Modeling of minor loops of a ferroïc single crystal

The characterizations of ferroïc materials are often realized by measuring the minor loops. The polarization P and/or the strain S are reported as a function of the ac electric field E and/or the ac mechanical stress σ. The slope of the loop depends on the real part of ε or d (respectively the dielectric constant and the piezoelectric coefficient) and its opening is related to the imaginary part of ε or d. To interpret the loop properties, we use a model describing the motion of the domain walls, model presented elsewhere in the congress. For a low value of the amplitude of E or s, the motion of the walls is vibration and jumps between pinning centers. For the vibration, ε and d do not depends on the amplitude of E or σ while they vary linearly with it for the contribution of the jumps. The variation of ε and d as a function of E or σ follows a hyperbolic law. The behavior is valid for the real part of these quantities but also for the imaginary part. For very low amplitudes, ε and d are constant because the jumps contribution is negligible and the vibration displacement is proportional to E or σ. In these conditions, the minor loop is an ellipse. The adding of the jumps at higher amplitude leads to a distortion of the loop.