11/02/2013

Marie Curie Actions

Project No: 230785  NANOMAGNETS - Mesoscale Quantum Dissipation with Applications to Nanotechnology      2009-2012

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:

 

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:

Quantum dissipation arising from quantum fluctuations and the quantum mechanics of macroscopic variables is important because of the ever decreasing size (mesoscale) of the nanoparticles used in technology. The most striking example occurs in information storage by magnetic nanoparticles, where the governing factor for magnetization reversal by macroscopic quantum tunnelling is spin size S. The S dependence, with associated large quantum effects, becomes ever more marked as one proceeds from single domain particles to molecular clusters to single molecule magnets to individual spins. Here in the context of a general investigation of mesoscale quantum mechanics of particles (separable and additive Hamiltonians) and spins it is proposed to generalize Wigner's quasi-phase space formulation of quantum mechanics without dissipation (originally used to calculate quantum corrections to classical statistical mechanics i.e. the quantum/classical borderline characteristic of the mesoscale), to systems with non-separable Hamiltonians (spins) including the effects of dissipation to the surrounding heat bath. The results, obtained by (a) matrix continued fraction methods of solution of the appropriate master equations (b) computer simulation and (c) quantum Kramers escape rate theory will be compared with suitable experimental observations of the reversal time of the magnetization and the dynamic susceptibilities of nanomagnets (molecular magnets, nanoclusters, and nanoparticles).

 

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

We have derived an evolution equation for the reduced density matrix for a spin system with a non-axially symmetric Hamiltonian in contact with a thermal bath using the projection operator technique. Hence we have obtained the master equation for the time evolution of the quasiprobability density function of spin orientations in the phase space representation of the polar and azimuthal angles for spin systems subject to a magnetic field. We have also elaborated a theory of magnetic relaxation of spin systems for both axially and non-axially symmetric magnetic anisotropy potentials. Moreover, we have derived recurrence equations for the statistical moments from the evolution equation for the density matrix and the corresponding master equation. We have also developed effective numerical algorithms and programs for evaluating the spin size effects and the temperature dependence of the reversal time of the magnetization, the switching fields and corresponding dynamic hysteresis loops of various spin systems such as molecular magnets, nanoclusters, etc.

Description of the main results achieved so far:

The nonlinear relaxation of quantum spins interacting with a thermal bath is treated via the respective evolution equations for the reduced density matrix and phase space distribution function in the high temperature and weak spin-bath coupling limits using the methods already available for classical spins. The solution of each evolution equation is written as a finite series (analogous to the Fourier series representation of the classical distribution function) of the polarization operators and spherical harmonics, respectively, where the coefficients of the series (statistical averages of the polarization operators and spherical harmonics) are found from entirely equivalent differential recurrence relations. Each system matrix has an identical set of eigenvalues and eigenfunctions. For illustration, the time behavior of the longitudinal component of the magnetization and its characteristic relaxation times are evaluated for a uniaxial paramagnet of arbitrary spin S and generalized to the non-axially symmetric problem of a paramagnet of arbitrary spin S subjected to a dc magnetic field of arbitrary direction and orientation.  We have evaluated effects of strong magnetic fields on the magnetization relaxation of spin systems with non-axially symmetric magnetic anisotropy potentials (biaxial, cubic, etc.). We have developed a comprehensive theory of the magnetization relaxation of spin systems as a function of S spanning the entire region between individual spins and single domain particles which is necessary for the analysis of both the experimental data and the coexistence of quantum and classical phenomena in spin systems (molecular magnets, nanoclusters, etc.). In particular, we have developed numerical algorithms for the estimation of the observables (the reversal time, dynamic susceptibility, etc.) of spin systems such as molecular magnets, nanoclusters, etc. We have estimated spin size effects and the temperature dependence of the switching fields and corresponding hysteresis loops of molecular magnets. We have also estimated spin size effects on the reversal time of the magnetization of various spin systems such as molecular magnets, nanoclusters, etc. Thus we have achieved all the anticipated objectives set out in our project.

 

We have submitted for publication 4 papers and have published  6 papers in primary physical journals. We have also given 8 talks and presented 6 posters at 11 conferences and workshops.

 

Publications:

We have published and have submitted for publication 10 papers in primary physical journals:

1.    Yu. P. Kalmykov, W. T. Coffey, and S. V. Titov, "Phase space Langevin equation for spin relaxation in a dc magnetic field", EPL 2009, v. 88, No. 1, p. 17002_(6 pages).  http://dx.doi.org/10.1209/0295-5075/88/17002

2.    Yu. P. Kalmykov, S. V. Titov, and W. T. Coffey, "Nonlinear longitudinal relaxation of a quantum superparamagnet with arbitrary spin value S: Phase space and density matrix formulations", Phys. Rev. B 2010, v. 81, No. 9, p. 094432_(14 pages).  http://link.aps.org/doi/10.1103/PhysRevB.81.094432

3.    Yu. P. Kalmykov, S. V. Titov, and W. T. Coffey, "Spin size effects in stochastic resonance in quantum uniaxial superparamagnets", Phys. Rev. B 2010, v. 81, No. 17, p. 172411. http://link.aps.org/PhysRevB.81.172411

4.    Yu. P. Kalmykov, S. V. Titov, and W. T. Coffey, "Quantum effects in stochastic resonance in uniaxial superparamagnets subjected to a dc bias magnetic field", J. Phys. Cond. Matter 2010, v. 22, No. 37, p. 376001_(7 pages).  http://stacks.iop.org/0953-8984/22/376001

5.    Yu. P. Kalmykov, B. P. Mulligan, S. V. Titov, and W. T. Coffey, "Master equation in phase space for a spin in an arbitrarily directed uniform external field ", J. Stat. Phys. 2010, v. 141, p. 589-606. http://www.springerlink.com/content/a76058036p637187/

6.   Yu. P. Kalmykov, W. T. Coffey, and S. V. Titov, "Statistical moment equations for stochastic spin dynamics in phase space: A uniaxial paramagnet subjected to a dc bias field of arbitrary orientation", Phys. Rev. B 2012, v. 86, No. 10, p. 104435_(12 pages).  http://link.aps.org/doi/10.1103/PhysRevB.86.104435

7.    W. T. Coffey, Yu. P. Kalmykov,  S. V. Titov, and William J. Dowling, "Longest relaxation time of relaxation processes for classical and quantum Brownian motion in a potential: escape rate theory approach", Advances in Chemical Physics, 2013, Vol. 152, pp. 111-309, Series Ed. S. A. Rice, Wiley, New York.

8.   Yu. P. Kalmykov, W. T. Coffey, and S. V. Titov, "Statistical moment equations for stochastic spin dynamics in phase space: A uniaxial paramagnet subjected to a dc bias field of arbitrary orientation", Phys. Rev. B 2012, v. 86, No. 10, p. 104435_(12 pages).  http://link.aps.org/doi/10.1103/PhysRevB.86.104435


9.    Yu. P. Kalmykov, W. T. Coffey, and S. V. Titov, "Dynamic magnetic hysteresis of nanomagnets with arbitrary spin value S", Phys. Rev. B. 2012, in preparation. 

10.  Yu. P. Kalmykov, W. T. Coffey, and S. V. Titov, "Nonlinear longitudinal relaxation of a uniaxial nanomagnet with arbitrary spin value S in a dc magnetic field of arbitrary orientation", Phys. Rev. B. 2012, in preparation. 

 

We have given 8 talks and presented 6 posters at 11 conferences and workshops.

 

1. W. T. Coffey, Yu. P. Kalmykov, B. P. Mulligan, and S. V. Titov (oral), "Phase-space description of spin dynamics", Annual German Physical Soc. Spring Meeting, Dresden, 22-27 March 2009, DY 13.4 Tue 15:30 ZEU 255, p. 25. http://www.dpg-verhandlungen.de/2009/dresden/dy13.pdf

2. W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov (poster), "Spin dynamics in phase-space", 5th International Workshop on Nanomagnetism & Superconductivity, Spain, Comaruga 2009, http://www.ub.edu/gmag/comaruga/ .

3. W. T. Coffey, Yu. P. Kalmykov, B. Mulligan, and S. V. Titov (poster), "Spin dynamics: quantum master equations in phase space for axially symmetric potentials", Tunneling and Scattering in Complex Systems - From Single to Many Particle Physics, Seminar and Workshop, Dresden, Germany, September 07 - 25, 2009.

4. B. P.J. Mulligan, W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov (oral), "Phase space master equations for the Lipkin-Meshkov Hamiltonian ", Annual German Physical Soc. Spring Meeting, Regensburg, 21-26 March 2010, DY 4.6 Mon 15:15 H47, http://www.dpg-verhandlungen.de/2010/regensburg/dy4.pdf

5. B. P.J. Mulligan, W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov (poster), "Spin dynamics: quantum master equation in phase space fora spin in a uniform external field ", Annual German Physical Soc. Spring Meeting, Regensburg, 21-26 March 2010, DY 6.9, Mon 16:00 Poster B2, http://www.dpg-verhandlungen.de/2010/regensburg/dy6.pdf

6. W. T. Coffey, Yu. P. Kalmykov, B. P. J. Mulligan, and S. V. Titov (poster)"Spin dynamics of a uniaxial paramagnet: Phase space formulation", Theoretical, Computational, and Experimental Challenges to Exploring Coherent Quantum Dynamics in Complex Many-Body Systems, May 9-12 2010, Dublin, Ireland. http://www.cecam.org/workshop-3-460.html

7. W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov (oral), “Nonlinear magnetic relaxation of quantum superparamagnets: Phase-space method”, 7th Int. Conf. on Fine Particle Magnetism (ICFPM-2010), Uppsala, Sweden, June 21-24, 2010, p. 124. http://www-conference.slu.se/icfpm2010/ 

8. W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov (oral), "Nonlinear magnetic relaxation of quantum superparamagnets: Phase-space description", 6th International Workshop on Nanomagnetism & Superconductivity, Spain, Comaruga 2010, p. 35 http://www.ub.edu/gmag/comaruga/ .

9. Yu. P. Kalmykov, B. P.J. Mulligan, S. V. Titov, and W. T. Coffey, (poster), "Spin dynamics in phase space", Annual German Physical Soc. Spring Meeting, Dresden, 13-17 March 2011, MA 19.84, Tue 10:45 Poster P2, http://www.dpg-verhandlungen.de/2011/dresden/ma19.pdf

10. Yu. P. Kalmykov, B. P. J. Mulligan, S. V. Titov, and W. T. Coffey, (oral), "Master equation in phase space for a spin in an arbitrarily directed uniform external field", Annual German Physical Soc. Spring Meeting, Dresden, 13-17 March 2011, MA 45.5, Wed 18:30 http://www.dpg-verhandlungen.de/2011/dresden/ma45.pdf

11. W. T. Coffey, Y. P. Kalmykov , S. V. Titov L. Cleary, and W. J. Dowling (oral), "Quantum master equation in phase space applied to the Brownian motion in a tilted periodic potential", Annual German Physical Soc. Spring Meeting, Dresden, 13-17 March 2011, DY 33.3, Thu 14:30 http://www.dpg-verhandlungen.de/2011/dresden/dy33.pdf

12. B. P. J. Mulligan, Yu. P. Kalmykov, S. V. Titov, and W. T. Coffey (poster), "Spin dynamics in phase space", Noise in Non-Equilibrium Systems: From Physics to Biology, International Workshop – April 11 - 14, 2011, On the occasion of Prof. Peter Hänggi's 60th birthday, http://www.mpipks-dresden.mpg.de/~nines11/

13. W. T. Coffey, L. Cleary, W. Dowling, Yu. P. Kalmykov, and S. V. Titov (oral), "Phase space master equations for quantum Brownian motion in a periodic potential: comparison of various kinetic models", Madrid Workshop on Open Quantum Systems, 3-5 October 2011, Madrid, p. 36. http://fama.iff.csic.es/con/MWOQS-2011/workshop-files/Book_of_Abstracts_MWOQS2011.pdf

14.   W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov (oral), "Statistical moment equations for stochastic spin dynamics in phase space: A uniaxial paramagnet subjected to a dc bias field of arbitrary orientation", Annual German Physical Soc. Spring Meeting: Magnetism Division, Regensburg, 11-16 March 2013, MA 43.10, Thu 17:15 http://www.dpg-verhandlungen.de/2013/regensburg/ma43.pdf