Because of the increasing demand for electrical energy storage and conversion applications, rechargeable energy storage components such as batteries, electrochemical capacitors, and dielectric capacitors have been intensively studied1,2,3. Dielectric capacitors have various advantages from the viewpoints of high power density, fast charge/discharge capability, long work lifetime, and high-temperature stability4,5. Therefore, they are the optimal option for applications in pulsed-discharge and power-conditioning systems, high-powered accelerators, and self-powered IoT devices6,7.

To achieve a high recoverable energy density for dielectric capacitors, the following polarization (P)-electric field (E) properties are advantageous: a smaller remanent polarization (Pr) and a larger maximum polarization (Pmax). Antiferroelectric (AFE) materials are a leading candidate, because their antipolar ordering of constituting atoms enables us to obtain zero Pr and a markedly large Pmax8,9. When an E is applied and its strength exceeds a threshold (ET), the antipolar polarization changes to a ferroelectric (FE) one with a robust polarization; given that the field is turned off, the AFE phase with zero Pr reappears. This reversible E-induced AFE-FE phase transition provides the characteristic double P-E loop10,11,12. However, there are limited reports of the perovskite oxides exhibiting the double P-E hysteresis loop at ambient conditions. This is because the E-induced phase transition is irreversible or cannot be achieved owing to ET greater than their breakdown fields.

NaNbO3 is one of the potential lead-free AFE perovskite oxides; it crystalizes in orthorhombic Pbcm (P phase) with the √2 × √2 × 4 superlattice of the primitive cell of simple perovskite structure in the temperature (T) range of −100 °C < T < 373 °C and shows a complicated phase transition behavior13,14,15. Though the antipolar ordering was reported in the 1960s and 1970s16,17 (Fig. 1a), the double P-E hysteresis loop was not obtained, because an application of E caused an irreversible phase transition and a resultant orthorhombic Pmc21 (FE) phase (Q phase with a 2 × √2 × √2 superlattice of the primitive cell) appears18,19 (Fig. 1b). Once the FE-Q phase is induced at fields exceeding ET, the FE-Q phase remains stable even after removing E, i.e., the AFE-P phase is not recovered18,20,21. This irreversible phase transition from the AFE-P to the FE-Q phase is due to a small free-energy difference between them.

Figure 1
figure 1

Crystal structures of NaNbO3: (a) AFE-P (Pbcm) phase, (b) FE-Q (Pmc21) phase and (c) AFE-R (Pnma) phase. The green, blue, and gray balls indicate Na+, Nb5+ and O2−, respectively. The data of the AFE-P, FE-Q and AFE-R phases are referenced from Ref.51 (ICDD No. 00–033–1270),52 (ICDD No. 01–079–7429) and15 (COD No. 4329527), respectively.

Inspired by materials design in PbZrO3-based AFEs22,23,24,25,26, the formation of solid solutions and chemical doping have been adopted for NaNbO320,27,28,29,30. Empirically, the simultaneous substitutions of smaller cations on the Na site and larger cations on the Nb site reduce the Goldschmidt tolerance factor and stabilize the AFE-P phase31. Zhang et al. have reported that 0.95NaNbO3–0.05SrSnO3 with a tolerance factor of 0.964, which is slightly smaller than that of NaNbO3 (0.967), exhibits reversible E-induced phase transitions as a result of an expansion of unit cell volume accompanied by disordered Na atoms32. NaNbO3–CaZrO3 is also a representative solid solution showing the double P-E hysteresis loop. Shimizu et al. have reported that the substitutions of Ca2+ on the Na site and Zr4+ on the Nb site lower the polarizability and reduce the tolerance factor while maintaining charge neutrality. This materials modification leads to a reversible E-induced phase transition for the 0.96NaNbO3–0.04CaZrO3 composition31.

It is interesting to note that chemically induced hydrostatic pressure (pchem) is an important degree of freedom for manipulating materials properties33. It is defined that pchem is the volume-averaged lattice internal force caused by chemical modifications. Various functional materials such as superconductors34,35, ferroelectrics33,36,37, and negative/zero thermal expansion materials38,39 etc. exhibit structures and properties tuned by pchem. A straightforward approach to apply pchem to the NaNbO3 lattice (pseudocubic unit cell volume Vpc = 5.96 × 10−2 nm3) is to form a solid solution with counterparts with different Vpc. The choice of perovskites with a smaller Vpc leads to lattice shrinkage, which can be regarded as an effect of positive pchem, while that with a larger Vpc results in a lattice expansion caused by negative pchem.

The formation of solid solutions with the counterparts having a larger Vpc such as SrSnO332 (Vpc = 6.56 × 10−2 nm340), CaZrO331,41 (6.46 × 10−2 nm342), CaHfO343 (6.39 × 10−2 nm344) and CaSnO345 (6.21 × 10−2 nm346) can be regarded as an application of negative pchem into the NaNaO3 lattice. The fine tuning of the compositions results in a reversible E-induced phase transition. In contrast, There are few reports on the reversible phase transition for the counterparts with a smaller Vpc47,48. These results suggest that a positive pchem is not suitable for the chemical modification.

For the above-mentioned systems, the Na site as well as the Nb site involve foreign atoms to satisfy charge neutrality. This type of modifications is likely to vary pchem but indeed disturbs the Nb-O-Nb covalent bonding, more generally, an orbital hybridization between niobium 4d and oxygen 2p, that plays a central role in ferroelectricity49,50. To elucidate the influence of pchem in a strict manner, the lattice needs to be modified only on the Na site while the Nb site remains intact.

In this paper, we report the impact of pchem on the crystal structure and polarization properties of NaNbO3 by a substitution only on the Na site. To apply pchem, the Na site is partially occupied by Ca2+ and Na vacancy (V) while the Nb site remains unchanged; this chemical tunning is termed ‘Ca modification’ with the following formula (CaxNa1−2xVx)NbO3. Our combined investigation by ceramic experiments and density functional theory (DFT) calculations shows that the Ca-modified ceramic sample with a reduced Vpc exhibits a double P-E hysteresis loop as a result of the stabilization of the AFE-P phase. We demonstrate that a positive pchem derives the AFE-P phase and delivers the reversible E-induced phase transition.


Figure 2 shows the scanning electron microscope (SEM) images of the Ca-modified NaNbO3 ceramics. All the samples have dense microstructures; the average grain size is 2.1 µm for x = 0.05 and increases with increasing x (7.9 µm for x = 0.20) (Fig. 2f). The increasing tendency of the grain size and the relative density with increasing x indicate that the Ca modification promotes grain growth. We note that the average grain sizes are much larger than a critical grain size of 0.27 μm below which the FE-Q phase is stabilized by intragranular stress53. We think that the grain size effect on the phase stability can be neglected in our samples.

Figure 2
figure 2

(a)–(e) Cross sectional SEM images of the Ca-modified samples with x of (a) 0.05, (b) 0.10, (c) 0.13, (d) 0.15, and (e) 0.20. (f) Composition dependence of the average grain size estimated by the intercept method using the lower magnification images shown in Fig. S1.

Figure 3a shows the XRD patterns of the Ca-modified samples along with that of the AFE-P phase as a reference. Throughout our manuscript, the Miller indices for a 1 × 1 × 1 pseudocubic cell are used. The formation of the perovskite phase was confirmed for all the samples. The intensity of 110 for x = 0.05 and 0.10 is relatively small compared with the calculated one of NaNbO3, which is likely to be due to a weak preferential orientation of grains in the vicinity of the sample surface. While the splits of the fundamental hkl reflections arising from the orthogonal crystal structure were observed for x = 0.05 and 0.10, they disappeared for x ≥ 0.15. Figure 3b shows an enlarged view of the XRD patterns in the range of 35.5° ≤ 2θ ≤ 42.5°. For x = 0.05 and 0.10, we found the 1 1 3/4 and 1/2 1/2 3/2 superlattice reflections specific for the AFE-P phase. This result indicates that the AFE-P phase with the antipolar ordering is retained for x ≤ 0.10. Though it is difficult to identify the phase for x ≥ 0.13, we consider that the AFE-R phase appears (Fig. 1c), which is one of the high-temperature phases of NaNbO3 with the space group Pmmn or Pnma with a 2 × 2 × 6 superlattice structure15. This is supported by the temperature dependence of relative permittivity, as discussed later. Figure 3c shows the composition dependence of the lattice parameter of the pseudocubic unit cell (apc) obtained from powder XRD patterns (Fig. S2). The ionic radius for 12 coordination is rNa = 0.139 nm for Na+54 and rCa = 0.134 nm for Ca2+54. In the range of 0 ≤ x ≤ 0.15, the apc decreases with increasing x and follows the Vegard’s law. This behavior is in contrast to that for Bi-doped NaNbO3 with Na vacancy28,55 where the cell volume increases with increasing the Bi content28. The apc deviates from the Vegard’s law at x = 0.15, because a solubility limit of Ca with Na vacancy exists at around x = 0.15. It is also notable that our samples have a high insulating property with an extremely low conductivity, e.g., 1.92 \(\times\) 10−11 S/cm for x = 0.10, see Fig. S4. This low conductivity indicates that the charge neutrality in the Ca-modified samples is satisfied by the formation of A-site vacancy rather than a partial reduction of Nb5+ to Nb4+ inevitably associated with a high conductivity caused by electron injection into the conduction band.

Figure 3
figure 3

(a) XRD patterns of the Ca-modified samples (CaxNa1−2xVx)NbO3 and (b) enlarged view in the range of 35.5° ≤ 2θ ≤ 42.5°. The solid triangles indicate the 1/2 1/2 3/2 reflections and the squares the 1 1 3/4 ones, both of which are characteristic for the AFE-P phase55. (c) Composition (x) dependence of the lattice parameter apc of the pseudocubic unit cell.

Figure 4 shows the temperature dependence of the relative dielectric permittivity (εr) and dielectric loss (tanδ). The temperature of the maximum εr (Tm) corresponds to the phase transition from the high-temperature AFE-R phase to the low-temperature AFE-P phase; Tm = ~ 360 °C for x = 0.014. The Tm decreases with increasing x and becomes lower than 25 °C for x ≥ 0.13. This Tm tendency with x is consistent with the results of the XRD measurements: the P phase is stabilized for x ≤ 0.10 whereas the R phase appears for x ≥ 0.13 at room temperature.

Figure 4
figure 4

Temperature dependence of the relative dielectric permittivity (εr) and loss (tanδ) of the Ca-modified samples at 1 kHz: (a) x = 0, (b) x = 0.005, (c) x = 0.05, (d) x = 0.10, (e) x = 0.13, and (f) x = 0.15.

Figure 5 shows the P-E and the current density (J)-E hysteresis properties. These data were obtained for the samples in the poled state, i.e., a unipolar or bipolar E was applied prior to the measurements. A typical ferroelectric P-E loop with a relatively large Pr of 37 μC cm−2 was observed for x = 0.005 (Fig. 5a). This is not consistent with the result of the XRD pattern (before the P-E measurement in Fig. 3) showing the AFE-P phase. The P-E loop for x = 0.05 looks like a ferroelectric polarization hysteresis while that for x = 0.10 exhibits an apparent double hysteresis. This seemingly inconsistent with the result of the XRD patterns arises from an irreversible phase transition from the AFE-P to the FE-Q phase for x ≤ 0.05, as described later. The sample with x = 0.13 exhibits a similar behavior with relaxor antiferroelectrics56; a pinched P-E loop with a small hysteresis and a dielectric dispersion around room temperature (Fig. S3) with a Tm below 25 °C (Fig. S3). It is reasonable to consider that the AFE-R phase is stabilized for x = 0.13. With further increasing x, the loop is slanted and eventually almost closed for x = 0.15.

Figure 5
figure 5

Bipolar P-E and J-E hysteresis loops of the Ca-modified samples (CaxNa1−2xVx)NbO3: (a) x = 0.005, (b) x = 0.05, (c) x = 0.10, (d) x = 0.13, (e) x = 0.15. Unipolar P-E hysteresis loops at 1 Hz of (f) x = 0.05 and (g) x = 0.10.

Figure 6a shows the first cycle of the bipolar P-E hysteresis loop for an as-prepared, unpoled sample (x = 0.05). With a positive sweep of E, the sample exhibits a jump of P at a ET and has a large P in State 1. This large P is retained even after removing E (State 2). With a negative E sweep followed by turning off the field, the sample displays an apparent negative P. As described later, the overall behavior of the first P-E loop can be explained by the irreversible phase transition from the AFE-P phase (State 0) to the FE-Q phase (States 1 and 2).

Figure 6
figure 6

(a) First cycle of the bipolar P-E hysteresis loop for the unpoled sample with x = 0.05. (b) Micro-XRD patterns of the Ca-modified samples (CaxNa1−2xVx)NbO3 with x = 0.005, 0.10 after the P-E measurements. The data for powders with the same compositions are also shown, which are denoted by ‘before’. The solid circles, triangles, and squares indicate 1 1 1, 1/2 1/2 3/2 and 1 1 3/4 reflections55, respectively, and the open rectangular and triangles indicate 0 1/2 3/2 and 1 1 1/2 reflection, respectively. The data of NaNbO3 in the P and Q phases are referenced from 00–033–127051 and 01–079–742952 in ICDD, respectively.

Figure 6b shows the XRD patterns after the P-E measurements (denoted by ‘after’) along with those for the powders with the same compositions (‘before’). For x = 0.005 (before), the 1 1 3/4 superlattice reflection specific for the AFE-P phase57 was observed. Moreover, the 1 1 1 reflection splits in the same manner as the reference of the P phase. After the measurements, the 1 1 3/4 reflection disappears while the 1 1 1/2 reflection of the FE-Q phase appears at 2θ = 36.2°52. These results show that once an E exceeding its ET is applied a phase transition from the P to the Q phase takes place and also that the Q phase remains stable even after removing E. The well-opened P-E loop for x = 0.005 (Fig. 5a) arises from the polarization switching of the FE-Q phase.

In contrast, the XRD data for x = 0.10 (after) represents the P-phase feature with the 1 1 3/4 and 1/2 1/2 3/2 reflections. These results along with the double hysteresis loop (Fig. 5c) clearly demonstrate that the sample (x = 0.10) displays a reversible E-induced P-Q phase transition. The J-E loop has two broad peaks in the higher E region as well as two sharp peaks in the lower E region. We define the E values for the former two peaks corresponding to the transition from the P to the Q phase as \({E}_{\mathrm{P}\to \mathrm{Q}}^{\prime}\) and \({E}_{\mathrm{P}\to \mathrm{Q}}^{\prime \prime}\) and those for the latter peaks corresponding to the transition from the Q to the P phase as \({E}_{\mathrm{Q}\to \mathrm{P}}^{\prime}\) and \({E}_{\mathrm{Q}\to \mathrm{P}}^{\prime \prime}\). The threshold fields of the E-indued phase transition, \({E}_{\mathrm{P}\to \mathrm{Q}}\) and \({E}_{\mathrm{Q}\to \mathrm{P}}\), for x = 0.05 and 0.10 are summarized in Table 1.

Table 1 Threshold fields of the E-induced phase transition between the AFE-P and the FE-Q phases estimated from the JE loops (Fig. 5).

With increasing x, \({E}_{\mathrm{P}\to \mathrm{Q}}\) rises while \({E}_{\mathrm{Q}\to \mathrm{P}}\) approaches zero. It is considered that the double P-E loop for x = 0.10 is a result of the small \({E}_{\mathrm{Q}\to \mathrm{P}}\). It is expected that a higher Ca-V content, i.e., a further increase in x, gives rise to a clearer double loop. However, for the samples with x ≥ 0.13, the polarization switching derived from the E-induced phase transition was not observed. This is because the AFE-R phase is stabilized and then the \({E}_{\mathrm{P}\to \mathrm{Q}}\) becomes higher than its breakdown field. It is also noted that the peaks of J for the P → Q transition at \({E}_{\mathrm{P}\to \mathrm{Q}}^{\prime}\) and \({E}_{\mathrm{P}\to \mathrm{Q}}^{\prime \prime}\) are sharp compared with those for the Q → P transition at \({E}_{\mathrm{Q}\to \mathrm{P}}^{\prime}\) and \({E}_{\mathrm{Q}\to \mathrm{P}}^{\prime \prime}\), especially for x ≤ 0.05. These phase transitions take place through nucleation and growth dynamics of FE (Q) domains in the AFE (P) matrix and AFE (P) domains in the FE (Q) one. We think that local random fields, as reported in NaNbO3–(Bi0.5Na0.5)TiO3 solid solutions58, play an important role in the E-induced phase transitions, but the details are unclear.

Figure 5f,g show the unipolar P-E hysteresis loops. The recoverable energy storage density (Urec) and the energy efficiency (η) are calculated by the following equations.

$${U}_{\mathrm{rec}}={\int }_{{P}_{\mathrm{r}}}^{{P}_{\mathrm{max}}}EdP,$$
$$\eta =\frac{{U}_{\mathrm{rec}}}{{U}_{\mathrm{rec}}+{U}_{\mathrm{loss}}}\times 100,$$

where Pmax, Pr, and Uloss are maximum polarization, remanent polarization, and energy loss, respectively1,4. The Urec and the η estimated from the hysteresis loops are 0.34 J cm−3 and 34% for x = 0.05 and 0.74 J cm−3 and 17% for x = 0.10. The larger Urec for x = 0.10 is attributed to the reversible E-induced phase transition between the AFE-P and the FE-Q phases.

Figure 7 exhibits the total energy Upc of NaNbO3 as a function of pseudocubic unit cell volume Vpc in the range of hydrostatic pressure p between −2.8 GPa and 2.8 GPa. It is interesting to note that the difference in Upc is relatively small; several meV at a smaller Vpc and ~ 10 meV at a larger Vpc. When the positive p is applied, i.e., the Vpc is compressed, the Upc of the AFE-P phase is slightly smaller. In contrast, at a negative p with an expanded Vpc, the FE-Q phase is markedly stabilized.

Figure 7
figure 7

Pseudocubic total energy (Upc) per the ABO3 formula unit as a function of pseudocubic unit cell volume (Vpc) for the AFE-P phase (Pbcm) and the FE-Q phase (Pmc21). The hydrostatic pressure (p) changes between −2.8 GPa and 2.8 GPa. Filled circles indicate the data obtained by DFT calculations and solid lines the fitting curves obtained by a nonlinear least square method using Eq. (3).

Figure 8a displays the free-energy difference of \({\Delta G = {G}_{{Pmc2}_{1}}-G}_{Pbcm}\) vs. p (\(G = {U}_{\mathrm{pc}}+p{V}_{\mathrm{pc}}\)). The ∆G curve is not smooth caused by the Pulay stress59,60,61; the plane-wave basis set is not complete with respect to changes in the volume. In the p range, p > 0.2 GPa, ∆G is positive and the AFE-P phase appears. In the range of p < 0.2 GPa, ∆G becomes negative and the FE-Q phase arises.

Figure 8
figure 8

Free energy difference (\(\Delta G {= G}_{Pmc{2}_{1}}-{G}_{Pbcm}\)), where \({G}_{Pmc{2}_{1}}\) and \({G}_{Pbcm}\) denotes the free energies of the FE-Q phase (Pmc21) and the AFE-P phase (Pbcm) one: (a) obtained by DFT calculations for NaNbO3 at zero kelvin and (b) aligned for the ceramic samples at 25 ℃. The AFE-P phase is stabilized in the region of ∆G > 0 while the FE-Q phase appears in the region of ∆G < 0. In reality, the Ca ions and the vacancies on the A site play a different role, while the FE and AFE orderings stem primarily from an orbital interaction between Nb-4d and O-2p. We assume that the effects of the Ca modification on \(\Delta G\) for our samples is simplified by the change in chemically induced p (pchem) in (b).


It is important to note that the ∆G feature in Fig. 8a is the result of the DFT calculations at zero kelvin under the periodic boundary condition. In reality, our experiments were performed at 25 °C. Additionally, our samples underwent the phase transitions during cooling from the high-temperature cubic phase to the low-temperature orthorhombic phases. Moreover, the samples were cooled down to room temperature under the constraints of a spatially fixed matrix of dense ceramics. We therefore think that the zero point of the chemically induced p (pchem) of the as-prepared NaNbO3 sample at 25 ℃ (Fig. 8b) is inevitably shifted from that in Fig. 8a.

Figure 8b schematizes the ∆G curve with respect to pchem for our samples, where the effects of the Ca modification is treated as the change in pchem. The as-prepared undoped sample is of the AFE-P phase and placed at zero pchem (I.). The electrical poling, an application of E, induces an irreducible phase transition to the FE-Q phase, which is accompanied by the marked change in lattice parameters. It is reasonable to consider that this electrical poling causes a shift of pchem to the region where the FE-Q phase arises (II.), probably because of the influence of intragranular stress53.

Here, we discuss the effect of the Ca modification on pchem. As shown in Fig. 3c, the increase in x (the Ca-V content) decreases apc. In other words, the Ca modification increases pchem as a result of a reduced Vpc, and the sample with x = 0.10 moves at III., where the AFE-P phase appears. It is likely that the electrical poling decreases pchem; the poled sample is still of the AFE-P phase (IV.). We show that the positive pchem by the Ca modification stabilizes the AFE-P phase even after the electrical poling and is thereby effective for realizing the reversible E-induced phase transition, as observed for the sample (x = 0.10).

In conclusion, we have investigated E-induced phase transitions in Ca-modified NaNbO3 ceramics with A-site vacancy [(CaxNa1−2xVx)NbO3]. We experimentally found that the Ca modification reduces a cell volume and the resultant positive pchem stabilizes the AFE-P phase for x ≤ 0.10. The reversible E-induced phase transition between the AFE-P and the Q phases occurs for the sample with x = 0.10, which results in a double hysteresis loop. The Urec of the sample (x = 0.10) with the reversible phase transition is more than two times as high as that of the sample (x = 0.05) with the irreversible one. Our study opens the way to utilizing pchem for NaNbO3-based antiferroelectric materials for energy storage applications.


Material preparation

Undoped NaNbO3 and Ca-modified NaNbO3 [(CaxNa1−2xVx)NbO3] ceramic samples with x = 0.005, 0.05, 0.10, 0.13, 0.15, and 0.20 were prepared by a solid-state reaction using Na2CO3 (99.99%), CaCO3 (99.99%), and Nb2O5 (99.99%) as raw materials. Na2CO3 powders were dried at 280 °C for at least 4 h before weighing. The raw materials were mixed by ball milling for 1 h in ethanol. The mixed powders were dried and calcined in an alumina crucible at 900 °C for 3 h in air. The resultant powders were pulverized by ball milling for 1 h and pressed into pellets with a diameter of 10 mm followed by sintering at 1250–1300 °C for 4–5 h in air for x = 0 and in an oxygen atmosphere for the other compositions. To suppress the volatilization of Na, the pellets were covered by powders with the same composition. The relative densities of the samples are as follows: 92.4% for NaNbO3 and 94.9% for x = 0.005, 98.2% for x = 0.05, 98.3% for x = 0.10, 98.9% for x = 0.13, 98.7% for x = 0.15, and 97.3% for x = 0.20. The samples obtained were cut and polished into disks with a thickness of 0.15–0.20 mm. An annealing for oxidation was performed at 1000 °C for 48 h in air. For the measurements of relative permittivity and P-E hysteresis loops, Au electrodes with a diameter of 1 mm and 2 mm, respectively, were sputtered on both sides of the disks. The lattice parameters were obtained by the Rietveld analysis for powder XRD patterns (Fig. S2) with a Rietveld refinement program Z-Rietveld (version 1.1.3). We adopted space group Pbcm (the AFE-P phase) and Pnma (the AFE-R phase). Structural refinement was conducted until the reliability index Rwp becomes minimum. The apc is calculated by \({a}_{\mathrm{pc}}= \sqrt[3]{(a\times b\times c)/8}\) for Pbcm and \({a}_{\mathrm{pc}}= \sqrt[3]{(a\times b\times c)/24}\) for Pnma, respectively, where a, b, and c are the refined lattice parameters of the orthorhombic unit cell. For the powder samples for the Rietveld analysis, the calcined powders were pulverized by ball milling for 1 h and then sintered at 1150–1300 °C for 4 h in air.

DFT calculations

DFT calculations were performed via the generalized gradient approximation62 with a plane wave basis set. The projector-augmented wave method63 was applied by the Vienna ab initio simulation package (VASP)64. We employed the gradient-corrected exchange-correlation functional of the Perdew-Burke-Ernzerhof revised for solids (PBEsol)65 and a plane-wave cut-off energy of 520 eV. The following two phases with different symmetries were considered: the AFE-P phase with space group Pbcm (Z = 8) and the FE-Q phase with space group Pmc21 (Z = 4). The adopted mesh size of the k-point sampling grid was 9 × 9 × 7 for Pbcm and 11 × 11 × 11 for Pmc21. We confirmed that the optimized structures and the resultant total energies were unchanged when the finer k-point mesh sizes were adopted.

To evaluate the phase stability, we calculated the total energy (U) as a function of the cell volume (V) and then analyzed by the Murnaghan equation of state66

$$U\left(V\right)={U}_{0}+\frac{{B}_{0}V}{{B}_{0}^{\mathrm{^{\prime}}}}\left[\frac{{\left({V}_{0}/V\right)}^{{B}_{0}^{\mathrm{^{\prime}}}}}{{B}_{0}^{\mathrm{^{\prime}}}-1}+1\right]-\frac{{B}_{0}{V}_{0}}{{B}_{0}^{\mathrm{^{\prime}}}-1} ,$$

where U0, B0, B0′, and V0 are the total energy, the bulk modulus and its first derivative with respect to the hydrostatic pressure (p) and V at p = 0. We converted the U and the V into the Upc and the Vpc of the pseudocubic unit cell, because the cell sizes of the two phases are different. Since the free energy (G) is expressed as G = Upc + pVpc, we can obtain the relation between G and p using the fitting parameters in Eq. (3). It should be noted that our DFT calculations predict the G feature at zero kelvin.