iournal of agnetism
m . __
l!B $$
ELSEVIER Journal of Magnetism and Magnetic Materials 135 (1994) 161-170
‘1’ _ 1 magnetic
materials
Cluster spin-glass model for amorphous Fe-Zr alloys near the critical concentration: a magnetization study
L.F. Kiss *, T. Kern&y, I. Vincze, L. Grhky
Research Institute for Solid State Physics, of the Hungarian Academy of Sciences, PO Box 49, H-1525 Budapest, Hungary (Received 16 February 1993; in revised form 29 November 1993)
Abstract
The magnetic properties of melt-quenched amorphous FeIoo_X Zr, (x = 7-12) alloys, were studied in the temperature and magnetic field ranges 15 K < T < 300 K and 0 < H,,, < 1.8 T, respectively. Assuming that Fe,,Zr, is an assembly of magnetic clusters with a blocking temperature distribution, the magnetization versus magnetic field curves were fitted using NCel’s theory for fine particles at several temperatures in order to obtain the evolution of cluster parameters with temperature. The same model is used for Fe,,Zr, and Fe,,Zr,, above the Curie temperature CT,). When NCel’s theory is modified to account for the interaction between the cluster moments, the experimental data are properly described by this frame. This model can explain both the low-temperature anomalies of susceptibility observed for x = 7-10 and the high-temperature superparamagnetic behaviour which is shown to be a characteristic feature of the whole a-Fe 100_xZrx alloy series. The evolution of magnetic cluster behaviour with temperature and concentration is discussed.
1. Introduction
A proper understanding of the magnetic prop- erties of amorphous Fe loo_xZrx alloys near the critical concentration (x, = 7) has been a great challenge for more than a decade [I]. Three dif- ferent types of magnetic behaviour are observed in this alloy series: (i) ferromagnet-like alloys (x = 12) characterized by the Curie temperature T,; (ii) re-entrant spin glasses (RSG, x = S-10) characterized by two transition temperatures, T, denoting a transition from paramagnetic (PM) to ferromagnetic (FM) state and Tf from ferromag- netic to spin-glass (SG) state; and (iii) spin glasses
* Corresponding author.
(n = 71, where only one transition from paramag- netic to spin-glass state occurs at Tg [l-5]. The Curie temperature T, decreases with decreasing Zr content, while Tf changes in the opposite way, coinciding with T, for x = 7. Below Tf or T,, the coercivity increases enormously both with de- creasing temperature and with decreasing Zr con- tent at constant temperature (i.e. approaching the critical concentration) [2,4,5]. In addition, time-dependent magnetization (magnetic viscos- ity) is observed below the spin-glass temperature CT,, T,) suggesting non-equilibrium effects in Fe- Zr glassy alloys, similar to those observed in amorphous Fe-Y [6]. In a-Fe,,,_,Zr, (7 <x I 12) the magnetization versus magnetic field curve shows no saturation even up to 20 T [2,7]. It is similar to that of a superparamagnetic system for
0304&X53/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00094-8
162 L.F. Kiss et al. /Journal of Magnetism and Magnetic Materials 135 (1994) 161-170
intermediate fields 0.2 T <H,,, < 2 T. The high- field susceptibility (xhf) is thus at least one order of magnitude higher than that for amorphous Co-Zr alloys of similar composition [3]. The large x,,r value is one of the characteristic features which are generally associated with Invar proper- ties and is observed in many other amorphous 19-111 and crystalline [12] alloy systems as well.
Several models attempt to account for the above-mentioned magnetic properties of Fe-Zr amorphous alloys. The model of Saito et al. [13]
assumes that the spin-glass state below Tf is due to the freezing of frustrated spin clusters of anti- ferromagnetic Fe spins distributed in a ferromag- netic matrix. However, Miissbauer spectra taken in external magnetic fields [14] do not support the existence of antiferromagnetic coupling between Fe spins. Kaul and co-workers [L-18] suggest that amorphous Fe,,,_,Zr, (x = 8-10) alloys consist of a ferromagnetic matrix (infinite cluster) plus finite ferromagnetic clusters separated and magnetically isolated by frustration zones. This picture is supported by bulk magnetic [15] and ferromagnetic resonance (FMR) data [16] and is further strengthened by measuring the effect of isothermal annealing on the magnetic behaviour of amorphous Fe-Zr alloys using FMR technique 117,181. The existence of a real (collinear) FM matrix in Fe-rich amorphous Fe-Zr alloys is questioned by Ryan and co-workers [2,19] in their
‘wandering-axis ferromagnet’ model in which the spin structure is locally ferromagnetic with small variations in neighbouring spin directions but the local ferromagnetic axis chacges direction over distances of the order of 25 A. According to this model a phase transition to an asperomagnetic state 1191 takes place at Tf (transverse spin freez- ing), similar to that in the theoretical model of Gabay and Toulouse [20]. The basic idea behind the above models is the existence of competing (ferromagnetic and antiferromagnetic) exchange interactions between neighbouring Fe spins caused by fluctuating interatomic distances due to the amorphous structure.
On the other hand, there are models which claim that the anomalous low-temperature mag- netic behaviour of Fe-rich amorphous Fe-Zr al- loys has nothing to do with spin freezing. Read et
al. [41 attribute the abrupt increase in coercivity below Tf to domain wall pinning by antiferromag- netic Fe clusters dispersed in a FM matrix. In this model the transition at Tf is simply caused by the breakdown of the kink point relation (xint = M/Hi,, B l/D, where D is the demagnetization factor of the sample and Hint is the internal magnetic field) due to the rapid decrease in xint with decreasing temperature. In this picture, Tf depends on D (i.e. the sample geometry) and consequently it does not reflect a real physical transition. Beck and Kronmiiller [5] suggest that the anomalous low-temperature properties of Fe-rich amorphous Fe-Zr alloys are determined by increasing local magnetic anisotropy energies of a normal ferromagnet with decreasing temper- ature. The physical origin of such a temperature dependence of the anisotropy is, however, not clear.
The resemblance of the magnetic properties of fine particles [21] to those of classical spin glasses has led to the assumption of a granular magnetic structure (magnetic clusters) in spin glasses. The assumption of a distribution for the cluster size has been demonstrated to reproduce the spin- glass behaviour in classical spin glasses [22]. Real fine-particle systems differ in many ways from the suppositions of NCel’s classical theory for fine particles [21] (i.e. single and temperature-inde- pendent cluster size, absence of cluster an- isotropy, temperature-independent saturation magnetization of the clusters, no interaction be- tween cluster moments). Nevertheless, their mag- netization versus magnetic field curves at differ- ent temperatures can be transformed into univer- sal curves showing the superparamagnetic be- haviour of these systems (see details in Section 3).
Similar behaviour is thus expected also in spin- glass systems, including amorphous Fe-Zr alloys.
In order to study the effect of magnetic clus- ters on the magnetic behaviour of Fe-rich Fe l,,O_XZrX amorphous alloys, we focused our attention mainly on the alloy at the critical con- centration (x, = 7) where no ferromagnetic ma- trix is expected to occur. Measurements were also made on samples near the critical concentration (X = 8, 9, 10 and 12) where the alloys approach FM-like behaviour with increasing X. It suggests
L.F. Kiss et al. /Journal of Magnetism and Magnetic Materials 135 (1994) 161-170 163
that the magnetic anomalies, which are also ob- served for x = 7, do not arise from the FM matrix (infinite cluster) but from what we suppose to be finite magnetic clusters. Two types of bulk mag- netic measurements were performed. First, the low-field magnetization (II,,, = 10 Oe) was mea- sured as a function of temperature (15 K < T <
300 K) both after zero-field cooling and after cooling in HcoO, = 10 Oe external field (‘field cooling’). Second, the magnetization versus exter- nal magnetic field (0 < H,,, < 1.8 T) was recorded at several selected temperatures. The magnetic data of a-Fe,,,_,Zr, below T, (x = 7) or Tf (x = 8) and above T, or T, (x = 8 and 12) are de- scribed in terms of a model according to which the material consists of an assembly of interacting magnetic clusters. The model is based on NCel’s theory for fine particles [21] assuming a blocking-temperature (i.e. cluster size) distribu- tion for the clusters and treating their interac- tions in an approximate way (see details in Sec- tion 3).
2. Experimental
The magnetization was measured by a Foner- type vibrating sample magnetometer with a sensi- tivity of 10P4 emu in an electromagnet with mag- netic fields up to 1.8 T. The temperature of the sample was varied using a continuous-flow He cryostat. The samples were produced by melt- spinning in ribbon form of cross section 1 mm X 12 km and their amorphous state was checked by X-ray diffraction and Miissbauer spectroscopy.
lo-15 ribbon pieces each 4 mm length with a total mass of about 2.5 mg were glued together by water-glass (sodium silicate). The magnetic field during the measurements was directed in plane and along the length of the ribbons.
3. Results and discussion
Fig. 1 displays the low-field magnetization at I&,, = 10 Oe as a function of temperature for x = 7 and 8 after cooling (from room tempera-
ture) in zero field (ZFC) and in H,,, = 10 Oe (FC). Fe,,Zr, and Fe,,Zr, show typical spin-
jTm
0 50 100 200 2
T (K)
0
Fig. 1. Magnetization M versus temperature T measured in an external magnetic field H,,, = 10 Oe for n = 7 and 8 of the a-Fe toO_xZrx alloy series. ZFC = zero-field cooling; FC = field cooling (H,,,, = 10 Oe).
glass-like (with a cusp at T,) and re-entrant spin- glass (RSG) behaviour (with transition tempera- tures Tf defined in Fig. 1 and Curie temperature T,), respectively, in accordance with literature data [l-.5]. Similar RSG behaviour of the M-T curves is found for x = 9 and 10 and only a seemingly normal ferromagnetic behaviour is ob- served for Fe,,Zr,,. The T,, Tf and T, values are summarized in Table 1 for the studied amor- phous Fe-Zr alloys. (The values of T, and Tf are slightly field dependent; Table 1 contains T, and Tf values measured at Hext = 10 Oe.)
Based upon Fig. 1 and considerations made by Beck [23] the internal susceptibility (xint = II~/H~,,~) must be of the same order of magnitude
as the external susceptibility (x,,, = M/H,,,) in
Table 1
Magnetic transition temperatures CT,, spin-glass temperature;
Tr, re-entrant spin-glass temperature; rc, Curie temperature) measured at H,,, = 10 Oe for a-Fe,DO_xZrx
x (at%) T, (K) Tr UQ T, W
7 105 _ _
8 61 174
9 _ 31 205
10 20 225
12 _ 1225
164 L.F. Kiss et al. /Journal of Magnetism and Magnetic Materials 135 (1994) 161-170
01,,,,,,~‘,1”,,1”“l”l
0.0 0.5 1.0 1.5 2.0
100
A
<
z
.T 50 (b)
H
Fig. 2. Magnetization A4 versus external magnetic field Hex, for x = 7 (a), 8 (b) and 12 (c) of the a-Fe,,,_,Zr, alloy series at selected temperatures as denoted . at eactl curve.
the Fe-rich a-Fe ,OO_xZrx alloys near the critical concentration (x = 8, 9, 10 and 12) and Xext is not demagnetization-controlled as in a real ferromag- net. This suggests that these alloys are not con- ventional ferromagnets even in the temperature range Tf < T < T,. Moreover, it also follows that Xint = Xext for susceptibility values smaller than X ext = 0.4-0.5 emu/g Oe, i.e. for the whole M-T curve of Fe,,Zr, and for temperatures T < Tf and T > T, in the case of the other compositions.
The magnetization as a function of external magnetic field is shown in Figs. 2(a)-2(c) for x = 7, 8 and 12 at several selected temperatures.
It agrees with previous experimental results [l-5].
3.1. The model of non-interacting cluster moments
On cooling, the magnetic moment of a cluster is thermally trapped by the magnetic anisotropy at a temperature (called the blocking tempera- ture, TB) above which the assembly of the clus- ters shows superparamagnetic behaviour [21]. The blocking temperature is given by T, = KV/25k, where K and I/ are the uniaxial anisotropy con- stant and the volume of the clusters, respectively [21,24], k denotes the Boltzmann constant, and the numerical factor of 25 is obtained in the case
/ 1 \FeezZr8 / \
I I
Fig. 3. Assumed normalized blocking temperature distribution function f(TB) for Fe,,Zr, (solid line) and Fe,,Zr, (broken line).
L.F. Kiss et al./Journal of Magnetism and Magnetic Materials 13.5 (1994) 161-170 165
of static measurements (V = lo-’ l/s, where v is frequency) using a Larmor frequency of v0 = lo9 Hz [25].
In order to account for the observed suscepti- bility-temperature relation shown in Fig. 1, a distribution in cluster volume (and hence in blocking temperature [f(Tn>] is assumed. The steep susceptibility increase with increasing tem- perature below T, or Tf on the ZFC curves is interpreted as the gradual disappearance of clus- ter blocking at their individual Te’s according to the distribution function f(T&.
The magnetic field (Hi,t = Hext = H) and tem- perature dependence of the magnetization M(H, T) are given for high fields, i.e. for mH s=
kT (m is the magnetic moment of a cluster), similar to Ref. [231, as
dT, +Ms /
Tmf(Td dTB+XhfH, (1)
where L(a) = coth a - l/a denotes the Langevin function, f(TB) is the distribution of the blocking temperatures satisfying lomf(T,J dTB = 1, m(T,)
=M,V(T,) = SOkT,/H, is the magnetic mo- ment of a cluster with the anisotropy field HA = 2K/M,, and MS is the saturation magnetization.
The first term in (1) describes the superparamag- netic behaviour of the clusters whose blocking temperatures are lower than the actual tempera- ture T. The second term gives the magnetization of the blocked clusters whose TB’s are higher than T and whose moments are aligned along a field high enough to overcome the anisotropy field of the clusters. The third term in (1) ac- counts for the rather large high-field susceptibil- ity (,yhf) observed in these alloys (Figs. 2a-c). The factors giving rise to ,yhf, which cannot be over- come even by a field as high as 20 T [2,7], may not cause the low-temperature anomalies of amorphous Fe-Zr alloys which cease to exist in fields as low as a few hundred Oe [l-5].
In terms of the model described above the initial susceptibility (H + 0) can be expressed as
x(T) =~=f(T,,~
dT, +Xhi,where C(T,) = n(TB>m2(TB>/3k = MpdTB)/3k is the Curie constant, with n(T,) denoting the number of clusters per gram. The first term in (2) gives the contribution to the susceptibility of the free (superparamagnetic) clusters whose T,‘s are below T, the actual temperature. The contribu- tion of the blocked clusters to x(T) can be ne- glected [22].
In order to determine the parameters by fit- ting, some assumptions about f(TB> should be made. Theoretically, f(T,) can be derived from (2) as
fcTB) =
&
B (TBX)/ *iym(T~X)>
but because of the T-dependent Curie constant and the large scattering of data points for Fe,3Zr,, a quantitative result for f(TB) could not be ob- tained. An additional problem arises in calculat- ing f(TB) of Fe,,Zr, by the simultaneous pres- ence of a ferromagnetic fraction in the alloy.
Therefore, inspired by the linearly increasing d/dT (TX) functions below T = 90 and 60 K for Fe,,Zr, and Fe92Zr,, respectively, we suggest blocking temperature distribution functions f(TB) as shown in Fig. 3 for these two alloys. The temperatures at which f(TB> attains its maxi- mum, are chosen in the vicinity of T, for Fe,,Zr, and Tf for Fe,,Zr,. The maximum blocking tem- perature TgaX, above which f(TB) is zero, is taken to be Tr = 120 K for Fe,,Zr, because Mossbauer results on the same alloy show clear paramagnetic behaviour above 115 K [26]. Such an estimation is not possible for Fe,,Zr, because of the ferromagnetic fraction present in this alloy, therefore in this case Ty is chosen to be Tr
= 80 K (by analogy to Fe,,Zr,). Slight changes in f(TB) have no significant effect on the results.
Expressions (1) and (2) can be fitted simulta- neously to the measured M(H) and x(T) curve, with the use of f(TB) depicted in Fig. 3 and by varying the three fitting parameters M,, HA and xhf. Derived quantities from the fitting parame-
ters are (8 the average magnetic moment of the free (superparamagnetic) clusters m,, (ii> the av- erage number of Fe spins in a cluster N, and (iii>
166 L.F. Kiss et al. /Journal of Magnetism and Magnetic Materials 135 (1994) 161-170
the average number of clusters per gram E. Since FessZr,, shows no re-entrant spin-glass tempera- ture, no f(Tn> can be postulated for it. In this case the second term is zero in (1) and the first term is replaced by A4,. L(E,H/kT) without av- eraging over f(TB> in the whole temperature range. Similar changes have to be considered in (2). Thus in case of FessZr,*, %I, takes the role of HA as one of the fitting parameters. The fitting procedure can only be made if xint(T) is known, which (as previously discussed) is the case for Fe,,Zr, over the whole temperature range inves-
cm!x!ob = 1 q EI!asG 1.5
_ _ Langevin function
=x=.x 121 K +**++ 141 K 00~00 162 K
Fed-7
(4
tigated and for the other amorphous Fe-Zr al- loys at temperatures T < Tf and T > T,. fl is found to be about 22000 and 32000 below T, or Tf for Fe,,Zr, and Fe,,Zr,, respectively, and decreases rapidly with temperature above T, for all the three alloys. xhf ranges between 0.9 and 1.3 X 10e3 emu/g Oe for all the fits, while HA assumes values of several thousand Oe for Fe,,Zr, and Fe,,Zr,. The fit is not sensitive for two of the parameters, A!, and xhf, meaning that large variation of HA (E,) has little effect on them and similar values could be obtained for MS
QQQQOb = 1 q Dooo 1.5
--- Langevin function .x.-n 184 K
++*++ 200 K
QQQQQb = 1
I / A&A 2 2.5
--- Langevin function .=.== 259 K
I++++ 267 K
a
(b)
Fig. 4. Universal function [M(H) -,yhrH]/Mz versus a = iii,H/kT for Fe,&, at T= 121 K (XI, 141 K (+I and 162 K (0) (a);
for Fe,,Zr, at T = 184 K (x) and 200 K (+) (b) and for FessZr,, at T = 259 K (x) and 267 K (+) Cc), together with the theoretical curves Ml/M, versus a = iii,H/kT for fine particles with interaction for interaction parameters b = iEi,AMs/kT = 1, 1.5, 2 and 2.5, and the Langevin function of the interaction-free model.
L.F. Kiss et al. /Journal of Magnetism and Magnetic Materials 13.5 (1994) 161-170 167
and xhf by simply drawing a straight line to the high-field portion of the M(H) curves. The fit is rather poor for intermediate magnetic fields be- tween 0.1 and 1.5 T. We show later that it can be significantly improved by taking into account the interactions between the cluster moments.
For an assembly of superparamagnetic parti- cles without interaction, M versus H/T is a universal function which is independent of tem- perature and particle anisotropy 1271. The M ver- sus H/T curves should also be corrected for the variation of M, with T [28] and for the magneti- zation contribution originating from Ahf. The cor- rected data ([M(H) - XhfHl/Ms versus M,H/T) for all the alloys investigated approach a univer- sal curve for T > T, and T,, where all the mag- netic clusters rotate freely. These quasi-universal curves allow us to determine ?‘& and F indepen- dently from the previous fitting procedure (using only the two insensitive parameters, M, and Ahf).
According to NCel’s model for superparamag- netic particles [29] 04 - ,yhf H )/MS for relatively high fields (mH x== kT) can be approximated as
MS M,H’ (3)
N and ??i, thus calculated from the high-field portion of the M(H) curve are much smaller and have much smoother temperature dependences (above T, or Tf> than those previously deter- mined essentially from a low-field fitting to the susceptibility.
3.2. Interaction between cluster moments
In the mean-field approximation H is replaced by H+AM’(H), where M’(H)=M(H)-X,,H
and A is the molecular field constant, so we get from (2) for H + 0 (when L(a) + a/3):
M(H) =M,
E,(H+AM’)
3kT
1 +
Xt,fH. (4)Expressing i+ii, in terms of the measured suscepti- bility A = M/H, we obtain:
(5)
where ??I: = 3kTCX - Xhf)/Ms is the magnetic moment without interaction (A = 0). Note that m, and A cannot be unambiguously determined from the measured A (i.e. if only low-field data are used).
For higher fields we have to solve the follow- ing two equations:
M’ kT
-=
MS
~ a’-
E-i,AM, i M’/M,=L(a’),
(6) (7)
where a’ = E,( H + A M’)/kT. The %?, H/kT = a and E,Am,/kT = b are dimensionless parame- ters for the magnetic field and the interaction, respectively. Figs. 4(a)-(c) show among others the theoretical M’/M, versus a curves for several b parameters, together with the Langevin function of the non-interacting case. The introduction of an interaction has a relatively high effect on the initial susceptibility but the congruency of the curves is only slightly affected at higher fields.
Supposing that the apparent large average magnetic moments (my> obtained from the first, low-field fitting are the consequence of the ne- glected interaction and using their more realistic values (??I,) from the second, high-field fitting, the molecular field constant A can be calculated according to (5). The dimensionless interaction parameter b can be readily obtained from A by definition.
Knowing the cluster parameters above T, or T,, the measured M(H) curves can be trans- formed into dimensionless master curves, (M - XhfH)/Ms versus a =Ei,H/kT. However, MS determined in the first fitting procedure was sys- tematically underestimated by several percent be- cause artificially large moments had to be used in the argument of the Langevin function to fit the susceptibility. (The larger the moment, the more rapidly the Langevin function saturates.) Correct- ing MS for this effect (MT) in (M - Xhf H)/M, and leaving a unchanged, we get the master curves shown in Figs. 4(a)-4(c) for Fe,,Zr,, Fe,,Zr, and Fe,,Zr,,, respectively. For compari- son, the theoretical M’ versus a curves for sev- eral values of the parameter b are also drawn.
The quantities derived from the fitting parame-
168 L.F. Kiss et al. /Journal of Magnetism and Magnetic Materials 135 (1994) 161-170
ters would be only slightly (and systematically) modified by this correction, therefore the original values will be used in the discussion. The mea- sured curves with calculated interaction parame- ters b between 1.7 and 2.7 lie in the stripe of the corresponding theoretical curves. The congruency of the measured curves with the theoretical ones for interacting fine particles is striking and indi- cates that the Fe-rich Fe,,,_,Zr, amorphous al- loys show superparamagnetic behaviour above the spin-glass temperature T, (x = 7) and the Curie temperature T, (x = 8-12). The only discrepancy at the shoulder of the curves (a = 2-6) may be attributed to the fact that the distribution of the moments was not taken into account in the calcu- lation of the theoretical curves.
Fig. 5 shows the temperature and concentra- tion dependence of the interaction parameters A and b for Fe,,Zr,, Fe,,Zr, and FessZr,, above their critical temperatures CT, or Tc). b, i.e. the ratio of the interaction energy to the thermal one, is the real measure of the interaction between the cluster moments (rather than h). It decreases with temperature and increases slightly with con- centration. The latter is consistent with the exper- imental fact that the magnetic properties of the Fe-rich Fe ,a0 ox Zr, amorphous alloy system ap-
- 100
z Fe&r12
: 50- Fe& r7 I
0
.z /I
x bib
Fig. 5. Temperature and concentration dependence of the interaction parameters A and b = E,AM, /kT for FeY3Zr7, Fe,*Zrs and FessZr,, (solid lines are only guides for the eye).
10000 ,
Fed
rl2\
lz 5000-
Fe92Zr8
a
I Fed r7
01
( I I1 ,I I I0 100 200 3’
T (K)
0
Fig. 6. Temperature and concentration dependence of the average number E of Fe spins in a cluster for Fe,,Zr,, Fe,,Zrs and FessZr,, (solid lines are only guides for the eye).
preach to those of a normal ferromagnet as x increases from x = 7 to 12.
The molecular field constant A increases with temperature for all the three alloys. A similar dependence of A on T was obtained by Beck [23]
based on fitting the M-H curves of crystalline Au,,Fe,, at different temperatures. Those curves can also be described by assuming that the alloy consists of finite superparamagnetic clusters.
Mossbauer studies on Fe,,Zr,, [30] and resis- tance fluctuation experiments in CuMn [31] also - support the observed A-T dependence.
The temperature and concentration depen- dence of the average number (p) of Fe spins in a cluster is shown in Fig. 6 for the three alloys above T, or T,. m is either practically constant (Fe,,Zr,) or decreases with temperature (the other two alloys). The increase in fl upon lower- ing the temperature can be associated with at least two mechanisms: (i) clusters polarize the spins in their vicinity, aided also by the slow increase in MS, and (ii) the coalescence of two or more small clusters into a big one. The large static moments found even about 60 K (Fe,,Zr,), 25 K (Fe,,Zr,) or 10 K (Fe,,Zr,,) above the respective critical temperatures CT, or T,) can be regarded as anomalous compared to the magnetic behaviour of normal ferromagnets near T, (e.g.
Ni [32]). Dynamic spin clustering is thought to
L.F. Kiss et al. /Journal of Magnetism and Magnetic Materials 135 (1994) 161-l 70 169
cause similar anomalies in this case but in a much narrower temperature range above T, [E = (T - T,)/T, I 0.041. The increase in @ with composi- tion reflects the compositional increase of the interaction mentioned above. Mijssbauer results on the same samples [26] confirm the existence of finite magnetic clusters, yielding a cluster size of approximately 2000 Fe atoms in Fe,,Zr, above Tg, which is similar to that found for N in this work.
4. Conclusions
It has been shown that melt-quenched amor- phous Fe,,,_,Zr, (X = 7, 8, 9, 10 and 12) alloys contain finite magnetic clusters of increasing av- erage size with Zr content X. These clusters are shown to be responsible for the superparamag- netic behaviour of the alloys observed above the spin-glass transition temperature T, (x = 7) or the Curie temperature Tc (x = 8, 9, 10 and 12) and also for the low-temperature anomalies of the dc susceptibility observed below T, (x = 7) or the re-entrant spin-glass transition temperature Tf (x = 8, 9 and 10). The interpretation is based on NCel’s theory for fine particles with a distribu- tion of blocking temperatures by taking into ac- count the interaction between the cluster mo- ments. The measured M(H) curves for three alloys (X = 7, 8 and 12) were transformed into universal master curves and were shown to fit to the theoretical curves of the model. The tempera- ture and concentration dependences of the clus- ter parameters correspond well to the results of other measurements.
Although this description uses a significant number of parameters, there is a well established feature of the experimental results: the magnetic properties of a-Fe,,_, Zr, (and of other widely different spin-glass alloys) are very similar to those of fine particles. It implies that a kind of granular magnetic structure (a not yet understood, per- haps fractal magnetic structure) is present which is not taken into account by the presently ac- cepted theories. This fact might have far-reaching consequences and should initiate new ap- proaches.
Acknowledgements
The authors are indebted to Dr I. Bakonyi, Dr J. Balogh and Dr D. Kaptis for valuable discus- sions, and to L. Bujdoso for preparing some sam- ples. They are also grateful to the referees of the paper who stimulated a thorough revision of the original version. This investigation forms part of the research programs OTKA-2933 and 4464.
References
111 121
131 [41 151 [61 [71
181 [91 1101 [ill 1121 [I31 [I41
H. Hiroyoshi and K. Fukamichi, Phys. Lett. 85A (1981) 242.
D.H. Ryan and J.M.D. Coey, Phys. Rev. B 35 (1987) 8630.
S.N. Kaul, Phys. Rev. B 27 (1983) 6923.
D.A. Read, T. Moyo and G.C. Hallam, J. Magn. Magn.
Mater. 44 (1984) 279.
W. Beck and H. Kronmiiller, Phys. Stat. Solidi. (b) 132 (1985) 449.
J.M.D. Coey, D. Givord, A. LiCnard and J.P. Rebouillat, J. Phys. F: Metal Phys. 11 (1981) 2707.
H. Hiroyoshi, K. Fukamichi, A. Hoshi and Y. Nakagawa, in: Proc. Int. Symp. on High Field Magnetism (Osaka, 1982), ed. M. Date (North-Holland, Amsterdam, 1983) p.113.
K. Shirakawa, S. Ohumna, M. Nose and T. Masumoto, IEEE Trans. Magn. 16 (1980) 910.
H. Hiroyoshi, K. Fukamichi, M. Kikuchi, A. Hoshi and T.
Masumoto, Phys. Lett. 65A (1978) 163.
K. Fukamichi, H. Hiroyoshi, M. Kikuchi and T. Ma- sumoto, J. Magn. Magn. Mater. 10 (1979) 294.
S.N. Kaul and M. Rosenberg, Phys. Rev. B 25 (1982) 5863.
J. Crangle and G.C. Hallam, Proc. R. Sot. (London) A 272 (1963) 119.
N. Saito, H. Hiroyoshi, K. Fukamichi and Y. Nakagawa, J. Phys. F: Metal Phys. 16 (1986) 911.
M. Ghafari, U. Gonser, H.-G. Wagner and M. Naka, Nucl. Instrum. Meth. 199 (1982) 197.
[15] S.N. Kaul, J. Phys.: Condens. Matter 3 (1991) 4027.
[16] S.N. Kaul and V. Siruguri, J. Phys.: Condens. Matter 4 (1992) 505.
[17] S.N. Kaul and Ch.V. Mohan, J. Appl. Phys. 71 (1992) 6090.
[18] S.N. Kaul and Ch.V. Mohan, J. Appl. Phys. 71 (1992) 6103.
[19] D.H. Ryan, J.O. Strom-Olsen, R. Provencher and M.
Townsend, J. Appl. Phys. 64 (1988) 5787.
1201 M. Gabay and G. Toulouse, Phys. Rev. Lett. 47 (1981) 201.
[21] L. Niel, Compt. Rend. 228 (1949) 664.
1221 E.P. Wohlfarth, Phys. Lett. 70A (1979) 489.
170 L.F. Kiss et al. /Journal of Magnetism and Magnetic Materials 135 (1994) 161-170 [23] P.A. Beck, Phys. Rev. B 32 (1985) 7255.
[24] B.D. Cullity, Introduction to Magnetic Materials (Ad- dison-Wesley, London, 1972) Q. 410.
[25] E. Kneller, in: Handbuch der Physik, ed. H.P.J. Wign (Springer-Verlag, Berlin, 1966) Q. 466.
[26] D. Kaptas, T. Kern&y, L.F. Kiss, J. Balogh, L. Granasy and I. Vincze, Phys. Rev. B 46 (1992) 6600.
[27] see Ref. 1241, Q. 411.
[28] see Ref. [25], Q. 480.
[29] see Ref. [25], Q. 479.
[30] S.N. Kaul, V. Siruguri and G. Chandra, Phys. Rev. B 45 (1992) 12343.
[31] M.B. Weissman, N.E. Israeloff and G.B. Alers, J. Magn.
Magn. Mater. 114 (1992) 87.
[32] J.S. Kouvel and M.E. Fisher, Phys. Rev. A 136 (1964) 1626.
