Prior to exploring the effect of strain on the electronic properties of bulk LaCoO₃, it is necessary to first calculate the electronic structures with effective on-site coulomb-exchange interaction parameter U_eff varying from 1.0 to 4.6 eV. Figure 1 presents the calculated bandgap of bulk LaCoO₃ (see Figure 2A) under different U_eff. It is found that the bulk LaCoO₃ exhibits metallic properties for U_eff < 1.8 eV. And, with the U_eff increasing beyond 1.8 eV, the bandgap occurs. When U_eff value of 2.8 eV was employed, a bandgap of 0.441 eV was observed which is in good consonance with experimental results (Chainani et al., 1992; Abbate et al., 1993; Arima et al., 1993). Therefore, we adopted the U_eff of 2.8 eV in the following discussions. As shown in Figure 2B, the degenerate 3d orbitals of the free Co atom are split into t_2g and e_g bands under the restriction of cubic symmetry of LaCoO₃. The valence bands and the conduction bands are attributed to t_2g and e_g, respectively. This indicates that six d electrons of trivalent Co ion fully occupied t_2g orbitals, agreeing well with non-magnetic ground states of LaCoO₃ (Abbate et al., 1993). Besides, there is significant hybridization O-2p and Co-3d, suggesting a strong covalent bond between Co and O. The empty states at the energy of 4.0 eV above Fermi level are the La 4f states. The Bader charge analysis suggests that the calculated charge transfer of La is +2.071e, close to its formal charges of +3e. The charge transfer of Co and O atom is +1.355e and −1.142e, respectively. This deviation of the charge transfer from the corresponding formal charges +3e and −2e further confirms the covalent feature of Co–O bond. In our DFT analysis for the effect of strain on the spin arrangement of LaCoO₃, the strain is achieved by equally changing lattice parameters along with a, b, and c directions. The strain is evaluated by γ = (a_s−a)/a×100%, where a_s and a are the lattice constants of strained and strain-free structure, respectively. The corresponding strain energy of LaCoO₃ is defined as ΔE = E_s−E, where E_s and E denote the total energy of strained and strain-free structure, respectively. Figure 2C shows the curve of the strain energy ΔE of LaCoO₃ versus γ. The minimum ΔE occurs at γ = 0, corresponding to the unstrained system.

To assess the strain effect on electronic configurations within a reasonable strain range, we first determined the ultimate strain of LaCoO₃ by calculating the strain–stress curve. It is evident that stress increases with strain and reaches a peak at 18%, as shown in Figure 2C. The ultimate strain of 18% suggests a good resistance to stress for LaCoO₃, which makes it possible to synthesize multicomponent perovskites by partially substituting La or Co, reflecting the diversity of perovskite-type oxides (Peña and Fierro, 2001; Smit et al., 2006). When the strain becomes larger than ultimate strain, the system is subject to attack by vacancy defects or high-temperature effects, etc., and the consequent collapse of crystal structures (Topsakal et al., 2010; Akhoundi et al., 2019).

Magnetic behaviors of materials arise from its spin arrangement. The results for the magnetic moment localized at Co atom (Mag_Co) under various strain (γ) are shown in Figure 3A and Supplementary Tables S1, S2. The main feature of the curve is that Mag_Co exhibits step-like change with strain in the calculated range from γ = −4.0 to 30%, reflecting low, intermediate, and high spin configurations. For strained LaCoO₃ with intermediate- and high-spin configurations, the magnetic moment of Co increases with strain. This positive correlation between magnetic moment and strain is a general behavior, which was observed in various materials (Kushima et al., 2010; Hsu et al., 2018; Zhao et al., 2020b). Given the slight variation in the magnetic moment over each terrace, we take the average values over the corresponding strain range as the DFT calculated Mag_Co, listed in Table 1. From Figure 3A, for γ < 1.0%, system manifests non-magnetic behavior. As strain increases (γ ≥ 1.0%), LaCoO₃ becomes ferromagnetic. Within γ = 1.0–4.0%, the values of Mag_Co maintain at around 1.406 μ_B. When γ ≥ 4.0%, the average Mag_Co is 2.561 μ_B. Owing to the dependence of magnetic moment on the U_eff parameter of the DFT+U method (Anisimov et al., 1991; Knížek et al., 2005), the calculated magnetic moments and actual magnetization values differ. Although a large U_eff value can be used to obtain the magnetic moment consistent with the experimental value, this would lead to the wrong estimation of the ground state energy (Tolba et al., 2018). To ensure the accuracy of results, we calculated the magnetic moments localized at Co atoms with U_eff = 3.0, 3.3, 3.5, 3.8, and 4.0 eV, for γ = 0.0, 2.0, and 8.0% (Supplementary Table S1). It is found that there is only a slight variation in the magnetic moment localized Co atom within the reasonable range of U_eff values (more information, see Supplementary Table S1), indicating the accuracy of our conclusion on the spin configuration transition. Furthermore, the previous theoretical investigation has shown that the strained LaCoO₃ has an intermediate-spin and high-spin configuration at a strain of 3 and 11%, respectively, which is consistent with the results in this work (Kushima et al., 2010).

To seek the physical origin of the step-like dependence of Mag_Co on the strain, we analyze the electronic structures by calculating the d-band center and corresponding orbital occupancy (Figures 3B,C; for detailed data, see Supplementary Tables S2, S3). It is observed that t_2g and e_g bands are spin-degenerate for γ < 1.0% (Figure 3B). However, in the case of γ ≥ 1.0%, the spin degeneracy is lifted, and the degenerate d bands split into two bands. The corresponding spin splitting is estimated by the gap of d-band center between the majority α-spin and minority β-spin, presented in Figure 3B. The spin splitting increases with γ. As compared with the t_2g band, the degree of spin splitting is significantly larger in e_g, which is accompanied by the change of the octahedral field splitting. The octahedral field splitting is assessed by the gap of d-band center between the t_2g and e_g with same spin states. It is noted that the predominant change of the octahedral field split concentrates in majority α–spin d orbitals. The majority α–spin t_2g and e_g band centers show step-like descent as γ increases. Especially, the majority α–spin e_g-band center becomes lower than the t_2g-band center when γ ≥ 4.0%. This indicates the smallest octahedral field splitting occurs within γ ≥ 4.0% compared with that in γ < 1.0% and 1.0% ≤ γ < 4.0%. These changes of crystal field splitting and spin splitting would affect orbital occupancy and accordingly magnetic moment. The fundamental origin of the change of octahedral field splitting is the change of the electrostatic interaction between Co and O with different Co–O bond length (d_Co–O). The low-spin, intermediate-spin, and high-spin configurations are in the range d_Co–O < 1.925 Å, 1.925 Å ≤ d_Co–O < 1.982 Å, and d_Co–O ≥ 1.982 Å, respectively (Table 1).

We further quantify the d band-filling numbers by integrating the projected densities of states with respect to energy up to the Fermi level. Neg,αfilling, Neg,βfilling, Nt2⁢g,αfilling, and Nt2⁢g,βfilling were used to denote t_2g and e_g filling in different spin states. The corresponding data are listed in Supplementary Table S3. Figure 3C reveals that Neg,αfilling increases with strain. In contrast, the minority β-spin t_2g bands exhibit a reverse filling trend. The rise of majority α–spin e_g band-filling number results in the enhancement of magnetic moment and the transition of spin configuration. Nt2⁢g,αfillingand Neg,βfilling under strain were observed to be roughly constant. This follows from 3d⁶ electron configuration of trivalent Co ion and the d-orbital splitting under the octahedral field. The projected density of states of Co atoms (Figures 4a–e) indicates that with strain increasing the minority β–spin t_2g states shift upward with relative to Fermi level, whereas electrons gradually fill majority α–spin e_g states. Based on the analysis of the octahedral field, spin splitting, and d-orbital filling, we obtained low-spin, intermediate-spin, and high-spin arrangements in the corresponding octahedral field, illustrated in Figures 4f–h.

Strain could be achieved through introduction of defect or epitaxial substrate, etc. To apply tensile strain in LaCoO₃, we used Ba with a larger radius than La to partially replace La atom of LaCoO₃ (Ba²⁺: 1.61 Å, La³⁺: 1.36 Å) (Shannon, 1976). Nine possible configurations were considered based on the Ba content and the locations of Ba atoms: La_0.875Ba_0.125CoO₃ (Ba_c), La_0.75Ba_0.25CoO₃ (Ba_c+b), La_0.75Ba_0.25CoO₃ (Ba_c+f1, Ba_c+f2, Ba_c+e1, Ba_c+e2), La_0.625Ba_0.375CoO₃ (Ba_c+f1+f3, Ba_{c+b +f1}), and La_0.5Ba_0.5CoO₃(Ba_c+f1+f2+f3) (see Figure 5). Here, c, b, f, and e denote corner, body center, face center, and edge center sites, respectively. Ba_c, Ba_c+b, and Ba_c+f1+f2+f3 denote Ba occupying the dispersive La sites to form Ba uniform-distributed structures (Figure 5). Other Ba substitutional defects distributed unevenly instead. The Ba substitutions can be formed due to the small substitutional energies of −0.356, −1.076, −1.083, −1.418, −1.415, −0.766, −1.076, 0.402, and 0.071 eV. The overall substitutional energies manifest the stability of the structure, reflecting the possibility of substituted perovskite compounds with the formula of A_1–xA′_xB₁–_xB′xO₃ (Peña and Fierro, 2001). In experiments, various R_1–xA_xCoO₃ (R = La, Pr, and Nd; A = Ba, Sr, and Ca), with x ranging from 0 to 0.5, have been successfully synthesized (Muta et al., 2002; Khalil, 2003; Masuda et al., 2003; Kun et al., 2013). For our investigated Ba substitution-modified LaCoO₃, the charge transfer of Ba is +1.552 to +1.577e, close to its formal charges of +2e (Table 2). The charge transfer of La approaches to its formal charge (+3e) with a value of about +2.0 e.

The consequence of the introduction of Ba substitution at La site causes the lattice expansions. In this work, the uniform distribution of Ba atoms promotes homogeneous elongation of lattice constants along with a, b, and c directions (Ba_c: a_s = b_s = c_s = 0.96%; Ba_c+b: a_s = b_s = c_s = 1.31%; Ba_c+f1+f3: a_s = b_s = c_s = 1.74%; Ba_c+f1+f2+f3: a_s = b_s = c_s = 2.05%) (Figure 6A). For the systems with uneven distributions of Ba atoms (Ba_c+f1, Ba_c+f2, Ba_c+e1, Ba_c+e2, and Ba_c+b+f1), especially Ba_c+b+f1, the structures derivate away from cubic symmetry (see Figure 6A), resulting in anisotropic physical properties of Co atoms. The lattice expansion-induced strain corresponds to the intermediate-spin strain range of defect-free LaCoO₃ (1.0–4.0%) (Figure 6A). The corresponding magnetic moments of each Co atoms are listed in Table 2 and Supplementary Table S4. For Ba_c+b+f1 substitutional defect modified LaCoO₃, the magnetic moments of Co atom are largely different, with an average value of 2.157 μ_b ranging from 1.606 to 2.761 μ_b (Supplementary Table S4 and Figure 6B). In the case of slight deviation from cubic structures (Ba_{c+f1/f2/e1/e2}), Co atoms have the same magnetic moments (Ba_c+f1: 1.834 μ_b; Ba_c+f2: 1.836 μ_b; Ba_c+e1: 1.834 μ_b; Ba_c+e2: 1.833 μ_b) (Figure 6B and Table 2). For the systems with uniform Ba distribution, Ba_c, Ba_c+b, Ba_c+f1+f3, and Ba_c+f1+f2+f3, eight Co atoms in each structure are equivalent and possess the same magnetic moment (Ba_c: 2.111 μ_b; Ba_c+b: 2.377 μ_b; Ba_c+f1+f3: 1.936 μ_b; Ba_c+f1+f2+f3: 2.197 μ_b). Although the strain induced by Ba introduction is within the strain range of intermediate spin of defect-free LaCoO₃, the magnetic moments localized at Co atoms deviate from ∼1.5 μ_b of the intermediate spin structures.

To uncover the fundamental origin of the difference of magnetic moments between Ba-substituted LaCoO₃ and defect-free intermediate-spin LaCoO₃, we further analyze the charge states of Co atoms in Ba substitution modified LaCoO₃. Figures 6C,D outline the total charge transfer of Co to adjacent O atoms, ΔQ^Co and the charge transfer of Co owing to the valence electronic configuration difference between Ba²⁺ and La³⁺, Δ⁢Qwo⁢_⁢strainCo, excluding the charge change resulting from lattice relaxation. The total charge transfer of Co atoms varies from +1.40 to +1.45e, differing from the value in strain-free LaCoO₃ (+1.335 e) (Figure 6C]. The Ba valence states indeed induced the variation in charge transfer of Co (Figure 6D). Specifically, for Ba_c, Ba_c+b, and Ba_c+f1+f2+f3 substitution modified LaCoO₃, Δ⁢Qwo⁢_⁢strainCo is 0.105, 0.126, and 0.111 e, respectively. As a comparison to these structures with uniform distribution of Ba atom, in the defective systems with Ba_c+f1, Ba_c+f2, Ba_c+e1, and Ba_c+e2 substitutions, Δ⁢Qwo⁢_⁢strainCo dramatically decreases by 67–75% with values of 0.035–0.041e. It is a clear indication that the Δ⁢Qwo⁢_⁢strainCo is proportional to Mag_Co (e.g., Ba_c+b: Δ⁢Qwo⁢_⁢strainCo = 0.126e, Mag_Co = 2.365 μ_b; Ba_c+f1: Δ⁢Qwo⁢_⁢strainCo = 0.035e, Mag_Co = 1.835 μ_b) (Figures 6B,D and Supplementary Figure S1). Correspondingly, for Ba_c+f1+f3 substitution modified LaCoO₃, the small Mag_Co of 1.916 μ_b is accompanied by a small Δ⁢Qwo⁢_⁢strainCo of 0.048 e. For Ba_{c,f1/f2/e1/e2} and Ba_c+f1+f3, the extremely small Δ⁢Qwo⁢_⁢strainCo of about 0.04e results in Mag_Co of ∼1.8 μ_b, which is well consistent with 1.5-μ_b magnetic moment of the pristine intermediate-spin configuration.

This effect of Δ⁢Qwo⁢_⁢strainCo on Mag_Co is further verified by the results of Ba_c+b+f1 substitutional defect-modified LaCoO₃. For example, the large Δ⁢Qwo⁢_⁢strainCo on Co², Co⁵, and Co⁸ in Ba_c+b+f1 substitution-modified LaCoO₃ (upper case digits on Co represent the different Co sites; see Supplementary Figure S2 for details), are accompanied by large Mag_Co of 2.059, 2.757, and 2.758 μ_b (Supplementary Table S4). Meanwhile, the small Mag_Co of 1.781, 1.783, 1.606, and 1.752 μ_b are observed on Co¹, Co⁴, Co⁶, and Co⁷ atoms associating with the corresponding small Δ⁢Qwo⁢_⁢strainCo of 0.037, 0.040, 0.037, and 0.020e. However, Co³ atom in Ba_c+b+f1-modified LaCoO₃ is an abnormal case. The magnetic moment localized at Co³ site is 2.761 μ_b despite the small Δ⁢Qwo⁢_⁢strainCo of 0.032e. Further analysis suggests that the charge transfer of Co³ site, using the charge state of Co atom of pristine strain-free LaCoO₃ as benchmark, is 0.036e, which is almost the same as Δ⁢Qwo⁢_⁢strainCo. Therefore, the valence state of Ba dominates the charge transfer in Co³ site, eventually leads to large magnetic moment localized at Co³ site. Overall, the Ba substitutional defect-modified LaCoO₃ possesses intermediate-spin configurations. The values of magnetic moments localized at Co atoms deviating from the 1.5 μ_b of pristine strained intermediate spin LaCoO₃ results from the different electronic configurations between Ba and La. And, the degree of deviation depends on the magnitude of the charge transfer of Co atoms following from the different valence states between Ba and La.

The projected density of states of Co atom reveals that the systems with Ba_c, Ba_c+b, and Ba_c+f1+f2+f3 substitutional defect hold similar orbital occupations, which differs from Ba_c+f1+f3 substitution modified LaCoO₃ (Figures 7A–D). The d-orbital filling exhibit that, for Ba_c+f1+f3, the minority β-spin t_2g filling increases accompanied by reducing majority α–spin e_g filling, compared with the defective LaCoO₃ including Ba_c, Ba_c+b, and Ba_c+f1+f2+f3 substituents (Supplementary Figure S3), which contributes to the small magnetic moment. The defective systems with a small magnetic moment, Ba_c+f1, Ba_c+f2, Ba_c+e1, and Ba_c+e2, have similar d-orbital occupation as Ba_c+f1+f3 (Supplementary Figure S4). All these features of d-orbital occupation are found in different Co atoms of Ba_c+f1+f3 substitution modified LaCoO₃ (Supplementary Figure S5). Ba substitutional defects induce a unit-like e_g occupancy resulting in a moderate interaction between Co active center and oxygen-related adsorbates, which contributes to the reduction of the OER/ORR thermodynamic overpotential. Therefore, Ba substitution defect modified LaCoO₃ is promising for oxygen redox catalytic reactions.