## 3.1 Structural properties

The space group of cubic HfO2 is Fm3m with crystalline structure of fluorite and the local symmetry is the . Moreover, it is face-centered cubic lattice with one Hf atom at the position (0, 0, 0) and the O atoms at the position (1/4, 1/4, 1/4). We have obtained the calculated structural parameters by minimizing the total energy. The obtained value is *a*=5.242 Å in the pure cubic HfO2, which agrees with the previous theoretical [29–35] and experimental data [36]. Meanwhile, the values are *a*=5.216 Å in the O vacancy cubic HfO2 and *a*=5.212 Å in the Hf vacancy cubic HfO2.

## 3.2 Electronic properties

Figures 1-3 show the calculated results for the band structure of the pure, O vacancy and Hf vacancy cubic HfO2 respectively, along with the high symmetry directions of the BZ. For the pure cubic HfO2, the band gap is direct at G with the energy of 2.94 eV and the others band gap energies with the valence-to-conduction band transition symmetry are 3.44 eV, 3.52 eV and 3.37 eV for X→G, R→G, M→G respectively. However, the band gap of 2.94 eV is smaller than experimental value 5.7eV [37] due to the well-known underestimation of conduction band state energies in ab initio calculations within DFT. For the O vacancy cubic HfO2, the Fermi surface extends into the conduction band and the band gap is indirect at X→R with the energy of 0.58 eV and the others band gap energies with the valence-to-conduction band transition symmetry are 1.18 eV, 0.79 eV and 0.59 eV for R→R, M →R, G→R respectively. For the Hf vacancy cubic HfO2, the Fermi surface extends into the valence band and the band gap is direct at G with the energy of 3.09 eV and the others band gap energies with the valence-to-conduction band transition symmetry are 3.49 eV, 3.52 eV and 3.41 eV for X→G, R→G, M→G respectively.

Figures 4-6 show the calculated results for the total and partial densities of states of the pure, O vacancy and Hf vacancy cubic HfO2 respectively. For the pure cubic HfO2, the lower valence bands are composed predominantly of O 2s character, and the upper valence bands are consist of O 2p states which show a hybridization character with Hf 5d, and the conduction bands are composed mostly of Hf 5d states which show a hybridization character with O 2p. The peak value of the lower valence bands is found at -16.58 eV and there are three peak values of the upper valence bands at -4.47 eV, -4.06 eV and -1.38 eV respectively. For the O vacancy cubic HfO2, the peak value of the lower valence bands is found at -19.10 eV and there are five peak values of the upper valence bands at -6.85 eV, -6.03 eV, -5.56 eV, -4.50 eV and -3.81 eV respectively, due to the O vacancy which causes three different O atoms according to the distance between the O vacancy and the O atom. In the figure 5, (a), (b) and (c) show the partial densities of states of the third-near, second-near and first-near O atoms according to the distance between the O vacancy and the O atom. Moreover, there is a new split peak at -0.36 eV due to the Hf 5d state. For the Hf vacancy cubic HfO2, the peak value of the lower valence bands is found at -16.09 eV and there are four peak values of the upper valence bands at -3.43 eV, -2.44 eV, -1.18 eV and -0.17 respectively. By comparing the results, we find that the O vacancy causes that the Fermi surface extends into the conduction band and the Hf vacancy causes that the Fermi surface extends into the valence band, but the width of band gaps decreases in the O vacancy and increases in the Hf vacancy.

## 3.3 Optical properties

The calculated optical properties of the pure, O vacancy and Hf vacancy cubic HfO2 at the equilibrium lattice constant are presented in Figures 7-15, for the energy range up to 30 eV. Because of the localization, we must use a scissors operator to deal with the underestimated band gaps. The minimum band gap becomes 5.7 eV which is in agreement with experiment by a rigid shift of 2.8 eV, 5.1 eV and 2.6 eV for the pure, O vacancy and Hf vacancy cubic HfO2 respectively.

Figures 7-9 show the real and imaginary parts of the dielectric function for the pure, O vacancy and Hf vacancy cubic HfO2 respectively. In the figure 7, there is a peak at 6.65 eV, with the value of 8.09, for the real part of the dielectric function, which originates from O-2p to Hf-5d. Figure 8 shows that there are three peaks at 5.78 eV, 8.29 eV and 10.84 eV, with the values of 7.94, 6.07 and 4.25 respectively, for the real part of the dielectric function. Figure 9 shows there is a peak at 2.27 eV, with the value of 23.94, for the real part of the dielectric function. The calculated static dielectric constants are 3.99, 4.28 and 12.32 for the pure, O vacancy and Hf vacancy cubic HfO2 respectively.

Figures 10-12 show the refractive index and the extinction coefficient for the pure, O vacancy and Hf vacancy cubic HfO2 respectively. The static refractive indexes are 2.00, 2.07 and 3.51 for the pure, O vacancy and Hf vacancy cubic HfO2 respectively. The value of 2.00 for pure cubic HfO2, the refractive index changes with energy reaching two peaks at 6.76 eV, with the value of 2.88, and 7.89 eV, with the value of 2.87. Then, it decreases to a minimum value of 0.04 at 20.86 eV. For O vacancy cubic HfO2, the refractive index changes with energy reaching three peaks at 5.86 eV, with the value of 2.85, 8.42 eV, with the value of 2.58, and 11.20 eV, with the value of 2.39. Then, it decreases to a minimum value of 0.002 at 23.50 eV. For Hf vacancy cubic HfO2, the refractive index changes with energy reaching five peaks at 2.42 eV, 7.08 eV, 9.71 eV, 16.35 eV and 18.64 eV, with the values of 5.09, 1.80, 2.01, 0.55 and 0.62. Then, it decreases to a minimum value of 0.01 at 21.65 eV.

Figures 13-15 show the complex conductivity function, energy-loss spectrum, absorption coefficient and optical reflectivity for the pure, O vacancy and Hf vacancy cubic HfO2 respectively. The optical properties can be computed from the complex dielectric function ε(ω).