Magnetization reversal across multiple serial barriers in a single ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticle

The depinning of nanoscale magnetic textures, such as domain walls, vortices, and skyrmions, is of paramount importance for magnetic storage and information processing. We measure the time-resolved magnetic switching statistics of an individual, non-single-domain ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticle using a micrometer-scale superconducting quantum interference device. A strong narrowing of the waiting-time distributions before reaching the final state is observed as compared to the exponential distribution expected for a single barrier. A model consisting of multiple barriers in series is proposed and used to understand the narrow waiting-time distributions and their evolution with magnetic field. The number of barriers is found to reduce as the thermodynamic switching field is approached and eventually, very close to it, an exponential distribution, resulting from a single barrier, is observed.

Figure 1

Scanning electron micrograph of the $\mu$-SQUID with a single ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticle $F\#1–3$ shown in (a)–(c) respectively.

Figure 2

(a) Measured $M\text{−}H$ loops of the ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticle $F\#1$ for a field at an in-plane angle $\theta =-{45}^{\circ }$ and at $T=4.0\mathrm{K}$. The loop has two major jumps (marked by arrows) in each field sweep direction due to vortex nucleation and annihilation. The inset of (a) shows the switching-field histograms at the vortex annihilation at positive field. (b) Measured $M\text{−}H$ loops for the ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticle $F\#2$ (at $T=4.2$ K, $\theta =0^{\circ }$) exhibiting two parallel paths marked by red and black arrows.

Figure 3

(a)–(c) show, respectively, the schematic of the free-energy landscape for fields close to the switching field to illustrate two parallel minimum energy paths, two serial barriers in a single path, and two parallel paths with two serial barriers in each. (d) shows the schematic of the energy landscape for two serial barriers at different applied fields close to the switching field $H_{\mathrm{sw}}$. This illustrates how the barriers disappear one by one as one tilts the energy landscape by increasing $H$.

Figure 4

A comparison of experimental CDF of not switching for an ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticle at 129.2 mT and that of a permalloy nanowire at 63 mT. The horizontal time axis is in units of respective ${\tau }_{\mathrm{eff}}$. The fits to three equal ($\tau /3={\tau }_{\mathrm{eff}}=25.73$ s) and unequal serial barriers (see Table 1) are shown for the nanoparticle data. The fits to the stretched exponential ($\beta =0.6$ and $\tau =0.65{\tau }_{\text{eff}}=3.23$ s) and three parallel barriers (see Table S1 in Supplemental Material [15]) are shown for the nanowire data. The green line shows an exponential with average switching time as ${\tau }_{\mathrm{eff}}$.

Figure 5

Experimental CDF of not switching vs time (dots) for ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticle $F\#1$ at $\theta =-{45}^{\circ }, T=4.0\mathrm{K}$ obtained at waiting fields close to (a) annihilation and (b) nucleation. (c) and (d) show the CDF obtained for $F\#2$ at $T=4.2$ K, $\theta =0^{\circ }$ and $F\#3$ at $T=2.0$ K, $\theta ={60}^{\circ }$ respectively. The purple lines are fits to the three unequal barrier models and the green dashed lines are fits to a single exponential.

Figure 6

Waiting-time histograms for ${\mathrm{Fe}}_3{\mathrm{O}}_4$ nanoparticle $F\#3$ at different waiting fields $H_w, T=2.0\text{K}$, and $\theta ={60}^{\circ }$. A large number ($\sim 400$) of switching data was acquired. The purple line in (a), (b) shows a fit to the PDF corresponding to an unequal barrier model while the green dashed line in (c) shows a fit to an exponential PDF. The inset is a switching-field histogram for this particle with 800 counts obtained at the same temperature, angle, and at a sweep rate of 0.3 mT/s.