**Marie Curie Actions (MCA) **

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

(1st April 2012 - 31st March 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:**

In the context of the project, we have developed accurate theoretical methods for evaluating the nonlinear ac stationary response of an individual single domain nanoparticle 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 also extended the methods to calculating the nonlinear ac stationary response of an assembly of magnetic nanoparticles in in superimposed dc and ac magnetic fields. Furthermore, we have developed effective methods of calculation of the DMH in an individual nanoparticle and assemblies of nanoparticles with randomly oriented easy axes and we have analyzed the dependence of the area of the DMH loops on the temperature, frequency, and ac and dc bias field magnitude and orientation. By generalizing these results to the calculation of the nonlinear susceptibility and energy absorption of ferrofluids, we have also elaborated efficient methods of calculation of nonlinear DMH in liquid suspensions of magnetic nanoparticles. Theoretical predictions have been compared with relevant experimental data comprising the magnetization dynamics of nanoparticles driven by a strong ac field with particular application in magnetic hyperthermia. Hence, we have achieved all the anticipated objectives set out in our project.

The results obtained may have 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 the medical uses, DMH can also be used to characterize the recording density, signal-to-noise ratio, etc., in a given nanogranular medium. Furthermore, by accounting for the effect of thermally activated 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 tumor. In addition to the above mentioned applications to ferromagnetic hyperthermia and data storage technologies we expect that our studies have substantially increased our knowledge concerning such promising trends in experimental techniques as microrheometry of complex (including biological) liquids employing small magnetic particles.

Objective of the research: Nonlinear response theory for nanomagnets is in increasing demand due to their fundamental importance and, equally, their wide implications for real devices, based on the superparamagnetism phenomenon. For example, magnetization relaxation of single-domain particles is a cornerstone in the physics of nanosystems and a key factor in the improvement of magnetic recording carriers and information storage. The same theoretical framework, properly refined, is essential for magnetic hyperthermia. Recently, spectroscopy of probe particles suspended in complex fluids has come to the fore as a unique tool for microrheology in chemical engineering and biomedicine. Furthermore, nonlinear phenomena such as resonant activation and suppression of noise have been observed. In applications, one needs to study the interaction of an external field with the system. From the mathematical point of view, the task is reduced to the solution of a stochastic nonlinear Langevin equation for the appropriate dynamical variables. Hitherto, the only well-developed theoretical approach to this problem has been linear response theory, which assumes that the energy of a particle in an external field is smaller than the thermal energy. In contrast, the theory of nonlinear response has been far less well developed because of its inherent mathematical and physical complexity. The calculation of nonlinear responses, even for systems described by a single coordinate, is a difficult task as there is no longer any connection between the step-on and the step-off responses and the ac one as the response now depends on the precise form of the stimulus. In effect, no unique response function valid for all stimuli exists unlike linear response. Available results have mainly emerged either by perturbation theory or by numerical simulations. We remark that the perturbation approaches are valid only for relatively weak external fields, while numerical results have the disadvantage that closed-form solutions are not available which impedes even qualitative predictions of the behavior of the system under investigation. The main objective of the project was to gain an understanding of the magnetodynamics of assemblies of fine magnetic particles of different nature (size, saturation magnetization, etc.) placed in solid or 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.

Work performed: In the context of the project, we have developed accurate theoretical methods to evaluating the nonlinear ac stationary response of an individual single domain nanoparticle 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 also extended the methods to calculating nonlinear magnetic relaxation of an assembly of magnetic nanoparticles in the presence of strong dc and ac magnetic fields. Furthermore, we have developed effective methods of calculation of the DMH in an individual nanoparticle and assemblies of nanoparticles with randomly oriented easy axes and we have analyzed the dependence of the area of the DMH loops on the temperature, frequency, and ac and dc bias field magnitude and orientation. By generalizing these results to the calculation of the nonlinear susceptibility and energy absorption of ferrofluids, we have also elaborated efficient methods of calculation of nonlinear DMH in liquid suspensions of magnetic nanoparticles. Theoretical predictions have been compared with relevant experimental data comprising the magnetization dynamics of nanoparticles driven by a strong ac field with particular application in magnetic hyperthermia.

Relevance for basic and applied science and for applications including industrial links: The results obtained may have 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 the medical uses, DMH can also be used to characterize the recording density, signal-to-noise ratio, etc., in a given nanogranular medium. Furthermore, by accounting for the effect of thermally activated 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 tumor. In addition to the above mentioned applications to ferromagnetic hyperthermia and data storage technologies we expect that our studies have substantially increased our knowledge concerning such promising trends in experimental techniques as microrheometry of complex (including biological) liquids employing small magnetic particles.

Results and degree to which the objectives were met: We have developed accurate analytical and numerical methods for treating energy absorption and dissipation to the surrounding heat bath in the dynamic magnetic hysteresis of solid and liquid suspensions of magnetic nanoparticles. The analytical and numerical results obtained concur with available experimental observations. Hence, we have achieved all the anticipated objectives set out in our project.

**Publications:**

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).

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).

3. W. T. Coffey, Yu. P.
Kalmykov, The Langevin Equation: with Applications in Physics, Chemistry and
Electrical Engineering, 3rd Edition, World Scientific, Singapore, 2012,
xxii+827 pp. http://www.worldscientific.com/worldscibooks/10.1142/8195

4. 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).

5. N. Wei, D. Byrne, W. T.
Coffey, Yu. P. Kalmykov, S. V. Titov, “Nonlinear frequency-dependent effects in the dc magnetization of uniaxial
magnetic nanoparticles in superimposed strong alternating current and direct
current fields”, J. Appl. Phys. 2014, v. 116, No. 17, p. 173903.

6. D. J. Byrne. T. Coffey, W.
J. Dowling, Yu. P. Kalmykov, S. V. Titov, On the Kramers very low damping
escape rate for point particles and classical spins in: Advances in Chemical
Physics, 2015, Vol. 156, pp. 393-459, Series Eds. S. A. Rice and A. R. Dinner,
Wiley, New York.

7. D. Byrne, W. T. Coffey, Yu. P. Kalmykov, S. V. Titov, J. E. Wegrowe, “Spin-transfer torque effects in the dynamic forced response of the magnetization of nanoscale ferromagnets in superimposed ac and dc bias fields in the presence of thermal agitation”, Physical Review B, vol. 91, p.174406 (2015).

Papers and chapters submitted
and in preparation:

8. Yu. P. Kalmykov, W. T. Coffey, S. V. Titov, "Spin relaxation in phase
space", in preparation to Advances in Chemical Physics.

9. Yu. P. Kalmykov, B.
Ouari, S. V. Titov, "Nonlinear
stationary ac response and dynamic magnetic hysteresis of antiferromagnetic
nanoparticles in superimposed ac and dc bias magnetic fields", in preparation to Physical Rview B.

10. Yu. P. Kalmykov, W. T. Coffey, S. V. Titov,
"Nonlinear ac stationary response and dynamic magnetic hysteresis of a
quantum uniaxial paramagnet", in preparation to Physical Review B

We have also given 12 talks at
sundry conferences:

1. H. El Mrabti, P. M.
Déjardin, S. V. Titov, and Yu. P. Kalmykov,"Damping dependence in dynamic
magnetic hysteresis of superparamagnets", 6th International School of
Nanosciences 2012, Le Tremblay-sur-Maldre, 24-29 June 2012, p. 29.

2. S. V. Titov and Yu. P.
Kalmykov (poster), "Damping effects on the magnetic dynamic hysteresis of
assemblies of single domain ferromagnetic particles", Nizhnii Novgorod,
Russia, Int. Workshop on Millimeter and Submillimeter Waves, Russia, 26
February-1 March 2013, Abstracts p. 123.

3. P.M. Déjardin (oral),
Relaxation time of interacting dipoles in the self-consistent field approximation,
5èmes Journées de Dynamique du Sud-Ouest, France, Perpignan, 4-5 June 2013, Abstracts
p. 17.

4. B. Ouari, P.M. Déjardin, L.
Méchernène, A. Benosman, S. V. Titov, and Y. P. Kalmykov (oral), Dynamic
magnetic hysteresis loops and nonlinear susceptibility of magnetic
nanoparticles having biaxial anisotropy, 5èmes Journées de Dynamique du
Sud-Ouest, France, Perpignan, 4-5 June 2013, Abstracts p. 18.

5. W. T. Coffey (oral), “Very
Low damping escape rates for classical spins”, Rome School on Open Systems and
the Quantum-Classical Boundary, Rome, Italy, COST Action MP1006
"Fundamental Problems in Quantum Physics", 8-12 Avril 2013, Abstracts
p. 27.

6. Y. P. Kalmykov, W. T. Coffey, S. V. Titov, J.
E. Wegrowe, D. Byrne (oral), Magnetization reversal in the presence of thermal
agitation and spin-transfer, 9th International Workshop on Nanomagnetism &
Superconductivity at the nanoscale, Spain, Comaruga, 1-5 July 2013, Abstracts
p. 38.

7. D.J. Byrne, W. T. Coffey,
Yu. P. Kalmykov, and S. V. Titov (oral), "Linear and nonlinear stationary
ac response of the magnetization of nanomagnets in the presence of thermal
agitation and spin-transfer torques", Annual German Physical Soc. Spring
Meeting: Magnetism Division, Dresden, 30 March- 4 April 2014, Abstracts MA
44.2, April 3, Thu 15:15.

8. Yu. P. Kalmykov, W. T.
Coffey, and N. Wei (oral), "On the nonlinear response of noninteracting
electric dipoles and single domain ferrofluid particles to strong alternating
and dc bias fields", Annual German Physical Soc. Spring Meeting: Magnetism
Division, Dresden, 30 March- 4 April 2014, Abstracts MA 9.9, March 31, Mon
17:15.

9. W. T. Coffey, W. Dowling,
Yu. P. Kalmykov, and S. V. Titov (oral), "On the very low damping escape
rate for point particles and classical spins", Annual German Physical Soc.
Spring Meeting: Magnetism Division, Dresden, 30 March- 4 April 2014, Abstracts
MA 29.12, April 2, Wed 18:00.

10. Y. P. Kalmykov, W. T.
Coffey, S. V. Titov, (oral), “Magnetization reversal time of magnetic nanoparticles
at very low damping”, 10th International Workshop on Nanomagnetism &
Superconductivity at the nanoscale, Spain, Comaruga, 30 June-4 July 2014,
Abstracts p. 26.

11. W. T. Coffey, N. Wei, S.
V. Titov, Y. P. Kalmykov, D. Byrne (oral), "Nonlinear frequency-dependent
effects in the dc magnetization of uniaxial magnetic nanoparticles in
superimposed strong alternating current and direct current fields", Annual
German Physical Soc. Spring Meeting: Magnetism Division, Berlin, 16 March - 20
March 2015, Abstracts MA 14.11, March 17, Tue 12:15.

12. W. T. Coffey, Y. P.
Kalmykov, V. Titov, D. Byrne, J.-E. Wegrowe (oral), "Nonlinear
frequency-dependent effects in the dc magnetization of uniaxial magnetic
nanoparticles in superimposed strong alternating current and direct current
fields", Annual German Physical Soc. Spring Meeting: Magnetism Division,
Berlin, 16 March - 20 March 2015, Abstracts MA 9.7, March 16, Mon 17:15