In search of Pca21 phase ferroelectrics

In recent years, hafnia-based ferroelectrics have attracted enormous attention due to their capability of maintaining ferroelectricity below 10 nm thickness and excellent compatibility with microelectronics flow lines. However, the physical origin of their ferroelectricity is still not fully clear, although it is commonly attributed to a polar Pca21 orthorhombic phase. The high-temperature paraelectric phases (the tetragonal phase or the cubic phase) do not possess a soft mode at the Brillouin zone center, thus the ferroelectric distortion has to be explained in terms of trilinear coupling among three phonon modes in the tetragonal phase. It is necessary to explore new materials with possible ferroelectricity due to the polar Pca21 phase, which in turn should be very helpful in evaluating the microscopic theory for ferroelectric hafnia. In this work, based on the idea of the Materials Genome Engineering, a series of hafnia-like ferroelectrics have been found, exemplified by LaSeCl, LaSeBr, LuOF and YOF, which possess adequate spontaneous polarization values and also relatively favorable free energies for the polar phase. Their common features and individual differences are discussed in detail. In particular, a promising potential ferroelectric material, Pca21 phase LuOF, is predicted and recommended for further experimental synthesis and investigation.

Nevertheless, a high coercive field also implies a high energy input for polarization switching. It also tends to cause endurance and leakage problems, because a high electric field is detrimental to the structural integrity of the film [27]. Note that there is a huge gap between the typical coercive fields of HZO and PZT. For the sake of device reliability, it would be ideal to develop certain ferroelectric materials with a coercive field of less than 1 MV cm −1 but still with a large band gap. The band gap of HfO 2 is nearly 6 eV [28,29], which seems sufficiently high. Nevertheless, the band gap of ZrO 2 is lower [30], and figure 1 shows that the conduction band of HZO is dominated by Zr states. It is still worth exploring similar ferroelectric materials with a lower coercive field and larger band gap than HZO.
On the other hand, from a materials chemistry point of view it is also difficult to explain the uniqueness of HfO 2 /ZrO 2 in these special Pca2 1 -class ferroelectrics. A well-known Aurivillius-phase ferroelectric, Bi 2 WO 6 , possesses Pca2 1 symmetry as well [31,32]. However, it is a natural superlattice whose ferroelectricity originates from the lattice mismatch between the [Bi 2 O 2 ] 2+ layer and the pseudo-perovskite layer, as revealed in common Aurivillius ferroelectrics [33]. The ferroelectric distortion in HfO 2 /ZrO 2 has an entirely different manifestation, involving the transition of certain O anions between two equivalent positions [34]. Recently, Yuan et al [35] proposed that the ferroelectricity of hafnia-based ferroelectrics stems from their special cation/anion ionic radius ratio, which points to the seven-coordination (7C) configuration. The so-called 7C theory predicts that HfO 2 and ZrO 2 should not be the only member of this category, and more binary or ternary compounds, such as SrI 2 and YSBr, are listed as potential ferroelectric candidates. In ternary compounds, distinct anions are used to replace the three-coordination O and four-coordination O in Pca2 1 HfO 2 , respectively. This enables great materials design flexibility and it is possible to select potential materials from a vast number of candidates to give the best performance in terms of ferroelectric polarization and coercive field as well as band gap.
In this work, we focus on the ionic Re-VI-VII ternary compounds where Re stands for a rare earth metal element while VI and VII are chalcogen and halide elements, respectively. Binary compounds are not taken into consideration because the 7C theory indicates that the number of binary compounds satisfying the 7C condition is relatively small (HfO 2 , ZrO 2 , SrI 2 , etc.). On the contrary, 7C is more easily realized through a mixture of VI and VII anions. Moreover, halides are even more ionic, which could potentially yield larger band gap values. Besides the prototype Pca2 1 ferroelectric phase, other related phases involve monoclinic P2 1 /c and the high-pressure Pbca-I phase, which are illustrated in figure 2. For comparative reasons, we also include some other trivalent cations, such as Al 3+ , Ga 3+ and In 3+ , in the list for our search. There are in total 63 candidates in our data set, as listed in tables 1 and 2. The free energies of their various phases, the possible ferroelectric polarization values, coercive fields as well as band gaps are under investigation.

Methods
Density functional theory calculations were performed using the Vienna Ab initio Simulation Package (VASP; version 5.4.4) [36,37]. The projector augmented-wave (PAW) method [38,39] was implemented, and the plane wave basis was truncated at a kinetic energy cutoff of 600 eV. The valence electrons for the PAW potentials were 2s and 2p for O, 2s and 2p for F, 3s and 3p for S, 3s and 3p for Cl, 3p, 3d and 4s for Sc, 4s and 4p for Se, 4s and 4p for Br, 4s, 4p, 4d and 5s for Y, 5s, 5p, 5d and 6s for La, 5p, 5d and 6s for Hf and 5p, 5d and 6s for Lu. The Perdew-Burke-Ernzerhof functional of the generalized gradient approximation (GGA) was used for the exchange-correlation energy [40]. The Brillouin zones were sampled using at least an 8 × 8 × 8 equal-spacing k-point mesh. On account of the band gap problem in GGA calculations, we switched to either the more time-consuming Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional [41,42] or the shell GGA-1/2 method [43][44][45][46] for rectified electronic band structures, and they are comprehensively compared in this work. The phonon dispersion was calculated jointly using the VASP code and the   I  InOF  I  AlOCl  II  GaOCl  II  InOCl  I  AlOBr  II  GaOBr  II  InOBr  II  AlSF  II  GaSF  II  InSF  I  AlSCl  II  GaSCl  II  InSCl  II  AlSBr  II  GaSBr  II  InSBr  I  AlSeF  II  GaSeF  II  InSeF  I  AlSeCl  II  GaSeCl  II  InSeCl  II  AlSeBr  II  GaSeBr  II  InSeBr  II  ScOF  I  YOCl  II  LaSF  I  ScOCl  I  YOBr  II  LaSeF  I  ScOBr  II  YSF  I  LuOCl  I  ScSF  I  YSeF  I  LuOBr  II  ScSCl  I  YSeCl  I  LuSF  I  ScSeF  I  LaOF  I  LuSCl  I  ScSeCl  I  LaOCl  I  LuSeF  I  ScSeBr  I  LaOBr  II  LuSeCl  I  LuSeBr  I PHONOPY code based on density functional perturbation theory [47]. The switching barriers were calculated using the climbing image nudged elastic band (NEB) method [48]. The spontaneous polarizations were calculated using the Berry phase method [49].

Results and discussion
As a primary screening criterion, a candidate material should be stable in the Pca2 1 structure with a finite spontaneous polarization. Among the 63 Re-VI-VII compounds, most candidates fail to pass the ferroelectric material screening for this structural reason, as demonstrated in table 1. For example, GaSCl and LaOBr cannot maintain the desired Pca2 1 phase, leading to layered structures after relaxation. On the other hand, compounds like GaOF and LaOCl do not collapse during structural relaxation, but the optimized structures show zero polarization, as shown in figures 3 and S1-S7.  On the other hand, table 2 lists the ten candidate materials that exhibit a finite polarization in their Pca2 1 phase. Among these, YOF has already been reported [35] but is included in our comparison due to its potentially excellent ferroelectric properties. Phonon instability is not observed in any of these ten compounds in their relaxed Pca2 1 phase. The phonon spectra are illustrated in figure S8. After the kinetic stability test, we calculated their electronic band structures using the HSE06 hybrid functional (figure 4) and shell GGA-1/2 (table 2). The results in figure 4 show that the band gaps are greater than 3 eV in all cases. Hence, they meet the minimum fundamental criterion with regard to a sufficient band gap. Nevertheless, YOF and LuOF are outstanding in that their band gaps exceed 6 eV (greater than 7 eV if evaluated through shell GGA-1/2), which is very beneficial for reduction of leakage current in the device.
The next significant step in the screening is the free energy comparison between the Pca2 1 phase and various other phases in these compounds. The example of HfO 2 informs us that the Pca2 1 phase does not have to be the ground state phase at zero temperature and atmospheric pressure, but under certain grain size and/or stress conditions it may be preferred to the ground state P2 1 /c phase [50]. Hence, we calculated the Helmholtz free energies of these ten compounds in Pca2 1 , P2 1 /c and Pbca-I phases. Relative to the lowest energy phase (P2 1 /c in all cases), there are four compounds that possess a relatively stable Pca2 1 phase, i.e. LaSeCl, LaSeBr, LuOF and YOF, shown in figures 5 and S9. An effective criterion for evaluating their Pca2 1 phase stability is the critical temperature for thermodynamic transition from the low-temperature P2 1 /c phase to the Pca2 1 phase, where a low temperature implies a better stability of the polar phase. Among these four compounds, LuOF has the lowest transition temperature, nearly 1200 K, while YOF has the highest transition temperature of ∼2000 K.
Of the four candidates, LuOF and YOF possess larger band gaps and are more promising. However, a sufficient spontaneous polarization (P S ) and a proper coercive field (E C ) are also essential factors. These ferroelectric candidates are special in that they possess multiple switching paths, but there is one path that exhibits the lowest switching barrier. Hence, to fix the P S one has to analyze the switching barriers along various paths. As illustrated in figure 6, all paths can be categorized into external paths (anions penetrating through the cation plane at z = 0) and internal paths (anions not penetrating through the cation plane), with two distinct P S values. The one with the lowest switching barrier is selected as the P S of the ferroelectric material, as listed in table 3. Among the ten candidates, YSeBr shows a large switching barrier (>1.9 eV) for all paths, thus it is excluded due to the expected enormous coercive field. The P S values of LaSeCl, LaSeBr, LuOF and YOF are predicted to be 20.01 µC cm −2 , 16.84 µC cm −2 , 36.83 µC cm −2 and 19 µC cm −2 , respectively. LuOF and YOF are promising, but their switching paths are distinct. The switching mode of YOF is similar to ferroelectric HfO 2 or HZO, i.e. the internal path [51]. The intermediate phase of HfO 2 during such switching is the tetragonal P4 2 /nmc phase. When it comes to YOF, the intermediate phase is a similar structure but without tetragonal symmetry, since its two anions are distinct. Therefore, it is temporarily called a P4 2 /nmc-like phase in this article. Nevertheless, LuOF prefers the external switching path with a Pbcm intermediate phase, as illustrated in figure 7. The possible reason for this is that Lu 3+ is smaller than Y 3+ , and Fanions may penetrate the z = 0 plane (made of Lu 3+ ) to prefer the external path. It is interesting to note that the preferred path of LuOF corresponds to the larger P S , which is quite unusual among Pca2 1 ferroelectrics [35]. Subsequently, we compare LuOF and YOF with HfO 2 in terms of their theoretical ferroelectric and electronic properties. The P S value of HfO 2 is the largest among the three, while YOF shows the smallest P S value. Our thermodynamic calculation shows that it is most convenient to obtain the Pca2 1 phase of LuOF. Moreover, the stability of the intermediate phase during polarization switching is also crucial for cycling endurance. If the intermediate phase is too stable the material may stick in the intermediate paraelectric phase rather than go to the polar phase.
To obtain the ferroelectric phase in practical situations, we estimated the energies of HfO 2 , LuOF and YOF in various phases under stress. For each compound, its P2 1 /c, Pca2 1 , Pbca-I and intermediate phase were investigated, as shown in figure 8. The P2 1 /c phase is the ground state for all three compounds, but it becomes less favorable for compressed crystals. It turns out that relatively small levels of compressive strain (4.95%, 5.78% and 4.23% for HfO 2 , LuOF and YOF, respectively) are sufficient to transform them into the  Pbca-I or Pca2 1 phases. These two phases are highly related, and experimental evidence has been reported concerning their inter-conversion during operations in hafnia-based ferroelectric capacitors [52]. Hence, maintaining compressive strain is still an important means to promote the ferroelectric phase in LuOF and YOF, as in HfO 2 . The compressive strain is effective in suppressing the P2 1 /c phase because its emergence accompanies a volume expansion based on the prototype cubic fluorite-like structure. There is yet another issue concerning whether strong compressive strain is always best, because some other phases may also be stabilized against the P2 1 /c phase. In this sense we find that LuOF has certain advantages. Both the Pbca-I phase and the Pca2 1 phase are desirable, while all other phases are to be suppressed. Consider, for example, if ultimate compression is carried out, exemplified by volume compression to 85% that of the ground state (magenta dotted line in figure 8). In all cases, the desired phases are stabilized, but their energy differences to the next lowest energy phase are different, being 0.1259 eV f.u. −1 (f.u. stands for formula unit) for LuOF, 0.1216 eV f.u. −1 for HfO 2 and 0.0661 eV f.u. −1 for YOF. In HfO 2 , compression also renders the tetragonal P4 2 /nmc phase stabilized. Hence, the situation is complicated in ferroelectric HfO 2 and HZO because the Pca2 1 phase has to compete with both the P2 1 /c phase and the P4 2 /nmc phase. On the other hand, due to their ternary nature, LuOF and YOF do not have any P4 2 /nmc phase. However, YOF has an intermediate P4 2 /nmc-like phase, which is energetically similar to polar phases under compression. The situation in LuOF is clearer because its intermediate phase Pbcm is high in energy even for compressed cases. Its Pbca-I and Pca2 1 phase energies are significantly lower than other phases as long as sufficient compression is afforded (figures 8 and S10).
Subsequently, we calculated the anion vacancy formation energies of the three compounds (table S1). The oxygen chemical potential was set to half the energy of an O 2 molecule. The general trend of difficulty in anion vacancy formation is HfO 2 < YOF < LuOF. For example, 6.13 eV for O in HfO 2 compared with 6.85 eV for O and 6.87 eV for F in LuOF. The capability of resisting vacancy formation in LuOF is very beneficial for leakage current reduction and cycling endurance. Hence, it is conjectured that LuOF could be a promising dielectric for use in nanoscale ferroelectric capacitors.
It is interesting to compare our material search strategy with several well-known published works. Early in 2010 and before the discovery of ferroelectric hafnia, Bousquet, Spaldin and Ghosez [53] explored the possibility of strain-induced ferroelectricity in simple rock salt binary oxides such as BaO and SrO. Their motivation stemmed from the observation that these materials possess large Born effective charges (both metal cations and O anions). Moreover, the polar transverse-optical phonon mode is known to be sensitive to strain. Therefore, strain engineering may potentially transform certain rock salt oxides into the polar ferroelectric phase. Among these, BaO and EuO were shown to be particularly attractive. The material search in that work was conducted under clear physical guiding principles and within a fixed family of atomic structures. The material search by the Ramprasad group [54] was a very extensive one, aimed at finding ferroelectric candidates from a series of 21 binary oxides, not limited to a specific structure. The strategy involved identifying a dynamically stable polar structure of the selected oxide with relatively low energy and large spontaneous polarization. Hence, the potential polar structure very likely occurs in a different polar space group than Pca2 1 , greatly extending the scope of the search. Five candidates were listed as promising ferroelectric materials, including Pna2 1 CaO 2 , Pna2 1 SrO 2 , Pna2 1 Ga 2 O 3 , Pna2 1 Al 2 O 3 and Pca2 1 TiO 2 . Some candidates are compared with Pca2 1 LuOF in table 4. The P S values of the predicted candidates are constantly smaller than that of HfO2, and LuOF is relatively remarkable in terms of its adequate spontaneous polarization as well a moderate switching barrier.
In our case, the theoretical foundation is the 7C theory, which explains the ferroelectricity in ionic compounds from the perspective of chemical coordination number. The target material candidates were extended through element substitution following the common understanding of the valency for each element. This has certain limitations but also benefits. The potential outcomes are constrained within the family of Pca2 1 structures. However, it fits our theoretical expectation because an important objective is to find new members of the Pca2 1 ferroelectric family, now only containing HfO 2 and ZrO 2 . Such a guiding principle was set not only to predict a technically useful ferroelectric material but also that additional similar ferroelectric materials to the Pca2 1 ionic ferroelectric family would be very useful for a deeper understanding of the surprising ferroelectricity observed in hafnia. For instance, the preference for an external switching path renders Pca2 1 LuOF a very attractive counter-example to HfO 2 . Of course, theoretical prediction can never replace experimental synthesis. The encouraging fact is that our chemical substitution strategy follows the normal element valency arguments and it is expected that the chemical compositions of the resulting materials will be reasonable. The possible experimental growth of LuOF thin film deserves intensive attention in particular.
There are several published experimental works reporting the synthesis of LuOF. Taoudi et al [55] reported that the crystal structure of LuOF was isomorphic to α-ZrO 2 , i.e. baddeleyite (P2 1 /c). The LuOF samples, as white powders, were thermally stable up to 700 • C. Müller and co-workers synthesized bulk LuOF from appropriate Lu 2 O 3 -LuF 3 mixtures in gas-tight platinum capsules through high temperature annealing at 1450 K for 2 h [56]. According to the x-ray powder diffraction analysis, LuOF has a monoclinic structure Table 4. Important parameters for some known or predicted ferroelectric candidates, from [54] (CaO2, SrO2, Al2O3, Ga2O3), [35] (HfO2 and YOF) and this work (LuOF). All the band gaps were calculated in this work, with the HSE06 hybrid functional or shell GGA-1/2.  [58] obtained microcrystalline powders of LuOF in conventional solid state reactions, which showed a monoclinic structure. To obtain the Pca2 1 phase, thin film LuOF is desired such that the monoclinic phase may be suppressed by (i) mechanical confinement and (ii) the strong bias field during capacitor annealing, which tends to promote polar phases. Hence, we propose that magnetron sputtering could be a viable approach for fabricating LuOF thin films. The target can be obtained through high-temperature sintering from Lu 2 O 3 and LuF 3 powders [56], followed by slow cooling. LuOF is not supposed to cause environmental problems, because F is firmly bound to the metal, and it has been reported that LuOF can be used as a scintillator [58] for high-energy photon detection. It is robust towards incorporation of moisture in the atmosphere [59]. However, in the context of microelectronic flow-lines, F is not an ordinary element. A price is paid to reach a wide band gap through the extremely electronegative F element. This does not affect the potential theoretical value of LuOF as an interesting Pca2 1 ferroelectric candidate that could strengthen our understanding of this family of ferroelectrics. Experimental attempts to prepare LuOF thin film through, for example, sputtering are highly recommended.

Conclusion
We investigated 63 possible hafnia-like compounds with the tentative Pca2 1 structure, and ten of them could well maintain spontaneous polarization as expected. According to the Helmholtz free energy results, four of them are likely to be stabilized under slight compressive stress. In particular, LuOF was further picked out due to its excellent all-round performance. It has a large band gap value (6.12 eV according to HSE06 calculation), high spontaneous polarization (P S = 36.83 µC cm −2 ) and a moderate ferroelectric switching barrier (0.203 eV). In addition, based on the results for energy variation under various stress levels as well as the anion vacancy formation energies, the advantages of LuOF over HfO 2 and YOF are further highlighted. Moreover, the favorable switching path of Pca2 1 LuOF is distinct from that of Pca2 1 HfO 2 . Adding new members like LuOF into the Pca2 1 ferroelectric family will promote the physical understanding of hafnia ferroelectrics.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).