Microscopic derivation of free energy under electric field in ferroelectric and ferroelectric heterostructures containing free carriers

We derive Free energy and the macroscopic charge distribution in ferroelectric and ferroelectric heterostructures from a microscopic quantum mechanical model, and attempt to bridge the macroscopic and the atomic/quantum mechanical pictures. We postulate that ferroelectric should be defined in a heterostructure, especially when it is nanometer-scale. In addition, the separation of the polarization energy into the pure electrostatic and the material reconfiguration energy is proposed.