Strong quartic anharmonicity and thermoelectric performance in antiperovskite Sr₃XN (X = P, As, Sb, and Bi)

Abstract

Antiperovskite (APV) materials have garnered significant attention due to their diverse physical properties. The self-consistent phonon (SCP) theory and the Boltzmann transport equation (BTE) were combined to calculate thermal and electronic transport properties of APV Sr₃XN (X = P, As, Sb, Bi) based on density functional theory (DFT). Pronounced quartic anharmonicity was identified in Sr₃PN and Sr₃AsN, attributed to the vibrations of Sr atoms. This distinctive characteristic induces four-phonon (4ph) scattering, significantly reducing the lattice thermal conductivity ({\kappa }_L), particularly at elevated temperatures such as 900 K, with values as low as 0.58 W·m⁻¹·K⁻¹ for Sr₃PN. Additionally, Sr₃BiN exhibits remarkable electrical conductivity and a high Seebeck coefficient, leading to a promising figure of merit (ZT) of up to 1.03, underscoring its potential as a thermoelectric material. These findings demonstrate that the ultra-low lattice thermal conductivity driven by 4ph scattering, coupled with excellent thermoelectric properties, makes these materials promising candidates for high-efficiency thermoelectric applications.

Graphical abstract

Keywords

antiperovskite / four-phonon scattering / self-consistent phonon / thermoelectric

1 Introduction

Perovskite (PV) materials initially attracted widespread interest due to their excellent physical properties [1–4]. Classical inorganic PVs crystallise in a cubic ABX₃ lattice, where A and B are cations, and X is a highly electronegativity anion (e.g., O, N, or S) [5]. Interest has since broadened to hybrid organic–inorganic perovskites — exemplified by CH₃NH₃PbBr₃ (MAPbBr₃) and MAPbI₃, which offer exceptional compositional tunability and functional versatility [6, 7]. More recently, antiperovskites (APVs) have emerged as an inverse-structured analogue with the general formula X₃AB, in which X is usually a transition-, alkali- or alkaline-earth-metal cation [8, 9]. APVs have garnered considerable attention because of their unique properties [8, 10], including their magnetic properties [11, 12], negative thermal expansion [13–15], ionic conductivity [16, 17], superconductivity [3, 18], optoelectronic, phase stability [9], and thermoelectric properties [19–23].

Interestingly, Sr-based APVs exhibit intriguing physical properties. For instance, Sr₃PN has been reported to exhibit remarkable optical properties, including a high light absorption coefficient, low reflectivity, and exceptional photoconductivity, with a notable extinction coefficient [24]. Additionally, Sr₃AsN is predicted to exhibit substantial mechanical robustness, energetic stability, and dynamic equilibrium within the visible spectrum [25]. Sr₃MN (M = Sb or Bi) has also attracted attention as a potential Pb-free material for thin-film solar cells [2], while Sr₃BiN exhibits a strong optical response in the ultraviolet regime [26]. Despite these promising attributes, there has been limited investigation of the thermal and electronic transport properties of these materials.

Anharmonicity describes deviations of a potential energy surface (PES) from an ideal harmonic shape near a system’s equilibrium configuration [27]. While the harmonic approximation considers only second-order terms in the PES expansion, anharmonicity arises from higher-order contributions, enabling phonon–phonon interactions [28]. In materials science, these anharmonic interactions significantly influence both thermal and electronic transport properties [29–34]. For example, enhanced phonon–phonon scattering shortens phonon lifetimes, thereby lowering lattice thermal conductivity and amplifying its sensitivity to temperature changes [21, 35, 36]. On the other hand, anharmonicity-induced phonon hardening can increase phonon mobility, thus improving heat transport characteristics [37, 38]. Furthermore, strong anharmonicity significantly modulates electron−phonon interactions, affecting phonon scattering processes [39, 40] and enhancing electron−phonon coupling — a key mechanism underlying superconductivity [41].

A comprehensive treatment of anharmonicity encompasses several methodological tiers. Initially, quasi-harmonic and free-energy-perturbation approaches incorporate thermal expansion effects while sampling higher-order terms through stochastic integration [42]. Perturbative methods, such as linearised Boltzmann-transport solvers exemplified by Phono3py [43] and ShengBTE [22, 44–48], evaluate three- and four-phonon collision integrals. At higher complexity, renormalised-phonon frameworks — including self-consistent phonon (SCPH) [49], the stochastic self-consistent harmonic approximation (SSCHA) [50] and the temperature-dependent effective-potential (TDEP) [51] — iteratively update the dynamical matrix using fnite-temperature force constants. Furthermore, approaches based on Green–Kubo and direct non-equilibrium molecular dynamics methodologies capture anharmonicity by computing thermal conductivity directly from heat-flux correlations obtained from classical or ab initio trajectories [52].

Four-phonon scattering processes, despite their computational complexity, have recently become essential for accurately describing thermal transport in strongly anharmonic materials or at elevated temperatures [53]. This additional scattering mechanism significantly reduces thermal conductivity relative to predictions based solely on three-phonon interactions. Studies on materials such as silicon [54], diamond [54], and single-layer graphene [55] have explicitly demonstrated the substantial reduction in anomalously high thermal conductivity due to four-phonon scattering. This additional scattering mechanism significantly reduces thermal conductivity beyond predictions based solely on three-phonon interactions. Studies on materials such as silicon [54], diamond [54], and single-layer graphene [55] have explicitly demonstrated the substantial reduction in anomalously high thermal conductivity due to four-phonon scattering. The pronounced impact of quartic interactions underscores the crucial relationship between lattice anharmonicity and phonon scattering processes, highlighting the necessity for continued exploration of quartic anharmonic effects to achieve more accurate thermal transport predictions.

This work provides a comprehensive investigation of four APV materials, previously predicted by high-throughput calculations [56] to exhibit promising thermoelectric performance, with a focus on their phase stability, lattice dynamics, and thermal and electronic transport properties. It emphasizes the significant role of quartic anharmonicity effects in achieving low lattice thermal conductivity ({\kappa }_L) and high figure-of-merit (ZT) values. A combined approach employing self-consistent phonon (SCPH) theory [49, 57], the Boltzmann transport equation (BTE), and four-phonon scattering mechanisms was used. Notably, Sr₃PN and Sr₃AsN exhibit pronounced anharmonicity, and the inclusion of four-phonon scattering substantially reduces their {\kappa }_L. Our analysis, supported by phonon evaluations at the theoretical model level, underscores the pivotal role of four-phonon processes in accurately assessing anharmonic effects. These insights have contributed significantly to achieving high ZT values, reinforcing the importance of quartic anharmonic interactions in designing of thermoelectric materials. Thus, our findings highlight the essential role of four-phonon processes in developing thermoelectric materials with exceptionally low lattice thermal conductivity, critical for optimizing material performance.

2 Methodology

Density functional theory (DFT) calculations [58] were conducted using the Vienna Ab initio Simulation Package (VASP) [59], utilizing a plane-wave basis set and the Perdew−Burke−Ernzerhof revised for solids (PBEsol) exchange-correlation functional [60]. The Sr (4s4p5s), X (s^2{p}^3), and N (s^2{p}^3) shells were treated as valence states. The PBEsol functional was chosen because it modifies the gradient expansion of PBE to better reproduce the exchange-correlation energy density in dense electron systems, providing more accurate predictions of lattice constants.

The electronic transport properties were calculated utilizing the AMSET code [61], with a band gap correction applied via the Heyd−Scuseria−Ernzerhof (HSE06) hybrid functional. HSE06 combines Hartree−Fock (HF) exact exchange within the PBE generalized gradient approximation, providing a more accurate estimation of band gaps that are typically underestimated by PBE. The electrical conductivity (\sigma), Seebeck coefficient (S), and charge carrier contribution to the thermal conductivity ({\kappa }_e) were obtained using the following equations [61]:

(1){\sigma }_{\alpha \beta }=e^2{\displaystyle \int {\displaystyle \sum _{\alpha \beta }(\epsilon )(-{\displaystyle \frac{\mathrm{\partial }{f}^0}{\mathrm{\partial }\epsilon }})\mathrm{d}\epsilon ,}}

(2)S_{\alpha \beta }={\displaystyle \frac{1}{eT}}{\displaystyle \frac{{\displaystyle \int {\displaystyle \sum _{\alpha \beta }(\epsilon )(\epsilon -{\epsilon }_F)(-{\displaystyle \frac{\mathrm{\partial }{f}^0}{\mathrm{\partial }\epsilon }})\mathrm{d}\epsilon }}}{{\displaystyle \int {\displaystyle \sum _{\alpha \beta }(\epsilon )(-{\displaystyle \frac{\mathrm{\partial }{f}^0}{\mathrm{\partial }\epsilon }})\mathrm{d}\epsilon }}}},

(3)\begin{array}{rl}{\kappa }_e^{\alpha \beta }=& \frac{1}{T}\{20\frac{[{\displaystyle \int {\displaystyle \sum _{\alpha \beta }(\epsilon )(\epsilon -{\epsilon }_F)(-{\displaystyle \frac{\mathrm{\partial }{f}^0}{\mathrm{\partial }\epsilon }}){]}^2\mathrm{d}\epsilon }}}{{\displaystyle \int {\displaystyle \sum _{\alpha \beta }(\epsilon )(-{\displaystyle \frac{\mathrm{\partial }{f}^0}{\mathrm{\partial }\epsilon }})\mathrm{d}\epsilon }}}\\ & -\int \sum _{\alpha \beta }(\epsilon )(\epsilon -{\epsilon }_F)(-\frac{\mathrm{\partial }{f}^0}{\mathrm{\partial }\epsilon })\mathrm{d}\epsilon \},\end{array}

where \alpha and \beta denote Cartesian coordinates. e, \epsilon, and f^0 represent the electron charge, energy, and Fermi–Dirac distribution, respectively. {\mathrm{\Sigma }}_{\alpha \beta }(\epsilon ) is the spectral conductivity. T is temperature, and {\epsilon }_F is the Fermi level for a given doping concentration.

High-standing convergence criteria were applied for structural optimization, as phonon properties are highly sensitive to the precision of the underlying electronic structure and forces between atoms. A kinetic energy cutoff of 600 eV and a Monkhorst−Pack k-mesh of 10 \times 10 \times 10 for Brillouin zone integration were used in a system containing five atoms. The thresholds were set to {10}^{-6}\mathrm{e}\mathrm{V}/Å for the Hellmann−Feynman forces acting on each atom and {10}^{-8}\mathrm{e}\mathrm{V} for the total energy. For all single-point self-consistent calculations, a kinetic energy cutoff of 600 eV was used, along with a k-mesh of 5 \times 5 \times 5 and a 2 \times 2 \times 2 supercell. A convergence criterion of {10}^{-8}\mathrm{e}\mathrm{V} for total energy was applied.

The frozen phonon approach [62] was employed to evaluate the harmonic interatomic force constants (IFCs). To obtain higher-order IFCs, a compressive-sensing method [63–65] was use to get higher order IFCs so ab initio molecular dynamics (AIMD) simulations were performed, using 2 \times 2 \times 2 supercells with a 2 \times 2 \times 2 Monkhorst−Pack k-point grid. Subsequently, 100 atomic structures from the trajectories were selected, and random displacements of 0.1 Å were introduced to reduce cross-correlations among sampled configurations. The resulting displacement-force datasets were used to determine the third order and fourth order. Cutoff radii of 5 Å and 4 Å were applied for the 3rd and 4th IFCs, respectively, to ensure convergence while maintaining computational efficiency. Combined this with the self-consistent phonon (SCPH) theory [66], the temperature-dependent phonon bands were obtained. Calculation details can be found in the supplementary.

The FourPhonon [67, 68] package was employed to iteratively solve the linearized phonon BTE, enabling the calculation of the {\kappa }_L. The lattice temperature-dependent {\kappa }_L is obtained as [44]

(4){\kappa }_l^{\alpha \beta }={\displaystyle \frac{1}{{\kappa }_B{T}^2\mathrm{\Omega }N}}{\displaystyle \sum _{\lambda }{f}_0(f_0+1)(\hslash {\omega }_{\lambda }{)}^2{\upsilon }_{\lambda }^{\alpha }{F}_{\lambda }^{\beta },}

where \lambda comprises both a phonon branch index and a wave vector, \alpha and \beta represent Cartesian coordinates, \hslash is the reduced Planck constant, {\kappa }_B represents the Boltzmann constant, T is the temperature, \mathrm{\Omega } is the unit cell volume, and N is the number of atoms. {\omega }_{\lambda } and {\upsilon }_{\lambda } are the angular frequency and the group velocity of phonon mode \lambda, respectively. At thermal equilibrium, in the absence of a temperature gradient or other thermodynamical forces, phonons are distributed according to Bose−Einstein statistics, f_0. For a certain mode \lambda, the linearized BTE is expressed as [69]

(5)F_{\lambda }={\tau }_{\lambda }^0({\upsilon }_{\lambda }+{\mathrm{\Delta }}_{\lambda }),

where {\tau }_{\lambda }^0 denotes the relaxation time of mode \lambda, and {\mathrm{\Delta }}_{\lambda } is the deviation of the distribution function, characterizing the phonon population. The deviation of the distribution function arises due to the competition between the driving force (temperature gradient) and scattering mechanisms. The iterative process begins with an initial guess for the {\mathrm{\Delta }}_{\lambda }, which is subsequently used to update the equations by refining F_{\lambda } at each step until the conductivity tensor converges. For the BTE calculations, the number of grid planes in the inner {\mathit{q}}^{\prime } space was tested, as shown in Fig. S7. The blue line represents the results for 3ph scattering, which clearly converge when the q-mesh is larger than 7. The red line corresponds to the 4ph scattering, which shows convergence when the q-mesh exceeds 6. Therefore, an 8\times 8\times 8 q-mesh was adopted for all calculations to ensure accuracy while maintaining reasonable computational cost.

For materials with strong anharmonicity, the quasiparticle (phonon‐gas) picture becomes incomplete: incoherent vibrational transport can make a sizable contribution to the lattice thermal conductivity and often raises {\kappa }_L above the value predicted by purely coherent (propagating) phonons. Within the unified framework, these incoherent contributions are captured by the off-diagonal terms of the heat-flux operator and lead to the following expression for {\kappa }_L [70–72]:

(6)\begin{array}{rl}{\kappa }_L=& \frac{{\hslash }^2}{k_B{T}^{2}V{N}_{\mathit{q}}}\sum _{\mathit{q}}\sum _{s,s^{\prime }}\frac{{\mathrm{\Omega }}_{\mathit{q}}^s+{\mathrm{\Omega }}_{\mathit{q}}^{s^{\prime }}}{2}{\mathit{v}}_{\mathit{q}}^{s,s^{\prime }}\otimes {\mathit{v}}_{\mathit{q}}^{s^{\prime },s}\\ & \cdot \frac{{\mathrm{\Omega }}_{\mathit{q}}^s{n}_{\mathit{q}}^s\left(n_{\mathit{q}}^s+1\right)+{\mathrm{\Omega }}_{\mathit{q}}^{s^{\prime }}{n}_{\mathit{q}}^{s^{\prime }}\left(n_{\mathit{q}}^{s^{\prime }}+1\right)}{4{\left({\mathrm{\Omega }}_{\mathit{q}}^{s^{\prime }}-{\mathrm{\Omega }}_{\mathit{q}}^s\right)}^2+{\left({\mathrm{\Gamma }}_{\mathit{q}}^s+{\mathrm{\Gamma }}_{\mathit{q}}^{s^{\prime }}\right)}^2}\left({\mathrm{\Gamma }}_{\mathit{q}}^s+{\mathrm{\Gamma }}_{\mathit{q}}^{s^{\prime }}\right).\end{array}

Here, \hslash, k_B, T, V, and N_{\mathit{q}} denote the reduced Planck constant, the Boltzmann constant, the absolute temperature, the unit-cell volume, and the number of sampled phonon wave vectors, respectively. {\mathrm{\Omega }}_{\mathit{q}}^s is the (possibly renormalized) phonon frequency of branch s at wave vector \mathit{q}, n_{\mathit{q}}^s is the Bose–Einstein occupation, {\mathrm{\Gamma }}_{\mathit{q}}^s is the scattering rate (linewidth, i.e., the inverse lifetime), and {\mathit{v}}_{\mathit{q}}^{s,s^{\prime }} is the generalized group-velocity tensor. The dyadic \otimes and the centered dot indicate a tensor product and a double contraction. The Lorentzian denominator encodes spectral broadening from finite lifetimes, while the off-diagonal (s\ne s^{\prime }) terms account for incoherent (diffuson-like) energy transfer that becomes prominent in strongly anharmonic or disordered crystals.

The thermoelectric efficiency depends on the thermoelectric ZT, which is defined as [73, 74]

(7)ZT={\displaystyle \frac{\sigma S^{2}T}{{\kappa }_e+{\kappa }_l}}.

3 Results and discussion

3.1 Phase stability and lattice dynamics

The APV crystallizes in a face-centered cubic structure with space group Pm\overline{3}m (No. 221), as illustrated in the inset of Fig. 1(e). As shown in Table 1, the optimized lattice constants for four compounds, namely Sr₃PN, Sr₃AsN, Sr₃SbN, and Sr₃BiN are 5.02, 5.05, 5.14, and 5.18 Å, respectively. These results are consistent with previous studies [2, 76–79], showing differences of less than 0.08 Å. Apparently, the lattice parameters of these compounds increase as the atomic radius of the X atom grows (P: 107 pm, As: 119 pm, Sb: 139 pm, Bi: 148 pm) and its electronegativity decreases (P: 2.19, As: 2.18, Sb: 2.05, Bi: 2.02). Table 1 also shows the computed dielectric tensors and Born effective charges, which are used to correct the dynamical matrices via the non-analytical term correction (NAC). For example, the dielectric tensor and Born effective charges for Sr₃PN are: {ϵ}^{\mathrm{\infty }}=9.05, Z^{\ast }(\mathrm{S}\mathrm{r}{)}_{||}=2.60, Z^{\ast }(\mathrm{S}\mathrm{r}{)}_{\perp }=2.71, Z^{\ast }(\mathrm{X})=-3.45, and Z^{\ast }(\mathrm{N})=-4.46.

The thermodynamic stability determines whether a compound can be synthesized. Therefore, an accurate evaluation of the stability of the APVs is essential. The thermodynamic stability can be assessed using the convex hull constructed from the formation energies of all relevant competing phases within a given chemical system. In this work, the formation energies and convex hull data of the Sr–P–N, Sr–As–N, Sr–Sb–N, and Sr–Bi–N ternary systems were obtained from the OQMD database [80]. The formation energy of a compound with composition x is defined as E_f^{\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}}(x)=E_{\mathrm{t}\mathrm{o}\mathrm{t}}^{\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}}(x)-{\sum }_i{\nu }_i{\mu }_i^{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}}, where E_{\mathrm{t}\mathrm{o}\mathrm{t}}^{\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}}(x) is the total energy of the compound, {\nu }_i is the stoichiometric coefficient of element i, and {\mu }_i^{\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}} is the total energy per atom of the elemental reference. The negative formation energies shown in Table 2 indicate that the compound can, in principle, form exothermically from its elements. The stability of each compound is further quantified by its energy above the convex hull, defined as E_{\mathrm{h}\mathrm{u}\mathrm{l}\mathrm{l}}=E_f^{\mathrm{p}\mathrm{h}\mathrm{a}\mathrm{s}\mathrm{e}}(x)-E_f^{\mathrm{h}\mathrm{u}\mathrm{l}\mathrm{l}}(x), where E_f^{\mathrm{h}\mathrm{u}\mathrm{l}\mathrm{l}}(x) represents the lowest formation energy achievable at the same composition by any combination of stable competing phases [81, 82]. As shown in Fig. S1, Sr₃SbN and Sr₃BiN lie on the convex hull (\mathrm{\Delta }{E}_H=0), indicating thermodynamic stability at 0 K. Sr₃PN and Sr₃AsN are 26 and 82 meV/atom above the hull, respectively, placing them within the commonly accepted metastability range (\lesssim 100\mathrm{m}\mathrm{e}\mathrm{V}/\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}) [56]. Compounds within this range are often experimentally accessible due to entropic stabilization or kinetic trapping. For Sr₃AsN, an orthorhombic Pnma polymorph is predicted, consistent with other antiperovskites [83]. This Pnma phase is 29 meV/atom lower in energy than the cubic Pm\overline{3}m structure. However, it exhibits a relatively low ZT value [4] and is therefore not considered in subsequent analysis. The cubic Pm\overline{3}m phase remains metastable with respect to the convex hull but may be stabilized at finite temperatures. To further assess their resistance to decomposition, the decomposition enthalpies were evaluated, defined as \mathrm{\Delta }{H}_{\mathrm{d}}=E_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}-{\sum }_i{\nu }_i{E}_i, where E_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}} and E_i denote the total energies of the compound and the lowest-enthalpy mixture of competing phases at the same composition, respectively [84]. Under this convention, indicates that a compound is stable relative to its decomposition products. For Sr₃PN, Sr₃AsN, Sr₃SbN, and Sr₃BiN, the calculated \mathrm{\Delta }{H}_{\mathrm{d}} values are 0.123, −0.251, −0.680, and −0.731 \mathrm{e}\mathrm{V}/\mathrm{f}.\mathrm{u}., respectively. The slightly positive value for Sr₃PN suggests that it is metastable at 0 K but may be kinetically or entropically stabilized at finite temperatures, consistent with its small energy above the convex hull (26 meV/atom). In contrast, the negative \mathrm{\Delta }{H}_{\mathrm{d}} values for Sr₃AsN, Sr₃SbN, and Sr₃BiN confirm their thermodynamic stability. Taken together, these results indicate that Sr₃PN and Sr₃AsN are near the stability boundary and represent plausible synthesis targets, whereas Sr₃SbN and Sr₃BiN are robust thermodynamic ground states [78].

To further assess the thermal stability of these APV materials, AIMD simulations were conducted to track changes in free energy over time. As shown in Fig. 1(c), the free energy stabilizes over time, and variations in free energy across different temperatures are minimal, confirming the thermal stability of the four APV materials under varying conditions.

Dynamic stability refers to a material’s ability to resist distortions and maintain its structure under dynamic perturbations. To evaluate this, lattice dynamics were analyzed using harmonic phonon dispersion relations, as represented by the yellow lines in Figs. 1(a)−(d). For Sr₃PN and Sr₃AsN, imaginary frequencies appear at the M(0.5,0.5,0) and R(0.5,0.5,0.5) points, indicating dynamic instability (). These unstable modes correspond to the irreducible representations M_2^{+} at M-pont and R_5^{-} at R-point, respectively. In contrast, the harmonic phonon dispersion of Sr₃SbN and Sr₃BiN reveals no imaginary frequencies, confirming their dynamic stability. Furthermore, the imaginary frequencies observed in Sr₃PN and Sr₃AsN are primarily induced by the vibrations of Sr atoms, highlighting the role of atomic contributions to dynamic instability.

To further explore the phonon modes, frozen phonon PES were calculated for the M_2^{+} and R_5^{-} modes at the M and R points, as shown in Figs. 1(e) and (f). Both Sr₃PN and Sr₃AsN exhibit soft modes at these points, characterized by double-well PES, confirming their dynamic instability. In this scenario, the minimum energy points are not located at the zero-tilt amplitude (Q_1=Q_2=0) but occur at non-zero coordinates, indicating structural distortions [85]. The mode diagram further shows that the vibrations at the M and R points are primarily driven by Sr atoms, consistent with the earlier findings that phonon soft modes are dominated by Sr contributions. To address these instabilities, SCP theory incorporates anharmonic effects into the lattice dynamics. This iterative approach refines the effective potential by accounting for higher-order interactions, renormalizing the phonon frequencies, and removing imaginary components, thereby dynamically stabilizing the structure. Additionally, SCP theory captures temperature-dependent stabilization, where anharmonic contributions at finite temperatures smooth the double-well PES, enabling accurate predictions of thermodynamic and vibrational properties. As a result, the imaginary frequencies in Sr₃PN and Sr₃AsN can be resolved using SCP theory, making it feasible to study these materials despite their thermodynamic metastability.

In addition, the stability and symmetry of APV structures can be predicted using the tolerance factor (t) [86–89]. The tolerance factor is given by t=(r_X+r_{Sr})/[\sqrt{2}(r_N+r_{Sr})] [90], where r_{Sr}, r_N, and r_X represent the ionic radii of the Sr cation, N anion, and X(P, As, Sb, Bi) anion, respectively. For Sr₃PN and Sr₃AsN, t values of 0.825 and 0.846 suggest significant lattice distortion, aligning with their observed dynamic instability [9]. In contrast, Sr₃SbN and Sr₃BiN exhibit t values of 0.904 and 0.928, respectively, which stabilize the lattice in a cubic structure due to the larger ionic radii of the Sb and Bi anions.

The inclusion of the quartic anharmonic term in SCP theory plays a crucial role in solving the self-consistent equation, providing the phonon frequencies and polarization vectors at different temperatures. As shown by the red lines in Figs. 1(a)−(d), the soft modes of Sr₃PN and Sr₃AsN shift, primarily due to the dominant contribution of the diagonal terms in the quartic term coefficient matrix. Off-diagonal terms can redistribute anharmonic effects among different modes. Although they may indirectly influence the soft modes, their impact is generally weaker compared to the direct contributions of the diagonal terms. This shift indicates that the anharmonic contributions have a substantial impact on phonon behavior. It is also observed that the low-lying modes of Sr₃PN and Sr₃AsN harden with increasing temperature. Moreover, with increasing temperature, the phonon frequencies over 250 ({\mathrm{c}\mathrm{m}}^{-1}) of Sr₃SbN and Sr₃BiN shift upward, while phonon self-energy corrections significantly affect medium and high-frequency optical phonons. The disappearance of imaginary frequencies dominated by Sr atoms further suggests notable anharmonicity in these materials. At 600 K, the harmonic and anharmonic phonon density of states reveal that collective motions involving Sr and X atoms primarily govern the acoustic and low-frequency optical modes, while N atoms control the behavior of the two high-frequency phonon branches. The inclusion of Born effective charges in self-consistent phonon (SCP) calculations leads to frequency splitting, particularly at the \mathrm{\Gamma } point, highlighting the importance of accurately modeling anharmonic interactions in lattice dynamics. This frequency splitting, known as longitudinal optical–transverse optical (LO-TO) splitting, arises from long-range Coulomb interactions associated with macroscopic electric fields generated by LO phonons. To accurately capture this effect, NAC must be incorporated when calculating phonon frequencies. NAC accounts for the contributions of Born effective charges and the dielectric constant, enabling precise predictions of phonon dispersion, as illustrated in Figs. 1(a)–(d).

The mechanical stability of a material is described by its strain energy (W), which is influenced by its elastic constants [91, 92]:

(8)W=E_{\mathrm{s}\mathrm{t}\mathrm{r}}-E_{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{r}}={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i,j}{C}_{i,j}{ϵ}_i{ϵ}_j,}

where C_{ij} represents the second-order elastic tensor and ϵ denotes strain. E_str and E_unstr refer to the strained and unstrained states, respectively. Further details about the elastic tensors are provided in the Supplementary Materials. The corresponding mechanical stability criteria can be expressed as [92]

(9)C_{11}-C_{12}>0,C_{11}+2C_{12}>0,C_{44}>0,

where the elastic constants C_{ij}, shown in Table 3, satisfy the mechanical stability criteria, confirming the mechanical stability of the materials. Additionally, various mechanical property parameters were examined, including the bulk modulus (B), shear modulus (G), Pugh’s ratio (B/G) and Young’s modulus (E), as presented in Table 3. The results indicate that all four APVs exhibit similar mechanical properties, with B of around 48 GPa, G around 40 GPa, and E around 93 GPa. These values suggest that all four APVs possess significant resistance to deformation and low compressibility. Moreover, based on the Pugh ratio (B/G) (approximately 1.2), these APVs exhibit brittle behavior, as the values fall below the critical threshold defined by the Pugh criterion [93].

3.2 Lattice thermal conductivity

Lattice thermal conductivity ({\kappa }_L) is a key property for evaluating the thermal transport capabilities of materials, especially in applications such as thermoelectrics and thermal barrier coatings. In this study, {\kappa }_L was estimated using the Boltzmann Transport Equation (BTE) with inputs from self-consistent phonon (SCP) frequencies and eigenvectors, {\nu }_q(T)=\mathrm{\partial }{\mathrm{\Omega }}_q(T)/\mathrm{\partial }q. The SCP framework enables the renormalization of second-order force constants at each temperature, leading to effective harmonic force constants.

The computed {\kappa }_L for the four cubic APVs are illustrated in Fig. 2, where both three-phonon {\kappa }_{3ph}^{\mathrm{S}\mathrm{C}\mathrm{P}} and combined three- and four-phonon scattering {\kappa }_{3,4ph}^{\mathrm{S}\mathrm{C}\mathrm{P}} processes were considered. At 900 K, {\kappa }_{3ph}^{\mathrm{S}\mathrm{C}\mathrm{P}} for Sr₃PN, Sr₃AsN, Sr₃SbN, and Sr₃BiN amount to 1.48, 1.10, 1.38, and 1.74 \mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot {\mathrm{K}}^{-1}, respectively. However, {\kappa }_{3,4ph}^{\mathrm{S}\mathrm{C}\mathrm{P}} shows significant reductions, with values decreasing to 0.57, 0.55, 0.84, and 1.09 \mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot {\mathrm{K}}^{-1}, respectively, at the same temperature. Comparing the two cases, {\kappa }_{3,4ph}^{\mathrm{S}\mathrm{C}\mathrm{P}} is significantly lower than {\kappa }_{3ph}^{\mathrm{S}\mathrm{C}\mathrm{P}}, with reductions at 300 K of 37.2%, 38.5%, 17.6%, and 18.1% for Sr₃PN, Sr₃AsN, Sr₃SbN, and Sr₃BiN respectively. As the temperature increases to 900 K, these reductions become even more pronounced, reaching 61.5%, 50%, 39.1%, and 37.4%, respectively. This suggests that, even in weakly anharmonic materials, 4ph scattering plays a critical role in reducing thermal conductivity. Compared to other APV materials, these APVs exhibit relatively low {\kappa }_L. For instance, experimental data show that Ca₃SnO has a {\kappa }_L of 17 \mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot {\mathrm{K}}^{-1} at 290 K [94], and calculations for SnMn₃N report 3.593 \mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot {\mathrm{K}}^{-1} at 300 K [95]. The low thermal conductivity in these four APVs can be attributed to strong anharmonicity, which suppresses heat transport by enhancing phonon-phonon interactions. Furthermore, the {\kappa }_L(\omega ) is primarily concentrated in the low-frequency region, accounting for 77.8%, 79.2%, 84.1%, and 82.3% of the total thermal conductivity for Sr₃PN, Sr₃AsN, Sr₃SbN, and Sr₃BiN, respectively. The significant role of four-phonon scattering in suppressing {\kappa }_L makes these APVs promising candidates for applications where low thermal conductivity is essential. The low {\kappa }_L observed in APVs, combined with their strong anharmonicity, highlights their potential for enhanced thermoelectric performance.

Furthermore, the lattice thermal conductivity {\kappa }_L including the off-diagonal contributions was also calculated, as shown in Fig. S2. The off-diagonal terms slightly increase {\kappa }_L, indicating a moderate contribution from incoherent phonon transport. At 900 K, the calculated {\kappa }_L^{3,4\mathrm{p}\mathrm{h},\mathrm{o}\mathrm{f}\mathrm{f}} values for Sr₃PN, Sr₃AsN, Sr₃SbN, and Sr₃BiN are 0.81, 0.73, 1.00, and 1.24 W·m⁻¹·K⁻¹, respectively, suggesting that the off-diagonal contributions are negligible.

Another notable observation is the unusually weak temperature dependence of {\kappa }_L within each cubic APV, particularly in Sr₃PN and Sr₃AsN. In general, lattice thermal conductivity ({\kappa }_L) follows a power law relationship, {\kappa }_L\propto T^{-\gamma }, where \gamma is approximately 1 for most materials. However, in Sr₃PN and Sr₃AsN, the observed \gamma values deviate significantly from this universal behavior, with \gamma values of 0.58 and 0.71, respectively. This suggests a much weaker temperature dependence in these materials compared to the expected T^{-1} trend. The presence of 4ph scattering further amplifies the temperature sensitivity, especially in materials with strong anharmonicity. In Sr₃PN and Sr₃AsN, anharmonic effects are so pronounced that imaginary phonon frequencies appear even at 0 K, leading to \gamma values of 1.02 and 0.94, respectively, under the influence of 4ph scattering. This increased temperature sensitivity results in a sharper reduction of thermal conductivity as temperature rises. In Sr₃SbN and Sr₃BiN, the \gamma values also increase due to 4ph scattering, from 0.8 to 1.04 in Sr₃SbN, and from 0.87 to 1.09 in Sr₃BiN. This heightened temperature sensitivity results in significant suppression of {\kappa }_L, highlighting the critical role of four-phonon scattering in limiting heat transport in these materials [96, 97].

3.3 Mode level phonon analysis

For a deeper understanding of the effects of 4ph scattering, a mode-level phonon analysis is necessary, starting with an examination of scattering rates (SRs). As shown in Fig. 4(a), the 4ph SRs are comparable in magnitude to the 3ph SRs, especially in materials with strong anharmonicity, such as Sr₃PN and Sr₃AsN. In fact, in the low-frequency region, the SRs for 4ph scattering even surpass those of 3ph scattering, underscoring the dominant role that 4ph processes play in these materials.

Similar to the behavior of 3ph scattering rates, the influence of 4ph SRs becomes more pronounced at higher temperatures, particularly for Sr₃SbN and Sr₃BiN, shown in Fig. 4(b). The temperature-dependent increase in SRs emphasizes the growing impact of 4ph processes as temperature rises, which further limits thermal conductivity. Furthermore, Sr₃AsN has the highest SRs among the materials studied, consistent with its lowest {\kappa }_L. This relationship highlights the critical role of phonon scattering in determining thermal conductivity, particularly in materials where anharmonicity is strong and 4ph processes become significant.

Before conducting calculations, it can be predicted that 4ph scattering will reduce the lattice thermal conductivity {\kappa }_L of the material, as the Umklapp process is a key factor influencing thermal resistance. In 4ph scattering processes, the Umklapp (U) process plays a dominant role, as shown in Fig. 4(c). There are three types of 4ph scattering, q+q_1\to q_2+q_3, q\to q_1+q_2+q_3 and q+q_1+q_2\to q_3, with the first type being the most prevalent, as shown in Fig. S4. The splitting and combination processes for 3ph scattering are illustrated in Fig. S3. In general, 3ph scattering leads to longer relaxation times. However, incorporating 4ph scattering into the calculations yields more accurate predictions of relaxation times, particularly at room temperature. This highlights the risk of overestimating relaxation times when only 3ph scattering is considered, underscoring the importance of accounting for 4ph scattering mechanisms [54]. Additionally, the scattering phase spaces for 3ph and 4ph processes were calculated, as shown in Fig. S5 and Fig. S6, respectively. The decomposed phase spaces and corresponding scattering rates exhibit similar trends, confirming that these four APVs exhibit large anharmonic scattering rates, which strongly affects thermal transport.

Phonon group velocity ({\upsilon }_{ph}) is a key parameter for analyzing the lattice thermal conductivity ({\kappa }_L) of materials, as it directly influences heat transport. Figure 5 is the {\upsilon }_{ph} at 300, 600, and 900 K. Notably, the {\upsilon }_{ph} values remain largely consistent across these temperatures, which reflects the similarity in the phonon spectrum. Observing low {\upsilon }_{ph} values across all materials supports the overall low {\kappa }_L. Although Sr₃PN exhibits the smallest group velocity, this does not directly result in the lowest {\kappa }_L. This divergence can be attributed to the role of phonon lifetime, which significantly affects {\kappa }_L and is inversely correlated with the SRs. The relatively high SRs observed in Sr₃AsN, which lead to shorter phonon lifetimes, may explain why Sr₃AsN has the lowest {\kappa }_L despite not having the smallest {\upsilon }_{ph}. This emphasizes the importance of both group velocity and phonon lifetime in determining thermal conductivity.

To gain a deeper understanding of the microscopic mechanisms contributing to the low {\kappa }_L, the Grüneisen parameter (\gamma) was computed, which measures anharmonicity in the material. As illustrated in Fig. 5(b), a notable observation is that, compared to the optical phonon modes, the acoustic phonon modes exhibit higher values of \gamma. This suggests that acoustic phonons experience stronger cubic anharmonicity, directly impacting heat transport by enhancing phonon−phonon scattering.

3.4 Electronic transport

The electronic band structure and density of states for Sr₃PN, Sr₃AsN, Sr₃SbN, and Sr₃BiN provide crucial insights into the semiconducting behavior of these materials. As shown in Fig. 6, all four materials exhibit characteristics of direct band gap semiconductors, with the conduction band minimum (CBM) and valence band maximum (VBM) both located at the \mathrm{\Gamma } point. The calculated band gap values using the PBEsol functional with spin−orbit coupling (SOC) are 0.38 eV for Sr₃PN, 0.35 eV for Sr₃AsN, 0.15 eV for Sr₃SbN, and 0.17 eV for Sr₃BiN. To improve the accuracy of band gap predictions, the HSE06 hybrid functional was employed. The refined band gap values are significantly higher, at 0.88 eV for Sr₃PN, 0.92 eV for Sr₃AsN, 1.19 eV for Sr₃SbN, and 1.25 eV for Sr₃BiN, reflecting the well-known tendency of PBEsol to underestimate band gaps. In terms of orbital contributions, the CBM is primarily dominated by Sr d orbital, while the VBM is mainly contributed by the p orbital of N and p orbital of X. This distribution of electronic states further characterizes the electronic structure of these APV materials and their potential for semiconducting applications.

In contrast to traditional perovskites (PVs), antiperovskites (APVs) exhibit a unique structural arrangement where cations occupy the positions typically reserved for anions in the octahedral structure. This distinctive arrangement, coupled with the abundance of X-site cations, imparts unconventional physical and chemical properties, particularly affecting d-spin states, band structure, and ion transport. These structural differences set APVs apart from traditional PVs, enabling unique electronic behavior. The valence bands of Sr₃XN (X = P, As, Sb, Bi) arise from the hybridization of Sr 4d-N 2p and Sr 4d-X p states, as shown in previous studies [4, 98]. This hybridization is a key factor in shaping the electronic structure and influencing the material’s conductive properties. Additionally, the presence of Sr s states is constrained by the symmetry of the p states originating from X and N atoms [98]. This restriction further defines the behavior of the electronic states, contributing to its distinctive electronic properties.

Figure 7 displays an overview of the electronic transport properties, encompassing both p-type and n-type materials. To model rational electronic transport behavior, the band gap obtained from the HSE06 hybrid functional was used as the input parameter for these materials. As the carrier concentration (n) increases, both the electrical conductivity (\sigma) and the electronic thermal conductivity ({\kappa }_e) increase, while the Seebeck coefficient (S) decreases. This behavior is attributed to the growing participation of charge carriers in the conduction (\sigma \approx ne\mu) due to the elevated carrier concentration n. Conversely, at higher temperatures, \sigma decreases while S increases. When comparing p-type and n-type materials, n-type materials demonstrate larger \sigma, whereas p-type materials exhibit a larger S. This trend is consistent with the greater band dispersion observed at the conduction CBM for n-type materials and the greater degeneracy at the VBM for p-type materials, as shown in Fig. 6. In terms of power factor, p-type materials achieve values of 0.62, 0.81, 0.83, and 1.03 W·m⁻¹·K⁻² for Sr₃PN, Sr₃AsN, Sr₃SbN, and Sr₃BiN, respectively. However, the n-type counterparts display even higher maximum power factors of 0.78, 1.13, 1.13, and 1.62 W·m⁻¹·K⁻², respectively. There are highest power factors (>0.8) at 900 K for these two materials.

The ultimate goal of the analysis is to achieve excellent thermoelectric properties, which are quantified by the ZT, as illustrated in Fig. 8, particularly at n_h at 900 K. This remarkable performance is driven by the combination of ultralow lattice thermal conductivity ({\kappa }_L) and a high power factor. The substantial electronic band dispersion at the CBM further enhances ZT for n-type materials, with ZT values of 0.62, 0.81, 0.83, and 1.03 at 900 K for Sr₃PN, Sr₃AsN, Sr₃SbN, and Sr₃BiN, respectively. For instance, Sr₃SbN and Sr₃BiN exhibit exceptional ZT values at n_e around 3\times {10}^{18} cm⁻³. Importantly, the highest ZT for n-type is significantly larger than p-type, suggesting that n-type variants are more promising for thermoelectric applications. In practical experiments, factors such as thermal radiation, air-induced thermal convection, and the impact of grain boundary scattering on carrier mobility (\mu) may lead to slight reductions in the calculated ZT values [99]. Nevertheless, n-type materials remain strong candidates for high-performance thermoelectric devices due to their superior thermoelectric properties.

4 Conclusion

To summarize, this study investigated into the thermal and electrical transport characteristics and the thermoelectric potential of APV materials ({\mathrm{S}\mathrm{r}}_{3}X\mathrm{N}; X = P, As, Sb, Bi). As previously explained, the significance of quartic anharmonicity has been emphasized in all the materials under consideration. Similar to 3ph scattering, the prominence of 4ph scattering is dictated by two crucial factors: the anharmonicity of the energy landscape and the available scattering phase space. The influence of 4ph interactions extends beyond phonon self-energy adjustments, as it also diminishes the temperature sensitivity of {\kappa }_L, ultimately impacting the overall thermoelectric performance. It is important to highlight that the computation of both three- and four-phonon scattering rates is a challenging task involving intricate interactions among matrix elements, eigenvectors, and frequencies, which collectively encompass various phonon modes. Moreover, the unique combination of highly dispersive band edges and electronic band degeneracy in Sr₃XN leads to outstanding electronic conductivity and a substantial Seebeck coefficient. The optimal power factor is realized in Sr₃BiN, ultimately resulting in the maximum ZT value of 1.03.

Beyond bulk APVs, emerging two-dimensional APV monolayers have been predicted to be structurally robust and to exhibit electronic transport favorable for thermoelectricity, highlighting 2D APVs as an intriguing platform worthy of attention [100–102]. Our study not only highlights the exceptional thermoelectric properties of APV Sr₃BiN but also underscores the importance of quartic anharmonicity for {\kappa }_L and ZT, which significantly impacts the thermoelectric performance of the material. This research provides valuable insights for designing and optimizing thermoelectric properties in strongly anharmonic materials moving forward.

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