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On the width of the critical regime in amorphous ferromagnets
Journal of Magnetism and Magnetic Materials 50 (1985) L247-L249 North-Holland, Amsterdam
L247
LETTER TO THE EDITOR ON THE WIDTH OF THE CRITICAL REGIME IN AMORPHOUS FERROMAGNETS
M. FAHNLE, R. MEYER * and H. K R O N M O L L E R Institut fiir Physik, Max-Planck-lnstitut fiir Metallforschung~ 7000 Stuttgart 80, Heisenbergstr. 1, Fed. Rep. Germany Received 26 December 1984; in revised form 11 April 1985
Amorphous ferromagnets exhibit an unusually large temperature range of deviations from the Curie-Weiss behavior in the paramagnetic regime. The question is discussed whether this may be interpreted as an extended critical behavior or as a non-critical phenomenon for which scaling ideas are not valid. Comparison is made with other disordered ferromagnets.
Amorphous ferromagnetic alloys exhibit a rather unusual temperature dependence of the paramagnetic zero field susceptibility x0(T) (for a review, see ref. [1]): Whereas in crystalline ferromagnets Xo(T) may be well described by a Curie-Weiss law (linear Xo l(T).plot) at high temperatures, there is a very large temperature range of curvature of this plot in amorphous systems (fig. 1). Furthermore, the effective exponent
,/(r) = ( T - T~)XodXol/dT = din( Xo])/dln(T/T¢ - 1)
(1)
introduced by Kouvel and Fisher [3] exhibits a non-monotonic temperature dependence (insert of fig. 1), in sharp contrast to the monotonic behavior of y(T) found for pure systems. For T ~ Tc the effective exponent, however, approaches the asymptotic critical value "Ycdescribing the asymptotic critical behavior
Xo(T-, T~)- ( T -
To) -~'c.
ferromagnet (~'c = 1.387), as predicted by the Harris criterion [5,6]. Larger values for the susceptibility exponent are found when fitting the experimental data by a power law or by a scaling form in an extended temperature range. However, these values correspond to average values of ,/(T) over a large temperature range (see fig. 1) and are not representative for the asymptotic critical exponent for T---> T~ (for a discussion, see refs. [1,4,7]). This peculiar behavior may be discussed from three different points of view:
40
/
,///// / /
35
o amorphous Fe20Nis6B24 ~ crystalline Ni
=~,<°30
25
///
20
* Present address: Institut Max von Lane-Paul Langevin, 156X, Centre de Tri, 38042 Grenoble Cedex, France.
Fez0 Niss B2~,
oo°°°°°
oo
/
7"
rm
o
oo°
x Y*/
10
When using eq. (1) for a determination of the asymptotic critical exponent (Yc is given by the inverse slope of the plot (XodXo]/dT) -] vs. T for T ~ T~) all experiments (see refs. [1,4]) yield values for ~'c which are close to the one obtained for the three-dimensional crystalline Heisenberg
~:~
/
15
(2)
/
x xx ~:~o~
0.00
/ ~
,.¢Ox:c ,
010
121 -0 l
I
0.20
""~ N, , ,.=_ 01 02 03 , I TIT~ -1
0.30 T/Tc-I
0.40
Fig. 1. Temperature dependence of Xo(T) and -/(T) for amorphous Fe20Ni56B24 [1,2] and for crystallineNi [3]. The data for Ni are multipliedwith a constant factor so that the broken line is parallel to a line through the last three data points of the amorphousmaterial.
0304-8853/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
L248
M. F6hnle et al. / Critical regime in amorphous ferromagnets
1. There is a small asymptotic critical regime (defined by the temperature range for which L. (fig. 1)) and a wide "non-critical" transition regime to the Curie-Weiss law. in which scaling ideas are not valid [1,2,4,8]. 2. There is a small asymptotic critical regime and a crossover to an extended critical regime at higher temperatures with a modified critical behavior
7(T)
X,,(T)- (T- T~)
(3)
~"
with T,~ < Tc and a new critical exponent 7". Such a behavior may be relevant for example for diluted systems close to the percolation limit [9] or for disordered ferromagnets close to the spin-glass transition, as discussed at the end of this letter. 3. There is a small asymptotic critical regime, followed by a continuous series of crossover regimes between many different fixed points. In this case scaling ideas should be valid beyond the small asymptotic critical regime, but nevertheless the susceptibility may not be described by the simple power law (3). For temperatures outside the asymptotic critical regime the Kouvel-Fisher exponent exhibits temperature dependent values in all three cases: Even if eq. (3) holds for a large temperature range with a temperature independent exponent y*, eq. (1) yields 7 ( T ) = ~,*(T-
T~)/(T- T/).
(4)
temperature range above the asymptotic critical regime for which the exponent y'(T) is temperature independent, i.e. there is no extended critical regime in which eq. (3) holds. It is not possible to distinguish further between cases 1 end 3, because both times the exponent T'(T) yields temperature dependent values outside the asymptotic critical regime. We prefer an interpretation according to case 1, and we therefore have described the nonmonotonic temperature dependence of y(T) in amorphous ferromagnets by the correlated molecular field theory [1,8]. The calculations were able to reproduce the behavior of 7(T) and r ' ( T ) in amorphous ferromagnets at least qualitatively. It should be noted that the non-monotonic temperature dependence of ~,(T) is not only found for the amorphous ferromagnet Fe20Ni56B24 discussed in this letter, but is a general feature of amorphous ferromagnets [1,4,7]. In contrast, homogeneous ferromagnetic systems exhibit a monotonic decrease of ~,(T) with increasing temperature. This has been shown experimentally for crystalline Ni (fig. 1) and for the ferromagnetic insulators KzCuC14 • 2H20 and CuRb2Br 4 . 2H20 [10], and it has been proved [4,11] for localized spin models with isotropic Ising and Heisenberg exchange interactions on a large variety of two- and threedimensional crystal structures by high temperature series expansions with Pad6 approximations. By Monte Carlo computer simulations [4,11] it has been demonstrated that the non-monotonic temperature dependence of "f(T) is a general feature of all site-disordered ferromagnets, regardless of
The effective exponent, "f(T), therefore is not suitable to distinguish between the three cases discussed above. We therefore define another critical exponent
[1]
T,(T)=[d~(XodXol/dT)
o o
1.8
o
o
oO
o
o o o
o
l] 1
&
(5)
o o o°
~
o
o
o
~
1,4
which also approaches the critical exponent 7c for T---, T~, but which yields the temperature independent value of 7" for case 2 and temperature dependent values for cases 1 and 3 beyond the asymptotic critical regime. Fig. 2 shows the results for the amorphous ferromagnet Fe20Nis6B24 [2]. Obviously there is no
1.0
I 0.1
I 0.2
I O.3
( T - T t ) I Tc
Fig. 2. The effective exponents ~,(T) (triangles) and y'(T) (circles) for amorphous Fe20Ni56B24 [2].
M. Fi~hnleet al. / Critical regime in amorphousferromagnets
the question whether the disorder occurs on an amorphous or on a crystalline structure. Experimentally, a non-monotonic behavior of 3'(T) was found for the chemically disordered crystalline ferromagnet Eu06Sr0.4S [12]. More experimental data on 3'(T) of homogeneously disordered crystalline ferromagnets are highly desirable (Aldred and Kouvel [13] have discussed FexCrl_ x and NixCul_ x, however, the behavior of these giant moment systems may not be comparable to the one of the systems discussed above). Results for 3"(T) have been published also for amorphous Fe10Ni70B16P4 [2], exhibiting the same non-monotonic temperature dependence as the material discussed in this letter. Furthermore, we have calculated 3"(T) for the site-disordered simple cubic ferromagnets discussed in ref. [11] and found similar results. We therefore assume that the nonmonotonic temperature dependence both of 3'(T) and 3"(T) is a general feature of site-disordered ferromagnets. The situation may be different for systems for which the critical temperature To is strongly reduced beyond the mean field value of k T c m = ½ s ( s -k 1 ) ~ l J I .
(6)
Here S is the spin quantum number, £ is the mean number of interacting neighbors and I JI is a typical value of the exchange interactions. In this case the asymptotic critical regime ( ( T - T c ) / T c << 1) may be very small, and a crossover may be observed for higher temperatures to a modified critical regime, for which eq. (3) holds. The range
L249
of validity of eq. (3) then is determined by l J I, namely k T << I J I, which is much larger than kT~. Examples are percolative systems close to the multicritical point, for which 3'c of eq. (2) corresponds to the thermal critical exponent and 3'* of eq. (3) is the percolation exponent [9], and disordered ferromagnetic systems with strong antiferromagnetic couplings close to the spin-glass transition (see also ref. [14]). The experiments discussed by Westerholdt and Sobotta [15] on the highly diluted antiferromagnetic insulator (Cr x All - x) 203 and on the ferromagnetic systems EuxSrl_xS0.3Seo. 5 and Fe~Aul-x close to the concentrations for spin-glass transitions possibly may be interpreted on this line. References [1] M. F~thnle, G. Herzer, H. Kronmi~ller, R. Meyer, M. Saile and T. Egami, J. Magn. Magn. Mat. 38 (1983) 240. [2] R. Meyer, thesis, University of Stuttgart (1983). [3] J.S. Kouvel and M.E. Fisher, Phys. Rev. 136 (1964) A1626. [4] M. F~ahnle, J. Magn. Magn. Mat. 45 (1984) 279. [5] A.B. Harris, J. Phys. C 7 (1974) 1671. [6] U. Krey, Phys. Lett. A51 (1975) 189. [7] S.N. Kaul, IEEE Trans. Magn. MAG-20 (1984) 1290, [8] M. F~mle and G. Herzer, J. Magn. Magn. Mat. 44 (1984) 274. [9] A. Coniglio, Phys. Rev. Lett. 46 (1981) 250. [10] E. Carrr, J.P. Renard and J. Souletie, to be published. [11] M. F~khnle, J. Phys. C 16 (1983) L819, C 18 (1985) 181. [12] H. Maletta, G. Aeppli and S.M. Shapiro, J. Magn. Magn. Mat. 31-34 (1983) 1367. [13] A.T. Aldred and J.S. Kouvel, Physica 86-88B (1977) 329. [14] U. Krey, Lecture Notes Phys. 192 (1983) 137. [15] K. Westerholdt and G. Sobotta, J. Phys. F13 (1983) 2371.
L247
LETTER TO THE EDITOR ON THE WIDTH OF THE CRITICAL REGIME IN AMORPHOUS FERROMAGNETS
M. FAHNLE, R. MEYER * and H. K R O N M O L L E R Institut fiir Physik, Max-Planck-lnstitut fiir Metallforschung~ 7000 Stuttgart 80, Heisenbergstr. 1, Fed. Rep. Germany Received 26 December 1984; in revised form 11 April 1985
Amorphous ferromagnets exhibit an unusually large temperature range of deviations from the Curie-Weiss behavior in the paramagnetic regime. The question is discussed whether this may be interpreted as an extended critical behavior or as a non-critical phenomenon for which scaling ideas are not valid. Comparison is made with other disordered ferromagnets.
Amorphous ferromagnetic alloys exhibit a rather unusual temperature dependence of the paramagnetic zero field susceptibility x0(T) (for a review, see ref. [1]): Whereas in crystalline ferromagnets Xo(T) may be well described by a Curie-Weiss law (linear Xo l(T).plot) at high temperatures, there is a very large temperature range of curvature of this plot in amorphous systems (fig. 1). Furthermore, the effective exponent
,/(r) = ( T - T~)XodXol/dT = din( Xo])/dln(T/T¢ - 1)
(1)
introduced by Kouvel and Fisher [3] exhibits a non-monotonic temperature dependence (insert of fig. 1), in sharp contrast to the monotonic behavior of y(T) found for pure systems. For T ~ Tc the effective exponent, however, approaches the asymptotic critical value "Ycdescribing the asymptotic critical behavior
Xo(T-, T~)- ( T -
To) -~'c.
ferromagnet (~'c = 1.387), as predicted by the Harris criterion [5,6]. Larger values for the susceptibility exponent are found when fitting the experimental data by a power law or by a scaling form in an extended temperature range. However, these values correspond to average values of ,/(T) over a large temperature range (see fig. 1) and are not representative for the asymptotic critical exponent for T---> T~ (for a discussion, see refs. [1,4,7]). This peculiar behavior may be discussed from three different points of view:
40
/
,///// / /
35
o amorphous Fe20Nis6B24 ~ crystalline Ni
=~,<°30
25
///
20
* Present address: Institut Max von Lane-Paul Langevin, 156X, Centre de Tri, 38042 Grenoble Cedex, France.
Fez0 Niss B2~,
oo°°°°°
oo
/
7"
rm
o
oo°
x Y*/
10
When using eq. (1) for a determination of the asymptotic critical exponent (Yc is given by the inverse slope of the plot (XodXo]/dT) -] vs. T for T ~ T~) all experiments (see refs. [1,4]) yield values for ~'c which are close to the one obtained for the three-dimensional crystalline Heisenberg
~:~
/
15
(2)
/
x xx ~:~o~
0.00
/ ~
,.¢Ox:c ,
010
121 -0 l
I
0.20
""~ N, , ,.=_ 01 02 03 , I TIT~ -1
0.30 T/Tc-I
0.40
Fig. 1. Temperature dependence of Xo(T) and -/(T) for amorphous Fe20Ni56B24 [1,2] and for crystallineNi [3]. The data for Ni are multipliedwith a constant factor so that the broken line is parallel to a line through the last three data points of the amorphousmaterial.
0304-8853/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
L248
M. F6hnle et al. / Critical regime in amorphous ferromagnets
1. There is a small asymptotic critical regime (defined by the temperature range for which L. (fig. 1)) and a wide "non-critical" transition regime to the Curie-Weiss law. in which scaling ideas are not valid [1,2,4,8]. 2. There is a small asymptotic critical regime and a crossover to an extended critical regime at higher temperatures with a modified critical behavior
7(T)
X,,(T)- (T- T~)
(3)
~"
with T,~ < Tc and a new critical exponent 7". Such a behavior may be relevant for example for diluted systems close to the percolation limit [9] or for disordered ferromagnets close to the spin-glass transition, as discussed at the end of this letter. 3. There is a small asymptotic critical regime, followed by a continuous series of crossover regimes between many different fixed points. In this case scaling ideas should be valid beyond the small asymptotic critical regime, but nevertheless the susceptibility may not be described by the simple power law (3). For temperatures outside the asymptotic critical regime the Kouvel-Fisher exponent exhibits temperature dependent values in all three cases: Even if eq. (3) holds for a large temperature range with a temperature independent exponent y*, eq. (1) yields 7 ( T ) = ~,*(T-
T~)/(T- T/).
(4)
temperature range above the asymptotic critical regime for which the exponent y'(T) is temperature independent, i.e. there is no extended critical regime in which eq. (3) holds. It is not possible to distinguish further between cases 1 end 3, because both times the exponent T'(T) yields temperature dependent values outside the asymptotic critical regime. We prefer an interpretation according to case 1, and we therefore have described the nonmonotonic temperature dependence of y(T) in amorphous ferromagnets by the correlated molecular field theory [1,8]. The calculations were able to reproduce the behavior of 7(T) and r ' ( T ) in amorphous ferromagnets at least qualitatively. It should be noted that the non-monotonic temperature dependence of ~,(T) is not only found for the amorphous ferromagnet Fe20Ni56B24 discussed in this letter, but is a general feature of amorphous ferromagnets [1,4,7]. In contrast, homogeneous ferromagnetic systems exhibit a monotonic decrease of ~,(T) with increasing temperature. This has been shown experimentally for crystalline Ni (fig. 1) and for the ferromagnetic insulators KzCuC14 • 2H20 and CuRb2Br 4 . 2H20 [10], and it has been proved [4,11] for localized spin models with isotropic Ising and Heisenberg exchange interactions on a large variety of two- and threedimensional crystal structures by high temperature series expansions with Pad6 approximations. By Monte Carlo computer simulations [4,11] it has been demonstrated that the non-monotonic temperature dependence of "f(T) is a general feature of all site-disordered ferromagnets, regardless of
The effective exponent, "f(T), therefore is not suitable to distinguish between the three cases discussed above. We therefore define another critical exponent
[1]
T,(T)=[d~(XodXol/dT)
o o
1.8
o
o
oO
o
o o o
o
l] 1
&
(5)
o o o°
~
o
o
o
~
1,4
which also approaches the critical exponent 7c for T---, T~, but which yields the temperature independent value of 7" for case 2 and temperature dependent values for cases 1 and 3 beyond the asymptotic critical regime. Fig. 2 shows the results for the amorphous ferromagnet Fe20Nis6B24 [2]. Obviously there is no
1.0
I 0.1
I 0.2
I O.3
( T - T t ) I Tc
Fig. 2. The effective exponents ~,(T) (triangles) and y'(T) (circles) for amorphous Fe20Ni56B24 [2].
M. Fi~hnleet al. / Critical regime in amorphousferromagnets
the question whether the disorder occurs on an amorphous or on a crystalline structure. Experimentally, a non-monotonic behavior of 3'(T) was found for the chemically disordered crystalline ferromagnet Eu06Sr0.4S [12]. More experimental data on 3'(T) of homogeneously disordered crystalline ferromagnets are highly desirable (Aldred and Kouvel [13] have discussed FexCrl_ x and NixCul_ x, however, the behavior of these giant moment systems may not be comparable to the one of the systems discussed above). Results for 3"(T) have been published also for amorphous Fe10Ni70B16P4 [2], exhibiting the same non-monotonic temperature dependence as the material discussed in this letter. Furthermore, we have calculated 3"(T) for the site-disordered simple cubic ferromagnets discussed in ref. [11] and found similar results. We therefore assume that the nonmonotonic temperature dependence both of 3'(T) and 3"(T) is a general feature of site-disordered ferromagnets. The situation may be different for systems for which the critical temperature To is strongly reduced beyond the mean field value of k T c m = ½ s ( s -k 1 ) ~ l J I .
(6)
Here S is the spin quantum number, £ is the mean number of interacting neighbors and I JI is a typical value of the exchange interactions. In this case the asymptotic critical regime ( ( T - T c ) / T c << 1) may be very small, and a crossover may be observed for higher temperatures to a modified critical regime, for which eq. (3) holds. The range
L249
of validity of eq. (3) then is determined by l J I, namely k T << I J I, which is much larger than kT~. Examples are percolative systems close to the multicritical point, for which 3'c of eq. (2) corresponds to the thermal critical exponent and 3'* of eq. (3) is the percolation exponent [9], and disordered ferromagnetic systems with strong antiferromagnetic couplings close to the spin-glass transition (see also ref. [14]). The experiments discussed by Westerholdt and Sobotta [15] on the highly diluted antiferromagnetic insulator (Cr x All - x) 203 and on the ferromagnetic systems EuxSrl_xS0.3Seo. 5 and Fe~Aul-x close to the concentrations for spin-glass transitions possibly may be interpreted on this line. References [1] M. F~thnle, G. Herzer, H. Kronmi~ller, R. Meyer, M. Saile and T. Egami, J. Magn. Magn. Mat. 38 (1983) 240. [2] R. Meyer, thesis, University of Stuttgart (1983). [3] J.S. Kouvel and M.E. Fisher, Phys. Rev. 136 (1964) A1626. [4] M. F~ahnle, J. Magn. Magn. Mat. 45 (1984) 279. [5] A.B. Harris, J. Phys. C 7 (1974) 1671. [6] U. Krey, Phys. Lett. A51 (1975) 189. [7] S.N. Kaul, IEEE Trans. Magn. MAG-20 (1984) 1290, [8] M. F~mle and G. Herzer, J. Magn. Magn. Mat. 44 (1984) 274. [9] A. Coniglio, Phys. Rev. Lett. 46 (1981) 250. [10] E. Carrr, J.P. Renard and J. Souletie, to be published. [11] M. F~khnle, J. Phys. C 16 (1983) L819, C 18 (1985) 181. [12] H. Maletta, G. Aeppli and S.M. Shapiro, J. Magn. Magn. Mat. 31-34 (1983) 1367. [13] A.T. Aldred and J.S. Kouvel, Physica 86-88B (1977) 329. [14] U. Krey, Lecture Notes Phys. 192 (1983) 137. [15] K. Westerholdt and G. Sobotta, J. Phys. F13 (1983) 2371.