Abstract
Calcia (CaO) stabilized cubic-$HfO_2$ is studied by density functional theory (DFT) with generalized gradient approximation (GGA). When a Ca atom is substituted for a Hf atom, an oxygen vacancy is produced to satisfy the charge neutrality. The lattice parameter of a $2{\times}2{\times}2$ cubic $HfO_2$ supercell then increases by $0.02\;{\AA}$. The oxygen atoms closest to the oxygen vacancy are attracted to the vacancy as the vacancy is positive compared to the oxygen ion. When the oxygen vacancy is located at the site closest to the Ca atom, the total energy of $HfO_2$ reaches its minimum. The energy barriers for the migration of the oxygen vacancy were calculated. The energy barriers between the first and the second nearest sites, the second and the third nearest sites, and the third and fourth nearest sites are 0.2, 0.5, and 0.24 eV, respectively. The oxygen vacancies at the third and fourth nearest sites relative to the Ca atom represent the oxygen vacancies in undoped $HfO_2$. Therefore, the energy barrier for oxygen migration in the $HfO_2$ gate dielectric is 0.24 eV, which can explain the origin of gate dielectric leakage.
