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The thermodynamics of magnetic aftereffect
ARTICLE IN PRESS
Physica B 343 (2004) 267–274
The thermodynamics of magnetic aftereffect E. Della Torre*, L.H. Bennett Electrical and Computer Engineering Department, The George Washington University, Washington, DC 20052, USA
Abstract Thermal magnetic aftereffect is due to the interaction between magnons (spin waves) and the change of equilibrium magnetization states. This non-equilibrium process is controlled by (i) the height of the energy barriers between magnetization states and (ii) by the chemical potential that determines how many magnons (thermal quanta) are available. Non-Arrhenius behavior is observed in a non-monotonic variation of decay rate with temperature observed in nanoparticulate media at low temperature. We attribute this behavior to quantum statistics and the nonuniform density of states in energy space. At low temperatures, Bose–Einstein condensation of the magnons is observed in nanoparticulate systems. In these materials, the decay rate increases to a maximum at low temperatures above the condensation temperature. A standard method of evaluating permanent magnet materials is to accelerate the decay rate, by applying a reverse holding field of the order of coercivity. We show that the problem with this technique is the ratio of the decay rate in zero field to the decay rate at the coercivity is a function of the material properties. This model shows excellent agreement with measurements of the variation of the thermal magnetic aftereffect at low temperatures in both nanograin iron powders and MP tapes. r 2003 Elsevier B.V. All rights reserved. Keywords: Magnetic aftereffect; Chemical potential; Thermal activation
1. Introduction To understand fully the temperature dependence of the magnetic aftereffect of small particles that have exhibited non-Arrhenius behavior [1], it is useful to elaborate their thermodynamic framework. For simplicity, we will consider an uncomplicated Ising model for the particles so that we can neglect many ramifications associated with a ferromagnetic thermodynamic system. In order to simplify the analysis, we will assume that the basic particles are all identical and perfectly ordered. *Corresponding author. Tel.: +1-202-994-0410; fax: +1202-994-0227. E-mail address: [email protected] (E. Della Torre).
We picture small particles as having only two equilibrium states, up or down. This is a good approximation in the regime of slow relaxation ðtimes > 1 s), but it is of course not what is happening in the Larmor frequency regime ðo107 s). In this time regime, the spins are precessing and, depending on the temperature, individual spins may reverse, creating a spin wave. In this paper, we will not address the problems associated with the distribution in size, shape and physical properties of the particles in the system. Any physical properties, such as saturation magnetization, anisotropy, and crystal lattice dimensions which vary with temperature, and the possibility of phase transitions will also be neglected in this paper. The distribution in
0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.08.108
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parameters can best be handled in the Preisach model framework [2] which has been shown to be successful in describing many properties of the system. The Preisach model, considers each mesoscopic magnetic particle, called a hysteron, to have a distribution in particle properties. This results in replacing the exponential decay of the magnetization by one that decays linearly with log-time.
2. Arrhenius law We will view the material as the one consisting of loosely coupled magnetic particles. These particles at finite temperatures will support magnons that are the wave-particle duals of spin waves. When one of these magnons interacts with the particle, if it has sufficient energy, it may reverse the magnetization of the particle. These magnons have spin one and thus are bosons, that is, they obey Bose–Einstein (BE) statistics. In the limit of high temperatures, BE statistics go over to Maxwell–Boltzmann (MB) statistics. If the energy of the magnon is less than that of the barrier energy to a reversal of the magnetization, then the collision is purely elastic. The frequency of particle reversal, f ; using magnons is the product of the attempt frequency fA and the number of magnons that have sufficient energy to overcome the energy barrier to reversal, nðE > EB ). We will now show that if the magnon energy distribution obeys MB statistics and the density of states in energy space is uniform, then the rate equation is given by the Arrhenius law. On the other hand, if the energy distribution obeys quantum statistics or the density of states in energy space is not uniform, non-Arrhenius behavior would be observed. In general, the number of particles that in a given collision have an energy greater than EB is obtained by integrating over all possible energies, that is Z N nðEÞdðEÞ dE; ð1Þ nðE > EB Þ ¼ EB
where nðEÞ is the expectation value of the number of particles in a state that has energy E; and dðEÞ is the density of states in energy space. We will now
assume that nðEÞ is given by 1 ; nðEÞ ¼ ðEzÞ=kT e þa
ð2Þ
where z is the chemical potential, ez=kT is the fugacity, and a is zero for MB statistics, þ1 for Fermi–Dirac (FD) statistics and 1 for BE statistics. If the density of states in energy space is a constant, d; then Eq. (1) becomes Z N dE nðE > EB Þ ¼ d ðEzÞ=kT þa EB e Z N E=kT e dE ¼ dez=kT : ð3Þ ðEzÞ=kT 1 þ ae EB This can be integrated by letting u ¼ eE=kT ; then for BE statistics we have Z N du nðE > EB Þ ¼ dkT ez=kT z=kT u eEB =kT 1 e ¼ dkT ln½1 eðEB zÞ=kT :
ð4Þ
Thus the frequency of reversals, f ; is given by f ¼ fA dkT ln½1 eðEB zÞ=kT :
ð5Þ
This, for bosons, is equivalent to the Arrhenius law if the density of states in energy space is a constant. If z is negative and large in magnitude, then BE statistics reduces to MB statistics. Thus using lnð1 þ xÞEx in the limit of small x reduces to the Arrhenius law, f ¼ fA dkT eðEB zÞ=kT :
ð6Þ
The chemical potential determines how many particles there are, since the total number of particles is given by Z N N¼d eðEzÞ=kT dE ¼ dkT ez=kT : ð7Þ 0
However, magnons are bosons and the density of states is proportional to the square root of the energy. It was shown [3] that for an assembly of identical particles, whose energy barrier is not a function of temperature, the rate of magnetization reversal, R; is given by N X IðnEB =kTÞenz=kT R ¼ CT 3=2 ; ð8Þ n3=2 n¼1 which is a generalization of the Arrhenius law for magnetization reversal due to thermal aftereffect.
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It is noted that this equation is valid for BE statistics. Normally, the summation can be approximated to very high accuracy by just the first term, which corresponds to MB statistics. The function, IðxÞ described in Ref. [3], is a monotonically decreasing function of x; and is given by pffiffiffih pffiffi i pffiffiffi p IðxÞ ¼ 1 erfð ðxÞ þ xex : ð9Þ 2 The function, IðxÞ; is plotted as a function of x in Fig. 1, and IðEB =kTÞ; as a function of temperature, in Fig. 2, which shows how for a particular energy barrier, IðxÞ variation with temperature affects the rate. It is noted that pffiffiffi p Ið0Þ ¼ ¼ 0:8862: ð10Þ 2 It was shown [3] that for nanograin iron powders, Bloch’s 32 power law holds accurately to over 150 K: The derivation of this equation follows the derivation of the 32 power variation of the saturation magnetization with temperature, except in that case B was zero since the material was saturated at all times. In the case of aftereffect measurements, the material after being saturated is DC-demagnetized, that is it is put into the maximum entropy state, hence B is a large negative number. As the aftereffect process continues, the entropy decreases as the holding field tries to saturate the material.
Fig. 2. Plot of IðEB =kTÞ as a function of temperature.
3. The chemical potential In this section, we discuss the relationship between the chemical potential and the entropy of the system. When the system is in the lowest entropy state, the chemical potential that drives it to equilibrium is largest. As the entropy increases, the chemical potential decreases and is a minimum when the entropy is a maximum. Since the chemical potential for bosons is negative, its magnitude increases as we approach equilibrium. In computing the entropy, we have to sum over all the sources of disorder. The chemical potential obeys [4] dz ¼ S dT þ v dp H:dM;
ð11Þ
where S; v; p; and T are the entropy, volume, pressure and temperature, respectively. Thus, under constant pressure dz S¼ : ð12Þ dT p;M
The probabilistic definition of entropy is X S ¼ k P ln P;
ð13Þ
all states
Fig. 1. Variation of IðxÞ with respect to its argument.
where k is Boltzmann’s constant and P is the probability of a well k being filled. We now use an Ising-like model for a magnetic material that has two stable states, that we will
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refer to as the up state and the down state. When a magnetic material is saturated then all the up states are filled, that is they have a probability of one, and the down states are empty, that is they have a probability of zero. Thus, the entropy of the saturated system is zero. On the other hand, when the material is demagnetized, it has the maximum entropy, and each state has a probability of 12 being filled and Eq. (13) becomes Smax ¼ kN ln 2;
ð14Þ
where N is the number of particles. For other values of magnetization, using these Ising magnets, we know the probability of finding up phases populated is ð1 þ MÞ=2; and the probability of finding a down phase populated is ð1 MÞ=2; where M is the irreversible magnetization. Thus, the entropy of N such Ising magnets is given by X Nk ð1 þ MÞ ð1 þ MÞ ln S ¼ k P ln P ¼ 2 2 N ð1 MÞ þ ð1 MÞ ln : ð15Þ 2 A plot of entropy per particle divided by k for a single particle as a function of magnetization is shown in Fig. 3. The peak of this curve occurs at M ¼ 0; and is Smax : The change in entropy with magnetization is also the well known source of the magnetocaloric effect.
Fig. 3. Normalized entropy as a function of normalized magnetization.
Since the entropy of the demagnetized phase does not depend upon temperature, from Eq. (12) we get zdemag ¼ Smax ðT TB Þ ¼ Nk ln 2ðT TB Þ;
ð16Þ
where TB the constant of integration is the temperature at which B goes to zero. Note that B is negative as it is for all bosons. (A similar analysis in Ref. [5] replaces z by TSai ; where they call Sai ; the activation entropy.) It is seen that as the bit size becomes smaller, the number of particles in a bit decreases, and so the negative chemical potential decreases in magnitude. For other magnetizations, z is given by Nk 1þM ð1 þ MÞ ln z ¼ SðT TB Þ ¼ 2 2 1M þ ð1 MÞ ln ð17Þ ðT TB Þ: 2 It is noted that the chemical potential is most negative in the demagnetized state, as expected. The demagnetized state occurs when a field is applied near the coercivity where the decay rate is fastest.
4. Comparison with experiment We will now use the rate as computed in Eq. (8) for a comparison with the experiment. We see that R is the product of three terms. The first two, IðEB =kTÞ as seen in Fig. 2, and T 3=2 are monotonically increasing functions of T: The third, the fugacity is either a constant or a monotonically decreasing function of temperature. In particular, if TB ¼ 0; B as in Eq. (16) is directly proportional to T; then eB=kT is a constant, hence R in Eq. (8), is a monotonically increasing function of T: However, by setting TB to a value greater than zero in Eq. (16), that is a positive BE condensation temperature, then eB=kT decreases with increasing T as shown in Fig. 4. From the plot of eB=kT as a function of temperature it is seen that below TB ; it is equal to one and above it, it is a monotonically decreasing function of T that approaches eS=kT
ARTICLE IN PRESS E Della Torre, L.H. Bennett / Physica B 343 (2004) 267–274
Fig. 4. Effect of a linear negative chemical potential variation with a Bose temperature of 15 K on the fugacity.
in the limit of high temperatures. The main variations are just above the BEC temperature. If TB ¼ 0; then e is a constant equal to eS=kT : Thus, there is a one-to-one connection between the occurrence of a Bose–Einstein condensation and the presence of a temperature peak in the aftereffect rate. Either in the case of the nanograin iron, or in the case of the MP tape, there is no other explanation for the peak, i.e., there are no phase changes, no changes in the easy direction of magnetization, no movement of interstitial atoms, or other significant crystallographic or magnetic changes which could provide even a perfunctory reason for the peak. The aftereffect decay in this case is exponential with time. Real materials have a linear log-time behavior, during a certain time window, due to the distribution of barrier energies. It is noted that each particle has the same entropy, hence the effect of the sum of the particles is the same. However, the Bose temperature of the particles may differ. If we assume that the Bose temperatures are uniformly distributed between TB DT and TB þ DT; then B will increase quadratically from 0 at T DT until T þ DT: After that, it will increase linearly. We shall compare the maximum decay rate as a function of temperature predicted by the theory above with the experiment for two materials: a sample of nanograin iron powders and a sample of metal particle, MP tape. The sample was first saturated and then a magnetic field in the opposite direction was applied to maximize the decay rate. This field incidentally maximizes the irreversible
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susceptibility and for materials with symmetrical hysteresis loops reduces the irreversible magnetization to zero. At this point, the energy barrier to reversal of the magnetization is zero and IðnEB =kTÞ can be replaced by Ið0Þ ¼ 0:8862: If we assume that if all the particles were identical and perfectly oriented, then the chemical potential would be linear, then since the particles are not identical, we expect that there will be some deviation from linearity. In Ref. [5], we fit the measured S dependence on T: Here we assume that the chemical potential is linear with a constant slope, dB=dT; and that intersects the temperature axis at T ¼ Tint but with a rounding given by DT: In this manner, we can test how well Eq. (3) describes the variation in R: The result is a much improved fit. A plot of fit of R to measurements as a function of temperature is shown in Fig. 5 for a nanograin iron powder sample [6]. A plot of B as a function of temperature is shown in Fig. 6. Figs. 7 and 8 are the corresponding figures for the MP tape sample [7]. It is noted that only three parameters were varied to obtain the fit. The parameters used in these fits are shown in Table 1. The fit to the data was obtained by minimizing the mean squared error, erms that is 8sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 2 = < s XN SðTÞ S meas ðTn Þ : erms ¼ min n¼1 : N ; Smeas ðTn Þ ð18Þ
Fig. 5. Fit of experimental coefficient as a function of temperature for the nanograin iron particles.
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Fig. 6. Required chemical potential to obtain the fit for the nanograin iron powder sample.
Fig. 8. Required chemical potential to fit the decay data for MP tape sample.
Table 1 Parameters used to fit the rate–temperature curves
Fig. 7. Experimental fit of measured decay coefficient vs. temperature for the MP tape sample.
The error of the fit for the two materials tested is shown in the table above. With this criteria, the error for the nanograin iron powder is quoted in the table. These errors are comparable to the other errors in this model. For example, the experimental error was about 5%. Also, the model assumes that both energy barrier and the attempt frequency are not a function of temperature. Since the saturation magnetization of iron varies the order of 3% in the temperature range from 0 K to room temperature, it is expected that the energy barrier will vary by that order of magnitude. It is noted that the curve of the decay coefficient with temperature is very sensitive to the shape of the chemical potential–temperature curve. Any
Material
Tint
DT
dz=k dt
Powder MP tape
35.5 38.6
15.2 37.5
6.21 4.41
deviation from linearity of the B–T curve, causes large changes in the R–T curve. Thus, the shape of the B–T curve, and in particular the intersection with the axis, is very accurate. It is noted that the slope of the B–T curve is the entropy. Since all the decay coefficients were measured for all temperatures using the same sequence of magnetic fields, they all start in the same magnetic state. Hence they have the same entropy. This entropy is near to the maximum possible since the holding field DC demagnetizes the material. (The only way that the entropy could be higher is if the material were AC-demagnetized, i.e. in the ground state with zero applied field.) As the aftereffect progresses, the entropy decreases due to the effect of the holding field attempting to magnetize the material. If the material were saturated, the entropy would be zero.
5. Initial decay rate In magnetic aftereffect measurements, the initial decay rate is usually all that can be observed
ARTICLE IN PRESS E Della Torre, L.H. Bennett / Physica B 343 (2004) 267–274
experimentally in a reasonable time. In this section, we will limit our consideration to two factors, that affect the initial decay rate: the change in the chemical potential and the change in the energy barrier. The two specific cases that we will consider are the zero-field case and the demagnetized case. In both cases, we start from positive saturation. In the zero-field case, we then reduce the field to zero. In the demagnetized case, we reduce the field to negative remanence coercivity. 5.1. Zero-field case Starting from saturation, the zero-field irreversible magnetization will still be at or near saturation. As the magnetization decays, the chemical potential will change and so the simplification in this section applies only for the beginning of the decay. At saturation, the entropy is zero or minimal and hence, the chemical potential will be zero or small. This will greatly increase the decay rate from that measured near the coercivity. However, the energy barrier will be much higher. The variation in the energy barrier with applied field in the case of incoherent reversal varies linearly with field, while for coherent rotation, such as in the Stoner–Wohlfarth model, varies quadratically. For linear variation, we may say that EB ¼ m0 mjH HC j;
MJ ðHÞ: This will be given by m mjH HC j z=kT R0 ¼ CT 3=2 I 0 ; e kT
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ð21Þ
where z is given by Eq. (16). It is noted that as the magnetization decays, the chemical potential will change. In the demagnetized case, H ¼ HC ; and z is given by Eq. (16). Thus, R0 jH¼HC ¼ CT 3=2 Ið0ÞeSmax ðTT0 Þ=kT ¼ CT 3=2 0:8862 eSmax ðTT0 Þ=kT :
ð22Þ
Comparison of Eq. (22) and Eq. (24) shows that there is little relationship between the decay rates since the first one is dependent on the remanence coercivity, which is a function of the anisotropy of the particle, while the other is dependent upon the entropy, which is a function of the number of particles per unit volume. To illustrate this point, if we assume for simplicity that the Arrhenius law were applicable, then the logarithm of the decay rate would be given by Eb ðHÞ z ln R ¼ : ð23Þ kT Using Eq. (17) and assuming that the magnetization varies as the error function of H; with a standard deviation of 0.3 times the coercivity, we get the variation of the numerator in Eq. (23) as shown in Fig. 9. We see that depending upon the
ð19Þ
where m is the moment of a typical particle. From Eq. (8), the rate at H ¼ 0; R0 is given by m mjH HC j R0 jH¼0 ¼ CT 3=2 I 0 kT H¼0 ¼ CT 3=2 Iðm0 mHC =kTÞ:
ð20Þ
This would be the initial rate if all the particles were identical. For a distribution of particle sizes, shapes and orientations, one has to use the Preisach approach. 5.2. The demagnetized case At other holding fields, the magnetization can be obtained from the descending major loop
Fig. 9. Plot of exponent of decay rate as a function of holding field.
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parameters, we can obtain a wide variety of patterns and the value at H ¼ HC is not strongly dependent on the value at H ¼ 0: For a real material, we expect a more complicated behavior because the Arrhenius law should be replaced by Eq. (8), distribution of parameters should be included, reversible magnetization component should be added, etc. These corrections are beyond the scope of this paper.
6. Discussion and conclusions We have shown that the chemical potential, which is a function of the magnetic state and temperature plays an important role in magnetic aftereffect predominantly in the demagnetized state. The barrier energy, also a function of the applied field, increases greatly from the demagnetized state to the saturated state. Often-used accelerated aftereffect measurements, to assess the long time behavior of permanent magnets, are not a very reliable measure of this behavior if the model used to characterize the behavior does not include the implications of the chemical potential. Non-Arrhenius behavior at low temperatures is attributed to two effects: distribution of energy states and Bose–Einstein statistics. For large particles, non-Arrhenius monotonic increase in decay rate with increasing temperature is due to nonuniform density of energy of states. On the other hand, for nano sized particles, the increase in decay rate with decreasing temperature is attributed to Bose condensation of magnons at positive temperatures.
This analysis was developed for a very uniform distribution of particles. For a real material with a normal distribution of parameters, this analysis can best be refined by a Preisach model, such as a modification of the Preisach–Arrhenius model.
Acknowledgements We thank L.J. Swartzendruber who first steered us in the direction of quantum statistics. We are grateful to William Boettinger and Robert Sekerka for several discussions on thermodynamics and statistical mechanics and to Charles W. Clark and James E. Williams for an important conversation on Bose–Einstein condensation. This work has been partially supported by NSF under Grant 99-70058.
References [1] U. Atzmony, Z. Levine, R.D. Mc Michael, L.H. Bennett, J. Appl. Phys. 79 (1996) 5456. [2] E. Della Torre, L.H. Bennett, R.A. Fry, O. Alejos, IEEE Trans. Magn. 38 (2002) 3409. [3] E. Della Torre, L.H. Bennett, IEEE Trans. Magn. 37 (2001) 1118. [4] L.D. Landau, E.M. Lifshitz, Statistical Physics, Pergamon Press, Oxford, New York, 1980. [5] R. Skomski, R.D. Kirby, D.J. Sellmyer, J. Appl. Phys. 85 (1999) 5069. [6] U. Atzmony, Z. Levine, R.D. McMichael, L.H. Bennett, J. Appl. Phys. 79 (1996) 5456. [7] L.J. Swartzendruber, L.H. Bennett1, E. Della Torre, H.J. Brown, Behavior of magnetic aftereffect in a metal particle recording material along a magnetization reversal curve, MRS Spring Meeting, San Francisco, April 13–17, 1998, paper L9.20.
Physica B 343 (2004) 267–274
The thermodynamics of magnetic aftereffect E. Della Torre*, L.H. Bennett Electrical and Computer Engineering Department, The George Washington University, Washington, DC 20052, USA
Abstract Thermal magnetic aftereffect is due to the interaction between magnons (spin waves) and the change of equilibrium magnetization states. This non-equilibrium process is controlled by (i) the height of the energy barriers between magnetization states and (ii) by the chemical potential that determines how many magnons (thermal quanta) are available. Non-Arrhenius behavior is observed in a non-monotonic variation of decay rate with temperature observed in nanoparticulate media at low temperature. We attribute this behavior to quantum statistics and the nonuniform density of states in energy space. At low temperatures, Bose–Einstein condensation of the magnons is observed in nanoparticulate systems. In these materials, the decay rate increases to a maximum at low temperatures above the condensation temperature. A standard method of evaluating permanent magnet materials is to accelerate the decay rate, by applying a reverse holding field of the order of coercivity. We show that the problem with this technique is the ratio of the decay rate in zero field to the decay rate at the coercivity is a function of the material properties. This model shows excellent agreement with measurements of the variation of the thermal magnetic aftereffect at low temperatures in both nanograin iron powders and MP tapes. r 2003 Elsevier B.V. All rights reserved. Keywords: Magnetic aftereffect; Chemical potential; Thermal activation
1. Introduction To understand fully the temperature dependence of the magnetic aftereffect of small particles that have exhibited non-Arrhenius behavior [1], it is useful to elaborate their thermodynamic framework. For simplicity, we will consider an uncomplicated Ising model for the particles so that we can neglect many ramifications associated with a ferromagnetic thermodynamic system. In order to simplify the analysis, we will assume that the basic particles are all identical and perfectly ordered. *Corresponding author. Tel.: +1-202-994-0410; fax: +1202-994-0227. E-mail address: [email protected] (E. Della Torre).
We picture small particles as having only two equilibrium states, up or down. This is a good approximation in the regime of slow relaxation ðtimes > 1 s), but it is of course not what is happening in the Larmor frequency regime ðo107 s). In this time regime, the spins are precessing and, depending on the temperature, individual spins may reverse, creating a spin wave. In this paper, we will not address the problems associated with the distribution in size, shape and physical properties of the particles in the system. Any physical properties, such as saturation magnetization, anisotropy, and crystal lattice dimensions which vary with temperature, and the possibility of phase transitions will also be neglected in this paper. The distribution in
0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.08.108
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parameters can best be handled in the Preisach model framework [2] which has been shown to be successful in describing many properties of the system. The Preisach model, considers each mesoscopic magnetic particle, called a hysteron, to have a distribution in particle properties. This results in replacing the exponential decay of the magnetization by one that decays linearly with log-time.
2. Arrhenius law We will view the material as the one consisting of loosely coupled magnetic particles. These particles at finite temperatures will support magnons that are the wave-particle duals of spin waves. When one of these magnons interacts with the particle, if it has sufficient energy, it may reverse the magnetization of the particle. These magnons have spin one and thus are bosons, that is, they obey Bose–Einstein (BE) statistics. In the limit of high temperatures, BE statistics go over to Maxwell–Boltzmann (MB) statistics. If the energy of the magnon is less than that of the barrier energy to a reversal of the magnetization, then the collision is purely elastic. The frequency of particle reversal, f ; using magnons is the product of the attempt frequency fA and the number of magnons that have sufficient energy to overcome the energy barrier to reversal, nðE > EB ). We will now show that if the magnon energy distribution obeys MB statistics and the density of states in energy space is uniform, then the rate equation is given by the Arrhenius law. On the other hand, if the energy distribution obeys quantum statistics or the density of states in energy space is not uniform, non-Arrhenius behavior would be observed. In general, the number of particles that in a given collision have an energy greater than EB is obtained by integrating over all possible energies, that is Z N nðEÞdðEÞ dE; ð1Þ nðE > EB Þ ¼ EB
where nðEÞ is the expectation value of the number of particles in a state that has energy E; and dðEÞ is the density of states in energy space. We will now
assume that nðEÞ is given by 1 ; nðEÞ ¼ ðEzÞ=kT e þa
ð2Þ
where z is the chemical potential, ez=kT is the fugacity, and a is zero for MB statistics, þ1 for Fermi–Dirac (FD) statistics and 1 for BE statistics. If the density of states in energy space is a constant, d; then Eq. (1) becomes Z N dE nðE > EB Þ ¼ d ðEzÞ=kT þa EB e Z N E=kT e dE ¼ dez=kT : ð3Þ ðEzÞ=kT 1 þ ae EB This can be integrated by letting u ¼ eE=kT ; then for BE statistics we have Z N du nðE > EB Þ ¼ dkT ez=kT z=kT u eEB =kT 1 e ¼ dkT ln½1 eðEB zÞ=kT :
ð4Þ
Thus the frequency of reversals, f ; is given by f ¼ fA dkT ln½1 eðEB zÞ=kT :
ð5Þ
This, for bosons, is equivalent to the Arrhenius law if the density of states in energy space is a constant. If z is negative and large in magnitude, then BE statistics reduces to MB statistics. Thus using lnð1 þ xÞEx in the limit of small x reduces to the Arrhenius law, f ¼ fA dkT eðEB zÞ=kT :
ð6Þ
The chemical potential determines how many particles there are, since the total number of particles is given by Z N N¼d eðEzÞ=kT dE ¼ dkT ez=kT : ð7Þ 0
However, magnons are bosons and the density of states is proportional to the square root of the energy. It was shown [3] that for an assembly of identical particles, whose energy barrier is not a function of temperature, the rate of magnetization reversal, R; is given by N X IðnEB =kTÞenz=kT R ¼ CT 3=2 ; ð8Þ n3=2 n¼1 which is a generalization of the Arrhenius law for magnetization reversal due to thermal aftereffect.
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It is noted that this equation is valid for BE statistics. Normally, the summation can be approximated to very high accuracy by just the first term, which corresponds to MB statistics. The function, IðxÞ described in Ref. [3], is a monotonically decreasing function of x; and is given by pffiffiffih pffiffi i pffiffiffi p IðxÞ ¼ 1 erfð ðxÞ þ xex : ð9Þ 2 The function, IðxÞ; is plotted as a function of x in Fig. 1, and IðEB =kTÞ; as a function of temperature, in Fig. 2, which shows how for a particular energy barrier, IðxÞ variation with temperature affects the rate. It is noted that pffiffiffi p Ið0Þ ¼ ¼ 0:8862: ð10Þ 2 It was shown [3] that for nanograin iron powders, Bloch’s 32 power law holds accurately to over 150 K: The derivation of this equation follows the derivation of the 32 power variation of the saturation magnetization with temperature, except in that case B was zero since the material was saturated at all times. In the case of aftereffect measurements, the material after being saturated is DC-demagnetized, that is it is put into the maximum entropy state, hence B is a large negative number. As the aftereffect process continues, the entropy decreases as the holding field tries to saturate the material.
Fig. 2. Plot of IðEB =kTÞ as a function of temperature.
3. The chemical potential In this section, we discuss the relationship between the chemical potential and the entropy of the system. When the system is in the lowest entropy state, the chemical potential that drives it to equilibrium is largest. As the entropy increases, the chemical potential decreases and is a minimum when the entropy is a maximum. Since the chemical potential for bosons is negative, its magnitude increases as we approach equilibrium. In computing the entropy, we have to sum over all the sources of disorder. The chemical potential obeys [4] dz ¼ S dT þ v dp H:dM;
ð11Þ
where S; v; p; and T are the entropy, volume, pressure and temperature, respectively. Thus, under constant pressure dz S¼ : ð12Þ dT p;M
The probabilistic definition of entropy is X S ¼ k P ln P;
ð13Þ
all states
Fig. 1. Variation of IðxÞ with respect to its argument.
where k is Boltzmann’s constant and P is the probability of a well k being filled. We now use an Ising-like model for a magnetic material that has two stable states, that we will
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refer to as the up state and the down state. When a magnetic material is saturated then all the up states are filled, that is they have a probability of one, and the down states are empty, that is they have a probability of zero. Thus, the entropy of the saturated system is zero. On the other hand, when the material is demagnetized, it has the maximum entropy, and each state has a probability of 12 being filled and Eq. (13) becomes Smax ¼ kN ln 2;
ð14Þ
where N is the number of particles. For other values of magnetization, using these Ising magnets, we know the probability of finding up phases populated is ð1 þ MÞ=2; and the probability of finding a down phase populated is ð1 MÞ=2; where M is the irreversible magnetization. Thus, the entropy of N such Ising magnets is given by X Nk ð1 þ MÞ ð1 þ MÞ ln S ¼ k P ln P ¼ 2 2 N ð1 MÞ þ ð1 MÞ ln : ð15Þ 2 A plot of entropy per particle divided by k for a single particle as a function of magnetization is shown in Fig. 3. The peak of this curve occurs at M ¼ 0; and is Smax : The change in entropy with magnetization is also the well known source of the magnetocaloric effect.
Fig. 3. Normalized entropy as a function of normalized magnetization.
Since the entropy of the demagnetized phase does not depend upon temperature, from Eq. (12) we get zdemag ¼ Smax ðT TB Þ ¼ Nk ln 2ðT TB Þ;
ð16Þ
where TB the constant of integration is the temperature at which B goes to zero. Note that B is negative as it is for all bosons. (A similar analysis in Ref. [5] replaces z by TSai ; where they call Sai ; the activation entropy.) It is seen that as the bit size becomes smaller, the number of particles in a bit decreases, and so the negative chemical potential decreases in magnitude. For other magnetizations, z is given by Nk 1þM ð1 þ MÞ ln z ¼ SðT TB Þ ¼ 2 2 1M þ ð1 MÞ ln ð17Þ ðT TB Þ: 2 It is noted that the chemical potential is most negative in the demagnetized state, as expected. The demagnetized state occurs when a field is applied near the coercivity where the decay rate is fastest.
4. Comparison with experiment We will now use the rate as computed in Eq. (8) for a comparison with the experiment. We see that R is the product of three terms. The first two, IðEB =kTÞ as seen in Fig. 2, and T 3=2 are monotonically increasing functions of T: The third, the fugacity is either a constant or a monotonically decreasing function of temperature. In particular, if TB ¼ 0; B as in Eq. (16) is directly proportional to T; then eB=kT is a constant, hence R in Eq. (8), is a monotonically increasing function of T: However, by setting TB to a value greater than zero in Eq. (16), that is a positive BE condensation temperature, then eB=kT decreases with increasing T as shown in Fig. 4. From the plot of eB=kT as a function of temperature it is seen that below TB ; it is equal to one and above it, it is a monotonically decreasing function of T that approaches eS=kT
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Fig. 4. Effect of a linear negative chemical potential variation with a Bose temperature of 15 K on the fugacity.
in the limit of high temperatures. The main variations are just above the BEC temperature. If TB ¼ 0; then e is a constant equal to eS=kT : Thus, there is a one-to-one connection between the occurrence of a Bose–Einstein condensation and the presence of a temperature peak in the aftereffect rate. Either in the case of the nanograin iron, or in the case of the MP tape, there is no other explanation for the peak, i.e., there are no phase changes, no changes in the easy direction of magnetization, no movement of interstitial atoms, or other significant crystallographic or magnetic changes which could provide even a perfunctory reason for the peak. The aftereffect decay in this case is exponential with time. Real materials have a linear log-time behavior, during a certain time window, due to the distribution of barrier energies. It is noted that each particle has the same entropy, hence the effect of the sum of the particles is the same. However, the Bose temperature of the particles may differ. If we assume that the Bose temperatures are uniformly distributed between TB DT and TB þ DT; then B will increase quadratically from 0 at T DT until T þ DT: After that, it will increase linearly. We shall compare the maximum decay rate as a function of temperature predicted by the theory above with the experiment for two materials: a sample of nanograin iron powders and a sample of metal particle, MP tape. The sample was first saturated and then a magnetic field in the opposite direction was applied to maximize the decay rate. This field incidentally maximizes the irreversible
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susceptibility and for materials with symmetrical hysteresis loops reduces the irreversible magnetization to zero. At this point, the energy barrier to reversal of the magnetization is zero and IðnEB =kTÞ can be replaced by Ið0Þ ¼ 0:8862: If we assume that if all the particles were identical and perfectly oriented, then the chemical potential would be linear, then since the particles are not identical, we expect that there will be some deviation from linearity. In Ref. [5], we fit the measured S dependence on T: Here we assume that the chemical potential is linear with a constant slope, dB=dT; and that intersects the temperature axis at T ¼ Tint but with a rounding given by DT: In this manner, we can test how well Eq. (3) describes the variation in R: The result is a much improved fit. A plot of fit of R to measurements as a function of temperature is shown in Fig. 5 for a nanograin iron powder sample [6]. A plot of B as a function of temperature is shown in Fig. 6. Figs. 7 and 8 are the corresponding figures for the MP tape sample [7]. It is noted that only three parameters were varied to obtain the fit. The parameters used in these fits are shown in Table 1. The fit to the data was obtained by minimizing the mean squared error, erms that is 8sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9 2 = < s XN SðTÞ S meas ðTn Þ : erms ¼ min n¼1 : N ; Smeas ðTn Þ ð18Þ
Fig. 5. Fit of experimental coefficient as a function of temperature for the nanograin iron particles.
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Fig. 6. Required chemical potential to obtain the fit for the nanograin iron powder sample.
Fig. 8. Required chemical potential to fit the decay data for MP tape sample.
Table 1 Parameters used to fit the rate–temperature curves
Fig. 7. Experimental fit of measured decay coefficient vs. temperature for the MP tape sample.
The error of the fit for the two materials tested is shown in the table above. With this criteria, the error for the nanograin iron powder is quoted in the table. These errors are comparable to the other errors in this model. For example, the experimental error was about 5%. Also, the model assumes that both energy barrier and the attempt frequency are not a function of temperature. Since the saturation magnetization of iron varies the order of 3% in the temperature range from 0 K to room temperature, it is expected that the energy barrier will vary by that order of magnitude. It is noted that the curve of the decay coefficient with temperature is very sensitive to the shape of the chemical potential–temperature curve. Any
Material
Tint
DT
dz=k dt
Powder MP tape
35.5 38.6
15.2 37.5
6.21 4.41
deviation from linearity of the B–T curve, causes large changes in the R–T curve. Thus, the shape of the B–T curve, and in particular the intersection with the axis, is very accurate. It is noted that the slope of the B–T curve is the entropy. Since all the decay coefficients were measured for all temperatures using the same sequence of magnetic fields, they all start in the same magnetic state. Hence they have the same entropy. This entropy is near to the maximum possible since the holding field DC demagnetizes the material. (The only way that the entropy could be higher is if the material were AC-demagnetized, i.e. in the ground state with zero applied field.) As the aftereffect progresses, the entropy decreases due to the effect of the holding field attempting to magnetize the material. If the material were saturated, the entropy would be zero.
5. Initial decay rate In magnetic aftereffect measurements, the initial decay rate is usually all that can be observed
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experimentally in a reasonable time. In this section, we will limit our consideration to two factors, that affect the initial decay rate: the change in the chemical potential and the change in the energy barrier. The two specific cases that we will consider are the zero-field case and the demagnetized case. In both cases, we start from positive saturation. In the zero-field case, we then reduce the field to zero. In the demagnetized case, we reduce the field to negative remanence coercivity. 5.1. Zero-field case Starting from saturation, the zero-field irreversible magnetization will still be at or near saturation. As the magnetization decays, the chemical potential will change and so the simplification in this section applies only for the beginning of the decay. At saturation, the entropy is zero or minimal and hence, the chemical potential will be zero or small. This will greatly increase the decay rate from that measured near the coercivity. However, the energy barrier will be much higher. The variation in the energy barrier with applied field in the case of incoherent reversal varies linearly with field, while for coherent rotation, such as in the Stoner–Wohlfarth model, varies quadratically. For linear variation, we may say that EB ¼ m0 mjH HC j;
MJ ðHÞ: This will be given by m mjH HC j z=kT R0 ¼ CT 3=2 I 0 ; e kT
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ð21Þ
where z is given by Eq. (16). It is noted that as the magnetization decays, the chemical potential will change. In the demagnetized case, H ¼ HC ; and z is given by Eq. (16). Thus, R0 jH¼HC ¼ CT 3=2 Ið0ÞeSmax ðTT0 Þ=kT ¼ CT 3=2 0:8862 eSmax ðTT0 Þ=kT :
ð22Þ
Comparison of Eq. (22) and Eq. (24) shows that there is little relationship between the decay rates since the first one is dependent on the remanence coercivity, which is a function of the anisotropy of the particle, while the other is dependent upon the entropy, which is a function of the number of particles per unit volume. To illustrate this point, if we assume for simplicity that the Arrhenius law were applicable, then the logarithm of the decay rate would be given by Eb ðHÞ z ln R ¼ : ð23Þ kT Using Eq. (17) and assuming that the magnetization varies as the error function of H; with a standard deviation of 0.3 times the coercivity, we get the variation of the numerator in Eq. (23) as shown in Fig. 9. We see that depending upon the
ð19Þ
where m is the moment of a typical particle. From Eq. (8), the rate at H ¼ 0; R0 is given by m mjH HC j R0 jH¼0 ¼ CT 3=2 I 0 kT H¼0 ¼ CT 3=2 Iðm0 mHC =kTÞ:
ð20Þ
This would be the initial rate if all the particles were identical. For a distribution of particle sizes, shapes and orientations, one has to use the Preisach approach. 5.2. The demagnetized case At other holding fields, the magnetization can be obtained from the descending major loop
Fig. 9. Plot of exponent of decay rate as a function of holding field.
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parameters, we can obtain a wide variety of patterns and the value at H ¼ HC is not strongly dependent on the value at H ¼ 0: For a real material, we expect a more complicated behavior because the Arrhenius law should be replaced by Eq. (8), distribution of parameters should be included, reversible magnetization component should be added, etc. These corrections are beyond the scope of this paper.
6. Discussion and conclusions We have shown that the chemical potential, which is a function of the magnetic state and temperature plays an important role in magnetic aftereffect predominantly in the demagnetized state. The barrier energy, also a function of the applied field, increases greatly from the demagnetized state to the saturated state. Often-used accelerated aftereffect measurements, to assess the long time behavior of permanent magnets, are not a very reliable measure of this behavior if the model used to characterize the behavior does not include the implications of the chemical potential. Non-Arrhenius behavior at low temperatures is attributed to two effects: distribution of energy states and Bose–Einstein statistics. For large particles, non-Arrhenius monotonic increase in decay rate with increasing temperature is due to nonuniform density of energy of states. On the other hand, for nano sized particles, the increase in decay rate with decreasing temperature is attributed to Bose condensation of magnons at positive temperatures.
This analysis was developed for a very uniform distribution of particles. For a real material with a normal distribution of parameters, this analysis can best be refined by a Preisach model, such as a modification of the Preisach–Arrhenius model.
Acknowledgements We thank L.J. Swartzendruber who first steered us in the direction of quantum statistics. We are grateful to William Boettinger and Robert Sekerka for several discussions on thermodynamics and statistical mechanics and to Charles W. Clark and James E. Williams for an important conversation on Bose–Einstein condensation. This work has been partially supported by NSF under Grant 99-70058.
References [1] U. Atzmony, Z. Levine, R.D. Mc Michael, L.H. Bennett, J. Appl. Phys. 79 (1996) 5456. [2] E. Della Torre, L.H. Bennett, R.A. Fry, O. Alejos, IEEE Trans. Magn. 38 (2002) 3409. [3] E. Della Torre, L.H. Bennett, IEEE Trans. Magn. 37 (2001) 1118. [4] L.D. Landau, E.M. Lifshitz, Statistical Physics, Pergamon Press, Oxford, New York, 1980. [5] R. Skomski, R.D. Kirby, D.J. Sellmyer, J. Appl. Phys. 85 (1999) 5069. [6] U. Atzmony, Z. Levine, R.D. McMichael, L.H. Bennett, J. Appl. Phys. 79 (1996) 5456. [7] L.J. Swartzendruber, L.H. Bennett1, E. Della Torre, H.J. Brown, Behavior of magnetic aftereffect in a metal particle recording material along a magnetization reversal curve, MRS Spring Meeting, San Francisco, April 13–17, 1998, paper L9.20.