Solitons and electroacoustic interactions in ferroelectric crystals. I. Single solitons and domain walls

This paper is devoted to the study of single solitons and one-wall motion in elastic ferroelectrics in the presence of electromechanical couplings. To that purpose, a rather simple microscopic model is devised for ferroelectric crystals presenting a molecular group (e.g., ${\mathrm{NaNO}}_2$). Continuum nonlinear coupled equations are deduced from this model and are given a Hamiltonian form, allowing for the analytical and numerical study of single ferroelectric solitons coupled to acoustic phenomena. For single solitons it is shown that the whole problem can be recast as a double sine-Gordon equation of which one solution is stable and can be interpreted as the motion of a ferroelectric wall with electromechanical couplings. The remaining mechanical equations then allow one to evaluate the stress field generated by ferroelectric solitons also as the corresponding elastic displacement, since strain compatibility conditions are satisfied in the present case. Numerical graphs illustrate the space-time evolution of a wall and the accompanying stress and displacement fields. Energies involved such as the wall energy, as well as the thickness of a moving wall, can be evaluated on account of electromechanical couplings. An interpretation of all results is finally given in terms of phase-transition phenomena including an incommensurate phase. To that purpose, a Landau-Ginzburg type of approach is formulated in which a Lifshitz invariant and electromechanical couplings are accounted for. The initiation of the ferroelectric phase takes place within the incommensurate phase, locally in the crystal, with the formation of domains and the motion of walls.