A detailed discussion is given of the predictions of the Landau-Devonshire theory for the behavior of thin ferroelectric films. It is assumed that the phase transition in the bulk is second order, and the theory is extended by the inclusion in the free-energy density of a term proportional to $|\nabla P{|}^2$ together with a boundary condition involving an extrapolation length δ. Expressions for the polarization profile $P\left(z\right)\mathrm{}$ are given in terms of Jacobi elliptic functions; some of these are in a clearer and simpler form than has been available previously. Interesting analytic results for the entropy in terms of elliptic integrals and elliptic functions are also presented. Graphical illustrations of the main results are included. Careful consideration is given to the nature of the phase transition, and it is concluded that it is second order in all cases.

• Received 21 March 2000

DOI:https://doi.org/10.1103/PhysRevB.63.144109