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Definition of the magnetic ground state using Preisach-based aftereffect models
Physica B 275 (2000) 28}33
De"nition of the magnetic ground state using Preisach-based aftere!ect models C.E. Korman!,*, E. Della Torre" !Institute for Magnetics Research, The George Washington University, Washington, DC 20052, USA "Department of Electrical Engineering and Computer Science, The George Washington University, Washington, DC 20052, USA
Abstract Preisach-based aftere!ect models are employed to de"ne the ground state of a magnetic system. Four di!erent types of processes due to thermal perturbations, AC demagnetization, Curie point demagnetization and DC demagnetization are reviewed and compared with each other. By modeling the time evolution of the magnetization and magnetic "eld on the Preisach plane, it is shown that the asymptotic state of the system due to thermal perturbations is the true magnetic ground state. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Ground state; Aftere!ect; Hysteresis; Magnetic; Preisach model
1. Introduction It is well understood that the physical origin of hysteresis is due to the multiplicity of a large number of metastable states exhibited by such systems. At constant temperature and applied magnetic "elds, large deviations of thermal perturbations cause hysteretic systems to move from higher-energy metastable states to lowerenergy metastable states. As time progresses, such motions become increasingly probable and result in slow changes in the output of the hysteretic system [1]. This phenomenon, which is generally referred to as &aftereffect', or &creep', results in the gradual temporal loss of memory of a hysteretic system. A hysteretic system with many meta-stable states will reach its lowest energy state when it is at thermal equilibrium at an ambient temperature and "xed applied magnetic "eld. This state results in the complete loss of memory of past magnetic "eld and temperature variations. Consequently, the lowest energy
* Correspondence address: Department of EECS, The George Washington University, Washington DC 20052, USA. Tel.: #1-202-994-4952; fax: #1-202-994-0227. E-mail address: [email protected] (C.E. Korman)
state is completely characterized by the statistics of the thermal perturbations. Since the e!ect of all past history is eliminated, one may also refer to the lowest energy state as the anhysteretic state. However, it is important to distinguish the lowest energy or anhysteretic state from a demagnetized state. For instance, under zero applied "eld the state of a magnetic material will relax to its lowest energy state of zero magnetization-albeit, after a very long time. Furthermore, one can point out that there are several di!erent methods to achieve this, such as, AC demagnetization, DC demagnetization and Curie point demagnetization. All of these processes result in zero magnetization in the presence of zero applied "eld which coincides with the magnetization of the lowest energy state of the material with no applied "eld. However, the fact that there is zero magnetization does not imply that this is the lowest energy state. This is because the state of the magnetic system still depends on the particular magnetization history employed to result in zero magnetization and random thermal perturbations will still force the system to decay to its lowest energy state. Therefore, the determination of the lowest energy state of a magnetic material exhibiting hysteresis is important from both the theoretical and practical point of view. The purpose of this work is to review and compare several types of demagnetization processes and de"ne the
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 6 9 1 - 2
C.E. Korman, E. Della Torre / Physica B 275 (2000) 28}33
lowest energy state of a hysteretic material in the framework of Preisach-based aftere!ect models [2,3]. Furthermore, the formalism of the Preisach model will also be employed to describe how a hysteretic system can be brought to the lowest energy state and compare this state with those corresponding to AC, Curie point and DC demagnetization.
2. Magnetic aftere4ect The phenomena of hysteresis and aftere!ect have traditionally been studied along two distinct lines. In phenomenological modeling of hysteresis the Preisach approach has been prominent [4], while the aftere!ect phenomenon has been studied by using thermal activation-type models (see, for instance Refs. [1,5]). In order to determine the lowest energy state, one has to look at the mechanisms by which a hysteretic system reaches thermal equilibrium and, consequently, &forgets' its past magnetization and temperature variation history. For constant applied "elds and temperature, random thermal #uctuations results in the gradual decay of the magnetization and lower the energy of the system towards its lowest state. This phenomenon and the state of a hysteretic system can be modeled by employing Preisachtype hysteresis models. Recently, several groups have proposed Preisach-based aftere!ect models which attempt to unify various hysteresis and aftere!ect phenomena under a common framework [2,3,6}8]. Since, a detailed review of these models is beyond the scope of this paper, we will only concentrate on their predictions regarding the asymptotic state of the system. In particular, we will focus on the Preisach model driven by stochastic inputs [6] and the Preisach}Arrhenius approach [3] to compute the lowest energy state as a function of applied "eld and temperature. 2.1. Preisach model driven by stochastic inputs During the past several years, Korman and Mayergoyz have formulated a uniform approach to the modeling of both hysteresis and aftere!ect. In this approach, the stochastic nature of aftere!ect is emphasized by driving Preisach-type hysteresis models with stochastic inputs, [2,6]. This approach has several inherent advantages compared to the more traditional thermal activation-type models. First, since the approach is based on Preisach-type models, the hysteretic nature (Preisach function) and the input history of the system are explicitly accounted for in the model. Second, the nature of thermal perturbations which gradually result in the decay or loss of &memory' of the hysteretic system are explicitly entered into the model. Third, thermal activation models are regarded as &noninteracting particle' models. As such, this model is an &interacting particle'
29
model and contains thermal activation-type models as a particular case when hysteresis is due to symmetric loops only. Fourth, thermal activation-type models are scalar in nature, whereas Preisach-type models have vector generalizations [9]. In this approach, Preisach-type hysteresis transducers are driven by stochastic inputs which model random thermal perturbations of the magnetic "eld. The stochastic inputs are modeled by zero mean stochastic processes which are superimposed on a constant applied input H . ! In the simplest case, the random thermal perturbations are modeled by discrete-time independent identically distributed (i.i.d.) random processes. The random perturbations may have arbitrary probability distributions but are usually assumed to be Gaussian. Since the input is a random process, the output of the Preisach model is a random process, as well. Consequently, it is of interest to determine the time evolution of the expected value of the output. The basic Preisach model of hysteresis is constructed by the superposition of rectangular hysteresis loop with switching up and down values of a and b, respectively. In previous work, it was shown that the expected value of the output is given by the following expression [2]:
PP
fM "fM # n =
awb
k(a, b)[0(a, b)! f(a, b)]rn da db, ab
(1)
where
PP
fM " lim fM " = n n?=
awb
k(a, b)f(a, b) da db.
(2)
Here, n denotes the discrete time index, fM is the exn pected value of the output of the Preisach transducer, fM is the asymptotic expected value of the output, and = k(a, b) is the Preisach function. The initial state of each rectangular loop is given by 0(a, b). This function on the Preisach plane denotes the past input history which is the &staircase' interface that separates the regions where the output of the loops are equal to #1 and !1, respectively. As a result of thermal perturbations the expected value of the output for each rectangular loop decays to its steady-state value f(a, b) with some geometric decay rate r . This stationary value only depends on the probabilab ity density function of the random thermal perturbations. Both the asymptotic expected value f(a, b) and the decay rate r can be computed explicitly in terms of this noise ab probability density function, as follows: :=o(x; H ) dx!:b o(x; H ) dx ! ~= ! f(a, b; H )" a , ! :=o(x; H ) dx#:b o(x; H ) dx a ! ~= !
P
r (H )" ab !
a o(x; H ) dx, ! b
(3)
(4)
30
C.E. Korman, E. Della Torre / Physica B 275 (2000) 28}33
where o(x; H ) denotes the probability density function of ! the input with a mean value of H . As a result of thermal ! perturbations, it is clear that all rectangular loops asymptotically &forget' their past input history, h(a, b). It is natural for the random thermal perturbations to be evenly distributed about the mean applied "eld. Usually, the random thermal perturbations are modeled by zero mean Gaussian random variables with standard deviation p. In this particular case, the standard deviation can be identi"ed directly from standard aftere!ect measurements [10,11]. Now consider a rectangular hysteresis loop with switching up "eld a and switching down "eld b. According to Eq. (3), if p is small then it is clear that the asymptotic expected value of all rectangular loops for which H ,(a#b)/2'H will be approximately #1 i ! and all rectangular loops for which H (H will be i ! approximately !1. This asymptotic expected value monotonically increases from !1 to #1 as a function of H and is equal to zero at H "H . In other words, all i i ! such loops on the Preisach plane have equal probability of being in the #1 and !1 states, respectively. Fig. 1 shows the asymptotic expected value of all rectangular loops on the Preisach plane. We denote this to be the lowest energy state since, by de"nition, the expected value of the output no longer varies due to thermal perturbations. Since the probability density function of the thermal perturbations depends on the ambient temperature, it is clear that the lowest energy state is only a function of temperature. One can also denote this to be the anhysteretic state, as well, since the state is completely independent of all possible past input and temperature variations. This is a crucial fact and will di!erentiate this state from those resulting from other types of demagnetization processes, such as Curie point, AC and DC demagnetizations. 2.2. The Preisach}Arrhenius model More recently, an alternate method to model aftereffect in the framework of Preisach models was developed by Della Torre et al. [3]. The approach is based on the well-known Arrhenius law to calculate the calculate the decay rate of the magnetization of individual hysterons. According to this model, the magnetization as a function of time can be computed by
PP
f (t)"
awb
k(a, b)Q(a, b, t) da db.
(5)
The function Q(a, b, t) denotes the time-dependent state function of each hysteron. According to the model this function obeys the following di!erential equation
A
B
dQ q #q ~ Q"Q , # ` = dt q q ` ~
(6)
Fig. 1. An example of a staircase interface and the lowest energy state on the Preisach plane.
where
A B
A B
a!h h!b , q "q exp . (7) ~ 0 h h & & For a constant applied "eld, it is easy to show that the steady-state solution of Eq. (6) is given as follows:
q "q exp ` 0
A
B
A
B
q !q a#b!2h ` ~ "tanh q q 2h ` ~ & h !h "tanh i . (8) h & Here, h"H#a m is the operative "eld, a is the movm m ing parameter, h "(a#b)/2, 1/q is an attempt frei 0 quency in the order of 1010 Hz and h is a #uctuation & "eld which is equal to k¹/k M<. In this formulation 0 < denotes the activation model which can be calculated from micromagnetic calculations. The initial value of Q is given by the magnetization history of the sample and is characterized by the staircase interface on the Preisach plane. Therefore, starting from an initial state where Q is discontinuous about the magnetization history staircase, as time progresses, the state becomes continuous over the Preisach plane. In particular, for large time the asymptotic value of the output is given by Q " =
A
PP
B
A
B
h !h i da db. (9) h awb & From Eqs. (8) and (9), it is clear that both models predict similar results for the asymptotic case. Both models predict that the asymptotic state of a rectangular loop is approximately #1 if h'h and !1 if h(h . Furtheri i more, as shown in Fig. 1, there is a smooth monotonic transition for the asymptotic state as a function of h . i Furthermore, in the low thermal perturbation limit as
m (h)" =
k(a, b) tanh
C.E. Korman, E. Della Torre / Physica B 275 (2000) 28}33
h goes to zero, Q approaches the sign function at h"h . & i This is the same results as that predicted by the previous model which can be seen from Eq. (3) as the noise standard deviation p goes to zero. A similar asymptotic result is also obtained in Bertotti's model from the thermodynamic point of view and it is a special case of the above two models [7]. Again it is important to emphasize that the asymptotic state predicted by the two models are independent of the past magnetization and temperature variation history. Since, by the very construction of the models, the asymptotic state has no memory of the past history, this state can be denoted as the anhysteretic state. Furthermore, since the time variation of the magnetization goes to zero under thermal perturbations, the asymptotic state must be the lowest energy state of the hysteretic system. Although the derivation of the two models have di!erent emphasis, both models have similar predictions regarding the lowest energy state of a hysteretic system.
3. Demagnetized states Thermal relaxations allow an hysteretic system to reach the lowest energy state for a given applied "eld and temperature. However, for most magnetic materials this may not be of practical interest since the relaxation to the lowest energy state takes a long time. This is especially the case when the sample is to be demagnetized to achieve zero magnetization and magnetic "eld. In this case, there are several alternative processes which can bring the hysteretic system to zero magnetization and "eld in a practical way, such as, AC, Curie point and DC demagnetization processes. However, none of these processes bring the hysteretic system to its lowest energy state. As a result, the hysteretic system will attempt to reach its lowest energy state due to thermal perturbations and the magnetization may temporarily deviate from zero. This may happen despite the fact that zero magnetization and magnetic "eld are established in the sample upon the completion of the demagnetization process. In this section we will compare the state of an hysteretic system after such demagnetization processes to the state due to thermal relaxation. It will clearly be shown that none of these states are the lowest energy state for the hysteretic system.
31
approaches the value given in Eq. (8). As a result, all loops along the diagonal line H "0 on the Preisach i plane will have zero magnetization. Since the asymptotic state of the rectangular loops is an odd function about H "0, according to Eqs. (2), (3), (8) and(9), both i Preisach-based aftere!ect models predict that fM "m "0, H "h "0. (10) = = i i The asymptotic state of the system for H "0 is shown in ! Fig. 2. It is important to note that, as a function of H , i there is a smooth transition on the Preisach plane from !1 to #1 in the (expected) value of rectangular loops. Only in the limiting case of small thermal perturbations does the transition become an abrupt one on the Preisach plane. 3.2. AC demagnetization In the case of AC demagnetization, a gradually decreasing alternating magnetic "eld is applied to the hysteretic system. If such an alternating "eld is centered around zero then the magnetic "eld and magnetization will eventually be forced to zero. Upon the completion of such a process, the state of the system on the Preisach plane is a sawtooth interface about the line H "0 (Fig. i 2). For a process with arbitrarily close consecutive minima and maxima, the sawtooth interface approaches an abrupt transition at H "0. Therefore, the resulting magi netization will become zero. However, as it is shown in Fig. 2, the AC demagnetized state is not the lowest energy state. Therefore, the hysteretic system will continue to relax to the lowest energy state as predicted by the aftere!ect models. 3.3. Curie point demagnetization Consider the demagnetization process where the magnetic material is heated above its Curie temperature and
3.1. Demagnetization due to afterewect Under zero applied "eld, thermal perturbations will eventually bring an hysteretic system to zero magnetization. In the framework of Preisach model driven by stochastic inputs, this will be achieved when the expected value of the rectangular loops approach the value given in Eq. (3). In the framework of the Preisach}Arrhenius model, this will be achieved when the state function
Fig. 2. Comparison of the staircase interface resulting from an AC demagnetization process to the lowest energy state on the Preisach plane.
32
C.E. Korman, E. Della Torre / Physica B 275 (2000) 28}33
Fig. 3. Comparison of the state resulting from a Curie point demagnetization process to the lowest energy state on the Preisach plane.
gradually cooled down under zero applied magnetic "eld. Once the material is cooled to room temperature, the resulting magnetization is zero. However, at this point the hysteretic system is still not at its lowest energy state. Fig. 3 shows the resulting state of the hysteretic system in the Preisach plane due to this process. It can be seen that all loops with a(0 will be in the #1 state, and all loops with b'0 will be in the !1 state. Furthermore, all loops with a'0 and b(0 will have equal probability of being in the #1 and !1 states since the system will show no preference to either one of these states upon the lowering of temperature below the Curie point. Therefore, all such loops will have an (expected) value of zero. Comparing this state to that due to aftere!ect, it is again clear that Curie point demagnetization does not result in the lowest energy state. Consequently, the system will continue to relax despite the fact that both the magnetization and the magnetic "eld are zero. 3.4. DC demagnetization The DC demagnetization is perhaps the simplest method to demagnetize a sample. For instance, from positive saturation, the magnetic "eld is reversed to such a value that once the "eld is increased and brought to zero the resulting magnetization is zero. Fig. 4 shows the reversal curve for such a process and Fig. 5 shows the corresponding state of magnetization on the Preisach plane which is a one-step staircase interface about the line H "0. Clearly, this is not the lowest energy state i and thermal perturbations will force the system to relax to the lowest energy state. This relaxation will "rst increase the magnetization in the positive direction, and then in the negative direction before relaxing back to zero magnetization. Similar results of nonmonotonic relaxation have been observed in Nd}Fe}B hard magnets exhibiting the spring e!ect by LoBue et al. [12] and
Fig. 4. A possible reversal curve for DC demagnetization.
Fig. 5. Comparison of the staircase interface resulting from a DC demagnetization process to the lowest energy state on the Preisach plane.
predicted by the thermodynamic generalization of the Preisach model [7]. More recently, Swartzendruber et al. have observed this e!ect in a metal perticle tape and modeled it by the Preisach}Arrhenius approach [13]. These observations have been made at nonzero applied "elds. At zero applied "elds, however, such a relaxation may be di$cult to observe for most practical recording materials due to the vary large energy barriers which must be surpassed by thermal perturbations.
4. Conclusion Two Preisach-based aftere!ect models were reviewed with special emphasis on their prediction of the temporal asymptotic state. It was shown that the resulting asymptotic state due to thermal relaxations is the lowest energy state of the hysteretic material. It was also shown that this state is independent of the magnetization and temperature variation history. The dependence of this lowest energy state on the Preisach function, temperature and
C.E. Korman, E. Della Torre / Physica B 275 (2000) 28}33
other material properties was brie#y discussed. In the case of zero applied magnetic "elds, the asymptotic state resulting from thermal perturbations was compared to those resulting from the AC, Curie point and DC magnetization processes. Employing the state of the rectangular loops on the Preisach plane, it was shown that none of these processes result in the lowest energy state. As a result, it was emphasized that the hysteretic system will continue to relax to its lowest energy state despite the fact that magnetization and the magnetic "elds are zero.
References [1] R. Street, J.C. Wooley, Proc. Phys. Soc. A 62 (1949) 562. [2] I.D. Mayergoyz, C.E. Korman, J. Appl. Phys. 69 (1991) 2128.
33
[3] E. Della Torre, L.H. Bennett, L.J. Swartzendruber, Mater. Res. Soc. Symp. Proc. 517 (1998) 291. [4] I.D. Mayergoyz, Mathematical Models of Hysteresis, Springer, New York, 1990. [5] S.H. Charap, J. Appl. Phys. 63 (1988) 2054. [6] C.E. Korman, I.D. Mayergoyz, IEEE Trans. Magn. 32 (5) (1996) 4204. [7] G. Bertotti, Phys. Rev. Lett. 76 (1996) 1739. [8] P.D. Mitchler, E. Dan Dahlberg, E.E. Wesseling, R.M. Roshko, IEEE Trans. Magn. 32 (1996) 3185. [9] G. Friedman, I.D. Mayergoyz, IEEE Trans. Magn. 28 (1992) 2262. [10] C.E. Korman, P. Rugkwamsook, IEEE Trans. Magn. 33 (5) (1997) 4176. [11] P. Rugkwamsook, C.E. Korman, IEEE Trans. Magn. 34 (4) (1998) 1863. [12] M. LoBue, V. Basso, G. Bertotti, K.H. Muller, IEEE Trans. Magn. 33 (1997) 3862. [13] L.J. Swartzendruber, L.H. Bennett, E. Della Torre, H.J. Brown, J.H. Judy, Mater. Res. Soc. Symp. Proc. 517 (1998) 360.
De"nition of the magnetic ground state using Preisach-based aftere!ect models C.E. Korman!,*, E. Della Torre" !Institute for Magnetics Research, The George Washington University, Washington, DC 20052, USA "Department of Electrical Engineering and Computer Science, The George Washington University, Washington, DC 20052, USA
Abstract Preisach-based aftere!ect models are employed to de"ne the ground state of a magnetic system. Four di!erent types of processes due to thermal perturbations, AC demagnetization, Curie point demagnetization and DC demagnetization are reviewed and compared with each other. By modeling the time evolution of the magnetization and magnetic "eld on the Preisach plane, it is shown that the asymptotic state of the system due to thermal perturbations is the true magnetic ground state. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Ground state; Aftere!ect; Hysteresis; Magnetic; Preisach model
1. Introduction It is well understood that the physical origin of hysteresis is due to the multiplicity of a large number of metastable states exhibited by such systems. At constant temperature and applied magnetic "elds, large deviations of thermal perturbations cause hysteretic systems to move from higher-energy metastable states to lowerenergy metastable states. As time progresses, such motions become increasingly probable and result in slow changes in the output of the hysteretic system [1]. This phenomenon, which is generally referred to as &aftereffect', or &creep', results in the gradual temporal loss of memory of a hysteretic system. A hysteretic system with many meta-stable states will reach its lowest energy state when it is at thermal equilibrium at an ambient temperature and "xed applied magnetic "eld. This state results in the complete loss of memory of past magnetic "eld and temperature variations. Consequently, the lowest energy
* Correspondence address: Department of EECS, The George Washington University, Washington DC 20052, USA. Tel.: #1-202-994-4952; fax: #1-202-994-0227. E-mail address: [email protected] (C.E. Korman)
state is completely characterized by the statistics of the thermal perturbations. Since the e!ect of all past history is eliminated, one may also refer to the lowest energy state as the anhysteretic state. However, it is important to distinguish the lowest energy or anhysteretic state from a demagnetized state. For instance, under zero applied "eld the state of a magnetic material will relax to its lowest energy state of zero magnetization-albeit, after a very long time. Furthermore, one can point out that there are several di!erent methods to achieve this, such as, AC demagnetization, DC demagnetization and Curie point demagnetization. All of these processes result in zero magnetization in the presence of zero applied "eld which coincides with the magnetization of the lowest energy state of the material with no applied "eld. However, the fact that there is zero magnetization does not imply that this is the lowest energy state. This is because the state of the magnetic system still depends on the particular magnetization history employed to result in zero magnetization and random thermal perturbations will still force the system to decay to its lowest energy state. Therefore, the determination of the lowest energy state of a magnetic material exhibiting hysteresis is important from both the theoretical and practical point of view. The purpose of this work is to review and compare several types of demagnetization processes and de"ne the
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 6 9 1 - 2
C.E. Korman, E. Della Torre / Physica B 275 (2000) 28}33
lowest energy state of a hysteretic material in the framework of Preisach-based aftere!ect models [2,3]. Furthermore, the formalism of the Preisach model will also be employed to describe how a hysteretic system can be brought to the lowest energy state and compare this state with those corresponding to AC, Curie point and DC demagnetization.
2. Magnetic aftere4ect The phenomena of hysteresis and aftere!ect have traditionally been studied along two distinct lines. In phenomenological modeling of hysteresis the Preisach approach has been prominent [4], while the aftere!ect phenomenon has been studied by using thermal activation-type models (see, for instance Refs. [1,5]). In order to determine the lowest energy state, one has to look at the mechanisms by which a hysteretic system reaches thermal equilibrium and, consequently, &forgets' its past magnetization and temperature variation history. For constant applied "elds and temperature, random thermal #uctuations results in the gradual decay of the magnetization and lower the energy of the system towards its lowest state. This phenomenon and the state of a hysteretic system can be modeled by employing Preisachtype hysteresis models. Recently, several groups have proposed Preisach-based aftere!ect models which attempt to unify various hysteresis and aftere!ect phenomena under a common framework [2,3,6}8]. Since, a detailed review of these models is beyond the scope of this paper, we will only concentrate on their predictions regarding the asymptotic state of the system. In particular, we will focus on the Preisach model driven by stochastic inputs [6] and the Preisach}Arrhenius approach [3] to compute the lowest energy state as a function of applied "eld and temperature. 2.1. Preisach model driven by stochastic inputs During the past several years, Korman and Mayergoyz have formulated a uniform approach to the modeling of both hysteresis and aftere!ect. In this approach, the stochastic nature of aftere!ect is emphasized by driving Preisach-type hysteresis models with stochastic inputs, [2,6]. This approach has several inherent advantages compared to the more traditional thermal activation-type models. First, since the approach is based on Preisach-type models, the hysteretic nature (Preisach function) and the input history of the system are explicitly accounted for in the model. Second, the nature of thermal perturbations which gradually result in the decay or loss of &memory' of the hysteretic system are explicitly entered into the model. Third, thermal activation models are regarded as &noninteracting particle' models. As such, this model is an &interacting particle'
29
model and contains thermal activation-type models as a particular case when hysteresis is due to symmetric loops only. Fourth, thermal activation-type models are scalar in nature, whereas Preisach-type models have vector generalizations [9]. In this approach, Preisach-type hysteresis transducers are driven by stochastic inputs which model random thermal perturbations of the magnetic "eld. The stochastic inputs are modeled by zero mean stochastic processes which are superimposed on a constant applied input H . ! In the simplest case, the random thermal perturbations are modeled by discrete-time independent identically distributed (i.i.d.) random processes. The random perturbations may have arbitrary probability distributions but are usually assumed to be Gaussian. Since the input is a random process, the output of the Preisach model is a random process, as well. Consequently, it is of interest to determine the time evolution of the expected value of the output. The basic Preisach model of hysteresis is constructed by the superposition of rectangular hysteresis loop with switching up and down values of a and b, respectively. In previous work, it was shown that the expected value of the output is given by the following expression [2]:
PP
fM "fM # n =
awb
k(a, b)[0(a, b)! f(a, b)]rn da db, ab
(1)
where
PP
fM " lim fM " = n n?=
awb
k(a, b)f(a, b) da db.
(2)
Here, n denotes the discrete time index, fM is the exn pected value of the output of the Preisach transducer, fM is the asymptotic expected value of the output, and = k(a, b) is the Preisach function. The initial state of each rectangular loop is given by 0(a, b). This function on the Preisach plane denotes the past input history which is the &staircase' interface that separates the regions where the output of the loops are equal to #1 and !1, respectively. As a result of thermal perturbations the expected value of the output for each rectangular loop decays to its steady-state value f(a, b) with some geometric decay rate r . This stationary value only depends on the probabilab ity density function of the random thermal perturbations. Both the asymptotic expected value f(a, b) and the decay rate r can be computed explicitly in terms of this noise ab probability density function, as follows: :=o(x; H ) dx!:b o(x; H ) dx ! ~= ! f(a, b; H )" a , ! :=o(x; H ) dx#:b o(x; H ) dx a ! ~= !
P
r (H )" ab !
a o(x; H ) dx, ! b
(3)
(4)
30
C.E. Korman, E. Della Torre / Physica B 275 (2000) 28}33
where o(x; H ) denotes the probability density function of ! the input with a mean value of H . As a result of thermal ! perturbations, it is clear that all rectangular loops asymptotically &forget' their past input history, h(a, b). It is natural for the random thermal perturbations to be evenly distributed about the mean applied "eld. Usually, the random thermal perturbations are modeled by zero mean Gaussian random variables with standard deviation p. In this particular case, the standard deviation can be identi"ed directly from standard aftere!ect measurements [10,11]. Now consider a rectangular hysteresis loop with switching up "eld a and switching down "eld b. According to Eq. (3), if p is small then it is clear that the asymptotic expected value of all rectangular loops for which H ,(a#b)/2'H will be approximately #1 i ! and all rectangular loops for which H (H will be i ! approximately !1. This asymptotic expected value monotonically increases from !1 to #1 as a function of H and is equal to zero at H "H . In other words, all i i ! such loops on the Preisach plane have equal probability of being in the #1 and !1 states, respectively. Fig. 1 shows the asymptotic expected value of all rectangular loops on the Preisach plane. We denote this to be the lowest energy state since, by de"nition, the expected value of the output no longer varies due to thermal perturbations. Since the probability density function of the thermal perturbations depends on the ambient temperature, it is clear that the lowest energy state is only a function of temperature. One can also denote this to be the anhysteretic state, as well, since the state is completely independent of all possible past input and temperature variations. This is a crucial fact and will di!erentiate this state from those resulting from other types of demagnetization processes, such as Curie point, AC and DC demagnetizations. 2.2. The Preisach}Arrhenius model More recently, an alternate method to model aftereffect in the framework of Preisach models was developed by Della Torre et al. [3]. The approach is based on the well-known Arrhenius law to calculate the calculate the decay rate of the magnetization of individual hysterons. According to this model, the magnetization as a function of time can be computed by
PP
f (t)"
awb
k(a, b)Q(a, b, t) da db.
(5)
The function Q(a, b, t) denotes the time-dependent state function of each hysteron. According to the model this function obeys the following di!erential equation
A
B
dQ q #q ~ Q"Q , # ` = dt q q ` ~
(6)
Fig. 1. An example of a staircase interface and the lowest energy state on the Preisach plane.
where
A B
A B
a!h h!b , q "q exp . (7) ~ 0 h h & & For a constant applied "eld, it is easy to show that the steady-state solution of Eq. (6) is given as follows:
q "q exp ` 0
A
B
A
B
q !q a#b!2h ` ~ "tanh q q 2h ` ~ & h !h "tanh i . (8) h & Here, h"H#a m is the operative "eld, a is the movm m ing parameter, h "(a#b)/2, 1/q is an attempt frei 0 quency in the order of 1010 Hz and h is a #uctuation & "eld which is equal to k¹/k M<. In this formulation 0 < denotes the activation model which can be calculated from micromagnetic calculations. The initial value of Q is given by the magnetization history of the sample and is characterized by the staircase interface on the Preisach plane. Therefore, starting from an initial state where Q is discontinuous about the magnetization history staircase, as time progresses, the state becomes continuous over the Preisach plane. In particular, for large time the asymptotic value of the output is given by Q " =
A
PP
B
A
B
h !h i da db. (9) h awb & From Eqs. (8) and (9), it is clear that both models predict similar results for the asymptotic case. Both models predict that the asymptotic state of a rectangular loop is approximately #1 if h'h and !1 if h(h . Furtheri i more, as shown in Fig. 1, there is a smooth monotonic transition for the asymptotic state as a function of h . i Furthermore, in the low thermal perturbation limit as
m (h)" =
k(a, b) tanh
C.E. Korman, E. Della Torre / Physica B 275 (2000) 28}33
h goes to zero, Q approaches the sign function at h"h . & i This is the same results as that predicted by the previous model which can be seen from Eq. (3) as the noise standard deviation p goes to zero. A similar asymptotic result is also obtained in Bertotti's model from the thermodynamic point of view and it is a special case of the above two models [7]. Again it is important to emphasize that the asymptotic state predicted by the two models are independent of the past magnetization and temperature variation history. Since, by the very construction of the models, the asymptotic state has no memory of the past history, this state can be denoted as the anhysteretic state. Furthermore, since the time variation of the magnetization goes to zero under thermal perturbations, the asymptotic state must be the lowest energy state of the hysteretic system. Although the derivation of the two models have di!erent emphasis, both models have similar predictions regarding the lowest energy state of a hysteretic system.
3. Demagnetized states Thermal relaxations allow an hysteretic system to reach the lowest energy state for a given applied "eld and temperature. However, for most magnetic materials this may not be of practical interest since the relaxation to the lowest energy state takes a long time. This is especially the case when the sample is to be demagnetized to achieve zero magnetization and magnetic "eld. In this case, there are several alternative processes which can bring the hysteretic system to zero magnetization and "eld in a practical way, such as, AC, Curie point and DC demagnetization processes. However, none of these processes bring the hysteretic system to its lowest energy state. As a result, the hysteretic system will attempt to reach its lowest energy state due to thermal perturbations and the magnetization may temporarily deviate from zero. This may happen despite the fact that zero magnetization and magnetic "eld are established in the sample upon the completion of the demagnetization process. In this section we will compare the state of an hysteretic system after such demagnetization processes to the state due to thermal relaxation. It will clearly be shown that none of these states are the lowest energy state for the hysteretic system.
31
approaches the value given in Eq. (8). As a result, all loops along the diagonal line H "0 on the Preisach i plane will have zero magnetization. Since the asymptotic state of the rectangular loops is an odd function about H "0, according to Eqs. (2), (3), (8) and(9), both i Preisach-based aftere!ect models predict that fM "m "0, H "h "0. (10) = = i i The asymptotic state of the system for H "0 is shown in ! Fig. 2. It is important to note that, as a function of H , i there is a smooth transition on the Preisach plane from !1 to #1 in the (expected) value of rectangular loops. Only in the limiting case of small thermal perturbations does the transition become an abrupt one on the Preisach plane. 3.2. AC demagnetization In the case of AC demagnetization, a gradually decreasing alternating magnetic "eld is applied to the hysteretic system. If such an alternating "eld is centered around zero then the magnetic "eld and magnetization will eventually be forced to zero. Upon the completion of such a process, the state of the system on the Preisach plane is a sawtooth interface about the line H "0 (Fig. i 2). For a process with arbitrarily close consecutive minima and maxima, the sawtooth interface approaches an abrupt transition at H "0. Therefore, the resulting magi netization will become zero. However, as it is shown in Fig. 2, the AC demagnetized state is not the lowest energy state. Therefore, the hysteretic system will continue to relax to the lowest energy state as predicted by the aftere!ect models. 3.3. Curie point demagnetization Consider the demagnetization process where the magnetic material is heated above its Curie temperature and
3.1. Demagnetization due to afterewect Under zero applied "eld, thermal perturbations will eventually bring an hysteretic system to zero magnetization. In the framework of Preisach model driven by stochastic inputs, this will be achieved when the expected value of the rectangular loops approach the value given in Eq. (3). In the framework of the Preisach}Arrhenius model, this will be achieved when the state function
Fig. 2. Comparison of the staircase interface resulting from an AC demagnetization process to the lowest energy state on the Preisach plane.
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C.E. Korman, E. Della Torre / Physica B 275 (2000) 28}33
Fig. 3. Comparison of the state resulting from a Curie point demagnetization process to the lowest energy state on the Preisach plane.
gradually cooled down under zero applied magnetic "eld. Once the material is cooled to room temperature, the resulting magnetization is zero. However, at this point the hysteretic system is still not at its lowest energy state. Fig. 3 shows the resulting state of the hysteretic system in the Preisach plane due to this process. It can be seen that all loops with a(0 will be in the #1 state, and all loops with b'0 will be in the !1 state. Furthermore, all loops with a'0 and b(0 will have equal probability of being in the #1 and !1 states since the system will show no preference to either one of these states upon the lowering of temperature below the Curie point. Therefore, all such loops will have an (expected) value of zero. Comparing this state to that due to aftere!ect, it is again clear that Curie point demagnetization does not result in the lowest energy state. Consequently, the system will continue to relax despite the fact that both the magnetization and the magnetic "eld are zero. 3.4. DC demagnetization The DC demagnetization is perhaps the simplest method to demagnetize a sample. For instance, from positive saturation, the magnetic "eld is reversed to such a value that once the "eld is increased and brought to zero the resulting magnetization is zero. Fig. 4 shows the reversal curve for such a process and Fig. 5 shows the corresponding state of magnetization on the Preisach plane which is a one-step staircase interface about the line H "0. Clearly, this is not the lowest energy state i and thermal perturbations will force the system to relax to the lowest energy state. This relaxation will "rst increase the magnetization in the positive direction, and then in the negative direction before relaxing back to zero magnetization. Similar results of nonmonotonic relaxation have been observed in Nd}Fe}B hard magnets exhibiting the spring e!ect by LoBue et al. [12] and
Fig. 4. A possible reversal curve for DC demagnetization.
Fig. 5. Comparison of the staircase interface resulting from a DC demagnetization process to the lowest energy state on the Preisach plane.
predicted by the thermodynamic generalization of the Preisach model [7]. More recently, Swartzendruber et al. have observed this e!ect in a metal perticle tape and modeled it by the Preisach}Arrhenius approach [13]. These observations have been made at nonzero applied "elds. At zero applied "elds, however, such a relaxation may be di$cult to observe for most practical recording materials due to the vary large energy barriers which must be surpassed by thermal perturbations.
4. Conclusion Two Preisach-based aftere!ect models were reviewed with special emphasis on their prediction of the temporal asymptotic state. It was shown that the resulting asymptotic state due to thermal relaxations is the lowest energy state of the hysteretic material. It was also shown that this state is independent of the magnetization and temperature variation history. The dependence of this lowest energy state on the Preisach function, temperature and
C.E. Korman, E. Della Torre / Physica B 275 (2000) 28}33
other material properties was brie#y discussed. In the case of zero applied magnetic "elds, the asymptotic state resulting from thermal perturbations was compared to those resulting from the AC, Curie point and DC magnetization processes. Employing the state of the rectangular loops on the Preisach plane, it was shown that none of these processes result in the lowest energy state. As a result, it was emphasized that the hysteretic system will continue to relax to its lowest energy state despite the fact that magnetization and the magnetic "elds are zero.
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[3] E. Della Torre, L.H. Bennett, L.J. Swartzendruber, Mater. Res. Soc. Symp. Proc. 517 (1998) 291. [4] I.D. Mayergoyz, Mathematical Models of Hysteresis, Springer, New York, 1990. [5] S.H. Charap, J. Appl. Phys. 63 (1988) 2054. [6] C.E. Korman, I.D. Mayergoyz, IEEE Trans. Magn. 32 (5) (1996) 4204. [7] G. Bertotti, Phys. Rev. Lett. 76 (1996) 1739. [8] P.D. Mitchler, E. Dan Dahlberg, E.E. Wesseling, R.M. Roshko, IEEE Trans. Magn. 32 (1996) 3185. [9] G. Friedman, I.D. Mayergoyz, IEEE Trans. Magn. 28 (1992) 2262. [10] C.E. Korman, P. Rugkwamsook, IEEE Trans. Magn. 33 (5) (1997) 4176. [11] P. Rugkwamsook, C.E. Korman, IEEE Trans. Magn. 34 (4) (1998) 1863. [12] M. LoBue, V. Basso, G. Bertotti, K.H. Muller, IEEE Trans. Magn. 33 (1997) 3862. [13] L.J. Swartzendruber, L.H. Bennett, E. Della Torre, H.J. Brown, J.H. Judy, Mater. Res. Soc. Symp. Proc. 517 (1998) 360.