Magnetic anisotropy of ferro- and Ferrimagnets

Journal of Magnetism and Magnetic Materials 19 (1980) 396-408 © North-Holland Publishing Company

SESSION 10: MAGNETOSTRICTION, ANISOTROPY MAGNETIC ANISOTROPY OF FERRO- AND FERRIMAGNETS G. AUBERT CNRS, Laboratoire Louis NOel, associ~ ~ l 'USMG, 166)(, 38042 Grenoble-C~dex, France

The thermodynamical definitions of a magnetic anisotropy energy density and of the other relevant macroscopic quantities are recalled. The experimental situation is analysed on this basis and it is shown how one can get information on both the anisotropy energy density and the anisotropy of magnetization from torque measurements. Experimental results and theoretical models are discussed in some selected examples: itinerant electron anisotropy in 3d metals and alloys, single ion anisotropy in insultators, induced anisotropy in binary alloys.

the experimental data should be corrected for thermal expansion and magnetostrictive effects before being compared with the theoretical predictions. Even if these "corrections" have often been called upon in order to justify some discrepancies, in most cases they are not as fundamental considering the present roughness of the models.

1. Introduction The anisotropic properties o f magnetic materials are of great importance from both the fundamental and the technological points o f view. The abundance o f existing literature on this subject makes it impossible to give a significant account o f all the different aspects involved and here we shall focus our attention on some specific results obtained mainly in our group during the last ten years. A more general and comprehensive review can be found in [ 1] where more than 300 references are quoted. We limit our subject to samples free to distort themselves at a constant normal pressure and the only controlled parameters are the temperature T and the external applied magnetic field Bo. Yet as usual, we shall call energy what should be called enthalpy. The problem of magnetostriction and, more generally, the situation in the presence of applied stresses is much more complicated. The most common approach which simply consists in superimposing the descriptions of a magnetic rigid crystal and o f a non-magnetic elastic one is not always valid [2]. The main reasons of this failure are the following ones: (a) due to the magnetic torque density, the stress tensor is no longer symmetrical, (b) the existence of energetic terms, linear in strains, makes it necessary to use a finite strain theory o f elasticity. We cannot completely avoid the problem here because in most theoretical models for magnetic anisotropy, the crystal is assumed to be a rigid one and

2. Macroscopic description o f magnetic anisotropy 2.1. The relevant thermodynamical potential

The equilibrium magnetization distribution o f a fixed (but free to distort itsel 0 sample in a constant external applied field B o = #oHo and at a constant temperature T corresponds to the minimum o f a thermodynamical potential G(T, Bo). From the classical equations of electromagnetism, it follows that the total work done by the sources of the external field and by the operator holding the sample in a quasistatic infinitesimal isothermal transformation is given by (6 W)T = f f f Bo " 6M1 dm where 6M 1 dm is the variation of the magnetic moment o f the mass element dm with respect to axes attached to this element. In the case o f a fixed and rigid sample, this expression can be written (6 W)T = f f f Bo " dM dr where dr is a volume element defined in the fixed axes and M the magnetization vector field. If the sample is fixed but deformable, we can also adopt the latter expression, if we consider virtual transformations o f a rigid sample whose shape and mass distribu396

G. Aubert /Magnetic anisotropy of ferro- and ferrimagnets tion are the actual equilibrium ones. From thermodynamics, we know that the isothermal reversible work equals the variation o f the free energy F that is (dF)T = (5 W)T and d F = - S dT + f f f Bo " dM dr. The thermodynamical potential G(T, B0) follows directly from the Legendre transform G = F - f f f Bo "6M dr since dG = - S dT - f f f M" dB o dr and obviously dG = 0 for fixed T and B o. Due to dipolar interactions, the free energy F cannot be expressed as a sum over the volume of the sample o f an energy density but this can be done if we separate the so-called demagnetizing field energy gd = 1120 fffspace I'1~2 dr = -~Uo fffsample M" H' dr, where H ' is the field created by the magnetic poles equivalent to the magnetization distribution. Then we can write: F = f f f E a dr + Ua and we call Ea the magnetic anisotropy energy density. This is a local macroscopic quantity which must particularly reflect the symmetry of the crystal structure and thereby is also said to be magnetocrytalline. This expression o f F is strictly valid for single domain samples. For multidomain (or unsaturated) ones, it does not take into account wall energies and cannot be used to describe the associated irreversible phenomena. In the case o f a deformable sample, the density E a corresponds to the actual equilibrium configuration of the sample free to distort at constant pressure and thereby may differ from that o f the hypothetical rigid crystal of most theoretical models. Coming back to G, we thus have:

a

=fff[Ea

-½Po M " H ' - P o M "

Uoldr

(1)

ducing a new density:

E'a=Ea

=f f f [a a - .o n . d M ] d r

PoM.H=Ea-PoMHM

(2)

and expressing E'a as a function of the set of variables T, HM and u, we get:

dE a - poH . dM = (aEa] \ aT ] HM, U dT

L \ ~ - ~ / T,u +

V.

×

L_

\ au /T, HM

At constant T and Bo, OHM and dot can be considered as independent virtual local variations and the condition ( d G ) T , B o - 0 leads to

1 (aga \~/ru t

M=---

'

(3)

/,tO

p ° M X H = u × \ au / T,HM For a practical application of formula (4) one has to transform the condensed notation 3E'a/aU into a more explicit form. If we make use of spherical coordinates for the direction u(O, (~) with the two other unit vectors: Uo = (au/aO) 0 and u~, = sin -1 O(au/a(p)o, we get MXH

and the equilibrium condition for a fixed sample at constant T and Bo

(dG)T, Bo

397

=

-sin -1 0 \a¢

+ U'q

\aO ]T, HM, d)

/T, HM, O u0

uo

(4')

-= 0,

where H = Ha + H ' is the so-called internal field (N.B., we made use of the reciprocity theorem: f f f M • dH' dr = f f f H' . d M dr since H ' is created by the poles equivalent to the distribution M).

2.3. The anisotropy constants Any phycical scalar density such as E'a can be written as a series expansion in the form: i

Ea =~Kn(T, 2.2. The local equilibrium equations They result from a straightforward manipulation o f the expression (dG)T, Bo. Let u be the unit vector o f the direction of M: M = Mu, dM = dMu + dot X M, where dotis an infinitesimal rotation. It follows: H dM = HM dM + (M × H) • d atwith HM = H . u. Intro-

Hm)fn(U),

n

where the angular functions fn(U) make up a complete set reflecting the symmetry of the crystal lattice and the K'n(T, HM) a r e coefficients characteristic of the material. From relations (3) and (2), we get:

M = ~ Mn (T, HM) fn (u) with n

398

G. Aubert /Magnetic anisotropy of ferro- and ferrimagnets

1 (oK; i M. :-2oo -GIT '

(s)

3. Experimental methods 3.1. General principles

and Ea = ~ Kn (T, HM) fn (U) with Kn = K'n + I~oMnHM • n

(6) The coefficents Kn(T, HM) are usually called the anisotropy constants o f the material. From eqs. (5) and (6), we get a relation between the variations of K n and M n with the field:

~H--MM! T = UoH M \ O-~M ] T "

(7)

The choice of the set of angular functions fn(U) is not unique and may raise some problems if an expansion to high orders is needed [3,4]. We simply recall the most common lower order expansions used for cubic crystals 22

22

22

222+

Ea=Ko+Kl(oqo~2+O~z~3+c~3oq)+KzoqOqO~ 3

...,

where the ai's are the direction cosines of u with respect to the quaternary cubic axes, and for hexagonal crystals Ea = Ko + KI sin 2 0 + K 2 sin 4 0 + K3 sin 6 0 + K 6 sin 6 0 cos 6~ + .... where 0 is the angle of u with the hexagonal c-axis and ~) the angle of its projection in the basal plane with a binary axis. It is to be noticed that every anisotropy constant has no specific meaning in itself since it depends on the particular choice of the set of basis functions fn(u). What is physically important is the maximum order of the expansion which is needed to describe correctly the experimental data, and the energy or magnetization differences between specified directions in the crystal. The more significant differences are obviously those which we shall call AEa or AM between the directions of higher symmetry of the lattice among which we generally find the so-called easy and hard directions of magnetization, i.e., the directions of minimum and maximum Ea, respectively. As we shall show later, these differences can easily be obtain from experiments and are also easier to derive from theoretical models than any set o f constants.

There are three main classes o f methods of studying magnetic anisotropy: (a) magnetization measurements along different directions, (b) torque measurements, (c) ferro- or ferri-magnetic resonance. The use of magnetization measurements relies directly on the expression (dG)T = - f f f M • dB o dr and leads to the difference between G(T, Bo) and Gref of some reference state. This method is not very sensitive in the case of small anisotropies, and makes it difficult to study the variation of anisotropy constants with the field and can be unsafe if the reference state is not carefully chosen (for instance a demagnetized state is not a good reference state because of the irreversibilities always present at low fields). The resonance techniques make use o f small deviations from equilibrium of the magnetization distribution, induced by a perpendicular r.f. field. They could be very powerful for anisotropy studies but are in fact vitiated by the intricate problems of energy losses and of angular momentum transfer• Therefore, they are more specific of this type of studies (linewidth, g f a c t o r , . . . ) than o f anisotropy [5]. 3.2. Torque measurements Indisputably, torque measurements and related techniques are the picked tools for anisotropy studies. They imply relation (4) and consequently the density/:a. The typical experimental arrangement consists of a device holding the sample and measuring the component along a z-axis o f the mechanical torque necessary to keep the sample fixed while a uniform external field B o is rotated in the xy-plane perpendicular to this z-axis. This mechanical torque is opposite to the magnetic torque F = f f f M × B o dr •

,t

=fff, oMX~d~--fffuoMXIr dr,

(8)

where the first term results from the intrinsic anisotropic properties o f the material and the second one from the dipolar interactions (shape anisotropy). The

G. Aubert /Magnetic anisotropy o f ferro- and ferrimagnets

latter term can easily be cancelled by a suitable choice of shape of the sample (for many reasons the sphere is the best one), but it can be interesting to measure magnetization with a torquemeter. The experiment is easy to analyse if the following conditions are fulfilled: (a) spherical sample (b) Bo high enough to make the sample single domain, (c) the xy-plane is a symmetry plane o f the crystal lattice. Then it results from symmetry that the sample is uniformly magnetized and that M lies like Bo in the x y plane. In this connection, let us recall an old mistake which concerns the determination o f K z in cubic crystals. The xy-plane is chosen to be a {I 11) plane of the cubic crystal, so i f M were in this plane, the torque measured along z should not depend on K1, the lower order constant involved being K2. But a (111)plane is not a symmetry plane and when Bo is rotated in this plane, M goes oscillating about it, which introduces K1 dependent terms in the expression of the torque. These terms can be of the same order of magnitude as the previous ones, thus leading to a complete misinterpretation of the experiment if they are omitted. They can be taken into account [6], but the analysis becomes intricate enough to remove all the practical interest of such measurements (their academic interest still remaining of course). Coming back to the situation specified by the above-mentioned conditions, the directions of M and B0 in the xy-plane depend for each one on a single angle. Let 0 be the angle of M with the x-axis and be the angle o f B o (N.B., these 0 and ~ are not to be confused with the spherical coordinates introduced in eq. (4')). The measured magnetic torque for a sample of volume V being rz, we have: 1

p F z = (M X Bo) z = M B o sin(4~ - 0)

(9)

and from eqs. (8) and (4), we also have: lp :#o(MXH.)z=(aE'at V z \ 30 ]r,i-tM "

(10)

It is also important to note that at constant field intensity Be, we have rigorously dG = Fz dq~. Now at a controlled T a n d for a chosen fixed value ofBo, Fz is

399

measured as a function of the experimentally known angle q~. So, if we consider two symmetry directions 1 and 2, easily detectable because for these directions: 4)1 = 0 l, Cz = 0z and the torque is zero, J'~ rz(qS) d~b = G2 - G1 = AG and from eq. (1) with H ' = - } M , we get: V - l A G = AE a - BoAM + ~ /aoA(M2). In most cases, the anisotropy of the magnetization AM is very small with respecto to the isotropic part and we can write A(M2) ~ 2 M A M so V - I AG ~ ,Sa~"a - taoHAM = 2~"a with H = Ho - ~ M. We thus find that torque measurements give directly &E'a as a function of T and of riM = H (symmetry directions) known from H o ifM(T, H) has been measured by a different method. At a given temperature, not too close to the Curie point &E'a is in general found to be a linear function of H. This is for instance the case for Ni, Fe, Y.I.G., below room temperature, whose AE'a exhibit a perfect linearity (withon our experimental accuracy) for a 1 T range of Bo. This means from eq. (7) that in this range (sufficiently low fields as compared to exchange) both AE a and A ~ can be considered as field independent. They can be obtained as functions of the temperature only from the experimental AE'a, AM by the slope of the variation with the field and AE a by the extrapolated value at H = 0. 3.3. Determination o f the anisotropy constants from torque measurements

If one takes any interest in the anisotropy constants, the procedure is a little more complicated. Indeed, Pz is measured as a function of q~while the anisotropy constants are defined on the basis of expansions of E'a, Ea and M as functions ofO. For a given value of q~, relation (9) allows to calculate 0 from the corresponding measured Pz(q~) i f M has been measured by a different method (N.B., in this process one can in most cases neglect the anisotropy of magnetization). Then we know Pz as a function o f 0 and relation (10) leads to the determination of the Kn by identifying the Fourier expansion of Pz(0), deduced from experiment through relation (9), with the formal Fourier expansion of (~E'a/~O)T,H M. Kn and Mn are obtained from the variations of K~ with the field as for z~xEa and AM. The maximum order at which the expansion of E'a can be cut is clearly indicated by the experiment itself since it corresponds to the maximum order of

G. Aubert /Magnetic anisotropy of ferro- and ferrimagnets

400

the significant Fourier coefficients of the expansion of Fz(O). Table 1 gives some examples taken from measurement [7] in the (110) plane of Ni and Fe cubic crystals, the x-axis being [100]. The only significant (from symmetry but of course checked by experiment) coefficients in the Fourier expansions of Pz are those of sin 2k~ or sin 2kO. For each example, the left column gives the coefficients for Pz(q~) as they are extracted directly from the experimental data while the right one gives those of Pz(O) as deduced through relation (9). We first notice that these coefficients can be very different in particular for the higher order ones, which (besides other intrinsic difficulties discussed in refs. [3,4]) makes a correct determination of high-order anisotropy constants very difficult. It is obvious that the ancient procedures, often met in literature, which involve extrapolations to infinite field to force 0 into 4~are t not reliable if only because the K n are field dependent. It appears from these examples, that a large number of anisotropy constants is needed to describe the anisotropy of Ni at low temperature while two (or three,) are enough at room temperature. In the case of Fe two constants only suffice at low temperature.

3.4. Some characteristics o f a torquemeter As an example, we give the main characteristics of the low temperature torquemeter built and used in our group. A more detailed description can be found elsewhere [7]. (a) Maximum B o : 2 T, homogeneity 10 - 4 (in 1 cm3). (b) Temperature range: 2 - 3 0 0 K, controlled within 0.01 K. (c) Maximum F z : 3 × 106 dyne - cm, reproducibility better then 10 -5. (d) Minimum detectable l'z : 1 dyne - cm. (e) Absolute calibration of the torque " 10 -3. The torque is measured for equidistant angular positions of the magnet (5 degrees steps). These positions are known with respect to the sample (angle ~b) with an accuracy better than 2 X 10 -3 degree by optical means and electronic feedback control. The experiment is automatically processed and the Fourier expansion of Fz(~b) is obtained up to the 35th order by a fast numerical method which takes advantage of the equidistance of steps. All the numerous checks of the reliability of the experiment and all the procedures to get the AXE'a,zSEa, zSM and

Table 1 Fourier coefficients of sin 2k4~(left column) and sin 2kO (right column) from measurements in the (110) plane with Bo = 1.9179 T. The coefficients are given in experimental units (1 e.u. = 58.68 erg/cm 3 for Ni and 71.35 erg/cm3 for Fe). 2k

Ni

4.20 K

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

-4935.59 -7553.34 - 1082.85 -775.05 195.04 -98.19 -15.52 3.55 13.01 11.82 8.86 5.17 2.20 0.04 -1.44 -1.88 - 1.17

Ni -5194.78 -7485.58 -458.48 -29.85 38.33 37.30 27.44 16.68 7.77 1.40 - 1.93 - 2.83 -2.53 - 1.69 -0.49 1.23 1.49

294.59 K

-260.27 -408.58 18.75 - 1.65 0.22 -0.03 -0.00 -0.00 -0.02 -0.00 -0.00 -0.00 0.00 0.01 0.01 0.01 -0.00

Fe -260.88 -408.10 20.72 0.29 -0.02 -0.01 -0.00 -0.00 -0.02 -0.00 -0.00 -0.00 0.00 0.01 0.01 0.01 -0.00

4.27 K

1951.53 2902.57 -38.5 3 - 35.47 0.89 0.65 -0.14 -0.03 0.01 -0.13 -0.03 0.01 -0.04 -0.13 0.07 -0.02 0.00

1939.66 2910.98 -2.34 0.01 -0.10 0.01 0.12 -0.02 0.01 -0.13 -0.03 O.01 -0.04 -0.14 0.06 0.02 0.00

G. Aubert /Magnetic anisotropy of ferro- and ferrimagnets the various anisotropy constants are run on a computer. The samples are spheres whose diameter is chosen as big as possible (~<9 ram) in order to increase the relative accuracy of the experiment which is obviously better for a high level of torque. These samples are cut by standard techniques and are spherical within 1 tam. Their orientations in the torquemeter are checked by X-rays.

Table 2 Fourier coefficients of expansions of Pz(0) relative to a spherical sample of Ni for T = 4.20 K and Bo = 1.9179 T given in experimental units (1 e.u. = 58.68 erg/cm3).

- 10700.48

a4

-46.00

a

p

73.95

4. Itinerant electron anisotropy: 3d metals and alloys

a12

4.1. Microscopic origin

a16

33.11

a2o

3.75

a24

-5.52

a28

-3.02

a32

2.28

The magnetic properties of 3d metals and alloys are well described by an itinerant electron model. Anisotropy results from spin-orbit coupling which slightly shifts the energy bands if the magnetization direction is varied. The introduction of S.O. coupli.ng in band calculations does not raise any fundamental problem except for the increase in computationnal difficulties. The situation is of course more simple if the direction of magnetization is taken to be a direction of high symmetry of the crystal lattice, so the calculations lead in fact to the differences z2xEa and AM rather than to any set of anisotropy constants. However, as z2xEa and AM are deduced from differences between much bigger terms, the results obtained from ab initio calculations remain questionable in the present state of the art, even for Ni which has been extensively studied. Though the problem has not been completely solved for the moment, interesting answers have been recently given to some irritating questions, particularly in the rather pathological case of Ni and Ni alloys.

4.2. The anomalous magnetic anisotropy o f Ni metal at low temperature As is clearly evidenced in table 1, the magnetic anisotropy of Ni metal at low temperature cannot be described by a few anisotropy constants [3,7,8]. Raising the temperature above some 50 K drastically reduces the number of significant constants. The clue of this problem was found by Gersdorf [9] after a critical look at the data we published in ref. [3] and which is reproduced in table 2. The a~k and b~k are the Fourier coefficients of the expansion

401

r

b~ b4 b~ b~ b'lo b'12 brl4 btl6 b'la b2o b22 t b24 b26 t b28 t b3o t b32 b34

-5194.78 -7485.5 8 -458.48 -29.85 38.33 37.30 27.44 16.68 7.77 1.40 -1.93 -2.83 -253 -1.69 -0.49 1.23 1.49

of Fz(0), deduced from measurements in the (100) plane (only sin 4kO terms present) and in the (110) plane, respectively. It appears that, for a sufficiently high order 1, a'l ~ 2b'l, which implies that these coefficients are generated by a singular component of the torque only different from zero near the (100) directions. Indeed these directions appear twice as often in a (100) plane as in a (I 10) one. If we denote this singular component as Fs(0 ) and simply try Ps(0) = a sin lr0/00 for l0 1< 0 o , where 0 o is the angle of M with the closest (100) direction o f the plane and Is(0 ) = 0 anywhere else, we find: (b~k)s = 4a (n/0o) sin 2kOo n (rr/Oo) z - (2k) 2 F

t

and (al)s = 2(bl)s which fits the data for l ~> 18 fairly well with a = 125 e.u and 0 o = 0.31 rd ~ 17.8 °. The suggested mechanism [9] for the origin of this singularity is the following one: the shifting of the

402

G. Aubert / Magnetic anisotropy of ferro- and ferrimagnets

bands due to S.O. coupling is a regular function o f the direction o f the magnetization, but, as the only contribution to the magnetic anisotropy arises from those states which lie near the Fermi level, a singularity can occur if the top o f some band crosses the Fermi level when the magnetization is rotated. The influence o f temperature is immediately understood as a consequence of the F e r m i - D i r a c statistics which takes into account an increasing number o f states around the Fermi level as the temperature is increased. From existing band calculations [10], there appears a good candidate located around the point X of the Brillouin zone. Here, we reproduce the position (in Ry) o f some particular bands at point X corresponding to the chosen direction of magnetization and between brackets at a point X in a perpendicular direction: Band 10, X5~ = --0.2124 ( - 0 . 2 1 4 9 ) , Band 9, Xs~ = - 0 . 2 2 0 3 ( - 0 . 2 1 6 0 ) , Band 8, X2, = - 0 . 2 2 5 5 ( - 0 . 2 2 7 2 ) . The position o f the Fermi level with respect to the same origin is/~ = - 0 . 2 3 7 in this calculation and the S.O. coupling constant ?, = 0.0067 Ry ~ 735 c m - 1 . The pocket of holes corresponding to Xs+ and its shift by rotation o f magnetization has been confirmed by de Haas-van Alphen experiments but the X2, pocket has never been found. The suggestion is then to assume that it is much closer to the Fermi level than has been calculated. Consequently, when the magnetization is in a (100) direction, the top o f the band would be slightly above the Fermi level at the corresponding X points but would sink below it when the magnetization is rotated a few degrees apart from (100). In the quantitative model proposed in [9], the X2, band is assumed to be parabolic with the energy e o o f its top depending on 0 as e0 - ta = C(O g 02). Neglecting the variation o f the Fermi level ta due to the redistribution of electrons, there results, at absolute zero, a singular anisotropy energy density of the form fa~ =

account by spreading the singularity over an angular interval 26, which gives:

rs(0): 5B

Ps = 5B(02o - 02) 3/20 for [01 ~< 0o. The non-zero temperature of the experiment and crystal imperfections can be roughly taken into

0+6

f

(0g x2) 3/2 x

0--6

and leads to

30BO3j3(2kOo) sin 2k6 (2k) 2 2k5

t

(b2k)s =

when J3 denotes a Bessel function of the first kind. The experimental b~ coefficients for l ~> 12 can be fitted by this expression within some tenths o f e.u. if we take B = 18013 e.u., 0 o = 0.307 rd ~ 17.6 ° and 6 = 0.84 rd ~ 4.8 °. The remaining normal part o f the torque is then described by only four anisotropy constants. The remarkable success of this interpretation should encourage efforts to be made for an improvement of band calculations in order to support a reliable model for magnetic anisotropy.

4.3. The thermal variations o f zXEa and zSM o f Ni and the influence o f alloyhlg Ni with other metals We now consider the thermal variations of the energy and magnetization differences Z2XEa and ZkM between the (111 ) and (100) directions o f a Ni crystal. As can be seen in ref. [11], -zS~2a is a very rapidly decreasing function o f T which cannot be fitted by any power law of the magnetization as it v~as suggested by old models (Akulov's or Zener's tenth power law for instance), while ~ goes through a maximum and does not vanish at absolute zero. It appears that the maximum of ZXMalmost coincides with the inflexion point of AxEa. As we found this coincidence in many cases not restricted to metals (in Y.I.G. for instance), we looked for a model independent justification which is in fact given by thermodynamics. Indeed, from definitions recalled in section 2, the entropy is given by:

-B(O~ - 02) 5/2 for 101 ~< 0o,

which gives a singular component of the torque:

1

~ T / HM, td and by standard manipulation o f thermodynamics we get the relation: 2

~t

\ 3T2 }HM,U -

T + (3M/OHM)T,u

ito% 1 ~

,u

G. Aubert /Magnetic anisotropy of ferro, and ferrimagnets where CM is the specific heat per unit volume of material at constant M. For small anisotropies, the isothermal susceptibility ×T can be considered as isotropic and Xr = (3M/OHM)7~,u• We thus obtain if AM < < M and for HM = 0 d2AEa ~ ACM dT 2

T

+

2P0 dMdAM X7 dT dT

The above-mentioned coicidence in the zeros of dAM/dT and of d2AEa/dT 2 simply means that the anisotropy of the specific heat ACM is in these cases negligible. In order to interpret the thermal variations of AE a and AM by an itinerant electron model of anisotropy and considering the difficulties encountered in ab initio band calculations, we have suggested a semiphenomenological approach of the problem [ 12]. Let g(e) be the total density of states for spin up or spin down electrons of a Ni atom. The effect of S.O. coupling is to introduce a dependence ofg(e) in the direction of M, that is Ag(e) between two different directions, (111 ) and (100) for instance. We simply use an effective field approximation for exchange (Stoner-Wohlfarth model). Calculations are a little tedious because o f the Fermi-Dirac statistics and allow us to deduce the phenomenological Ag(e) from the experimental AE a and AM. The most striking result of this model is that the variations of AE a can be fitted within l0 -3 between 0 and 300 K by a single function ALla = A/~a c~T/sinh aT, with a = 0.018 K -1 and A~a = - 3 . 1 3 × 10 -2 K/at (fig. 1). This mathematical form is a direct consequence of the model and means that, in the vicinity of the Fermi level, Ag(e) is simply a harmonic function with a "period" in energy 2~TE/a = 762 cm -1 , which is precisely what can be expected from S.O. coupling effects. From A~a , we deduce an amplitude o f about 10 -2 eV -1/at for the oscillations of Ag(e), which gives an idea of the accuracy with which band calculations should be made. The problem is a little more complicated for AM, due to a possible orbital contribution. The same model is able to support a rough interpretation of the effect of alloying as long as an assumption of rigid bands is valid. It works quite well in the case of N~i-Cu alloys [12] and allows to understand in a satisfactory way the very drastic effect of alloying Ni with R ta which we report elsewhere [13].

403

"52

% al <3

la.l

0

'

460

'

260

'

TEMPERATURE(K) Fig. 1. Thermal variation of the anisotropy of energy of Ni metal fitted by a single function aT/sinh aT.

5. Single ion anisotropy in insulators

5.1. Microscopic origin It is a good approximation for insulators to consider that the magnetic moments are localized on single ions whose quantum states would be characterized, in the absence of any interactions and in the Russell-Saunders approximation, by three good quantum numbers corresponding to the angular momentum operators L, S and J = L + S. The exchange interactions couple the spins Si and S / o f the neighboring ions i and ] and in most cases these interactions can be considered as isotropic (mechanisms for an anisotropic exchange have been suggested and used to interpret the properties of some rare-earth compounc~s). Thus the magnetic anisotropic properties result from the Coulomb interaction of the electronic charge clouds with the electrostatic field arising from neighboring ions, the so-called crystal field. This electronic density distribution involves the angular momentum L so, as for metals, the S.O. couplings plays a vital role in making the energy of the system o f spins coupled by the isotropic exchange interaction to depend on their directions with respect to the crystal lattice whose symmetry properties are reflected in those of the crystal field.

5.2. The single ion anisotropy model Let Jf = ~£c + J£o be the Hamiltonian operator of the system where J£c is the crystal field Hamiltonian

G. A ubert /Magnetic anisotropy of ferro- and ferrimagnets

404

operator which can be written as a sum over the magnetic ions o f individual operators h c corresponding to the interaction o f the individual magnetic ions with tile crystal field at their site. The form of these individual operators can be deduced from group theory considerations and they involve only a few parameters (for instance only one for a 3d ion in a site of cubic symmetry). These so-called crystal field parameters can be evaluated by direct calculation in some simple cases (point charge model) or taken as phenomenological ones. The very important fact is the extreme simplification of the problem which results from symmetry considerations. Tire other part Jr0 o f the Hamiltonian describes the Zeeman interaction with the applied field, the exchange interaction and other ones that can be a possible source of anisotropy. Among these, the actual magnetic dipolar interaction leads to an extremely small anisotropy but pseudo-dipolar or pseudo-quadrupolar interactions have been introduced whose microscopic origin will not be discussed here. If Jfo can be considered as isotropic, we are left with what is called a single ion anisotropy model, as all anisotropic effects results from Jfc = ~ i o n s ( h c ) i o n • We can then simplify the problem in using for Jfo an effective field approximation considering that each ion interacts with a magnetic field n + H e f f. H e f f depends on the magnetization and its simplest form is the Weiss molecular field. Jf is then a sum o f single ion operators h = PogJPB J " ( n + neff) + h c and the problem has become solvable besides of course some computationnal difficulties. The procedure is to be continued schematically as follows: (a) diagonalization o f h for a specified direction o f magnetization, which gives the energies er of 2J + 1 states coming from the original fundamental multiplet (N.B., in some cases, it is necessary to take into account the first excited one); (b) calculation o f the sums over states for the individual ions ~'ion = ~

e-#er,

F

where/3 = 1/kBT ; (C) evaluation of an effective thermodynamical potential: 1

Gef f -

~ In ~'ion • /3 ions

The magnetization for the chosen direction is given by M =

l__ t.to

(OGeff/on)~,Heff

i f G e f f is relative to the unit volmne.

The energy density E'a does not exactly equal Geff because in such a calculation, we treat n e f f as an external applied field and so we count the corresponding interactions twice. In fact, E'a = Gef f + ! ½PO M "Hef f. This difference between E a and Geff is often ignored in the literature and can lead to significant errors particularly if AM is not much smaller than M. 5.3. A typical example: the y t t r i u m iron garnet The only magnetic ions of Y.I.G. are Fe 3+ ions whose electronic configuration is 3d s, which gives a fundamental multiplet 6Ss/2. This multiplet whith L = 0 is not directly affected by the crystal field arising from the neighbouring 0 2 - ions but tile higher energy multiplets with L 4= 0 are affected and their slight admixture with the fundamental one can be described by a so called spin Hamiltonain whose form can be predicted by group theory and which involves phenomenological parameters. The 40 Fe 3+ ions o f formula 8(Y3FesO12 ) are distributed on 24 tetrahedral sites labelled 24d and 16 octahedral ones labelled 16a. These sites have a local cubic symmetry distorted along (100) or (111 ) directions respectively and their cubic axes have specified orientations with respect to the cubic lattice (space group Oh1°). For each ion, the Hamiltonian has the form: h = .og

+

s + la

+

+

~S(S + I ) ( 3 S 2 + 3S - 111 + D [S 2 - 1S(S + 1)1 + 1@0 F [3 5 Sz4 - 30 S(S + 1) Sz2 + 25 S~ - 6S(S + 1) + 3S2(S + l f ] , where ~, 77, ~"are the local quaternary axes o f the undistorted cubic site and z the axis of the distorsion. g is isotropic and has the spin alone value. This Hamiltonian operates on the fundamental multiplet 6Ss/z. In the case of Y.I.G. the crystal field part of the Hamiltonian can be considered as a perturbation and standard perturbation theory can be used (N.B., it may be necessary to calculate some second order terms). We thus have 6 adjustable parameters for

G. A u bert / Magnetic anisotropy of ferro- and ferrimagn ets

anisotropy that are a, D and F for the 24d and 16a sites, respectively. Their order o f magnitude can be estimated from paramagnetic resonance of Fe 3+ ions substituted in Y.Ga.G. [ 14]. This type of calculation has been pioneered by Wolf [ 15]. Paradoxically, the main problem does not arise from the anisotropic part of the Hamiltonian but from the effective field one. It is indeed impossible to fit the anisotropy experimental data by simply using a molecular field Heff = wM. We reproduce on fig. 2 our data which confirms what was already known for ZXEa,apart from an improved accracy, but shows a very strange behavior of zXM (N.B., AM is so small that, to our knowledge, it has never been measured elsewhere). It is to be noticed that, even if AM is very small, it must be explained by the model under consideration and it is a much more difficult test to fit simultaneously both zXEa and 2xM than to fit zXEa alone. The solution is given here by a theorem due to Callen and Shtrickman [16]. This theorem states that the functional dependences of higher moments (S n) on the first one
g2

2 ~O

405

is ferrimagnetic, we naturally need the sublattice magnetizations which can be obtained by NMR experiments [17]. Using this theorem, the procedure is now the following: we calculate the isotropic sublattice magnetizations Md and M a as functions of the sublattice effective fields Heffd and Heffa and evaluate these fields at every temperature from the experimental Md and M a using molecular field theory expressions (i.e., Brillouin functions). These values of the effective fields are then used for numerical calculation of z2xEa and AM, always using molecular field theory expressions. The difference with the pure molecular field treatment is clearly evidenced by the thermal variation o f these "mathematical" fields: when T is lowered from the Curie temperature, they begin to increase like the usual molecular fields but go through a maximum and approach zero at absolute zero! This is not so surprising after all, since we know that molecular field theory fits the magnetization behaviour only qualitatively. By this method it is possible to interpret the anisotropic properties of Y.I.G., particularly the strange behavior of AM which results from the fact that the net magnetization is the difference between the two sublattice magnetizations while the energy is the sum of the sublattice energies and the interaction energy between sublattices (not to be counted twice!). A more detailed account of this subject will be published elsewhere.

6. Anisotropy induced in binary alloys by magnetic annealing

l,.-

Z-

6.1. Microscopic origin v O

y )--

"6 hi
-i

TEMPERATURE(K)

Fig. 2. Anisotropy of energy and of magnetization of Y.I.G. between the <111 ) and <100) directions.

When a magnetic alloy is heated at a temperature high enough to allow atomic diffusion but lower than its Curie temperature, a type of ordering may develop in the alloy due to the existence of a spontaneous magnetization which introduces a preferential direction, whence the name of directional ordering. With no applied magnetic field, the sample is divided into magnetic domains with different directions of magnetization so, if one wishes to develop the same directional ordering throughout the sample, one must apply an external magnetic field high enough to saturate the sample during the whole process of

406

G. A ubert /Magnetic anisotropy o f ferro- and ferrimagnets

ordering, this being called magnetic annealing. Obviously, this ordering may affect the magnetocrystalline anisotropy of the material by changing the values of the anisotropy constants but above all it causes new anisotropic terms to appear since the synnnetry properties are modified. These new terms are referred to as induced anisotropy. The microscopic origin of the induced anisotropy is to be found in the rearrangement of the atoms in the crystal lattice since the interaction between neighboring atoms depends on the kind of atom (which is quite obvious) and on the direction of magnetization through the same mechanisms as those evoked for the magnetocrystalline anisotropy. The first quantitative model was proposed by N6el [18] in which each pair, AA, BB or AB of neighboring atoms of the kind A or B of the binary alloy, is attributed an energy depending on the angle of the magnetization with the direction of the pair. This dependence is taken as a phenomenological expansion in a series of Legendre polynomials. In spite of various attempts to improve the theoretical models, the present situation is that none of the proposed ones can be considered as satisfactory but N4el's original pair model is still remarkable for its deep physical insight.

6.2. Experimental methods Measurement of an induced anisotropy relies on the same techniques as every type o f magnetic anisotropy but, as it is necessary to vary the magnetization direction during the measurement, it is necessary to prevent the atomic diffusion at the same time. This can be done by a magnetic annealing at some temperature T' followed by a quenching to a temperature low enough to prevent diffusion. Measurement can then be made at a temperature T not too high in order that during the time of the experiment, no appreciable diffusion can occur. Many difficulties may arise from the quenching process and they are discussed in further details elsewhere [19,20]. As for magnetocrystaUine anisotropy, a formal expansion of the induced anisotropy energy density can be deduced from symmetry considerations. For a polycrystalline sample with a random distribution of the crystallites, the lower order term is: Eu = - K u cos 2 (u, u') where (u, u') is the angle between the actual direction u of the magnetization

at the measurement temperature T and its direction u' at the annealing temperature T'. The induced anisotropy constant Ku depends on T, T' and the alloy composition if the magnetic annealing has been long enough (we shall say "infinitely" long) to reach the thermodynamical equilibrium at T' but may depend on the history of the sample if this is not the case, whence all the problems introduced by the quenching process. For a cubic crystal, the lower order terms are usually written as: =

FG i

_

G

,

i~-j

where c~i and c~; are the direction cosines o f u and u', respectively, with respect to the quaternary axes of tile crystal lattice. The same comments as on Ku apply to F and G. The minus signs are introduced to make Ku, F or G generally positive because for T = T' they obviously are and they seldom change sign for T < T'. Experimental results can more often be described by these lower order terms only. The Fourier analysis of torque measurements that we have recalled in section 3 allows a determination of both F and G together with the usual anisotropy constants for instance by measurements in the (110) plane, provided successive magnetic annealings along two different directions at least of this plane are made. The actual situation is not so simple as we shall briefly show with an example.

6.3. The 5 0 - 5 0 Ni Co alloy In the (110) plane with [001] as origin, the mag magnetic torque is of the form I'dO ) = £k~>1 b~k sin 2kO in the absence of an induced anisotropy. If there is one, resulting from a magnetic annealing in some direction, we have additionnal terms of second order: ['uz(0) = bu sin 20 + au cos 20, where au' and bu' depend on F, G and the direction of magnetization during the annealing. If this is one of the symmetry directions of the plane, we have b'u = F, au = 0 for (100),b" = -(-~F+ aG), 1 au, = 0 for (110) and bu, : 1 t -1 gG, au - ~w/2G for (111 ). Experimental results show that the b~k for k ~> 2 are not significantly affected by the magnetic annealing, which means that the usual anisotropy constants do not vary significantly and thus that b~ itself does not depend on annealing. This fact should leave us with an extremely

G. Aubert / Magnetic anisotropy o f ferro- and ferrimagnets

simple problem but experimental values o f the coefficients of order two of the torque appear to be unreliable. We have shown [20] that one can get rid of this difficulty by an adapted method of analyzing it the data. Let us call a~ and b : the experimental coefficients of cos 20 and sin 20 in the total torque, a~ = i ii t ¢ au and b 2 = b2 + bu. We represent each experimental result in a plane by a point of cartesian coordinates a~ and b~ - b; (N.B., b; is known from higher order • I 2 I coefficients i.e. b2 = 3 b4 + ...). Points A, B or C correspond to magnetic annealing at T' and for an "infinitely" long time along <100 >, <110 >or <111 > respectively. Such a diagram is shown on fig. 3 for two sets of experiments ABC and A'B'C' performed at the same T and T' but with different apparatus (one of them allowing annealing is situ) and with different reference states• These states are represented by D and D' and are presumed to correspond to a perfect disorder after respectively slow cooling from 650°C in a rotating magnetic field and after fast quenching from 860°C. It appears that the triangles ABC and A'B'C' which should coincide do not but can be brought into a fairly good coincidence by a translation. If, for the moment, we forget their positions in the plane and consider only their common dimensions, we must have:

AB=3F+¼G, C H = ~'v~Gand H A :

HB: F+-~G.

A J( / //

///

I

[ I

/ //

d-~._. -100

=

I

407

The points O 1 and O'1 on AB and A'B', respectively, are constructed as OIA = O'1 A' = F and we have CO1 = C'O'I = ~. Now, if there were no problem, the points O1, O'1, D and D' should coincide with O. The fact that they do not coincide means that during the history of the sample, different processes generating directional anisotropy have occured and that only part of their effects are wiped out by the magnetic annealings at T'. Nevertheless, the equality of the triangles indicates that the remaining part of these directional anisotropies does not affect significantly the development of a directional ordering by magnetic annealing. A complete analysis of our results can be found in [20]. Here we simply mention some values: for T = 20°C and T' = 450°C, G = 1.614 × 104 erg/cm 3, F = 4.016 X 103 erg/cm 3 and G/F = 4.02. The ratio G/F should be exactly 4 and independent of T and T' for an fcc binary alloy following the Ndel's pair model. In fact, we find that G/F is an increasing function of both T a n d T', for instance at T = 400°C and T' = 590°C, we find G = 6.813 × 103 erg/cm 3, F = 1.288 × 103 erg/cm 3 and G/F = 5.29.

Acknowledgements These results would not have been obtained without the enduring efforts of my fellow workers whom I had the pleasure to guide during their thesis work: P. Escudier, L. Frazao, P.M. de Groot, B. Michelutti and R. Pechart. I am very indebted to Dr. R. Gersdorf for stimulating discussions.

IH'°o References

i

a' o, C

H

-200

,B

-16o Fig. 3. Representation (experimental units) of different states of a 5 0 - 5 0 Ni-Co alloy after magnetic annealings at 450°C (the temperature of measurement is 20°C).

[ 1] M.I. Darby and E.D. Isaac, IEEE Trans• Magnetics 10 (1974) 259. [2] P.M. de Groot, Th~se 3~me Cycle, Grenoble (1975). [3] G. Aubert, Y. Ayant, E. Belorizky and R. Casalegno, Phys. Rev. B14 (1976) 5314. [4] R. Gersdorf and G. Aubert, Physica 95B (1978) 135. [5] G. Aubert, J. Appl. Phys. 39 (1968) 504. [6] G. Aubert, Th~se, Grenoble (1966). [7] P. Escudier, Th~se, Grenoble (1973) and Ann. de Phys. 9 (1975) 125. [8] J.J.M. Franse, J. de Phys. Colloq. 32 (1971) C1-186. [9] R. Gersdorf, Phys. Rev. Lett. 40 (1978) 344.

408

G. Aubert / Magnetic anisotropy o f ferro- and ferrimagnets

[ 10] C.S. Wang and J. Callaway, Phys. Rev. B9 (1974) 4897. [11] G. Aubert and P. Escudier, in: Proc. ICM Moscow (1973) I 215. [12] G. Aubert and B. Michelutti, Physica 86-88B (1977) 295. [13] G. Aubert and B. Michelutti, J. Magn. Magn. Mat. 1 5 - 1 8 (1980) 575. [14] S. Geschwind, Phys. Rev. 121 (1961) 363. [15] P.W. Wolf, Phys. Rev. 108 (1957) 1152.

[16] H.B. Callen and S. Shtrikman, Solid State Commun. 3 (1965) 5. [17] R. Gonano, E. Hunt and H. Meyer, Phys. Rev. 156 (1967) 521. [18] L. Neel, J. Phys. Radium 15 (1954) 225. [19] G. Aubert, R. Pechart and M. Ritz, J. de Phys. Colloq 32 (1971) C1-96. [20] R. Pechart, Th~se, Grenoble (1972).