Figure 1: The temperature dependences of transverse dielectric (ɛ₁₁ and ɛ₂₂) and shear piezoelectric (d₁₅ and d₂₄) properties for single-domain relaxor–PT crystals.

(a–c) Dielectric permittivities for the rhombohedral PMN–0.28PT, orthorhombic PMN–0.32PT and tetragonal PZN–0.15PT crystals, respectively. In the PMN–0.28PT crystals, a rhombohedral–tetragonal and a tetragonal–cubic phase transitions exist at 368 and 405 K, respectively. In the PMN–0.32PT crystal, an orthorhombic–tetragonal and a tetragonal–cubic phase transitions occur at 360 and 430 K, respectively. In the PZN–0.15PT crystal, a tetragonal–cubic phase transition exists at 468 K. (d–f) Enlarged low-temperature sections (20–300 K) for a–c, respectively. (g) The measured shear piezoelectric coefficients for the three relaxor–PT crystals as a function of temperature. (h) A schematic plot showing the major finding of this work for the enhancement of dielectric/piezoelectric properties in relaxor–PT crystals (the temperature-dependent piezoelectric/dielectric responses of the classical ferroelectrics is inferred from phenomenological theory and first-principle calculations of Pb(Zr_0.5Ti_0.5)O₃ (refs 50, 51)). For comparison, the corresponding temperature dependences of the longitudinal dielectric permittivity ɛ₃₃ of single-domain PMN–0.28PT, PMN–0.32PT and PZN–0.15PT crystals are shown in Supplementary Fig. 1. For orthorhombic crystals, ɛ₁₁ and ɛ₂₂ are two independent tensors. The permittivity ɛ₁₁ versus temperature of the orthorhombic PMN–0.32PT crystal is given in Supplementary Fig. 2. For the PZN–0.15PT crystal, the temperature dependence of the reciprocal of relative dielectric constant is given in Supplementary Fig. 3, for identifying its relaxor characteristic.