Electro-optic properties of ${\mathrm{ZrO}}_2$, ${\mathrm{HfO}}_2$, and ${\mathrm{LiNbO}}_3$ ferroelectric phases: A comparative density functional study

We report the Pockels electro-optic properties of ${\mathrm{ZrO}}_2$ and ${\mathrm{HfO}}_2$ orthorhombic $Pbc{2}_1$ and rhombohedral $R3m$ ferroelectric phases, and we compare them to the well-known rhombohedral $R3c$ ${\mathrm{LiNbO}}_3$ Pockels material from density functional theory calculations using the crystal suite of programs. Specifically, three essential processes are explicitly investigated: The electronic, the ionic (or vibrational), and the piezoelectric contributions. Our calculations reveal that the ionic part coming from the low-frequency phonon modes contributes the most to the electro-optic coefficients of rhombohedral ${\mathrm{LiNbO}}_3$ and of orthorhombic ${\mathrm{ZrO}}_2$ and ${\mathrm{HfO}}_2$. Additionally, the low-frequency phonon modes exhibit zero contribution to the Pockels coefficients of ${\mathrm{ZrO}}_2$ and ${\mathrm{HfO}}_2$ rhombohedral $R3m$ phase.

Figure 1

The crystallographic structures of (a) ${\mathrm{LiNbO}}_3$ (space group $R3c$), (b) rhombohedral ${\mathrm{ZrO}}_2\text{−}{\mathrm{HfO}}_2$ (space group $R3m$), and (c) orthorhombic ${\mathrm{ZrO}}_2\text{−}{\mathrm{HfO}}_2$ (space group $Pbc{2}_1$).

Figure 2

EO coefficients (unclamped) $r_{33}$ and $r_{13}$ of LNO and $o\text{−}{\mathrm{ZrO}}_2$ at different wavelengths using B3LYP functional.

Figure 3

Figure of merit ($n^3{r}_{ijk}/\epsilon$) of ${\mathrm{HfO}}_2$ and ${\mathrm{ZrO}}_2$ compared to LNO using $r_{33}$ and $r_{13}$ coefficients.

Figure 4

$r_{33}$ EO coefficient of $r\text{−}{\mathrm{ZrO}}_2$ computed at different angles.