**Structure and electronic properties**

The unit cell of both CLMO and CLBMO are orthorhombic (*Pnma*) structures with several MnO6 octahedral as illustrated in Fig. 1 (a) and (b), respectively. CLMO unit cell consists of 2 Ca atoms, 2 La atoms, 4 Mn atoms, and 12 O atoms with the formula Ca2La2Mn4O12. CLBMO unit cell has one Bi atom substituting one La atom so that there is one Bi atom and one La atom each in the unit cell with the formula Ca2LaBiMn4O12. The calculation results in the ferromagnetic configuration as the most stable structure of the two compounds. The lattice constants of CLMO are *a* = 6.1785 Å, *b* = 8.2549 Å and *c* = 6.1763 Å with volume of 315.01 Å3, whereas the lattice constants of CLBMO are *a* = 5.9423 Å, *b* = 8.0680 Å and *c* = 6.0619 Å with volume of 290.62 Å3. Despite the ionic radius of Bi3 + being almost identical to that of La3 + 39,40, substituting one Bi atom for one La atom in the unit cell effectively reduces the lattice constants. CLMO shows the Mn–O bond length of 2.0624 Å, which is longer than CLBMO’s of 1.9423 Å. Likewise, CLMO has an Mn–O –Mn angle of 172.81o, which is larger than that of CLBMO of 160.32o, thus, the more tilted MnO6 octahedra. These results are consistent with those obtained in the previous experiment 41. Noted that, the magnetic moments of Mn3 + and Mn4 + are 3 and 4 µB, respectively, neglecting some orbital contribution 42. Therefore, the local magnetic moment per Mn atom was obtained from the calculation which is 3.6 µB for CLMO and slightly decreased to 3.5 µB for CLBMO, presumably confirming the mixing of Mn3 + and Mn4+.

Figure 2 (a) shows the electronic spin-polarized band structure of CLMO and CLBMO, respectively, in which the Fermi energy lies upwards in the conduction band for the spin-up state (↑). It implies a slight overlap between the upper valence and the lower conduction band of electrons, confirming both are in the metallic-like state. For the spin-down state (↓), there is a bandgap of ~ 2.1 eV either for CLMO or CLBMO, which shows the two have insulator properties as well. The upper valence band of CLBMO has slightly shifted downwards away from the Fermi level than that of CLMO, reducing the probability of conduction electrons hopping from the valence to the conduction energy band. As in the transition metal oxide perovskites, in the case of manganite-based material, around the Fermi level is filled with Mn−3d and O−2p bands. Due to the octahedral crystal field, the Mn−3d splits into a higher energy level of the two-fold degenerated *e*g and a lower energy level of the three-fold degenerated *t*2g orbitals. The Mn-*t*2g orbitals have low overlap with O−2p orbitals and are strongly localized. On the other hand, Mn-*e*g orbitals are more diffuse and directed towards O−2p orbitals, and their overlap is large enough for effective hopping42.

The spin-up DOS of CLMO and CLBMO are dominated by the orbital hybridizations, mainly in the low energy range below the Fermi energy as shown in Fig. 2 (b). The sharp increase of DOS in higher energy levels is mostly contributed to by the La−4p state. It concludes that, around Fermi energy, CLMO has the DOS which is dominated by the hybridization of Mn−3d – O−2p. Whereas in CLBMO, the presence of Bi−6s forms the hybridization of Bi−6s – O−2p around Fermi energy as well, which competes with the inherent hybridization of Mn−3d – O−2p. The DOS around Fermi energy further determines the two compounds' thermoelectric and electronic transport properties.

**Thermoelectric parameters**

The thermoelectric parameters *S* (*µ*V/T), *σ*/*τ* (✕1020 Ω−1·m − 1·s − 1), *к*e/*τ* (✕1015 W/m·K·s), *PF*/*τ* (✕1011 W/m·K2·s) and carrier concentration *n* (✕1022 cm − 1) of CLMO and CLBMO obtained from the DFT-BTE calculation are shown in Tables 1 and 2, respectively. CLMO has a higher *σ*/*τ* than CLBMO and according to the Wiedemann-Franz law, resulting in *к*e/*τ* which is also higher in the temperature range of 300 to 800 K. While CLBMO has higher absolute *S* than those of CLMO for the temperature range of 300 to 800 K. CLMO also shows a higher concentration of charge carriers than CLBMO. We will then use this calculated data to determine the DOS effective mass.

**Table 1**

The thermoelectric parameters *S* (*µ*V/T), *σ*/*τ* (✕1020 Ω−1·m − 1·s − 1), *к*e/*τ* (✕1015 W/m·K·s), *PF*/*τ* (✕1011 W/m·K2·s) and carrier concentration *n* (✕1022 cm − 1) of CLMO obtained from the DFT-BTE calculation.

**Table 2**

The thermoelectric parameters *S* (*µ*V/T), *σ*/*τ* (✕1020 Ω−1·m − 1·s − 1), *к*e/*τ* (✕1015 W/m·K·s), *PF*/*τ* (✕1011 W/m·K2·s) and carrier concentration *n* (✕1022 cm − 1) of CLBMO obtained from the DFT-BTE calculation.

Table 3 shows the *κ*ph of CLMO and CLBMO which are calculated with relaxation time being generated by the software. The *к*ph of CLBMO is about two times smaller than CLMO at a temperature of 300 to 800 K. In the electron − phonon scattering process, the phonon lifetime is reduced when the charge carrier concentration is high43. The electron-phonon interaction significantly reduces *κ*ph when the charge carrier concentration is above 1019 cm − 3. At such a high concentration of charge carriers, the scattering rate of electron-phonon interactions exceed the scattering rate of phonon–phonon interactions for low-frequency phonons. So, with the order of 1022 carrier concentration, CLMO and CLBMO are the cases. Hence, this is an advantage for CLMO and CLBMO as thermoelectric materials require low *κ*ph since most of the heat in the lightly doped manganates is carried by phonons with the lowest frequencies.

**Table 3**

The phonon thermal conductivity *к*ph (W/m.K) of CLMO and CLBMO was obtained from the DFT-BTE calculation.

**Thermoelectric optimization**

In Fig. 2 (a) we see that both CLMO and CLBMO have a symmetrical parabolic dispersion band, especially at the *k*-point of gamma which has the highest symmetry. This confirms that the RSPB is suitable for determining the DOS effective mass as the basis for calculating the *PF* optimization. Optimization calculations using the RSPB model typically rely on effective mass, which usually requires experimental data on the thermoelectric parameters. To ensure comparability and validity, we also perform optimization calculations using the RSPB model, with the input thermoelectric parameters obtained from experiments conducted on CaMnO3 (CMO) and Ca0.97Bi0.03MnO3 (CBMO)28. In addition to use the RSPB model, we are also investigating the effect of the scattering mechanism by applying thermoelectric optimization based on scattering-dependent single-parabolic band (TOSSPB) model44. Figure 3 illustrates the optimization of CMO and CBMO using acoustic deformation phonon (ADP) and polarized optical phonon (POP) scattering mechanisms, as well as the RSPB model that does not consider these mechanisms. Based on the optimization curve of ADP, which is like that of RSPB, it can be inferred that ADP is the dominant scattering mechanism, whereas POP has minimal impact. Thus, the optimization of the RSPB model involves using input thermoelectric parameters from the DFT-BTE calculation, while considering only ADP as the scattering mechanism. To achieve optimal power factor, the CMO parent compound requires increasing the carrier concentration through doping, while a 3% Bi doping is necessary to achieve optimal power factor in CBMO.

Figure 4(a) presents the *PF/τ* optimization curves of CLMO and CLBMO at temperatures of 400 and 800 K as a function of charge carrier concentration calculated by Eq. (5) with the input data from Tables 1 and 2. It can be noticed on the curves that the heavily doped manganates CLMO and CLBMO put *PF/τ* values obtained from DFT-BTE calculation (shown by small triangles) on the right side of the optimum *PF/τ*. The optimum *PF/τ* of CLMO and CLBMO are 43.12 and 50.63 ✕ 1010 W/m·K2·s at a temperature of 400 K, respectively, while for a higher temperature of 800 K are 78.32 and 71.06 ✕ 1010 W/m·K2·s, respectively. The optimal PF/τ for CLMO and CLBMO is attained at charge carrier concentrations of 3.6308 and 12.0226 ✕ 1021 at 400 K, respectively. Meanwhile, at 800 K, the optimal PF/τ for CLMO and CLBMO is 9.7857 and 17.6441 ✕ 1021, respectively.

Regarding power factor, it can be inferred that CLBMO still exhibits satisfactory thermoelectric performance, with optimization levels of 56% and 69% at temperatures of 400 and 800 K, respectively. In contrast, CLMO is unsuitable for use as a thermoelectric material, with optimization levels of only 16% and 30% at temperatures of 400 and 800 K, respectively. In Fig. 4 (b) of the Pisarenko plot, *S* which gives the optimum *PF/τ* for CLMO and CLBMO is between 165–169 µV/K for temperatures of 400 and 800 K. Figure 4 (c) shows carrier mobility *µ* of CLMO and CLBMO which are 1.76 and 1.02 ✕ 1014 cm2/V·s2 at 400 K, respectively, and are 3.21 and 0.89 ✕ 1014 cm2/V·s2 at 800 K, respectively. CLBMO has lower *µ* than CLMO either at temperatures of 400 or 800 K, due to the lower *σ*. To reach its optimum value, less doping reduction is required from CLBMO compared to CLMO.

**Corrected figure of merit**

The most common implementations of the Boltzmann transport equations assume the scattering mechanism to be in a CRTA model. While constant *τ* factor is taken, consequently, Eqs. (2) and (3) return the quantities *σo* = *σ/τ* and *κ*e,0 = *κ*e/*τ* (in the Seebeck coefficient τ cancels out). In the mean to equalize *κ*e and *κ*ph in terms of the physical unit, we use *τ* obtained by Cohn *et al*.45 for general Ca1-xLaxMnO3. Unlike the lightly doped manganite, whose *κ*e is about two orders of magnitude smaller than *κ*ph28,46, conversely, by including the reasonable *τ* in the calculation, both CLMO and CLBMO have *κ*e which are two up to three orders of magnitude bigger than their *κ*ph. Therefore, the total *к* of both compounds is thoroughly determined by *κ*e, while *κ*ph is neglected in the calculation. The dimensionless figure of merit *ZT* is a parameter of thermoelectric material efficiency. The higher the *ZT*, the more efficient it is as a thermoelectric material. Assuming *τ* is replaced by the relaxation times of each *σ* and *κ*e in the case of heavily doped CLMO and CLBMO, and by eliminating *κ*ph, *ZT* can be expressed as:

$$ZT=\frac{{S}^{2}{\sigma }_{0}{\tau }_{\sigma }T}{{к}_{e,0}{\tau }_{ke}}$$

(8)

where *τσ* and *τκe* are the relaxation time of *σ* and *κ*e, respectively. A comparison of the calculated values of *τκe*/*m* and *τσ*/*m*, where *m* is the carrier (electron) mass, showed that both these parameters are of the same order-of-magnitude, but that the thermal and electrical relaxation times are not in general equal 47. Accordingly, the *ZT* from Eq. (8) can be expressed as a function of a ratio of *τσ*/*τκe*:

$$ZT=\left(\frac{{S}^{2}{\sigma }_{0}T}{{к}_{e,0}}\right)\left(\frac{{\tau }_{\sigma }}{{\tau }_{ke}}\right)$$

(9)

When *σ* and *κ*e are written using the following approximation form47:

$$\sigma \cong \left(\frac{n{e}^{2}{\tau }_{\sigma }}{m}\right){\delta }_{ir}$$

(10)

$${к}_{e}\cong \left(\frac{{\pi }^{2}n{{k}_{B}}^{2}{\tau }_{ke}T}{3m}\right){\delta }_{ir}$$

(11)

where *n*, *e*, *k*B, *m*, *δ*ir and *T* are carrier concentration, electron charge, Boltzmann constant, electron mass, Kronecker delta, and temperature in Kelvin, respectively, so by using Wiedemann-Franz law *κ*e= *LσT*, Eqs. (10) and (11), the relationship between the Lorenz number, *L*, and *τ* is obtained:

$$L=\frac{1}{3}{\left(\pi {k}_{B}/e\right)}^{2}\frac{{\tau }_{e,0}}{{\tau }_{\sigma }}$$

(12)

where *L*o = (1/3) (*πk*B/*e*)2 is the degenerate limit. The value of *L* itself degenerates with the increase in *S* which depends on temperature changes10,26. Using Eqs. (9) and (12), the corrected *ZT* regarding the Lorenz number is given by the equation:

$${ZT}_{c}=\left(\frac{{S}^{2}{\sigma }_{0}T}{{к}_{e,0}}\right)\left(\frac{1}{L/{L}_{0}}\right)$$

(13)

whereas *L* can be calculated using the equation of 26:

$$L={L}_{0}.\left\{2+\frac{\left({\pi }^{2}/3\right)-2}{{\left[1+{\left(2\pi /{n}_{r}\right)}^{3/2}\right]}^{2/3}}\right\}$$

(14)

where *L*0= 0.7426 ✕ 10 − 8 WΩK − 2 and *nr* are obtained from Eq. (7).

Figure 5 shows *ZT*c which goes hand in hand with the constant relaxation time approximation *ZT* (*ZT*CRTA). CLBMO produces *ZT*c and *ZT*CRTA values of 0.327 and 0.273 at a temperature of 800 K, respectively, and exhibits a greater difference with increasing temperature. Unlike the case with CLMO, which does not appear to be a significant difference between *ZT*c and *ZT*CRTA as the temperature increase. CLMO only shows a *ZT*c and *ZT*CRTA of 0.063 and 0.060 at a temperature of 800 K, respectively. That means, the *ZT* of CLBMO can be adjusted up by decreasing *τκe* or increasing *τσ*, and in another way, it may be also by decreasing the Lorenz number.