Thermal and electric field stability of square-net domain patterns in barium titanate

We use polarized light microscopy in situ with an externally applied electric field to induce square-net birefringence patterns in top-seeded single crystals of barium titanate above the Curie temperature ($T_C$). The patterns occur under a wide range of electric field magnitudes larger than $\sim 0.2$ kV/mm and at temperatures up to $6{}^{\circ }\mathrm{C}$ above $T_C$. We mapped this behavior on a pseudophase diagram showing the region in electric field and temperature space where this domain configuration is stable. We observed the electric field induced transformation into the periodically ordered structure from both the cubic structure above $T_C$ and the tetragonal structure below $T_C$. Synchrotron x-ray reciprocal space maps indicate the presence of ferroelastic domains perpendicular to the electric field direction at a range of electric field magnitudes similar to those required to induce the periodic domain ordering we observe using polarized light microscopy. Combined with a simple model of the optical retardation, we show that the periodic domain ordering responsible for the square-net birefringence occurs only at the surfaces of the crystals and that these domain structures are unexpectedly invariant with respect to the electric field that induces them.

Figure 1

(a) $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_3$ observed at 0.4 kV/mm and at a sample holder temperature of $142.7{}^{\circ }\mathrm{C}$. (i) Close to the electrode the sample is cool and domains are polarized parallel to the surface. (ii) Square-net pattern is observed in an area between the electrode and sample holder. (iii) Domains polarized parallel to the electric field are observed at high temperature near the boundary to the sample holder.

Figure 2

Method for tracking square-net patterns and calibrating the temperature scale. (1) An image from the video is loaded. For each image the sample holder temperature and applied electric field has been recorded. (2) Squares on the loaded image are detected. (3) Using the temperature map constructed from the phase transition temperature a temperature is assigned to each detected square. (4) Temperature and electric field values for the square-net pattern are recorded on the phase diagram.

Figure 3

Pseudophase diagram for the field induced phase transition. Black points marked the observed transitions between “phases.” Periodic domain ordering is observed between the cubic to tetragonal phase transition at sample holder temperatures from 138 to $145{}^{\circ }\mathrm{C}$. Below this temperature region the sample remains in the tetragonal phase which at high fields align the polarization with the field direction. Above $145{}^{\circ }\mathrm{C}$ the cubic state transits into a tetragonal domains polarized along the electric field direction for high electric field magnitudes.

Figure 4

Results from reciprocal space maps (RSMs) taken at varying electric fields just above $T_C$. Above: $d$ spacing of the present [1 0 0] planes for field strengths between 0 and 0.8 kV/mm. Below: Amount of the sample in the respective phases.

Figure 5

(a)–(c) Simulations of the birefringence patterns at parallel to perpendicular oriented domain width ratios of 1, 1.5, and 3. (d) and (e) Images of square-net pattern at 300 and at 800 V/m. We note that (d) and (e) are at the same position of the crystal. (f) Possible domain structure responsible for the square-net pattern. Blue regions are slab structures as described in [9]. White region corresponds to bulk domains with net polarization along the electric field direction.