A macroscopic theory for the existence of the hysteresis and butterfly loops in ferroelectricity

We present in this paper a minimal macroscopic theory leading to the hysteresis loop and the butterfly loop in ferroelectricity. This theory is based on the notion that as domains switch under the action of an external electric field the number of dipoles aligned in the direction of the field also alters and it obeys a rate law which depends non-linearly on the field. In addition, we specify the simplest constitutive relations of the stress and the electric displacement which together with the rate law are sufficient to yield the hysteresis loop and the butterfly loop. Comparison with experimental data is exhibited for the case corresponding to the hysteresis loop.