A DFT+U study of the structural, electronic, magnetic, and mechanical properties of cubic and orthorhombic SmCoO₃ Open Access

Solid oxide fuel cells (SOFC) are considered potential alternatives to today’s power generators. In order to expand the applicability of these devices, their reliability and cost need to be improved, goals which can be achieved by lowering the operating temperatures.^1,2 However, this has proven to be challenging, since the efficiency of traditional SOFC is compromised when the operating temperatures are lowered. Some of the problems that have been encountered are cathodic polarization, increase in the resistance in the cell, or reduced catalytic activity, among others.³ Many of these issues relate to the cathode’s performance, which is required to have high catalytic activity for the oxygen reduction reaction (ORR), and which should also facilitate oxygen conduction to the electrolyte.³ To this end, cobalt-based perovskites have gained much attention as cathodic materials, especially SmCoO₃, since it shows high catalytic efficiency and good mixed conductivity at intermediate operating temperatures.^4–13 Interestingly, both theoretical and experimental data on SmCoO₃-based perovskites are rather scarce, including its magnetic and mechanical properties, because more research has focused on the related LaCoO₃ material.

SmCoO₃ is found in two different crystal structures; orthorhombic and cubic (Fig. 1). The orthorhombic structure is the most stable phase at low temperatures, whereas the cubic structure is stabilized when the material is doped and at high temperatures.¹⁴ The temperature also influences the electronic and magnetic properties of SmCoO₃. The electronic behavior has been studied experimentally, and a semiconductor to metal transition has been reported at temperatures higher than 500 K.^15–17 This transition is suggested to be due to a change in the cobalt spin state.^15–19 At the same time, the Co spin state is heavily influenced by the local structure, dopant substitution, and applied external pressure, and it is normal to find different spin domains or the co-existence of two spin states in the same sample.¹⁹ To add further complexity, the spin consequently affects the magnetic structure and vice versa,¹⁹ which at the same time will influence the chemistry of SmCoO₃. Thus, there is a clear interest in understanding which spin state is coupled to a certain magnetic structure and which magnetic structure is the most likely to be responsible for the metallic state observed at high temperatures.

Mechanical stability of SOFC cathodes is also of importance for the performance and stability of the fuel cell, especially for industrial scalability, a fact that is often overlooked in fundamental studies.²⁰ SOFC materials should have thermal compatibility with the electrolyte, good mechanical strength, and be tough.^1,21 It is further undesirable to have large mechanical deformation at operating temperatures, and an understanding of the mechanical properties such as bulk and Young’s moduli is useful.^1,20,21 To the best of our knowledge, there are no reported experimental values for the mechanical properties or elastic constants of SmCoO₃, neither for the cubic nor the orthorhombic structure. For the determination of the elastic constants, large single crystals with very high purity are required, and this is difficult to achieve for many perovskites.^22,23

Thus, despite the availability of some experimental data on SmCoO₃, detailed information is lacking, including the antiferromagnetic arrangement and the Co spin state at high temperatures, or the precise values of the main mechanical properties. Ab initio calculations represent a powerful tool to provide insight into these properties and, in this paper we have used density functional theory (DFT) calculations to study the magnetic structure and the mechanical properties for the two polymorphs of SmCoO₃, taking into account the effect of temperature by simulating the thermal expansion of the material.

The Vienna Ab initio Simulation Package (VASP version 5.3.5) code has been used for all calculations.^24–27 We have used DFT to perform spin-polarized simulations with the PBE functional to describe the correlation-exchange interaction.^28,29 The project-augmented wave (PAW) method was used to describe the ion-electron interaction,³⁰ and we considered the following valence electrons for the atomic species involved: Sm (5s²5p⁶6s²), Co (4s²3d⁷), and O (2s²2p⁴). Sm f-electrons were included in the pseudopotential as core electrons as the samarium ion in SmCoO₃ is +3 charged. The kinetic energy cut-off for the plane-wave basis set was set to 500 eV after testing it for both crystal structures. Structure optimizations were performed with the conjugate gradient method, considering electronic and ionic convergence criteria of 1 × 10⁻⁵ eV and 1 × 10⁻³ eV Å⁻¹, respectively. The tetrahedron method was used for the smearing,³¹ and was applied together with a 4 × 4 × 4 Γ-centered Monkhorst-Pack grid for the cubic model and 8 × 8 × 6 for the orthorhombic model, to describe the reciprocal space.³² The bulk models used throughout this paper are the 2 × 2 × 2 Pm-3m cubic cell and the 2 × 1 × 1 Pnma orthorhombic cell, both containing 40 atoms (Fig. 1). Finally, in order to properly describe the electronic and magnetic structures of SmCoO₃, the Hubbard approximation was employed as described in the Dudarev approximation.³³ Hence, we have applied a U_eff = 3 eV to the Co d-electrons, as done in our previous publications.³⁴ 

From the two reported crystal structures of SmCoO₃, the orthorhombic phase (Pnma) is known to be more stable than the cubic phase (Pm-3m) at low temperatures.^35–37 However, when SmCoO₃ is used for SOFC (normally doped and operating at high temperatures), the cubic phase becomes the more stable.^38,39 Our calculations indicate that, as observed experimentally, the orthorhombic phase is 0.13 eV atom⁻¹ lower in energy than the cubic phase. Both structures show good agreement with the experimental structural data, as shown in Table I, indicating that the Hubbard parameter used for the cubic phase is transferable to the orthorhombic system. The only noticeable difference with experiment is the lattice parameter b for the orthorhombic system, which is ∼1.7% larger than the experimental value. This discrepancy has also been seen in previous theoretical work on orthorhombic cobalt perovskites.⁴⁰ This larger b parameter induces deviations in the bond lengths that are parallel to the b direction. Fortunately, the largest elongation for the Co–O bonds is only 0.03 Å, whereas the Sm–O bonds are stretched by a maximum of 0.14 Å.

The magnetic properties of lanthanide cobaltate perovskites are attributed to the Co³⁺ atoms.^17,18 Co³⁺ is a d⁶ center, which is hexa-coordinated with O²anions in an octahedral configuration in the cubic structure, and in a distorted octahedron in the orthorhombic structure. According to crystal field theory (CFT), the octahedral crystal field splits the five d-orbitals between the t_2g (d_xy, d_yz, and d_xz) and the e_g (d_x2−y2 and d_z2), with the former being lower in energy than the latter. Depending on the occupancy of these orbitals, we distinguish between three different spin states for Co³⁺: (a) low spin state (LS, $t2g6eg0$⁠) with S = 0; (b) intermediate spin state (IS, $t2g5eg1$⁠) with S = 1; and (c) high spin state (HS, $t2g4eg2$⁠) with S = 2, all of them schematically represented in Fig. 2.^15,16 We are assuming that the occupation of these orbitals will obey Hund’s rule, i.e., that the most stable configurations should be those that maximize the spin multiplicity. For example, HS is S = 2 if and only if all unpaired electrons are either α or β. However, due to the nature of DFT, it becomes very complicated to determine pure intermediate or high spin states, and they normally appear as a combination. In the HS case, we can have three α electrons and one β, which means S = 1, or two α and two β, which describes a singlet state, which explains why in Table II, Co magnetization can be found between 0 and $4μB$⁠, depending on the spin state.

For LS states, the system becomes diamagnetic. However, when Co³⁺ centers are found in IS or HS, we can distinguish four different paramagnetic structures, as shown in Fig. 3: three antiferromagnetic (AFM) structures, labeled AAFM, CAFM, and GAFM, and one ferromagnetic, labeled as FM.

Magnetic moments for samarium and oxygen ions are negligible in relation to their cobalt neighbors and they have consequently been omitted. The Bader charges (Table II) on Sm and O show negligible variance with magnetic moment and analysis also showed that the magnetic structure seems to have little to no impact on the charge of the different atoms in the material. Noticeably though, the orthorhombic phase shows higher charges throughout compared to the cubic phase, which could imply that the orthorhombic phase is more ionic than the cubic phase (Table II).

From the projected density of state (PDOS), we observe that for the SmCoO₃ diamagnetic (DM) cubic structure (Fig. 4(a)) the Co t_2g orbitals describe the valence band, in combination with the O 2p band, whereas the conduction band is defined by the Co e_g orbitals. The bandgap between the conduction and valence bands is 0.68 eV. This picture agrees completely with the crystal field theory prediction for the LS state of Co ions. However, the orthorhombic structure also gives rise to an insulator material, with a larger bandgap of 1.20 eV (Fig. 4(b)). In this case, however, the CFT cannot be used to describe the electronic structure as t_2g and e_g appear at the same energies, suggesting a distortion of the octahedral environment of the cobalt centers. Therefore, both valence and conduction bands are described by a combination of t_2g and e_g orbitals, and the O 2p band. For the diamagnetic structures, both materials show $μCo=0μB$⁠.

In the AFM structures, the formal splitting of the Co³⁺ d-orbitals is not as clear as it was for the DM. One fact that is common for almost all structures regardless of the crystallographic phase is that SmCoO₃ becomes metallic, as there is no bandgap at the Fermi level (Fig. 5). This behavior has previously been observed in related perovskites.⁴⁰ Specifically, the orthorhombic phase becomes metallic when it has an A- or CAFM structure, but it remains a semiconductor when the material has a GAFM structure. This suggests that the experimentally observed insulator to metal transition would not involve the GAFM magnetic structure. Furthermore, since all three magnetic structures show non-split t_2g- and e_g-orbitals, electronic conduction goes through the d-metal band, without orbital preference.

Conversely, all cubic antiferromagnetic systems show metallic behavior. The electronic structures for AAFM and GAFM show that conduction is mainly through the t_2g Co orbitals, whereas for the CAFM it is mainly through the e_g, with a small contribution of the t_2g. Spin density plots for the calculated magnetic structure for the cubic and orthorhombic systems are included in Figures 6 and 7. Examining these, it can be seen that the spin ordering presented in Fig. 3 has been preserved in the calculations.

Since the ferromagnetic structure has a non-zero total spin, α and β PDOS are not symmetric (Fig. 8). The cubic FM structure has metallic behavior, where the t_2g combined with the O 2p defines the β conduction and valence band, and e_g with the O 2p describes the α bands. For orthorhombic FM, a pure half-metallic structure was found, agreeing with previous experimental results found for other perovskites, such as FM LaCoO₃, where FM ordering has been assigned to an IS state.^40,42,43 Again, Co d-states are mixed t_2g and e_g, but the occupied states in spin-up direction at the Fermi level are mainly O 2p, with a Co d bandgap of 0.82 eV in the β spin.

The magnetic moments for the cobalt magnetization in the cubic structures are slightly lower compared to the orthorhombic phase (Table III). The differences in different magnetic structures between the phases are between 0.6 and 0.7 μ_B for AAFM and CAFM, and nearly 1.0 μ_B for FM. As a consequence, for AAFM, CAFM, and FM magnetic structures, the cubic phase presents Co centers in a mixed LS and IS state, but they are clear IS states for the orthorhombic phase.

The most remarkable difference is found for the GAFM structure. In both phases, Co magnetization is between 0.6 and 1 μ_B higher compared to the rest of the magnetic structures, and they are very similar for both polymorphs, 2.94 μ_B for the orthorhombic and 2.88 μ_B for the cubic phase. As a result, in the GAFM structure, Co spin states are mixed IS and HS.

To evaluate the evolution of the magnetic structure with temperature (Fig. 9), we have mimicked the volume expansion using the isotropic thermal expansion coefficient $(αT)$ for the orthorhombic SmCoO₃, $αT= 2.17$ × 10⁻⁵ K⁻¹.⁶ The linear thermal expansion coefficient is related to the lattice parameters, and thus supercell volume, through ⁴⁴ 

where l is the lattice parameter, l₀ is the initial lattice parameter (here the reference is the ground state), and T is the temperature. The same thermal expansion coefficient was assumed for the cubic phase, since to the best of our knowledge no thermal expansion coefficient has been reported for this phase. In addition, we assumed that the thermal expansion was linear with respect to the temperature. For the calculations, the expanded volumes were kept fixed, whereas the ion positions were allowed to relax.

Examining Fig. 9, it can be seen that, as predicted experimentally, the diamagnetic structure is the ground state for both polymorphs, and the antiferromagnetic structures are higher in energy at low temperatures. At 500 K the CAFM and the DM states become nearly degenerated for the orthorhombic phase. However, due to the small relative energy differences, we believe that there would be coexistence of both states. These results are in line with the experimental transition found above 500 K.

For the cubic structure, however, we observe a clear crossing between the DM and the CAFM at around 1100 K in which the latter becomes almost 1 eV more stable than the former. This behavior is indeed in full agreement with the experimental results, which as previously noted, predicts an antiferromagnetic structure for the metallic state. It is important to highlight here that at these high temperatures the phase transition from orthorhombic to cubic is known to occur, and according to our results, the insulator to metal transition is coupled with it.¹⁴ 

The stabilization of the CAFM structure seems to be related with the change in Co magnetic moment due to temperature. As depicted in Fig. 10, the magnetic moment of cobalt is initially $1.72μB$⁠, and slightly increases with temperature until 1000 K. After that point, we can observe a drastic change in the Co magnetic moment, which reaches $2.93μB$ at 1273 K. This increase is related to the deformation of the system, which is now pseudo-cubic, in agreement with a previous high temperature synthesis of SmCoO₃.¹⁴ 

Mechanical properties are obtained from the elastic constants (C_ij). C_ij were computed in VASP, using the finite difference technique to calculate the Hessian matrix. In the finite difference technique, displacements of each ion are made in the direction of each Cartesian coordinate in the lattice, with the Hessian being determined from the atomic displacements. To reduce computational effort, only non-equivalent symmetry displacements are considered. The elastic tensor is then calculated by distorting the original lattice and derived using the strain-stress relationship, with the elastic constants calculated according to Eq. (2).^22,45,46 Due to symmetry operations, cubic materials have three independent elastic constants,⁴⁷ whereas orthorhombic systems have nine,⁴⁸ 

The calculated C_ij can be found in Table IV. Generally speaking, C_ii relate to the material’s response to a uniform pressure that is applied perpendicularly to each cell face. A distortion affecting two non-equal axes is represented by C_ij and its equivalents.^45,49 Also, specifically for cubic materials, the Cauchy relation (C₁₂ = C₄₄) should hold,⁴⁵ although real materials do not always obey it.^50,51 For example, in our cubic SmCoO₃, this violation is observed since C₁₂ = 75.89 GPa, and C₄₄ = 100.72 GPa (Table IV). This is in disagreement with previously published molecular dynamics-derived data,²³ but it may be due to short-comings in the interatomic potential model, as the shell model is known to favor solutions that do not violate the Cauchy relation.^51,52

In addition, the elastic constants have to adhere to the general rules for mechanical stability. These relations differ depending on the symmetry of the crystal. In the case of cubic crystals, C_ij should accomplish:⁴⁵ $C11−C12> 0$⁠, $C11+2C12> 0$⁠, and $C44> 0$⁠, whereas for orthorhombic crystals:⁴⁹ $C11+C22> 2C12$⁠, $C22+C33> 2C23$⁠, $C11+C33> 2C13$⁠, $Cii> 0$⁠, and $C11+C22+C33+2C12+2C13+2C23> 0$⁠. The elastic constants obtained here fulfill these conditions for both materials.

From the elastic constants, mechanical properties were calculated using Equations (3)–(13) which are collected in Table V.⁵³ Equations (3), (5), (7), and (10) are valid for cubic systems since C₁₁, C₁₂, and C₄₄ are the complete set of independent elastic constants.⁴⁷ Orthorhombic crystals, on the other hand, have a larger set of independent elastic constants, and their mechanical properties are calculated using Equations (4), (6), (8), and (11)–(13).⁴⁹ For clarity, superscripts c and o are used to indicate the cubic and orthorhombic crystal, respectively. Furthermore, the bulk and shear moduli have been calculated using the Voigt approximation.^45,47,49,54

The bulk modulus (B) (Eqs. (3) and (4)) is the material’s ability to resist a uniform compression (i.e., fracture resistance). It has been shown experimentally that for ABO₃ perovskites, B is dependent on the lattice volume, with larger cell volumes leading to smaller B through an inverse relationship.⁴⁵ Our calculated bulk moduli follow this behavior. B^O is 126.67 GPa, whereas B^C = 204.46 GPa, with the volume for the orthorhombic cell being larger than the cubic. Comparison with an experimental study made on LaCoO₃ at room temperature, whose orthorhombic bulk modulus is 122 GPa, shows that our results show a good match with general cobaltate perovskites.⁵⁵ The scale of B can also be related to the material’s hardness, with higher values indicating harder materials.⁴⁵ Therefore, our values imply that the cubic structure should be harder than the orthorhombic. However, the geometry of the structures has to be taken into account, and thus, the shear modulus (G) is required to make any comment on the relative hardness of the two materials,

G (Eqs. (5) and (6)) is the relation between shear stress and shear strain (i.e., resistance to plastic deformation). G was found to be 137.58 GPa for the cubic and 90.15 GPa for the orthorhombic phase. A large shear modulus is related to a larger resistance against elastic shear strain and surface penetration, which in turn is proportional to hardness.⁴⁵ According to our results, cubic SmCoO₃ is more resistant to surface penetration than the orthorhombic phase, reinforcing the suggestion that the cubic phase is harder than the orthorhombic one,

The B/G ratio gives empirical information on a material’s plastic properties. A material with a B/G ratio larger than 1.75 is expected to be ductile, whereas a B/G smaller than 1.75 describes a brittle material. The B/G for SmCoO₃ calculated here is found to be 1.49 for the cubic structure and 1.41 for the orthorhombic phase, which puts both polymorphs in the brittle category. No previous reports on SmCoO₃’s brittleness have been found, but a study on the related cobaltite perovskite, LaCoO₃, showed that it is brittle,⁵⁶ thus making it plausible that SmCoO₃ should be brittle as well.

Poisson’s ratio $(σ)$ (Eqs. (7) and (8)) is defined as the ratio of lateral to longitudinal strain in the elastic region while being under uniform uniaxial stress, and relates to the change in a material during uniaxial stress.⁵³ It could be used to provide information on interatomic forces. Covalent materials have $σ< 0.1$⁠, whereas for ionic materials σ is higher than 0.25.²² σ for the cubic structure is here calculated to be 0.14, whereas σ for the orthorhombic structure is 0.21, indicating that the latter structure is more ionic than the cubic phase, although each structure has both ionic and covalent character,⁵⁷ 

From B and G, Young’s modulus (E), commonly known as the elastic modulus, can be obtained through Eq. (9).⁴⁷ E represents the slope of the elastic region in a stress-strain curve. The expression for E is the same for both structures, but their values differ.^47,49 Cubic SmCoO₃ was found to have an E of 337.11 GPa, whereas E for the orthorhombic phase was only 218.60 GPa. The smaller E for the orthorhombic system shows that this phase is more receptive to physical changes than its cubic counterpart, which fits with the reasoning above regarding their relative hardnesses,

Elastic anisotropy (A) is measured for the (100), (010), and (001) shear planes, which in the cubic symmetry (Eq. (10)) are identical, but different in the orthorhombic structure (Eqs. (11)–(13)).⁴⁹ This property describes a material’s tendency to form micro-cracks.²² An elastic anisotropy value of 1 indicates an isotropic crystal, whereas deviation indicates shear anisotropy. Here, all A ≠ 1, indicating that the behavior of the two crystals will be dependent on the stress direction,

From the elastic constants and calculated mechanical properties, it can then be concluded that the values of the cubic B, G, and E are higher than their orthorhombic counterparts, and that the orthorhombic SmCoO₃ is more sensitive to deformation than the cubic phase. These conclusions are in line with the hardness results, which indicate that the cubic polymorph is harder than its orthorhombic counterpart, with a lesser degree of anisotropy. However, comparing the ratio-dependent properties, σ and B/G, we found that both structures exhibit similar brittleness and ductility, which is expected considering that they have the same chemical composition. Consequently, both SmCoO₃ phases are prone to fracture, which should be taken into account under SOFC operation.

In this work, we present a DFT+U study of the magnetic and mechanical properties of the cubic and the orthorhombic SmCoO₃ material. We have paid special attention to the magnetic and mechanical properties, which are important when we are considering them as SOFC materials. The electronic ground state for both phases was found to be diamagnetic with the other paramagnetic structures higher in energy. In addition, as also shown experimentally, the orthorhombic phase was lower in energy than the cubic one. In order to introduce the effect of temperature, we mimicked the effect of thermal expansion by increasing the lattice parameter of the bulk material, using the thermal expansion coefficient for orthorhombic SmCoO₃. Hence, we were able to determine the coexistence of both the CAFM and the diamagnetic structure in the case of the orthorhombic phase at high temperatures, and a clear stabilization of the CAFM structure for the cubic phase around 1100 K, all in full agreement with the experimental data. These transitions were linked to a large increase in the Co spin state, as well as to a distortion of the unit cell.

Furthermore, we evaluated the most important mechanical properties based on our calculated elastic constants. We found that there were clear differences between the two SmCoO₃ polymorphs, relating to anisotropy, bulk, shear, and Young’s modulus. It was found that the orthorhombic SmCoO₃ exhibits lower hardness than the cubic phase, and that it also has a larger degree of anisotropy. Conversely, both phases exhibited the same ionic/covalent character $(σ)$⁠, and brittleness (B/G). We consider that these results could be helpful in adopting SmCoO₃-based perovskites for SOFC use.

The authors acknowledge the Engineering and Physical Sciences Research Council (EPSRC) for financial support (Grant Reference Nos. EP/K016288/1 and EP/K001329/1). We also acknowledge the use of the UCL Legion High Performance Computing Facility (Legion@UCL) in the completion of this work. Finally, via our membership of the UK’s HPC Materials Chemistry Consortium, which is funded by EPSRC (No. EP/L000202), this work made use of the facilities of ARCHER, the UK’s national high-performance computing service, which is funded by the Office of Science and Technology through EPSRC’s High End Computing Programme. N.H.d.L. thanks the Royal Society for an Industry Fellowship. E.O. gratefully acknowledges EPSRC funding of the Centre for Doctoral Training in Molecular Modelling and Materials Science (No. EP/G036675/1).