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Theory of magnetostriction in amorphous ferromagnets: Influence of non-local elastic behaviour
Journal of Magnetism and Magnetic Materials 88 (1990) 7-12 North-Holland
THEORY OF MAGNETOSTRICTION IN A M O R P H O U S FERROMAGNETS: INFLUENCE O F NON-LOCAL ELASTIC BEHAVIOUR T. BEUERLE and M. F,4,HNLE Max-Planck.Institut flit Metallforschung, Institut far Physik, Heisenbergstr. 1, 7000 Stuttgart 80, Fed. Rep. Germany Received 6 February 1990; in revised form 28 March 1990
In former phenomenological theories of magnetostriction in amorphous ferromagnets, a local versiori of elasticity theory has been applied, although the atomic scale of the structural fluctuations requires in principle a non-local treatment. In this paper the dominant non-local contributions are calculated. It is shown that they may be neglected in strongly magnetostrictive systems but mzy become relevant for the discussion of weakly magnetostrictive alloys.
1. Outline of the problem In a series of recent papers [1-12], a phenomenological theory of amorphous ferromagnets has been developed, which allows the calculation of the technologically very important magnetoelastic properties of these materials from statistical information about the spatially fluctuating material tensers. The equations are able to determine the local magnetostrictive strains [7], the effective magnetostriction constant [1-5] and its modification by magnetic field [8-10] and external stress [11] or by magnetic field annealing [12], and to elucidate the magnetostriction mechanism in amorphous ferromagnets [1-7]. All these calculations have been performed within the framework of the local linearized theory of elasticity, although the material tensors of amorphous ferromagnets exhibit spatial fluctuations on an atomic scale. It is well known that continuum theories can be extended to atomic scale, but in this case, in principle, a non-local theory should be applied [13]. This becomes obvious from the fact that the force acting on an atom at site r depends on the locations r ' of at least all the neighbouring atoms. Extending the continuum theory to atomic scale means that we introduce volume elements of atomic size which contain one rather than very many atoms. An integral formulation 3f the total elastic energy thus requires the
introduction of a non-local coupling term between volume elements at different sites. In refs. [1-I2], these non-localities have been neglected in line with the conjecture [14] that a local approximation suffices for general, qualitative considerations for which no very precise numbers are demanded. Indeed, the calculation of atomic-level strains and stresses in amorphous materials resulting from the lack of atomic long-range order yields results [14,15] in rather good agreement with those from atomistic calculations. It is the aim of the present paper to take into account the leading non-local contributions to the elastic energy when calculating the effective magnetostriction constant of amorphous ferromagnets.
2. Non-local contributions to the elastic energy According to Krrner and Datta [13], the total K)t
~.., x f..Lo L Jt ~,~
g
It I U I
I-IUI.,~ILI
111~
I~;;i 11~1
U•
written as
f f avav' %,.,(,.,
/) (1)
where
s(r)
represents the displacement field and is a non-local material tensor. For a simplification of eq. (1), it is usually assumed that
%k;(r,
r t)
0304-8853/90/$03.50 © 1990 - Elsevier Science Publishe:s B.V. (North-Holland)
8
T. Beuerle, M. Fiihnle / Theory of magnetostricti~n in amorphousferromagnets
the quantity Vs varies on a scale r0 which is large compared to the interaction range R of the atomic forces, i.e. the interaction range of the kernel C~jkt(r, r'). This is not strictly fulfilled for the magnetostrictive strains ha amorphous ferromagnets, which exhibit spatial fluctuations on an atomic scale [7], so that r0 = R. Because we do not demand a high degree of accuracy but just want toget a feeling for the order of magnitude of non-local effects, we nevertheless adopt the above assumption, which allows us to evaluate formally the non-local ,,tensor according to
c,j~,(~,/)
= c, jk,(,,,)~(,'- ,") + F,k,m(,) O,,~(..- .,') + A,jk,,m,(r) OmO,8(r - r') + . . . ,
(2)
yielding
-
½f dV [Ci2ktci2ct, t + 2F, jkt,,qj +A,,~,~.(O~,,)(O.,~,)].
3,~ct,,
is the tensor of rigid rotations and c~kt denotes the permutation tensor. This clearly demonstrates that the Grad ~ terms describing the effect of torque stresses are related to the non-locality of the system [17].
3. Balance-of-force equation and statistical input The following calculation is in line with the procedure described in full detail in refs. [2,4], and we therefore discuss only the broad outline. First, the total energy of the system as a function of the deformation variables is completed by the inclusion of the strain-dependent part of the magnetic anisotropy energy, given by
ag = f dV B~jkt(r)c,j(r)yk" h.
(3)
Ctjkt(r) = f dV" ci~k,(r, r'),
(4)
-
)~.
(5)
r')~(r-
r'),,
(6)
A,jk;~.(r) = ½f dV' c,4,,(r, r ' ) ( r -
.~. = "(o;~;- 0~s;)
~(m~ )
with the local material tensors
F,,~,~(,) = - j d r ' ~,j,,
where
and the strain field ~,: = ½(~,sz + Off,). The first term in eq. (3) represents the local contribution and is the only term which is taken into account in refs. [1-12], whereas the other two terms are the leading non-local corrections. To derive eq. (3) the Vs-terms in eq. (1) and ,ha;,. ao.~..,.,:.,oo ¢...~:~t. ..... 1. . . . , . _ _ eq. "~' inserted into e q (1)) are combined in such a way that the elastic energy may be completely expressed in terms of the strain tensor c and Grad ~. It is also possible (see ref. [16] and the appendix) to combine the various terms in such a way that the elastic energy may be expressed in terms of ~,
(7)
lqer¢ :he quantity B, jkt(r) represents the mag,netoelastic tensor of the amorphous material at site r, and the vector T denotes the direction cosines of the magnetization. Because we consider only the satura6cr~ magnetostriction, these direction cosines are constant throughout the whole sample. The rotation-dependent part of the magnetic anisotropy energy density responsible for the generally very small reorientation mechanism of magnetostriction [1-4.] is neglected in the following. The total energy is then minimized x~th respect to the displacement field s, thereby automatically fulfilling the compatibility conditions. The resulting balance-of-force equation reads mag
with r~..ag r
~,j ~,)= B.,~,(~)v~,..,
(9)
and
= c , ~ , , ( , ) - (0~G~,~(~)) (~0)
T. Beuerle, M. F~ihnle/ Theory of magnetostriction in amorphousferrornagnets When keeping in ~ k l only the loca~ term C~jkt, eq. (8) reduces to the formerly used local balanceof-force equation (e.g. eq. (12) of ref. [4]). Eq. (8) represeats a stochastic differential equation because of the spatially fluctuating material tensor Cijkl(r), Fijklm(r), Aijklmn(r) and Bijkl(r). It is solved iteratively by a Green's function perturbation approach based on the definition of a hypothetical homogeneous reference medium, which may be chosen arbitrarily. We define this refererLce medium as the one without non-locality and with "all spatially fluctuating tensor components Cq, t(r) replaced by their volume averages. We thus deal with an elastically homogeneous isotropic local medium which is characterized by the tensor (~.~t = (C, jkl(r)), grid for which the Green's function is well known [4,18]. The displacement field s(r) of the actual material then may be represented by a Green's function perturbation series with the deviations =
-
as perturbation parameters. In the local version of the theory, the deviations are totally given by the spatial fluctuations
whereas in the present theory the deviations inelude all the non-local terms described by the tensors Fijklm(r) and Aokl~n(r). Altogether, the displacement field s(r)may be written in functional form,
s(,) = $[ Bijkt(r), Cijkl(r), Fijkl,n(r), A,/klm,,(r)],
(11) which allows in principle the calculation of s(r) if the spatial dependences of all natefial tensors are given. In practice~ the only available information about the material tensors is a statistical one, i.e. the best we can do is to make some plausible assumptions about the n-point correlation functions of these material tensors. We therefore must be satisfied with the calculation of certain volume
9
To obtain reasonable appro:dmations for the correlation functions, ,he pelycrystalline model of amorphous ferromagnets is used [1-12,19], which takes care of the very strong topological and chemical shert-range order of generally uniaxial symmetry wkich is characteristic for many amorphous ferremagnets. Accordingly, the amorphous material is conceived of as consisting of very small "grains" (basically an atom under consideration and its nearest-neighbour atoms) which exhibit identical material tensors with hexagonal symmetry. The statistics of the w.~terial tensors is then totally defined by the statistics of the grain orientation. In the following, we perform the calculations only up to the first order of the perturbation series. We then need as statistical information only the isotropic distribution function density 1
P(cos 0, q ) = 4¢r for the angles 0, ¢ characterizing the grain odentatior;, as well as the two-point correlation function for the grain orientation (defined by eq. (20) of ref. i4]), for which we insert g~2'(r,
/r'
)
=
exp(
--
I r - 1 r2p/ r 6
~-)
(12)
with the structural correlation length r0. It should be noted that in the local version of the theory [2,4], we did not need any ansatz for the correlation functions, just the normalization condition g~2)(r, r ) = 1, as long as we considered macroscopicaUy iso~'ropic systems. As typical ,'alues for the material tensors B, jk~ and C~i~,t of the grains, we insert those of crystalline Co or Gd. For the tensors ~r u k~,~ and Aijklmn no information ;s available in the literature. We therefore go back to the definition of these material tensors via eqs. (5,6) and insert a plausible ansatz for the non-local ~ensor c,/~t(r, r'), namely
_
1
(13)
Here R is the interaction range of the interatomic forces, which in the t ase of ame,-phous ferromagnets is comparable t~ the structural correlation
10
T.
Beuerle, M. Fa'hnle / Theoryof magnetostrictionin amorphousferromagnets
length r0. Note that our ansatz depends only on I r - r' I which corresponds to the assumption of purely central interaction forces between the atoms. This seems to be rather well justified for metallic amorphous ferromagnets, where partially covalent binding forces may only arise from the metalloid atoms. As shown in the appendix, the contributions of the Grad to terms to the elastic energy do not vanish for purely central interaction forces, in contrast to the statement of refs. [2,4]. Inserting eq. (13) into eqs. (5), (6) yields
Gk,,.(r) =0, Aijkt
.(r)=
2 (14)
4. Results
With the above discussed statistical input we can calculate (c~j(r)) and hence the effective magnetostriction tensor according to
= Aijkl.~k.~lxerf
(15)
and the effective magnetostriction constant ),eft h~.s¢f ~ 2(X~¢,,-,,22,, ),
(16)
yielding
The second terms, respectively, are due to the spatial fluctuations of the tensor C, jkt, and they constitute the only first-order conection within the framework o~ the local theory [2,4]. The last terms, respectively, describe the first-order correction originating from the non-locality of the material, and we assume R -- r0. For the Co parameters, which we conceive as representative for strongly magnetostrictive amorphous ferromagnets, the first-order correction terms are only of the order 10% of the dominating zeroth-order term. Obviously, in this case rather accurate values of h~ff may be obtained from eq. (19), and both the spatial fluctuations mad the non-locality play only a minor role. In contrast, for the Gd parameters, the zerothorder result is very small because of a partial cancellation of various contributions to the volume averages in eq. (19). As a result, the spatial fluctuations of Cijki(r ) and hence [2,4] the effect of elastic couplings between neighbouring grains is much stronger, and the non-locality term exceeds the other terms by about 50%. In this case, both the spatial fluctuations as weil as the effect of non-locality are very important, and the perturbation calculation should be extended to higher order to get accurate results.
5. Conclusions
Xelf = - 7 . 0 4 x 10-5 + 0.50 x 10-5 + 0.75 x 10-5 RE
4
(17)
when inserting the material parameters of Co, and esff
R 2
= 3.63 x 10 .6 + 3.89 x 10 .6 + 5.84 x 10 . 6 -
d
for the material parameters of Gd. Here the first terms, respectively, represent the zeroth-order approximation corresponding to the hypothetical elastically isotropic, homogeneous and local medium, given by ~eff, z e r o t h o r d e r =
_
- l"
(19)
We have shown that non-local corrections play only a minor role for the case of strongly magnetostrictive amorphous ferromagnets with x~ff around 10-5 as in, for instance, FeNiB-alloys. The situation may be different for the technologically very important, weakly magnetostrictive Co-base alloys (~ff is around 10- 7). As outlined, for instance, in refs. [2,4,12,19], there are two possibilities to achieve low values of ," ~e f t. in the first case, which is represented by our Gd parameters, the low effective magnetostriction arises from a partial cancellation of various contributions to the volume averages in eq. (19), whereas the material is strongly magnetostrictive on a local basis (see also ref. [7]). Then the non-locality of the system becomes essential. In contrast, if the small magnetostriction constant has nothing to do with partial
T. Beuerle, M. Fiihnle / Theory of magnetostricf:on in amorphous ferromagnets
cancellation, but results from a small local magnetostriction comparable in magnitude to 2k~rf, the non-locality may also be neglected. It should be noted that so far it was not possible to decide from any experiment or existing theory whether the weakly magnetostrictive Co-base alloys belong to the first or to the second case.
Acknowledgement The authors are indebted to J. Furthmiiller and R. PaweUek for helpful discussions.
Appendix In eq. (3) the elastic energy has been expressed by c and Grad c. A spatially fluctuating tensor field
,,j(,) = -I-[o, sj(,) + O,s,(,.)] implies the existence of a fluctuating tensor field
,,,, (,.) = -I [O, sj(,.)
-
a/,, (,)]
of rigid rotations. Because a rigid rotation of the whole body is not relevant, the rotation field 60 may contribute to the elastic energy only via its gradient, Grad o. Accordingly, we can rearrange the various terms in eq. (3) in such a way that the elastic energy is expressed by strain q p the gradient of rotation 1
Kij = 2Cjkl ~iO~kt
giving rise to torque stresses [17], and the symmetric part of the gradient of strain,
x;,, = ½(a c,, + a c,, + ak%),
witt E u L := ~AiqkpmnEqrnj~pnl,
Gi:~ ~== -- -~ 2 Fi jk pmCpm 1
(A.2) (a.3)
and
nijmkl =
2 -- "~Aijkpmn~pn t.
(A.4)
Tiffs clearly demonstrates that the torque stresses are related to the non-locality of the system [17]. Inserting for the non-local tensor Cukt(r, r') our ansatz (13) w,hich corresponds to the assumption of purely central interaction forces, the F- and G-teasors vanish but all the other tensors are non-zero, so that there are still contributions of the Grad ~0 terms, in contrast to the statement of refs. [2,4].
References J. Furthmiiller, M. Fiihnle and G. Herzer, J. Phys. F 16 (1986) L255. [:! J. Furthmiiller, M. Fiihnle and G. Herzer, J. Magn. Magn. Mat. 69 (1987) 79, 89. 13i M. F~ihnle, J. Furthmiiller and G. Herzer, J. de Phys. 46 (1988) C6-1329. [4i M. F~ihnle, J. Furthmiiller, R. Pawellek and G. Herzer, in: Physics of Magnetic Materials, eds. W. Gor7_kowski, H.K. Lachowicz and H. Szymczak (World Scientific, Singapore. 1989) p. 228. [5] M. F~ihnle, C. Els~isser, J. Furthmiiller, R. Pawellek, E.H. Brandt and M.C. Btihrn, Physica B 161 (1989) 225. [6] M. Fahnle and J. Furthmiiiler, .I. Magn. Magn. Mat. 72 (1988) 6. R. Pawellek, J. FurthmBller and M. Hihnle, J. Magn. Magn. Mat. 75 (1988) 225. [s] R. Pawellek and M. F~hnle, Phys. Stat. Sol. (a) 111 (1989) 617. [9] M. Fahnle, R. Pawellek and H. Kronmiiller, Phys. Stat. Sol. (a) 112 (1989) 189. [!07 R. Pawellek, M. Hihnle and H. Kronmiiller, Phvsica B ~x,A
concomitant to the existence of a gradient field of rotations. Altogether, the elastic energy reads
11
kAl*..,s),
~..~..
[1~,! M. F~i.hnle and J. Furthmiiller, Phys. Stat. Sol. (a~ 116 (1989) 819.
[12i M. Fahnle and J. Furthmiiller, Phys. Stat Sol. {a) 117 (1990) K71.
= ½f dV [
[13] E. Kr~Sner and B.K. Datta, Z. Phys. 196 (1966) 203.
+
+ AijktrnnXumXktn + 2 Fokt,~,jX kt. + 2Gijm¢ijgk¢ + 2 H i j m k t X i j m X k t ] ,
(AJ)
[14] V. Vitek and T. Egami, Phys. Stat. Sol. (b) 144 (1987) 145. [151 R. Pawellek and M. Fahnle, J. Phys.: Condens. Matter 1 (1989) 7257. [161 E.L. Aero and E.V. Kuvshinskii, Soy. Phys.-Solid State 2 (1960) 1272.
12
I". Beuerle, M. Fiihnle / Theory of magnetostriction in amorphous ferromagnets
[17] F. Hehl and E. KrSner, Z. Naturforsch. 20 (1965) 336. [18] E. KriSner, in: Modelling Small Deformations of Polycrystals, eds. J. Gittus and J. Zarke (Elsevier, Barking, Essex, England, 1986) p. 229.
[19] R.C. O'Handley and N.J. Grant, Proc. Conf. on Rapidly Quenched Metals, eds. S. Steeb and H. Warlimont (North-Holland, Amsterdam, 1985) p. 1125.
THEORY OF MAGNETOSTRICTION IN A M O R P H O U S FERROMAGNETS: INFLUENCE O F NON-LOCAL ELASTIC BEHAVIOUR T. BEUERLE and M. F,4,HNLE Max-Planck.Institut flit Metallforschung, Institut far Physik, Heisenbergstr. 1, 7000 Stuttgart 80, Fed. Rep. Germany Received 6 February 1990; in revised form 28 March 1990
In former phenomenological theories of magnetostriction in amorphous ferromagnets, a local versiori of elasticity theory has been applied, although the atomic scale of the structural fluctuations requires in principle a non-local treatment. In this paper the dominant non-local contributions are calculated. It is shown that they may be neglected in strongly magnetostrictive systems but mzy become relevant for the discussion of weakly magnetostrictive alloys.
1. Outline of the problem In a series of recent papers [1-12], a phenomenological theory of amorphous ferromagnets has been developed, which allows the calculation of the technologically very important magnetoelastic properties of these materials from statistical information about the spatially fluctuating material tensers. The equations are able to determine the local magnetostrictive strains [7], the effective magnetostriction constant [1-5] and its modification by magnetic field [8-10] and external stress [11] or by magnetic field annealing [12], and to elucidate the magnetostriction mechanism in amorphous ferromagnets [1-7]. All these calculations have been performed within the framework of the local linearized theory of elasticity, although the material tensors of amorphous ferromagnets exhibit spatial fluctuations on an atomic scale. It is well known that continuum theories can be extended to atomic scale, but in this case, in principle, a non-local theory should be applied [13]. This becomes obvious from the fact that the force acting on an atom at site r depends on the locations r ' of at least all the neighbouring atoms. Extending the continuum theory to atomic scale means that we introduce volume elements of atomic size which contain one rather than very many atoms. An integral formulation 3f the total elastic energy thus requires the
introduction of a non-local coupling term between volume elements at different sites. In refs. [1-I2], these non-localities have been neglected in line with the conjecture [14] that a local approximation suffices for general, qualitative considerations for which no very precise numbers are demanded. Indeed, the calculation of atomic-level strains and stresses in amorphous materials resulting from the lack of atomic long-range order yields results [14,15] in rather good agreement with those from atomistic calculations. It is the aim of the present paper to take into account the leading non-local contributions to the elastic energy when calculating the effective magnetostriction constant of amorphous ferromagnets.
2. Non-local contributions to the elastic energy According to Krrner and Datta [13], the total K)t
~.., x f..Lo L Jt ~,~
g
It I U I
I-IUI.,~ILI
111~
I~;;i 11~1
U•
written as
f f avav' %,.,(,.,
/) (1)
where
s(r)
represents the displacement field and is a non-local material tensor. For a simplification of eq. (1), it is usually assumed that
%k;(r,
r t)
0304-8853/90/$03.50 © 1990 - Elsevier Science Publishe:s B.V. (North-Holland)
8
T. Beuerle, M. Fiihnle / Theory of magnetostricti~n in amorphousferromagnets
the quantity Vs varies on a scale r0 which is large compared to the interaction range R of the atomic forces, i.e. the interaction range of the kernel C~jkt(r, r'). This is not strictly fulfilled for the magnetostrictive strains ha amorphous ferromagnets, which exhibit spatial fluctuations on an atomic scale [7], so that r0 = R. Because we do not demand a high degree of accuracy but just want toget a feeling for the order of magnitude of non-local effects, we nevertheless adopt the above assumption, which allows us to evaluate formally the non-local ,,tensor according to
c,j~,(~,/)
= c, jk,(,,,)~(,'- ,") + F,k,m(,) O,,~(..- .,') + A,jk,,m,(r) OmO,8(r - r') + . . . ,
(2)
yielding
-
½f dV [Ci2ktci2ct, t + 2F, jkt,,qj +A,,~,~.(O~,,)(O.,~,)].
3,~ct,,
is the tensor of rigid rotations and c~kt denotes the permutation tensor. This clearly demonstrates that the Grad ~ terms describing the effect of torque stresses are related to the non-locality of the system [17].
3. Balance-of-force equation and statistical input The following calculation is in line with the procedure described in full detail in refs. [2,4], and we therefore discuss only the broad outline. First, the total energy of the system as a function of the deformation variables is completed by the inclusion of the strain-dependent part of the magnetic anisotropy energy, given by
ag = f dV B~jkt(r)c,j(r)yk" h.
(3)
Ctjkt(r) = f dV" ci~k,(r, r'),
(4)
-
)~.
(5)
r')~(r-
r'),,
(6)
A,jk;~.(r) = ½f dV' c,4,,(r, r ' ) ( r -
.~. = "(o;~;- 0~s;)
~(m~ )
with the local material tensors
F,,~,~(,) = - j d r ' ~,j,,
where
and the strain field ~,: = ½(~,sz + Off,). The first term in eq. (3) represents the local contribution and is the only term which is taken into account in refs. [1-12], whereas the other two terms are the leading non-local corrections. To derive eq. (3) the Vs-terms in eq. (1) and ,ha;,. ao.~..,.,:.,oo ¢...~:~t. ..... 1. . . . , . _ _ eq. "~' inserted into e q (1)) are combined in such a way that the elastic energy may be completely expressed in terms of the strain tensor c and Grad ~. It is also possible (see ref. [16] and the appendix) to combine the various terms in such a way that the elastic energy may be expressed in terms of ~,
(7)
lqer¢ :he quantity B, jkt(r) represents the mag,netoelastic tensor of the amorphous material at site r, and the vector T denotes the direction cosines of the magnetization. Because we consider only the satura6cr~ magnetostriction, these direction cosines are constant throughout the whole sample. The rotation-dependent part of the magnetic anisotropy energy density responsible for the generally very small reorientation mechanism of magnetostriction [1-4.] is neglected in the following. The total energy is then minimized x~th respect to the displacement field s, thereby automatically fulfilling the compatibility conditions. The resulting balance-of-force equation reads mag
with r~..ag r
~,j ~,)= B.,~,(~)v~,..,
(9)
and
= c , ~ , , ( , ) - (0~G~,~(~)) (~0)
T. Beuerle, M. F~ihnle/ Theory of magnetostriction in amorphousferrornagnets When keeping in ~ k l only the loca~ term C~jkt, eq. (8) reduces to the formerly used local balanceof-force equation (e.g. eq. (12) of ref. [4]). Eq. (8) represeats a stochastic differential equation because of the spatially fluctuating material tensor Cijkl(r), Fijklm(r), Aijklmn(r) and Bijkl(r). It is solved iteratively by a Green's function perturbation approach based on the definition of a hypothetical homogeneous reference medium, which may be chosen arbitrarily. We define this refererLce medium as the one without non-locality and with "all spatially fluctuating tensor components Cq, t(r) replaced by their volume averages. We thus deal with an elastically homogeneous isotropic local medium which is characterized by the tensor (~.~t = (C, jkl(r)), grid for which the Green's function is well known [4,18]. The displacement field s(r) of the actual material then may be represented by a Green's function perturbation series with the deviations =
-
as perturbation parameters. In the local version of the theory, the deviations are totally given by the spatial fluctuations
whereas in the present theory the deviations inelude all the non-local terms described by the tensors Fijklm(r) and Aokl~n(r). Altogether, the displacement field s(r)may be written in functional form,
s(,) = $[ Bijkt(r), Cijkl(r), Fijkl,n(r), A,/klm,,(r)],
(11) which allows in principle the calculation of s(r) if the spatial dependences of all natefial tensors are given. In practice~ the only available information about the material tensors is a statistical one, i.e. the best we can do is to make some plausible assumptions about the n-point correlation functions of these material tensors. We therefore must be satisfied with the calculation of certain volume
9
To obtain reasonable appro:dmations for the correlation functions, ,he pelycrystalline model of amorphous ferromagnets is used [1-12,19], which takes care of the very strong topological and chemical shert-range order of generally uniaxial symmetry wkich is characteristic for many amorphous ferremagnets. Accordingly, the amorphous material is conceived of as consisting of very small "grains" (basically an atom under consideration and its nearest-neighbour atoms) which exhibit identical material tensors with hexagonal symmetry. The statistics of the w.~terial tensors is then totally defined by the statistics of the grain orientation. In the following, we perform the calculations only up to the first order of the perturbation series. We then need as statistical information only the isotropic distribution function density 1
P(cos 0, q ) = 4¢r for the angles 0, ¢ characterizing the grain odentatior;, as well as the two-point correlation function for the grain orientation (defined by eq. (20) of ref. i4]), for which we insert g~2'(r,
/r'
)
=
exp(
--
I r - 1 r2p/ r 6
~-)
(12)
with the structural correlation length r0. It should be noted that in the local version of the theory [2,4], we did not need any ansatz for the correlation functions, just the normalization condition g~2)(r, r ) = 1, as long as we considered macroscopicaUy iso~'ropic systems. As typical ,'alues for the material tensors B, jk~ and C~i~,t of the grains, we insert those of crystalline Co or Gd. For the tensors ~r u k~,~ and Aijklmn no information ;s available in the literature. We therefore go back to the definition of these material tensors via eqs. (5,6) and insert a plausible ansatz for the non-local ~ensor c,/~t(r, r'), namely
_
1
(13)
Here R is the interaction range of the interatomic forces, which in the t ase of ame,-phous ferromagnets is comparable t~ the structural correlation
10
T.
Beuerle, M. Fa'hnle / Theoryof magnetostrictionin amorphousferromagnets
length r0. Note that our ansatz depends only on I r - r' I which corresponds to the assumption of purely central interaction forces between the atoms. This seems to be rather well justified for metallic amorphous ferromagnets, where partially covalent binding forces may only arise from the metalloid atoms. As shown in the appendix, the contributions of the Grad to terms to the elastic energy do not vanish for purely central interaction forces, in contrast to the statement of refs. [2,4]. Inserting eq. (13) into eqs. (5), (6) yields
Gk,,.(r) =0, Aijkt
.(r)=
2 (14)
4. Results
With the above discussed statistical input we can calculate (c~j(r)) and hence the effective magnetostriction tensor according to
(15)
and the effective magnetostriction constant ),eft h~.s¢f ~ 2(X~¢,,-,,22,, ),
(16)
yielding
The second terms, respectively, are due to the spatial fluctuations of the tensor C, jkt, and they constitute the only first-order conection within the framework o~ the local theory [2,4]. The last terms, respectively, describe the first-order correction originating from the non-locality of the material, and we assume R -- r0. For the Co parameters, which we conceive as representative for strongly magnetostrictive amorphous ferromagnets, the first-order correction terms are only of the order 10% of the dominating zeroth-order term. Obviously, in this case rather accurate values of h~ff may be obtained from eq. (19), and both the spatial fluctuations mad the non-locality play only a minor role. In contrast, for the Gd parameters, the zerothorder result is very small because of a partial cancellation of various contributions to the volume averages in eq. (19). As a result, the spatial fluctuations of Cijki(r ) and hence [2,4] the effect of elastic couplings between neighbouring grains is much stronger, and the non-locality term exceeds the other terms by about 50%. In this case, both the spatial fluctuations as weil as the effect of non-locality are very important, and the perturbation calculation should be extended to higher order to get accurate results.
5. Conclusions
Xelf = - 7 . 0 4 x 10-5 + 0.50 x 10-5 + 0.75 x 10-5 RE
4
(17)
when inserting the material parameters of Co, and esff
R 2
= 3.63 x 10 .6 + 3.89 x 10 .6 + 5.84 x 10 . 6 -
d
for the material parameters of Gd. Here the first terms, respectively, represent the zeroth-order approximation corresponding to the hypothetical elastically isotropic, homogeneous and local medium, given by ~eff, z e r o t h o r d e r =
_
(19)
We have shown that non-local corrections play only a minor role for the case of strongly magnetostrictive amorphous ferromagnets with x~ff around 10-5 as in, for instance, FeNiB-alloys. The situation may be different for the technologically very important, weakly magnetostrictive Co-base alloys (~ff is around 10- 7). As outlined, for instance, in refs. [2,4,12,19], there are two possibilities to achieve low values of ," ~e f t. in the first case, which is represented by our Gd parameters, the low effective magnetostriction arises from a partial cancellation of various contributions to the volume averages in eq. (19), whereas the material is strongly magnetostrictive on a local basis (see also ref. [7]). Then the non-locality of the system becomes essential. In contrast, if the small magnetostriction constant has nothing to do with partial
T. Beuerle, M. Fiihnle / Theory of magnetostricf:on in amorphous ferromagnets
cancellation, but results from a small local magnetostriction comparable in magnitude to 2k~rf, the non-locality may also be neglected. It should be noted that so far it was not possible to decide from any experiment or existing theory whether the weakly magnetostrictive Co-base alloys belong to the first or to the second case.
Acknowledgement The authors are indebted to J. Furthmiiller and R. PaweUek for helpful discussions.
Appendix In eq. (3) the elastic energy has been expressed by c and Grad c. A spatially fluctuating tensor field
,,j(,) = -I-[o, sj(,) + O,s,(,.)] implies the existence of a fluctuating tensor field
,,,, (,.) = -I [O, sj(,.)
-
a/,, (,)]
of rigid rotations. Because a rigid rotation of the whole body is not relevant, the rotation field 60 may contribute to the elastic energy only via its gradient, Grad o. Accordingly, we can rearrange the various terms in eq. (3) in such a way that the elastic energy is expressed by strain q p the gradient of rotation 1
Kij = 2Cjkl ~iO~kt
giving rise to torque stresses [17], and the symmetric part of the gradient of strain,
x;,, = ½(a c,, + a c,, + ak%),
witt E u L := ~AiqkpmnEqrnj~pnl,
Gi:~ ~== -- -~ 2 Fi jk pmCpm 1
(A.2) (a.3)
and
nijmkl =
2 -- "~Aijkpmn~pn t.
(A.4)
Tiffs clearly demonstrates that the torque stresses are related to the non-locality of the system [17]. Inserting for the non-local tensor Cukt(r, r') our ansatz (13) w,hich corresponds to the assumption of purely central interaction forces, the F- and G-teasors vanish but all the other tensors are non-zero, so that there are still contributions of the Grad ~0 terms, in contrast to the statement of refs. [2,4].
References J. Furthmiiller, M. Fiihnle and G. Herzer, J. Phys. F 16 (1986) L255. [:! J. Furthmiiller, M. Fiihnle and G. Herzer, J. Magn. Magn. Mat. 69 (1987) 79, 89. 13i M. F~ihnle, J. Furthmiiller and G. Herzer, J. de Phys. 46 (1988) C6-1329. [4i M. F~ihnle, J. Furthmiiller, R. Pawellek and G. Herzer, in: Physics of Magnetic Materials, eds. W. Gor7_kowski, H.K. Lachowicz and H. Szymczak (World Scientific, Singapore. 1989) p. 228. [5] M. F~ihnle, C. Els~isser, J. Furthmiiller, R. Pawellek, E.H. Brandt and M.C. Btihrn, Physica B 161 (1989) 225. [6] M. Fahnle and J. Furthmiiiler, .I. Magn. Magn. Mat. 72 (1988) 6. R. Pawellek, J. FurthmBller and M. Hihnle, J. Magn. Magn. Mat. 75 (1988) 225. [s] R. Pawellek and M. F~hnle, Phys. Stat. Sol. (a) 111 (1989) 617. [9] M. Fahnle, R. Pawellek and H. Kronmiiller, Phys. Stat. Sol. (a) 112 (1989) 189. [!07 R. Pawellek, M. Hihnle and H. Kronmiiller, Phvsica B ~x,A
concomitant to the existence of a gradient field of rotations. Altogether, the elastic energy reads
11
kAl*..,s),
~..~..
[1~,! M. F~i.hnle and J. Furthmiiller, Phys. Stat. Sol. (a~ 116 (1989) 819.
[12i M. Fahnle and J. Furthmiiller, Phys. Stat Sol. {a) 117 (1990) K71.
= ½f dV [
[13] E. Kr~Sner and B.K. Datta, Z. Phys. 196 (1966) 203.
+
+ AijktrnnXumXktn + 2 Fokt,~,jX kt. + 2Gijm¢ijgk¢ + 2 H i j m k t X i j m X k t ] ,
(AJ)
[14] V. Vitek and T. Egami, Phys. Stat. Sol. (b) 144 (1987) 145. [151 R. Pawellek and M. Fahnle, J. Phys.: Condens. Matter 1 (1989) 7257. [161 E.L. Aero and E.V. Kuvshinskii, Soy. Phys.-Solid State 2 (1960) 1272.
12
I". Beuerle, M. Fiihnle / Theory of magnetostriction in amorphous ferromagnets
[17] F. Hehl and E. KrSner, Z. Naturforsch. 20 (1965) 336. [18] E. KriSner, in: Modelling Small Deformations of Polycrystals, eds. J. Gittus and J. Zarke (Elsevier, Barking, Essex, England, 1986) p. 229.
[19] R.C. O'Handley and N.J. Grant, Proc. Conf. on Rapidly Quenched Metals, eds. S. Steeb and H. Warlimont (North-Holland, Amsterdam, 1985) p. 1125.