Ground-state magnetovolume effects in alloys

74

Journal of Magnetism and Magnetic Materials 45 (1984) 74 78 North-Holland, Amsterdam

G R O U N D - S T A T E M A G N E T O V O L U M E E F F E C T S IN ALLOYS Robert J O Y N T and Volker H E I N E Cavendish Laborato(v, Madinglev Road, Cambridge CB3 0HE, UK

We present a general theory for the magnetic contribution to the lattice parameters of ferromagnetic alloys of type A • B 1 ~. This contribution is obtained as a function of x. Calculations are given for the F e - C o alloys and are in reasonable agreement with experiment. General trends in transition metal alloys are also found to be in agreement with the theory. The connection of the magnetovolume effect with the local moments on the various atoms is stressed. The effect is seen to be a sensitive probe of local magnetic order.

1. Introduction

It is very appropriate to dedicate a paper on the magnetovolume effect to Professor Wohlfarth. He, more than anyone else, is responsible for the viewpoint that these effects are particularly important for the microscopic understanding of metallic magnetism. The early work in the field, both experimental and theoretical, was very succintly summarized by him [1]. Also, he is responsible for one of the two theories of the magnetic contribution to the thermal expansion of weak magnets, a very active controversy at the present time. On this topic, see Wohlfarth [2,3] as well as Moriya and Usami [4]. Our concern here is with ground-state volumes, particularly those of magnetic alloys. These tend to attract attention because of the Invar effect many of them have large negative magnetic thermal expansion. The natural way to understand this effect is to begin at zero temperature and that is what we propose to do. The aim is to calculate the magnetic contribution to the lattice parameter in the ground state as a function of the concentrations of the constituents. We believe that this throws light on the microscopic picture of metallic magnetism, particularly in judging the accuracy of common models. It also helps to confirm the importance of the magnetovolume effect as an accurate measure of the size of local magnetic moments, and thereby as a reliable probe for local

symmetry-breaking in ferromagnets above the crttical temperature. The method used is very similar in spirit to that of Wohlfarth, but we stress the connection to modern band-theoretical methods, in particular to the Local-Density Functional formalism (LDF). The result is a very simple picture of the groundstate volume associated with magnetic effects. It can be tested against experiment - both those carried out in the laboratory and those carried out in the circuits of the computer.

2. Theory Our approach is essentially that of Holden, Heine and Samson [5] whom we shall collectively denote by HHS. We adopt their model Hamiltonian

H=Hh+He=(Vo/V)Bhh+(Vo/V)Vh,.,

(1)

where H b is the kinetic (i.e. band) energy and H~ is the Coulomb interaction, and h b and h~ are their volume independent counterparts. V0 is a reference volume. The value of fi is about 5/3, coming from the volume dependence of d-band hopping matrix elements [6]. The s - p hybridization will reduce this value (the free electron result would be 2/3). Janak and Williams [7] deduce a value of about 4 / 3 from their L D F calculations. They also find that y is very close to zero, which means that the

0304-8853/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

R. Joynt, l~ Heine / Magnetovolume effects in allqvs

Coulomb interaction is really intra-atomic, to a good approximation. Deviations come from the fact that (1) is already a renormalized Hamiltonian, and renormalizations of the parameters appearing in h~, such as those of the Kanamori type, will depend slightly on the volume. Nevertheless, we shall set y = 0 for simplicity. HHS took the thermal expectation value of (1) at zero temperature and at finite temperature. Taking the difference of these two allowed them to deduce a relationship between the local moment, the magnetic disorder energy, and the thermal expansion. Rather than do this at two different temperatures, however, we can consider the ground-state expectation values at different values of U, the Hubbard parameter which characterizes the strength of H~. As a theoretical procedure, there is no problem with this, of course, but we also note that it would also make an interesting computer experiment. Since we are only concerned with ground-state energies, the Hartree-Fock cum L D F picture on which the discussion is based is likely to be valid, to a very good approximation, since there is no need to consider collective excitations. Thus the present theory has a somewhat more firm basis than the HHS theory, In the latter, excitation energies are considered, and the separation of these into band and magnetic parts is more difficult. Only in an adiabatic approximation for the magnetic disorder can this done. This sort of approximation holds when the characteristic frequency of the disorder is much smaller than the frequencies associated with band energies. For highly disordered configurations this is dubious. We are interested in the value of the pressure

P = - O ( E ) / O V = -(OF~/~V) = f l ( H b ) / V .

(2)

Now let the interaction U be changed from some small value, say U0, that gives a nonmagnetic ground state through the Stoner criterion value, to the physical value associated with the magnetic ground state. The hopping matrix elements are held fixed. (Hb), to a good approximation, depends on U only through the population of the states. Thus there is no change in the pressure until the system magnetizes. Since the volume is determined by the zero of (2), this also means that there is no change in the lattice parameter. This is

75

an example of a dictum of Anderson [8] that continuations of this kind are analytic except at a value where the system changes symmetry. This result immediately explains why, in L D F calculations which do not use spin-polarized functionals, very good lattice constants are obtained for nonmagnetic metals such as A1, and even for almost ferromagnetic materials such as Pd, but rather poor results are obtained for Fe, Co, Ni. See Janak and Williams [7], to be referred to as JW. This can be corrected by the use of spin-polarized functionals, as in Andersen [9]. The difference in pressure between the magnetized state and any of its nonmagnetic counterparts is

p(u)-p(Vo)

=

-

(3)

where the expectation values are taken at the indicated values of the Hubbard parameter. In a rigid-band model, this has the explicit form

V(V)-P(Uo)

Here p is the paramagnetic density of states for one spin as a function of the band energy, ~t is the paramagnetic fermi energy, and c fr, ~ , are the fermi energies after polarization. In this model the last two quantities are fixed uniquely once m, the magnetization, is specified. In the weak limit, where U ~ 1 / p ( ¢ f ) and ~tT - ~ = k << p(~r)/p'(Ef), this reduces to

V[ P( U ) - P(Uo) ] = ,SmZ/4p( ~.r ).

(5)

This last is the form investigated by JW. Eq. (5) is similar to an equation of Inoue and Shimizu [10], which contained an adjustable parameter, and to an expression of Poulsen et al. [11], for a more specialized model. We think that the derivation given here makes the underlying physics clear, but, more importantly, we can quickly generalize it to alloy systems. Let the alloy be of the AB type where one or both of the sites has a magnetic moment, say ma and m u. The atoms will be characterized as well by UA and UB. The separation of kinetic and Coulomb terms proceeds in the same way as be-

76

R. Joynt, K Heine / Magnetovolume ef/ects m alho's

fore, and the same argument leads to P ( V ~ , U ~ ) - P(O,O)

= (/~///)[ CAf~ir"PA(()( dc +Cu

f,'~pB(E)~ dc (I

(1

--cAf~ PA(e)edc-CBf~?o,(e)(d(

]

,

(6)

where the C ' s are the concentrations and 0a, P t~ are local densities of states per atom. The weak limit of this expression is P ( UA,UB) -- P(O,O) =

( B/V )

× [CAm2/4pA(c,) + C,m~/4O,,(~f) ] .

(7)

We contend that this is the correct weighting of the moments, in contrast to weighting by concentration, as in Inoue and Shimizu [9].

3. Fe-Co alloys We would now like to compare our restuh to experimental data. The proper way to do this is to do a band calculation for each alloy, and evaluate the above expressions in the appropriate way. For this preliminary work, we will instead use band structures which have already been calculated, as well as some simple extrapolations and approximations. This has the disadvantage of less accuracy, but the advantage of making the trends, and their origin, more clear. Fortunately, there is no shortage of measurements of lattice constants of ferromagnetic alloys, and, for some of these alloys, the computer calculations necessary to evaluate eq. (6) have been done. We will concentrate on Fe, Co 1 ,, since this has interesting Invar-like properties, not too m a n y metallurgical difficulties, and reliable band calculations for x = 0, ½, 1. It has the additional advantage that the bulk moduli of iron and cobalt are quite similar, so that the pressure alone can be considered as giving rise to the dependence of volume on x. The disadvantage of these materials

is that the shape of the bands does depend on the value of U to some extent, a dependence which has been neglected in arriving at eq. (6). However, this is probably not of much quantitative importance. The procedure is as follows. Step one. Use eq. (3) to compute how much of the volume of pure iron and pure bcc cobalt are due to magnetic effects. Eq. (3) gives the magnetic pressure, which works against the bulk modulus to give the volume change. The claim is that, because of the arguments of the last section, the separation of magnetic and nonmagnetic volumes is well-defined, This yields the nonmagnetic lattice constants, which are listed in the table, along with some intermediate quantities in the calculation. The kinetic energies are only approximate, since they were obtained by a graphical integration of densities of states given by Schwarz et al. [12]. The case of Co is complicated by the fact that there is a change of phase to hcp at x = 0.7, so we use a linear extrapolation to get the 'observed' lattice constant for bcc Co. Step two. In nonmagnetic alloys, Vegard's law, which says that the lattice constant is a linear function of x, is obeyed rather well. Together with the information from step one, we c o m p u t e the nonmagnetic lattice parameter as a function of .v. Since Vegard's law is assumed in this fashion, the present theory only aims to explain deviations from this law. It is by no means a complete account of alloy volumes. Step three, Use formula (6) to find the magnetic pressure for F e - C o : x = 0.5. We take the partial densities of states from Schwarz and Salahub [13], and use their values of the magnetic moments as well. Various intermediate quantities which appear in the calculation are given in the table. This gives a prediction for the observed lattice parameter which agrees with the experimental results quoted by Pearson [14], to within the accuracy of the approximations, as can be seen in the figure. The agreement would be improved by reducing fi slightly, which would take hybridization into account, but we have avoided this in order to stress the fact that the calculation is free of adjustable parameters. Step four. We would like to extend this to arbitrary x. If one has band calculations for all x,

R. Jovnt V. Heine / Magneto(~olume effects in alloys

Table 1 Values of parameters used in the calculations, m is a local moment size. KE is the change in kinetic energy in the system produced by the formation of the local moment, a is the lattice parameter and K is the bulk modulus

2.861 •

X I•

X

~2.85 C 0 o

Fe Co

Fe - Co

~2.84

77

X

(bcc)

Fe ~ ~ ( Fe Lo~ Co

0

._l

M (#B)

KE (eV/atom)

102Aa/a

K

2.14 1.65

0.94 0.44

4.02 1.67

1730 1950

2.72 1.69

1.27 0.44

3.51

1840

(kbar)

2.83 I

0

I

I

20

I

l

l

40 at

%

l

60

80

Co

Fig. 1. Lattice parameters of Fe 1_ , - C o , . The squares are the measured values, and the crosses are calculated values. References and methods of calculations are described in the text.

then step three can simply be repeated as often as one wishes. These are not available. So we make the simplest possible approximation. This is to take the experimental values of the magnetic moments as a function of x from Bardos [15] and use these to obtain kinetic energies, again using the F e - C o densities of states. This gives values for the magnetic pressure as a function of x, and the resulting lattice parameters are shown in fig. 1. We stress that the use of the F e - C o bands in this way is an extremely crude approximation. It nevertheless gives the right trend for small x. The iron m o m e n t is increasing rapidly in this region for interesting reasons which are discussed, for example, in ref. [16]. This is enough to cancel the effects of the dilution of the iron until about x = 0.3. For x > 0.5, the iron m o m e n t is essentially constant since all its nearest neighbors are Co atoms. This gives a reversion to linear behavior.

4. General case The overall experimental situation for the lattice constants of transition metal alloys has been reviewed by Shiga [15]. He finds that Vegard's law describes the behavior of the nonmagnetic alloys well. For many magnetic systems A] B X the

equation a ( x ) = aA(1 - x ) + aBX + rlt, I

(8)

fits the data, where a A, a B and /7 (which he calls C) are parameters, and I/x] is the average of the magnitudes of the local moments. For ferromagnetically aligned alloys ]/~] is proportional to the macroscopic magnetization. Expression (8) agrees nicely with the picture presented here. a A and a B have the physical interpretation of being the lattice parameters associated with hypothetical samples of nonmagnetic pure A and pure B. Thus a A should have the same value for all c o m p o u n d s in which A is a component. Shiga does find this to be true, particularly for iron. The parameter F also can be interpreted. Consider first substances which have a large total m o m e n t for all x. Then, for small relative changes in the m o m e n t s ~ m A , ~ rn B, eq. (6) simplifies. The change in the magnetic contribution to the pressure becomes 8 P = ( 5 A / 3 V ) [ C A S m A + CBSmB].

(9)

As above, A is the change in kinetic energy which results from flipping one spin: A = c~ - ~ . Therefore one finds the pressure change proportional to 8[l~P = CASmA = CBSrnB, the change in average local moment. F is just 5A/9V~, where K is the bulk modulus. This quantity measures the response of the lattice parameter to a change in magnetization. The second case is that of small local moments. The most interesting instances of this are those alloys which have a nonmagnetic to ferromagnetic transition as a function of x, as for example fcc Ni~ ~ - M n , . This is ferromagnetic for x < 0.3. In these cases the transition region is described by eq.

78

R. Jovnt, V. Heine / Magnetovolurne effects in alh~vs

(7), so that the pressure is quadratic in the local moment size. So we expect the expression (8) to break down, as indeed it does in Ni Mn, and in the similar case fcc C o - M n .

5. Discussion and conclusion

A number of shortcomings of these calculations should be pointed out. The first, as already mentioned, is the use of the band structure of an ordered alloy, i.e. F e - C o , to discuss the disordered iron-rich alloys. This can in principle be remedied by CPA calculations. Only trends and estimates can really be obtained by the rigid-band hypothesis that we have employed. A second drawback is the assumption that the bulk modulus can be taken as constant, or interpolated, across large concentration ranges. In fact this quantity is sensitive to magnetic effects as well. In principles one should look for the zero of eq. (3), with the nonmagnetic contributions to the pressure added in as well. Again, this should be done in conjunction with a genuine band calculation. Finally the effect of zero-point magnon fluctuations has been ignored. If this is taken into account, then the volume dependence of the spin-wave stiffness enters. Not much is known about this dependence [3]. However, this sort of effect is expected to be small in magnitude compared with changes in moment size [5]. The ideas contained in this paper are generalizations of concepts which are quite old. Simply stated, a metallic magnet reduces its Coulomb energy by increasing its band energy. The cost of the latter is in turn reduced by expansion, while the former is more or less unaffected by this expansion. The extension of this scheme to alloys requires only the notions of a local density of

states, and the associated local moment. This extension does seem to be successful, both as a rough quantitative guide and in explaining general trends. The importance of this is that it gives one confidence in the magnetovolume effect as an indicator of local (atomic) magnetic ordering of the electrons. Many recent controversies in the subject of ferromagnetism have tended to revolve around this issue of local order. This is true of both strong and weak ferromagnetism, particularly in the paramagnetic phase. The confirmation in alloys of simple models of the magnetovolume effect gives us confidence in applying these models to these more complicated situations.

References [1] E.P. Wohlfarth, J. Phys. (_'2 (1969) 1683. [2] E.P. Wohlfarth, Solid Stale Commun. 35 (1980) 68. [3] E.P. Wohlfarth, in: Magnetoelasticity in Transition Metals and Alloys, ed. M, Shimizu (North-Holland, Amsterdam, 1982) p. 203. [4] T. Moriya and K. Usami, Solid State Commun. 34 (1980l 95. [5] A.J. Holden, V. Heine and J.H. Samsom, J. Phys. F 14 (1984) 1005. [6] V. Heine, Phys. Rev. 153 (1967) 637. [7] J.F. Janak and A.R. Williams, Phys. Rev. BI4 (1976) 4199. [8] P.W. Anderson, Basic Notions of Condensed Malter Physics (Benjamin, New York, 1984) chap. 3. [9] O.K. Andersen, J. Madsen, U.K. Paulsen and J. Kollar, Physica 86 88B (1977) 249. [10] J. lnoue and M. Shimizu, J. Phys. F 13 (1984) 2677. [111 U.K. Poulsen, J. Kollar and O.K. Andersen, J. Phys. F 6 (1976) L241. [12] K. Schwarz, P. Mohn, P. Blaha and J. Kubler, preprint. [13] K. Schwarz and D.R. Salahub, Phys. Rev. B25 (1982) 3427. [14] W.B. Pearson, A. Handbook of Lattice Spacings and Structures of Metals (Pergamon, New York, 1958) p. 505. I15] M. Shiga, in AlP Conf. Proc, 18, pt. 1, ed. C. Graham (AIP, 1974) p. 463. [16] A.R. Williams, V . L Moruzzi, C.D. Gelatt, J. Kubler and K. Schwarz, J. Appl. Phys. 53 (1982) 2019.