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The Landau theory was applied to treat the phase diagrams for a multiferroic with two second order phase transitions taking into account the coupling of the primary order parameters with strain. Two order parameters are coupled biquadratically which corresponds to the magnetoelectric materials. The coupling with strain is assumed to be linear in strain and quadratic in order parameters. Three ordered phases are discussed. Analytic relationships were obtained for the phase transition temperatures and for elastic modulus changes through the phase transitions. Strong influence of the coupling with strain on the phase diagrams was shown. IntroductionRecently the renewed attention was focused on multiferroic materials (single-phase and composite ones) because of their promising applications as multifunctional devices (see [1,2] and references therein). The most interesting case is when two coupling order parameters are related to magnetic and electric orderings leading to the magnetoelectric effect. Experimental studies of multiferroic materials showed that their properties can be remarkably influenced by interaction with strains. Such an interaction becomes apparent, for instance, in elastic anomalies through the phase transitions [3][4][5][6][7][8][9][10] and in the impact of substrate-induced or epitaxial strain on the ferroic phase transitions and orientations of magnetic moments [11][12][13]. While many efforts were made to reveal the role of strain in the multiferroic materials, some effects of coupling of the magnetic and electric order parameters with strain were still not discussed properly.

Second order ferroelectric phase transition for a thin film with arbitrary boundary conditions is considered within the framework of the Landau theory. General phaseplane portraits for the relevant Euler-Lagrange equation are constructed and different types of trajectories are analyzed. Full set of order parameter profiles is found. The method of reconstructing the admissible order parameter profiles for particular film thickness and extrapolation lengths is discussed.

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