M_{o} are concerned, similar to ours (`HF TB') with M_{s} = −1.49 μ_{B} and M_{o} = 3.19 μ_{B}, where the offdiagonal operator as well as the number operators are taken into account. In order to estimate M_{o} well, one should include the orbitaldependent exchange potential, which is taken into account in neither the LDA nor the OP method. The values in parentheses (Table 1) of the neutron measurement by Wedgwood (1972) are from an analysis by Severin et al. (1993). The calculation was also performed neglecting the spinoffdiagonal operators in the present model (`spindiagonal HF'); the obtained M_{s} and M_{o} are apparently similar to those obtained by the HF TB method. Problems with the scaled HF and the spindiagonal HF method will be discussed below.

Fig. 1 presents the U 3d → 5f XMCD spectrum ΔF(ω) = F_{−}(ω) − F_{+}(ω), calculated with use of the unoccupied Bloch states and energy eigenvalues obtained for HF TB on the basis of Fermi's `Golden rule'; F_{±} represents the spectrum for the incident positive (+) and negative (−) helicities. The Lorentzian convolution with FWHM of 4.0 eV is adopted, which represents the U 3d corehole lifetime broadening. The calculated XMCD spectral shape is found to be in good agreement with the measurement by Collins et al. (1995), with dispersive features in the M_{5} region. The M_{5} to M_{4} intensity ratio R_{XMCD}, determined by combining the socalled L_{z} (Thole et al., 1992) and S_{z} (Carra et al., 1993) XMCD sum rules, is expressed as
where 〈L_{z}〉 = M_{o}/μ_{B} and 〈S_{z}〉 = M_{s}/(2μ_{B}), and 〈T_{z}〉 is the of the z component of the magnetic dipole operator, given by
s_{i} and r_{i} are the spin and the position vectors of the ith 5f electron, respectively. The HF TB method, taking into account the of the offdiagonal operators as well as the number operators, gives 〈T_{z}〉 = −0.36 and R_{XMCD} = 0.169, which is in reasonable agreement with the observed value of 0.13 ± 0.03 (Collins et al., 1995). The spindiagonal HF TB method, on the other hand, gives 〈T_{z}〉 = −0.22 (∼60% of the HF TB value) and R_{XMCD} = 0.292. The magnitudes of M_{s} and M_{o} (or 〈S_{z}〉 and 〈L_{z}〉) are rather insensitive to whether the spinoffdiagonal operators are included or not, since L_{z} and S_{z} are expressed by the number operators specified by m and σ. T_{z} is, on the other hand, expressed in terms of spinoffdiagonal operators as well as number operators and its Furthermore, R_{XMCD} is more sensitively influenced by the extent of spinoffdiagonal mixing in the Bloch wavefunction. It has been shown that R_{XMCD} can be a severe test of wavefunction and that the HF TB method is a promising method to calculate magnetic quantities including wave functions.
4. Co/Pd and Co/Cu superlattices
In this section, the relation between M_{o} or M_{o}/M_{s} of transitionmetal atoms and various atomic environments will be discussed. For this purpose, superlattices of Co atoms in facecentred cubic (f.c.c.) Pd or Cu hosts have been chosen, as shown in Fig. 2, for which M_{s} and M_{o} of Co will be calculated. In Fig. 2(a), linear chains of Co atoms (solid circles) are periodically embedded in the [110] direction in a threedimensional Pd or Cu (open circles) matrix. In Fig. 2(b), Co chains are in the [100] direction. In Fig. 2(c), twodimensional sheets of Co atoms are stacked in the [001] direction, i.e. forming a kind of multilayer. In Fig. 2(d), Co atoms form a threedimensional In Fig. 2, the lattice constant of the pure host metal, Pd or Cu, is assumed. The parameters in H_{2} for pure metals, where s, p and d orbits are taken into account, are determined according to Papaconstantopoulos (1986) in the paramagnetic state; these parameters are used in the calculation in the ferromagnetic state. The Co 3d majority spin states are almost filled; thus the neglect of the spinoffdiagonal operators, as discussed in the preceding section, is found to be a good approximation. The calculation for f.c.c. Co gives M_{s} = 1.52 μ_{B} and M_{o} = 0.11 μ_{B}, and that for h.c.p. Co gives M_{s} = 1.58 μ_{B} and M_{o} = 0.13 μ_{B}. The parameters in the superlattices are estimated according to Andersen et al. (1978) and the Co–Pd (Cu) transfer integrals are assumed to be the arithmetic means of the Co–Co and Pd–Pd (Cu–Cu) values. For details of the calculations, the reader is referred to the work of Okutani & Jo (2000).
In Table 2, the calculated M_{o} and M_{s} of Co for the superlattices shown in Fig. 2 are presented along with the choice of quantization axes. The number of 3d electrons is, compared with the bulk Co, increased by ∼0.2 in the model superlattices and M_{s} is decreased by ∼0.2 μ_{B}. The following observations regarding M_{s} and M_{o} of Co atoms can be made.
Result (i) arises from the fact that the Co majority spin state is almost filled and the number of minorityspin 3d electrons determines M_{s}. Results (ii), (iii), (iv) and (v) provide a picture of the environment dependence of M_{o} in the Pd and Cu matrixes. In the case of the Pd matrix, the lattice parameter is considerably larger compared with that of bulk Co, and pure Co with such a lattice constant has M_{o} of 0.36 μ_{B} as a result of a narrowing of the 3d state in the present model. A comparison between this value and those in Table 2(a) is suggestive. The effective Co–Co and the Co–Pd are similar to each other. The key factor in the enhancement of M_{o} is, in this case, the lattice parameter, while the atomic environment dependence plays a minor role. The magnitude of 0.36 μ_{B} can even be reduced a little by surrounding Pd atoms. In the case of the Cu matrix, on the other hand, M_{o} of pure Co enlarged to the Cu lattice parameter is only 0.16 μ_{B}, which is much smaller than the values given in Table 2(b). Since the effective Co–Cu is much smaller than the Co–Co in the Cu matrix and also than in pure Co metal, the neighbouring Cu atoms cause the enhancement of M_{o} of Co through a narrowing of the 3d state. This seems to be sensitively reflected in the strong environment dependence of M_{o}: in the multilayer (see Fig. 2c), a Co atom is surrounded by four Co atoms and M_{o} is somewhat smaller (0.7∼0.4 μ_{B}) compared with the case in which the Co is surrounded by no Co atoms (Figs. 2b and 2d), where M_{o} is comparable to M_{s}.
The contribution to the magnetic anisotropy energy (MAE) (ΔE) is known to be composed of differences in the band energies (ΔE_{b}) and the magnetic dipole–dipole interaction energies (ΔE_{dd}) between two orientations (Szunyogh et al., 1995). The band energy is lower for the magnetization direction with larger M_{o}; M_{s} is, in the present case, insensitive to the direction. The calculated result for the Co/Cu multilayer [see Table 2(b) for the lattice shown in Fig. 2(c)] shows that M_{o} is much larger for the direction perpendicular to the Co layer compared with that for the parallel direction. If it is assumed that Cu atoms in the Co/Cu multilayers play a role similar to that of Au atoms in Co/Au multilayers and that the main contribution to ΔE is ΔE_{b} in Co/Cu, the result is qualitatively consistent with that of the firstprinciples calculation for Co/Au performed by Újfalussy et al. (1996).
Recently, enhancements of M_{o}, determined by XMCD measurements, have been reported for Co/Pd (Wu et al., 1992), Co/Cu (Tischer et al., 1995) and Co/Pt multilayer systems (Nakajima et al., 1998). In these cases, and in a recent study of nanoscale Fe clusters (Edmonds et al., 1999), the reported enhancement factor, compared with the bulk metals, is at most double. If the monolayer sandwich of the present study is modified to two monolayers in the calculation, M_{o} of Co is considerably reduced. For a quantitative discussion, a confirmation of the atomic structure, between theory and experiment, will be needed.
5. Concluding remarks
In summary, M_{s} and M_{o} of the U 5f state in US and of the Co 3d state in various Co/Pd and Co/Cu superlattices have been calculated with an HF approximation on the basis of a tightbinding model, including the full atomic orbitals in valence states and the multipole interactions between 5f or d electrons. In US, it is stressed that inclusion of the of the spinoffdiagonal operators in addition to the number operators is crucially important when describing the Bloch state via an analysis of the U M_{4,5} XMCD spectrum. Calculations assuming ferromagnetism reveal an enhancement of M_{o} of Co in both the Co/Pd system and the Co/Cu system compared with bulk Co. The enhancement in the Co/Pd system arises from the large lattice constant in the Pd matrix, while that in Co/Cu results from the small of the Co–Cu pair compared with the Co–Co pair.
The magnitudes of M_{o} and MAE, i.e. the perpendicular or parallel anisotropy in the magnetic multilayers, are subjects closely related to each other. At present, the firstprinciples calculation of MAE has been performed only for limited systems, although the 3delectron number dependence (of magnetic atoms) of MAE has been discussed on the basis of simplified models (see for example DorantesDávia & Pastor, 1998). It is expected that in the future these firstprinciples calculations will be developed further and that realistic calculations of MAE in various systems, including magnetic multilayers, will be made.
In itinerant 3d the thermodynamical properties, e.g. the temperature dependence of magnetization, have been the main subject of theoretical studies (Moriya, 1985). So far, the temperature dependence of only M_{s} has been discussed; the temperature dependence of M_{o} has been neglected. In systems with enhanced M_{o}, the temperature dependence not only of M_{s} but also of M_{o} (or M_{o}/M_{s}) should be the subject of interest, especially as the latter may provide new information regarding electronic structure. Recent measurements of M_{o} in Co clusters on Au(111) (Dürr et al., 1999) have addressed this subject from an experimental point of view.
Acknowledgements
The author thanks T. Oguchi, A. Tanaka, T. Shishidou and M. Okutani for discussions and collaboration. This work is partially supported by a GrantinAid for Scientific Research from the Ministry of Education, Science, Sports and Culture.
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