Ab initio energetics of $\text{La}B{\text{O}}_3\left(001\right)$ ($B=\text{Mn}$, Fe, Co, and Ni) for solid oxide fuel cell cathodes

$\text{La}B{\text{O}}_3$ ($B=\text{Mn}$, Fe, Co, and Ni) perovskites form a family of materials of significant interest for cathodes of solid oxide fuel cells (SOFCs). In this paper ab initio methods are used to study both bulk and surface properties of relevance for SOFCs, including vacancy formation and oxygen binding energies. A thermodynamic approach and the density functional theory plus $U$ method are combined to obtain energies relevant for SOFC conditions ($T\approx 800°\text{C}$, $P{\text{O}}_2\approx 0.2\text{atm}$). The impact of varying $U_{\text{eff}}$ $\left(U_{\text{eff}}=U-J\right)$ on energy and electronic structure is explored in detail and it is shown that optimal $U_{\text{eff}}$ values yield significantly better agreement with experimental energies than $U_{\text{eff}}=0$ (which corresponds to the standard generalized gradient approximation). $\text{La}B{\text{O}}_3$ oxygen vacancy formation energies are predicted to be in the order $\text{Fe}>\text{Mn}>\text{Co}>\text{Ni}$ (where the largest implies most difficult to form a vacancy). It is shown that (001) $B{\text{O}}_2$ terminated surfaces have 1–2 eV lower vacancy formation energies and therefore far higher vacancy concentrations than the bulk. The stable surface species at low temperature are predicted to be the superoxide $\text{O}_2{}^{-}$ for $B=\text{Mn}$, Fe, Co and a peroxide $\text{O}_2{}^{2-}$ with a surface oxygen for $B=\text{Ni}$. Entropy effects are predicted to stabilize the monomer oxygen surface state for all $B$ cations at higher temperatures. Overall oxygen coverage of the (001) $B{\text{O}}_2$ surface is predicted to be quite low at SOFC operating conditions. These results will aid in understanding the oxygen reduction reaction on perovskite SOFC cathodes.

Figure 1

(Color online) An internally relaxed $\text{La}B{\text{O}}_3$ ($B=\text{Mn}$, Fe, Co, and Ni) $2\times 2\times 2$ bulk structure with the lattice constant fixed to that of a fully relaxed ideal cubic perovskite. This structure is used to approximate the $\text{La}B{\text{O}}_3$ perovskites under SOFC condition.

Figure 2

(Color online) Formation energies (per ${\text{O}}_2$) of nontransition metal oxides (CaO, MgO, ${\text{Li}}_2\text{O}$, ${\text{Al}}_2{\text{O}}_3$, ${\text{SiO}}_2$, and ${\text{Na}}_2\text{O}$) with PBE (o) (cutoff energy of 400 eV) and PW91 $\left({\text{O}}_{\text{s}}\right)$ (cutoff energy of 250 eV) PAW potentials as a function of the experimental formation enthalpies extracted from Ref. 25. Each solid represents the best fit to each set of data. A $1.36\text{eV}/{\text{O}}_2$ (consistent with Ref. 25) and a $0.33\text{eV}/{\text{O}}_2$ energy correction are obtained for PBE (o) and PW91 $\left({\text{O}}_{\text{s}}\right)$, respectively. The $0.33\text{eV}/{\text{O}}_2$ correction to the ${\text{O}}_2$ energy is applied in this work.

Figure 3

(Color online) Calculated $\text{La}B{\text{O}}_3$ ($B=\text{Mn}$, Fe, Co, and Ni) formation energies of $\frac{1}{2}{\text{La}}_2{\text{O}}_3+B{\text{O}}_x+\frac{1}{2}\left(\frac{3}{2}-x\right){\text{O}}_2\Rightarrow \text{La}B{\text{O}}_3$ vs experimental formation reaction enthalpies (at 298 K) taken from Refs. 42, 43 at (a) $U_{\text{eff}}=0\text{eV}$ and (b) optimal $U_{\text{eff}}$. The root mean square root error of the $U_{\text{eff}}=0\text{eV}$ data is 0.672 eV while that of those at optimal $U_{\text{eff}}$ is 0.211 eV.

Figure 4

(Color online) Convergence tests for (a) ${\text{LaMnO}}_3$ surface and (b) surface oxygen vacancy formation energies with respect to the number of layers in asymmetric slabs. In terms of the tradeoff between the energetic convergences and the amount of computation time, we adopt eight-layer asymmetric slabs to perform surface reaction energetic calculations.

Figure 5

(Color online) (a) A schematic illustration of the (001) eight-layer slab for $\text{La}B{\text{O}}_3$ surface reaction simulations. In order to avoid the energy fluctuations coming from the relaxation of the other terminations, we fix the first two layers of the bottom termination (not involved in the surface reactions) to the bulk coordinates and allow atomic relaxation for the rest of layers. Surface reactions (oxygen vacancies and oxygen adsorption) are simulated only on the unfixed termination. (b) Top view of (001) $B{\text{O}}_2$ surface and locations of oxygen adsorption sites: $B$ site and bridge site.

Figure 6

(Color online) The calculated ${\mu }_{\text{O}}^{\text{eff}}-\frac{1}{2}\left(E_{{\text{O}}_2}^{\text{VASP}}+\Delta h_{{\text{O}}_2}^0\right)$ vs $T$ at $P\left({\text{O}}_2\right)=0.2\text{atm}$ for oxygen monomers and oxygen dimers based on Eqs. (3, 4a).

Figure 7

(Color online) (a) ${\text{LaMnO}}_3$ (AAFM), (b) ${\text{LaMnO}}_3$ (FM), (c) ${\text{LaFeO}}_3$ (GAFM), (d) ${\text{LaFeO}}_3$ (FM), (e) ${\text{LaCoO}}_3$ (FM), and (f) ${\text{LaNiO}}_3$ (FM) bulk DOS plots. From top to bottom: projected $p$ PDOS, projected $t_{2g}$ PDOS, projected $e_g$ PDOS, and TDOS. DOS is in units of states per eV per $2\times 2\times 2$ unit cell.

Figure 8

(Color online) (a) ${\text{LaMnO}}_3$, (b) ${\text{LaFeO}}_3$, (c) ${\text{LaCoO}}_3$, (d) ${\text{LaNiO}}_3$ bulk, and (001) $B{\text{O}}_2$ surface oxygen vacancy formation energies $\left(E_{\text{vac}}\right)$ vs $U_{\text{eff}}$. All the $E_{\text{vac}}$ values shown in the figures have been corrected with the ${\text{O}}_2$ correction.

Figure 9

(Color online) (a) ${\text{LaMnO}}_3$ (AAFM), (b) ${\text{LaMnO}}_3$ (FM), (c) ${\text{LaFeO}}_3$ (GAFM), (d) ${\text{LaFeO}}_3$ (FM), (e) ${\text{LaCoO}}_3$ (FM), and (f) ${\text{LaNiO}}_3$ (FM) perfect vs with a vacancy bulk TDOS and surface PDOS at $U_{\text{eff}}=0\text{eV}$ and at optimal $U_{\text{eff}}$. The surface PDOSs are taken from the top two relaxed surface layers of the slabs and are normalized to give the same integral as the TDOSs.

Figure 10

(Color online) Three types of (001) $B{\text{O}}_2$ surface oxygen adsorption energies (per O) vs $U_{\text{eff}}$: O-$B$, O-bridge, and ${\text{O}}_2\text{−}B$ for (a) ${\text{LaMnO}}_3$, (b) ${\text{LaFeO}}_3$, (c) ${\text{LaCoO}}_3$, and (d) ${\text{LaNiO}}_3$. All the $E_{\text{ad}}$ values shown in the figures are corrected with the ${\text{O}}_2$ correction $\left(0.33\text{eV}/{\text{O}}_2\right)$.

Figure 11

(Color online) $\text{La}B{\text{O}}_3$ bulk and $B{\text{O}}_2$ surface vacancy formation energies vs transition metal types ($B=\text{Mn}$, Fe, Co, and Ni) (a) at $U_{\text{eff}}=0\text{eV}$ and (b) at optimal $U_{\text{eff}}$’s. The experimental oxygen vacancy formation enthalpy data (with error bars) are taken from Refs. 71, 72 for ${\text{LaMnO}}_3$, Ref. 73 for ${\text{LaFeO}}_3$, and Ref. 74 for ${\text{LaCoO}}_3$. The fitted oxygen vacancy formation reaction is $2B^x+{\text{O}}^x\leftrightarrow 2B^{\prime }+V_{\text{O}}^{••}+\frac{1}{2}{\text{O}}_2$ (Kröger-Vink notation).

Figure 12

(Color online) Surface $\Delta G_{\text{vac}}^{\ast }$ for ${\text{LaMnO}}_3$ (black), ${\text{LaFeO}}_3$ (red), ${\text{LaCoO}}_3$ (blue), and ${\text{LaNiO}}_3$ (green) at (a) $U_{\text{eff}}=0\text{eV}$ and (b) at optimal $U_{\text{eff}}$ as a function of temperature at $P\left({\text{O}}_2\right)=0.2\text{atm}$. Shaded areas specify the possible $B{\text{O}}_2$ surface oxygen vacancy formation energy range between the ferromagnetic state (given by the line) and the ground state antiferromagnetic state as the other boundary.

Figure 13

(Color online) $\text{La}B{\text{O}}_3$ $B{\text{O}}_2$ surface oxygen adsorption energies vs transition metal types ($B=\text{Mn}$, Fe, Co, and Ni) (a) at $U_{\text{eff}}=0\text{eV}$ and (b) at optimal $U_{\text{eff}}$.

Figure 14

(Color online) $\Delta G_{\text{ad}}^{\ast }$ (per $B$ site) vs temperature ($\text{O-}B$: solid black line, ${\text{O}}_2\text{−}B$: dotted blue line, and O-bridge: dashed orange line at $P\left({\text{O}}_2\right)=0.2\text{atm}$ at the optimal $U_{\text{eff}}$ for (a) ${\text{LaFeO}}_3$ (FM), (b) ${\text{LaFeO}}_3$ (GAFM), and (b) ${\text{LaCoO}}_3$ (FM) $B{\text{O}}_2$ surfaces.

Figure 15

(Color online) Surface $B$-site coverage based on the ideal Langmuir adsorption model with the calculated $\Delta G_{\text{O}\_\text{ad}}^{\ast }$ $\left[P\left({\text{O}}_2\right)=0.2\text{atm}\right]$ at the optimal $U_{\text{eff}}$ for (a) ${\text{LaFeO}}_3$ (FM), (b)${\text{LaFeO}}_3$ (GAFM), and (c) ${\text{LaCoO}}_3$. The results suggest that $B{\text{O}}_2$ surface $B$ sites are likely to be covered by adsorbed ${\text{O}}_2$ dimers at low temperatures. The adsorbed ${\text{O}}_2$ dimmer would be desorbed upon heating and surface $B$-site coverage becomes small (almost bare surface).