Self-organized criticality of magnetic avalanches in disordered ferrimagnetic material

We observe multiple steplike jumps in a Dy-Fe-Ga-based ferrimagnetic alloy in its magnetic hysteresis curve at 2 K. The observed jumps are found to have a stochastic character with respect to their magnitude and the field position, and the jumps do not correlate with the duration of the field. The distribution of jump size follows a power law variation indicating the scale invariance nature of the jumps. We have invoked a simple two-dimensional random bond Ising-type spin system to model the dynamics. Our computational model can qualitatively reproduce the jumps and their scale-invariant character. It also elucidates that the flipping of antiferromagnetically coupled Dy and Fe clusters is responsible for the observed jumps in the hysteresis loop. These features are described in terms of the self-organized criticality.

Figure 1

Field-cooled (FC) and zero-field-cooled (ZFC) magnetization ($M$) data as a function of temperature ($T$) measured under 100 Oe of magnetic field for ${\mathrm{DyFe}}_{2.5}{\mathrm{Ga}}_{0.5}$.

Figure 2

(a)–(d) Full magnetization loop recorded at different temperatures for ${\mathrm{DyFe}}_{2.5}{\mathrm{Ga}}_{0.5}$. The inset depicts the temperature variation of the coercive field.

Figure 3

(a) The fifth leg (0 to 70 kOe in the first quadrant, after returning from $-70$ kOe) of the magnetization loop recorded at 2 K for different sweep rates of the magnetic field ($H$). (b) The full loop at 2 K after the sample's being cooled in different magnetic fields. Inset of (b) depicts the fifth leg of the hysteresis loop recorded at 2 K for different cycles.

Figure 4

Magnetization as a function of time for different protocols measured at 2 K after the application of a magnetic field of (a) 12 kOe and (b) 13 kOe. Before the measurement, the sample was first cooled to 2 K, and it was kept at this temperature for the rest of the measurements.

Figure 5

Panels (a) and (b), respectively, show the variation of electrical resistivity ($\rho$) as a function of the field at 2 and 7 K. The inset of (a) represents the variation of electrical resistivity as a function of temperature. The inset of (b) shows the variation of $\rho$ as a function of $H$ at 125 K.

Figure 6

(a) Histogram of the observed jumps in the experimental magnetization vs field data. (b) Number of occurrences as a function of jump size in $log\text{−}log$ scale. (c) Jump size as a function of the applied field. (d) Individual jumps are plotted as a function of the jump number.

Figure 7

Model schematic shows a lattice structure in two dimensions with Fe and Dy randomly distributed in a $3\text{:}1$ ratio. Identical and nonidentical atoms are linked via ferromagnetic and antiferromagnetic interactions, respectively. Disorder is introduced by nonmagnetic Ga replacing Fe according to fraction $f_{\mathrm{Ga}}$.

Figure 8

(a) Hysteresis loop of random-bond Ising model for disorder fraction $f_{\mathrm{Ga}}=$ 0.2 at $\mathcal{L}=64$. Inset: Many small jumps occur at a large field in the hysteresis loops. (b) Magnetization profiles for four distinct disorder fractions $f_{\mathrm{Ga}}=0$, 0.1, 0.2, and 0.3 are illustrated when the external field ($h$) is reduced. (c)–(e) Snapshots of spin configurations before and after large jumps in the magnetization at field strengths of $h=$ 1.5, 2.5, and $h=$ 5 (as in a), respectively. Red, brown, green, yellow, and blue represent $\mathrm{Fe}(+$), $\mathrm{Fe}(-$), $\mathrm{Dy}(+$), $\mathrm{Dy}(-$), and nonmagnetic Ga, respectively. The magnetic field ($h$) is applied in the $+ve$ direction. The snapshots are magnified in the middle to provide a detailed insight into cluster flipping.

Figure 9

(a) Distribution of magnetization jumps showing occurrence ($N$) as a function of $\mathrm{\Delta }m$ for disorder fraction $f_{\mathrm{Ga}}=0.2$ at system size $\mathcal{L}=32$. The data points correspond to an average of over 200 samples. (b) The power law distribution of $N\left(\mathrm{\Delta }m\right)$ vs $\mathrm{\Delta }m$ decays with exponent $\alpha =1\pm 0.05$. (c) Magnetization jump sizes $\left(\mathrm{\Delta }m\right)$ of a single hysteresis loop demonstrate jump sizes ranging from small to large. (d) Jump size is plotted as a function of the external field $h$.

Figure 10

Panels (a) and (b) show powder x-ray diffraction (PXRD) patterns of ${\mathrm{DyFe}}_3$ and ${\mathrm{DyFe}}_{2.5}{\mathrm{Ga}}_{0.5}$ samples, respectively. The insets show the perspective view of the crystal structures.

Figure 11

Isothermal field variation of magnetization at 5 K for the compound ${\mathrm{DyFe}}_{2.5}{\mathrm{Ga}}_{0.5}$. All five legs are highlighted.

Figure 12

Snapshots of the spin configurations $\mathrm{Fe}(+$), $\mathrm{Fe}(-$), $\mathrm{Dy}(+$), $\mathrm{Dy}(-$), and nonmagnetic Ga is shown with increasing external field $h$ for disorder fraction $f_{\mathrm{Ga}}=0.2$ at $\mathcal{L}=64$.

Figure 13

Probability distributions of magnetization jump size derived from the hysteresis loop at system size $\mathcal{L}=64$ with disorder fraction (a) $f_{\mathrm{Ga}}=0,$ and (b) 0.2, respectively. The intermittent bumps in the histogram plot are due to the lack of enough sample averages. In Fig. 9 we have calculated the standard deviation which shows a larger errorbar for intermittent bump.