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PHYSICAL REVIEW B 82, 100413共R兲 共2010兲

Nonlinear magnetization relaxation of superparamagnetic nanoparticles in superimposed ac and dc magnetic bias fields Serguey V. Titov,1 Pierre-Michel Déjardin,2 Halim El Mrabti,2 and Yuri P. Kalmykov2 1Institute

of Radio Engineering and Electronics, Russian Academy of Sciences, 1, Vvedenskii Square, Fryazino 141190, Russia Université de Perpignan Via Domitia, 52, Avenue de Paul Alduy, 66860 Perpignan Cedex, France 共Received 9 July 2010; revised manuscript received 9 August 2010; published 24 September 2010兲

2LAMPS,

The nonlinear ac response of the magnetization M共t兲 of a uniaxially anisotropic superparamagnetic nanoparticle subjected to both ac and dc bias magnetic fields of arbitrary strengths and orientations is determined by averaging Gilbert’s equation augmented by a random field with Gaussian white-noise properties in order to calculate exactly the relevant statistical averages. It is shown that the magnetization dynamics of the uniaxial particle driven by a strong ac field applied at an angle to the easy axis of the particle 共so that the axial symmetry is broken兲 alters drastically leading to different nonlinear effects due to coupling of the thermally activated magnetization reversal mode with the precessional modes of M共t兲 via the driving ac field. DOI: 10.1103/PhysRevB.82.100413

PACS number共s兲: 75.40.Gb, 05.40.Jc, 75.50.Tt, 76.20.⫹q

Fine ferromagnetic particles are characterized by thermal instability of their magnetization M共t兲 resulting in spontaneous change in their orientation from one metastable state to another by surmounting energy barriers, giving rise to superparamagnetism which is very important in information storage and rock magnetism as well as in biomedical applications.1,2 Due to the large magnitude of the magnetic dipole moment 共⬃104 – 105 B兲 giving rise to a relatively large Zeeman energy even in moderate external magnetic fields, the magnetization reversal process has a strong-field dependence causing nonlinear effects in the dynamic susceptibility and field-induced birefringence,2,3 stochastic resonance,4–6 dynamic hysteresis,7,8 etc. However, nonlinear response to an external field represents an extremely difficult task even for dilute systems because it always depends on the precise nature of the stimulus. Thus no unique response function valid for all stimuli exists unlike in linear response.9 The nonlinear magnetic response of an individual superparamagnetic nanoparticle in the presence of the thermal agitation can be evaluated2,3,9 by calculating the relevant statistical averages from Gilbert’s 共or Landau-Lifshitz兲 equation augmented by a random field h共t兲 with Gaussian white-noise properties, accounting for thermal fluctuations of M共t兲 due to the heat bath, viz.,10 ˙ 共t兲 + h共t兲兴其. tM共t兲 = ␥兵M共t兲 ⫻ 关− MV共t兲 − M

共1兲

Here ␥ is the gyromagnetic ratio, is the damping parameter, and V共M , t兲 is the free energy per unit volume. This is made up of the nonseparable Hamiltonian of the magnetic anisotropy U共M兲 and Zeeman energy densities, the latter arising from external magnetic dc and ac fields H0 + H cos t of arbitrary strengths and orientations. Now the nonlinear ac stationary response has hitherto been calculated for uniaxial superparamagnets either 共i兲 by assuming the energy of a particle in external fields is much less than the thermal energy kT so that the response may be evaluated via perturbation theory 共e.g., Refs. 3 and 11兲 or 共ii兲 by assuming that strong external fields are directed along the easy axis of the particle so that axial symmetry is preserved 共e.g., Refs. 2 and 12兲. Thus the results are very restricted. In particular, the conventional assumption of axial symmetry is 1098-0121/2010/82共10兲/100413共4兲

hardly realizable in nanoparticle systems under experimental conditions because the easy axes of the particles are randomly oriented in space. Furthermore, many interesting nonlinear phenomena 共such as damping dependence of the response and interplay between precession and thermoactivation3兲 cannot be included because in axial symmetry no dynamical coupling between the longitudinal and transverse 共or precessional兲 modes of motion exists. In contrast, discarding the above assumptions we shall now present an exact nonperturbative method for the nonlinear magnetization relaxation of superparamagnetic particles with an arbitrary anisotropy potential U in a strong ac driving field superimposed on a strong dc bias field of arbitrary orientations. Moreover, taking as example uniaxial superparamagnets, we shall demonstrate that for arbitrary orientations of ac and dc bias fields 共so breaking the axial symmetry兲, the magnetization dynamics changes substantially leading to different nonlinear effects which cannot be treated via perturbation theory. We remark in passing that nonlinear effects in relaxation processes of superparamagnetic nanoparticles are closely related to those in nonlinear dielectric relaxation and the dynamic Kerr effect in molecular liquids and liquid crystals,11,13 harmonic mixing in a cosine potential,14 the nonlinear impedance of Josephson junctions,15 ac-driven vortices in superconductors,16 etc. Thus our approach can also be applied to nonlinear effects in these. When the magnitude of the ac field H共t兲 is so large that the Zeeman energy of a particle is comparable to or higher than kT, one is faced with an intrinsically nonlinear problem which of course cannot be treated by perturbation theory and which we solve as follows. First we transform the stochastic Gilbert Eq. 共1兲 to an infinite hierarchy of stochastic differential-recurrence relations which on averaging over their realizations using the properties of white noise yield differential-recurrence relations for the statistical moments 具Y l,m典共t兲 共the expectation values of the spherical harmonics Y l,m兲, viz.,9,17

Nt具Y l,m典共t兲 = 兺 el,m,l+r,m+s共t兲具Y l+r,m+s典共t兲,

共2兲

s,r

where N = 0共␣ + ␣−1兲, ␣ = ␥ M s is a dimensionless damping constant, 0 =  M S / 共2␥兲 is the free-rotational diffusion time

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of M共t兲, M S is the saturation magnetization,  = v / 共kT兲, and v is the volume of the particle. Equation 共2兲 was derived in Ref. 16 and we may now solve it exactly as follows. By introducing column vectors cn共t兲 共n = 1 , 2 , 3 , . . .兲 with c0 = 具Y 00典 = 1 / 冑4, which are formed from the statistical moments cl,m共t兲 = 具Y l,m典共t兲, Eq. 共2兲 becomes the matrix recurrence equation

Ntcn共t兲 = q−n cn−1共t兲 + qncn共t兲 + q+n cn+1共t兲 + 关p−n cn−1共t兲 + pncn共t兲 + p+n cn+1共t兲兴共eit + e−it兲, 共3兲 ⫾ where the supermatrices qn , q⫾ n and pn , pn are generated from the coefficients el,m,l+r,m+s共t兲. We remark in passing that the explicit form of the vectors cn共t兲 depends on the type of the free-energy density V共 , 兲; for uniaxial superparamagnets, cn共t兲 are given by Eq. 共7兲 below. Since we are solely concerned with the stationary ac response, which is independent of the initial conditions, we require the steady-state solution of Eq. 共3兲 only. In the steady-state response, symmetry under time translation is retained under the discrete time transformation t → t + 2 / . Thus we may seek all the cn共t兲 ⬁ ckn共兲eikt. in the form of the time Fourier series cn共t兲 = 兺k=−⬁ On substituting this series into Eq. 共3兲, we have the recurrence relations for the Fourier amplitudes ckn共兲, namely,

qkn共k兲ckn共兲

+

k q+n cn+1 共兲

+

where the matrix continued fraction S1 is defined by the recurrence equation − Sn = − 关Qn + Q+n Sn+1Qn+1 兴−1 .

The vector C1 contains all the Fourier amplitudes required for the nonlinear stationary response. We emphasize that so far our matrix continued fraction solution, Eq. 共6兲, is valid for an arbitrary anisotropy potential U. Next we shall apply the above general method to the particular case of uniaxial superparamagnets subjected to the ac and dc bias ac fields H0 + H cos t applied in arbitrary directions, where the free energy can be written in dimensionless form as

V = sin2 − 0共␥1 sin cos + ␥2 sin sin + ␥3 cos 兲 − cos t共␥1⬘ sin cos + ␥2⬘ sin sin + ␥3⬘ cos 兲. Here ␥1 , ␥2 , ␥3 and ␥1⬘ , ␥2⬘ , ␥3⬘ are the direction cosines of the vectors H0 and H, respectively, K is the anisotropy constant, = K, 0 = H0M S, and = HM S. Now the vectors cn共t兲 ⫾ and the supermatrices qn , q⫾ n and pn , pn are given by

k−1 k+1 k+1 + p−n 关cn−1 共兲 + cn−1 共兲兴 + pn关ck−1 n 共兲 + cn 共兲兴

cn共t兲 = 共4兲

where qn共k兲 = −ikNI + qn and I is the identity matrix. Now Eq. 共4兲 can be transformed into the matrix recurrence relations

共5兲

p+n =

where the column vectors R and Cn and the tridiagonal supermatrices Qn and Q⫾ n are defined as

冢 冣 冢冣 ]

C n共 兲 =

]

q+n =

]

c−2 n 共兲

c−1 n 共兲 c0n共兲 c1n共兲 c2n共兲

R=

−1

冑4

p−1 q−1 p−1

,

o

b2n−1

o , o

o d2n , o o

q−n =

Z2n Y2n , o Z2n−1

冊 冊

a2n b2n , d2n−1 a2n−1

pn =

V2n o , W2n−1 V2n−1

qn =

冊

X2n W2n . Y2n−1 X2n−1

共7兲

Here o and 0 are zero matrices and vectors of appropriate dimensions, respectively. The tridiagonal submatrices al, bl, and dl have the dimensions 共2l + 1兲 ⫻ 共2l + 1兲, 共2l + 1兲 ⫻ 共2l + 3兲, and 共2l + 1兲 ⫻ 共2l − 1兲, respectively. Their matrix elements are given by

0

,

冉 冊 冉 冉 冊 冉 冉 冊 冉

p−n = 共n ⬎ 1兲,

具Y 2n,2n典共t兲 具Y 2n−1,−2n+1典共t兲 ] 具Y 2n−1,2n−1典共t兲

Q1C1 + Q+1 C2 = R, Q−n Cn−1 + QnCn + Q+n Cn+1 = 0

冢 冣 具Y 2n,−2n典共t兲 ]

k q−n cn−1 共兲

k−1 k+1 + p+n 关cn+1 共兲 + cn+1 共兲兴 = 0,

共6兲

C1 = S1 · R,

,

− + 共al兲n,m = ␦n−1,mal,−l+m + ␦n,mal,−l+m−1 + ␦n+1,mal,−l+m−2 ,

0 ]

− + 共bl兲n,m = ␦n,mbl,−l+m−1 + ␦n+1,mbl,−l+m−2 + ␦n+2,mbl,−l+m−3 ,

关Qn兴l,m = ␦l−1,mpn + ␦l,mqn共m兲 + ␦l+1,mpn ,

− + 共dl兲n,m = ␦n−2,mdl,−l+m+1 + ␦n−1,mdl,−l+m + ␦n,mdl,−l+m−1 ,

⫾ ⫾ ⫾ 关Q⫾ n 兴l,m = ␦l−1,mpn + ␦l,mqn + ␦l+1,mpn .

where

The exact solution of Eq. 共5兲 for C1 can be now given using matrix continued fractions, viz., 100413-2

an,m = − i

m␥3⬘ , 4␣

bn,m = −

␥3⬘n 4

冑

共n + 1兲2 − m2 , 共2n + 1兲共2n + 3兲

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NONLINEAR MAGNETIZATION RELAXATION OF…

冑 冑 冑

1

共n + m + 1兲共n + m + 2兲 , 共2n + 1兲共2n + 3兲

␥⬘共n + 1兲 dn,m = 3 4 + = dn,m

−Im[χ1 (ω)]

共␥1⬘ − i␥2⬘兲n 8

n2 − m2 , 共2n + 1兲共2n − 1兲

共␥1⬘ − i␥2⬘兲共n + 1兲 8

+

␥ ⬘c k 共 兲 3 3 10

k k 共␥1⬘ + i␥2⬘兲c1−1 共兲 − 共␥1⬘ − i␥2⬘兲c11 共兲

冑2

册

.

共8兲

Here we shall assume from now on that the vectors H0 and H are parallel and they lie in the XZ plane of the laboratory coordinate system so that ␥1 = ␥1⬘ = sin , ␥2 = ␥2⬘ = 0, and ␥3 = ␥3⬘ = cos , where is the angle between H0 and the Z axis is taken as the easy axis of the particle. For nonparallel ac and dc fields, results will be presented elsewhere. For a weak ac field, → 0, 11共兲 = 3m11共兲 / defines the normalized linear dynamic susceptibility and our results agree in all respects with the benchmark linear-response solution.18 The plots of the magnetic loss spectrum −Im关11共兲兴 vs N are shown in Fig. 1. Here the lowfrequency behavior of 11共兲 can be described by a single Lorentzian, viz.,

11共兲 1−␦ ⬇ + ␦, 1 1 + i 1共0兲

共9兲

where is the longest relaxation time in the absence of an ac external field and ␦ is a parameter 共here is associated with the reversal time of the magnetization;19 simple analytic equations for and ␦ are given in Refs. 9 and 20兲. Equation 共9兲 is also plotted in Fig. 1. Now is related to the characteristic frequency max, where −Im关11共兲兴 reaches a maximum, and/or the half-width ⌬ of Re关11共兲兴 of the Lorentzian as

⬇

−1 max ⬇

⌬ . −1

−2

10

−3

10

−4

4

σ =10 α =0.01 ξ0 =0 ξ =0.01 (a)

3

1

5

2

1

(b)

4 1

5 −3

10

10

−1

ωτN

10

1

3

10

FIG. 1. 共Color online兲 −Im关11共兲兴 vs N for various values of the angle and 共a兲 0 = 0 共no dc bias field兲 and 共b兲 0 = 3 共strong dc bias field兲. Solid lines: matrix continued fraction solution; filled circles: Eq. 共9兲 with calculated using the method of Ref. 20.

single high-frequency Lorentzian band. The third or ferromagnetic resonance 共FMR兲 peak 共due to excitation of transverse modes with frequencies close to the precession frequency pr of the magnetization兲 appears only at low damping and strongly manifest itself at high frequencies. Moreover, for = 0, when the axial symmetry is restored, the FMR peak disappears 关curves 1 in Fig. 1共a兲兴 because the transverse modes no longer take part in the relaxation process so that this peak is a signature of the symmetry-breaking action of the applied field. In strong ac fields, ⬎ 1, pronounced nonlinear effects occur 共see Fig. 2兲. In particular, the low-frequency band of −Im关11共兲兴 can no longer be approximated by a single Lorentzian. Nevertheless, Eq. 共10兲 may still be used in order to estimate an effective reversal time of the magnetization . 共We remark that may also be evaluated from the spectra of the higher order harmonics12 because the low-frequency parts of their spectra are also dominated by the magnetization reversal兲. The behavior of max 共and, hence, 兲 as functions of the ac field amplitude depends on whether or not a dc field is applied. For a strong dc bias, 0 ⬎ 1, the lowfrequency peak shifts to lower frequencies reaching a maximum at ⬃ 0 thereafter decreasing exponentially with increasing . In other words, as the dc field increases, the reversal time of the magnetization initially increases and then having attained its maximum at some critical value ⬃ 0 decreases exponentially 共see Fig. 2兲. For weak dc bias 0 ⱕ 0 ⬍ 1, the low-frequency peak shifts monotonically to higher frequencies. As seen in Fig. 2共a兲, as the ac field am-

共10兲

For zero dc bias field, 0 = 0, is independent of the angle while in a strong dc field, e.g., for 0 = 3, substantially depends on 关Fig. 1共b兲兴. In addition, a far weaker second relaxation peak appears at high frequencies. This relaxation band is due to the “intrawell” modes which are virtually indistinguishable in the frequency spectrum appearing as a

2 1:ψ=0 2:ψ=π/6 3:ψ=π/4 4:ψ=π/3 5:ψ=π/2

σ =10 α = 0.01 ξ0 = 3 ξ =0.01 3

−5

共n − m兲共n − m − 1兲 共2n + 1兲共2n − 1兲

冑冋

10

1

10

− + − + − + an,m = −共an,−m 兲ⴱ, bn,m = −共bn,−m 兲ⴱ, and dn,m = −共dn,−m 兲ⴱ. The ma⫾ trices qn , qn consist of the five submatrices Vl, Wl, Xl, Yl, and Zl 共they also appear in the linear response and are defined explicitly, e.g., in Ref. 9, Chap. 9兲. Having determined k 共兲 from Eq. 共6兲, we can evaluate the the amplitudes cl,m ⬁ Re关mk1共兲eikt兴, where magnetization M H共t兲 = v M S兺k=1

mk1共兲 = 4

10 −1 10 −2 10 −3 10 −4 10

10

−1

10

−2

10

−3

10

−4

4

3

10

−2

10

−3

10

−4

α =0.01 σ =10 ξ0=3 ψ = π/4

(a)

1: ξ=0,01 2: ξ=1 3: ξ=3 4: ξ=5

2

1

1

+ = bn,m

0

共␥1⬘ − i␥2⬘兲 冑共n + m + 1兲共n − m兲, =−i 8␣

−Im[χ1(ω)]

+ an,m

ξ0=3 ξ =1 σ =10 ψ = π/4

2 4

3

(b)

1

1: α=0.01 2: α=0.1 3: α=1 4: α=10 −5

10

10

−3

10

−1

ωτΝ

1

10

10

3

FIG. 2. 共Color online兲 −Im关11共兲兴 vs N 共a兲 for various values of the ac field parameter and 共b兲 for various values of damping ␣.

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plitude increases, the FMR peak decreases and also broadens showing pronounced nonlinear saturation effects characteristic for a soft spring. This effect is very similar to that already known in atomic and molecular spectroscopy.21 Thus we see that the intrinsic damping dependence of the ac nonlinear response for the oblique field configuration 关see Fig. 2共b兲兴 serves as a signature of the coupling between the longitudinal and precessional modes of the magnetization. Hence, it should be possible to determine the damping coefficient ␣ from measurements of nonlinear response characteristics,2 e.g., by fitting the theory to the experimental dependence of 11共兲 on the angle and the ac and dc bias field strengths, so that the sole fitting parameter is ␣, which can be determined at different temperatures T, yielding its temperature dependence. This is important because a knowledge of ␣ and its T dependence allows for the separation of the various relaxation mechanisms.9 We now estimate the parameter range, where the nonlinear effects appear. For cobalt nanoparticles with mean diameter a ⬃ 10 nm and saturation magnetization M S ⬃ 1460 G, the field parameter for T ⬃ 30 K is on the order of unity for H ⬃ 6 kT / 共a3M S兲 ⬃ 5 Oe. Furthermore, an ac field of this order of magnitude is easily attained in measurements of the nonlinear response of magnetic nanoparticles; e.g., Bitoh et al.22 have measured the nonlinear susceptibility of cobalt nanoparticles in the temperature range 4.2–280 K using an ac magnetic field ⬃5 – 30 Oe. As far the characteristic time N is concerned, for ␥ ⬇ 2 ⫻ 107 rad/ Oe s and ␣ ⬃ 0.1, we have at room temperature N ⬃ 10−8 s. To conclude we have developed a nonperturbative approach in terms of matrix continued fractions for the nonlinear relaxation of a uniaxial superparamagnetic particle for arbitrary strengths and orientations of the dc bias and ac driving fields. We have shown that the nonlinear ac stationary response to a strong ac field applied at an angle to the easy axis of the particle 共so that the axial symmetry is bro-

Q. A. Pankhurst et al., J. Phys. D 42, 224001 共2009兲. Yu. L. Raikher and V. I. Stepanov, Adv. Chem. Phys. 129, 419 共2004兲. 3 J. L. García-Palacios and P. Svedlindh, Phys. Rev. Lett. 85, 3724 共2000兲. 4 L. Gammaitoni et al., Rev. Mod. Phys. 70, 223 共1998兲. 5 Yu. L. Raikher et al., Phys. Rev. E 56, 6400 共1997兲. 6 Yu. L. Raikher and V. I. Stepanov, Phys. Rev. Lett. 86, 1923 共2001兲; Yu. L. Raikher et al., J. Magn. Magn. Mater. 258-259, 369 共2003兲. 7 I. Klik and Y. D. Yao, J. Appl. Phys. 89, 7457 共2001兲; Yu. L. Raikher and V. I. Stepanov, J. Magn. Magn. Mater. 320, 2692 共2008兲. 8 P. M. Déjardin et al., J. Appl. Phys. 107, 073914 共2010兲. 9 W. T. Coffey et al., The Langevin Equation, 2nd ed. 共World Scientific, Singapore, 2004兲. 10 W. F. Brown Jr., Phys. Rev. 130, 1677 共1963兲; IEEE Trans. Magn. 15, 1196 共1979兲. 11 W. T. Coffey et al., Phys. Rev. E 71, 062102 共2005兲. 12 P. M. Déjardin and Yu. P. Kalmykov, J. Appl. Phys. 106, 123908 共2009兲; J. Magn. Magn. Mater. 322, 3112 共2010兲. 1 2

ken兲 is very sensitive to both the ac field orientation and amplitude owing to the coupling induced by the symmetry breaking driving field between the precession of the magnetization and its thermally activate reversal over the saddle point. In particular, the pronounced damping and ac field dependence of the nonlinear response 11共兲 can be used to determine the damping coefficient ␣ just as for higher harmonic responses.3 We emphasize that these nonlinear effects in 11共兲 cannot be treated via perturbation theory. Our calculations, since they are valid for ac fields of arbitrary strengths and orientations, allow one both to predict and interpret quantitatively nonlinear phenomena in magnetic nanoparticles such as nonlinear magnetic susceptibility, nonlinear stochastic resonance and dynamic hysteresis, nonlinear ac field effects on the switching field curves, etc., where perturbation theory and the assumption that axial symmetry is preserved are no longer valid 共these results will be published elsewhere兲. For practical applications 共e.g., in magnetic nanoparticle hyperthermia1兲, in order to account for the polydispersity of the particles of a real sample and the fact the easy axes of particles are randomly distributed in space, one must also average the nonlinear response functions mk1共兲 over appropriate distribution functions2 共averaging of mk1共兲 over particle volumes and orientations can be readily accomplished numerically using Gaussian quadraturs23兲. Here, only uniaxial superparamagnetic particles have been treated. Particles with nonaxially symmetric anisotropies 共cubic, biaxial, etc.兲 can be considered in like manner. Finally, our results can be adapted to other nonlinear phenomena such as nonlinear dielectric relaxation and the dynamic Kerr effect in molecular liquids and liquid crystals.11,13 The support of the work by the Agence Nationale de la Recherche, France 共Project No. ANR-08-P147-36兲 and by EGIDE, France 共Program ECO-NET, Project No. 21394NH兲 is gratefully acknowledged.

H. Watanabe and A. Morita, Adv. Chem. Phys. 56, 255 共1984兲; J. L. Dejardin et al., ibid. 117, 275 共2001兲. 14 H. J. Breymayer et al., Appl. Phys. B: Lasers Opt. 28, 335 共1982兲. 15 W. T. Coffey et al., Phys. Rev. B 62, 3480 共2000兲; Phys. Rev. E 61, 4599 共2000兲. 16 V. A. Shklovskij and O. V. Dobrovolskiy, Phys. Rev. B 78, 104526 共2008兲. 17 Yu. P. Kalmykov and S. V. Titov, Phys. Rev. Lett. 82, 2967 共1999兲. 18 Yu. P. Kalmykov and S. V. Titov, Fiz. Tverd. Tela 共St. Petersburg兲 40, 1642 共1998兲 关Phys. Solid State 40, 1492 共1998兲兴; W. T. Coffey et al., Phys. Rev. B 64, 012411 共2001兲. 19 W. T. Coffey et al., Phys. Rev. Lett. 80, 5655 共1998兲. 20 Yu. P. Kalmykov, J. Appl. Phys. 96, 1138 共2004兲. 21 R. Karplus and J. Schwinger, Phys. Rev. 73, 1020 共1948兲. 22 T. Bitoh et al., J. Phys. Soc. Jpn. 64, 1311 共1995兲; J. Magn. Magn. Mater. 154, 59 共1996兲. 23 Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun 共Dover, New York, 1972兲. 13

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PHYSICAL REVIEW B 82, 100413共R兲 共2010兲

Nonlinear magnetization relaxation of superparamagnetic nanoparticles in superimposed ac and dc magnetic bias fields Serguey V. Titov,1 Pierre-Michel Déjardin,2 Halim El Mrabti,2 and Yuri P. Kalmykov2 1Institute

of Radio Engineering and Electronics, Russian Academy of Sciences, 1, Vvedenskii Square, Fryazino 141190, Russia Université de Perpignan Via Domitia, 52, Avenue de Paul Alduy, 66860 Perpignan Cedex, France 共Received 9 July 2010; revised manuscript received 9 August 2010; published 24 September 2010兲

2LAMPS,

The nonlinear ac response of the magnetization M共t兲 of a uniaxially anisotropic superparamagnetic nanoparticle subjected to both ac and dc bias magnetic fields of arbitrary strengths and orientations is determined by averaging Gilbert’s equation augmented by a random field with Gaussian white-noise properties in order to calculate exactly the relevant statistical averages. It is shown that the magnetization dynamics of the uniaxial particle driven by a strong ac field applied at an angle to the easy axis of the particle 共so that the axial symmetry is broken兲 alters drastically leading to different nonlinear effects due to coupling of the thermally activated magnetization reversal mode with the precessional modes of M共t兲 via the driving ac field. DOI: 10.1103/PhysRevB.82.100413

PACS number共s兲: 75.40.Gb, 05.40.Jc, 75.50.Tt, 76.20.⫹q

Fine ferromagnetic particles are characterized by thermal instability of their magnetization M共t兲 resulting in spontaneous change in their orientation from one metastable state to another by surmounting energy barriers, giving rise to superparamagnetism which is very important in information storage and rock magnetism as well as in biomedical applications.1,2 Due to the large magnitude of the magnetic dipole moment 共⬃104 – 105 B兲 giving rise to a relatively large Zeeman energy even in moderate external magnetic fields, the magnetization reversal process has a strong-field dependence causing nonlinear effects in the dynamic susceptibility and field-induced birefringence,2,3 stochastic resonance,4–6 dynamic hysteresis,7,8 etc. However, nonlinear response to an external field represents an extremely difficult task even for dilute systems because it always depends on the precise nature of the stimulus. Thus no unique response function valid for all stimuli exists unlike in linear response.9 The nonlinear magnetic response of an individual superparamagnetic nanoparticle in the presence of the thermal agitation can be evaluated2,3,9 by calculating the relevant statistical averages from Gilbert’s 共or Landau-Lifshitz兲 equation augmented by a random field h共t兲 with Gaussian white-noise properties, accounting for thermal fluctuations of M共t兲 due to the heat bath, viz.,10 ˙ 共t兲 + h共t兲兴其. tM共t兲 = ␥兵M共t兲 ⫻ 关− MV共t兲 − M

共1兲

Here ␥ is the gyromagnetic ratio, is the damping parameter, and V共M , t兲 is the free energy per unit volume. This is made up of the nonseparable Hamiltonian of the magnetic anisotropy U共M兲 and Zeeman energy densities, the latter arising from external magnetic dc and ac fields H0 + H cos t of arbitrary strengths and orientations. Now the nonlinear ac stationary response has hitherto been calculated for uniaxial superparamagnets either 共i兲 by assuming the energy of a particle in external fields is much less than the thermal energy kT so that the response may be evaluated via perturbation theory 共e.g., Refs. 3 and 11兲 or 共ii兲 by assuming that strong external fields are directed along the easy axis of the particle so that axial symmetry is preserved 共e.g., Refs. 2 and 12兲. Thus the results are very restricted. In particular, the conventional assumption of axial symmetry is 1098-0121/2010/82共10兲/100413共4兲

hardly realizable in nanoparticle systems under experimental conditions because the easy axes of the particles are randomly oriented in space. Furthermore, many interesting nonlinear phenomena 共such as damping dependence of the response and interplay between precession and thermoactivation3兲 cannot be included because in axial symmetry no dynamical coupling between the longitudinal and transverse 共or precessional兲 modes of motion exists. In contrast, discarding the above assumptions we shall now present an exact nonperturbative method for the nonlinear magnetization relaxation of superparamagnetic particles with an arbitrary anisotropy potential U in a strong ac driving field superimposed on a strong dc bias field of arbitrary orientations. Moreover, taking as example uniaxial superparamagnets, we shall demonstrate that for arbitrary orientations of ac and dc bias fields 共so breaking the axial symmetry兲, the magnetization dynamics changes substantially leading to different nonlinear effects which cannot be treated via perturbation theory. We remark in passing that nonlinear effects in relaxation processes of superparamagnetic nanoparticles are closely related to those in nonlinear dielectric relaxation and the dynamic Kerr effect in molecular liquids and liquid crystals,11,13 harmonic mixing in a cosine potential,14 the nonlinear impedance of Josephson junctions,15 ac-driven vortices in superconductors,16 etc. Thus our approach can also be applied to nonlinear effects in these. When the magnitude of the ac field H共t兲 is so large that the Zeeman energy of a particle is comparable to or higher than kT, one is faced with an intrinsically nonlinear problem which of course cannot be treated by perturbation theory and which we solve as follows. First we transform the stochastic Gilbert Eq. 共1兲 to an infinite hierarchy of stochastic differential-recurrence relations which on averaging over their realizations using the properties of white noise yield differential-recurrence relations for the statistical moments 具Y l,m典共t兲 共the expectation values of the spherical harmonics Y l,m兲, viz.,9,17

Nt具Y l,m典共t兲 = 兺 el,m,l+r,m+s共t兲具Y l+r,m+s典共t兲,

共2兲

s,r

where N = 0共␣ + ␣−1兲, ␣ = ␥ M s is a dimensionless damping constant, 0 =  M S / 共2␥兲 is the free-rotational diffusion time

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of M共t兲, M S is the saturation magnetization,  = v / 共kT兲, and v is the volume of the particle. Equation 共2兲 was derived in Ref. 16 and we may now solve it exactly as follows. By introducing column vectors cn共t兲 共n = 1 , 2 , 3 , . . .兲 with c0 = 具Y 00典 = 1 / 冑4, which are formed from the statistical moments cl,m共t兲 = 具Y l,m典共t兲, Eq. 共2兲 becomes the matrix recurrence equation

Ntcn共t兲 = q−n cn−1共t兲 + qncn共t兲 + q+n cn+1共t兲 + 关p−n cn−1共t兲 + pncn共t兲 + p+n cn+1共t兲兴共eit + e−it兲, 共3兲 ⫾ where the supermatrices qn , q⫾ n and pn , pn are generated from the coefficients el,m,l+r,m+s共t兲. We remark in passing that the explicit form of the vectors cn共t兲 depends on the type of the free-energy density V共 , 兲; for uniaxial superparamagnets, cn共t兲 are given by Eq. 共7兲 below. Since we are solely concerned with the stationary ac response, which is independent of the initial conditions, we require the steady-state solution of Eq. 共3兲 only. In the steady-state response, symmetry under time translation is retained under the discrete time transformation t → t + 2 / . Thus we may seek all the cn共t兲 ⬁ ckn共兲eikt. in the form of the time Fourier series cn共t兲 = 兺k=−⬁ On substituting this series into Eq. 共3兲, we have the recurrence relations for the Fourier amplitudes ckn共兲, namely,

qkn共k兲ckn共兲

+

k q+n cn+1 共兲

+

where the matrix continued fraction S1 is defined by the recurrence equation − Sn = − 关Qn + Q+n Sn+1Qn+1 兴−1 .

The vector C1 contains all the Fourier amplitudes required for the nonlinear stationary response. We emphasize that so far our matrix continued fraction solution, Eq. 共6兲, is valid for an arbitrary anisotropy potential U. Next we shall apply the above general method to the particular case of uniaxial superparamagnets subjected to the ac and dc bias ac fields H0 + H cos t applied in arbitrary directions, where the free energy can be written in dimensionless form as

V = sin2 − 0共␥1 sin cos + ␥2 sin sin + ␥3 cos 兲 − cos t共␥1⬘ sin cos + ␥2⬘ sin sin + ␥3⬘ cos 兲. Here ␥1 , ␥2 , ␥3 and ␥1⬘ , ␥2⬘ , ␥3⬘ are the direction cosines of the vectors H0 and H, respectively, K is the anisotropy constant, = K, 0 = H0M S, and = HM S. Now the vectors cn共t兲 ⫾ and the supermatrices qn , q⫾ n and pn , pn are given by

k−1 k+1 k+1 + p−n 关cn−1 共兲 + cn−1 共兲兴 + pn关ck−1 n 共兲 + cn 共兲兴

cn共t兲 = 共4兲

where qn共k兲 = −ikNI + qn and I is the identity matrix. Now Eq. 共4兲 can be transformed into the matrix recurrence relations

共5兲

p+n =

where the column vectors R and Cn and the tridiagonal supermatrices Qn and Q⫾ n are defined as

冢 冣 冢冣 ]

C n共 兲 =

]

q+n =

]

c−2 n 共兲

c−1 n 共兲 c0n共兲 c1n共兲 c2n共兲

R=

−1

冑4

p−1 q−1 p−1

,

o

b2n−1

o , o

o d2n , o o

q−n =

Z2n Y2n , o Z2n−1

冊 冊

a2n b2n , d2n−1 a2n−1

pn =

V2n o , W2n−1 V2n−1

qn =

冊

X2n W2n . Y2n−1 X2n−1

共7兲

Here o and 0 are zero matrices and vectors of appropriate dimensions, respectively. The tridiagonal submatrices al, bl, and dl have the dimensions 共2l + 1兲 ⫻ 共2l + 1兲, 共2l + 1兲 ⫻ 共2l + 3兲, and 共2l + 1兲 ⫻ 共2l − 1兲, respectively. Their matrix elements are given by

0

,

冉 冊 冉 冉 冊 冉 冉 冊 冉

p−n = 共n ⬎ 1兲,

具Y 2n,2n典共t兲 具Y 2n−1,−2n+1典共t兲 ] 具Y 2n−1,2n−1典共t兲

Q1C1 + Q+1 C2 = R, Q−n Cn−1 + QnCn + Q+n Cn+1 = 0

冢 冣 具Y 2n,−2n典共t兲 ]

k q−n cn−1 共兲

k−1 k+1 + p+n 关cn+1 共兲 + cn+1 共兲兴 = 0,

共6兲

C1 = S1 · R,

,

− + 共al兲n,m = ␦n−1,mal,−l+m + ␦n,mal,−l+m−1 + ␦n+1,mal,−l+m−2 ,

0 ]

− + 共bl兲n,m = ␦n,mbl,−l+m−1 + ␦n+1,mbl,−l+m−2 + ␦n+2,mbl,−l+m−3 ,

关Qn兴l,m = ␦l−1,mpn + ␦l,mqn共m兲 + ␦l+1,mpn ,

− + 共dl兲n,m = ␦n−2,mdl,−l+m+1 + ␦n−1,mdl,−l+m + ␦n,mdl,−l+m−1 ,

⫾ ⫾ ⫾ 关Q⫾ n 兴l,m = ␦l−1,mpn + ␦l,mqn + ␦l+1,mpn .

where

The exact solution of Eq. 共5兲 for C1 can be now given using matrix continued fractions, viz., 100413-2

an,m = − i

m␥3⬘ , 4␣

bn,m = −

␥3⬘n 4

冑

共n + 1兲2 − m2 , 共2n + 1兲共2n + 3兲

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NONLINEAR MAGNETIZATION RELAXATION OF…

冑 冑 冑

1

共n + m + 1兲共n + m + 2兲 , 共2n + 1兲共2n + 3兲

␥⬘共n + 1兲 dn,m = 3 4 + = dn,m

−Im[χ1 (ω)]

共␥1⬘ − i␥2⬘兲n 8

n2 − m2 , 共2n + 1兲共2n − 1兲

共␥1⬘ − i␥2⬘兲共n + 1兲 8

+

␥ ⬘c k 共 兲 3 3 10

k k 共␥1⬘ + i␥2⬘兲c1−1 共兲 − 共␥1⬘ − i␥2⬘兲c11 共兲

冑2

册

.

共8兲

Here we shall assume from now on that the vectors H0 and H are parallel and they lie in the XZ plane of the laboratory coordinate system so that ␥1 = ␥1⬘ = sin , ␥2 = ␥2⬘ = 0, and ␥3 = ␥3⬘ = cos , where is the angle between H0 and the Z axis is taken as the easy axis of the particle. For nonparallel ac and dc fields, results will be presented elsewhere. For a weak ac field, → 0, 11共兲 = 3m11共兲 / defines the normalized linear dynamic susceptibility and our results agree in all respects with the benchmark linear-response solution.18 The plots of the magnetic loss spectrum −Im关11共兲兴 vs N are shown in Fig. 1. Here the lowfrequency behavior of 11共兲 can be described by a single Lorentzian, viz.,

11共兲 1−␦ ⬇ + ␦, 1 1 + i 1共0兲

共9兲

where is the longest relaxation time in the absence of an ac external field and ␦ is a parameter 共here is associated with the reversal time of the magnetization;19 simple analytic equations for and ␦ are given in Refs. 9 and 20兲. Equation 共9兲 is also plotted in Fig. 1. Now is related to the characteristic frequency max, where −Im关11共兲兴 reaches a maximum, and/or the half-width ⌬ of Re关11共兲兴 of the Lorentzian as

⬇

−1 max ⬇

⌬ . −1

−2

10

−3

10

−4

4

σ =10 α =0.01 ξ0 =0 ξ =0.01 (a)

3

1

5

2

1

(b)

4 1

5 −3

10

10

−1

ωτN

10

1

3

10

FIG. 1. 共Color online兲 −Im关11共兲兴 vs N for various values of the angle and 共a兲 0 = 0 共no dc bias field兲 and 共b兲 0 = 3 共strong dc bias field兲. Solid lines: matrix continued fraction solution; filled circles: Eq. 共9兲 with calculated using the method of Ref. 20.

single high-frequency Lorentzian band. The third or ferromagnetic resonance 共FMR兲 peak 共due to excitation of transverse modes with frequencies close to the precession frequency pr of the magnetization兲 appears only at low damping and strongly manifest itself at high frequencies. Moreover, for = 0, when the axial symmetry is restored, the FMR peak disappears 关curves 1 in Fig. 1共a兲兴 because the transverse modes no longer take part in the relaxation process so that this peak is a signature of the symmetry-breaking action of the applied field. In strong ac fields, ⬎ 1, pronounced nonlinear effects occur 共see Fig. 2兲. In particular, the low-frequency band of −Im关11共兲兴 can no longer be approximated by a single Lorentzian. Nevertheless, Eq. 共10兲 may still be used in order to estimate an effective reversal time of the magnetization . 共We remark that may also be evaluated from the spectra of the higher order harmonics12 because the low-frequency parts of their spectra are also dominated by the magnetization reversal兲. The behavior of max 共and, hence, 兲 as functions of the ac field amplitude depends on whether or not a dc field is applied. For a strong dc bias, 0 ⬎ 1, the lowfrequency peak shifts to lower frequencies reaching a maximum at ⬃ 0 thereafter decreasing exponentially with increasing . In other words, as the dc field increases, the reversal time of the magnetization initially increases and then having attained its maximum at some critical value ⬃ 0 decreases exponentially 共see Fig. 2兲. For weak dc bias 0 ⱕ 0 ⬍ 1, the low-frequency peak shifts monotonically to higher frequencies. As seen in Fig. 2共a兲, as the ac field am-

共10兲

For zero dc bias field, 0 = 0, is independent of the angle while in a strong dc field, e.g., for 0 = 3, substantially depends on 关Fig. 1共b兲兴. In addition, a far weaker second relaxation peak appears at high frequencies. This relaxation band is due to the “intrawell” modes which are virtually indistinguishable in the frequency spectrum appearing as a

2 1:ψ=0 2:ψ=π/6 3:ψ=π/4 4:ψ=π/3 5:ψ=π/2

σ =10 α = 0.01 ξ0 = 3 ξ =0.01 3

−5

共n − m兲共n − m − 1兲 共2n + 1兲共2n − 1兲

冑冋

10

1

10

− + − + − + an,m = −共an,−m 兲ⴱ, bn,m = −共bn,−m 兲ⴱ, and dn,m = −共dn,−m 兲ⴱ. The ma⫾ trices qn , qn consist of the five submatrices Vl, Wl, Xl, Yl, and Zl 共they also appear in the linear response and are defined explicitly, e.g., in Ref. 9, Chap. 9兲. Having determined k 共兲 from Eq. 共6兲, we can evaluate the the amplitudes cl,m ⬁ Re关mk1共兲eikt兴, where magnetization M H共t兲 = v M S兺k=1

mk1共兲 = 4

10 −1 10 −2 10 −3 10 −4 10

10

−1

10

−2

10

−3

10

−4

4

3

10

−2

10

−3

10

−4

α =0.01 σ =10 ξ0=3 ψ = π/4

(a)

1: ξ=0,01 2: ξ=1 3: ξ=3 4: ξ=5

2

1

1

+ = bn,m

0

共␥1⬘ − i␥2⬘兲 冑共n + m + 1兲共n − m兲, =−i 8␣

−Im[χ1(ω)]

+ an,m

ξ0=3 ξ =1 σ =10 ψ = π/4

2 4

3

(b)

1

1: α=0.01 2: α=0.1 3: α=1 4: α=10 −5

10

10

−3

10

−1

ωτΝ

1

10

10

3

FIG. 2. 共Color online兲 −Im关11共兲兴 vs N 共a兲 for various values of the ac field parameter and 共b兲 for various values of damping ␣.

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plitude increases, the FMR peak decreases and also broadens showing pronounced nonlinear saturation effects characteristic for a soft spring. This effect is very similar to that already known in atomic and molecular spectroscopy.21 Thus we see that the intrinsic damping dependence of the ac nonlinear response for the oblique field configuration 关see Fig. 2共b兲兴 serves as a signature of the coupling between the longitudinal and precessional modes of the magnetization. Hence, it should be possible to determine the damping coefficient ␣ from measurements of nonlinear response characteristics,2 e.g., by fitting the theory to the experimental dependence of 11共兲 on the angle and the ac and dc bias field strengths, so that the sole fitting parameter is ␣, which can be determined at different temperatures T, yielding its temperature dependence. This is important because a knowledge of ␣ and its T dependence allows for the separation of the various relaxation mechanisms.9 We now estimate the parameter range, where the nonlinear effects appear. For cobalt nanoparticles with mean diameter a ⬃ 10 nm and saturation magnetization M S ⬃ 1460 G, the field parameter for T ⬃ 30 K is on the order of unity for H ⬃ 6 kT / 共a3M S兲 ⬃ 5 Oe. Furthermore, an ac field of this order of magnitude is easily attained in measurements of the nonlinear response of magnetic nanoparticles; e.g., Bitoh et al.22 have measured the nonlinear susceptibility of cobalt nanoparticles in the temperature range 4.2–280 K using an ac magnetic field ⬃5 – 30 Oe. As far the characteristic time N is concerned, for ␥ ⬇ 2 ⫻ 107 rad/ Oe s and ␣ ⬃ 0.1, we have at room temperature N ⬃ 10−8 s. To conclude we have developed a nonperturbative approach in terms of matrix continued fractions for the nonlinear relaxation of a uniaxial superparamagnetic particle for arbitrary strengths and orientations of the dc bias and ac driving fields. We have shown that the nonlinear ac stationary response to a strong ac field applied at an angle to the easy axis of the particle 共so that the axial symmetry is bro-

Q. A. Pankhurst et al., J. Phys. D 42, 224001 共2009兲. Yu. L. Raikher and V. I. Stepanov, Adv. Chem. Phys. 129, 419 共2004兲. 3 J. L. García-Palacios and P. Svedlindh, Phys. Rev. Lett. 85, 3724 共2000兲. 4 L. Gammaitoni et al., Rev. Mod. Phys. 70, 223 共1998兲. 5 Yu. L. Raikher et al., Phys. Rev. E 56, 6400 共1997兲. 6 Yu. L. Raikher and V. I. Stepanov, Phys. Rev. Lett. 86, 1923 共2001兲; Yu. L. Raikher et al., J. Magn. Magn. Mater. 258-259, 369 共2003兲. 7 I. Klik and Y. D. Yao, J. Appl. Phys. 89, 7457 共2001兲; Yu. L. Raikher and V. I. Stepanov, J. Magn. Magn. Mater. 320, 2692 共2008兲. 8 P. M. Déjardin et al., J. Appl. Phys. 107, 073914 共2010兲. 9 W. T. Coffey et al., The Langevin Equation, 2nd ed. 共World Scientific, Singapore, 2004兲. 10 W. F. Brown Jr., Phys. Rev. 130, 1677 共1963兲; IEEE Trans. Magn. 15, 1196 共1979兲. 11 W. T. Coffey et al., Phys. Rev. E 71, 062102 共2005兲. 12 P. M. Déjardin and Yu. P. Kalmykov, J. Appl. Phys. 106, 123908 共2009兲; J. Magn. Magn. Mater. 322, 3112 共2010兲. 1 2

ken兲 is very sensitive to both the ac field orientation and amplitude owing to the coupling induced by the symmetry breaking driving field between the precession of the magnetization and its thermally activate reversal over the saddle point. In particular, the pronounced damping and ac field dependence of the nonlinear response 11共兲 can be used to determine the damping coefficient ␣ just as for higher harmonic responses.3 We emphasize that these nonlinear effects in 11共兲 cannot be treated via perturbation theory. Our calculations, since they are valid for ac fields of arbitrary strengths and orientations, allow one both to predict and interpret quantitatively nonlinear phenomena in magnetic nanoparticles such as nonlinear magnetic susceptibility, nonlinear stochastic resonance and dynamic hysteresis, nonlinear ac field effects on the switching field curves, etc., where perturbation theory and the assumption that axial symmetry is preserved are no longer valid 共these results will be published elsewhere兲. For practical applications 共e.g., in magnetic nanoparticle hyperthermia1兲, in order to account for the polydispersity of the particles of a real sample and the fact the easy axes of particles are randomly distributed in space, one must also average the nonlinear response functions mk1共兲 over appropriate distribution functions2 共averaging of mk1共兲 over particle volumes and orientations can be readily accomplished numerically using Gaussian quadraturs23兲. Here, only uniaxial superparamagnetic particles have been treated. Particles with nonaxially symmetric anisotropies 共cubic, biaxial, etc.兲 can be considered in like manner. Finally, our results can be adapted to other nonlinear phenomena such as nonlinear dielectric relaxation and the dynamic Kerr effect in molecular liquids and liquid crystals.11,13 The support of the work by the Agence Nationale de la Recherche, France 共Project No. ANR-08-P147-36兲 and by EGIDE, France 共Program ECO-NET, Project No. 21394NH兲 is gratefully acknowledged.

H. Watanabe and A. Morita, Adv. Chem. Phys. 56, 255 共1984兲; J. L. Dejardin et al., ibid. 117, 275 共2001兲. 14 H. J. Breymayer et al., Appl. Phys. B: Lasers Opt. 28, 335 共1982兲. 15 W. T. Coffey et al., Phys. Rev. B 62, 3480 共2000兲; Phys. Rev. E 61, 4599 共2000兲. 16 V. A. Shklovskij and O. V. Dobrovolskiy, Phys. Rev. B 78, 104526 共2008兲. 17 Yu. P. Kalmykov and S. V. Titov, Phys. Rev. Lett. 82, 2967 共1999兲. 18 Yu. P. Kalmykov and S. V. Titov, Fiz. Tverd. Tela 共St. Petersburg兲 40, 1642 共1998兲 关Phys. Solid State 40, 1492 共1998兲兴; W. T. Coffey et al., Phys. Rev. B 64, 012411 共2001兲. 19 W. T. Coffey et al., Phys. Rev. Lett. 80, 5655 共1998兲. 20 Yu. P. Kalmykov, J. Appl. Phys. 96, 1138 共2004兲. 21 R. Karplus and J. Schwinger, Phys. Rev. 73, 1020 共1948兲. 22 T. Bitoh et al., J. Phys. Soc. Jpn. 64, 1311 共1995兲; J. Magn. Magn. Mater. 154, 59 共1996兲. 23 Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun 共Dover, New York, 1972兲. 13

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