Thermodynamics of ABO 3 -Type Perovskite Surfaces

The ABO3-type perovskite manganites, cobaltates, and ferrates (A= La, Sr, Ca; B=Mn, Co, Fe) are important functional materials which have numerous high-tech applications due to their outstanding magnetic and electrical properties, such as colossal magnetoresistance, half-metallic behavior, and composition-dependent metal-insulator transition (Coey et al., 1999; Haghiri-Gosnet & Renard, 2003). Owing to high electronic and ionic conductivities. these materials show also excellent electrochemical performance, thermal and chemical stability, as well as compatibility with widely used electrolyte based on yttrium-stabilized zirconia (YSZ). Therefore they are among the most promising materials as cathodes in solid oxide fuel Cells (SOFCs) (Fleig et al., 2003) and gas-permeation membranes (Zhou, 2009). Many of the above-mentioned applications require understanding and control of surface properties. An important example is LaMnO3 (LMO). Pure LMO has a cubic structure above 750 K, whereas below this temperature the crystalline structure is orthorhombic, with four formula units in a primitive cell. Doping of LMO with Sr allows one to increase both the ionic and electronic conductivity as well as to stabilize the cubic structure down to room temperatures necessary conditions for improving catalytic performance of LMO in electrochemical devices, e.g. cathodes for SOFCs. In optimal compositions of


Introduction
The ABO 3 -type perovskite manganites, cobaltates, and ferrates (A= La, Sr, Ca; B=Mn, Co, Fe) are important functional materials which have numerous high-tech applications due to their outstanding magnetic and electrical properties, such as colossal magnetoresistance, half-metallic behavior, and composition-dependent metal-insulator transition (Coey et al., 1999;Haghiri-Gosnet & Renard, 2003). Owing to high electronic and ionic conductivities. these materials show also excellent electrochemical performance, thermal and chemical stability, as well as compatibility with widely used electrolyte based on yttrium-stabilized zirconia (YSZ). Therefore they are among the most promising materials as cathodes in solid oxide fuel Cells (SOFCs) (Fleig et al., 2003) and gas-permeation membranes (Zhou, 2009). Many of the above-mentioned applications require understanding and control of surface properties. An important example is LaMnO 3 (LMO). Pure LMO has a cubic structure above 750 K, whereas below this temperature the crystalline structure is orthorhombic, with four formula units in a primitive cell. Doping of LMO with Sr allows one to increase both the ionic and electronic conductivity as well as to stabilize the cubic structure down to room temperatures -necessary conditions for improving catalytic performance of LMO in electrochemical devices, e.g. cathodes for SOFCs. In optimal compositions of bb 3 1-x x 493 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 MnO 2 (LaO-MnO 2 ) 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.
is -2.7 eV for eight-layers (LaO-MnO 2 ) 4 slab and -2.2 eV for the symmetrical seven-layer MnO 2 -(LaO-MnO 2 ) 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 2Γ2 × 2Γ2 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 494 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] MnO 2 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 (x b =1/8) were very well reproduced .

Thermodynamic analysis of surface stability
As was mentioned above, understanding of many surface related phenomena requires preliminary investigation of the relative stabilities of various crystalline surfaces. Usually (especially for high-temperature processes such as catalysis in electrochemical devices), determining the structure with the lowest internal energy is not sufficient. The internal energy characterizes only systems with a constant chemical composition, while atomic diffusion and atomic exchange between environment and surfaces occur at high temperatures. Thus, we have to take into account the exchange of atoms between the bulk crystal, its surface, and the gas phase, into our analysis of surface stability. Such processes are included into the Gibbs free energies at the thermodynamic level of description. Therefore, we have to calculate the surface Gibbs free energy (SGFE) Ω i for the LMO and LSM surfaces of various orientations and terminations. The SGFE is a measure of the excess energy of a semi-infinite crystal in contact with matter reservoirs with respect to the bulk crystal (Bottin et al., 2003;Heifets et al., 2007aHeifets et al., , 2007bJohnston et al., 2004 ;Mastrikov et al., 2009;Padilla & Vanderbilt, 1997, 1998Pikunov et al., 2008;Pojani et al., 1999;Reuter & Scheffler, 2001b. The SGFEs are functions of chemical potentials of different atomic species. The most stable surface has a structure, orientation and composition with the lowest SGFE among all possible surfaces.

Method of analysis for LMO surfaces
Introducing the chemical potentials  La ,  Mn , and  O for the La, Mn, and O atomic species, respectively, the SGFE per unit cell area  i corresponding to the i termination is defined as where i slab G is the Gibbs free energy for the slab terminated by surface i, N i La , N i Mn , and N i O denote numbers of La, Mn, and O atoms in the slab. Here we assume that the slab is symmetrical and has the same orientation, composition, and structure on both sides. The SGFE per unit area is represented by The thermodynamic part of the description below follows the well known chemical thermodynamics formalism developed originally by Gibbs in 1875 (see Gibbs, 1948) for www.intechopen.com perfect bulk and surfaces and extended by Wagner & Schottky, 1930(also Wagner, 1936 for point defects. The chemical potential  LaMnO3 of LMO (in the considered orthorhombic or cubic phase) is equal to the sum of the chemical potentials of each atomic component in the LMO crystal: 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 μ La , μ Mn , and μ O , leaving only two of them as independent variables. We use in following μ O as one of the independent variables because we consider oxygen exchange between the LaMnO 3 crystal and gas phase and have to account for strong dependence of this chemical potential on T and pO 2 . As another independent variable, we use μ Mn . We will simplify the equation for the SGFE and eliminate the chemical potentials  La and  LaMnO3 by substituting this expression for the LMO bulk chemical potential: where i A,a are the Gibbs excesses in the i-terminated surface of components a with respect to the number of ions in A type sites (for ABO 3 perovskites) of the slabs (Gibbs,1948;Johnston et al., 2004) where E j is the static component of the crystal energy, E j vibr is the vibrational contribution to the crystal energy, v j volume, and s j 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 LaMnO 3 is ~64 Å 3 . Then the largest pv j 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 g j , 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 simulations. While it might be important to check vibrational contributions in some cases, here we will neglect it. Besides, facilities in computer codes for calculations of vibrational spectra of crystals and slabs appeared only within a few last years and such calculations are still very demanding and practically possible only for relatively small unit cells. Therefore, we approximate the Gibbs free energies with the total energies obtained from DFT calculations: what resembles the expression for the Gibbs free energy of surface formation. Here E slab stands for the total energy of a slab and replaces the Gibbs free energy of the slab. The equilibrium condition (5)   The range of values of the chemical potentials which consistent with existence and stability of the crystal (LMO here) itself is determined by the set of the following conditions. To prevent La and Mn metals from leaving LMO and forming precipitates, their chemical potentials must be lower in LMO than in corresponding bulk metals. These conditions mean: Note, however, that sometimes compositions are fixed by bringing the multinary crystals into coexistence with less complex sub-phases. If the SGFE becomes negative, surface formation becomes energetically favorable and the crystal will be destroyed. Therefore, the condition for sustaining a crystal structure is for SGFE to be positive for all potential surface terminations. Therefore, one more set of conditions on the chemical potentials of the crystal components can be written as where i corresponds to the surface with the lowest SGFE.

Method of analysis for LSM surfaces
In LSM we have to re-define the SGFEs, because there are now four components in this material (instead of three in LMO) with Sr atoms substituting a fraction of La atoms in the perovskite A sub-lattice. The SGFE definition for LSM can be written as Let us denote concentration of Sr atoms in the bulk of LSM as where bulk Sr N is the average number of Sr atoms per crystal unit cell in the bulk. Then for LSM becomes the average number of La atoms per LSM unit cell in the bulk. The chemical potential of a LSM formula unit is Equilibrium between LSM surface and its bulk means that We will continue using approximation (9) in the following, replacing the Gibbs free energies of bulk and slab unit cells by their total energies. The conditions (32, 33) impose restrictions on four chemical potentials of all LSM components and reduces the number of independent components to three. We have chosen to keep the chemical potentials of O, Mn and La as independent variables. Then the chemical potential of the Sr atom can be expressed as and its deviation (analogous to eqs. (10-12) and keeping in mind approximation (9)) as www.intechopen.com do not change with respect to LMO. We still have to remember only that N A does not coincide any more with N La in LSM . N A refers only to the number of A-sites in the perovskite unit cell, but not to the number of La atoms. Since we excluded chemical potential for Sr, only the excesses for La atoms will be required. For the calculation of excesses of La atoms we have to account for mixing of La and Sr atoms in A-site of the perovskite lattice. Using eqs.(7,31), the excess of La atoms for surface i can be expressed as: Then SGFEs for LSM reads ,, , The conditions of LSM crystal stability include the same bounds which work for LMO. However, we have to add conditions preventing precipitations of several new materials and express all conditions through three chemical potentials for La, Mn and O atoms. Precipitation of Mn, La, and Sr metals will be avoided if and (1 ) where Gibbs free energy of LSM formation is where www.intechopen.com Similarly, precipitation of LaMnO 3 and SrMnO 3 perovskites will be prevented, if Here the Gibbs free energy of SrMnO 3 formation is defined as Lastly, spontaneous formation of surfaces does not occur, if condition (29) is satisfied as well.

Determination of the chemical potential of oxygen atom
As mentioned above, an exchange of O atoms between surfaces and environment occurs at all surfaces, especially at high temperatures. Moreover, such an exchange is a key factor in many electrochemical and catalytic processes. Therefore, oxygen in the studied crystal (for instance, LMO or LSM, in this Chapter) has to be considered in equilibrium with oxygen gas in atmosphere beyond the crystal surface. The equilibrium in exchange with O atoms means equality of oxygen chemical potentials in a crystal and in the atmosphere: Chemical potentials are hardly available experimentally. It is much more convenient to operate with gas temperatures and pressures determining the oxygen chemical potential. At the same time, the Gibbs free energies of crystals are insensitive to temperature and the pressure (within approximations accepted in our present description). Therefore, we can use the dependence of oxygen gas chemical potential 2 gas O  to express the Gibbs free energies for surfaces through temperature and oxygen gas partial pressure. Oxygen gas under the considered conditions can be treated (to a very good approximation) as an ideal gas. Therefore, dependence of its chemical potential from pressure can be expressed by the standard expression (as done by Johnston et al., 2004 andScheffler, 2001b)  data from the standard thermodynamic tables (Chase, 1998;Linstrom & Mallard, 2003), following Johnston et al., 2004 andScheffler, 2001b. These data are collected in Table 1. For this we define an isolated oxygen molecule E O2 as the reference state. Changes in the chemical potential for oxygen atom can be written as Here 0 2 (, ) is the change in the oxygen gas Gibbs free energy at the pressure p 0 and temperature T with respect to its Gibbs free energy at T 0 =298.15 K 00 0 0 222 (9). The formation heat for La and Mn oxides under standard conditions can also be found in thermodynamic tables (Chase, 1998;Linstrom & Mallard, 2003). Equation (52) allows us to estimate the standard oxygen gas enthalpy. Since we define the total energy of an oxygen molecule as a zero for chemical potential and enthalpy calculations, the correction for the enthalpy could to be defined as Using the experimental standard entropy for oxygen (Chase, 1998;Linstrom & Mallard, 2003) as 0 2 O S  205.147 J·mol -1 K -1 , the correction to the oxygen chemical potential can be calculated as

Thermodynamic consideration of oxygen adsorption and vacancy formation
Let us consider formation of relevant oxygen species and point defects in the bulk and at the LaMnO 3 surface. We use the same approximation as in the previous sections: we neglect the changes of vibrational entropy in the solid, thus only states comprising gaseous O 2 exhibit the temperature-dependent Gibbs free energy contribution. In this approximation, differences between the Gibbs energies for bulk crystals or slabs (including defects and adsorbates) can be replaced with the differences in the total energy calculated from DFT, while variation of oxygen chemical potential for gaseous O 2 is taken from experimental data.
Here we accounted for the fact that we use a symmetrical slab with an oxygen vacancy at each side of the slab. The total energy of such a slab is written as

Stability of LMO surface terminations: Plane-wave DFT simulations
Based on the results of plane-wave calculations and theoretical considerations described in Section 3, the phase diagrams characterizing stability of different LMO surfaces have been drawn in Figure 1. These diagrams were built for both low-temperature orthorhombic and high-temperature cubic phases. For O-terminated [011] and LaO+O [001] surfaces it was not possible to keep the cubic structure during lattice relaxation. Therefore, we used  i values for the orthorhombic phase for both phase diagrams in Figure 1, as it was done, for instance, by Bottin et al., 2003. The calculated input data used for drawing this figure are collected in Tables 2 and 3 with different degrees of oxidation, one should treat the obtained data with some precaution. Thus, we highlighted by solid lines the boundaries where metal oxides La 2 O 3 and Mn 2 O 3 with metals in oxidation state 3+ (lines 2 and 5) begin to precipitate in the perovskite. In these oxides, metal oxidation numbers coincide with the oxidation states for the same metals in LaMnO 3 . Right hand side of the diagrams in Figure 1 contains a family of chemical potentials of O atoms (50) as functions of temperature and partial pressure of oxygen gas. This part of the figures allows us to translate easily-measurable external parameters (temperature and oxygen gas pressure) into oxygen chemical potential, which is one of the variables determining explicitly the SGFE. Using this part of the figures, we can relate points on the phase diagrams with the conditions under which experiments and/or industrial processes occur. To do this, one can just to draw a vertical line for a given temperature    www.intechopen.com Thermodynamics of ABO 3 -Type Perovskite Surfaces 507 and its crossings with lines corresponding to different gas pressures creates a pressure scale for this particular temperature. This can replace the axis for oxygen chemical potential. Alternatively, moving along the lines for chemical potential at a particular gas pressure, we can study the phase behavior with temperature. Figure 1 shows such a consideration for T=1200 K which is a typical condition for SOFC operations. We marked on these phase diagrams the most important range of oxygen gas partial pressures (between pO 2 =0.2 atm. and 1 atm). Oxygen-rich conditions with a larger O chemical potential correspond to higher oxygen gas partial pressures and/or lower temperatures; in turn, oxygen-pure conditions with the lower O chemical potentials correspond to smaller oxygen gas partial pressures and/or higher temperatures. Consistent positioning of these experimental curves with respect to our computed stability diagram requires also the correction described by Eq. (54). When drawing the right side of Figure 1, we used the correction of -0.77 eV (Table 3) calculated as average of a series of different oxides. It was calculated using the same set of oxides, which precipitation is considered in our plane-wave modeling. Both the values and the scattering (±0.37 eV) of calculated corrections are much larger than in similar studies (Heifets et al., 2007a(Heifets et al., , 2007bJohnston et al., 2004;Reuter & Schefer, 2001a) for non-magnetic oxides (e.g. SrTiO 3 ).
Here we consider manganese oxides which are spin-polarized solids. Besides, we included several Mn oxides with various oxidation states. This is a typical situation where DFT calculations face well known problems. The scattering of the correction magnitudes provides an estimate of uncertainty in positioning of the chemical potentials for O atoms on the right side of the phase diagrams. Figure 2 shows cross sections of the phase diagrams at T = 1200 K and pO 2 = 0.2 atm., i.e. in the range of typical SOFC operational conditions. Correspondingly, at the cross sections of the diagrams (Figure 2), the stability region lies between lines 2 and 6. This figure helps to clarify behavior of the SGFEs for surfaces with various terminations. As it can be seen from Figures 1 and 2

Stability of LMO surface terminations: LCAO simulations
Calculations performed within the LCAO approach combined with hybrid B3LYP functional were also employed in order to draw the phase diagram for stability of different LMO surface terminations (Figure 3). These calculations were carried out for a cubic phase and A-AFM magnetic ordering, where spins have the same orientations in the planes parallel to the surfaces of the slabs, but have opposite directions in neighbor planes. The comparison of stability shown in this figure includes only two primary candidates for the stable surfaces: LaO-and MnO 2 -terminated (001) surfaces. The stability range is limited by lines 2, 3, and 5, which correspond to precipitation of La 2 O 3 , MnO, and Mn 2 O 3 . These are substantially different oxides than suggested above in computations performed with planewave BS and PW91 functional. Indeed, the gap between precipitation of La 2 O 3 and Mn 2 O 3 shifted down significantly. Now the boundary between stability regions for LaO-and MnO 2 -terminated surfaces crosses the gap where LMO is stable, while PW91-GGA calculations described above and by Mastrikov et al., 2009Mastrikov et al., , 2010 suggested that only the MnO 2 -terminated surface was stable. In calculations with hybrid B3LYP functional the MnO 2 -terminated surface seems to be stable, up to SOFC operational temperatures (1200 K) under ambient oxygen gas partial pressures (pO 2 =0.2 atm.). Above this temperature LaOterminated surface gradually becomes more stable in the larger range in LMO crystal stability region until at ~1900 K it becomes the only stable surface. A precipitation of MnO or La 2 O 3 has to occur while LMO crystal is heated.
Positioning the family of O atom chemical potential curves on the right side of Figure 3 was done in the same way as for Figure 1, but using LCAO calculations with hybrid B3LYP functional. The averaged correction 0 O  (54) in this case is noticeably smaller ( -0.40 eV) than it was for PW91-GGA functional. However, deviations of this correction from its average value (±0.3 eV) is still large. This fact likely comes from the DFT difficulties, taking place even within hybrid functionals for spin-polarized systems. For diamagnetic systems, for instance SrTiO 3 , such deviation drops down, from ~0.25 eV in LDA calculations (Johnston et al., 2004) to ~0.03 eV in calculations (Heifets et al., 2007b) with the hybrid functional. LaO -1 -0.5 6.32 6.46 [001] MnO 2 1 0.5 -0.42 -0.43 Table 4. Parameters defining the surface Gibbs free energies Ω i (Eq. 13) and used to build diagram in Figure 3. The same as Table 2, but for the cubic phase of LMO only and produced with LCAO approach and hybrid B3LYP functional.

Stability of surface terminations for LSM: LCAO simulations
Since the SGFEs for LSM surfaces depend now on three variables, it is a little more complicated to draw corresponding phase diagrams. Therefore, we have drawn only several sections for the most interesting parts of the phase diagram for bulk concentration of Sr atoms x b = 1/8. Thus, Figure 4 shows the section of surface stability phase diagram under ambient oxygen gas partial pressure pO 2 =0.2 atm. and three various temperatures: a) 300 Kroom temperature (RT), b) 1100 K, which is approximately the SOFC operational temperature, and c) 1500 K, which is close to sintering temperatures. We compared several terminations of LSM (100)  Sr O  -terminated surface with respect to the MnO 2 -terminated surface. However, as soon as Sr concentration x s at the 1 xx ss La Sr O  -terminated surface becomes 0.5 or larger due to Sr segregation, such a surface becomes unstable. For better understanding changes in the surface stability with temperature, we have drawn two additional cross-sections along the precipitation lines for SrO and LaMnO 3 at pO 2 =0.2 atm. These cross-sections are presented in Figure 5. It can be seen here that upon heating the MnO 2 -terminated surface leaves the stability region and becomes replaced by the La 0.75 Sr 0.25 O-terminated surface. As heating continues, precipitation of La 2 O 3 or MnO begins. This is consistent with experimental observations by Kuo et al., 1989. A similar degradation process without Sr doping would require stronger overheating or very strongly reducing conditions. Detailed LCAO hybrid functional calculations of oxygen atom adsorption are necessary (see preliminary results in , in order to check PW91-GGA prediction (discussed in previous subsection) that the MnO 2 -terminated surface is stabilized by adsorbed oxygen atoms.

Oxygen adsorption and vacancy formation in LMO
As shown above, the MnO 2 -terminated (001) surface of LaMnO 3 appears to be the most stable one. Therefore, we optimized the atomic structure of surface oxygen vacancies, as well as O atoms and O 2 molecules adsorbed at different sites on this surface. For a comparison we also optimized the structure of oxygen vacancies in the LaMnO 3 bulk and at the LaO-terminated [001] surface. These simulations were performed using plane wave BS and PW91 functional. Details of the atomic position optimization are described by Mastrikov et al., 2010. In this Chapter, we limit our discussion only to the energies of different adsorbed species and vacancies and thermodynamic consideration of the relevant processes. Note that some adsorbed species have tilted geometry.  Table  5. For a classification of different molecular oxygen species we considered atomic charges and the O-O bond length. The data in Table 5 suggest that atomic adsorption of O atoms is energetically more preferable than adsorption of O 2 molecule. In both cases the best adsorption site for both O atom and O 2 molecule on MnO 2 -terminated surface is on top of surface Mn ion. Oxygen vacancies have smaller formation energy on MnO 2 -terminated surface than in the bulk suggesting vacancy segregation towards this surface. In contrary, much more energy is required to create an oxygen vacancy on LaO-terminated surface.  6. Spectrum of possible "one-particle" states, where "particles" are O atoms (right panel) and O 2 molecules (left panel). Each level in these panels corresponds to relative energies (∆E r ) of different molecular and atomic species occurring during oxygen incorporation reaction on the MnO 2 [001]-terminated surface of LaMnO 3 , cf. Table 5. The axes on the left and right give the energy ∆E r relative to resting O 2 molecule away from the surface (on the left) or an atom in such O 2 molecule (on the right). In the ground state of the crystal all lattice sites in crystal bulk (states X) and I surface (states VIII) are occupied (red levels) and the rest of the "one-particle" states vacant. The numbers at levels correspond to the numbers assigned to respective states in Table 5. The highest level on the right panel corresponds to a free (not in a molecule) O atom away from the crystal. The central panel shows the experimental T-and pO 2 -dependence of the Gibbs energy of gaseous O 2 (Table 1 and Eq.(50)), its energy scale refers to an O 2 molecule on the left and to an O atom in an O 2 molecule on the right. The labels m on the lines represent the pressure: pO 2 =10 m atm. The arrows indicate various Gibbs reaction energies due to moving of a "particle" between crystal and gas: red = formation of adsorbed superoxide 2 O  on defect-free surface; green = formation of adsorbed Oatop Mn on defect-free surface; black = incorporation of oxygen into a surface oxygen vacancy.  The collected energies allow us to draw the diagram shown in Figure 6. This diagram is based on a standard model of "non-interacting particles", where "particles" are O atoms and O 2 molecules in different positions. The energy levels drawn at the side panels represent single-particle energies corresponding to bringing a particle to a given position at the surface or in the bulk. The left hand panel refers to bringing a free gas-phase O 2 molecule to the crystal surface. Similarly, the right hand panel refers to taking an O atom from a free O 2 molecule and placing it on the crystal surface. These processes include also placing of an atom or a molecule into surface vacancies: this is a process inverse to the formation of a vacancy. Therefore, to place the corresponding energy level (at right hand panel), one has to use the vacancy formation energy with the opposite sign. A similar logic was applied in placing the energy level for bringing an oxygen atom into vacancy in the bulk. Such an O atom in a vacancy becomes actually a regular O atom in the crystal lattice (wherever, in the bulk or in the surface). Therefore, the energies of such states can be considered as those for an O atom in the bulk or on the surface. In the ground state of the crystal all lattice sites (states VIII, IX and X) are occupied and all other states vacant. The variation of the oxygen chemical potential is drawn in the central panel as a function of temperature for several gas partial pressures. These curves are drawn in the same way as similar lines on the right hand side in Figure 1, including the offset defined by Eq. (54). Because the energy scale at the left panel is twice as large as at the right panel, the same curves represent variations either in the chemical potentials for an O 2 molecule, if they are referred to the left panel, or for O atom, if they are referred to the right one. In such an arrangement, the diagram in Figure 6 can be used to represent the Gibbs energies for reactions of exchange with O atoms or O 2 molecules between oxygen gas and both crystal bulk and surfaces. For example, red arrow represents an adsorption of an O 2 molecule atop surface Mn ion in the tilted position (configuration I) from oxygen gas under partial pressure pO 2 =1 atm. and T=1000 K. The Gibbs free energy of corresponding reaction can be obtained by subtracting the energy of the initial state from that of the final state. For the reaction described by the red arrow this energy indeed corresponds to the adsorption energy for O 2 molecule. Similarly, the green arrow describes the adsorption of O atom atop Mn ion in MnO 2 -terminated surface. Lastly, the black arrow describes incorporation of an O atom into a surface oxygen vacancy. In the latter case, an arrow with opposite direction corresponds to the formation of a surface oxygen vacancy, as it can be confirmed by a comparison with Eqs. (60, 61). The diagram in Figure 6 is very suitable way of a graphical representation of the exchange between a gas and a crystal with various species and the analysis of corresponding processes. For a given temperature and oxygen partial pressure this diagram allows one to read the Gibbs reaction energy of a process and thus to obtain its mass action constant. As an example, let us discuss some processes under typical fuel cell conditions of T = 1000 K and pO 2 = 1 atm. The formation of molecular adsorbates (superoxide I = red arrow, and peroxide II) is endergonic by ∆ r G  +2 eV per O 2 since the entropy loss overcompensates the electronic energy gain. Even the formation of adsorbed atomic O -(species VI, green arrow) is still slightly endergonic, by ∆ r G  +0.5 eV per O, what leads the low adsorbate coverage under SOFC conditions. Only the oxygen incorporation into a surface vacancy (black arrow) is strongly exergonic, by ∆ r G  -1.7 eV per O (i.e. the inverse process, surface oxygen vacancy formation, is endergonic by +1.7 eV). Also, changes in temperature and/or partial pressure can change the sign of the reaction energy. To give an example: while oxygen atom adsorption is exothermic here, it changes from exergonic at low temperatures and/or high partial pressures to endergonic at higher temperatures and/or lower pressures.

Conclusions and perspectives
In this Chapter, combining ab initio calculations of the atomic and electronic structure with chemical thermodynamics, we have described how to predict surface stabilities with different orientations and terminations for ABO 3 perovskite materials and solid solutions. We considered also adsorption and formation of vacancies in the bulk and at the surfaces. The input data for such thermodynamic analyses are available from standard DFT calculations. We neglected vibrational contributions to the Gibbs free energies in present simulations, since there are good arguments that these contributions are quite small. However, the latest versions of the computer codes used here (VASP and CRYSTAL) are capable to perform the calculations of phonon spectra, which can be used to include vibration contributions into thermodynamic potentials. Despite currently such surface phonon calculations are computationally very expensive, with development of new algorithms and faster computers such analysis in near future could become a routine practice. We have applied the described techniques to experimentally well studied SOFC cathode materials (LMO and LSM surface. At elevated temperatures only the latter one can be found within an entire region of LSM stability which definitely should affect the LSM cathode performance due to our prediction that oxygen vacancies easily segregate from LMO bulk towards MnO 2terminated surface but not to the LaO-terminated one (Mastrikov et al, 2010). Thermodynamic consideration of oxygen adsorption and formation of surface vacancies allowed us to move beyond the usual analysis of purely DFT electronic energy differences in these processes at zero K and to describe changes in the Gibbs free energies under realistic environmental conditions (high temperature and partial pressure of oxygen gas). In particular, we have shown that an oxygen adsorption from the gas phase is exergonic at low temperatures, but becomes endergonic at SOFC operational temperatures. Consideration of the energy differences between O atoms and O 2 molecules in the gas phase and at surfaces makes it possible to determine theoretically preferred adsorption sites, adsorption energies and formation energies for vacancies (e.g. Kotomin et al., 2008, Mastrikov et al., 2010. Thus the obtained data can be used for analysis of the kinetics of chemical reactions and investigation of their mechanisms. For example, we performed such an analysis (Mastrikov et al., 2010) in the study of oxygen reduction and incorporation into the LMO surfaces. Note also that under realistic experimental conditions in a multinary crystal typically only one component is reversibly exchangeable so that only one chemical potential can be varied in-situ. Typically the others are varied under preparation conditions, and then soestablished sublattice stoichiometry is frozen under experimental conditions. This complex interplay of in-situ and ex-situ parameters has been discussed by Maier, 2003. Various questions, where thermodynamic approach is necessary, are still open. One example is Sr segregation towards LSM surfaces which was experimentally observed e.g. by Fister et al., 2008 andHerger et al., 2008. In the study by Herger et al., 2008, the Sr segregation energy was estimated to be ~0.16 eV while preliminary theoretical estimate gave the segregation energy ~0.54 eV (Piskunov et al., 2008). Another question is, how vacancy formation and oxygen adsorption energies depend on surface Sr concentration. Lastly, our present simplified simulations assume that LMO and LSM have large flat surfaces. However, one can expect in reality much more rough surface structure containing many facets and steps. This is important also because SOFC cathode materials are polycrystalline. Therefore, there is a necessity to check, which kinds of steps and facets are the most stable ones and likely to exist at LMO and LSM surfaces and how their presence affects adsorption of oxygen atoms and molecules, vacancy formation and reactions occurring at the surfaces.