Inverse‐Perovskite Ba₃ BO (B = Si and Ge) as a High Performance Environmentally Benign Thermoelectric Material with Low Lattice Thermal Conductivity

Abstract

High energy‐conversion efficiency (ZT) of thermoelectric materials has been achieved in heavy metal chalcogenides, but the use of toxic Pb or Te is an obstacle for wide applications of thermoelectricity. Here, high ZT is demonstrated in toxic‐element free Ba₃ BO (B = Si and Ge) with inverse‐perovskite structure. The negatively charged B ion contributes to hole transport with long carrier life time, and their highly dispersive bands with multiple valley degeneracy realize both high p‐type electronic conductivity and high Seebeck coefficient, resulting in high power factor (PF). In addition, extremely low lattice thermal conductivities (κ _lat) 1.0–0.4 W m⁻¹ K⁻¹ at T = 300–600 K are observed in Ba₃ BO. Highly distorted O–Ba₆ octahedral framework with weak ionic bonds between Ba with large mass and O provides low phonon velocities and strong phonon scattering in Ba₃ BO. As a consequence of high PF and low κ _lat, Ba₃SiO (Ba₃GeO) exhibits rather high ZT = 0.16–0.84 (0.35–0.65) at T = 300–623 K (300–523 K). Finally, based on first‐principles carrier and phonon transport calculations, maximum ZT is predicted to be 2.14 for Ba₃SiO and 1.21 for Ba₃GeO at T = 600 K by optimizing hole concentration. Present results propose that inverse‐perovskites would be a new platform of environmentally‐benign high‐ZT thermoelectric materials.

High thermoelectric performance is demonstrated in inverse‐perovskite Ba₃ BO (B = Si and Ge). The Ba₃ BO shows high power factor due to highly dispersive valence bands with multiple valley degeneracy, and also exhibits low lattice thermal conductivity due to the soft framework of O−Ba₆ octahedra with ionic O−Ba bonds. Inverse‐perovskite oxides would be a potential high‐ZT thermoelectric material without toxic elements.

1. Introduction

Due to the recently increasing energy crisis, there has been increasing attention to thermoelectric technology for power generation using waste heat energy.^{[ 1} , ² , ^{3 ]} The efficiency of thermoelectric energy conversion is governed by the dimensionless figure of merit (ZT), defined as ZT = S ²·σ·T·κ ^–1, where T is the absolute temperature, S is the Seebeck coefficient, σ is the electronic conductivity, and κ is the thermal conductivity of the thermoelectric materials.^{[ 4} , ⁵ , ^{6 ]} The product S ² σ is known as the power factor (PF), and the κ includes the contributions from electronic (κ _ele) and lattice (κ _lat) heat conduction. Therefore, high ZT thermoelectric materials should exhibit large S and high σ to obtain high PF, as well as low κ to create a large temperature gradient. So far, the high ZT has been demonstrated mainly in heavy metal chalcogenides, such as Bi₂Te₃, PbTe, and GeTe,^{[ 7} , ⁸ , ^{9 ]} which possess low κ _lat, but the use of toxic elements, such as Pb and Te, is not preferred for wide applications of thermoelectricity. There are many efforts on the development of environmentally benign thermoelectric materials, such as oxides, silicides, and sulfides,^{[ 10} , ¹¹ , ¹² , ¹³ , ^{14 ]} but further exploration of novel material systems is demanded for improving the thermoelectric performance.

We herein focus on inverse‐perovskite oxides as potential environmentally benign thermoelectric materials without toxic elements. The inverse‐perovskite oxides are represented by chemical formula of A ₃ BO with formal ion charges of A ²⁺ (alkaline earth = Ca²⁺, Sr²⁺, Ba²⁺), B ^4– (the p‐block 14 group = Si⁴⁻, Ge⁴⁻, Sn⁴⁻, Pb⁴⁻), and O²⁻.^{[ 15} , ¹⁶ , ¹⁷ , ¹⁸ , ¹⁹ , ^{20 ]} It crystallizes in the inverse‐perovskite structure that has an inverted cation and anion sites in comparison to the normal perovskite oxide ABO₃ (A ²⁺ B ⁴⁺O²⁻) such as SrTiO₃ (Figure 1 ). In the normal perovskite structure, B ⁴⁺ cation occupies the body‐centered site of the pseudo‐cubic unit cell and O²⁻ anion occupies the face‐centered sites, forming a B–O₆ octahedron. A ²⁺ cation occupies the vertex sites of the unit cell. On the other hand, in the inverse‐perovskite structure, O²⁻ anion occupies the body‐centered site and the A ²⁺ cation occupies the face‐centered sites, forming an O−A ₆ octahedron. B ⁴⁻ anion occupies the vertex sites. ZT of perovskite SrTiO₃ is usually limited to ∼0.1 due to its high κ _lat ≈10 W m⁻¹ K⁻¹ at room temperature (RT),^{[ 21 ]} originating from the hard framework of the Ti–O₆ octahedra with the strong Ti–O bonds. In contrast, the inverse‐perovskite structure is constructed from the soft framework of the O–A ₆ octahedra because of the larger A ²⁺ ion than B ⁴⁺ ion and the consequent longer O–A bonds. From another point of view, normal perovskite SrTiO₃ is formed by the high‐density packing structure of the light element O²⁻ ions, while inverse‐perovskite A ₃ BO is formed by the high‐density packing structure of heavier A ²⁺ ions. These largely contrasting structural characteristics let us expect a large reduction of κ _lat in inverse‐perovskite A ₃ BO. The most distinctive feature of inverse‐perovskite A ₃ BO is that the B ion is negatively charged, which actively contributes to hole conduction; the localized O 2p state forms valence band maximum (VBM) in conventional oxides including normal perovskite oxides, while spatially spread p orbital of large‐size B ⁴⁻ anion (ion radius: > 2 Å of B ⁴⁻, 1.4 Å of O²⁻) forms VBM in inverse‐perovskite oxide,^{[ 22 ]} which can realize high hole mobility and σ. Our work is hence motivated by the expectation that the inverse‐perovskite oxide would be a potential candidate for high ZT thermoelectric materials.

Figure 1.

Schematic illustration of crystal structures and phonon transport in inverse‐perovskite A ₃ BO (left) and normal perovskite ABO₃ (right). The normal perovskite structure of ABO₃ (e.g., SrTiO₃) is built with the hard framework of B−O₆ octahedron with short B−O bonds, providing high‐density packing structure of the light element O²⁻ ions. In contrast, the inverse‐perovskite structure of A ₃ BO (e.g., Ba₃ BO (B = Si and Ge)) is constructed from the soft framework of O−A ₆ octahedron with long O−A bonds, providing the high‐density packing structure of heavy A ²⁺ ions. The lattice thermal conductivity (κ _lat) of normal perovskite ABO₃ is usually high, while the largely contrasting structure characteristics are expected to lead the large reduction of κ _lat in inverse‐perovskite A ₃ BO.

The A ₃ BO with B = Sn and Pb adopts a high‐symmetry cubic structure (Pm‐3m) and exhibits unique Dirac electronic structures with high carrier mobility, being expected as a new class of topological crystalline insulators and superconductors.^{[ 22} , ²³ , ²⁴ , ²⁵ , ^{26 ]} However, their narrow bandgaps (theoretical bandgaps < 0.2 eV) limit their thermoelectric properties because the compensation by the coexistence of electrons and holes leads to low S, resulting in low PF at high temperatures.^{[ 27 ]} On the other hand, theoretical studies proposed that bandgap is sensitive to structural distortion, which enhances the thermoelectric properties of inverse‐perovskite A ₃ BO.^{[ 28} , ²⁹ , ^{30 ]} By replacing the B site with smaller Si and Ge ions, the cubic structure is distorted to an orthorhombic lattice in agreement with a smaller tolerance factor ($t=\frac{r_B+r_A}{2^{1/2}(r_{\mathrm{O}}+r_A)}$) of an inverse‐perovskite structure, where r_A , r_B , r _O are ionic radii for A, B, O ions.^{[ 28} , ^{31 ]} The orthorhombic Ca₃SiO (Ca₃GeO) with t = 0.937 (0.948) takes the space group of Imma and increases the bandgap to 0.7–0.9 eV.^{[ 28} , ^{32 ]} The thermoelectric properties were experimentally measured for Ca₃SiO and Ca₃GeO bulk polycrystals, which show low κ _lat = 1.0–1.9 W m⁻¹K⁻¹ at RT but the obtained ZT are limited to less than 10⁻⁵ because the properties were measured with cold‐pressed bulks with large amount of CaO impurity (24%–32% in the Ca₃SiO sample and 8%–10% in the Ca₃GeO sample).^{[ 33 ]}

Here, we synthesized high‐purity bulk polycrystals of highly distorted Ba₃ BO (B = Si and Ge) with t = 0.908 and 0.918, which crystalize in orthorhombic inverse‐perovskite structures (space group: Pnma) with the pronounced tilting and twisting of the O−Ba₆ octahedra (Figure 2a). The theoretical bandgaps are 0.86 eV for Ba₃SiO and 0.80 eV for Ba₃GeO. The undoped samples showed p‐type degenerate conduction with hole concentrations ≈4.8 × 10¹⁸ cm⁻³ for Ba₃SiO and ≈6.2 × 10¹⁹ cm⁻³ for Ba₃GeO at RT. The bulk samples exhibited low κ _lat of 1.00 W m⁻¹ K⁻¹ for Ba₃SiO and 0.77 W m⁻¹ K⁻¹ for Ba₃GeO, which are lower than 1.7–2.0 W m⁻¹ K⁻¹ of Bi₂Te₃ and PbTe bulks at RT. Ba₃SiO and Ba₃GeO bulks exhibited relatively high ZT = 0.16 and 0.35 at RT, respectively, and the ZT value increased up to 0.84 for Ba₃SiO bulk at T = 623 K and 0.65 for Ba₃GeO bulk at T = 523 K. We systematically investigated the electronic and phonon transport properties of Ba₃ BO by using first‐principles calculations to clarify the underlying physical mechanisms for their low κ _lat and the potential of thermoelectric ZT.

Figure 2.

Crystal structure and electronic structure analyses of inverse‐perovskite Ba₃ BO (B = Si and Ge). a) Crystal structures of Ba₃SiO (Ba₃GeO) with space group of Pnma. The top and the bottom panels are the side view of (101) plane and the top view of (010) plane, respectively. b,c) Electronic band structures and d,e) partial density of states for b,d) Ba₃SiO and c,e) Ba₃GeO. The right panels of (b,c) are enlarged views of valence band maximum.

2. Results and Discussion

2.1. Crystal Structure and Electronic Structure Analyses

The bulk polycrystals of Ba₃ BO (B = Si and Ge) were synthesized by high‐temperature solid‐state reactions of 2Ba + Si(Ge) +BaO → Ba₃SiO(Ba₃GeO). From X‐ray diffraction (XRD) measurements, a small amount of BaO impurity (7.3 mol%) is detected for the Ba₃GeO bulk, and the weak unidentified diffraction peaks are observed for the Ba₃SiO bulk (Figure S1, Supporting Information, and CCDC 2291770 and 2291771 are the supplementary crystallographic data for this paper). Microstructure analysis by a field‐emission scanning electron microscopy (FE‐SEM) shows that the bulks are composed of sintered grains with an average grain size of ≈10 µm with some pores (Figure S2, Supporting Information), resulting in the sintered density of 80–87%. Energy dispersive X‐ray spectroscopy (EDS) mapping confirms the uniformity of the chemical composition of Ba, Si (Ge), and O over the grain region. The orthorhombic lattice parameters estimated by Rietveld analysis of XRD patterns are a = 7.581 Å, b = 10.706 Å, c = 7.543 Å for Ba₃SiO and a = 7.592 Å, b = 10.732 Å, c = 7.559 Å for Ba₃GeO. Lattice volume is expanded from 612.261 ų of Ba₃SiO to 615.896 ų of Ba₃GeO because of the slightly larger ion radius of 2.08 Å for Ge⁴⁻ anion than 2.04 Å of Si⁴⁻ anion.^{[ 31 ]} The pseudo‐cubic lattice parameters of the orthorhombic unit cell for Ba₃SiO (Ba₃GeO) are b/2 = 5.353 Å (5.366 Å) and $\sqrt{a^2+c^2}$ = 5.347 Å (5.357 Å), indicating the O−Ba₆ octahedra are slightly elongated along the b‐axis (Figure 2a). The orthorhombic distortion splits the Ba sites to the non‐equivalent Ba1 and Ba2 sites. The O−Ba₆ octahedra are tilted and twisted around all three octahedral axes, where the apical O−Ba2−O and basal O−Ba1−O bond angles for Ba₃SiO (Ba₃GeO) are 157^o (158^o) and 158^o (160^o), showing distinct deviations from 180^o of the cubic lattice.

Figure 2b‐e summarizes electronic band structures and density of states (DOSs) of Ba₃ BO calculated by the VASP^{[ 34} , ^{35 ]} code with Heyd–Scuseria–Ernzerhof (HSE) hybrid functional.^{[ 36 ]} The conduction band minimum (CBM) and VBM located around the Γ point (left panels of Figure 2b,c), where the difference in the k vector between direct and indirect bandgap (E _g) is small, as indicated by the blue and the red arrows. The E _g are calculated to be 0.86 eV for Ba₃SiO and 0.80 eV for Ba₃GeO, which are a little larger than the experimentally measured E _g of 0.62 eV and 0.58 eV (See diffuse reflectance spectra in Figure S3, Supporting Information), due to the E _g overestimation by HSE hybrid functional. The atomic charges estimated by the Bader charge analysis are Ba^+1.15 ₃Si^–2.02O^−1.45 and Ba^+1.14 ₃Ge^–1.99O^−1.45, confirming the anionic states of Si and Ge ions. The CBM is mainly contributed by the Ba 5d state with only one single valley at the Γ point, while the VBM arises primarily from the Si 3p (Ge 4p) state (Figure 2d,e). Specifically, the valence bands around the Γ point are composed of one flat band and four highly dispersive bands nearly degenerating within the 0.15 eV energy range below the Fermi level (right panels of Figure 2b,c). On the other hand, O 2p state located at a deeper energy level contributing little to carrier transport.

2.2. Carrier Transport Properties

Figure 3a,b shows the temperature (T) dependence of σ and S for Ba₃ BO bulks. The Ba₃GeO bulk exhibits higher σ = 151 S cm⁻¹ than 28 S cm⁻¹ of Ba₃SiO bulk at T ≃ 300 K (Figure 3a). The metallic T dependence of σ is observed for the Ba₃GeO bulk over the whole T range. On the other hand, σ of Ba₃SiO bulk shows metallic T dependence at high T ≥ 510 K, while σ decreases at T < 500 K. Both the samples show positive S over the whole T range (Figure 3b), indicating the majority carrier is hole. The S = +444 µV K⁻¹ for Ba₃SiO is larger than +255 µV K⁻¹ for Ba₃GeO at T ≃ 300 K. The S value linearly increases to +507 µV K⁻¹ at T = 630 K for Ba₃SiO and +372 µV K⁻¹ at T = 624 K for Ba₃GeO with increasing T. Note that Hall effect measurement was difficult to perform because these samples are sensitive to air and sample degradation occurs during the transfer to the measurement system.

Figure 3.

Carrier transport properties of Ba₃ BO (B = Si and Ge). a,b) Temperature (T) dependences of a) electronic conductivity (σ) and b) Seebeck coefficient (S) of Ba₃ BO bulks. c,d) Calculated electronic transport coefficients: c) σ _calc. and d) S _calc. as a function of carrier concentration (n) at T = 300 K and 600 K. The solid lines are the polynominal fitting to the data points of σ _calc. vs n and S _calc. vs n. The measured S (S _meas.) of Ba₃ BO bulks are plotted in (d), and the dotted lines indicate the n estimated from the S _meas. in the S _calc. vs n relations. e,f) Weighted carrier mobility (µ _w) vs. T plots fitted by ${\mu }_{\mathrm{w},\mathrm{total}}(T)=\exp (\frac{q{E}_{\mathrm{b}}}{k_{\mathrm{B}}T}){\mu }_{\mathrm{w},\mathrm{in}-\mathrm{grain}}(T)$, where µ_{w,in − grain}(T) is obtained by the Matthiessen's rule, ${\mu }_{\mathrm{w},\mathrm{in}-\mathrm{grain}}^{-1}={\mu }_{\mathrm{w},\mathrm{imp}.}^{-1}+{\mu }_{\mathrm{w},\mathrm{opt}.}^{-1}$, for e) Ba₃SiO bulk and f) Ba₃SiO bulk. The red lines show the total mobility (µ _w,total) and the blue lines show the in‐grain µ _w without GB scattering (µ _w,in‐grain). The green and the purple dotted lines show the ionized impurity scattering limited mobility (µ _w,imp.) and the optical phonon scattering limited mobility (µ _w,opt.), respectively.

We then calculated the carrier lifetime (τ _e) and obtained the carrier concentration (n) dependences of σ _calc. and S _calc. at T = 300 K and 600 K (Figure 3c,d) by density functional perturbation theory (DFPT)^{[ 37 ]} as implemented in Quantum ESPRESSO package^{[ 38} , ^{39 ]} and PERTURBO code.^{[ 40 ]} It is seen that σ _calc. increases while S _calc. decreases with n as usually observed due to the well‐known competitive relationship. The σ _calc. of Ba₃SiO is nearly one order of magnitude higher than that of Ba₃GeO, which originates from the smaller τ _e in Ba₃GeO due to strong electron–phonon scattering (Figure S4, Supporting Information). Here, we need to compare σ _calc. and the measured σ but σ _calc. is a function of n. We, therefore, first estimated experimental n from the measured S (S _meas.) using the calculated S _calc. vs n relation as a reference. We estimated n to be ≈4.8 × 10¹⁸ cm⁻³ (≈5.0 × 10¹⁸ cm⁻³) for the Ba₃SiO bulk and ≈6.2 × 10¹⁹ cm⁻³ (≈2.7 × 10¹⁹ cm⁻³) for the Ba₃GeO bulk at T = 300 K (600 K) as shown by the yellow circles in Figure 3d, indicating that the n of Ba₃SiO is one order of magnitude lower than that of Ba₃GeO. The n exhibits a weak T dependence for both the Ba₃SiO and the Ba₃GeO bulks, indicating degenerate hole conduction, which is supported by X‐ray photo‐emission spectroscopy spectra near the VBM (Figure S5, Supporting Information) because the Fermi level locates 0.1 eV below the VBM. We finally estimated σ _calc. at the estimated n as the crossing points of the solid and dotted lines in Figure 3c.

Table 1 compares the experimentally measured and theoretically calculated carrier transport properties of Ba₃ BO. Prior to explaining the detailed results, we like to summarize that the experimentally obtained results are consistent with the calculated ones both at T = 300 K and 600 K for Ba₃GeO while three times differences are found at T = 300 K for Ba₃SiO. For Ba₃GeO, the σ _calc. (n at 300 K) = 195 S cm⁻¹ and σ _calc. (n at 600 K) = 35 S cm⁻¹ are almost consistent with the measured σ (σ _meas.) = 151 S cm⁻¹ at 300 K and 21 S cm⁻¹ at 600 K, respectively. Accordingly, carrier mobility µ = σ_meas./en and ${\tau }_{\mathrm{e}}=m_{\mathrm{band}}^{\ast }\mu /e$ ($m_{\mathrm{band}}^{\ast }$is band effective mass) in Table 1 show similar consistency. On the other hand, for Ba₃SiO, although the σ _calc. (n at 600 K) = 28 S cm⁻¹ is consistent with σ _meas. = 37 S cm⁻¹ at T = 600 K, but σ _calc. (n at 300 K) = 89 S cm⁻¹ is three‐times higher than σ _meas. = 28 S cm⁻¹ at T = 300 K. Accordingly, the estimated µ = 46.6 cm² V⁻¹ s⁻¹ is consistent with µ _calc. = 35.0 cm² V⁻¹ s⁻¹ at T = 600 K, while they show three‐times difference (µ = 36.4 cm² V⁻¹ s⁻¹ and µ _calc. = 115.7 cm² V⁻¹ s⁻¹) at 300 K. We here need to recognize that the calculated results reflect the transport properties of the ideal single crystal while the experimental ones include carrier scattering by defects and grain boundaries (GB), giving the significant discrepancy at lower temperatures.

Table 1.

Summary of experimentally measured and theoretically calculated carrier transport properties for Ba₃ BO (B = Si and Ge) at T = 300 K and 600 K. $m_{\mathrm{band}}^{\ast }$ is the band effective mass and $m_{\mathrm{DOS}}^{\ast }$ is the density of state effective mass, calculated by BoltzTraP2 code. n is the carrier concentration obtained from the measured S in calculated S _calc. vs n relation (Figure 3d). σ _meas. is the experimentally measured electronic conductivity (Figure 3a). σ _calc. is the calculated electronic conductivity obtained from the calculated σ _calc. vs n relation at the estimated n (Figure 3c). µ and µ _calc. are the carrier mobility obtained by µ = σ_meas./en and µ_calc. = σ_calc./en. τ _e and τ _e,calc. are the carrier life time obtained by ${\tau }_{\mathrm{e}}=m_{\mathrm{band}}^{\ast }\mu /e$ and ${\tau }_{\mathrm{e},\mathrm{calc}.}=m_{\mathrm{band}}^{\ast }{\mu }_{\mathrm{calc}.}/e$.

	$m_{\mathrm{band}}^{\ast }$	$m_{\mathrm{DOS}}^{\ast }$	T [K]	n [cm−3]	σ meas. [S cm−1]	σ calc. [S cm−1]	µ [cm2 V−1 s−1]	µ calc. [cm2 V−1 s−1]	τ e [fs]	τ e,calc.[fs]
Ba3SiO	0.85m 0	2.46m 0	300	4.8×1018	28	89	36.4	115.7	17.6	55.9
			600	5.0×1018	37	28	46.6	35.0	22.5	16.9
Ba3GeO	0.72m 0	2.24m 0	300	6.2×1019	151	195	15.2	19.6	6.2	8.0
			600	2.7×1019	21	35	4.8	8.1	2.0	3.3

To separate the single‐crystalline‐like carrier transport in crystalline grains and the effect of GBs, we employ the Seto model,^{[ 42 ]} $\mu (T)=\exp (\frac{q{E}_{\mathrm{b}}}{k_{\mathrm{B}}T}){\mu }_{\mathrm{in}-\mathrm{grain}}(T)$, where µ_{in − grain}(T) is the in‐grain carrier mobility and the GB contribution is expressed as $\exp (\frac{q{E}_{\mathrm{b}}}{k_{\mathrm{B}}T})$ (E _b is the GB barrier height and k _B is the Boltzmann constant). As we could not perform Hall effect measurements, we estimate µ(T) as weighted mobility from σ and S by ${\mu }_{\mathrm{w}}=\frac{3h^3\sigma }{8\pi e{(2m_e{k}_{\mathrm{B}}T)}^{3/2}}[\frac{exp[\frac{|S|}{k_{\mathrm{B}}/e}-2]}{1+exp[-5(\frac{|S|}{k_{\mathrm{B}}/e}-1)]}+\frac{\frac{3}{{\pi }^2}\frac{|S|}{k_{\mathrm{B}}/e}}{1+exp[5(\frac{|S|}{k_{\mathrm{B}}/e}-1)]}]$, where m _e is the free electron mass.^{[ 41 ]} The µ _w is related to the drift mobility µ by ${\mu }_{\mathrm{w}}\approx \mu {(\frac{m_{\mathrm{DOS}}^{\ast }}{m_{\mathrm{e}}})}^{3/2}$, where $m_{\mathrm{DOS}}^{\ast }$ is density of state effective mass. The µ _w of Ba₃SiO bulk exhibits a negative T coefficient at T ≥ 440 K, while it decreases with a decrease of T at low T region ≤ 440 K (the black circles in Figure 3e). The µ _w of Ba₃GeO bulk increases with a decrease of T in a wide range of T ≥ 327 K but it levels off at T < 327 K (the black circles in Figure 3f). Then, µ_{w,in − grain}(T) is modeled by Matthiessen's rule, ${\mu }_{\mathrm{w},\mathrm{in}-\mathrm{grain}}^{-1}={\mu }_{\mathrm{w},\mathrm{imp}.}^{-1}+{\mu }_{\mathrm{w},\mathrm{opt}.}^{-1}$. The impurity scattering mobility is expressed as ${\mu }_{\mathrm{w},\mathrm{imp}.}^{-1}=\mathrm{A}$ (temperature independent) in the degenerate regime. The optical phonon scattering mobility is expressed as ${\mu }_{\mathrm{w},\mathrm{opt}.}^{-1}=1/(\mathrm{B}(\exp (\frac{\hslash {\omega }_0}{kT})-1))$ (B is a constant), where ℏω₀ is the longitudinal optical phonon energy. These parameters are obtained by least‐squares fitting of the total mobility ${\mu }_{\mathrm{w},\mathrm{total}}(T)=\exp (\frac{q{E}_{\mathrm{b}}}{k_{\mathrm{B}}T}){\mu }_{\mathrm{w},\mathrm{in}-\mathrm{grain}}(T)$ to the experimental µ_w(T). In Figure 3e,f, the solid red line shows the µ _w,total providing good agreement with experimental µ _w over a wide T range. At high T region, the µ _w,in‐grain (blue lines) is dominated by optical phonon scattering (the purple dotted lines), where the ℏω₀ values were optimized to be 260 meV for Ba₃SiO and 210 meV for Ba₃GeO. The µ _w,in‐grain increases and approaches the µ _w,imp. (green dotted lines), when T is reduced to ≈300 K. The µ _w,in‐grain is higher than µ _w,total especially at lower T range for Ba₃SiO bulk, indicating that the carrier transport is limited by GB scattering. The µ _w,in‐grain is 3.2 times higher than µ _w,total at T ≃ 300 K, which is consistent with the µ _calc./µ = 3.2 obtained in Table 1. For Ba₃GeO, the µ _w,in‐grain is nearly the same with µ _w,total in a wide range of T ≥ 327 K but the difference becomes a little larger at T < 327 K. The µ _w,in‐grain is 1.3 times higher than µ _w,total at T ≃ 300 K, in consistence with the µ _calc./µ = 1.3. The E _b is estimated to be 28 meV for Ba₃SiO while E _b = 8 meV is much smaller for Ba₃GeO. The higher density carriers in Ba₃GeO bulk may screen the GB background charges and reduce the GB barrier heights. Although the Ba₃ BO bulks have a relatively poor polycrystalline nature with low sintered densities 80–87% and the carrier transport of Ba₃SiO bulk is limited by GB scattering at low T region, the mobility analysis suggests that Ba₃ BO possess potentially high carrier mobility.

Next, effective masses m ^* are estimated as m ^* determines S in the simple free electron model by $S=\frac{k_{\mathrm{B}}}{e}(\frac{3}{2}\ln m_{\mathrm{DOS}}^{\ast }+\ln 2{(\frac{2\pi k_{\mathrm{B}}T}{h^2})}^{\frac{3}{2}}+r+2-\ln n)$, where $m_{\mathrm{DOS}}^{\ast }$ is the density‐of‐states effective mass at VBM. Here the band effective masses ($m_{\mathrm{band}}^{\ast }$= $N{e}^2\frac{{\tau }_{\mathrm{e}}}{\sigma }$) at VBM are calculated to be slightly large at 0.85m ₀ for Ba₃SiO and 0.72m ₀ for Ba₃GeO (Table 1). However, the calculated carrier lifetime (τ_e,calc.) is long at 55.9 fs for Ba₃SiO and 8.0 fs for Ba₃GeO, resulting in the relatively high µ _calc. of 115.7 and 19.6 cm² V⁻¹ s⁻¹ at T = 300 K, respectively. The dispersive bands at VBM with relatively small $m_{\mathrm{band}}^{\ast }$ and long τ_e contribute to high µ ($=\frac{e{\tau }_{\mathrm{e}}}{m_{\mathrm{band}}^{\ast }}$) and σ (= µne). However, the lifetime calculations were performed at the rigid band scheme (no ion dynamics) and polaron effect is not considered but it can reduce the real µ. On the other hand, $m_{\mathrm{DOS}}^{\ast }$ are calculated to be 2.46m ₀ for Ba₃SiO and 2.24m ₀ for Ba₃GeO. The large $m_{\mathrm{DOS}}^{\ast }$ originates from the high valence band degeneracy as explained for Figure 2b,c, which contributes to the large S. Therefore, the negatively‐charged B ion contributes to hole transport with long carrier life time, and the dispersive valence bands (small $m_{\mathrm{band}}^{\ast }$) with the high valley degeneracy (large m _DOS ^*) are suitable for realizing high PF (= S ² σ).

2.3. Thermal Transport Properties

Next we discuss thermal transport properties by separating electronic and lattice contributions. Figure 4a summarizes the T dependence of total κ (κ _total) and electronic κ (κ _ele) of Ba₃SiO and Ba₃GeO bulks. The κ _ele is calculated by Wiedemann‐Franz law as κ _ele = LTσ, where L is the Lorenz number calculated from $L={(\frac{k_{\mathrm{B}}}{e})}^2(\frac{(r+\frac{7}{2})F_{r+5/2}(\eta )}{(r+\frac{3}{2})F_{r+\frac{1}{2}}(\eta )}-{[\frac{(r+\frac{5}{2})F_{r+3/2}(\eta )}{(r+\frac{3}{2})F_{r+1/2}(\eta )}]}^2)$. Here, the reduced Fermi energy η is obtained based on the free carrier model using the measured S as $S=\frac{k_{\mathrm{B}}}{e}(\frac{(r+\frac{5}{2})F_{r+3/2}(\eta )}{(r+\frac{3}{2})F_{r+1/2}(\eta )}-\eta )$ with the Fermi integral defined as $F_n(\eta )=\underset{0}{\overset{\infty }{\int }}\frac{{\chi }^n}{1+e^{\chi -\eta }}d\chi$, where γ = −1/2 is the scattering factor.^{[ 43 ]} The Ba₃SiO and Ba₃GeO bulks showed low κ _total of 1.02 W m⁻¹K⁻¹ for Ba₃SiO and 0.84 W m⁻¹K⁻¹ for Ba₃GeO at T = 300 K. The κ _total values of Ba₃SiO and Ba₃GeO bulks decrease to 0.69 W m⁻¹K⁻¹ and 0.43 W m⁻¹K⁻¹ as the T rises to 623 K. The estimated κ _ele was less than 0.1 W m⁻¹K⁻¹, where the maximum κ _ele was 0.04 W m⁻¹K⁻¹ at T = 523 K for Ba₃SiO and 0.07 W m⁻¹K⁻¹ at T = 300 K for Ba₃GeO, indicating the small electronic contribution to κ _total. Then, the lattice κ (κ _lat) is extracted by subtracting the electronic contribution from the κ _total, i.e. κ _lat = κ _total – κ _ele. The T dependences of κ _lat are summarized in  Figure 4b, where those of the normal perovskite SrTiO₃ bulk^{[ 44 ]} as well as representative chalcogenides of Bi₂Te₃ and PbTe bulks^{[ 45} , ⁴⁶ , ^{47 ]} are superimposed for comparison. The κ _lat decreases from 1.00 W m⁻¹ K⁻¹ at T = 300 K to 0.66 W m⁻¹ K⁻¹ at T = 623 K for Ba₃SiO and it decreases from 0.77 W m⁻¹ K⁻¹ at RT to 0.41 W m⁻¹ K⁻¹ at T = 623 K for Ba₃GeO. The κ _lat at T = 300 K is much lower than 8.2 W m⁻¹ K⁻¹ of SrTiO₃ bulk^{[ 44 ]} and also even lower than ≈1.7 W m⁻¹ K⁻¹ of Bi₂Te₃ bulk^{[ 45 ]} and ≈2.0 W m⁻¹ K⁻¹ of PbTe bulk,^{[ 47 ]} while it is comparable to κ _lat of state‐of‐the‐art chalcogenide thermoelectric materials, such as 0.7 W m⁻¹K⁻¹ of SnSe bulk,^{[ 48 ]} 0.6 W m⁻¹K⁻¹ of Cu₂Se bulk,^{[ 49 ]} and 0.6–0.8 W m⁻¹K⁻¹ of GeTe bulk.^{[ 7 ]}

Figure 4.

Phonon transport properties of Ba₃ BO (B = Si and Ge). a) Temperature (T) dependences of total thermal conductivity (κ _total) and electronic thermal conductivity (κ _ele) for Ba₃ BO bulks. b) T dependence of lattice thermal conductivity (κ _lat) for Ba₃ BO bulks, compared with the reported κ _lat of normal perovskite SrTiO₃ bulk^{[ 44 ]} as well as PbTe and Bi₂Te₃ bulks.^{[ 45} , ⁴⁶ , ^{47 ]} Calculated κ _lat of Ba₃ BO and SrTiO₃ models are also shown by the solid lines. c,d) Anharmonic phonon dispersions (left panel) and partial phonon density of states (DOSs) projected on each element (right panel) for c) Ba₃ BO and d) SrTiO₃ at T = 300 K obtained by the self‐consistent phonon (SCPH) approximation. e) Comparison of κ _lat spectra for Ba₃ BO (top panel) and SrTiO₃ (bottom panel) at T = 300 K. Frequency‐dependent cumulative κ _lat normalized by total κ _lat is also shown for each panel. f) Phonon group velocity, ν _ph (top panel), and phonon lifetime, τ _ph (bottom panel) in terms of the phonon frequency.

2.4. Origin of Low Lattice Thermal Conductivity in Inverse‐Perovskite

We compared the phonon transport properties of Ba₃ BO with the normal perovskite SrTiO₃ to discern the distinguishing characteristics of inverse perovskites. First, we performed the simple phonon gas model analysis using ${\kappa }_{\mathrm{lat}}=\frac{1}{3}{C}_{\mathrm{v}}·v_{\mathrm{s}}·l_{\mathrm{ph}}=\frac{1}{3}{C}_{\mathrm{v}}·v_{\mathrm{s}}^2·{\tau }_{\mathrm{ph}}$, where C _v is the specific heat per volume, v _s is the sound velocity, l _ph is the phonon mean free path, and τ _ph is the phonon lifetime (Table S1, Supporting Information). The v _s, measured by ultrasonic pulse echo method at RT, are 2317 m s⁻¹ (1981 m s⁻¹) for Ba₃SiO (Ba₃GeO), which is less than a half of 5241 m s⁻¹ for the normal perovskite SrTiO₃. In addition, the estimated τ _ph of 0.16 ps (0.16 ps) for Ba₃SiO (Ba₃GeO) is a half of 0.32 ps of SrTiO₃. The smaller v _s and lower τ _ph lead to the shorter l _ph (= v _s τ _ph) of 0.38 nm (0.32 nm) for Ba₃SiO (Ba₃GeO) than 1.70 nm for SrTiO₃, resulting in the low κ _lat. The bulk modulus was calculated to be 92.8 GPa (80.0 GPa) and the Debye temperature was 220 K (187 K) for Ba₃SiO (Ba₃GeO). The estimated Grüneisen parameter was relatively large at 1.3–1.4, which is comparable to low κ _lat thermoelectric materials such as Bi₂Te₃ and PbTe,^{[ 50 ]} Therefore, both the low sound velocity and the strong phonon scattering are responsible for the intrinsically low κ _lat in Ba₃ BO.

To further elucidate the underlying mechanism responsible for the low κ _lat in Ba₃ BO, we conducted first‐principles anharmonic lattice dynamics (ALD) calculations based on the density functional theory (DFT), as implemented in the ALAMODE code.^{[ 52} , ^{53 ]} Figure 4c,d compares the anharmonic phonon dispersions (left panels) and the partial phonon DOSs projected on each element (right panels) for Ba₃ BO and SrTiO₃ models at T = 300 K. Ba₃ BO exhibits flatter phonon bands and all phonon bands only exist at a low frequency below 9 THz (left panel of Figure 4c). The phonon DOSs of the Ba₃SiO reveal that the vibrations of Ba atoms predominantly contribute to the lower frequency phonon bands with the cut‐off frequency ≈4.3 THz, while the Si and O atoms primarily contribute to higher frequency phonon branches (right panel of Figure 4c). For Ba₃GeO, the phonon bands of heavier Ge ion shift to lower frequency and have interaction with the phonon bands of Ba ion. On the other hand, SrTiO₃ exhibits largely dispersive phonon bands even at higher frequencies (left panel of Figure 4d), with extensive dispersion at high frequencies of 4–14 THz, originating from the cooperative Ti and O atomic vibration (right panel of Figure 4d). The dispersion is considerably larger than observed in the low‐frequency phonon bands primarily attributed to Sr atomic vibrations at frequencies below ≈4.3 THz.

We then calculated κ _lat by solving the Peierls–Boltzmann transport equation (PBTE) within the relaxation time approximation. The calculated κ _lat (averaged along a,b,c‐axes) as a function of T for Ba₃ BO and SrTiO₃ are compared in Figure 4b. The calculated κ _lat of Ba₃SiO (Ba₃GeO) are 1.21 W m⁻¹ K⁻¹ (0.86 W m⁻¹ K⁻¹) and that of SrTiO₃ is 8.46 W m⁻¹ K⁻¹ at T = 300 K, in consistence with the experimentally measured values. Figure 4e compares the κ _lat spectra at T = 300 K with respect to phonon frequency for Ba₃ BO and SrTiO₃ models. The frequency‐dependent cumulative κ _lat normalized by total κ _lat are also shown in the panels. For Ba₃ BO, the κ _lat spectra peak at ≈1 THz and phonons in the low‐frequency region below ≈4.3 THz contribute mostly to κ _lat. The acoustic and optical modes are hybridized when the q point is far from the Γ point, making it difficult to distinguish the acoustic and optical contributions clearly. However, if we consider a cut‐off frequency for acoustic phonons, at which the acoustic and optical branches cross (Figure 4c), at ≈1.4 THz for Ba₃ BO, the contribution of low‐frequency acoustic phonons to κ _lat is ≈55%. The low‐frequency acoustic phonons and mid‐frequency optical phonons within 4.3 THz contribute to the ≈91% of total κ _lat, indicating that heat conduction primarily arises from the vibrational motion of Ba atoms (also Ge atoms in Ba₃GeO). In contrast, for SrTiO₃, the low‐frequency acoustic phonons and mid‐frequency optical phonons within 4.3 THz contribute to only the ≈31% of total κ _lat, i.e., not only the low‐frequency Sr atom vibrations but also high‐frequency phonons associated with Ti and O atomic vibrations contribute greatly to heat conduction.

Figure 4f compares the phonon group velocity, ν _ph (top panel) and phonon lifetime, τ _ph (bottom panel) in terms of the phonon frequency. Ba₃ BO exhibit much lower ν _ph than SrTiO₃ across all frequency ranges, primarily due to the presence of flat phonon band branches (left panel of Figure 4c). On the other hand, τ _ph is almost similar at low frequency (<3.5 THz) for both Ba₃ BO and SrTiO₃, but the value of Ba₃ BO becomes smaller at higher frequency region (3.5–5 THz). For SrTiO₃, the τ _ph becomes smaller at high frequency (>6 THz), but ν _ph is still large, reflecting the widely spread optical phonon bands. Therefore, the large ν _ph for phonons associated not only with Sr atomic vibration but also with Ti and O atomic vibrations provides a large contribution to high κ _lat in SrTiO₃. The τ _ph of Ba₃ BO becomes further smaller in the higher frequency region. The low κ _lat of Ba₃ BO predominantly originates from the low ν _ph for phonons associated with Ba atomic vibration, and the phonons associated with Si and O atomic vibrations have a negligible contribution to κ _lat due to the very short τ _ph. Ba₃ BO shows more phonon bands than SrTiO₃ (left panel of Figure 4c), because it has the local structure distortion with lower crystalline symmetry, resulting in the splitting of degenerated phonon bands. These broad frequency shifts enhance the phonon‐phonon scattering probability that leads to a large τ _ph reduction in Ba₃ BO. The inverse and the normal perovskite structures are constructed from the network of O−Ba₆ and Ti−O₆ octahedra. Then, the bonding energies of O−Ba in Ba₃ BO and Ti−O in SrTiO₃ as a measure of bonding strengths were estimated through the chemical bonding analysis using the crystal orbital Hamilton population (COHP)^{[ 53 ]} performed by the LOBSTER code.^{[ 54 ]} For the Ba₃ BO case, the −iCOHP values (the integrated −COHP up to the Fermi level, corresponding to the bond strength) averaged for O−Ba bonds are as small as 0.276 eV per bond for Ba₃SiO and 0.283 eV per bond for Ba₃GeO, indicating ionic interaction between the Ba atom and O atom in O−Ba₆ octahedra of the Ba₃ BO. On the other hand, the Ti−O bonds of SrTiO₃ have more than 10 times larger −iCOHP values of 3.48 eV per bond, originating from the strong covalent interaction between the Ti atom and the O atom in SrTiO₃ lattice. The inverse‐perovskite Ba₃ BO has a similar crystal structure to the perovskite SrTiO₃, but the ionic nature of the O−Ba bond softens the octahedra framework and thus the contribution of high‐frequency optical phonons is negligible for heat transport in Ba₃ BO. Note that the related phenomenon of low κ _lat and strong phonon scattering is observed in layered BaAgSb with weakly ionic bonded Ba atoms.^{[ 55 ]}

2.5. Thermoelectric Properties

Figure 5a,b summarizes the T dependences of PF (= S ² σ) and ZT (= S ² σT/κ) of the Ba₃ BO bulks. The calculated PF (PF_calc. = S ² σ _calc.) and ZT (ZT _calc. = S ² σ _calc. T/κ _calc.) are also superimposed in the panel. The PF of Ba₃SiO and Ba₃GeO bulks are limited to 5.5 and 9.8 µW cm⁻¹ K⁻² at T ≃ 300 K, respectively, because of GB scattering (Figure 5a), but, as a consequence of their low κ, Ba₃SiO and Ba₃GeO show relatively high ZT = 0.16 and 0.35 at T ≃ 300 K, respectively (Figure 5b). On the other hand, they potentially show higher PF_calc. = 17.3 and 12.4 µW cm⁻¹ K⁻² by eliminating GB scattering, and ZT _calc. would be increased up to 0.44 and 0.40 for Ba₃SiO and Ba₃GeO, respectively. The PF of Ba₃SiO largely increases to 11.6 µW cm⁻¹ K⁻² when T increases to 464 K, and then decreases to 9.0 µW cm⁻¹ K⁻² at high T = 630 K. The ZT value increases continuously up to 0.84 at T = 623 K, which is slightly higher than ZT _calc. = 0.65 at T = 600 K due to higher σ than σ _calc. On the other hand, Ba₃GeO exhibits the maximum PF = 10.8 µW cm⁻¹ K⁻² at low T = 327 K, and the PF decreases continuously down to 5.9 µW cm⁻¹ K⁻² at T = 523 K, where the maximum ZT = 0.65 was obtained. The ZT _calc. of Ba₃GeO increases continuously up to 0.74 at T = 600 K, but the PF and ZT suddenly decrease at T ≥ 548 K due to the decrease of σ (Figure 3a) and the increase of κ (Figure 4a). We speculate that these σ and κ changes of Ba₃GeO would originate from the transition to higher symmetric inverse‐perovskite structure at high T because it has slightly high tolerance factor = 0.918. The high T crystal‐structure characterization is necessary for this conclusion.

Figure 5.

Thermoelectric properties of Ba₃ BO (B = Si and Ge). a,b) Temperature (T) dependences of a) power factor (PF) and b) dimensionless figure of merit (ZT) of Ba₃ BO bulks. c,d) Carrier concentration (n) dependences of c) calculated PF (PF_calc. = S ² σ _calc.) and d) calculated ZT (ZT _calc. = S ² σ _calc. T/κ _calc.) at T = 300 K and 600 K. The arrows indicate the maximum PF_calc. and ZT _calc. at the optimal n (n _opt.). The yellow circles in a‐d indicate the PF_calc. and ZT _calc. obtained from the experimental n (n _exp.) estimated in Figure 3d.

Note that we tried hole doping to obtain optimum ZT by potassium ion (K⁺) substitution for Ba₃SiO. The σ is increased by K doping and the two‐orders of magnitude increase of carrier concentration is observed in (Ba_2.6K_0.4)SiO. However, a considerably large amount of K dopant is necessary to increase the carrier concentration and also carrier mobility is largely suppressed by such heavy K doping. Therefore, further exploration of efficient acceptor dopants is necessary to optimize their ZT. Instead, we estimate the maximum ZT of Ba₃ BO with optimized n theoretically. We here calculate PF_calc. and ZT _calc. as a function of n (Figure 5c,d). The PF_calc. vs n and ZT _calc. vs n plots have maxima with respect to n, and the maximum values (max PF_calc. and max ZT _calc.) are obtained at optimal n (n _opt.) as indicated by the arrows in Figure 5c,d. The n _opt. are estimated to be 4.0 (8.1) × 10¹⁹ cm⁻³ and 1.6 (1.6) × 10²⁰ cm⁻³ for Ba₃SiO and Ba₃GeO at T = 300(600) K. Table 2 summarizes the theoretical thermoelectric properties of Ba₃ BO with n _opt. at T = 300 and 600 K. The max ZT _calc. of Ba₃SiO is predicted to be 0.98 and 2.14 at T = 300 and 600 K, respectively, where the max PF_calc. are much increased to 48.0 and 28.9 µW cm⁻¹ K⁻² by tuning n to n _opt.. The max ZT _calc. of Ba₃GeO are predicted to be 0.43 and 1.21 at T = 300 K and 600 K, respectively, where the max PF_calc. is increased to 15.8 and 11.2 µW cm⁻¹ K⁻². The higher ZT of Ba₃SiO compared with Ba₃GeO accounts for its higher PF. Figure 6 compares the present maximum predictions for Ba₃SiO and Ba₃GeO with state‐of‐the‐art thermoelectric materials. The max ZT _calc. value of Ba₃SiO is much higher than those of eco‐friendly thermoelectric materials^{[ 56} , ⁵⁷ , ⁵⁸ , ⁵⁹ , ⁶⁰ , ⁶¹ , ⁶² , ⁶³ , ⁶⁴ , ⁶⁵ , ⁶⁶ , ⁶⁷ , ⁶⁸ , ⁶⁹ , ⁷⁰ , ⁷¹ , ⁷² , ⁷³ , ⁷⁴ , ⁷⁵ , ⁷⁶ , ⁷⁷ , ⁷⁸ , ⁷⁹ , ⁸⁰ , ⁸¹ , ^{82 ]} as seen in Figure 6a. Although higher ZT has been reported for the thermoelectric materials with heavy toxic elements of Pb, Te, Se, and Sb,^{[ 49} , ⁸³ , ⁸⁴ , ⁸⁵ , ⁸⁶ , ⁸⁷ , ⁸⁸ , ⁸⁹ , ⁹⁰ , ⁹¹ , ⁹² , ⁹³ , ⁹⁴ , ⁹⁵ , ⁹⁶ , ⁹⁷ , ⁹⁸ , ⁹⁹ , ¹⁰⁰ , ¹⁰¹ , ¹⁰² , ¹⁰³ , ¹⁰⁴ , ¹⁰⁵ , ¹⁰⁶ , ¹⁰⁷ , ¹⁰⁸ , ¹⁰⁹ , ¹¹⁰ , ¹¹¹ , ¹¹² , ¹¹³ , ¹¹⁴ , ¹¹⁵ , ¹¹⁶ , ¹¹⁷ , ¹¹⁸ , ¹¹⁹ , ¹²⁰ , ¹²¹ , ¹²² , ¹²³ , ¹²⁴ , ¹²⁵ , ¹²⁶ , ¹²⁷ , ¹²⁸ , ¹²⁹ , ¹³⁰ , ¹³¹ , ¹³² , ¹³³ , ¹³⁴ , ¹³⁵ , ^{136 ]} the max ZT _calc. value of Ba₃SiO is comparable in the same temperature range (Figure 6b). For fair discussion, we like to note that these predictions are of ideal single crystals so the real maximum values would be reduced a bit by the impurity doping and consequent electron scattering. The present results, nonetheless, demonstrate the potential of inverse‐perovskite Ba₃ BO as a high‐performance environmentally benign thermoelectric material that can be alternative to currently practical ones composed of heavy and toxic elements.

Table 2.

Summary of theoretical thermoelectric properties for Ba₃ BO (B = Si and Ge) with optimal n (n _opt.) at T = 300 K and 600 K.

	T [K]	n opt. [cm−3]	σ calc. [S cm−1]	S calc. [µV K−1]	PFcalc. [µW cm−1 K−2]	κ calc. [W m−1 K−1]	κ ele,calc. [W m−1 K−1]	κ lat,calc. [W m−1 K−1]	ZT calc.
Ba3SiO	300	4.0×1019	726.7	+257	48.0	1.46	0.25	1.21	0.98
	600	8.1×1019	411.2	+265	28.9	0.81	0.21	0.60	2.14
Ba3GeO	300	1.6×1020	503.5	+177	15.8	0.86	0.25	0.86	0.43
	600	1.6×1020	197.2	+238	11.2	0.43	0.13	0.43	1.21

Figure 6.

Comparison of maximum ZT as a function of temperature (T) for Ba₃ BO (B = Si and Ge) with respect to a) eco‐friendly thermoelectric materials including sulfides (Cu₂S,^{[ 56} , ⁵⁷ , ⁵⁸ , ^{59 ]} SnS,^{[ 60 ]} Cu_0.1TiS₂,^{[ 61 ]} (Cu,Fe)S₂,^{[ 62 ]} Cu₇Sn₃S₁₀,^{[ 63 ]} Cu₅FeS₄,^{[ 64 ]} Cu₂₆Ta₂Sn_5.5S₃₂,^{[ 65 ]} Cu₂ZnSnS₄ ^{[ 66 ]}), silicides (Si_0.8Ge_0.2,^{[ 67 ]} Mg₂Si,^{[ 68} , ⁶⁹ , ^{70 ]} Mg₂(Si,Sn),^{[ 71 ]} MnSi_1.75,^{[ 72 ]} CrSi₂,^{[ 73 ]} β‐FeSi₂,^{[ 74 ]} Sr_0.92Y_0.08Si₂ ^{[ 75 ]}), oxides (Sr_0.9La_0.1Ti_0.9Nb_0.1O₃,^{[ 76 ]} Sr_0.93La_0.07Ti_0.93Nb_0.07O₃,^{[ 77 ]} Sr_0.775La_0.15□_0.075TiO_{3− δ} ,^{[ 78 ]} Zn_0.96Al_0.02Ga_0.02O,^{[ 79 ]} Na_1.7Co₂O₄,^{[ 80 ]} Ca_2.8Ag_0.05Lu_0.15Co₄O_{9+ δ} ,^{[ 81 ]} Ca_2.95Tb_0.05Co₄O₉Bi_0.25 ^{[ 82 ]}), and b) state‐of‐the‐art thermoelectric materials with heavy (toxic) elements: Bi_0.5Sb_1.5Se₃,^{[ 83} , ⁸⁴ , ⁸⁵ , ^{86 ]} AgSeTe₂,^{[ 87} , ^{88 ]} GeTe,^{[ 89} , ⁹⁰ , ⁹¹ , ⁹² , ⁹³ , ⁹⁴ , ⁹⁵ , ^{96 ]} SnSe,^{[ 97} , ⁹⁸ , ⁹⁹ , ¹⁰⁰ , ¹⁰¹ , ¹⁰² , ¹⁰³ , ^{104 ]} SnTe,^{[ 105} , ¹⁰⁶ , ¹⁰⁷ , ¹⁰⁸ , ¹⁰⁹ , ¹¹⁰ , ¹¹¹ , ^{112 ]} PbCh (Ch = S, Se, Te),^{[ 113} , ¹¹⁴ , ¹¹⁵ , ¹¹⁶ , ¹¹⁷ , ¹¹⁸ , ¹¹⁹ , ¹²⁰ , ^{121 ]} Cu₂ Ch (Ch = Se, Te),^{[ 49} , ¹²² , ¹²³ , ¹²⁴ , ¹²⁵ , ¹²⁶ , ¹²⁷ , ¹²⁸ , ^{129 ]} Half Heusler FeNbSb and FeTaSb,^{[ 130} , ¹³¹ , ^{132 ]} BiCuOSe,^{[ 133} , ^{134 ]} Tetrahedrite Cu₁₂Sb₄S₁₃.^{[ 135} , ^{136 ]} Green, purple, and orange symbols in (a) indicate maximum ZT values for sulfide, silicide, and oxide thermoelectric materials, respectively.

3. Conclusion

In summary, we demonstrated the high ZT in bulk polycrystals of the p‐type inverse‐perovskite Ba₃ BO (B = Si and Ge) without toxic elements. The valence band around the Fermi level arises from the p state of the negatively charged B anion with large ion size, and the hole transport with long carrier life time and their highly dispersive bands with multiple valley degeneracy realize both high σ and high S, simultaneously. In addition, the bulks exhibited low κ _lat of 1.00 W m⁻¹ K⁻¹ for Ba₃SiO and 0.77 W m⁻¹ K⁻¹ for Ba₃GeO at RT, which is significantly lower than 8.2 W m⁻¹ K⁻¹ of normal perovskite SrTiO₃ bulk,^{[ 44 ]} and even lower than 1.7−2.0 W m⁻¹ K⁻¹ of Bi₂Te₃ and PbTe bulks.^{[ 45} , ⁴⁶ , ^{47 ]} The low κ _lat of Ba₃ BO originates from the low ν _ph for phonons associated with Ba atomic vibration, and the phonons associated with Si and O atomic vibrations have a negligible contribution to κ _lat due to the very short τ _ph. The crystal structure of Ba₃ BO is constructed from the highly distorted O−Ba₆ octahedra framework with weak O−Ba ionic bonds, which provides extremely low ν _ph and strong phonon scattering. As a consequence of high PF and low κ _lat, the Ba₃SiO and Ba₃GeO exhibited rather high ZT of 0.16 and 0.35 at RT, respectively. The ZT value increased continuously up to 0.84 at T = 623 K for Ba₃SiO and 0.65 at T = 523 K for Ba₃GeO. In addition, based on first‐principles carrier and phonon transport calculations, we predicted that a higher ZT could be obtained by optimizing hole concentration in Ba₃ BO. Specifically, the maximum ZT potentially increases to 2.14 for Ba₃SiO and 1.21 for Ba₃GeO at T = 600 K. The present results indicate that inverse‐perovskites would be a new platform of environmentally benign high ZT thermoelectric materials.

4. Experimental Section

Bulk Synthesis

The Ba₃SiO and Ba₃GeO bulk polycrystals were synthesized by solid‐state reactions of a stoichiometric mixture of Ba, Si or Ge, and BaO via a reaction of 2Ba + Si(Ge) +BaO → Ba₃SiO(Ba₃GeO). First, fresh Ba metal (purity 99.99%, Sigma–Aldrich) was finely cut into small pieces of grains.^{[ 137 ]} The Ba grain, Si (purity 99.9%, Kojundo Chemical Lab.) or Ge powders (purity 99.9%, Kanto Chemical), and BaO powder (purity 99.9%, Kanto Chemical) were mixed and then pressed into 10‐mmϕ pellet. The obtained pellet was wrapped in Ta foil and then sealed in an Ar‐filled stainless tube. The sealed stainless tube was heated at an optimized temperature of 750 °C for 10 h for Ba₃SiO and 700 °C for 10 h for Ba₃GeO. The product was reground and densified to 10‐mmϕ pellet again, and then it was wrapped in Ta foil and then sealed in an Ar‐filled stainless tube. The sealed stainless tube was heated again at 900 °C for 10 h for Ba₃SiO and 700 °C for 10 h for Ba₃GeO. The bulk densities are 4.30 g cm⁻³ for Ba₃SiO and 4.35 g cm⁻³ for Ba₃GeO. The sintered densities are estimated to be 87.0% and 80.4%, respectively. The chemical compositions of the bulk samples measured with EDS are Ba_3.2SiO_1.6 and Ba_3.3GeO_1.5. The deviation from the stoichiometric composition would come from the coexistence of impurity phases, such as BaO, and the oxidation during sample transfer to measurement chamber. All the synthesis processes were performed in a glovebox with a dry inert Ar gas (the dew point < −100 °C, the oxygen concentration < 1 ppm).

Crystal Structure Analysis

Crystalline phases were determined by XRD with the Bragg−Brentano geometry with a Cu Kα radiation source at RT. The lattice parameters were determined by the Pawley method using the TOPAS ver. 4.2 program (Karlsruhe, Germany: Bruker AXS GmbH). Rietveld analysis, where the fundamental parameter (FP) method was employed, was performed for crystal structure refinement. The microstructure of the bulks was evaluated using a field‐emission scanning electron microscopy (FE‐SEM; JSM‐7600F, JEOL) equipped with an energy‐dispersive spectrometer (EDS). The electronic structures were characterized by X‐ray photoemission spectroscopy (XPS) performed at the undulator beamline BL‐2A of the Photon Factory, High Energy Accelerators Research Organization (KEK). The binding energy was calibrated with the Fermi level of an evaporated reference Au film. Diffuse reflectance (R) spectra were measured at RT with a spectrophotometer in the wavelength (λ) range of 200−2400 nm. The obtained R spectra were converted using the Kubelka−Munk function (1−R)²/(2R) = α/S _f, where α and S _f denote the optical absorption coefficient and the scattering factor, respectively, to obtain the quasi‐optical absorption spectra.

Electronic and Thermal Properties

σ and S were simultaneously measured by the four‐probe method (ZEM‐3, ADVANCE RIKO, Inc.) under a He atmosphere. The κ was obtained from κ = D·C·ρ, where the thermal diffusivity (D) along the out‐of‐plane direction in the bulk was measured in an Ar atmosphere by a laser flash diffusivity method (LFA 457, NETZSCH) and the heat capacity (C) was measured by differential scanning calorimetry (DSCvesta, Rigaku Corp.), and the sample density (ρ) was determined by the dimensions and mass of the samples. The sound velocity (v _s) is obtained by $v_{\mathrm{s}}={(\frac{1}{3}[\frac{2}{v_{\mathrm{t}}^3}+\frac{1}{v_{\mathrm{l}}^3}])}^{-1/3}$, where v _t and v _l are the transverse and longitudinal sound velocities measured by ultrasonic pulse‐echo method (1077DATA, KARL DEUTSCH) at RT. A detail of the phonon gas model analysis is described in the caption of Table S1 (Supporting Information).

Density Functional Theory Calculation

The electronic structure calculations were performed for Ba₃ BO models by DFT conducted using the projector augmented wave (PAW) method as implemented in the VASP code.^{[ 34} , ^{35 ]} Ba [5d6s6p], Si [3s3p], Ge [4s4p], and O [2s2p] orbitals were included as valence states. The variable‐cell structure relaxations were performed by the generalized gradient approximation (GGA) Perdew–Burke–Ernzerhof (PBE) functional^{[ 138 ]} with a plane wave cut‐off energy of 550 eV, a Γ‐centered k‐mesh with the k‐spacing of 0.2 Å⁻¹, as well as the convergence criteria of 10⁻⁶ eV for the energy and 0.01 eV Å⁻¹ for the force. The relaxed lattice parameters are a = 7.762 Å, b = 10.844 Å, c = 7.569 Å for Ba₃SiO and a = 7.761 Å, b = 10.888 Å, c = 7.608 Å for Ba₃GeO, in consistence with the experimentally obtained values within 3% differences. The electronic band structures and DOSs were obtained by the HSE hybrid functional.^{[ 36 ]} The carrier effective masses were calculated by BoltzTraP2 code.^{[ 139 ]} The carrier transport properties of Ba₃ BO were calculated by using DFT and DFPT as implemented in the Quantum ESPRESSO package.^{[ 38} , ^{39 ]} The GBRV ultrasoft pseudopotentials^{[ 140 ]} were employed with the kinetic energy cutoff of 40 Ry (320 Ry) for wavefunctions (charge density). The k‐mesh density of 6×4×6 was used for the DFT calculations, and the 2×2×2 q points were used for the phonon and electron‐phonon calculations within DFPT. To compute the transport coefficients using dense k and q grids, the maximally localized Wannier functions (MLWFs) were constructed from the isolated 12 Kohn–Sham states below the VBM. The Wannierization was performed using the Wannier90 code,^{[ 141 ]} where the Si (Ge) p orbitals were used as initial projections and the outer energy window of [−2.0, 0] eV relative to the VBM. The n, S, κ _ele, σ, and τ _e values at T = 300 K and 600 K were calculated using the PERTURBO code.^{[ 41 ]} The electron–phonon coupling coefficients were interpolated to the dense 120×80×120 k and q points and then used to solve the Boltzmann transport equation within the relaxation time approximation (RTA). The carrier lifetimes were computed from the imaginary part of the Fan–Migdal self‐energy, where the summation over the q points was performed by randomly sampling 10⁶ q points from a uniform distribution. It was confirmed that the transport coefficients reached converged with the above parameters.

The phonon transport calculations for Ba₃ BO were performed using the ALAMODE code.^{[ 51} , ^{52 ]} A 2×2×2 supercell (160 atoms) was used for the calculation of harmonic interatomic force constants (IFCs) and the anharmonic IFCs. The harmonic IFCs were determined by the finite‐displacement approach^{[ 142} , ^{143 ]} and the anharmonic IFCs up to sixth‐order were estimated by the compressive sensing lattice dynamics. The temperature‐induced renormalized harmonic IFCs at T = 300 K were computed using the self‐consistent phonon (SCPH) theory,^{[ 52 ]} and were employed in the subsequent phonon transport calculations. All allowed interactions were included for the harmonic IFCs, the third‐order IFCs inside the cutoff radii of 12 bohr, and fourth‐, fifth‐, and sixth‐order IFCs inside the cutoff radii of 8 bohr. The DFT calculations to obtain the force were performed using the GGA‐PBE functional with a plane‐wave energy cutoff of 400 eV, a Γ‐centered 2×2×1 k‐mesh and an energy convergence criterion of 10⁻⁸ eV. κ _lat was calculated by solving the Peierls–Boltzmann transport equation under the RTA with a 7×7×5 q point mesh, which provides sufficient accuracy confirmed by the convergence tests (Figure S6, Supporting Information). The non‐analytic correction was included to the dynamical matrix by the mixed‐space approach,^{[ 144 ]} in which the Born effective charges of constituent elements and the dielectric constants were obtained by DFPT.^{[ 37 ]}

CCDC 2291770 and 2291771 contain the supplementary crystallographic data for this paper. These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via www.ccdc.cam.ac.uk/data_request/cif.

Conflict of Interest

The authors declare no conflict of interest.

Supporting information

Supporting Information

He X., Kimura S., Katase T., Tadano T., Matsuishi S., Minohara M., Hiramatsu H., Kumigashira H., Hosono H., Kamiya T., Inverse‐Perovskite Ba₃ BO (B = Si and Ge) as a High Performance Environmentally Benign Thermoelectric Material with Low Lattice Thermal Conductivity. Adv. Sci. 2024, 11, 2307058. 10.1002/advs.202307058

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.