Distribution function of random electric fields in disordered ferroelectrics thin films

We present the calculation of first moment $E_0$ and variance $\Delta E$ of distribution function of random fields in a ferroelectric of finite size. This defines completely the distribution function in gaussian limit. Specific calculations have been performed for the case of slab-shaped ferroelectric thin film. We have shown that $E_0$ and $\Delta E$ can be expressed through the integrals from first and second degree of Green's function of such confined geometry ferroelectric in $k$ - space. To obtain the Green's function, we solve the differential equation minimizing Landau free energy of a ferroelectric with respect to the boundary conditions on its surfaces. We show, that the distribution function of random fields in the finite-size ferroelectric differs from that of the unbounded bulk material. For example, both $E_0$ and $\Delta E$ depends on film thickness $L$. Knowledge of this distribution function permits to calculate the observable physical properties of ferroelectric thin films made from ferroelectric relaxors. Our method of calculation of $E_0(L)$ and $\Delta E(L)$ can be easily generalized for ferroelectric of arbitrary shape.