Low temperature: magnetisation jumps

As the temperature is decreased below the freezing temperature, the mag- netisation curves depart from the H/T scaling. This is shown in the inset of figure 7.3 where the magnetisation is plotted as a function of internal field. Indeed, at the lower fields the curves at 700mK, 600mK and 500mK, increase their magnetisation at a much slower pace than for a cooperative paramagnet. The departure from the H/T scaling marks the onset of out- of-equilibrium dynamics and the start of a dramatic change of the behaviour of the system. Near zero field the magnetisation increases almost linearly with a tiny slope that depends on the temperature, with higher slope for higher temperature. After that, a temperature dependent field is reached where the magnetisation jumps abruptly to a value close to that at 700mK. These jumps are very sudden and can hardly be resolved by our instrument recording 1 point every ∼2s.

After the jump, the magnetisation remains almost constant, evolving into a quasi-plateau of magnetisation value ranging from 2.4µB/Dy at 400mK to

2.8 µB/Dy at 100mK. Note that the jumps are huge, given that the change

in magnetisation accounts for about half of the magnetisation at saturation for this field direction. As the field is increased, the transition from the quasi-plateau to the Kagome plateau happens in either a smooth fashion

0 1 2 3 0.1 0.3 0.5 0.7 M( � B /Dy) �₀H(T) 100mK 200mK 300mK 400mK 500mK 600mK 700mK 0 1.5 3.3 0 0.2 0.4 0.6 M( �B /Dy) �₀(H-DM)/T (T/K) 0 1.5 3.3 0 0.2 0.4 0.6 M( B /Dy) 0(H-DM)/T (T/K)

Figure 7.3: Low field magnetisation for [111] field sweeps between zero field and the Kagome plateau for different temperatures ranging between 100−700mK. The curves were prepared by a zero field cooling from above 700mK, and they were measured at a sweep rate of 0.1T/min. The curves fall clearly out-of-equilibrium with a sub-Curie susceptibility. For the lowest temperatures, the magnetisation jumps abruptly. Inset: Hi/T cooperative

paramagnetic scaling. The jumps for the lower temperatures have negative slope as a function of the internal field indicating that they are triggered events.

or through a second jump depending on the temperature. As a general feature, both the intermediate plateaux and the jumps become sharper with decreasing temperature.

In the inset of figure 7.3 one can notice that the jumps present a negative slope in the Hi/T scaling, indicating that they are triggered events. In a

triggered event there is an ignition threshold (an energy barrier) above which the material starts a process that cannot be stopped even if the triggering parameter at a later moment is below that threshold. In the case of the magnetisation jump, triggering occurs at an internal field Hj which, in a

simple picture1_{, can be associated with a given energy µH}

j. Even though

the internal field during and after the jump becomes lower than the triggering field (manifested by a negative slope) the magnetisation continues increasing

1_{Strictly, the free energy should be counted, but because of the clear out-of-equilibrium}

after the jump.

The triggering in Dy2Ti2O7 is similar to the ignition of a piece of paper

where an initial fire has to be set in order to start the burning process. Once the first fire is set, the fire front propagates fast, giving no time to the paper to diffuse the heat away. Furthermore, the heat coming from the fire feedbacks positively into the burning process by increasing the probability of thermally overcome energy barriers due to the high temperature. This process is called deflagration and in the case of magnetism was observed and characterised in a series of publications on specially prepared molecular magnets [75, 76, 77, 78, 79] for which a theory was proposed by Garanin and Chudnovsky [80]. The details of the deflagration process in Dy2Ti2O7 will

be analysed in chapter 8.

Now, we want to expand on the experimental quantities extracted from the jumps of figure 7.2. These jumps are characterised by the field at which they occur and by the magnetisation value attained on the plateau immedi- ately after the jump, two values that are strictly reproduced over runs. The jump field is temperature dependent, happening at increasingly higher fields for lower temperatures: (400mK, 0.1T), (300mK, 0.13T), (200mK, 0.15T) and (100mK, 0.16T). The tendency is to a unique real triggering field at zero temperature, and presumably the different triggering fields are only due to the thermal contribution to the threshold energy for different temperatures. Focusing in the microscopic origin of the jumps, we can assume that the flip of the apical spins precedes that of the basal spins given that the spin projection over the applied field is three times larger for the first, and knowing that at the Kagome plateau all apical spins are collinear with the magnetic field direction. At 100mK and at the threshold field for that temperature, the change in magnetic energy due to an apical spin flip from down to up is 2µHj ∼2.2K. Taking into account the fact that the energy to break the ice

rule by flipping an apical spin from a spin ice state is 4Jef f = 4.4K (2.2K per

tetrahedron) in the nearest neighbour model, we see that the energy scale for the jump is consistent with a deconfined defect picture where breaking the rules in only one tetrahedron is allowed at a cost of 2Jef f = 2.2K. As an

activated process, this can only happen either at the boundaries or at defects in the bulk.

This is not the first appearance of this energy scale. In reference [50] the relaxation time measured in the susceptibility experiments discussed in section 2.4 was fitted with an exponential decay (Ahrrenius law) indicative of an activated process. The best fit in the region 2.5−5K was obtained for the characteristic energy 2Jef f = 2.2K, corresponding to the energy of

single topological defects. In that case, it was argued that this energy scale came from the fractionalization of defects giving half of the energy to each

of the topological defects created by a single spin flip and by the strong screening due to the proliferation of monopoles. In this case, however, we believe that screening cannot be invoked given that the monopole density at these temperatures is extremely low.