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Although the magnetisation jumps were absent from the magnetisation stud- ies by Sakakibaraet al. [29] (figure 2.11) using a Faraday force magnetometer, they were observed in indirect measurements of the magnetisation by means of neutron diffraction experiments on Dy2Ti2O7 by Fennell et al. [14, 81].

We could solve this apparent inconsistency by measuring the dependence of the out-of-equilibrium dynamics of Dy2Ti2O7 on the magnetic field sweep

rate.

In figure 7.4 the magnetisation as a function of internal magnetic field at 200mK is shown for magnetic field sweep rates spanning a factor of about thirty. The curves in this figure can clearly be separated into two groups: those showing one or more jumps, with sweep rates v greater or equal to 0.025T/min, and those showing a continuous growth of the magnetisation, from 0.01T/min to 0.003T/min. This difference already reconciles the pre- vious experiments from Fennell et al. and Sakakibara et al., assigning the discrepancy to the magnetic field sweep rate dependence of the magnetisa- tion. Presumably the latter experiment was performed at a much slower field sweep rate than the former, and then missed the jumps. Also in the same figure, the orange curve is formed by the points reached after a series of field coolings at different fields.

In the group of curves that show a magnetisation jump, the height re- mains unaltered with a magnetisation value around 2.8µB/Dy. In the other

group, the slope of the continuous curve seems to be quite similar. The main difference within this last group comes from the behaviour of the curves at low fields (below 0.15T). Given that at a slower sweep rate the system has more time to relax, the curves increase linearly from zero field, with a larger slope for the slower sweep rates. Eventually, all slow curves tend to merge into a single curve at field above∼0.15T. This common curve has a similar slope the field coolings curve (orange curve).

Another feature that should be noted in the same figure is the abrupt change in slope close to 0.25T joining the slow sweep rate curve below this field with the Kagome plateau above it. This behaviour will be found to be more marked later in the discussion of the results obtained with the plastic magnetometer.

0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 M( �B /Dy) �₀(H-DM)(T) T=200mK 0.003T/min 0.006T/min 0.01T/min 0.025T/min 0.05T/min 0.1T/min Field Coolings

Figure 7.4: Low field [111]-magnetisation for different field sweep rates at a temperature of 200mK and after a zero field cooling. The orange curve is formed by the initial points reached after a series of field coolings run at different fields. This curve have a very similar slope to the slow sweep rate curves.

We can introduce now a theory for the mechanism of the low field mag- netisation based on the dynamics of the monopolar excitations introduced in section 2.3.4. If the magnetisation proceeds via field-driven motion of these excitations, the maximal rate at which the system can respond should be given by (dM/dt)max ∼ ρm10µB/ms, where ρm denotes the density of

thermally activated unbound monopoles that is independent of field for very weak fields and strongly suppressed at low temperature; and the use of the single spin flip time scale of 1ms is based on the work described in section 2.4. In this case, even the slowest experimental sweep rate in figure 7.4 re- quires processes whereM changes much faster than (dM/dt)max to maintain

equilibrium for T Tequil. As a result, the system enters a strongly out-of-

equilibrium regime where the magnetisation remains very small despite the presence of the applied magnetic field.

In this picture, even though the density of monopoles is low as in the case of the field in [100], the effect of a single monopole can be large because it can flip a string of spins increasing the magnetisation O(L) as is swept to the sample surface.

when they flip to align with the applied field, is quickly transported away by the vibrational degrees of freedom. In spin ice, presumably, there will be a range of parameters where the area density available for filament creation would be low enough for their Zeeman energy release to be absorbed by the rest of the system: we identified this with the slow-sweep regime. As we will argue below, in this picture, the fast-sweep regime corresponds to the situation where this is no longer the case, and a thermal runaway is induced. The increased rate at which Zeeman energy is dumped into the system as the field sweep rate increases overtaxes the ability of the lattice (the phonons) to equilibrate the local lattice temperature with that of the rest of the bath. As a result, the sample heats up locally, leading to the creation of more (and more easily unbound) monopoles. These in turn dump more energy as they move in the field direction and thermal runaway is ‘ignited’ above a critical sweep rate.

In this scenario, the slow sweep rate curve would be given by an energy landscape that is tilted by the applied magnetic field, similar to what happens to glasses (see figure 5.2). In order to agree with experiment, this landscape should be very shallow given that it is only seen at the lowest temperatures. This mechanism may also be helped by a localised heating produced by the spin flips which, though not being enough for triggering a cascade may influence the turning of spins in the close vicinity.