We study the decay rate of a solution of a homogeneous elliptic equation in the exterior of the ball |X| ≤ r.First, note two well-known results. Let u(z) be an analytic or harmonic function outside the disk |z| ≤ r. If u(z) decays at infinity faster than any power of |z|, then u(z) ≡ 0.
Thin films of Pb(Zr 0.52 Ti 0.48 )O 3 (PZT) were prepared by excimer laser ablation on a Pt/Ti/SiO 2 /Si substrate and were crystallized by subsequent annealing at 750 • C for 90 min. Crystalline phases in the PZT films were investigated by X-ray diffraction analysis. The microstructure and composition of the films were studied by transmission electron microscopy and energy dispersive X-ray spectroscopy, respectively. It is found that the films consist almost entirely of the perovskite phase, but a thin layer of the pyrochlore phase exists at the surface of the films. Electrical properties of these films were evaluated by measuring P-E hysteresis loops and dielectric constants. The remanent polarization and the coercive field of the films were 23.9 µC/cm 2 and 60.5 kV/cm, respectively, while the dielectric constant and loss values measured at 1 kHz were approximately 950 and 0.04, respectively. The effect of the microstructure on the electrical properties of the PZT thin films is discussed.
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