- Email: [email protected]
Non-equilibrium behavior in a disordered magnetic system
~
N ELSEVIER
Journal of Magnetism and Magnetic Materials 152 (1996) 54-60
Journalof magnetism and magnetic ,4~ materials
Non-equilibrium behavior in a disordered magnetic system S. Koutani, G. Gavoille * Laboratoire de Cristallographie et Moddlisation des Matdriaux Mindraux et Biologiques, URA-CNRS 809, Facult~ des Sciences, BP 239, F-54506 Vandoeuvre-lbs-Nancy C£dex, France
Received 15 February 1995
Abstract We have studied the non-equilibrium magnetic state of a thin film of cobalt ferrite with a nanocrystalline structure which shows a frozen disordered magnetic state with extended ferrimagnetic correlations at low temperature. The magnetization has been measured in a wide range of temperatures and magnetic fields, and the irreversible susceptibility and the relaxation rate of the magnetization along the hysteresis loop show a scaling behavior where the scaling field is a temperature-dependent magnetic field H o. The magnetic field H o and the free energy barrier height in spin glasses show a similar temperature dependence. Our measurements, together with extensive studies of the spin glass phase, show that disordered magnetic systems share many common features. We have also observed that the relaxation of the magnetization that follows a magnetic field change has a behavior which is qualitatively similar to that observed in ageing experiments in spin glasses.
1. Introduction We have recently reported on the equilibrium magnetic properties of cobalt ferrite thin films with a nanocrystalline structure [1]. Depending on the preparation, the films exhibit more or less extended ferrimagnetic correlations but no long-range magnetic order at low temperature. We report here on the non-equilibrium magnetic state of the film which has a divergent linear susceptibility but no bulk magnetization in the low-temperature equilibrium state. Magnetization measurements, including irreversible susceptibility and relaxation studies, have been carried out in a wide range of temperatures and magnetic fields. As the most extensive studies of disordered magnetic systems are concerned with mean field (long-range interactions) spin glasses, it seems
* Corresponding author. Email: [email protected];fax: + 33-8340-6492.
very attractive to compare the behavior of our system with that of spin glasses since, apart from the absence of conventional long-range magnetic order, they differ strongly in the range of the interactions and by the nature of the low-temperature spin configuration. Most of the investigations of the spin glasses have been carried out in very small magnetic fields [2], and despite the different magnetic orders and experimental procedures the two systems share many common features.
2. Experimental The magnetization was measured with a VSM operating between 4 and 300 K. The magnetic field was supplied by an electromagnet whose response to a magnetic field change of 1 kOe is less than 5 s, whatever the magnitude of the field. The temperature range of the experiments, 140-300 K, was chosen in
0304-8853/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0304-8853(95)00451-3
S. Koutani, G. Gavoille / Journal of Magnetism and Magnetic Materials 152 (1996) 54-60 L
such a way that an equilibrium state is obtained in the largest available magnetic field (20 kOe). This state is the reference state for all the measurements. As we are concerned only with the irreversible process, the measured magnetization was not corrected for the diamagnetism of the glass substrate and of the sample holder. In order to obtain reliable results, the following procedure was used. All magnetization curves were obtained starting from the reference state. The magnetic field was changed by steps of 1 kOe every 25 s, and the magnetization in a given magnetic field was measured 25 s after the field was set up. For the study of the magnetization relaxation in a given magnetic field, the clock was started as soon as the field was stabilized, i.e. 5 s after the setting of the field.
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We have measured the irreversible susceptibility along the descending branch of the hysteresis loop in negative magnetic fields - H ( H > 0). The irreversible susceptibility at point A in Fig. 1 was obtained as follows. We measure the magnetization M ( - H - h) for positive h along the branch AI' and M( - H + h) along the branch AD. The linear terms given by a polynomial fitting procedure of M ( - H ) - M(-Hh) and of M ( - H + h ) - M ( - H ) , respectively, yields the total XT and the reversible Xr~v susceptibilities. The data of the irreversible susceptibility XL~ = X r - Xrev are reported in Fig. 2 for
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Fig. 2. Irreversiblesusceptibility versus magnetic fields at different temperatures. different temperatures. Within the accuracy of the data, the maximum of the irreversible susceptibility does not depend on the temperature and it appears for a temperature dependent magnetic field Ho(T). This suggests that the irreversible susceptibility may be expressed in terms of a reduced variable which has a fixed point when the temperature changes. Actually, the data corresponding to the different temperatures are on a master curve if the susceptibility is plotted against ( H - H o ) / H o (Fig. 3)+ The temperature dependence of H 0 is given in Fig. 4; the data may be tentatively fitted by the expression
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Always starting from the reference state, we have measured the time decay of the magnetization along the descending branch of the hysteresis loops (point A in Fig. 1) for 1000 s after the stabilization of the
S. Koutani, G, Gavoille / Journal of Magnetism and Magnetic Materials 152 (1996)54-60
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Fig. 5. Time decay of the magnetization along the hysteris loop at 140 K. From top to bottom: H = 2, 3, 4, 6, 8, 10 and 12 kOe. the relaxation rate S = - d M / d l n t, w h i c h is reported in Fig. 6 for different temperatures. Note that S and Xirr s h o w a m a x i m u m for the same magnetic
magnetic field. Within the accuracy of our measurements, the magnetization has a logarithmic time decay in any magnetic field at any temperature. The data obtained at 140 K are reported in Fig. 5. This short time relaxation is then well characterized by 7 7
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Fig. 6. Relaxation rate of the magnetization versus magnetic field at different temperatures.
S. Koutani, G. Gavoille/ Journal of Magnetism and Magnetic Materials 152 (1996) 54-60 3!
field H 0, but the magnitude of Sm~x depends on the temperature. A plot of S / H o versus the normalized magnetic field ( H - H o ) / H o is given in Fig. 7. The scatter of the data is greater than in Fig. 3, but if we take into account the fact that the current data are less accurate than the previous ones, we may infer that the scaling holds for S / H o as well as for Xirr-
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~io
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shows a logarithmic time decay in our time window, M d begins to increase before decaying at long time. The time corresponding to the maximum of Md(t) increases with increasing magnetic field. The relaxation of the same macroscopic state, characterized by the magnetization, strongly depends on the history of the sample. M d is reached in increasing magnetic fields, whilst M r is reached in decreasing ones.
S. Koutani, G. Gavoille / Journal of Magnetism and Magnetic Materials 152 (1996)54-60
58
3.4. Relaxation of the magnetization along a demagnetization curve The relaxation of the magnetization has been studied along the curve AD in Fig. 1, with point A corresponding to the magnetic field H 0. The data obtained at 140 K are reported in Fig. 9. The time window of our experiments is very small compared to the largest relaxation time of the system, but we would suggest that the curves have the same shape, increasing at short times and decreasing at long times with a crossover time tc(H o - H ) which increases when H 0 - H increases.
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4. Discussion Slow and non-exponential relaxation processes are usually interpreted as resulting from a complex free-energy landscape in the configuration space, which consists of disjoint regions separated by free energy barriers. A system prepared in a non-equilibrium state at time t = 0 relaxes toward thermodynamic equilibrium. This non-equilibrium relaxation process may be interpreted as the system populating new regions of the phase space when t increases. If we assume a thermally activated process following an Arrhenius law, the populated region O ( t w) at time t w is bounded by free-energy barriers higher than A(tw)=kTln(tw/to), where t o is a microscopic time. If the probability of escaping from l'2(t w) between times t w and t w + t is very small, then the system is said to be confined in g2(t w) for time t, and is in equilibrium within 12(tw); such a state is referred to as a quasi-equilibrium state [3]. The non-equilibrium relaxation may be studied by looking at the response to the application of a small probing magnetic field at t w. This has been extensively done in spin glasses since the pioneering work by Lundgren et al. [4], who showed that the response shows a maximum at t = t W, interpreted as the crossover between the quasi-equilibrium and the non-equilibrium dynamics. The relaxation of the system prepared in a remanent state by the removal of a large magnetic field is characterized by a continuous decay of the magnetization. After the application of a small positive field at t w, the total magnetization at t w + t first increases with t before decreasing; the crossover between the two regimes appears at t = t w [5]. This results from the opposite signs of the relaxation rate of the remanent magnetization and of the response to the probing magnetic field which dominates at times shorter than t w. Our measurements of the relaxation of the remanent magnetizations show that the relaxation rates of M d and M r have opposite signs at short times although the sign of the non-equilibrium relaxation rate is positive in both cases. The difference arises from the fact that M d is obtained in increasing magnetic fields, while M r is obtained in decreasing ones. We may expect that the equilibrium response to a magnetic field change dominates at short times, so we may infer from the relaxation of M d that the
s. Koutani, G. Gavoille /Journal of Magnetism and Magnetic Materials 152 (1996) 54-60
system is initially confined by free-energy bamers whose heights increase with H since the maximum of M d ( - H , t ) shifts to longer times when H increases from 0 to H 0. The relaxation of the magnetization along the demagnetization curve AD shows a similar behavior with a crossover time tc increasing with H 0 - H . This may also be interpreted as the confinement of the initial state with free-energy bartiers of increasing heights when the magnetic field is removed. In low magnetic fields, the magnetization changes by jumps between nearly similar states of positive magnetization while in large magnetic fields, it is between nearly similar states of negative magnetization. The crossover between the two regimes occurs near H 0, where the jumps may be between states related or nearly related by time reversal symmetry. The magnetic field affects not only the heights of the barriers, but also the magnetization of the relevant states. The magnetization processes cannot be described by a distribution of two-level systems with magnetic field-independent magnetizations. The system moves in the phase space as the magnetic field changes; some regions are populated, while others are depopulated as a consequence of the dependence of the free-energy landscape in the magnetic field. The equilibrium states M(20 kOe) and M ( - 2 0 kOe) are related by time reversal symmetry, so that H 0 may be considered as the magnetic field needed to overcome the lowest time reversal barrier [6,7]. Two metastable states, say a and b, may be characterized by their Hamming distance, which can be written for a system of N Ising spins located at sites i as:
111
]
dab = "~ 1 -- -~ }Zm~imbi ,
(2)
where m a is the thermal average of the magnetization in state a. At 0 K, dab reaches its maximum value dab = 1 if the states a and b are related by time reversal symmetry. Note that in Ising spin glasses with long-range interactions, the height of the barriers is an increasing function of dab [6,7]. Our measurements probe only the low effective energy barriers in a given magnetic field, and the magnitudes of Xirr and S along the hysteresis loop are measures of the similarity between the relevant metastable states.
59
For small H, i.e. in a state obtained by the removal of a large magnetic field, and for large H, the only relevant states are nearly similar, whereas they are very dissimilar near H o where large magnetization jumps are observed. H 0 may be considered as the magnetic field needed to suppress the large energy barriers between very dissimilar states. It is interesting to note that the mean field theory of spin glasses predicts that the states become increasinly similar when the magnetic field increases, and that they are macroscopically similar along the Almeida-Thouless line [8] where the irreversibility vanishes. The scaling analysis of the irreversible susceptibility and of the relaxation rate of the magnetization can be compared with the experimental results obtained by Lederman et al. [9] on AgMn spin glass. They studied the relaxation of the thermoremanent magnetization using the following procedure. The sample is cooled in a small magnetic field from above Tg to a temperature T below Tg and the magnetic field is removed at a time t w after the temperature quench. They observed that ageing at ( T - A T 1, t,<) and at ( T - A T 2, tw2) leads to the same relaxation curve. From their data, the temperature dependence of the barrier height is given by --~
= -a(T) A + b(T).
(3)
On the other hand, the derivative of Eq. (1) with respect to T yields a similar equation for H 0 but with temperature-independent coefficients. The bartier height may be probed in ageing experiments, as well as by the application of a magnetic field. It must be noted, however, that the magnetic field perturbs the state of the system, and the similarities between the two kinds of experiments call for further theoretical investigations on disordered systems.
5. Conclusions We have studied some aspects of the non-equilibrium behavior of a disordered magnetic system in a wide range of magnetic fields and temperatures. We have emphasized the fact that our observations may be related to the structure of the spin glass phase investigated by numerical calculations or by
60
S. Koutani, G. Gavoille / Journal of Magnetism and Magnetic Materials 152 (1996) 54-60
experiments in a zero magnetic field. Our sample is far from being a mean field spin glass since the interactions are of short range. Since the sample exhibits extended ferrimagnetic correlations, our results show that disordered systems share common properties eventually related not only to the complexity of the free-energy landscape in the configuration space, but also to the response to temperature and magnetic field changes. The scaling behavior of the irreversible susceptibility and of the short time relaxation rate of the magnetization show that the temperature renormalizes the barrier heights.
References [1] S. Koutani and G. Gavoille, J. Magn. Magn. Mater. 138 (1994) 237; S. Koutani, Thesis, Nancy, 1993. [2] For a review, see J. Ferre and N. Bontemps, Mater. Sci. Forum 50 (1989) 21. [3] R.G. Palmer, Adv. in Phys. 31 (1982) 669. [4] L. Lundgren, P. Svedlindh, P. Norblad and O. Beckman, Phys. Rev. Lett. 51 (1983) 911. [5] P. Norblad, P. Svedlindh, P. Granberg and L. Lundgren, Phys. Rev. B 35 (1987) 7150. [6] K. Nemoto, J. Phys. A: Math. Gen. 21 (1988) L287. [7] D. Vertechi and M.A. Virasoro, J. Physique 50 (1989) 2325. [8] M. Lederman, R. Orbach, J. Hammann and M. Ocio, J. Appl. Phys. 69 (1991) 5234. [9] M. Lederman, R. Orbach, J. Hamman, M. Ocio and E. Vincent, Phys. Rev. B 44 (1991) 7403.
N ELSEVIER
Journal of Magnetism and Magnetic Materials 152 (1996) 54-60
Journalof magnetism and magnetic ,4~ materials
Non-equilibrium behavior in a disordered magnetic system S. Koutani, G. Gavoille * Laboratoire de Cristallographie et Moddlisation des Matdriaux Mindraux et Biologiques, URA-CNRS 809, Facult~ des Sciences, BP 239, F-54506 Vandoeuvre-lbs-Nancy C£dex, France
Received 15 February 1995
Abstract We have studied the non-equilibrium magnetic state of a thin film of cobalt ferrite with a nanocrystalline structure which shows a frozen disordered magnetic state with extended ferrimagnetic correlations at low temperature. The magnetization has been measured in a wide range of temperatures and magnetic fields, and the irreversible susceptibility and the relaxation rate of the magnetization along the hysteresis loop show a scaling behavior where the scaling field is a temperature-dependent magnetic field H o. The magnetic field H o and the free energy barrier height in spin glasses show a similar temperature dependence. Our measurements, together with extensive studies of the spin glass phase, show that disordered magnetic systems share many common features. We have also observed that the relaxation of the magnetization that follows a magnetic field change has a behavior which is qualitatively similar to that observed in ageing experiments in spin glasses.
1. Introduction We have recently reported on the equilibrium magnetic properties of cobalt ferrite thin films with a nanocrystalline structure [1]. Depending on the preparation, the films exhibit more or less extended ferrimagnetic correlations but no long-range magnetic order at low temperature. We report here on the non-equilibrium magnetic state of the film which has a divergent linear susceptibility but no bulk magnetization in the low-temperature equilibrium state. Magnetization measurements, including irreversible susceptibility and relaxation studies, have been carried out in a wide range of temperatures and magnetic fields. As the most extensive studies of disordered magnetic systems are concerned with mean field (long-range interactions) spin glasses, it seems
* Corresponding author. Email: [email protected];fax: + 33-8340-6492.
very attractive to compare the behavior of our system with that of spin glasses since, apart from the absence of conventional long-range magnetic order, they differ strongly in the range of the interactions and by the nature of the low-temperature spin configuration. Most of the investigations of the spin glasses have been carried out in very small magnetic fields [2], and despite the different magnetic orders and experimental procedures the two systems share many common features.
2. Experimental The magnetization was measured with a VSM operating between 4 and 300 K. The magnetic field was supplied by an electromagnet whose response to a magnetic field change of 1 kOe is less than 5 s, whatever the magnitude of the field. The temperature range of the experiments, 140-300 K, was chosen in
0304-8853/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0304-8853(95)00451-3
S. Koutani, G. Gavoille / Journal of Magnetism and Magnetic Materials 152 (1996) 54-60 L
such a way that an equilibrium state is obtained in the largest available magnetic field (20 kOe). This state is the reference state for all the measurements. As we are concerned only with the irreversible process, the measured magnetization was not corrected for the diamagnetism of the glass substrate and of the sample holder. In order to obtain reliable results, the following procedure was used. All magnetization curves were obtained starting from the reference state. The magnetic field was changed by steps of 1 kOe every 25 s, and the magnetization in a given magnetic field was measured 25 s after the field was set up. For the study of the magnetization relaxation in a given magnetic field, the clock was started as soon as the field was stabilized, i.e. 5 s after the setting of the field.
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We have measured the irreversible susceptibility along the descending branch of the hysteresis loop in negative magnetic fields - H ( H > 0). The irreversible susceptibility at point A in Fig. 1 was obtained as follows. We measure the magnetization M ( - H - h) for positive h along the branch AI' and M( - H + h) along the branch AD. The linear terms given by a polynomial fitting procedure of M ( - H ) - M(-Hh) and of M ( - H + h ) - M ( - H ) , respectively, yields the total XT and the reversible Xr~v susceptibilities. The data of the irreversible susceptibility XL~ = X r - Xrev are reported in Fig. 2 for
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Fig. 2. Irreversiblesusceptibility versus magnetic fields at different temperatures. different temperatures. Within the accuracy of the data, the maximum of the irreversible susceptibility does not depend on the temperature and it appears for a temperature dependent magnetic field Ho(T). This suggests that the irreversible susceptibility may be expressed in terms of a reduced variable which has a fixed point when the temperature changes. Actually, the data corresponding to the different temperatures are on a master curve if the susceptibility is plotted against ( H - H o ) / H o (Fig. 3)+ The temperature dependence of H 0 is given in Fig. 4; the data may be tentatively fitted by the expression
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(1)
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3.2.
H (kOe)
I'
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Fig. 1. Magnetizationcurves obtained in the experiments.
Always starting from the reference state, we have measured the time decay of the magnetization along the descending branch of the hysteresis loops (point A in Fig. 1) for 1000 s after the stabilization of the
S. Koutani, G, Gavoille / Journal of Magnetism and Magnetic Materials 152 (1996)54-60
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Fig. 5. Time decay of the magnetization along the hysteris loop at 140 K. From top to bottom: H = 2, 3, 4, 6, 8, 10 and 12 kOe. the relaxation rate S = - d M / d l n t, w h i c h is reported in Fig. 6 for different temperatures. Note that S and Xirr s h o w a m a x i m u m for the same magnetic
magnetic field. Within the accuracy of our measurements, the magnetization has a logarithmic time decay in any magnetic field at any temperature. The data obtained at 140 K are reported in Fig. 5. This short time relaxation is then well characterized by 7 7
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Fig. 6. Relaxation rate of the magnetization versus magnetic field at different temperatures.
S. Koutani, G. Gavoille/ Journal of Magnetism and Magnetic Materials 152 (1996) 54-60 3!
field H 0, but the magnitude of Sm~x depends on the temperature. A plot of S / H o versus the normalized magnetic field ( H - H o ) / H o is given in Fig. 7. The scatter of the data is greater than in Fig. 3, but if we take into account the fact that the current data are less accurate than the previous ones, we may infer that the scaling holds for S / H o as well as for Xirr-
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The study of the relaxation of the remanent magnetization led us to consider two remanent states. The first state, M d, is obtained from the path I A D in Fig. 1, and the second state, M r results from path I'CR. Owing to the symmetry properties of the hysteresis loop, M d ( - H ) = - M r ( H ) , we need to consider only positive magnetizations. The data at 140 K obtained with the previously described procedure are reported in Fig. 8; striking differences between the relaxations of M d and M r are observed. While M r
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shows a logarithmic time decay in our time window, M d begins to increase before decaying at long time. The time corresponding to the maximum of Md(t) increases with increasing magnetic field. The relaxation of the same macroscopic state, characterized by the magnetization, strongly depends on the history of the sample. M d is reached in increasing magnetic fields, whilst M r is reached in decreasing ones.
S. Koutani, G. Gavoille / Journal of Magnetism and Magnetic Materials 152 (1996)54-60
58
3.4. Relaxation of the magnetization along a demagnetization curve The relaxation of the magnetization has been studied along the curve AD in Fig. 1, with point A corresponding to the magnetic field H 0. The data obtained at 140 K are reported in Fig. 9. The time window of our experiments is very small compared to the largest relaxation time of the system, but we would suggest that the curves have the same shape, increasing at short times and decreasing at long times with a crossover time tc(H o - H ) which increases when H 0 - H increases.
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Fig. 9. Time dependence of the magnetization along the demagnetization curve AD (Fig. 1) at 140 K. From top to bottom: H = 5, 4, 2, 1 and 0 kOe.
4. Discussion Slow and non-exponential relaxation processes are usually interpreted as resulting from a complex free-energy landscape in the configuration space, which consists of disjoint regions separated by free energy barriers. A system prepared in a non-equilibrium state at time t = 0 relaxes toward thermodynamic equilibrium. This non-equilibrium relaxation process may be interpreted as the system populating new regions of the phase space when t increases. If we assume a thermally activated process following an Arrhenius law, the populated region O ( t w) at time t w is bounded by free-energy barriers higher than A(tw)=kTln(tw/to), where t o is a microscopic time. If the probability of escaping from l'2(t w) between times t w and t w + t is very small, then the system is said to be confined in g2(t w) for time t, and is in equilibrium within 12(tw); such a state is referred to as a quasi-equilibrium state [3]. The non-equilibrium relaxation may be studied by looking at the response to the application of a small probing magnetic field at t w. This has been extensively done in spin glasses since the pioneering work by Lundgren et al. [4], who showed that the response shows a maximum at t = t W, interpreted as the crossover between the quasi-equilibrium and the non-equilibrium dynamics. The relaxation of the system prepared in a remanent state by the removal of a large magnetic field is characterized by a continuous decay of the magnetization. After the application of a small positive field at t w, the total magnetization at t w + t first increases with t before decreasing; the crossover between the two regimes appears at t = t w [5]. This results from the opposite signs of the relaxation rate of the remanent magnetization and of the response to the probing magnetic field which dominates at times shorter than t w. Our measurements of the relaxation of the remanent magnetizations show that the relaxation rates of M d and M r have opposite signs at short times although the sign of the non-equilibrium relaxation rate is positive in both cases. The difference arises from the fact that M d is obtained in increasing magnetic fields, while M r is obtained in decreasing ones. We may expect that the equilibrium response to a magnetic field change dominates at short times, so we may infer from the relaxation of M d that the
s. Koutani, G. Gavoille /Journal of Magnetism and Magnetic Materials 152 (1996) 54-60
system is initially confined by free-energy bamers whose heights increase with H since the maximum of M d ( - H , t ) shifts to longer times when H increases from 0 to H 0. The relaxation of the magnetization along the demagnetization curve AD shows a similar behavior with a crossover time tc increasing with H 0 - H . This may also be interpreted as the confinement of the initial state with free-energy bartiers of increasing heights when the magnetic field is removed. In low magnetic fields, the magnetization changes by jumps between nearly similar states of positive magnetization while in large magnetic fields, it is between nearly similar states of negative magnetization. The crossover between the two regimes occurs near H 0, where the jumps may be between states related or nearly related by time reversal symmetry. The magnetic field affects not only the heights of the barriers, but also the magnetization of the relevant states. The magnetization processes cannot be described by a distribution of two-level systems with magnetic field-independent magnetizations. The system moves in the phase space as the magnetic field changes; some regions are populated, while others are depopulated as a consequence of the dependence of the free-energy landscape in the magnetic field. The equilibrium states M(20 kOe) and M ( - 2 0 kOe) are related by time reversal symmetry, so that H 0 may be considered as the magnetic field needed to overcome the lowest time reversal barrier [6,7]. Two metastable states, say a and b, may be characterized by their Hamming distance, which can be written for a system of N Ising spins located at sites i as:
111
]
dab = "~ 1 -- -~ }Zm~imbi ,
(2)
where m a is the thermal average of the magnetization in state a. At 0 K, dab reaches its maximum value dab = 1 if the states a and b are related by time reversal symmetry. Note that in Ising spin glasses with long-range interactions, the height of the barriers is an increasing function of dab [6,7]. Our measurements probe only the low effective energy barriers in a given magnetic field, and the magnitudes of Xirr and S along the hysteresis loop are measures of the similarity between the relevant metastable states.
59
For small H, i.e. in a state obtained by the removal of a large magnetic field, and for large H, the only relevant states are nearly similar, whereas they are very dissimilar near H o where large magnetization jumps are observed. H 0 may be considered as the magnetic field needed to suppress the large energy barriers between very dissimilar states. It is interesting to note that the mean field theory of spin glasses predicts that the states become increasinly similar when the magnetic field increases, and that they are macroscopically similar along the Almeida-Thouless line [8] where the irreversibility vanishes. The scaling analysis of the irreversible susceptibility and of the relaxation rate of the magnetization can be compared with the experimental results obtained by Lederman et al. [9] on AgMn spin glass. They studied the relaxation of the thermoremanent magnetization using the following procedure. The sample is cooled in a small magnetic field from above Tg to a temperature T below Tg and the magnetic field is removed at a time t w after the temperature quench. They observed that ageing at ( T - A T 1, t,<) and at ( T - A T 2, tw2) leads to the same relaxation curve. From their data, the temperature dependence of the barrier height is given by --~
= -a(T) A + b(T).
(3)
On the other hand, the derivative of Eq. (1) with respect to T yields a similar equation for H 0 but with temperature-independent coefficients. The bartier height may be probed in ageing experiments, as well as by the application of a magnetic field. It must be noted, however, that the magnetic field perturbs the state of the system, and the similarities between the two kinds of experiments call for further theoretical investigations on disordered systems.
5. Conclusions We have studied some aspects of the non-equilibrium behavior of a disordered magnetic system in a wide range of magnetic fields and temperatures. We have emphasized the fact that our observations may be related to the structure of the spin glass phase investigated by numerical calculations or by
60
S. Koutani, G. Gavoille / Journal of Magnetism and Magnetic Materials 152 (1996) 54-60
experiments in a zero magnetic field. Our sample is far from being a mean field spin glass since the interactions are of short range. Since the sample exhibits extended ferrimagnetic correlations, our results show that disordered systems share common properties eventually related not only to the complexity of the free-energy landscape in the configuration space, but also to the response to temperature and magnetic field changes. The scaling behavior of the irreversible susceptibility and of the short time relaxation rate of the magnetization show that the temperature renormalizes the barrier heights.
References [1] S. Koutani and G. Gavoille, J. Magn. Magn. Mater. 138 (1994) 237; S. Koutani, Thesis, Nancy, 1993. [2] For a review, see J. Ferre and N. Bontemps, Mater. Sci. Forum 50 (1989) 21. [3] R.G. Palmer, Adv. in Phys. 31 (1982) 669. [4] L. Lundgren, P. Svedlindh, P. Norblad and O. Beckman, Phys. Rev. Lett. 51 (1983) 911. [5] P. Norblad, P. Svedlindh, P. Granberg and L. Lundgren, Phys. Rev. B 35 (1987) 7150. [6] K. Nemoto, J. Phys. A: Math. Gen. 21 (1988) L287. [7] D. Vertechi and M.A. Virasoro, J. Physique 50 (1989) 2325. [8] M. Lederman, R. Orbach, J. Hammann and M. Ocio, J. Appl. Phys. 69 (1991) 5234. [9] M. Lederman, R. Orbach, J. Hamman, M. Ocio and E. Vincent, Phys. Rev. B 44 (1991) 7403.