Introduction

In 2011, Boescke et al.1 discovered that Si-doped hafnia thin films have ferroelectricity; due to their high compatibility with conventional CMOS processes2, compared with conventional perovskite materials, hafnia thin films still have good ferroelectricity with decreasing thickness, declining by a few nanometers3,4. Due to the advantages of hafnia over other materials, the ferroelectricity of hafnia has good prospects for applications in ferroelectric field effect transistors (FeFET)5 and non-volatile memories (NVM)6, and piezoelectricity can also be applied to actuators7 and nanogenerators8.

Although the ferroelectricity of hafnia has promising applications, the origin of its ferroelectricity is still unclear, and recent studies generally attribute the ferroelectricity of hafnia thin films to the presence of the space group Pca21 phase (O-phase)9,10. However, the stable phase of hafnia is the monoclinic phase (M-phase) at room temperature11, the tetragonal (T-phase) and cubic phases (C-phase) at high temperatures12, the antiferroelectric phase (OA-phase) at high pressures13, and the O-phase is metastable. We know from first-principles computations that, without considering the effects of temperature on the crystals, the lowest energy of the hafnia crystals is the M-phase14,15, and the O-phase of hafnia does not behave as a stable phase in the absence of external factors; therefore, it is crucial to find factors that favor the O-phase.

According to studies conducted over the last decade, many factors cause hafnia crystals to be in the ferroelectric phase. Crystal point defects have a nonnegligible influence on the ferroelectricity of hafnia, among which dopants and oxygen vacancies (VO) in hafnia crystals have been extensively studied. The effects of dopants on the crystals of hafnia have been systematically investigated by Rohit Batra et al. through first-principles computations16, and the results show that dopants with high ionic radii and low electronegativities clearly favor the ferroelectric phase in hafnia, and positive trivalent dopants seem to be the most favorable for ferroelectricity; the thin films experimentally doped with Zr, Y, and La perform well regarding ferroelectricity17,18,19,20,21. In recent studies, Chen et al. verified that the concentration of \({{\rm{V}}}_{{\rm{O}}}\) has a nonnegligible effect on both the wake-up and fatigue effects of hafnia thin films through simulations22; He, Ma, and Liu et al. have revealed that charged \({{\rm{V}}}_{{\rm{O}}}\) can stabilize the O-phase and reduce the energy barrier from the nonpolar to the polar phase by first-principles computations23,24.

Another important factor is the effect of stress on the ferroelectricity of hafnia. In the past, many theoretical calculations and experiments have shown that in-plane compressive stresses (2D) are favorable for the formation of the O-phase25,26,27 and high pressures (3D) are favorable for the formation of the OA-phase13,28. Studies on uniaxial strain (1D) are lacking. Recently, an experimental study has shown that external surface compressive stress affects both the remnant polarization and fatigue effect of hafnia thin films29. The effect of stress on hafnia shows the important role of confinement in the ferroelectricity of the films, while the anomalous size effect is responsible for the different results from perovskite. This phenomenon indicates that epitaxial growth favors the formation of the ferroelectric phase.

The applied electric field (\({{\bf{E}}}_{{\bf{e}}}\)) is deeply involved in both the wake-up and fatigue effects of hafnia. Recently, Cheng et al. observed in an experimental study that the \({{\bf{E}}}_{{\bf{e}}}\) transforms the OA-phase into the O-phase (wake-up effect) and the O-phase back into the OA-phase (fatigue effect) with increasing electric field cycling10. It has been shown that the wake-up and fatigue effects of hafnia thin films occur due to the redistribution of \({{\rm{V}}}_{{\rm{O}}}\) under the action of the \({{\bf{E}}}_{{\bf{e}}}\)22. In conclusion, the impacts of the \({{\bf{E}}}_{{\bf{e}}}\) on the wake-up and fatigue effects should be realized through multiple pathways and is an important factor that cannot be neglected in the study of hafnia ferroelectric thin films.

Although many previous calculations showed that a single factor can promote the formation of the O-phase, it does not stabilize the O-phase or requires harsh conditions30,31. Seeking the joint action of multiple factors to stabilize the O-phase may be a more feasible method than using a single factor. While the joint action of multiple factors may make the study more complicated than the action of a single factor, it makes it easier to reach the conditions for stabilizing the O-phase25,32,33.

Based on these interesting phenomena, the combined effect of \({{\rm{V}}}_{{\rm{O}}}\), uniaxial strain and \({{\bf{E}}}_{{\bf{e}}}\) on hafnia-based crystals are investigated in this paper. Our calculations show that the increase in the concentration and charge state of \({{\rm{V}}}_{{\rm{O}}}\) and the uniaxial compression strain could reduce the energy difference between the O-phase (or OA-phase) and the M-phase, and the system finally favor the OA-phase. Besides, the \({{\bf{E}}}_{{\bf{e}}}\) in the same direction with the spontaneous polarization (\({{\bf{P}}}_{{\bf{s}}}\)) of the O-phase could promote the formation of O-phase against with M-phase, while only the amplitude greater than 4 \({\rm{MV}}\cdot {{\rm{cm}}}^{-1}\) is required to stabilize the O-phase. Therefore, the effects of above single factor cannot explain the existence of O-phase in the fabricated hafnia-based films. By calculating the systems under the coupling action of the \({{\rm{V}}}_{{\rm{O}}}\), strain and \({{\bf{E}}}_{{\bf{e}}}\), we are surprised to find that the O-phase could be stabilized easily in mild conditions.

Results and discussions

Oxygen vacancies

We refer to existing crystal structures for preliminary structure optimization34,35,36. The structural and parameters of five hafnia crystals after optimization are shown in Fig. 1 and Table 1 (HZO shown in Supplementary Fig. 5 and Table 1). The parameters of the structures of hafnia are in good agreement with the results from previous studies14,15,26,37. For the choice of the density function, because there have been studies comparing the results of the generalized gradient approximation (GGA) and local density approximation (LDA) density functions, there is only a quantitative difference (not qualitative) between the two for the calculation of crystals; the GGA density function is relatively consistent with the experiments for the calculation of crystal energy and structure11,38,39. Therefore, in this paper, the GGA density function is used uniformly in the calculation.

Fig. 1: Crystal structures of hafnia.
figure 1

The (001) projections of each crystal structure of hafnia, Hf, tri-, and tetra-coordinated O (\({{\rm{O}}}_{3},{{\rm{O}}}_{4}\)) atoms are shown in green, blue, and red colors, respectively. \({{\bf{P}}}_{{\bf{s}}}\) and \({{\bf{E}}}_{{\bf{e}}}\) denote the spontaneous polarization and the applied electric field, respectively.

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Table 1 Structural parameters of each hafnia crystal unit cell and \({P}_{{\rm{s}}}\).
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While neutral \({{\rm{V}}}_{{\rm{O}}}\) is more easily formed and exists more frequently than charged \({{\rm{V}}}_{{\rm{O}}}\), the \({{\bf{E}}}_{{\bf{e}}}\) make it possible to change the \({{\rm{V}}}_{{\rm{O}}}\) charge state40,41. We first investigate neutral \({{\rm{V}}}_{{\rm{O}}}\) and then charged \({{\rm{V}}}_{{\rm{O}}}\) on the basis of neutral \({{\rm{V}}}_{{\rm{O}}}\). In the five hafnia crystals studied, both \({{\rm{O}}}_{3}\) and \({{\rm{O}}}_{4}\) are present in the M, OA, and O-phases, while only \({{\rm{O}}}_{4}\) are present in the T and C-phases. Therefore, we first perform calculations for hafnia crystals with tri- or tetra-coordinated \({{\rm{V}}}_{{\rm{O}}}\) (\({{\rm{V}}}_{{\rm{O}}3}\), \({{\rm{V}}}_{{\rm{O}}4}\)) at a concentration of 1.56\(\%\) to select the type of \({{\rm{V}}}_{{\rm{O}}}\) that is the most favorable for the energy of crystals. Figure 2a shows the final results for the effects of different \({{\rm{V}}}_{{\rm{O}}}\) types for each crystal. For the M, OA, and O- phases, \({{\rm{V}}}_{{\rm{O}}4}\) is more energetically favorable than \({{\rm{V}}}_{{\rm{O}}3}\), and \({{\rm{V}}}_{{\rm{O}}3}\) is not effective for reducing the energy difference between the O/OA-phases and the M-phase; OA and O- phases are sensitive to the selection of the \({{\rm{V}}}_{{\rm{O}}}\) type. The possible reason for this result is the relatively stable structure formed by the \({{\rm{O}}}_{3}\) with the near-neighbor hafnium atoms at localized positions in the crystals. In previous studies, increasing the \({{\rm{V}}}_{{\rm{O}}}\) concentration was generally achieved by fixing the number of \({{\rm{V}}}_{{\rm{O}}}\) and shrinking the supercell, but this resulted in a nonuniform distribution of \({{\rm{V}}}_{{\rm{O}}}\), making the interaction between \({{\rm{V}}}_{{\rm{O}}}\) obvious. Based on this problem, we consider the effect of the \({{\rm{V}}}_{{\rm{O}}}\) distribution mode on the energies of the crystals and compare the effects of uniform and nonuniform (Supplementary Fig. 1) distributions of \({{\rm{V}}}_{{\rm{O}}4}\) in the supercell at a concentration of 3.12% on the energy of each crystal. Figure 2b shows the effects of the \({{\rm{V}}}_{{\rm{O}}}\) distribution mode on the energies of hafnia crystals. All crystal structures other than the C-phase exhibit nonuniform distributions of \({{\rm{V}}}_{{\rm{O}}}\), increasing the energies of the crystals, while the C-phase energy has a small decrease due to the nonuniform distribution of \({{\rm{V}}}_{{\rm{O}}}\). A possible reason for this phenomenon is that the high symmetry of the C-phase causes a large coupling of the crystal relaxation induced by the nonuniform distribution of \({{\rm{V}}}_{{\rm{O}}}\), and the reduced energy is greater than the increased energy of the \({{\rm{V}}}_{{\rm{O}}}\) interaction. In previous studies, it has been proposed that the \({{\bf{E}}}_{{\bf{e}}}\) effect uniformly distributes \({{\rm{V}}}_{{\rm{O}}}\) in the film20. Considering the effect of the subsequent electric field on \({{\rm{V}}}_{{\rm{O}}}\), it is best to choose the uniformly distributed \({{\rm{V}}}_{{\rm{O}}4}\) when studying \({{\rm{V}}}_{{\rm{O}}}\).

Fig. 2: Effect of VO on crystals.
figure 2

a Effect of \({{\rm{V}}}_{{\rm{O}}}\) type on crystals energy, \({E}_{x}^{{\rm{\alpha }}}\) is the supercell energy of the crystal \({\rm{\alpha }}\) when the \({{\rm{V}}}_{{\rm{O}}}\) concentration is x, \({E}_{x4}^{{\rm{M}}}\) is the M-phase supercell energy for a \({{\rm{V}}}_{{\rm{O}}4}\) concentration of x; b effect of distribution mode on crystal energy at 3.125% of \({{\rm{V}}}_{{\rm{O}}4}\) concentration, \({E}_{{\rm{n}}}^{{\rm{\alpha }}}\) and \({E}_{{\rm{u}}}^{{\rm{\alpha }}}\) indicate the nonuniform and uniform distribution of energy of the \({{\rm{V}}}_{{\rm{O}}4}\) in crystal \({\rm{\alpha }}\), respectively; c effect of uniformly distributed \({{\rm{V}}}_{{\rm{O}}4}\) concentration on crystals energy with reference to M-phase (\({E}_{x}^{{\rm{M}}}\)); d effect of \({{\rm{V}}}_{{\rm{O}}}\) concentration on the contribution of electrons to the dielectric constant (\({\varepsilon }_{{\rm{e}}}\)) and the \({P}_{{\rm{s}}}\) of the O-phase, the solid and dashed lines indicate \({\varepsilon }_{{\rm{e}}}\) and -\({P}_{{\rm{s}}}\), respectively.

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Based on the above studies, we select \({{\rm{V}}}_{{\rm{O}}4}\) in \(2\times 2\times 2\) hafnia crystal supercells with concentrations of 0%, 1.56%, 3.12%, and 6.25% that we uniformly distribute for the subsequent research in this paper (Supplementary Figs. 1 and 2). At the same time, we used molecular dynamics to check the rationale for the conformation of the \({{\rm{V}}}_{{\rm{O}}}\) (Supplementary Fig. 3). These \({{\rm{V}}}_{{\rm{O}}}\) concentrations are homogeneous in geometry, reducing some of the effects caused by the nonuniform distribution of \({{\rm{V}}}_{{\rm{O}}}\). Although combinations of different \({{\rm{V}}}_{{\rm{O}}}\) types may yield some interesting results, the main focus of this paper is the differences between the different types of \({{\rm{V}}}_{{\rm{O}}}\). Therefore, this phenomenon is not considered.

Figure 2c shows that the energies of the OA-, O-, C-, and T-phases decrease as the concentration of \({{\rm{V}}}_{{\rm{O}}}\) increases, but is always greater than that of the M-phase. When the \({{\rm{V}}}_{{\rm{O}}}\) concentration is less than 3.12%, the energy of the T-phase is most affected by \({{\rm{V}}}_{{\rm{O}}}\), followed by that of the C-phase; however, when the \({{\rm{V}}}_{{\rm{O}}}\) concentration is >3.12%, the effects of \({{\rm{V}}}_{{\rm{O}}}\) on the T and C-phases are weakened. The energy of the OA and O-phases decrease linearly with increasing \({{\rm{V}}}_{{\rm{O}}}\) concentration, which is in good agreement with previous studies21. Supplementary Fig. 4 shows that the b axis of all crystals, except for that of the M-phase crystal, decreases to varying degrees with increasing \({{\rm{V}}}_{{\rm{O}}}\) concentration. This phenomenon shows that \({{\rm{V}}}_{{\rm{O}}}\) and uniaxial strain mutually promote the stability of the O-phase.

Based on the different contributions of the crystal dielectric constant and the density functional perturbation theory of the dielectric constant, we can express the dielectric constant in Eq. 5 as \({\varepsilon }_{{\rm{r}}}={\varepsilon }_{{\rm{i}}}\left(x,y\right)+{\varepsilon }_{{\rm{e}}}(x,y)\), where \({\varepsilon }_{{\rm{i}}}\) and \({\varepsilon }_{{\rm{e}}}\) denote the contributions of ions and electrons to the dielectric constant of the crystal, respectively. Therefore, \({\varepsilon }_{{\rm{i}}}\) and \({\varepsilon }_{{\rm{e}}}\) are calculated separately at different \({{\rm{V}}}_{{\rm{O}}}\) concentrations. Figure 2d shows the \({{\rm{\varepsilon }}}_{{\rm{e}}}\) of hafnia crystals with different \({{\rm{V}}}_{{\rm{O}}}\) concentrations. The \({\varepsilon }_{{\rm{e}}}\) is clearly proportional to the \({{\rm{V}}}_{{\rm{O}}}\) concentration, which is due to the linear increase of unbonded electrons in the crystal with increasing \({{\rm{V}}}_{{\rm{O}}}\) concentration. The \({{\rm{V}}}_{{\rm{O}}}\) have the greatest effect on the \({\varepsilon }_{{\rm{e}}}\) of the C-phase, probably due to the high symmetry of the crystals which increases the electronic degrees of freedom. Since calculating \({\varepsilon }_{{\rm{i}}}\) using supercells is very computationally intensive, we calculate \({\varepsilon }_{{\rm{i}}}\) at 0% and 6.25% \({{\rm{V}}}_{{\rm{O}}}\) concentrations for the cubic phase. Our calculations show that \({\varepsilon }_{{\rm{i}}}\) is only 4% greater when the concentration of \({{\rm{V}}}_{{\rm{O}}}\) is 6.25% than when it is 0%. Therefore, in this paper, we consider that \({{\rm{V}}}_{{\rm{O}}}\) affects only \({\varepsilon }_{{\rm{e}}}\).

When studying the effects of \({{\rm{V}}}_{{\rm{O}}}\) on the \({P}_{{\rm{s}}}\) of crystals, we only consider the \({P}_{{\rm{s}}}\) changes resulting from atomic and electron displacements due to the absence of \({{\rm{V}}}_{{\rm{O}}}\); we neglect the \({P}_{{\rm{s}}}\) resulting from macroscopic symmetry breaking due to the absence of \({{\rm{V}}}_{{\rm{O}}}\). For the nonpolar phase, our calculations show that the \({{\rm{V}}}_{{\rm{O}}}\) concentration has a small effect on \({P}_{{\rm{s}}}\) and does not exhibit a clear pattern. We think that the effect of \({{\rm{V}}}_{{\rm{O}}}\) on the \({P}_{{\rm{s}}}\) of the nonpolar phase can be effectively explained by statistics and probability; the surface (001) exhibits central symmetry, and the probability of small \({{\bf{P}}}_{{\bf{s}}}\) occurring along the [010] or [\(0\bar{1}0\)] direction due to \({{\rm{V}}}_{{\rm{O}}}\) is equal. Therefore, the macroscopic \({P}_{{\rm{s}}}\) of the nonpolar phase that arises due to \({{\rm{V}}}_{{\rm{O}}}\) should be zero in a large system. However, the (001) crystal plane of the O-phase does not exhibit central or vertical symmetry along the [010] direction; thus, we infer that the \({{\bf{P}}}_{{\bf{s}}}\) in the [010] direction should have some relationship with the \({{\rm{V}}}_{{\rm{O}}}\) concentration. Figure 2d, Supplementary Table 2 and Supplementary Fig. 6 show that within a certain range of \({{\rm{V}}}_{{\rm{O}}}\) concentrations, the \({P}_{{\rm{s}}}\) of the O-phase increases with increasing \({{\rm{V}}}_{{\rm{O}}}\) concentration. The most likely reason for this phenomenon is that \({{\rm{V}}}_{{\rm{O}}}\) has an asymmetric force on the nearest \({{\rm{O}}}_{3}\).

Next, we further consider the effect of the charged \({{\rm{V}}}_{{\rm{O}}}\) on the hafnia crystals when the \({{\rm{V}}}_{{\rm{O}}}\) concentration is 6.25%. Figure 3a shows that when the positive charge of \({{\rm{V}}}_{{\rm{O}}}\) increases, the energies of the OA- and O-phases decrease rapidly relative to the M-phase, and the energy of the OA-phase decreases even more rapidly; the competing phase of the O-phase changes from the M-phase to the OA-phase due to the increase in the positive charge of the \({{\rm{V}}}_{{\rm{O}}}\), causing the energy difference between the O-phase and the stable-phase to first decrease and then increase.

Fig. 3: Effect of charged VO on crystals.
figure 3

a Energy, and b \(\triangle {P}_{{\rm{s}}}\) of the O-phase; Eα is the supercell energy of the crystal \({\rm{\alpha }}\) when the \({{\rm{V}}}_{{\rm{O}}}\) concentration is 6.25%, EM is the M-phase supercell energy, Es is the stable-phase supercell energy, the reference value of \(\triangle {P}_{{\rm{s}}}\) is the \({P}_{{\rm{s}}}\) when the charged \({{\rm{V}}}_{{\rm{O}}}\) is zero.

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By combining Figs. 2c and 3a and using the M-phase as a reference, we express the effects of \({{\rm{V}}}_{{\rm{O}}}\) on the OA and O-phases as follows:

$${E}^{{\rm{\alpha }}}={E}_{0}-{av}-{bq}$$
(1)

where \(v\) and \(q\) denote the number of \({{\rm{V}}}_{{\rm{O}}}\) and positive charges in the \(2\times 2\times 2\) supercell system, respectively, and \(0\le q\le 2v(0\le q\le 128\cdot x)\); \(a\) and \(b\) represent the effect coefficient of neutral \({{\rm{V}}}_{{\rm{O}}}\) and charge on energy, respectively. This result can be verified in the existing research (Supplementary Fig. 10)23,24. The Supplementary Fig. 9 feature further details. When the OA-phase becomes a stable structure, the concentration and positive charge of \({{\rm{V}}}_{{\rm{O}}}\) increase to inhibit the O-phase. Therefore, \({{\rm{V}}}_{{\rm{O}}}\) is most favorable to the O-phase when the boundary between the M-phase and OA-phase is selected. Figure 3b, Supplementary Figs. 7 and 8 show the effect of changing the charged \({{\rm{V}}}_{{\rm{O}}}\) on the \({P}_{{\rm{s}}}\) of the O-phase. With increasing positive charge, the \({P}_{{\rm{s}}}\) of the O-phase decreases at an accelerated rate, because of the relaxation of hafnium atoms and \({{\rm{O}}}_{3}\), while the \({{\rm{O}}}_{4}\) weakens the trend of decreasing \({P}_{{\rm{s}}}\).

Uniaxial strain and oxygen vacancies coupling

Only the \({{\bf{P}}}_{{\bf{s}}}\) of the O-phase of the five hafnia crystals changes with strain; that is, only the O-phase has the piezoelectric effect, not the other nonpolar phases. Therefore, studies on uniaxial strain have previously focused on the piezoelectric properties of the O-phase42,43; studies on uniaxial strain-driven phase transitions in hafnia are lacking. From the volume comparison of each crystal of hafnia it can be roughly inferred that compression and a small volume are favorable for the stabilization of the O- and OA-phases. Previous studies have shown that the formation of O- and OA-phases is favored at high pressures11. Although uniaxial strain induces in-plane stresses in thin films, this effect can make the calculation of in-plane stresses very difficult due to the effects of the substrate and the complex boundary conditions; thus, we do not consider the relaxation of the crystal in the (010) plane when studying uniaxial strain in this paper.

We consider the trend of the energy of different hafnia crystals with uniaxial strain at different \({{\rm{V}}}_{{\rm{O}}}\) concentrations in this section; and in this paper, uniaxial strain refers specifically to the change in the b-axis in the [010] direction, Supplementary Fig. 11 shows the effect of different uniaxial strains on hafnia crystals. First, we consider the effect of uniaxial strain on each hafnia crystal in the absence of neutral \({{\rm{V}}}_{{\rm{O}}}\). Figure 4a shows the following points: a) a large b-axis is favorable for the presence of the M-phase, and as the b-axis decreases (uniaxial compression), the OA-phase gradually stabilizes; b) although the O-phase is energetically more favorable than the M-phase in the large uniaxial compression state, it cannot stabilize because of the competition from the OA-phase; c) as the uniaxial compression strain increases, although the energy gap between the T- or C-phases and M-phase decreases, it is always much more energetically favorable than the M-phase and does not show a stabilizing tendency.

Fig. 4: Effects of uniaxial strain and VO coupling on crystal energies.
figure 4

a, b Variation of crystal energy with uniaxial strain at different \({{\rm{V}}}_{{\rm{O}}}\) concentrations, the energy is referenced to the strain-free M-phase (\({E}_{x}^{{{\rm{M}}}_{0}}\)); c effect of uniaxial strain on crystal energy in different \({{\rm{V}}}_{{\rm{O}}}\) charge states; d energy difference between O-phase and OA-phase. The strain is referenced to the \({{\rm{V}}}_{{\rm{O}}}\)-free and strain-free O-phase, \(i\) in \(x(i)\) denotes the \({{\rm{V}}}_{{\rm{O}}}\) charge state and defaults to 0 if not specified.

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Then, the uniaxial strain of the crystals is calculated for different \({{\rm{V}}}_{{\rm{O}}}\) concentrations and charge states. As shown in Fig. 4b and Supplementary Fig. 12: the addition of \({{\rm{V}}}_{{\rm{O}}}\) does not affect the trend of the energy of each crystal of hafnia; but with increasing \({{\rm{V}}}_{{\rm{O}}}\) concentration, a) there is a significant decrease in the b axis for all crystal structures, except for the M-phase, b) the uniaxial compression strain of the stable OA-phase decreases, c) the differences in energy between each crystal structure and the M-phase decrease, and d) the O-phase does not show stability. Figure 4c, Supplementary Figs. 13 and 14 show that the M-phase is relatively unstable and changes the relationship between uniaxial strain and its crystal energy when the crystals have large positive charges; this phenomenon should occur due to the coupling between uniaxial strain and charged defect-induced tensile and compressive stresses in the (010) plane. Finally, we calculate the energy difference between the O-phase and OA-phase with uniaxial strain at different \({{\rm{V}}}_{{\rm{O}}}\) concentrations and charges. Figure 4d and Supplementary Fig. 16 show that when the crystals are uncharged, the energy difference between the O and OA-phases decreases as the b axis decreases, and the increase in \({{\rm{V}}}_{{\rm{O}}}\) concentration stabilizes the OA-phase; when the crystal is positively charged, as the charge increases, the uniaxial strain gradually becomes more favorable to the OA-phase.

To clearly understand the effects of \({{\rm{V}}}_{{\rm{O}}}\) and uniaxial strain on crystal energies, we fit the energies of hafnia crystal supercells under the influences of the number of \({{\rm{V}}}_{{\rm{O}}}\) (\(v\)), the amount of charge (\(q\)) and the b axis strain \(\epsilon\) (reference value is the b-axis of a perfect O-phase supercell: \(\epsilon =(y-{b}^{{\rm{o}}})/{b}^{{\rm{o}}}\)).

$${E}^{{\rm{\alpha }}}(v,q,\epsilon )=A\left(v,q\right){\left(\epsilon -B\left(v,q\right)\right)}^{2}+C(v,q)$$
(2)

where \(A\left(v,q\right)\), \(B\left(v,q\right)\) and \(C\left(v,q\right)\) denote the sensitivity of the crystal energy to uniaxial strain, strain in the b-axis under different conditions, and energy of the crystal in the absence of external stress, respectively. The Supplementary Tables 3 and 4 and 5 feature further details about this topic.

Above we calculate the effects of uniaxial strain on the energies of hafnia crystals. Next, we will consider the effects of uniaxial strain on the \({P}_{{\rm{s}}}\) and dielectric constant of crystals (Supplementary Figs. 15 and 17). The effects of \({{\rm{V}}}_{{\rm{O}}}\) and uniaxial strain on the O-phase \({P}_{{\rm{s}}}\) of hafnia are shown in Fig. 5a. The figure clearly shows that the \({P}_{{\rm{s}}}\) of the O-phase increases as the b-axis decreases, which is in good agreement with previous studies42,44; uniaxial compression can increase the contribution of \({{\rm{V}}}_{{\rm{O}}}\) to \({P}_{{\rm{s}}}\). Then, we calculate the effects of charged defects on the \({P}_{{\rm{s}}}\) of the O-phase at different strain states. Figure 5b shows that as the compressive strain increases, the decrease in charged defects on \({P}_{{\rm{s}}}\) first decreases and then increases, and the extreme point moves to the lower left position with the increase in charged defects. To intuitively represent the effects of \({{\rm{V}}}_{{\rm{O}}}\) and uniaxial strain on the \({P}_{{\rm{s}}}\) of the O-phase, \({P}_{{\rm{s}}}\) is expressed as follows:

$${P}_{{\rm{s}}}\left(v,q,\epsilon \right)=D\left(\epsilon ,v\right)-F(\epsilon ,q)$$
(3)

where \(D\left(\epsilon ,v\right)\) indicates the effect of uniaxial strain and neutral \({{\rm{V}}}_{{\rm{O}}}\) on \({P}_{{\rm{s}}}\), and \(F\left(\epsilon ,q\right)\) indicates the reduction in charged defects on \({P}_{{\rm{s}}}\) at different strain states. The Supplementary information feature further details.

Fig. 5: Effect of uniaxial strain on the spontaneous polarization.
figure 5

a O-phase \({P}_{{\rm{s}}}\), the curve is a fit of the uniaxial strain to the mean value of the \({P}_{{\rm{s}}}\) at different \({{\rm{V}}}_{{\rm{O}}}\) concentrations; b changes in \({P}_{{\rm{s}}}\) for different charge states.

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Mechanical-electrical-chemical coupling

In previous work, numerous experiments and theoretical calculations have confirmed that the \({{\bf{E}}}_{{\bf{e}}}\) have an important impact on the wake-up effect of hafnia thin films22,45. Equation 5 clearly shows that the \({{\bf{E}}}_{{\bf{e}}}\) is energetically very favorable to the stabilization of the O-phase when the direction of \({{\bf{P}}}_{{\bf{s}}}\) is the same. In this study, only one particular direction of the \({{\bf{E}}}_{{\bf{e}}}\) is considered, i.e., the direction of parallel O-phase \({{\bf{P}}}_{{\bf{s}}}\), which can make a greater difference between polar and nonpolar phases under the action of the \({{\bf{E}}}_{{\bf{e}}}\). The maximum magnitude of the \({{\bf{E}}}_{{\bf{e}}}\) is specified to be 4 \({\rm{MV}}\cdot {{\rm{cm}}}^{-1}\), which can effectively meet the experimental requirements and ensure that the crystals do not degrade. In addition, we use Eq. 5 to calculate the energies of the crystals without considering the transformation of the nonpolar phases into the O-phase (i.e., wake-up effect) or the polarization reversal of the O-phase due to the \({{\bf{E}}}_{{\bf{e}}}\). Under the action of the \({{\bf{E}}}_{{\bf{e}}}\), Eq. 5 shows that the energy reduction of the crystals is determined by the dielectric constant, the amplitude and direction of \({{\bf{P}}}_{{\bf{s}}}\), and the \({{\bf{E}}}_{{\bf{e}}}\). For the nonpolar phases, the larger the dielectric constant and the \({{\bf{E}}}_{{\bf{e}}}\) are the more obvious the energy reduction in the crystals. For the polar phase, \({{\bf{P}}}_{{\bf{s}}}\) and the \({{\bf{E}}}_{{\bf{e}}}\) play a decisive role. When the direction of \({{\bf{P}}}_{{\bf{s}}}\) and the \({{\bf{E}}}_{{\bf{e}}}\) are the same, the energy decreases; the energy increases when the directions are opposite.

Figure 6a shows that although \({{\rm{V}}}_{{\rm{O}}}\) can effectively reduce the energy difference between the O-phase and M-phase in the stress-free state, the \({{\bf{E}}}_{{\bf{e}}}\) with an amplitude of more than 4 \({\rm{MV}}\cdot {{\rm{cm}}}^{-1}\) is still required to stabilize the O-phase at an \({{\rm{V}}}_{{\rm{O}}}\) concentration of 6.25%. Given the results of this calculation and combined with the previous calculations of uniaxial strain, the combined effect of \({{\rm{V}}}_{{\rm{O}}}\), uniaxial strain, and \({{\bf{E}}}_{{\bf{e}}}\) on hafnia crystals will be considered in the next sections.

Fig. 6: Effect of mechanical-electrical-chemical coupling on crystal energies.
figure 6

Effect of the \({{\bf{E}}}_{{\bf{e}}}\) on the energy of hafnia crystals at \({{\rm{V}}}_{{\rm{O}}}\) concentrations of 0 and 6.25% and a stress-free, b \(y\)=9.75, c y = 9.95, d \(y\)=10.15. (Calculated using Eq. 5). The same \({{\rm{V}}}_{{\rm{O}}}\) concentration and stress-free M-phase are used as energy references (\({E}_{x}^{{{\rm{M}}}_{0}}\)), the solid and dashed lines indicate an \({{\rm{V}}}_{{\rm{O}}}\) concentration of 0% and 6.25%, respectively.

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To observe the effects of uniaxial strain and \({{\rm{V}}}_{{\rm{O}}}\) concentration on the stabilization of the O-phase by the \({{\bf{E}}}_{{\bf{e}}}\), as shown in the subplot of Fig. 6, we consider the effects of the \({{\bf{E}}}_{{\bf{e}}}\) on the hafnia crystals at different uniaxial strains and at \({{\rm{V}}}_{{\rm{O}}}\) concentrations of 0 and 6.25%. By analyzing the subplot of Fig. 6, we can draw the following conclusions. As the uniaxial compressive strain increases (\(y\) decreases), a) the effects of \({{\rm{V}}}_{{\rm{O}}}\) on the reductions in the relative energies of crystals increase; the expression from the graph is that as the \(y\) value decreases, the energy difference of the same crystal at different \({{\rm{V}}}_{{\rm{O}}}\) concentrations increases; b) the amplitude of the \({{\bf{E}}}_{{\bf{e}}}\) stabilizing the O-phase decreases and does not change with the increase in \({{\rm{V}}}_{{\rm{O}}}\) concentration; c) the phase competing with the O-phase changes from the M-phase to the OA-phase, and this transition is advanced with the increase in \({{\rm{V}}}_{{\rm{O}}}\) concentration; and d) when the \({{\rm{V}}}_{{\rm{O}}}\) concentration increases, the amplitude of the \({{\bf{E}}}_{{\bf{e}}}\) in the stable O-phase first decreases and then increases.

The above calculations show the energy trends of each crystal of hafnia in the presence of neutral \({{\rm{V}}}_{{\rm{O}}}\), uniaxial compressive strain, and the \({{\bf{E}}}_{{\bf{e}}}\). However, to clearly show influencing trends of the three factors, we plot the phase diagram of each crystal of hafnia and HZO under the three factors in Figs. 7a and 8a to clarify the role of the three factors. We can find that it is difficult to stabilize the O-phase with a single or double factor, while a triple factor can stabilize the O-phase much easily. Then, we consider the effect of the charge state on the crystal with the \({{\rm{V}}}_{{\rm{O}}}\) concentration of 1.56%. Figures 7b and 8b show that when the competing phase of the O-phase is the M-phase, the increase in the charged defect promotes the stabilization of the O-phase; however, when the competing phase is the OA-phase the increase in the charged defect suppresses the O-phase, and the charged defects of HZO are more strongly inhibited than those of hafnia for the O-phase. This phenomenon is mainly caused by \(C\left(v,q\right)\) because the energy differences between the OA and O-phases gradually increase with increasing of charge, and the rate of increase in HZO is greater than that in hafnia.

Fig. 7: Phase diagram of hafnia.
figure 7

Under the action of \({{\rm{V}}}_{{\rm{O}}}\) (a concentration, b charge state (x = 1.56%)), uniaxial strain, and \({{\bf{E}}}_{{\bf{e}}}\). The reference for the strain is the perfect O-phase.

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Fig. 8: Phase diagram of HZO.
figure 8

Under the action of \({{\rm{V}}}_{{\rm{O}}}\) (a concentration, b charge state (x = 1.56%)), uniaxial strain, and \({{\bf{E}}}_{{\bf{e}}}\). The reference for the strain is the perfect O-phase.

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Analyzing Figs. 7 and 8 and Supplementary Fig. 18, the main function of strain and VO are to transform the competing phase of the O-phase from M-phase to OA-phase, and the simultaneously \({{\bf{E}}}_{{\bf{e}}}\) further induce the phase transition from OA-phase to O-phase. The \({{\rm{V}}}_{{\rm{O}}}\) concentration would not significantly affect the electric field required to wake up the OA-phase to O-phase, but greatly affect the strain required to transform M-phase to O-phase and OA-phase. This indicates that the \({{\rm{V}}}_{{\rm{O}}}\) and strain must be carefully tailored for inducing both ferroelectric and antiferroelectric properties. The strain should be preferred to \({{\rm{V}}}_{{\rm{O}}}\) in stabilizing O-phase. Normally, the electric field activation of over 1 \({\rm{MV}}\cdot {{\rm{cm}}}^{-1}\) seems to be necessary to stabilize O-phase in pure hafnia. The required electric field can be reduced below 0.7 \({\rm{MV}}\cdot {{\rm{cm}}}^{-1}\) in the HZO system, which indicates that suitable do** can help form O-phase against both M-phase and OA-phase.

For experiments, because of the interdependence of the factors and the advantage of strain for stabilizing the O-phase, strain should be prioritized in the experimental preparation of hafnia ferroelectric thin films, and then the \({{\rm{V}}}_{{\rm{O}}}\) and the \({{\bf{E}}}_{{\bf{e}}}\) should be designed sequentially. For example, in preparation of hafnia ferroelectric thin films using atomic-layer-deposition method. The suitable strain-induced substrate or cap** layer needs to be considered first. Besides, it might be useful to control the \({{\rm{V}}}_{{\rm{O}}}\) concentration of around 0.6% when the strain is ~−1.5%. Suitable chemical precursors and the plasma participation are available to adjust the \({{\rm{V}}}_{{\rm{O}}}\). The \({{\bf{E}}}_{{\bf{e}}}\) of over 1.5 \({\rm{MV}}\cdot {{\rm{cm}}}^{-1}\) can be carried out to transform the OA-phase to O-phase. Furthermore, when the uniaxial compressive strain decreases, the \({{\rm{V}}}_{{\rm{O}}}\) concentration and the \({{\bf{E}}}_{{\bf{e}}}\) should increase accordingly. Although the selections of \({{\rm{V}}}_{{\rm{O}}}\), uniaxial strain, and \({{\bf{E}}}_{{\bf{e}}}\) are specific and do not consider the complex boundaries and mechanical conditions present in the experiment, we believe that this phenomenon does not change the effect of a particular factor on hafnia-based crystals. This section shows that it is feasible to stabilize the hafnia-based ferroelectric phase by multifactor coupling, and the coupling of multiple factors has obvious advantages over a single factor. We compare the phase diagrams qualitatively with the experimental results in the Supplementary Fig. 19.

In summary, the individual and combined effects of the \({{\rm{V}}}_{{\rm{O}}}\), uniaxial strain, and the \({{\bf{E}}}_{{\bf{e}}}\) on the crystal energy of hafnia-based are systematically investigated. The increase of \({{\rm{V}}}_{{\rm{O}}}\) could reduce the energy differences between ferroelectric and monoclinic phase, but could not render the ferroelectric phase as the most stable one. Since the \({{\bf{P}}}_{{\bf{s}}}\) shows a dependency on the concentration and charge state of \({{\rm{V}}}_{{\rm{O}}}\), it indicates that there is a coupling effect between the \({{\rm{V}}}_{{\rm{O}}}\) and \({{\bf{E}}}_{{\bf{e}}}\). When both \({{\rm{V}}}_{{\rm{O}}}\) and uniaxial strain are present, the uniaxial strain can independently stabilize the antiferroelectric phase and promote the stabilization of the ferroelectric phase, and the increase in the \({{\rm{V}}}_{{\rm{O}}}\) concentration and charge state reduces the strain demand in stabilizing the ferroelectric phase. In addition, the uniaxial compressive strain increases the \({{\bf{P}}}_{{\bf{s}}}\) of the ferroelectric phase, which will enhance the effect of the \({{\bf{E}}}_{{\bf{e}}}\) on the phase stability. These indicate that the stabilization of ferroelectric phase in hafnia is a typical mechanical-electrical-chemical coupling situation. When considering \({{\rm{V}}}_{{\rm{O}}}\), uniaxial strain and \({{\bf{E}}}_{{\bf{e}}}\) simultaneously, it will achieve the purpose of stabilizing the ferroelectric phase easily. This work provides an explanation for the typical wake-up effect and theoretical guidance to obtain and stabilize ferroelectric phase in hafnia.

Methods

Computational condition

In this paper, we prioritize the effect of \({{\rm{V}}}_{{\rm{O}}}\) before considering the impacts of uniaxial strain and \({{\bf{E}}}_{{\bf{e}}}\) on each crystal of hafnia. The energies of each crystal of hafnia under \({{\rm{V}}}_{{\rm{O}}}\) and the uniaxial strain conditions are calculated by first-principles computations. The structures of the hafnia considered in this study include the M-phase, OA-phase, O-phase, T-phase, and C-phase. The structures of hafnia are drawn and first-principles computations (DFT) are obtained using VESTA software and the Vienna Ab initio Simulation Package (VASP)46,47. For electronic-structure calculations, we use the projector-augmented wave methodology and the Perdew-Burke-Ernzerhof exchange-correlation functional48,49. The ionic contribution to the dielectric constant of the crystals is calculated using a unit cell to conserve computational resources, while all other calculations are obtained by a \(2\times 2\times 2\) supercell. For the calculations of the supercell and unit cell, the Brillouin zone k-points are chosen as 5 × 5 × 5 and 10 × 10 × 10 centered meshes50, respectively. A basis set of plane waves with kinetic energies reaching 650 eV is used to represent the wave functions, and the convergence criterion for the ionic minimization during crystal structure optimization is an atomic force smaller than 0.03 \({\rm{eV}}{{\text{\AA }}}^{-1}\); for the electronic minimization, an energy difference of smaller 10−6 eV is chosen as the criterion. The calculation of neutral \({{\rm{V}}}_{{\rm{O}}}\) allows for crystal shape variation, and the calculation of charged \({{\rm{V}}}_{{\rm{O}}}\) keeps the shape constant. Uniaxial strain is performed in the direction parallel to the \({{\bf{P}}}_{{\bf{s}}}\) of the ferroelectric phase (the [010] direction). Considering the influences of the substrate and the complex boundary conditions, structural optimizations related to uniaxial strain all keep the crystal shape constant.

When calculating the effect of the \({{\bf{E}}}_{{\bf{e}}}\) on each crystal of hafnia, we have determined Eq. 5 by referring to Eq. 4 and combining it with the findings of our study51, to obtain the energy of each crystal supercell under all conditions.

$$E={E}_{{\rm{DFT}}}-\varOmega {\bf{P}}\cdot {{\bf{E}}}_{{\bf{e}}}$$
(4)
$${E}^{\alpha }={E}_{{\rm{DFT}}}^{\alpha }(x,y)-{E}_{{\rm{DFT}}}^{{{\rm{M}}}_{0}}(x)-{\varOmega }^{\alpha }(x,y)[({\varepsilon }_{0}({\varepsilon }_{{\rm{r}}}^{\alpha }(x,y)-1){{\bf{E}}}_{{\bf{e}}}+{{\bf{P}}}_{{\bf{S}}}^{\alpha }(x,y)]\cdot {{\bf{E}}}_{{\bf{e}}}$$
(5)

where \(x\) denotes the \({{\rm{V}}}_{{\rm{O}}}\) concentration and \(x=v/N({\rm{O}})\) (\(v\): the number of \({{\rm{V}}}_{{\rm{O}}}\); \(N\left({\rm{O}}\right)=64\): total number of oxygen atoms in the perfect supercell); \(y\) denotes the length of the supercell in the [010] direction; \({E}_{{\rm{DFT}}}^{{\rm{\alpha }}}\) denotes the energy of the crystal \(\alpha\) obtained by DFT calculation; \({E}_{{\rm{DFT}}}^{{{\rm{M}}}_{0}}(x)\) denotes the lowest energy of the M-phase at the \({{\rm{V}}}_{{\rm{O}}}\) concentration of \(x\); \({\varOmega }^{{\rm{\alpha }}}\) denotes the volume of crystal \({\rm{\alpha }}\); \({\varepsilon }_{{\rm{r}}}^{{\rm{\alpha }}}\) denotes the dielectric constant of crystal \({\rm{\alpha }}\) in the [010] direction; \({{\bf{P}}}_{{\bf{s}}}^{{\boldsymbol{\alpha }}}\) denote the \({{\bf{P}}}_{{\bf{s}}}\) of the crystal \({\rm{\alpha }}\); and \({E}^{{\rm{\alpha }}}\) denotes the energy of the \({\rm{\alpha }}\)-crystal supercell. Density functional perturbation theory (DFPT) and modern polarization theory (Berry-phase method) calculations are performed using VASP to obtain the dielectric constant and the \({{\bf{P}}}_{{\bf{s}}}\) of hafnia, respectively52,53,54,55.