Ima j Appl Math 1968 Wohlfarth 359 74

/ . Inst. Maths Applies (1968) 4, 359-374

Band Model and Ferromagnetismf

E. P. WOHLFARTH

Department of Mathematics, Imperial College, London

[Received February 5, 1968]

The subject is reviewed from an elementary point of view. Topics treated include:band structure of ferromagnetic metals, the band model at 0°K, magnetization at lowtemperatures and at higher temperatures, the Curie point, the susceptibility aboveTa interactions and spin waves in metals. The literature cited goes up to and includes1966.

1. Introduction

ALTHOUGH the model of energy bands has been used for almost 40 years to describethe relatively simple properties of metallic materials and semiconductors, the bandmodel of feiromagnetism in metals and alloys has become topical only in the pastfew years, apparently for the following reason. Although the theoretical basis waslaid as long ago as 30 years by Slater (1936) and Stoner (1938, 1939), the variousexperimental results show only now quite clearly that the model is by and largeappropriate for a satisfactory description of the complex field of ferromagnetism inmetals. It may not be accidental that Herring (1966) has recently published a bookcontaining a very extensive survey of the theoretical fundamentals and in particularon the quantum-mechanical basis of the model, and that some very sophisticatedinvestigations are being carried out at present in many laboratories in order to consoli-date this basic work. Naturally this does not mean that all problems in this field cannow be considered as solved, but the great interest shown in the band model promisesfurther progress. The view here presented differs so greatly from that of Herring as toeliminate any risk of overlapping.

In Section 2 a short description of the band structure of ferromagnetic metals isgiven, with iron taken as an example. Section 3 discusses the properties of the bandmodel at absolute zero and Section 4 those at low temperatures. The elevated tem-perature ranges, including the Curie point, are treated in Sections 5 and 6. Section 7discusses the susceptibility above the Curie point. The first fundamental problemdeals with the interactions between the electrons in Section 8, and the second importantproblem relates to the spin waves in metals, the properties of which are described inSections 9 and 10.

2. Band Structure of the Ferromagnetic Metals

Iron, cobalt and nickel atoms have, outside closed shells, a certain number of 3dand 45 electrons, whose energy levels form energy bands in the transition metals.

•f Translated from Magnetismus, Proceedings Dresden Conference, 13-15 October, 1966, VEBDeutscher Verlag fiir Grundstoffindustrie, Leipzig 1967, with permission of the publishers.

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360 E. P. WOHLFARTH

The structure of these bands determine the description of the ferromagnetic propertiesof these metals. Calculation of the band structures is nowadays carried out by variousapproximate methods. Perhaps the most attractive method for ferromagnetic metalsis the tight binding approximation, because the atomic character of the wave functionshere appears directly in the calculation. For the simpler case, in which only a singleelectron per atom is essentially involved, the wave function of which in the free atomhas the form <£(r), the wave function in the metal has the Bloch form (Mott &Jones, 1936)

W) l^-Rl4>(r-Rt). (1)

Here N is the number of atoms, and the summation must extend over all lattice pointsR;; k is the wave vector. The problem then concerns the relation between the electronenergy e^ and the vector k in the whole region of the Brillouin zone, and also thecalculation of the state density N(e), which in principle is obtained from the surfaceintegral (Mott & Jones, 1936)

w 87i3JVk e k

but in practice must be computed. The energy ek is obtained as the result of a per-turbation calculation, in which the perturbation is the difference AV between theatomic and the metallic potential energy. It has the form

Here the summation extends over the next neighbours RX,EO is the energy in thefree atom, and A and B are negative integrals of the form

A=U(T)AV<t>(r)dx,(4)

Energy curves and N(e) curves for cubic lattices appear in Mott & Jones (1936). Forthe more complex case of the transition metals it is necessary to use in (1), insteadof the single atomic wave function <j>, a sum of such functions. Each term must bemultiplied by a function of k. These functions are then eliminated, and ek is obtainedby solving the equation

\Hnm(k)-ev8nm\ = 0. (5)

H^Qi) depends on simple trigonometric functions of the angles akx, aky and akz

(a being the lattice spacing) and on overlap integrals of the form (4), correspondingto the atomic wave functions <f>m(r) and $n(r). The N(e) curve can again be obtainednumerically after solution of (5).

The integrals in Hnm(k) can be calculated by two different methods. The first method,which we (Fletcher & Wohlfarth, 1951) have used for nickel, uses values of the atomicwave functions and of the atomic potential functions for direct numerical calculationof the integrals. If only ^-functions are introduced in the calculation, not more thanthree independent integrals are necessary to describe the rf-band for nickel (Slater &

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BAND MODEL AND FERROMAGNETISM 361

Koster, 1954). The calculated N(e) curve has a pronounced maximum at the higherenergy values, and the paramagnetic Fermi energy is quite close to this maximum.As will be shown below, this is a necessary but not sufficient condition for ferro-magnetism in metals. The second method for calculation of the integrals was usedby us (Cornwell & Wohlfarth, 1961) for iron. Here the integrals are not calculateddirectly, but the whole method has the form of an interpolation scheme (Slater &Koster, 1954). For this, values of £k, which are calculated by another method, areneeded for a relatively small number of k states in order to allow calculation of thosevalues of the integrals which represent the et values with the greatest accuracy. Once

4 5 r

-0 7 -0 6 -05 -0-4 -0 3 -0 2 ~0'l 0 0-1

Fio. 1. N(t) curve for iron (Cornwell & Wohlfarth, 1961).

the integrals are thus known it is possible to determine eu from a much greater numberof k-states and thus calculate the N(e) curve with greater accuracy. For the case ofiron 8u values were already available from Wood (1962), which it was possible torepresent fairly adequately by 30 overlap integrals. These integrals contained in thesense of the whole method s-, p- and ^-functions, and overlapping of the functionsextended beyond the first, second and in some cases even third neighbours. Figure 1shows the calculated N(e) curve for iron. Here the paramagnetic Fermi energy liesagain close to the highest maximum. Although it can therefore be seen even now thatthe phenomenon of ferromagnetism in iron and nickel is closely connected with theoccurrence of maxima in A^e) curves, the condition for the ferromagnetism is a littlemore complex; for more recent N(e) calculations for palladium using a different method(Freeman et al, 1966) have shown that here too the Fermi energy lies in the vicinityof a maximum. However, palladium is not ferromagnetic but it has a very highparamagnetic susceptibility, and it can be stated in the sense of this paper thatpalladium is "almost" ferromagnetic.

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362 E. P. WOHLFARTH

3. Band Model at Absolute Zero

The following must be assumed at absolute zero (Stoner, 1938, 1951):(1) The carriers of the magnetism are holes in the rf-band.(2) The interaction effects between the carriers can be described by a molecular

field approximation. The structure of the bands was described in Section 2, and themolecular field approximation is discussed in Section 8.

Let now £ be the relative magnetization of the ferromagnetic, i.e. the ratio betweenthe magnetization and its maximum value. It is then convenient to represent themolecular field energy per particle in the form —kQ%, where 6' is a characteristictemperature for the material. The molecular field energy per atom is then

Em = -ink9'C2, (6)where n is the number of particles (holes) per atom. If the magnetization C = 0, thenthe levels are filled up to the paramagnetic Fermi energy eo, so that (N(e) is the statedensity per spin)

JoIf C > 0, then ± bands are involved which contain n± particles and are filled up toe± with

Joor

= I N(e)de,

(8)

in£=\ iV(e)de= N(e)de.Jto J«~

The energy of these particles is then

EE=\ eN(e)de-\eN(e)de, (9)J to J «-

i.e., EE is according to (8) a complicated function of £. The total energy'per atom at0°K is therefore EE + Em, and the equilibrium condition then is

d(EE+Ejjdt = 0;

hence, in accordance with (6), (8) and (9),

AE= e+-e- = 2k9'C- (10)The quantity AE is known as exchange splitting. For stability of the ground state it isnecessary to have the value

Therefore, the condition for ferromagnetism at 0°K is that the right-hand side of (11)at£ = 0 is less than 0 i.e.

W > nl2N(eo). (12)(12) is the "Stoner criterion". As stated above, the high values of N(e) thereforefacilitate metallic ferromagnetism. For the case that (12) applies, the equilibrium

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value of C is given by (10) and the e± values depend on £ in accordance with (8).This equilibrium value changes between 0(n+ = vr = \ri) and l(n+ = n, n~ = 0) asa function of the dimensionless quantity 2N(Eo)k6' jn.

If C < 1, then the application of a high magnetic field H should produce a para-process with initial susceptibility xu with (Wohlfarth, 1962)

Measurements of Xi for iron, after a correction for orbital effects, give approximately1-6x10-5 cm-3 (Stoelinga & Gersdorf, 1966) or approximately 2-5xlO-5cm-3

(Freeman et al., 1966); this agrees roughly with the band calculations (see Section 2).Thus for iron n+>n~ # 0; this metal is therefore described as weakly magnetic.For nickel, Xi is approximately zero after appropriate correction (Freeman etal, 1966)so that here N(e~) K 0, n~ « 0. Nickel is therefore described as strongly magnetic.These differences between iron and nickel have been known for some time (Stoner,1951), and the high field measurements supply proof for these older ideas.

As stated above, palladium is strongly paramagnetic, so that here it follows that

kff < n/2N(eo). (14)

The susceptibility of palladium and other metals at 0°K is then given from (13) by

where

X0 = 2fi2BN{e0) (16)

is the ordinary Pauli spin susceptibility and the factor [1—(k9' /nfiQxo]'1 is known asexchange enchancement. For palladium this factor is, from (Freeman et al., 1966),equal to 12-4, that is to say it is extremely large. In this case it can be expected that thesusceptibility in high magnetic fields should depend relatively strongly on the field(Wohlfarth, 1966b). The formula (15) is generalized below (Sections 6 and 9) to finitetemperatures and finite wave vectors.

4. Magnetization at Low Temperatures

In order to be able to describe the properties of metallic ferromagnetics at finitetemperatures, it is first assumed that the formulae of Fermi statistics are applicable(Stoner, 1938). This assumption was discussed by Cornwell (1965). With the aid of thisassumption and the molecular field approximation we obtain instead of (8) for

, 00 the relations

= fCJo

- l

(17)

where kTt] is the paramagnetic chemical potential.The equations (17) are very complex but can be solved for low temperatures

(Thompson et al., 1964). If the equilibrium value of C at 0°K is equal to 1, it is easy to

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364 E. P. WOHLFARTH

show that the temperature-dependence of C at low temperatures has the followingform:

: = l--/(T)e-A / k T ,n

= f°°Jo

where TJ§ = ^± (T = 0) and A is an energy gap between the Fermi energy in the + spinband and the highest level in the - spin band. The function I(T) was calculated by us(Thompson et al., 1964) from the band calculation (Fletcher & Wohlfarth, 1951) fornickel and has roughly the value

-/(r) = 2-6xicrsr*. (19)n

Experimental results by Foner & Thompson (1959) for nickel show, however, thatthe change of ( at low temperatures follows in the main a simple T* law. This lawfollows from the theory of the spin waves in metals, which will be further discussedin Section 9. The results of this Section must therefore be regarded as terms additionalto the spin wave law, because at low temperatures the spin wave excitations and theband electron excitations are additive in their effects on the magnetization (Comwell,1965; Cornwell & Wohlfarth, 1960). The measured additional term (Foner &Thompson, 1959) for nickel then is

l -C = 2-7xl(r5T*e-4 4 0 / T , (20)

so that the energy gap A becomes approximately 0-04 eV. It must be reported, however,that other measurements show more or less large deviations (Rode & Hermann,1964) from those in Foner & Thompson (1959), that is to say this value of the im-portant quantity A should not be considered to be exact. It is hoped that A can bemeasured indirectly, e.g. with neutrons.

If the equilibrium value of £ at 0°K is smaller than 1, (17) can be also solved at lowtemperatures. Asymptotic series expansion then shows that (Thompson et al., 1964;Cornwell & Wohlfarth, 1960)

i- = l -CT 2 , (21)U

where Co is the value of C at 0°K and the constant C depends in a complex way on thedetails of the N(e) curve. Reliable experimental results bearing on (21) are not yetknown. It should be remembered that the spin wave law here gives the most importantcomponent of the change in magnetization; measurements of the additional terms aretherefore extremely difficult.

5. Magnetization at Elevated Temperatures

Here the complicated equations (17) must be calculated numerically and the resultsnaturally depend on the value of function N(e). Calculations for the magnetization

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are available for the functions N(e) ~ e* (Stoner, 1938), N(e) = const. (Wohlfarth,1951) and for a semi-empirical N(e) curve for nickel (Shimizu et al., 1965).

Figure 2 shows the results of these calculations. The curves 1 and 2 were calculatedfor the Ni—N(e) curve for two values of 8', which did not differ greatly from oneanother. The broken curve applies to N(e) ~ e+ and a 8' value which can be takenfrom the susceptibility measurements. Further, curve 1 corresponds to the calculatedcurve for N(e) = const, and a 8' value from susceptibility measurements. All the curveslie very close together but always far below the measured curve for nickel, which is alsodrawn in Fig. 2. Two explanations exist for this discrepancy. Firstly, spin waveeffects naturally are not considered in these calculations; but at high temperaturesthese can no longer be assumed to be additive, and the magnitude of these effectscannot therefore be assessed at present (see also Section 6).

I O r -

Cos

Exp.

0 5T/Te

1-0

Fio. 2. Magnetization curves for nickel (Shimizu et al, 1965).

Secondly, the assumption (6) for the molecular field energy is doubtful because, asdescribed in Section 8, this assumption is correct only if the interactions between twoparticles are independent of their wave vectors. A small dependence leads to anextended law of the form (Wohlfarth, 1953)

£B=-i^2(l+W), (22)where A is a small constant. Extended solutions of (17) calculated on the basis of thislaw (Hunt, 1953; Shimizu et al, 1965) show that the difference, as illustrated in Fig. 2,between theory and experiment can be eliminated. The same situation is met incalculations of the specific heat. Hunt (1953) shows that for nickel with N{e) x e*a value of A = 0 1 is sufficient to explain the changes in magnetization and specificheat up to the Curie point. However, as stated above, the influence of the spin wavesis here still unknown, but as regards the specific heat this influence should be small.

6. The Curie PointIf the Curie point is determined by the vanishing of the magnetization £ of the band

electrons, then it can be defined simply as the temperature Tc at which the susceptibility

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366 E. P. WOHLFARTH

X(T) becomes infinite. It follows from equations (17) that %(T) can be calculated easilyby writing

C < 1, nBHlkT < 1,

where H is the external field. We then obtain

with

= X(T)H l '

N(e)de

de, (25)

that is to say the equations (15) and (16) have here been generalized to finite tempera-tures. The function/ = J\T) is the Fermi function from (17), with tj± = t]. The Curiepoint is therefore given by the relation

dede = 1, (26)

that is to say the relationship between Tc and 9' is extremely complicated. Equation (26)generalizes equation (12). As an example, for a constant function N(e) = n/(Wohlfarth, 1951), if follows that

T =

lnand thus is a non-linear function. A general survey shows (Wohlfarth, 1964) that 9'/Tc

is about 3 for nickel and about 8 for iron.The model therefore does not require that 6' and Tc are equal, as is the case in

classical statistics (see e.g. (27) with eo -* 0). Even so, the applicability of equation(26) has been questioned repeatedly (Mott, 1944). It is also not unreasonable to assume(Mott, 1964) that if (as will be described in Section 8) 9' depends mainly on intra-atomic Coulomb forces, these do not collapse at the Curie point so that Tc, hereperhaps determined by interatomic effects, has nothing to do with 9'. In this case,local atomic moments are likely to exist also above Tc. This has in fact been observedby Collins (1965) in an iron-nickel alloy. With the aid of neutron diffraction it wasfound in this case that the atomic moment between Tc and approximately 2TC amountsto roughly half the moment at low temperatures and depends only slightly on thetemperature. However, a simple localized model would suggest that the moment hasthe same value above and below Tc, and it has therefore been suggested (Doniach, 1967)that these results are to be explained as arising from scattering between neutrons anddamped metallic spin waves. If that is so, it cannot be considered that these measure-ments settle the Tc problem, but the whole question should still be regarded as open.

Attempts have nevertheless been made (Wohlfarth, 1964) to use a number ofexperimental results for nickel and iron on the basis of the above model in order tocalculate roughly values of the exchange splitting AE (10). Some of these results, butnot all of them, concern the Curie point, so that the fairly good agreement betweenthe various AE values may perhaps suggest that (26) is reasonable. The values of AE

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for nickel and iron, which are obtained semi-empirically in this way, are (Wohlfarth,1964):

nickel AE = 0-35±005 eV,iron AE= 14±002eV.

New experimental and theoretical results concerning the Curie point are thereforenecessary in all cases in order to solve the whole Tc complex.

7. Susceptibility above Tc

In the temperature range very close to Tc (e.g. between Tc and rc+20°K) it wasfound by many investigators (e.g. Kouvel & Fisher, 1964; Colville & Arajs, 1965;Noakes et al, 1966) that the susceptibility x(r) follows the law

X = C(.T-Tcy\ (28)

where y usually lies in the neighbourhood of 1-33 but sometimes a little lower; C isa constant. For the molecular field model y is equal to 1-00 but for a statistical localmodel (e.g. Heisenberg model) y is very likely equal to ^. The band model describedcannot therefore include (28) with y ^ 1 without taking into account short rangeorder effects.

In the higher temperature regions the model has again a simple form. The suscepti-bility is then given by (24) and (25), and the Curie point by (26). It is convenient toexpress a molar susceptibility XA a s a function of the temperature in the followingmanner (Rhodes & Wohlfarth, 1963):

g(X) = ~ T N(e) /(e) deI P° N(e)(29)

de.

Here MA is the saturation moment per mole at 0°K, MB the molar Bohr magneton,R the gas constant and £ the relative magnetization at 0°K. If XA is expressed formally,by a Curie-Weiss law, then the Curie-Weiss constant CA(T, Tc) is a function of Tc andthe temperature T at which CA is measured. Formally, we can then define an effectivenumber of the magnetic carriers per atom, qc, by

R. (30)

In the case of a localized model qc = q,, where q, is defined as the number of carriersat absolute zero, i.e.

q, = MAIMB. (31)

However, for the band model the calculations show that qjq^ must be greater than1 (Rhodes & Wohlfarth, 1963). In the case

we haveCA = nMl/R (32)

and1, = «Co>

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368 E. P. WOHLFARTH

where n is the number of particles per atom and £0 the value of £ at absolute zero.Since here also £0 is proportional to the Curie point (Thompson et al., 1964), we there-fore have in this range qjq, ~ \jTc-^co. The relationship between qjq, and Tc isinteresting also in other cases. Figure 3 shows this relationship for some metals and

1200

FIG. 3. Ratio qc\q, for metals and alloys (Rhodes & Wohlfarth, 1963). (1) Localized model, (2) bandmodel.

200 400 60O 800 1000 I2O0

FIG. 4. Ratio qjq, for metal borides (Cadeville, 1965).

alloys (Rhodes & Wohlfarth, 1963) and Fig. 4 for various metal borides (Cadeville,1965). In short, these results mean that if qjq, = 1 for all Tc then we have a localizedmodel; ifqdq,^ 1, then we have a band model. No statement can therefore be madeabout iron.

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If Tc < T4, eo/k, it is shown by equation (24) and the asymptotic expansion of (25)that (Wohlfarth, 1966a)

riV-iT'-Tlr1. (33)

8. Interactions

After the whole temperature range has now been described more or less satisfactorilyby the band model, two main points remain—the problem of interactions and that ofspin waves. The important interactions are exchange forces and correlation forces.Exchange forces have real meaning only in the case of the Hartree-Fock approxima-tion and relate to forces between parallel spins. However, as is now shown, themolecular field constant calculated in this way is much too large compared with theexperimental values. The Hartree-Fock method is therefore incomplete, and thecorrelation forces between the antiparallel spins must also be included; but this ismuch more difficult and will be discussed only briefly below.

As a model calculation (Wohlfarth, 1953), the exchange for the wave functions oftype (1) was calculated, i.e. for

^r1) = 4=Een"-"^(r1-R/). (1)

Here k is the wave vector and n the position vector; the summation must extend overall lattice points R,. The exchange energy for all t spins is

^t=-iIZ^ (34)i j

with an analogous equation for the I spins. In the Hartree-Fock approximation thetotal exchange energy is then the sum

J = J,+Ji. (35)The energy JtJ is given by

^ 4Z' • " ' (36)

I(l,m,n,p)= \\—<t>*(Tl--R,)4>*(r2-ILj<KTi--Rn)<Kr2--Rp)dTldT2.J J r12

If all forces are considered either intra-atomic (R, = Rm = Rn = Rp = 0) or(owing to the decay of the <f> function or the screening by conduction electrons)interatomic between nearest neighbours, then we obtain/^ = (l/N)I0 + smaller termswhich depend on k(, kj. Here IQ are the intra-atomic Coulomb integrals,

h = [ [ ^ W'i)l2 l ^ ) ! 2 < i d*2- (37)

Without the smaller terms in Jy, the total energy per atom then depends on themagnetization C as follows:

J = -i/o(rt+2+n-2), n± = * « ( l ± 0 ,see (8). The energy per atom is therefore

J= - i J o . i H 2 [ ( l + 0 2 + ( l_02] 1

i.e. J = const, -inke'p, k9' = ±/!/0;J K '

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370 E. P. WOHLFARTH

this agrees with the molecular field approximation (6). The smaller terms in (36) thenlead to (22).

The value of 70 is very high for the rf-functions, approximately 20 eV, that is to say,much higher than the band width W (Herring, 1966). It has been suggested (Herring,1966; Mott, 1964) that the conduction electrons should reduce 7<j greatly by screening,so that 70 becomes smaller than the band width. If this is not so, then 7o must bedecreased by correlation; we shall return to this below. The most important integralswhich occur in the smaller terms of (36) describe screened Coulomb forces betweennearest neighbours and therefore give much smaller values. Even smaller are theexchange integrals (Heisenberg) between nearest neighbours. It is still doubtfulwhether these terms can be neglected in metals as they usually are. A further additionalterm, which does not occur in the above model calculation but nevertheless must beimportant, concerns the intra-atomic exchange energy between 2 {/-functions withdifferent quantum numbers. However, these terms (known as Hund rule terms) arealso relatively small (Herring, 1966).

The most important question therefore is how to decrease 70 by correlation. Thisproblem has not yet received a general solution; but in recent years some work hasappeared which, although carried out on the basis of very rough assumptions, showsin which direction progress is likely to be made (Kanamori, 1963; Hubbard, 1963,1964a,b,c; Gutzwiller, 1964).

The correlation which matters here concerns antiparallel spins, that is to say, theenergy 7o is so large only because the antiparallel spins are not correlated. In accordancewith Hubbard (1963) we consider a lattice with almost one electron per atom whichall have t spins. If another t spin electron enters the lattice, it will naturally beviolently repelled by all t spins by the Pauli principle, so that the t spin band is notinfluenced by interactions. Now, in addition, an entering \. spin is repelled by the tspins. Consequently, the I spin electron can occupy only a small number of latticesites, and the width of the I spin band is much smaller (in the end almost zero) thanthe width of the t spin band. The exchange splitting AE is therefore

AE « \W = nlat, (39)

where Iar is the effective value of 7o and Wthe band width. For a single band and anelectron number n = 1, therefore,

Iat«W- (40)

The effective value of the interaction is therefore determined only by the band structurein this extreme case. If 70 is not quite as large, it can be shown (Kanamori, 1963) that

7efr * 70/(l + do), (7-i *tlW. (41)

For small values 7o (screening by conduction electrons?) therefore 1^ x 70, and forlarge IQ the Ten is given by (40).

For a real metal we can calculate roughly as follows (Hubbard, 1964c): for nickel,taken as example, the band calculations described (Fletcher & Wohlfarth, 1951) showthat only the three uppermost bands of the five J-bands are significant. Since, however,7o derives only from the overlap of atomic functions with the same quantum number,only $ of the integrals is important. For nickel also n x 0-6 so that

AE xix0-6xifVx^W x 0-3-0-4 eV. . (42)

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This value is in good agreement with the semi-quantitative value which was obtainedin Section 6 (Wohlfarth, 1964). With the smaller additional terms in (36) A£ dependsslightly on the wave vector, and in recent calculations (Edwards, 1966) the temperature-dependence of /efr is also discussed.

9. Spin Waves in Metals

With very simple assumptions (Herring, 1966) the dispersion curve for spin waves inmetals can be determined as follows. The assumptions are:

(1) Only a band with the band energy 6k plays a part. However, it is questionablewhether a single band is sufficient for ferromagnetism, as will be shown in Section 10.

(2) The interaction has the form of a constant kO' = inlen. Dependence on thewave vector may also be involved, as was shown in Section 8, and in this case theresults of this Section must be corrected (Herring, 1966).

FIG. 5. Schematic ftco, diagram.

The Hamilton operator is therefore

y c*+ c* _ ,c iC (43)

Here c is a destruction operator and c* a generation operator, k the wave vectorof the electrons, q a spin reversal wave vector and a a spin operator. With the operator

Sq(k) = cJ+q -ck + , (44)

which means that an electron with + spin in the state k reverses its spin and is thusexcited into the state k+q, the equation of motion

ih$Q(k) = [Sq(k), H] (45)

can be transformed with the R.P.A. approximation so that the following equationis obtained for the spin wave energy hcoq

/k~/fc+qI V +q 1 ^ ] (46)

where /u is a Fermi distribution function for the energy ek. Equation (46) generalizesequation (26), because 2k9'/n is equal to 7^.

25

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372 E. P. WOHLFARTH

Equation (46) has two solutions: first the single-particle solution

(47)

which is shown by the shading in Fig. 5. The figure relates to the strongly magneticcase (nickel), in which the minimum excitation denned in (18) is referred to as A.For A = 0 a similar diagram can be plotted. At q = 0 the excitation energy is equalto the exchange splitting AE = nl^.

The second solution of the integral equation (46) gives the spin wave energy hcoQ

itself, and this is also shown schematically in Fig. 5. This branch generally meets thesingle-particle solution at a q vector which is known as qtw Beyond this (at q > <7nuu)the spin waves are damped.

The solution of (46) is very complex for realistic ek. However, if q is small comparedwith the value at the edge of the Brillouin zone, it can be shown by series expansionthat for a cubic metal

(48)with

r/+J. f- f+_f- i /(49)

The relation (48) leads, among other things to the T* law for the magnetization,similar to the Heisenberg model of magnetism. Values of D were measured with theaid of neutron diffraction for many metals and alloys, and the results were compiledby us (Wohlfarth, 1966c) and discussed. In the present paper only one aspect of D,i.e. the question of the sign of this quantity will be dealt with.

10. The Coefficient D

The coefficient D is determined by equation (49), and it is quite clear that itscalculation for realistic e^ is also complicated. At present only some few calculationsfor this problem exist.

The first (Herring, 1966) of these calculations concerned the well known approxima-tion ek = (ffl/2m*)&. Secondly it was shown (Doniach & Wohlfarth, 1965) thatfor the case in which the equilibrium value of the relative magnetization C is smallthe value of D is proportional to (. However, the ferromagnetic ground state becomesunstable if D has a zero point. The proportionality between D and { is therefore veryreasonable, because C->0 also means instability in accordance with the Stonercriterion (12). It can also be shown simply (Thompson et ai, 1964) that for £ <3 1 theCurie point Tc is likewise proportional to £. (But see Section 6.)

It can therefore be stated that

D = kTjiW), (50)where a is the lattice spacing and/(n) is a function of the electron density n per atom;for Ek ~ k2 we have/(n) « n~*. Since for the Heisenberg model with local spin 5

D = const. kTjfll(S+1) (51)

(Hatherly et ai, 1964), it may perhaps be assumed that the correlation between D andTc has more general validity than can properly be expected. The experimental correla-tion (Wohlfarth, 1966c) is also fairly reasonable.

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A new calculation of the quantity D was carried out by us (Katsuki & Wohlfarth,1966) for the simple cubic lattice, for which e* is given by (3). Figure 6 shows therelationship between DjWa2 and n, where W is the band width. The parameter isIett/W. It is seen that D either has two zero points or (for /efr/fP=0-85) alwaysremains negative, so that ferromagnetism is impossible. The first zero point is againdetermined by the Stoner criterion. The zero point at higher n values is also connected

2 0

0-2 04 06 0 8

FIG. 6. Coefficient D (Katsuki and Wohlfarth, 1966).

with a collapse of the ferromagnetic ground state. At even higher n values the spinorder is antiferromagnetic or even more complex (Perm, 1966). As it seems(Kanamori, 1963) that IarlW is never greater than approximately 1, it is doubtfulwhether a single non-degenerate band can be ferromagnetic at all. In spite of this,the results of this section show the direction in which a solution of the problem ofdegenerate bands can be attempted.

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