Structural and dielectric properties of amorphous ${\mathrm{ZrO}}_2$ and ${\mathrm{HfO}}_2$

Zirconia $\left({\mathrm{ZrO}}_2\right)$ and hafnia $\left({\mathrm{HfO}}_2\right)$ are leading candidates for replacing ${\mathrm{SiO}}_2$ as the gate insulator in complementary metal-oxide semiconductor technology. Amorphous versions of these materials ($a\text{−}{\mathrm{ZrO}}_2$ and $a\text{−}{\mathrm{HfO}}_2$) can be grown as metastable phases on top of a silicon buffer; while they tend to recrystallize during subsequent annealing steps, they would otherwise be of considerable interest because of the promise they hold for improved uniformity and electrical passivity. In this work, we report our theoretical studies of $a\text{−}{\mathrm{ZrO}}_2$ and $a\text{−}{\mathrm{HfO}}_2$ by first-principles density-functional methods. We construct realistic amorphous models using the “activation-relaxation” technique of Barkema and Mousseau. The structural, vibrational, and dielectric properties of the resulting models are analyzed in detail. The overall average dielectric constant is computed and found to be comparable to that of the monoclinic phase.

Figure 1

$a\text{−}{\mathrm{ZrO}}_2$ distribution of coordination numbers resulting from the activation-relaxation (ART) simulation. Zr and O atoms are indicated by filled and open bars, respectively. The cutoff radius defining nearest neighbors is $3\mathrm{\AA }$.

Figure 2

(a) $a\text{−}{\mathrm{ZrO}}_2$ phonon density of states (DOS) vs frequency. (b) $a\text{−}{\mathrm{ZrO}}_2$ infrared activity (phonon DOS weighted by ${\tilde{Z}}_{\lambda }^2∕{\omega }_{\lambda }^2$) vs frequency.

Figure 3

$a\text{−}{\mathrm{HfO}}_2$ distribution of coordination numbers resulting from the activation-relaxation (ART) simulation. Hf and O atoms are indicated by filled and open bars, respectively. The cutoff radius defining nearest neighbors is $3\mathrm{\AA }$.

Figure 4

(a) $a\text{−}{\mathrm{HfO}}_2$ phonon density of states (DOS) vs frequency. (b) $a\text{−}{\mathrm{HfO}}_2$ infrared activity (phonon DOS weighted by ${\tilde{Z}}_{\lambda }^2∕{\omega }_{\lambda }^2$) vs frequency.

Figure 5

Monoclinic ${\mathrm{HfO}}_2$ infrared activity. The spectrum has been convoluted with a Gaussian of width $5{\mathrm{cm}}^{-1}$.

Figure 6

(a) Scatter plot of isotropically averaged atomic $Z^{*}$ values (vertical axis) vs atom type and coordination number (horizontal axis) for $a\text{−}{\mathrm{ZrO}}_2$. Circles and diamonds denote O and Zr atoms, respectively. (b) Same but with “dielectric activity” [see Eq. (5) of Ref. 3] plotted vertically.

Figure 7

(a) Scatter plot of isotropically averaged atomic $Z^{*}$ values (vertical axis) vs atom type and coordination number (horizontal axis) for $a\text{−}{\mathrm{HfO}}_2$. Circles and diamonds denote O and Hf atoms, respectively. (b) Same but with “dielectric activity” [see Eq. (5) of Ref. 3] plotted vertically.