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Nonlinear response of a uniaxially anisotropic superparamagnetic particle subjected to a dc bias and strong ac field is evaluated. For the third harmonic, a non-monotonic dependence of the response amplitude on the strength of the oscillating field (noise-induced resonance) is found. Introduction Studies on stochastic resonance (SR), the number of which multiplies impressively during last years, have revealed a variety of remarkable properties of noisy dynamic systems. In particular, noise-induced resonance (NIR) [1] has been found for both symmetrical [1] and non-symmetrical [2] potentials. Unlike the ordinary stochastic resonance (see the review [3]), the description of NIR does not involve the signal-to-noise ratio and is given in terms of higher harmonic generation. The phenomenon manifests itself as occurrence of “dips” on the amplitudes of higher harmonics plotted as the functions of the noise power or external field strength. Under imposing of a bias field, complementary odd/even harmonics turn up in the system spectrum. Both these effects are now under intense investigation [4,5] in view of their application to detection/enhancement of weak signals. Assemblies of superparamagnetic particles provide a very clear physical example of an essentially noise-affected system. Conventional SR in them, i.e., the response to a weak magnetic field on the thermofluctuational background, has been studied in Refs. [6–10]. In Refs. [11,12] a step has been made to the case of the finite-amplitude excitation. However, NIR-like phenomena have not been addressed. In the original papers [1,2] the existence of NIR had been established only in a weak-field limit, i.e., in terms of power series expansions. The basic demerit of this method is that when evaluating the amplitude of an l-th harmonic, all the contributions of the order (ξ ) are neglected; here ξ is the ratio of the intensity of the driving field amplitude to the reference thermal (noise) energy. Obviously, the higher the field the greater the expansion results deviate from the true ones. Here we describe a magnetic response of a superparamagnetic particle subjected to a dc (bias) field and an ac harmonic field of an arbitrary strength. The novelty is twofold: (1) we prove the NIR-like effects in superparamagnetic systems and (2) we move far beyond the limits of a small parameter approach. Solution of the micromagnetic Fokker-Planck equation We assume a particle to be single-domain so that its magnetic state is that of a magnetic dipole of the magnitude μ = IV, where I is the magnetization of the particle material, and V its volume. Being well below the Curie temperature, we treat I as a constant. Then all the motions of the dipole are reduced to its rotations, and are described by a unit vector e or the set of spherical coordinates θ and φ, with the polar axis pointing along the particle easy axis. Performing any angular motion, the dipole μ = μe effectively experiences the bulk magnetic anisotropy (a symmetrical two-well potential), a dc bias field co-aligned with the particle easy axis (a one well potential) and a ac field of frequency ω imposed along the same axis. The driving frequency is assumed to be small in comparison with that of the Larmor precession in the effective internal (anisotropy + bias) field. Therefore, in the magnetodynamic equation (taken in the Gilbert form) only the relaxation part must be retained. The system resides in a thermal bath of temperature T, the Boltzmann constant is set to unity. Thermally-affected Gilbert magnetodynamics of a single-domain particle is described by the micromagnetic Fokker-Planck equation derived by Brown [14]. It is written for the orientation distribution function W(e,t) of the particle magnetic moment and reads ( ) ( ) { } 1 1 div 2 W E W W E W t β λ τ − ∂ = ∇ ×∇ + ∇ + ∆ ∂ e , (1) where ∆ is the Laplacian on the surface of a unit sphere, E is the free energy density. The parameter τ = βI(1 + λ)/2γλ (2) is the characteristic relaxation time, where β = V/T, γ is the gyromagnetic ratio for electrons, and η is the damping parameter of the Larmor precession used in the Gilbert equation, so that the combination λ = γηI makes a dimensionless dissipation constant. Interactions between different particles of the system as well as any memory effects are neglected. With Eq. (1), as in any statistical approach, the physically observed quantities are the averages fn(t) = 〈Pn(cos θ)〉(t) = 0 π ∫ Pn(cos θ) W(θ,t) sinθ dθ . (3) In particular, the issue of prime interest is f1(t), that is the dimensionless magnetization of the system. Under superimposed constant uniform magnetic field H0 and an ac field H1 = H1 cosωt (both applied along the polar axis) the energy density takes the form E = –K cos 2 θ – (H0+ H1 cosωt) I cosθ , (4) where K is the uniaxial anisotropy constant. Allowing the magnitudes of both ac and dc fields to be large enough so that the Zeeman energies of the magnetic moment are comparable with or higher than the thermal energy T, one faces an essentially nonlinear problem. To solve it, we use the method developed in Refs. [15,16]. Substituting the energy (4) in Eq. (1) and expanding W(θ,t) in a series of Legendre polynomials Pn(cosθ), one arrives at the set of differential-recurrence equations [12]: τ dt d fn(t) + 2 1 n(n + 1) fn(t) = e(t) [an fn–1(t) + bn fn+1(t)] + σ[cn fn–2(t) + dn fn(t) + gn fn +2(t)], (5) where the following notations for the coefficients are used ( 1) 2(2 1) n n n n a b n + = − = + , ( 1)( 1) (2 1)(2 1) n n n n c n n + − = − + , ( 1) (2 1)(2 3) n n n d n n + = − + , ( 1)( 2) (2 1)(2 3) n n n n g n n + + = − + + , (6) and the dimensionless material parameters are e(t) = ξ0 + ξ cosωt, ξ0 = βIH0, ξ = βIH1 , σ = βK . (7) Here we solely concern ourselves with the steady ac response, which is independent of the initial conditions, so that one needs only the stationary solution of Eq. (3). In view of that, we present the sought for asymptotic (with respect to time) relaxation functions in the form |
