The present status of the temperature dependence of magnetocrystalline anisotropy, and the l(l+1)2 power law

J. Phys. Chem. Solids.

Pergamon

Printed in Great Britain.

Press 1966. Vol. 27, pp. 1271-1285.

THE PRESENT STATUS OF THE TEMPERATURE DEPENDENCE ANISOTROPY,

OF MAGNETOCRYSTALLINE AND THE Z(Z+1)/2 POWER LAW H. B. CALLEN*

Department

of Physics?,

University

of Pennsylvania,

Philadelphia, Pa.

and E. CALLENS Department

of Physics, Osaka University,

Osaka, Japan

(Received 9 February 1966) Abstract-The present status of the theory of the temperature dependence of magnetocrystalline anisotropy in ferromagnetic insulators is reviewed and summarized. The Z(Z+1)/2 power law for the behavior at low temperatures is derived in a general fashion and the extension to arbitrary temperatures is also given. It is then shown that the Z(Z+1)/2 power law is not directly applicable to antiferromagnets, but that contrary to current opinion, quantum corrections to the ground state introduce significant alterations. We next discuss the magnetic field dependence of the anisotropy energy and finally, the Z(Z+1)/2 power law for the temperature dependence of magnetostriction. 1. INTRODUCTION

THE 10th power law for the magnetocrystalline anisotropy of a cubic crystal has had a devious and convoluted history. Starting with a perceptive and profound analysis by AKULOV(~) in 1936 the theorem has survived a succession of alternative proofs of increasing opacity, a sequence of disproofs and counter-examples, and a panoply of experimental tests which range from complete agreement to outright contradiction. Amid this welter of confusion several important extensions of the theorem were achieved, most notably by Van Vleck and by Keffer, and the theorem now stands forward in greatly increased generality and scope. We here recapitulate this complex history, offer a restatement of the most general proof of the theorem in terms more readily accessible to most physicists and attempt a synthesis of the experimental and theoretical situations. For an excellent review of all aspects of anisotropy other than the temperature * Supported in part by the U.S. Office of NavaI Research. t In association with the Laboratory for Research on the Structure of Matter, University of Pennsylvania. $ Permanent Address: U.S. Naval Ordnance Laboratory, White Oak, Silver Spring, Md.

dependence the reader is referred to J. I(ANAMORI.t2) To describe the history of the theorem we first recall that the magnetocrystalline anisotropy is defined in terms of the dependence of the free energy on the direction of the magnetization. For a cubic ferromagnet (or ferrimagnet, or NCel antiferromagnet) the free energy is usually expanded in a power series in the direction cosines MI, Q, as, of the magnetization direction relative to the crystallographic axes. Symmetry dictates that this power series take the form F = Ks( T) + KI( T) [a12as2 + a$a$ + I&( T)a12a22a32+ . . . Akulov that

+ a$a12] (1.1)

showed by a simple classical argument

KI(T)/KI(O)

2: I-1OSm -N [l -6m]10 2:[m(T)]10

(1.2) where m( 2”) is the reduced magnetization ( = m(T)/ M(0)) and where am(T) = 1 -m(T). The Akulov result applies only at temperatures sufficiently low that 6m ,< 1. The Akulov derivation is based on the

1271

1272

H. B. CALLEN

assumption of independent classical spins, each of which has an energy of the form of equation (1 .l) with Kr(T) replaced by K{(O) and ~(2 replaced by the direction cosines ~2 of that individual spin. Each spin fluctuates, and a simple statistical calculation, which we repeat in Section 2, then gives the first equality in equation (1.2) (to first order in am). It is important to note that although this derivation is classical, and although it arbitrarily assumes independence of the spins, nevertheless the number’ 10 arises not from a particular model but rather from the peculiar structural combination of direction cosines in equation (1.1); a combination which is completely dictated by the symmetry of the crystal. Thus the Akulov derivation, specialized as it may have been, did identify the source of the 10th power law and did strongly suggest its generality. Akulov compared the theoretical result with data of Honda et ~1. on iron, and concluded that the law was quite accurately obeyed up to 0.65 Tc. TIn the following year Van Vleck laid the groundwork for a quantum theory of magnetic anisotropy. He first showed that an appropriately anisotropic effective spin Hamiltonian could result from a combination of spin-orbit coupling and crystal field splitting. As contrasted to the single-ion terms assumed by Akulov, the Van Vleck mechanisms are two-spin interactions, of pseudo-dipolar or pseudo-quadrupolar structure. Van Vleck observed that the fourth degree magnetic anisotropy (the Kl term of equation (1.1)) can be achieved by carrying the pseudo-dipolar interactions to second order in perturbation theory, or by carrying the pseudo-quadrupolar terms to first order. In another part of Van Vleck’s fundamental 1937 paper he confronted the problem of the temperature dependence of Kl(T). Employing a molecular field treatment of the exchange interaction, and using the pseudo-dipolar form of the Hamiltonian, Van Vleck found a second power law rather than Akulov’s tenth power. Similarly the pseudo-quadrupolar terms in the Hamiltonian were found to give a 6th power law (shifting to the 5th power with increasing temperature), in disagreement with Akulov’s 10th power theorem. After a long period of quiescence, interest was revived in 1954 by zENER,t4)who generalized the

and E. CALLEN Akulov 10th power law to the Z(Z+ 1)/2 power law. That is, Zener showed that if the angular dependent factors in equation (1.1) are regrouped in terms of spherical harmonics Ylm(a), and if KZ(T) is the coefficient of the terms of given 1 (KI(T) being a linear combination of the Kt(T)), then ~l( T) E &(1+1)/s In particular,

(1.3)

he concluded that

KJ(T) = KI(T)+-~$&(T) KG(T)

= I&(T) z rn21

2: ml0

(1.4) (1.5)

This deeply insightful conclusion has been proved correct on a rigorous basis, as will be seen below, whereas Zener’s original discussion depended on the more-or-less plausible association of the statistical density matrix with the solution of a random-walk diffusion equation. In 1955 KEFFER(~)recognized that the paradox posed by the disagreement of Akulov’s theorem and Van Vleck’s molecular field calculations arose from the violence that molecular field theory does to spin correlations. He therefore recalculated the anisotropy of pseudo-quadrupolar origin, substituting a cluster theory for molecular field theory. This cluster theory, incidentally, had been rather casually introduced by Van Vleck in his 1937 paper, in a section entitled “Alternative perhaps-improved calculation” and had been later rediscovered and more fully exploited by OGUCHI.@) At any rate Keffer did now find that pseudo-quadrupolar interaction would give a 10th power law at low temperatures, changing rapidly to Van Vleck’s 6th power dependence as increasing temperature disrupts the spin correlations. Keffer also rederived the Zener Z(Z+ 1)/2 generalization, using the classical single-ion mechanism and molecular field theory. At this point it was clear that the single-ion terms in the Hamiltonian were completely consistent with the Z(Z+1)/2 power law, and Keffer’s cluster calculation was generally accepted as establishing the theorem for pseudo-quadrupolar interactions. It might be expected that the pseudodipolar terms should then have posed no residual mystery. But, in fact, an almost comical series of over-damped oscillations ensued.

STATUS

OF TEMPERATURE

DEPENDENCE

OF MAGNETOCRYSTALLINE

Peculiarly, in the very paper in which Keffer established the Z(Z+l)jZ law for the single-ion and pseudo-quadrupolar terms and identified the crucial role of spin correlations, he also briefly remarked that a spin-wave analysis of the dipolar terms would probably give a power even lower than two! &4SUYA(7) then carried out just such a spin wave calculation@) for the pseudodipolar terms and found a 16th power! CHARAP and WEISS(g) found an error of a factor of 2 in Kasuya’s calculation and concluded that the 8th power was correct. Finally KEFFER and OCZCHI(~@ added certain terms omitted by Charap and Weiss, regained the 10th power and stressed that correlations ensure the 10th power law in all cases. Meanwhile VAN VLECK,(~~) in a penetrating paper delivered at the Grenoble Conference on Magnetism, co-ordinated all elements in the puzzle. In a simple intuitive discussion he clearly exhibited the symmetry-basis of the theorem, he showed that its validity for two-spin terms requires the rigid correlation of the two spins over the relevant temperature region, he observed that the molecular field theory destroys this correlation whereas the cluster theory maintains it and finally he gave a general proof of the a(l+ 1)/2 power law; an equivalent, but more explicit proof will be elaborated in Section 3. With the theoretical situation for insulating ferromagnets thus clarified, attention turned to antiferromagnets. It might have been supposed that the quantum corrections in the ground state of an ~tiferromagnet would vitiate the I(E+ 1)/Z power law. But a series of investigations have claimed to demonstrate that this is not so, and that the anisotropy strictly obeys an 1(1-l-1)/2 power dependence on the sublattice magnetization. OGUCHI(~~) first undertook a direct spin-wave calculation of the temperature dependence of the anisotropy in an uniaxial antiferromagnet. For the real magnetic dipole-dipoIe mechanism he found an ms dependence appropriate to 1 = 2. However, when he added the single-ion mechanism he found an m2-s dependence. This was generally ascribed to a small numerical error, particularly after a subsequent re-calculation by KANAMORI and TACHIKI.(“~~) MeanwhiIe PINCUS(~~) suggested a simple general argument for the proposition that the Z(Z+ 1)/2 power law applies rigorously to antiferromagnets. This argument, buttressed by the

ANISOTROPY

1273

Oguchi and Kan~ori-Tac~i calculations, has been widely accepted and antiferromagnets are generally considered to be equivalent to ferromagnets in this respect. We do not agree. In Section 5 we show that, despite the above calculations, there are important quantum mod~cations of the 1(1+1)~2 power in antiferromagnets. The tortuous development of theory has had its counterpart in the experimental area. Whereas Akulov had concluded that early experiments on iron agreed with the 10th power law below 065 I’,, Zener identified “essentially perfect agreement” with the data of BOZoRTH(15)and POTTFX@) over the entire ferromagnetic temperature range. GRAHAM@~)now measured &(T) for iron at low temperature and found an 8th power; a conclusion which may be moderated by the effect of thermal expansion.(rs* 19) Also nickel was estimatedfrsf to obey roughly a 50th power. These data had the unfortunate effect of casting the 10th power law into general disrepute. However, the derivation of the 10th power law is restricted to materials with localized spins and 3d metals are almost surely exempted. For various insulating magnetic materials and even for rare earth metals the experimental data, which we shall review below, is in excellent agreement with the Z(1+ 1)/2 power law. As we shall also see, the theory of the temperature dependence of the anisotropy can be extended to arbitrary temperatures and it again stands in excellent agreement with the data for insulating materials. In Section 6 we discuss the magnetic field dependence of magnetic anisotropy and in Section 7 we review briefly the application of these same concepts to the problem of saturation magnetostriction.

2. AEULOV’S CLASSICAL SINGLE-SPIN PROOF Because of its simplicity and because it contains the root of all subsequent treatments, we first briefly recall the classical treatment of singlespin anisotropy as given by ~uLov.(~) The physical idea described by this theory is shown in pictorial form in Fig. 1. It is assumed that each spin has an intrinsic direction-dependent energy, arising by interaction with the crystal field. The energy density for a

H.

1274

B.

CALLEN

given spin is of the form E = ~O+&(0)[&‘22+

and. E. CALLEN

whence, from equation (2.2) %‘2%s2-t-Vs2V12] -t . . . (2.1)

where ~1, ~2, vs are the direction cosines of the spin considered. The free energy is then given by perturbation theory, to first order in the interaction constants Kl(O), as F = ~O+xj(0)(V~2V22+Y22Ys2+Vs2~~2)+... (2.2) where the averages are to be calculated in the isotropic or “unperturbedlf distribution function. This free energy is then to be compared with the phenomenological equation (1.1) which defines the tempera~r~-dependent anisotropy constant &(T). It is convenient to first consider the magnetization to be along the [OOl] axis, so that each spin fluctuates through a small random angle in the JNTMNSIC

40011= Fo-+-&( 0) <@ >

(2.4)

Since the magnetization is we(T) = (cosS> = (i-@/2)

(2,51

this becomes Flrooj = Fo(T)+~&(O)C%Q)

(2.6)

which is to be compared with equation (l.l), which states &ol~ = Ko(T)

(2.7)

To eliminate the unknown functions k_o(T), and Fe(T) we now consider the magnetization to. be along a [llO] direction. Choosing a u”’‘axis along this [tlO] direction and y’ along [OOl] we have

whence FIG. 1, Temperature dependence of anisotropy energy. When the magnetization is in an easy direction the anisotropy energy is averaged over the Auctuations of the spin direction as shown; this average v&x is shown by the point A’. The energy at A’ is greater than the zero temperature value at A. In a hard direction the energy is similarly decreased from B to B’. The dotted anisotropy curve (T > 0°K) is more nearly isotropic than the solid curve (T = O”K), implying a decrease in anisotropy coefficient with increasing temperature, or decreasing magnetization. From t%L.EN E. R. and CALLEN H. B., J. P&. C&m. Solids 16, 310 (1960). vicinity

of this direction. Thus vs = sin@ sin# N Bsincj =

case

2:

l-82/2

= Fi)( T> f ~K(O)(l -2Sm)

(2.9)

In this case equation (1.1) gives FillOl = fiTo(T)+ a&(T)

(2.10)~

The free energy difference ~~]~a~-~~~~~~ can be calculated by subtracting equations (2.7) and fZ.lO), or by subtracting equations (2.6) and (2.9), whence ~~lo~-~~ll

= ~~1(~)=~~1{0)~1-

lOSm> (2.11>

which is the theorem enunciated in equation (1.2).

~1 = sin@ cosrj N B cOs#

v3

&IO] = Pa(T) + $K11(0){ I- 8’2 >

(2.3)

3. THE GENERAL THEORY Aithough Akulov’s classical single-ion proof. of the 10th power law in cubic crystals appears.

STATUS

OF TEMPERATURE

DEPENDENCE

OF MAGNETOCRYSTALLINE

to be very specialized, it actually contains the basis for generalization to higher anisotropy constants, to arbitrary crystal symmetry, to a quantum mechanical treatment, to two-ion as well as single-ion mechanisms and to arbitrary temperatures. Most of these generalizations have been made, more or less explicitly, as discussed in the Introduction. We here summarize the present theory indicating each of the above generalizations in turn. Consider an insulating ferromagnet with all spins in equivalent sites, an assumption which simplifies the notation without materially affecting the analysis. We first assume each individual spin to have a direction-dependent energy density of the form

4s) = t-o+2 KZ(O)~Z(S) 1

(3.1)

In the classical case S denotes a unit vector along the spin direction, whereas in the quantum mechanical case S denotes the conventional spin operators. K&O) is a constant and y&S) is a normalized polynomial of lth degree in the components of S. The form of gsp1(S) is dictated by crystal symmetry and it is convenient to write yz(S) as a linear combination of spherical harmonics (or of their operator analogues,(a*) which have identical rotation properties). 9pE(S) =

2 Qzmyl*(s)

(3.2)

m Then the constants aim are dictated by the crystal symmetry. Finally, the sum over 2 is restricted to even values because S is odd under time reversal. The free energy of the system is, to first order in the anisotropy constants

F = Fo+x W(0)<92(s)> Z =Fo+c

RI(O) 2 aP
as the polar axis: Y&S)

= c

where the average is to be computed in the isotropic or “unperturbed” distribution function (or density matrix). If the unperturbed distribution function is such that the average magnetization is along the direction EL it is convenient to re-express the spherical harmonic Yzm{S) in terms of rotated spherical harmonics, defined with respect to u

b&#,m’)Y@(S’)

(3.4)

m’

where the b&,m’) sines of a. Then

depend

< YP(S) > = c

on the direction

b&m’)


co-

>

nt ’

= b&&O) ( YPfs’) >

(3‘5)

where we have used the fact that the unperturbed distribution is cylindrically symmetric around a, so that (Ytm’(S’)) = 0 unless m’ = 0. Furthermore the familiar “addition theorem” for spherical harmonics(a1922) identifies bl(m,O)as Up(a), the classical spherical harmonics which are functions of the angles 8 and # of a (relative to the z-axis). Thus (YP(S) and returning

> = Yz”(a) (YzO(S) > to equation

(3.6)

(3.3)

F = Fo+? ~(0) (YzO(s’>>[zagmYp(a)] m

=Fo+c 40) (YzO(S)>yi(a) which identifies the temperature tropy coefficients as

dependent

(3.7) aniso-

This is the fundamental equation for the temperature dependence of the anisotropy constants. However, it is prudent to recast it slightly. From equation (3.8) it is clear that Yea must be normalized to unity in the ground state of the system. As this is not the usual normalization it is therefore preferable to rewrite the equation as _KU) =

(3.3)

1275

ANISOTROPY

K,(O)

(XO(S’)h < KO(S’) >o

(3.9)

the subscripts referring to the temperature at which the statistical averages are to be computed. The equation is now independent of the normalization which may, for convenience, be attributed to the Yzm(S’). With this fundamental equation it remains only to evaluate the average (Yz*(S’) > by direct statistical mechanical calculation. We first prove the Z(j+l)/Z power law in the quantum mechanical case. For this purpose let prnf

H.

1276

33. CALLEN

be the probability of the state in which S-a = m’; that is the pnz is the diagonal matrix element (m’Ipim’>. Then
=

5 (m’] X”(S’)jm’)pm, mJ= --s

+

=

ps-1

(3.11)

Ps + Ps-1 i?oting that (Sj Y;sIS) = (Yrs)a through by this quantity, Q(T) -=p ‘Q(0)


=-Ps + < X0 >o


dividing

ps-1

(3.12) ps+ps-1

(S,l,S-


1,o/s,r,s,s-


=

1) (3.13)

where(jl,js, ml, msljr,js, j, m) denotes a ClebschGordan coefficient. The recursion relations for the Clebsch-Gordan coefficients evaluates* the ratio appearing in equation (3.13), as 1 --I@+ 1)jZS. Inserting this ratio in equation (3.12) then gives KI(T)

<%‘)T

Q(o)


_=e=

1_

qt+1>

Ps-1

2s

ps+ps-1

-I___

(3.14) Finally the quantity Ps_I/(Ps+Ps_~) pressed in terms of the monetization

Eliminating p,+.l/(p,s + ps+l) sought for result

finally

(3.15)

gives

the

(Y?>r, < V >o = 1 (1’1) -[l 2

-m(T)]

N [m( T)]l(l+l)‘z

Extension of the proof to two-spin interaction mechanisms is almost immediate. The Hamiltonian of the system is the sum of two-spin interactions, analogous to equation (3.1) H = Ho-Q

Ps-1

The ratio of matrix elements appearing in the right-hand member is evaluated by the WignerEckart theorem@) (S-l~~s~S-1)

!F

= ____ = I--- 1 ps-l s ps+ Ps-1 < Y1° >o

(3.16)

and

PS

+ (s--1]Y~o(s-l>

m(T)

Q(T) p=----@)

&“is>p+l s

(81

(S-1~Y~~~S-l~~

is most easily done by noting that Yla is just Sea, the component of the spin along a. Thus, by equation (3.13),

(3.10)

At sufficiently low temperatures only the two states m’ = S,S- 1 have appreciable probability, so that
and E. CALLEN

can be exm(T). This

* In the recurrence relation given by MEW~ZBACHER,(~~) equation (17.64), insertjl = S, js = I, ml = S, ms = 0, j = S, m = S- 1 and take the lower signs. This gives one equation; a second equation is obtained by inserting jl=S, ja=I, ml=S, ma= -1, j=S, m=S and taking the upper signs. Eliminating (S, E, S, --II&‘, I, S, S- 1 > from these two equations then gives the desired relation.

Q(O) 2 c40t(Sa,S,) 1

(3.17)

lid

Here the functions 9&S’&*) are symmetric in Sz and Sj, and can be expanded in functions analogous to the spherical harmonics YZ(Sb,Sj) = 2 &=Y?(&Sg) m

(3.18)

The significance of the notation Yz~(S~,S~) is that this function transforms under co-ordinate rotations isomorphically with the classical spherical harmonics. To fully define Yrm(Sa,S,r) one should also specify its transformation properties under a partial rotation (i.e. a rotation of only one of the two spins), but such partial rotations will not prove to be of interest to us and we can therefore suppress the additional associated indices in Y~~(&,S,). The formalism applicable to the singleion case now carries through directly. In addition, and this is the crucial point emphasized by VAN VLECK,(~~) it must be assumed that the only two states of the two-ion complex which have appreciable probability at low temperatures are the ground state, with total spin 2S and a component = 2S and the excited state with total spin 2S and a component 2S- 1. This excited state is obtained from the ground state by a co-ordinate rotation (of both spins), as discussed above. Consequently, the two-spin complex acts precisely like a single ion of spin 2S and the analysis in

STATUS

OF TEMPERATURE

DEPENDENCE

OF MAGNETOCRYSTALLINE

terms of the Wigner-Eckart theorem transcribes identically. The assumption that the interacting spins are rigidly coupled at very low temperatures is justified by spin wave theory. The very low-lying excited states of the system are spin waves of essentially infinite wavelength, and these do not induce angular deviations between near-by spins. The above consideration suggests the limiting temperature above which the Z(Z+1)/2 power law will fail, for a given two-ion mechanism. Let R be the range of this two-ion mechanism, and let Kn = 27~/R. If I is the energy of a spin wave of wave vector k~, then for kBT > c(kR) the correlation between the interacting spins will be destroyed by thermal spin waves. In this connection it may be noted that anisotropy which arises from true dipole-dipole interactions is relatively long range (as opposed, for instance, to short range pseudo-dipolar coupling arising from anisotropic exchange), and that the Z(Z+1)/2 power law is then restricted to vanishingly low temperatures. In addition to the temperature bound discussed above there exists a second bound which arises from the assumption that only the lowest states (of the single ion or of the two-spin complex) are appreciably occupied. This leads to the requirement that 1 -m(T)

< l/S

(3.19)

In no case can one expect an Z(Z+ 1)/2 power law for temperatures greater than those satisfying the inequality (3.19). If the spin pairs of equation (3.17) are allowed to assume internal angles the anisotropy energy is quite different. For example consider Van Vleck’s short-range pseudo-quadrupolar perturbation. He -

(&.rt3>2(s~+3)2

(3.20)

&FFERt5) showed that if only the long wavelength spin waves states are excited this anisotropy energy varies as K~(T) N N <9'4(&,%)) - ml0

(3.21)

If, on the other hand, one evaluates (Hg) in a density matrix which ignores correlation, such as molecular field theory, then each spin factor in

equation (3.20) transforms and

ANISOTROPY

1277

separately as a Yss(&),

K~(T) N
(3.22)

This was the source of the 6th power estimate(s) of the pseudo-quadrupolar anisotropy. As the above two limits may restrict the validity of the Z(Z+1)/2 law to quite low temperatures it is of interest to attempt the evaluation of equation (3.9) for KZ(T)/Q(O) at arbitrary temperatures.

4. TEMPERATURE DEPENDENCE OF XL(T) AT ARBITRARY TEMPERATURE Direct evaluation of equation (3.9) for arbitrary temperatures has been given most persuasively for the single-ion mechanism and we shall devote most of our attention to this case. In fact there is evidence that the single-ion mechanism is dominant in many non-conducting magnetic materials, particularly those of cubic symmetry. This conclusion was reached theoretically by YOSIDA and TAcHIKI(~~)and experimentally by FOLEN and RADO,@) each in the case of ferrimagnetic spinels. Folen and Rado measured the anisotropy in a series of magnesium-iron ferrites and found the anisotropy to be linear in the concentration, indicative of the single-ion mechanism. In order to compute (Yrs(S’)) we require the statistical distribution function (or reduced oneion density matrix). CALLEN and SHTRIKMAN(~~) have shown that at least for a wide class of statistical models (including all many-body “collective excitation” theories) this distribution is simply exponential in m’ = S’* a. pm, = exp[Xm’]/tr

exp[Xm’]

(4.1)

Here X is an as-yet undefined function of temperature. It will be recognized that molecular field theory is a particular theory which conforms to equation (4.1), with X then identified as proportional to (m’ )/kBT. We shall not make this assumption and we stress that the calculation of this section is not a molecular field calculation (although molecular field theory gives precisely the same result, of course). The essential physical content of Callen and Shtrikman’s demonstration of equation (4.1) is as follows. In any collective excitation theory the excitations (“renormalized spin waves”) are

1278

W.

3.

CALLEN

independent, and each contributes ~dependen~y to the probability of a spin deviation on the ith spin. The probability of the ith spin carrying nl+ ns spin deviations is then the product of the probabilities of ptl and ~2 separately, or P,+~~-~~ = which implies the exponential ~&“-n,‘ps-?&,, form of equation (4.X). We then have, for the quantum mechanical case,

and

E.

CALLEN

In the low temperature limit m(T) is close to unity and it follows from equation (4.8) that X is then huge. In this limit of large X, 1$+&X) can be expanded to first order in l/X and one finds fa+&X)

= @s,2(x)]@+l’/s

flow T)

or

Kt(T) = K$o)[f$ T)]xz+l)i ’ (Yra(S’))

YrOlm’~exp~*~‘],~exp~X~‘] -s (4.2) and for the classical case (with m’ = cos y) < UP(F)

=~
> = < YP(m’) >

=: / Yrs(m’) exp[Xm’]dm’/

i exp[Xm’]dm’

(4.3)

(4.9)

(4.10)

which constitutes Keffer’s proof of the &-f-1)/2 power law. Equations (4.7) and (4.8) can be considered as a pair of parametric equations, from which X can be eliminated, to express KZ(T)/KJ(O) in terms of m(T) at arbitrary temperatures. We indicate this symbolically by writing the inverse of (4.8) as X = fs,s-l(m) and equation (4.7) as

-1

-1

Keffer considered the classical average in equation (4.3) in his proof of the I($-+ 1)/Z power law, observing that the integral is expressible in terms of hyperbolic Bessel functions :(a)

In Fig. 2, taken from Ref. 26, we show ~~(T)~~~(~~ as a function of m, for 1 = 2 and 4.

1 s -1

Y rsfm’) expfXm‘]&z’

= &r&X)

(4.4)

and

For E = 1 it may be noted that r)s&X) is identical to the familiar Langevin function, for the magnetization is FIG. 2. ~a~~ti~at~~n dependence of anisotropy energy of single-ion origin, for anguIar hamonks I = 2 and 4 and for various spin qalues, including the classica hit mf.

STATUS

OF TEMPERATURE

- ---- -- -- --DEPENDENCE

- - MAGNETOCRYSTALLINE Or

Returning to the quantum mechanical case, of equation (4.2), we again have two parametric equations in X

-8s

(4.12)

and m(T) =

2(732’)YlOlm'> exp[Xm’]/

5 exp[Xm’]

-s

-S

WOLF@~) has eliminated eXpreSSing

Kl(T)/K@)

aS

fUndiOn

of

m(T),

DATA

-THEORY

50

I50

300

350 300 TEMPERATURE

350 1%)

400

450

500

>50

FIG. 3. Temperature variation of the first anisotropy constant, Kl; of YIG. The magnetization has be-& eliminated bv means of the analysis due to Wolf (Ref. 27) shown in Fig. 2, and calculated sublattice magne&ations. There are two adjustable phenomenological parameters in the theory. Figure from Ref. 28.

> Te

(4.14)

with a continuous transition between those limits. An exact calculation@@ for I= 2 and S = 1 shows that, for the single ion mechanism, the low temperature m3 law is a fairly good representation of the exact result over almost all of the ferromagnetic range. Evaluation of equation (3.9) for two-ion terms is in less satisfactory condition. All that can be said with rigor is that the low and high temperature limits are rnl(Z+l)/s and ml; the transition between these limits is governed by range considerations as discussed after our equation (3.20). A cluster theory calculation@) for the I = 2 spherical tensor, for many spin values, suggests that the transition to uncorrelated behavior occurs at very low temperatures, for all separation of spin pairs, so that m2 is a fairly good representation of this correlation function almost everywhere.

IN ANTIFERROMAGNETS

5. ANISOTROPY

0

T

KZ(0)

(4.13)

completely analogously to equation (4.11). These results are also given in Fig. 2, for I = 2 and 4, and for several spin values.’ A comparison of the single-spin theory with experiment has been given by RODRIGUEet aZ.(ss) Figure 3 shows the calculated and the measured anisotropy constant Kl(T) of yttrium iron garnet *--EXPERIMENTAL

as a function of temperature. The calculated anisotropy constant was obtained by Rodrigue et al. from the functions shown in Fig. 2 and from the measured sublattice magnetization curves. The agreement is excellent over the entire temperature range. Equations (4.12) and (4.13), which reduce to m(T)z(zS1)/s at low temperatures, also simplify in the paramagnetic region to

w(T) - mz(T,W,

X from these sums, a

1279

ANISOTROPY

As discussed in the Introduction, it is widely accepted that the 2(1+1)/2 power law applies rigorously to antiferromagnets if m(T) is taken as the sublattice magnetization. We shall show, however, that the quantum corrections to the antiferromagnetic ground state introduce significant alterations in the Z(Z+1)/2 power law, robbing it both of its model-independent generality and of its attractive simplicity. First consider the case of a single-ion mechanism, and let us review the disarmingly simple argument of PINCUS.(14) He introduces a hypothetical state of complete which

sublattice

appears

of the argument.

only

saturation in

Denote

the

(the

N6el

intermediate

the sublattice

state), stages

magnetic

1280

H.

B. CALLEN

and E. CALLEN

E(Z). This factor, which is greater than unity, depends upon the deviation from unity of the sublattice magnetization in the antiferromagnetic ground state, 1 -m(O). In the classical limit the Z(E+-1)/2 power law re-emerges, but for finite 61) spin values, particularly for higher angular harmonics, the value of t predicted by equation and at T = 0",then, (5.8) can be large. For example, in a cubic crystal the lowest anisotropy coefficient occurs at 1 = 4, z(t+1v2= [m(o)]z"+1'/2 (5.2) and in simple cubic structure l-m(O) r 0*0785. Taking S = 1, this argument would suggest 6 = 4.2. Now, we do rsot assert that equation (5.9) is whence, dividing, more correct than equation (5.3); in fact we shall m( q uz+l)z show below that the terms of 0(1 -m)s which were KdT) -= (5.3) neglected in equation (5.6) completely alter the m(O) Kr(0) situation. Rather the purpose of the discussion above is to show that deriving equation (5.6) to This, of course, is the familiar Z(Z+1)/2 law. first order in (1-m) does not determine the power However, recalling equation (3.16) we note that law unambi~ously. It is therefore important to equation (5.1) should more properly be written contrast the two conflicting conclusions drawn in the form above from this equation. Let us suppose that V+1) ‘QtT) equation (5.6) were to be rigorously true. Let -= - ----2--[l-m(T)]+O[l-m(T)]s+... 1-m(O) -N0.08, and suppose 1-m(T) -NO-09. Then Q(d) (5.4) 1-m(T)/m(O) 2 O-01, which is of the same order as [l -m(T)]s. Consequently an expansion and similarly at T = O", correctly in 1 - m( T)~m(O) cannot be obtained by keeping 1 -m(O) and l-m(T) each to the same w-l) 4T) -= ---yy-[l-m(O)]+O[l-m(0)]2$... order and this is precisely the procedure followed Q(O) in equations (5.1)-(5.3). In contrast, the conclusion (545) (5.9) is an exact consequence of equation (5.6). Dividing, and ignoring for the moment the terms However, the numbers cited above indicated of order [I - m]2, i~ediately that the terms of order Cl- m(T)]2 in the numerator of equation (5.6) and the terms KE(T) I-#(Z+l)[l-m(T)] of order [l -m(O)]2 in the denominator, can alter ---T-= (5.6) the coefficient of [l- m(T)/m(O)] in K~(T)/K~(O). G(O) 1 -W-ll)[l -m(O)] These terms are not determined by symmetry alone m(T) m(O) and hencethe power law for KI(T)/KE(O) of an = 1 -+(Z+ 1) l-antif~~agnet is not determined by symmetry 1 -@(I+ l)[l -m(O)] [ m(O)

moment

and the anisotropy

constant

in this state

by M@) and KI(O) respectively. Then Pincus observes, that, by analogy with a ferromagnet,

z1

1

1

1

m(T) l/Z%-lfE(I)

(5.7)

c~~d~ut~o~.

To illustrate the above assertion we now turn uniaxial crystal with a (5.8) to a particular model-a E f m(0) I single-ion mechanism underlying the anisotropy. We carry out a spin-wave calculation to second where order in 1-m and we then investigate the power law for K2(T)l~2(0). Focussing attention on a m(O) @) = (5.9) particular spin on the “up” sublattice, the 1 -@(Z-i- l)[l -m(O) ] relevant term in the perturbative Hamiltonian is This alternative form of derivation thus comes to the spherical tensor@@ a distinctly different conclusion, to the effect that Y20 N 3(S+-S(Sf 1) (5.10) the Z(E+ 1)/2 power is to be corrected by the factor

STATUS

OF TEMPERATURE

DEPENDENCE

Transforming to Bose annihilation operators equation (5.10) becomes Yso N S(2S+

l)-3(2S-

OF MAGNETOCRYSTALLINE

and creation

l)a+a-t3u+a+au (5.11)

The second term can be related sublattice magnetization:

at once to the

St = S-a+a

(5.12)

whence m(T) = 1

(5.13)

Equation (5.9) would result from (5.13) and (5.11) if the last term in (5.11) were neglected, To evaluate this last term one uses the Fourier and Bogolubov transformations@@ and after a few straightforward steps one finds (u+u+aa > = 2S2[1-

m( T)]2

(5.14)

Unfortunately this expression does not vanish for spin l/2, although the left-hand member obviously should; this is a direct result of the fact that simple spin wave theory ignores all kinematical effects (i.e. limitations to the lowest 2S+ 1 states of the associated bosons). Nevertheless (5.14) is almost certainly a reasonable estimate of the n~agnitude of (u+&aa> for S > l/2. Then, from equation (5.11) we find to first order in I1 - w~i~691~

‘a(T)


‘Q(O)

< Y2O)o

---c-=-z

l-35

+ ... (5.15)

m(T)

=

35

I 1

(5.16)

No>

where (2~-1)~-6~~+~(1~4~) ’ = (2S-l)S--656+36(1+28)

f5’17)

and s/s

z 1 -m(O)

(5.18)

For S -+ co the value of e approaches unity. However, in contrast to equation (5.9) we now find is: < 1 for finite S.

ANISOTROPY

1281

NMR measurements have suggested that the spin defect 8 in the ground state of antiferromagnets is appreciably smaller than the value predicted by spin wave theory.@11 It may be possible to shed some light on this unresolved discrepancy by measurement of the temperature dependence of the anisotropy, particularly for cubic crystals and for small spin values. Utilizing antiferromagnetic resonance such measurements have been performed, by JOHNSON and NETH~COT.(3z) Their measurements were performed on MnFs, and conform well to the Oguchi 2.9 power relation for KB(T)/KZ(O)over the large range T/TN 6 l/3. However, recent reexamination of the Johnson and Nethercot data by BURGIEL and STRANOBERG@~) suggests that a 3.4 power is a better representation at the lowest temperatures. Burgiel and Strandberg also extend the measurements close to the NCel temperature and find that for l/3 < T/TN < 1 a 2.88 power is descriptive. However, as we have remarked, although a power law may be a rough representation of the data at ~termediate temperatures, it has no theoretical validity. It should be borne in mind that our analysis in this section is not highly relevant to MnFs in any case, as our spin wave analysis presupposes a single ion perturbation and 2(EFFER(34) has shown that the paramagnetic susceptibility of this material has an anisotropy which can be accounted for as arising 90% from real magnetic dipolar forces and only 10% from a single-ion term. 6. MAGNETIC FlELD DEPENDENCE OF THE ANISOTROPY Throughout this paper we have written m(2’) for the (sublattice) magnetization. Thii is the magnetization in the absence of the anisotropic perturbation. If, however, the unperturbed Hamiltonian contains an applied magnetic field, this unperturbed magnetization is itself a function of both T and H. Consequently the formalism of this paper remains valid throughout if we merely replace m(T) by m(T,H). In this way the anisotropy becomes field-dependent. In the case of a classical spin the magnetic field dependence of KZ( T,H)/K~(O) is obtained directly by expansion@@ of equation (4.11), K;tyoT

= f,,l,,(f3,2-l(mtT,H)))

(6.1)

H. B. CALLEN

1282

In the paramagnetic

region this results in

w(T,H)

31X1(T)Hl

KZ(O)

1: (2Z+ 1) ! !kP(O)

(6.2) =

where X(T) is the susceptibility, M(0) is the saturation magnetization at zero temperature and (2Z+ l)!! denotes 1 x 3 x 5 . . . (21-l- 1). Below the Curie temperature the field dependence is slightly more complicated cz(T,H)

=

Kl(T,o)

KZ(0)

3

m(Tyo)P -fw1-(5/~0)[f5,24,2] 1+

2f.5,2

-

3m2(

Xexg

T,O)

M(0)

(6.8) Each of the bracketed functions has value Z(Z+ 1)/2 at T = 0”. However the Z = 2 curve has an initial slope of 6, whereas the 1 = 4 curve has an initial IO

Kr (0)

+ 3 fl-r&o)-

and E. CALLEN

I

I

I

I

I

I

I

1

I

[(Z-t 1)/~0+~(~,0)1&+1&0) 1+ 2fs,z(xo) - 3m2( T,O)

6m 3

“gc)

8 7

where xs is to be eliminated equation fs,s(xs)

by the parametric

= m(T,O) = coth x0-- l/x0

m~T.O1[I-f.,tl-~~X~[fr,*+frlrl 1+2f~lt-3m’

1

1

(6.4)

In equation (6.3) we have also written 8m = m(T,H)-m(T,O)

(6.5)

and we have not equated this to X. H because of the non-analytic behavior of X(T,H) as H -+ 0. Experimentally, however, one simply replaces

Sm = X,,,(T,H)*H

(6.6)

with Xexp(T,H) a measured (and therefore wellbehaved) function; it corresponds to the theoretical X(T,H) averaged over a small but nonzero range of H. Elimination of X0 in terms of the experimental magnetization, through equation (6.5), is facilitated by Fig. 1. In Fig. 4 we show the quantities in brackets in the following two equations for the “forced anisotropy coefficients”.

&j~AKE?,o = 3

m( T,O) - [3/x0+ m( T,O)]fs,z 1+2&p

- 3m2( T,O)

Xexp M(O)

(6.7)

FIG. 4. Functions occurring in the theory of magnetic field dependence of the anisotropy energy of uniaxial and cubic crystals. See equations (6.7) and (6.8) of the text. slope of 345. Hence the Z = 2 contribution to the “fractional forced anisotropy” is very much larger than is the Z = 4 contribution and uniaxial crystals should show a much larger effect than cubic crystals. Unfortunately, the only measurements@@ of the forced anisotropy exist for 3d metals, to which none of our analysis applies.

STATUS

OF TEMPERATURE

DEPENDENCE

OF MAGNETOCRYSTALLINE

3. ~G~TOS~I~ION

a uniaxial

Besides the isotropic exchange energy, the Zeeman energy and the anisotropy energy of equation (2-l), the Hamiltonian also contains magnetoelastic terms and the elastic energy: H = H~+&+HA+H~,+H,

(7.1)

The magnetoelastic energy, coupling spins to strains, gives rise to magnetostriction. As magnetostriction has been the subject of a recent extensive analysis,(2s) only a suggestive discussion is given here, to stress the intimate similarity with anisotropy. The magnetoelastic terms are of the form

Hm, = - 1 Bock, 1

crystal.

ANISOTROPY

Under

these

X(T) = X(O)&,,(X)

1283

circumstances, (7.5)

where X is shown as a function of m(T,H) in Fig. 1. Comparison of equation (7.5) with an appropriate m~netost~~on coefficient of dysprosium is shown in Fig. 5. In this comparison, taken from CLARK et al.@‘) the magnetization (and suscep-

(7.2)

and the elastic energy is H, = gc,s

(7.3)

Because the magnetoelastic energy is linear in strain and the elastic energy quadratic, the minimum energy occurs at non-zero strain. Rotating to the magnetization axis to evaluate the first order free energy, again only the cylindrically symmetric Y&S’) remains, and aF/& = 0 implies Er=

c

1

L?
(7.4)

The spherical tensor may be in the space of a’ single spin, in which case Bj represents the variation of a one-ion anisotropy energy by the strain, or it may be in the space of two spins, in which case Bz represents the variation of a two-ion magnetic energy, such as isotropic or anisotropic exchange, or a two-ion anisotropy energy. In any case the thermal average ( Yz* ) is precisely the same quantity we have considered in connection with the anisotropy energy. This identification was first made by KITTEL and VAN VLECK,@@ who pointed out that the magnetostriction coefficient should also follow the 1(Z+ 1)/2 power law at low temperatures. For definiteness, we assume that the magnetostri~tion arises from a single-ion source (which is not always the case), and that the spin is sufficiently large as to be approximated by classical statistics, and we consider only the lowest (I = 2) magnetostriction coefficients. There are generally two such coefficients, hlls and X111, in a cubic crystal and four in

10-f a_ lo”0

I 100

200

I

300

TEMPERATURE %

Rd. 5. Temperature variation of an appropriate magnetostrictian coefficient of dysprosium metal. The points are the experimental data of Clark et al. (Ref. 37) and the solid line is theoretical, adjusted at one point. There is good agreement over three decades. Note that the low temperature rns law, shown dashed, is accurate only at low temperatures. In the parsmagnetic region, where the moment is small, the theory reduces to 3/S m2 for this 1 = 2 coefficient.

tibility in the paramagnetic range) are taken from experiment@@ and the only adjustable constant is h(0). It will be seen that equation (7.5) agrees extremely well with experiment over the entire range of measurement, both in the paramagnetic region and below the NCel temperature (with a basal plane magnetic field present to suppress the antiferromagnetic spiral and induce ferromagnetic ordering). On the other hand, the low-temperature law ma applies accurately only at low temperatures. In a ferrimagnet with aligned sublattice magnetization there will ordinarily be a separate magnetoelastic coupling coefficient for each type of ion in each crystallographic environment. For 1 = 2 magnetostrictions, a single-ion source and a classical spin, and Iabelling the sublattice reduced

1284

H. B. CALLEN

magnetization

by m,(T,H);

and E. CALLEN magnetostriction,

h( T,H) = 2 &#)f&fsis-l(mn(

T,H)))

(7.6)

The field dependence or forced magnetostriction has been discussed in Section 6 in connection with anisotropy energy. Application of equation (7.6) to the saturation magnetostriction (that is, to h(T,O) measured with sufficient magnetic field

experiment

and comparison

of theory

and

is excellent*

REJXRENCFS I. AKULOV N., Z. Phys. 100,197 (1936). 2. KANAMORI J., Mugnetim (editors G. T. Rado and H. Suhl) Chapter 4. Academic Press N.Y. (1963). 3.

VAN VLECKJ. H., Phys. Rew. 52, II78 (1937); see also footnote 13 of Phys. Rew. 78, 266 (1950).

TEMPERATURE (OK)

FIG. 6. Magnetostriction

of YIG, as a function of temperature. hl = 3/2 koo and ha = 3/2 X111. Points are experiments& and lines are theoretical, based upon experimental sublattice magnetization data. Figure from Ref. 39.

strength to eliminate domains, but not large enough to alter m(T)), has been made to YIG by CALLEN et al.(sQ) and to other garnets by CLARK et al.(4*) Figure 6 shows the YIG results. Because there are two magnetic sublattices, each macroscopic Xroo rz 2/3frr and magnetostriction coefficient, hrrr = 2/3h2 depends upon two magnetoelastic coefficients, which are treated as adjustable parameters in Fig. 6. Thus the formalism of magnetostriction is essentially identical to that of anisotropy. Appropriate experimental data is available in the case of

4. 5. 6. 7. 8. 9.

&NER C., Phys. Rev. 96, 1335 (1954). KEPFERF., Phyr. Rev. 100, 1692 (1955). OGUCHI T., Prog. Theor. Phys. Kyoto 13,148 (1955). &iSUYA T., J: Phys, Sot. Japan 11, 944 (1956). See also PAL L., Acta Phys. Hung. 3, 287 (1954).

CHAMP S. H. and Wmss P. R., Phys. Rev. 116, 1372 (1959); erratum, Phys. Rev. 121, 1863 (1961). 10. KEFFZRF. and OCUCHI T., Phys. Rev. 117, 718 (1960). If. VAN VLECK J. H., J. Ph.w. Radium, Paris 20, 124 (1959). 12. OGUCHI T., Phys. Rev. 111,1063 (1958). J. and TACHIKI M., J. Phyr. Sot. Japan 13. KANAMORI 17,1384 (1962).

STATUS

OF TEMPERATURE

DEPENDENCE

OF MAGNETOCRYSTALLINE

14. PINCUSP., Phys. Rev. 113,769 (1959). 15. BOZORTHR. M., Fewomagnetism p. 568,720. D. Van Nostrand, New York (1951). 16. POTTERH. H., PYOC.R. Sot. 146,362 (1954). 17. GRAHAMC. D., Phys. Rew. 112, 1117 (1958). 18. CUR W. J., Phys. Rev. 109,1971 (1958). 19. MITSEKA. I., Soviet Phys. Solid State 5,1312 (1964). 20. CALLENE. R. and CALLENH. B., Phys. Rev. 129,

578 (1963). 21. WIGNER E. P., Gruppen Theorie. Friedrich Vieweg und Sohn, Braunschweig (1931). C. EC-T,

Revs. Mod. Phys. 2, 305 (1930). 22. MERZBACHER E., Quantum Mechanics. Wiley, New York (1961). 23. YOSIDA K. and TACHIKI M., Prog. Theor. Phys. Kyoto 17, 331 (1957); WOLF W. P., Phys. Rev. 108,1152 (1957). 24. FOLEN V. J. and RADO G. T., J. Appl. Phys. 29,

438 (1958). 25. CALLENH. B. and SH.TRIKXXN S., Solid St. Commun.

3, 5 (1965). 26. CALLENE. and CALLENH. B., Phys. Rev. 139, A455

(1965). 27. WOLF W. P., Phys. Rev. 108,1152 (1957).

ANISOTROPY

1285

28. RODRIGUEG. P., MEYER H. and JONESR. V., J. Appl. Phys. 31, 3768 (1960). 29. CALLENE. R., J. Appl. Phys. 33,832 (1962). 30. MATTIS D. C., The Theory of Magnetism p. 169. Harper and Row, New York (1965). 31. Section 43 of Spin Waves, Frederic Keffer, Hundb. Phys. volume~18 (to be published). 32. TOHNSONF. M. and NETHERCOT A. H.. TR.. Phvs. Rev. 114,705 (19S9). 33. BURGIELJ. C. and STRANDBERG M. W. P., J. Phys.

Chem. Solids 26, 877 (1965). 34. KEFFERF., Phys. Rev. 87,608 (1952). 35. VEERMANJ., FRANSEJ. J. and RATHENAUG. W., J. Phys. Chem. Solids 24, 947 (1963). 36. KITTEL C. and VAN VLECKJ. H., Phys. Rew. 118, 1231 (1960).

37. CLARK A. E., DE SAVAGEB. F. and BOZORTHR., Phvs. Rev. 138. A216 096% 38. BEHI&NDTD., L~GVOLD‘s. anh SPEDDINGF., Phys. Rev. 109,1544 (1958). 39. CALLENE. R., CLARKA. E., DE SAVAGEB., COLEMAN W. and CALLENH. B., Phys. Rev. 130,173s (1963). 40. CLARKA. E., DE SAVAGEB. F. and CALLENE. R.,

J. Appl. Phys. 35, 1028 (1964).