Two Mechanisms for the Curie Weiss Susceptihili ty and General

Supplement of the Progress of Theoretical Physics, No. 69, 1980 323

Two Mechanisms for the Curie· Weiss Susceptihili ty and General Spin Fluctuations in Magnetic Systems

Toru MORIYA

Institute for Solid State Physics University of Tokyo, Tokyo 106

(Received July 23, 1980)

Simple arguments are given to explain in a unified way the two mechanisms for the Curie-Weiss susceptibility in the mutually opposite extreme cases, the local moment limit and the weakly ferromagnetic limit. The behaviors of spin fluctuations in these extremes and in the intermediate situation are illustrated.

§ I. Introduction

The Curie-Weiss (hereafter abbreviated to CW) law for the magnetic susceptibility as observed in great many magnetic substances has been quite

familiar to physicists for nearly a century and has been interpreted in terms of a local magnetic moment picture initiated by Langevin1> and Weiss2> and sub

stantiated by Van Vleck8> in quantum mechanical terms in great detaiL For a long time experimental observations of the CW law have been regarded as signals for the existence of the local moments.

Around 1960, weakly ferromagnetic metals such as ZrZn2 and Sc8In have been discovered by Matthias et aL 4> Very small magnetic moments per atom

at T=O K and the spin density map in ferromagnetic ZrZn2 as observed by neutron diffraction5> indicated that the local moment picture was not valid for

this class of materials. Nevertheless the observed magnetic susceptibilities above

Ta obeyed the CW law very precisely up to ""10 T 0 • These experimental observations have motivated the study of itinerant electron magnetism beyond the Hartree-Fock approximation (HF A) and dynamical HF A or the random phase approximation (RPA), since the classical Stoner theory of itinerant

electron magnetism based on the HF A could not explain the CW law in any consistent way.

In early 1970's a new mechanism for the CW law in weakly ferromagnetic metals has been presented independently by Murata and Doniach6>

and by Moriya and Kawabata.n The former is a phenomenological mode

mode coupling theory of spin fluctuations while the latter is the quantum statistical mechanical theory of spin fluctuations renormalized in a self-con

sistent fashion going one step beyond the conventional HF-RPA. In these

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324 T. Moriya

theories, the CW law arises from the coupling between relatively long wave modes of spin fluctuations which tends to suppress the usual growth in the amplitude of long wave spin fluctuations in metals with increasing tempera

ture.

The self-consistent renormalization (SCR) theory of spin fluctuationsn has been further developed extensively in subsequent years leading to a number of new predictions as to the temperature and magnetic field dependences of various physical quantities which have been borne out by subsequent experimental investigations on weakly ferro- and antiferromagnetic metals.81

One of the most important consequences of the SCR theory for weakly ferroand antiferromagnetic metals was that many of the physical properties so far attributed to the nature of the local moment system were shown to be shared by the mutually interacting spin fluctuation modes, having local character in

the reciprocal or wave vector space, i.e., in the opposite limit to the local

moment case.

This fact has given an important clue to the resolution of classical con

troversy as to choosing between the localized and itinerant models, i.e., the problem has been reduced to that of spin fluctuations in a very general sense encompassing the local moment limit and weakly ferro- and antiferromagnetic

limit as mutually opposite extremes.

Recently a unified theory of spin fluctuations has been developed with the use of the functional integral method interpolating between the abovementioned two extremes81 ' 91 and the meaning of the CW law has been discussed

from a general point of view.

The purpose of the present note is to point out that some of our general

conclusions can be derived simply from the fluctuation-dissipation theorem and some additional arguments on the coupling between the spin fluctuation

modes without having recourse to the functional integral formalism. We also

give a simple physical explanation for the new mechanism for the CW law

and visualize, to some extent, the contrast between the two mechanisms and the intermediate situation by illustrating examples based on the general theory. In this sense the contents of the present note include hardly new results,

though some improvements in detail are made over the previous results. We rather hope that this note will help better understanding of the generalized mechanism for the CW susceptibility. In § 2 a simple physical explanation is given for the new mechanism for the CW law in weakly ferromagnetic metals. In § 3 we give general considerations on the magnetic susceptibility and the spin fluctuations and discuss on a general expression connecting two mechanisms for the CW law. In § 4 we conclude with a brief summary and discussions. Throughout this paper we use the units It= kB = 1 and the sus

ceptibility is expressed in units of g2tLB 2, g being the gyro-magnetic ratio and

!LB the Bohr magneton.

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Two Mechanisms for the Curie- Weiss Susceptibility 325

§ 2. A simple explanation for the new mechanism for the Curie-Weiss susceptibility

We give here simple physical arguments to show the origin of the CW susceptibility in weakly ferromagnetic metals. We express the free energy of the system in terms of spatially-varying spin density S (r) and its Fourier transform Sq as follows:

where v 0 is the atomic volume, N 0 the number of atoms in the crystal, Xoq the magnetic susceptibility for the non-interacting electron system, I the intraatomic exchange energy J divided by N 0, and ro the constant. The first term is the increase in the kinetic energy due to the excitation of spin fluctuations, the second term is the exchange energy which is assumed to be local, and we take terms up to the fourth order in the spin density, which is approximated to be local and gives the mode-mode coupling. The magnetic susceptibility is given from the free energy as follows:

(2)

Since we have the following relation from the fluctuation-dissipation theorem:10>

(3)

we can obtain <S~oc) and Xq by solving Eqs. (2) and (3), provided the qdependence of Xoq is given from the band structure. Since the uniform susceptibility diverges at T=Ta we have

(4)

This means that when <Sfoc) is proportional to T we get the CW law. This is possible when (1/Xoq) -21 is very small for q~O and increases relatively rapidly with increasing q. In such a case Xq for not so small q is almost temperature-independent, since the first two terms in Eq. (2) dominate over the last term, and thus the corresponding < 1Sql2) increases linearly with T, giving rise to the linear increase in T of <Sfoc). As was stressed previously, 11l

the CW law in this case is obeyed by Xq with small q only. For example, if we assume

Xoo/Xoq = 1 + Al,

we get

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326 T. Moriya

with

(J = 1j2IX0 , a= 2IX00 , (5)

where qc is the cutoff wave vector. So far as (A/ a(J) u2qc is large <S~ac) is almost linear in T.

As was stated already, the important point is that the CW law in this case arises from the linear increase in T of <Siac) in contrast with the local moment mechanism where <Sfoc) is constant. In order to understand these two different mechanisms in a unified way let us consider the spin fluctuations from a general point of view in the following section.

§ 3. General considerations on the magnetic susceptibility of ferromagnetic metals

Let us consider a tight-binding model for brevity and describe the spin density at the site j in the following Fourier series:

S 1 =No-1 ~ e'q·R,Sq, q

(6)

where N 0 is the number of atoms in the crystal. The Fourier transform of the space-time correlation function is related with the dynamical susceptibility through the fluctuation-dissipation theorem10l as follows:

2 I aP ( · ) -----=- m XJL w + zs , 1-e-"'IT

yqaP(w) = s_== dt e'"'t<Sqa(t+t')S'!..q(t'))

2 TlmxqaP(w+is), (s~+O), 1-e-"'1

(7)

where a and {3 denote the x, y, z components. Integrating Eq. (7) for j = l and a= {3 over w we get

(8)

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where Wm = 2nmT, m being an integer. These relations are exact and to be satisfied by any correct theoretical results. However, these relations are also

useful in developing approximation methods as will be discussed in what follows.

The dynamical susceptibility is a response of the system to the infinitesimal field hqei"t oscillating both in space and in time. For the non-interacting electron system calculation is straightforward and the result is well--known. We now take account of the effect of electron-electron interaction through the fluctuating exchange field produced by the spin density fluctuations. We first

calculate the dynamical susceptibility of the non-interacting system under the

influence of the fluctuating field. The result may be written in terms of the correlation functions of the fluctuating field which can be related with the space-time correlation function for the spin density. Thus the susceptibility in this sense may be written as

x(Q,!2; {Yq(w)}). (9)

The induced moment <S (Q, t)) can be calculated as follows by taking account of the molecular field induced by the external field hQeuu. We have

where we consider the intra-atomic exchange interaction only. Thus the dynamical susceptibility is given as follows:

x(Q, !2) = x[Q, !2;{Yq(w)} J (10) 1-2Ix[Q, !2;{Yq(w)} J

When an explicit expression for X [Q, !2; {Yq (w)}] 1s obtained in a certain approximate way, we can calculate the dynamical susceptibility by solving

Eqs. (7) and (10) simultaneously. This type of theory has actually been developed previously with the use

of a diagrammatical method in the temperature Green function formalism12' and also by using the equations of motion method.w An integral equation for

X (Q, !2) has been obtained and solved for some special cases; in the weakly

ferromagnetic limit the result reduced to tha:t of the SCR theory while in

the case of larger amplitude of spin fluctuations sloppy spin waves above T 0

were also described. Since it is very hard to solve this problem in general, we here make a

further drastic simplification of assuming X (q, w) to be a function of <Sfoc) only. This approximation may be reasonable at least in the two limiting

cases, i.e., when the thermal s_pin fluctuations are of long wave character and

when the spin fluctuations have strongly local character. In the former case only small q components of the spin fluctuations contribute to the tempera

ture-dependent part of <Sfoc) and we can rely on the long wave approxima-

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328 T. Moriya

tion. In the latter case we may start with the random distribution of the local spin fluctuations which have the average squared amplitude <Sfoc). Ab

breviating <S~c) to SL2, for brevity, and assuming a cubic symmetry for the

system we have

We now define the following reduced quantities:

a= 1/2Ix co, o) = [1-2Ix co, o; SL2) J !2Ix co, o; S£2),

rJqm (SL) = 1- [x (q, iwm; Sl) lx (0, 0; S£2)],

as functions of S£2• Then Eq. (6) reduces to

SL2= 3T .I; .I; 1-rJqm(SL) 2INl m q () +rJ«,.(SL)

(11)

(12)

(13)

By solving Eqs. (12) and (13) for SL2 we can determine all the components of the dynamical susceptibility.

In actual practice we have to calculate X (q, w, SL2) in some approximate way as has been studied in a recent series of work8J,w, 15' within a static

approximation in the functional integral formalism. However, we can draw some general conclusions before making actual calculations. We first note that at Ta we have ()= 0 and thus

Ta= 2INo(SL2)a (.L:;(1-r5qm) )-1,

3 m r5qm a

1-2Ix[O, 0; (S.z/)a] =0, (14)

where the suffix C means the value at Ta and < ) the average over q.

If one uses a long wave approximation in the first expression of Eq. (12) replacing SL2 by the uniform mode and regard it to be small, making an ex

pansion in it, and also approximate X (q, w) in the second expression of Eq.

(12) by Xo (q, w), we get the results of the SCR theory from Eqs. (12), (13) and (14).

To proceed further, we make a static or high temperature approximation

at this stage leaving only the m = 0 term in the summation. We have

S 2 _ 3T ( 1-rJ; L- 2INo tJ+rJ '

(15)

where rJ is a function of q. We first note that the Curie temperature is given by

Ta (16)

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Two Mechanisms for the Curie-Weiss Susceptibility 329

and we have

From Eq. (12) we also have

When the temperature variation of SL2 is relatively slow, we can expand the left-hand side of Eq. (18) in Si (T) - Si (Ta). Defining

- [ax(O, 0; T, SL2)/8SL2]a =l_Nor=No/3To, 2

- [ax(O, O; T, SL2)/8T]a=v/2I,

we get the following equation from Eqs. (17) /"'.J (19) :

(19)

(20)

where r= 2/3T0 is the longitudinal stiffness constant for the spin density which measures the stiffness against the change in amplitude of the spin fluctuation. 9>

We now show that this equation contains the above-mentioned two types of the CW law in the opposite two extremes. First in the weakly ferro

magnetic limit, where iJ is small and T 0 is not small in general, we have for iJ4:._r5 and iJ4:._1

iJ= (2Nol/3To) [SL2 (T) -SL2 (Ta)] +v(T-Ta)

:::::[ ( 1 ~ 6 )To-1 +v] (T-Ta). (21)

Since v is usually small the Curie constant is inversely proportional to the longitudinal stiffness constant.

In the opposite limit or the local moment case we have r-HXJ or T0~0 and SL is fixed. Then Eq. (20) becomes

(22)

Expanding the left-hand side of Eq. (22) in terms of rJ/iJ for T'J;>T0, we get

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330 T. Moriya

(23)

Note that the Curie constant is modified by the factor 1/(1-(o")). (rJ) 1s usually quite small in magnetic insulators, where we have <rJ) = T 0/2CU, C and U being the Curie constant and the intra-atomic exchange energy, respectively. In magnetic metals <rJ) is usually expected to be larger than in insulators. 9>

These results are essentially the same as those derived previously with the use of the functional integral formalism. The slight difference between Eq. (20) and the corresponding one in Ref. 8) [Eq. ( 4 · 24)] or Ref. 9) is due to different ways of expansion around T 0 and the present result is considered to be more reasonable, since it leads to correct results in the local

moment limit while the previous one does only asymptotically.

In order to see the magnetic susceptibility in the intermediate range more explicitly we take as an example the following distribution function for rJ:

For brevity we assume P (rJ) to be temperature-independent. Then Eq. (20) reduces to

z 1 +(O)z

with

z=(Jj(rJ),

t=2T/(rJ/To,

to=2To/(r5)2T 0 • (25)

When () and (rJ) is small compared with 1 this equation may be approximated by the previous result :9>

(26)

Note that the important parameter here is t 0 • When t 0 is large we have local moment-like situation while the weakly ferromagnetic limit is realized for t0 --'>0. The same is true for Eq. (25). The result of numerical calculation for Eq. (25) is shown in Fig. 1. We first see that the result depends rather slightly on (rJ). The CW law is best obeyed both in the weakly

ferromagnetic limit (t0 : small) and the local moment limit (t0 --'>oo). In the intermediate range the linearity of 1/r.-T curves is generally worse, as is naturally expected from the entirely different origins of the CW law in the opposite extremes. In a rough sense, however, the susceptibility may be regarded to obey the CW law approximately even in the intermediate range and the

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(a) (b)

Fig. 1. Reduced inversed susceptibility tJJ<o) against reduced temperature for various values of ta=2Tal<a)"To: (a) for <a>=0.4 and (b) for <a)=O.Ol.

Curie constant is the sum of the contributions arising from the two mechanisms. s>. 9> This fact may easily be seen from Fig. 1 as follows: We assume that T 0 is fixed. Then from Eq. (16) <S~oc)a is fixed for a fixed value of <0"). Thus the change in t 0 from oo~O corresponds to that in T 0 from 0-4oo (or r from oo~O). Figure 1 clearly shows that the decrease of the longitudinal stiffness constant r tends to increase the Curie constant. This is due to the increase of <S~oc) with temperature.

In order to visualize the nature of spin :fluctuations in the two extremes and also in the intermediate range we show in Fig. 2 the values for <1Sql 2)

as functions of temperature for various values of q or O"q/<0"> for several typical examples. Figure 2 (a) represents the weakly ferromagnetic case where only small q-components are temperature-independent at high temperatures and <S~oc) as shown by the dashed line increases almost linearly with temperature. Figures 2 (d) and (f) are examples for the local moment limit where <S~oc) is constant. All the q-components here become almost temperatureindependent at high temperatures. Figure 2 (b) represents a typical intermediate case and Figs. 2 (c) and (e) are examples closer to the local moment limit. Note that the Curie constant is rather sensitive to the longitudinal stiffness constant even near the local moment limit. It should be mentioned here that this effect might be somewhat exaggerated in the present approach, where

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332 T. Moriya

the local moment condition is satisfied only on an average even in the limit of r----'>oo, since the approximation in this limit corresponds to the spherical model rather than the real Heisenberg model. In actual practice, it seems

<if) =0.2 50

tc = 1.0 600 (To!Tc= 50)

40

!-\-' ~ 0

~ 400 ~ 30

"" <-<"" "'-- ;;r cK v v 20 .., -. "' 200 "'

10

10 15 T/Tc 10 15 T/Tc (a) (b)

(<f) = 0.2 <if)= 0.2

tc = 1 o tc = oo

(TofTe = 5) (To= Ol 20 20

8' 0 ._,u

0 z z - "' "' 0.5 ..._ "'--cff. CT

!!]. 10 v 10 1.0 v -. -.

"' "' 2.0

10 15 T/Tc 5 10 15 T/Tc (c) (d)

(11) = 0.01 <if>= O.Ql 600

tc = 10.0 600 tc = oo

(Tot Tc = 2000) (T0 = Ol

,_,u ._,u 0 0

z 400 z 400 -"" "" ""-- "'--cK cr

Vl v v -. -. "' 200 "' 200

0 0 1 10 15 T/Tc 5 10 15 T/Tc

(e) (f)

Fig. 2. Temperature-dependence of the average squared amplitude of the spin fluctuation components with various wave vectors for various values of (11) and to (or To/To). The dashed lines indicate the corresponding values for the local spin fluctuation: 2J(Sfoe)/Ta.

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to be important to consider the S£2-dependence of (J and the loss of energy due to the formation of charge density fluctuations as has been discussed in recent investigations,m although general conclusions as discussed here are not altered.

§ 4. Conclusions and discussion

We have given very simple arguments to show the relation between the nature of spin fluctuations and the uniform magnetic susceptibility, leading to a general expression as obtained previously. In particular, we have shown examples to visualize the contrast between the two mechanisms for the CurieWeiss susceptibility arising respectively from quite different behaviors of spin fluctuations in the mutually opposite extremes. Examples for the spin fluc

tuations in the intermediate case are also illustrated. Although these examples are restricted to the cases where (Sfoc) changes rather slowly with temperature, our general expressions up to Eq. (18) cover quite general situations

including the case where (Sfoc) changes rapidly with temperature. The latter case has also been discussed previously.s>,g>,I6>

In any case we see that the physical parameters of fundamental importance here are the average squared amplitude of the local spin fluctuation

(Sfoc), the longitudinal stiffness constant r, and ((J), a measure for the nonlocal character of the spin fluctuations. The form of the distribution function P ((J) is also important. In other words, the general problem of itine

rant electron magnetism is parametrized here in a simple fashion. We expect that these arguments will not only help better understanding

of the generalized mechanism for the CW susceptibility but also call attention

to the vital importance of the very general spin fluctuations in the understanding of magnetism. The' general expressions as discussed here and else

where are based on an adiabatic approximation. Although the results seem to clarify the physics associated with the CW law, considerations of the

dynamical effects are essential for a more quantitative and really satisfactory description of magnetism in general. Such a study is highly awaited.

It is the author's great pleasure to dedicate this little piece of work to

Professor Ryogo Kubo for his sixtieth birthday.

References

1) P. Langevin, J. de Phys. 4 (1905), 678. 2) P. Weiss, J. de Phys. 6 (1907), 667. 3) ]. H. van Vleck, Theory of Electric and Magnetic Susceptibilities (Clarendon Press, Ox

ford, 1932) . 4) B. T. Matthias and R. M. Bozorth, Phys. Rev. 100 (1958), 604.

B. T. Matthias, A M. Clogston, H. J. Williams, E. Corenzwit and R. C. Sherwood, Phys. Rev. Letters 7 (1961), 7.

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334 T. Moriya

5) S. ]. Pickart, H. A. Alperin, G. Shirane and R. Nathans, Phys. Rev. Letters 12 (1964), 444. 6) K. K. Murata and S. Doniach, Phys. Rev. Letters 29 (1972), 285. 7) T. Moriya and A. Kawabata, J. Phys. Soc. Japan 34 (1973), 639; 35 (1973), 669; Proc.

Int. Conf. on Magnetism, Moscow, 1973, IV, 5. 8) For a review see T. Moriya, ]. Magn. Magn. Mat. 14 (1979), 1. 9) T. Moriya and Y. Takahashi, J. Phys. Soc. Japan 45 (1978), 397; J. de Phys. 39 (1978),

C6-1466. 10) R. Kubo, J. Phys. Soc. Japan 12 (1957), 893. 11) T. Moriya, Physica 86"-'SSB (1977), 356. 12) T. Moriya, J. Phys. Soc. Japan 40 (1976), 933. 13) T. Moriya, Physica 91B (1977), 235. 14) T. Moriya and H. Hasegawa, J. Phys. Soc. Japan 48 (1980), 1490. 15) K. Usami and T. Moriya, ]. Magn. Magn. Mat. 20 (1980), 171. 16) Y. Takahashi and T. Moriya, J. Phys. Soc. Japan 46 (1979), 1451.

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Two Mechanisms for the Curie Weiss Susceptihili ty and General