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The magnetic phase transition in amorphous ferromagnets and in spin glasses
240
Journal
of Magnetism
and Magnetic
Materials
North-Holland
INVITED
38 (1983) 240-252 Publishing
Company
PAPER
THE MAGNETIC PHASE TRANSITION 1N AMORPHOUS GLASSES M. FAHNLE,
G. HERZER,
Itnrrtur fir Phwh, Mu.x Plud
H. KRONMijLLER,
R. MEYER,
FERROMAGNETS AND IN SPIN
M. SAILE
Imtrtut fir Mer~rllfor~~/ru~~,~. 7000 Srurt,q~rr X0. Fed. Rep. (;
and T. EGAMI
The magnetic phase transition in materials with exchange disorder (amorphous fcrromagnets. spin gla~c.\) 1s dihcubacd. In the critical temperature range the behavior of amorphous ferromagnetic transition mctalLmctalloid glasses i> found to be similar to the one derived for a three-dimensional homogeneous H&e&erg fcrromagnet. The most prominent diffcrencc between disordered and homogeneous materials is manifested in a large temperature range of dc\iations from the mean field behavior beyond the critical region, as observed experimentally for the temperature depcndcncc of the linear susceptibility of amorphous ferromagnets and of the nonlinear susceptibi!ity of spin glasses. A molecular field theory with correlations in space and time is developed, which relates the deviations from the mean field behavior to the interplay between the temperature dependent thermal correlations in the spin bystem and the spatial fluctuations of the material. Application to dynamical processes (kinetic critical slowing down) is discussed.
1. Introduction Phase transitions in disordered materials have attracted much interest in the last few years. Special emphasis was given to systems with random distribution of magnetic and nonmagnetic atoms (site percolation problem, for a review see ref. [l]), random anisotropy [2,3], random magnetic field [3,4] and random exchange bonds [5,6]. In this paper we consider the last case, i.e. systems with spatial fluctuations of the exchange interactions. Roughly speaking, the phase transition in a magnetic material takes place if the thermal energy is comparable to the exchange coupling energy, and from this point of view it becomes clear that materials with exchange disorder should exhibit an unusual behavior near the phase transition. It will be shown both theoretically and experimentally that as a result of the exchange fluctuations there may be deviations from the mean field behavior in the temperature range, which is much larger than the range of real critical behavior in crystalline materials. If there are no or only few antiferromagnetic 0304-8853/83/0000-0000/$03.00
0 1983 North-Holland
couplings, such exchange disorder systems represent amorphous ferromagnets, which are distinguished by a parallel alignment of the magnetic moments. Examples are quenched or sputtered alloys preferably of composition T,,M,, (T = transition metal, M = metalloid as C, P, B, Si). Assuming that the exchange interaction JI,( R,,) between spins at sites i and j depends on the interatomic distance, R,,, the topological disorder in these materials (lack of atomic long range order) is a first source of exchange fluctuations. Furthermore, in materials with several kinds of transition metal atoms, for instance Fe and Ni, there is a different exchange interaction between Fe and Fe, Ni and Fe, and Ni and Ni, and, therefore, the more or less random arrangement of Fe and Ni atoms, the chemical disorder, is a further source of exchange fluctuations. When increasing the number of antiferromagnetic bonds in the system, there is a transition to the spin glass state with spatially random orientation of the magnetic moments [7]. Prototype spin glasses are the disordered alloy systems Fe Au or CuMn with low concentrations of magneticatoms. -
M. Fiihnle et (11./ Phase frmsitiotl in umorphous ferromugnets and spit! glasses
In these materials the size and the sign of the RKKY exchange interactions oscillate on a scale, which is smaller than the intermoment distance, thus producing both ferromagnetic and antiferromagnetic couplings if the magnetic moments are distributed randomly. It has been shown theoretically [5,8] that in the ferromagnets with spatially random exchange interactions the critical behavior is identical to the one of the corresponding homogeneous material, if the specific heat exponent a of the homogeneous materials is negative. For positive values of (Y, however, the critical exponents describing the phase transition should be modified (Harris criterion [5]). Intuitively this simple rule may be well understood, taking in mind that a distribution of exchange interactions means a distribution of local critical temperatures. For positive values of OLthe specific heat diverges locally if the temperature approaches the local critical temperature. The spins then are heated up locally, and the heat is transferred to the neighborhood via exchange couplings, thus changing the critical behavior of the whole system. The ferromagnetic transition metal-metalloid glasses discussed in this paper may be well represented by the three-dimensional Heisenberg model [9], for which the exponent (Yof the homogeneous material is negative. It will be shown in section 2 that our experimental results for the susceptibility exponent y are indeed consistent with the Heisenberg value of y = 1.387 for T + T,, in agreement with the theoretical prediction. The difference between amorphous and crystalline materials is manifested in a totally different temperature dependence of the susceptibility outside the real critical region of the phase transition. Specifically, in amorphous materials the temperature dependent effective exponent y(T) as introduced by Kouvel and Fisher [lo] exhibits a non-monotonic temperature dependence (section 2), in sharp contrast to the case of crystalline materials which are distinguished by a monotonically decreasing y(T). In sections 3 and 4 we develop a theory, which is able to explain this unusual behavior of amorphous ferromagnets. It is well known that in general the renormalization group theory gives exact results only for homogeneous four-dimen-
241
sional systems. To get information about real three-dimensional systems approximate calculations must be performed for non-integral dimensions d = 4 - z with c -X 1, and then the results have to be extrapolated to d = 3 (C = 1). This procedure is justified for the critical temperature range of homogeneous systems by the empirical experience that the so-obtained critical exponents agree well with theoretical observations [ll]. However, to us it seems to be extremely difficult to extend these calculations to the temperature range outside the real critical region (instead of T -+ T,) in amorphous (instead of homogeneous) three dimensional (instead of d = 4 - E) systems. An attempt in this direction was made by Sobotta and Wagner [12] using a special type of renormalization theory with an e-expansion to O(e). However, their theory yields a monotonically decreasing effective exponent y(T) and thus fails to reproduce the experimentally observed non-monotonic behavior of y(T), which is the most important feature of the phase transition in amorphous ferromagnets. We therefore did not try to approach the problem by a renormalization group analysis, but we have developed in a phenomenological approach a mean-field theory with correlations in space and time, which is able to explain the experimental observations. Results of the theory are given in section 4. In section 5 we extent our correlated mean field theory to the case of spin glasses and show that a similar large temperature range of deviations from the mean field behavior may occur in these materials.
2. The paramagnetic ferromagnets
susceptibility
of amorphous
To illustrate the unusual behavior of amorphous ferromagnets outside the real critical temperature range, fig. 1 shows the temperature dependence of the inverse paramagnetic zero field susceptibility, x;‘(T), for the amorphous ferromagnetic alloy Fe,,Ni,,B,, (details of the experiments as well as the Arrott plots M* vs. H/M, which deviate strongly from the straight lines of crystalline materials are published elsewhere). For comparison, data for crystalline Ni as obtained by Kouvel
f / 70 *
4or
/’
35
/’
30
1.1-. ,’
t
/I’
1’
/I
/’
J
“//’ I
0.00
0.05
7’
0.10
/I’
x crystalline
Ni
o amorphous
Fe,0NI,,B2L
I
I
I
I
I
0.15
020
0.25
0.30
0.35
T/T,
-1
O.&O
-
Fig. 1. Temperature dependence of the paramagnetic zero field susceptibility for amorphous Fe,,NiS,B,, and crystalline Ni. The data for Ni have been multiplied by a constant factor in such way that the Curie-Weiss line of Ni is parallel to a straight line through the last three data points for the amorphous material. The dashed line represents the Curie-Weiss line of Ni.
and Fisher [lo] are also presented in the figure. For crystalline Ni there is an appreciable curvature of the plot in a small temperature range, and for high temperatures the usual Curie-Weiss behavior (mean field behavior) is observed. This temperature range of curvature of the plot is much larger for the amorphous material, and there is no Curie-Weiss behavior in the whole temperature range 0 < T/T, - 1 G 0.35. For T approaching T, the temperature dependence of the susceptibility is usually described by x;‘(T) - (T - T,)y with y = 1.3866 for a Heisenberg ferromagnet [13]. Of course this simple power law is only valid for the real critical temperature range of the phase transition. It is not able to describe the gradual transition from the critical behavior to the high-temperature mean field behavior. To characterize the temperature dependence of the susceptibility in the whole temperature range, Kouvel and Fisher [lo] have introduced an effective exponent y(T) defined by
Thereby,
d ln( T/T,
T
0
1.6
amorphous
Fe,,
- 1)
= (T-
in a Heisenberg
q)x,,z. system
Ni,,
Bz4
1.5
‘,3-\ , / , , , crystalline
1.2
0
0.05
0.10
0.15
Ni
0.20
0.25
0.30 T/T,
dlnx;l Y(T) =
exponent y(T) should be y = 1.3866 for T -+ T,, and for very large temperatures y(T) should approach the mean field value of 1. Within the experimental accuracy the values of y(T,) are close to the Heisenberg value of 1.3866, both in crystalline Ni and in amorphous as shown in fig. 2. We thus observe Fe,,Ni,,B,,, the same type of critical behavior in the homogeneous and in the disordered Heisenberg system, in agreement with the theoretical prediction [5,8]. However, whereas the effective exponent y(r) decreases monotonically in the crystalline material, this quantity initially increases in the amorphous system and runs through a broad maximum before decreasing again to the mean field value of 1, obviously a totally different behavior (fig. 2)! Similar results for x;‘(T) and y(T) are found by Gaunt et al. [14] for Fe,zNi,,Cr,,P,,Bc with y(T,) = 1.35 + 0.03, by Kaul [15-171 for with y(T,) = 1.33 + 0.05, Fe,,Ni,,Pi,B, FeJONiAOP,,$ with y(T,) = 1.31 k 0.02 and Fe,,Ni,,Cr,,P,,$ with y(c) = 1.33 f 0.05, by Kaul and Rosenberg [18] for Fe,,Ni,,B,,Si with y(T,) = 1.35 f 0.04, by Dey and Luborsky [19] for Fe,,Ni,,B,,Si with y(c) = 1.36 + 0.04, and by Poon and Durand [20] for amorphous Gd,, Au 20 with y(T,) = 1.29 + 0.05. Yamada et al. [21] as
-I
0.35
(
-
(1)
this
effective
Fig. 2. The temperature dependent effective exponent y(T) as defined by eq. (1) for amorphous Fe,,Ni,,B,, and crystalline Ni.
M. Fiihnle et ul. / Phuse irunsition in umorphous ferrcjmugners und spin glusses
well as Mizoguchi and Yamauchi [22] have carried out an extensive analysis for the critical region only. Yamada et al. [21] obtained y(T,)= 1.3 + 0.05 for Fe,,P,,C,, a value close to that found by Poon and Durand [20] for Gd,,Au,,, whereas Mizoguchi and Yamauchi [22] obtained y( T,) = 1.34 f 0.025 for CO,~B,,P,,. Other authors tried to fit the temperature dependence of xii(T) in a large temperature range approximately by one unique value of y. Because this y most probably is some kind of average value of the temperature dependent exponent y(T) they have to use numbers which are appreciably larger than the Heisenberg value: to describe the temperature dependence of x;‘(T) for Fe,,Ni,,Cr,,P,,$ outside the real critical region, Figueroa et al. [23] used y = 1.71 f 0.06, which corresponds to the maximum value of y(T) found by Gaunt et al. [14] for the same alloy. Meyer and Kronmtiller [24] performed an approximate fit for x;‘(T) in a large temperature range, yielding y = 1.63 + 0.07 for Fe,,Ni,,B,,P,. Beckman et al. [25] found y = 1.58 & 0.1 for Fe,,Ni,,P,,$Al,, and Malmh;ill et al. [26] used a unique value of y = 1.7 + 0.1 for the amorphous ferromagnetic alloy Fe,,Ni,,P,,$Si,. Most recently Maletta et al. [27] found the non-monotonic behavior of y(T) for the first time in a disordered crystalline alloy, namely in the alloy system Eu,,,Sr,,S, with y(T,) = 1.38 (Heisenberg value) and a maximum value of y = 2.2. As will be shown in the next section, the nonmonotonic temperature dependence of y(T) is an effect of exchange fluctuations, which are present both in amorphous and in disordered crystalline alloys. Therefore, it is not surprising that the same non-monotonic behavior of the effective exponent y(T) is found in both systems. Tranchita and Claus [28] as well as Aldred and Kouvel [29] observed unusual large but temperature independent values of y for diluted disordered CuNi and FeCr alloys. However, in these materials the long range ferromagnetic order is due to the interaction of large polarization clouds induced by Ni-rich (Fe-rich) local regions. It is not surprising that such giant moment systems show a different behavior near the phase transition. The values of the exponent /3 for various amorphous systems are close to those values ob-
243
served f r crystalline ferromagnets, although they are cons’stently slightly larger (for a review see ref. [17]). Fu thermore, the scaling law y = p(6 - 1) is satisfied 1 approximately in most amorphous materials [17]. Obviously in the critical temperature range the behavior of amorphous ferromagnets is very similar to that one of corresponding homogeneous materials. The most prominent difference between amorphous and crystalline materials is manifested in the temperature dependence of x;‘(T) and y(T) beyond the real critical temperature range.
3. Correlated molecular field theory In this section we describe our correlated molecular field theory [30-341, which is able to explain the unusual behavior of amorphous ferromagnets beyond the critical region as discussed in section 2 (a theory which is basically very similar to our theory has been developed later by Kaneyoshi [35]). By means of theoretical investigations it has been shown [36,37] that spatially random magnetostatic and magnetocrystalline fluctuations (anisotropy fluctuations) in transition metal base amorphous ferromagnets as well as internal stresses (which determine the properties of the low temperature magnetization curve of these material,s [9,38]) play only a minor role outside the critical region (their influence on the critical behavior is discussed for instance in refs. [l-3]). The unusual temperature dependence of the susceptibility is really an effect of the spatial fluctuations of the exchange interactions. It is therefore sufficient to consider the Heisenberg Hamiltonian Ei=
-c
C.@;(f)s;(l), i
(2)
J’
which contains the fluctuating exchange interactions J,, and the time dependent spin operators $(t) and 4(t). To illustrate the basic physical idea of our theory we consider first the case of a crystalline Ising ferromagnet. At zero temperature all spins are parallel because of the ferromagnetic exchange interactps. When increasing the temperature single spins are turned around because of the
interaction with the random thermal agitations of the heat bath (phonon system), but at each time most of the spins are parallel because of the ferromagnetic exchange interactions (fig. 3). Thereby the thermal average ($) of a spin at site i is non-zero (ferromagnetic region), but there are thermal fluctuations &$(t) of the spin. For temperatures much larger than the Curie temperature T, the thermal energy is much larger than the exchange interaction energy. Then the spins react more or less independently on the thermal agitations of the heat bath, and there are no correlations between the spin fluctuations S$,( t) and 8$(t) at different sites i andj (fig. 3). The thermal average ($,) is zero (paramagnetic region). When approaching T, there is a competition between the tendency to order spins because of the ferromagnetic exchange interactions and to disorder spins because of the random thermal agitations of the heat bath. As a consequence, the system creates large clusters of spins (fig. 3) in each of which the ferromagnetic ground state is preserved for a short time (intracluster excitations are neglected). The spins in the clusters are strongly coupled together and react cooperatively, more or less like superparamagnetic particles. The mean size of the clusters is given by the thermal correlation length t(T), which diverges when approach-
ing T,. It is well known, for example, from renormalization group calculations (for a review, see refs. [39,40]) that it is the nonlinear interaction between these critical correlated spin fluctuations of very large size, which is responsible for the curvature of the xOl(T) plot in the real critical region of the phase transition of crystalline materials. In amorphous materials, however, there is a curvature of the plot in a temperature range, for which the thermal correlation length t(T) is of the order of a few nearest neighbor distances only, far away from the validity of the renormalization group analysis of the real critical spin fluctuations. Therefore, one could tentatively assume that in amorphous materials the correlated spin fluctuations play only a minor role in this temperature range, and we want to neglect them totally for the moment by applying a mean field theory (molecular field theory). Indeed, within the framework of a molecular field theory the interaction energy of a spin at site i with all the spins at sitesj is approximated by the interaction of this spin with the thermal averages (3;). Obviously the application of such a molecular field approximation means the neglection of the correlations between the spin fluctuations at different sites, because the molecular field Hamiltonian
en,= - c CJ,,(s;>W
I/I T = T_
Fig. 3. Schematic representation of snapshot Ising ferromagnet at various temperatures.
pictures
of an
(3)
contains only the fluctuations of the spins at sites i, whereas the time-fluctuations of the spins at sitesj are averaged out. In crystalline materials the situation is very simple because due to the long range order the thermal averages of all spins at sitesj are the same. In amorphous materials, however, there are spatial fluctuations of the local magnetization because of the spatially random exchange interactions. Two different methods to take into account these exchange fluctuations are discussed in the literature (for a review, see ref. [41]). In the so-called inhomogeneous molecular field theory spatial fluctuations of the magnetization with all wavelengths are allowed by inserting into the molecular field Hamiltonian indeed the local inhomogeneous quantity (3,) as in eq. (3). Within the framework of this
M. Fiihnle
et ul. / Phuse tmnsition
inhomogeneous theory an increase of the critical temperature is obtained [37,41,42] as shown in fig. 4 as a result of the spatial fluctuations of the exchange interactions. In contrast, high temperature expansions, the Bethe-Peierls-Weiss approximation and the constant-coupling approximation yield a decrease [43] of T, for spatially random exchange interactions, a result which has been confirmed recently by Monte Carlo computer simulations [44]. Furthermore, the curvature of the x;‘(T) plot is opposite in sign to the curvature of the experimentally obtained plot, which is also shown schematically in fig. 4. Handrich [45] has argued that it does not make sense to totally neglect on the one hand the thermal fluctuations of the spins at sites j by performing a molecular field approximation, but to take into account accurately the spatial fluctuations of the local magnetization on the other hand. He has inserted into the molecular field Hamiltonian the volume average (S) of the magnetization instead of the local inhomogeneous magnetization (the symbol means the volume average). Within the framework of this homogeneous molecular field theory the usual Curie-Weiss behavior is obtained in the paramagnetic region, and the critical temperature is not affected by the exchange fluctuations (fig. 4). Obviously
both the homogeneous
and the inho-
0.7 0.6 -
T/T,-1 Fig. 4. Results for x0(T) as obtained by the homogeneous molecular field theory (full line) and the inhomogeneous theory (dashed line). The dashed-dotted line is a schematic representation of the experimental data. For the calculations we have inserted the material parameters of Fe,,Ni,P,,$ as discussed in section 4 with A = 0.5 and I, = 1.5d.
in umorphow
ferromugnets
und spin glusses
245
mogeneous molecular field approximation are not able to reproduce the correct physical behavior. However, each theory which interpolates between these two extreme cases, i.e. which exhibits characteristic features of the homogeneous theory near T, and characteristic features of the inhomogeneous theory far away from T, must necessarily reproduce the correct physical behavior (fig. 4). What is the physical justification for the application of such an interpolating theory? So far we have totally neglected the influence of correlated spin fluctuations because we have applied a molecular field theory. These correlated spin fluctuations have the form of clusters, in which the spins show a strong cooperative behavior (fig. 3). For T approaching T, the mean size of the clusters diverges, and then the spin system as a whole reacts cooperatively and does not feel anything from the spatially random exchange fluctuations. Therefore, the main features of the phase transition in amorphous Heisenberg systems are very similar to those in crystalline materials: in both cases we observe a sharp transition temperature T,, and in both cases the susceptibility exponent y for T + T, is determined by the nonlinear interactions between the critical spin fluctuations. When increasing the temperature the mean size of the clusters decreases. Then the nonlinear interaction between the correlated spin fluctuations is negligibly small, and in crystalli-ne materials we observe the usual Curie-Weiss behavior. Nevertheless, there are still correlated spin fluctuations of smaller size (noncritical correlated spin fluctuations), in which the spins show a strong cooperative behavior. These noncritical spin fluctuations must be taken into account in amorphous ferromagnets as long as the thermal correlation length t(T) is larger than or comparable to the correlation length of the exchange interactions, which is about the nearestneighbor distance. The exchange fluctuations try to produce spatial fluctuations of the magnetization. Because of the strong cooperative behavior of the spins in the clusters, however, there are no magnetization fluctuations with wavelengths shorter than the mean size of the clusters. All the other magnetization fluctuations are averaged out. For T + T, the quantity t(T) diverges, and then we have about the same value of the magnetization
in the whole sample as in the homogeneous molecular field theory. For very large temperatures the thermal correlation length t(T) is very small and there are spatially random fluctuations of the magnetization as in the inhomogeneous molecular field theory. It is this transition between a homogeneous situation for T + T, and an inhomogeneous situation far away from T,, which is responsible for the unusual behavior of amorphous ferromagnets. The large temperature range of curvature of the xi’(T) plot thus is not a real critical phenomenon, but it is the result of the interplay between the temperature dependent magnetic correlations in the spin system and the spatial fluctuations of the exchange interactions. For a quantitative formulation we insert into the molecular field Hamiltonian a spatially correlated magnetization, which in a continuum approach ((4.) + S(r’); M(r’) = gpBS(r’) with g = Lande’s g-factor and pa = Bohr’s magneton; JI, + J(r, r’)) is given by (see refs. [30-341)
namics of the whole clusters rather than by the high-frequency intracluster excitations, and it is therefore limited by the finite “lifetime” of the clusters. For a dynamical generalization of the theory we assume that there are no relevant magnetization fluctuations with wavelengths shorter than the mean size t(T) of the clusters and with frequencies higher than the inverse “lifetime” of the clusters. We then insert into the molecular field Hamiltonian a magnetization with correlations in space and time [32-341
Mb’,
M( %rr
=/
- Y”, [(T))M(r”)d3Y.
(4)
Here f( r’ - r”, t(T)) is an integral kernel which averages out the irrelevant fluctuations of the magnetization M(r”). For T -+ T, the integral kernel is a slowly varying function in space; and, therefore, the correlated magnetization is about constant in the whole sample as in the homogeneous molecular field theory. In contrast, for very high temperatures the kernel f( r’ - r”, E(T)) is well localized at r’, and we have the spatial fluctuations of the magnetization as in the inhomogeneous molecular field theory. To determine the dynamical evolution of the spin system we have to distinguish between two different time scales: on the one hand there are high-frequency single spin excitations within the clusters, which are irrelevant for the macroscopic evolution of the whole system because they are averaged out. On the other hand the cluster as a whole may be turned around spin by spin by the random thermal agitations of the heat bath. The larger the clusters, the larger is the so-defined “lifetime” of the clusters. Macroscopically the time evolution of the system is determined by the dy-
- r”, t - t’)
r-z
x M(
t’)dt’ d3/‘.
r”,
(5)
Here the integral kernel g(r’ - r”, t - t’) averages out the irrelevant fluctuations of the magnetization with high frequencies and short wavelengths. As a possible ansatz for the Fourier transform of g(r’ - r”, t - t’) we insert the real part of the dynamical susceptibility x(k, w) g(k,
f(+
t>corr =/I ‘+rg( rl
Re[x(k,
a> =A
w>l,
(6)
where X(/C, w) is calculated with the conventional mean field theory [46] with a local expansion of the exchange interaction “”
w’=
_i~~~:A
+ CkZ
(7)
The quantity 7 determines the time scale for single spin excitations, and A is the Ginzburg-Landau parameter which has the form [41,47] A =A’(T/T,‘-
1).
(8)
a = A’T/Tco, A’ = L
3kT.O
(9) s(s+
1).
(10)
4‘wrd* Here c is the concentration of magnetic ions in the alloy, S denotes the spin quantum number, and T’,‘,O is given by (V = volume of the sample) kT,‘=fS(S+l); X Jl vv
J( r,
r’)d3r’ d3
r”.
.
M. Fiihnle et ul. / Phase trunsition
For the stiffness
X
I/vv .I(
r,
parameter
in umorphous
C we insert
r’)( r - r’)2d3r d3r’.
(12)
The connection between the integral kernel g( r’ r”, t=- t’) of the dynamical theory (eq. (5)) and the kernel f( r’ - r”, E(T)) of the static theory (eq. (4)) is given by 03) with the correlation length size of the spin clusters)
(describing
the mean
t=Jc/A.
#.l
_
).tr,
<)
_
is the Fourier transform correlation function
of the
4;t2 exd- Ir’ -,J-“l/C) ) 1+--r
1
(15)
which exhibits the general features of the integral kernel f discussed above. By evaluating the Brillouin function of the molecular field theory [30,34,41,47], we then obtain for T > T, the following linear equation of motion for M(r, t) ay$M(r,
247
and spin glusses
geneous and instantaneous molecular field theory is reproduced. This theory still is a mean field theory which explicitly neglects the correlations between the thermal spin fluctuations. However, it takes into account implicitly the effect of the correlated spin fluctuations by the introduction of the correlated magnetization M( r’, t ),,,, into the molecular field. Of course the theory is not able to describe correctly the real critical behavior for T + T,. The theory holds as long as the nonlinear interaction between the spin fluctuations [39] is not the dominant one. It improves the conventional theory for a range of temperatures, in which the correlated spin fluctuations are small enough but still important, and it approaches the conventional theory for very large temperatures.
(14)
Apparently f(k) Ornstein-Zernike f(
ferromugnets
t) = -aM(r,
J(r,
r’)M(r’,
t) t) corrd3r’ + H( r,
t
).
06) Here the second term on the r.s. is the correlated molecular field with M(r’, t),,,, given by eq. (5), and H(r, t) is the external magnetic field. This equation is a linear integrodifferential equation with spatially random fluctuating integral kernel, which may be solved approximately by special statistical methods discussed in full detail in refs. [30,34]. By inserting into the molecular field expression of eq. (16) the local, instantaneous magnetization M(r’, t) instead of the correlated magthe conventional inhomonetization M( r’, t),,,
4. Results of the theory Inserting typical values for the material parameters A’, C and Tco (specifically, we have used the material parameters of amorphous Fe,Ni,oP1,Bh [37,30]) we obtain from eq. (16) for the temperature dependence of the paramagnetic zero field susceptibility x0(T) = a~/aNI,,, the results shown in fig. 5. Obviously there is a large temperature range of curvature of the plot due to the effect of exchange fluctuations j( r, r’) = J( r, r’) - J(r, in agreement with the experimental observation (fig. 1). The parameter A is a measure of the relative fluctuation strength of the exchange interactions and is given by
(17)
A=/=, S(r)
=/j(r,
(18)
r’)d3r’//J(r,d3r’.
of the exWe assume a weak spatial correlation change fluctuations S(r) described by the correlation function r,,
( r - r’) = A2 exp[ - (r - r’)‘//,‘]
.
09)
To obtain the results shown in fig. 5 we have inserted for the correlation length 1, = 1.5d, where d is the nearest-neighbor distance. The larger the
(A’X0)_’ t 0.4 -
/ /= A 0 0.3 0.4
0.3 -
0.5 0.6
0
0.1
0.2
0.3 T/T:-1
Fig. 5. Results for ,y~ ‘(T)
0.4 -
for various values of A.
value of A (fig. 5) and/or the value of the correlation length I,, the larger is the temperature range of curvature of the plot. In our theory there is an enhancement of T, (defined by the divergence of x0(T)) for A > A, =/T 2C/AI, , as shown in fig. 5. In contrast, high temperature series expansions [43] and Monte Carlo computer simulations [44] yield a decrease of T, for spatially random fluctuations of the exchange interactions 4,. For spatially correlated fluctuations no theoretical results are known. In our theory this enhancement of T, for A > A, is removed when replacing in the integral kernel g(k, w) (eqs. (6)-(7)) the quantity A by the unknown susceptibility x;‘(T). The replacement means that the temperature dependence of the correlation length t(T) is no longer introduced ad hoc as in eq. (14) but is calculated self-consistently according to t(T) = /c/xol( T) . This relation between the temperat‘ure dependence of ((7’) and x;‘(T) holds exactly only for a mean field theory, but it is fulfilled approximately also for the critical temperature range of a three-dimensional Heisenberg ferromagnet (the critical exponents for x0 and 5 are y = 1.3866 and Y = 0.7054). The curvature of the xi’(T) plot obtained by this self-consistent modification of our
theory is smaller than in the original theory, and we have to insert appreciably larger values for A and/or I, to yield a similar large temperature range of curvature. The temperature dependent effective exponent y(T) as defined by eq. (1) starts at the mean field value of 1 and then runs through a maximum with increasing temperature (fig. 6) in qualitative good agreement with experimental observations (fig. 2). Of course our mean field theory is not able to reproduce y = 1.3866 for T + T,, because we neglect explicitly the nonlinear interactions between the critical spin fluctuations. The maximum of y(T) is more pronounced and it increases and shifts to lower values of T with increasing values of A (fig. 6) and/or I,. When performing the self-consistent version of our theory this maximum shifts slightly to larger values of T with increasing I, (fig. 7) and/or A. However, the maximum of y then is smaller and occurs at higher values of T (fig. 7) than in the original theory. According to fig. 5 one could tentatively try to describe the temperature dependence of x;‘(T) for instance for A = 0.5 and (T/T,’ - 1) > 0.3 by a Curie-Weiss law x;‘(T) - (T - T,), thus defining a paramagnetic Curie temperature Tp > T, for the amorphous material. However, when increasing the temperature further and further, the theoretical plot runs through an inflection point and approaches very gradually the dashed Curie-Weiss line with Tp = T,. For (T/T,” - 1) = 1.65 (4.35) the relative deviations still are 5% (1%). Therefore the definition of a paramagnetic Curie temperature T,
T/T,'-l-
Fig. 6. Results for y(T)
for various values of A. with I, = 1.5d.
M. Fiihnle et ul. / Phase irunsirion
c
in umorphour
1.8
1,=3Sd
1.7
,/'Y. '. i/ '.\
1.6
249
und sprn giusses
1.81
1.9
t
ferromogneis
A =0.45 j 1,=1.5d
1.6 l.s-p,
0
0.2
0.4
0.6 0.8 T/Tco-1 -
1.0
Fig. 7. Results for y(T) as obtained by the self-consistent version of our theory for various values of /,, with A = 0.5.
+......* 1.4~: /*** \ **.....***_ \ 5.*... I ;** \ *... \ a.... .r..!T' \ 1.3;1 *.*......** '\ I:' W... \\ -... 1.2$ '. ',_T'JT' Ii --._ 1.1 5 --_ 1.0 0
I 0.1
I 0.2
I 0.3
I 0.4
0.3
T/T," -1 -
for amorphous ferromagnets is more or less arbitrary, and it does make sense only if specifying simultaneously the temperature range used for the linear fit of x;‘(T). It is a feature of the effective exponent y(T) = (T- T,) x0 dx;‘/dT that it deviates rather strongly from the mean field value of 1 for (r/r,’ - 1) > 0.3, even though for instance for A G 0.5 the curvature of the xi’(T)-plot is very small in this temperature range (compare figs. 6 and 5). To remove this disadvantage we introduce another effective exponent y’(T) by
(20) This new temperature dependent exponent y’(T) exhibits the same qualitative features as y(T) (and the same qualitative behavior when changing A and/or I,), but the maximum of y’(T) is larger and occurs at lower values of T, and for high temperatures the absolute values of y’(T) are smaller than the values of y(T) (fig. 8). As an example for the application of the dynamical theory we calculate the relaxation of the magnetization when suddenly switching off an external magnetic field [32-341. Within the framework of the conventional molecular field theory an exponential relaxation of the magnetization is obtained. The relaxation time is temperature dependent and diverges when approaching T, according to 7 - (T - q)-A~~ with A,, = 1, i.e. for T + T,
Fig. 8. Comparison between the two effective exponents y(T) (eq. (1)) and v’(T) (eq. (20)) as obtained by the nonself-consistent version of our theory.
the system needs more and more time to react on an external perturbation (critical slowing down). More sophisticated theories, which take into account explicitly the nonlinear interactions between the critical spin fluctuations once more yield an exponential relaxation of the magnetization with time, but now the exponent A,, is larger, even larger than the static susceptibility exponent y(“kinetic” critical slowing down). For instance, a high-temperature series expansion of Yahata [48] gives A,, = 1.4 for the kinetic Ising model on a simple cubic lattice. With our correlated molecular field theory [32-341 we obtain A,, = 1.5 for T + T,, both in crystalline and amorphous materials, in good agreement with Yahata’s value of A MM = 1.4. For amorphous ferromagnets the values of the relaxation time 7 are enhanced as an effect of the exchange fluctuations. For details of the calculations, see ref. [34].
5. The nonlinear susceptibility of spin glasses When increasing the number of antiferromagnetic bonds in the system, there is a transition from the ferromagnetic ground state to the non-
magnetic spin glass state. For a symmetric distribution of exchange interactions J,, (the case of preferably ferromagnetic or antiferromagnetic couplings is discussed in ref. [49]) there is a spatially random orientation of the spins in the spin glass ground state, which may be described by the following general formula m=
s,,.
(21)
It is well known that experimentally the spin glass state may be characterized [50] by a cusp in the linear susceptibility x0(T) at the freezing temperature T,. So far, however, it is totally unclear whether this cusp is an indication of a real equilibrium phase transition with strong spin correlations as in the case of ferromagnets, or if the relaxation times in the system are so large that the cusp at T, is only a nonequilibrium phenomenon (for a review of this problem, see ref. [6]). For instance, it is argued that the spin glass ground state is highly degenerate and that the system moves steadily and slowly from one ground state to the other, so that there are no spin correlations at all when averaging over an infinitely long time. Indeed Monte Carlo simulations yield an equilibrium phase transition to the spin glass state only at zero temperature (Tf = 0) for spin glass models with short-range interactions, both for Ising, XYand Heisenberg spin glasses in 2 and 3 dimensions. However, experimentally one does find a cusp at nonzero T,, and there is theoretical [51] and experimental evidence that there are strong temperature dependent spin correlations near T,, at least during the time of the measurement. The most striking evidence for these temperature dependent spin correlations near T, is manifested in the experimentally observed [52] divergence of the nonlinear susceptibility, which (for T > T,) may be written as [53] x2(T)
- 1/T3~(s,s,)?.
(22)
‘.I
This divergence of X2(T) presence of correlated spin size of which is very large described by a temperature length t(T). Assuming that
clearly indicates the fluctuations, the mean near T, and may be dependent correlation the results of the above
discussed Monte Carlo simulations are valid for the real spin glass systems in nature, this must be interpreted as a nonequilibrium phenomenon where the system remains in a specific valley of the phase space for the time of the measurement. The correlated thermal spin fluctuations around this valley then once more have the form of spin clusters of size t(T), in each of which the specific ground state configuration is preserved for a short time. By taking into account anisotropic energy terms in the Hamiltonian this “trapping” effect of the system in a specific valley of the phase space is strongly enhanced [51]. If the time of the measurement is smaller than the “trapping” time of the system in one specific phase space valley, we obviously have the same interplay between the temperature dependent thermal correlations in the spin system and the spatial randomness of the exchange interactions as in the case of amorphous ferromagnets. Of course in the case of spin glasses this interplay has no effect on the temperature dependence of the linear susceptibility (23)
When inserting the general spin glass formula, eq. (21) into eq. (23) a simple Curie law, x0(T) - l/T, is derived, which is indeed observed in spin glasses without a preferable sign of the exchange couplings (in systems with preferably ferromagnetic or antiferromagnetic exchange couplings strong deviations from the Curie-law occur [49]). However, one might observe this interplay for the temperature dependence of the nonlinear susceptibility x2(T), which contains the squares of the correlation functions instead of the correlation functions (eq. 22). Within the framework of a simple molecular field theory Suzuki [54] has derived xz( T) - (T T,)-u. with y, = 1, and he has pointed out that the exponent y, is of central importance for the theory of spin glasses. Experimentally, Barbara et al. [52] found a large value of y, = 3.6 &-0.6 in a very large temperature range. In contrast, Monod and Bouchiat pointed out that y, might depend on temperature [55], and they found [56] ys-values
M. Ftihnle
et al. /
Phuse trunsition
between 1 and 2 for T, < T G 2Tr. Deviations from a simple power law dependence of x1(T) were also reported by Chikazawa et al. [57]. We think that a possible temperature dependence of y, in a very large temperature range has nothing to do with a real critical phenomenon, but is once more the result of the interplay between the temperature dependent thermal correlations in the spin system and the spatial randomness of the exchange interactions. Because of the close analogy to the case of amorphous ferromagnets we therefore have generalized our correlated molecular field theory to the case of spin glasses [58,59]. We have shown that the temperature dependence of the nonlinear susceptibility x z (T) should be described by a temperature dependent effective exponent y,(T) = d ln(x;‘(T)/d ln(T/T, - l), which has the same qualitative features (figs. 6-8) as the corresponding effective exponent y(T) (eq. (1)) in amorphous ferromagnets. For details of the theory, see refs. [58,59]. Amorphous ferromagnets and spin glasses thus show to some extent a universal behavior when we compare the linear susceptibility x0(T) of amorphous ferromagnets and the nonlinear susceptibility x1(T) of spin glasses.
Acknowledgement One of the authors (M.F.) is indebted to Prof. S.-K. Ma from the Department of Physics and the Institute for Pure and Applied Physical Sciences at University of California, San Diego, La Jolla, for helpful discussions.
References 111D. Stauffer, Phys. Rep. 54 (1979) 2. VI A. Aharony and E. Pytte, Phys. Rev. Lett. 45 (1980) 1583. I31 A. Aharony, J. Magn. Magn. Mat. 31-34 (1982) 1432. I41 Y. Imry and S.-K. Ma, Phys. Rev. Lett. 35 (1975) 1399. 151 A.B. Harris, J. Phys. C 7 (1974) 1671. WI K. Binder and W. Kinzel, Lecture Notes in Physics 149 (1981) 124.
I71 K. Fischer, Phys. Stat. Sol. (b) 116 (1983) 357. PI U. Krey, Phys. Lett. 51A (1975) 189. I91 H. Kronmtiller, M. F%hnIe, M. Domann, H. Grimm,
R. Grimm and B. Groger, J. Magn. Magn. Mat. 13 (1979) 53. WI J.S. Kouvel and M.E. Fisher, Phys. Rev. 136 (1964) A1626.
in mnorphous ferronwgnets
[ll]
D.J. Wallace,
und spin glusses
in: Phase Transitions
251
and Critical
Phenom-
ena, vol. 6, eds. C. Domb and M.S. Green (Academic Press, London, New York, San Francisco, 1976) p. 293. [12] G. Sobotta and D. Wagner, J. Magn. Magn. Mat. 15-18 (1980) 257. [13] J.C. Le Guillou and J. Zinn-Justin, Phys. Rev. Lett. 39 (1977) 95. 1141 P. Gaunt, SC. Ho, G. Williams and R.W. Cochrane, Phys. Rev. B23 (1981) 251. [15] S.N. Kaul, Phys. Rev. B23 (1981) 1205. [16] S.N. Kaul, Phys. Rev. B24 (1981) 6550. [17] S.N. Kaul, Phys. Rev. B22 (1980) 278. [18] S.N. Kaul and M. Rosenberg, Phil. Mag. B44 (1981) 357. [19] S. Dey and F.E. Luborsky, unpublished. [20] S.J. Poon and J. Durand, Phys. Rev. B16 (1977) 316. [21] K. Yamada, Y. Ichikawa, Y. Endoh and T. Masumoto, Solid State Commun. 16 (1975) 1335. [22] T. Mizoguchi and K. Yamauchi, J. de Phys. 35 (1974) C4-287. [23] E. Figueroa, L. Lundgren, 0. Beckman and S.M. Bhagat, Solid State Commun. 20 (1976) 961. [24] R. Meyer and H. Kronmiiller, Phys. Stat. Sol. (b) 109 (1982) 693. 1251 0. Beckman, K. Gramm, L. Lundgren, K.V. Rao and H.S. Chen, Solid State Commun. 39 (1981) 777. WI R. Malmh%ll, G. Beckstrom, K.V. Rao, S.M. Bhagat, M. Meichle and M.B. Salamon, J. Appl. Phys. 49 (1978) 1727. 1271 H. Maletta, G. Aeppli and S.M. Shapiro, J. Magn. Magn. Mat. 31-34 (1983) 1367. I281 C.J. Tranchita and H. Claus, Solid State Commun. 27 (1978) 583. 1291 A.T. Aldred and J.S. Kouvel, Physica 86-88B (1977) 329. 1301 G. Herzer, M. Fahnle, T. Egami and H. Kronmtiller, Phys. Stat. Sol. (b) 101 (1980) 713. 1311G. Herzer, M. F%hnle, T. Egami and H. Kronmtiller, J. Appl. Phys. 52 (1981) 1794. ~321 M. F&hnle, G. Herzer, T. Egami and H. Kronmiiller, J. Magn. Magn. Mat. 24 (1981) 175. WI M. Fahnle, G. Herzer, T. Egami and H. Kronmtiller, J. Appl. Phys. 53 (1982) 2326. [341 M. Fahnle, Phys. Stat. Sol. (b) 106 (1981) 519. I351 T. Kaneyoshi, J. Phys. Sot. Japan 51 (1982) 73. [361 H. Kronmtiller and M. F%hnle, Phys. Stat. Sol. (b) 97 (1980) 513. Phys. Stat. Sol. (b) 98 [371 M. Fahnle and H. Kronmtiller, (1980) 219. 1381 M. F&hnle and H. Kronmtiller, J. Magn. Magn. Mat. 8 (1978) 149. [391 S.-K. Ma, Modern Theory of Critical Phenomena (Benjamin, Reading, 1976). und Kritische WI W. Gebhardt and U. Krey, Phasentibergange Ph&nomene (Vieweg, Braunschweig-Wiesbaden, 1980). 1411 M. F&hnle, Phys. Stat. Sol. (b) 99 (1980) 547. [421 S. Kobe and K. Handrich, Sov. Phys. Solid State 13 (1971) 734. I431 H.A. Brown, Phys. Stat. Sol. (b) 114 (1982) K189. WI M. F&hnle, J. Phys. Cl6 (1983) L819.
[45] 1461 [47] [48] [49]
K. Handrich, Phys. Stat. Sol. 32 (1969) K55. M. Suzuki and R. Kubo, J. Phys. Sot. Japan 24 (1968) 51. D. Wagner and E.P. Wohlfarth, J. Phys. F 9 (1979) 717. H. Yahata, J. Phys. Sot. Japan 30 (1971) 657. K.V. Rao, M. Fahnle, E. Figueroa, 0. Beckman and L. Hedman, Phys. Rev. B27 (1983) 3104. [50] J. Mydosch, Lecture Notes Phys. 149 (1981) 87. [51] R.E. Walstedt and L.R. Walker, Phys. Rev. Lett. 47 (1981) 1624. 1521 B. Barbara, A.P. Malozemoff and Y. Imry. Phys. Rev. Lett. 47 (1981) 1852.
[53] Y. Ueno and T. Oguchi, Progr. Theoret. Phys. 63 (1980) 342. [54] M. Suzuki, Progr. Theoret. Phys. 58 (1977) 1151. [55] P. Monod and H. Bouchiat, unpublished. [56] P. Monod and H. Bouchiat, J. Phys. Lett. 43 (1982) L45. [57] S. Chikazawa, S. Taniguchi, H. Matsuyama and Y. Miyako. J. Magn. Magn. Mat. 31-34 (1983) 1355. 1581 M. F%hnle and T. Egami, Solid State Commun. 44 (1982) 533. [59] M. Fahnle and T. Egami, J. Appl. Phys. 53 (1982) 7693.
Journal
of Magnetism
and Magnetic
Materials
North-Holland
INVITED
38 (1983) 240-252 Publishing
Company
PAPER
THE MAGNETIC PHASE TRANSITION 1N AMORPHOUS GLASSES M. FAHNLE,
G. HERZER,
Itnrrtur fir Phwh, Mu.x Plud
H. KRONMijLLER,
R. MEYER,
FERROMAGNETS AND IN SPIN
M. SAILE
Imtrtut fir Mer~rllfor~~/ru~~,~. 7000 Srurt,q~rr X0. Fed. Rep. (;
and T. EGAMI
The magnetic phase transition in materials with exchange disorder (amorphous fcrromagnets. spin gla~c.\) 1s dihcubacd. In the critical temperature range the behavior of amorphous ferromagnetic transition mctalLmctalloid glasses i> found to be similar to the one derived for a three-dimensional homogeneous H&e&erg fcrromagnet. The most prominent diffcrencc between disordered and homogeneous materials is manifested in a large temperature range of dc\iations from the mean field behavior beyond the critical region, as observed experimentally for the temperature depcndcncc of the linear susceptibility of amorphous ferromagnets and of the nonlinear susceptibi!ity of spin glasses. A molecular field theory with correlations in space and time is developed, which relates the deviations from the mean field behavior to the interplay between the temperature dependent thermal correlations in the spin bystem and the spatial fluctuations of the material. Application to dynamical processes (kinetic critical slowing down) is discussed.
1. Introduction Phase transitions in disordered materials have attracted much interest in the last few years. Special emphasis was given to systems with random distribution of magnetic and nonmagnetic atoms (site percolation problem, for a review see ref. [l]), random anisotropy [2,3], random magnetic field [3,4] and random exchange bonds [5,6]. In this paper we consider the last case, i.e. systems with spatial fluctuations of the exchange interactions. Roughly speaking, the phase transition in a magnetic material takes place if the thermal energy is comparable to the exchange coupling energy, and from this point of view it becomes clear that materials with exchange disorder should exhibit an unusual behavior near the phase transition. It will be shown both theoretically and experimentally that as a result of the exchange fluctuations there may be deviations from the mean field behavior in the temperature range, which is much larger than the range of real critical behavior in crystalline materials. If there are no or only few antiferromagnetic 0304-8853/83/0000-0000/$03.00
0 1983 North-Holland
couplings, such exchange disorder systems represent amorphous ferromagnets, which are distinguished by a parallel alignment of the magnetic moments. Examples are quenched or sputtered alloys preferably of composition T,,M,, (T = transition metal, M = metalloid as C, P, B, Si). Assuming that the exchange interaction JI,( R,,) between spins at sites i and j depends on the interatomic distance, R,,, the topological disorder in these materials (lack of atomic long range order) is a first source of exchange fluctuations. Furthermore, in materials with several kinds of transition metal atoms, for instance Fe and Ni, there is a different exchange interaction between Fe and Fe, Ni and Fe, and Ni and Ni, and, therefore, the more or less random arrangement of Fe and Ni atoms, the chemical disorder, is a further source of exchange fluctuations. When increasing the number of antiferromagnetic bonds in the system, there is a transition to the spin glass state with spatially random orientation of the magnetic moments [7]. Prototype spin glasses are the disordered alloy systems Fe Au or CuMn with low concentrations of magneticatoms. -
M. Fiihnle et (11./ Phase frmsitiotl in umorphous ferromugnets and spit! glasses
In these materials the size and the sign of the RKKY exchange interactions oscillate on a scale, which is smaller than the intermoment distance, thus producing both ferromagnetic and antiferromagnetic couplings if the magnetic moments are distributed randomly. It has been shown theoretically [5,8] that in the ferromagnets with spatially random exchange interactions the critical behavior is identical to the one of the corresponding homogeneous material, if the specific heat exponent a of the homogeneous materials is negative. For positive values of (Y, however, the critical exponents describing the phase transition should be modified (Harris criterion [5]). Intuitively this simple rule may be well understood, taking in mind that a distribution of exchange interactions means a distribution of local critical temperatures. For positive values of OLthe specific heat diverges locally if the temperature approaches the local critical temperature. The spins then are heated up locally, and the heat is transferred to the neighborhood via exchange couplings, thus changing the critical behavior of the whole system. The ferromagnetic transition metal-metalloid glasses discussed in this paper may be well represented by the three-dimensional Heisenberg model [9], for which the exponent (Yof the homogeneous material is negative. It will be shown in section 2 that our experimental results for the susceptibility exponent y are indeed consistent with the Heisenberg value of y = 1.387 for T + T,, in agreement with the theoretical prediction. The difference between amorphous and crystalline materials is manifested in a totally different temperature dependence of the susceptibility outside the real critical region of the phase transition. Specifically, in amorphous materials the temperature dependent effective exponent y(T) as introduced by Kouvel and Fisher [lo] exhibits a non-monotonic temperature dependence (section 2), in sharp contrast to the case of crystalline materials which are distinguished by a monotonically decreasing y(T). In sections 3 and 4 we develop a theory, which is able to explain this unusual behavior of amorphous ferromagnets. It is well known that in general the renormalization group theory gives exact results only for homogeneous four-dimen-
241
sional systems. To get information about real three-dimensional systems approximate calculations must be performed for non-integral dimensions d = 4 - z with c -X 1, and then the results have to be extrapolated to d = 3 (C = 1). This procedure is justified for the critical temperature range of homogeneous systems by the empirical experience that the so-obtained critical exponents agree well with theoretical observations [ll]. However, to us it seems to be extremely difficult to extend these calculations to the temperature range outside the real critical region (instead of T -+ T,) in amorphous (instead of homogeneous) three dimensional (instead of d = 4 - E) systems. An attempt in this direction was made by Sobotta and Wagner [12] using a special type of renormalization theory with an e-expansion to O(e). However, their theory yields a monotonically decreasing effective exponent y(T) and thus fails to reproduce the experimentally observed non-monotonic behavior of y(T), which is the most important feature of the phase transition in amorphous ferromagnets. We therefore did not try to approach the problem by a renormalization group analysis, but we have developed in a phenomenological approach a mean-field theory with correlations in space and time, which is able to explain the experimental observations. Results of the theory are given in section 4. In section 5 we extent our correlated mean field theory to the case of spin glasses and show that a similar large temperature range of deviations from the mean field behavior may occur in these materials.
2. The paramagnetic ferromagnets
susceptibility
of amorphous
To illustrate the unusual behavior of amorphous ferromagnets outside the real critical temperature range, fig. 1 shows the temperature dependence of the inverse paramagnetic zero field susceptibility, x;‘(T), for the amorphous ferromagnetic alloy Fe,,Ni,,B,, (details of the experiments as well as the Arrott plots M* vs. H/M, which deviate strongly from the straight lines of crystalline materials are published elsewhere). For comparison, data for crystalline Ni as obtained by Kouvel
f / 70 *
4or
/’
35
/’
30
1.1-. ,’
t
/I’
1’
/I
/’
J
“//’ I
0.00
0.05
7’
0.10
/I’
x crystalline
Ni
o amorphous
Fe,0NI,,B2L
I
I
I
I
I
0.15
020
0.25
0.30
0.35
T/T,
-1
O.&O
-
Fig. 1. Temperature dependence of the paramagnetic zero field susceptibility for amorphous Fe,,NiS,B,, and crystalline Ni. The data for Ni have been multiplied by a constant factor in such way that the Curie-Weiss line of Ni is parallel to a straight line through the last three data points for the amorphous material. The dashed line represents the Curie-Weiss line of Ni.
and Fisher [lo] are also presented in the figure. For crystalline Ni there is an appreciable curvature of the plot in a small temperature range, and for high temperatures the usual Curie-Weiss behavior (mean field behavior) is observed. This temperature range of curvature of the plot is much larger for the amorphous material, and there is no Curie-Weiss behavior in the whole temperature range 0 < T/T, - 1 G 0.35. For T approaching T, the temperature dependence of the susceptibility is usually described by x;‘(T) - (T - T,)y with y = 1.3866 for a Heisenberg ferromagnet [13]. Of course this simple power law is only valid for the real critical temperature range of the phase transition. It is not able to describe the gradual transition from the critical behavior to the high-temperature mean field behavior. To characterize the temperature dependence of the susceptibility in the whole temperature range, Kouvel and Fisher [lo] have introduced an effective exponent y(T) defined by
Thereby,
d ln( T/T,
T
0
1.6
amorphous
Fe,,
- 1)
= (T-
in a Heisenberg
q)x,,z. system
Ni,,
Bz4
1.5
‘,3-\ , / , , , crystalline
1.2
0
0.05
0.10
0.15
Ni
0.20
0.25
0.30 T/T,
dlnx;l Y(T) =
exponent y(T) should be y = 1.3866 for T -+ T,, and for very large temperatures y(T) should approach the mean field value of 1. Within the experimental accuracy the values of y(T,) are close to the Heisenberg value of 1.3866, both in crystalline Ni and in amorphous as shown in fig. 2. We thus observe Fe,,Ni,,B,,, the same type of critical behavior in the homogeneous and in the disordered Heisenberg system, in agreement with the theoretical prediction [5,8]. However, whereas the effective exponent y(r) decreases monotonically in the crystalline material, this quantity initially increases in the amorphous system and runs through a broad maximum before decreasing again to the mean field value of 1, obviously a totally different behavior (fig. 2)! Similar results for x;‘(T) and y(T) are found by Gaunt et al. [14] for Fe,zNi,,Cr,,P,,Bc with y(T,) = 1.35 + 0.03, by Kaul [15-171 for with y(T,) = 1.33 + 0.05, Fe,,Ni,,Pi,B, FeJONiAOP,,$ with y(T,) = 1.31 k 0.02 and Fe,,Ni,,Cr,,P,,$ with y(c) = 1.33 f 0.05, by Kaul and Rosenberg [18] for Fe,,Ni,,B,,Si with y(T,) = 1.35 f 0.04, by Dey and Luborsky [19] for Fe,,Ni,,B,,Si with y(c) = 1.36 + 0.04, and by Poon and Durand [20] for amorphous Gd,, Au 20 with y(T,) = 1.29 + 0.05. Yamada et al. [21] as
-I
0.35
(
-
(1)
this
effective
Fig. 2. The temperature dependent effective exponent y(T) as defined by eq. (1) for amorphous Fe,,Ni,,B,, and crystalline Ni.
M. Fiihnle et ul. / Phuse irunsition in umorphous ferrcjmugners und spin glusses
well as Mizoguchi and Yamauchi [22] have carried out an extensive analysis for the critical region only. Yamada et al. [21] obtained y(T,)= 1.3 + 0.05 for Fe,,P,,C,, a value close to that found by Poon and Durand [20] for Gd,,Au,,, whereas Mizoguchi and Yamauchi [22] obtained y( T,) = 1.34 f 0.025 for CO,~B,,P,,. Other authors tried to fit the temperature dependence of xii(T) in a large temperature range approximately by one unique value of y. Because this y most probably is some kind of average value of the temperature dependent exponent y(T) they have to use numbers which are appreciably larger than the Heisenberg value: to describe the temperature dependence of x;‘(T) for Fe,,Ni,,Cr,,P,,$ outside the real critical region, Figueroa et al. [23] used y = 1.71 f 0.06, which corresponds to the maximum value of y(T) found by Gaunt et al. [14] for the same alloy. Meyer and Kronmtiller [24] performed an approximate fit for x;‘(T) in a large temperature range, yielding y = 1.63 + 0.07 for Fe,,Ni,,B,,P,. Beckman et al. [25] found y = 1.58 & 0.1 for Fe,,Ni,,P,,$Al,, and Malmh;ill et al. [26] used a unique value of y = 1.7 + 0.1 for the amorphous ferromagnetic alloy Fe,,Ni,,P,,$Si,. Most recently Maletta et al. [27] found the non-monotonic behavior of y(T) for the first time in a disordered crystalline alloy, namely in the alloy system Eu,,,Sr,,S, with y(T,) = 1.38 (Heisenberg value) and a maximum value of y = 2.2. As will be shown in the next section, the nonmonotonic temperature dependence of y(T) is an effect of exchange fluctuations, which are present both in amorphous and in disordered crystalline alloys. Therefore, it is not surprising that the same non-monotonic behavior of the effective exponent y(T) is found in both systems. Tranchita and Claus [28] as well as Aldred and Kouvel [29] observed unusual large but temperature independent values of y for diluted disordered CuNi and FeCr alloys. However, in these materials the long range ferromagnetic order is due to the interaction of large polarization clouds induced by Ni-rich (Fe-rich) local regions. It is not surprising that such giant moment systems show a different behavior near the phase transition. The values of the exponent /3 for various amorphous systems are close to those values ob-
243
served f r crystalline ferromagnets, although they are cons’stently slightly larger (for a review see ref. [17]). Fu thermore, the scaling law y = p(6 - 1) is satisfied 1 approximately in most amorphous materials [17]. Obviously in the critical temperature range the behavior of amorphous ferromagnets is very similar to that one of corresponding homogeneous materials. The most prominent difference between amorphous and crystalline materials is manifested in the temperature dependence of x;‘(T) and y(T) beyond the real critical temperature range.
3. Correlated molecular field theory In this section we describe our correlated molecular field theory [30-341, which is able to explain the unusual behavior of amorphous ferromagnets beyond the critical region as discussed in section 2 (a theory which is basically very similar to our theory has been developed later by Kaneyoshi [35]). By means of theoretical investigations it has been shown [36,37] that spatially random magnetostatic and magnetocrystalline fluctuations (anisotropy fluctuations) in transition metal base amorphous ferromagnets as well as internal stresses (which determine the properties of the low temperature magnetization curve of these material,s [9,38]) play only a minor role outside the critical region (their influence on the critical behavior is discussed for instance in refs. [l-3]). The unusual temperature dependence of the susceptibility is really an effect of the spatial fluctuations of the exchange interactions. It is therefore sufficient to consider the Heisenberg Hamiltonian Ei=
-c
C.@;(f)s;(l), i
(2)
J’
which contains the fluctuating exchange interactions J,, and the time dependent spin operators $(t) and 4(t). To illustrate the basic physical idea of our theory we consider first the case of a crystalline Ising ferromagnet. At zero temperature all spins are parallel because of the ferromagnetic exchange interactps. When increasing the temperature single spins are turned around because of the
interaction with the random thermal agitations of the heat bath (phonon system), but at each time most of the spins are parallel because of the ferromagnetic exchange interactions (fig. 3). Thereby the thermal average ($) of a spin at site i is non-zero (ferromagnetic region), but there are thermal fluctuations &$(t) of the spin. For temperatures much larger than the Curie temperature T, the thermal energy is much larger than the exchange interaction energy. Then the spins react more or less independently on the thermal agitations of the heat bath, and there are no correlations between the spin fluctuations S$,( t) and 8$(t) at different sites i andj (fig. 3). The thermal average ($,) is zero (paramagnetic region). When approaching T, there is a competition between the tendency to order spins because of the ferromagnetic exchange interactions and to disorder spins because of the random thermal agitations of the heat bath. As a consequence, the system creates large clusters of spins (fig. 3) in each of which the ferromagnetic ground state is preserved for a short time (intracluster excitations are neglected). The spins in the clusters are strongly coupled together and react cooperatively, more or less like superparamagnetic particles. The mean size of the clusters is given by the thermal correlation length t(T), which diverges when approach-
ing T,. It is well known, for example, from renormalization group calculations (for a review, see refs. [39,40]) that it is the nonlinear interaction between these critical correlated spin fluctuations of very large size, which is responsible for the curvature of the xOl(T) plot in the real critical region of the phase transition of crystalline materials. In amorphous materials, however, there is a curvature of the plot in a temperature range, for which the thermal correlation length t(T) is of the order of a few nearest neighbor distances only, far away from the validity of the renormalization group analysis of the real critical spin fluctuations. Therefore, one could tentatively assume that in amorphous materials the correlated spin fluctuations play only a minor role in this temperature range, and we want to neglect them totally for the moment by applying a mean field theory (molecular field theory). Indeed, within the framework of a molecular field theory the interaction energy of a spin at site i with all the spins at sitesj is approximated by the interaction of this spin with the thermal averages (3;). Obviously the application of such a molecular field approximation means the neglection of the correlations between the spin fluctuations at different sites, because the molecular field Hamiltonian
en,= - c CJ,,(s;>W
I/I T = T_
Fig. 3. Schematic representation of snapshot Ising ferromagnet at various temperatures.
pictures
of an
(3)
contains only the fluctuations of the spins at sites i, whereas the time-fluctuations of the spins at sitesj are averaged out. In crystalline materials the situation is very simple because due to the long range order the thermal averages of all spins at sitesj are the same. In amorphous materials, however, there are spatial fluctuations of the local magnetization because of the spatially random exchange interactions. Two different methods to take into account these exchange fluctuations are discussed in the literature (for a review, see ref. [41]). In the so-called inhomogeneous molecular field theory spatial fluctuations of the magnetization with all wavelengths are allowed by inserting into the molecular field Hamiltonian indeed the local inhomogeneous quantity (3,) as in eq. (3). Within the framework of this
M. Fiihnle
et ul. / Phuse tmnsition
inhomogeneous theory an increase of the critical temperature is obtained [37,41,42] as shown in fig. 4 as a result of the spatial fluctuations of the exchange interactions. In contrast, high temperature expansions, the Bethe-Peierls-Weiss approximation and the constant-coupling approximation yield a decrease [43] of T, for spatially random exchange interactions, a result which has been confirmed recently by Monte Carlo computer simulations [44]. Furthermore, the curvature of the x;‘(T) plot is opposite in sign to the curvature of the experimentally obtained plot, which is also shown schematically in fig. 4. Handrich [45] has argued that it does not make sense to totally neglect on the one hand the thermal fluctuations of the spins at sites j by performing a molecular field approximation, but to take into account accurately the spatial fluctuations of the local magnetization on the other hand. He has inserted into the molecular field Hamiltonian the volume average (S) of the magnetization instead of the local inhomogeneous magnetization (the symbol means the volume average). Within the framework of this homogeneous molecular field theory the usual Curie-Weiss behavior is obtained in the paramagnetic region, and the critical temperature is not affected by the exchange fluctuations (fig. 4). Obviously
both the homogeneous
and the inho-
0.7 0.6 -
T/T,-1 Fig. 4. Results for x0(T) as obtained by the homogeneous molecular field theory (full line) and the inhomogeneous theory (dashed line). The dashed-dotted line is a schematic representation of the experimental data. For the calculations we have inserted the material parameters of Fe,,Ni,P,,$ as discussed in section 4 with A = 0.5 and I, = 1.5d.
in umorphow
ferromugnets
und spin glusses
245
mogeneous molecular field approximation are not able to reproduce the correct physical behavior. However, each theory which interpolates between these two extreme cases, i.e. which exhibits characteristic features of the homogeneous theory near T, and characteristic features of the inhomogeneous theory far away from T, must necessarily reproduce the correct physical behavior (fig. 4). What is the physical justification for the application of such an interpolating theory? So far we have totally neglected the influence of correlated spin fluctuations because we have applied a molecular field theory. These correlated spin fluctuations have the form of clusters, in which the spins show a strong cooperative behavior (fig. 3). For T approaching T, the mean size of the clusters diverges, and then the spin system as a whole reacts cooperatively and does not feel anything from the spatially random exchange fluctuations. Therefore, the main features of the phase transition in amorphous Heisenberg systems are very similar to those in crystalline materials: in both cases we observe a sharp transition temperature T,, and in both cases the susceptibility exponent y for T + T, is determined by the nonlinear interactions between the critical spin fluctuations. When increasing the temperature the mean size of the clusters decreases. Then the nonlinear interaction between the correlated spin fluctuations is negligibly small, and in crystalli-ne materials we observe the usual Curie-Weiss behavior. Nevertheless, there are still correlated spin fluctuations of smaller size (noncritical correlated spin fluctuations), in which the spins show a strong cooperative behavior. These noncritical spin fluctuations must be taken into account in amorphous ferromagnets as long as the thermal correlation length t(T) is larger than or comparable to the correlation length of the exchange interactions, which is about the nearestneighbor distance. The exchange fluctuations try to produce spatial fluctuations of the magnetization. Because of the strong cooperative behavior of the spins in the clusters, however, there are no magnetization fluctuations with wavelengths shorter than the mean size of the clusters. All the other magnetization fluctuations are averaged out. For T + T, the quantity t(T) diverges, and then we have about the same value of the magnetization
in the whole sample as in the homogeneous molecular field theory. For very large temperatures the thermal correlation length t(T) is very small and there are spatially random fluctuations of the magnetization as in the inhomogeneous molecular field theory. It is this transition between a homogeneous situation for T + T, and an inhomogeneous situation far away from T,, which is responsible for the unusual behavior of amorphous ferromagnets. The large temperature range of curvature of the xi’(T) plot thus is not a real critical phenomenon, but it is the result of the interplay between the temperature dependent magnetic correlations in the spin system and the spatial fluctuations of the exchange interactions. For a quantitative formulation we insert into the molecular field Hamiltonian a spatially correlated magnetization, which in a continuum approach ((4.) + S(r’); M(r’) = gpBS(r’) with g = Lande’s g-factor and pa = Bohr’s magneton; JI, + J(r, r’)) is given by (see refs. [30-341)
namics of the whole clusters rather than by the high-frequency intracluster excitations, and it is therefore limited by the finite “lifetime” of the clusters. For a dynamical generalization of the theory we assume that there are no relevant magnetization fluctuations with wavelengths shorter than the mean size t(T) of the clusters and with frequencies higher than the inverse “lifetime” of the clusters. We then insert into the molecular field Hamiltonian a magnetization with correlations in space and time [32-341
Mb’,
M( %rr
=/
- Y”, [(T))M(r”)d3Y.
(4)
Here f( r’ - r”, t(T)) is an integral kernel which averages out the irrelevant fluctuations of the magnetization M(r”). For T -+ T, the integral kernel is a slowly varying function in space; and, therefore, the correlated magnetization is about constant in the whole sample as in the homogeneous molecular field theory. In contrast, for very high temperatures the kernel f( r’ - r”, E(T)) is well localized at r’, and we have the spatial fluctuations of the magnetization as in the inhomogeneous molecular field theory. To determine the dynamical evolution of the spin system we have to distinguish between two different time scales: on the one hand there are high-frequency single spin excitations within the clusters, which are irrelevant for the macroscopic evolution of the whole system because they are averaged out. On the other hand the cluster as a whole may be turned around spin by spin by the random thermal agitations of the heat bath. The larger the clusters, the larger is the so-defined “lifetime” of the clusters. Macroscopically the time evolution of the system is determined by the dy-
- r”, t - t’)
r-z
x M(
t’)dt’ d3/‘.
r”,
(5)
Here the integral kernel g(r’ - r”, t - t’) averages out the irrelevant fluctuations of the magnetization with high frequencies and short wavelengths. As a possible ansatz for the Fourier transform of g(r’ - r”, t - t’) we insert the real part of the dynamical susceptibility x(k, w) g(k,
f(+
t>corr =/I ‘+rg( rl
Re[x(k,
a> =A
w>l,
(6)
where X(/C, w) is calculated with the conventional mean field theory [46] with a local expansion of the exchange interaction “”
w’=
_i~~~:A
+ CkZ
(7)
The quantity 7 determines the time scale for single spin excitations, and A is the Ginzburg-Landau parameter which has the form [41,47] A =A’(T/T,‘-
1).
(8)
a = A’T/Tco, A’ = L
3kT.O
(9) s(s+
1).
(10)
4‘wrd* Here c is the concentration of magnetic ions in the alloy, S denotes the spin quantum number, and T’,‘,O is given by (V = volume of the sample) kT,‘=fS(S+l); X Jl vv
J( r,
r’)d3r’ d3
r”.
.
M. Fiihnle et ul. / Phase trunsition
For the stiffness
X
I/vv .I(
r,
parameter
in umorphous
C we insert
r’)( r - r’)2d3r d3r’.
(12)
The connection between the integral kernel g( r’ r”, t=- t’) of the dynamical theory (eq. (5)) and the kernel f( r’ - r”, E(T)) of the static theory (eq. (4)) is given by 03) with the correlation length size of the spin clusters)
(describing
the mean
t=Jc/A.
#.l
_
).tr,
<)
_
is the Fourier transform correlation function
of the
4;t2 exd- Ir’ -,J-“l/C) ) 1+--r
1
(15)
which exhibits the general features of the integral kernel f discussed above. By evaluating the Brillouin function of the molecular field theory [30,34,41,47], we then obtain for T > T, the following linear equation of motion for M(r, t) ay$M(r,
247
and spin glusses
geneous and instantaneous molecular field theory is reproduced. This theory still is a mean field theory which explicitly neglects the correlations between the thermal spin fluctuations. However, it takes into account implicitly the effect of the correlated spin fluctuations by the introduction of the correlated magnetization M( r’, t ),,,, into the molecular field. Of course the theory is not able to describe correctly the real critical behavior for T + T,. The theory holds as long as the nonlinear interaction between the spin fluctuations [39] is not the dominant one. It improves the conventional theory for a range of temperatures, in which the correlated spin fluctuations are small enough but still important, and it approaches the conventional theory for very large temperatures.
(14)
Apparently f(k) Ornstein-Zernike f(
ferromugnets
t) = -aM(r,
J(r,
r’)M(r’,
t) t) corrd3r’ + H( r,
t
).
06) Here the second term on the r.s. is the correlated molecular field with M(r’, t),,,, given by eq. (5), and H(r, t) is the external magnetic field. This equation is a linear integrodifferential equation with spatially random fluctuating integral kernel, which may be solved approximately by special statistical methods discussed in full detail in refs. [30,34]. By inserting into the molecular field expression of eq. (16) the local, instantaneous magnetization M(r’, t) instead of the correlated magthe conventional inhomonetization M( r’, t),,,
4. Results of the theory Inserting typical values for the material parameters A’, C and Tco (specifically, we have used the material parameters of amorphous Fe,Ni,oP1,Bh [37,30]) we obtain from eq. (16) for the temperature dependence of the paramagnetic zero field susceptibility x0(T) = a~/aNI,,, the results shown in fig. 5. Obviously there is a large temperature range of curvature of the plot due to the effect of exchange fluctuations j( r, r’) = J( r, r’) - J(r, in agreement with the experimental observation (fig. 1). The parameter A is a measure of the relative fluctuation strength of the exchange interactions and is given by
(17)
A=/=, S(r)
=/j(r,
(18)
r’)d3r’//J(r,d3r’.
of the exWe assume a weak spatial correlation change fluctuations S(r) described by the correlation function r,,
( r - r’) = A2 exp[ - (r - r’)‘//,‘]
.
09)
To obtain the results shown in fig. 5 we have inserted for the correlation length 1, = 1.5d, where d is the nearest-neighbor distance. The larger the
(A’X0)_’ t 0.4 -
/ /= A 0 0.3 0.4
0.3 -
0.5 0.6
0
0.1
0.2
0.3 T/T:-1
Fig. 5. Results for ,y~ ‘(T)
0.4 -
for various values of A.
value of A (fig. 5) and/or the value of the correlation length I,, the larger is the temperature range of curvature of the plot. In our theory there is an enhancement of T, (defined by the divergence of x0(T)) for A > A, =/T 2C/AI, , as shown in fig. 5. In contrast, high temperature series expansions [43] and Monte Carlo computer simulations [44] yield a decrease of T, for spatially random fluctuations of the exchange interactions 4,. For spatially correlated fluctuations no theoretical results are known. In our theory this enhancement of T, for A > A, is removed when replacing in the integral kernel g(k, w) (eqs. (6)-(7)) the quantity A by the unknown susceptibility x;‘(T). The replacement means that the temperature dependence of the correlation length t(T) is no longer introduced ad hoc as in eq. (14) but is calculated self-consistently according to t(T) = /c/xol( T) . This relation between the temperat‘ure dependence of ((7’) and x;‘(T) holds exactly only for a mean field theory, but it is fulfilled approximately also for the critical temperature range of a three-dimensional Heisenberg ferromagnet (the critical exponents for x0 and 5 are y = 1.3866 and Y = 0.7054). The curvature of the xi’(T) plot obtained by this self-consistent modification of our
theory is smaller than in the original theory, and we have to insert appreciably larger values for A and/or I, to yield a similar large temperature range of curvature. The temperature dependent effective exponent y(T) as defined by eq. (1) starts at the mean field value of 1 and then runs through a maximum with increasing temperature (fig. 6) in qualitative good agreement with experimental observations (fig. 2). Of course our mean field theory is not able to reproduce y = 1.3866 for T + T,, because we neglect explicitly the nonlinear interactions between the critical spin fluctuations. The maximum of y(T) is more pronounced and it increases and shifts to lower values of T with increasing values of A (fig. 6) and/or I,. When performing the self-consistent version of our theory this maximum shifts slightly to larger values of T with increasing I, (fig. 7) and/or A. However, the maximum of y then is smaller and occurs at higher values of T (fig. 7) than in the original theory. According to fig. 5 one could tentatively try to describe the temperature dependence of x;‘(T) for instance for A = 0.5 and (T/T,’ - 1) > 0.3 by a Curie-Weiss law x;‘(T) - (T - T,), thus defining a paramagnetic Curie temperature Tp > T, for the amorphous material. However, when increasing the temperature further and further, the theoretical plot runs through an inflection point and approaches very gradually the dashed Curie-Weiss line with Tp = T,. For (T/T,” - 1) = 1.65 (4.35) the relative deviations still are 5% (1%). Therefore the definition of a paramagnetic Curie temperature T,
T/T,'-l-
Fig. 6. Results for y(T)
for various values of A. with I, = 1.5d.
M. Fiihnle et ul. / Phase irunsirion
c
in umorphour
1.8
1,=3Sd
1.7
,/'Y. '. i/ '.\
1.6
249
und sprn giusses
1.81
1.9
t
ferromogneis
A =0.45 j 1,=1.5d
1.6 l.s-p,
0
0.2
0.4
0.6 0.8 T/Tco-1 -
1.0
Fig. 7. Results for y(T) as obtained by the self-consistent version of our theory for various values of /,, with A = 0.5.
+......* 1.4~: /*** \ **.....***_ \ 5.*... I ;** \ *... \ a.... .r..!T' \ 1.3;1 *.*......** '\ I:' W... \\ -... 1.2$ '. ',_T'JT' Ii --._ 1.1 5 --_ 1.0 0
I 0.1
I 0.2
I 0.3
I 0.4
0.3
T/T," -1 -
for amorphous ferromagnets is more or less arbitrary, and it does make sense only if specifying simultaneously the temperature range used for the linear fit of x;‘(T). It is a feature of the effective exponent y(T) = (T- T,) x0 dx;‘/dT that it deviates rather strongly from the mean field value of 1 for (r/r,’ - 1) > 0.3, even though for instance for A G 0.5 the curvature of the xi’(T)-plot is very small in this temperature range (compare figs. 6 and 5). To remove this disadvantage we introduce another effective exponent y’(T) by
(20) This new temperature dependent exponent y’(T) exhibits the same qualitative features as y(T) (and the same qualitative behavior when changing A and/or I,), but the maximum of y’(T) is larger and occurs at lower values of T, and for high temperatures the absolute values of y’(T) are smaller than the values of y(T) (fig. 8). As an example for the application of the dynamical theory we calculate the relaxation of the magnetization when suddenly switching off an external magnetic field [32-341. Within the framework of the conventional molecular field theory an exponential relaxation of the magnetization is obtained. The relaxation time is temperature dependent and diverges when approaching T, according to 7 - (T - q)-A~~ with A,, = 1, i.e. for T + T,
Fig. 8. Comparison between the two effective exponents y(T) (eq. (1)) and v’(T) (eq. (20)) as obtained by the nonself-consistent version of our theory.
the system needs more and more time to react on an external perturbation (critical slowing down). More sophisticated theories, which take into account explicitly the nonlinear interactions between the critical spin fluctuations once more yield an exponential relaxation of the magnetization with time, but now the exponent A,, is larger, even larger than the static susceptibility exponent y(“kinetic” critical slowing down). For instance, a high-temperature series expansion of Yahata [48] gives A,, = 1.4 for the kinetic Ising model on a simple cubic lattice. With our correlated molecular field theory [32-341 we obtain A,, = 1.5 for T + T,, both in crystalline and amorphous materials, in good agreement with Yahata’s value of A MM = 1.4. For amorphous ferromagnets the values of the relaxation time 7 are enhanced as an effect of the exchange fluctuations. For details of the calculations, see ref. [34].
5. The nonlinear susceptibility of spin glasses When increasing the number of antiferromagnetic bonds in the system, there is a transition from the ferromagnetic ground state to the non-
magnetic spin glass state. For a symmetric distribution of exchange interactions J,, (the case of preferably ferromagnetic or antiferromagnetic couplings is discussed in ref. [49]) there is a spatially random orientation of the spins in the spin glass ground state, which may be described by the following general formula m=
s,,.
(21)
It is well known that experimentally the spin glass state may be characterized [50] by a cusp in the linear susceptibility x0(T) at the freezing temperature T,. So far, however, it is totally unclear whether this cusp is an indication of a real equilibrium phase transition with strong spin correlations as in the case of ferromagnets, or if the relaxation times in the system are so large that the cusp at T, is only a nonequilibrium phenomenon (for a review of this problem, see ref. [6]). For instance, it is argued that the spin glass ground state is highly degenerate and that the system moves steadily and slowly from one ground state to the other, so that there are no spin correlations at all when averaging over an infinitely long time. Indeed Monte Carlo simulations yield an equilibrium phase transition to the spin glass state only at zero temperature (Tf = 0) for spin glass models with short-range interactions, both for Ising, XYand Heisenberg spin glasses in 2 and 3 dimensions. However, experimentally one does find a cusp at nonzero T,, and there is theoretical [51] and experimental evidence that there are strong temperature dependent spin correlations near T,, at least during the time of the measurement. The most striking evidence for these temperature dependent spin correlations near T, is manifested in the experimentally observed [52] divergence of the nonlinear susceptibility, which (for T > T,) may be written as [53] x2(T)
- 1/T3~(s,s,)?.
(22)
‘.I
This divergence of X2(T) presence of correlated spin size of which is very large described by a temperature length t(T). Assuming that
clearly indicates the fluctuations, the mean near T, and may be dependent correlation the results of the above
discussed Monte Carlo simulations are valid for the real spin glass systems in nature, this must be interpreted as a nonequilibrium phenomenon where the system remains in a specific valley of the phase space for the time of the measurement. The correlated thermal spin fluctuations around this valley then once more have the form of spin clusters of size t(T), in each of which the specific ground state configuration is preserved for a short time. By taking into account anisotropic energy terms in the Hamiltonian this “trapping” effect of the system in a specific valley of the phase space is strongly enhanced [51]. If the time of the measurement is smaller than the “trapping” time of the system in one specific phase space valley, we obviously have the same interplay between the temperature dependent thermal correlations in the spin system and the spatial randomness of the exchange interactions as in the case of amorphous ferromagnets. Of course in the case of spin glasses this interplay has no effect on the temperature dependence of the linear susceptibility (23)
When inserting the general spin glass formula, eq. (21) into eq. (23) a simple Curie law, x0(T) - l/T, is derived, which is indeed observed in spin glasses without a preferable sign of the exchange couplings (in systems with preferably ferromagnetic or antiferromagnetic exchange couplings strong deviations from the Curie-law occur [49]). However, one might observe this interplay for the temperature dependence of the nonlinear susceptibility x2(T), which contains the squares of the correlation functions instead of the correlation functions (eq. 22). Within the framework of a simple molecular field theory Suzuki [54] has derived xz( T) - (T T,)-u. with y, = 1, and he has pointed out that the exponent y, is of central importance for the theory of spin glasses. Experimentally, Barbara et al. [52] found a large value of y, = 3.6 &-0.6 in a very large temperature range. In contrast, Monod and Bouchiat pointed out that y, might depend on temperature [55], and they found [56] ys-values
M. Ftihnle
et al. /
Phuse trunsition
between 1 and 2 for T, < T G 2Tr. Deviations from a simple power law dependence of x1(T) were also reported by Chikazawa et al. [57]. We think that a possible temperature dependence of y, in a very large temperature range has nothing to do with a real critical phenomenon, but is once more the result of the interplay between the temperature dependent thermal correlations in the spin system and the spatial randomness of the exchange interactions. Because of the close analogy to the case of amorphous ferromagnets we therefore have generalized our correlated molecular field theory to the case of spin glasses [58,59]. We have shown that the temperature dependence of the nonlinear susceptibility x z (T) should be described by a temperature dependent effective exponent y,(T) = d ln(x;‘(T)/d ln(T/T, - l), which has the same qualitative features (figs. 6-8) as the corresponding effective exponent y(T) (eq. (1)) in amorphous ferromagnets. For details of the theory, see refs. [58,59]. Amorphous ferromagnets and spin glasses thus show to some extent a universal behavior when we compare the linear susceptibility x0(T) of amorphous ferromagnets and the nonlinear susceptibility x1(T) of spin glasses.
Acknowledgement One of the authors (M.F.) is indebted to Prof. S.-K. Ma from the Department of Physics and the Institute for Pure and Applied Physical Sciences at University of California, San Diego, La Jolla, for helpful discussions.
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