Lead-based halide perovskite solar cell materials have received much attention because of their excellent properties such as suitable direct band gap, high optical absorption, long diffusion length, and long carrier lifetime^[1−5]. Since halide perovskites were reported in 2009, they have sparked research interest, with solar cell power conversion efficiency (PCE) increasing at an impressive rate from 3.8% to the current 25.7%^{[6, 7]}. Despite the excellent photovoltaic properties and the rapid development of lead halide perovskites, there are two unavoidable challenges. The first challenge is that lead halide perovskites are highly susceptible to degradation when encountering water or air, leading to instability. The other challenge is the toxicity of lead^[8−10]. In particular, the search for lead-free optoelectronic semiconductors that can match the performance of lead-based halide perovskites is very challenging.

Recently, Tang et al. and coauthors proposed an ion-type inversion strategy to design a new class of antiperovskite semiconductors with optoelectronic properties similar to those of lead-based halide perovskites^{[11, 12]}. Antiperovskite X₃BA (X is a cation, A and B are different-sized anions) has the same perovskite-type crystal structure as the conventional perovskite ABX₃ (A and B are different-sized cations, X is an anion), but the positions of the anions and cations at the lattice sites are swapped. Among them, nitride antiperovskites X₃NA with the anion N³⁻ located in the center of the octahedron have attracted extensive research interest due to the unique features of the nitrogen element^[13−15]. So far, at least 16 experimental syntheses of X₃NA (X₃NE (X = Sr, Ba; E = Sb, Bi)^{[16, 17]}; Mg₃NPn (Pn = As, Sb)^[18]; Ca₃NM (M = P, As, Sb, Bi, Ge, Sn, Pb, Tl)^{[19, 20]}; and A₃NAs (A = Mg, Ca, Sr, Ba)^[21]) have been reported. The physical properties of X₃NA have been widely studied, including optoelectronic^{[22, 23]}, superconductivity^[24−27], magnetoresistance^{[28, 29]}, magnetostriction^[30], magnetic and magnetocaloric^[31], thermoelectricity^{[32, 33]}, negative thermal expansion^{[34, 35]}, Dirac semi-metallic characteristic^{[36, 37]}, and topological insulators characteristic^[38]. Among nitride antiperovskites X₃NA (X = Sr²⁺, Ca²⁺, Mg²⁺; A = P³⁻, Sb³⁻, Bi³⁻), Sr₃PN have suitable band gaps ~1.2 eV, but others have band gaps that are too large for solar cell applications^{[13−15, 39]}. For example, Ca₃NP and Mg₃NP have wide direct band gaps of 2.1 eV and 2.3 eV, respectively^[39]. In addition to compositional engineering, strain and pressure are also important ways to modulate the electronic and optical properties of materials by tuning interatomic distance without changing the material composition. At present, a variety of pressure-induced phenomena have been found in perovskites， such as structural phase transition^[40−42], semiconductor-to-metal transition^{[43, 44]}, bandgap optimization^[44−47], emergence of photoluminescence^{[40, 48]}, crystalline to amorphous transitions^[49], and so on. However, the effect of pressure on the physical properties of antiperovskite materials has been relatively unexplored, except for a few studies reporting pressure-induced structural phase transitions (e.g., Ni₃CMg)^[41] and semiconductor-to-metal transition (e.g., Ca₃NP)^[44]. In particular, studies exploring the structure-property relationships of antiperovskite compounds under pressure are rare.

In this work, taking two large-bandgap antiperovskites X₃NP (X²⁺ = Ca, Mg) as examples, we studied the effect of pressure on their electronic, optical, and mechanical properties using first-principles calculations. We first investigated the structural stability and equation of state of X₃NP under pressure. We then calculated the trend of the optoelectronic properties of X₃NP with the pressure. The results reveal that the electronic (i.e., band gap, carrier effective masses), dielectronic, exciton binding energy, and optical properties of Mg₃NP show relatively small fluctuations upon compression, but those of Ca₃NP are more sensitive to pressure. The different pressure-dependent optoelectronic properties in Mg₃NP and Ca₃NP were analyzed in detail. Finally, the mechanical properties of X₃NP under compression were investigated.

From previous studies^[44], it was found that the main competing phases of X₃NP (X = Ca, Mg) are one-dimensional (1D) and three-dimensional (3D) Pnma phases at pressure below 40 GPa, as shown in Figs. 1(a) and 1(c). It can be seen that the 3D Pnma phase consists of highly distorted [NX₆] octahedra that are connected by means of corner-sharing, while the 1D Pnma phase shows edge-sharing octahedra. The calculated total energy and phonon spectra confirm the thermal and dynamic stability of the 3D Pnma phase of X₃NP (X = Ca, Mg), as shown in Figs. 1(d) and 1(e).

Considering whether the increase in pressure leads to a phase transition in X₃NP (X = Ca, Mg), we calculated the total enthalpy (H = E + PV) differences (ΔH = H_tot(1D) − H_tot(3D)) of Pnma between 1D and 3D. From 0 to 20 GPa, ΔH is greater than 0 for Ca₃NP and Mg₃NP up to 40 GPa. Moreover, the phonon spectra of the two compounds from 0 to 20 GPa (see Figs. 1(d) and 1(e), and Fig. S3) do not exhibit any imaginary phonon modes, indicating that they are all thermally and dynamically stable in the 3D phase throughout this pressure range.

As shown in Fig. 2, the lattice constants and volumes of X₃NP (X²⁺ = Ca, Mg) decrease linearly with increasing pressure from 0 to 20 GPa. Specifically, Ca₃NP (Mg₃NP) shortens by 7.4% (5.2%), 5.9% (4.5%), and 6.1% (4.9%) in the a, b, and c directions, respectively, as shown in Table S1. The volume reduction of 18.3% (13.9%) indicates that Ca₃NP is more compressive than Mg₃NP. The compression in X₃NP (X = Ca, Mg) shows a rate of decline a > b > c direction, as shown in Figs. 2(a) and 2(c), implying compressibility a > b > c axis. It is obvious that the trend of lattice parameters with pressure is contrary to that of conventional halide perovskites and chalcogenide perovskites^{[51, 52]}. For example, for the same orthorhombic phase Pnma, the highest compressibility is along the b-axis in inorganic halide perovskites and along the c-axis in chalcogenide perovskites. The equations of state (EOS) of both Ca₃NP and Mg₃NP are presented in Figs. 2(b) and 2(d), respectively. The pressure versus volume curves were fitted compressive using a third-order Birch-Murnaghan EOS fit. It is worth mentioning that the bulk modulus and the elastic constants in three directions of Ca₃NP are lower than those of Mg₃NP, the ambient pressure bulk moduli of Ca₃NP and Mg₃NP are 61.76 and 89.57 GPa, respectively, as shown in Figs. 2(b) and 2(d), which make the compressibility of Ca₃NP higher than that of Mg₃NP. These results are close to the values reported in Table S5.

The band structures and partial density of states (PDOS) of X₃NP (X = Ca, Mg) at different pressures based on the HSE06 method are shown in Figs. S5 and S6. First, it can be seen that X₃NP (X = Ca, Mg) maintains a direct band gap at Γ-point from 0 to 20 GPa. Second, the band gap of X₃NP (X = Ca, Mg) almost linearly decreases with increasing pressure, as shown in Table S4 and Fig. 3(a). Specifically, the band gap of Ca₃NP decreases with increasing pressure from 2.136 eV at 0 GPa to 1.767 eV at 20 GPa, a decrease of 17.3%. In contrast, Mg₃NP decreases only 2.79% from 2.366 eV at 0 GPa to 2.300 eV at 20 GPa. In the following, we will analyze in detail the reasons for the apparently different pressure-dependent bandgap variations of Ca₃NP and Mg₃NP.

The PDOS of Ca₃NP shows that the conduction band minimum (CBM) is mainly contributed by the cation Ca d orbital, and anion and cation s orbitals, while the valence band maximum (VBM) consists mainly of the p orbitals of P and N, as shown in Figs. S5(c) and S5(d). As the pressure increases, CBM of Ca₃NP shift down, leading to a significant decrease in the band gap, as shown in Figs. S5(a) and S5(b) and Figs. S7(a) and S7(d). In Mg₃NP, the CBM is contributed mostly by the cation and anion s orbitals and the VBM is mainly contributed by the p orbitals of P/N, as shown in Figs. S6(c) and S6(d). Unlike Ca₃NP, the CBM and VBM of Mg₃NP move only slightly with increasing pressure, and the upward shift of VBM is more pronounced, which causes a slight decrease in the band gap, as shown in Figs. S6(a) and S6(b) and Figs. S7(e)−S7(h). According to PDOS in Fig S5 and Fig S6, VBM is a P+N p state and CBM is a cation+anion s and cation d state, under pressure, CBM drops in energy due to s-d repulsion, which is large for Ca than for Mg because Ca 3d orbital energy is closer to the conduction band edge, so the Ca band gap drops faster than Mg. The VBM drops due to p-p repulsion between cation and anions but is also pushed up due to p-p repulsion between P and N, so its change is relatively small. Therefore, compared to Ca₃NP, the small band gap reduction of Mg₃NP with increasing pressure could be attributed to the low d orbital contribution and the low lattice constant compressibility.

Carrier effective mass (m^*) is another important descriptor that can be used to assess the electronic properties of photovoltaic materials. The hole and electron effective masses of X₃NP (X= Ca, Mg) under different pressure are shown in Fig. 3(b) and Table S3. It can be seen that the average hole m_h (electron m_e) effective masses of Ca₃NP and Mg₃NP are 0.636 (0.530) and 0.568 (0.336) m₀ at 0 GPa, respectively. Compared to the p-d orbital hybridization in Ca₃NP, the CBM and VBM of Mg₃NP are contributed by s-p and p-p orbital hybridization^[53]. Therefore, Mg₃NP has relatively dispersed bands near the Fermi level and small carrier effective masses. As the pressure increases, the average m_e of Mg₃NP and Ca₃NP show small fluctuations, and the average m_h of Ca₃NP shows a monotonic decreasing trend, while the average m_h of Mg₃NP has small variation. A gradual decrease in the m_h of Ca₃NP is attributed to the increase in the P and N p-p repulsion and increased bandwidth of the valence band by the pressure, as seen in Table S4.

Static dielectric constants can also be used to evaluate carrier transport properties because they can effectively limit defects and impurities scattering^{[54, 55]}. The static permittivity (εst) is contributed by ions (εion) and electrons (ε∞), where the electronic permittivity has a strong correlation with the electronic properties^[11]. The dielectric constants of X₃NP (X = Ca, Mg) at 0 GPa show strong anisotropy (e.g., for Ca₃NP, εstxx = 23.8, εstyy = 24.0, εstzz = 22.3; for Mg₃NP, εstxx = 29.7, εstyy = 27.7, εstzz = 23.6), as shown in Table S6, which are higher than that of typical halide perovskite CsPbBr₃ (εion = 14.6, ε∞ = 4.6, εstd = 19.2)^{[11, 56]}. The ε∞ of X₃NP (X = Ca, Mg) increases slightly with increasing pressure, which is caused by the reduced band gap^[57], as shown in Table S4 and Fig. 3(c). However, the εion of X₃NP (X = Ca, Mg) decreases gradually with increasing pressure, which is due to the reduced N-X bond length and increased frequency of lowest-lying optical phonon modes^[58]. That is, the reduced εion is due to increased confinement of the N ions under pressure, which limits the ion movement.

The exciton binding energy Eb of X₃NP (X = Ca, Mg) is obtained based on the calculated effective mass and electronic dielectric constant. The small exciton binding energy is an essential basis for determining the ideal solar cell material because the smaller exciton binding energy (E_b) enables the rapid separation of photogenerated carriers^[59]. The E_b is calculated using the Wannier model^[60]:

1Eb=Ryμ*ϵ2,

where R_y = 13.56 eV is the atomic Rydberg energy, µ* is the reduced exciton mass (1/µ* = 1/m_e + 1/m_h), and ε is the electronic dielectric constant. It can be seen that the Eb of Ca₃NP and Mg₃NP are 69 and 37 meV at 0 GPa, respectively. As the pressure increases, the Ebof Ca₃NP shows a monotonic decreasing trend and Mg₃NP shows small fluctuations, as shown in Table S4 and Fig. 3(d). The large decrease of the exciton binding energy of Ca compound is due to the increased Ca d component in CBM whereas the VBM is anion p state. The slight increase of the exciton binding energy of the Mg compound is due to the volume confinement.

The optical absorption in the visible region is key for optoelectronic application. The optical absorption of X₃NP (X = Ca, Mg) is anisotropic along different directions at different pressures, as shown in Fig. S8. The optical absorption coefficients (α ~ 10⁵ cm⁻¹) of X₃NP (X = Ca, Mg) in the visible region (see Fig. 4(a)) are comparable to the experimentally derived CH₃NH₃PbI₃^[61]. In the range of pressure from 0 to 20 GPa, Ca₃NP has a more pronounced redshift than Mg₃NP and an overall higher absorption coefficient in the visible region, as shown in Fig. 4(a). This corresponds exactly to the magnitude and size of their band gap variation, with the suitable band gap being more efficient in optical absorption.

The spectroscopic limited maximum efficiency (SLME) method is very effective in evaluating PV performance in terms of band gap and optical absorption strength, and identifying and analyzing promising PV materials^{[62, 63]}. The maximum conversion efficiency is closely related to the thickness of the film, as shown in Fig. 4(b). We show that the maximum SLME of 19.629% (15%) and 27.5% (16%) can be reached at 0 and 20 GPa for Ca₃NP (Mg₃NP) with a film thickness of 0.5 µm, respectively. The optical absorption of Mg₃NP is changed only slightly, which makes the SLME only increase by 1%. When Ca₃NP is at 20 GPa, SLME is comparable to conventional lead-based halide perovskites MAPbI₃ (~30.90%)^[12] and CsPbI₃ (~29.4%)^[64]. At 20 GPa, the smaller band gap and the redshift of the optical absorption coefficient are the main mechanisms for the increase in theoretical efficiency; that is, improved SLME is due to increase absorption (current) due to the reduced band gap.

The mechanical properties of a material have a very large impact on the actual solar cell preparation^{[65, 66]}. We examine the mechanical stability of X₃NP (X = Ca, Mg) under different pressure by calculating the elastic constants (C_ij). For orthorhombic structure (SG Pnma) with nine independent elastic constants and X₃NP (X = Ca, Mg) satisfying mechanical stability. The values of the elastic constants for both at different pressure are shown in Table S5, and the mechanical stability criteria for SG Pnma are shown in the Supplemental Material, Sec. Ⅲ. The elastic constants of both have significant elastic anisotropy. The principal elastic constants of X₃NP (X = Ca, Mg) are C₁₁, C₂₂, and C₃₃, and they reflect the strong resistance of the crystal to deformation along the a-, b- and c-directions, respectively. The C₁₁, C₂₂, and C₃₃ of Ca₃NP (Mg₃NP) were 103.3 (176.1), 128.7 (175.4), and 113.2 (188.7) GPa, respectively. Here, in terms of uniaxial strain, the Ca₃NP and Mg₃NP crystals have the highest compressibility along the b- and c-axes, respectively, while the lowest compressibility is along the c- and b-axes, respectively, as shown in Table S5. Furthermore, the elastic constants of Ca₃NP (Mg₃NP) satisfy C₄₄ = 45.5 GPa (81.4 GPa) < (C₁₁, C₂₂, C₃₃), which means that both are susceptible to shear deformation^[67]. The elastic constant of X₃NP (X = Ca, Mg) has a tendency to increase in the pressure range from 0 to 20 GPa, except for a decrease of 4.45 GPa for the elastic constant C₅₅ of Mg₃NP. Ca₃NP and Mg₃NP are driven by the pressure with a rapid linear increase of C₂₂ and C₁₁, respectively, and their values exceed the other elastic constants, as shown in Fig. 5. The chalcogenide perovskite AZrS₃ (A = Mg, Ca, Sr and Ba) reported by Majumdar^[52] and Rong^[68]et al., also has an orthorhombic structure where the elastic constants increase with decreasing lattice constants. Therefore, the increase in the elastic constants should be caused by the shortening of the lattice constants and bond lengths due to the increase in pressure. The increased elastic constant is due to short bond length or increased covalency.

Meanwhile Young's modulus (Y), bulk modulus (B), shear modulus (G) and Poisson's ratio (υ) are calculated by Voigt-Reuss-Hill (VRH)^{[69, 70]} approximation based on the corresponding elastic constants, as summed in Table S5, specific relationships in Supplemental Material, Sec. III. Y, B, and G represent the material's resistance to against longitudinal tension, resistance to fracture and resistance to plastic deformation, respectively^[71]. X₃NP (X = Ca, Mg) increases with increasing pressure in the interval from 0 to 20 GPa for Y, B and G, and the increase in modulus is approximately the same for both. The increase in pressure causes the bond length to decrease, resulting in a progressive increase in Y for X₃NP (X = Ca, Mg). This happens because larger strains are usually required to break smaller bond lengths, and thus obtain elastic constants and larger Young's modulus^[11]. In addition, the elastic properties of perovskites are mainly determined by the strength of the chemical bond X−B^{[72, 73]}. As the radius of the cation at the X-site decreases [r(Ca²⁺) > r(Mg²⁺)], the elastic constants and Y of the corresponding compounds increase with the shortening of the X-N bond length. This implies that Mg₃NP has a larger C_ij and Y compared to Ca₃NP, which is confirmed in Table S5. X₃NP (X = Ca, Mg) has the largest increase in B, while G is the least sensitive to pressure.

To better represent the orientation-dependent Y, we plotted the (100), (010) and (001) planar 2D projections of Y in polar coordinates for X₃NP (X = Ca, Mg) under different pressures, as shown in Fig. 6. The three planes of X₃NP (X = Ca, Mg) under different pressure exhibit some common features, namely, apparent anisotropy, Y at each angle increases with increasing pressure, anisotropy becomes more prominent with increasing pressure, obvious axial symmetry in shape and clear angular characteristics. The Y of some angles in the plane remains unchanged or increases by a small amount under increasing pressure, while other angles increase substantially, which leads to the enhanced anisotropy. Taking Ca₃NP (100) plane as an example, the Young's modulus is almost constant at 0° and 180°, and increases rapidly at 90° and 270° along with the strengthening of the pressure, as illustrated in Fig. 6(a). In addition, the Young's modulus distributions in the same plane of X₃NP (X = Ca, Mg) are different due to the different bond lengths of Ca−N and Mg−N.

It is worth mentioning that B/G is called Pugh’s ratio. The value of B/G equal to 1.75 is used as the boundary value between ductility and brittleness^{[74, 75]}. The B/G values of Ca₃NP and Mg₃NP showed brittleness (less than 1.75) at 0−5 GPa and 0−15 GPa and ductility (higher than 1.75) at 10−20 GPa and 20 GPa, respectively, and the B/G increased with the increase of pressure, as shown in Table S5. In addition, the Poisson's ratio (υ) proposed by Frantsevich's rule can also be used as an indication of toughness and brittleness, with Poisson's ratio equal to 0.26 as the critical value, greater than 0.26 for toughness and less than 0.26 for brittleness^[75]. In this paper, the ductility and brittleness of X₃NP (X = Ca, Mg) under different pressures are evaluated by B/G and ν methods and the results are consistent with each other. This demonstrates that X₃NP (X = Ca, Mg) shifts from brittleness to ductility as pressure increases.

In summary, we have investigated the electronic, optical, and mechanical properties of antiperovskite X₃NP (X²⁺ = Ca, Mg) in the pressure range 0−20 GPa by first-principles calculations. Our results show that X₃NP (X = Ca, Mg) shows an anisotropic compressibility with a > b > c axis, which is different from the typical Pnma phase in halide perovskites and chalcogenide perovskites. Meanwhile, Ca₃NP (B₀ = 61.76 GPa) has higher compressibility than Mg₃NP (B₀ = 89.57 GPa) due to its small bulk modulus B₀. The electronic and optical properties of Mg₃NP show small fluctuations upon compression, but those of Ca₃NP are more sensitive to pressure. For example, as the pressure increases from 0 to 20 GPa, the band gap of Ca₃NP decreases from 2.136 to 1.767 eV, the ionic dielectric constant is reduced by 7.4%, 5.9%, and 6.1% in the a, b, and c directions, respectively, the exciton binding energy decreases from 69 to 50 meV, the optical absorption is significantly red-shifted, and the SLME sharply increases from 19.6% to 27.5%. The different pressure-dependent optoelectronic properties in Mg₃NP and Ca₃NP are mainly attributed to their different lattice compressibility and band-edge orbital hybridization. Ca₃NP has lower unoccupied 3d orbital energy, which leads to strong s-d coupling at the CBM edge. Finally, the mechanical properties of X₃NP are increased upon compression due to the reduction in bond length, while inducing a brittle-to-ductile transitions for Ca₃NP and Mg₃NP at 10 and 20 GPa, respectively. Therefore, our research provides theoretical guidance and insights for future experimental tuning of the physical properties of antiperovskite semiconductors by pressure.