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Theory and experiment in metallic magnetism
Journal of Magnetism and Magnetic Materials 45 (1984) 1-8 North-Holland, Amsterdam
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T H E O R Y AND E X P E R I M E N T IN M E T A L L I C M A G N E T I S M E.P. W O H L F A R T H Department of Mathematics, Imperial College, London SW7 2BZ, UK
A brief historical survey is followed by a discussion of the present state of the subject with special reference to the following topics: saturation magnetization, Curie temperature, low temperature variations, static and dynamic susceptibility, high magnetic field effects, other results discussed here. The future outlook is also considered,
1. Historical introduction
The proper treatment of metallic magnetism has always been more difficult than that for nonmetals. The localized models more appropriate for the latter are the outcome of somewhat older developments in quantum mechanics and statistical mechanics and are thus more deeply embedded in the thought processes of those engaged in this art. In addition, it is much easier to envisage the physical situation in this case than when dealing with the properties of itinerant electrons in metallic systems. Basically, the energy levels of localized electrons are more easily understood and so is the idea of Heisenberg exchange, one of the earliest useful examples of quantum mechanics. The steps from a given exchange integral J to a calculated and, indeed, observable ferromagnetic Curie temperature T~ may present difficult problems in statistical mechanics, but at least it is clear that T~ - J. The same intrinsic simplicity arises when discussing antiferromagnets or materials with even more complicated localized spin structures. Again, when measuring the spin wave energies of such materials with neutrons the spin wave stiffness D obtained by this technique also bears a simple relation to the exchange integral, D - J. Furthermore, in normal cases one can be fairly sure that the spin waves represent the only elementary magnetic excitations in these systems and that the same D determines simply the temperature decrease of the static magnetization M(T) via the " m o s t famous
law in magnetism"
i(O)
i(O)
(1)
The ingenious pioneering work of Heisenberg and Bloch (for an account given by those who were there, see Sommerfeld and Bethe [1]) has rightly had an overwhelming effect in this field, but has also greatly influenced the degree of progress in understanding metallic magnetism. Historically, the pioneers of what is now called itinerant magnetism (Mort, Slater, Stoner) did not develop this subject very much later than those who first worked on localized electrons. However, it has become obvious from the fact that the electrons responsible for the magnetic order are now distributed in complicated energy bands that much of the physical simplicity of the localized models has no place in discussions of metallic magnetism. This energy band structure not only determines the static properties, such as M(T), arising from the corresponding single particle excitations, but these excitations occur concurrently with the spin wave excitations in ferromagnets or with other spin fluctuations such as paramagnons. The very fact that spin waves have a natural place in itinerant systems, although already implied by Slater [2] in 1937, had to wait for Herring and Kittel [3] to be stressed again. Not only are the static properties of ferromagnets and antiferromagnets now determined by two types of elemen-
0304-8853/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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E.P. Wohlfarth / Theory and experiment in metallic magnetism
tary excitations, but these may also interact (electron magnon interactions). Further, the spin waves and other spin fluctuations are expressed in terms of dynamic susceptibilities x(q, ~) which depend sensitively on the electronic structure. Hence simplifying assumptions have often been made regarding the form of X and the complications of the electronic structure may then not have been described adequately, as will be discussed further. An apparently simpler question has formed a continuing historical thread throughout this subject and is as controversial now as ever: Is the Curie temperature, which is so easily envisaged in the localized model, determined entirely by single particle excitations, or by spin waves or other spin fluctuations, or is this temperature an outcome of even more complicated processes? This question will also be discussed. The history of itinerant magnetism was summarized earlier [4]. It seems not surprising, from the complicated problems just outlined, that it took so long (roughly from 1939 to about 1960) for this model to begin to gain acceptance. What happened to bring this about was less a theoretical breakthrough, but more the development of successful experiments concerned with the Fermi surface of ferromagnetic metals. We wish, therefore, to stress on this occasion that a deep respect for, and understanding of experimental data, backed of course by theory and computation, provides the best chance of understanding metallic magnetism. Thus we will discuss a number of such data in a background of many of the other papers at this meeting. As will be seen, quite a few of the results of experiments can be equally well explained on the basis of localized models and this did not make it any easier to sell the itinerant model in the 1950's! Having thus discussed the experimental situation principally for metallic ferromagnets we will try to forecast where future research in this fascinating subject might be most useful. In this summary of outstanding useful investigations we will also stress another continuing feature of the subject of itinerant magnetism. For real metallic systems not only is the energy band structure complicated, but the materials themselves exhibit a number of metallurgical features, such as
concentration fluctuations on an atomic scale, antistructure atoms or the presence of small clusters. We can not hope to have a universal theory under these circumstances since different materials are affected to different degrees. However, the magnetic properties of metallic systems being frequently so sensitive to these metallurgical features makes careful measurements particularly useful tests for structural investigations of real materials.
2. Experiment and theory On this occasion it may be permitted to approach the choice of topics somewhat subjectively!
2.1. Saturation magnetization at 0 K, M The fact that the ferromagnetic metals have non-integral values of the saturation magnetization when expressed i n / % was, historically speaking, an early indication that an itinerant electron description was more appropriate in this case than a straight localized description where integral values should occur. The variation of the saturation magnetization in alloying these metals with nonferromagnetic metals is a difficult theoretical problem. In the simplest case it may be calculated on the basis of the Landau theory of phase transitions which gives [5] M ( c ) - I c 0 - cl l/z,
(2)
where c is the concentration and c o is the critical concentration for the disappearance of ferromagnetism. Plots of M : vs. c which are linear have frequently been observed [6], e.g. for N i - A I alloys near c o = 74.5% Ni but in a larger number of cases the index in (2) is closer to 1 than to ½, i.e. M varies linearly with c; an historically important case is that of N i - C u alloys. The reasons for this breakdown of Landau theory are thought to be heterogeneities of the magnetization on a local atomic scale [7], i.e. a L a n d a u - G i n z b u r g model must be applied.
2.2. Curie temperature T,. Here we are in media res! There is a "folklore" that the Curie temperatures of Fe, Co and Ni
E.P. Wohlfarth / Theory and experiment in metallic magnetism
calculated on the basis of Stoner theory, i.e.
l fo~N(,) ~
Ld, =1,
(3)
where I is the effective intraatomic Coulomb interaction and f ( O the Fermi function, gives values of about 5 times larger than those observed; see, for example, ref. [8]. For iron this is probably true since the Hund's rule interaction is very pronounced but for Ni, without any important Hund's rule interaction, we have stated before [9] that a value of I comparable with a photoemission value for the exchange splitting leads to Curie temperature in reasonable agreement with that observed. This proposal has had a measure of support from Hasegawa [10]. The other outstanding problem regarding the Curie temperature concerns that of ferromagnetic alloys. Whereas both the Stoner theory and that of Moriya involving spin fluctuations [11] give relation (2) for the magnetization (obviously, since we are at T = 0) they give different results for the Curie temperature ( T > 0). Thus •(c)
-Ic0
-
cl",
(4)
but whereas ~-= ½ for Stoner it equals 3 for Moriya. Similar differences arise when considering the pressure dependence of T¢ [12]. For the latter, values of • close to ½ are sometimes, but not always observed, although the data are not for several reasons overwhelmingly reliable. In many cases the pressure dependence of T¢ follows laws which can only be interpreted if the heterogeneities of the magnetization are again taken into account and ref. [12] makes this plausible, see also ref. [6]. Hence it is not possible to use a relation like (4) to decide unambiguously between different theories of metallic magnetism. This is unfortunate since crucial experiments are rare in this field!
2.3. Low temperature dependence of magnetization M(T) This is one of the oldest problems in the subject. It is concerned with the form of the terms additional to the spin wave term (1). There are
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many candidates for this A M / M : (i) T 5/2 . . . . : h i g h e r order spin wave terms; (ii) T 4 m a g n o n - m a g n o n interactions; (iii) T 2 Stoner excitations, weak ferromagnets; (iv) T n e x p ( - A / k T ) : dto., stong ferromagnets, A = Stoner gap; (V) T 4/3 Moriya theory of spin fluctuations, and others. Any attempts to fit observed data to such laws is fraught with danger: The data must be very accurate, a fit with several parameters is always difficult to defend and a good fit, however convincing, may not be any more so than that to a different law. Nevertheless, this problem is so important that there is a vast literature of papers concerned with it. The situation in nickel is particularly complicated (see the handbook article [9]) since the form of the additional term is not clear and since the spin wave stiffness D itself is not the same as that from neutron scattering data. This is an outstanding problem in metallic magnetism. Although we have fully recognized this danger we have nevertheless fitted the term (iv) to very accurate data (2 × 10 -5 accuracy for M(T)) for the amorphous ferromagnets Fe~ Ni 80- x B18Si 2 and FesoB20 [13]. The coefficient n was taken to be equal to 3 compatible with the (badly determined) band structure. The Stoner gap A was found to be small (20 to 60 K) but to vary systematically with the iron concentration. The coefficient of the exponential also varies systematically and in agreement with expectations [13]. The resulting spin wave stiffness is now (it was not before!) compatible with that from neutron scattering data. Hence we have claimed to have obtained experimental evidence of a rare kind: Single particle and collective excitations coexist peacefully!
2.4. Static and dynamic susceptibility above Tc, x(T) Another classical key problem in metallic magnetism concerns the relatively common observation that the Curie-Weiss law is frequently obeyed over wide ranges of temperatures above T~. On the face of it this would imply the presence of local
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E.P. Wohlfarth / Theory and experiment in metallic" magnetism
moments and hence the inapplicability of a straight Stoner model with Fermi statistics. As was discussed by Edwards [14], Staunton [15] and others pronounced local moments (not localized electrons) may well occur in metallic iron and the observed Curie-Weiss constant then corresponds to a small number of such moments per atom. This small number contrasts with the (incorrect) large number demanded by theories of pronounced short range order above T~ [14]. The situation in nickel is, however, completely different. Shimizu (see ref. [9] for a review) has rightly pointed out that the Curie-Weiss law is, in fact, not all that well obeyed in any case and that the resulting observations are not really incompatible with the Stoner model. A similar case was made out for some weak itinerant ferromagnets (ZrZn 2, N i - P t . . . . ) in terms of the complicated local features in the density of states curve [16]. For example, the Curie-Weiss constant of ZrZn2 just above T~. agrees almost perfectly with the Stoner model [16]. The influence of spin fluctuation effects on the Curie-Weiss law and constant have also been discussed [11] and it seems that this important question has not yet been answered. The same goes for the full explanation of the Rhodes-Wohlfarth plot [17]; this is an empirical plot against T,. of the ratio of magnetic moments measured above and below T~. The original interpretation [17] was based on the weak itinerant ferromagnetism of materials with low Curie temperatures and thus low magnetic moments at OK. Alternative explanations [11] concentrate more on the existence of local moments above T,.. These problems are even more acute when discussing the dyanmic susceptibility of metallic ferromagnets above T~. For nickel the pioneering neutron measurements of Lowde and Windsor ([18], see also the handbook article [9] for a review) were shown to be not incompatible with band calculations at temperatures above and below Tc. We can not resist the temptation to quote from this article as follows: "Perhaps the later neutron data focusing attention on the persistence of spin waves above T,. should be reconsidered on this earlier basis"; however, this controversial subject will be discussed in detail at this meeting by Shirane and Mook [19] and we do not wish to interfere between the neutron experts!
2.5. High magnetic' field effects These effects are of central importance in this subject and some aspects have been reviewed before [20].
2.5.1. "High field" susceptibility' at 0 K, Xo The existence of this effect, first proposed by us in 1949 [21], arises from the fact that if the effective intraatomic Coulomb interaction I is sufficiently small so that I N ( e v ) - 1 = i - 1 << 1 then the magnetization in zero field and at 0 K is susceptible to field variations such that X0 = / ~ U ( c v ) / ( i -
1),
(5)
where N(c v) is the electronic density of states at the Fermi energy; in addition, if T~. were to be given entirely by single particle excitations then
(6)
Xo = t*2 N( t"F)TF/T', 2 " where
(7) t F
in terms of derivatives of N(Ev). Relation (6) has been used to discuss high pressure effects [12] which are sensitive to a breakdown of this relation arising from heterogeneities of the magnetization. The existence of X0 is an indication that the itinerant electron model provides a good description of the magnetic properties of a substance since a straight localized model (without unaligned localized spins) can not encompass such an effect. Large values of X0 have often been observed. From (4) and (6)
Xo ~ I c 0 - el ',
(8)
and it is the non-applicability of this relation which is such a good indicator of alloy heterogeneity; for example, in F e - A l alloys X0 has a broad maximum near co rather than a divergence [221.
2.5.2. High field effects in nickel near T [23], see
also ref [20] These measurements seem to us to be very important in supporting the idea that in nickel single particle excitations are present with consid-
E.P. Wohlfarth / Theory and experiment in metallic magnetism
erable weight near T~. They were carried out in high fields up to 32 T and at temperatures up to 700 K (T~ = 627 K). The magnetization M(H, T) was fitted to the Landau expression [5], [24]
H=A(T)M+B(T)M
3,
(9)
valid outside regimes of critical behaviour. The coefficient A ( T ) is given on the basis of Fermi statistics, and with the coefficients such that A(T~) = 0, by the relation
A ( T ) = alo + a,2T 2 + a14T4 +
.
.
(10)
.
which was found to be applicable up to T ~ from 550 to 700 K and, with T 4 included, even above (up to 1700 K, using other data for the paramagnetic susceptibility; this fit verifies, incidentically, that Shimizu [9] was right in doubting the applicability of the Curie-Weiss law in nickel). The expression (10) breaks down below 550 K, presumably since M ( T ) is then too large. Of even greater interest is the coefficient B ( T ) which is given by [51, [24]
B ( T ) = a30 + a32 T2.
(11)
This expression is completely different from that based on classical statistical mechanics [20], namely
B ( T ) = T/3CM2(O),
(12)
where C is the Curie-Weiss constant. Hence a test of (11) and (12) provides a rare opportunity to distinguish models. For nickel fig. 1 shows that (11) applies very well from 550 to 700 K. The o
30 0
~,20
T2
I
1
I
2
I
3
I
4
I
5
T2 105K2 Fig. 1. Variation of the Landau coefficient B with T for nickel near the Curie point [23].
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coefficient a32 is small but clearly finite; it is given by the fine structure of the density of states curve via formulae like (7) only even more complicated, We will return to B ( T ) shortly. The analysis of [23] is, however, even more incisive: Using the parameters ai i the saturation magnetization at 0 K (where the formulae are expected to be inaccurate) came out as 62.6 e m u / g r , only 7% above the correct value 58.6! The exchange splitting obtained on this basis is 0.33 eV in good agreement with photoemission data [25] and compatible with the observed Curie temperature as already pointed out. Hence it must be concluded that the high field magnetic properties of nickel in a wide range about Tc are quantitatively described in terms of the prevalent single particle excitations. The relevance to the spin wave controversy [19] needs to be discussed. However, it must be stressed that Ponomarav and Moreva [26] tested the model for single crystal terbium and found the expected result that it did not apply whereas one based on localized electrons did.
2.5.3. Brief reference to Arrott plots The relation (9) has been used to analyze M(H, T) data via plots of M 2 vs. H / M . The results for a wide range of alloys, both crystalline and amorphous, were summarized in ref. [6]. It seems clear that deviations from straight line Arrott plots provide valuable evidence for the extent of spatial variations of the magnetization. These can be discussed on the basis of L a n d a u - G i n z b u r g theory where A and B are spatially dependent and the fluctuations of M ( r ) are limited by a positive gradient term in the free energy. In extreme cases this approach breaks down and a model involving superparamagnetic clusters is more useful in describing these fluctuations. We recall that pressure data are also useful indicators of such effects. In all cases it is a question of the relative importance of these origins of fluctuations of the magnetization arising from metallurgical effects and other origins such as spin fluctuations [11]. 2.5. 4. First-order transitions [20] As was first pointed out by us in 1962 [27] paramagnetic metals and alloys may in high fields and at 0 K undergo metamagnetic or first-order
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E.P. Wohlfarth / Theory and experiment in metallic magnetism
transitions provided the fine structure of the density of states curve is such that A(0) = a~0 > 0 and B(0) = a30 < 0. The analysis showed that the effect is most likely if the density of states curve has a local region of positive curvature at the Fermi energy and if IN(~v) is just below 1. The resultant critical field is proportional to {a3o/la~o]}1/2 and the effect has been further discussed for Pd [28], where this field H c was computed to be 300 T. For Y Co 2 there is experimental evidence for this effect with H ~ - 100 T but band structure calculations [29] give larger values. For TiBe 2 some extremely accurate band calculations [30] illustrate well the sensitivity of the effect to the density of states fine structure. However, a "close look at low temperature magnetization data" [31] implies that the metamagnetic transition expected at about 5 T [20] and observed to be of this order [32], is more complicated than forecast and that the scale of the fine structure effect even lies below that of the band calculations of refs. [30]. For all three examples, Pd, YCo 2 and TiB% the paramagnetic susceptibility vs. temperature curves have maxima at relatively low temperatures and these could also arise from the density of states features leading to metamagnetism [27], [20]. A further range of materials where the effect has been observed are the dichalcogenides Co(SxSel_x) 2 [33]. There are as yet no corresponding accurate band calculations. For ferromagnetic materials first order transitions may occur as a function of temperature where B(T) changes sign, i.e. at a critical temperature given by (11) as T~* = la3o/a3211/2. This was the interpretation given to explain [34] the firstorder transitions in ErCo 2, H o C o 2 and DyCo 2 and may also apply to other materials such as MnAs for which it would replace that of Bean and Rodbell [35]. The latter used for B(T) relation (12) and brought about a sign change via appropriate magnetoelastic terms'in the free energy expansion. The physics of the two models is thus completely different. Band calculation for MnAs have now been published [36] and will have to be analyzed to distinguish these approaches. We have discussed high field effects particularly since they seem to provide unusually rich results relevant to metallic magnetism.
2.6. Other results The other papers of this Conference and in these Proceedings cover various outstanding and controversial problems in the subject of metallic magnetism. The neutron scattering data [19] regarding the existence or otherwise of spin wave excitations above T~ are particularly important since if they really and incontrovertibly existed the simplest theoretical approaches to metallic magnetism would have to be abandoned. We have stressed our belief that the high field data [23] for nickel are relevant to this controversy in pointing to the prevalence of single particle excitations in this temperature range. Unfortunately no corresponding data for iron are available. However, the case for a simple Stoner model for nickel has been strengthened to some extent by the beautiful photoemission data obtained up to and beyond T~ and discussed by Kisker [37]. Again one would hope to have equivalent data for iron, it being understood that the Curie temperature is much higher. The polarization of photoemitted electrons has been measured as a function of energy [38] and the very valuable result was obtained that the existence of a maximum followed by a sign change gave strong evidence for strong itinerant ferromagnetism with a measurable Stoner gap. This method has been used most recently [39] to give some degree of confirmation to the proposal [13] that a range of iron rich amorphous ferromagnets are also strong itinerant. Among other papers the one on de H a a s - v a n Alphen experiments on itinerant ferromagnets [40] is particularly interesting as it enables the magnetism of the classical weak itinerant materials Ni3A1 and ZrZn 2 to be discussed. It thus seems that experimental techniques of a very high sophistication are being applied to the problems of metallic magnetism and this historical development can only be applauded.
3. Future outlook
In the more immediate future it would be hoped that the controversies regarding spin waves above Tc may be settled amicably [19] and the same goes for that regarding giant short range order there
E.P. Wohlfarth / Theory and experiment in metallic magnetism
[41]. Looking further ahead a number of problems will require solution and these arise naturally from the account given here and in other talks at this meeting. Firstly, band calculations of an astonishing accuracy are now possible and a real breakthrough must be made in computing effects such as those of spin fluctuations for real materials to the same degree of sophistication. We recall our earlier discussion [42] regarding paramagnons in palladium and related materials where the relatively small effect of these fluctuations on physical properties was very tentatively ascribed to band structure and degeneracy effects. Unfortunately the early work of Diamond [43] has never been pursued further. Secondly, the continuing refinement of neutron scattering, photoemission, de H a a s - v a n Alphen, high pressure, high field and related facilities will give an opportunity for obtaining ever more reliable data on metallic materials of greater complexity. Once the obsession with iron and nickel has abated it will be possible to look at the other materials in greater detail. If one were to be asked to predict one series of materials with a promising scientific future it would be transition metal compounds such as MnX(X = As,Sb,Bi). This is not to forget the actinides whose glowing future we predicted earlier [4]. Finally, we repeat yet again the importance of metallurgical defects such as concentration fluctuations. The account of this effect given in [6] shows clearly how sensitive many magnetic properties are to such defects and this dependence must be studied more systematically, e.g. by measuring properties as a function of metallurgical treatment. To repeat again, where these influences are severe it is certainly not possible, even if it were on more fundamental grounds, to have a universal theory of metallic magnetism.
Acknowledgements It gives me particular pleasure on this occasion to thank all my friends and colleagues for their great continuing kindness and support. I wish to thank Bal/lzs Gyorffy, Franqois Gautier and Her-
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bert Capellmann for organizing this meeting and David Edwards not only for editing the Proceedings.
References [1] A. Sommerfeld and H. Bethe, Handbuch der Physik 24/2 (1933) 333. [2] J.C. Slater, Phys. Rev. 52 (1937) 198. [3] C. Herring and C. Kittel, Phys. Rev. 81 (1951) 869. C. Herring, Phys. Rev. 85 (1952) 1003, 87 (1952) 60; Proc. Roy. Soc. A371 (1980) 67. [4] E.P. Wohlfarth, Inst. P. Conf. Ser. 55 (1980) 161. [5] E.P. Wohlfarth, Proc. ICM Moscow 2 (1974) 28. [6[ J. Schneider, A. Handstein and K. Zfiv~ta, J. Magn. Magn. Mat. 42 (1984) 73. [7] H. Yamada and E.P. Wohlfarth, Phys. Lett. 51a (1975) 65. [8] T. Jarlborg and M. Peter, J. Magn. Magn. Mag 42 (1984) 89. [9] E.P. Wohlfarth, Physica l19B (1983) 203; Europhys. News 14 (7) (1983) 7; Handbook Ferromagnetic Materials 1 (1980) 1. [10] H. Hasegawa, J. Phys. F13 (1983) 1915, 1926. [11] T. Moriya, J. Magn. Magn. Mat. 14 (1979) 1. [12] E. P. Wohlfarth, Phys. Sol. High Press. (North-Holland, Amsterdam, 1981) p. 175. [13] E. Babi6, Z. Morohni6 and E.P. Wohlfarth, Phys. Lett. 95A (1983) 335. [14] D.M. Edwards, J. Magn. Magn. Mat. 36 (1983) 213. [15] J.B. Staunton, B.L. Gyorffy, A.J. Pindor, G.M. Stocks and H. Winter, J. Magn. Magn. Mat. 45 (1984) 15. [16] E.P. Wohlfarth and P.F. de Ch~tel, Physica 48 (1970) 477. E.P. Wohlfarth, Comments Solid State Phys. 6 (1975) 123. [17] R. Rhodes and E.P. Wohlfarth, Proc. Roy. Soc. 273 (1963) 247. E.P. Wohlfarth, J. Magn. Magn. Mat. 7 (1978) 113. [18] R.D. Lowde and C.G. Windsor. Adv. Phys. 19 (1970) 813. [19] G. Shirane, J. Magn. Magn, Mat. 45 (1984) 33. [20] E.P. Wohlfarth, J. Appl. Phys. 50 (1979) 7542; J. Magn. Magn. Mat, 20 (1980) 77; High Field Magnetism, ed. M. Date (North-Holland, Amsterdam, 1983) p. 69. [21] E.P. Wohlfarth, Proc. Roy. Soc. 195 (1949) 434, 444. [22] Y. Muraoka, H. Wada, M. Shiga and Y. Nakamura, Physica l19B (1983) 174. [23] B.K. Ponomarev and V.G. Thiessen, Sov. Phys. JETP 46 (1977) 173. [24] D.M. Edwards and E.P. Wohlfarth, Proc. Roy. Soc. A303 (1968) 127. M.J. Besnus, Y. Gottehrer and G. Munschy, Phys. Stat. Sol. (b) 49 (1972) 597. [25] D.E. Eastman, F.J. Himpsel and J.A. Knapp, J. Magn. Magn. Mat. 15-18 (1980) 1136. [26] B.K. Ponomarev and N.I. Moreva, Sov. Phys. JEPT 42 (1976) 690. [27] E.P. Wohlfarth and P. Rhodes, Phil. Mag. 7 (1962) 1817.
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E.P. Wohlfarth / Theory and experiment in metallic" magnetism
[28] T. Jarlborg and A.J, Freeman, Phys. Rev. B23 (1981) 3577. [29] K. Schwarz and P. Mohn, J. Phys. F14 (1984) L129. [30] T. ,larlborg, P. Monod and M. Peter, Solid State Commun. 47 (1983) 889. [31] F. Acker, R. Huguenin, M. Pelizzone and J.L. Smith, to be published. [32] J.M. van Ruitenbeek, A.P.J. van Deursen, L.W.M. Schreurs, R.A. de Groot, A.R. de Vroomen, Z. Fisk and J.L. Smith, J. Phys, F, to be published. [33] K. Adachi, A. Matsui, Y. Omata, H. Molymoto, M. Motokawa and M. Date, J. Phys. Soc. Japan 47 (1979) 675. [34] D. Bloch, D.M. Edwards, M. Shimizu and J. Voiron. J, Phys. F5 (1975) 1217.
[35] [36] [37] [38] [39] [40] [41] [42]
[43]
C.P. Bean and D.S. Rodbell, Phys. Rev. 126 (1962) 104. R. Podlucky, J. Magn. Magn. Mat. 43 (1984) 204. E. Kisker, J. Magn. Magn. Mat. 45 (1984) 23. W. Eib and S.F. Alvarado, Phys. Rev, Lett. 37 (1976) 444. E.P. Wohlfarth, Phys. Rev. Lett. 38 (1977) 524. R. Ailenspach, E. Colla, D. Mauri, M. kandolt and E.P. Wohlfarth, Phys. Left., to be published. G.G. Lonzarich, J. Magn. Magn. Mat. 45 (1984) 43. H. Capellmann, to be published. P.F. de Chhtel and E.P. Wohlfarth, Comments Solid State Phys. 5 (1973) 133, 141. see also A.H. MacDonald, Phys. Rev. B24 (1981) 1130. W. Joss and G.W. Crabtree, Phys. Rev. B, to be published. J.B. Diamond, Intern. J. Magnetism 2 (1972) 241.
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T H E O R Y AND E X P E R I M E N T IN M E T A L L I C M A G N E T I S M E.P. W O H L F A R T H Department of Mathematics, Imperial College, London SW7 2BZ, UK
A brief historical survey is followed by a discussion of the present state of the subject with special reference to the following topics: saturation magnetization, Curie temperature, low temperature variations, static and dynamic susceptibility, high magnetic field effects, other results discussed here. The future outlook is also considered,
1. Historical introduction
The proper treatment of metallic magnetism has always been more difficult than that for nonmetals. The localized models more appropriate for the latter are the outcome of somewhat older developments in quantum mechanics and statistical mechanics and are thus more deeply embedded in the thought processes of those engaged in this art. In addition, it is much easier to envisage the physical situation in this case than when dealing with the properties of itinerant electrons in metallic systems. Basically, the energy levels of localized electrons are more easily understood and so is the idea of Heisenberg exchange, one of the earliest useful examples of quantum mechanics. The steps from a given exchange integral J to a calculated and, indeed, observable ferromagnetic Curie temperature T~ may present difficult problems in statistical mechanics, but at least it is clear that T~ - J. The same intrinsic simplicity arises when discussing antiferromagnets or materials with even more complicated localized spin structures. Again, when measuring the spin wave energies of such materials with neutrons the spin wave stiffness D obtained by this technique also bears a simple relation to the exchange integral, D - J. Furthermore, in normal cases one can be fairly sure that the spin waves represent the only elementary magnetic excitations in these systems and that the same D determines simply the temperature decrease of the static magnetization M(T) via the " m o s t famous
law in magnetism"
i(O)
i(O)
(1)
The ingenious pioneering work of Heisenberg and Bloch (for an account given by those who were there, see Sommerfeld and Bethe [1]) has rightly had an overwhelming effect in this field, but has also greatly influenced the degree of progress in understanding metallic magnetism. Historically, the pioneers of what is now called itinerant magnetism (Mort, Slater, Stoner) did not develop this subject very much later than those who first worked on localized electrons. However, it has become obvious from the fact that the electrons responsible for the magnetic order are now distributed in complicated energy bands that much of the physical simplicity of the localized models has no place in discussions of metallic magnetism. This energy band structure not only determines the static properties, such as M(T), arising from the corresponding single particle excitations, but these excitations occur concurrently with the spin wave excitations in ferromagnets or with other spin fluctuations such as paramagnons. The very fact that spin waves have a natural place in itinerant systems, although already implied by Slater [2] in 1937, had to wait for Herring and Kittel [3] to be stressed again. Not only are the static properties of ferromagnets and antiferromagnets now determined by two types of elemen-
0304-8853/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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E.P. Wohlfarth / Theory and experiment in metallic magnetism
tary excitations, but these may also interact (electron magnon interactions). Further, the spin waves and other spin fluctuations are expressed in terms of dynamic susceptibilities x(q, ~) which depend sensitively on the electronic structure. Hence simplifying assumptions have often been made regarding the form of X and the complications of the electronic structure may then not have been described adequately, as will be discussed further. An apparently simpler question has formed a continuing historical thread throughout this subject and is as controversial now as ever: Is the Curie temperature, which is so easily envisaged in the localized model, determined entirely by single particle excitations, or by spin waves or other spin fluctuations, or is this temperature an outcome of even more complicated processes? This question will also be discussed. The history of itinerant magnetism was summarized earlier [4]. It seems not surprising, from the complicated problems just outlined, that it took so long (roughly from 1939 to about 1960) for this model to begin to gain acceptance. What happened to bring this about was less a theoretical breakthrough, but more the development of successful experiments concerned with the Fermi surface of ferromagnetic metals. We wish, therefore, to stress on this occasion that a deep respect for, and understanding of experimental data, backed of course by theory and computation, provides the best chance of understanding metallic magnetism. Thus we will discuss a number of such data in a background of many of the other papers at this meeting. As will be seen, quite a few of the results of experiments can be equally well explained on the basis of localized models and this did not make it any easier to sell the itinerant model in the 1950's! Having thus discussed the experimental situation principally for metallic ferromagnets we will try to forecast where future research in this fascinating subject might be most useful. In this summary of outstanding useful investigations we will also stress another continuing feature of the subject of itinerant magnetism. For real metallic systems not only is the energy band structure complicated, but the materials themselves exhibit a number of metallurgical features, such as
concentration fluctuations on an atomic scale, antistructure atoms or the presence of small clusters. We can not hope to have a universal theory under these circumstances since different materials are affected to different degrees. However, the magnetic properties of metallic systems being frequently so sensitive to these metallurgical features makes careful measurements particularly useful tests for structural investigations of real materials.
2. Experiment and theory On this occasion it may be permitted to approach the choice of topics somewhat subjectively!
2.1. Saturation magnetization at 0 K, M The fact that the ferromagnetic metals have non-integral values of the saturation magnetization when expressed i n / % was, historically speaking, an early indication that an itinerant electron description was more appropriate in this case than a straight localized description where integral values should occur. The variation of the saturation magnetization in alloying these metals with nonferromagnetic metals is a difficult theoretical problem. In the simplest case it may be calculated on the basis of the Landau theory of phase transitions which gives [5] M ( c ) - I c 0 - cl l/z,
(2)
where c is the concentration and c o is the critical concentration for the disappearance of ferromagnetism. Plots of M : vs. c which are linear have frequently been observed [6], e.g. for N i - A I alloys near c o = 74.5% Ni but in a larger number of cases the index in (2) is closer to 1 than to ½, i.e. M varies linearly with c; an historically important case is that of N i - C u alloys. The reasons for this breakdown of Landau theory are thought to be heterogeneities of the magnetization on a local atomic scale [7], i.e. a L a n d a u - G i n z b u r g model must be applied.
2.2. Curie temperature T,. Here we are in media res! There is a "folklore" that the Curie temperatures of Fe, Co and Ni
E.P. Wohlfarth / Theory and experiment in metallic magnetism
calculated on the basis of Stoner theory, i.e.
l fo~N(,) ~
Ld, =1,
(3)
where I is the effective intraatomic Coulomb interaction and f ( O the Fermi function, gives values of about 5 times larger than those observed; see, for example, ref. [8]. For iron this is probably true since the Hund's rule interaction is very pronounced but for Ni, without any important Hund's rule interaction, we have stated before [9] that a value of I comparable with a photoemission value for the exchange splitting leads to Curie temperature in reasonable agreement with that observed. This proposal has had a measure of support from Hasegawa [10]. The other outstanding problem regarding the Curie temperature concerns that of ferromagnetic alloys. Whereas both the Stoner theory and that of Moriya involving spin fluctuations [11] give relation (2) for the magnetization (obviously, since we are at T = 0) they give different results for the Curie temperature ( T > 0). Thus •(c)
-Ic0
-
cl",
(4)
but whereas ~-= ½ for Stoner it equals 3 for Moriya. Similar differences arise when considering the pressure dependence of T¢ [12]. For the latter, values of • close to ½ are sometimes, but not always observed, although the data are not for several reasons overwhelmingly reliable. In many cases the pressure dependence of T¢ follows laws which can only be interpreted if the heterogeneities of the magnetization are again taken into account and ref. [12] makes this plausible, see also ref. [6]. Hence it is not possible to use a relation like (4) to decide unambiguously between different theories of metallic magnetism. This is unfortunate since crucial experiments are rare in this field!
2.3. Low temperature dependence of magnetization M(T) This is one of the oldest problems in the subject. It is concerned with the form of the terms additional to the spin wave term (1). There are
3
many candidates for this A M / M : (i) T 5/2 . . . . : h i g h e r order spin wave terms; (ii) T 4 m a g n o n - m a g n o n interactions; (iii) T 2 Stoner excitations, weak ferromagnets; (iv) T n e x p ( - A / k T ) : dto., stong ferromagnets, A = Stoner gap; (V) T 4/3 Moriya theory of spin fluctuations, and others. Any attempts to fit observed data to such laws is fraught with danger: The data must be very accurate, a fit with several parameters is always difficult to defend and a good fit, however convincing, may not be any more so than that to a different law. Nevertheless, this problem is so important that there is a vast literature of papers concerned with it. The situation in nickel is particularly complicated (see the handbook article [9]) since the form of the additional term is not clear and since the spin wave stiffness D itself is not the same as that from neutron scattering data. This is an outstanding problem in metallic magnetism. Although we have fully recognized this danger we have nevertheless fitted the term (iv) to very accurate data (2 × 10 -5 accuracy for M(T)) for the amorphous ferromagnets Fe~ Ni 80- x B18Si 2 and FesoB20 [13]. The coefficient n was taken to be equal to 3 compatible with the (badly determined) band structure. The Stoner gap A was found to be small (20 to 60 K) but to vary systematically with the iron concentration. The coefficient of the exponential also varies systematically and in agreement with expectations [13]. The resulting spin wave stiffness is now (it was not before!) compatible with that from neutron scattering data. Hence we have claimed to have obtained experimental evidence of a rare kind: Single particle and collective excitations coexist peacefully!
2.4. Static and dynamic susceptibility above Tc, x(T) Another classical key problem in metallic magnetism concerns the relatively common observation that the Curie-Weiss law is frequently obeyed over wide ranges of temperatures above T~. On the face of it this would imply the presence of local
4
E.P. Wohlfarth / Theory and experiment in metallic" magnetism
moments and hence the inapplicability of a straight Stoner model with Fermi statistics. As was discussed by Edwards [14], Staunton [15] and others pronounced local moments (not localized electrons) may well occur in metallic iron and the observed Curie-Weiss constant then corresponds to a small number of such moments per atom. This small number contrasts with the (incorrect) large number demanded by theories of pronounced short range order above T~ [14]. The situation in nickel is, however, completely different. Shimizu (see ref. [9] for a review) has rightly pointed out that the Curie-Weiss law is, in fact, not all that well obeyed in any case and that the resulting observations are not really incompatible with the Stoner model. A similar case was made out for some weak itinerant ferromagnets (ZrZn 2, N i - P t . . . . ) in terms of the complicated local features in the density of states curve [16]. For example, the Curie-Weiss constant of ZrZn2 just above T~. agrees almost perfectly with the Stoner model [16]. The influence of spin fluctuation effects on the Curie-Weiss law and constant have also been discussed [11] and it seems that this important question has not yet been answered. The same goes for the full explanation of the Rhodes-Wohlfarth plot [17]; this is an empirical plot against T,. of the ratio of magnetic moments measured above and below T~. The original interpretation [17] was based on the weak itinerant ferromagnetism of materials with low Curie temperatures and thus low magnetic moments at OK. Alternative explanations [11] concentrate more on the existence of local moments above T,.. These problems are even more acute when discussing the dyanmic susceptibility of metallic ferromagnets above T~. For nickel the pioneering neutron measurements of Lowde and Windsor ([18], see also the handbook article [9] for a review) were shown to be not incompatible with band calculations at temperatures above and below Tc. We can not resist the temptation to quote from this article as follows: "Perhaps the later neutron data focusing attention on the persistence of spin waves above T,. should be reconsidered on this earlier basis"; however, this controversial subject will be discussed in detail at this meeting by Shirane and Mook [19] and we do not wish to interfere between the neutron experts!
2.5. High magnetic' field effects These effects are of central importance in this subject and some aspects have been reviewed before [20].
2.5.1. "High field" susceptibility' at 0 K, Xo The existence of this effect, first proposed by us in 1949 [21], arises from the fact that if the effective intraatomic Coulomb interaction I is sufficiently small so that I N ( e v ) - 1 = i - 1 << 1 then the magnetization in zero field and at 0 K is susceptible to field variations such that X0 = / ~ U ( c v ) / ( i -
1),
(5)
where N(c v) is the electronic density of states at the Fermi energy; in addition, if T~. were to be given entirely by single particle excitations then
(6)
Xo = t*2 N( t"F)TF/T', 2 " where
(7) t F
in terms of derivatives of N(Ev). Relation (6) has been used to discuss high pressure effects [12] which are sensitive to a breakdown of this relation arising from heterogeneities of the magnetization. The existence of X0 is an indication that the itinerant electron model provides a good description of the magnetic properties of a substance since a straight localized model (without unaligned localized spins) can not encompass such an effect. Large values of X0 have often been observed. From (4) and (6)
Xo ~ I c 0 - el ',
(8)
and it is the non-applicability of this relation which is such a good indicator of alloy heterogeneity; for example, in F e - A l alloys X0 has a broad maximum near co rather than a divergence [221.
2.5.2. High field effects in nickel near T [23], see
also ref [20] These measurements seem to us to be very important in supporting the idea that in nickel single particle excitations are present with consid-
E.P. Wohlfarth / Theory and experiment in metallic magnetism
erable weight near T~. They were carried out in high fields up to 32 T and at temperatures up to 700 K (T~ = 627 K). The magnetization M(H, T) was fitted to the Landau expression [5], [24]
H=A(T)M+B(T)M
3,
(9)
valid outside regimes of critical behaviour. The coefficient A ( T ) is given on the basis of Fermi statistics, and with the coefficients such that A(T~) = 0, by the relation
A ( T ) = alo + a,2T 2 + a14T4 +
.
.
(10)
.
which was found to be applicable up to T ~ from 550 to 700 K and, with T 4 included, even above (up to 1700 K, using other data for the paramagnetic susceptibility; this fit verifies, incidentically, that Shimizu [9] was right in doubting the applicability of the Curie-Weiss law in nickel). The expression (10) breaks down below 550 K, presumably since M ( T ) is then too large. Of even greater interest is the coefficient B ( T ) which is given by [51, [24]
B ( T ) = a30 + a32 T2.
(11)
This expression is completely different from that based on classical statistical mechanics [20], namely
B ( T ) = T/3CM2(O),
(12)
where C is the Curie-Weiss constant. Hence a test of (11) and (12) provides a rare opportunity to distinguish models. For nickel fig. 1 shows that (11) applies very well from 550 to 700 K. The o
30 0
~,20
T2
I
1
I
2
I
3
I
4
I
5
T2 105K2 Fig. 1. Variation of the Landau coefficient B with T for nickel near the Curie point [23].
5
coefficient a32 is small but clearly finite; it is given by the fine structure of the density of states curve via formulae like (7) only even more complicated, We will return to B ( T ) shortly. The analysis of [23] is, however, even more incisive: Using the parameters ai i the saturation magnetization at 0 K (where the formulae are expected to be inaccurate) came out as 62.6 e m u / g r , only 7% above the correct value 58.6! The exchange splitting obtained on this basis is 0.33 eV in good agreement with photoemission data [25] and compatible with the observed Curie temperature as already pointed out. Hence it must be concluded that the high field magnetic properties of nickel in a wide range about Tc are quantitatively described in terms of the prevalent single particle excitations. The relevance to the spin wave controversy [19] needs to be discussed. However, it must be stressed that Ponomarav and Moreva [26] tested the model for single crystal terbium and found the expected result that it did not apply whereas one based on localized electrons did.
2.5.3. Brief reference to Arrott plots The relation (9) has been used to analyze M(H, T) data via plots of M 2 vs. H / M . The results for a wide range of alloys, both crystalline and amorphous, were summarized in ref. [6]. It seems clear that deviations from straight line Arrott plots provide valuable evidence for the extent of spatial variations of the magnetization. These can be discussed on the basis of L a n d a u - G i n z b u r g theory where A and B are spatially dependent and the fluctuations of M ( r ) are limited by a positive gradient term in the free energy. In extreme cases this approach breaks down and a model involving superparamagnetic clusters is more useful in describing these fluctuations. We recall that pressure data are also useful indicators of such effects. In all cases it is a question of the relative importance of these origins of fluctuations of the magnetization arising from metallurgical effects and other origins such as spin fluctuations [11]. 2.5. 4. First-order transitions [20] As was first pointed out by us in 1962 [27] paramagnetic metals and alloys may in high fields and at 0 K undergo metamagnetic or first-order
6
E.P. Wohlfarth / Theory and experiment in metallic magnetism
transitions provided the fine structure of the density of states curve is such that A(0) = a~0 > 0 and B(0) = a30 < 0. The analysis showed that the effect is most likely if the density of states curve has a local region of positive curvature at the Fermi energy and if IN(~v) is just below 1. The resultant critical field is proportional to {a3o/la~o]}1/2 and the effect has been further discussed for Pd [28], where this field H c was computed to be 300 T. For Y Co 2 there is experimental evidence for this effect with H ~ - 100 T but band structure calculations [29] give larger values. For TiBe 2 some extremely accurate band calculations [30] illustrate well the sensitivity of the effect to the density of states fine structure. However, a "close look at low temperature magnetization data" [31] implies that the metamagnetic transition expected at about 5 T [20] and observed to be of this order [32], is more complicated than forecast and that the scale of the fine structure effect even lies below that of the band calculations of refs. [30]. For all three examples, Pd, YCo 2 and TiB% the paramagnetic susceptibility vs. temperature curves have maxima at relatively low temperatures and these could also arise from the density of states features leading to metamagnetism [27], [20]. A further range of materials where the effect has been observed are the dichalcogenides Co(SxSel_x) 2 [33]. There are as yet no corresponding accurate band calculations. For ferromagnetic materials first order transitions may occur as a function of temperature where B(T) changes sign, i.e. at a critical temperature given by (11) as T~* = la3o/a3211/2. This was the interpretation given to explain [34] the firstorder transitions in ErCo 2, H o C o 2 and DyCo 2 and may also apply to other materials such as MnAs for which it would replace that of Bean and Rodbell [35]. The latter used for B(T) relation (12) and brought about a sign change via appropriate magnetoelastic terms'in the free energy expansion. The physics of the two models is thus completely different. Band calculation for MnAs have now been published [36] and will have to be analyzed to distinguish these approaches. We have discussed high field effects particularly since they seem to provide unusually rich results relevant to metallic magnetism.
2.6. Other results The other papers of this Conference and in these Proceedings cover various outstanding and controversial problems in the subject of metallic magnetism. The neutron scattering data [19] regarding the existence or otherwise of spin wave excitations above T~ are particularly important since if they really and incontrovertibly existed the simplest theoretical approaches to metallic magnetism would have to be abandoned. We have stressed our belief that the high field data [23] for nickel are relevant to this controversy in pointing to the prevalence of single particle excitations in this temperature range. Unfortunately no corresponding data for iron are available. However, the case for a simple Stoner model for nickel has been strengthened to some extent by the beautiful photoemission data obtained up to and beyond T~ and discussed by Kisker [37]. Again one would hope to have equivalent data for iron, it being understood that the Curie temperature is much higher. The polarization of photoemitted electrons has been measured as a function of energy [38] and the very valuable result was obtained that the existence of a maximum followed by a sign change gave strong evidence for strong itinerant ferromagnetism with a measurable Stoner gap. This method has been used most recently [39] to give some degree of confirmation to the proposal [13] that a range of iron rich amorphous ferromagnets are also strong itinerant. Among other papers the one on de H a a s - v a n Alphen experiments on itinerant ferromagnets [40] is particularly interesting as it enables the magnetism of the classical weak itinerant materials Ni3A1 and ZrZn 2 to be discussed. It thus seems that experimental techniques of a very high sophistication are being applied to the problems of metallic magnetism and this historical development can only be applauded.
3. Future outlook
In the more immediate future it would be hoped that the controversies regarding spin waves above Tc may be settled amicably [19] and the same goes for that regarding giant short range order there
E.P. Wohlfarth / Theory and experiment in metallic magnetism
[41]. Looking further ahead a number of problems will require solution and these arise naturally from the account given here and in other talks at this meeting. Firstly, band calculations of an astonishing accuracy are now possible and a real breakthrough must be made in computing effects such as those of spin fluctuations for real materials to the same degree of sophistication. We recall our earlier discussion [42] regarding paramagnons in palladium and related materials where the relatively small effect of these fluctuations on physical properties was very tentatively ascribed to band structure and degeneracy effects. Unfortunately the early work of Diamond [43] has never been pursued further. Secondly, the continuing refinement of neutron scattering, photoemission, de H a a s - v a n Alphen, high pressure, high field and related facilities will give an opportunity for obtaining ever more reliable data on metallic materials of greater complexity. Once the obsession with iron and nickel has abated it will be possible to look at the other materials in greater detail. If one were to be asked to predict one series of materials with a promising scientific future it would be transition metal compounds such as MnX(X = As,Sb,Bi). This is not to forget the actinides whose glowing future we predicted earlier [4]. Finally, we repeat yet again the importance of metallurgical defects such as concentration fluctuations. The account of this effect given in [6] shows clearly how sensitive many magnetic properties are to such defects and this dependence must be studied more systematically, e.g. by measuring properties as a function of metallurgical treatment. To repeat again, where these influences are severe it is certainly not possible, even if it were on more fundamental grounds, to have a universal theory of metallic magnetism.
Acknowledgements It gives me particular pleasure on this occasion to thank all my friends and colleagues for their great continuing kindness and support. I wish to thank Bal/lzs Gyorffy, Franqois Gautier and Her-
7
bert Capellmann for organizing this meeting and David Edwards not only for editing the Proceedings.
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E.P. Wohlfarth / Theory and experiment in metallic" magnetism
[28] T. Jarlborg and A.J, Freeman, Phys. Rev. B23 (1981) 3577. [29] K. Schwarz and P. Mohn, J. Phys. F14 (1984) L129. [30] T. ,larlborg, P. Monod and M. Peter, Solid State Commun. 47 (1983) 889. [31] F. Acker, R. Huguenin, M. Pelizzone and J.L. Smith, to be published. [32] J.M. van Ruitenbeek, A.P.J. van Deursen, L.W.M. Schreurs, R.A. de Groot, A.R. de Vroomen, Z. Fisk and J.L. Smith, J. Phys, F, to be published. [33] K. Adachi, A. Matsui, Y. Omata, H. Molymoto, M. Motokawa and M. Date, J. Phys. Soc. Japan 47 (1979) 675. [34] D. Bloch, D.M. Edwards, M. Shimizu and J. Voiron. J, Phys. F5 (1975) 1217.
[35] [36] [37] [38] [39] [40] [41] [42]
[43]
C.P. Bean and D.S. Rodbell, Phys. Rev. 126 (1962) 104. R. Podlucky, J. Magn. Magn. Mat. 43 (1984) 204. E. Kisker, J. Magn. Magn. Mat. 45 (1984) 23. W. Eib and S.F. Alvarado, Phys. Rev, Lett. 37 (1976) 444. E.P. Wohlfarth, Phys. Rev. Lett. 38 (1977) 524. R. Ailenspach, E. Colla, D. Mauri, M. kandolt and E.P. Wohlfarth, Phys. Left., to be published. G.G. Lonzarich, J. Magn. Magn. Mat. 45 (1984) 43. H. Capellmann, to be published. P.F. de Chhtel and E.P. Wohlfarth, Comments Solid State Phys. 5 (1973) 133, 141. see also A.H. MacDonald, Phys. Rev. B24 (1981) 1130. W. Joss and G.W. Crabtree, Phys. Rev. B, to be published. J.B. Diamond, Intern. J. Magnetism 2 (1972) 241.