Introduction

Ferroelectric materials are technologically important and promising for nonvolatile memories owning to their fast switchable polarization and relatively low energy consumption1,2. However, the size effect in conventional perovskite ferroelectrics has suppressed their further down-scaling when silicon semiconductor process is coming to a sub-ten nanometer era3. Recently, the discovery of robust scale-free ferroelectricity in MO2 (M = Hf, Zr) based fluorite thin films sheds light on next-generation ferroelectric-based nano-electronic memories4,5,6.

The fluorite-based ferroelectrics are distinct from conventional perovskite ferroelectrics because of their complex origin of ferroelectricity in relation to metastable ferroelectric phase and defects. Both HfO2 and ZrO2 show stable monoclinic phase in their bulk form. While, some high-temperature or high-pressure phases, such as cubic and tetragonal, can be stabilized and exist metastably at room temperature in the fluorite structure via doping and nano-structuring7. However, none of the above structures are reported to be polar until the discovery of polar orthorhombic phase in Mg-doped ZrO2 at cryogenic temperature8. Since then, tremendous MO2 (M = Hf, Zr) based thin films doped with different dopants were reported to show ferroelectric orthorhombic phase on either Si or oxide substrates as well as with noble metal or metal nitride electrode9. Moreover, Y-doped HfO2 single crystal and epitaxial rhombohedral Hf0.5Zr0.5O2 thin film ferroelectrics prepared by state-of-the-art methods have further advanced the understanding of the polymorphic nature and phase control of HfO2- and ZrO2-based fluorite oxides10,11.

Compared with pure HfO2 and ZrO2 that have limited or nearly no polarization6,12,13, dramatically enhanced ferroelectricity can be induced in doped cases generally due to the stabilization of polar orthorhombic phase in either ultrathin film or small grains from an average structure point of view9,10,14,15,16,17. It is widely accepted that defect plays an important role in not only the induced/enhanced ferroelectricity but also the fatigue and bipolar-cycling performance18. Recently, it is also suggested that a small rhombohedra distortion in the polar orthorhombic phase by structural defects can effectively improve the ferroelectricity14. However, there is still no clear point to bridge the relation between point defects and ferroelectricity at the atomic level in fluorite ferroelectrics. Lee et al.4 recently pointed out that flat polar phonon bands with localization of electric dipoles lead to the scale-free ferroelectricity in HfO2, which as well explains the generally high coercive field and makes it distinct from the conventional ferroelectrics with spread dipoles. This atomic level understanding of the ferroelectricity indicates that the mechanism of the defect-enhanced ferroelectricity may be also significantly different from that of the conventional ferroelectrics where defects either negatively act as leaky center or positively improve ferroelectric polarization via correlation between defect-dipoles and intrinsic polarization19,20,21,22,23,24,25. Actually, even in non-ferroelectric metal oxides, defect-dipoles with relatively large dipole moment can lead to high-performance colossal dielectric permittivity (>10000) behavior26,27,28, collective nonlinear electric polarization in either bulk ceramics or ultrathin films29,30. In this consideration, we believe that it is hard to neglect the effect of defect-dipole polarization on the macroscopic polarization of fluorite-based ferroelectric system where dominant monoclinic phase and slight amount of metastable polar orthorhombic phase coexists. Therefore, how the defect affects the ferroelectricity and how it correlates with its nearby environment at the atomic level remains an important question.

Here, by using the first principal density functional theory (DFT) calculation, similar as that of the HfO2 system, we confirmed that the flat polar phonon bands in orthorhombic-phase (o-phase) ZrO2 with localized electric dipoles to give rise to the scale-free ferroelectricity. In order to figure out the bridge between defects and ferroelectricity at the atomic level, by taking oxygen vacancy-only (VO -only) La-doped MO2 (M = Hf, Zr) as examples, we report a dramatically enhanced flattening of polar phonon bands. The polar phonon bands in La-MO2 (M = Hf, Zr) becomes significantly flattened at an optimal doping level, compared with pure ones. The dramatically flattened polar phonon bands fundamentally enhances the polar states in orthorhombic phase, which is consistent with the enhanced ferroelectricity in La-doped HfO2 and also imply a similarly enhanced ferroelectricity in doped ZrO2. We further found that point-defect-dipoles polarize surrounding host ions, which forms partially localized electric dipoles and partially diffuse the sharp zero-width domain walls. The localized electric dipoles and spread dipoles together lead to an enhanced polar state should provide an atomic-level insight for the origin of ferroelectricity in doped fluorite ferroelectrics. However, the lower energy barrier with larger polarization of VO-only doped MO2 suggest that VO and local lattice distortion should be considered to further optimize their ferroelectric properties. This research is believed to provide a missing link between defects and induced ferroelectricity in fluorite ferroelectrics, which would inspire their property-optimization.

Results and discussion

Transformation of phonon band structure in pure fluorite MO2 (M = Hf and Zr)

Both HfO2 and ZrO2 mainly have three kinds of crystal phases31, including monoclinic phase (m-phase, space group P21/c) at room temperature, tetragonal phase (t-phase, space group P42/nmc) and cubic phase (c-phase, space group Fm3m) at high temperature32. Orthorhombic phase (o-phase, space group Pca21) was discovered under extreme conditions (high pressure and temperature)33,34. To determine the origin of polar orthorhombic phase in fluorite MO2 (M = Hf and Zr), we calculated the free energies of the four phases after structural relaxation. Figure 1a show the free energy difference ΔE between m-phase and other phases by ΔE = Eother-phase−Em-phase (in unit meV/f.u.). The lowest energy phase (m-phase), which exists at room temperature, is the most stable phase for MO2 (M = Hf, Zr). On the contrary, t-phase and c-phase just exist at very high temperature. The smaller ΔE of the o-phase than t-phase suggests more metastability of o-phase. It should be noted that among the four phases, only o-phase is polar.

Fig. 1: The origin of the ferroelectricity in fluorite MO2 (M = Hf, Zr).
figure 1

a Calculated difference of total energy ΔE for crystal m-phase, o-phase, t-phase, and c-phase for fluorite MO2 (M = Hf, Zr). ΔE =Eother-phase-Em-phase (in unit meV per formular unit MO2, meV/f.u.), green and red spheres indicate Hf/Zr and O, respectively. b, c Phonon dispersion spectra of c-HfO2 and c-ZrO2 and their symmetry-lowering steps. The blue dots labeled Γ15, X5, and X’2 indicate the primary instability modes in Γ and X wave vectors. d The phonon modes Γ15, X5, and X’2 correspond to transformations to polar-, antipolar-, and t-phase, respectively. The Γ15 and X5 modes condense in phase with equal magnitude to generate an orthorhombic structure that is periodically sandwiched between dead spacer layers. The arrows denote the polarization due to the displacement of oxygen atoms. e, f The above two inset images show the orthorhombic structures with polarization up and polarization down respectively (indicated by the dashed rectangular region) and periodically sandwiched between spacer layer. The green spheres indicate Hf/Zr, while the red spheres indicate oxygen.

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To analyze the symmetry-lowering process, we calculated the phonon spectra of c-phase HfO2 and ZrO2 (c-HfO2 and c-ZrO2) as shown in Fig. 1b, c. The blue dots labeled Γ15, X5, and X’2 indicate the primary modes in Γ and X wave vectors where the first two respectively correspond to polar and antipolar phonon modes that combine in-phase with equal magnitude to generate an orthorhombic structure that is ferroelectrically active and periodically sandwiched between dead spacer layers. The structurally instable X’2 modes at ω = −6.80 THz in HfO2 and ω = −5.75 THz in ZrO2 involve the antiparallel x-axis displacements of neighboring oxygen atoms in the yz plane, which transforms c-phase to t-phase through the condensation of X’2 mode with zero net polarization (Fig. 1d). This fact significantly suggests the main contribution from anion oxygen displacement rather than cation in ZrO2 compared with conventional perovskite ferroelectric where ferroelectricity is originated from displacement of cations. This result suggests that the symmetry of the c-phase is lowered to o-phase and can generate mixed antiferroelectric and ferroelectric states in both HfO2 and ZrO2, which is consistent with the symmetry lowering process reported in HfO24.

The transition energy barrier between the two ferroelectric configurations (Pdown and Pup) via spacer layer has been calculated by nudged elastic band (NEB) method35 as shown in Fig. 1e. The flatness of phonon bands demonstrates nearly zero group velocity and domain walls propagate slowly by hopping the large energy barrier (1.35 eV in HfO2 and 1.16 eV in ZrO2) as shown in Fig. 1e, which is two orders of magnitude higher compared with the energy barrier (0.02 eV) of conventional perovskite ferroelectrics with diffused domain walls4. The spontaneous polarization (Ps) in o-phase MO2 (M = Hf, Zr) was calculated by Berry phase and NEB method between polarization up and down o-phase. The energy-polarization curve (E-P) conforms to the double-well potential of Landau-Ginsberg-Devonshire theory. The PS values of HfO2 is 70.00 µC·cm^-2, which is close to other theoretical calculations35. Compared with HfO2, though o-ZrO2 shows a slightly lower PS value of 64.70 µC·cm^-2, its but much lower energy barrier indicates a lower coercive field compared with HfO2.

We further uncovered the phonon band structure of the o-phase with polar flat phonon band structures highlighted in blue color as shown in Fig. 2. Both o-HfO2 and o-ZrO2, and o-Hf0.5Zr0.5O2 show a small negative frequency at Γ point, which should be partially due to the inaccuracy of the calculation and partially due to that o-phase is metastable since o-phase only exists under some extreme conditions, including thin films down to nanometer scale, high temperature and pressure (more detail see Supplementary Figs. 1–3). It should be noted that there is an obvious signature of flat polar phonon bands indicated by the two blue lines in Fig. 2a, b compared with c-phase cases (Fig. 1b, c). There are several distinct flat polar phonon bands (blue line) from Γ to Y wave vectors by the partially shown vibration modes with obvious oxygen displacement where the superposition of the two bands at Γ and Y gives rise to the localized electric dipoles similarly as shown in Fig. 1d. Here, by taking the bands with frequency difference Δω between Γ to Y wave vectors within 0.5 THz as flat phonon bands, nearly ~29.7%, ~27.8%, and ~35.1% of the phonon bands are flat among the band structures in HfO2, ZrO2, and Hf0.5Zr0.5O2, respectively. The substantial amount of flat polar phonon bands indicates an increased possibility of the coherent polar modes, which should be intrinsically responsible for its zero-width domain wall and scale-free ferroelectricity17,35. These results suggest that this kind of flat polar phonon behavior is a universal phenomenon in fluorite-based oxide.

Fig. 2: Flat polar phonon bands in o-phase MO2 (M = Hf, Zr).
figure 2

a Phonon spectra of o-HfO2 where the flat polar phonon eigenmodes from Γ (~9.66 THz) and Y (~9.71 THz) wave vectors were indicated, respectively. b Phonon spectra of o-ZrO2 where the flat polar phonon eigenmodes from Γ (~9.07 THz) and Y (~9.42 THz) wave vectors were indicated, respectively. c Phonon spectra of o-Hf0.5Zr0.5O2 where the flat polar phonon eigenmodes from Γ (~11.58 THz) and Y ( ~ 11.70 THz) wave vectors were indicated, respectively. The blue lines indicate the flat polar phonon band where the frequency difference Δω between Γ to Y points is within 0.5 THz. The red spheres and aqua green spheres indicate oxygen and host cation, respectively.

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Fig. 3: Point-defect-induced flattening of phonon band structures in La-doped fluorite MO2 (M = Hf, Zr).
figure 3

a Mapping of reported typical maximum remnant polarization values in pure and doped fluorite-based metal oxides MO2 (M = Zr, and Hf) with different dopants6,14,36,37,38,39,40,41,42,43,44. b, c Comparison of the absolute values of the frequencies (Δω< 0.5 THz) of the polar bands from Γ to Y wave vectors for pristine and the doped cases with the same size supercells. d Phonon spectra of La12.5%Hf87.5%O2–6.25% where the flat polar phonon modes from Γ (~15.08 THz) and Y (~15.10 THz) wave vectors indicated by e and f, respectively. g Phonon spectra of La12.5%Zr87.5%O2–6.25% where the flat polar phonon modes from Γ (at 15.02 THz) and Y (at 15.05 THz) wave vectors at ~15.1 THz were indicated by h and i, respectively. Vo-nonpolar and Vo-polar mean the site of VO at nonpolar region (spacer layer) and polar region (ferroelectric layer), respectively. j Evolution of energy barrier ΔE and spontaneous polarization (PS) values in doped MO2 (M = Hf, Zr).

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Defective and stressed fluorite MO2 (M = Hf and Zr)

As we mentioned before, for pure fluorite MO2 (M = Zr and Hf), it is either difficult to see obvious ferroelectricity or the ferroelectricity is relatively weaker compared with doped cases. As shown in Fig. 3a, the reported highest values of remnant polarization (Pr) in the rare-earth doped cases or solid solution of Hf05Zr0.5O2 are generally at least two times larger than that of the pure ones followed by a stabilized o-phase6,14,36,37,38,39,40,41,42,43,44. Though ZrO2 thin film has been reported to have Pr as high as over 50 μC·cm^-2 after annealing in N2 atmosphere, but the high leaky current makes it hardly to neglect the substantial contribution of defects like oxygen vacancy in addition to texture growth14,37. Therefore, the polar o-phase in MO2 (M = Zr, and Hf) can be stabilized by VO and additional dopants, such as La, Si, Al, Y, consequently, inducing or enhancing ferroelectricity6,10,39. Both Hf0.5Zr0.5O2 and ZrO2 show stable o-phase and enhanced ferroelectricity in atomically thin films deposited by ALD on Si substrate5,6. To understand the effect of point-defects on the ferroelectricity in fluorite binary oxides, it is necessary and important to investigate the flat polar phonon bands in the doped cases.

In this paper, we mainly focus on La-doped MO2, which exhibits relatively high remnant polarization15. As the La concentration increases, the energy difference ΔE of each formula unit between o-phase and m-phase decreases gradually, indicating an enhanced stabilization of o-phase (Supplementary Fig. 4a). This is consistent with previous studies that a suitable concentration of dopant enhances the stability of ferroelectric o-phase. We calculated the phonon spectra of LaxM1-xO2-δ (x = 6.25%-50%, M = Hf and Zr), in which oxygen vacancies were introduced for charge neutrality. The coercive field above mega-volt (MV) level in ZrO2 and HfO2-based ferroelectric system should be large enough to manipulate the modes below 25 THz compared with that below 10 THz in conventional perovskite ferroelectric with only kV level coercive field (Supplementary Fig. 4b).

In order to more intuitively distinguish the degree of dispersion under different doping concentrations, Δω of each phonon line was calculated (Supplementary Figs. 5 and 6). We set Δω < 0.5 THz as flat phonon band and calculated the total number of flat phonon bands in MO2 (M = Hf, Zr). Since the main ionic displacement is from oxygen and there is no obvious cation displacement, we will mainly consider the modes associated with oxygen displacement to contribute to the ferroelectricity in doped fluorite o-MO2. To our surprise, all the La-doped MO2 systems show significant flattened phonon band structures (more details see Supplementary Figs. 5 and 6). We have calculated supercells with sizes of 1 × 1 × 1, 1 × 2 × 1, and 2 × 2 × 1 for comparisons. The supercell size of the 1 × 1 × 1 is the same as that of the VO-only-doped cases, while 1 × 2 × 1 supercell is the same as that of the 25% La-doped HfO2, 2 × 2 × 1 supercell is the same as that of the 12.5% La-doped HfO2, 2 × 2 × 2 supercell is the same as that of the 6.25% La-doped HfO2. These are similar as the cases in ZrO2. As shown in Fig. 3b, the distribution frequency of the flat phonon bands (Δω< 0.5 THz) shows no obvious difference between pure o-HfO2, VO-only doped cases and 50% La doped case (more details see Supplementary Fig. 7). However, more modes become flat for La doped HfO2 in the doping range of 6.25% to 25%. Figure 3c shows similar significantly flattened bands were observed in 12.5% and 25% La-doped ZrO2 compared with that of the pure o-ZrO2, VO-only doped cases and 50% La-doped case (Supplementary Fig. 8). With this in mind, the dramatically flattened phonon bands would suggest a quasi-phonon-glass behavior and much lower thermal conductivity in specific direction45,46. Here, Vo at both polar layer (Vo-nonpolar) and nonpolar layer (Vo-nonpolar) are considered for the Vo-only doped case. Thus, it may also be expected to induce a giant thermal rectification effect to cater for the thermal dissipation in ferroelectric MO2 based integrated circuits. This result is expected to extend the research of doping-induced thermal rectifying effect and would leads to potential applications for thermal rectifying materials and devices. These results suggest that VO is helpful to transform nonpolar phase to polar phase and stabilize the o-phase, which is somehow consistent with the reversible oxygen vacancy related phase transition and ferroelectricity in hafnia-based ferroelectric devices47,48,49. The much of flat polar phonon bands with La-VO defect pairs compared with VO-only doped case suggests that the synergistic effect of defect-complex on the flattening of the polar phonon bands.

To confirm the polarization contribution of flat polar phonon modes, we calculated their corresponding lattice vibration modes that show net polarization after the condensation of modes from Γ to Y wave vectors. We mainly present two typical vibration modes for each spectrum by taking the 12.5% La-doped MO2 (M = Hf, Zr) as examples where blue lines indicate the flat polar phonon bands (Fig. 3d). There is no obvious polar phonon band in the non-flat ones, which means that the main polarization contribution is from flat polar phonon bands. The selected typical flat polar phonon modes (blue dots) point to their lattice vibration modes. It is obvious that the two flat polar phonon modes at Γ and Y wave vectors not always condense in-phase with equal magnitude to generate an orthorhombic structure that is periodically sandwiched between spacer layers, leading to a net polarization rather than an antipolar behavior like that shown in Fig. 1d–f. In o-MO2 (M = Hf, Zr). The antipolar spacer layers lead to a sharp/zero-width domain walls. While in doped HfO2, such as La12.5%Hf87.5%O2–6.25%, the two typical flat polar phonon modes at Γ (~15.08 THz) (Fig. 3e) and Y (~15.10 THz) (Fig. 3f) points visibly not tend to condense in-phase with equal magnitude to generate a periodic antipolar/polar structure. Similar spread dipoles were also observed in doped ZrO2, such as La12.5%Zr87.5%O2–6.25% with the two typical flat polar phonon modes at Γ (~15.02 THz) (Fig. 3h) and Y (~15.05 THz) (Fig. 3i). Therefore, doping engineering would easily lead to spread dipoles and partially diffuse domain walls compared with pure ones with fully localized dipoles and zero-width domain walls. We also similarly confirmed the partially diffused domain walls by analyzing the flat polar phonon modes in La-doped MO2 with other doping levels (Supplementary Figs. 5 and 6 and Table 1–2).

The effect of point-defect-driven flattening of polar phonon bands in doped fluorite MO2 (M=Hf, Zr) is believed to increase the flat polar phonon bands and thus enlarge the possibility of polar modes and magnitude of dipole moment, leading to a stabilized o-phase and an enhanced ferroelectricity. We further studied the double-well energy landscape of the doped LaxM1-xO2. The energy barrier ΔE (eV/f.u.) first increases when x reaches to 12.5% and then decreases when x increases to 25% accompanied with a dramatically decreased PS. The PS of 12.5% La-doped HfO2 slightly decreases from 70.0 μC·cm^-2 to 65.8 μC·cm^-2, while, the PS increases from 64.7 μC·cm^-2 to104.5 μC·cm^-2 for 12.5% La-doped ZrO2. For Vo-only doped case, the energy barrier obviously decreases at doping level of 12.5%. For example, the 12.5% VO-doping at polar layer (VO-polar) in HfO2 system decreases the energy barrier from 1.345 eV to 1.007 eV whilst maintaining a high PS even above 70.4 μC·cm^-2. The much lower energy barrier ΔE and higher PS for VO-polar-only doped case suggest a better ferroelectric performance compared with the case where VO locates at the antipolar layer (VO-nonpolar) as well as La-doped ones. However, both the more flattened polar phonon modes for La-doped HfO2 with doping level in the range of 6.25%-12.5% and for VO-only doped ZrO2 is consistent with the generally higher ferroelectric performances experimentally observed in doped MO26,14,36,37,38,39,40,41,42,43,44. Since La-doped ZrO2 shows similar enhanced flattened polar phonon bands, it is reasonable to predict high-performance ferroelectricity in 6.25–12.5% La-doped ZrO2 and VO-only doped ZrO2.

The above results suggest a key role of VO to achieve high-performance ferroelectricity in fluorite MO2 compared with the La dopant. However, for the origin of the enhanced flattened bands in doped structure, we think there are three main reasons. First, there is no standard definition of flat phonon bands. We take the band with frequency difference Δω between Γ to Y wave vectors within 0.5 THz as flat phonon band, which is actually a kind of softening of the phonon bands and have been reported in defect-engineered materials50,51,52. Therefore, the doping-induced disorder would lead to the softening of phonon mode and flatten the phonon band, which enhances the flattening of the phonon bands. Second, in the doped cases, due to the local symmetry breaking, the lattice vibration mode becomes different from that in pure case. Especially the antipolar phonon band no longer conserves the antipolar state and results even in a net polarization. In this case, it may become even quite polar that can be evidenced by the local defect-dipolar states with net total polarization in each direction detailed in Fig. 5a. Finally, in La-doped structure, the longer bond-length of La-O (2.2–2.8 Å) than that of Zr–O (2–2.2 Å) would be able to decrease the elastic interaction between neighboring oxygen and cation around the point defect that is consistent with literature4. By taking 12.5% La-doped ZrO2 as an example (Supplementary Figure 9), doping significantly decrease the spring interaction between neighboring ions around the point-defects. Therefore, more phonon bands become flat and polar due to the longer interaction between some host ions around the point defects. From a thermodynamic point of view, the energy difference between o-phase and m-phase decreases with La doped. The flattened bands reflect the stability of ferroelectric phase with La doping, and also imply that the o-phase is more accessible with La doping. As a whole, we think the point-defect-induced local symmetry breaking, stabilized o-phase, and the weaker interaction between the host ions around the point-defects would be responsible for the flattened phonon bands in the La-doped MO2 (M = Zr, Hf).

Since both defects and epitaxial strain can affect the ferroelectricity, we also analyzed the effect of epitaxial strain on the phonon band structures. Fig. 4a, b show that there is no obvious variation in the distribution of the flat polar phonon band frequencies (Δω) of ±4% uniaxially and biaxially strained o- HfO2 in different directions (More details see Supplementary Figs. 10 and 11). Similar results were observed in strained o-ZrO2 as shown in Fig. 4c, d. As we expected, the results under biaxial strains were basically consistent with the results under uniaxial strains. Above all, within 5% strain, the flatness of phonon band in o-phase is stable and robust. These results suggest that point-defect plays a more important role in the flattening of the polar phonon band and scale-free ferroelectricity. The dramatically enhanced flattened polar phonon bands suggest point-defects to be effective to engineer the ferroelectricity in fluorite-based ferroelectrics.

Fig. 4: Effect of strain on the phonon band structure of the fluorite o-MO2 (M = Hf, Zr).
figure 4

a–d Comparison of the absolute values of the frequencies (Δω < 0.5 THz) of the polar bands from Γ to Y wave vectors for pristine and the doped cases with the same size supercells in pure o-phase HfO2 and ZrO2 under different strain condition.

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Defect-dipolar states in fluorite MO2 (M = Hf and Zr)

Point defects can induce local distortion due to mismatched size relative to host ions18,29,53,54,55. In conventional perovskite metal oxides, point-defect induced defect-dipoles and their correlation with intrinsic polarization are general due to the symmetry-conforming property of point defects56. Based on multi-scale simulations, we also reported that ferroelectric-like polarization hysteresis can be obtained in Mg-doped TiO2 due to defect-driven local symmetry breaking29. In non-ferroelectric SrTiO3, ferroelectric polarization behavior can also experimentally be observed to arise from nano-polar-regions associated with point defects30,55. Similar behavior of strap waist hysteresis loop in conventional perovskite ferroelectrics20 has also been observed in ZrO257, suggesting that the collective behavior of defect-dipoles also works in fluorite MO2 (M=Hf, Zr). Therefore, it is necessary to figure out how the point defect interacts with the nearby environment or how the defect-dipoles affect the surrounding electric polarization.

As might be expected, M (M = Hf, Zr) and oxygen ions especially the latter displaced three-dimensionally around the point defect. The lowest energy state was identified as the local structures illustrated in Fig. 5a for both systems. As shown in Fig. 5b, c, the magnitude of the ionic displacements relative to undoped o-MO2 was obviously correlated with the distance from La-VO defect complex, with the largest exceeding 0.6 Å and 1.0 Å by oxygen ion in 12.5% La-doped HfO2 and ZrO2, respectively. The ionic displacement becomes more obvious in 25% La-doped MO2 (Supplementary Fig. 12). While, VO-only doped cases with VO at polar layer (Fig. 5b-c) and antipolar layer (Supplementary Fig. 13) generally show much lower ionic displacement. This result suggests that defect-induced dipole is nucleated by the presence of La and its neighboring VO which then drives surrounding oxygen to move-off-center with large displacement, resulting in a “super” dipole and large local lattice distortion. This process is similarly as that in Mg-doped TiO2 where Mg-VO defect cluster drive the local symmetry breaking to form a “super” dipole29. The large lattice distortion may provide an explanation for the generally higher polarization but higher energy barrier and coercive field compared with VO-only doped cases.

Fig. 5: Theoretical DFT calculation of local defect-dipolar states.
figure 5

a The local structures in 12.5% La-doped MO2 (M = Hf, Zr) around the La-VO defect center show significant ionic relaxations with partially diffused domain walls, with both oxygen and M displacement relative to pure one. The pink and blue arrows reflect the ionic displacement within the crystal structures. b, c Ionic displacements of doped MO2 relative to pure one as a function of radial distance from the VO defect, indicating that the majority of ionic displacement occurs in the vicinity of La-VO defect pairs, within 10 Å scope from the VO. The result of VO-only doped MO2 where VO at polar layer is provided as comparisons. d Schematic illustration of the polar modes with sharp domain walls in pure MO2. e Schematic illustration of the normal polar modes with localized electric dipoles and sharp domain walls together with the point-defect induced polar modes with spread electric dipoles and diffused domain walls in La-doped MO2. The red sphere and aqua green sphere indicate oxygen and host cation, respectively. The polar phonon bands were indicated in d and e relative to the sharpness of the domain walls. The red, green, and yellow spheres indicate oxygen, host cation, and La dopants, respectively.

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For pure case, the zero-width sharp domain wall of the pure case originally comes from the condensation of polar and antipolar phonon bands in phase with equal magnitude to generate an orthorhombic structure that consists of alternating spacer layers and ferroelectric layers with up and down polarization respectively. Therefore, the generation of strict antipolar mode and its equal magnitude with polar mode would be two main pre-requisite conditions that are critical to induce the zero-domain walls. With La doping, local symmetry breaking emerges around the ‘La-VO-La’ defect like that in Mg-doped TiO2 that we previously reported29, which is believed to break the two main pre-requisite conditions for zero-width domain walls. The point-defect dipoles polarize the surrounding Zr and oxygen ions to give rise to spread electric dipoles and distort the dead/antipolar layer, in which case the antipolar modes are hard to conserve strict antipolar state and maintain the equal magnitude with polar mode. This is why the point-defect-induced local symmetry breaking areas show net total polarizations in x, y, and z directions (Fig. 5a).

Therefore, the point-defect induced lattice distortion greatly tuned the sharpness of domain walls. As shown in Fig. 5d, e, the polarization in pure o-MO2 is mainly from flat polar phonon modes with localized electric dipoles and sharp domain walls. While in doped cases, flat polar phonon modes give rise to generally diffused domain walls, which is responsible for the stabilized/enhanced ferroelectricity in doped fluorite MO2 (M = Hf, Zr). The increased number of polar flat bands would be mainly related to the number of polar states or polarizability of the lattice. This is consistent with the recently reported reversible oxygen migration and phase transitions in hafnia-based ferroelectric devices where ferroelectricity is unmistakably intertwined with oxygen vacancy49.

However, the much larger lattice distortion introduced by La-VO pairs accompanied with much larger energy barrier compared with VO-only doped case suggests that moderate local lattice distortion and concentration of VO are vital for high-performance ferroelectricity. To further understand the scale-free ferroelectricity, we calculated the zero-energy domain wall (Supplementary Fig. 14). For pure HfO2 and ZrO2, the negative domain wall energy for the antiferroelectric Pbca and ferroelectric Pca21 suggests that the antiferroelectric Pbca is more stable. However, compared with the pure one, for HfO2, 6.25% La doping and polar oxygen vacancies would both help to stabilize Pca21 phase. While, for ZrO2, 12.5% La doping concentration is the best, but the Pbca phase is still a more stable phase. Therefore, the scale-free ferroelectricity is still guaranteed in the doped cases. These results would inspire the understanding and exploration of the defect-induced ferroelectricity in similar fluorite MO2-based thin films and other doping-induced ferroelectric systems, such as doped AlN, GaN, and ZnO where similarly MV level of coercive field is generally the case6,7,58,59,60,61,62,63. For memory devices, the application of fluorite MO2 would be expected to further consider the effect of diffused and sharp domain walls on the ferroelectric stability of the memory devices.

In this work, we have confirmed instability modes in the phonon band of fluorite MO2 (M = Hf, Zr), leading to the transformation of c-phase to o-phase and t-phase. The appearance of o-phase is accompanied by flat polar phonon bands with extremely localized electric dipoles, which explains the scale-free ferroelectricity in fluorite MO2 ultrathin film compared with the spread dipoles in conventional perovskite ferroelectrics. By analyzing La-doped MO2 and VO-only doped MO2, all the doped cases show significantly flattened polar phonon bands. Their significantly enhanced flatness of polar phonon bands compared with of the pure ones, indicating an increased population of localized electric dipoles in the doped-MO2. Moreover, there is no obvious polar phonon band in the non-flat ones, which means that the main contribution to the polarization is from flat polar phonon bands. The quasi-phonon-glass dynamics in doped MO2 with significantly flattened phonon bands may suggest a giant anisotropic thermal rectification effect in the film and inspire further design in highly integrated circuits. There is no obvious variation in the flatness of polar phonon bands in epitaxially strained MO2 compared with the La-doped ones, which suggests that the point defect plays a more vital role in the induced/enhanced ferroelectricity in fluorite MO2. Different from the localized electric dipoles with sharp zero-width domain walls in pure o-MO2, the point-defect-induced local symmetry breaking in doped cases lead to collective electric polarization across the point-defect, which changes the very sharp zero-width domain walls with localized electric dipole observed in pure case into diffused domain walls with spread electric dipoles. The La-Vo defect pairs induce significantly higher local lattice distortion with significantly flattened polar phonon bands and generally higher energy barrier compared with VO-only doped case, suggesting that a moderate local lattice distortion and VO are vital for achieving high-performance ferroelectricity. Our results would advance the fundamentally understanding of the induced ferroelectricity in fluorite ferroelectrics. The defect-driven flattened polar phonon band may also inspire the investigation of the ferroelectric origin in doping-induced ferroelectric systems from the atomic-level, such as doped AlN, GaN, and ZnO.

Methods

Computational methods

All calculations are carried out using the projector augmented-wave (PAW)64 method as implemented in the Vienna Ab-initio Simulation Package (VASP)65. The Perdew–Burke–Ernzerhof (PBE)66,67 functional under the generalized gradient approximation(GGA) is applied to deal with the exchange and correlation interactions. The Zr(4 s,4p,4d,5 s), La(5 s,5p,5d,6 s) and O(2 s, 2p) electron orbitals are treated as valence states with following atomic valence configurations: Zr(4s24p44d25s2), La(5s25p65d16s2), and O(2s22p6). The cutoff energy of plane wave is set to be 600 eV which is large enough for energy convergency. To model the effect of concentration and figure out the defect states with lowest energy configuration, 1 × 1 × 1, 2 × 2 × 1, and 2 × 2 × 2 supercells based on unperturbed rhombohedra MO2 were constructed. The 7 × 7 × 7 Monkhorst-Pack68 k-points mesh is applied for the first Brillouin zone of the 12 atom unit cells and 3 × 3 × 3 Monkhorst-Pack k-points grid for 2 × 2 × 2 96-atom supercells for pure and doped MO2 for phonon spectra calculation. The force tolerance for the structure optimization and total energy were respectively converged to 10–2 eV Å−1 and 10–6 eV. The phonon dispersion was calculated by density functional perturbation theory (DFPT) using Phonopy69,70. The energy barrier for the polarization switching was calculated by the nudged elastic band method (NEB) and the result is consistent with the double-well potential based on Landau-Ginzburg-Devonshire (LGD) theory. Berry phase calculations were performed to calculate the spontaneous polarization on o-phase cells. The polarization quantum has been taken into consideration for data correction.