The $phi(kpp)sim kpp$ relation is called polarization structure. By density functional calculations, we study the polarization structure in ferroelectric perovskite PbTiO$_3$, revealing (1) the $kpp$ point that contributes most to the electronic polarization, (2) the magnitude of bandwidth, and (3) subtle curvature of polarization dispersion. We also investigate how polarization structure in PbTiO$_3$ is modified by compressive inplane strains. The bandwidth of polarization dispersion in PbTiO$_3$ is shown to exhibit an unusual decline, though the total polarization is enhanced. As another outcome of this study, we formulate an analytical scheme for the purpose of identifying what determine the polarization structure at arbitrary $kpp$ points by means of Wannier functions. We find that $phi(kpp)$ is determined by two competing factors: one is the overlaps between neighboring Wannier functions within the plane {it perpendicular} to the polarization direction, and the other is the localization length {it parallel} to the polarization direction. Inplane strain increases the former while decreases the latter, causing interesting non-monotonous effects on polarization structure. Finally, polarization dispersion in another paradigm ferroelectric BaTiO$_3$ is discussed and compared with that of PbTiO$_3$.
تحميل البحث