Effect of nonlinear superparamagnetic response on susceptibility curves for nanoparticle assemblies

F. Tournus, A. Hillion, A. Tamion, and V. Dupuis

Phys. Rev. B 87, 174404 – Published 3 May 2013

Abstract

We examine the effect of the applied magnetic field amplitude on zero field-cooled/field-cooled (ZFC/FC) curves, through the nonlinear susceptibility of superparamagnetic particles (i.e., at thermondynamic equilibrium, but taking into account the magnetic anisotropy). This nonlinear effect is shown to be the first to manifest itself when going away from the linear response regime (i.e., when the magnetic moment is simply proportional to the applied field), largely before the modification of the macrospin switching energy due to the external field. We demonstrate that it has a significant impact on ZFC/FC curves, especially for the low-temperature behavior of the FC curve, even in usual experimental conditions. We then show how this nonlinearity can be taken into account, in an easy way, to obtain a better modeling of the susceptibility curves and consequently a more reliable determination of the nanoparticles’ magnetic properties. The theoretical considerations are confronted in a series of experimental measurements on Co nanoparticles.

Received 21 February 2013

DOI:https://doi.org/10.1103/PhysRevB.87.174404

Authors & Affiliations

F. Tournus^*, A. Hillion, A. Tamion, and V. Dupuis

Institut Lumière Matière, UMR5306 Université Lyon 1-CNRS, Université de Lyon 69622 Villeurbanne cedex, France

^*florent.tournus@univ-lyon1.fr

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Vol. 87, Iss. 17 — 1 May 2013

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Images

Figure 1
Relative error on the equilibrium magnetic moment $m_{eq}$ , as a function of the dimensionless parameter $ξ$ , for different approximations: the linear response approximation (red [gray] dotted line), the Langevin approximation (blue [gray] dashed line), and the true nonlinear response truncated to the third-order susceptibility (black solid line). The calculations are done for particles of 5-nm diameter, with $M_{S} = 1.35 \times 10^{6}$ A/m, $K_{eff} = 100$ kJ/m $^{3}$ (which corresponds to $T_{B} ≃ 18$ K and $H_{A} ≃ 1500$ Oe), at a temperature of 25 K (i.e., $σ ≃ 19$ ).Reuse & Permissions
Figure 2
(a) Numerical calculation of the reduced third-order susceptibility $α$ (red [gray] solid line) as a function of $σ$ , compared to the low- $σ$ and high- $σ$ analytical approximations (blue [gray] dashed line and green [gray] dash-dotted line). The extreme cases, corresponding to a Langevin or a Ising function, are indicated with horizontal dotted lines. In a ZFC/FC measurement, the superparamagnetic regime (i.e., thermodynamic equilibrium) is observed typically for $σ \leq 25$ , whereas the region $σ > 25$ (hatched region) corresponds to the blocked regime. (b) Closer view, for $σ$ between 0 and 10, of the numerically determined $α (σ)$ value (red [gray] solid line) and the low- $σ$ and high- $σ$ analytical approximations (blue [gray] dashed line and green [gray] dash-dotted line). The two black squares indicate the $σ_{\min}$ and $σ_{\max}$ values and materialize the range where a linear interpolation will be used for $α (σ)$ .Reuse & Permissions
Figure 3
ZFC/FC curves simulations using the progressive crossover model (PCM), for a single MAE (a) or for a MAE distribution due to a log-normal particle-size distribution (b). Different approximations for $m_{eq}$ are compared (the analytical expressions are summarized in the Appendix): the linear response approximation (red [gray] dotted line), the Langevin approximation (blue [gray] dashed line), and the true nonlinear response truncated to the third-order susceptibility (black solid line). The calculations are done for particles with $M_{S} = 1.35 \times 10^{6}$ A/m, $K_{eff} = 100$ kJ/m $^{3}$ (which corresponds to $H_{A} ≃ 1500$ Oe), with a 25-Oe applied magnetic field (i.e., $h ≃ 0.017$ ). $τ_{0}$ is fixed to $10^{- 10}$ s and the temperature sweeping rate is $r_{T} = 0.033$ K/s. The particle diameter is equal to 5 nm (single MAE case, case a), while it follows a log-normal distribution (case b) with a median diameter of 3 nm and a dispersion parameter of 0.3. For the single MAE case, the curves corresponding to the abrupt transition model (ATM, dash-dotted line) are also shown. In the case of a significant MAE distribution (as in case b) the ATM and PCM are equivalent.Reuse & Permissions
Figure 4
Normalized ZFC/FC curves measured for a sample of Co nanoparticles embedded in a gold matrix, with different applied field amplitudes (a). Simulated ZFC/FC curves, taking into account the third-order susceptibility in the superparamagnetic contribution, for different applied magnetic fields (b). In insert, the experimental curves with a 50-Oe applied field are compared to the curves simulated with the third-order susceptibility approximation and the linear response approximation. The nanoparticles have a log-normal size distribution (median diameter of 2.75 nm and dispersion parameter of 0.26) and a $K_{eff} = 162$ kJ/m $^{3}$ . A residual field of 0.7-Oe is included.Reuse & Permissions
Figure 5
Experimental normalized $Δ m = m_{FC} - m_{ZFC}$ curves for the sample of Co nanoparticles embedded in a gold matrix, with the different applied field amplitudes.Reuse & Permissions

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