Low-temperature polymorphs of $\mathrm{Zr}{\mathrm{O}}_2$ and $\mathrm{Hf}{\mathrm{O}}_2$: A density-functional theory study

The total energy and equation of state of the monoclinic, tetragonal, cubic, orthorhombic-I (Pbca) and orthorhombic-II (cotunnite) phases of zirconia and hafnia are determined using density functional theory (DFT) in the local density (LDA) and generalized-gradient (GGA) approximations. It is found that GGA corrections are needed to obtain low-temperature phase transitions under pressure that are consistent with experiment, i.e., $(\text{monoclinic}\to \text{orthorhombic}\text{−}\mathrm{I}\to \text{cotunnite})$. The GGA values of the bulk modulus of the cotunnite phase are found to be 251 and $259\mathrm{GPa}$ for $\mathrm{Zr}{\mathrm{O}}_2$ and $\mathrm{Hf}{\mathrm{O}}_2$ respectively, highlighting the similarity of these two compounds. We introduce a population analysis scheme in which atomic radii are adapted to the actual charge distribution in the material. The results indicate that the effective atomic radius of Hf is smaller than that of Zr, a drastic manifestation of the relativistic lanthanide contraction. The population analysis demonstrates that ionicity: (i) decreases from the monoclinic to the cotunnite phase, and (ii) is larger for $\mathrm{Hf}{\mathrm{O}}_2$ than for $\mathrm{Zr}{\mathrm{O}}_2$. The bandgap and heat of formation are also larger for monoclinic $\mathrm{Hf}{\mathrm{O}}_2$ than for $\mathrm{Zr}{\mathrm{O}}_2$ by $0.60\mathrm{eV}$ and $0.60\mathrm{eV}$/ formula unit, respectively. The tetragonal phase, which often exists as a metastable phase at ambient conditions, has a bandgap larger than the monoclinic phase by 0.35 and $0.65\mathrm{eV}$ for $\mathrm{Zr}{\mathrm{O}}_2$ and $\mathrm{Hf}{\mathrm{O}}_2$, respectively.

Figure 1

The unit cells for the following structures of $\mathrm{Zr}{\mathrm{O}}_2$ and $\mathrm{Hf}{\mathrm{O}}_2$, using $\mathrm{Hf}{\mathrm{O}}_2$ as an example: monoclinic, orthorhombic-I, tetragonal, and cotunnite (orthorhombic-II).

Figure 2

The total energy at zero pressure versus volume at zero pressure for each phase of $\mathrm{Zr}{\mathrm{O}}_2$ and $\mathrm{Hf}{\mathrm{O}}_2$ in the LDA and GGA. For each oxide and method the leftmost point corresponds to the cotunnite phase and is followed by the cubic, tetragonal (“tet”), orthorhombic-I (“orc1”), and monoclinic (“mc”) phases. The dashed lines connect phases accessible at room temperature under compression (cotunnite, orthorhombic-I, and monoclinic).

Figure 3

The computed total energy versus volume in the five structures of Fig. 1, for: (a) $\mathrm{Zr}{\mathrm{O}}_2$, LDA; (b) $\mathrm{Zr}{\mathrm{O}}_2$, GGA; (c) $\mathrm{Hf}{\mathrm{O}}_2$, LDA; (d) $\mathrm{Hf}{\mathrm{O}}_2$, GGA. Here $\cot =\text{cotunnute}$, $\mathrm{orc}1=\text{orthorhombic}\text{−}\mathrm{I}$, $\mathrm{cub}=\text{cubic}$, $\mathrm{tet}=\text{tetragonal}$, and $\mathrm{mono}=\text{monoclinic}$.

Figure 4

The total density of states within the generalized-gradient approximation for (a) $\mathrm{Zr}{\mathrm{O}}_2$ and (b) $\mathrm{Hf}{\mathrm{O}}_2$ for the five structures indicated.