Role of geometrical symmetry in thermally activated processes in clusters of interacting dipolar moments

O. Hovorka, J. Barker, G. Friedman, and R. W. Chantrell

Phys. Rev. B 89, 104410 – Published 12 March 2014

Abstract

Thermally activated magnetization decay is studied in ensembles of clusters of interacting dipolar moments by applying the master-equation formalism, as a model of thermal relaxation in systems of interacting single-domain ferromagnetic particles. Solving the associated master equation reveals a breakdown of the energy barrier picture depending on the geometrical symmetry of structures. Deviations are most pronounced for reduced symmetry and result in a strong interaction dependence of relaxation rates on the memory of system initialization. A simple two-state system description of an ensemble of clusters is developed, which accounts for the observed anomalies. These results follow from a semianalytical treatment, and are fully supported by kinetic Monte Carlo simulations.

Received 9 September 2013
Revised 20 February 2014

DOI:https://doi.org/10.1103/PhysRevB.89.104410

Authors & Affiliations

O. Hovorka^1,2,*, J. Barker¹, G. Friedman³, and R. W. Chantrell¹

¹Department of Physics, The University of York, York, YO10 5DD, United Kingdom
²Engineering and the Environment, University of Southampton, Southampton, SO17 1BJ, United Kingdom
³Electrical and Computer Engineering, Drexel University, Philadelphia, Pennsylvania 19104, USA

^*Corresponding author: o.hovorka@soton.ac.uk

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Vol. 89, Iss. 10 — 1 March 2014

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Images

Figure 1
Relaxation in an ensemble of $10^{6}$ four-spin identical chains oriented along the $z$ axis with a spherically random distribution of anisotropy axes: (a) Magnetization decay in external field $\vec{h} = \vec{0}$ and (b) the corresponding $f (ε)$ according to Eqs. (1, 2, 3, 4, 5, 6) where the $ε$ axis is in the units of $k_{B} T$ . Initialization is in a field oriented parallel ( ${\vec{h}}_{z}^{*}$ ) and perpendicular ( ${\vec{h}}_{x}^{*}$ ) with respect to $\hat{z}$ . In (a) calculations are based on Eqs. (1) and (6) (lines) validated by the kinetic Monte Carlo calculations (symbols). Assumed is the model system Eq. (2) for the parameter set discussed in the text.
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Figure 2
The cluster structures forming ensembles. (C) chains oriented along the $z$ axis with $N_{s} = 2 - 9$ spins; (R) rings with $N_{s} = 3 - 9,$ and (T) triangles with $N_{s} = 4 - 9$ lying in the $x y$ plane; (A) 3D structures of size $N_{s} = 4 - 9$ taken from Ref. [49].
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Figure 3
Dipolar interaction dependence of: the mean eigenbarrier $\bar{ε} = \int ε f (ε) d ε$ (relative to the noninteracting case $I = 0$ ) for an ensemble of (a) four-spin chains as in Fig. 1 and (b) clusters of four-spins arranged into an equilateral-triangle-based pyramid; and of the initialization-dependent energy barrier $δ e^{*}$ and correction $Δ$ defined in Eq. (7) for ensembles of (c) four-spin chains and (d) pyramids in (a)–(b). Initialization is in ${\vec{h}}_{z}^{*}$ ( $⋄$ ) and ${\vec{h}}_{x}^{*}$ ( $\circ$ ), $I_{0}$ defined in the text. The solid lines in (a)–(b) correspond to the fits by Eq. (7). In (c) and (d), the solid and dotted lines are only guiding lines while the dashed lines corresponds to the mean energy barrier $〈 δ e 〉$ obtained over the entire ensemble.
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Figure 4
Definition and validation of Eq. (7). (a) The two-state system model of an ensemble which incorporates the dependence on initialization and leads to Eq. (7). (b) The data collapse generated by fitting Eq. (7) to all ensembles of structures in Figs. 2 (total 512 points), validating Eq. (7). Subfigures (c), (d), (e), and (f) show exponential plots of the fit coefficients $a_{1}$ and $a_{2}$ for ensembles of structures $C$ , $R$ , $T,$ and $A$ , respectively. The filled and open symbols relate to initialization in ${\vec{h}}_{z}^{*}$ and ${\vec{h}}_{x}^{*}$ , and $N_{s} = 2 (\circ), 3 (□), 4 (▿), 5 (⋄), 6 (▵), 7 (☆), 8 (hexa), 9 (penta)$ .
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Figure 5
The relative $\bar{ε}$ vs $δ e^{*}$ for ensembles of $10^{6}$ individual structures in Fig. 2 and initializations in ${\vec{h}}_{z}^{*}$ and ${\vec{h}}_{x}^{*}$ . $N_{s} = 2 (\circ), 3 (□), 4 (▿), 5 (⋄), 6 (▵), 7 (☆), 8 (hexa$ , filled symbols $), 9 (penta)$ and the symbol size grows with the increasing interaction strength $I$ . In every subplot the axis division unit equals 0.5. The guide lines: $\bar{ε} = δ e^{*}$ (dotted) and $\bar{ε} = δ e^{*} = 0$ (dash-dotted).
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Figure 6
The upper estimates of the relative $max (δ e_{corr}) / δ e^{*}$ as a function of the asymmetry measure $J$ for the cluster types $C$ , $R$ , $T$ , and $A$ shown in (a), (b), (c), and (d), respectively. The filled and open symbols relate to initialization in ${\vec{h}}_{z}^{*}$ and ${\vec{h}}_{x}^{*}$ , and $N_{s} = 2 (\circ), 3 (□), 4 (▿), 5 (⋄), 6 (▵), 7 (☆), 8 (hexa), 9 (penta)$ .
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