Dynamic behavior of magnetic avalanches in the spin-ice compound ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$

Avalanches of the magnetization, that is to say an abrupt reversal of the magnetization at a given field, have been previously reported in the spin-ice compound ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. This out-of-equilibrium process, induced by magnetothermal heating, is quite usual in low-temperature magnetization studies. A key point is to determine the physical origin of the avalanche process. In particular, in spin-ice compounds, the origin of the avalanches might be related to the monopole physics inherent to the system. We have performed a detailed study of the avalanche phenomena in three single crystals, with the field oriented along the [111] direction, perpendicular to [111], and along the [100] directions. We have measured the changing magnetization during the avalanches and conclude that avalanches in spin ice are quite slow compared to the avalanches reported in other systems such as molecular magnets. Our measurements show that the avalanches trigger after a delay of about 500 ms and that the reversal of the magnetization then occurs in a few hundreds of milliseconds. These features suggest an unusual propagation of the reversal, which might be due to the monopole motion. The avalanche fields seem to be reproducible in a given direction for different samples, but they strongly depend on the initial state of magnetization and on how the initial state was achieved.

Figure 1

Magnetization $M$ vs the applied field $H$ along the [001] direction for sample 1, starting from different field-cooled states [from the bottom to the top: $H_{\mathrm{FC}}=-1400$ (green) to +1000 Oe (black) in steps of 400 Oe]. The field sweeping rate was 35 $\mathrm{Oe}{\mathrm{s}}^{-1}$ ($=0.21$ $\mathrm{T}{\min }^{-1}$) and the starting temperature before the avalanche was 75 mK. Also shown is the isothermal $M$ vs $H$ measured at 900 mK (red circles). The avalanches are clearly seen as a sudden increase in magnetization at certain critical applied fields. An example of the definition of $M_{\mathrm{start}}$ and $H_{\mathrm{aval}}$ is shown for the 600-Oe field-cooled curve.

Figure 2

Magnetization $M$ vs the applied field $H$ along the [111] direction for sample 2, starting from five different field-cooled states $H_{\mathrm{FC}}=-800$, $-400$, 0, 400, and 800 Oe (from the bottom brown, green, black, blue, and red respectively). The field sweeping rate is 35 $\mathrm{Oe}{\mathrm{s}}^{-1}$ and the starting temperature before the avalanche was 75 mK. Also shown is the isothermal $M$ vs $H$ measured at 670 (green circles) and 900 mK (red circles). The inset is a closeup showing the end of an avalanche for the $-400$ Oe field-cooled curve and shows that the magnetization increases very rapidly until it reaches its 900 mK equilibrium value, then it continues to increase but at a much slower rate, until falling out of equilibrium and becoming frozen when the sample temperature dips below approximately 600 mK.

Figure 3

Internal avalanche field $H_{\mathrm{i}\mathrm{aval}}$ as a function of the initial magnetization $M_{\mathrm{start}}$ for sample 1 with the field along the [001] direction (green) and along [111] (blue), for sample 2 and 2* along the [111] direction (orange and black respectively) and sample 2 perpendicular to [111] (red), and for sample 3 measured with the field along the [111] direction (light blue).

Figure 4

Magnetization as a function of the applied field along the [111] direction at 82 mK for sample 2 at the various indicated field ramp rates, starting from a zero-field-cooled (ZFC) state. The field at which the avalanche occurs is nearly independent of the ramp rate. Inset: Closer inspection of the approach to the avalanche field.

Figure 5

$M$ (blue circles) and $T_{\mathrm{thermo}}$ (red dotted line) vs time $t$ during an avalanche for sample 1 with $H\parallel$ [111], $H_{\mathrm{FC}}=0$ Oe at 76 mK. The corresponding applied field $H$ is indicated on the top axis. $t=0$ matches with the starting of the field ramp.

Figure 6

Magnetization $M$ vs the applied field $H$ along the [111] direction for sample 2, where the initial starting magnetization $M_{\mathrm{start}}=0$ has been arrived at by using two different methods. There are actually four curves, two for each method. For the green and red curves $M_{\mathrm{start}}=0$ was attained by the conventional ZFC from 900 mK. For the blue and black curves the $M_{\mathrm{start}}=0$ was attained using the avalanche quenched protocol [39].

Figure 7

Magnetization $M$ vs the applied field $H$ along the [111] direction for sample 2*, (ellipsoid) starting at $+3900$ Oe for three different field ramping rates. The sample was first avalanche quenched from 0 to $+3900$ Oe, and then allowed to cool so that the starting temperature before ramping was below 110 mK. Also shown (red) is the equilibrium $M\left(H\right)$ vs $H$ measured at 900 mK.

Figure 8

Relaxation of the magnetization $M$ vs time $t$, in a semilogarithmic scale, after ZFC, in an applied field of 400 Oe, at temperatures between 300 and 650 mK for sample 2. The field was along the [111] direction. Over this temperature range and for fields below 600 Oe the relaxation is more or less normal, in the sense that it is roughly exponential, and there was no self-heating of the sample, and no avalanche.

Figure 9

Magnetization $M$ vs time $t$ in a semilogarithmic scale, after ZFC, in an applied field of 1200 Oe for sample 2.

Figure 10

Bottom: $dM/dt$ vs time $t$ in a semilogarithmic scale, after ZFC, in an applied field of 1200 Oe for the data shown in Fig. 9. The inset shows a closeup of the low-temperature data showing a small peak for the 350-mK data, and below this temperature, no other peaks could be discerned. Top: The temperature during the relaxation measured on a thermometer 20 cm above the sample. The peaks indicate self-heating of the sample during rapid magnetization changes.

Figure 11

Magnetization $M$ vs time $t$ in a semilogarithmic scale, after ZFC, in an applied field of 1700 Oe, at temperatures between 75 and 210 mK for sample 2.

Figure 12

$M/H$ vs time $t$ in a semilogarithmic scale, after ZFC, at 500 mK, for fields $H$ between 50 and 1200 Oe, for sample 3.

Figure 13

The effective relaxation time $\tau$ vs temperature $T$ in a semilogarithmic scale for applied fields between 10 and 1400 Oe, for sample 2. The lines are just guides for the eyes. The green shaded region shows the zone in which the sample is self-heating, resulting in a reduction of the relaxation time. Inset: Relaxation curves $M$ vs $t$ at 800 Oe showing the way of defining $\tau$ (see text).

Figure 14

$dM/dt$ vs time $t$ data shown in Fig. 10 plotted on a linear time scale. The upward curvature of the magnetization as a function of time during an avalanche is clearly seen. The lines are fits to the 375- and 350-mK data.