Reduction of thermal noise  in system of magnetic nanoparticles under asymmetric stochastic resonance conditions

 

A.G. Isavnin, I.I.Mirgazov

Kazan (Volga region) Federal University, Russia

 

Stochastic resonance effect is previous quite sharp increase and subsequent gradual decay of response of a multistable system to a weak periodic modulation as internal noise of the system goes up [1]. Such a phenomenon is well explored both theoretically and experimentally within wide range of applications, from global climatic changes to nervous processes in live organisms [2]. Generally external periodic signal is assumed to be weak, so without noise any transitions between stable states of the system are impossible. At a certain noise intensity regular part of dynamics of the system increases, that is the transformation of stochastic energy into coherent one occurs. The two-state theory says that for a bistable system the response reaches its maximum when mean time of transitions between stable states becomes comparable to a half of period of modulation. At greater noise intensities the coherency of signal and noise fails and the response goes down.

Monodomain magnetic particles with «easy axis» anisotropy are bistable elements with steady states associated with two opposite directions of easy axis. Such nanoparticles are technologically important materials and their unique properties continuously attract attention of scientists. In previous papers stochastic resonance was considered in fine magnetic particles for thermal [3,4] and tunneling [5,6] modes of magnetization reversal. Obtained results displayed specific non-monotonous dependence of the response to radiofrequency signal on the noise level. Values of signal-to-noise ratio and components of dynamic magnetic susceptibility derived within the two-state approximation were verified by means of numerical calculations [7,8], based on a Fokker-Planck equation with periodic drift term.

Present paper considers the influence of additional external magnetic field applied at arbitrary angle  to the easy axis on output noise power as stochastic resonance rises.

Energy of the uniaxial monodomain particle under such circumstances is .     (1)

The first here term describes interaction of the magnetic moment of superparamagnetic particle with anisotropy field (K is anisotropy constant, v is the particle’s volume, q is the angle between magnetization vector and easy axis), the second term is associated with permanent magnetic field Í that distorts symmetry of the double-well potential.

We, like in [4], associate input signal with external radiofrequency field and input noise with temperature T of the sample. Then output of the system is change of the magnetic moment.

The external periodic signal is assumed to be weak enough, so it alone cannot lead to changes of direction of the particle’s magnetic moment. This implies the condition m0MH1v < DU holds. Thermal activation of the system increases rate of switches of the particle’s magnetic moment and that leads to possibility to surmount the potential barrier.

Further the discrete-orientations approximation is used in the calculations. Therefore the magnetic moment of the particle is allowed to be in just two states corresponding two minima of the double well. So the stable states are for x=Mcosq at q1 , q2 . The two-state theory used trough this paper implies that instead of continuous diffusion of the particle’s magnetic moment over a sphere we consider its stochastic switches between two directions. Advantage of such approximation is possibility of using the master equation for transition rates that yields analytical solution. The master equation is [1,4]:

 .                (2)

Here n± is the probability of discrete variable x=Ìcosq to take value x± = Ì1,2. W±(t) are escape rates from the states corresponding to stable directions of the magnetic moment at angles q1, q2 to the easy axis. Such rates are described with Kramers-type formula [9, 4]:

 ,    .     (3)

Here ,  are heights of the potential barriers. As it was shown in [3],  , provided , so for simplification of our calculations let us assume . Here  is dimensionless amplitude of external modulation on the easy axis. Let us also denote .  ,                                                            (4)

is the attempt frequency, and  [10],  is the gyromagnetic ratio,  is a damping  constant from the Hilbert equation [9].

Power spectrum of the system as Fourier transform of the autocorrelation function displays Lorentzian background associated with stochastic dynamics and d - spike describing regular motion of the vector M at the frequency W of external signal: .          (5)

Here ,  is Kramers rate of escape from the lower minimum of the asymmetric nonmodulated potential.   is phase shift between response of the system and external periodic signal.

Output power of the system can be obtained by means of integrating (9) over w from 0 to ¥ [11]:

.           (6)

Here the first term describes output noise power, and the second one is associated with output signal of the system. It is obvious that the noise power diminishes to the same value as the signal power increases. So there is transformation of chaotic motion energy into coherent one.

The following reduced result (ratio of output noise power of the modulated system to output noise power of the non-modulated system)

  ,                                                (7)

reveals non-monotonous dependence on temperature with a distinct minimum, see Figure 1.

dif_alpha.png

Figure 1. Pnm/Pn vs temperature. Parameters of the system:  Ω=1E5 ñ-1,  v=5.5E-25 ì3,  M=1.72E6 A/ì,   Í1=3000 À/ì, H=10000 À/ì,  a) α=900, b) α=800, c) α=700

 

 

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