Thermal noise suppression in system of superparamagnetic nanoparticles by applying permanent magnetic field transverse to easy axes

A.G. Isavnin, I.I.Mirgazov

Kazan (Volga region) Federal University, Russia

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 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.

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 output noise power as stochastic resonance rises.

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 Н₁. 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 Н₁ 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 Нcos(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=Мcosq to take value x_± = ±М₀. 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 10⁹ - 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.

Output power of the system can be obtained by means of integrating (9) over w from 0 to ¥ [10]:

. (10)

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 power of the modulated system to output power of the non-modulated system)

P_nm / P_n = 1 - W²A²/(2(W²+W²)) , (11)

reveals non-monotonous dependence on temperature with a distinct minimum.

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