First principles band structure calculations for rare earth-transition metal compounds: magnetization, hyperfine parameters and magnetocrystalline anistropy

Journal of Magnetism and Magnetic Materials 99 (1991) 55-70 North-Holland

Topical review: E.P. Wohlfarth lecture

First principles band structure calculations for rare earth-transition metal compounds: magnetization, hyperfine parameters and magnetocrystalline anisotropy *

R. Coehoorn

Philips Research Laboratories, P.O. Box 80000, 5600 JA Eindhoven, Netherlands

Received 6 December 1990; in revised form 16 May 1991

First principles self-consistent band structure calculations using the augmented spherical wave method were used to calculate (i) the average and local magnetic moments in Y,Fe,,B, YrCo,,B and in binary Y-Fe and Y-Co compounds, (ii) 57Fe-hyperfine parameters for Y,Fe,,B and (iii) the Ai crystal field parameters for R,Fe,,B compounds (R = rare earth). The effects of volume changes and of TM-B (TM = transition metal: Fe, Co) interactions on the magnetization are discussed. The calculated isomer shifts and quadrupole splittings for Y,Fe,,B are in good agreement with experimental data, but the calculated hyperfine fields are systematically slightly too small. The crystal field parameters A! for the two R sites in R,Fe,,B and related compounds have been calculated from the aspherical charge density of the valence electrons of the R atoms themselves. The results are in good agreement with experiment. We propose a simple model, based on Miedema’s “macroscopic atom” model for cohesion in metals, from which trends in the magnetocrystalline anisotropy can be understood qualitatively.

1. Introduction

Since the discovery of Nd,Fe,,B based permanent magnet materials in 1984 [1,2] ternary rare earth transition metal compounds have been studied very intensively. Well-characterized sam- ples, including single crystals, have been investigated by means of a large number of experimental techniques [3-51. A large amount of reliable experimental information about these compounds is now available, in particular about the R,TM,,B and the binary R-TM compounds (R = rare earth, TM = transition metal: Fe, Co or Ni). Many of these compounds have large unit cells with a large number of inequivalent TM atoms. In the R,Fe,,B structure, for example, 6 different Fe sites are present and the unit cell contains 68 atoms. The complexity of these compounds rep-

* Lecture presented at Lucas Automotive, Birmingham, UK, 20 September 1990 in memorial to Professor E.P. Wohl- farth.

resents a challenge to first principles electronic structure theory. Only recently have realistic ab initio self-consistent band structure calculations become technically possible due to the advent of fast supercomputers.

In this paper we present results of band structure calculations on rare earth transition metal compounds, with the emphasis on the magnetization, hyperfine parameters and the contribution of the rare earth atoms to the magnetocrystalline anisotropy. The magnetization (section 3) and magnetocrystalline anisotropy (section 5) are important intrinsic properties for permanent magnet applications. Studies of hyperfine spectra, in particular by means of “Fe-MGssbauer spectroscopy and ‘55Gd-Mossbauer spectroscopy, play an important role in the search for novel permanent magnet materials, because experiments can be carried out on polycrystalline powders which are often more readily available than single crystals. 155Gd-M6ssbauer spectroscopy is frequently used to obtain information about crystal fields [6].

0304~8853/91/$03.50 0 1991 - Elsevier Science Publishers B.V. All rights reserved

56 R. Coehoom / First principles band structure calculations for RE-TM compounds

57 Fe-Mossbauer spectroscopy yields information about local magnetic moments (from the hyperfine fields), but the technique can also be used to obtain quantitative information about the amount of secondary phases. This type of information is very useful in investigations of the microstruc- ture, which determines coercivity. In section 4 we present results of the first calculation of all hyperfine parameters on the six Fe sites in Y,Fe,,B: hyperfine field, isomer shift and quadrupole splitting.

The results of first principles band structure calculations can be of help in the search for novel materials in different ways. First one can use the quantitative predictions of magnetic properties as a guideline in the experimental work. It is then important to know the accuracy of the calculations. Therefore we have compared theoretical and experimental properties for a large number of systems: R,Fe,,B, R,Co,,B and binary R-Fe and R-Co compounds, with R = Y and Gd. We have also calculated the equilibrium volumes and the volume dependence of the magnetization. If one wants to predict properties of compounds which have not (yet> been prepared, one should be able to predict the correct lattice parameters.

The increased insight into the origin of certain properties can lead to a second, more indirect, way of using the results of the calculations. As an example we show that from the improved under- standing of the 4f single ion magnetocrystalline anisotropy we can set up a simple model based on Miedema’s “macroscopic atom” model for cohesion in metals [7], which describes trends in the crystal field parameters in a series of compounds with the same crystal structure (section 5). Band structure calculations on a large series of compounds with the Nd,Fe,,B structure were used to support the model.

2. Calculational method

The results which we shall present in this paper have been obtained using the augmented spherical wave method [8]. Exchange and correlation were treated within the Local Spin Density Functional (LSDF) approximation, using the form

given by Von Barth and Hedin [9], with the parameters given by Janak [lo]. The calculations were scalar relativistic, including mass-velocity and Darwin terms. Spin-orbit interaction was neglected.

The ASW method uses the Atomic Spheres Approximation (ASA). The crystal is subdivided into overlapping ‘Wigner-Seitz’ spheres, cen- tered around the atomic positions, with a total volume that equals the volume of the solid. The ASA is usually a very good approximation in densely packed metals. Its numerical efficiency is an advantage over full potential calculations. The differences with respect to the calculated magnetization which result form different choices of the sphere radii have been discussed in ref. [ll] for the case of binary Y-Fe compounds. In most cases variations of the radii within reasonable limits yield variations in the calculated magnetic moments at the experimental lattice parameters of at most 0.05pa per magnetic atom. Through- out this paper we have used Wigner-Seitz sphere radii in the ratio r%: rT : rB = 1.35 : $00 : 0.74 resulting in rR = 1.9 A and rT = 1.4 A. These values are close to the radius of these atoms in the elemental rare earth (R = Y, Gd) and transition metals (T = Fe, Co). The radii are determined by the radius ratio’s and by the demand that the total volume of the unit cell equals the sum of the volumes of the Wigner-Seitz spheres. The radius ratio between Y and Fe is the same as we have used previously [ll]. We emphasize that the same radius ratio’s were used for alZ compounds studied; no variation of these radii was used to opti- mize the agreement between theory and experiment.

We would like to stress that although the potential inside each sphere is spherically sym- metric, the charge density which is calculated after each iteration step is not spherically sym- metric. The resulting nonspherical charge density can be used to study crystal field effects and the electric field gradient at the nucleus (which leads to the quadrupole splitting in Mijssbauer spectroscopy).

We furthermore note that within the ASW method core and valence electrons are treated in different ways. Core electron wave functions are

R. Coehoom / First principles band structure calculations for RE-TM compounds 51

assumed not to overlap with core electron wave functions on neighbouring atoms. Their wave functions and energies are recalculated in an ‘atomic’ calculation within each iteration step, in order to obtain a self-consistent charge density. This is of importance for the calculation of the isomer shift and hyperfine fields.

Calculations for R,TM i4 B compounds were performed using a mesh of 6 k-points in the irreducible part of the Brillouin zone. For the large unit cell involved this number proved to be sufficient to obtain a numerical accuracy of the calculated moments of 0.05~~. This followed from a comparison with the results of additional calculations with as many as 18 k-points. Further calculational details are similar to those for binary Y-Fe compounds [ 111.

3. Calculation of magnetic moments

In this section calculated total and local magnetic moments in Y,Fe,,B, Y,Co,,B and binary Y-Fe and Y-Co compounds are compared with experimental data. Yttrium is chemically very similar to the trivalent R atoms. Since it does not possess a partially filled 4f shell band structure calculations for these compounds are possible. Upon replacing Y by a rare earth atom, the contribution of the TM sublattice to the magnetization is expected to increase slightly due to the R-TM exchange interaction. For calculations of this effect we refer to a recent paper by Brooks et al. [12].

3.1. Y,Fe,,B and Y,Co,,B

In table 1 calculated total and local moments for Y,Fe,,B and Y,Co,,B are given which were calculated at the experimental lattice parameters. For the notation of the crystallographically nonequivalent sites, we have used the choice made by Herbst et al. [13]. Compared with the choice made by Shoemaker et al. [14], which we used in an earlier paper [15], the notation for transition metal S(i,> and 8(j,) sites is reversed. For both compounds the calculated magnetization agrees well with the experimental value. The table shows that the local moments on the Fe and Co sites agree well with results of earlier Linearized Muf- fin Tin Orbitals (LMTO) calculations [16,171, except for the moment at the j, sites in Y,Fe,,B. In view of the similarity of the ASW and LMTO methods the discrepancy with respect to the moments on the j, sites is a bit surprising. This issue needs further attention. It should be noted that in our ASW calculations, as well as in the LMTO calculations, spin-orbit interaction, and hence the orbital moments, has been neglected.

In fig. la a comparison is given between the calculated local moments of Y,Fe,,B and the experimental moments obtained from neutron diffraction experiments by Givord et al. for Y,Fe,,B and Nd,Fe,,B [181. From hyperfine field measurements by Fruchart et al. [19] and from band structure calculations in which Y has been replaced by Gd [20] it follows that the Fe local moments depend only slightly on the R atoms. Therefore, large differences between moments on

Table 1

Results of ASW calculations of Y,TM,,B compounds, at V = V,,,. Local moments are equal to the integrated spin density in the atomic spheres. Results of LMTO calculations for YaFe,,B [16] and Y2Co,,B [17] are given within parentheses

Total moment Local moments

&a /f.u.) (pa /atom) &alc m=Pt TM sites Y sites B

16(k,) a’ 16(k,) S(il) 86,) 4(c) 4(e) b, 4(0 4(g) 4(f)

YaFer,B 30.6 30.5 2.11 2.31 2.22 2.51 2.40 2.01 - 0.43 - 0.39 -0.15

(2.14) (2.24) (2.10) (2.83) (2.55) (2.20) Yz%J3 19.2 19.4 1.17 1.41 1.46 1.43 1.59 0.96 - 0.33 - 0.32 -0.11

(1.21) (1.44) (1.51) (1.38) (1.75) (1.09)

a) 1 B neighbour; b, 2 B neighbours.

58 R. Coehoom / First principles band structure calculations for RI-TM compounds

2 2.5 _o z

IL” E

2.0

(4 V, Fe+ /

/’ /’

/’ /‘e

kl q

/’ /’ c-0

/I I, I I I I I /

2.0 2.5

qExpt (Q

t lb) v, COMB

t 1.5 ,.

Im

i

/‘&II k,

2 /’

/’ ‘j, 0

Y / zo

E” k, /’ c’ /

/’

1.0 /’ j/e

,‘a /. - /I I, I I I I I I

1 .o 1.5

mCFP’ 0~~)

Fig. 1. (a) Calculated local moments in Y,Fe,,B, as a function of the moments obtained from neutron diffraction for Y,Fer,B (closed circles) and Nd,Fe,,B (open squares) [18]. (b) Calcu- lated local moments in Y,Co,,B, as a function of the mo-

ments obtained from neutron diffraction [Zl].

corresponding Fe sites in the Y and Nd compounds, as reported by Givord et al. for the 4(c) sites, for example, in our opinion indicate an error in the analysis of the neutron diffraction data. Fig. la shows a fair correlation between the calculated and experimental local moments. The calculated moments tend to be about 0.2~~ lower than the experimental moments. This is probably due to the fact that the experimental moments do not contain a contribution due to the negative spin density in the interstitial regions of the crystal.

Neutron diffraction data for Y,Co,,B are, as far as we know, not available. The calculated moments are therefore compared with experi-

mental moments for Nd,Co,,B [21] (see fig. lb). In view of the experimental uncertainty of the neutron diffraction data, which is 0.2-0.3~,, the agreement between theory and experiment is good.

One of the interesting aspects of the magnetic properties of R,Fe,,B compounds is that the magnetization of the Fe-sublattice is higher than one would expect from the binary Y-Fe compounds: for Y,Fe,,B, Eexpt = 2.18~.,, whereas for Y,Fe,, (with an almost equal Y: Fe ratio) m expt = 2.08~~. On the other hand, for Y,Co,,B the magnetization of the Co-sublattice is lower than one would expect from binary Y-Co compounds: for Y,Co,,B, Eiexpt = 1.39j+, whereas for Y,Co,,, EeWt = 1.62~~. This difference between Y,Fe,,B and Y&o,,B can be understood from a consideration of two effects of the presence of B atoms on the magnetization.

The first effect of the presence of B atoms is that the average TM-TM distance in 2-14-1 compounds is slightly larger than in the binary Y-TM compounds. From the total density of states (DOS) of Y,Fe,,B in fig. 2 it can be seen that the Fermi level is just at the upper edge of the spin up 3d band. At a slightly compressed volume Y,Fe,,B would be a weak ferromagnet. On the other hand, Y,Co,,B is a strong ferromagnet (see fig. 2). Generally the magnetization is more stable against a change of volume in strong ferromagnets than in weak ferromagnets (see the explicit calculations for Y-Fe and Y-Co compounds in

30”““““‘,S’,“’ -10 -5 0 5

energy (eV) Fig. 2. Total density of states of Y,Fe,,B. The dashed-dotted

lines indicate the Fermi level positions in Y,Co,,B.

R. Coehoorn / First principles band structure calculations for RE-TM compounds 59

Y2Fel $ 3

Fe(h) j t 1 Wk2) I

t 1 Y(f) j tl

Energy (eV)

Fig. 3. Partial density of states for nonequivalent sites in Y2Fe,,B.

section 3.2 and the more quantitative discussion of this subject in ref. [ll]). B addition then leads to a larger increase of the magnetization due to expansion of the lattice in Y,Fe,,B than in

Y&o,,B* The second effect of B on the magnetization is

the decrease of the moments of the Fe or Co atoms which have one (16(k,) sites) or two (4(e) sites) B neighbours, due to hybridization of B(2p) states with TM(3d) states. This effect is largest for the 4(e) sites, as expected, and larger in Y&o,,B than in Y,Fe,,B (see fig. 1).

B-Fe hybridization can be studied from the local densities of states of Y,Fe,,B in fig. 3. The local DOS of 4(e) and 16(k,) Fe-atoms shows two peaks at about - 8 and - 9 eV. These states are mainly of B-2s character. Interactions between 2s states on B aioms, which form pairs at a distance of about 3.1 A, in the z = 0 and l/2 planes of the unit cell, can take place via their TM(qe)) and

TM(l@k,))) common neighbours. The figure shows that the other Fe atoms hardly interact with B. This conclusion is also expected to be valid for hybridization with B-2p states (which form the main part of the broad band in the B-partial density of states above - 7 eV), although this cannot be seen easily from fig. 3.

3.2. Y-Fe and Y-Co binary compounds

In order to obtain a more general insight into the accuracy of first principles calculations of magnetic moments we have performed calculations for binary Y-Fe and Y-Co compounds. For Y-Fe compounds a detailed analysis of the. results has been given previously [ 11,22,23]. We refer to these papers, and to the compilation by Villars and Calvert [24] for the lattice parameters and atomic positions. “Experimental” volumes for hypothetical compounds YFei2, YFe, and YCo,, were obtained by interpolation, using the volume of structurally similar compounds.

An important distinction between Y-Fe and Y-Co compounds is that Y-Fe compounds are weak ferromagnets (the spin up 3d band is not completely occupied) whereas the Y-Co compounds are strong ferromagnets for Y concentrations below 20 at%. As an example we show in fig. 4 the total DOS of Y,Fe,, and YCo,. Strong ferromagnetism, as is found for the Y-Co compounds, has two important consequences. First, the volume dependence of the magnetization is much smaller than for the Y-Fe compounds (see fig. 5). The relation between the density of states at the Fermi level and the volume dependence of the magnetization can be understood from the

-10 -5 0 5 -10 -5 0 5

Fig. 4. Total density of states for Y,Fe,, (rhombohedral) and YCo,, calculated at the theoretical equilibrium volume.

R. Coehoorn / First principles band structure calculations for R&TM compounds

1 .o 0.5 L

/x

‘&O23

I , , ,

0.90 0.95 1 .oo 0.90 0.95 1 .oo

V/Lpt VlVexpt

Fig. 5. Calculated volume dependence of the average magnetic moment per transition metal atom in Y-Fe and Y-Co compounds.

Crosses denote the moment at the calculated equilibrium volume.

Stoner-Wohlfarth model for the formation of magnetic moments, as discussed for Y-Fe compounds in ref. [ll]. A second consequence of strong ferromagnetism is that the moments depend only weakly on the structural details. In the case of Co compounds the local moments are all close to msPi, = 1~50~~. This is apparent from fig. 6, in which the local magnetic moments in Y-Co compounds (calculated at V,,,> are given as a function of the magnetic moments at corresponding sites in Y-Fe compounds. The relative insen-

,.5,..,ij ‘[ ,,,I / 1 I I I / I I I I I I I Ii I 1 .o 1.5 2.0 2.5

Magnetic moment &/Fe atom)

Fig. 6. Calculated Co magnetic moments on different

nonequivalent sites in binary Y-Co compounds, as a function

of the calculated Fe moments at the same sites in the corre-

sponding Y-Fe compounds.

sitivity of the Co moment to its surroundings follows from the strong ferromagnetic character of the moments and from the fact that in metals all atoms are very close to charge neutrality. If we assume that there are on average 0.5 sp electrons per Co-atom, which do not contribute to the magnetic moment, then 8.5 3d electrons are left. The spin up 3d band contains 5 electrons and consequently there are 3.5 electrons in the spin down 3d band, resulting in a moment of 1.5~~.

The equilibrium volume I/talc was obtained from the variation of the total energy with the unit cell volume. Its value, expressed with respect to the experimental equilibrium volume, is indicated by crosses in fig. 5. For Y-Fe and Y-Co compounds the calculated equilibrium volume V ca,c is systematically 6-7 and 4-6% smaller, respectively, than Vex,,. This point of disagreement is generally attributed to a failure of the local spin density approximation, within which exchange and correlation within 3d-transition metals is treated incorrectly [25].

In fig. 7 the calculated magnetization is given

at KX,, and at Kale, as a function of the Y concentration. The figure shows that the decrease of the magnetization with increasing Y-concentration is described well by the calculations. It


0.6 I I ‘. :

I I : :,

I I I :

0 10 20 30

atomic percent Y

Fig. 7. Calculated average moments at the experimental and calculated volume, and experimental moments, for Y-Fe and Y-Co compounds. Calculated data for the stable compounds

are connected by broken and dotted lines.

is interesting to note that in the case of Y-Fe compounds the decrease of the magnetization is not gradual. Y,Fe,, has a larger magnetization than one would expect from the general trend in the series. This was also found from the calculations. The effect is related to the relatively large volume per atom of the compound [ill.

For Fe, and for Y-Fe compounds, the spin moments that are obtained at I& are 0.06- 0.22~~ smaller than the total experimental moments ZIexpt, and the moments that were obtained at V& range from being almost identical

to zexpt (for YFe,, and Y,Fe,,) up to at most 0.14~~ larger than Eiexpt. If we assume that the average orbital moment is 0.05-O.lOp, per Fe atom, then the moments calculated at l&C are in better agreement with the experimental spin mo-

ments than the moments which were calculated

at Kxpt * For Co, and for Y-Co compounds, the mo-

ments at I/talc and at I/expt are smaller than the experimental values, for Y concentrations below 20%. This can be attributed to the neglect of the orbital moments. For Co the experimental value

Of morb is 0.15~~. The discrepancy between Eexpt and E,,,,(I/,,,,) suggests that for Y&o,, as well as for YCo, E,, = 0.2~~ per Co atom. For YCo, high values of Sorb are evident from neutron diffraction [26].

We conclude that the magnetization of Y-Fe and Y-Co compounds can be predicted from first principles theory with an accuracy of about 0.2~~ per TM atom. Calculations of the orbital moments, by the inclusion of spin-orbit interaction, are necessary to improve the accuracy. For Y-Fe compounds the moments that are calculated at the experimental volume are significantly too high. Calculations for the calculated volume, which is about 6% less than the experimental volume, in many cases yield values for the avtr- age spin moments which are in better agreement with experiment. However, this is not always .the case: for hypothetical YFe12, for which the magnetization can be obtained rather accurately by an extrapolation of experimental data for YFe,,_,V, compounds [27], the calculated magnetization at the calculated volume is definitely too small.

4. Hypertine parameters at Fe nuclei in Y,Fe,,B

4.1. Hyperfine fields

Hyperfine fields yield information about the local moments on the six crystallographically different Fe sites. It is often assumed that hyperfine fields are proportional with the 3d moments. However, recently we have shown from first principles calculations that for Y-Fe compounds‘ and Y,Fe,,B, the assumption of a constant proportional factor between hyperfine. fields and the 3d moments can lead to errors of about 0.3~~. in estimates of the moments from the hyperfine fields [28]. In this section we discuss the case of

62 R. Coehoorn / First principles band structure calculations for RE-TM compounds

40 I _ V, Fed

/ /

/ /

/ /

F / Y / 8 30-

k2 ’ *

2 / /j, c

j2 * ‘+’ *

/ kl / -o-

- / e _ /

/

20” I I I ’ 1 I I I 20 30 40

-B,, exp’(T)

Fig. 8. Calculated Fermi contact contributions to the hyperfine fields in Y,Fe,,B, as a function of the experimental

(total) hyperfine fields [30-321.

Y,Fe,,B. We have calculated the Fermi-contact contribution to the hyperfine field, B,, by means of the relativistic expression given by Bliigel et al. [29]. In the nonrelativistic case B, is proportional to the spin density at the nucleus, but in the relativistic case the spin density within the Thom- son radius rT (= Ze2/4re,mc2) is most important. For Fe or = 72 X lo- l5 m, which is much larger than the nuclear radius (4 x lo-l5 m).

Fig. 8 shows a comparison of the calculated Fermi-contact contributions to the hyperfine field, B,, with the experimental effective hyperfine fields [30-321. The assignment of subspectra due to atoms on 4(c) and 4(e) sites cannot be made on the basis of the Mossbauer spectra only. A good agreement with the calculated hyperfine parameters is obtained if the highest of the two hyperfine fields is assigned to the 4(c) site. Apart from a systematic underestimation of the hyperfine field, which was also found for Y-Fe compounds and for Fe [28], a fair correlation between theoretical and experimental data is obtained for five of the six sites. For Fe the discrepancy between theory and experiment was already noticed by Bhigel et al. [29] and by Ebert et al. [33]. The disagreement becomes even worse if the positive orbital contribution is taken into account (a few T for Fe [28,33]). Bliigel has attributed the discrepancy to a failure of the LSDF approximation. The calculated hyperfine field at the high moment site (8Cj2)) seems too low (in absolute value). This

might indicate that the (positive) orbital contribution to the hyperfine field, which was neglected in the calculation, is much smaller for this site than for the other sites.

4.2. Isomer shifts

The isomer shift at the six Fe-sites in Y,Fe,,B has been obtained from the charge density p(O) at the nucleus, using the following expression:

IS = -0.22(p(O) -PrJO)), (1)

where the isomer shift IS is given in mm/s and where the charge density is given in units (electrons/a:). The reference material is bee-Fe at 0 K. Experimental data which have been obtained with respect to Fe at room temperature should be corrected by adding +0.118 mm/s. The propor- tionality factor (0.22) in eq. (1) has been taken in accordance with the value used by Eriksson et al. [34]. It is proportional to the difference between the radii of the excited state and the ground state of the nucleus. For the evaluation of the charge density at the nucleus we have used the method described by Eriksson et al. [34].

In fig. 9 calculated and experimental [30-32,351 values for the isomer shift are shown. Most recent experimental determinations of isomer shifts are in good mutual agreement. The spread in the experimental values is indicated in the figure by the length of the error bar. Only for the 4(c) sites there are appreciable differences. We find a value

ISca~c(mm/s) / /

, O.lO- ” &

/ I

-o.lo~ey , I

0.10

k2 g-c ISexpt(mm/s)

-07 -O.lO- /

/ /

/

Fig. 9. Calculated isomer shifts for Fe atoms in Y,Fe,,B, as a function of the experimental values [30-32,351. The length of the error bars corresponds to the spread in the data given by

different authors.


of -0.08 mm/s for this site, in agreement with the results of Denissen [30] and Onodera et al. [31]. In view of the numerical precision which is needed to calculate the effect (relative charge density differences of the order lo-‘1 the agreement between theory and experiment is good. For Fe atoms at the S<j,> site the distance to the neighbouring atoms is relatively large, which is expected to lead to a relatively low electron density at the nucleus. This is confirmed by the high IS for this site.

4.3. Quadrupole splittings

A comparison of calculated and experimental quadrupole splittings (QS) can tell us how well the asphericity of the charge density around a certain Fe atom is described by the calculation. In the second place it can be used if an assignment of the different subspectra in the Mossbauer spectrum is a matter of debate. This is the case, for example, for the 4(c) and 4(e) subspectra in R,Fe,,B compounds. Grandjean et al. [35] have used a different assignment from several other authors [30-321.

The quadrupole splitting QS is related to the electric field gradient V,, (= a*V/aZ*> at the Fe nucleus by

QS=eQ1/,,/2, (2) where Q is the quadrupole moment of the 57Fe nucleus in the excited state, and where in the case of Y,Fe,,B (uniaxial magnetocrystalline anisotropy) the z-direction is parallel to the c-axis. V,, is determined by the asphericity of the charge density p(r, 0, 4) around the nucleus:

v,, = & drp(r) 1 i 3 cos%(r) - 1

I r3 . (3)

0

We have calculated V,, from the charge density of the 4p and 3d valence orbitals of the Fe atoms. Possible contributions to V,, from the region outside the Wigner-Seitz sphere of the Fe atom for which V,, is calculated have been neglected. Calculations of Vz, for elemental hcp metals and for Gd atoms in GdT,Si, compounds (T = 3d, 4d, 5d transition metal atom) have shown

/

Fig. 10. Calculated quadrupole splittings for Fe atoms in Y*Fe,,B, as a function of the experimental values [30-32,351. The length of the error bars corresponds to the spread in the

data given by different authors.

that this approximation is fairly accurate [36]. We refer to ref. [36] for the calculational details of this procedure.

Fig. 10 shows a comparison of calculated and experimental values of the quadrupole splittings in Y,Fe,,B. The error bars indicate the spread in the experimental data given by different authors [30-32,351. The agreement is satisfactory. The calculations support the assignment of 4(c) and 4(e) subspectra given in refs. [30-321.

The good agreement between calculated and experimental hyperfine parameters for Y,Fe,,B strongly supports the statement that even for crystallographically very complicated systems first principles band structure calculations yield realistic results; this could be helpful in the interpreta- tion of the rather complex hyperfine spectra.

5. Magnetocrystalline anisotropy

The large magnetocrystalline anisotropy of Nd 2 Fe i4 B permanent magnets originates mainly from the rare earth single ion anisotropy. This contribution is due to the interaction between the aspherical electrostatic potential at the rare earth atoms and the aspherical charge density of the rare earth 4f shell. The interaction energy depends on the orientation of the 4f magnetic moment, because a rotation of the magnetic moment leads to a rotation of the 4f charge cloud, due to spin-orbit interaction.

64 R. Coehoorn / First principles band structure calculations for R&TM compounds

In principle, it should be possible to predict the electrostatic potential on rare earth atoms from ab initio self-consistent band structure calculations. Very few calculations of this type have been reported in the literature. Frequently one assumes that crystal fields on rare earth atoms originate from the charges on the other atoms in the lattice. Practical calculations are then often performed within the point charge model, in which the charge density outside the central atom is replaced by an effective set of point charges on the neighbouring atoms. Sometimes a satisfactory description of crystal field effects can be obtained using point charges [37-391. However, recent non-self-consistent band structure calculations of crystal field parameters for Nd,Fe,,B by Zhong and Ching [40] and self-consistent band structure calculations of electric field gradients in GdT,Si, compounds by Coehoorn et al. [36] indicate that in rare earth-transition metal intermetallic compounds the asphericity of the valence electron charge density of the rare earth atoms itself forms the dominant contribution to the lowest order crystal field parameter Ai, and not the charge density on neighbouring atoms. In this section we report results of calculations of the crystal field parameter A; for R,Fe,,B compounds. In section 5.1 the calculational method is discussed. In section 5.2 results are given, and in section 5.3 a simple model is presented which describes trends in the 4f-anisotropy.

5.1. Calculational method

We restrict ourselves here to a discussion of the interaction between the quadrupole moment of the 4f shell with the quadrupole component of the electrostatic field due to all other electrons. This leads to the lowest order contribution to the magnetocrystalline anisotropy, which in many cases dominates the higher order terms in the temperature range which is of interest for practical applications around and above room temperature [31.

Under the condition that the R-TM exchange interaction leads to a splitting of the MJ energy levels of the 4f shell which is much larger than the bare crystal field splitting of these levels

(which is often a good approximation) the first order anisotropy constant K, is given by [41,421

K,(T=O) = -~cuJ(r2)4f(2J2-.l)A;, (4)

where a,(r2)4f(2J2 -.I) is equal to the axial quadrupole moment of the 4f shell of a free rare earth atom in the state MJ =.I. The sign of the second order Stevens coefficient (Ye indicates whether to second order the 4f-charge cloud is ‘discus-like’ (a, < 0, e.g. Nd), or ‘rugbyball-like’ (aJ > 0, e.g. Sm). Values of LYE for all rare earth atoms in their ground state have been tabulated by Hutchings [43]. Eq. (4) reduces the calculation of the rare earth contribution to the magnetocrystalline anisotropy to the calculation of the crystal field parameter Ai, because all other quantities in eq. (4) are well known from atomic theory.

The crystal field parameter Ai is a measure of the lowest order deviation from sphericity of the electric potential at a lattice site with hexagonal or tetragonal point symmetry. The corresponding term in the spherical harmonics expansion of the potential is proportional to (3 cos2B - l), where 13 is the angle with the c-axis. It is convenient to split Ai into two terms, A;(lat) an A$al), due to the charge density outside (‘lattice contribution’) and inside (‘valence electron contribution’) the rare earth Wigner-Seitz sphere, respectively. We will concentrate our discussion on A;(val). Within a Gd Wigner-Seitz sphere the charge density due to one single electron wave function YJJ can be decomposed in the following way:

lV12

= Ia,1211Crs12

+ l ap, I 2 I +bp, I 2 + other 6p contributions

+ I adxy I 2 I $Ldxy I 2 + other 5d contributions

+ a$a&s**,, + a&*&p*l

+ other cross terms.

The total charge density is characterized by the coefficients a*aj, summed over all occupied states. We remark that the radial part of the atomic-like wavefunctions $i in the Wigner-Seitz sphere is not fixed, but that it depends on the


energy of the eigenstate P. For the technical details we refer to the paper by Williams et al. [8]. In the cases i = j the coefficients can be called ‘occupation numbers’: ns, nP,, nPY, nPZ,

‘d,y, etc. We will use an abbreviated notation for p and d occupation numbers: n,, n,, n,, n,,,, etc. The first term in the charge density does not contribute to Ai, because of its spherical symmetry. The cross terms in the charge density (i #j) also do not contribute to A;, except for the sdzz terms, which have been neglected. The p and d contributions to Ai are given by [15]

-e2 1 &(val, P) = G 3 An,&

0

and

-e2 1 5.2. Crystal fields in R, Fe,, B compounds A;(val, d) = G 7 An,S,,

0 It has been found experimentally that for R,Fe,,B compounds At is similar for most R atoms. For Nd,Fe,,B the value of Ai is not significantly different from the value of Ai for the compounds with R = Tb, Dy, Ho and Er [44]. Therefore it can be expected that the results of a calculation of A; for Gd,Fe,,B can be compared directly with the experimental results which have been obtained for the R atoms which where mentioned above. We have treated the 4f states as band states, with the spin down band completely occupied, and the spin up band completely empty, and with a 4f moment which was oriented antiparallel to the Fe3d moments.

respectively, where

(5)

with 1= p, d, and where r < (r,) is the smaller (larger) of r and R. The functions p,(R), p,(R) and p4&r), are the radial charge densities of 6p, 5d and 4f electrons, respectively, normalized over the Wigner-Seitz sphere. The parameters An,, and And, which give the degree to which the p and d shells are prolate or oblate, are defined in terms of the occupation numbers:

An,=$z,+n,) -n,,

and

(8)

An, = n,2_,,2 + nxy - i( n,, + nyz) - n,z. (9)

It is customary to express A; in units of Kac2, where a, is the Bohr radius, and to express S, integrals in units of ai3. The prefactors of An,S, in eqs. (5) and (6) can then be replaced by 62.7 X

lo3 and 44.8 x lo3 Ka,, respectively. The radial charge densities of rare earth 6p, 5d and 4f shells, and the occupation numbers which are needed to evaluate A:(val>, can be obtained easily from

Table 2 Results of ASW calculations of A;(val) for Gd,Fe,,B. Units: ai for S; Ka;’ for Ai(val)

4(f) sites 4(g) sites

spin t spin 1 spin f spin _1

An, - 0.020 -0.016 -0.015 - 0.015

A”, - 0.018 - 0.036 - 0.021 - 0.047

% 0.091 0.083 0.091 0.082

sd 0.065 0.076 0.064 0.075 At(val, p) 114 83 86 77 &val, d) 52 122 60 158 A$total) 371 381

ASW calculations, which use an atomic-like basis set within the Wigner-Seitz sphere.

Results of calculations of A;(val) in Gd,Fe,,B are given in table 2. The table shows that the 6p and 5d contributions to A: are of the same order of magnitude. The average numbers of p and d electrons in each orbital are about 0.12 and 0.13 electrons, respectively, for spin up electrons, and 0.15 and 0.20 electrons, respectively, for spin down electrons. The values of An, and An, which are given in the table result from deviations from these averages of about 10% for each individual p or d orbital. For the (f) and (g) sites we calculate that A$al) is equal to 371 and 381 Ku:, respectively. Experimental (site-averaged) values for R = Nd are A; = 300 Ku: [44] and At = 350 Ku; [451. The experimental value of Ai agrees fairly


well with the calculated valence electron contribution to Ai, which suggests that the lattice contribution &(lat) is only of minor importance. More evidence for the statement that the lattice contribution to the asphericity of the potential on rare earth atoms in metals is small compared to the valence electron contribution has been obtained from a comparison of calculated and experimental electric field gradients at Gd nuclei in a large number of intermetallic compounds, including Gd,Fe,,B [l&46]. We expect that this is the general situation in metals, where charges are screened very effectively (with small screening lengths), resulting in effectively almost neutral atoms. Remember that from a comparison between calculated and experimental electric field gradients on Fe nuclei in Y,Fe,,B (section 4.3) we also found that the neglect of the lattice contribution to I’,, gives a good first approximation.

For the calculation of A@al) we have used the ASW method, which employs the Atomic Spheres Approximation (ASA). Recently, Daalderop [52] has investigated some of the consequences of the use of this approximation, for two specific systems. A comparison of self-consistent full potential (FLAPW) calculations for Gd and GdCo, with warped muffin-tin calculations (in which the on-site potential and is spherically symmetrized) showed that the calculated on-site crystal field parameter (&!(val>) differs in the two cases by about 15%. Further investigations of this point should establish in a more general way to what extent the ASA affects the calculated values of &!(val).

5.3. A simple model

We would like to have a simple model to predict trends in the asphericity of the rare earth valence electron charge density, which as we have seen is the main origin of the magnetocrystalline anisotropy. In a study of electric field gradients (V,,) on Gd nuclei in GdT,Si, compounds (T = 3d, 4d, 5d transition metal) we found that the dependence of V,, on the T atom could well be understood from the trend in the asphericity of the valence electron charge density which is ex-

petted within Miedema’s “macroscopic atom” model for cohesion in metals [7]. In this subsec- tion we will formulate the model and show that it is also applicable to R,Fe,,B compounds.

Within the macroscopic atom model, alloying is viewed as bringing together atomic cells, which possess some properties of the elemental metals: the electronegativity 4*, the volume per atom, and the charge density at the boundary of the Wigner-Seitz cell (indicated by the parameter n ws in the model). Upon alloying, the discontinu- ity of the charge density at the cell boundary between neighbouring atoms must be eliminated. A possible process, which leads to an aspherical charge density in the cell, is the charge redistribu- tion between orbitals with the same 1, but with different azimuthal Cm> quantum numbers. The model parameters II,, have been given for all metallic elements [7]. They are strongly corre- lated with the actual charge densities at the edge of the Wigner-Seitz-cells in elemental metals [47].

Based on this model of alloy or compound formation, we come to the following qualitative prediction for Ai of a certain rare earth atom:

1. If the position of its neighbours with the highest values of n,, is within or close to the plane perpendicular to the c-axis containing the rare earth atom, An, and An, are expected to be positive (A! < 0). 2. On the other hand, An, and An, are expected to be negative if these neighbours are situated on or close to the line parallel to the c-axis through the central rare earth atom (A! >

0).

The latter situation is shown schematically in fig. 11 (“metallic model”). Note that the model which we employ is entirely different from the point charge model (“ionic model” in fig. 11). Both models yield different predictions for trends in A; in a series of compounds with identical crystal structures in which the type of neighbour atoms of the rare earth atoms is varied. This can be concluded from fig. 12, which shows that in a transition metal series the value of n,, is not proportional to the electronegativity parameter

+*9 which is the key parameter in the ionic model. For Ru, Rh and Pd, for example, $* is


Ionic model (point charges) Metallic model .y:. . . . . . :.,:...:,. l-----l ;,:: “.‘,‘.

.;;“e”tra(;: j.i: :. ,,..., 5 . . . . . . ._ .. _; ..

.::.~:.,,:,.:::‘. ,’ ; _. ..‘..::;‘.‘. m ‘:ieutril : : .... ‘neutral’.

:, .’ ., _.._, ‘_. ._ .‘. ‘: .‘. :::: . . . . . . :_ ‘,.l::::,:: . . . . . . .

:.;: :. >:; . . . . . ..~. 1 ,. . . . .$eutral$ L-l ::, .: ::.:,.. ..x;: . . ., ._.

Fig. 11. Schematic view of the charge distribution in two different models which lead to a positive value of the crystal field parameter A! on a rare earth atom. Both figures show a cross section of the YZ-plane with the crystal. The squares denote Wigner-Seitz cell boundaries. Wigner-Seitz cells have been taken as being cubic for simplicity. The central Wigner- Seitz cell contains a rare earth atom. Within the “ionic” (point charge) model PI! follows from charge differences between the neighbouring atoms. Within the “metallic model” differences in the electron density at the cell boundary of the central atom (indicated in the figure with different densities

of points) are important.

almost the same, whereas the IZ,, values differ greatly.

Striking evidence of the usefulness of the nws- based model comes from a study of the effect on At of substitutions of the Fe atoms on 4(c) sites in R,Fe,,B compounds by other atoms with ap- proximately the same size. Yang et al. [48] have found that Si substituted for Fe in Nd,Fe,,B

‘b N

‘.Gd

I I I I I I 4 5 6

Electronegativity (6)

Fig. 12. Charge density at cell boundary (dimensionless Miedema parameter n,,,J, versus Miedema’s electronegativity

parameter 4 * .

Nd2 Fel$SiB

0 boron

0 iron

@ silicon at 4(c)

@ neodymium

Fig. 13. Unit cell of Nd,Fe,sSi,B, with the Si atoms on the 4(c) sites.

strongly prefers this site, and that there is a drastic increase of the anisotropy field at room temperatures for 5% Si substitutions: from about 9 to about 12.5 T. Taking into account the decrease of the magnetization upon substitution, we estimate from their data that Ai increases by about 30%. Higher order contributions to the anisotropy field have been neglected. This increase of A; can be understood qualitatively from the large difference between n,,(Fe) and n,,(Si), and from the position of the 4(c) sites with respect to the Nd atoms. As shown in fig. 13, the 4(c) sites are situated within the planes at z = 0 and l/2, which also contain the rare earth atoms. The Nd atoms at the 4(f) sites have 2 nearest neighbours at 4(c) sites, at a distance of 3.40 A. The Nd atoms at 4(g) sites have two nearest neighbours at 4(c) sites at a distance of 3.11 A. Substitutions of Fe atoms on the 4(c) sites by Si leads to a decrease of the IZ,~ parameter of neighbouring atoms in the x/y plane around the rare earth, and thereby to an increase of Ai.

We have investigated the problem in more detail by performing self-consistent band structure calculations of A; for Gd,Fe,,M,B compounds, in which the M atoms occupy the 4(c)


800 e

/I vacancy I\ I \

\ \

\ \

600 \ \

\ cu

1 ‘sgi,

400 Ag

Gd2 Fe13 M1 B

or 4(f)

ov 4(g)

I I I I I

0 1 2 3 4 5 6 7

n ws (M-atom)

Fig. 14. Calculated A! crystal field parameters for the 4(f)

and 4(g) sites (open and filled circles) in Gd2Fe13M,B com-

pounds, as a function of the nws parameter of the M atom.

Triangles have been used for atoms which are much bigger

than Fe and Si. The dashed line has been drawn as a guide to

the eye.

sites. The positions of the atoms were taken the same in all calculations. For M = Si this is expected to be a good approximation, because the lattice parameters of Y,(Fe,,,,Si,,,)B are only 0.5% smaller than those of Y,Fe,,B [48]. In the case of other M atoms the structure is hypothetical: experimentally the M atoms occupy other Fe sites preferentially, occupy Fe sites almost ran- domly, or can hardly be substituted for Fe. Fig. 14 shows the calculated values of A! for the (f) and (g) sites, plotted as a function of the Miedema parameter nws. In the figure results of calculations for M elements whose atomic volume is close to the atomic volumes of Fe and Si are indicated with filled and open circles. The figure shows a strong correlation between A: and nws of the M atoms. For the late 4d metals (Ru, Rh and Pd) we found a strong variation of A& which would be difficult to explain within the point charge model because for these elements the electronegativity is almost the same (see fig. 11).

The value of Ai for M = Si is 33% higher than the value for M = Fe (the unsubstituted compound), which is in good agreement with the experimental result (see above). For Ni and Pd, Ai is slightly higher than expected on the basis of their nws values. A similar discrepancy was also found in our study of GdT,Si, compounds. For our purposes the use of a smaller value of n,, for these elements would be better.

What happens if atoms are introduced on the 4(c) sites which are bigger than Fe and Si? We have performed calculations for M = MO, Nb and Ag, without changing lattice parameters, atomic positions or Wigner-Seitz sphere radii. Fig. 14 (triangles) shows that the same trend in A; as a function of nws was found, but that the calculated values of Ai(val> are about 100 Kai2 (for Ag) to about 200 KaL2 (for MO) smaller than expected on the basis of their value of nws, and on the basis of the general trend in fig. 12. Apparently the effective value of n,, is larger than the nor- mal value if an atom is placed on a site which is too small. This is exactly what one expects, because the charge density at the edge of the Wigner-Seitz cell increases under compression. For atoms for which nws is small (Ag), this effect is smaller than for atoms with a large value of n ,,_, such as MO, as is evident from the calculations.

We have drawn broken lines through the points in fig. 14 which extrapolate to Ai = 715 Kai2 for n = 0. It is tempting to interpret this as the at&age value of Ai for a compound with vacancies on all 4(c) sites. A band structure calculation of Ai for a system with empty spheres (spheres without a nucleus) on the 4(c) sites confirms this picture. We find values of 520 and 880 (average 700) Kai2 for Ai on the (f) and 9g) sites, respectively. The effect of vacancies on 4(c) sites on the crystal field of R(4(g)) atoms is the highest, because the (c)-(g) distance is smaller than the (c)-(f) distance.

6. Summary and concluding remarks

Calculations of the electronic structure and magnetic properties of rare earth transition metal


compounds yield accurate predictions of magnetic moments, hyperfine parameters and the rare earth contribution to the single ion anisotropy. The accuracy of the average predicted moments is of the order of 0.2~~ per transition metal atom. Calculations of hyperfine parameters for Y,Fe,,B confirm the accuracy of the calculation. Calculations of all hyperfine parameters yield realistic results, and could be helpful in the inter- pretation of hyperfine spectra for complex systems.

A calculation of crystal field parameters for R,Fe,,B compounds on the basis of the charge density of the 5d and 6p valence electrons yields a value for Ai which is in good agreement with the experimental value. Also the drastic increase of Ai upon 5 at% Si substitution for Fe can be explained well by the calculations. The results have led to a simple model, which can be used to understand and predict trends in Ai. This model could be very useful in the search for novel compounds for permanent magnet applications because it leads to general predictions concerning the crystal structure of compounds of interest. In system in which rare earth planes are present, sandwiched in between planes consisting of atoms with a much higher value of nWs (e.g. Fe, Co), the rare earth atoms at the interface are expected to have a positive value of Ai. This explains the trend in the magnetocrystalline anisotropy of R-Fe multilayer systems [49-511, where R atoms with a negative value of the second order Stevens coefficient LYE (Nd, Tb, Dy) lead to a high uniaxial magnetocrystalline anisotropy. Conversely, systems which contain chains of rare earth atoms parallel to the c-axis, surrounded by atoms with higher values of nWs, are expected to have high negative values of Ai. In these cases one should use Sm (aJ positive) to obtain a high magnetocrystalline anisotropy.

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Documents

First principles band structure calculations for rare earth-transition metal compounds: magnetization, hyperfine parameters and magnetocrystalline anistropy