M₃OI (M = K, Rb) compounds exhibit simple anti-perovskite structures, following the general formula ABX₃. As illustrated in Figure 1a, the crystal structures of K₃OI and Rb₃OI crystallize within the Pm3m space group. In these structures, the alkali metals (M = K, Rb) occupy the X-sites within the unit cell, while the oxygen (O) ions occupy the B-sites. This arrangement results in M₆O octahedra formed by the oxygen anions and the alkali metal cations [25, 26]. The iodine (I) atoms are situated at the A-sites, which are located at the corners of the unit cell. Figure 1b depicts the crystal structure of anti-perovskite derivatives K₄OI₂ and Rb₄OI₂, which bear resemblance to the (La, Ba)CuO₄ structure. These compounds belong to the tetragonal crystal system and crystallize in the space group I4/mmm. In this structure, two distinct alkali metal sites are presented, as noted in previous studies [27]. Supplementary Figure S1 highlights the layered nature of this arrangement, where alternating M₃OI and MI layers are stacked along the z-axis(c), mirroring the structural arrangement seen in Na₄OI₂ [28].

The optimized lattice constants for these four structures are summarized in Table 1. K₃OI has been successfully synthesized in experiments [29]. Additionally, Table 1 provides bond lengths for each structure, along with key mechanical properties such as elastic constants, bulk modulus (B), Young’s modulus (E), shear modulus (G), and the Debye temperature (Θ) [30]. The force constant for M₃OI (M = K, Rb) adheres to the mechanical stability criterion in Equation 3, while M₄OI₂ (M = K, Rb) meets the mechanical stability criterion in Equation 4. Therefore, all four structures are mechanically stable [31]. C 11 − C 12   >   0   ;   C 11 + 2 C 12   >   0   ;   C 44   >   0 (3) C 11   >   C 12   |   ;   2 C 13 2 < C 33 C 11 + C 12   ; C 44   >   0   ;   C 66 >   0 (4)

Debye temperature serves as an indicator of the bond strength of materials, with a positive correlation between these two parameters. Among the studied materials, K₃OI exhibits the highest Debye temperature of 233.3 K, followed by K₄OI₂ of 187.3 K and Rb₃OI of 174.2 K. The lowest Debye temperature of 144.2 K is observed in Rb₄OI₂. This trend is consistent with the bond length of them. We find that all the Debye temperatures are significantly below the room temperature of 300 K, suggesting that the M₃OI and M₄OI₂ compounds exhibit pronounced anharmonic behavior.

The charge density in Figure 2 shows a typical covalent bond charge distribution between the alkali metal M and the oxygen (O) element. Specifically, the strong covalent bond between the alkali metal M and O forms a stable M₆O octahedral framework structure. In contrast, the iodine (I) element exhibits a strong electronic localization phenomenon, with weaker bonding to other atoms. By comparing the charge densities in Figures 2a,b for K₃OI and Rb₃OI, and in Figures 2c,d for K₄OI₂ and Rb₄OI₂, it can be observed that the Rb-O bond is weaker than the K-O bond, which is consistent with the results of bond length in Table 1.

The phonon dispersion spectra show that K₃OI, K₄OI₂, Rb₃OI, and Rb₄OI₂ exhibit no imaginary frequencies under the harmonic approximation in Figure 3, indicating that these materials are thermodynamically stable. The high-frequency optical branches are primarily contributed by the vibration modes of the oxygen (O) atoms, while the low-frequency optical and acoustic branches are dominated by the heavier iodine (I) and alkali metal (K, Rb) elements. The phonon dispersions for all four structures exhibit strong phonon-optical coupling within the low-frequency optical and acoustic branch regions, which is typically accompanied by peaks in the phonon density of states (PDOS). A comparison between Figures 3a,c reveals that the highest optical branch frequency for K₃OI is 9 THz, which is higher than the 8 THz for Rb₃OI. K₃OI exhibits stronger phonon dispersion, particularly in the acoustic branch, with the phonon group velocity for K₃OI (0–1.44 THz) being significantly higher than for K₄OI₂ (0–1.19 THz). Similarly, Figures 3b,d show that the highest optical branch frequencies for K₄OI₂ and Rb₄OI₂ are 9 THz (K₄OI₂) and 8 THz (Rb₄OI₂), respectively, while the highest frequencies for the acoustic branches are 1.11 THz (K₄OI₂) and 1.00 THz (Rb₄OI₂). These materials have lower maximum acoustic branch frequencies compared to strong anharmonic materials such as Na2TlSb [32] (∼1.4 THz) and SnSe [33] (∼1.45 THz). Generally, when atoms of a similar structure are replaced with heavier elements, the phonon modes tend to shift to lower frequencies, which reduces the phonon group velocity and consequently suppresses the thermal conductivity. In the M₄OI₂, the enhanced phonon-optical coupling relative to M₃OI provides more phonon scattering channels for the material. Notably, the phonon dispersion for Rb₃OI shows a pronounced softening of the TA branch at the M and R points, a phenomenon commonly observed in cubic perovskite structures. This ferroelectric-like vibration behavior is typically attributed to vibrations in the octahedral framework [34, 35]. Figure 4 illustrates the TA branch vibration modes at the high-symmetry M and R points for K₃OI and Rb₃OI. At the M point, the I atoms in K₃OI exhibit in-plane motion along the positive and negative z-axis (c) directions, while in Rb₃OI, the I atoms undergo in-plane motion within the xy (ab) plane with some rotational motion, and the O atoms exhibit a rattling behavior similar to that in octahedral structures. Figures 4b,d show that at the R point, the K atoms in K₃OI undergo opposing movements around the axis within the xy plane, leading to octahedral distortion, while in Rb₃OI, the I atoms exhibit in-plane motion along the positive and negative y-axis directions. In addition to the octahedral distortion [36], the rattling behavior within the octahedral cage is often considered a source of strong anharmonicity in anti-perovskite materials [37].

We employed the diffusion-like model of dual-phonon [38] with shengBTE to calculate the κ_L of these materials in the temperature range of 100–900 K. as shown in Figure 5a and Supplementary Figure S2. The lattice thermal conductivity decreases with increasing temperature. We find that Rb₄OI₂ along the zz (c)direction has the lowest thermal conductivity with and without considering the contribution of the diffuson-like phonon. At 300 K, when the two-phonon channel is neglected, it is 0.18 W m^-1 K⁻¹, lower than the 0.20 W m^-1 K⁻¹ reported for SnSe [22]. On the other hand, the K₃OI and Rb₃OI exhibit isotropic thermal conductivity while that for K₄OI₂ and Rb₄OI₂ show significant anisotropy. After considering the dual-phonon channel, the lattice thermal conductivity of K₃OI and Rb₃OI are 0.89 W m^-1 K⁻¹ and 0.52 W m^-1 K⁻¹, respectively. The lattice thermal conductivity of K₄OI₂ and Rb₄OI₂ is 0.71 W m^-1 K⁻¹ and 0.47 W m^-1 K⁻¹ along the xx direction, and 0.42 W m^-1 K⁻¹ and 0.30 W m^-1 K⁻¹ along the zz direction, respectively. At room temperature, the κ_L of these materials is comparable to the industry standard PbTe [39] and significantly lower than most oxide thermoelectric materials [40–43]. At higher temperatures, κ_L remains below 1 W m^-1 K⁻¹, which is a characteristic feature of high-performance thermoelectric materials.

Heat capacity curve in Figure 5b shows that the specific heat capacity increases gradually with temperature in the range of 100–400 K. After 400 K, the heat capacities of the materials tend to converge. Over the entire temperature range, K₃OI and Rb₄OI₂ exhibit the highest and the lowest specific heat capacity, respetively. At 300 K, the specific heat capacities of K₄OI₂ and Rb₃OI are 1.112 × 10⁶ Jm^-3K⁻¹ and 1.105 × 10⁶ Jm^-3K⁻¹, respectively, while the heat capacity curves of the two materials nearly overlap beyond 400 K. Cumulative lattice thermal conductivity, as shown in Supplementary Figure S3, indicates that at 300 K, the lattice thermal conductivity is primarily contributed by phonon modes with an average free path (MFP) below 10 nm. ShengBTE results reveal that the high-frequency optical branches of these materials significantly contribute to the lattice thermal conductivity due to the strong phonon-optical coupling.

To further investigate the physical mechanisms for the thermal transport of these materials, Figure 6 plots the phonon relaxation time, phonon group velocity, scattering phase space, and Grüneisen parameter. As seen in Figure 6b, at 300 K, the phonon group velocity of K₃OI and K₄OI₂ along the xx direction is higher than that for Rb₃OI and Rb₄OI₂ in the acoustic branches and low-frequency optical branches (0-1.64 THz). In addition, the phonon group velocity of the M₃OI structure is higher than that of the M₄OI₂. The difference in the phonon group velocities between K₃OI and R₃IO or K₄OI₂, is mainly attributed to the increased effective atomic mass. This result aligns well with the phonon dispersion spectrum in Figure 3. The ultra-low lattice thermal conductivity of K₄OI₂ and Rb₄OI₂ in the zz direction, and their remarkable anisotropic structural properties, can be further explained by the softer bonds [44] and the lower phonon group velocity in the zz direction, as seen in Supplementary Figure S4.

Figure 6a presents the phonon relaxation times of these materials. Within the frequency range of the acoustic phonon, K₃OI (0-1.64 THz) and K₄OI₂ (0-1.42 THz) has overall higher phonon relaxation times compared to Rb₃OI (0-1.22 THz) and Rb₄OI₂ (0-1.11 THz), respectively. Moreover, the phonon relaxation times of K₄OI₂ and Rb₄OI₂ structures in the acoustic branch frequency range are larger than those of K₃OI and Rb₃OI structures. Next, we explored the P3 scattering phase space and Grüneisen parameter, as shown in Figures 6c,d. It indicates that K₃OI has the smallest scattering phase space, followed by K₄OI₂, and Rb₄OI₂ exhibits the largest phase space in the low-frequency acoustic branch frequency range (0-1 THz). This phenomenon is attributed to the phonon dispersion becoming more confined in this frequency range (0-1 THz) as the average relative atomic mass increases, which makes it easier for phonon scattering to satisfy the energy conservation law. In the strong phonon-optical coupling frequency range, K₃OI exhibits smaller P3 compared to K₄OI₂, and Rb₃OI is smaller than Rb₄OI₂ due to the increased number of low-frequency optical branches, which strengthens the phonon-optical coupling and increases the scattering channels. Figure 6d displays the Grüneisen parameter at 300 K. The distribution is concentrated in the acoustic and low-frequency optical branch regions, with Rb₃OI showing a distinctly different parameter distribution. Its peak value occurs at ∼1THz, while the other three structures have their peaks near zero frequency. Rb₄OI₂ has a higher Grüneisen parameter than K₄OI₂, and K₄OI₂ is higher than K₃OI. Although Rb₃OI has the lowest Grüneisen parameter in the acoustic branch (0-0.6 THz), its Grüneisen values in the 0.8–1.8 THz range are significantly largest, leading to the highest overall Grüneisen parameter and strongest phonon anharmonicity, and then the lowest phonon relaxation times. Additionally, the Grüneisen parameter distribution in Rb₃OI is influenced by the softening behavior of the transverse acoustic (TA) branch in the corresponding frequency range, which weakens the coupling with the low-frequency optical branch and leads to weaker phonon-optical coupling and a shift of the Grüneisen peak.

The electronic band structures were calculated using the hybrid functional HSE06. As shown in Figure 7. The electronic band gaps are 2.18 eV (K₃OI), 2.40 eV (K₄OI₂), 1.61 eV (Rb₃OI), and 1.92 eV (Rb₄OI₂), all of which are direct band gaps. The valence bands are primarily composed of the oxygen (O) orbitals, while the conduction bands are derived from the combined contributions of the alkali metals (K and Rb) and halogen (I) elements. The flat electronic band structure of the valence band indicates that it possesses a larger effective carrier mass, resulting in a lower carrier mobility. On the other hand, the significant band dispersion in the conduction band at the high-symmetry Γ point suggests that carriers near the conduction band exhibit a smaller effective mass and higher mobility. Notably, the valence band demonstrates a distinct multi-valley structure [45], which is often associated with high Seebeck coefficients of thermoelectrics [46].

The electronic transport properties of materials were studied using the AMSET software based on the electronic band structure. Figure 8 presents the computed electrical conductivity (σ), Seebeck coefficient (S), electronic thermal conductivity (κ_e), and power factor (PF) at different temperatures (300 K, 600 K, and 900 K). From Figures 8a,b, the electrical conductivity of n-type semiconductors is one order of magnitude higher than that of the corresponding p-type semiconductors. It is noteworthy that the electrical conductivity of the anisotropic material along the zz direction is the lowest, with a value ∼1 S/m at a carrier concentration of 10¹⁹ cm^-3.

The Seebeck coefficient (S) of the p-type semiconductors is significantly higher than that of n-type ones, which can be attributed to the multi-valley phenomenon in the valence band of the electronic structure. For instance, at a carrier concentration of 10¹⁹ cm^-3, the Seebeck coefficient (S) of Rb₃OI ranges from 527 to 684 µVK^-1 between 300 K and 900K, which is higher than the 600 µVK^-1 of Rb₃AuO [7]. And it is notably higher than the typical range (200–300 µVK^-1) found in thermoelectric power materials [47]. Additionally, S decreases with increasing temperature as the carrier concentration decreases, which is consistent with the trend for the most semiconductor thermoelectric materials due to carrier excitation by heat.

According to the Wiedemann-Franz law, electrical conductivity is proportional to the electronic thermal conductivity. When doping concentration is low, the electronic thermal conductivity can be neglected. However, at higher doping concentrations, the electronic thermal conductivity becomes comparable to or even greater than the lattice thermal conductivity. The PF of the material is calculated based on the S² σ relationship. Due to the larger Seebeck coefficient (S) of the p-type semiconductors, the power factor of p-type materials is higher than that of n-type semiconductors, with the maximum value reaching 0.7 mWm^-1K⁻² for Rb₃OI at 600 K, with a doping concentration of 4 × 10²¹ cm^-3.

Subsequently, we computed the ZT values for these materials in Figure 9. At 900 K, Rb₃OI exhibits the highest ZT value of 1.91 for p-type doping at the carrier concentration of 1 × 10²¹ cm^-3. The carrier scattering rate of R₃OI with a hole doping concentration of 1 × 10²¹ cm^-3 were shown in Supplementary Figure S5. We found that the carrier scattering rate increased with the rise in temperature due to the thermal excitation of carriers and the scattering rate of carriers was mainly contributed by POP scattering.

Next, Rb₄OI₂ shows the maximum ZT value of 1.44 in the zz direction, with a p-type doping concentration of 7 × 10²⁰ cm^-3. In this system, the thermoelectric performance of the p-type semiconductor is superior to that of the n-type, which is attributed to the higher PF resulting from the larger Seebeck coefficient (S). Comparing with the ZT values of some inorganic halide perovskites of CsPbI₃ (0.45), CsSnI₃ (0.95), and CsGeI₃ (1.05) [48], Rb₃OI and Rb₄OI₂ have better thermoelectric performance. The results indicate that Rb₃OI and Rb₄OI₂ have promising potential for advanced thermoelectric device applications.