Theoretical study of the flexoelectric effect based on a simple model of ferroelectric material

By the flexoelectric effect is called a linear relation between polarization and gradient of deformation. The aim of the present work is to estimate the flexoelectric coefficient of ferroelelectrics on the basis of a microscopic model. The well known model is used: the ion of charge e spaced in potential double-wells interacts with the other ions of the same type in neighboring cells with springs whose spring constant is beta. Limiting our consideration on the regime of the order-disorder mechanism we find that flexoelectric coefficient is the product of susceptibility epsilon(0)chi(r) (epsilon(0) vacuum permitivity) and the coefficient betaa(2)/e, where a is a lattice constant. It can be shown that our results are not in contradiction with Tagantsev's expression [Phys. Rev., B34, 5883, (1986)] for flexoelectric coefficient: chia/e. However the latter result was deduced by the more phenomenological way or concerns the displacement type of ferroelectrics only.