scaling exponents in our switching studies when all nucleation, motion and merging events were included in the dataset and no cutoffs in the tip bias were applied. The resulting event size distributions and corresponding power law fits are shown in figure5.18. We find that the exponents are lower when no bias cutoffs are applied, reflecting the overall larger event size occurring at tip biases higher than the upper cutoff values. This comparison highlights the importance of the consideration of different dynamics regimes, and could potentially be used as a means to distinguish between them.
Figure 5.18: (a) Event size distributions and power law fit for PZT-Nuc Vtip>
0, Vtip<0 and PZT-Mot, Vtip>0 in the creep regime. The power law fitting on PZT-Nuc, Vtip<0 was limited to events triggered by tip bias within the same bias window as for PZT-Nuc, Vtip >0. The measurement series are shifted for clarity.
(b) Fitted exponents for box widths of 60-310 nm, clustering aroundτ = 2.24. The histogram of fitted exponents is displayed on the right.
5.7 Constant driving force
To compare avalanche statistics in the creep and depinning regimes, we next focused on switching in PZT-Nuc under a scanning tip biased at two different constant voltages of -3.5 V and -5.0 V. Figure5.19shows the resulting maps of the local time at which the polarisation at individual pixels reversed in units of the switching scan number. In the -3.5 V case, we observe only very slow switching dynamics. In particular, the pre-existing written domain walls appear to be strongly pinned in their initial positions, with only small displacements occurring in the first few switching scans where the domain walls are exploring more favourable configurations of the potential landscape in their immediate proximity, as shown in figure5.19(a). We also observe very few nucleation events, occurring stochastically throughout the measurement series. The newly nucleated domains expand slowly under the applied tip bias as can be seen from close ups of some of the domains in figure5.19(b). The stochastic nucleation of new domains throughout the measurement series, their ultraslow outward growth with time and the rapidly saturating growth
Figure 5.19:Maps of switching time in measurements on written domain structures on PZT-Nuc at (a, b) -3.5 V and (c) -5.0 V. (b) shows close-ups of nucleated domains illustrating their slow growth. Time is expressed in units of the switching scan number, where each scan takes∼15 minutes.
of the initially written domain walls qualitatively suggest that the growth is occurring in the creep regime2.
At a higher tip bias of -5V, the polarisation reversal is much more rapid and polarisation reversal events are much larger, although some down-oriented domains remain by the end of the measurement as seen by the white areas in the middle of figure5.19(c).
To study differences in the growth of nucleated domains at both tip biases, the size of individual domains expressed as their equivalent disc
2It is tempting to compare the values of the tip bias in the constant bias measurements shown here and in the measurements shown earlier where the tip bias is increased incrementally. However, these measurements were performed with separate tips which can lead to changes in the effective field applied to the sample. Moreover, the domain writing process can affect the tip and adsorbates at the surface of the sample can accumulate at the tip, further affecting the effective tip field significantly. Direct comparisons of switching dynamics at a given voltage between measurement series are therefore difficult, though within a given measurement series, the effective tip field usually does not appear to change significantly.
5.7 Constant driving force
Figure 5.20: Equivalent disc radii of nucleated domains and corresponding event maps where the switching scans are all performed at the same DC tip bias. (a) At a tip bias of -3.5 V, the domains grow logarithmically with time, consistent with creep motion. (b) At a tip bias of -5.0 V, the radius of nucleated domains grows linearly with time, indicative of a depinned domain wall motion. The inset shows the corresponding map of event types, where events in red, blue and green correspond to nucleation, motion and merging of domains respectively
radius are extracted and shown in figure5.20. In the Vtip=−3.5 V case, the domain wall radius increases logarithmically with time, while in the Vtip
= -5.0 V case, the radii of nucleated domains expand linearly with time.
We believe the logarithmic domain growth observed at low tip bias can be seen as a marker of ultraslow dynamics. As the tip bias is applied, the domain wall initially explores favourable surrounding configurations, allowing larger switching events to occur. However, when the tip bias is low compared to the characteristic barrier height, the switching event size rapidly decreases, and this more rapid growth is followed by a slower expansion due to thermal activation. The effects of individual events are also clearly seen in the dynamics. At high tip bias, where we can presume the applied field strength exceeds the characteristic barrier height, the growing domains expand at a constant rate into the surrounding region, and the effects of individual switching events cannot be distinguished in the dynamics.
We note here that past studies of ferromagnetic domain walls in the framework of disordered elastic systems ([98,164]) under a constant uniform field showed a constant, field-dependent velocity, and thus a linear expansion of the domain with time, in all the dynamic regimes of creep, depinning, and flow. However, these measurements were carried out by magneto-optic Kerr microscopy with optical resolution over very large segments of domain wall, and over relatively long times. In studies of ferroelectric domain walls, in contrast, logarithmic domain growth with time was observed under a static tip [1,101,165], but in this case related to the highly spatially inhomogeneous electric field. It is possible that in the scanning biased tip configuration, a long enough series of measurements, and significantly larger scan areas could allow a more uniform average domain wall dynamics to be observed even in the creep regime, as in the ferromagnetic studies. Such measurements are
challenging, however, as they would require a prohibitively large number of PFM scans, leading to significant wear of the AFM tip.
From the distribution of event sizes at -5.0 V, we obtain an exponent of τ = 2.10±0.05 over 2.5 decades of event sizes in the depinning regime. In the creep regime however, we find much higher exponent values ofτ = 3.36±0.16, emphasising the role of very small switching events, although we note that the fitting is more difficult, and our confidence in the exponent value is far lower in this case. The size distributions and corresponding power law fits are shown in figure 5.21.
Figure 5.21:Probability distributions of polarisation reversal event sizes at -3.5 V (a) and -5.0 V (b) and their corresponding power law fits. In the former, switching events are predominantly very small and their distribution is confined to low values, making the fit unreliable.