Compliant substrate controlling the phase diagrams of multiferroic multilayers

For a ferroelectric/ferromagnetic multilayer sandwiched between substrates, the tunability of the internal strains, ferroelectric states and phase diagrams by adjusting the substrate thickness are investigated. Based on the continuum mechanics and phenomenological theory, the thermodynamic potentials of the restraint ferroelectric/elastic and ferroelectric/ferromagnetic multilayer are constructed, respectively. Due to the interaction of different layers mediated by the mechanical coupling, the total strains in different layers are solved as functions of the misfit strains, electric polarizations of the ferroelectric slabs and magnetic polarizations of the ferromagnetic slabs. More importantly, the phase diagrams of multiferroic multilayer sandwiched between substrates are comprehensively investigated based on the modified thermodynamic potential. From our results, it can be found that the phase diagrams of the multiferroic multilayers can be effectively shifted by changing the multilayer-substrate thickness ratio, and the phase diagram configuration of the multiferroic multilayer can be dramatically changed by controlling the co-effect of the multilayer-substrate thickness ratio and other tunable quantities, such as multilayer-substrate misfit strains, in-multilayer thickness ratio and in-multilayer misfit strains.