**Marie Curie Actions (MCA) **

Project No:
230785 DMH - Nonlinear dynamic hysteresis of nanomagnetic
particles with application to data storage and medical
hyperthermia (2012-2015)

**
Project Co-ordinator:**

Yuri P. Kalmykov, LAMPS, Université de Perpignan Via Domitia, 52, Avenue Paul Alduy, 66860, Perpignan Cedex, France

Tel. +(33)-468662062; FAX
+(33)-468662234;

e-mail:
kalmykov@univ-perp.fr

WEB: http://lamps.univ-perp.fr/kalmykov/

**
Partners: **

**
**

**
Yuri P. Kalmykov and Pierre-Michel Déjardin**,
LAMPS, Université
de Perpignan Via Domitia, 52, Avenue Paul Alduy, 66860, Perpignan Cedex, France

**
William T.
Coffey**,
Department of Electronic and Electrical Engineering, Trinity College, Dublin 2,
Ireland

**
Serguey V. Titov**,
Institute
of Radio Engineering and Electronics of the Russian Academy of Sciences,
Vvedenskii Square 1, Fryazino, Moscow Region, 141190, Russian Federation

**
**

**Summary description of the project objectives:**

The main topic of research of the project is theoretical studies of the
magnetodynamics of single-domain particles driven by a strong ac field. Due to the large magnitude of the
magnetic dipole moment (~10^{4}–10^{5} Bohr magnetons) 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, stochastic resonance, dynamic hysteresis, etc. However, the 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.^{ }The
subject of prime interest to us is the dynamic magnetic hysteresis (DMH), i.e.,
the magnetic response of nanoparticles to an ac field of arbitrary amplitude
with/without a dc bias field. The DMH phenomenon has many applications, two of
the most important being: magnetic moment switching (under pulsed fields) and
heat generation (under oscillating fields). The first is important from the
viewpoint of magnetic data storage, the second - for the development of
magnetically induced hyperthermia not exclusively for medical applications.
Besides these pparently, DMH can also be used to characterize the recording density,
signal-to-noise ratio, etc., in a given nanogranular medium. By accounting for
the effect of thermally driven magnetization reversal (superparamagnetism), we
can model switching processes for any desired field-
temperature-thermomagneticprotocol, e.g., heat-assisted or hybrid magnetic
recording techniques. As far as biomedical applications are concerned, one of
the most promising thermal approaches which has recently received considerable
attention is *local magnetic hyperthermia*.
This utilizes ac magnetic field energy absorption by nanosized ferromagnetic
particles syringed into the tumour. In spite of numerous publications, progress
in this method is hampered by the lack of a reliable understanding of the laws
governing the nonlinear magnetodynamics of liquid suspensions of magnetic
nanoparticles with coupling between magnetic and mechanical rotations.
Presently, two aspects of DMH theory require unification. The first concerns
DMH in an individual nanoparticle and in an assembly of nanoparticles in solid
suspensions. The nonlinear ac
stationary response hitherto has been calculated for *uniaxial* superparamagnets either (i) 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* or (ii) by
assuming that strong external fields are directed along the easy axis of the
particle so that *axial symmetry *is
preserved. Thus the results are in reality very restricted. In particular, the
conventional assumption of axial symmetry is 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 (such as the damping dependence of the response and the interplay
between precession and thermoactivation) cannot be included because in axial
symmetry no dynamical coupling between the longitudinal and transverse (or
precessional) modes of motion exists. The second set of problems includes DMH in
nanoparticles suspended in liquid or viscoelastic matrices. There the relevant issue
is the interplay between the internal (magnetic relaxation) and external
(viscous dissipation due to mechanical rotation) losses and their combined
effect on heat generation.

The objective of the planned advanced theoretical and numerical
simulation studies is the magnetodynamics of assemblies of fine magnetic
particles of different nature (size, etc.) placed in liquid environment subjected
to magnetic fields. The diversity of physical properties of particles and
liquid matrices, as well as that of the magnetic agitation modes renders the
scope of the research very wide. In particular, we shall study magnetodynamics
and energy absorption in suspensions of nanoparticles with magnetic hysteresis
under strong ac fields. In our studies we shall use numerical simulations. In
addition to the abovementioned applications to ferromagnetic hyperthermia of
cancers and data storage technologies we expect that our studies will substantially
increase our knowledge of such promising trends in experimental techniques as
microrheometry of complex (including biological) liquids employing small
magnetic particles. At present, we already have the basis of a nonperturbative approach to nonlinear ac
stationary responses of superparamagnetic nanoparticles with various magnetic
anisotropy potentials (uniaxial, biaxial, cubic) driven by strong ac and dc
electric fields.

**Description of the work performed since the beginning of the project:**

We have already
elaborated the basis of a nonperturbative approach to nonlinear ac
stationary responses of superparamagnetic nanoparticles with various
magnetic anisotropy potentials (uniaxial, biaxial, cubic) driven by
strong ac and dc electric fields. In particular, we have calculated the
nonlinear ac stationary response of an individual single domain
(superparamagnetic) particle with various magnetic anisotropy
potentials (uniaxial, biaxial, etc) of both surface and volume origin
in the presence of strong dc and ac magnetic fields both in the high
and low damping limits. We have generalizedf the theory to nonlinear
magnetic relaxation of an assembly of noninteracting superparamagnetic
particles in the presence of strong dc and ac magnetic fields.

**Description of the main results achieved so far: **

The nonlinear stationary ac response of the magnetization of assemblies consisting of (i) noninteracting uniaxial superparamagnetic nanoparticles with aligned easy axes and (ii) randomly oriented nanoparticles subjected to superimposed ac and dc bias magnetic fields of arbitrary strength and orientation is calculated by averaging Gilbert’s equation augmented by a random field. The magnetization dynamics of uniaxial particles driven by a strong ac field applied at an angle to the easy axis of the particle (so that the axial symmetry is broken) alters drastically leading to new nonlinear effects due to coupling of the thermally activated magnetization reversal mode with the precessional modes via the driving ac field. In particular, the high frequency response reveals significant nonlinear effects in the precessional motion with significant consequences for the dynamic hysteresis and ultra-fast switching of the magnetization following an ultrafast change in the applied field. It is demonstrated that both the area of the dynamic magnetic hysteresis (DMH) loop and the volume power loss of an assembly of uniaxial superparamagnetic nanoparticles with a random distribution of easy axes are very sensitive to the damping at low, intermediate, and high frequencies. In particular, a dynamical regime which is resonant in character occurs in the vicinity of the ferromagnetic resonance (FMR) frequency for low to moderate values of the AC field amplitude. The resonant regime is characterized by a diamagnetic-like response of the particles, resulting from a phase lag of the stationary nonlinear magnetization with respect to the applied field greater than .

We have have published 3 papers in primary physical journals. We have given 2 talks and presented 1 poster at 3 conferences and workshops.

**Publications:**

*We have published 3 papers in primary physical journals:*

1. H. El
Mrabti, P. M. Déjardin, S. V. Titov, and Yu. P. Kalmykov, "*Damping dependence in dynamic magnetic
hysteresis of single domain ferromagnetic particles*",*
**Phys. Rev. B.* 2012, v. 85, No. 9, p. 094425_(6 pages). http://link.aps.org/doi/10.1103/PhysRevB.85.094425

2. W. T.
Coffey and Yu. P. Kalmykov, *"Thermal
fluctuations of magnetic nanoparticles: Fifty years after Brown",* *J. Appl.
Phys.** * 2012, v. 112, No. 12, p. 121301_(47 pages). http://dx.doi.org/10.1063/1.4754272

3. B. Ouari, S. V. Titov, H. El Mrabti, and Yu. P.
Kalmykov, "*Nonlinear susceptibility
and dynamic hysteresis loops of magnetic nanoparticles with biaxial anisotropy*",* **J.
Appl. Phys .* 2013, v. 113, No. 5, p. 053903_(9 pages) http://dx.doi.org/10.1063/1.4789848

*We have given presented a poster at a conference *