Models for the magnetic ac susceptibility of granular superferromagnetic $\mathrm{Co}\mathrm{Fe}∕{\mathrm{Al}}_2{\mathrm{O}}_3$

The magnetization and magnetic ac susceptibility, $\chi ={\chi }^{\prime }-i{\chi }^{″}$, of superferromagnetic systems are studied by numerical simulations. The Cole-Cole plot, ${\chi }^{″}$ versus ${\chi }^{\prime }$, is used as a tool for classifying magnetic systems by their dynamical behavior. The simulations of the magnetization hysteresis and the ac susceptibility are performed with two approaches for a driven domain wall in random media. The studies are motivated by recent experimental results on the interacting nanoparticle system ${\mathrm{Co}}_{80}{\mathrm{Fe}}_{20}∕{\mathrm{Al}}_2{\mathrm{O}}_3$ showing superferromagnetic behavior. Its Cole-Cole plot indicates domain-wall motion dynamics similarly to a disordered ferromagnet, including pinning and sliding motion. With our models, we can successfully reproduce the features found in the experimental Cole-Cole plots.

Figure 1

Cole-Cole plots, ${\chi }^{″}\mathrm{vs}$ ${\chi }^{\prime }$, (a) analytically obtained for a noninteracting monodisperse ensemble of SPM particles with ${\mu }_0{M}_s^2∕3K=1$ and $KV∕k_{B}T=1$ (see text); (b) numerical result for a polydisperse ensemble with a log-normal distribution (circles) and a Maxwell distribution (diamonds) of particle volumes with ${\mu }_0{M}_s^2∕3K=1$, $K∕k_{B}T=1$, ${\tau }_0=1$, $\mathrm{\Delta }V=0.9$, and $⟨V⟩=1$; and (c) shows experimentally obtained curves on the SSG system ${[\mathrm{Co}\mathrm{Fe}\left(0.9\mathrm{nm}\right)∕{\mathrm{Al}}_2{\mathrm{O}}_3\left(3\mathrm{nm}\right)]}_{10}$ at three different temperatures, $T=50$, 55, and 60 K (Ref. 57). The particle sizes follow a Gaussian distribution with $⟨V⟩=11.5{\mathrm{nm}}^3$ and $\mathrm{\Delta }V=0.95{\mathrm{nm}}^3$. The frequency range is indicated in the figure.

Figure 2

$M$ vs $H$ curves from simulations of the adiabatic approach (i) with $T=0.1$, $H_0=1.85$, $L_z=8.0$, $\mu =0.24$, $\theta =0.83$, and $\beta =0.66$ at different frequencies $f=1.6\times {10}^{-7}$ (a), $1.6\times {10}^{-3}$ (b), $7\times {10}^{-3}$ (c), and $8\times {10}^{-2}$ (d). Note that all quantities are measured in dimensionless units, as described in the text. Lines are guides to the eye.

Figure 3

(a) ac susceptibility, ${\chi }^{\prime }$ and ${\chi }^{″}$, vs ac frequency, $f$, obtained with model (i) with the same parameters as in Fig. 2. (b) Same data, but plotted in the Cole-Cole presentation, ${\chi }^{″}$ vs ${\chi }^{\prime }$. The solid line represents a least-squares fit of the low-frequency data to a circle and the arrow shows the direction of increasing $f$.

Figure 4

Experimental Cole-Cole plot taken from Ref. 31 showing ${\chi }^{″}$ vs ${\chi }^{\prime }$ obtained on the SFM granular system ${[\mathrm{Co}\mathrm{Fe}\left(1.4\mathrm{nm}\right)∕{\mathrm{Al}}_2{\mathrm{O}}_3\left(2\mathrm{nm}\right)]}_{10}$. The susceptibility was measured at ac amplitudes ${\mu }_0{H}_0=50$ (a) and $5\mu T$ (b) at 10 mHz $⩽f⩽1$ kHz at $T=380$ (1), 350 (2), 320 (3), and 260 K (4). Transition fields are marked by arrows (Ref. 31).

Figure 5

$M$ vs $H$ curves for the nonadiabatic approach (ii) with $H_0=1.85$ at different frequencies, $f=0.0016$ (a), 0.08 (b), and 0.48 (c) for an infinite system (in this case one defines $M=Z$). (d) shows the magnetization curve for $f=0.0016$ but a finite system $\left(L_z=8.0\right)$ so that the DW touches the boundaries.

Figure 6

Real and imaginary part of the ac susceptibility vs frequency, calculated within approach (ii) for infinite systems (a) (the real part is shifted and scaled) and the corresponding Cole-Cole plot (b). In (c) ${\chi }^{\prime }$ and ${\chi }^{″}$ are plotted for the finite system $\left(L_z=8.0\right)$ and Cole-Cole plot (d). The inset in (d) shows the high-frequency behavior in more detail. The arrows show the direction of increasing frequencies. All simulations were performed with $H_0=1.85$.