Phase coexistence and Landau expansion parameters for a $0.70\mathrm{Pb}\left(\mathrm{M}{\mathrm{g}}_{1/3}\mathrm{N}{\mathrm{b}}_{2/3}\right){\mathrm{O}}_3\text{−}0.30\mathrm{PbTi}{\mathrm{O}}_3$ single crystal

Multidomain relaxor-based ferroelectric single crystals $\left(1\text{−}x\right)\mathrm{Pb}\left(\mathrm{M}{\mathrm{g}}_{1/3}\mathrm{N}{\mathrm{b}}_{2/3}\right){\mathrm{O}}_3\text{−}x\mathrm{PbTi}{\mathrm{O}}_3$ (PMN-PT) have extraordinarily large electromechanical properties, but the origin of their giant piezoelectric properties is still not well understood. The Landau-like phenomenological theory is a feasible tool to study domain structures and their correlation with piezoelectric effects, but so far no expansion coefficients have been measured due to the phase mixture complication involved. In this work, the Landau free-energy expansion parameters for $0.70\mathrm{Pb}\left(\mathrm{M}{\mathrm{g}}_{1/3}\mathrm{N}{\mathrm{b}}_{2/3}\right){\mathrm{O}}_3\text{−}0.30\mathrm{PbTi}{\mathrm{O}}_3$ (PMN-0.30PT) single crystal were determined from the temperature-dependent polarization-electric field $\left(P\text{−}E\right)$ hysteresis loops along ${\left[001\right]}_{\mathrm{C}}$ and ${\left[011\right]}_{\mathrm{C}}$ directions, and the rhombohedral $\left(R\right)$ to tetragonal $\left(T\right)$ phase-transition temperature. Using the obtained parameters, ferroelectric and dielectric properties were quantitatively calculated and compared with experiments. Good agreement was achieved in the temperature regions of $T$ and $R$ phases, but deviations were found in the cubic-phase temperature region since the contribution of polar nanoregions was not considered. In the phase-coexistence region from 73 to $93{}^{\circ }\mathrm{C}$, the polarization and dielectric constant can be quantitatively explained with the volume fractions of the coexisting $R$ and $T$ phases predicted by the canonical distribution. These obtained parameters can help theoretical studies and simulations of these relaxor-based ferroelectric single crystals to reveal the correlation between domain microstructures and the origin of giant electromechanical properties of multidomain PMN-PT single crystals.

Figure 1

Phase diagram of PMN-$x\mathrm{PT}$ system given by Noheda et al. [8].

Figure 2

Temperature-dependent polarization and dielectric constant at different frequencies. Inset figures are hysteresis loops along ${\left[001\right]}_{\mathrm{C}}$ at temperatures of 64, 75, and $100{}^{\circ }\mathrm{C}$.

Figure 3

Theoretical fittings of ${\left[001\right]}_{\mathrm{C}}$-oriented PMN-0.30PT hysteresis loops at temperatures of (a) $132{}^{\circ }\mathrm{C}$, (b) $138{}^{\circ }\mathrm{C}$, and (c) $140{}^{\circ }\mathrm{C}$. Single-domain state at $132{}^{\circ }\mathrm{C}$ was verified by using the polarizing light microscopy as inset in (a).

Figure 4

Hysteresis loops of ${\left[011\right]}_{\mathrm{C}}$-oriented PMN-0.30PT at (a) $128{}^{\circ }\mathrm{C}$, (b) $132{}^{\circ }\mathrm{C}$, and (c) $137{}^{\circ }\mathrm{C}$. Single-domain state at $137{}^{\circ }\mathrm{C}$ was verified by using the polarizing light microscopy as inset in (a).

Figure 5

Experimental and theoretical calculated strains for PMN-0.30PT and PMN-0.31PT single crystals.

Figure 6

(a) Schematic of arbitrary spatial polarization as a function of angles $\theta$ and $\phi$, (b) energy surface of coexisting $R$ and $T$ phases, and (c) the angular variation of free energy of $R$ and $T$ phase at 75, 85, and $93{}^{\circ }\mathrm{C}$.

Figure 7

Distribution fractions for tetragonal and rhombohedral phases from 73 to $125{}^{\circ }\mathrm{C}$ with volumes containing $n$ unit cells (u.c.) and $n=20000$, 35 000, and 50 000.

Figure 8

Comparison between theoretical calculations and experimental results. (a) Dielectric constant ${\varepsilon }_{33}$ and (b) spontaneous polarization $P_3$.

Figure 9

$P\text{−}E$ hysteresis loop at $132{}^{\circ }\mathrm{C}$. Inset is the illustration of the sample with crossed analyzer/polarizer pairs and the applied electric field direction.

Figure 10

Electric-field-dependent phase change at high temperature of $132{}^{\circ }\mathrm{C}$ with a dc electric field applied along ${\left[001\right]}_{\mathrm{C}}$ observed at the polarizer angle of 0 and 45°, respectively.

Figure 11

$P\text{−}E$ hysteresis loop at $128{}^{\circ }\mathrm{C}$. Left inset is the illustration of the sample #2 with crossed analyzer/polarizer pairs and the applied electric field direction. Right inset is the polarization state from $6T$ state to $1O$ state.

Figure 12

Electric-field-dependent phase change at high temperature of $128{}^{\circ }\mathrm{C}$ with a dc electric field applied along the ${\left[001\right]}_{\mathrm{C}}$ direction observed at polarizer angle of 45 and 0°, respectively.

Figure 13

$P\text{−}E$ hysteresis loop at $137{}^{\circ }\mathrm{C}$.

Figure 14

Electric field dependence of domain structures at $137{}^{\circ }\mathrm{C}$ with a dc electric field applied along ${\left[011\right]}_{\mathrm{C}}$ observed at 45 and $0^{\circ }$, respectively.