Nonlinear response of fine superparamagnetic particles to the sudden change of a strong uniform DC magnetic field

Journal of Magnetism and Magnetic Materials 241 (2002) 400–414

Nonlinear response of fine superparamagnetic particles to the sudden change of a strong uniform DC magnetic field W.T. Coffeya, Yu.P. Kalmykovb,*, S.V. Titovc b

a Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland Centre d’Etudes Fondamentales, Universit!e de Perpignan, 52 Avenue de Villeneuve, 66860 Perpignan Cedex, France c Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Fryazino, Moscow Region 141190, Russian Federation

Received 13 September 2001

Abstract The nonlinear response of an assembly of superparamagnetic particles due to a sudden change both in the magnitude and in the direction of a strong external uniform DC magnetic field is evaluated. The desired system of moments (the expectation values of the spherical harmonics /Yl;m SðtÞ), which governs the kinetics of the magnetization M of an individual particle, is derived by averaging the stochastic Gilbert equation augmented by a random field over its realizations. As an example, the nonlinear transient response of particles possessing cubic anisotropy is considered. Here, the solution of the moment system is obtained using the matrix continued fraction method. The spectrum of the appropriate relaxation function and the relaxation time of the magnetization are calculated for typical values of the anisotropy, dissipation, and nonlinearity parameters. In general, the relaxation time and spectrum depend strongly on the dissipation parameter due to the coupling between the transverse and longitudinal modes. This behavior is particularly pronounced in the nonlinear regime. r 2002 Elsevier Science B.V. All rights reserved. PACS: 75.50.Tt; 76.20.+q; 05.40.Jc Keywords: Superparamagnetic particles; Cubic anisotropy; Gilbert’s equation; Nonlinear response; Magnetization relaxation

1. Introduction A single domain ferromagnetic particle is characterized by an internal anisotropy potential having two local states of equilibrium with a potential barrier between them. If the particles are small (B10 nm) (so that the potential barriers are relatively low), the magnetization vector MðtÞ may cross over the barriers between one potential well and the other due to thermal fluctuations [1]. The resulting thermal instability of the magnetization gives rise to the phenomenon of superparamagnetism [2], because each particle behaves

*Corresponding author. Tel.: +33-468-662-062; fax: +33-468-662-234. E-mail address: [email protected] (Y.P. Kalmykov). 0304-8853/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 0 9 5 1 - 9

W.T. Coffey et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 400–414

401

(as far as the static susceptibility is concerned) like an enormous paramagnetic atom having a magnetic moment B104–105 Bohr magnetons. Research on the thermal fluctuations and relaxation of the magnetization of single-domain ferromagnetic particles is of current interest for the purpose of improving the characteristics of magnetic carriers of information [3]. The theory of thermal fluctuations of the magnetization of a single domain ferromagnetic particle having been initiated by Ne! el [1], who used the transition state theory of chemical reaction rates to estimate the longest thermal relaxation time, was further developed by Brown [4] using Langevin’s approach to the theory of Brownian motion. Brown took as the Langevin equation Gilbert’s equation augmented by a random field [5]. He then derived from this Langevin equation the Fokker–Planck equation for the distribution function W ðM; tÞ of the orientations of M (see, for example, Refs. [6–8]). Similar diffusion equations describing the Brownian motion of a particle in the high damping noninertial limit are frequently used in the study of dielectric relaxation in molecular and liquid crystals [9], dynamic Kerr effect in liquids [10], and so on. In view of the large magnitude of the magnetic dipole moment (104–105 BM), the interaction (Zeeman) energy of the particle even in a weak moderate external magnetic field H0 may be of the order of the thermal energy kT (k is the Boltzmann constant, and T is the temperature). The relatively large Zeeman energy means that one must take into account nonlinear effects when analyzing the relaxation of the magnetization due to sudden changes both in the magnitude and in the direction of an external DC magnetic field [11–13]. If the characteristic time of the change of the field is much shorter than that of the magnetization relaxation time, one can consider these changes as instantaneous. The problem then may be posed in an obvious manner like the calculation of the after-effect function. However, so far the theory has been successfully developed for the linear response of superparamagnetic fine particles only, where the change in the particle energy due to the variations of an external magnetic field is far less than kT. The calculation of the nonlinear response in strong external fields is a much more difficult task because of the dependence of the nonlinear response upon the form of the stimulus. Hence a universal after-effect function, which may describe all kinds of responses as in linear response no longer exists. Thus, as far as the nonlinear response is concerned, results have hitherto only been derived for a weak AC magnetic field using perturbation theory (see, for example, Refs. [14–17]). A limited degree of success in avoiding the restriction to a weak AC magnetic field has been achieved in recent papers [18–21] for systems of uniaxial particles: in Refs. [18–20] the nonlinear AC response has been evaluated numerically in strong AC magnetic fields while the first treatment of the transient nonlinear relaxation response to sudden changes of a strong magnetic field has been given in Ref. [21]. The purpose of the present paper is to extend this study by investigating the dynamics of the magnetization of superparamagnetic particles with a cubic magnetocrystalline anisotropy when both the magnitude and the direction of a strong external DC magnetic field are suddenly altered (the results of our investigation of the AC response of superparamagnetic particles with a cubic anisotropy subjected to a strong external AC magnetic field will be given in a forthcoming paper).

2. Nonlinear transient response of the magnetization: the Langevin equation approach We suppose that both the magnitude and the direction of an external spatially uniform DC magnetic field is suddenly altered at time t ¼ 0 from HI to HII : We are interested in the transient relaxation of the system of noninteracting superparamagnetic fine particles starting from an equilibrium state I with the distribution function WI (tp0) to another equilibrium state II with the distribution function WII (t-N). The initial distribution function in equilibrium state I is the Boltzmann distribution, viz. WI ¼ ZI1 eb½UðMdHI Þ :

ð1Þ

402

W.T. Coffey et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 400–414

Having switched off the magnetic field, the distribution function in the new equilibrium state II is also given by the Boltzmann distribution, viz. 1 b½UðMdHII Þ e : WII ¼ ZII

ð2Þ

Here U is the free energy density, b ¼ n=kT; n is the particle volume, ZN (N ¼ I; II) are the relevant partition functions, the magnetization vector may be written as M ¼ MS u ¼ MS ðiuX þ juY þ kuZ Þ;

ð3Þ

where MS is the saturation magnetization of the particle, and u is a unit vector whose components are uX ¼ sin y cos j;

uY ¼ sin y sin j;

uZ ¼ cos y;

ð4Þ

with y and j the polar and azimuthal angles (see Fig. 1). The dynamics of the magnetization MðtÞ can be described by the normalized relaxation function: /Mr SðtÞ  /Mr SII /r  uSðtÞ  /r  uSII f ðtÞ ¼ ¼ ; ð5Þ /Mr SI  /Mr SII /r  uSI  /r  uSII where Mr is the projection of magnetization in the direction of a unit vector r: r ¼ ivX þ jvY þ kvZ ;

ð6Þ

(vX ; vY ; vZ are the direction cosines) and the angle brackets denote the averaging of the realizations of the relevant random variable over WN ; Z 2p Z p dj ðr  uÞWN sin y dy: ð7Þ /r  uSN ¼ 0

0

Thus the problem of calculating the transient response is truly nonlinear, because the alteration of a strong external DC magnetic field both in magnitude and in direction is arbitrary. In the diffusion model, the dynamics of the magnetization vector MðtÞ of a single-domain particle are similar (apart from the precession term) to the Brownian rotation of a macromolecule in a liquid in the high

Fig. 1. Geometry of the problem.

W.T. Coffey et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 400–414

403

friction (or noninertial) limit and may be completely described by the Langevin equation for the magnetization. Brown took as the Langevin equation Gilbert’s equation augmented by a random field [4,5]    d d MðtÞ ¼ g MðtÞ HðtÞ þ hðtÞ  Z MðtÞ ; ð8Þ dt dt where g is the gyromagnetic ratio, Z is the damping parameter specifying the dissipative coupling between the spin and its thermal bath. The magnetic field HðtÞ acting on the particle may consist of externally applied magnetic fields, the crystalline anisotropy field and a random Gaussian white noise field hðtÞ with the properties hi ðtÞ ¼ 0;

hi ðtÞhj ðt0 Þ ¼ 2Zb1 dij dðt  t0 Þ;

ð9Þ

(here the overbar means the statistical average over an ensemble of particles which all have at time t the same magnetization M). The random white noise field takes into account the thermal fluctuations of the magnetization of an individual particle. We remark that ‘‘memory’’ and surface effects are not included in Eq. (8). It is also assumed that the internal magnetization of particles is homogeneous. This assumption has been discussed elsewhere (see, for example, Refs. [7,22]). Furthermore, the description of the relaxation processes in the context of Eq. (8) does not take into account effects such as macroscopic quantum tunneling (a mechanism of magnetization reversal suggested in Ref. [3]). These effects become important at very low temperatures and necessitate an appropriate quantum mechanical treatment. Gilbert’s equation augmented by a random field (8) is a vector stochastic differential equation with multiplicative noise terms which poses an interpretation problem as discussed in Refs. [23–25]. Here one must use the Stratonovich definition of a stochastic differential equation involving the average of multiplicative noise terms [26,27]. That definition is the mathematical idealization of the magnetic relaxation process because the white noise arising in the Gilbert equation is in reality the limit of a colored noise process. Thus, it is unnecessary to transform the Langevin Eq. (8) to an Ito# equation. Moreover, one can apply the methods of ordinary analysis. In magnetic relaxation, the relevant quantities are averages involving the spherical harmonics Yl;m defined as [28] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2l þ 1Þðl  mÞ! imj m m e Pl ðcos yÞ; Yl;m ðy; jÞ ¼ ð1Þ ð10Þ 4pðl þ mÞ!  Yl;m ðy; jÞ ¼ ð1Þm Yl;m ðy; jÞ;

ð11Þ

where the Pm l ðcos yÞ are the associated Legendre functions jmjpl: The derivation of the stochastic equation for Yl;m from Gilbert’s Eq. (8) and the averaging of the equation so obtained over an ensemble of particles (realizations), having at time t the same magnetization MðtÞ; was given in Ref. [29]. This averaged equation for the expected values of the Yl;m is tN

d b lðl þ 1Þ /Yl;m S ¼  /ðgrad VII þ a1 ½u grad VII Þ  grad Yl;m S  /Yl;m S; dt 2 2

ð12Þ

where VII ¼ U  ðM  HII Þ; ; tN ¼

bMS ð1 þ a2 Þ 2ga

ð13Þ

is the characteristic free diffusion relaxation time, a ¼ gZMS is the dimensionless dissipation parameter characterizing the dissipative coupling to the heat bath. The r.h.s. of Eq. (12) can be expressed [29] in terms

W.T. Coffey et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 400–414

404

of the angular momentum operators [28]   q q q 2 7ij 7 þ i cot y L ¼ D; LZ ¼ i ; L7 ¼ e ; qj qy qj

ð14Þ

where D is the Laplacian on the surface of unit sphere. One has [29] * d lðl þ 1Þ b 2 /Yl;m S þ ½L ðVYl;m Þ  VL2 Yl;m  Yl;m L2 V : tN /Yl;m S ¼  dt 2 4 rffiffiffiffiffiffi ib 3  fY 1 ½ðLZ Vþ ÞðLþ Yl;m Þ  ðLþ Vþ ÞðLZ Yl;m Þ 4a 2p 1;1 + 1 ½ðLZ V ÞðL Yl;m Þ  ðL V ÞðLZ Yl;m Þg ; þY1;1

ð15Þ

where we have used the following representation for the expansion of VII in terms of spherical harmonics N X R N X 1 X X VII ¼ Vþ þ V ; Vþ ¼ vR;S YR;S ; V ¼ vR;S YR;S : R¼1 S¼0

R¼1 S¼R

By making use of the results of the theory of angular momentum [28], one may essentially simplify the solution of the problem under consideration because the action of the angular momentum operators on Yl;m is given by [28] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LZ Yl;m ¼ mYl;m ; L7 Yl;m ¼ l ðl þ 1Þ  mðm71ÞYl;m71 ; L2 Yl;m ¼ l ðl þ 1ÞYl;m :

ð16Þ

On using Eqs. (16) and the relations [28,29] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lþl ð2l þ 1Þð2l1 þ 1Þ X1 /l; 0; l1; 0jl2 ; 0S/l; m; l1; m1 jl2 ; m þ m1 S pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Yl;m Yl1 ;m1 ¼ Yl2 ;mþm1 ; 4p 2l2 þ 1 l ¼jll j 2

1

Dl2 ¼2 1 Yl;7m Y1;71

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l1 8pð2l þ 1Þðl  mÞ! X ð2L þ 1ÞðL þ m  1Þ! YL;7ðm1Þ ; ðm > 0Þ ¼ 3ðl þ mÞ! ðL  m þ 1Þ! L¼me l;m

DL¼2

(/l1 ; m1 ; l2 ; m2 jl; mS are the Clebsch-Gordan coefficients [10] and el;m ¼ 1; if the indexes l and m are of the same order of evenness and el;m ¼ 0 otherwise), we can transform Eq. (15) to [29]: X dl 0 ;m7s;l;m /Yl 0 ;m7s SðtÞ; ð17Þ tN /Y’ l;m SðtÞ ¼ l 0 ;s

where dl 0 ;m7s;l;m

( rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N ½l 0 ðl 0 þ 1Þ  rðr þ 1Þ  lðl þ 1Þ ð2l þ 1Þð2l 0 þ 1Þ X pffiffiffiffiffiffiffiffiffiffiffiffiffi : vr;7s p 2 2r þ 1 r¼s sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r1 X i ð2r þ 1Þðr  sÞ! ðL þ s  1Þ! /l; 0; l 0 ; 0jr; 0S /l; m; l 0 ; m8sjr; 8sS þ a ðr þ sÞ! ðL  s þ 1Þ! L¼se

lðl þ 1Þdl;l 0 ds;0 b þ ð1Þm ¼ 4 2

r;s

DL¼2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /l; 0; l 0 ; 0jL; 0Sððm7sÞ ðL þ sÞðL  s þ 1Þ/l; m; l 0 ; m8sjL; 8sS ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 8s ðl 0 7m þ sÞðl 0 8m  s þ 1Þ/l; m; l ; m8s71jL; 8s71SÞ :

ð18Þ

W.T. Coffey et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 400–414

405

Here it is assumed that sX0; and the summation is carried out over those values of indexes for which the Clebsch-Gordan coefficients /l1 ; m1 ; l2 ; m2 jl; mS [29] are meaningful. We now have from Eq. (17) the set of recurrence equations for the relaxation functions cl;m ðtÞ ¼ /Yl;m SðtÞ  /Yl;m SII ; viz. XX d dl 0 ;m7s;l;m cl 0 ;m7s ðtÞ: ð19Þ tN cl;m ðtÞ ¼ dt s l0 This set must be solved subject to the initial conditions cl;m ð0Þ ¼ /Yl;m SI  /Yl;m SII :

ð20Þ

In order to derive Eq. (19), we remark that the equilibrium averages /Yl 0 ;m7s SN (N ¼ I; II) satisfy the recurrence relation: XX dl 0 ;m7s;l;m /Yl 0 ;m7s SN ¼ 0; ðN ¼ I; IIÞ: ð21Þ l0

S

The normalized nonlinear relaxation function from Eq. (5) can then be expressed in terms of cl;m ðtÞ as pffiffiffi 2vZ c1;0 ðtÞ þ ðvX þ ivY Þc1;1 ðtÞ  ðvX  ivY ÞðvX  ivY Þc1;1 ðtÞ pffiffiffi : ð22Þ f ðtÞ ¼ 2vZ c1;0 ð0Þ þ ðvX þ ivY Þc1;1 ð0Þ  ðvX  ivY Þc1;1 ð0Þ We remark that Eq. (19) may also be obtained from the corresponding Fokker–Planck equation for the distribution function W of the orientations of M (that equation was derived for the first time by Brown [4] from Gilbert’s Eq. (8)) q 1 W¼ fb½a1 u  ðgrad V grad W Þ þ divðW grad V Þ þ DW g; qt 2tN

ð23Þ

where u is the unit vector directed along M: As has been shown in Ref. [29], the Langevin equation and Fokker–Planck equation approaches are equivalent.

3. Solution of Eq. (19) Eqs. (17)–(21) are valid for arbitrary anisotropy (uniaxial, cubic, and so on). Here, we consider cubic anisotropy and we suppose, for simplicity, that the field HI and HII are parallel to the Z-axis of the particle, i.e. vX ¼ vY ¼ 0; vZ ¼ 1 in Eq. (22). For t > 0; the free-energy density VII of the particle with cubic anisotropy in the field HII is [30] rffiffiffi K p VII ¼ ðsin4 y sin2 2j þ sin2 2yÞ  HII MS cos y ¼ 2HII MS Y1;0 4 3 rffiffiffiffiffiffiffiffi 2K pffiffiffi K 10p K ð24Þ  pY4;0  ½Y4;4 þ Y4;4  þ ; 15 15 7 5 where K is the anisotropy constant, which may take both positive and negative values (for cubic crystals of Fe-type and Ni-type, respectively). The coefficients dl 0 ;m0 ;l;m in Eq. (19) with VII defined by Eq. (24) are given by Eq. (18) and are listed in the Appendix (see also Refs. [23,25,30]). Eq. (19), which is, from a mathematical point of view, an infinite-dimensional system of simultaneous linear differential equations with constant coefficients, can be solved by applying the matrix continued fraction approach [26,27]. The essence of this approach consists in the transformation of the multiterm recurrence Eq. (19) into a tridiagonal vector recurrence equation, the exact solution of which is given in terms of matrix continued fractions. In order to apply this approach to the problem under consideration, we introduce the

W.T. Coffey et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 400–414

406

vector Cn ðtÞ: 0

c4n ðtÞ

1

Bc C B 4n1 ðtÞ C Cn ðtÞ ¼ B C @ c4n2 ðtÞ A

ðn ¼ 1; 2; 3; yÞ:

ð25Þ

c4n3 ðtÞ This vector has 4 subvectors 0 1 c4ni;4ðn1þdi0 Þ ðtÞ Bc C B 4ni;4ðn2þdi0 Þ ðtÞ C c4ni ðtÞ ¼ B C; @ A ^

ði ¼ 0; 1; 2; 3Þ

ð26Þ

c4ni;4ðn1þdi0 Þ ðtÞ Eq. (19) can then be transformed to a tridiagonal vector recurrence equation of the form d þ Cn ðtÞ ¼ Q n Cn1 ðtÞ þ Qn Cn ðtÞ þ Qn Cnþ1 ðtÞ dt ðn ¼ 1; 2; 3; yÞ;

tN

ð27Þ

 with C0 ðtÞ ¼ 0: The matrices Qn ; Qþ n ; Qn are given in the Appendix. By invoking the general method for * 1 ðoÞ in the form solving matrix recursion Eqs. (27) [27], we obtain the exact solution for the spectrum C [23,26] ( ! ) N n Y X þ * 1 ðoÞ ¼ tN D1 ðoÞ C1 ð0Þ þ Q Dk ðoÞ Cn ð0Þ ; ð28Þ C k1

n¼2

k¼2

where I

Dn ðoÞ ¼ iotN I  Qn  Qþ n

I iotN I  Qnþ1  Qþ nþ1

I Q iotN I  Qnþ2 & nþ2

is a matrix continued fraction, the tilde denotes the Fourier transform: Z N * FðoÞ ¼ F ðtÞeiot dt:

ð29Þ Q nþ1

ð30Þ

0

The initial value vector Cn ð0Þ in Eq. (28) can also be calculated in terms of matrix continued fractions (see Appendix A). * 1 ðoÞ from Eq. (28), we may calculate the spectrum of the relaxation function from Having determined C Eq. (22) and the integral relaxation time of the transient nonlinear response (area under the curve of the relaxation function), viz. Z N c*1;0 ð0Þ : ð31Þ t¼ f ðtÞ dt ¼ c1;0 ð0Þ 0 The relaxation time t may equivalently be defined in the context of the Fokker–Planck Eq. (23) converted to the Sturm–Liouville problem as P k ck =lk t¼ P ; ð32Þ k ck

W.T. Coffey et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 400–414

407

where lk and ck are the eigenvalues and their corresponding weight coefficients (amplitudes), as the function c1;0 ðtÞ can be given by X c1;0 ðtÞ ¼ ck elk t : ð33Þ k

The integral nonlinear relaxation time t Eq. (32) contains contributions from all the eigenvalues. In general, it is impossible to evaluate t analytically from Eq. (32) as knowledge of all the lk and ck is required. The approach developed does not attempt to calculate t by explicitly calculating the eigenvalue spectrum as required by Eq. (32) rather it yields t in terms of matrix continued fractions. However, as has been shown in many examples (e.g., Refs. [13,27]), the results given by Eqs. (31) and (32) are completely equivalent. As far as a physical interpretation is concerned in many cases the relaxation time t is determined by the slowest low-frequency relaxation mode governing transitions of the magnetization over the barriers from one potential well into another. The characteristic frequency of this overbarrier relaxation mode is determined by the smallest eigenvalue l1 : Thus l1 is the reciprocal time constant associated with the long time behavior of the relaxation function which is only determined by the slowest low-frequency relaxation mode. The behavior of the relaxation time t and the inverse of the smallest eigenvalue l1 is sometimes similar. However, if different time scales are involved, the behavior of these can be quite different [13,30–32].

4. Results and discussion The exact solution in terms of matrix continued fractions (see Eqs. (28) and (29)) is well suited to numerical calculations. All the matrix continued fractions and the associated series converge very rapidly, so that 10–30 downward iterations for the computation of these matrix fractions and 10–30 terms in the series (Eq. (28)) are sufficient for an accuracy of at least six significant figures in most cases. The results strongly depend on the dissipation parameter a due to the coupling of the longitudinal and transverse relaxation modes. However, this dependence qualitatively corresponds to that obtained for the linear

Fig. 2. lnðt=tN Þ as a function of s (so0) for various values of hI ¼ hII for a sudden reversal of the direction of the DC magnetic field (HI ¼ HII ), a ¼ 0:1:

408

W.T. Coffey et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 400–414

Fig. 3. lnðt=tN Þ as a function of s (so0) for a sudden decrease in the magnitude of the DC magnetic field: h ¼ hII ¼ hI =2; a ¼ 0:1:

Fig. 4. lnðt=tN Þ as a function of s (s > 0) and h ¼ hII ¼ hI =2 for a sudden decrease in the magnitude of the DC magnetic field, a ¼ 0:1:

response which is investigated in detail in Refs. [23,25,30]. Theoretical and experimental estimates of a give values of the order of B0.01–0.1 [7,22,33]. Here, the calculation was made for a ¼ 0:1: The evolution of the relaxation time t as a function of the anisotropy parameter s (so0) for various values of the field parameter h ¼ hI is illustrated by Fig. 2 for a sudden reversal of the direction of a strong uniform external DC magnetic field (HI ¼ HII ; i.e. hII ¼ hI ). The evolution of the relaxation time t as a function of s is illustrated by Fig. 3 (so0) and Fig. 4 (s > 0) for a sudden change of the magnitude of DC magnetic field (h ¼ hII ¼ hI =2). In these figures, the initial decrease of t at small s is due to the effect of the high-frequency transverse modes, which are the characteristic frequencies close to the precession frequency of the magnetization vector M (at high damping a > 1; this initial decrease of t does not exist). The

W.T. Coffey et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 400–414

409

dependence of t on the anisotropy parameter s for small h and large s has an activation character. This leads to an exponential growth of the relaxation time t as the height of the potential barrier s increases as is predicted by transition state theory. However, as HII increases, the relaxation time t decreases with increasing s (see Figs. 2–4) in contrast to the transition state theory. This effect, first reported in Ref. [13] in an analysis of the linear response of an assembly of uniaxial particles in the low-temperature limit, is due to the depletion of the population of the upper potential well [31,32]. This effect also exists in cubic crystals [30]. In particular, for values of the parameter h above a certain critical level hc ; the integral relaxation time t no longer has an activation character at large s (curve 4 in Fig. 2). At this critical value of the bias parameter, the relaxation switches from being dominated by the slowest overbarrier mode to being dominated by the fast intrawell relaxation modes [31]. Thus the relaxation time t decreases as the height of the potential barrier increases. Hence, the relaxation time exponentially diverges from the greatest relaxation time or the inverse of the smallest eigenvalue l1 for values of the field in excess of the critical value. * The calculated absolute value of the relaxation function spectrum jfðoÞj is represented in Fig. 5 (so0) and Fig. 6 (s > 0) for a step switch-on of the DC magnetic field (hI ¼ 0; hII ¼ h). Two dispersion bands are * visible in the spectrum jfðoÞj: The characteristic frequency and half-width of the low-frequency band are determined by the inverse of the smallest eigenvalue l1 : The high-frequency band, which becomes more pronounced at smaller values of the damping constant, arises from the excitation of high-frequency longitudinal modes and transverse modes. The approach developed in the present paper allows us to evaluate also the linear response characteristics of a system of superparamagnetic particles with cubic anisotropy to infinitesimally small changes in the strength of the strong DC field HI ; i.e. for hII ¼ hI  k; at k-0: In this particular case, the relaxation function f ðtÞ from Eq. (5) coincides with the normalized longitudinal dipole equilibrium correlation function C8 ðtÞ; viz. lim f ðtÞ ¼ C8 ðtÞ ¼

k-0

/cos yð0Þcos yðtÞSI  /cos yð0ÞS2I : /cos2 yð0ÞSI  /cos yð0ÞS2I

* N Þ as a function of log10 ðotN Þ and s for a sudden switch-on (hI ¼ 0; hII ¼ 0:2) of the DC magnetic field for various s: Fig. 5. lnðjfj=t s ¼ 5 (curve 1), s ¼ 15 (curve 2), s ¼ 30 (curve 3), and s ¼ 50 (curve 4); a ¼ 0:1:

410

W.T. Coffey et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 400–414

* N Þ as a function of log ðotN Þ for a sudden switch-on (hI ¼ 0; hII ¼ 0:3) of the DC magnetic field for various s: s ¼ 1 Fig. 6. lnðjfj=t 10 (curve 1), s ¼ 5 (curve 2), s ¼ 10 (curve 3), and s ¼ 1 (curve 4); a ¼ 0:1:

Having determined the one-sided Fourier transform of C8 ðtÞ; one can calculate the linear integral relaxation time tint ; viz. Z N * C8 ðtÞ dt; tint ¼ C8 ð0Þ ¼ 0

and the linear dynamic longitudinal susceptibility w8 ðoÞ of the system [30]: Z N C8 ðtÞeiot dt; w8 ðoÞpC8 ð0Þ  io 0

and, hence, the linear response to a small AC field HðtÞ ¼ H expðiotÞ superimposed on the DC bias field HI since the (longitudinal) Z-component of the magnetization MZ ðtÞ is defined as MZ ðtÞ ¼ vMs N0 ½/cosySðtÞ  /cos ySI  ¼ w8 ðoÞHðtÞ: The linear response of the system under consideration has been studied in detail in Ref. [30]. It should also be noted that just as in uniaxial particles [32], the absolute value of the overall nonlinear integral relaxation time may differ substantially from the linear integral relaxation time tint of the magnetization in the states I and II (here tint in the states I and II characterizes the linear response of the spins to the small change in the strength of the field parameter hN ; i.e. tint in the state I is evaluated for hII ¼ hI  k at k-0 while tint in the state II is calculated for hI ¼ hII þ k at k-0). In the calculations, we have assumed that all the particles are identical; in order to account for polydispersity, it is necessary to average over the appropriate distribution function (e.g., over the particle volumes, see for detail Ref. [15]). Furthermore, the neglect of interparticle interactions in the present model suggests that the results we have obtained are applicable for systems, where the effects of the dipole–dipole interactions may be ignored, such as individual nanoparticles and dilute solid suspensions of nanoparticles. * To summarize we have presented in this paper a general method for the calculation of the spectrum fðoÞ and relaxation time t of the nonlinear transient response function f ðtÞ (Eq. (5)) of an assembly of

W.T. Coffey et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 400–414

411

superparamagnetic particles due to a sudden change of a strong external DC magnetic field. For single* and t can be calculated from Eqs. (20) and (28) using domain magnetic particles with cubic anisotropy, fðoÞ the matrix continued fraction technique for all values of the nonlinearity, anisotropy and dissipation parameters. The method we have proposed may be useful in the construction and interpretation of experiments, which probe nonlinear response.

Acknowledgements We thank Prof. J.L. De! jardin for useful comments. The support of this work by the Enterprise Ireland Research Collaboration Fund 2001, USAF, EOARD (contract F61773-01-WE407) and the Russian Foundation for Basic Research (project no. 01-02-16050) is gratefully acknowledged. S.V.T thanks the Royal Irish Academy for the award of the Senior Visiting Fellowship in 2001.

Appendix A  The matrices Qn ; Qþ n ; Qn in Eq. (28) are defined as 0 B4n þ F4n P4n A4n þ E4n B T T B B4n þ f4n F4n A4n1 þ E4n1 B4n1 þ F4n1 Qn ¼ B B p PT T T B4n1 þ f4n1 F4n1 A4n2 þ E4n2 @ 4n 4n T T T D4n p4n1 P4n1 B4n2 þ f4n2 FT4n2

0

J4n

0

0

0

1

D4n1 P4n2

J4n1 D4n2

0 J4n2

0 0

C C C; A

B4n3 þ F4n3

P4n3

D4n3

J4n3

B B Q n ¼B @ 0 B B B Qþ ¼ n B @

D4n

1

C C C; B4n2 þ F4n2 C A A4n3 þ E4n3 P4n1

ðA:1Þ

ðA:2Þ

j4nþ4 JT4nþ4

DT4nþ3

p4nþ2 PT4nþ2

BT4nþ1 þ f4nþ1 FT4nþ1

0

j4nþ3 JT4nþ3

DT4nþ2

p4nþ1 PT4nþ1

0

0

j4nþ2 JT4nþ2

DT4nþ1

0

0

0

j4nþ1 JT4nþ1

The exception is Q 1 which degenerates to a column vector, viz. 0 1 J4 BD C B 3C Q C; 1 ¼B @ P2 A

1 C C C; C A

ðA:3Þ

ðA:4Þ

B1  The symbol T in Eqs. (A.1)–(A.3) denotes transposition. The dimensions of the matrices Qn ; Qþ n ; Qn are accordingly ð8n  2Þ ð8n  2Þ; ð8n  2Þ ð8n þ 6Þ; ð8n  2Þ ð8n  10Þ: The factors pn ; jn and fn are given by

2n  11 ; pn ¼  2n þ 9

jn ¼ 

n4 ; nþ1

fn ¼ 

n1 : nþ1

ðA:5Þ

W.T. Coffey et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 400–414

412

Submatrices A4n ; A4n1 ; A4n2 ; A4n3 ; B4n1 ; B4n2 ; B4n3 ; D4n1 ; P4n1 ; P4n2 have the form 1 0 0 ? 0 x4ni;4ðn1þdi0 Þ xþ 4ni;4ðn1þdi0 Þ C B  C B x4ni;4ðn2þd Þ x4ni;4ðn2þdi0 Þ xþ ? 0 4ni;4ðn2þdi0 Þ i0 C B C B X4ni ¼ B C x ? 0 0 x 4ni;4ðn3þd Þ i0 Þ 4ni;4ðn3þd i0 C B C B ^ ^ ^ & ^ A @ 0 0 0 ? x4ni;4ðn1þdi0 Þ

ðA:6Þ

(i ¼ 0; 1; 2; 3). The dimension of these submatrices is ½2ðn þ d0i Þ  1 ½2ðn þ d0i Þ  1: Submatrices B4n ; D4n ; J4n ; P4n ; D4n2 ; D4n3 ; J4n1 ; J4n2 ; J4n3 ; P4n3 are defined as 1 0 þ 0 0 ? 0 x4ni;4ðn1þdi0 Þ C B C B x4ni;4ðn2þdi0 Þ xþ 0 ? 0 4ni;4ðn2þdi0 Þ C B C B  þ ðA:7Þ X4ni ¼ B x4ni;4ðn3þdi0 Þ x4ni;4ðn3þdi0 Þ x4ni;4ðn3þdi0 Þ ? 0 C C B C B ^ ^ ^ & ^ A @  0 0 0 ? x4ni;4ðn1þdi0 Þ (i ¼ 0; 1; 2; 3). The dimension of the submatrix (A.7) is ½2ðn þ d0i Þ  1 ½2ðn þ d0i Þ  3: Submatrices E4n ; E4n1 ; E4n2 ; E4n3 ; F4n1 ; F4n2 ; F4n3 have the form of (A.6) with x7 i ¼ 0: Submatrix F4n has the form of (A.7) with x7 ¼ 0: The elements of the submatrices so defined are given by i al;m ¼ s

a7 l;m ¼

9ðl 2  1Þððl þ 1Þ2  1Þ  15m2 ð6lðl þ 1Þ  5  7m2 Þ lðl þ 1Þ ;  2 ð4l 2  9Þð4ðl þ 1Þ2  9Þ

15s

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl8mÞðl7m þ 4Þðl 2  ðm71Þ2 Þðl 2  ðm72Þ2 Þðl 2  ðm73Þ2 Þ ; 2ð4l 2  9Þð4ðl þ 1Þ2  9Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ismð3l  5  7m Þ l 2  m2 ; ¼ að4l 2  9Þ 4l 2  1 2

bl;m

2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl8mÞðl8m  4Þðl 2  ðm71Þ2 Þðl 2  ðm72Þ2 Þðl 2  ðm73Þ2 Þ ; 4l 2  1

b7 l;m

3is ¼7 2að4l 2  9Þ

pl;m

sð2l þ 9Þðl 2  l  2  7m2 Þ ¼ ð2l  5Þð2l  1Þð2l þ 3Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl 2  m2 Þððl  1Þ2  m2 Þ ; ð2l  3Þð2l þ 1Þ

sð2l þ 9Þ 2ð2l  5Þð2l  1Þð2l þ 3Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl8m  5Þðl8m  4Þðl8m  3Þðl8mÞ½l 2  ðm72Þ2 ½l 2  ðm71Þ2  ; ð2l  3Þð2l þ 1Þ

p7 l;m ¼ 

dl;m

7ism ¼ að4ðl  1Þ2  1Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl 2  m2 Þððl  1Þ2  m2 Þððl  2Þ2  m2 Þ ; 4ðl  1Þ2  9

W.T. Coffey et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 400–414

413

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is ðl8m  6Þðl8m  5Þyðl8m  1Þðl8mÞðl7m þ 1Þ ; ¼8 2 2að4ðl  1Þ  1Þ 4ðl  1Þ2  9 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7sðl þ 1Þ ððl  3Þ2  m2 Þððl  2Þ2  m2 Þððl  1Þ2  m2 Þðl 2  m2 Þ jl;m ¼ ; ð2l  5Þð2l  3Þð2l  1Þ ð2l  7Þð2l þ 1Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sðl þ 1Þ ðl8m  7Þðl8m  6Þyðl8m  1Þðl8mÞ 7 jl;m ¼ 2ð2l  5Þð2l  3Þð2l  1Þ ð2l  7Þð2l þ 1Þ

7 dl;m

el;m ¼ 

fl;m

ishN m ; a

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 2  m2 : ¼ shN ðl þ 1Þ 4l 2  1

Here s ¼ bnK=4; xN ¼ bnMs HN ; hN ¼ xN =2s: The initial value vectors Cn ð0Þ in Eq. (29) are calculated in the following manner. We introduce the vector 0 N 1 r4n B rN C B 4n1 C RN C n ¼B N @ r4n2 A rN 4n3 with

0

1 /Y4ni;4ðn1þdi0 Þ SN B /Y C 4ni;4ðn2þdi0 Þ SN C B rN C 4ni ðtÞ ¼ B @ A ^

ði ¼ 0; 1; 2; 3Þ;

/Y4ni;4ðn1þdi0 Þ SN where the index N ¼ I; II corresponds to the fields HI and HII : Next, we transform Eq. (21) to the matrix recursion formula þ N N N Q n Rn1 þ Qn Rn þ Qn Rnþ1 ¼ 0;

n ¼ 0; 1; 2; 3; y

The solution of this equation has the form 1 N  N  N N N RN n ¼ Dn ð0ÞQn Rn1 ¼ pffiffiffiffiffiffiDn ð0ÞDn1 ð0ÞyD1 ð0ÞQ1 : 4p pffiffiffiffiffiffi Here, we have allowed for the fact that RN 0 ¼ 1= 4p: So we can write the initial value vector as Cn ð0Þ ¼ RIn  RII n:

References [1] [2] [3] [4]

L. N!eel, Ann. Geophys. 5 (1949) 99. H.B. Braun, H.N. Bertram, J. Appl. Phys. 75 (1994) 4609. C.P. Bean, J.D. Livingston, Suppl. J. Appl. Phys. 30 (1959) 1205. W.F. Brown Jr., Phys. Rev. 130 (1963) 1677.

414 [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

W.T. Coffey et al. / Journal of Magnetism and Magnetic Materials 241 (2002) 400–414 T.L. Gilbert, Phys. Rev. 100 (1956) 1243. W.F. Brown Jr., IEEE Trans Magn. 15 (1979) 1196. Yu.L. Raikher, M.I. Shliomis, Adv. Chem. Phys. 87 (1994) 595. L.J. Geoghegan, W.T. Coffey, B. Mulligan, Adv. Chem. Phys. 100 (1997) 475. G. Moro, P.L. Nordio, Z. Phys. BFCondens. Matter 64 (1986) 217. J.L. Dejardin, Dynamic Kerr Effect, World Scientific, Singapore, 1996. A. Aharoni, Phys. Rev. 177 (1969) 763. D.A. Garanin, V.V. Ischenko, L.V. Panina, Teor. Mat. Fiz. 82 (1990) 242. W.T. Coffey, D.S.F. Crothers, Yu.P. Kalmykov, J.T. Waldron, Phys. Rev. B 51 (1995) 15947. E.K. Sadykov, A.G. Isavnin, Fiz. Tverd. Tela (St.Petersburg) 38 (1997) 2104 [Phys. Solid State 38 (1996) 1160]. Yu.L. Raikher, V.I. Stepanov, Phys. Rev. B 55 (1997) 15005. Yu.L. Raikher, V.I. Stepanov, A.N. Grigirenko, P.I. Nikitin, Phys. Rev. B. 56 (1997) 6400. J.L. Garcia-Palacios, P. Svedlindh, Phys. Rev. Lett. 85 (2000) 3724. Yu.L. Raikher, V.I. Stepanov, Phys. Rev. Lett. 86 (2001) 1923. Yu.L. Raikher, V.I. Stepanov, Fiz. Tverd. Tela (St. Petersburg) 43 (2001) 270 [Phys. Sol. State 43 (2001) 279]. J.L. D!ejardin, Yu.P. Kalmykov, P.M. D!ejardin, Adv. Chem. Phys. 117 (2001) 275. Yu.P. Kalmykov, S.V. Titov, Fiz. Tverd. Tela (St. Petersburg) 42 (2000) 893 [Phys. Sol. State 42 (2000) 918]. I. Klik, L. Gunther, J. Stat. Phys. 60 (1990) 473. Yu.P. Kalmykov, S.V. Titov, W.T. Coffey, Phys. Rev. B. 58 (1998) 3267. Yu.P. Kalmykov, S.V. Titov, Fiz. Tverd. Tela (St. Petersburg) 40 (1998) 1642 [Phys. Solid State, 40 (1998) 1492]. Yu.P. Kalmykov, S.V. Titov, ZhETF 115 (1999) 101 [JETP 88 (1999) 58] H. Risken, The Fokker-Planck Equation, Springer, Berlin, 1989. W.T. Coffey, Yu.P. Kalmykov, J.T. Waldron, The Langevin Equation, World Scientific, Singapore, 1996. R.N. Zare, Angular Momentum. Understanding Spatial Aspects in Chemistry and Physics, Wiley, New York, 1989. Yu.P. Kalmykov, S.V. Titov, Phys. Rev. Lett. 82 (1999) 2967. Yu.P. Kalmykov, Phys. Rev. B 61 (2000) 6205. D.A. Garanin, Phys. Rev. E 54 (1996) 3250. Yu.P. Kalmykov, J.L. D!ejardin, W.T. Coffey, Phys. Rev. E. 55 (1997) 2509. W.T. Coffey, D.S.F. Crothers, J.L. Dormann, Yu.P. Kalmykov, E.C. Kennedy, W. Wernsdorfer, Phys. Rev. Lett. 80 (1998) 5655.