Phase diagram of ferroelectric thin film with diluted surface

We study the influence of the surface of the ferroelectric films on the properties of the intrinsic ferroelectric polarization state in frame of statistical Transverse Ising Model (TIM). The quality of the surface is modelled by the dilution parameter q$q$ that reflects the relative weight of the paraelectric impurities vs. ferroelectric ions. Using the effective field approach the variation of the average value of spontaneous polarization as function of temperature for different film thicknesses has been investigated as function of dilution parameter q$q$. The phase diagram of the system and crossover from the ferroelectric to the paraelectric phase is discussed.