**2. Computational details**

The employed thermodynamic analysis relies on the energies obtained by DFT computations of the electronic structure of slabs terminated by given surfaces using the above-mentioned two types of basis sets. All calculations are performed with spin-polarized electronic densities, complete neglect of spin polarization results in considerable errors in material properties (Kotomin et al, 2008)).

The plane wave calculations were performed with VASP 4.6.19 code (Kresse & Hafner, 1993; Kresse & Furthmüller, 1996; Kresse et al., 2011), which implements projector augmented wave (PAW) technique (Bloechl, 1994; Kresse & Joubert, 1999), and generalized gradient approximation (GGA) exchange-correlation functional proposed by Perdew and Wang (PW91) (Perdew et al., 1992) . Calculations were done with the cut-off energy of 400 eV. The core orbitals on all atoms were described by PAW pseudopotentials, while electronic

compared to [011] and others). However, the [001] surfaces have, in turn, two different terminations: LaO or MnO2. We will compare stabilities of these terminations under different environmental conditions (temperature and partial pressure of oxygen gas). Another important question to be addressed is, how Sr doping affects relative stabilities of the LMO surfaces? These issues directly influence the SOFC cathode performance. Answering these questions requires a thermodynamic analysis of surfaces under realistic SOFC operational conditions which is in the main focus of this Chapter. Such a thermodynamic analysis is becoming quite common in investigating structure and stability of various crystal surfaces (Examples of thermodynamic analyses of binary and ternary compounds can be found in Reuter & Scheffler, 2001a, 2001b; Bottin et al., 2003; Heifets et

The thermodynamic analysis requires careful calculations of energies for two-dimensional slabs terminated by surfaces with various orientations and terminations. The required energies could be calculated using *ab initio* methods of the atomic and electronic structure based on density functional theory (DFT). In this Chapter, we present the results obtained using two complementary *ab initio* DFT approaches employing two different types of basis sets (BS) representing the electronic density distribution: plane waves (PW) and linear combination of atomic orbitals (LCAO). Both techniques were used to calculate the atomic and electronic structures of a pure LMO whereas investigation of the Sr influence on the

After studying the stabilities of various surfaces, the next step is investigating the relevant electrochemical processes on the most stable surfaces. For this purpose, we have to evaluate the adsorption energies for O2 molecules, O atoms, the formation energies of O vacancies in the bulk and at the stable perovskite surfaces. These energies, together with calculated diffusion barriers of these species and reactions between them, allow us to determine the mechanism of incorporation of O atoms into the cathode materials. However, such mechanistic and kinetic analyses lie beyond the scope of this Chapter (for more details see e.g. Mastrikov et al., 2010). Therefore, we limit ourselves here only to the thermodynamic characterization of the initial stages of the oxygen incorporation reaction, which include formation of stable adsorbed species (adsorbed O atoms, O2 molecules) and formation of oxygen vacancies. The data for formation of both oxygen vacancies and adsorbed oxygen

The employed thermodynamic analysis relies on the energies obtained by DFT computations of the electronic structure of slabs terminated by given surfaces using the above-mentioned two types of basis sets. All calculations are performed with spin-polarized electronic densities, complete neglect of spin polarization results in considerable errors in

The plane wave calculations were performed with VASP 4.6.19 code (Kresse & Hafner, 1993; Kresse & Furthmüller, 1996; Kresse et al., 2011), which implements projector augmented wave (PAW) technique (Bloechl, 1994; Kresse & Joubert, 1999), and generalized gradient approximation (GGA) exchange-correlation functional proposed by Perdew and Wang (PW91) (Perdew et al., 1992) . Calculations were done with the cut-off energy of 400 eV. The core orbitals on all atoms were described by PAW pseudopotentials, while electronic

stability of different (001) surfaces was performed within LCAO approach.

atoms and molecules have been collected using plane wave based DFT.

al., 2007a, 2007b, Johnson et al., 2004).

**2. Computational details** 

material properties (Kotomin et al, 2008)).

wavefunctions of valence electrons on O atoms and valence and core-valence electrons on metal atoms were explicitly evaluated in our calculations.

We found that seven- and eight-plane *slabs* infinite in two (x-y) directions are thick enough to show convergence of the main properties. The periodically repeated slabs were separated along the z-axis by a large vacuum gap of 15.8 Å. All atomic coordinates in slabs were allowed to relax. To avoid problems with a slab dipole moment and to ensure having identical surfaces on both sides of slabs, we employed the symmetrical seven-layer slab MnO2(LaO-MnO2)3 in our plane-wave simulations, even though it has a Mn excess relative to La and a higher oxygen content. Such a choice of the slab structure however only slightly changes the calculated energies. For example, the energy for dissociative oxygen adsorption on the [001] MnO2-terminated surface

$$2O\_2 + 2Mn\_{Mn}^{x} \rightleftharpoons 2O\_{ad}^{\cdot} + 2Mn\_{Mn}^{\bullet} \tag{1}$$

is -2.7 eV for eight-layers (LaO-MnO2)4 slab and -2.2 eV for the symmetrical seven-layer MnO2-(LaO-MnO2)3 slab. The use of symmetrical slabs also allows decoupling the effects of different surface terminations and saving computational time due to the possibility to exploit higher symmetry of the slabs. The simulations were done using an extended 2√2 × 2√2 surface unit cell and a 2 × 2 Monkhorst-Pack k-point mesh in the Brillouin zone (Monkhorst & Pack, 1976). Such a unit cell corresponds to 12.5% concentration (coverage) of the surface defects in calculations of vacancies and adsorbed atoms and molecules.

The choice of the magnetic configuration only weakly affects the calculated surface relaxation and surface energies (Evarestov, et. al., 2005; Kotomin et al, 2008; Mastrikov et al., 2009). Relevant magnetic effects are sufficiently small (≈0.1eV) as do not affect noticeably relative stabilities of different surfaces; these values are much smaller than considered adsorption energies and vacancy formation energies. As for slabs the ferromagnetic (FM) configuration has the lowest energy, we performed all further plane-wave calculations with FM ordering of atomic spins.

The quality of plane-wave calculations can be illustrated by the results for the bulk properties (Evarestov, et. al., 2005; Mastrikov et al., 2009). In particular, for the lowtemperature orthorhombic structure the A-type antiferromagnetic (A-AFM) configuration (in which spins point in the same direction within each [001] plane, but opposite in the neighbor planes) is the energetically most favorable one, in agreement with experiment. The lattice constant of both the cubic and orthorhombic phase exceeds the experimental value only by 0.5%. The calculated cohesive energy of 30.7 eV is also close to the experimental value (31 eV).

In our *ab initio* LCAO calculations we use DFT-HF (*i.e.*, density functional theory and Hartree-Fock) hybrid exchange-correlation functional which gave very good results for the electronic structure in our previous studies of both LMO and LSM (Evarestov et al., 2005; Piskunov et al., 2007). We employ here the hybrid B3LYP exchange-correlation functional (Becke, 1993). The simulations were carried out with the CRYSTAL06 computer code (Dovesi, et. al., 2007), employing BS of the atom-centered Gaussian-type functions. For Mn and O, all electrons are explicitly included into calculations. The inner core electrons of Sr and La are described by small-core Hay-Wadt effective pseudopotentials (Hay & Wadt, 1984) and by the nonrelativistic pseudopotential (Dolg et al., 1989), respectively. BSs for Sr and O in the form of 311d1G and 8–411d1G, respectively, were optimized by Piskunov et al., 2004. BS for Mn was taken from (Towler et al., 1994) in the form of 86–411d41G, BS for La is

Thermodynamics of ABO3-Type Perovskite Surfaces 495

perfect bulk and surfaces and extended by Wagner & Schottky, 1930 (also Wagner, 1936)

equal to the sum of the chemical potentials of each atomic component in the LMO crystal:

<sup>3</sup> 3

3 3 *bulk LaMnO LaMnO*

 *LaMnO La Mn O* 

Owing to the requirement for the surface of each slab to be in equilibrium with the bulk LMO, the chemical potential is equal to the specific bulk crystal Gibbs free energy accordingly to

Eq. (4) imposes restrictions on *μLa*, *μMn*, and *μO*, leaving only two of them as independent variables. We use in following *μO* as one of the independent variables because we consider oxygen exchange between the LaMnO3 crystal and gas phase and have to account for strong dependence of this chemical potential on *T* and *pO2*. As another independent variable, we use *μMn*. We will simplify the equation for the SGFE and eliminate the chemical potentials

*LaMnO3* by substituting this expression for the LMO bulk chemical potential:

to the number of ions in A type sites (for ABO3 perovskites) of the slabs (Gibbs,1948;

*i i i a A a a A bulk*

Here A type of sites are occupied solely by La atoms in LMO, so NA=NLa for LMO. This will become somewhat more complicated in solid solutions such as LSM (see the next subsection). *bulk NA* is the number of A-sites in unit cell in the bulk. *bulk Na* is the number of *a*

the crystal energy, *vj* volume, and *sj* entropy. All these values are given per formula unit in *j*-type (=La,Mn, LMO…) crystals. We can reasonably assume that the applied pressure is not higher than ~100 atm. in practical cases. The volume per lattice molecule in LaMnO3 is ~64 Å3. Then the largest *pvj* term in Eq.(13) can be estimated as ~ 5 meV. This value is much smaller than the amount of uncertainty in our DFT computations and, therefore, can be safely neglected. As it is commonly practiced, we will neglect the very small vibration contributions to *gj*, including contributions from zero-point oscillations to the vibrational part of the total energy. This rough estimate is usually valid, but can be broken if the studied material has soft modes. The same consideration is valid for slabs used in the present

*<sup>N</sup> N N*

<sup>1</sup> , <sup>2</sup> [ ] *<sup>i</sup> i i bulk i i G N slab <sup>A</sup>gLaMnO A Mn A O*

A,a are the Gibbs excesses in the *i*-terminated surface of components *a* with respect

*LaMnO3* of LMO (in the considered orthorhombic or cubic phase) is

*g* (5)

(4)

(6)

(7)

*vibr* is the vibrational contribution to

 

3 , ,

*bulk*

*A*

*N* 

 *Mn O*

*vibr j j <sup>j</sup> j j <sup>g</sup> E E Ts vp* (8)

for point defects. The chemical potential

*La* and 

where Γ*<sup>i</sup>*

Johnston et al., 2004) :

atoms in unit cell in the bulk.

, 1 2

The Gibbs free energies per unit cell for crystals and slabs are defined as

where *Ej* is the static component of the crystal energy, *Ej*

provided in the CRYSTAL code's homepage (Dovesi, et. al., 2007) in form 311-31d3f1, to which we added an *f*-type polarization Gaussian function with exponent optimized in LMO (α=0.475). The reciprocal space integration was performed by sampling the Brillouin zone with the 4×4 Monkhorst-Pack mesh (Monkhorst & Pack, 1976). In our LCAO calculations, nine-layer symmetrical slabs (terminated on both sides by either [001] MnO2 or La(Sr)O surfaces) were used. The calculations were carried out for cubic phases and for A-AFM magnetic ordering of spins on Mn atoms. All atoms have been allowed to relax to the minimum of the total energy. This approach was initially tested on bulk properties as well, the experimentally measured atomic, electronic, and low-temperature magnetic structure of pure LMO and LSM (*xb*=1/8) were very well reproduced (Piskunov et al., 2007).
