Oxygen vacancy formation and migration in double perovskite Sr₂CrMoO₆: a first-principles study

Abstract

The perovskite-type oxide ABO₃ with mixed ionic and electronic conductivity (MIEC) is a promising candidate for electrode materials of the intermediate-temperature solid oxide fuel cells (IT-SOFCs). The strontium molybdate SrMoO₃ shows high electrical conductivity but low oxygen ionic conductivity due to the stoichiometric features of oxygen. To enhance the oxygen ions diffusion rate, B-site Mo⁴⁺ cation was partially substituted by trivalent transition metals (TM) to generate the oxygen-deficient double perovskite Sr₂BMoO_6−δ. In this study, we present the theoretical investigation of the oxygen vacancy properties of Sr₂CrMoO₆ (SCM) using density functional theory (DFT) with on-site Coulomb potential U for the Cr 3d electrons. The oxygen vacancy formation and migration were investigated for the B-site-ordered SCM, as well as the SCM with Cr/Mo antisite defects (ADs). The oxygen ion migration was optimized by the climbing image nudged elastic band (CINEB) method to identify the minimum energy pathway. The B-site Cr/Mo ADs allow various TM–O bond types in the double perovskite SCM. The calculated formation energy and migration barrier suggests that the B-site Cr/Mo ADs are favorable to enhance the oxygen ionic conductivity of the SCM. These results elucidate the influence of B-site Cr and Mo cations on the oxygen vacancy formation and migration in the double perovskite SCM, which can be useful for investigating Cr-doped SrMoO₃ as an IT-SOFCs anode.

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1. Introduction

Solid oxide fuel cells (SOFCs) are a promising electrochemical power source due to their high power conversion efficiency and fuel flexibility.¹ To realize widespread application, the high operating temperature (typically 900–1200 °C) needs to be reduced to the intermediate temperature (IT) range.² However, the interfacial polarization resistance of the SOFCs electrodes will rapidly increase with decreased operating temperature, which would significantly impact device performance.³ To overcome these drawbacks, a great deal of effort has been devoted to explore novel electrode materials with mixed electronic and ionic conductivity (MIEC).⁴ The MIEC property extends the electrochemical reaction from the surface area to the bulk of an electrode material that could greatly improve the electrochemical reaction rate.⁵

Perovskite-type oxide ABO₃ is the widely used electrode material of SOFCs due to its high temperature stability. Some MIEC perovskites, e.g., La_0.6Sr_0.4Co_0.8Fe_0.2O_3−δ (LSCF) and Ba_0.5Sr_0.5Co_0.8Fe_0.2O_3−δ (BSCF), exhibit outstanding performance as an SOFCs cathode.^6–12 In perovskites, the ionic conductivity is mainly related to the oxygen ion diffusion via a vacancy-hopping mechanism. The self-diffusion rate D_O of oxygen ions can be generally expressed as follows:¹³

D_O = C_VD_V (1)

where C_V and D_V are the oxygen vacancy concentration and oxygen vacancy diffusion rates, respectively. C_V depends on the formation energy, E_form, of the oxygen vacancy, and the D_V is determined by the migration energy barrier, E_barrier, according to the Arrhenius equation as follows:

D_V = Ae^{−E_barrier/k_BT} (2)

where A is a pre-exponential factor and k_B is the Boltzmann constant. Therefore, the oxygen ionic conductivity at the identical working temperature predominantly depends on the E_form and E_barrier of the oxygen vacancy.

The Mo-based double perovskites have been demonstrated to exhibit stability under a reducing atmosphere and can be used as the anode of IT-SOFCs. SrMoO₃ possesses high electrical conductivity (around 10⁴ S cm⁻¹) but low ionic conductivity due to the oxygen-stoichiometric characteristic.¹⁴ To improve its ionic conductivity, Mo⁴⁺ should be partially substituted by other low valent metal cations.^15–20 Bivalent alkaline metals and transition metals (TM) were employed to substitute the Mo cations to generate double perovskites (e.g., Sr₂MgMoO₆, Sr₂NiMoO₆ and Sr₂CoMoO₆), which were used as the sulfur tolerant anode of IT-SOFCs.^21–24 Moreover, the trivalent Fe cation doped SrMoO₃ shows excellent catalytic capacity and oxygen ionic conductivity as a consequence of the mixed-valence of Mo⁵⁺/Mo⁶⁺ and Fe³⁺/Fe⁴⁺, which were employed as both the anode and cathode for IT-SOFCs.^25–28 Likewise, the Cr-doped SrMoO₃ is gradually attracting attention in the field of IT-SOFCs anodes and rechargeable Li–air battery cathodes.^29,30 However, the ionic conductivity of Cr-doped SrMoO₃, to the best of our knowledge, was still not investigated from a quantum mechanical perspective. We herein present a computational study of the double perovskite SCM with a focus on the oxygen vacancy formation and migration process using density functional theory (DFT) plus on-site Coulomb potential. The oxygen vacancy E_form and E_barrier were investigated for the SCM as well as the SCM with B-site antisite defects (ADs). These calculations provide theoretical insight into the oxygen ionic conductivity feature of the double perovskite SCM, which can be useful for the Cr-doped SrMoO₃ in IT-SOFCs anode applications.

2. Computational details

All DFT calculations in this study were performed using the Quantum ESPRESSO software package with a collinear spin-polarized projector-augmented wave (PAW) approach.³¹ The exchange correlation effects were described by the generalized gradient approximation (GGA) according to the Perdew–Burke–Ernzerhof (PBE) functional.³² A 40 Ryd (∼540 eV) cutoff energy was applied for the wave function expansion. The Γ-point centered 4 × 4 × 4 Monkhorst–Pack k-mesh grid was used for Brillouin zone integration.³³ All lattice parameters, including atom positions and cell volumes, were relaxed according to the force convergence threshold of 10⁻⁴ Ryd Bohr⁻¹ on each atom. It is a well-known fact that the pure DFT approach produces a large self-interaction error for strongly correlated TM oxides due to the large intra-atomic exchange interactions of the localized d-orbital electrons. To eliminate this self-interaction error, we employed the so-called DFT+U method by introducing exact Coulomb (U) and exchange (J) interactions for the d-orbital electrons. The Dudarev potential parameter U_eff (U_eff = U − J, 3 eV) was applied for the on-site electronic correlation of the Cr 3d states.^34,35 Due to the delocalized feature of Mo 4d electrons, we did not employ U_eff for the Mo cations, which has been theoretically validated previously for Sr₂FeMoO₆ (SFM).^27,28

To model the double perovskite SCM crystal, a 40-atom rock-salt supercell was employed in our study and is displayed in Fig. 1. For the oxygen-deficient SCM crystal, an oxygen vacancy is generated following the Kröger-Vink reaction:³⁶

where B represents the B-site elements and is the oxygen vacancy. Upon forming the oxygen vacancy, the cation B^× is reduced to B′ by accepting the leftover electrons from the removed oxygen atom. Given that the material is always electroneutral, the oxygen vacancy E_form can be directly obtained from the electronic energy difference through the following simplified approximation:³⁷

E_form = E_defect − E_perfect + 1/2E_O2 (4)

where E_O2 is the energy of a free oxygen molecule in its triplet state. The GGA–PBE method overestimates the binding energy of oxygen molecule; therefore, the Lee et al. method to correct the ab initio calculated O₂ energy was used in the present study.^38,39

Fig. 1 Schematic crystal diagram of SCM. Green spheres denote the A-site Sr, blue and magenta spheres are B-site Cr and Mo, respectively, and red spheres represent O.

Moreover, the ionic charge was calculated using the Bader charge method to elucidate the electron rearrangement after vacancy formation.^40,41 The minimum energy path (MEP) of the oxygen ion migration was optimized using the climbing-image nudged elastic band (CINEB) method. The migration E_barrier was then computed according to the energy profiles of the migration states.^42,43

3. Results and discussion

3.1 SCM electronic properties

We began by optimizing the geometrical parameters of SCM using a spin-polarized DFT+U approach. The SCM was reported to be tetragonal with a slight distortion at 2 K temperature and underwent a phase transition to the cubic symmetry with the space group Fm3m at room temperature (RT).^44,45 In this study, we adopted the lattice parameters of the RT phase to model the SCM crystal. The calculated lattice parameters by DFT+U are listed in Table 1. The estimated lattice constants were pretty close to the experimental values with a small 0.29% error. Moreover, the optimized bond lengths for Mo–O and Cr–O were also nearly equal because of the similar Cr³⁺ and Mo⁵⁺ cationic radii. These results reveal the adequate ability of the DFT+U method for the double perovskite prediction.

Table 1 Calculated and measured lattice parameters for the SCM

	SCM
	DFT+U	Experiment^a
a Ref. 45.
a (Å)	7.8052	7.8278
rMo–O (Å)	1.9462	1.9569
rCr–O (Å)	1.9564	1.9569

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Furthermore, we studied the electronic properties of the SCM based on the DFT+U relaxed structure. The density of states (DOS), as shown in Fig. 2a, reveals the half-metallic features of SCM. In the spin-up channel, the calculated ground state exhibited a band gap near the E_F. The hybridization between Cr t_2g state with O 2p state formed the valence band maximum (VBM), whereas the Mo e_g state was the main component of the conduction band minimum (CBM). For the spin-down channel, the VBM was predominantly made up of O 2p states, whereas the CBM was mainly composed of the hybridization between Cr t_2g, Mo t_2g and O 2p states, which crossed through the E_F. This hybridized orbital in the spin-down channel finally resulted in SCM electronic conductivity. This electronic result is analogous to that of the SFM in terms of the half-metallic electronic structure. The electronic structure difference for the SCM and SFM is that the doped Fe³⁺ cation had a more significant contribution than that of the doped Cr³⁺ cation on the hybridized orbital near the E_F in the spin-down state.²⁷

Fig. 2 PDOS (a) and charge density (b) for SCM with spin-polarized DFT+U approach (U_Cr = 3 eV).

As discussed above, the DOS at the spin-up VBM revealed the 3d electrons spin alignment of the Cr³⁺ cation (3t³_2ge⁰_g4s⁰). The DOS crossed the E_F in spin-down channel indicated that the spin state of the Mo 4d electrons was antiparallel to that of Cr 3d electrons. The contour plot of electron density at the (001) plane of SCM is illustrated in Fig. 2b. As can be seen that the Cr cations with localized 3d electrons exhibited the highest electron density in the SCM supercell. It can be noted that the electronic distribution between Cr and O was different from that between Mo and O suggesting distinguished bond types in the MoO₆ and CrO₆ octahedrons. In the CrO₆ octahedron, the B-site Cr cation and oxygen ions are bridged by the strong ionic bond through the electron transfer between Cr and O. This strong interaction leads to the high vacancy formation energy and migration barrier for the SCM. Nevertheless, the interaction of Mo–O is not a pure ionic bond in the MoO₆ octahedron. This covalence-involved chemical bond is beneficial to the MIEC property, which has also been reported in the SFM investigation.²⁸

3.2 Oxygen vacancy formation in SCM

The formation energy E_form of oxygen vacancies in the SCM was evaluated. Removal of a neutral oxygen atom from the 40-atom SCM supercell constructs the oxygen-deficient Sr₂CrMoO_6−δ (δ = 0.25), as described in Section 2. The oxygen vacancy concentration in this defective supercell is higher than that of the experimental results. However, increasing the supercell size to meet the experimental vacancy concentration is very time consuming, and the 40-atom supercell has been proven in previous publications to be adequate to study the oxygen vacancy properties of double perovskites.^37,46 As the B-site Cr³⁺ and Mo⁵⁺ cations have the very close ionic radii, it is reasonable to assume the existence of the ADs in SCM. We herein also considered the oxygen vacancy formation in the SCM supercell with 25% Cr/Mo ADs. The different oxygen vacancies studied in this study (Cr––Mo, Cr––Cr, Mo––Mo) are displayed in Fig. 3.

Fig. 3 Different oxygen vacancy types in (a) B-site-ordered and (b) B-site Cr/Mo ADs SCM.

First, we relaxed the atomic positions of the oxygen-deficient SCM supercell based on the oxygen-stoichiometric lattice constant. As one oxygen vacancy leaves two electrons behind in the perovskite, accommodating the leftover electrons influences the electronic properties of the adjacent B-site cations. The Bader charge, q, was calculated to study the reaccommodation of the leftover electrons from the oxygen vacancy, which are listed in Table 2. It can be found that the variance of q_Cr is more significant than that of the q_Mo upon the oxygen vacancy formation. In order to visualize this electron rearrangement between Cr and Mo in SCM, the charge density difference was calculated based on the DFT+U ground state, which is depicted in Fig. 4. As can be seen that the leftover electrons from the oxygen vacancy changed the adjacent Cr 3d electron density as a consequence of the strong ionic bond between Cr and O. In addition, the charge density on the other neighboring oxygen ions also exhibited an increase suggesting the broad influence of the oxygen vacancy through the mixed TM–O bonds. As the Mo 4d state electrons are able to travel over the whole crystal through the hybridized orbital,²⁷ the average charge density of the SCM supercell was increased with the oxygen vacancy formation, which corresponds to the oxygen-deficient SCM high energy. As a result, the oxygen vacancy E_form(Cr––Mo) in the SCM was calculated to be 4.13 eV by the spin-polarized DFT+U approach.

Table 2 Calculated Bader charge q and oxygen vacancy E_form in the SCM^a

	B-site-ordered SCM		ADs SCM
	Without	Cr– –Mo	Without	Cr– –Mo	Cr– –Cr	Mo– –Mo
a The Bader charge values refer to the Cr and Mo cations that are bound to the oxygen vacancy.
qCr	1.78	1.20	1.78	1.19	1.21	1.66
qMo	2.08	2.03	2.08	1.93	1.93	1.93
Eform (eV)	—	4.13	—	4.21	3.37	4.54

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Fig. 4 Top view (a) and contour plot (b) on the (001) plane of the charge density difference (Δρ = ρ_perfect − ρ_defecient − ρ_O) for Cr––Mo in the B-site-ordered SCM.

To investigate the oxygen vacancy formation in the ADs SCM, we first relaxed the oxygen-stoichiometric supercell with 25% Cr/Mo ADs by the DFT+U approach. The ground state energy of the ADs SCM was found to be 0.104 eV per formula unit (f.u.) higher than that of the original SCM. The ADs formation energy is smaller than that in the SFM supercell (0.21 and 0.16 eV per f.u. for ferromagnetic and AFM arrangements),²⁸ which is as well responding to the experimental ADs concentration of 35% in the SCM.⁴⁵ As the oxidization states of Cr and Mo cations in the ADs SCM are identical with those in the B-site-ordered SCM, the calculated Bader charges for Cr and Mo are not surprisingly nearly equal to that in the ADs SCM. For the oxygen vacancy between the Cr and Mo cations, the calculated oxygen vacancy properties were similar to that in the B-site-ordered SCM. The leftover electrons from the oxygen vacancy were deallocated to the neighboring Cr/Mo ADs. Moreover, the calculated Bader charge revealed that the left electrons were transferred to the adjacent Cr cations as the result of the lower energy of the 3d orbital of the Cr cation. The E_form(Cr––Mo) in the ADs SCM was calculated to be 4.21 eV, which is close to the 4.13 eV value in the ordered SCM. For the case of the oxygen vacancy type Cr––Cr, the electron distribution between two neighboring Cr cations was distinguished between Cr and Mo. Due to the antisite Cr³⁺ cation, the Cr–O–Cr bond strength was relatively weak compared with other bonds containing the Mo⁵⁺ cation (e.g., Cr–O–Mo or Mo–O–Mo). Therefore, the calculated E_form(Cr––Cr) was the lowest one among all these vacancy types, as presented in Table 2. The vacancy formation between two Mo cations is most difficult considering the calculated 4.54 eV formation energy. As the Mo⁵⁺ cation has one intrinsic t_2g electron, the leftover electrons from the oxygen vacancy generated the additional Coulomb repulsion with the intrinsic 4d electron. This repulsion increases the total energy of the oxygen-deficient crystal leading to a high formation oxygen vacancy energy. However, we noted that the Bader charge of the neighboring Mo cations did not change significantly during the oxygen formation process resulting from the delocalization feature of Mo 4d electrons.

The oxygen vacancy property for the SCM is analogous to that of the SFM. The oxygen vacancy E_form in SFM is reported to be 3.94 eV, which is slightly smaller than that in the SCM.²⁸ Due to the different electronic features of Fe³⁺ (t³_2ge²_g) and Cr³⁺ (t³_2ge⁰_g) cations, the calculated Bader charge for Fe³⁺ cation was 1.65, which is lower than that of Cr³⁺ cation. This lower charge value indicates a higher Fe–O bond covalency, which is consistent with the lower formation energy. However, as the same oxidization state was present for Fe and Cr in these double perovskites, the formation energies of different vacancy types were predicted to follow the trend of Cr––Cr < Cr––Mo < Mo––Mo, which is the same as that of Fe––Fe < Fe––Mo < Mo––Mo in the ADs SFM.²⁸

3.3 Oxygen vacancy migration in SCM

The migration energy barrier is another essential factor for the oxygen ion diffusion rate. Due to the different electronic properties of B-site Cr and Mo cations (as shown in Fig. 2), the migration barrier in the SCM mainly depends on oxygen migration around the Cr sublattice. The MEP of the oxygen ion migration was optimized by the CINEB method and the energy profiles for oxygen migration around the Cr or Mo sublattice are displayed in Fig. 5. The E_barrier in the SCM was predicted to be 1.86 eV (Fig. 5a) around the Cr sublattice, which is a rather large value compared with other MIEC perovskites (e.g., LaFeO₃, LaCoO₃, and BSCF).^8,46,47 Although the E_barrier around the Mo sublattice had a small 0.39 eV value (Fig. 5b), it is unfeasible for the long-range oxygen ion transport in the B-site-ordered rock-salt SCM, which is merely around the Mo sublattice. The oxygen migration process in the SCM involves the Mo–O bond breaking at the initial state and forming at the final state. In order to explore this bonding alteration, we calculated the magnetic moments of the adjacent Cr and Mo cations, depicted in Fig. 5c and d, which could be used to reveal the charge-transfer process during the oxygen ion migration. The calculated μ_Cr is 2.55 μ_B for the initial and final states and achieves the maximum 2.65 μ_B value at the intermediate state of the migration pathway, which is distinguished from B-site single perovskites such as LaFeO₃.⁴⁶ For the LaFeO₃, the magnetic moment of the axial Fe cation remained nearly constant throughout the whole migration process. The increased magnetic moment of the axial Cr cation indicated that additional electrons have been transferred to the Cr e_g state through the hybridized orbital to form the provisional Cr and O bonds in the transition states. On the other hand, the magnetic moments of the adjacent Mo cations varied accordingly during this migration process. Three Mo cations are coordinated to the axial Cr, which is surrounded by the migrating oxygen, as shown in Fig. 5e. The magnetic moments of the Mo cations adjacent to the initial and final vacant sites exhibited an abrupt switch, respectively, which corresponded to the Mo–O bond breaking and formation during the migration. The change of the magnetic moment for the c-direction Mo suggests that the c-direction Mo cation also plays an important role in the electron transfer in this transition state. Due to the localization feature of Cr 3d electrons, the intrinsic t_2g electrons are incompetent to solely bond with the jumping oxygen. The variation of the electron distribution of Cr and its neighboring Mo cations affected the total energy during the migration process, which led to the high migration barrier for the B-site double perovskite SCM.

Fig. 5 The oxygen ion migration in the B-site-ordered SCM. Energy profiles of the oxygen migration around the Cr sublattice (a) and Mo sublattice (b). Magnetic moments for the adjacent Cr (c) and Mo (d) during the oxygen migration around the Cr sublattice. B-site Cr and Mo cations distribution (e) relating to the oxygen migration around the Cr sublattice.

Considering the high ADs concentration, the oxygen ion migration pathway is also optimized in the ADs SCM and the calculated energy profile is presented in Fig. 6. The long-range oxygen transport in the ADs SCM is merely around the Mo sublattice as the consequence of the Cr/Mo ADs. The E_barrier in the ADs SCM was calculated to be 1.64 eV along the migration direction from the oxygen site of Cr–O–Mo to that of Mo–O–Mo, whereas the E_barrier along the opposite direction was found to be 1.23 eV. This asymmetry of E_barrier can be related to the different E_form of these two vacancy types. Although the energy barrier is still high in the ADs SCM, it can be noted that the high concentration of B-site Cr/Mo ADs in the SCM is constructive to enhance the oxygen ionic conductivity by providing the oxygen vacancy site with smaller formation energy as well as the migration pathway with a reduced energy barrier.

Fig. 6 Energy profile of oxygen migration in the SCM with 25% Cr/Mo ADs.

4. Conclusions

The double perovskite SCM was investigated using a DFT approach plus the on-site Coulomb potential U for the Cr 3d electrons, with the focus on the oxygen ion transport property in SCM. The calculated electronic structure revealed the half-metallic features of SCM. The calculated energy of the SCM supercell with 25% Cr/Mo ADs demonstrated the high concentration of ADs in these double perovskites. The oxygen vacancy formation and migration were investigated for both the B-site-ordered SCM and B-site Cr/Mo ADs SCM. The leftover electrons from oxygen vacancy changed the electronic distribution in the SCM crystal leading to the high vacancy formation energy. The existence of Cr/Mo ADs generates different B–O bond types in the SCM. The calculated formation energies for the different oxygen vacancies in the ADs SCM were predicted to follow the trend of Cr––Cr < Cr––Mo < Mo––Mo. The migration barrier for the oxygen ions was estimated to be rather large compared with other MIEC perovskites used as the SOFCs cathode. However, the high ADs concentration in the double perovskite SCM is favorable to enhance the oxygen ionic conductivity. These calculations provided theoretical insight into the effect of the B-site Cr and Mo cations on the oxygen vacancy formation and migration in the SCM, which can be useful for the study of the Cr-doped SrMoO₃ in the application of IT-SOFCs anode materials.

Acknowledgements

This study was supported by the KAKENHI Scientific Research program, Japan (C, Grant Number 15K05597).

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