Nonlinear susceptibility and dynamic hysteresis loops of magneticnanoparticles with biaxial anisotropyBachir Ouari, Serguey V. Titov, Halim El Mrabti, and Yuri P. Kalmykov Citation: J. Appl. Phys. 113, 053903 (2013); doi: 10.1063/1.4789848 View online: http://dx.doi.org/10.1063/1.4789848 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v113/i5 Published by the American Institute of Physics. Related ArticlesCarrier density dependence of the magnetic properties in iron-doped Bi2Se3 topological insulator J. Appl. Phys. 113, 043926 (2013) Structural variants and the modified Slater-Pauling curve for transition-metal-based half-Heusler alloys J. Appl. Phys. 113, 043709 (2013) d0 magnetism in semiconductors through confining delocalized atomic orbitals Appl. Phys. Lett. 102, 022422 (2013) Influence of Ga doping on the Cr valence state and ferromagnetism in Cr: ZnO films Appl. Phys. Lett. 102, 022414 (2013) Half-metallic ferromagnetism in wurtzite ScM (M=C, Si, Ge, and Sn): Ab initio calculations Appl. Phys. Lett. 102, 022404 (2013) Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

Nonlinear susceptibility and dynamic hysteresis loops of magneticnanoparticles with biaxial anisotropy

Bachir Ouari,1,2 Serguey V. Titov,3 Halim El Mrabti,1 and Yuri P. Kalmykov1

1Univ. Perpignan Via Domitia, Laboratoire de Math�ematiques et Physique, EA 4217, F-66860 Perpignan,France2Macromolecular Research Laboratory, Physics Department, Abou Bekr Belkaid University,BP 119 Chetouane, Tlemcen, Algeria3Kotelnikov’s Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, VvedenskiiSquare 1, Fryazino, Moscow Region 141190, Russian Federation

(Received 29 August 2012; accepted 15 January 2013; published online 4 February 2013)

The nonlinear ac susceptibility and dynamic magnetic hysteresis (DMH) of a single domain

ferromagnetic particle with biaxial anisotropy subjected to both external ac and dc fields of

arbitrary strength and orientation are treated via Brown’s continuous diffusions model [W. F.

Brown, Jr., Phys. Rev. 130, 1677 (1963)] of magnetization orientations. The DMH loops and

nonlinear ac susceptibility strongly depend on the dc and ac field strengths, the polar angle between

the easy axis of the particle, the external field vectors, temperature, and damping. In contrast to

uniaxial particles, the nonlinear ac stationary response and DMH strongly depend on the azimuthal

direction of the ac field and the biaxiality parameter D. VC 2013 American Institute of Physics.

[http://dx.doi.org/10.1063/1.4789848]

I. INTRODUCTION

Fine Single domains ferromagnetic nanoparticles have

been an active focus of research for over 60 years prompted

by both fundamental aspects of their magnetization and

increasing interest by industry especially for data storage.1,2

Other important current applications include medical appli-

cations, e.g., in hyperthermia.3 Such nanoparticles are char-

acterized by instability of the magnetization M(t) due to

thermal agitation resulting in spontaneous change of its ori-

entation from one metastable state to another by surmounting

anisotropy-Zeeman energy barriers resulting in the phenom-

enon of superparamagnetism.4 Due to the large magnitude of

the magnetic dipole moment of single-domain nanoparticles

so that the Zeeman energy is relatively large even in weak

external magnetic fields, the magnetization reversal process

has a strong field dependence causing nonlinear effects in

the dynamic susceptibility, stochastic resonance, dynamic

magnetic hysteresis (DMH), etc.

The determination of the nonlinear magnetic susceptibil-

ity and DMH for arbitrary ac field strengths amounts to the

calculation of the nonlinear response of the magnetization in

the presence of the thermal agitation. This calculation is usu-

ally treated by the Langevin equation which is Gilbert’s5 (or

the Landau-Lifshitz6) equation augmented by a random field

h(t) with Gaussian white noise properties, accounting for

thermal fluctuations of the magnetization MðtÞ of an individ-

ual particle,

_MðtÞ ¼ cMðtÞ � � @V

@Mtð Þ � a

cMS

_MðtÞþh tð Þ� �

; (1)

where c is the gyromagnetic ratio, a is the dimensionless damp-

ing parameter, MS is the saturation magnetization, and V is

the free energy per unit volume comprising the non-separableHamiltonian of the magnetic anisotropy and Zeeman energy

densities. Here jMj ¼ MS is assumed to be constant so that the

only the orientation of M varies. Equation (1) is then used to

derive the accompanying Fokker-Planck equation7 governing

the time evolution of the probability density function Wðu; tÞof magnetization orientations on a unit sphere (u is a unit vec-

tor along M) and the relevant FPE is

2sN

@W

@t¼ b

au � ðrV �rWÞ þ r � ðrW þ bWrVÞ; (2)

where r ¼ @=@u is the gradient operator on the unit sphere,

sN ¼ s0ðaþ a�1Þ is the characteristic free diffusion time,

s0 ¼ bMS=ð2cÞ, b ¼ v=ðkTÞ, v is the volume of the particle,

k is Boltzmann’s constant, and T is the absolute temperature.

Various treatments of the nonlinear ac stationary

response have been effected using numerical solutions of the

governing dynamical equations (1) and/or (2). In particular,

efficient numerical algorithms for the calculation of the non-

linear ac stationary response of uniaxial superparamagnetic

nanoparticles have been proposed (see, e.g., Refs. 8–10) by

assuming that the ac driving field HðtÞ is directed along the

easy axis of the particle. However, in this axially symmetric

configuration, many interesting nonlinear effects cannot be

treated and understood because no dynamical coupling

between the longitudinal and transverse modes of motion

exists. Recently,11–13 these results have been generalized to

determine the nonlinear ac stationary response for a uniaxial

particle driven by a strong ac field applied at an angle to the

easy axis of the particle so that the axial symmetry is broken

by the Zeeman energy. In the above papers, only the axially

symmetric uniaxial anisotropy potential has been treated

considerably simplifying the analysis. However, the results

cannot be applied to particles with inherent nonaxially sym-

metric anisotropy, such as biaxial anisotropy,

bVbiaxð#;uÞ ¼ r sin2 #þ D sin2 # cos2 u: (3)

0021-8979/2013/113(5)/053903/9/$30.00 VC 2013 American Institute of Physics113, 053903-1

JOURNAL OF APPLIED PHYSICS 113, 053903 (2013)

Here, D and r are the biaxiality and barrier parameters,

respectively (D ¼ 0 corresponds to uniaxial anisotropy), #and u are polar and the azimuthal angles of the spherical

coordinate system. We treat this problem because biaxial an-

isotropy may yield an appreciable contribution to the free

energy density of magnetic nanoparticles.14 In particular, Eq.

(3) describes the magnetic anisotropy energy of a spheroidal

single-domain particle with easy anisotropy axis inclined at

a certain angle to the particle axis of symmetry15 as well that

of elongated particles, where easy- and hard-axis anisotropy

terms are present.16 Furthermore, biaxial anisotropy consti-

tutes a generic model for treatment of the thermally assisted

magnetization reversal in the presence of a spin-transfer tor-

que.17 Biaxial anisotropy generates azimuthally nonuniform

energy distributions with two saddle points that lead to new

effects, viz., strong intrinsic dependence of the magnetic

characteristics on the damping arising from coupling of the

longitudinal and transverse relaxation modes.18,19 Further-

more, in contrast to uniaxial particles, the nonlinear ac sta-

tionary response strongly depends on the azimuthal direction

of the ac field. A method of solution of the general nonlinear

problem [i.e., nonlinear ac stationary response for superpara-

magnets with arbitrary anisotropy and including the gyromag-

netic term in the Fokker-Planck equation (2)] has been given

by Titov et al.20 Here, using this approach, we calculate the

nonlinear magnetic susceptibilities and DMH of nanoparticles

with biaxial anisotropy subjected to a strong ac driving field

superimposed on a strong dc bias field. We shall demonstrate

that for biaxial anisotropy, the magnetization dynamics alter

substantially leading to new temperature and damping de-

pendent nonlinear effects at low, mid, and high frequencies.

II. BASIC EQUATIONS

The free-energy density V of a single-domain ferromag-

netic nanoparticle with biaxial anisotropy in superimposed

homogeneous external magnetic dc and ac fields H0 þH cos xt of arbitrary strengths and orientations is given by

bVð#;u; tÞ ¼ bVbiaxð#;uÞ � ðn0 þ n cos xtÞ� ðc1 sin# cos uþ c2 sin# sin uþ c3 cos#Þ;

(4)

where n0 ¼ bH0MS and n ¼ bHMS are the dc and ac external

field parameters, respectively, c1 ¼ sin w cos /; c2 ¼ sin wsin/; c3 ¼ cos w are the direction cosines of the ac external

field, w is the angle between H and the Z axis taken as the

easy axis of the particle, and / is the azimuthal angle (here

we have assumed that the vectors H0 and H are parallel). As

shown in Ref. 20, the solution of the Gilbert-Langevin equa-

tion (1) for an arbitrary anisotropy potential can be reduced

to that an infinite hierarchy of differential-recurrence equa-

tions for the statistical moments hYlmiðtÞ governing the dy-

namics of the magnetization. Here Ylmð#;uÞ is a spherical

harmonic of order l and rank m defined as

Ylmð#;uÞ ¼ ð�1Þmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2lþ 1Þðl� mÞ!

4pðlþ mÞ!

seimuPm

l ðcos#Þ;

where Pml ðxÞ are the associated Legendre functions. By using

the generalized differential-recurrence equations specialized

to biaxial nanoparticles, we have the ac stationary solution

for the magnetization,

MðtÞ ¼ MShcosHiðtÞ

¼ MS

ffiffiffiffiffiffi4p3

rfc3hY1;0iðtÞ �

ffiffiffi2p

Re½ðc1 � ic2ÞhY1;1iðtÞ�g;

in the direction of the ac driving field H, viz.,

MðtÞ ¼ MS

X1k¼�1

mk1ðxÞeikxt; (5)

where H is the angle between the vectors H and M,

mk1ðxÞ ¼

ffiffiffiffiffiffi4p3

r "c3ck

1;0ðxÞ

þðc1 þ ic2Þck

1;�1ðxÞ � ðc1 � ic2Þck1;1ðxÞffiffiffi

2p

#; (6)

the ckl;mðxÞ are the coefficients in the time Fourier series of

the averaged spherical harmonics, viz.,

hYlmiðtÞ ¼X1

k¼�1ck

l;mðxÞeikxt: (7)

The coefficients ckl;mðxÞ, thus, the magnetization MðtÞ, can

be evaluated using matrix continued fractions (see Appen-

dix). In calculating the nonlinear ac response, we shall con-

sider the parameter range r� 1 and n � 1 corresponding to

the low temperature limit and strongly nonlinear regime.

Henceforth, we shall assume that the vectors H0 and H lie in

the XZ (/ ¼ 0) or YZ (/ ¼ p=2) planes of the laboratory

coordinate system so that the direction cosines are

c1 ¼ sin w; c2 ¼ 0; c3 ¼ cos w

or

c1 ¼ 0; c2 ¼ sin w; c3 ¼ cos w;

respectively. Now, the DMH loop represents a parametric

plot of the steady-state time-dependent magnetization as a

function of the AC field, i.e., MðtÞ vs. HðtÞ ¼ H cos xt,where the time interval t coincides with the period of oscilla-

tion of that field for a given value of H. Here, we also calcu-

late the normalized area of the DMH loop An defined as9

An ¼1

4MSH

þMðtÞdHðtÞ ¼ � p

4Imðm1

1Þ; (8)

which is the energy loss per particle and per cycle of the AC

field.

III. NONLINEAR SUSCEPTIBILITY AND DMH LOOPS

For a weak ac field, n ! 0, vðxÞ ¼ 6m11ðxÞ=n defines

the normalized linear dynamic susceptibility and our results

053903-2 Ouari et al. J. Appl. Phys. 113, 053903 (2013)

agree in all respects with the benchmark linear response so-

lution.18,19 The results indicate that a marked dependence of

vðxÞ on r (inverse temperature), D (biaxiality), n (ac field),

n0 (dc field), a (damping), and angles w and / exists and that

three distinct dispersion bands appear in the spectrum of

vðxÞ (see curve 1 in Fig. 1(b)). The low frequency behavior

of vðxÞ is dominated by the barrier-crossing mode. In addi-

tion, a weaker second relaxation peak appears at high fre-

quencies. This relaxation band is due to the “intrawell”

modes, which are virtually indistinguishable in the frequency

spectrum appearing as a single high-frequency band. At low

dc fields, the amplitude of this band is far weaker than that

of the first band. However, in a strong dc magnetic field

n0 � 1, this band can dominate the spectrum because as n0

increases, the magnitude of the low frequency band

decreases. Just as in the absence of the dc bias field, there is

an inherent geometric dependence of vðxÞ on the damping

parameter a arising from the coupling of the longitudinaland transverse relaxation modes. This coupling appears in

the dynamical equation of motion, where the longitudinal

component of the magnetization is entangled with the trans-

verse components resulting in the appearance of a third fer-

romagnetic resonance (FMR) peak in the spectrum of vðxÞdue to excitation of transverse modes with characteristic fre-

quencies close to the precession frequency of the magnetiza-

tion. The FMR peak appears only for low damping ða� 1Þ

and strongly manifests itself at high frequencies. As adecreases, the half-width of the FMR peak decreases.

In strong ac fields, n> 1, pronounced nonlinear effects

occur (see Figs. 1–3 illustrating the dependence of the non-

linear response on the biaxiality D, damping a, angle w, dc

bias field n0, and the ac field parameter n). In particular, the

low frequency band can no longer be approximated by a sin-

gle Lorentzian indicating that a variety of modes participate

in the magnetization reversal process. However, the charac-

teristic frequency xmax of the maximum magnetic loss may

still be used in order to estimate an effective reversal time of

the magnetization s, i.e., s x�1max. The behavior of xmax

(and, hence, s) as functions of the ac field amplitude depends

on whether or not a dc field is applied. For strong dc bias,

n0 > 1, the low frequency peak shifts to lower frequencies

reaching a maximum at n n0 thereafter xmax decreasing

with increasing n. 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

n n0 then decreases (see Fig. 3). For weak dc bias field,

n0 � 1, the low frequency peak shifts monotonically to

higher frequencies. Furthermore, as seen in Fig. 1, with

increasing n, the magnitude of the FMR peak at the preces-

sion frequency xpr decreases and also broadens showing

pronounced nonlinear saturation effects characteristic of a

soft spring. Furthermore, for low damping, a second weak

FIG. 1. The imaginary part of the suscep-

tibility �Im½vðxÞ� vs. the dimensionless

frequency xs0 for barrier parameter

r ¼ 10, biaxiality parameter D ¼ 10, dc

field parameter n0 ¼ 5, angles w ¼ p=4

and / ¼ 0, and various values of damp-

ing a (a) and ac field amplitude n (b).

FIG. 2. �Im½vðxÞ� vs. xs0 for biaxiality

parameter D ¼ 5, dc field parameter

n0 ¼ 5, the ac field parameter n ¼ 5, angle

/ ¼ 0, and various angles w (a) and barrier

parameters r (b).

053903-3 Ouari et al. J. Appl. Phys. 113, 053903 (2013)

resonance peak [owing to phase locking of the nonlinear os-

cillatory (precessional) motion of the magnetization MðtÞ]appears at frequencies xpr=2 (Fig. 2). This effect is similar

to that already existing in nonlinear parametric oscillator sys-

tems driven by an ac external force.21 We remark, however,

that the high-frequency behavior (x� xpr) remains virtually

unchanged. Indeed, the behavior of the imaginary part of sus-

ceptibility �Im½vðxÞ� for x� xpr is independent of n (see

Fig. 1). Moreover, in contrast to uniaxial particles, the FMRpeak always appears for low damping in the spectrum�Im½vðxÞ� at w ¼ 0 because the transverse modes alwaystake part in the longitudinal relaxation process. In Fig. 1(b),

the peak height of the first band firstly (at moderate fields)

increases with increasing the ac field amplitude and then

reduces at higher ac fields. This result implies that these

changes in the ac field magnitude may modulate the heat pro-

duction (specific power loss) in a magnetic nanoparticle

working as a hyperthermia source. In general, however, the

most effective way to tune the heat production is to vary the

magnitude of the dc field10 (see Fig. 3(b)). Our calculation

indicates that with increasing n0 the magnitude of the low fre-

quency band decreases due to the depletion of the population

in the shallowest potential well. Finally, the appearance of the

FMR band is a signature of the onset of dynamic hysteresis at

high frequencies, a feature which is absent for high damping.

In Figs. 4–8, normalized DMH loops, i.e.,

mðtÞ ¼ MðtÞ=MS vs: hðtÞ ¼ HðtÞ=H ¼ cos xt;

for various values of the ac field amplitude n, damping a,

and the dimensionless barrier parameter r / 1=T are pre-

sented. Clearly in Fig. 4, in contrast to uniaxial particles, the

DMH strongly depends on the azimuthal direction of the ac

field H and not only on the polar angle w between the easy

axis and H. Moreover, the coercivity, the remanent magnet-

ization, and the saturation magnetization strongly depend on

the barrier height (inverse temperature) parameter r, so that

considerable variation in the size and shape of loops exists

for different values of r at low frequencies. For xs0 ¼ 10�2,

a ¼ 10�1, n0 ¼ 0, and r � 5, all the hysteresis loops are

large, retaining a large fraction of the saturation field when

the driving field is removed. For r � 2, we have narrow hys-

teresis loops, which implies that a small amount of energy is

used up in repeatedly reversing the magnetization. Figures

5–10 show that the DMH loops and the normalized loop area

An depend strongly on the biaxial parameter D, the oblique

angle w, the damping a and the dimensionless frequency xs0

indicating that the relaxation of the magnetization is mostly

caused by thermal fluctuations. At high frequencies, e.g., at

xs0 ¼ 1 (see Fig. 8), the dynamic hysteresis is due to the

resonant absorption in the FMR band. Thus, the DMH aris-

ing from a high-frequency periodic signal may be evaluated

permitting quantitative analysis of ultrafast switching of the

magnetization. The rotation of the hysteresis loop in Fig.

8(c) is due to the sensitivity of the FMR frequency xpr to the

biaxially parameter (due to this dependence, the phase shift

between MðtÞ and HðtÞ may vary substantially in the vicinity

FIG. 3. �Im½vðxÞ� vs. xs0 for barrier pa-

rameter r ¼ 10, ac field parameter n ¼ 5,

damping a ¼ 0:05, angles w ¼ p=4 and

/ ¼ 0 and various biaxiality parameters

D (a) and dc field amplitudes n0 (b).

FIG. 4. DMH loops at the dimensionless

frequency xs0 ¼ 0:01 for biaxiality pa-

rameter D ¼ 10, damping a ¼ 0:1 and

various values of the barrier parameter r.

Figures (a) and (b) correspond to the ori-

entation of the external fields in the XZ(angle / ¼ 0) or YZ (/ ¼ p=2) planes,

respectively; the oblique angle w ¼ p=4

and the ac field parameter n ¼ 2r.

053903-4 Ouari et al. J. Appl. Phys. 113, 053903 (2013)

of xpr). This phenomenon can be used to determine the biax-

iality factor D for magnetic materials. Clearly, at low fre-

quencies (Fig. 7) the DMH loop area increases as the

biaxiality parameter increases. This is also true when either

the damping or the angle w is decreased. This effect is shown

in Figs. 5–9. Although being relatively weak for r ¼ 10

(Fig. 5(c)), it can become quite pronounced in the superpara-

magnetic regime r ¼ 2 (Fig. 5(a)). To interpret this, two

hypotheses can be made. The first, as already discussed, is

that low damping implies a longer response time. The sec-

ond, which is required to explain the temperature depend-

ence, is that the precession actually takes place about the

direction of the total field, including thermal contributions.

Then the random nature of this noise-induced precession

may hinder the magnetic response, with the consequence

that the area is increased.

FIG. 5. DMH loops at xs0 ¼ 0:01 for

damping a ¼ 0:1, ac field parameter n¼ 10, dc field parameter n0 ¼ 0, angles

w ¼ p=4 and / ¼ 0 and various biaxial-

ity D and barrier r parameters.

FIG. 6. DMH loops at xs0 ¼ 0:01 for barrier parameter r ¼ 5, damping a ¼ 0:1, ac field parameter n ¼ 10, dc field parameter n0 ¼ 0 and various biaxiality

parameters D, / ¼ 0, and angle w ¼ 0; p=4; p=3; and p=2.

053903-5 Ouari et al. J. Appl. Phys. 113, 053903 (2013)

FIG. 7. DMH loops at xs0 ¼ 0:01 for

barrier parameter r ¼ 5, ac field parame-

ter n ¼ 10, dc field parameter n0 ¼ 0,

angles w ¼ p=4 and / ¼ 0, and various

biaxiality parameters D and damping

a ¼ 0:05; 0:1; 1; 10.

FIG. 8. DMH loops for barrier parameter

r ¼ 5, damping a ¼ 0:1, ac field parame-

ter n ¼ 10, dc field parameter n0 ¼ 0,

angles w ¼ p=4 and / ¼ 0, and various

biaxiality parameters D and dimension-

less frequency xs0 ¼ 10�3, 10�2, 1, and

100.

FIG. 9. Normalized area of the DMH loop

An as functions of the biaxiality parameter

D for damping a ¼ 0:1, dc field parameter

n0 ¼ 0, frequency xs0 ¼ 10�2, / ¼ 0,

and various barrier parameters r and

angles w.

053903-6 Ouari et al. J. Appl. Phys. 113, 053903 (2013)

IV. CONCLUSION

A rigorous treatment of the nonlinear susceptibility and

the dynamic hysteresis loops of a single domain particle with

biaxial anisotropy have been given using Brown’s model. In

carrying out all the calculations, the system was initially

assumed to possess a biaxial anisotropy and no approxima-

tions were used except those already inherent in Brown’s

model. The present calculation is the first attempt to treat an

anisotropy other than uniaxial in this manner. We remark

that a detailed treatment of the same problem restricted to

uniaxial anisotropy for both an ac field and dc field of arbi-

trary strength applied at angles to the easy axis of magnetiza-

tion was given by Titov et al.20 and El Mrabti et al.12 They

evaluated the dependence of both the nonlinear dynamic sus-

ceptibility and DMH on the ac and dc field strength, temper-

ature and field angles. In our biaxial anisotropy problem just

as the uniaxial problem above, the intrinsic a dependence of

the response represents a signature of the coupling between

the longitudinal and precessional modes of the magnetization

(Fig. 1). Moreover, again as in the nonlinear response of uni-

axial particles, the FMR peak appears only for low damping

ða� 1Þ and strongly manifests itself at high frequencies.

We emphasize that it should again be possible as in the pre-

ceding case to determine a from measurements of nonlinear

response characteristics, e.g., by fitting the theory to the ex-

perimental dependence of observables on the ac and dc bias

field angles and strengths. As before in such measurements,

the sole fitting parameter is a, which can be determined

at different temperatures T, so yielding its temperature de-

pendence. This is significant because a knowledge of a and

its T dependence allows separation of the various relaxation

mechanisms while in strong ac fields, n> 1, pronounced

nonlinear effects occur.

Numerical calculations of the nonlinear dynamic sus-

ceptibility and the DMH show that a marked dependence of

these quantities on both the biaxiality parameter D and the

azimuthal angle / exists. This phenomenon is a general fea-

ture of nonaxially symmetric systems arising as usual in

from coupling of the longitudinal and transverse relaxation

modes. It is especially important here because unlike uniax-

ial anisotropy the biaxial case has intrinsic azimuthal angle

dependence arising from the D term in Eq. (3). In other

words, the nonaxially symmetric anisotropy generates an azi-

muthally nonuniform energy distribution with saddle points

leading to strong intrinsic dependence of magnetic character-

istics on D arising from the mode coupling due to D. As far

as the DMH is concerned at low frequencies, the DMH loops

exhibit a pronounced damping dependence due to coupling

of the thermally activated magnetization reversal mode to

the precessional modes of the magnetization via the driving

ac field. Again the frequency dependence of the DMH for

fixed angles w and / and temperature is very significant. The

underlying reason is that the remagnetization time is highly

sensitive to the frequency of the applied field. For example,

under a strong ac driving field, the Arrhenius dependence of

the relaxation time on temperature s er, which accurately

accounts for the linear regime, is modified because the strong

ac field drastically reduces the effective response time so

facilitating new re-magnetization regimes, which can never

be attained for weak ac fields. The dc bias component under

the appropriate conditions efficiently tunes this effect either

by enhancing or blocking the action of the ac field. This

description of the dynamic magnetic hysteresis is valid in a

wide field-frequency-temperature range and is free of over-

simplifications such as linear response or high-barrier

approximations so rendering it a useful tool both for verifica-

tion of existing solutions and extension to novel applications

involving magnetic nanoparticles, e.g., low-frequency mag-

netic hyperthermia and precise focusing of the heat gener-

ated by them. Once again, we see that for small ac fields, the

relaxation of the magnetization is mostly caused by thermal

fluctuations, so that the magnetic response time retains a

strong dependence on temperature. Nonaxially symmetric

dynamical hysteresis has valuable potential applications in

magnetic recording because the phenomenon can be used by

industry especially in the field of data storage to characterize

the recording density, signal-to-noise ratio, etc. Clearly, as

the effect of thermally driven magnetization reversal (super-

paramagnetism) is fully accounted for in the discussions we

have given, one would then be able to model the switching

processes for any desirable field-temperature protocol, e.g.,

FIG. 10. Normalized area An of the DMH

loop vs. the biaxiality parameter D for

barrier parameter r ¼ 5, ac field parame-

ter n ¼ 10, dc field parameter n0 ¼ 0,

angles w ¼ p=4 and / ¼ 0, and various

frequencies xs0 (a) and damping a (b).

053903-7 Ouari et al. J. Appl. Phys. 113, 053903 (2013)

heat-assisted or hybrid magnetic recording techniques. The

description of the angular dependence of the dynamic mag-

netic hysteresis which we have obtained could be used to an-

alyze the switching times in magnetic nanoparticles allowing

for both imperfect orientational distributions of the particle

easy axes and nonuniformity of the head field inside the re-

cording layer.

ACKNOWLEDGMENTS

We thank W. T. Coffey, P.-M. D�ejardin, L. M�echernene,

and A. Layadi for useful comments. One of the authors

(B.O.) acknowledges fruitful discussions with H. Aourag,

director of scientific research and technological development

in Algeria. The support of the work by the European Com-

munity (Programme FP7, project DMH, No. 295196) is

gratefully acknowledged.

APPENDIX: MATRIX CONTINUED FRACTIONSOLUTION OF EQ. (5)

As shown in Ref. 20, the stationary ac response can be

calculated from the formally exact matrix continued fraction

solution,

C1 ¼

�

c�21 ðxÞ

c�11 ðxÞc0

1ðxÞc1

1ðxÞc2

1ðxÞ�

0BBBBBBBBBBB@

1CCCCCCCCCCCA¼ �1ffiffiffiffiffiffi

4pp S1 �

�

0

p�1q�1p�10

�

0BBBBBBBBBBB@

1CCCCCCCCCCCA; (A1)

where the infinite matrix continued fraction S1 is defined by

the recurrence equation,

Sn ¼ �½Qn þQþn Snþ1Q�nþ1 ��1;

the elements of the three-diagonal supermatrixes Qn and Q6n

are defined as

½Qn�l;m ¼ dl�1ma2n b2n

d2n�1 a2n�1

� �þ dlm

X2n W2n

Y2n�1 X2n�1

� �

þ dlþ1ma2n b2n

d2n�1 a2n�1

� �;

½Qþn �l;m ¼ dl�1mo d2n

o o

� �þ dlm

Z2n Y2n

o Z2n�1

� �

þ dlþ1mo d2n

o o

� �;

½Q�n �l;m ¼ dl�1mo o

b2n�1 o

� �þ dlm

V2n o

W2n�1 V2n�1

� �

þ dlþ1mo o

b2n�1 o

� �;

(dlm is Kronecker’s delta) and the column vectors ck1ðxÞ, p�1 ,

and q�1 are defined as

ck1ðxÞ ¼

ck2;�2ðxÞ

ck2;�1ðxÞck

2;0ðxÞck

2;1ðxÞck

2;2ðxÞck

1;�1ðxÞck

1;0ðxÞck

1;1ðxÞ

0BBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCA

; q�1 ¼

�ffiffiffiffiffiffiffiffiffiffi3=10

pD

02rþ Dffiffiffi

5p

0

�ffiffiffiffiffiffiffiffiffiffi3=10

pD

ðc1 � ic2Þn0ffiffiffi6p

c3n0ffiffiffi3p

�ðc1 þ ic2Þn0ffiffiffi6p

0BBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCA

;

p�1 ¼

0

0

0

0

0ðc1 � ic2Þn

2ffiffiffi6p

c3n

2ffiffiffi3p

�ðc1 þ ic2Þn2ffiffiffi6p

0BBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCA

:

Here o and 0 are zero matrices and vectors of appropriate

dimensions, respectively. The five submatrices Vl, Wl, Xl

Yl, and Zl also appear in the linear response of biaxial par-

ticles and are defined explicitly in Ref. 18. The tridiagonal

submatrices al ; bl, and dl coincide with those for uniaxial

particles and are given explicitly in Ref. 20. For D ¼ 0, the

above solution reduces to that for uniaxial particles.20 Here

ckl;mðxÞ are the Fourier coefficients in the Fourier time series

Eq. (7). Clearly Eq. (A1) contains all the Fourier amplitudes

required for the nonlinear stationary response. Having deter-

mined the amplitude ck1;0ðxÞ and ck

1;61ðxÞ from Eq. (A1), we

can evaluate the magnetization given by Eq. (5).

Thus the nonlinear ac stationary response of biaxial

superparamagnetic particles driven by a strong ac field

applied at an angle to the easy axis of the particle can be

evaluated using matrix continued fractions. The calculations,

since they are valid for ac fields of arbitrary strength, pro-

vide a rigorous theoretical basis for comparison with experi-

ments on nonlinear response in strong ac fields, where the

results of perturbation theory are no longer valid, e.g., non-

linear magnetic susceptibility, nonlinear stochastic reso-

nance, and dynamic hysteresis (these results will be

published elsewhere).

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