Dipole-dipole and exchange interaction effects on the magnetization relaxation of two macrospins: Compared

Abstract

The magnetization response including dipole–dipole interactions of a pair of macrospins following the sudden alteration of a dc magnetic field is calculated from the stochastic Gilbert-Landau-Lifshitz equation by reducing the overall task to an infinite hierarchy of differential-recurrence relations in the time domain for the statistical moments (averaged products of spherical harmonics in this case), which is exactly solved in the frequency domain by matrix continued fractions. The greatest relaxation time and dynamic susceptibility are then compared with the corresponding results for two exchange-coupled spins using the same exact method. For a given interaction parameter dipole–dipole coupling in the circularly symmetric configuration considered has a much more pronounced effect on the magnetization dynamics than exchange coupling principally manifested by a greater increase in the relaxation time.

Introduction

The Debye theory [1], [2] of dielectric relaxation of polar fluids and the magnetization relaxation of single-domain ferromagnetic particles, i.e., macrospins, due to Néel and Brown [3] (rooted in the Debye-Fröhlich [1], [4] model of orientational relaxation of permanent electric dipoles over a potential barrier in a solid) which have much in common, disregard dipole–dipole interactions. Subsequently various attempts to include that interaction have been made beginning with Budó in 1949 [5], [6], who treated the dielectric relaxation of molecules containing rotating polar groups. This work was in essence continued by Zwanzig [7] and Zhou and Bagchi [8] by investigating the relaxation at high temperatures of a lattice of permanent dipoles including their mutual dipole–dipole interaction. The most significant result of Zwanzig’s calculation was that dipolar interactions with the lattice give rise to additional fast new relaxation times increasing dielectric loss at high frequencies. Further examples are generalizations of the Onsager [9], [10] theory of the static permittivity [11] of polar fluids to include dynamic effects, which attempt to treat the long-range dipole–dipole interaction between the dipoles of a polar fluid via a cavity model. Yet another approach, introduced also for dielectric relaxation, is based on Berne’s forced rotational diffusion model [12], [13], [14], where the rotational Brownian motion of interacting electric dipoles is treated self-consistently via the appropriate Fokker-Planck equation. Berne’s method was later extended [15] to magnetic relaxation of macrospins weakly coupled by magnetic dipole–dipole interactions.

In magnetization relaxation (in using the Néel-Brown model [3], [16]) it is necessary to assess the effect of interactions between macrospins, automatically requiring microscopic theories of the magnetization dynamics invariably leading to complex N-body problems. Nevertheless, before one can even attempt to study the magnetization relaxation time of many interacting spins, one must first try to understand the effect of interactions on the magnetization dynamics of a pair of interacting spins coupled via exchange and/or dipole–dipole interactions, including the magneto-crystalline anisotropy and Zeeman terms, which may be solved exactly using matrix continued fractions.

Hitherto interactions between macrospins have been treated using many disparate methods, which for the most part entirely rely on numerical simulation or approximate discretization methods so that closed integral expressions for the various characteristic relaxation times etc. are generally not available. For example, Lyberatos et al. [17] and Andersson et al. [18] studied the relaxation of interacting magnetic moments using a Monte Carlo method. Berkov and Gorn [19] numerically solved the set of Langevin equations for an assembly of spherical single-domain particles with uniaxial anisotropy coupled by dipole–dipole interactions. This was accomplished without any assumptions regarding the interaction strength and mainly concerns the temperature and frequency dependence of the imaginary part of the ac susceptibility of the assembly. They also reported crossover-type behavior between low and high anisotropy assemblies with a non-monotonic temperature dependence of the susceptibility peak as a function of temperature and concentration for small damping. In [20] a pair of coupled dipoles was treated using Langevin dynamics simulations, while thermally activated magnetization reversals in systems of interacting classical magnetic moments obtained both by Monte-Carlo methods and Langevin dynamics simulations were compared in [21], [22]. Moreover, Denisov and Trohidou [23] by solving the Fokker-Planck equation for the distribution function of two-dimensional ensembles of ferromagnetic nanoparticles derived an evolution equation for the magnetization in certain limiting cases only, while approximate analytic equations for the magnetization relaxation time were also proposed in Refs. [24], [25], [26]. Rodé et al. [27] numerically solved the Fokker-Planck equation for two interacting particles so calculating the time decay of the magnetization, yielding a simple relaxation time expression in terms of an effective volume, varying from close to the actual one for very weak interactions to twice that for strong interactions, so that the magnetic stability and coercive force increase with interaction strength. Chen et al. [28] also gave an analytic solution for the relaxation time for two interacting single-domain particles obtaining good agreement with numerical results from the Fokker-Planck equation. Again, via the latter equation Solomon [29] and Yoshimori and Korringa [30] treated the relaxation of two coupled spins in a magnetic field. Other authors (see, e.g., [31], [32], [33]) used a self-consistent potential in the Fokker-Planck equation to handle the magnetic dipole–dipole interactions, e.g. Zubarev and Iskakova [31] who concluded (cf [7]) for the linear response that interactions lead to a discrete set of microscopic relaxation mechanisms with a corresponding increase in the global time scale. Ivanov et al. [32] have obtained formulas for the static and dynamic susceptibility that compare favorably with numerical simulations and experimental data. Yet another method [33] based on the nonlinear response of magnetic dipoles to external alternating fields, yields a global time scale in agreement with the predictions of mean field theory. This calculation has been recently extended in [34] to yield an analytical formula for the dynamic susceptibility of isotropic dipolar assemblies with long-range interactions using the forced rotational diffusion equation beyond the mean field approximation. Finally, a formally exact method of determining the dynamics of a pair of exchange coupled macrospins was developed in Titov et al. [35] by averaging the stochastic Gilbert-Landau-Lifshitz equation for the magnetization over its realizations, so reducing the task to solving a hierarchy of differential-recurrence relations for the statistical moments (averaged spherical harmonics). In turn the formally exact solution of the hierarchy yielding interalia all the characteristic times of the system may be accomplished in the frequency domain by matrix-continued fractions. Thus, we have in essence the complete solution of the exchange-coupled two-spin problem in external fields in the analytic manner previously formulated by us [2] for non-interacting dipoles. The relaxation time so calculated for two equivalent spins in a uniaxial potential and a uniform magnetic field which may be altered in step-like fashion compares favorably with the Langer’s general asymptotic theory of the decay of metastable states [36] for strong system - bath coupling.

The purpose of the present work is to demonstrate how the two-dipole problem (a system with four degrees of freedom) may be treated analytically via the formally exact solution of the Langevin equations for the desired observables, which are the characteristic relaxation times and decay functions. Here we extend the exact method of [35] to study the effect of magnetic dipole–dipole interaction on the magnetization relaxation. (In [35] exchange interactions alone were treated). The dipole–dipole coupling is very useful for structural studies (because it depends only on known physical constants and the dipole separation) and for its effect on spin relaxation. Thus, magnetic dipole–dipole interactions are exactly treated using the two-spin model representing a system with more than two configurational degrees of freedom and the results are compared with those of the two-spin system with exchange interaction only. Now unlike exchange interactions, dipole–dipole interactions are anisotropic. However, as a first step towards including this anisotropy only parallel easy axes (also parallel to the direction of the applied dc field) are analysed, because the resulting circularly symmetric Hamiltonian drastically simplifies the calculations. For other orientations of the anisotropy axes which of course is the most interesting case the calculations are much more complicated and will lead to varying results arising from the anisotropic nature of the dipole–dipole interaction.

Our calculations are effected by first rewriting the (vector) stochastic Gilbert-Landau-Lifshitz (Langevin) equation governing the time-dependent magnetization as scalar Langevin equations for the products of the spherical harmonics specifying the orientation of each of the spins and then (using the theory of angular momentum in the manner of [2]) averaging them over their realizations in configuration space in an infinitesimal time (taken following Einstein [2] as shorter than any characteristic time of the system but long compared to the time of an adiabatic collision) given a sharp set of initial orientations so yielding the time evolution equation of the sharp values. Next by postulating an appropriate spatial distribution of the sharp values and then ensemble averaging over this distribution one has a hierarchy of equations for the statistical moments yielding the observables via rapidly convergent matrix continued fractions [2]. Hence, one has in analytic fashion the relaxation time for effectively all values of the interaction, anisotropy, and applied field parameters as well as other relevant observables (spectra of the relaxation functions, the complex susceptibility, etc.). The observables so calculated will then be compared with the corresponding ones for exchange interaction available in Ref. [35]. We emphasize that uniaxial anisotropy with the external field applied parallel to the easy Z-axis is supposed. The advantage of this particular anisotropy potential is that (although obviously subject to many symmetry restrictions) it ultimately results in a (tractable) recurrence relation (for the observables) in three indexes only (see Eqs. (6), (12) below). Moreover, it provides a benchmark solution with which calculations of the observables in the general non circularly symmetric case must agree.