The physics of magnetic materials from ab-initio calculations

Current Opinion in Solid State and Materials Science 4 (1999) 499–504

The physics of magnetic materials from ab-initio calculations M. Alouani*, H. Dreysse´ ´ ( IPCMS-GEMME), UMR 46 du CNRS-Universite´ Louis Pasteur, 23 rue du Loess, Institut de Physique et Chimie des Materiaux 67037 Strasbourg Cedex, France

Abstract Recent advances on ab-initio computation of physical properties of magnetic materials are reviewed. The emphasis is put forward regarding the new development of the electronic structure methods, namely the calculation of magnetic anisotropy energy, X-ray magnetic dichroism, non-collinear magnetism, spin density wave, and spin fluctuations in materials. These theoretical advances have lead to new levels of understanding of magnetic materials. In particular, new results on magnetic anisotropy, surface and interface magnetism, and magnetic alloys will be briefly discussed.  2000 Elsevier Science Ltd. All rights reserved.

1. Introduction In the last few years, with the advance in the computer technology, it has become clear that a precise solution of the Kohn-Sham equations [1,2] is needed to produce accurate results concerning the electronic structure and magnetic properties of materials. Accordingly, ab-initio methods have been extended to treat correctly the potential of solids, the relativistic effects, and the spin polarization [3–7]. Thus new methods have been used to compute the magnetic properties of materials [4,5,7–9]. In particular, important advances have been made in the determination of magneto-crystalline anisotropy [10–12], the non-collinear magnetism [13], the X-ray magnetic circular dichroism [7,12], and the magneto-optics [14]. To be able to calculate these properties both the spin–orbit coupling and the spin polarization have to be incorporated in the band structure method. On the other hand, the discovery of the sum rules that permit the determination of the spin and orbital moments from the integrated XMCD spectra [15–17], have made the X-ray magnetic circular dichroism (XMCD) spectroscopy an interesting tool for studying magnetic properties of materials [18,19]. This is so because XMCD can probe the magnetic properties of any specific atom and orbital of the material. XMCD can be also used to explore the magneto-crystalline anisotropy (MCA) by determining the orbital moment anisotropy. On the theoretical level, Bruno connected the orbital moment anisotropy to the MCA in the special case of the 3d transition metals [20] in *Corresponding author. Tel.: 133-38-810-7006. E-mail address: [email protected] (M. Alouani).

which there are no holes in the spin-up band and the crystalline field parameter is much smaller than the spin– orbit coupling. Recently, van der Laan showed that the magneto-crystalline anisotropy is directly related to the anisotropic part of the spin–orbit coupling rather than to the orbital moment [21]. A general relation that strictly relates the MCA to the anisotropy of the orbital moments is still lacking, however. These advances in XMCD spectroscopy have triggered off further interest in the ab-initio description of magnetism. Freeman and coworkers developed a slab LAPW method to study the XMCD of transition metals and thin films [4,5]. The method was used to check the validity of the XMCD sum rules, and it was found that the orbital moment obtained from the sum rule is within 10% from the direct calculation, and the spin moment is much worse and can be off by up-to 50% in the case of Ni(001) surface. Using a different formalism based on the fullpotential relativistic linear muffin-tin orbital method, Alouani, Wills and Wilkins studies the XMCD of ironnitrides [7]. They showed that the XMCD intensity is directly proportional to the spin magnetic moment and that the spin and orbital moments obtained from the XMCD sum rules are in good agreement with the direct calculation. Later, this formalism was used by Galanakis et al. to study the transition metal binary alloys and the Heusler alloys [12]. The same method was also extended to study the magneto-optical properties of materials [14]. Using multiple scattering theory Ankudinov and Rehr studied the XMCD in Gd [22], and Brouder, Alouani, Bennemann studied the K-edge of Fe [23]. In the next section we review advances in ab-initio methods which permit the quantitative determination of the

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physical properties of magnetic materials. In the third section we discuss recent calculations of non-collinear magnetism and the use of linear response theory to study spin fluctuation and magnons. In the fourth section we show some new results of the determination of the magnetic anisotropy energy (MAE) using the local spin density approximation (LSDA) as well as the generalized gradient approximation. In the fifth section we show the application of these ab-initio methods to transition-metal alloys and thin layers, and discuss some new results for these materials. Finally, we show that disorder is an important ingredient for the correct determination of the magnetic properties of binary alloys.

2. Methodological developments Based on the work of Callaway [24], Ebert was the first to implement the relativistic effects in ab-initio methods, and to explore the magnetic properties of spin polarized electrons. In particular, Ebert implemented a fully relativistic linear muffin-tin method in the atomic-sphere approximation [3], and later extended it to determine the magnetooptical properties and the X-ray magnetic dichroism of transition metals and their binary alloys [6]. This allowed him to make the first calculation of the magnetic X-ray dichroism at the Fe K edge that was found in good ¨ et al. agreement with the experimental results of Schutz [25]. Later, Halilov and Uspenskii used a similar method to determine the effect of the spin–orbit coupling and the spin polarization on the optical conductivity tensor of 3d ferromagnetic transition metals. However, the agreement with experiment was limited probably due to their use of the atomic sphere approximation. During the same period ¨ Kubler and coworkers [26] used the Augmented spherical wave method (ASW) in the atomic sphere approximation to study the magneto-optical Kerr effect of Fe and Ni. The agreement with experiment was much better than in the calculation of Halilov and Uspenskii [27,28] due to the better calculation of the matrix elements involved in the interband transitions. The ASW method was later used to determine the magneto-optical spectra of uranium compounds [26] (US, USe, and UTe). While the diagonal conductivity tensor elements were well reproduced, the off-diagonal elements and the Kerr angle did not agree well with the experimental data. Oppeneer and Antonov also used the relativistic LMTO-ASA to study the magneto-optical properties of the so-called Heusler alloys and good agreement with the experimental data was achieved [29]. Usually spin–orbit coupling is included in the band structure calculation using perturbation theory [3]. In some cases the full Dirac Hamiltonian is solved self-consistently [30]. It has been argued that the latter description is more appropriate since the basis set has no restriction; especially when dealing with heavy elements where spin–orbit

coupling is very strong [6]. Nonetheless, perturbation theory is more widely used in the Literature [7,22,23].

3. Non-collinear magnetism The non-collinear magnetism of materials is becoming a very active research field. A recent review can be found in Ref. [13]. New experimental results on multilayers have clearly established the existence of non-collinear magnetic arrangements in metallic multilayers like the Fe / Cr or FeCo / Mn systems. In this context, two angular dependences of the interlayer coupling between two ferromagnetic films separated by a metallic spacer have been proposed. The phenomenological parabolic law proposed by Slonczweski was found in a ‘d’ band tight-binding calculation [31]. A more recent study [32], using a ‘spd’ tightbinding scheme, has shown that the bilinear-quadratic expression for the interlayer coupling energy fits nicely the calculated values. With the computer power available today, such semi-empirical Hamiltonians are the only way to determine self-consistently the direction and magnitude of each local moment magnetic of large physical systems of interest (few hundreds of atoms have to be taken in account, at least) [32,33]. However even for these type of calculations the convergence process is very slow and extremely CPU time consuming. Ab-initio approaches can be used to deal with smaller systems, or periodic systems with small unit cells. For instance spiral spin-density waves in gamma-Fe have been revisited by Bylander and Kleinmann [34,35] who have shown that, unlike LSDA, the Generalized Gradient Approximation (GGA) results cannot be made to agree with experiments [34]. However, using an expression derived by Kleinmann [36] for the exchange-correlation energy density functional for Spiral Spin Density Waves (SSDW), the ground state of gamma-Fe is found to be a SSDW. Non-collinear order in small clusters has been expected for a long time. For unsupported clusters, Oda et al. [37] have shown that non-collinear magnetism can exist in Fe 3 or Fe 5 . Stronger effects are expected for transition metal based clusters, which display antiferromagnetic, or similar order in the bulk, and in which frustrations appear in the cluster geometry [38]. Ab-initio calculations of non-collinear magnetism were also used to explain the origin of the invar effect in iron-nickel alloys [39]. It was found that the magnetic structure is characterized by a continuous transition from the ferromagnetic state at low densities to a disordered non-collinear configuration at high densities. This noncollinear behavior gives rise to an anomalous volume dependence of the binding energy. Linear response theory was also used to describe spin fluctuations in materials [40]. With this method the dynamical spin susceptibility can be determined and compared to experimental results. This allows the determination of incommensurate anti-

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ferromagnetic state by determining the q vector, which maximize the spin susceptibility.

4. Magnetic anisotropy Spectacular results have been obtained recently in the ground state determination of the magnetic anisotropy energy (MAE) of materials. Thin films of Ni on Cu(001) have been found experimentally to display an unexpected set of magnetic phases. In contrast to a simple description, the magnetization has been found to be in-plane i.e., lying in the (001) plane for the first few ad-layers of Ni. For increasing Ni thickness it rotates perpendicular to the interface plane, and finally, for much higher Ni coverage, the magnetization rotates back in plane due to a stronger shape anisotropy. This behavior is linked to a tetragonal distortion of the Ni films. The lattice mismatch between Ni and Cu is small enough to accommodate a perfect 2D growth, but large enough to lead to a contraction of the Ni(001) films. Using the spin-polarized Korringa-Kohn-Rostoker (KKR) method, Uiberackeret et al. [41] have described Ni films to a thickness up to 15 layers. Considering a uniform relaxation, the spin reorientation occurs at about seven layers of Ni, in good agreement with the experimental results (for the value of the lattice relaxation and for the critical thickness of Ni). These band structure frameworks compute the magnetic anisotropy energy as the sum of individual terms. In the pre-cited work [41], the different

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contributions of the MAE are clearly identified for the Ni / Cu(001) system. The internal contribution (which acts as a bulk term) is counterbalanced by the surface and interface term which favour in-plane orientation. When the dipolar term is added a subtle balance occurs. The description of the Ni films proposed by Uiberacker et al. has been confirmed by another work [42] where all Ni planes are allowed to relax. Using the Vienna ab-initio simulation package the equilibrium distances found are in agreement with Ref. [41], with an additional surface inward relaxation of 3%, which should not affect the trends obtained in Ref. [41]. In addition, Spisak and Hafner have demonstrated that the films structure is probably more complex: the formation of a surfactant over-layer of Cu on top of the Ni films is energetically favoured, in agreement with LEED data. When determining the magnetic anisotropy properties, atomic relaxation plays a key role. For that reason allelectron full-potential approaches are necessary; the interatomic distances have not to be assigned a priori. The FP-LMTO method has lead to very interesting results. A presentation of such a framework can be found in the contribution of Eriksson and Wills in Ref. [10]. For instance, Galanakis et al. [11] have performed a systematical study of magnetic anisotropy energy of FCT Fe 0.5 Pd 0.5 alloy vs. the lattice parameters a and c; their results compare nicely with experiments in thin films. This method has been used to determine the MAE of many transition metals alloys within the local spin density approximation and the generalized gradient approximation [43]. Fig. 1 shows the MAE of many transition metals

Fig. 1. The local spin density approximation (LSDA) (filled circles) and the generalized gradient approximation (GGA) (empty circles) calculated magnetic-anisotropy energy (MAE) of XY, XPt 3 and Mn(V)Au 4 (X5Fe, Co, Mn, Y5Pd, Pt, Au) ordered alloys compared to experimental results [43] (open triangles). The easy axis for the L1 0 structure alloys and the V(Mn)Au 4 is the [001] axis and for XPt 3 the [111]. In the case of the FePt, CoPt, FeAu ] and VAu 4 alloys the theory always favours the perpendicular axis. The other binary alloys show different behavior depending on the type of approximation to the exchange-correlation potential.

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calculated within the LSDA and the GGA compared to experimental results [43].We can deduce directly from this figure that both LSDA and GGA produce the same tendencies as we pass from one system to another. But there are systems like MnPt 3 , CoPt 3 and MnAu 4 , where the two functionals present strong deviations. For all the other binary alloys, the MCA values calculated within the two approximations differ by less than 1 meV, but when the values are close to zero, as is the case for FePd, it is possible that the LSDA and GGA predict a different magnetization axis. The MCA results obtained using the LSDA and GGA are, in most cases different, which led us to the conclusion that there is no general rule favouring either LSDA or GGA as the better description of the MAE of magnetic alloys. The calculated orbital moment anisotropy is similar for both LSDA and GGA and cannot explain the differences in the calculation of the MAE. Nevertheless, from this study it seems that the LSDA results are slightly in better agreement with the available experimental results. To this claim further experimental data are needed. These results indicate the present limitations of ab-initio band calculations. Physical systems of interest usually include a large number of non-equivalent atoms (few tens at least). The need of an ab-initio method with a better CPU time scaling with the number of atoms is necessary. The recent developments of Beiden et al. [44] and Petit et al. [45] allow new interesting possibilities. In this new approach [45] a local interaction zone (LIZ) for solving the quantum mechanical problem is considered, while the Poisson equation is solved in the whole space, using a screened reference medium [46]. The KKR matrices become sparse and thus a LIZ of more than thousands atoms can be considered. This real-space scheme has been tested only for simple crystallographic arrangements but a priori it could be used for much larger systems. The implementation of a full-potential version, which exists already for the standard KKR method [47] will make this approach an attractive one.

5. Thin films and alloys Ab-initio band structure methods provide a nice tool to elucidate the behavior of adsorbed atoms, and are now widely used due the availability of efficient numerical codes 1 . It is largely recognized that inter-diffusion occurs during the growth of transition metal on a substrate. A recent paper [48] on the initial growth of Co on Cu(001) combines an experimental study and a FP-LAPW calcula1

The availability and the free diffusion of the LMTO and the LAPW numerical codes to the scientific community has intensified the use of the ab-initio methods. For the diffusion of the LMTO code see the web home page: / / http: / / www.mpi-stuttgart.mpg.de /ANDERSEN / index.html and for the LAPW code called the Wien97 code see the home page http: / / www.tuwien.ac.at / theochem / wien97 /

Fig. 2. One and two Fe layers slabs on W substrate (upper drawings) and surface view of one and two layers of Fe on W(110) substrate. The a W is the lattice constant of bulk fcc W. The calculated interlayer distances between the W and Fe d[Fe-W(I)] is 3.51 a.u., and that for the two layer system d[Fe-W(I)] is 3.68 a.u., and the surface Fe and Fe(I) distance d[F(S)-Fe(i)] is 3.65 a.u. The vacuum spacing between the two slabs for the mono-layer and for the bi-layer systems are is 3.5 a W and 5.5 a W , respectively. The surface Fe atom is represented by a filled circle, and the Fe(I) by an empty circle.

tion to show that Co atoms occupying substitutional sites in the Cu substrate act as pinning centres for subsequent island nucleation. The description of magnetic nanostructures on a noble metal Ag (001) has been also investigated by means of KKR-GF, illustrating the large number of possible magnetic arrangements [49]. For 4d and 5d elements, unexpected enhancement of the local magnetic moments is obtained. The adsorption on the (001) surface has been widely investigated (see for instance Refs. [38,50,51]). Experimental results involving middle series transition metals, particularly Mn, are still not clearly understood and a dense activity is developed [52]. The (111) cubic surface attracts also interest since it displays subtler behavior due to frustrations of antiferromagnetic coupling [52]. Fig. 2 shows the Fe / W(110) surface and relaxed Fe-Fe and Fe-W layers [53]. The FP-LMTO is capable of obtaining the correct spin alignment of the Fe layers by calculation the MAE (see Table 1). The determination of the electronic structure of semiinfinite ordered alloy requires the same techniques used in the previously reported films studies. The only change is a Table 1 Calculated Fe / W(110) magnetic anisotropy energy (MAE) or the one, two, and three Fe layer systems [53]. The MAE is decomposed intro magnetic surface anisotropy (MSA), magneto-elastic anisotropy (MEA), and shape anisotropy (SA) due to the interactions of the spins. In the case of the two-layer system it is the shape anisotropy rotates the magnetization in plane

MSA1MEA SA MAE

1 ML

2 ML

3 ML

3.35 0.08 3.43

20.05 0.26 0.21

1.45 0.43 1.88

M. Alouani, H. Dreysse´ / Current Opinion in Solid State and Materials Science 4 (1999) 499 – 504

Fig. 3. Theoretical XMCD spectrum for the Fe L 2,3 -edge of the ordered compressed fct FePd alloy and of the disordered fcc alloy compared with the experimental result for the FePd thick film deposited on MgO(001) at 623 K see Ref. [58] for more details. Copyright 2000, with permission from Elsevier Science.

larger number of non-equivalent atoms and, in some cases, the multiplicity of numerical solutions [54]. For semiinfinite disordered alloys the Coherent Potential Approximation [55] is definitely the right approach. Two contributions deserve special notice [56,57]. In Ref. [56], Turek et al. predicted a new class of magnetic materials. Taking two bulk non-magnetic transition metals as V, Ru, Rh and Pd, they showed that the (001) surfaces of the RuV, RhV and PdV binary alloys in the bcc structure, are magnetic over a broad concentration range. The magnetic moments are mainly located at V surface sites and are as large as 1 m B for alloy with 75% V concentration. For FePd the disorder is shown to play a major role for the determination of the magnetic properties. Fig. 3 presents the calculated Fe L 2,3 XMCD spectrum of the ordered compressed face centred tetragonal (fct) alloy (c /a50.954) and of the disordered face centred cubic (FCC) alloy, together with the experiment of the fct alloy. We notice that the spectrum of the disordered alloy is in a better agreement with experiment than that of the ordered alloy. This is clear evidence that the disorder in the FePd alloy is important and should be taken into account for the determination of its physical properties. The theory for the ordered alloy underestimates the L3 /L2 branching ratio, L3 /L2 51.12 (theory) and 1.32 (experiment), while theory for the disordered alloy gives a branching ratio of 1.25 in much better agreement with experiment [58]

6. Conclusion To conclude, we have shown that the new developments of ab-initio methods have allowed a quantitative determi-

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nation of physical properties of magnetic materials. For the magnetic anisotropy energy the MAE results obtained using the LSDA and GGA are in most cases different which led us to the conclusion that there is no general rule favouring either LSDA or GGA as the better description of the MAE of magnetic alloys. For the non-collinear magnetism of materials we have seen that ab-initio methods are becoming available and can explain some interesting physics like the invar effect in the iron-nickel alloys [39]. Nonetheless, for systems with many atoms per unit cell the semi-empirical tight-binding method is still highly employed, and it has been able to explain the non-collinear magnetic arrangements in metallic multilayers like Fe / Cr or FeCo / Mn systems [32]. In addition, many physical properties of thin films are now becoming possible to understand by means of ab-initio calculations. Namely, adsorbed atoms on surfaces [38,50,51], pinning centres for cluster nucleation [48], as well as magnetic state of surfaces [49,52].

Acknowledgements We acknowledge the TMR network on ‘Ab-initio calculations of magnetic properties of surfaces, interfaces and multilayers’ (Contract No ERB4061PL951423).

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