Theoretical stability of the polarization in a thin semiconducting ferroelectric

The size effect in ferroelectrics is examined by considering the semiconductivity of the ferroelectric. This approach is applied to a theoretical investigation of the stability of the spontaneous polarization in a ferroelectric film on a semiconductor using a homogeneous Ginzburg-Landau theory. The band structure in the ferroelectric/insulator/semiconductor is rigorously incorporated in the theory, as if for a conventional semiconductor heterostructure such as GaAs/(Ga,Al)As. The carriers generated in the ferroelectric are found to change drastically the stability of the spontaneous polarization and its size effect. Numerical and simple analytical results are presented for ${\mathrm{BaTiO}}_3$ and, mostly, for ${\mathrm{PbTiO}}_3.$ The spontaneous polarization in a single-domain ferroelectric on a semiconductor is shown to be bistable and induce a semiconductor space charge $D.\mathrm{}$ The value of $D$ is a tiny fraction of the spontaneous polarization and agrees with experimentally estimated values. Namely, the experimental quasiequilibrium $D$ values are mostly intrinsically limited by the heteroband structure, not by the technological difficulties. Moreover, the $D$ value for a finite insulator thickness was found to be mainly determined by the insulator thickness and the semiconductor properties of the ferroelectric, and was relatively insensitive to the ordinary ferroelectric properties and the trap densities at the interface. The theory forecasts possible limitations of the transistors using this structure. Nonetheless, a long retention of the semiconductor space-charge $D$ by a very thin ferroelectric can be realized by the advancement of the technology. Additionally, the stability of very thin ferroelectrics on semi-insulating substrates and the switching of ferroelectrics on parent materials of the high-$T_c$ superconductors are successfully explained.