The dielectric property of the cubic hafnia is investigated in terms of the polarizability density and the dielectric constant density defined by one of the authors. We study it by using the cluster model embedded in point charges which represent the surrounding atoms. It is shown that the cubic hafnia show complicated responses to external electric fields, in particular, rotational ones. We stress that the nanosize material should be studied in the local and tensor quantity analysis to describe rotational responses correctly. We also show that the choice of models, in particular, termination condition, is important for these analyses, by comparing the point charge model and the hydrogen termination model.

In this work, we investigate the electronic contribution to local dielectric property of La 2 O 3 and HfO 2 using cluster models. The relation between the coordinate number of metal atoms and their bonding energy shows a hint that hafnia takes the cubic structure by the incorporation of La 2 O 3 in HfO 2 . The local properties of polarizability and dielectric constant of La 2 O 3 and HfO 2 are closely similar to each other. It is considered to be one of the reasons why the incorporation of lanthanum atoms does not lower the permittivity of HfO 2 . We confirm this by the study of the local dielectric property of the HfLaO x cluster model. We compare the dielectric properties around an oxygen atom and that between the oxygen atom and a next metal atom. Our results show that the contribution to the dielectric response from the bond regions is not so large.

We investigate the electronic contribution to local dielectric property in terms of the local polarizability density and dielectric constant density, for the cubic, tetragonal, and monoclinic structure of HfO 2 and compare their dielectric properties with those of SiO 2 . We show appropriate termination conditions of our cluster models to realize the condensed property of dielectric; point charge conditions for HfO 2 whose bond is ionic, while hydrogen termination conditions for SiO 2 whose bond has covalent property. We show that local parts of materials have complicated responses to external electric fields, in particular, rotational ones. Hence, nanosize materials should be studied in the local and tensor quantity analysis to describe rotational responses correctly. It is clarified that the electronic contribution to local polarizability and dielectric constant densities is almost independent of the structures of HfO 2 crystals. We show that the electronic contribution to dielectric response of HfO 2 is significantly large compared to those of SiO 2 . In addition, it is found that the average value of dielectric constant around O atoms is larger than other regions in both HfO 2 and SiO 2 . #

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