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Dynamical study of remanent magnetization in nanoparticles
Journal of Magnetism and Magnetic Materials 237 (2001) 90–96
Dynamical study of remanent magnetization in nanoparticles G. Grinstein*, G.A. Held, R.H. Koch IBM Corporation, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA Received 3 January 2001; received in revised form 25 July 2001
Abstract For collections of noninteracting nanoparticles, we study the reduced static remanent magnetization, mR ; produced by the removal of a saturating magnetic field. We show that, except for special cases such as easy uniaxial anisotropy, mR depends on both the ramp-down rate of the field and the energy dissipation rate of the spin dynamics. Using the Landau–Lifshitz equation, we illustrate this result with explicit dynamical calculations of mR for cubic and for mixed cubic–uniaxial anisotropies. r 2001 Elsevier Science B.V. All rights reserved. PACS: 75.60.Lr; 75.10.Hk; 05.40.a; 75.10.b Keywords: Magnetic nanoparticles; Spin models; Remanent magnetization; Spin dynamics; Landau–Lifshitz equation
1. Introduction The drive to design magnetic storage devices with ever increasing bit densities and access speeds has resulted in a heightened need for more powerful techniques for the numerical simulation of classical magnetic systems [1]. The standard starting point for such simulations is the equation of Landau and Lifshitz (LL) [2], or, equivalently [3–6], of Gilbert (G) [7]. These equations capture the two essential ingredients of spin dynamics: the precession of magnetic moments around their local magnetic fields, and the simultaneous relaxation of the system towards an energy minimum. For systems operating at temperatures not much lower than the excitation energies of the system (which *Corresponding author. Tel.: +1-914-945-1604; fax +1-914945-2141. E-mail address: [email protected] (G. Grinstein).
are set by the anisotropy, exchange, and/or dipolar energies), the model must also include the effects of thermal fluctuations. These are accounted for by the addition to the LL equation of a random noise term [8,4–6]. The strength of this term is usually chosen to be related to the phenomenological damping constant, a; and the temperature (T), so that the noise kicks the system up in energy at the same average rate that the damping term pulls it down. In the long-time limit, a steady state obeying Boltzmann statistics for the energy function that describes the system is thereby achieved [8,9]. Equilibrium properties at nonzero temperature are therefore governed entirely by the energy function, and so are independent of the damping. Thus, although the strength of the damping can affect the dynamical properties of magnetic systems significantly [10–12], equilibrium properties at fixed T > 0 must be independent of a: They
0304-8853/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 4 9 2 - 9
G. Grinstein et al. / Journal of Magnetism and Magnetic Materials 237 (2001) 90–96
must also be independent of the gyromagnetic ratio (gÞ governing the speed of precession, and thus can be calculated as correctly through Monte Carlo algorithms as through the LLG equations. At temperatures well below the characteristic excitation energies of the system, however, the damping strength and other dynamical features can play an important role in determining the metastable or stable state into which the system relaxes, thereby affecting even its static behavior. Examples of this phenomenon have been discussed [10–13], notably in the context of the remanent coercivity of magnetic systems, though they have received less attention than the more obvious dynamical effects associated with damping. In this paper, we investigate another example of this phenomenon, by studying the static remanent magnetization of collections of model singledomain magnetic particles. Specifically, we consider the response of the magnetization of the particles to the removal, at fixed low temperature, of a uniform applied saturating magnetic field. This is a familiar experimental situation (for a review, see Refs. [14,15]); indeed, these calculations were motivated by a desire to understand our low-temperature (B225 K) remanence measurements of magnetic nanoparticle dispersions [16]. We take the applied field, h~ðtÞ; to decay exponentially in time from its initial saturating value h~0 : h~ðtÞ ¼ h~0 et=th ; where th is the ramp-down time. It is natural to assume that the remanent magnetization is independent of th [14,15]. However, as we demonstrate here, this assumption is not valid if th is comparable to the damping time. We assume that the system is at low enough temperature that energy barriers are insurmountable on any reasonable experimental time scale, so that the temperature is effectively zero. We then calculate the static remanent magnetization at long times, for collections of particles with two different symmetries: easy cubic anisotropy, and mixtures of uniaxial and cubic anisotropy. The results can be understood qualitatively through simple arguments that should apply to different anisotropies as well. While we treat only collections of particles dilute enough to be noninteracting, the extension of the qualitative arguments to dispersions dense
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enough to have significant dipolar interactions is straightforward. For each of the symmetries studied, we find that the long-time (static) limit of the reduced remanent magnetization, mR ; i.e., the ratio of the lowtemperature remanent magnetization to the saturation magnetization, is uniquely defined only when th is large when compared with the system’s characteristic relaxation time. When th is comparable to or less than the relaxation time, the final static state depends significantly on both the damping constant and th : The mechanism for this dependenceFthe ability of the precessing system to surmount apparent energy barriers that cannot be overcome in the absence of precession (or, equivalently, when the damping is too large)Fis reviewed below. While it is obviously difficult to vary the damping constant experimentally for a given dispersion of particles, the predicted effect should be observable in remanence experiments in which th is varied.
2. Noninteracting particles and the Landau–Lifshitz equation Following previous studies [8,17], we model each single-domain particle in a collection by a ~ ðtÞ; described by the magnetic moment vector M LL equation [2] ~ ~ ðM ~ r ~ EÞ ~ =qt ¼ g M r ~ E* þ gaM * : ð1Þ qM M M ~j jM Here g is the gyromagnetic ratio, and a is the dimensionless damping constant. The energy ~ Þ includes the anisotropy and *M function E* ¼ Eð Zeeman energies of the particle, as well as any interactions with other particles. Assuming that the volume, V; and saturation magnetization, Ms ; of the nanoparticles are uniform over the collection, we define the (dimensionless) unit vector S~ ~; whereupon Eð ~ Ms V S ~ Þ can be *M through M ~Þ; and Eq. (1) rewritten as written as EðS g ~ ga ~ ~ r ~EÞ: ð2Þ S rS~E þ S ðS qS~=qt ¼ S Ms V Ms V In the simplest example, each particle has a single easy axis, randomly oriented in space, and the
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collection is sufficiently dilute that interparticle interactions can be neglected. It is convenient (especially for generalizing to more complicated anisotropies), to take the easy axis for every particle to be the z-axis, with the direction of the applied field therefore varying randomly from particle to particle. Then, for a given particle, ~Þ ¼ Ku VS 2 Ms V h~ðtÞ S ~: EðS ð3Þ z
Here Ku is the uniaxial anisotropy energy per unit volume, and the field h~ is applied in a random direction. To study the situation where a saturating field is applied, we choose h~0 to satisfy Ms jh~0 jbKu : It is well known that if h~ðtÞ is independent of t; then the precessional term of ~Þ; while the Eq. (2) conserves the energy EðS damping term produces motion straight downhill in energy, with both terms preserving the unit ~: Thus, the particle’s spin precesses magnitude of S about the local field (i.e., about the direction ~ ~E), simultaneously moving monodefined by r S tonically downhill in energy until reaching a local ~ ~E ¼ 0; where it remains until energy minimum, r S the field is altered. In the presence of an applied saturating magnetic field, that minimum lies essentially in the field direction. When the field is removed, there are two obvious limits to consider: (1) th -N; so that the field is removed arbitrarily slowly; in practice this means that th is significantly longer than the typical relaxation time, tR BMs =ðgaKu Þ of the system (roughly 1010 2106 s for the examples discussed below). (2) th ¼ 0; so that the field is removed instantaneously; in practice this means that the relaxation time significantly exceeds th : The first limit is the more common one for experiments on nanoparticles. In this case, the position of the energy minimum moves slowly in time as the field is reduced, and the spin simply follows that minimum precisely [18]. Little preces~ and r ~ ~E point very nearly in sion occurs, since S S the same direction. Thus, the spin moves steadily from the field direction to the nearest accessible easy axis direction. In our simple uniaxial example, this means the easy directionFeither þz# or # zFlocated in the same hemisphere as the field. As is well known [18] (and easily verified), this produces, independent of a; a reduced remanent
magnetization mR ¼ 1=2 for a noninteracting collection of uniaxial particles. In the second limit, the only energy remaining when the field is instantaneously removed is the anisotropy energy. Since the dynamics cannot increase the particle’s energy, the spin cannot cross the hard XY plane, and so spirals down to # the easy directionFþz# or zFclosest to the field direction. The result, again, is the familiar mR ¼ 1=2; independent of a: Moreover, for pure uniaxial anisotropy, mR is equal to 1/2 for any rate of field removal between these two limits. To understand this, consider a particle with the saturating field applied in an arbitrary direction in the upper half plane. Initially, then, the spin points in this direction. As the field is reduced, however, the direction of the net field experienced by the spin rotates progressively towards the þz# axis. In precessing around this upward moving net field direction, the spin cannot cross the XY plane, and so always terminates in the þz# direction. This yields mR ¼ 1=2:
3. Noninteracting particles with easy cubic anisotropy For spherical particles with cubic crystal structure and negligible surface anisotropy, the energy EðS~Þ takes the form ~; EðS~Þ ¼ Kc VðS 4 þ S 4 þ S4 Þ Ms V h~ðtÞ S ð4Þ x
y
z
where Kc ð> 0Þ is the (easy) cubic anisotropy strength. Again, it is instructive to consider the limits th -N and th -0: As in the uniaxial case, extremely slow field removal implies that the spin simply tracks the evolution of the global minimum as it moves from the field direction to the nearest easy cubic direction. It has been shown [19] that the resulting reduced remanent magnetization, averaged over all possible orientations of pthe pffiffiffi ffiffiffi saturating field, is mR ¼ ð6= 2pÞtan1 ð1= 2Þ B0:83; independent of a: (This value exceeds the 0.5 of the uniaxial case because, for cubic symmetry, the ‘‘nearest easy direction’’ is, on average, closer in orientation to the applied field than it is for the uniaxial case.)
G. Grinstein et al. / Journal of Magnetism and Magnetic Materials 237 (2001) 90–96
The case of instantaneous field removal, th ¼ 0; is, however, quite different from the corresponding situation with uniaxial symmetry. To see this, imagine the one-eighth of the unit sphere confined # þ y; # and þz# axes. This surface is between the þx; divided by energy barriers into three equivalent regions, each containing one of the axes. The region associated with a given axis consists of all the points on the surface closer to that axis than to the other two. If there were no precession (i.e., if a-N), so that spins moved straight downhill in energy, then each spin would simply relax to the nearest easy direction, giving mR B0:83: When a is finite, however, a spin that starts close enough to one of the energy barriers can precess across that barrier, ending up at an axis other than the closest one. The reason this can occur is that the energy varies along the barriers, rather than staying constant. Hence the spin can cross a barrier without ever raising its total energy, provided the energy gain required to surmount the barrier by moving perpendicular to it is compensated by the energy loss associated by simultaneously moving parallel to the barrier from a high-energy region to a low-energy one. In other words, these nonuniform barriers can be surmounted by trajectories that cross them at low enough angles.1 This is in sharp contrast to the uniaxial case, where the energy along the XY barrier is uniform, making the barrier impenetrable to spins whose energy must always decrease. In consequence, a spin whose initial orientation is close to one of the barriers will be able to cross that barrier with nonzero probability, thus giving rise to values of mR less than 0.83 in the th ¼ 0 limit. Similar logic leads one to anticipate mR values less than 0.83 for all values of th and a less than infinity. In general, one expects mR to 1 This phenomenon has an intuitive analogy to the notion of crossing the Continental Divide. Suppose one starts just below the ridge that constitutes the Divide in a region where that ridge has a significant slope. Then one can cross the ridge while decreasing one’s altitude monotonically, by choosing a path that runs mostly parallel to the ridge in the downhill direction, but has a small component perpendicular to the ridge that takes one over the top. Of course, such a path does not follow the local altitude gradient, so water, which follows that gradient, does not flow over the Divide.
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decrease with decreasing th ; since smaller th allows for more precession. Decreasing a retards the rate of energy decay, and so should also allow more precession and a concomitantly reduced mR : As a continues to decrease, however, the probability of spins precessing across an energy barrier more than once grows larger, whereupon the tendency for mR to decrease monotonically with decreasing a may cease to hold.2 Fig. 1, a plot of mR as a function of th for several different values of a; shows that these expectations are indeed fulfilled. For a ¼ 0:01; a realistic value for a variety of materials [20–23], mR decreases steadily from close to 0.83 at th ¼ 106 s to about 0.77 at th ¼ 1010 s: Note that mR decreases as a decreases from 1.0 to 0.01, but then increases slightly as a is reduced further to 0.003. This reflects the multiple barrier crossings discussed above. The curves shown in Fig. 1 were produced by solving the LL equation numerically, using the Euler method [24]. We discretized the unit sphere with a roughly uniform mesh of 5000 points in order to average over all directions of the applied field. The parameters used were Kc =Ms ¼ 0:23 Oe;3, h0 ¼ 5:68 Oe; g ¼ 1:76 107 Oe1 s1 =ð1 þ a2 Þ; and a time step of 3 1011 s: For each direction, the system was allowed to evolve for 1,500,000 time steps (i.e. 45 ms), which was long enough for the system to reach a local energy minimum. The value of Kc =Ms used in this simulation is characteristic of Ni-rich cubic Fe–Ni alloys, for which Kc can be quite small [25]. Note, however, that the solution to Eqs. (2) and (4) is unchanged when t is rescaled by a multiplicative factor and Kc =Ms ; th ; and h0 are rescaled by its inverse. Thus, for any given values of Kc and Ms ; resultant values of mR may be extracted from Fig. 1 by simply rescaling the abscissa.
2
Figure 5 of the paper by Usov and Peschany (Ref. [13]), provides a vivid illustration of how sensitive the behavior of magnetic systems can be to small changes in a; particularly when a51: Also see, Refs. [10–12]. 3 Note that the solution to Eqs. (2) and (4) depends only on the ratio Kc =Ms ; and not on either the individual values of these parameters, or on V:
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Fig. 1. Reduced remanent magnetization, mR ; versus rampdown time, th ; for Eq. (4) with parameters given in text. Symbol definitions: ð.Þ a ¼ 1:0; ðmÞ a ¼ 0:1; ðKÞ a ¼ 0:01; ð\Þ a ¼ 0:003:
Fig. 2. Reduced remanent magnetization, mR ; versus rampdown time, th ; for Eq. (5) with parameters given in text, and Ku =Kc ¼ 5=2: Symbol definitions: ð.Þ a ¼ 1:0; ð~Þ a ¼ 0:4; ðmÞ a ¼ 0:1; ðKÞ a ¼ 0:01; ð\Þ a ¼ 0:003:
4. Noninteracting particles with both easy cubic and easy uniaxial anisotropies
For any aoN; one therefore expects mR to decrease from 1/2 at th ¼ N to some lower value at th ¼ 0: As with purely cubic symmetry, values of mR should decrease with decreasing a as well, at least for a values sufficiently large that the probability of spins precessing across the XY plane more than once is small. Verification of these expectations is shown in Fig. 2, which displays the results of solving the LL equation with an energy given by Eq. (5), with all parameter values the same as in Fig. 1, and with Ku ¼ 5Kc =2: The situation is somewhat more complicated for Ku =Kc o2; for in this case there are local minima # in addition to along the cubic axes 7x# and 7y; # A spin whose initial the global minima along 7z: orientation is in a direction sufficiently close to one of these local minima can therefore be expected to relax to that minimum. This gives rise to values of mR that are larger than 1/2 but less than 0.83, at least in the large th limit. For smaller th ; spins can precess across barriers such as the XY plane, thereby ending up at a distant minimum. In consequence, mR decreases with decreasing th ; and can even achieve values of mR less than 1/2 for sufficiently small th : As usual, values of mR tend to decrease with decreasing a; at least until a becomes very small. These features are illustrated in Figs. 3 and 4, which show the results of numerical solution of the LL equation for the same
Nanoparticles with several different sources of anisotropy are described by energy functions more complex than either (3) or (4). For example, a nonspherical particle with a well defined crystal structure will exhibit both shape and magnetocrystalline anisotropy. The simplest case is that of a particle with easy cubic anisotropy and a prolate shape, the particle’s long axis coinciding with one of the cubic axes. The resulting energy function is ~Þ ¼ Ku VS 2 Kc VðS4 þ S4 þ S 4 Þ EðS z x y z ~: Ms V h~ðtÞ S
ð5Þ
When h~ðtÞ ¼ 0; the energy landscape of this model is governed by the ratio Ku =Kc : It is straightforward to show [14] that for Ku =Kc > 2; the global energy minima at 7z# are the only minima on the entire unit sphere. One might therefore expect mR to be fixed at 1/2, just as in the easy uniaxial problem. Except when Ku =Kc is so large that the cubic term of Eq. (5) is negligible, however, this turns out to be false. Unlike in the uniaxial case, the energy in zero field varies as one moves along the energy barrier in the XY plane. Thus, the precession present for all finite values of a enables spins to cross this barrier, reducing mR below 1/2.
G. Grinstein et al. / Journal of Magnetism and Magnetic Materials 237 (2001) 90–96
Fig. 3. Reduced remanent magnetization, mR ; versus rampdown time, th ; for Eq. (5) with parameters given in text, and Ku =Kc ¼ 3=2: Symbol definitions: ð.Þ a ¼ 1:0; ðmÞ a ¼ 0:1; ðKÞ a ¼ 0:01; ð\Þ a ¼ 0:003:
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interacting systems, such as dispersions of nanoparticles dense enough to have interparticle dipolar forces comparable to the anisotropy energies of the individual particles [14,16,26–30]. In general, the reduced remanent magnetizations of interacting systems can be expected to decrease as the rate with which the saturating field is removed increases, and as a decreases, at least for sufficiently large a: While testing the dependence of mR on a is not experimentally feasible, the predicted dependence of mR on the field rampdown time th should be observable, since magnetic field pulses can be turned on and off on subnanosecond time scales. It is hoped that this paper will stimulate experimental work along these lines.
Acknowledgements We are grateful to Snorri Ingvarsson for helpful discussions concerning the experimental determination of damping constants.
References
Fig. 4. Reduced remanent magnetization, mR ; versus rampdown time, th ; for Eq. (5) with parameters given in text, and Ku =Kc ¼ 1: Symbol definitions: ð.Þ a ¼ 1:0; ðmÞ a ¼ 0:1; ðKÞ a ¼ 0:01; ð\Þ a ¼ 0:003:
parameter values as in Fig. 1, and with Ku =Kc ¼ 3=2 and Ku =Kc ¼ 1; respectively.
5. Systems with Different Symmetries and/or Interactions The principles discussed above for cubic and combined uniaxial–cubic symmetry are applicable to other symmetries as well. They also apply to
[1] D. Weller, A. Moser, IEEE Trans. Magn. 35 (1999) 4423. [2] L.D. Landau, E.M. Lifshitz, Phys. Z. Sowjet Union 8 (1935) 153. [3] R.L. Conger, F.C. Essig, Phys. Rev. 104 (1956) 915. [4] J.L. Garc!ia-Palacios, F.J. L!azaro, Phys. Rev B 58 (1998) 14 937. [5] D.A. Garanin, Phys. Rev. B 55 (1997) 3050. [6] G. Grinstein, R.H. Koch, submitted for publication. [7] T.L. Gilbert, Phys. Rev. 100 (1955) 1243. [8] W.F. Brown, Phys. Rev. 130 (1963) 1677. [9] G.E. Uhlenbeck, L.S. Ornstein, Phys. Rev. 36 (1930) 823. [10] L. He, et al., J. Magn. Magn. Mater. 155 (1996) 6. [11] Q. Peng, H.N. Bertram, J. Appl. Phys. 81 (1997) 4384. [12] M. Igarashi, et al., IEEE Trans. Magn. 35 (1999) 2721. [13] N.A. Usov, S.E. Peschany, J. Magn. Magn. Mater. 174 (1997) 247. [14] J.L. Dormann, D. Fiorani, E. Tronc, Adv. Chem. Phys. 98 (1997) 283, and references therein. [15] K. O’Grady, R.W. Chantrell, Magnetic Properties of Fine Particles, Elsevier, Amsterdam, 1992, p. 93. [16] G.A. Held, G. Grinstein, H. Doyle, S. Sun, C.B. Murray, Phys. Rev. B 64 (2001) 012 408. [17] L. N!eel, Ann. G!eophys. 5 (1949) 99. [18] E.C. Stoner, E.P. Wohlfarth, Trans. Roy. Soc. A 240 (1948) 599. [19] R. Gans, Ann. Phys. 15 (1932) 28.
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[20] W.K. Hiebert, A. Stankiewicz, M.R. Freeman, Phys. Rev. Lett. 79 (1997) 1134. [21] G. Ju, et al., Phys. Rev. B 62 (2000) 1171. [22] T.J. Silva, et al., J. Appl. Phys. 85 (1999) 7849. [23] F. Schreiber, et al., Solid State Commun. 93 (1995) 965. [24] W.H. Press, et al., Numerical Recipes in C, 2nd Edition, Cambridge University Press, New York, 1992.
[25] R. Bozorth, Ferromagnetism, Van Nostrand, New York, 1951. [26] T. Jonsson, et al., Phys. Rev. Lett. 75 (1995) 4138. [27] T. Jonsson, et al., Phys. Rev. Lett. 81 (1998) 3976. [28] C. Djurberg, et al., Phys. Rev. Lett. 79 (1997) 5154. [29] W. Luo, et al., Phys. Rev. Lett. 67 (1991) 2721. [30] H. Mamiya, et al., Phys. Rev. Lett. 80 (1998) 177.
Dynamical study of remanent magnetization in nanoparticles G. Grinstein*, G.A. Held, R.H. Koch IBM Corporation, Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA Received 3 January 2001; received in revised form 25 July 2001
Abstract For collections of noninteracting nanoparticles, we study the reduced static remanent magnetization, mR ; produced by the removal of a saturating magnetic field. We show that, except for special cases such as easy uniaxial anisotropy, mR depends on both the ramp-down rate of the field and the energy dissipation rate of the spin dynamics. Using the Landau–Lifshitz equation, we illustrate this result with explicit dynamical calculations of mR for cubic and for mixed cubic–uniaxial anisotropies. r 2001 Elsevier Science B.V. All rights reserved. PACS: 75.60.Lr; 75.10.Hk; 05.40.a; 75.10.b Keywords: Magnetic nanoparticles; Spin models; Remanent magnetization; Spin dynamics; Landau–Lifshitz equation
1. Introduction The drive to design magnetic storage devices with ever increasing bit densities and access speeds has resulted in a heightened need for more powerful techniques for the numerical simulation of classical magnetic systems [1]. The standard starting point for such simulations is the equation of Landau and Lifshitz (LL) [2], or, equivalently [3–6], of Gilbert (G) [7]. These equations capture the two essential ingredients of spin dynamics: the precession of magnetic moments around their local magnetic fields, and the simultaneous relaxation of the system towards an energy minimum. For systems operating at temperatures not much lower than the excitation energies of the system (which *Corresponding author. Tel.: +1-914-945-1604; fax +1-914945-2141. E-mail address: [email protected] (G. Grinstein).
are set by the anisotropy, exchange, and/or dipolar energies), the model must also include the effects of thermal fluctuations. These are accounted for by the addition to the LL equation of a random noise term [8,4–6]. The strength of this term is usually chosen to be related to the phenomenological damping constant, a; and the temperature (T), so that the noise kicks the system up in energy at the same average rate that the damping term pulls it down. In the long-time limit, a steady state obeying Boltzmann statistics for the energy function that describes the system is thereby achieved [8,9]. Equilibrium properties at nonzero temperature are therefore governed entirely by the energy function, and so are independent of the damping. Thus, although the strength of the damping can affect the dynamical properties of magnetic systems significantly [10–12], equilibrium properties at fixed T > 0 must be independent of a: They
0304-8853/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 4 9 2 - 9
G. Grinstein et al. / Journal of Magnetism and Magnetic Materials 237 (2001) 90–96
must also be independent of the gyromagnetic ratio (gÞ governing the speed of precession, and thus can be calculated as correctly through Monte Carlo algorithms as through the LLG equations. At temperatures well below the characteristic excitation energies of the system, however, the damping strength and other dynamical features can play an important role in determining the metastable or stable state into which the system relaxes, thereby affecting even its static behavior. Examples of this phenomenon have been discussed [10–13], notably in the context of the remanent coercivity of magnetic systems, though they have received less attention than the more obvious dynamical effects associated with damping. In this paper, we investigate another example of this phenomenon, by studying the static remanent magnetization of collections of model singledomain magnetic particles. Specifically, we consider the response of the magnetization of the particles to the removal, at fixed low temperature, of a uniform applied saturating magnetic field. This is a familiar experimental situation (for a review, see Refs. [14,15]); indeed, these calculations were motivated by a desire to understand our low-temperature (B225 K) remanence measurements of magnetic nanoparticle dispersions [16]. We take the applied field, h~ðtÞ; to decay exponentially in time from its initial saturating value h~0 : h~ðtÞ ¼ h~0 et=th ; where th is the ramp-down time. It is natural to assume that the remanent magnetization is independent of th [14,15]. However, as we demonstrate here, this assumption is not valid if th is comparable to the damping time. We assume that the system is at low enough temperature that energy barriers are insurmountable on any reasonable experimental time scale, so that the temperature is effectively zero. We then calculate the static remanent magnetization at long times, for collections of particles with two different symmetries: easy cubic anisotropy, and mixtures of uniaxial and cubic anisotropy. The results can be understood qualitatively through simple arguments that should apply to different anisotropies as well. While we treat only collections of particles dilute enough to be noninteracting, the extension of the qualitative arguments to dispersions dense
91
enough to have significant dipolar interactions is straightforward. For each of the symmetries studied, we find that the long-time (static) limit of the reduced remanent magnetization, mR ; i.e., the ratio of the lowtemperature remanent magnetization to the saturation magnetization, is uniquely defined only when th is large when compared with the system’s characteristic relaxation time. When th is comparable to or less than the relaxation time, the final static state depends significantly on both the damping constant and th : The mechanism for this dependenceFthe ability of the precessing system to surmount apparent energy barriers that cannot be overcome in the absence of precession (or, equivalently, when the damping is too large)Fis reviewed below. While it is obviously difficult to vary the damping constant experimentally for a given dispersion of particles, the predicted effect should be observable in remanence experiments in which th is varied.
2. Noninteracting particles and the Landau–Lifshitz equation Following previous studies [8,17], we model each single-domain particle in a collection by a ~ ðtÞ; described by the magnetic moment vector M LL equation [2] ~ ~ ðM ~ r ~ EÞ ~ =qt ¼ g M r ~ E* þ gaM * : ð1Þ qM M M ~j jM Here g is the gyromagnetic ratio, and a is the dimensionless damping constant. The energy ~ Þ includes the anisotropy and *M function E* ¼ Eð Zeeman energies of the particle, as well as any interactions with other particles. Assuming that the volume, V; and saturation magnetization, Ms ; of the nanoparticles are uniform over the collection, we define the (dimensionless) unit vector S~ ~; whereupon Eð ~ Ms V S ~ Þ can be *M through M ~Þ; and Eq. (1) rewritten as written as EðS g ~ ga ~ ~ r ~EÞ: ð2Þ S rS~E þ S ðS qS~=qt ¼ S Ms V Ms V In the simplest example, each particle has a single easy axis, randomly oriented in space, and the
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collection is sufficiently dilute that interparticle interactions can be neglected. It is convenient (especially for generalizing to more complicated anisotropies), to take the easy axis for every particle to be the z-axis, with the direction of the applied field therefore varying randomly from particle to particle. Then, for a given particle, ~Þ ¼ Ku VS 2 Ms V h~ðtÞ S ~: EðS ð3Þ z
Here Ku is the uniaxial anisotropy energy per unit volume, and the field h~ is applied in a random direction. To study the situation where a saturating field is applied, we choose h~0 to satisfy Ms jh~0 jbKu : It is well known that if h~ðtÞ is independent of t; then the precessional term of ~Þ; while the Eq. (2) conserves the energy EðS damping term produces motion straight downhill in energy, with both terms preserving the unit ~: Thus, the particle’s spin precesses magnitude of S about the local field (i.e., about the direction ~ ~E), simultaneously moving monodefined by r S tonically downhill in energy until reaching a local ~ ~E ¼ 0; where it remains until energy minimum, r S the field is altered. In the presence of an applied saturating magnetic field, that minimum lies essentially in the field direction. When the field is removed, there are two obvious limits to consider: (1) th -N; so that the field is removed arbitrarily slowly; in practice this means that th is significantly longer than the typical relaxation time, tR BMs =ðgaKu Þ of the system (roughly 1010 2106 s for the examples discussed below). (2) th ¼ 0; so that the field is removed instantaneously; in practice this means that the relaxation time significantly exceeds th : The first limit is the more common one for experiments on nanoparticles. In this case, the position of the energy minimum moves slowly in time as the field is reduced, and the spin simply follows that minimum precisely [18]. Little preces~ and r ~ ~E point very nearly in sion occurs, since S S the same direction. Thus, the spin moves steadily from the field direction to the nearest accessible easy axis direction. In our simple uniaxial example, this means the easy directionFeither þz# or # zFlocated in the same hemisphere as the field. As is well known [18] (and easily verified), this produces, independent of a; a reduced remanent
magnetization mR ¼ 1=2 for a noninteracting collection of uniaxial particles. In the second limit, the only energy remaining when the field is instantaneously removed is the anisotropy energy. Since the dynamics cannot increase the particle’s energy, the spin cannot cross the hard XY plane, and so spirals down to # the easy directionFþz# or zFclosest to the field direction. The result, again, is the familiar mR ¼ 1=2; independent of a: Moreover, for pure uniaxial anisotropy, mR is equal to 1/2 for any rate of field removal between these two limits. To understand this, consider a particle with the saturating field applied in an arbitrary direction in the upper half plane. Initially, then, the spin points in this direction. As the field is reduced, however, the direction of the net field experienced by the spin rotates progressively towards the þz# axis. In precessing around this upward moving net field direction, the spin cannot cross the XY plane, and so always terminates in the þz# direction. This yields mR ¼ 1=2:
3. Noninteracting particles with easy cubic anisotropy For spherical particles with cubic crystal structure and negligible surface anisotropy, the energy EðS~Þ takes the form ~; EðS~Þ ¼ Kc VðS 4 þ S 4 þ S4 Þ Ms V h~ðtÞ S ð4Þ x
y
z
where Kc ð> 0Þ is the (easy) cubic anisotropy strength. Again, it is instructive to consider the limits th -N and th -0: As in the uniaxial case, extremely slow field removal implies that the spin simply tracks the evolution of the global minimum as it moves from the field direction to the nearest easy cubic direction. It has been shown [19] that the resulting reduced remanent magnetization, averaged over all possible orientations of pthe pffiffiffi ffiffiffi saturating field, is mR ¼ ð6= 2pÞtan1 ð1= 2Þ B0:83; independent of a: (This value exceeds the 0.5 of the uniaxial case because, for cubic symmetry, the ‘‘nearest easy direction’’ is, on average, closer in orientation to the applied field than it is for the uniaxial case.)
G. Grinstein et al. / Journal of Magnetism and Magnetic Materials 237 (2001) 90–96
The case of instantaneous field removal, th ¼ 0; is, however, quite different from the corresponding situation with uniaxial symmetry. To see this, imagine the one-eighth of the unit sphere confined # þ y; # and þz# axes. This surface is between the þx; divided by energy barriers into three equivalent regions, each containing one of the axes. The region associated with a given axis consists of all the points on the surface closer to that axis than to the other two. If there were no precession (i.e., if a-N), so that spins moved straight downhill in energy, then each spin would simply relax to the nearest easy direction, giving mR B0:83: When a is finite, however, a spin that starts close enough to one of the energy barriers can precess across that barrier, ending up at an axis other than the closest one. The reason this can occur is that the energy varies along the barriers, rather than staying constant. Hence the spin can cross a barrier without ever raising its total energy, provided the energy gain required to surmount the barrier by moving perpendicular to it is compensated by the energy loss associated by simultaneously moving parallel to the barrier from a high-energy region to a low-energy one. In other words, these nonuniform barriers can be surmounted by trajectories that cross them at low enough angles.1 This is in sharp contrast to the uniaxial case, where the energy along the XY barrier is uniform, making the barrier impenetrable to spins whose energy must always decrease. In consequence, a spin whose initial orientation is close to one of the barriers will be able to cross that barrier with nonzero probability, thus giving rise to values of mR less than 0.83 in the th ¼ 0 limit. Similar logic leads one to anticipate mR values less than 0.83 for all values of th and a less than infinity. In general, one expects mR to 1 This phenomenon has an intuitive analogy to the notion of crossing the Continental Divide. Suppose one starts just below the ridge that constitutes the Divide in a region where that ridge has a significant slope. Then one can cross the ridge while decreasing one’s altitude monotonically, by choosing a path that runs mostly parallel to the ridge in the downhill direction, but has a small component perpendicular to the ridge that takes one over the top. Of course, such a path does not follow the local altitude gradient, so water, which follows that gradient, does not flow over the Divide.
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decrease with decreasing th ; since smaller th allows for more precession. Decreasing a retards the rate of energy decay, and so should also allow more precession and a concomitantly reduced mR : As a continues to decrease, however, the probability of spins precessing across an energy barrier more than once grows larger, whereupon the tendency for mR to decrease monotonically with decreasing a may cease to hold.2 Fig. 1, a plot of mR as a function of th for several different values of a; shows that these expectations are indeed fulfilled. For a ¼ 0:01; a realistic value for a variety of materials [20–23], mR decreases steadily from close to 0.83 at th ¼ 106 s to about 0.77 at th ¼ 1010 s: Note that mR decreases as a decreases from 1.0 to 0.01, but then increases slightly as a is reduced further to 0.003. This reflects the multiple barrier crossings discussed above. The curves shown in Fig. 1 were produced by solving the LL equation numerically, using the Euler method [24]. We discretized the unit sphere with a roughly uniform mesh of 5000 points in order to average over all directions of the applied field. The parameters used were Kc =Ms ¼ 0:23 Oe;3, h0 ¼ 5:68 Oe; g ¼ 1:76 107 Oe1 s1 =ð1 þ a2 Þ; and a time step of 3 1011 s: For each direction, the system was allowed to evolve for 1,500,000 time steps (i.e. 45 ms), which was long enough for the system to reach a local energy minimum. The value of Kc =Ms used in this simulation is characteristic of Ni-rich cubic Fe–Ni alloys, for which Kc can be quite small [25]. Note, however, that the solution to Eqs. (2) and (4) is unchanged when t is rescaled by a multiplicative factor and Kc =Ms ; th ; and h0 are rescaled by its inverse. Thus, for any given values of Kc and Ms ; resultant values of mR may be extracted from Fig. 1 by simply rescaling the abscissa.
2
Figure 5 of the paper by Usov and Peschany (Ref. [13]), provides a vivid illustration of how sensitive the behavior of magnetic systems can be to small changes in a; particularly when a51: Also see, Refs. [10–12]. 3 Note that the solution to Eqs. (2) and (4) depends only on the ratio Kc =Ms ; and not on either the individual values of these parameters, or on V:
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Fig. 1. Reduced remanent magnetization, mR ; versus rampdown time, th ; for Eq. (4) with parameters given in text. Symbol definitions: ð.Þ a ¼ 1:0; ðmÞ a ¼ 0:1; ðKÞ a ¼ 0:01; ð\Þ a ¼ 0:003:
Fig. 2. Reduced remanent magnetization, mR ; versus rampdown time, th ; for Eq. (5) with parameters given in text, and Ku =Kc ¼ 5=2: Symbol definitions: ð.Þ a ¼ 1:0; ð~Þ a ¼ 0:4; ðmÞ a ¼ 0:1; ðKÞ a ¼ 0:01; ð\Þ a ¼ 0:003:
4. Noninteracting particles with both easy cubic and easy uniaxial anisotropies
For any aoN; one therefore expects mR to decrease from 1/2 at th ¼ N to some lower value at th ¼ 0: As with purely cubic symmetry, values of mR should decrease with decreasing a as well, at least for a values sufficiently large that the probability of spins precessing across the XY plane more than once is small. Verification of these expectations is shown in Fig. 2, which displays the results of solving the LL equation with an energy given by Eq. (5), with all parameter values the same as in Fig. 1, and with Ku ¼ 5Kc =2: The situation is somewhat more complicated for Ku =Kc o2; for in this case there are local minima # in addition to along the cubic axes 7x# and 7y; # A spin whose initial the global minima along 7z: orientation is in a direction sufficiently close to one of these local minima can therefore be expected to relax to that minimum. This gives rise to values of mR that are larger than 1/2 but less than 0.83, at least in the large th limit. For smaller th ; spins can precess across barriers such as the XY plane, thereby ending up at a distant minimum. In consequence, mR decreases with decreasing th ; and can even achieve values of mR less than 1/2 for sufficiently small th : As usual, values of mR tend to decrease with decreasing a; at least until a becomes very small. These features are illustrated in Figs. 3 and 4, which show the results of numerical solution of the LL equation for the same
Nanoparticles with several different sources of anisotropy are described by energy functions more complex than either (3) or (4). For example, a nonspherical particle with a well defined crystal structure will exhibit both shape and magnetocrystalline anisotropy. The simplest case is that of a particle with easy cubic anisotropy and a prolate shape, the particle’s long axis coinciding with one of the cubic axes. The resulting energy function is ~Þ ¼ Ku VS 2 Kc VðS4 þ S4 þ S 4 Þ EðS z x y z ~: Ms V h~ðtÞ S
ð5Þ
When h~ðtÞ ¼ 0; the energy landscape of this model is governed by the ratio Ku =Kc : It is straightforward to show [14] that for Ku =Kc > 2; the global energy minima at 7z# are the only minima on the entire unit sphere. One might therefore expect mR to be fixed at 1/2, just as in the easy uniaxial problem. Except when Ku =Kc is so large that the cubic term of Eq. (5) is negligible, however, this turns out to be false. Unlike in the uniaxial case, the energy in zero field varies as one moves along the energy barrier in the XY plane. Thus, the precession present for all finite values of a enables spins to cross this barrier, reducing mR below 1/2.
G. Grinstein et al. / Journal of Magnetism and Magnetic Materials 237 (2001) 90–96
Fig. 3. Reduced remanent magnetization, mR ; versus rampdown time, th ; for Eq. (5) with parameters given in text, and Ku =Kc ¼ 3=2: Symbol definitions: ð.Þ a ¼ 1:0; ðmÞ a ¼ 0:1; ðKÞ a ¼ 0:01; ð\Þ a ¼ 0:003:
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interacting systems, such as dispersions of nanoparticles dense enough to have interparticle dipolar forces comparable to the anisotropy energies of the individual particles [14,16,26–30]. In general, the reduced remanent magnetizations of interacting systems can be expected to decrease as the rate with which the saturating field is removed increases, and as a decreases, at least for sufficiently large a: While testing the dependence of mR on a is not experimentally feasible, the predicted dependence of mR on the field rampdown time th should be observable, since magnetic field pulses can be turned on and off on subnanosecond time scales. It is hoped that this paper will stimulate experimental work along these lines.
Acknowledgements We are grateful to Snorri Ingvarsson for helpful discussions concerning the experimental determination of damping constants.
References
Fig. 4. Reduced remanent magnetization, mR ; versus rampdown time, th ; for Eq. (5) with parameters given in text, and Ku =Kc ¼ 1: Symbol definitions: ð.Þ a ¼ 1:0; ðmÞ a ¼ 0:1; ðKÞ a ¼ 0:01; ð\Þ a ¼ 0:003:
parameter values as in Fig. 1, and with Ku =Kc ¼ 3=2 and Ku =Kc ¼ 1; respectively.
5. Systems with Different Symmetries and/or Interactions The principles discussed above for cubic and combined uniaxial–cubic symmetry are applicable to other symmetries as well. They also apply to
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