Stabilization of Ferroelectric Hafnia and Zirconia through Y2O3 doping

TL;DR

This study uses density functional theory to show that Y2O3 doping stabilizes ferroelectric phases in hafnia and zirconia, potentially enabling new ferroelectric materials.

Contribution

It demonstrates that Y2O3 doping stabilizes ferroelectric phases in hafnia and zirconia using large supercell DFT simulations, a novel computational approach.

Findings

Y2O3 doping stabilizes ferroelectric orthorhombic phase

Doping favors polar phase over monoclinic phase

Applicable to both hafnia and zirconia

Abstract

We investigate the possible stabilization of ferroelectricity in bulk Y2O3-doped hafnia and zirconia. We use density functional theory (DFT) with large random supercells of hafnia and zirconia and study the relative phase stability of the centrosymmetric cubic and monoclinic phases compared with the polar orthorhombic phase. We find that Y2O3-doping stabilizes the polar ferroelectric phase over the monoclinic baddeleyite phase in both hafnia and zirconia.

Tables

Table 1. Table 1: The total direct piezoelectric tensor e (C/m 2 ) and converse piezoelectric tensor d (pm/V) in the ferroelectric phase of 0% (pure), 6.25%, and 12.5% Y-doped hafnia at 0 K. The doping structure is in 2 × 2 × 2 2\times 2\times 2 supercell. Indices are given in Voigt notation.

Index	0% Y in Ref [19]	0% Y	6.25% Y	12.5% Y
e31	-1.31	-1.39	-0.98	-0.62
e32	-1.33	-1.34	-1.17	-0.69
e33	-1.44	-1.46	-1.65	-0.92
e15	-0.20	-0.23	-0.003	0.14
e24	0.64	0.70	0.77	0.76
d31	-1.71	-2.01	-0.69	-0.92
d32	-1.77	-1.65	-1.46	-0.86
d33	-2.51	-2.46	-3.79	-1.91
d15	-2.03	-2.44	-0.03	2.03
d24	6.74	7.42	9.31	10.2

Equations

$$\begin{gathered} E_{\text{form}}=(E({\text{Y}}_x{\text{Hf}}_{1-x}{\text{O}}_{2-0.5x})-(1-x)E({\text{HfO}}_2) \\ -0.5xE({\text{Y}}_2{\text{O}}_3))/3, \end{gathered}$$

$$\begin{gathered} \text{P}={\text{P}}_0\pm n{\text{P}}_q\pm {\text{P}}_Z \\ =\frac{|e|}{V}\sum _i{Z}_i^{\ast }{\text{r}}_i\pm \frac{|e|}{V_{prim}}{\text{R}}_{prim}\pm \frac{|e|}{V}{Z}_i^{\ast }, \end{gathered}$$

$${\text{R}}_{prim}=n_1{\text{a}}_1+n_2{\text{a}}_2+n_3{\text{a}}_3,$$

Full text

Solid solution of Y2O3 in zirconia (ZrO2) or hafnia (HfO2) increases the stability of the cubic fluorite structure by about 2000 K [1, 2, 3]. Beautiful crystals of cubic zirconia are a commodity and Y-stabilized zirconia (YSZ) also has many industrial uses. Pure zirconia (ZrO2) is centrosymmetric, and exists as monoclinic baddeleyite (space group $P2_{1}/c$) under ambient conditions, transforms to tetragonal $P4_{2}/nmc$ at approximately 1350 K, and to cubic fluorite $Fm\overline{3}m$ at about 2650 K. It has been extensively studied using DFT [4] and machine learning ab initio molecular dynamics. Although the ground state of zirconia and hafnia is the centrosymmetric baddeleyite structure, hafnia and zirconia thin films show ferroelectricity [5, 6]. Ferroelectricity has been observed experimentally in bulk Y-doped hafnia, and the polar phase was attributed to metastable formation due to fast quenching [7]. Ferroelectric crystals up to 50 mm long were grown [7, 8]. Computations performed in these two papers did not find the ferroelectric phase to be stabilized, but this may be due to the small supercells of 48 atoms used there. Here we use large supercells and explore the relative stability of the cubic, tetragonal, and polar orthorhombic structures as functions of Y2O3 content. To maintain charge-balance there is one oxygen vacancy for every two substituted Hf or Zr. Throughout we give the percentage Y2O3 expressed as molar percentage Y replacement of Hf or Zr.

We built a series of $3\times 3\times 3$ supercells of the 12-atom orthorhombic primitive cell (a total of 108 formula units) at different Y concentrations using the special quasi-random structures (SQS) method (Fig. 1) [9, 10]. Our justification for using this approach is that at high temperatures, such as in the liquid phase, the atomic arrangement should be close to random, and with cooling the crystals do not order, since it involves diffusion of the highly charged Zr4+ or Hf4+ ions. Some oxygen redistribution is possible giving short-range ordering.

A different approach was used in Ref. [7], where all possible configurations in a very small supercell were exhaustively simulated, using the lowest energy configurations. In that work the lowest formation energy of 12.5% Y-doped hafnia is 0.031 eV/atom in the ferroelectric phase, which is 0.026 eV/atom and 0.007 eV/atom lower than the formation energies in our 11.1% and 14.8% supercells, where we assume random distributions of dopants and vacancies. Expressed as kB T, where kB is the Boltzmann constant, the corresponding temperatures are 302 K and 81 K, so at high temperatures random arrangements as we assume seem highly probable.

We also generated smaller $2\times 2\times 2$ supercells using the SQS method, and fully relaxed them by DFT. The energy differences and lattice constants computed with the $2\times 2\times 2$ supercells are consistent with the results in the $3\times 3\times 3$ supercells (Figs. S1, S2) for Y content up to 12.5%. We find differences at higher concentrations (Figs. 2(c) and S1), which shows that smaller supercells give inaccurate results at higher doping concentrations.

We computed the formation energies of doped hafnia and zirconia for the orthorhombic, monoclinic, and cubic phases with different Y2O3 concentrations (Fig. 2). In each case we started with the fully symmetric pure HfO2 or ZrO2 structures, placed the defects obtained by SQS as described above, and then relaxed the structures until forces and stresses were zero. The defects break the symmetry in any finite supercell, but we refer to each by the symmetry of the pure starting structures. The formation energies are defined as

[TABLE]

where the monoclinic P21/c ground state of pure HfO2 and cubic Ia$\overline{3}$ ground state of pure Y2O3 are used [11]. For both hafnia and zirconia, we find that the polar orthorhombic phase becomes more stable than the monoclinic and cubic phases at a Y molar concentration of 14.8%. The formation energies of Y-doped zirconia at ferroelectric phase are higher than those in Y-doped hafnia, suggesting that the ferroelectric phase would be harder to form in zirconia compared with hafnia.

We find a large change in volume in the polar phase in hafnia at a Y concentration of 14.8% (Fig. 2(a)), caused by the dramatically decreased lattice constant a (see also Fig. S2). The fluorite lattice acquires a tetragonal strain with a small c/a of $1.01\sim 1.015$ for Y concentrations less than 11.1%, but it becomes more cubic with Y content higher than 11.1% (Fig. S2). This can be an artifact of our finite supercells–the infinite (or large) cubic solid solution would have cubic symmetry. The fluorite phase shows lower formation energy at 11.1% Y-doped hafnia and zirconia at zero temperature. The polar orthorhombic phase exhibits lower formation energy from 14.8% to 22.2% Y-doped hafnia and in a small region around 14.8% Y concentration in zirconia. The small formation energy difference between cubic and polar orthorhombic phases are discussed at the end of this paper.

For high Y concentration (x=14.8%, 18.5%, or 22.2%), the energy difference between polar and cubic phases is smaller than energy difference between polar and monoclinic (Figs. 2(c)), so that baddeleyite is destabilized with high Y-contents, and the ferroelectric phase competes with the cubic phase at those high Y concentrations. The stabilization of cubic fluorite with Y2O3 is consistent with the known HfO2-Y2O3 and ZrO2-Y2O3 phase diagrams [12, 13, 14, 15, 16].

We also computed the phonon dispersion in Y-doped ferroelectric hafnia and ordered approximations of cubic zirconia (see Supplement). For hafnia phonon computations we used $2\times 2\times 2$ supercells for 6.25% and 12.5% Y-doped models. For these compositions, as discussed above, agreement is good compared with the larger supercells. Long-range interactions were included using the computed effective charges (see Supplement) and the non-analytical correction [17, 18].

The Y-doped structures show no imaginary frequencies (Fig. 3). We thus find that Y-doping in hafnia stabilizes the orthorhombic polar phase energetically and vibrationally as the ground state. No additional strains, defects, or appeal to kinetics is necessary.

We computed piezoelectric constants in pure hafnia, 6.25%, and 12.5% Y-doped hafnia within the $2\times 2\times 2$ supercells. The computed elastic moduli, direct and converse piezoelectric tensors in pure hafnia (see Table 1 and Supplement Table S1) are consistent with previous values [19]. We find that piezoelectricity in Y-doped hafnia is modest at 0 K, but it will increase at temperatures approaching phase transitions as observed in other materials [20, 21].

In hafnia thin films, the sign of the longitudinal piezoelectric coefficient is tunable via epitaxial strain [22]. We find the sign of e33 or d33 is negative for Y concentration up to 12.5%. The sign of e15 or d15 switches from negative to positive at Y dosage of 12.5%.

We computed the electric polarization of orthorhombic polar (Pca21), monoclinic, and cubic hafnia at different Y doping concentrations using the Berry’s phase approach based on modern theory of polarization [23, 24, 25] and using Born effective charges. The electric polarization is defined as follows

[TABLE]

[TABLE]

where $\textbf{{P}}_{0}$ is initial polarization, $\textbf{{P}}_{q}$ is the polarization lattice or quantum, $\textbf{{P}}_{Z}$ is the Born effective charge polarization quantum (see beloow), e is the electron charge, V is the volume of unit cell, $Z^{*}$ is the Born effective charge, $\textit{{r}}_{i}$ denotes the Cartesian position of atom i, sum runs over all the atoms in unit cell, R is lattice vector in primitive cell, $\textbf{{a}}_{1},\textbf{{a}}_{2},\textbf{{a}}_{3}$ are the lattice basis vectors in primitive cell. Since Eq. (2) involves a sum over the entire system, one must carefully group canceling or nearly canceling contributions. The Born polarization quantum is introduced by the Born effective charge of atoms in the border, corner, or face sites of the cell, to obtain a meaningful result from an infinite sum involving both diverging positive and negative terms.

We computed the effective charges in ferroelectric phase using density functional perturbation theory (DFPT) (see Supplement Table S2). Since the $3\times 3$ effective charge tensors depend on site symmetry, phase, and atomic environment, we use average values here in order to obtain approximate but robust results for different compositions, supercells, and dopant arrangements. The average Born effective charges for Hf and O are 5.26 and -2.63, respectively. Using these Born effective charges to calculate the electric polarization in polar phase per formula (2) gives 0.58 C/m2, close to the polarization we calculated via the direct Berry’s phase computation value of 0.55 C/m2. Our polarization agrees with that previously computed [26, 27].

In doped hafnia, the oxygen vacancy sites were first restored by placing oxygen atoms at the same positions as in pristine hafnia. All the atomic positions in doped hafnia were adjusted under periodic boundary conditions to closely match the pure structure. Then, we applied same amounts of polarization lattice and Born effective charge polarization quantum identically to the pristine case. Compared with pure hafnia, the electric polarization of the orthorhombic polar phase in doped hafnia decreases with increasing Y concentration, but rises again at higher Y content (Fig. 4). The minimum polarization of 0.43 C/m2 occurs at 11.1% Y. For the monoclinic and cubic phases, Y doping induces only a relatively small -less than 0.09 C/m2 - polarization. According to the formation energy (Fig. 2(c)), the polar and cubic phase becomes more stable than monoclinic phase as Y dosage reaches 14.8%. At those high Y concentrations (x =14.8%, 18.5%, or 22.2%), the energetically favored polar and cubic phases can reach $0.46\sim 0.54$ C/m2 and $0\sim 0.06$ C/m2, respectively. So, for hafnia with critical Y doping, considerable electric polarization should be practical in pure polar phase, but mixing with cubic phase will dramatically decrease the electric polarization, which probably explains the small polarization of 0.06 C/m2 measured in previous experiments [7].

Besides, we clarify that in 14.8% doped hafnia (see Fig. 2(c)), the lower formation energy at orthorhombic polar phase than that at cubic phase doesn’t clearly reveal that the polar phase is more stable than the cubic phase, given their small formation energy difference. Those energies are obtained from fully relaxed structures, which are initialized using the SQS configurations via Monte Carlo sampling until pair and triplet probabilities out to 4 Å and 3.8 Å. We tested larger SQS cutoffs using 14.8% Y-doped hafnia as an example, including pair and triplet probabilities out to 4 Å and 3.8 Å, 4.5 Å and 4.3 Å, 5 Å and 4.8 Å, respectively (Fig. S4). It turns out the formation energy difference between polar and cubic phases can be either positive or negative, depending on the SQS cutoffs. Nevertheless, the lower formation energies of both polar and cubic phases compared to the monoclinic phase are consistently maintained in these tests. The testing formation energies are marked in Fig. 2(c) as error bars. These results demonstrate the strong competition between polar and cubic phases in doped hafnia.

Next we discuss why we found the stability of ferroelectric hafnia, whereas earlier computations did not [7, 28, 29]. First, the previous calculation used 16 formula unit cell with 48 atoms [7], which is too small to simulate a system with the doping concentration as high as 12.5%. Second, Y-doping is expected to generate oxygen vacancies in hafnia, and the distribution of these vacancies is associated with the substituted Y. Experimental studies have shown that in hafnia thin films, the oxygen migration is even reversible at the interface between the hafnia-based capacitor and the top electrode, regardless of whether the electrode is oxygen reactive or not [30]. In experiments on hafnia bulk [7], fast quenching is typically used to prevent the diffusion of oxygen vacancies. However, in earlier computational studies [28, 29], the oxygen vacancies were either neglected or assumed to be located near the dopants. In contrast, our SQS-based structures allow for greater configurational flexibility. Upon fully structural relaxation, we observe that oxygen vacancies are positioned either close to Y (at distances of approximately 2-2.3 Å) or farther away (at distances greater than 3.5 Å). Crucially, the ferroelectric phase in hafnia exists at high temperatures, which redistributes the oxygen vacancy and promotes a more random spatial distribution. Besides, in previous experiments [31], pressure in Y-doped hafnia triggers the phase transition from orthorhombic polar phase to tetragonal phase at 12% Y-doped hafnia. But the formed tetragonal phase is not metastable upon pressure release, which might be attributed to the high Y concentration. In our calculations (Fig. S2(c)), the tetragonal strain in cubic phase emerges only at low Y concentrations, and disappears when the Y concentration exceeds 11.1%.

In summary, we find that the orthorhombic polar ferroelectric phase is stabilized by Y2O3-doping in bulk hafnia and zirconia. Further experiments to grow ferroelectric crystals and ceramics are encouraged, possibly leading to a new class of bulk ferroelectrics to join perovskites, with potentially broad uses, especially in extreme environments and at high temperatures, given the thermal and environmental stability of hafnia and zirconia. Besides, we find that the electric polarization of the orthorhombic polar phase in doped hafnia decreases with increasing Y concentration, but rises again as Y concentration reaches 14.8%. The negative longitudinal piezoelectric effect is maintained in doped hafnia bulk.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.