itinerant electron magnetism

10 Journal of Magnetism and Magnetic Materials 31-34 (1983) 10-19

I T I N E R A N T E L E C T R O N M A G N E T I S M

T t r u M O R I Y A

Institute for Solid State Physics, University of Tokyo, Roppongi, Minato-ku, Tokyo 106, Japan

A brief survey is given on the evolution of the theory of itinerant electron magnetism, in particular at finite temperatures, with special emphasis on the latest developments in the theory of spin fluctuations. The present status of our understanding of this problem and points of current controversy are summarized.

1. Introduction - itinerant vs. localized models

The theory of itinerant electron magnetism has started with the Bloch theory of ferromagnetism of an electron gas based on the Hartree-Fock (HF) approximation [1] with the familiar picture of exchange- split up and down spin bands. Wigner [2] subsequently pointed out the importance of the electron-electron correlation which suppresses and eliminates the possibil- ity of ferromagnetism in the electron gas.

The actual occurrence of ferromagnetism in transition metals has been considered to be due to the atomic character of d-band electrons around each atomic site and mainly intra-atomic exchange interactions. Thus, Slater [3] discussed the ferromagnetism of Ni on the basis of the tight-binding d-bands with intra-atomic exchange interaction only, by using the Hartree-Fock (HF) approximation. This model and its simplified ver- sion with a non-degenerate band have been used widely since then (the latter is currently called the Hubbard model). The finite temperature properties of the itinerant electron model have been studied by Stoner [4] within the HF approximation.

As one can easily see this type of theory can be consistent with the non-integral Bohr magneton num- bers per atom of the observed spontaneous magnetiza- tions of Fe, Co and Ni, their large cohesive energies and large coefficients of the T-linear terms in the low temperature specific heat, etc., which are not compatible with the localized electron or Heisenberg picture. How- ever, at finite temperatures the Stoner theory encounters a number of serious difficulties. For example the Curie-Weiss susceptibility above T c as observed in all ferromagnetic metals cannot be explained by the Stoner theory in any consistent way. The theoretical value of T c consistent with the observed spontaneous magnetization at T = 0 is always too high compared with experiment and the temperature dependence of magnetization M ( T ) below T c does not fit with experimental results for Fe or Ni. The specific heat anomaly around T c is calculated to be too small to explain the observations. These difficulties of the Stoner theory are better explained by the Heisenberg or localized model. Since simple theories based, respectively, on the itinerant and

localized models had both merits and demerits in a complementary way, the famous controversy over these two models has lasted for half a century.

Around 1950's to early 1960's it has become very clear that the d-electrons should be treated as localized electrons in magnetic insulators and as correlated itinerant electrons in transition metals. Anderson [5] has shown that in the magnetic insulator compounds d-electrons are localized owing to the Mort mechanism of strong correlation [6] and the relatively weak electron transfer or delocalization mechanism gives rise to the (kinetic) superexchange interaction between the local moments. On the other hand the observations of the Fermi surfaces associated with the d-electrons in magnetic transition metals and the success of the band calculations in reproducing them together with their magnetic properties in the ground states have convinced us of the necessity of using the itinerant model in describing the magnetism in transition metals.

As a matter of fact theoretical efforts since 1950's have been concentrated on finding a way of reconciling the above mentioned two mutually opposite pictures into a unified one, taking account of the effect of electron-electron correlation in the itinerant electron model [7]. There have been two main streams of theoretical investigations in this general direction. One was to improve the Stoner theory taking account of the electron correlation and the other was to study local moments in metals. Owing to recent investigations the theories along these two lines seem to be converging into a unified picture. A brief explanation of these developments is the purpose of the present note.

2. Hartree-Fock-RPA theories and their limitations

The drawbacks of the Stoner theory at finite temperatures arises from its insufficient approximation to the thermal excitations, i.e., the thermal excitations of electron-hole pairs (or of electrons across the Fermi level) are treated within the one-electron approximation. Thus in the Stoner theory ferromagnetism is destroyed by the reduction of magnetization or the band splitting due to thermally blurred Fermi distribution and above T c we have a simple paramagnetic metal.

0 3 0 4 - 8 8 5 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 0 3 . 0 0 © 1983 Nor th -Ho l l and

T. Moriya / Itinerant electron magnetism 11

As everybody knows today it is essential to correct this picture by taking account of the interaction between an excited electron and a hole which gives rise to the spin waves as collective excitations and exchange- enhanced spin fluctuations. The RPA (random phase approximation) theories of spin waves and more general spin fluctuations have been developed by Herring [8] and by Izuyama, Kim and Kubo [9]. Quantitative calculations of spin waves in d-metals based on the calculated band structures have been performed by several people, notably by Cooke et al. [10], and fair agree- ments with the results of neutron scattering experiments at low temperatures have been obtained for Fe and Ni.

Thus the RPA theory was successful in describing the elementary excitations from the ground state. The success, however, was limited to the ground state properties only. In particular RPA has turned out to be always poor above T c. This is quite natural since the RPA theory assumes the HF or Stoner equilibrium state which is a poor approximation except for T---, 0; at finite temperatures we have to calculate the thermal equilibrium state by taking account of the above mentioned collective modes and exchange-enhanced spin fluctuation.

Let us look at this problem from a somewhat different point of view. The Stoner condition for ferromagnetism is given by

2 J x ° > 1, (1)

where J is the intra-atomic exchange energy and X0 ° the uniform susceptibility for non-interacting electrons. Similarly the condition for the appearance of the spin density wave with wave vector q is

2Jx ° > 1. (2)

In a ferromagnet X ° is the largest of all the q-components and if eq. (2) is satisfied for a certain range of q values satisfying q < qm, the corresponding spin density wave states have lower energies than the Pauli paramagnetic s 9 "te This indicates that the paramagnetic state with the spin density fluctuations with q < qm has lower energy than the Stoner paramagnetic state. In particular, when eq. (2) is satisfied in the entire Bril- louin zone or at least the condition

2Jx°o,: = 2 J N - I ~ _ , X ° > 1 (3) q

is satisfied, the paramagnetic state with randomly oriented local moments has lower energy than the Stoner paramagnetic state. Eq. (3) is nothing but the Anderson or Wolff condition for the occurrence of the local moment [11] as applied to pure metals. The local moment theories of ferro- and antiferromagnetic metals has been developed since 1960's and will be discussed in the next section.

One might think that the Stoner theory is valid for weakly ferromagnetic metals, i.e., near the critical

boundary for ferromagnetism. However, it has turned out that this is not the case. Although eq. (2) is satisfied for small q only, long wave components of spin fluctuations, satisfying I 1 - 2Jx°l ,~ 1, are strongly exchange- enhanced and dominate the thermodynamical properties. This effect, which is neglected in the Stoner theory, has been studied extensively in late 1960's and 1970's and will be summarized briefly in section 4.

3. Local moment in metals

When there are local moments in Anderson's sense ferro- or antiferromagnetism is destroyed by thermal excitations of spin rotations and we have disordered local moments above T c. In other words ferro- and antiferromagnetism arises owing to the couplings among the local moments which were studied by Alexander and Anderson [12] and by Moriya [13] by generalizing the Anderson kinetic exchange mechanism in insulators to a pair of virtual bound states [14] in metals. After a systematic study [13] it was pointed out that the sign of the interaction between the neighboring local moments is primarily determined by the occupied fraction of the localized d-orbitals; when each atomic d-shell is nearly half-filled the coupling between the local moments is antiferromagnetic, while the coupling is ferromagnetic when the occupied or empty fraction of each d-shell is small. The sign of the induced spin polarization is also governed by the same rules. The same rules were obtained also for the Wolff-Clogston tight-binding model [15].

Let us explain in a very simple manner how these rules do come out [16]. We show in fig. 1 sketches of the densities of exchange-split virtual bound states for pairs of neighboring atoms. The vertical direction is taken for energy and the horizontal lines represent the Fermi level. Fig. la and b represent the half-filled case with the local moments parallel and antiparaUel with each other, respectively. The gain in kinetic energy due to

a b

c d Fig. 1. Mechanism for the interaction between a pair of local moments associated with virtual bound states.

12 T. Moriya / Itinerant electron magnetism

covalent transfer between two atoms is given by the second order perturbation and may be evaluated by

- b2]E]n,.(1 - n 2 o ) / a E ~ z , (4)

where b is the transfer energy, n j , the occupation number of electrons with spin o in the j th atomic orbital, and AE~2 is the average of the excitation energy. As one can clearly see the antiparallel pair has lower energy than the parallel pair since the former has a larger number of intermediate states as shown by the shaded area. On the, other hand when the virtual states are nearly filled as shown in fig. lc and d, the difference in energy denominators is more important and apparently the parallel pair has lower energy.

These simple rules have been remarkably successful in interpreting magnetic properties of a number of magnetic metals, alloys and intermetallic compounds in a qualitative manner. Furthermore, if one evaluates the effective interatomic exchange energy by properly estimating the width of the virtual bound state and the transfer integral from the observed or calculated d-band width, fairly good values are obtained for the Curie or N6el temperatures. These simple arguments, I believe, contain essential physics in understanding ferro- and antiferromagnetism in a substantial part of d-metals, their alloys and intermetallic compounds. This line of approach has been advanced further by Schrieffer et al. [17] with the use of a functional integral method which has provided a mathematical means of deducing the Curie law of magnetic susceptibility from the Anderson model. The method was also applied to a pair of local moments and it was shown that the static and saddle point approximations correspond to the above-discussed local Har t ree-Fock theory.

The interacting local moment picture was proposed earlier by Friedel et al. [18] from a perturbational point of view. They associated the electron occupation number of d-bands with the spatial extension of the local moment and obtained similar conclusions to the above as to the occurrence of ferro- and antiferromagnetism. This idea has been elaborated by Blandin and Lederer [19] into a Hartree-Fock theory of non-local susceptibilities.

Both of these lines of theories have been refined later on the basis of the Hubbard model with the use of functional integral methods and the coherent potential approximation (CPA), etc., notably by Cyrot [20] and by Capellmann [21]. In particular the theory of interaction between a pair of local moments has been extended to include the influence of the surrounding medium with randomly oriented local moments [22,23], and a recent calculation by Moriya and Hasegawa [23,24] corroborates the above-mentioned rules for the sign of interaction vs. occupation.

As a matter of fact in mid 1960's we hopefully thought that possible advances in mathematical treat-

merits along these local moment ideas should clarify an essential part of the magnetism of transition metals. However, the story was not necessarily that simple. Namely weakly ferromagnetic metals have been discovered and the necessity of dealing with the spin density fluctuations from a more general point of view has been realized.

4. Weakly ferro- and antiferromagnetic metals

In late 1950's to early 1960's Matthias et al. [25] have discovered weakly ferromagnetic metals ZrZn 2 and S%In, having very small spontaneous magnetic moments and low Curie temperatures. The magnetic susceptibilities above T¢ show very good Curie-Weiss behavior and the effective moments deduced from the Curie constants assuming the local moment mechanism were much larger than the spontaneous moment per atom. This fact is evidently incompatible with the local moment picture. Edwards and Wohlfarth [26] applied the Stoner theory to this class of materials without real success; it was not possible to explain the Curie-Weiss susceptibility and other experimental data in any consistent way.

As already mentioned we evidently have to improve the Stoner theory by taking account of the influence of exchange-enhanced spin fluctuations to the thermodynamical quantities. Such theories were first developed for nearly ferromagnetic metals by Doniach and Engels- berg [27], Berk and Schrieffer [28] and B~al-Monod, Ma and Fredkin [29]. These so-called paramagnon theories make use of the RPA spin fluctuations in calculating the renormalized free energy and were successful in the low temperature limit.

At finite temperature the coupling between different models of spin fluctuations, which is neglected in RPA,

SMALL

SATU- RATED

SPIN FLUCTUATIONS LOCAL IN LOCAL IN q-SPACE REAL SPACE

Nearly FerrOmagnetic Metals(a) Weakly FerromGgnetlc(b) & Antlferronmgnetlc(c) Metals

MnS1 ~-Mn CeFe 2

Cr CrB 2 Fe3Pt

Y-Mn MnP CoS 2

NI Co Fe

MnPt 3 FePd 3

Locol Moment SYstems(d)

(o) Pd, HfZn 2,TIBe 2,YRh6B 4,Cesn 5,NI-Pt Glloys, etc.,.

(b) Sc31n, ZrZn 2, NI3AI, Fe.5Co.5SI, LoRh6B 4, CeRh-$B 2, NI-Pt olloys, etc...

(c) B-Mn, V3Se 4, V3S 4, V5Se 8, etc... (d) Mognettc Insolotor Compounds, 4f-Meto]s, Heusler

Alloys, Fig. 2. Aclas~ficafion of magnetic substances according to the nature of spin fluctuations.


q=K

q=O

C

z /

is~ 2 -~5/

b

d

f

Fig. 3. Energy variation due to the spin polarization.

becomes essentially important. In particular i t is essential to calculate the spin fluctuations and the renormalized thermal equilibrium state simultaneously in a self- consistent fashion. Along this line a phenomenological mode-mode coupling theory by Murata and Doniach [30], a quantum statistical mechanical theory of self- consistent renormalization by Moriya and Kawabata [31] and many other theories have been developed in 1970's and consistent descriptions were given on the weakly and nearly ferro- and antiferromagnetic metals. This self-consistent renormalization theory has provided not only a new mechanism for the Curie-Weiss susceptibility but also predicted a number of qualitatively new properties as to the temperature and magnetic field dependences of specific heat, nuclear magnetic relaxa- tion rate, electrical resistivity, etc. and these predictions have been borne out by a number of experimental studies. For details we refer to a review paper [23].

We may summarize the important properties of spin fluctuations in weakly ferromagnetic metals as follows:

(1) The mean square of the local amplitude of spin density, S 2, decreases with increasing temperature below T c and increases above T c, resulting in a new mechanism for the CW susceptibility. We discuss this problem later from a more general point of view.

(2) Above T c we have a strong short range order or the spin correlation has a large spatial extension, since the thermally excited spin fluctuations are mainly long wave components. Large q-components are hardly excited since they have large distributions in energy which are extended to the order of the band widths. This

mechanism of short range order is specific to metals and is quite different from that in insulator magnets.

These properties of spin fluctuations in weakly ferromagnetic metals are in strong contrast with those of local moment systems. Nevertheless these fluctuations have many common physical consequences with the local moment system, including the CW law. This is a quite remarkable fact suggesting that many of the physical properties which were previously attributed solely to the local moment system are in actuality the consequences of more general spin density fluctuations rang- ing between the local moment limit and the weakly ferromagnetic limit and thus many ferromagnetic metals may be in the intermediate range. This finding has given motivation for us to develop a unified theory of spin fluctuations interpolating between these two mutually opposite extremes. Almost continuous distribution of the Rhodes-Wohlfarth ratio [33] (effective moment deduced from the Curie constant/saturation moment) between 1 and e~ also indicates the necessity of such an interpolation theory.

5. Unified picture of spin fluctuations

Let us now consider spin fluctuations from a very general point of view. They may be characterized by (1) the local amplitude of its mean square, SL:, which can in general vary with temperature and (2) the extension of spatial correlation or the degree of short range order. We may classify magnetic substances according to the nature of spin fluctuations in them. Fig. 2 shows such a classification [34]. In the local moment case the spin fluctuation is localized in real space and its local amplitude is large and fixed, while in the weakly ferro- and antiferromagnetic limit the spin fluctuation is localized in the reciprocal or q-space and its local amplitude is small and variable.

In order to get idea about how to deal with the intermediate case interpolating between these two mutually opposite extremes, we first sketch in fig. 3 dia- grams for the energy vs. amplitude of spin fluctuation for various cases for q = 0 and K, K being the zone boundary vector. The slope at the origin is negative when eq. (2) is satisfied. The dashed line indicates the energy for the system with randomly oriented local moments. When its slope is negative at the origin, as is the case when eq. (3) is satisfied, we have local moments in Anderson's sense. Fig. 3a shows the local moment case where the curves have deep minima and the size of the local moment is almost fixed. Fig. 3b corresponds to the weakly ferromagnetic case where energy minimum at finite ISql 2 exists only for very small q. Fig. 3c, d and e represent some typical examples for the intermediate case. In the cases c, d and f we have local moments in Anderson's sense but their amplitudes are not rigid as in the case a. In the case e we have no local moment. From various experimental results we expect that Fe, Ni and


CoS 2 belong to the cases d, e and f, respectively. The thermodynamical properties of these inter-

mediate materials may be evaluated by calculating the functional average over the free energy functional ~[(Sq)] as shown conceptually in fig. 3. The local moment theories as mentioned in the previous section, which correspond to the saddle point approximation to the functional integral theory, apply only when the dashed line in fig. 3 has a minimum. For general purposes we have to develop a theory beyond this limitation. Such a functional integral theory covering all of these cases in a unified way has been developed by Moriya and Takahashi [35]. They consider the following form for the functional representing all the cases in fig. 3 in an approximate way:

~YJ = -- ½ E Vq( S2L )ISq[ 2 "~- ½NoL( s 2 ) q

= - ½ E v A s ? _ ) ( s ; s , ) + j,t

with

S~= No'E (ISqI=>, ~,Vq( S~)=0. (5) q q

This is for the paramagnetic phase. For the ferro- and antiferromagnetic phases see ref. [23]. In this model functional the mode-mode couplings are taken into account in an average. In actual practice the Stratono- vi tch-Hubbard method in the functional integral for- malism has been employed. This approximation in the local moment limit, where S 2 = const, reduces to the spherical model approximation and in the weakly ferromagnetic limit to the Murata-Doniach result which is a static approximation to the Moriya-Kawabata approach. Thus we have an interpolation theory which is reasonable at least qualitatively in the both ends. As for the method of calculating the functional g' from a given band structure we have obtained, by using an expansion method starting from a coherent potential approximation, a closed form expression corresponding to eq. (5) which leads to known correct results in both the limits of small and large amplitude of spin fluctuation or the weak and strong couplings [23,24].

6. Consequences of the unified theory

6.1. Magnetic susceptibility and the Curie temperature

Here we summarize very briefly the results of the unified theory which as a matter of fact have also been shown to be derived without having recourse to the functional integral method [36,37]. The wave vector-de- pendent susceptibility Xq can be written as

I / X q = l / / X q ( S 2) - 2 J = 1/Xo + Vo( S~ ) - Vq( S~ ),

(6) with

= 3 rNExq , (7) q

where J is the intra-atomic exchange energy and ~(q(S~.) is the susceptibility of non-interacting electrons under the influence of the exchange field produced by the spin fluctuations and is approximated to be a function of S~ only. Thus we see that the effective exchange interaction constant is given by

Vq(S2) = ~ q ( S 2 ) ~ q ( S 2 ) , (8)

where ( ) means the average in the Brillouin zone. We need to solve eqs. (6) and (7) for S~ and then to calculate Xq from eq. (7). In particular, the Curie temperature is given as follows by setting 1/X o = 0 in eqs. (6) and (7).

r c = s ~ / 3 ( [ Vo(S~) - vq (s~)] - ' > . (9)

We see that this reduces to the result of the spherical model approximation when S~ is fixed. In the opposite limit of weakly ferromagnetic case 2q(S~) reduces to Xq(0) = X °, and the result of static approximation to the SCR theory is reproduced. Thus we see that the eq. (9) gives a reasonable general expression for the Curie temperature.

The uniform magnetic susceptibility above T c is given by eq. (6) with q -- 0 and we may approximate it as follows so far as the temperature variation of S~(T) is relatively small:

1 ( 2 J ) 2 X0 3T ° [ S [ ( T ) - S [ ( T c ) ] , (10)

with

1 / 3 t o = - [ ~, (11) where l / T o may be called the longitudinal stiffness constant for the spin fluctuation. Using eqs. (6) and (7) in eq. (10) we get

1 ( 2 j ) 2 1 ~q { T Tc }. (12) X0 To No l /X0 + V o - Vq V o - Vq

It is interesting to point out that this single expression contains the two mechanisms for the CW susceptibility: When T o --* 0 the coefficient of 1 / T o in the right hand side vanishes, leading to the spherical model expression for the magnetic susceptibility of a local moment system. In the opposite limit of large T o, l / x o is small. For weakly ferromagnetic metals we may expect l /Xo "~- (Vo - Vq) except for very small q in a fairly wide temperature range. Therefore l / x 0 in the denominator of the first term of eq. (12) may be neglected, leading to the Curie-Weiss law with the Curie constant T o / 4 J 2 ( V o - Vq)- 1 = 3ToTc/4j2S2L(Tc).

In general intermediate cases we have to solve eq. (12). We leave details for ref. [37] and show here only


simple approximate results. We introduce the following reduced quantities:

y = S2(Tc) /3TcXo ,

~o = ( T o / T c ) ( V o / 2 J ) 2, t= T /Tc , (13)

1/r o being the reduced longitudinal stiffness constant, and approximate (V 0 - Vq) in the denominators of eq. (12) by ~V 0 - Vq) = V 0. Then we have

r 0 y 2 + ( l + r 0 ) Y = t - - 1 . (14)

We show in fig. 4 the plots for y vs. T / T c for various values of ~'o and experimental results for comparison.

s~ a

S 2

T T

Fig. 5. Typical examples for the temperature dependence of S 2 mean square amplitude of the local spin fluctuation. The arrows indicate T c.

6.2. Temperature variation of the local amplitude of spin fluctuation

Temperature variation of S2(T) is significant when the longitudinal stiffness is small. We sketch in fig. 5 some typical examples for the temperature variation of S2(T). Below T c, S2(T) usually decreases with increasing temperature up to the value

S 2 ( T c ) = ~/CSL2(0). (15)

When the longitudinal stiffness constant is large or r o = 0 S 2 is constant as shown by line a in fig. 5 and this is the local moment case; The curve d shows the weakly ferromagnetic case where we have ~/c = 3 / 5 [39]. b and c are examples in the intermediate case and e and f will be discussed shortly.

In order to obtain the values of */c for the intermediate case we have to carry out numerical calculations by using band structures and it is not easy to get reliable results except for systematic trends.

The temperature dependence of S 2 above T c, on the other hand, can be obtained more easily from the ex- pressions in this section. In particular when the temper-

ature variation of S 2 is relatively small we get from eqs. (10) and (13)

S[(T)/S[(T¢) = I + roy. (16)

6. 3. Temperature-induced local moments

When S2L(T) increases rapidly with temperature as shown by the lines e and f in fig. 5, it may eventually saturate at around a certain temperature T*, since there is an upper bound S 2 for the value of S 2 associated with the number of electrons per atom. We ha~,e a relatively large value for ~'0 below T* and ~0 --- 0 above T*. The Curie constant changes from SL2(Tc)(I + % ) / 3 to S~l/3 (short range order is neglected) and thus the 1 / X - T plot is expected to have a break at around T*. This type of behavior is expected, for example, when ~k has a form as in fig. 3f and has actually been observed in CoS 2 and CoSe 2. The unusual magnetic and thermal properties of ferromagnetic semiconductor FeSi seem to be explained on the basis of this concept [39]. For details we refer to ref. [23].

6. 4. Thermal expansion and Invar phenomena

,o , r e l . , f il/ ?

0.4 ,, ~

0.2 '~~~c~x~ i 1 2 3 T/Te 4 I 2 3 T/Tc 4

Fig. 4. Reduced inverse susceptibility, y = S2(Tc)/3TcXo, vs. reduced temperature for various values of the inverse reduced longitudinal stiffness constant ~'0; calculated and experimental results [37].

The temperature variation of S 2 is naturally expected to reflect itself in the thermal expansion through the magne to-vo lume coupling. Such a theory has been developed by Moriya and U s a m i [40,41]. In weakly ferromagnetic metals magnetic contribution to the thermal expansion is negative below T c and is smaller than the Stoner-Wohlfar th result by a multiplicative factor of 2 /5 , this result seems to be consistent with observations. Above T c we have a positive magnetic contribution to the thermal expansion which is a new effect absent in the Stoner-Wohlfar th theory and has actually been observed in ZrZn 2, MnSi [42], etc. This positive contribution is also expected to be significant in nearly ferromagnetic metals and seems to have been observed in Ni-A1 alloys.

This line of consideration has also been extended to the fcc F e - N i Invar alloys. A simple rigid band CPA calculation of */c for fcc d-metals shows that it decreases remarkably with increasing number of d-holes as


20

15 ̧

@ ~ I01

T0=0

1.5 ' 2.0

4,0

- - - - - - ~ 5 q/qc

T0=I

' / / T/To= 1.0 1.3 1.5 2.0 4.0

~ o'.s q/q c

T O =10

T/Tc = 1.0 2.0 4.0 6.0

0.5 q/q¢

Fig. 6. Wave vector dependence of ~ISql2)/S~.(Tc) at various temperatures for various values of r0.

one approaches the critical boundary for the stability of ferromagnetic ground state, while for bcc d-metals ~c varies only a little with varying number of d electrons. Estimated order of magnitude for the spontaneous volume striction is also roughly reasonable, although the Invar effect is obtained in a too wide concentration range. This poor concentration dependence was not improved by a more detailed calculation taking account of the difference in potentials for Ni and Fe [41], suggesting possible importance of the short range order in these alloys in particular those with high Ni con- centrations. Subsequent calculations based on the same idea but different approaches also support these results [43,44].

6.5. Short range order (SRO)

The spatial spin correlation or the short range order (SRO) can be calculated from the present theory by Fourier-transforming (ISql 2 ) = TXq. For this purpose we need to know xq(S~) or Vq(S~). We will not go into details here but refer to ref. [37] and show in fig. 6 some examples for (ISql2)/S~(Tc) v s . q/q¢, q¢ being the radius of a sphere having the same volume as the Brillouin zone, for various values of reduced temperature T / T c. Here we simply assumed

V o - Vq= [ Cq2 for q<qm, (17) [ Cq2m for q > qm"

The short range order is larger when qm/qc is larger. One of the remarkable results may be that when the longitudinal stiffness is small or % is large, SRO tends to persist in a wide temperature range above T c. This is consistent with the result of recent neutron scattering experiment on MnSi [45].

When the static approximation is poor for large q components of spin fluctuations just as in the case of weakly ferromagnetic metals, we need to introduce a wave vector cut-off which naturally enhances the short range order as will be discussed in the next section. For details we refer to ref. [37].

7. Sloppy spin waves and short range order in Fe and Ni

Neutron inelastic scattering experiments on Ni by Mook et al. [46] in 1973 have discovered that spin wave-like excitations with not too small wave vectors persisted even above T c and the excitation vanished beyond a certain energy (wave-vector) boundary. This boundary succeeds a similar boundary at low temperatures which is interpreted as a lower bound for the Stoner excitations. Essentially the same phenomenon has been observed in Fe by Lynn [47].

These experimental results together with the observa- tion of forbidden magnon scattering of polarized neu- trons by Sokoloff et al. [48] and Lowde et al. [48], are very clear indications of the failure of the RPA theory and the importance of the mode-mode coupling in these systems.

Several theories have been presented to explain the sloppy spin waves on the basis of the strong short range order. In particular, Korenman, Murray and Prange [49], using the Landau-Lifshits equation, have calculated the frequencies and widths of the sloppy spin waves assuming that the spin density has a fixed local amplitude and its direction varies very gradually from place to place. The exchange stiffness constant was calculated by using a long wave approximation or a local band theory. After analysing the experimental results on Fe they estimated the average angle between a pair of neighboring moments above T c to be 36 °.


However, such a strong SRO has not been explained from theoretical point of view. Edwards [50] and Shastry et al. [51] have pointed out that if one use the Heisen- berg model for Fe, the observed specific heat anomaly and the magnetic susceptibility are not compatible with the strong SRO as postulated by Korenman and Prange. This conclusion seems to be true from a more general point of view if one stays within the static approximation and assume that all the spin fluctuation components in the Brillouin zone are more or less excited [37].

Recently Brown et al. [52], in a series of polarized neutron measurements of ([Sq[ 2) in various magnets, have reported the existence of strong SRO in Fe. They observed only very small amount of inelastic scattering with low energy transfer from spin fluctuations with relatively large q.

A possible explanation for the strong SRO may be given by considering the dynamical properties of spin fluctuations, in a somewhat similar way to the case of weakly ferromagnetic metals. The energy distributions for some of large q-components of spin fluctuation have wide extension spreading into high energy side and thermal excitations of these mode are quite small. In such a case long wave components dominate the thermal spin fluctuations even considerably above T c and we may expect a strong SRO. If this is the case, the local moment picture should be poor and we should rather extend a mode-mode coupling theory of dynamical susceptibilities [38]. Although the static approximation should also be poor in this case, we may give a rough qualitative discussion by introducing a wave vector cut-off just as in the case of weakly ferromagnetic metals. Such a discussion has actually been given recently and it was pointed out that moderately strong

SRO in Fe and Ni can be compatible with the observed magnetic susceptibility [37].

8. Numerical calculations based on band structures

A unified theory of spin fluctuations discussed in the preceding sections has a microscopic foundation, i.e., we can calculate the free energy functional in the form given by eq. (5) from a given band structure. Although we have already discussed in the preceding sections main physical consequences of the theory by using a relatively small number of parameters deduced from experimental data, it is quite important to relate these physical results with band structures. This seems to be particularly true for the intermediate case where the above-mentioned parametrization gives only crude semiqnantitative results. In other words the parametrization in the intermediate case is far from satisfactory as compared with the cases of local moment limit and weakly ferro- and antiferromagnetic limits, where the number of necessary parameters is less than the number of experimental methods to determine them and the theories can well be tested without having recourse to the first principle calculations based on the band structures; as a matter of fact although there are only a very limited number of first principle calculations for insulator magnets we understand these systems extremely well thanks to the aid of well-established spin Hamilto- nians.

There are about two lines of approaches for the calculation of free energy functional from a given band structure, approaches from the long wave limit and those from the local limit. We will not go into details of these approaches, referring to review articles and origi- nal papers, but tabulate recent calculations as follows.

Long wave approximations

Reference Amplitude variation

Non-local character (SRO)

Applied to

Hertz and Klenin [53] 0 /, Takahashi and Moriya [39] 0 zx Capeliman [54] × 0 You et al. [55] × 0

Lin-Chung and Holden [56] x 0 Wang et al. [57] × 0 Takahashi and Tano [58] 0 zx

Ni[x] FeSi [X, specific heat] Fe, Ni [Tc] Fe [energies of special spin configurations] Fe [arc, X, SRO] Fe, Ni [Tc, X, SRO] C°S2, C°Se2 [Tc, X]

Approaches from local limit (based on CPA)

Moriya and Hasegawa [23,24] 0 0

Hubbard [59] 0 x Hasegawa [60] 0 × Usami and Moriya [61] 0 zx Oguchi et al. [62] x 0

s.c. tight-binding band [interaction between moments, Mott transition] Fe, Ni [Tc, X, M(T)] Fe, Ni [Tc, X, M(T)] Fe, Co, Ni [Tc, X, M(T)] Fe [T c, X, SRO]


l[tx the above tables (1 attd ~< meatt that the effect is and is not taken into account, respectively. A means that the effect is taken into account empirically or with crude

(1) Temperature-induced local moments Semiquantitative explanations of the experimental

results have been given for the anomalous temperature dependences of magnetic susceptibility and specific heat in nearly ferromagnetic semiconductor FeSi and anoma- Ious magnetic susceptibilities in CoS~ aud CoSec.

(2) Metal-insulator transition An explicit description of the Mott transition has

been given with the correct results in the insulator limit (Heisenberg model with the Anderson superexchange) as we~t~ as ~n ~ e metallic weak coupling ~imi~ {RPh, ~n the limit of weak correlation).

~3) Curie lemperalures and magnelic suscepfibililies in Fe, Co and Ni

Calculated values for T c are smaller than the Stoner values but fflvers'tfied depenffmg on &eta't~s of apprord- raations, ke. 600-2700 K [or Fe and 300-14~0 K for Ni. The suscept/bil/ties obey the Cur/e-Weiss law ap- proximately but quantitative explanation for the Curie conslan~ts seems W be hard withou~t consideration of ~the quantum corrections.

(4) Short range order Explicit calculations of the spatial spin correlation or

SRO in Fe and Ni have been made by several authors by using approximations of the effective Heisenberg model and the results were essentially the same: there is no strong short range order in contrast with the recent neutron scattering results. These negative results seem to be consistent with the arguments in the preceding section. Although the existing numerical results seem to be still too crude to draw any final conclusions even within the static approximation, the limitation of the static approximation as discussed in the preceding section should be kept in mind.

To conclude we may say that numerical calculations based on the static approximation have semiquantitative significance and would be particularly useful for some systematic study of a series of materials or comparisons among them. For really quantitative purposes it seems that we have to take account of the dynamical nature of spin fluctuations.

9. Conclusions

We may now safely conclude that the finite temperature properties of magnetic materials are described in terms of spin density fluctuations which show various different characteristics, depending on materials, rang- ing between the local moment limit and weakly ferro-

ma~ctetic {imit as showtt kt f~. 2. MaAv ifinerattt e~ec- tron magnets including Cr, Mn, Fe, Co and Ni seem to be in the intermediate range.

factory approaches in both of these extreme cases including the dynamical properties of spin fluctuations. In the microscopic theories in these limits necessary infor- mation on the band structure, etc., is properly parame- trized so that we can determine the values for the parameters from experiments. Comparison between the- oties aad experimettts are Setterally satisfactory.

In the intermediate cases, where most of the itinerant magnets belong to, theories so far developed are mostly based on the adiabatic approximation. Although many important properties including the CW susceptibility of general origins and deviations from it, temperature-in- 6uoed ~oca~ momertts, ~herma~ expartsio~ 6~ae to mag- neto-volume coupling, etc., are described qualitatively or semiquantitatively, it seems to be necessary for quan~i~a@~e purposes Xo ~.ake accounX oi ~he dynam~ca~ effects. Such a consideration' seems to be required also for the explanation of the observed strong short range or&er and tl'te sp'm wave-}h~e modes, of stfm Protractions above T c "m Ni and Fe.

Dynamical properties of the mutually interacting modes of spin fluctuations in the inlermedia~e ease seem Io be one of Lhe mos~ imporlmaX subject to be clarified by future theoretical and experimental investigations.

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