The structural relaxation, electronic structures, formation energies and transition energy levels of monoclinic HfO2 with neutral and charged oxygen vacancies have been studied using the first principles calculation based on density-functional theory and generalized gradient approximation. The results show that oxygen vacancies with different charge states can be formed in m-HfO2 under both oxygen-rich and oxygen-poor conditions. Especially, lower formation energy is obtained in poor oxygen environment. In the presence of oxygen vacancies with different charge states, extra levels can be observed at different positions in the band gap. And the most stable charge states are obtained for varying Fermi levels in the HfO2 band gap. It is found that oxygen vacancy in m-HfO2 has a negative-U behavior.
