POSSIBLE  WAYS  TO  CONTROL  INTENSITY  OF  STOCHASTIC  RESONANCE  IN  MAGNETIC  NANOPARTICLES

A.G. Isavnin

Kazan State University, Russia

 

1. Introduction

Effect of stochastic resonance means 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 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.

Single-domain 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 transverse to the easy axis on dynamics of the particle’s magnetic moment as stochastic resonance rises. As well, it considers effects that emerge when source of periodic signal or the sample moves towards each other.

2. Single-domain magnetic particle under conditions of stochastic resonance

Hereafter it is assumed that stochastic resonance is in thermal, overbarrier mode. Energy of the uniaxial monodomain particle under such circumstances is

 .                                                                 (1)

Here the first 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 Í1. Double well potential (1) has maximum at q2=p/2 and minima at q1=arcsin(m0MH1/(2K)), q3=p-arcsin(m0MH1/(2K)). The height of potential barrier separating two minima is

DU = E(q2) - E(q1)  =  Kv - m0MH1v + m02M2H12v/(4K) .                                       (2)

As Í1 increases, the stable orientations of the magnetization vector shift to the direction of the permanent transverse field. Bistability of the system vanishes at H1=2K/(m0M) and just one minimum remains at q=p/2.

Let us 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. Under modulation of the particle with weak external field Ícos(Wt), applied along the easy axis, double-well potential

 ,                        (3)

begins to swing slightly - in one half of the modulation period 2p/W the right minimum becomes higher and the left one becomes lower, in the next half of period - vice versa. 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 m0MHv < 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. It is convenient to define discrete variable x=Mcosq describing projection of the magnetization vector to the easy axis. This variable can take only two values:

 .                                                            (4)

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]:

 .             (5)

Here n± is the probability of discrete variable x=Ìcosq to take value x± = ±Ì0. W±(t) is escape rate from ± state corresponding to stable directions of the magnetic moment at angles q1, q3 to the easy axis. Such rate is described with Kramers-type formula [9, 4]:

 .                                              (6)

The prefactor of Kramers formula a0 [3] is commonly near ferromagnetic resonance frequency and for iron fine particles has value of order 108 - 1010 s-1.

Probability density in the discrete orientations approximation reads:

 .                                    (7)

The potential is symmetric with respect to x=0, so x+ = -x- = M0.     Solution of the master equation (5) is:

 .          (8)

Here W=2a0exp(-DU/(kT)) is doubled Kramers escape rate of the system from either minimum of the symmetric non-modulated potential (1), A=m0M0Hv/(kT) is dimensionless amplitude of external modulation, f=arctan(W/W) is phase shift between response of the system and external periodic signal. Probability n+(t0) is equal to 1 if initial orientation of the magnetization is +M0 , and   0 if at t = t0  x = -M0.

Power spectrum of the system as Fourier transform of the autocorrelation function

 ,             (9)

displays Lorentzian background associated with stochastic dynamics and d - spike describing regular motion of the vector M at the frequency W of external signal.

One of the main indexing values of stochastic resonance is output signal-to-noise ratio (SNR). Using (9) and assuming the amplitude of external alternating field to be small such ratio can be written as:

 .           (10)

The temperature dependence of the SNR on T has bell-shaped distinct maximum. The SNR goes up as permanent transverse field Í1 rises [10].

3. DYNAMIC  MAGNETIC  SUSCEPTIBILITY

The SNR determines fraction of regular part of the system dynamics but contains no information about phase of output signal. In the study of stochastic resonance there is another approach, namely to consider the effect as amplification of a weak signal [11]. Following this way one should also find phase lag between output and input signals. As in general case the magnetic moment of particle does not change its direction coherently with external alternating field, it is real part of dynamic magnetic susceptibility that could be an appropriate index to describe amplitude amplification. Let us introduce complex susceptibility c = Rec - iImc and suppose for simplicity the initial phase of the modulation to be zero. Then  expression for time-dependent magnetization is

 .                                                              (11)

Asymptotic probabilities associated with stationary mode can be derived from (8) at  t0  tending to minus infinity:

 ,      n - (t) = 1 - n + (t) .                             (12)

Taking into account (7), one can obtain asymptotic averaged value  áx(t)ñ :

 .                                                   (13)

Obviously, real part of the susceptibility that describes co-phase motion of M and H, and imaginary part that describes motion with lag = p/2  are given with following expressions:

 ,                                         (14)

 .                                            

And absolute value of the dynamic susceptibility is

 .                          (15)

The real part is presented in Fig.2 as a function of temperature of the system. As in the case of zero permanent field [4,7], the curves have the specific bell-shaped form, typical for stochastic resonance. The auxiliary permanent field Í1 , applied transverse to the easy axis results in changing the potential barrier (2) and increases the response of system to external signal [12]. More precise results may be derived with non-constant value of prefactor α0 in (6) as analyzed in [13]. 

4. CASE  OF  MOVING  SAMPLE

Now let us consider the effects of motion. Suppose the sample with nanoparticles is moving towards the source of the external rf modulation with velocity V<<c (here c is velocity of light in space). Then, due to Doppler effect, we shall observe frequency of the external signal is to go up:

 .                                                                                          (16)

This “new” frequency W’ can be now used in (10), (14) and (15) instead of W. If the motion is in the opposite direction, then V in the denominator of (16) changes its sign and we have the frequency of external signal going down. As it was shown in [4,7], at lower frequencies one can observe higher values of SNR and components of the dynamic susceptibility c. Of course, the same effects will occur if the sample is fixed and the source of periodic modulation is moving.

Experimental scheme of measurement of the dynamic magnetic susceptibility can be found for example in [14]. Observation of dynamics of magnetic moment for individual particle is presented in [15]. For an ensemble of fine particles it is convenient to use frozen ferrofluids. By diluting ferrofluid the desired concentration of magnetic clusters may be produced. It means that the strength of interactions between nanoparticles is possible to tune. By freezing ferrofluid while permanent magnetic field applied one can get solid dispersion of fine particles with their easy axes aligned in the same direction.

5. CONCLUSION

This paper considers effect of thermal stochastic resonance in the system of single domain magnetic nanoparticles with uniaxial anisotropy. In addition, the influence of a permanent magnetic field applied transverse to the easy axis is examined. Such a field leads to a shift of stable directions of the magnetic moment in space and diminish the height of potential barrier separating two minima. As a result, the signal to noise ratio and dynamic magnetic susceptibility reveals higher values. It is important for observation of stochastic resonance and for amplification of weak periodic fields.

Another interesting effect presented in this paper is the Doppler frequency shift that also can vary the intensity of stochastic resonance.

So, there are two mores mode to tune the bistable magnetic system for amplification of external weak periodic signal. Such regimes seem to be quite useful in cases with difficulties to vary thermal noise intensity (e.g. near Curie temperature, etc.). The results of present investigation allow not only to enhance weak signals, but choose and set parameters of the system to avoid this enhancement when it is unwelcome.

 

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