magnetic anisotropy of quasi-one-dimensional transition metal compounds

Journal of Magnetism and Magnetic Materials 123 (1993) 153-158 North-Holland Aklm

Magnetic anisotropy of quasi-one-dimensional transition metal compounds

C. Pine t tes and C. Lacroix

Laboratoire Louis Ndel, CNRS, 166X, 38042 Grenoble-Cedex 9, France

Received 15 July 1992

The quasi-one-dimensional RCo 1 , compounds are described using a simplified tight-binding model. Due to the coexistence of quasi-localized and delocalized states having a different orbital character, a large magnetic anisotropy is found, which strongly depends on the band filling. The results are in qualitative agreement with the experimental values for the RCo 1 , compounds.

1. Introduction

Magnetic anisotropies of 3d itinerant systems are difficult to estimate starting from first principle calculations. Generally, in cubic systems the anisotropy is small because the orbital magnetic moment is quenched by the crystal field. It is usually admitted that in cubic systems anisotropy can be large only if there are degenerate energy levels close to the Fermi energy in the absence of spin-orbit coupling [1,2].

However, if the symmetry is lower, other contributions arise, which can be quite large: in particular this is the case of surfaces and thin films [3-5]. For such 2-dimensional systems there is a large contribution that can be attributed to the peculiarities of the 2-dimensional band structure; a smaller contribution comes from the crystal-field splitting between orbitals liying in plane (xy, x 2 _ y 2 ) and orbitals pointing out of plane (xz , yz, 3z ~ - r2 ) .

In this paper we present a calculation for quasi-l-dimensional systems: examples of such systems are the compounds RCo1_ , (R = La, Pr, Nd). In such compounds, the Co-Co distance within a chain is more than 2 times smaller

Correspondence to: C. Pinenes, Laboratoire Louis N~el, CNRS, 166X, 38042 Grenoble-Cedex 9, France._

than the distance between chains. A strong anisotropy energy, 49 K/Co , favoring the plane perpendicular to the chain, has been measured in LaCo 1 _, [6].

In the next section we present our model for the 1-dimensional itinerant Co system. In section 3 the expression of the anisotropy energy and orbital momentum are derived from a perturba- tive treatment of the spin-orbit coupling. In the last section the results are discussed in connexion with RCo I _, compounds.

2. Description of the model

Our model is based on a band calculation for Ni and Fe done by Weinert and Freeman using the Local Spin Density Approximation [7]. Here we are interested in a Co chain which must not differ much from Ni or Fe. A similar model has been studied by Labb6 [8] for describing orbital effects in the 1-dimensional system V3Si.

If the s -d hybridisation is neglected, the following band Hamiltonian is then obtained for the d electrons:

H = E Am-k-Era(k) - ° " T Ck+mo-Ckmo -, ( 1 ) k,m,o"

0304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

154 C. Pinettes, C. Lacroix / Magnetic anisotropy o f RCo I ~ compounds

where • m ( k ) and Am are the m-band kinetic energy and crystal-field parameter respectively, U is the Coulomb repulsion and Ix the magnetic moment per atom. The chain is directed along the z axis. Then m = 1 corresponds to the d~2 band; m = 2 and 3 correspond to the degenerate d ~ and dy~ bands; m = 4 and 5 correspond to the planar orbitals d,y and d,2 ,.:. The different band energies are related by:

Vdd ,n"

Vddo- • ~ ( k ) = • 3 ( k ) = • l ( k ) - -

and

Vdd8 e4(k ) = e s ( k ) = e l ( k ) rdd¢ ~ '

where Vdd,,, rdd.rr and Vdd ~ are the hopping inte- grals [9].

Moreover, we use the following relation between Vdd ~, gdd ~ and Vdd ~ which are in agreement with the effective values obtained by Wein- ert and Freeman: Vdd.~/rdmr = - 1 / 2 and Vdds/rddtr = 1/10. For W 1, the d~2 bandwidth, we take W~ = 2.7 eV, which is between the values for Ni and Fe given by Weinert and Freeman. Thus our model consists of 3 bands with different widths. In particular, the degenerate band corresponding to m = 4, 5 is very narrow (W 4 = 0.27 eV).

We have verified that the shape of the density of states does not affect the anisotropy energy. We have considered two different shapes for the density of states:

- the true 1-dimensional tight binding density of states which diverges at the band edges: p°(e) =

'1 2 2 (1/ar)~/-W - • , where W m is the m-bandwidth; . 4 m .

- elhptlc density of states: p°m(e)= (8/rrWm 2) × 71 2 e2

V ~ W m -

In both cases, the anisotropy energies have the same order of magnitude except when the Fermi level is close to the band edges, where non-physi- cal discontinuities are observed with the 1-dimensional density of states. Consequently, we have limited our study to elliptic densities of states.

We introduce also a crystal-field parameter A which splits the two planar orbitals (m = 4, 5)

from the three other orbitals. Thus in Hamilto- n ian(1)we put A 1 = A 2 = A 3 = 0 and A 4 = A 5 = J . Weinert and Freeman have shown that A > 0 for a pure Ni or Fe chain (A = 0.4 eV). However, for RCo I ~ systems the crystal field on Co atoms is mainly due to the R ions, which will probably give a negative A, since electrons in planar orbitals will be more strongly attracted by the R 3+ ions than the axial orbitals. In the following A will be considered as a parameter that can take either positive or negative values.

The exchange splitting is written here in its simplest form: o-UIX/2, Ix being the total magnetic moment calculated self-consistently. In principle the Coulomb repulsion between electrons depends on the orbital occupied by the two electrons, and this could be important for the calculation of orbital effects; however we have verified that it is not the case for the anisotropy energy, and we have neglected this effect in the following. The Coulomb repulsion U obtained by Weinert and Freeman for pure Ni or Fe chain is of the order of U ~ 1 eV for the different d-bands.

3. Anisotropy energy, orbital momentum and spin anisotropy

The anisotropy energy is due to the spin-orbit coupling which for 3d electrons can be considered as a perturbation:

V = A ~_,LiS i = V : + V • i

with

(2)

A V = = A E L T S ~, V i = - E ( L , + s ; - + L ~ S ? ) ,

i 2 i

V: and vV both contribute to the second order in A, however contributions will be calculated in a different way. For A we take the value given by Bruno in ref. [3]: A = 0.05 eV.

We present first the calculation for the general case and then for the case where the Fermi level is close to a degenerate point.

3.1. General case

If the Fermi level is far from a degenerate point one has the following results:

C. Pinettes, C Lacroix / Magnetic anisotropy of RCo 1 _ ~ compounds 155

(i) When the magnetic moment is along the z direction, V z lifts the degeneracy of the m = 2, 3 and m = 4, 5 bands and gives a contribution to the energy:

aE ' i ' = - ~ [ 4 + p~(e f ) ] , (3)

where p~(ef) and p~(ef) are the density of states at the Fermi level of the m = 2, 3 and m = 4, 5 bands with spin or respectively (p~,(e)= p°m(e + o~U/~/2)).

Another contribution comes from V ± by second order perturbation:

AEII 2 = A2~ ( U ( 2 , 4) + 3U(1 , 2)) , (4) ~r

where

U ( m , m') = Y'~ f(e~,(k)) - f (em,~(k))

where E,~(k)= e ra (k ) -~rUlz /2 and f(E) is the Fermi function.

(ii) On the contrary, if the magnetic moment is perpendicular to the chain, both V z and V ± contribute in a similar way:

AE±=A2~'. , f l tI '~I2 4) + J ~ ( 2 , 4)) IT

+ 3 ( I~ (1 , 2) + J ~ ( 1 , 2)) + ½U(2, 2)

+ 2 U ( 4 , 4 ) ] , (5)

where U(m, m') is defined as above and

f ( em(k) ) - f ( e m , ( k ) ) J~(m, m') = E

k

The anisotropy coefficient is then calculated as K 1 = AE± -AEII.

In the same way the orbital magnetic moment L A can be calculated to the first order in A. When the magnetic moment is along the z direction we obtain:

L,, = a[ P2* ( ef ) - Pz* ( e l ) + 2( P4* ( e l ) - P4* ( , f ) )]

(6)

and when the magnetic moment is perpendicular to the chain we have:

L ± = A 2~ [2~rJ~(2, 4) + 3~rJ~(1, 2)] . (7) Or

The orbital momentum average and anisotropy are then calculated as L = (L , + L ± ) /2 and L A

= L I I - L ± . The spin-orbit coupling also induces aniso-

tropic modifications 8Sii and 8S± of the spin magnetic moment. This modification is of the order A z and it is found to be very small (less than 0.01/±B/at). Thus we neglect these terms in the following.

We should point out that our calculations are justified if the energy difference between the ground state and the excited state is large enough: it is valid as long as the spin-orbit interaction couples states with energy difference larger than A. In particular, it is not valid for too small values of U / W 1 or when the Fermi level is close to a degenerate point. This case is studied below.

3.2. Calculation close to a degenerate point

We consider here the case where the Fermi level is exactly on a degenerate point in the absence of spin-orbit coupling, which arises for special values of the band filling. We have verified that the Fermi level never crosses such an isolated degenerate point for band filling around n = 9 (i.e. for a Co chain). Then we have calculated the anisotropy energy for two cases.

(i) We have first considered the case n = 2.5 and 7.5 (with A = 0): in these cases, if the splitting is large enough, the five bands cross exactly at the Fermi level ( k f = at/2, E l = - U l x / 2 for n = 2.5, ef = + Uix/2 for n = 7.5). Of course this is not a realistic case and such a situation never occurs when considering realistic band structures. In this case, both V z and V ± lift the degeneracy of the five bands around this point and give a contribution to K 1 of the form: K 1 =AA2+ BA z In A, where A and B depend on the different bandwidths, densities of states and their derivatives at the Fermi level.

156 C Pinettes, C. LacroLr / Magnetic anisotropy oJ'RCo I ~ compounds

1.2 1.2

0 . 6

Ef E

0

a

0.6

ef E

0

-0 .6 -0 .6

-2-- . . . . . .

0 x 0 x

k k Fig . 1. (a ) Energies of the d-orbitals as a function of the wave vector k for A = 0 with bandsplitting U # = 0 . 5 W I a n d A = 0. T h e

position of the Fermi level corresponds to n = 5.4. The dashed lines correspond to the down-spin bands and the solid lines to the up-spin bands; (b) energies of the 3 bands crossing at s~ when A 0 ( m = 1, s p i n ? ; m = 2, 3, spin $ ). The spin-orbit coupling lifts the degeneracy: dashed line = $ spin band, solid lines = mixing of T and $ spin states. The other bands are unchanged and have

been omitted for clarity. Magnetic moment is along the z-axis.

(ii) We have also considered the case shown in fig. 1, which corresponds to n = 5.4. In this case, the m = 2, 3 down spin bands intersect the m = 1 up spin band at the Fermi level. We also find an anisotropy energy of the form: Kj =AA2÷ BA 2 In A. This logarithmic term is found only in l -dimensional systems, in which the spin-orbit coupling produces a real gap between two of the bands (see fig. 1). In higher dimensions this would not be the case.

For these two cases we find large positive values for KI: for n = 5.4 we obtain K~ = + 3 0 K/a t , and for n = 2.5, K~ = + 5 0 K / a t with the parameters W 1 = 2 .7 e V , U = 0 . 8 e V , A = 0 .05

eV. Nevertheless, these results strongly depend on the value of A because of the A 2 In A term: in particular, for A = 0.01 eV, K~ vanishes for n = 2.5, whereas it is still large and positive for n = 2 and 3.

4. Results

4.1. Anisotropy energy

Results are shown in fig. 2 for both cases A = 0 and - 0 . 4 eV. The other parameters are: W ~ = 2 . 7 eV, U = 0 . 8 eV, A = 0 . 0 5 eV. Fig. 2

shows large anisotropy energies, one order of magnitude larger than the anisotropy energies measured on bulk materials: we obtain ] K 1 I > 10 K / a t with our model, whereas I K1 [ -< 1 K / a t in bulk materials [11]. Moreover, the sign of K l depends on the band filling and the crystal field parameter: the anisotropy in linear chains can favour the z axis of the chain as well as the plane perpendicular to it. As in many experimental cases, it may be noticed that the change in sign of K t would be in agreement with an ionic model [12], but this is quite fortuitous.

1 5

~ I 0

:= 5

._= 0

d 5

- 1 0 I I I I

0 2 4 6 8 0 n

Fig . 2. Anisotropy coefficient K E as a function of the band filling for ,3 = 0 (dashed line) and d = 0.4 eV (solid line).

W e t a k e U = 0 . 3 W I and units are in A 2 / W ~ (per unit cell) .

C. Pinettes, C. Lacroix / Magnetic anisotropy of RCo 1 _~ compounds 157

We have verified that the anisotropy coefficient K~ is almost independent on U as soon as U is large enough. Calculations have been made in the range 0.8 eV < U < 3 eV. Thus we have limited our study to the case U = 0.8 eV. We have also verified that the anisotropy energy does not depend much on the amplitude of a for realistic values of ~1 (0.2 eV < [ A I < 1 eV). On the other hand, we have considered both signs for A: because of the symmetry of the up and down spin bands, the d > 0 curve can be deduced from the A < 0 curve by an e lec t ron-hole symmetry. Then we can notice that the sign of A plays an important role on the sign and the magnitude of K~ for a given band filling n. Finally, by compar- ing the curves of fig. 2, we can see that when taking into account the crystal-field splitting, the anisotropy energy increases about by a factor of 2.

The observation that the anisotropy does not depend much on U implies that (see eqs. (4 and (5)) the contribution of the non-spin-flip terms, J'~(m, m'), is much larger than the contribution of the spin-flip terms, U(m, m'). Thus AE± -AEI I 2 is negative. On the other hand, the energy term, AEH1, gives a positive contribution to K x. This contribution strongly depends on the density of states at the Fermi level: in particular, it becomes very large when the Fermi level is close to the maximum of the density of states (i.e. for 2 < n < 3). Consequently the anisotropy energy is positive when the Fermi level is in the narrow band corresponding to m = 4, 5, and it is negative when the density of states at the Fermi level is small.

In RCOl_ ~ compounds, the metallic Co atoms have in fact less than nine 3d-electrons and the anisotropy favours the plane perpendicular to the chain. We clearly see on fig. 2 that a negative K 1 for n < 9 can be obtained only if A < 0. This is in agreement with the sign of A expected in the RCol_ ~ compounds as explained in section 2. With the parameters used in fig. 2, the value of K~ is of the order: K~ = - 2 0 K / C o , which is smaller than the anisotropy measured in the RCol_~ compounds, but our model is too simple to make quantitative comparisons. Moreover, the d:2 bandwidth W 1 in such compounds is not

50 , , , ,

...~m 3 0 "~

.~_

,-.] - 3 0

d - 5 0 i i l i

0 2 4 6 8 1 0 n Fig. 3. Orbital momen tum average L (dashed line) and anisotropy L A (solid line) as a function of the band filling. We take U = 0 . 3 W 1 and ~ = - 0 . 4 eV. Units are in A / W 1

(P- B/a t ) .

known precisely as well as the value of h. Thus, this calculation is satisfactory since it provides the correct order of magnitude of the anisotropy energy for such Co compounds.

4.2. Orbital momentum

We have reported in fig. 3 the orbital momentum average L and anisotropy L A. We have limited out study to the case zl = - 0 . 4 eV which, as shown above, gives anisotropy energy in good agreement with experiment.

For Co chains, we obtain L > +0.1/~B/Co which is in agreement with the value of L measured in LaCOl_~, L = + 0 . 3 ~ B / C o [10]. The sign and the order of magnitude of the orbital contribution to the magnetization L is close to that of pure bulk Co materials [11,13] and even to that of Co thin films [3]. This means that there is an important isotropic contribution to the magnetization which is not strongly affected by the band structure.

We obtain a large orbital momentum anisotropy. Moreover, fig. 2 shows that there is a strong connection between the anisotropy energy and the orbital momentum anisotropy: one has L A = +_aK~, where the + sign corresponds to n > 5 and the - sign to n < 5 and c t = 4 ~ B / A . The same relation was obtained by Bruno [3] in the

158 C. Pinettes, C. Lacroix / Magnetic anisotropy o f RCo I ~ compounds

case of 2-dimensional systems. The orbital anisotropy for Co chains is: L A = - - 0 .1 / X B / CO.

This anisotropy is two order of magnitude larger than for pure Co, where it is of the order of L A = 10-3/xB/at [11].

5. Conclusion

Due to the peculiarities of the 1-dimensional band structure, the anisotropy of Co chains is much larger than in bulk Co: strong anisotropy energies and orbital momentum anisotropy have been obtained as in the case of Co films. This anisotropy is strongly dependent on the filling of the 3d-band and on the sign of the crystal-field parameter 3. But we have shown also that it does not depend much on the value of the Coulomb energy U and of A, as soon as they are large enough.

Owing to the simplicity of the model, this calculation is satisfactory since it provides the correct order of magnitude for the anisotropy energy of quasi-linear compounds like the RCo I -~ compounds. A better agreement with experimental values can be obtained if a more realistic band structure will be used. Nevertheless, our calculation shows that the main effect is due to the existence of states with different band characters:

quasilocatized states for the planar orbitals,and delocalized states for the axial orbitals.

Acknowledgement

We thank R. Ballou for useful discussions. The present work has been supported by Eu-

ropean Commission with its Research and Devel- opment Program BIREM BREU-0068.

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magnetic anisotropy of quasi-one-dimensional transition metal compounds

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