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Based on the transverse Ising model and using decoupling approximation to the Fermi-type Green's function, we study the phase transition properties of the epitaxial ferroelectric film with one substrate. A general recursive equation of the ferroelectric thin film with two n-layer materials is obtained, which enables us to study the phase transition properties for any number layers for epitaxial ferroelectric thin film. With the help of this equation, we analyze the effect of the exchange interaction and the transverse field in the phase diagram, the influence to the polarizations and Curie temperature numerically. The results show that epitaxial ferroelectric film are able to induce a strong increase or decrease of Curie temperature to different exchange interactions and transverse fields within the epitaxial film layers. The theoretical results are in reasonable accordance with experimental data of different ferroelectric thin film.

A novel flexible metamaterial narrow-band-pass filter is designed and proved to be reliable by both numerical simulations and experimental measurements. The unit cell of the designed structure consists of circle ring resonators on top of a thin dielectric layer backed by a metallic mesh. The investigations on the distribution of the surface current and magnetic field as well as the analysis of the equivalent circuit model reveal that the magnetic resonance response between layers induced by the reverse surface current contributes to the high quality factor band-pass property. Importantly, it is a flexible design with a tunable resonance frequency by just changing the radius of the circle rings and can also be easily extended to have the multi-band-pass property. Moreover, this simplified structure with low duty cycle and ultra-thin thickness is also a symmetric design which is insensitive to the polarization and incident angles. Therefore, such a metamaterial narrow-band-pass filter is of great importance in the practical applications such as filtering and radar stealth, and especially for the conformal structure applications in the infrared and optical window area.