Individual Barkhausen pulses of ferroelastic nanodomains

Ferroelectric materials, upon electric field biasing, display polarization discontinuities known as Barkhausen jumps, a subclass of a more general phenomenon known as crackling noise. Herein, we follow at the nanoscale the motion of 90 degree needle domains induced by an electric field applied in the polarization direction of the prototypical ferroelectric BaTiO3, inside a transmission electron microscope. The nature of motion and periodicity of Barkhausen pulses leads to real-time visualization of distinctive interaction mechanisms of the domains with each other but without coming into contact, a mechanism that has not been observed before, or/and with the lattice where the domain walls appear to be moving through the dielectric medium relatively freely, experiencing weak Peierls-like potentials. Control over the kinetics of ferroelastic domain wall motion can lead to novel nanoelectronic devices pertinent to computing and storage applications.

Polarization switching in ferroelectric crystals is akin to first order phase transition induced by an applied electric field (1). The re-orientation of the domains during switching is a response to assume the equilibrium conditions that minimise the free energies of the domain and domain wall formation and growth. The motion towards the equilibrium state is hindered by potential barriers that lead to non-monotonous polarization change discontinuities referred to as Barkhausen effect (1,2,3,4,5,6). In ceramic (5) and single crystal BaTiO3 (3,7) the origins of this effect is placed on nucleation of spike-like domains, pinning of domain walls due to defects in the lattice (2,6), domain coalescence (8), and transition of the needle-like domains to parallelepiped domains when they fully cross the crystal in the direction along which the field is applied (7). Most electrical studies of Barkhausen jumps attempt to resolve single events from macroscopic properties of ferroelectrics (such as switching current or charge) while in optical observations the temporary position and motion of domain walls is probed. The various methods may reveal different origins of the pulses. This is associated with varying measurement timescales, different boundary conditions of the ferroelectric and physical dimensions of the material. At nanoscale dimensions traditional electrical measurements become exponentially more difficult due to miniscule charges associated with individual switching while optical measurements are diffraction limited.
Kinetically-controlled mechanisms of nanodomain switching and the way their interaction affects their motion when the dimensions become small have not been reported yet.
In the following, we stabilize the domain structure of a thin single crystal BaTiO3 to fully contain 90º ferroelastic nanodomains and we track their growth by applying a well-oriented external electric field. We locally probe the Barkhausen jumps for different potential barriers and at different electric field frequencies.
To achieve this, we prepared lamellae specimen along the pseudocubic [001]PC orientation from a single crystal BaTiO3 and placed it on a microelectromechanical system (MEMS) chip patterned with six electrodes; four used for temperature control and two for electric field biasing. The geometry of the device (Fig. S1 and Materials and Methods in supplementary materials) permits application of homogeneous electrical field across the probed sample area (9). First, the system was heated from room temperature over the ferroelectric-to-paraelectric transition temperature (TC) before cooling at 130 ºC for the biasing experiments (the full heating profile and image series is shown in Supplementary Movie S1). Sequential bright field (BF) scanning transmission microscopy (STEM) images of the temperature-induced process are depicted in Fig.   1, A to D. The TC was determined by the appearance and disappearance of the domains and it was measured to be 148 ºC on heating and 137 ºC on cooling. A systematic overestimation of the measured temperature of up to 18 °C (usually TC ~ 131 ºC from macroscopic measurements (10)) is attributed to error in the local temperature control of these devices. It is noted that heating above Tc was required to reduce the strain, which inevitably occurs due to the FIB preparation procedure (11). At room temperature, the specimen is dominated by unusually stable 180º walls (12), Fig.   1A, whereas during heating, a transition to the 90º domain structure at T ~ 50 ºC (13) is induced, Fig.1B. We note that this behaviour is not associated with the strain from clamping the lamella from both sides on the MEMS chips, as the same process is seen in a free-standing lamella as shown in Fig. S2. 130 °C, differential phase contrast (DPC) imaging (14) was employed. The DPC signal strength and direction is directly proportional to the magnitude and direction of the local polarization (14). is domain width and is slab's thickness (15). The same proportionality stays true for the ferroelastic domain structure (16,17,18,19). Barkhausen jumps is associated to both electric and mechanical compatibility (22,23,24). Two possibilities can be distinguished, the first one, where the needle domain's vertex terminates next to the crossing needle domain's body, and the second, in which equilibrium is achieved if two or more strongly charged needle domain vertices come in close contact (22). The observed Barkhausen jumps occur in between those equilibria. We note that experiments for an increasing maximum range of the applied bias were additionally performed and similar domain behaviour was observed (Supplementary Movies S3 to S8). although at around 3000 -3300 s negative potential is slowly returned to zero volts, the domain annihilates completely (marked with the dashed green arrow in the Fig. 3A). In short, the domain experiences larger applied electric field and stays intact, but on the decreasing field it slowly decreases in size and disappears, which again shows a relaxation event with a time constant of several minutes. The observed sluggish relaxation processes can be attributed to dielectric viscosity (2). According to observations from electrical measurements, the domain structure can relax up to several hours after poling. We may therefore witness here an individual event responsible for ageing and creep in ferroelectric materials (25,26,27).
When the lower frequency domain motion is compared to higher frequency ones, we notice a timedelay of the switching process when the potential is brought back to zero. During ultra-low frequency measurements, the domain remains at the same position for a significant time before it eventually switches.
By numerically differentiating the domain length vs. time data (Fig. 3A), the speed of the needle domains can be determined. Figure 3B shows the velocity as a function of applied voltage for the three distinct domains. Essentially, the speed is proportional to the switching current (i.e. ~∆ domain-domain interaction one, it is determined that the domain tip velocity for the defectmediated domain is more smeared out (i.e. the vertex is almost always moving). In contrast, the velocity of domain-domain interaction tip is accompanied with spikes and, at most times, the velocity is close to zero. This indicates that the potential well is deeper for domain-domain associated pinning as compared to pinning on the defects within the lattice (28,29). Additionally, in the case of the low frequency measurement (green line), the velocity spikes can be seen in the deep negative potential part representing Barkhausen jumps due to domain-domain interaction and slow relaxation processes simultaneously. Overall, the observed evolution and motion of ferroelectric single needle domains under application of electric field in the polar direction is characteristic of a forward domain growth process (30). However, we have previously shown that such movements follow Rayleigh-like square-law behaviour (9) and, therefore, pinning mechanisms typically studied for lateral domain wall movements are also applicable in this high field regime. To further assess the mechanisms of pinning of 90º needle nanodomain wall motion induced by external electric fields, we discuss their dynamic behaviour on the basis of the schematics in Fig. 4. It was recently shown that the equilibrium positions of 90º needle domains in BaTiO3 are adopted due to the redistribution of strain and electric displacements fields on their tips (22). Our results show that, in the case of a herringbone domain pattern, this effect causes separation of crossing domains that terminate at a certain distance from the perpendicular crossing domain without ever coming into contact (Fig.   4A). Moreover, the depolarizing and strain fields caused by the abrupt changes in the spontaneous polarization at the needle tip form a large potential barrier that manifests as well-defined Barkhausen jumps. The positions of these jumps are associated with the annihilation of the periodic, perpendicular domains, and the domain-domain induced pinning motion leads to hysteretic behaviour. A different mechanism is encountered when parallel needle domains are free to move within the lattice structure (Fig. 4B). In this case, the Barkhausen pulses are rare events, and are most likely dominated by Peierls potential barriers due to lattice potential and point defects in the non-perfect crystal (28,29). The associated shallow potential barrier does not affect perpendicular domain-domain motion but is dominating the jump frequency during the motion of single domains across the lattice. Therefore, at the moderate electric field and frequencies used herein, classical Barkhausen jumps mostly originate from domain-domain interactions and much less from interaction of domains with point defects.

Materials and Methods
Sample preparation Single crystal BaTiO3 (MTI Corporation) was focused ion beam prepared and it was transferred to a 4 heating and 2 biasing electrode microelectromechanical (MEMS) chip provided by DENSsolutions. The Pt contacts were ion beam deposited and the lamella was thinned to a final thickness of 268 nm. A detailed description on the device fabrication can be found in Ref. [9]. The average temperature on the heating element of the MEMS chip is calculated from its electrical resistance and DENSsolutions software was used to control the temperature. Electrical bias to the sample was applied using a Keithley SMU-2450 sourcemeter. Transmission electron microscopy Imaging was performed on a double spherical aberration (Cs) corrected ThermoFischer Scientific Themis 60-300 operated at 300 kV in STEM mode using 100 pA beam current, 70 µm C2 aperture, and a beam convergence angle of 28 mrad. Serial imaging during standard biasing experiments was done in bright field (BF) mode with a collection angle of 79 mrad. The pixel size was 3.2 nm and 2.98 s was the frame time. For the low frequency biasing experiment, the pixel size was 1.1 nm and the frame time was 8.22 s. DPC imaging was performed using a four segment detector and the imaging conditions were 200 pm pixel size and 34.3 s frame time. The camera length was 91.1 mm. Beam centering and gain/offset equilibration of the segments were done in the vacuum region prior to the experiment. Image processing Images of the standard biasing experiment were cropped to 764x764 pixel size. Images of the low frequency field measurement were not cropped and remained at the 2048x2048 pixel size. To enhance domain wall contrast in the BF images, postprocessing using ImageJ software was performed. It involved Fourier filtering of the horizontal scan noise of the images, subtraction of background (50 pixel rolling ball radius with sliding paraboloid enabled) followed by contrast and brightness adjustment.
DPC experimental signal processing involved the same 1024x1024 pixel region from the original set of 4 images (corresponding to the segments A, B, C, and D). The cropped region was further downsized with ImageJ to 256x256 pixel size essentially averaging out the noise. Further data processing was undertaken using a custom script written in Mathematica v.11.2. A 256x256 array was created, where each element represented a vector in the xy plane corresponding to the beam deflection. Beam deflection along the x axis is proportional to the differential signal D-B and correspondingly along the y axis it is proportional to C-A. Each element was corrected for the detector rotation with respect to the scan direction (+17°). Using this array, the vector field plots were created (Fig. 1e-f). The vector field plot consists of 32x32 arrows, which are proportional to the direction and magnitude of the polarization.  At 75 °C, the sample is mostly dominated by 90° domain walls, nevertheless some character from zig-zag domain walls is still seen. (C) Domain structure at 100 °C where the specimen is dominated by 90° domain walls. The appearance of zig-zag wall vertices seems to be associated with crossing 90° domain walls. Scale bar is 250 nm. The experiment was performed on ThermoFisher Talos F200S microscope operated at 200 kV in STEM mode and imaged with the high angle annular dark field (HAADF) detector. The sample was mounted on a standard grid and a Gatan single tilt heating holder was used for heating the sample.   Fig. 2 due to the measured domain (green false color) nucleating with antiparallel polarization compared to the ones in Fig. 2. The dark spots in the BF-STEM images are surface contamination, which grew upon lengthy in situ biasing and heating experiments. These spots seem to be located at the surface inactive layers and hence do not affect the domain motion.