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Micromagnetic background of magnetization processes in inhomogeneous ferromagnetic alloys
Journal of Magnetism and Magnetic Materials 112119921 1-5 North-Holland
iI4"
Invited paper
Micromagnetic background of magnetization processes in inhomogeneous ferromagnetic alloys H . Kronmiiller and T. Reininger
h~stitut fiir Physik, Max-Planck-lnstitut fiir Metallforschung, Heisenbergstrasse 1, 7000 Stuttgart 80, Germany
Using micromagnetic concepts, the characteristic parameters of the hysteresis loops of soft magnetic materials (Xt~, Hc, a R) are analyzed by comparing different types of magnetization processes. It is shown that the dominance of one or the other of the magnetization processes depends o , the microstructure as well as on intrinsic material parameters and pretreatments.
1. I n t r o d u c t i o n
During the last decade new soft magnetic materials have been developed where, in addition to high permeabilities, extraordinary electrical and ,:lastic properties have also been tailored [1-4]. The chemical permalloy-based alloys, Sendust or FeSi, are characterized by a low crystalline anisotropy and a low magnetostriction. Since these crystalline alloys i-ave a low critical flow stress, plastic deformations are a serious source of deterioration of soft magnetic properties. Only a few crystalline alloys obey the condition of vanishing magnetostriction and vanishing magnetocrystalline anisotropy. Therefore amorphous alloys have become of importance because these alloys a priori are characterized by vanishing anisotropy and consequently only the condition of zero magnetostriction has to be fulfilled. Suitable amorphous alloys are cobalt-based alloys w~ich are widely used as soft magnetic materials with a broad spectrum of applications. There exists a characteristic difference between crystaliine and amorphous alloys concerning the magnetization processes: Domain Wall (DW) displacements govern the magnetization
process in crystalline materials, whereas in amorphous alloys the type of magnetization processes - DW displacements or rotations - can be chosen according to appropriate pre-treatments. Accordingly, in crystalline materials, permeability and coercive field are exclusively determined by DW pinning, whereas in amorphous alloys DW pinning effects become irrelevant if rotation processes determine the magnetization process. In both cases, however, the microstructure of the ferromagnetic alloys is the decisive property which determines the characteristic parameters of the hysteresis loop. Fig. 1 shows schematically the correlation between microstructure, domain patterns and magnetization processes. The correlations between these properties together with the intrinsic material properties determine the sus-
Nicrostructure
Hognetization Correspondence to: Prof. H. KronmiJller, Max-Planck-lnstitut fiir Metallforschung, Institut fiir Physik, Heisenbergstrasse 1, Postfach 800665, W-7000-Stuttgart 80, Germany. Tel.: +49711-68601: telefax: +49-711-6874371.
Processes
Domain Pattern
Fig. l~ Schematic representation of the correlation between magnetic properties and microstructure, magnetization processes and domain patterns.
1~304-8853/~2/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
2
H. Kronmiiller, T. Reininger / Magnetization processes in inhomogeneous ferromagnets
ceptibility, X0, the coercive field, H c, and the Rayleigh constant, a R, which are relateo by Rayleigh's law for the magnetization M = go f-/+ tzoa~H', being valid at fields H _< H c.
2. Micromagnetism and magnetization processes
The magnetic ground state of a ferromagnetic material is characterized by the minimum of the total Gibbs free energy and a vanishing average magnetization ( M ) = 0 [5]. Under the influence of an applied magnetic field, Hext, the condition of minimum Gibbs free energy is satisfied by DW disp~.acements a n d / o r rotations of the spontaneous magnetization M s . Domain wall displacements are induced by the magnetostatic force per unit area of the DW [6] PH -- 2 HMscos ~0
( 1)
exerted by the internal field, H, acting under an angle ~b0 with respect to the positive magnetization within the domains. BY eq. (1) we may define characteristic magnetization processes considered in detail previously [6-8]. (1) In ideal materials the displacement of DWs is stopped if the interaal field
H=H~xt-N(M)
(2)
(N = demagnetizing factor) vanishes due to the increasing demagnetizing field N ( M ) . The unsheared susceptibility than leads to the wellknown result X = 1 / N and X - ' ~ for N -, 0. (2) If microstructural inhomogeneities oppose the displacement of the DWs by a force PM the equilibrium position of the DW is determined by PM + PH = 0. The coercive field then is determined by the maximum microstructural force, Pr~ax, and the susceptibility by the derivative d PM/dZ of the interaction force at the position of the DW (corresponding to the curvature of the interaction potential). This gives [6,7] He=
2M s cos ~b0 '
(2 M s cos ~b0) 2 1 &0 = D dPM/dZ [ .~=o'
where D denotes the DW distance. The calculation of p~ax and d PM/dZ requires special considerations depending whether we deal with an individual DW-single defect interaction or with a statistical problem where many defects are involved [9,10]. (3) If the magnetic field is oriented perpendicular to the DW plane, PH = 0 holds, and the magnetization process occurs by reversible rotations. In a uniaxial material with an ideal laminar domain pattern as in thin ribbons [11] we have Xo
=
Ms 2K l
,
Hc-0.
Here we deal with the most rigorously defined magnetization process which may be tailored by appropriate induced anisotropy constants. (4) In the case of an enserrble of randomly distributed uniaxial single-domain particles or grains in a polycrystalline mater~al the magnetization occurs predominantly by spontaneous nucleation processes. The average coercive field in this case has been determined by Stoner and Wohlfarth [8]. Applying similar concepts for the average susceptibility and taking into account the combined magnetization process for misaligned grains - reversible rotations + reversion by nucleation - we obtain for grains with diameters larger than the DW width 6 B
Hc = 0.48(2K1/Ms),
Xo = 1.76Ms/2K1.
(5) Eq. (5) represents the average values of H c and X0 in the field range 0 < H < H c. (5) The magnetization occurs by DW bowing if the DWs are strongly pinned. In the case of linear pinning centres, as the connections of grain boundaries, straight dislocation or dislocation tangles, one-dimensional stray-field-free DW vaulting takes place. The coercive field is charactC~lLCU by ux~ m:staomty nctu of the vaulted DW and the susceptibility by the reversible bowing of the DW, giving [12,13] 2~, B
!
Hc = M s cos ~b0 d ' (3)
(4)
2 Ms2 cos2&o d 2 X0
3
YB
D '
(6)
H. Kronmiifier, T. Reininger / Magnetization processes in inhomogeneous ferromagnets
where y~ corresponds to the specific wall energy and d to the average distance between the pinning centres (grain diameters).
3
%#oH)
Xo#o Hc J~
x=60 c/ / /
x _-20 / rat/ X =50
~000
/
3. Experimental results and discussion
3000
In order to analyze the magnetization process of a certain magnetic material the study of the characteristic parameters of the hysteresis loop, X0, Hc, a R, as a function of temperature or any other intrinsic or microstructural parameter has been found to be rather helpful. In particular such products as XoHc, a RH c or XZO/aR may be used to characterize the magnetization process. As an example we may consider the product )oHc . According to the preceding section we find XoHc/Ms = 0, 0.845, ~(d/D) for the ideal rotations, the ensemble of grains and the wall bowing process, respectively. In the case of the statistical pinning theory the result is [6,7,10]
"2-6-BB/]'
Ms " = ~ 2 cos &0-~- In
Xo
(7a)
8 [ ( D )
a~tt°Hc-
3err
(7b)
-~B
'
6"10s+ 3
o
'
o
% \ \ k \\°" k \ k x \ \ \ \ \ \ \ \ \ \ \ \ k
\\\\k~
~
.x....,x.. x
2.10"3f 2
x/"
./ / /
X0 -------
/ /
t,
2
/
6
8
I,(
0
10
20
30
40 50 60 70 80 ( r -'to ) ---,,- [HPu]
Fig. 2. ¥ o H c / M s and #oHc(~R vs. Xo for plastically deformed nickel single crystals. Characteristic properties Xo, H o txp taken from ref. [14].
/
/
10"2, i
/
/
nx=30
~
DX=40
5.10-~
/
/
/
~3
Fe~oNi4oP~4B2o
]000 / /
/ /
/
0
500
1000
~() 20 3'0 t.o s'o
Xo
x(%)
Fig. 3. X o A ' ? / M s and i.toHca R vs. X0 for magnetostrictive amorphous Feso_xNi,,B20 alloys with x varying from x = 0 to = 60 [9].
where the right-hand side of eq. (7) is independent of the type of interaction mechanism. In the particular case where the DWs interact each with one planar defect we find, using previous results [14] 4
x ° H c / M s - 9rr
H' X° l u° H~aR ,,,, ...... o.... ~ ......
O
/mx=o
cos d,,,(,SB/D),
(S)
In figs. 2-5 we present plots o[ the above quantities for several materials and for different types of varying intrinsic or microstructural material parameters. (i) Fig. 2 shows the results for plastically deformed nickel single crystals [15]. In this case the dislocation density increases according to a ( z %)2 law (~" = flow stress, % = critical flow stress). The DWs interact with a large number of dislocations and therefore eqs. (7) should apply. The experimental results in fact prove this assumption since ¥,)Hc/Ms=const. and a.~.HcCXV. for a constant ratio 8B/D. A quantitative analysis gives 6B/D = 2 × 10 -3 and D = 25 ~m which is compatible with experimental results for the domain width in nickel [16], (ii) Fig. 3 presents results for the magnetostrictice amorphous alloys Fes0_xNixB20 [9]. The constancy of the product XoHc/Ms and the approxi-
4
H. Kronmiiller, T. Reininger / Magnetization processes in inhomogeneousferromagnets
mate linearity of aRH c vs. X0 again prove the validity of the statistical pinning theory. From the temperature dependence of Hc, however, in contrast to crystalline nickel, the active pinning centres are found to be quasidislocadon dipoles [9,10]. A quantitative analysis gives ~B/D -- 5 X 10 -3, with ~a = 100 nm and D = 20 ~m. (iii) In the case of the nearly nonmagnetostrictiue amorphous alloy Co 7~Fe !Mo ~Mn 4Si 14B9 the characteristic parameters X0 and H c have been measured as a function of temperature for different values of induced anisotropies. In eq. (7a) then /Ja corresponds to the varying parameter which is given by ~B='rr~A/K,,. Assuming a temperature dependence of the exchange constant according to A =AoMsZ(T) and, using the measured temperature dependences of the induced anisotropy K u and of M s, the plot X0 Hc vs. M2/K~/2D should lead to a straight line, intersecting the origin. Actually such a relation is not obeyed by the experimental results for XoHo It is therefore suggested that in the extremely magnetically soft materials, X0 and H c are determined by interactions of different wavelengths. As proposed by Reininger [17], a statistical short-range potential, as well as a long-range potential resulting from the surface roughness determine H c. Accordingly H c ,s composed of a short-range term, H~ ~, and a long-range term,
3500
<
Co7~F%MoI Mn~Sil~Bg
3000
/
t~
I
~o 2500 1500 I
/.
oooI 5oo 0
~o
/
2000
tx
xx
x
° /
•/
o x~...~ /
,--,
10 ,
"X,. H,/Ms
%
(1100K
t..=l
x
o Xglot R
O
-r"
.,,,o..'--~-~ (1200K) o /
(1050K
(1100K) o / x
(850K1 o o
o (900K)
x 1870K)
(g~Sx0K} ~
xQ
0.I
. . . . . . . . . . . . . . . . . 100
10
theoretical value for domnin walt displacemerits , 1000
0.1
....... 10 ~ He C A l m ]
Fig. 5. The quantities XoHc/Ms and X2/aR of nanocrystalline Fe60Co30Zr m alloys for different nanocrystalline structures. (xa) corresponds to the amorphous alloy [21].
H~', whereas X0 is exclusively determined by the statistical short-range potential. Taking care of these facts the expression xoH c = X~'(H~ ~ + HcIr) has been determined by Reininger giving Xonc
G(
/xoMg KI/2-----~
1 -1- o:
Xo Ku 2 2 ' ~0Ms
(9)
where previous results for /t/~, have been used [10]. Fig. 4 gives plots according to eq. (9) showing a linear behaviour for a = 1.5 x 10 -5 [T2m3/j] [17]. A further quantitative analysis gives H ~ / H ~ ~ ratios varying between 0.2 and 0.3 in the temperature range 100 K < T < 400 K. (iv) In nanocrystalline materials it is well known [18-20] that H c and X0 vary drastically with the grain diameters. Starting from amorphous Fe60Co30Zrlv alloys, nanocrystalline structures have been produced by annealing [21]. The grain diameters varied between d = 10 nm at TA = 850 K and 150 nm for TA = 1200 K. Fig. 5 shows the plots of XoHc/Ms a n d X2/OIR as a function of H c (crystallization temperature). In
/ L
0
.
,
.
50 Js2
D-K •
100 r
K ' X o sR
"11
~
w2 L. +O~"
150 Js
200
250
]1 T 2" m
~12 .j~/21
100
Fig. 4. ~¢oHc for the nonmagnetostrictive C071FelMo 1 Mn4Sil4B, ~ alloys [17] as a function of J~/DK~/2 (1+ sr 2 aK~xo/J~ ), J~ =/zoM s, for different induced anisotropies K u (121 25, & 55, ><.92, • 105 J / m - 3 ) .
(1) Below 950 K (d < 25 nm) XoHc = 0.2 holds. which is compatible with the statistical pinning theory, and ~B/D = 0.2. Here it should be noted that in this stage H c varies according to a d 6 law. (2) Above 950 K the product XoHc increases sharply up to a value of 3 at Ta = 1050 K (d > 50 rim).
H. Kronrniiller, T. Reininger / Magnetization processes in inhomogeneous ferromagnets
According to our results of section 2 there exists nc conventional magnetization process which is compatible with this large value of XoHc/Ms. Obviously, the range of larger grain sizes (d > 50 nm) is not adequately described by current theories. One reason for this failure may be the assumption of magnetically isolated grains. Actually these are coupled by exchange as well as long-range dipolar interactions. Dipolar fields locally may enhance the external field and thus lead to much larger rotational initial susceptibilities than those corresponding to ,~0 of eq. (5). The transition from a DW mechanism to a rotation mechanism at grain sizes of ---40 nm (Tn 1000 K) is due to the increasing pinning of DWs at grain boundaries due to the decreasing effective wall width. Here it should be noted that the effective wall width in nanocrystalline materials decreases according to a 1/d 3 law [19] thus favouring rotational processes for increasing grain sizes.
References [1] R. Boll and H.-R. Hilzinger, IEEE Trans. Magn. MAG-19 (1983) 1946. [2] H. Fujimori, in: Amorphous Metallic Alloys, ed. F.E. Luborsky (Butterworth, London, 1983) p. 300.
5
[3] F E. Luborsky and J.U Walter. IEEE Trans. Magn. MAG-13 (1977) 953, 1635. [4] G. Herzer, IEEE Trans. Magn. MAG-26 (19t~0) 1397. [5] W.F. Brown, Jr., Micromagnetics (Intersciencc, New York, 1963). [6] H. Tr~iuble, in: Moderne Probleme der Metallphysik, vol. II, ed. A. Seeger (Springer, Berlin, Heidelberg, New York, 1966)p. 157. [7] H. Kronmiillcr, Intcrn. J. Nondcstructivc Testing 3 (1972) 315. [8] E.C. Stoner and E.P. Wehlfarth, Philos. Trans. R. Soc. (London) 240 (1948) 599. [9] H. Kronmiiller, M. Fiihnle, M. Domann, H. Grimm, R. Grimm and B. Gr6ger, J. Magn. Magn. Mater. 13 (1979) 53. [10] H. Kronmiiller, J. Magn. Magn. Mater. 24 (1981) 159. [lll H. Kronmiiller, N. 1V:oser and T. Reininger, An. F[s. B 86 (1990) 1. [121 A. Mager, Ann. Phys. (Leipzig) 11 (1952) 15, [131 H. Kronmiiller and H.-R. Hilzinger, J. Magn. Magn. Mater. 2 (1976) 3. [141 H.-R. Hilzinger and H. Kronmiiiler, Phys. Lett. A 51 (1975) 59. [151 E. Kdster, Phys. Stat. Sol. 19 (1967) 655. [161 G. Schrdder and H. Kronmiiller, Phys. Star. Sol. (a) 34 (1976) 623. [171 T. Reininger, Dr. rer. nat. Thesis, Universit~it Stuttgart (1990). [181 Y. Yoshizawa, S. Oguma and K. Yamauchi, J. Aplgl. Phys. 64 (1988) 6044. [19] G. Herzer, Mater. Sci. Eng. A 133 (1991) 1. 1201 H.-R. Hilzinger, Mater. Sci. Forum 62-64 (1990) 515. [211 H.-Q. Guo, T. Reininger, H. Kronmiiller, M. Rapp and V.H. Skumryev, Phys ~tJt. Sol. (a) 127 (1992) 519.
iI4"
Invited paper
Micromagnetic background of magnetization processes in inhomogeneous ferromagnetic alloys H . Kronmiiller and T. Reininger
h~stitut fiir Physik, Max-Planck-lnstitut fiir Metallforschung, Heisenbergstrasse 1, 7000 Stuttgart 80, Germany
Using micromagnetic concepts, the characteristic parameters of the hysteresis loops of soft magnetic materials (Xt~, Hc, a R) are analyzed by comparing different types of magnetization processes. It is shown that the dominance of one or the other of the magnetization processes depends o , the microstructure as well as on intrinsic material parameters and pretreatments.
1. I n t r o d u c t i o n
During the last decade new soft magnetic materials have been developed where, in addition to high permeabilities, extraordinary electrical and ,:lastic properties have also been tailored [1-4]. The chemical permalloy-based alloys, Sendust or FeSi, are characterized by a low crystalline anisotropy and a low magnetostriction. Since these crystalline alloys i-ave a low critical flow stress, plastic deformations are a serious source of deterioration of soft magnetic properties. Only a few crystalline alloys obey the condition of vanishing magnetostriction and vanishing magnetocrystalline anisotropy. Therefore amorphous alloys have become of importance because these alloys a priori are characterized by vanishing anisotropy and consequently only the condition of zero magnetostriction has to be fulfilled. Suitable amorphous alloys are cobalt-based alloys w~ich are widely used as soft magnetic materials with a broad spectrum of applications. There exists a characteristic difference between crystaliine and amorphous alloys concerning the magnetization processes: Domain Wall (DW) displacements govern the magnetization
process in crystalline materials, whereas in amorphous alloys the type of magnetization processes - DW displacements or rotations - can be chosen according to appropriate pre-treatments. Accordingly, in crystalline materials, permeability and coercive field are exclusively determined by DW pinning, whereas in amorphous alloys DW pinning effects become irrelevant if rotation processes determine the magnetization process. In both cases, however, the microstructure of the ferromagnetic alloys is the decisive property which determines the characteristic parameters of the hysteresis loop. Fig. 1 shows schematically the correlation between microstructure, domain patterns and magnetization processes. The correlations between these properties together with the intrinsic material properties determine the sus-
Nicrostructure
Hognetization Correspondence to: Prof. H. KronmiJller, Max-Planck-lnstitut fiir Metallforschung, Institut fiir Physik, Heisenbergstrasse 1, Postfach 800665, W-7000-Stuttgart 80, Germany. Tel.: +49711-68601: telefax: +49-711-6874371.
Processes
Domain Pattern
Fig. l~ Schematic representation of the correlation between magnetic properties and microstructure, magnetization processes and domain patterns.
1~304-8853/~2/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
2
H. Kronmiiller, T. Reininger / Magnetization processes in inhomogeneous ferromagnets
ceptibility, X0, the coercive field, H c, and the Rayleigh constant, a R, which are relateo by Rayleigh's law for the magnetization M = go f-/+ tzoa~H', being valid at fields H _< H c.
2. Micromagnetism and magnetization processes
The magnetic ground state of a ferromagnetic material is characterized by the minimum of the total Gibbs free energy and a vanishing average magnetization ( M ) = 0 [5]. Under the influence of an applied magnetic field, Hext, the condition of minimum Gibbs free energy is satisfied by DW disp~.acements a n d / o r rotations of the spontaneous magnetization M s . Domain wall displacements are induced by the magnetostatic force per unit area of the DW [6] PH -- 2 HMscos ~0
( 1)
exerted by the internal field, H, acting under an angle ~b0 with respect to the positive magnetization within the domains. BY eq. (1) we may define characteristic magnetization processes considered in detail previously [6-8]. (1) In ideal materials the displacement of DWs is stopped if the interaal field
H=H~xt-N(M)
(2)
(N = demagnetizing factor) vanishes due to the increasing demagnetizing field N ( M ) . The unsheared susceptibility than leads to the wellknown result X = 1 / N and X - ' ~ for N -, 0. (2) If microstructural inhomogeneities oppose the displacement of the DWs by a force PM the equilibrium position of the DW is determined by PM + PH = 0. The coercive field then is determined by the maximum microstructural force, Pr~ax, and the susceptibility by the derivative d PM/dZ of the interaction force at the position of the DW (corresponding to the curvature of the interaction potential). This gives [6,7] He=
2M s cos ~b0 '
(2 M s cos ~b0) 2 1 &0 = D dPM/dZ [ .~=o'
where D denotes the DW distance. The calculation of p~ax and d PM/dZ requires special considerations depending whether we deal with an individual DW-single defect interaction or with a statistical problem where many defects are involved [9,10]. (3) If the magnetic field is oriented perpendicular to the DW plane, PH = 0 holds, and the magnetization process occurs by reversible rotations. In a uniaxial material with an ideal laminar domain pattern as in thin ribbons [11] we have Xo
=
Ms 2K l
,
Hc-0.
Here we deal with the most rigorously defined magnetization process which may be tailored by appropriate induced anisotropy constants. (4) In the case of an enserrble of randomly distributed uniaxial single-domain particles or grains in a polycrystalline mater~al the magnetization occurs predominantly by spontaneous nucleation processes. The average coercive field in this case has been determined by Stoner and Wohlfarth [8]. Applying similar concepts for the average susceptibility and taking into account the combined magnetization process for misaligned grains - reversible rotations + reversion by nucleation - we obtain for grains with diameters larger than the DW width 6 B
Hc = 0.48(2K1/Ms),
Xo = 1.76Ms/2K1.
(5) Eq. (5) represents the average values of H c and X0 in the field range 0 < H < H c. (5) The magnetization occurs by DW bowing if the DWs are strongly pinned. In the case of linear pinning centres, as the connections of grain boundaries, straight dislocation or dislocation tangles, one-dimensional stray-field-free DW vaulting takes place. The coercive field is charactC~lLCU by ux~ m:staomty nctu of the vaulted DW and the susceptibility by the reversible bowing of the DW, giving [12,13] 2~, B
!
Hc = M s cos ~b0 d ' (3)
(4)
2 Ms2 cos2&o d 2 X0
3
YB
D '
(6)
H. Kronmiifier, T. Reininger / Magnetization processes in inhomogeneous ferromagnets
where y~ corresponds to the specific wall energy and d to the average distance between the pinning centres (grain diameters).
3
%#oH)
Xo#o Hc J~
x=60 c/ / /
x _-20 / rat/ X =50
~000
/
3. Experimental results and discussion
3000
In order to analyze the magnetization process of a certain magnetic material the study of the characteristic parameters of the hysteresis loop, X0, Hc, a R, as a function of temperature or any other intrinsic or microstructural parameter has been found to be rather helpful. In particular such products as XoHc, a RH c or XZO/aR may be used to characterize the magnetization process. As an example we may consider the product )oHc . According to the preceding section we find XoHc/Ms = 0, 0.845, ~(d/D) for the ideal rotations, the ensemble of grains and the wall bowing process, respectively. In the case of the statistical pinning theory the result is [6,7,10]
"2-6-BB/]'
Ms " = ~ 2 cos &0-~- In
Xo
(7a)
8 [ ( D )
a~tt°Hc-
3err
(7b)
-~B
'
6"10s+ 3
o
'
o
% \ \ k \\°" k \ k x \ \ \ \ \ \ \ \ \ \ \ \ k
\\\\k~
~
.x....,x.. x
2.10"3f 2
x/"
./ / /
X0 -------
/ /
t,
2
/
6
8
I,(
0
10
20
30
40 50 60 70 80 ( r -'to ) ---,,- [HPu]
Fig. 2. ¥ o H c / M s and #oHc(~R vs. Xo for plastically deformed nickel single crystals. Characteristic properties Xo, H o txp taken from ref. [14].
/
/
10"2, i
/
/
nx=30
~
DX=40
5.10-~
/
/
/
~3
Fe~oNi4oP~4B2o
]000 / /
/ /
/
0
500
1000
~() 20 3'0 t.o s'o
Xo
x(%)
Fig. 3. X o A ' ? / M s and i.toHca R vs. X0 for magnetostrictive amorphous Feso_xNi,,B20 alloys with x varying from x = 0 to = 60 [9].
where the right-hand side of eq. (7) is independent of the type of interaction mechanism. In the particular case where the DWs interact each with one planar defect we find, using previous results [14] 4
x ° H c / M s - 9rr
H' X° l u° H~aR ,,,, ...... o.... ~ ......
O
/mx=o
cos d,,,(,SB/D),
(S)
In figs. 2-5 we present plots o[ the above quantities for several materials and for different types of varying intrinsic or microstructural material parameters. (i) Fig. 2 shows the results for plastically deformed nickel single crystals [15]. In this case the dislocation density increases according to a ( z %)2 law (~" = flow stress, % = critical flow stress). The DWs interact with a large number of dislocations and therefore eqs. (7) should apply. The experimental results in fact prove this assumption since ¥,)Hc/Ms=const. and a.~.HcCXV. for a constant ratio 8B/D. A quantitative analysis gives 6B/D = 2 × 10 -3 and D = 25 ~m which is compatible with experimental results for the domain width in nickel [16], (ii) Fig. 3 presents results for the magnetostrictice amorphous alloys Fes0_xNixB20 [9]. The constancy of the product XoHc/Ms and the approxi-
4
H. Kronmiiller, T. Reininger / Magnetization processes in inhomogeneousferromagnets
mate linearity of aRH c vs. X0 again prove the validity of the statistical pinning theory. From the temperature dependence of Hc, however, in contrast to crystalline nickel, the active pinning centres are found to be quasidislocadon dipoles [9,10]. A quantitative analysis gives ~B/D -- 5 X 10 -3, with ~a = 100 nm and D = 20 ~m. (iii) In the case of the nearly nonmagnetostrictiue amorphous alloy Co 7~Fe !Mo ~Mn 4Si 14B9 the characteristic parameters X0 and H c have been measured as a function of temperature for different values of induced anisotropies. In eq. (7a) then /Ja corresponds to the varying parameter which is given by ~B='rr~A/K,,. Assuming a temperature dependence of the exchange constant according to A =AoMsZ(T) and, using the measured temperature dependences of the induced anisotropy K u and of M s, the plot X0 Hc vs. M2/K~/2D should lead to a straight line, intersecting the origin. Actually such a relation is not obeyed by the experimental results for XoHo It is therefore suggested that in the extremely magnetically soft materials, X0 and H c are determined by interactions of different wavelengths. As proposed by Reininger [17], a statistical short-range potential, as well as a long-range potential resulting from the surface roughness determine H c. Accordingly H c ,s composed of a short-range term, H~ ~, and a long-range term,
3500
<
Co7~F%MoI Mn~Sil~Bg
3000
/
t~
I
~o 2500 1500 I
/.
oooI 5oo 0
~o
/
2000
tx
xx
x
° /
•/
o x~...~ /
,--,
10 ,
"X,. H,/Ms
%
(1100K
t..=l
x
o Xglot R
O
-r"
.,,,o..'--~-~ (1200K) o /
(1050K
(1100K) o / x
(850K1 o o
o (900K)
x 1870K)
(g~Sx0K} ~
xQ
0.I
. . . . . . . . . . . . . . . . . 100
10
theoretical value for domnin walt displacemerits , 1000
0.1
....... 10 ~ He C A l m ]
Fig. 5. The quantities XoHc/Ms and X2/aR of nanocrystalline Fe60Co30Zr m alloys for different nanocrystalline structures. (xa) corresponds to the amorphous alloy [21].
H~', whereas X0 is exclusively determined by the statistical short-range potential. Taking care of these facts the expression xoH c = X~'(H~ ~ + HcIr) has been determined by Reininger giving Xonc
G(
/xoMg KI/2-----~
1 -1- o:
Xo Ku 2 2 ' ~0Ms
(9)
where previous results for /t/~, have been used [10]. Fig. 4 gives plots according to eq. (9) showing a linear behaviour for a = 1.5 x 10 -5 [T2m3/j] [17]. A further quantitative analysis gives H ~ / H ~ ~ ratios varying between 0.2 and 0.3 in the temperature range 100 K < T < 400 K. (iv) In nanocrystalline materials it is well known [18-20] that H c and X0 vary drastically with the grain diameters. Starting from amorphous Fe60Co30Zrlv alloys, nanocrystalline structures have been produced by annealing [21]. The grain diameters varied between d = 10 nm at TA = 850 K and 150 nm for TA = 1200 K. Fig. 5 shows the plots of XoHc/Ms a n d X2/OIR as a function of H c (crystallization temperature). In
/ L
0
.
,
.
50 Js2
D-K •
100 r
K ' X o sR
"11
~
w2 L. +O~"
150 Js
200
250
]1 T 2" m
~12 .j~/21
100
Fig. 4. ~¢oHc for the nonmagnetostrictive C071FelMo 1 Mn4Sil4B, ~ alloys [17] as a function of J~/DK~/2 (1+ sr 2 aK~xo/J~ ), J~ =/zoM s, for different induced anisotropies K u (121 25, & 55, ><.92, • 105 J / m - 3 ) .
(1) Below 950 K (d < 25 nm) XoHc = 0.2 holds. which is compatible with the statistical pinning theory, and ~B/D = 0.2. Here it should be noted that in this stage H c varies according to a d 6 law. (2) Above 950 K the product XoHc increases sharply up to a value of 3 at Ta = 1050 K (d > 50 rim).
H. Kronrniiller, T. Reininger / Magnetization processes in inhomogeneous ferromagnets
According to our results of section 2 there exists nc conventional magnetization process which is compatible with this large value of XoHc/Ms. Obviously, the range of larger grain sizes (d > 50 nm) is not adequately described by current theories. One reason for this failure may be the assumption of magnetically isolated grains. Actually these are coupled by exchange as well as long-range dipolar interactions. Dipolar fields locally may enhance the external field and thus lead to much larger rotational initial susceptibilities than those corresponding to ,~0 of eq. (5). The transition from a DW mechanism to a rotation mechanism at grain sizes of ---40 nm (Tn 1000 K) is due to the increasing pinning of DWs at grain boundaries due to the decreasing effective wall width. Here it should be noted that the effective wall width in nanocrystalline materials decreases according to a 1/d 3 law [19] thus favouring rotational processes for increasing grain sizes.
References [1] R. Boll and H.-R. Hilzinger, IEEE Trans. Magn. MAG-19 (1983) 1946. [2] H. Fujimori, in: Amorphous Metallic Alloys, ed. F.E. Luborsky (Butterworth, London, 1983) p. 300.
5
[3] F E. Luborsky and J.U Walter. IEEE Trans. Magn. MAG-13 (1977) 953, 1635. [4] G. Herzer, IEEE Trans. Magn. MAG-26 (19t~0) 1397. [5] W.F. Brown, Jr., Micromagnetics (Intersciencc, New York, 1963). [6] H. Tr~iuble, in: Moderne Probleme der Metallphysik, vol. II, ed. A. Seeger (Springer, Berlin, Heidelberg, New York, 1966)p. 157. [7] H. Kronmiillcr, Intcrn. J. Nondcstructivc Testing 3 (1972) 315. [8] E.C. Stoner and E.P. Wehlfarth, Philos. Trans. R. Soc. (London) 240 (1948) 599. [9] H. Kronmiiller, M. Fiihnle, M. Domann, H. Grimm, R. Grimm and B. Gr6ger, J. Magn. Magn. Mater. 13 (1979) 53. [10] H. Kronmiiller, J. Magn. Magn. Mater. 24 (1981) 159. [lll H. Kronmiiller, N. 1V:oser and T. Reininger, An. F[s. B 86 (1990) 1. [121 A. Mager, Ann. Phys. (Leipzig) 11 (1952) 15, [131 H. Kronmiiller and H.-R. Hilzinger, J. Magn. Magn. Mater. 2 (1976) 3. [141 H.-R. Hilzinger and H. Kronmiiiler, Phys. Lett. A 51 (1975) 59. [151 E. Kdster, Phys. Stat. Sol. 19 (1967) 655. [161 G. Schrdder and H. Kronmiiller, Phys. Star. Sol. (a) 34 (1976) 623. [171 T. Reininger, Dr. rer. nat. Thesis, Universit~it Stuttgart (1990). [181 Y. Yoshizawa, S. Oguma and K. Yamauchi, J. Aplgl. Phys. 64 (1988) 6044. [19] G. Herzer, Mater. Sci. Eng. A 133 (1991) 1. 1201 H.-R. Hilzinger, Mater. Sci. Forum 62-64 (1990) 515. [211 H.-Q. Guo, T. Reininger, H. Kronmiiller, M. Rapp and V.H. Skumryev, Phys ~tJt. Sol. (a) 127 (1992) 519.