Temperature-dependent elasticity of $\mathrm{Pb}\left[{\left(\mathrm{M}{\mathrm{g}}_{0.33}\mathrm{N}{\mathrm{b}}_{0.67}\right)}_{1\text{−}x}\mathrm{T}{\mathrm{i}}_x\right]{\mathrm{O}}_3$

Relaxor ferroelectric materials, such as $\mathrm{Pb}\left[{\left(\mathrm{M}{\mathrm{g}}_{0.33}\mathrm{N}{\mathrm{b}}_{0.67}\right)}_{1\text{−}x}\mathrm{T}{\mathrm{i}}_x\right]{\mathrm{O}}_3$ (PMN-PT) with generic stoichiometry, undergo a ferroelectric-to-paraelectric phase transition as a function of temperature. The exact transition characterized by Curie temperature ($T_c$) varies as a function of chemistry ($x$), i.e., the concentration of Ti. In this study, we investigated the structural phase transition by exploring the temperature dependence of the single-crystal elastic properties of $\mathrm{Pb}\left[{\left(\mathrm{M}{\mathrm{g}}_{0.33}\mathrm{N}{\mathrm{b}}_{0.67}\right)}_{0.7}\mathrm{T}{\mathrm{i}}_{0.3}\right]{\mathrm{O}}_3$, i.e., $x\approx 0.3$. We used resonant ultrasound spectroscopy to determine the elasticity at elevated temperatures, from which $T_c=398\pm 5\mathrm{K}$ for PMN-PT ($x\approx 0.3$) was determined. We report the full elastic constant tensor ($C_{ij}=\{C_{11},C_{12},C_{44}$}), acoustic attenuation ($Q^{-1}$), longitudinal ($V_P$) and shear ($V_S$) sound velocities, and elastic anisotropy of PMN-PT as a function of temperature for $400<T<871\mathrm{K}$. Temperature trends of the elastic constants $C_{11},C_{44}$ and bulk modulus indicate that at $T>T_c$ the material first stiffens and reaches maxima in the vicinity of the Burns temperature ($T_b\sim 673\mathrm{K}$), followed by a more typical gradual softening of the elastic constants. Similar temperature-dependent anomalies are also observed with anisotropy and $Q^{-1}$, with minima in the vicinity of $T_b$. We used the temperature dependence of $C_{ij}$, $Q^{-1}$, $V_P,V_S$, and anisotropy to infer the evolution of polar nanoregions as the material evolved from $T>T_c$.

Figure 1

(a) Temperature dependence of the first ten mechanical resonance frequencies. (b) Temperature evolution of MRS from 290 to 823 K. For the PMN-PT sample ($x\approx 0.3$) explored in this study, a distinct transition in the temperature dependence of MRS is observed at around $\sim 400\mathrm{K}$, i.e., ($T_c)$. Mode displacement plots [62] of the first five resonance modes of the PMN-PT sample are also shown and indicated by arrows with the corresponding modes in (b).

Figure 2

Temperature dependence of $C_{11}$, $C_{12}$, $C_{44}$, and acoustic energy loss ($Q_{\mathrm{av}}^{-1}$) of the same sample measured in two different temperature cycles. Also shown are the elastic constants for PMN $(x=0.0)$ [14, 58]. The two gray and white shaded regions shown in the figure indicates the ferroelectric-to-paraelectric transition with $T_c$ for PMN and for PMN-PT. The $T_c$ for PMN is $\sim 270\mathrm{K}$, whereas the $T_c$ for PMN is $\sim 398\mathrm{K}$.

Figure 3

(a) The plot shows the temperature dependence of the sound velocity anisotropy: $A{V}_P$, $A{V}_{S1}$, and $A{V}_{S2}$. We define the anisotropy $A{V}_P=\frac{200\times \left(V_{P\max }-V_{P\min }\right)}{\left(V_{P\max }+V_{P\min }\right)}$ and $A{V}_S=\frac{200\times \left(V_{S\max }-V_{S\min }\right)}{\left(V_{S\max }+V_{S\min }\right)}.$ The inset shows the inflection in $A{V}_P$ at the Burns temperature, $T_b$; (b)–(d) are the stereographic plots of the directional dependence of sound velocities: $V_p$, $V_{s1}$, and $V_{s2}$ at 412 K.

Figure 4

(a) Temperature-composition ($T-x$) phase diagram for PMN-PT with a stoichiometry $\mathrm{Pb}\left[{\left(\mathrm{M}{\mathrm{g}}_{0.33}\mathrm{N}{\mathrm{b}}_{0.67}\right)}_{1\text{−}x}\mathrm{T}{\mathrm{i}}_x\right]{\mathrm{O}}_3$. The phase diagram is subdivided based on the space-group symmetry of the PMN-PT crystals. “$C$” refers to the cubic space-group symmetry ($Pm\overline{3}m$), “$R$” refers to rhombohedral space-group symmetry (R3m), “$T$” refers to tetragonal space group symmetry (P4mm), and “MPB” refers to the morphotropic phase boundary, which is attributed to monoclinic phases ($Bm$ and $Pm$) [13]. The region “PC” refers to pseudocubic, where $a=b=c$ and $\alpha =\beta =\gamma \lesssim 90$ [43]. The phase-transition temperatures shown in the figure are from previous studies including x-ray diffraction (XRD) [10, 11, 43, 63, 64], dielectric measurements (DM) [11, 23, 63], and molecular-dynamics (MD) simulations [46]. The shaded region extended up to $x=0.5$ indicates the MPB determined from MD [46]. The inset shows the crystal structure of a PMN-PT phase with cubic space-group symmetry ($Pm\overline{3}m$). Dashed lines indicate the compositional variation of Burns temperature ($T_b$) [5, 7]. (b) Bulk ($K$) and shear ($G$) moduli as a function of $x$. Room temperature $K$ and $G$ are connected with dashed lines. Color scale indicates the temperatures. (c) Bulk ($K$) and shear ($G$) moduli of PMN ($x=0$) [58] and PMN-PT ($x=0.3$) as a function of temperature. $T_c$ for PMN and PMN-PT ($x=0.3$) is also marked in the plot. (d) $\mathrm{\Delta }K$ and shear $\mathrm{\Delta }G$ as a function of temperature. The blue and red lines represent fits $\left[A{(T-T_f)}^{\kappa }\right]$ to the $\mathrm{\Delta }K$ and $\mathrm{\Delta }G$ with $T_f=380\mathrm{K}$ [22]. The fitted parameter $A$ for $\mathrm{\Delta }K$ and $\mathrm{\Delta }G$ are $-499$ and $-2974\mathrm{GPa}$, respectively. The fitted parameter $K$ for $\mathrm{\Delta }K$ and $\mathrm{\Delta }G$ are $-0.96$ and $-1.36$, respectively. The two gray and white shaded regions shown in the figure indicate the ferroelectric-to-paraelectric transition with the $T_c$ for PMN and for PMN-PT.