**3. 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*<sup>i</sup>* 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., 2007a, 2007b; Johnston et al., 2004 ; Mastrikov et al., 2009; Padilla & Vanderbilt, 1997, 1998; Pikunov et al., 2008; Pojani et al., 1999; Reuter & Scheffler, 2001b, 2004). 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.

#### **3.1 Method of analysis for LMO surfaces**

Introducing the chemical potentials *La*, *Mn*, and *<sup>O</sup>* for the La, Mn, and O atomic species, respectively, the SGFE per unit cell area *i* corresponding to the *i* termination is defined as

$$\mathbf{L}\boldsymbol{\Omega}^{i} = \frac{\mathbf{1}}{2} [\mathbf{G}\_{\text{slab}}^{i} \text{ - } \mathbf{N}\_{\text{La}}^{i} \boldsymbol{\mu}\_{\text{La}} \text{ - } \mathbf{N}\_{\text{Mn}}^{i} \boldsymbol{\mu}\_{\text{Mn}} \text{ - } \mathbf{N}\_{\text{O}}^{i} \boldsymbol{\mu}\_{\text{O}}] \,\tag{2}$$

where *<sup>i</sup> Gslab* is the Gibbs free energy for the slab terminated by surface *i*, *Ni La, Ni Mn,* and *Ni <sup>O</sup>* 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

$$
\alpha \rho^i = \frac{\Omega^i}{A} \tag{3}
$$

The thermodynamic part of the description below follows the well known chemical thermodynamics formalism developed originally by Gibbs in 1875 (see Gibbs, 1948) for

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

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*<sup>i</sup>* 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., 2007a, 2007b; Johnston et al., 2004 ; Mastrikov et al., 2009; Padilla & Vanderbilt, 1997, 1998; Pikunov et al., 2008; Pojani et al., 1999; Reuter & Scheffler, 2001b, 2004). The SGFEs are functions of chemical potentials of different atomic species. The most stable surface has a structure, orientation and composition with the lowest

pure LMO and LSM (*xb*=1/8) were very well reproduced (Piskunov et al., 2007).

*La*, *Mn*, and

1

2

*i*

where *<sup>i</sup> Gslab* is the Gibbs free energy for the slab terminated by surface *i*, *Ni*

*La Mn O* (2)

[ - - - ],

*<sup>O</sup>* 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

*i*

*<sup>i</sup> ii i i GN N N slab La Mn O* 

> *i A*

The thermodynamic part of the description below follows the well known chemical thermodynamics formalism developed originally by Gibbs in 1875 (see Gibbs, 1948) for

*<sup>O</sup>* for the La, Mn, and O atomic species,

*La, Ni*

*Mn,* and

corresponding to the *i* termination is defined as

(3)

**3. Thermodynamic analysis of surface stability** 

SGFE among all possible surfaces.

*Ni*

**3.1 Method of analysis for LMO surfaces**  Introducing the chemical potentials

respectively, the SGFE per unit cell area

SGFE per unit area is represented by

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:

$$
\mu\_{\text{LaMnO}\,\text{3}} = \mu\_{\text{La}} + \mu\_{\text{Mn}} + \text{3}\,\mu\_{\text{O}} \tag{4}
$$

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

$$\text{l.} \mu\_{LaMnO3} = \text{g}^{bulk}\_{LaMnO3} \tag{5}$$

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

$$\mathbf{L}\boldsymbol{\Omega}^{i} = \frac{\mathbf{1}}{2} \left[ \mathbf{G}\_{\text{slab}}^{i} - \mathbf{N}\_{\text{A}}^{i} \mathbf{g}\_{\text{LaMn}O3}^{\text{bulk}} \right] - \boldsymbol{\Gamma}\_{\text{A},\text{Mu}}^{i} \boldsymbol{\mu}\_{\text{Mu}} - \boldsymbol{\Gamma}\_{\text{A},\text{O}}^{i} \boldsymbol{\mu}\_{\text{O}} \tag{6}$$

where Γ*<sup>i</sup>* 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 ABO3 perovskites) of the slabs (Gibbs,1948; Johnston et al., 2004) :

$$\Gamma\_{A,a}^{i} = \frac{1}{2} \left( N\_a^i - N\_A^i \frac{N\_a^{bulk}}{N\_A^{bulk}} \right) \tag{7}$$

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* atoms in unit cell in the bulk.

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

$$\mathbf{g}\_{j} = \mathbf{E}\_{j} \mathbf{\bar{\tau}} \mathbf{\bar{E}}\_{j}^{vibr} - T\mathbf{s}\_{j} + p\mathbf{v}\_{j} \tag{8}$$

where *Ej* is the static component of the crystal energy, *Ej vibr* is the vibrational contribution to 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

Thermodynamics of ABO3-Type Perovskite Surfaces 497

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:

deviations of the chemical potentials (10-12) transform the conditions to

 

where the formation energies of oxides are defined by

*<sup>f</sup> g LaMnO* has meaning of the Gibbs free energy of LaMnO3 formation from La,

*bulk Mn Mn*

*bulk La La* 

*bulk*

Similarly, precipitation of oxides does not occur, if the chemical potentials of atoms in LMO

2 3 2 3 *bulk La O La O*

*Mn O MnO*

3 4 3 4 *bulk Mn O Mn O*

2 3 2 3 *bulk Mn O Mn O*

> <sup>2</sup> <sup>2</sup> *bulk Mn O MnO*

<sup>3</sup> 3 ( ) *bulk*

( ) *bulk*

*g LaMnO g La O* (27)

2

2

(28)

 

 

 

*Mn O <sup>f</sup>*

 

 *y*

Exclusion of La chemical potential and expressing of these conditions through the

*Mn <sup>O</sup> f xy x*

3 2 <sup>3</sup> 2 3 2( ) ( ) *bulk bulk Mn O f f*

( ) <sup>2</sup>

*<sup>y</sup> g MO g xg <sup>E</sup>*

*bulk bulk bulk f xy MxOy M O*

. <sup>2</sup>

*bulk bulk MxOy M O*

*<sup>y</sup> E E xE*

 

 

*g* (17)

*g* (18)

*g* (19)

*g* (20)

*g* (21)

*g* (22)

*g* (23)

*Mn* 0 (24)

*g Mn O* (26)

*g LaMnO* (25)

Here 3 ( ) *bulk*

and

and

and

Mn and O2 in their standard states.

are smaller than in the oxides:

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:

$$\mathcal{S}\_{\rangle} \approx \mathbf{E}\_{\rangle} \tag{9}$$

Then, replacing the chemical potentials of La and Mn atoms by their deviations from chemical potentials in the most stable phases of respective elementary crystals,

$$
\Delta\mu\_{La} = \mu\_{La} - \text{g}\_{La}^{bulk} \approx \mu\_{La} - E\_{La}^{bulk} \tag{10}
$$

and

$$
\Delta\mu\_{\text{Mu}} = \mu\_{\text{Mu}} - \text{g}\_{\text{Mu}}^{bulk} \approx \mu\_{\text{Mu}} - \text{E}\_{\text{Mu}}^{bulk} \tag{11}
$$

and chemical potential of O atoms by its deviation from the energy of an oxygen atom in a free, isolated O2 molecule ( 2 / 2 *total EO* ),

$$
\Delta\mu\_{\rm O} = \mu\_{\rm O} - \frac{E\_{O\_2}^{total}}{2} \tag{12}
$$

we can determine the surface Gibbs free energy from

$$
\Omega \Omega^i = \phi^i - \Gamma^i\_{A,Mn} \Delta \mu\_{Mn} - \Gamma^i\_{A,O} \Delta \mu\_{O^\*} \tag{13}
$$

We can express the constant *i* in Eq. (13) as

$$\begin{split} \boldsymbol{\phi}^{i} &= \frac{\mathbf{1}}{2} \Big[ \mathbf{G}^{i}\_{\text{slab}} - \mathbf{N}^{i}\_{\text{A}} \mathbf{g}^{\text{bulk}}\_{\text{LaMnO}\text{\%}} \Big] - \boldsymbol{\Gamma}^{i}\_{\text{A},\text{Mn}} \mathbf{g}^{\text{bulk}}\_{\text{Mt}} - \frac{\mathbf{1}}{2} \boldsymbol{\Gamma}^{i}\_{\text{A},\text{O}} \mathbf{E}\_{O2} \approx \\ &\approx \frac{\mathbf{1}}{2} \Big[ \mathbf{E}^{i}\_{\text{slab}} - \mathbf{N}^{i}\_{\text{A}} \mathbf{E}^{\text{bulk}}\_{\text{LaMnO}\text{\%}} \Big] - \boldsymbol{\Gamma}^{i}\_{\text{A},\text{Mn}} \mathbf{E}^{\text{bulk}}\_{\text{Mt}} - \frac{1}{2} \boldsymbol{\Gamma}^{i}\_{\text{A},\text{O}} \mathbf{E}\_{O2} \Big]. \end{split} \tag{14}$$

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) can be rewritten as

$$
\Delta\mu\_{La} + \Delta\mu\_{Mn} + \Theta\Delta\mu\_O = \Delta\mathbf{g}\_f^{bulk} \text{(LaMnO}\_3\text{)}\tag{15}
$$

where

$$\begin{split} \text{Ag}^{\text{bulk}}\_{f}(\text{LaMnO}\_{3}) &= \text{g}^{\text{bulk}}\_{\text{LaMnO}3} - \text{g}^{\text{bulk}}\_{\text{La}} - \text{g}^{\text{bulk}}\_{\text{Mn}} - \frac{3}{2} \text{E}\_{\text{O2}} \approx \\ &\approx \text{E}^{\text{bulk}}\_{\text{LaMnO}3} - \text{E}^{\text{bulk}}\_{\text{La}} - \text{E}^{\text{bulk}}\_{\text{Mn}} - \frac{3}{2} \text{E}\_{\text{O2}} \, . \end{split} \tag{16}$$

Here 3 ( ) *bulk <sup>f</sup> g LaMnO* has meaning of the Gibbs free energy of LaMnO3 formation from La, Mn and O2 in their standard states.

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:

$$
\mu\_{\text{Mtr}} < \text{g}^{\text{bulk}}\_{\text{Mtr}} \tag{17}
$$

and

496 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

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

Then, replacing the chemical potentials of La and Mn atoms by their deviations from

*La La La La La*

*Mn Mn Mn Mn Mn*

and chemical potential of O atoms by its deviation from the energy of an oxygen atom in a

*O O*

, , . *<sup>i</sup> ii i*

1 1 , 2 2

what resembles the expression for the Gibbs free energy of surface formation. Here *E*slab

 

<sup>3</sup> ( ) <sup>2</sup>

*g LaMnO g gg E*

*bulk bulk bulk bulk f LaMnO La Mn O*

3 . <sup>2</sup>

*i b i ulk ibulk i slab A LaMnO A Mn Mn AO O*

[]

*EE E E N*

*slab A A LaMnO Mn A Mn O O*

*G N g g E*

 

1 1 2 2

*i i i bulk i i bulk*

stands for the total energy of a slab and replaces the Gibbs free energy of the slab.

*La Mn O <sup>f</sup>*

*bulk bulk*

*bulk bulk*

 

> 

2 2

 *Mn O A O* (13)

3 , , 2

3 , , 2

*g LaMnO* (15)

<sup>3</sup> 3 () *bulk*

3 3 2

*E EE E*

*bulk bulk bulk LaMnO La Mn O*

3 2

*total O*

*E*

chemical potentials in the most stable phases of respective elementary crystals,

*A Mn*

[ ]

in Eq. (13) as

free, isolated O2 molecule ( 2 / 2 *total EO* ),

We can express the constant

The equilibrium condition (5) can be rewritten as

we can determine the surface Gibbs free energy from

*i* *<sup>j</sup> <sup>j</sup> g E* (9)

*g E* (10)

*g E* (11)

(12)

(14)

(16)

calculations:

and

where

$$\mu\_{La} < \mathcal{g}\_{La}^{bulk} \tag{18}$$

Similarly, precipitation of oxides does not occur, if the chemical potentials of atoms in LMO are smaller than in the oxides:

$$2\,\mu\_{La} + 3\,\mu\_O < \mathcal{g}\_{La2O3}^{bulk} \tag{19}$$

$$
\mu\_{\rm Mn} + \mu\_{\rm O} < \mathcal{g}\_{\rm MnO}^{bulk} \tag{20}
$$

$$2\,\mathrm{\mathfrak{H}}\_{\mathrm{Mn}} + 4\,\mu\_{\mathrm{O}} < \mathrm{g}^{bulk}\_{\mathrm{Mn}\mathrm{3O}4} \tag{21}$$

$$2\,\mu\_{\text{Mn}} + 3\,\mu\_{\text{O}} < g\_{\text{Mn2O3}}^{\text{bulk}}\tag{22}$$

and

$$
\mu\_{\text{Mn}} + \text{2}\,\mu\_{\text{O}} < \text{g}^{\text{bulk}}\_{\text{MnO2}} \tag{23}
$$

Exclusion of La chemical potential and expressing of these conditions through the deviations of the chemical potentials (10-12) transform the conditions to

$$
\Delta\mu\_{Mn} < 0 \tag{24}
$$

$$
\Delta\mu\_{\text{Mu}} + \text{3}\Delta\mu\_{\text{O}} > \Delta\text{g}\_f^{\text{bulk}} \text{(LaMnO}\_3\text{)}\tag{25}
$$

$$
\lambda \Delta \mu\_{\text{Mn}} + y \Delta \mu\_{\text{O}} < \Delta \text{g}\_{f}^{\text{bulk}} \text{(} \text{Mn}\_{\text{x}} \text{O}\_{\text{y}} \text{)}\tag{26}
$$

and

$$2\Delta\mu\_{\text{Mu}} + 3\Delta\mu\_{\text{O}} > 2\Delta\text{g}\_{f}^{\text{bulk}} \text{(LaMnO}\_{3}\text{)} - \Delta\text{g}\_{f}^{\text{bulk}} \text{(La}\_{2}\text{O}\_{3}\text{)}\tag{27}$$

where the formation energies of oxides are defined by

$$\begin{split} \left( \mathbf{A} \mathbf{g}\_f^{bulk} \{ M\_x O\_y \} \right) &= \mathbf{g}\_{M \mathbf{x} \cup \mathbf{O}\_Y}^{bulk} - \mathbf{x} \mathbf{g}\_M^{bulk} - \frac{\mathcal{Y}}{2} E\_{O2} \approx \\ &\approx E\_{M \mathbf{x} \cup \mathbf{O}\_Y}^{bulk} - \mathbf{x} E\_M^{bulk} - \frac{\mathcal{Y}}{2} E\_{O2} \,. \end{split} \tag{28}$$

Thermodynamics of ABO3-Type Perovskite Surfaces 499

*Sr Sr Sr Sr Sr*

still have to remember only that NA does not coincide any more with NLa in LSM . NA 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

, 1 1 (1 ) 2 2

2 2 ( ) *<sup>O</sup> <sup>i</sup> <sup>i</sup> <sup>i</sup> bulk ii i bulk bulk slab A LSM A La La A Mn Mn A O*

0

 

<sup>3</sup> ( ) (1 ) <sup>2</sup>

*<sup>M</sup> <sup>O</sup> <sup>f</sup> <sup>x</sup> <sup>y</sup> <sup>n</sup> x*

2 3 2 3 ( ) *bulk La O <sup>f</sup>*

(1 ) (3 ) ( ) () *bulk bulk <sup>b</sup> La Mn <sup>b</sup> <sup>O</sup> b f <sup>f</sup> x*

 

 *y*

> 

*bulk bulk bulk bulk bulk g LSM x x f bb E E EE E LSM Sr La Mn O* (41)

( ) *bulk*

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.

*b f La Mn O x*

 

*<sup>N</sup> N NN N <sup>x</sup>*

 *A La La A Mn Mn A O*

*<sup>i</sup> <sup>A</sup> <sup>O</sup>* and ,

*bulk i i ii i La A La La bulk A La b A*

*A*

,, , *<sup>i</sup> <sup>i</sup> ii i*

*N* 

The expressions for the excesses ,

expressed as:

where

and

where

Then SGFEs for LSM reads

1

where Gibbs free energy of LSM formation is

Precipitation of oxides is avoided, if

Precipitation of Mn, La, and Sr metals will be avoided if

(1 ) <sup>3</sup> ( ) *bulk*

*bulk bulk*

*g E* (34b)

*<sup>i</sup> <sup>A</sup> Mn* do not change with respect to LMO. We

*E*

*Mn* 0 (38)

*La* (39)

*g Mn O* (42)

*g La O* (43)

*x g LSM x g SrO* (44)

*g LSM* (40)

2

<sup>2</sup> ,, ,

*G N E EE* (37)

(35)

*<sup>O</sup>* (36)

 

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

$$
\Omega \succeq 0 \tag{29}
$$

where *i* corresponds to the surface with the lowest SGFE.

### **3.2 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

$$\Omega \mathcal{Q}^{i} \equiv \frac{1}{2} \left[ G^{i}\_{\text{slab}} - \mathcal{N}^{i}\_{\text{La}} \mu\_{\text{La}} - \mathcal{N}^{i}\_{\text{Sr}} \mu\_{\text{Sr}} - \mathcal{N}^{i}\_{\text{Mn}} \mu\_{\text{Mn}} - \mathcal{N}^{i}\_{\text{O}} \mu\_{\text{O}} \right] \tag{30a}$$

Let us denote concentration of Sr atoms in the bulk of LSM as

$$\alpha\_b = \frac{N\_{Sr}^{bulk}}{N\_A^{bulk}}\tag{30b}$$

where *bulk NSr* is the average number of Sr atoms per crystal unit cell in the bulk. Then for LSM

$$N\_{La}^{bulk} = (1 - \boldsymbol{\pi}\_b) N\_A^{bulk} \tag{31}$$

becomes the average number of La atoms per LSM unit cell in the bulk. The chemical potential of a LSM formula unit is

$$
\mu\_{\rm LSM} = (1 - \chi\_b)\mu\_{\rm La} + \chi\_b\mu\_{\rm Sr} + \mu\_{\rm Mn} + \Im\mu\_O \tag{32}
$$

Equilibrium between LSM surface and its bulk means that

$$\mu\_{LSM} = \mathbf{g}\_{LSM}^{bulk} \tag{33}$$

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

$$
\mu\_{Sr} = \frac{1}{\varkappa\_b} \left( \mathbb{E}\_{\rm LSM} - (1 - \chi\_b)\mu\_{La} - \mu\_{Mn} - \mathfrak{Z}\mu\_O \right) \tag{34a}
$$

and its deviation (analogous to eqs. (10-12) and keeping in mind approximation (9)) as

$$
\Delta\mu\_{Sr} = \mu\_{Sr} - \mathbf{g}\_{Sr}^{bulk} \approx \mu\_{Sr} - E\_{Sr}^{bulk} \tag{34b}
$$

The expressions for the excesses , *<sup>i</sup> <sup>A</sup> <sup>O</sup>* and , *<sup>i</sup> <sup>A</sup> Mn* do not change with respect to LMO. We still have to remember only that NA does not coincide any more with NLa in LSM . NA 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:

$$\Gamma^{i}\_{A,La} = \frac{1}{2} \left( \mathcal{N}^{i}\_{La} - \frac{\mathcal{N}^{bulk}\_{La}}{\mathcal{N}^{bulk}\_{A}} \mathcal{N}^{i}\_{A} \right) = \frac{1}{2} \left( \mathcal{N}^{i}\_{La} - (1 - \mathbf{x}\_{b}) \mathcal{N}^{i}\_{A} \right) \tag{35}$$

Then SGFEs for LSM reads

$$
\Omega \Omega^i = \phi^i - \Gamma^i\_{A, \text{Id}} \Delta \ \mu\_{\text{La}} - \Gamma^i\_{A, \text{Ma}} \Delta \mu\_{\text{Ma}} - \Gamma^i\_{A, \text{O}} \Delta \mu\_{\text{O}} \tag{36}
$$

where

498 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

Note, however, that sometimes compositions are fixed by bringing the multinary crystals

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

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

[ ] *<sup>i</sup> <sup>i</sup> iii i GN N N N slab La Sr Mn*

where *bulk NSr* is the average number of Sr atoms per crystal unit cell in the bulk. Then for

(1 ) 3 *LSM b b La Sr Mn O*

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

<sup>1</sup> ( ) *LSM* (1 ) <sup>3</sup> *Sr <sup>b</sup> La Mn O*

 

*bulk LSM*

 *x x* 

*LSM*

*E x*

and its deviation (analogous to eqs. (10-12) and keeping in mind approximation (9)) as

*bulk Sr b bulk A N*

*N* (30b)

(1 ) *bulk bulk N xN La b A* (31)

*g* (33)

 

> 

*Mn O <sup>O</sup>* (30a)

(32)

(34a)

 *La Sr* 

0 *<sup>i</sup>* (29)

conditions on the chemical potentials of the crystal components can be written as

perovskite A sub-lattice. The SGFE definition for LSM can be written as

becomes the average number of La atoms per LSM unit cell in the bulk.

*x*

into coexistence with less complex sub-phases.

**3.2 Method of analysis for LSM surfaces** 

LSM

where *i* corresponds to the surface with the lowest SGFE.

1 2

The chemical potential of a LSM formula unit is

Equilibrium between LSM surface and its bulk means that

*b*

*x*

Let us denote concentration of Sr atoms in the bulk of LSM as

$$\boldsymbol{\phi}^{i} = \frac{1}{2} \left( \mathbf{G}\_{slab}^{i} - \mathbf{N}\_{A}^{i} \mathbf{E}\_{LSM}^{bulk} \right) - \boldsymbol{\Gamma}\_{A,La}^{i} \mathbf{E}\_{La}^{bulk} - \boldsymbol{\Gamma}\_{A,Ma}^{i} \mathbf{E}\_{Mn}^{bulk} - \boldsymbol{\Gamma}\_{A,O}^{i} \frac{\boldsymbol{E}\_{O}}{2} \tag{37}$$

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

$$
\Delta\mu\_{Mn} < 0\tag{38}
$$

$$
\Delta \mu\_{La} < 0 \tag{39}
$$

and

$$(\mathbf{1} - \mathbf{x}\_b)\boldsymbol{\Delta\mu}\_{\mathrm{Ld}} + \boldsymbol{\Delta\mu}\_{\mathrm{Mn}} + \mathbf{3}\boldsymbol{\Delta\mu}\_{\mathrm{O}} > \boldsymbol{\Delta\mathrm{g}}\_f^{\mathrm{bulk}} \text{(LSM)}\tag{40}$$

where Gibbs free energy of LSM formation is

$$\Delta \mathbf{g}\_f^{bulk} \{LSM\} = E\_{LSM}^{bulk} - \boldsymbol{\pi}\_b E\_{Sr}^{bulk} - (1 - \boldsymbol{\pi}\_b) E\_{La}^{bulk} - E\_{Mn}^{bulk} - \frac{\mathfrak{Z}}{2} E\_{O\_2} \tag{41}$$

Precipitation of oxides is avoided, if

$$
\lambda \ge \Delta \mu\_{\text{Mn}} + y \Delta \mu\_{\text{O}} < \Delta \text{g}\_f^{\text{bulk}} \text{(Mn}\_x \text{O}\_y\text{)}\tag{42}
$$

$$2\Delta\mu\_{La} + 3\Delta\mu\_O < \Delta\text{g}\_f^{\text{bulk}} \text{(La}\_2\text{O}\_3\text{)}\tag{43}$$

$$(\mathbf{1} - \mathbf{x}\_b)\Delta\mu\_{\rm La} + \Delta\mu\_{\rm Mn} + (\mathbf{3} - \mathbf{x}\_b)\Delta\mu\_{\rm O} > \Delta\mathbf{g}\_f^{\rm bulk} \\ \text{(LSM)} - \mathbf{x}\_b \Delta\mathbf{g}\_f^{\rm bulk} \text{(SrO)}\tag{44}$$

where

Thermodynamics of ABO3-Type Perovskite Surfaces 501

data from the standard thermodynamic tables (Chase, 1998; Linstrom & Mallard, 2003), following Johnston et al., 2004 and Reuter & Scheffler, 2001b. These data are collected in Table 1. For this we define an isolated oxygen molecule *EO2* as the reference state. Changes

2

<sup>1</sup> ( , ) ln 2 {}

0 00 0

(, ) (, ) ( , )

*gas*

<sup>2</sup>(, ) *gas GO T p* is the change in the oxygen gas Gibbs free energy at the pressure *p*0 and

22 2 2

*H H Tp T p Tp T p TS T S*

(, ) ( , ) (, ) ( , )

*T*, K ΔGO2gas (T,p0), eV T, K ΔGO2gas (T,p0), eV 100 -0.07 1000 -1.10 200 -0.17 1100 -1.23 250 -0.22 1200 -1.36 298.15 -0.27 1300 -1.49 300 -0.27 1400 -1.62 400 -0.38 1500 -1.75 500 -0.50 1600 -1.88 600 -0.61 1700 -2.02 700 -0.73 1800 -2.16 800 -0.85 1900 -2.29 900 -0.97 2000 -2.43

Table 1. Variations in the Gibbs free energy for oxygen gas at standard pressure (p0=1 atm.) with respect to its value at 0 K. Data are taken from thermodynamic tables (Chase, 1998;

quantum-mechanical computations. This correction can be estimated from computations of metal oxides and metals, in a way similar to Johnston et al., 2004. Enthalpy of an MxOy oxide

> 0 00 0 <sup>2</sup> , 2 *MxOy M O <sup>f</sup> MxOy*

Here enthalpies of the oxide, <sup>0</sup> *hMxOy* , and of the metal, <sup>0</sup> *hM* , can be approximated by the total energies for these materials calculated at 0 K on the same grounds as for approximation

in Eq. (50) is a correction which matches experimental data and the results of

*<sup>y</sup> h hh <sup>x</sup> <sup>H</sup>* (52)

*gas gas gas gas OO O O*

0 00 0 0 0 0

0 0

*p*

(50)

(51)

*O O*

<sup>2</sup> <sup>0</sup>

*<sup>p</sup> <sup>G</sup> T p kT*

in the chemical potential for oxygen atom can be written as

<sup>1</sup> (,) (, ) <sup>2</sup>

 

Here <sup>0</sup>

Linstrom & Mallard 2003).

The <sup>0</sup> *O* 

can be written as

*O O O*

*Tp Tp E*

temperature *T* with respect to its Gibbs free energy at *T0*=298.15 K

2 22

*GGG Tp Tp T p*

*gas gas gas OOO*

$$\Delta \mathbf{g}\_f^{bulk} \text{(SrO)} = \mathbf{E}\_{SrO}^{bulk} - \mathbf{E}\_{Sr}^{bulk} - \frac{\mathbf{E}\_{O\_2}}{\mathbf{2}} \tag{45}$$

Similarly, precipitation of LaMnO3 and SrMnO3 perovskites will be prevented, if

$$\begin{split} \frac{1}{1 - \mathbf{x}\_b} \Delta \mathbf{g}\_f^{\text{bulk}} \text{(LSM)} - \frac{\mathbf{x}\_b}{1 - \mathbf{x}\_b} \Delta \mathbf{g}\_f^{\text{bulk}} \text{(SrMnO}\_3\text{)}< \\ &< \Delta \mu\_{\text{Mn}} + \Delta \mu\_{\text{La}} + \mathfrak{Z} \Delta \mu\_O < \Delta \mathbf{g}\_f^{\text{bulk}} \text{(LaMnO}\_3\text{)}. \end{split} \tag{46}$$

Here the Gibbs free energy of SrMnO3 formation is defined as

$$\Delta \mathbf{g}\_f^{\text{bulk}} \text{(SrMnO}\_3\text{)} = \mathbf{E}\_{SrMnO3}^{\text{bulk}} - \mathbf{E}\_{Sr}^{\text{bulk}} - \mathbf{E}\_{Mn}^{\text{bulk}} - \frac{3}{2} \mathbf{E}\_{O2} \text{ }. \tag{47}$$

Lastly, spontaneous formation of surfaces does not occur, if condition (29) is satisfied as well.

#### **3.3 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:

$$
\mu\_O = \frac{1}{2} \mu\_{O2}^{\otimes \text{as}} \tag{48}
$$

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 <sup>O</sup>* 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 and Reuter & Scheffler, 2001b)

$$
\mu\_{O2}^{\rm gas}(T, p) = \mu\_{O2}^{\rm gas}(T, p^0) + kT \ln \left(\frac{p}{p^0}\right) \tag{49}
$$

where *k* is the Boltzmann constant. Here *p0* is the reference pressure which we can take as the standard pressure (1 atm.). The temperature dependence of <sup>0</sup> <sup>2</sup>(, ) *gas O T p* includes contributions from molecular vibrations and rotations, as well as ideal-gas entropy at pressure *p*0. We can evaluate the temperature dependence of <sup>0</sup> <sup>2</sup>(, ) *gas O T p* using experimental

3 ( ).

Lastly, spontaneous formation of surfaces does not occur, if condition (29) is satisfied as well.

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

> > *gas*

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

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 and Reuter &

> 0 2 2 <sup>0</sup> ( , ) ( , ) ln *gas gas*

*p*

 

*<sup>p</sup> T p T p kT*

where *k* is the Boltzmann constant. Here *p0* is the reference pressure which we can take as

contributions from molecular vibrations and rotations, as well as ideal-gas entropy at

*Mn La O f*

3 3 2 <sup>3</sup> ( ) . <sup>2</sup>

Similarly, precipitation of LaMnO3 and SrMnO3 perovskites will be prevented, if

<sup>1</sup> () ( ) 1 1

*<sup>x</sup> g LSM g SrMnO*

*bulk b bulk f f b b*

*x x*

Here the Gibbs free energy of SrMnO3 formation is defined as

**3.3 Determination of the chemical potential of oxygen atom** 

the dependence of oxygen gas chemical potential 2

Scheffler, 2001b)

surfaces through temperature and oxygen gas partial pressure.

*O O*

pressure *p*0. We can evaluate the temperature dependence of <sup>0</sup>

 

the standard pressure (1 atm.). The temperature dependence of <sup>0</sup>

equality of oxygen chemical potentials in a crystal and in the atmosphere:

*O*

<sup>2</sup> ( ) <sup>2</sup> *bulk <sup>O</sup> bulk bulk f SrO Sr*

*E g SrO E E* (45)

3

*bulk bulk bulk bulk g SrMnO <sup>f</sup> E EE E SrMnO Sr Mn <sup>O</sup>* (47)

*bulk*

*g LaMnO*

3

*<sup>O</sup>* (48)

*<sup>O</sup>* to express the Gibbs free energies for

(49)

<sup>2</sup>(, ) *gas O* 

*T p* using experimental

<sup>2</sup>(, ) *gas O* 

*T p* includes

(46)

data from the standard thermodynamic tables (Chase, 1998; Linstrom & Mallard, 2003), following Johnston et al., 2004 and Reuter & Scheffler, 2001b. These data are collected in Table 1. For this we define an isolated oxygen molecule *EO2* as the reference state. Changes in the chemical potential for oxygen atom can be written as

$$\begin{aligned} \Delta \mu\_O(T, p) &= \mu\_O(T, p \ ) - \frac{1}{2} E\_{O2} = \\ &= \frac{1}{2} \{ \Delta G\_{O2}^{\text{gas}}(T, p^0) + kT \ln \left( \frac{p}{p^0} \right) + \delta \mu\_O^0 \end{aligned} \tag{50}$$

Here <sup>0</sup> <sup>2</sup>(, ) *gas GO T p* 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 *T0*=298.15 K

$$\begin{split} \mathsf{AG}\_{O2}^{\mathrm{gas}}(T, p^{0}) &= \mathsf{G}\_{O2}^{\mathrm{gas}}(T, p^{0}) - \mathsf{G}\_{O2}^{\mathrm{gas}}(T^{0}, p^{0}) = \\ &= \mathsf{H}\_{O2}^{\mathrm{gas}}(T, p^{0}) - \mathsf{H}\_{O2}^{\mathrm{gas}}(T^{0}, p^{0}) - \mathsf{TS}\_{O2}^{\mathrm{gas}}(T, p^{0}) + \mathsf{T}^{0}\mathsf{S}\_{O2}^{\mathrm{gas}}(T^{0}, p^{0}) \end{split} \tag{51}$$


Table 1. Variations in the Gibbs free energy for oxygen gas at standard pressure (p0=1 atm.) with respect to its value at 0 K. Data are taken from thermodynamic tables (Chase, 1998; Linstrom & Mallard 2003).

The <sup>0</sup> *O* in Eq. (50) is a correction which matches experimental data and the results of quantum-mechanical computations. This correction can be estimated from computations of metal oxides and metals, in a way similar to Johnston et al., 2004. Enthalpy of an MxOy oxide can be written as

$$\mathrm{M}\_{\mathrm{MxOy}}^{0} = \mathrm{x}\mathrm{h}\_{\mathrm{M}}^{0} + \frac{\mathcal{Y}}{2}\mathrm{h}\_{\mathrm{O}}^{0} + \Delta\mathrm{H}\_{f,\mathrm{MxOy}}^{0} \tag{52}$$

Here enthalpies of the oxide, <sup>0</sup> *hMxOy* , and of the metal, <sup>0</sup> *hM* , can be approximated by the total energies for these materials calculated at 0 K on the same grounds as for approximation

Thermodynamics of ABO3-Type Perovskite Surfaces 503

*<sup>O</sup>*(, ) *T pO* due to *T*, *p*O2 is described by Eq. (50). Similarly, the vacancy formation energy

3 O <sup>O</sup> <sup>2</sup> 3 O ( ) : V :V ( , ) *surf surf*

3 O <sup>3</sup> <sup>2</sup> <sup>3</sup> <sup>1</sup>

*<sup>f</sup> <sup>O</sup> <sup>O</sup> LaMnO E LaMnO <sup>E</sup> LaMnO V E E* (59)

*slab ELaMnO* is the total energy of the slab without defects.

*T pO* (58)

(60)

(62)

values

*<sup>f</sup> <sup>f</sup> g LaMnO <sup>E</sup> LaMnO*

The Gibbs free energies of adsorption can be written in a similar way:

: V : 2 <sup>2</sup> ( ) *surf slab slab*

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 <sup>3</sup> : 2 *slab E LaMnO VO* ,

> <sup>3</sup> <sup>3</sup> : : *surf surf*

*surf slab slab*

1 : : 2 2

given relative to half an O2 molecule, but not with respect to gaseous O atoms.

**4.1 Stability of LMO surface terminations: Plane-wave DFT simulations** 

*ads ads ads <sup>O</sup> ads g LaMnO O E LaMnO O*

 <sup>2</sup> 3 3 2, ads 2, ads : : *surf surf ads ads <sup>O</sup> g LaMnO O E LaMnO O*

Here *µO2=2µO*, and we have to take account for two adsorbed O or O2 on the symmetrical slab. It is important to remember that the adsorption energy (60) for atomic O species is

In the present Chapter we will describe the vacancy formation energies and the adsorption energies of O atoms and O2 molecules obtained with plane wave BS and PW91 functional.

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

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. Optimized geometries for the slabs can be found in Mastrikov et al., 2009. The surface stability regions in the diagrams are limited by the lines 2, 6 and 4. These lines correspond to boundaries where coexistence occurs of LMO with La2O3, MnO2 and Mn3O4 , respectively. Because of the DFT deficiencies in describing the relative energies for materials

possible to keep the cubic structure during lattice relaxation. Therefore, we used *<sup>i</sup>*

 3 3 2, ads 2, ads <sup>2</sup> <sup>3</sup> <sup>1</sup> : : 2 2 <sup>2</sup> *surf slab E LaMnO O E LaMnO O E E ads slab <sup>O</sup> LaMnO* (63)

<sup>2</sup> 3 3 <sup>3</sup>

*ads ads ads <sup>O</sup> LaMnO E EE LaMnO O LaMnO O E* (61)

<sup>2</sup> 

where

and

for the surface vacancy can be presented as

because the slab has two vacancies. 3

**4. Results and discussions**

(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

$$
\hbar \delta \mathbf{h}\_{O2}^{0} = \hbar \mathbf{\hat{O}\_{O2}} - \mathbf{E}\_{O2} \tag{53}
$$

Using the experimental standard entropy for oxygen (Chase, 1998; Linstrom & Mallard, 2003) as <sup>0</sup> *SO*<sup>2</sup> 205.147 J·mol-1K-1, the correction to the oxygen chemical potential can be calculated as

$$\begin{split} \delta \mu\_O^0 &= \frac{1}{2} \{ \delta \mathfrak{h}\_{O2}^0 - T^0 S\_{O2}^{\rm gas}(T^0, p^0) \} \\ &= \frac{1}{y} \{ \mathfrak{h}\_{M\_xO\_y}^0 - \mathfrak{x} \mathfrak{h}\_M^0 - \Delta H\_{f, M\_xO\_y}^0 \} - \frac{1}{2} \{ E\_{O2} + T^0 S\_{O2}^{\rm gas}(T^0, p^0) \} \end{split} \tag{54}$$

#### **3.4 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 LaMnO3 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 O2 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 O2 is taken from experimental data. The Gibbs free energy of reaction for removal of a neutral O atom (1/2 O2) from the bulk (i.e. formation of one neutral V and allocation of the left-b <sup>O</sup> ehind two electrons mainly on two nearest Mn) is defined as

$$\Delta \mathbf{g}\_f^{bulk} \text{(LaMnO}\_3 \colon \mathbf{V}\_\odot \text{)} \quad \mathbf{E}^{bulk} \text{(LaMnO}\_3 \colon \mathbf{V}\_\odot \text{)} + \mu\_\text{O} \text{(T}, p\text{O}\_2\text{)} - \mathbf{E}\_{LaMnO3}^{bulk} \tag{55}$$

where 3 O : V *bulk E LaMnO* is the total energy per bulk supercell with an oxygen vacancy. This definition can be re-written as

$$
\Delta \mathbf{g}\_f^{bulk} \left( \mathrm{LaMnO}\_3; \, \mathrm{V}\_\mathrm{O} \right) \, = \Delta E\_f^{bulk} \left( \mathrm{LaMnO}\_3; \, \mathrm{V}\_\mathrm{O} \right) + \Delta \mu\_\mathrm{O} (\mathrm{T}, p\mathrm{O}\_2) \tag{56}
$$

where

$$\Delta E\_f^{\text{bulk}} \Big( \text{LaMnO}\_3 : V\_O \Big) = \mathbb{E}^{\text{bulk}} \Big( \text{LaMnO}\_3 : V\_O \Big) + \frac{1}{2} \mathbb{E}\_{O\_2} - E\_{\text{LaMnO}3}^{\text{bulk}} \tag{57}$$

is the formation energy of a neutral oxygen vacancy with respect to the calculated energy 2 1 <sup>2</sup> *EO* for oxygen atom in the molecule. The variation of oxygen chemical potential

<sup>2</sup> *<sup>O</sup>*(, ) *T pO* due to *T*, *p*O2 is described by Eq. (50). Similarly, the vacancy formation energy for the surface vacancy can be presented as

$$
\Delta \mathbf{g}\_f^{surf} \left( \mathrm{LaMnO}\_3; \mathrm{V}\_\mathrm{O} \right) \ = \Delta \mathbf{E}\_f^{surf} \left( \mathrm{LaMnO}\_3; \mathrm{V}\_\mathrm{O} \right) + \Delta \mu\_\mathrm{O} (\mathrm{T}, p\mathrm{O}\_2) \tag{58}
$$

where

502 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

(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

0 0

Using the experimental standard entropy for oxygen (Chase, 1998; Linstrom & Mallard, 2003) as <sup>0</sup> *SO*<sup>2</sup> 205.147 J·mol-1K-1, the correction to the oxygen chemical potential can be

1 1 ( ) ( ( , )) <sup>2</sup>

*M O M fMO O O*

3 O 3 O <sup>O</sup> <sup>2</sup> <sup>3</sup> ( : V ) :V ( , O ) *bulk bulk bulk <sup>f</sup> LaMnO g LaMnO E E LaMnO*

where 3 O : V *bulk E LaMnO* is the total energy per bulk supercell with an oxygen vacancy.

3 O <sup>O</sup> <sup>2</sup> 3 O ( ) : V :V ( , O ) *bulk bulk*

3 3 <sup>2</sup> <sup>3</sup>

*<sup>f</sup> O O <sup>O</sup> LaMnO E LaMnO V LaMnO V E E E* (57)

*<sup>f</sup> <sup>f</sup> g LaMnO <sup>E</sup> LaMnO*

: :

*bulk bulk bulk*

is the formation energy of a neutral oxygen vacancy with respect to the calculated energy

<sup>2</sup> *EO* for oxygen atom in the molecule. The variation of oxygen chemical potential

1

2

*T p* (55)

*T p* (56)

0 0 0 0 00

*h hx H E TS T p*

, 2 2

*h h O O* 2 2 *EO*2 (53)

*gas*

(54)

0 0 0 00 2 2

*O O O*

*y*

 

<sup>1</sup> ( ( , )) <sup>2</sup>

*h TS T p*

*gas*

*x y x y*

**3.4 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 LaMnO3 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 O2 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 O2 is taken from experimental data. The Gibbs free energy of reaction for removal of a neutral O atom (1/2 O2) from the bulk (i.e. formation of one neutral V and allocation of the left-b <sup>O</sup> ehind two electrons mainly on

enthalpy could to be defined as

two nearest Mn) is defined as

This definition can be re-written as

where

2 1

calculated as

$$\Delta E\_f^{surf} \left( \text{LaMnO}\_3; \text{V}\_\text{O} \right) \ = \frac{1}{2} \left( E^{slab} \left( \text{LaMnO}\_3; \text{2V}\_\text{O} \right) + \text{E}\_{O\_2} - \text{E}\_{LaMnO3}^{slab} \right) \tag{59}$$

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 <sup>3</sup> : 2 *slab E LaMnO VO* , because the slab has two vacancies. 3 *slab ELaMnO* is the total energy of the slab without defects. The Gibbs free energies of adsorption can be written in a similar way:

$$
\Delta \mathbf{g}\_{\rm ads}^{\rm surf} \left( \text{LaMnO}\_3 \colon \text{O}\_{\rm ads} \right) = \Delta E\_{\rm ads}^{\rm surf} \left( \text{LaMnO}\_3 \colon \text{O}\_{\rm ads} \right) - \Delta \mu\_O \tag{60}
$$

$$\Delta \operatorname{E}\_{\text{ads}}^{\text{surf}} \left( \operatorname{LaMnO}\_3 \colon \text{O}\_{\text{ads}} \right) \quad = \frac{1}{2} \Big( \operatorname{E}^{\text{slab}} \left( \operatorname{LaMnO}\_3 \colon \text{2O}\_{\text{ads}} \right) - \operatorname{E}\_{\text{O}\_2} - \operatorname{E}\_{\text{LaMnO}}^{\text{slab}} \Big) \tag{61}$$

and

$$\Delta \text{g}^{surf}\_{\text{ads}} \left( \text{LaMnO}\_3 \colon \text{O}\_{2,\text{ads}} \right) = \Delta E^{surf}\_{\text{ads}} \left( \text{LaMnO}\_3 \colon \text{O}\_{2,\text{ads}} \right) - \Delta \mu\_{O2} \tag{62}$$

$$\Delta \operatorname{E}\_{\mathrm{ads}}^{\mathrm{surf}} \Big( \mathrm{LaMnO}\_{3} \colon \mathrm{O}\_{2,\text{ads}} \Big) = \frac{1}{2} \Big( \mathrm{E}\_{\mathrm{slab}} \Big( \mathrm{LaMnO}\_{3} \colon \mathrm{2O}\_{2,\text{ads}} \Big) - \mathrm{2E}\_{\mathrm{O}\_{2}} - \mathrm{E}\_{\mathrm{LaMnO}}^{\mathrm{slab}} \Big) \tag{63}$$

Here *µO2=2µO*, and we have to take account for two adsorbed O or O2 on the symmetrical slab. It is important to remember that the adsorption energy (60) for atomic O species is given relative to half an O2 molecule, but not with respect to gaseous O atoms.

In the present Chapter we will describe the vacancy formation energies and the adsorption energies of O atoms and O2 molecules obtained with plane wave BS and PW91 functional.

## **4. Results and discussions**

### **4.1 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 *<sup>i</sup>* 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. Optimized geometries for the slabs can be found in Mastrikov et al., 2009.

The surface stability regions in the diagrams are limited by the lines 2, 6 and 4. These lines correspond to boundaries where coexistence occurs of LMO with La2O3, MnO2 and Mn3O4 , respectively. Because of the DFT deficiencies in describing the relative energies for materials

Thermodynamics of ABO3-Type Perovskite Surfaces 505

Fig. 1. Phase diagrams calculated with plane wave BS and PW91 GGA functional: The regions of stability of LaMnO3 surfaces with different terminations (LaO- and MnO2 terminated [001] surfaces without and with adsorbed O atom, O2- and O-terminated [011] surfaces) for both orthorhombic (a) and cubic (b) phases as functions of manganese and oxygen atoms chemical potential variations. Parameters for all lines on the left side of the figures are collected in Table 2. The encircled numbers point to lines, where metals or their oxides begin to precipitate: (1) metal La, (2) La2O3, (3) MnO, (4) Mn3O4, (5) Mn2O3, (6) MnO2, and (7) metal Mn. The right side of the figures contains a family of Δ*μ*O as functions of temperature at various oxygen gas pressures according to Eq. (50) and Table 1. The labels *m*  on the lines specifies the pressure according to: *p*O2 = 10*m* atm. Reprinted with permission

from Mastrikov et al., 2010. Copyright 2010 American Chemical Society.

(a)

(b)


Table 2. Parameters defining the surface Gibbs free energies Ω*i* (Eq. 13) as function of O and Mn chemical potentials : excesses , *<sup>i</sup> La O* and , *<sup>i</sup> La Mn* of O and Mn atoms in the surfaces with respect to La atoms (7), and free energy of formation *<sup>i</sup> La* (14) at Δ*μ*O = Δ*μ*Mn = 0 *e*V for the LaMnO3 surfaces under consideration. These results produced with plane wave BS and PW91 functional. Values of *<sup>i</sup> La* without brackets are for the orthorhombic phase, values in brackets are for the cubic phase. Reprinted with permission from Mastrikov et al., 2010 . Copyright 2010 American Chemical Society.


Table 3. The chemical potential correction (*e*V), Eq.(54), calculated for different oxides for both employed modeling techniques: (i) plane wave BS and PW91 functional and (ii) LCAO approach based on Gaussian-type atom-centered BS and hybrid B3LYP functional . The last line gives the average correction used in plotting the oxygen chemical potentials of the phase diagrams in Figures 1, 3, and 6.

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 La2O3 and Mn2O3 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 LaMnO3. 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

(J/m2) orientation termination

LaMnO3 surfaces under consideration. These results produced with plane wave BS and

brackets are for the cubic phase. Reprinted with permission from Mastrikov et al., 2010 .

La2O3 -0.41 -0.64 Mn2O3 -0.87 -0.53 MnO -0.52 -0.59 MnO2 -1.14 -0.09 Mn3O4 -0.90 -0.15 average -0.77 -0.40 Table 3. The chemical potential correction (*e*V), Eq.(54), calculated for different oxides for both employed modeling techniques: (i) plane wave BS and PW91 functional and (ii) LCAO approach based on Gaussian-type atom-centered BS and hybrid B3LYP functional . The last line gives the average correction used in plotting the oxygen chemical potentials of the

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 La2O3 and Mn2O3 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 LaMnO3. 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

Crystal Calculated <sup>0</sup>

[001] LaO -1 -0.5 5.38 (5.42) 5.67 (5.70) [001] LaO+O -0.875 -0.5 4.90 5.16 [001] MnO2 1 0.5 -3.02 (-2.93) -3.18 (-3.08) [001] MnO2+O 1.125 0.5 -3.16 (-3.53) -3.33 (-3.72) [110] LaMnO -1 0 6.13 (6.05) 4.56 (4.51) [110] O2 1 0 -1.16 (-1.26) -0.86 (-0.94) [110] O 0 0 1.09 0.81 Table 2. Parameters defining the surface Gibbs free energies Ω*i* (Eq. 13) as function of O and Mn chemical potentials : excesses , *<sup>i</sup> La O* and , *<sup>i</sup> La Mn* of O and Mn atoms in the surfaces with

without brackets are for the orthorhombic phase, values in

*O* 

Plane wave BS + PW91 LCAO + B3LYP

*i La* (*e*V/unit cell)

*La* (14) at Δ*μ*O = Δ*μ*Mn = 0 *e*V for the

*i La*

surface , *<sup>i</sup> La O* , *<sup>i</sup> La Mn*

respect to La atoms (7), and free energy of formation *<sup>i</sup>*

Copyright 2010 American Chemical Society.

phase diagrams in Figures 1, 3, and 6.

*La* 

PW91 functional. Values of *<sup>i</sup>*

Fig. 1. Phase diagrams calculated with plane wave BS and PW91 GGA functional: The regions of stability of LaMnO3 surfaces with different terminations (LaO- and MnO2 terminated [001] surfaces without and with adsorbed O atom, O2- and O-terminated [011] surfaces) for both orthorhombic (a) and cubic (b) phases as functions of manganese and oxygen atoms chemical potential variations. Parameters for all lines on the left side of the figures are collected in Table 2. The encircled numbers point to lines, where metals or their oxides begin to precipitate: (1) metal La, (2) La2O3, (3) MnO, (4) Mn3O4, (5) Mn2O3, (6) MnO2, and (7) metal Mn. The right side of the figures contains a family of Δ*μ*O as functions of temperature at various oxygen gas pressures according to Eq. (50) and Table 1. The labels *m*  on the lines specifies the pressure according to: *p*O2 = 10*m* atm. Reprinted with permission from Mastrikov et al., 2010. Copyright 2010 American Chemical Society.

Thermodynamics of ABO3-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 *pO2* =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

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, 2007b;

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

Figure 2 shows cross sections of the phase diagrams at *T* = 1200 K and *p*O2 = 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

As it can be seen from Figures 1 and 2, under fuel cell operational conditions in both LMO phases the MnO2-terminated [001] surface is the most stable one. In the orthorhombic phase it is the clean MnO2-terminated surface, whereas in the high-temperature cubic phase the most stable surface contains adsorbed O atoms. This indicates that under identical conditions higher O adsorbate coverage is expected for the cubic LMO phase. Modeling with plane-wave BS and PW91 functional suggest that, when LMO crystal is heated, precipitation of La2O3 or Mn3O4

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 MnO2-terminated (001) surfaces. The stability range is limited by lines 2, 3, and 5, which correspond to precipitation of La2O3, MnO, and Mn2O3. These are substantially different oxides than suggested above in computations performed with planewave BS and PW91 functional. Indeed, the gap between precipitation of La2O3 and Mn2O3 shifted down significantly. Now the boundary between stability regions for LaO- and

Johnston et al., 2004; Reuter & Schefer, 2001a) for non-magnetic oxides (e.g. SrTiO3).

clarify behavior of the SGFEs for surfaces with various terminations.

occurs, depending on chemical potentials variations during heating.

**4.2 Stability of LMO surface terminations: LCAO simulations** 

and/or higher temperatures.

the right side of the phase diagrams.

Fig. 2. Surface Gibbs Free Energies <sup>i</sup> for LaMnO3 in (a) orthorhombic and (b) cubic phases as functions of Δ*μ*Mn at *T* = 1200 K and *p*O2 = 0.2 atm Line numbers are the same as in Figure 4.1. The red lines indicate the most stable surface in the stability window between the precipitation lines for La2O3 and Mn2O3. Reprinted with permission from Mastrikov et al., 2010. Copyright 2010 American Chemical Society.


**Mn, eV**

**LaMnO**

2 5 4

for LaMnO3 in (a) orthorhombic and (b) cubic phases

3

**O2**

0

1

2

**<sup>i</sup>**

**, eV/unit cell**

3

4

5

6

0

1

3

2

3

**i, eV/unit cell**

4

5

6

as functions of Δ*μ*Mn at *T* = 1200 K and *p*O2 = 0.2 atm Line numbers are the same as in Figure 4.1. The red lines indicate the most stable surface in the stability window between the precipitation lines for La2O3 and Mn2O3. Reprinted with permission from Mastrikov et al.,

**LaO/O**

**MnO2**


**Mn, eV**

6

**LaO**


**Mn, eV**


**Mn, eV**

**LaO/O LaO O2**

6

**LaMnO**

2 5

4

Fig. 2. Surface Gibbs Free Energies <sup>i</sup>

2010. Copyright 2010 American Chemical Society.

0

1

**O MnO2 /O**

2

3

**i, eV/unit cell**

4

5

6

0

1

**O**

**MnO2 /O**

**pO2=0.2 atm, T=1200 K**

**MnO2**

2

3

**i, eV/unit cell**

4

5

6

(a)

(b)

 

 

> **pO2=0.2 atm, T=1200 K**

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 *pO2* =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, 2007b; Johnston et al., 2004; Reuter & Schefer, 2001a) for non-magnetic oxides (e.g. SrTiO3).

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 *p*O2 = 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, under fuel cell operational conditions in both LMO phases the MnO2-terminated [001] surface is the most stable one. In the orthorhombic phase it is the clean MnO2-terminated surface, whereas in the high-temperature cubic phase the most stable surface contains adsorbed O atoms. This indicates that under identical conditions higher O adsorbate coverage is expected for the cubic LMO phase. Modeling with plane-wave BS and PW91 functional suggest that, when LMO crystal is heated, precipitation of La2O3 or Mn3O4 occurs, depending on chemical potentials variations during heating.

### **4.2 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 MnO2-terminated (001) surfaces. The stability range is limited by lines 2, 3, and 5, which correspond to precipitation of La2O3, MnO, and Mn2O3. These are substantially different oxides than suggested above in computations performed with planewave BS and PW91 functional. Indeed, the gap between precipitation of La2O3 and Mn2O3 shifted down significantly. Now the boundary between stability regions for LaO- and

Thermodynamics of ABO3-Type Perovskite Surfaces 509

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 *xb*= 1/8. Thus, Figure 4 shows the section of surface stability phase diagram under ambient oxygen gas partial pressure *pO2*=0.2 atm. and three various temperatures: a) 300 K room 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) surfaces: 2 1 , , *x x s s MnO La Sr O* in the last case concentrations of Sr atoms in the surface layer were varied: *xs*= 0.25, 0.5, 0.75 and 1 (which simulates a segregation effect). Only three terminations appear at the shown sections: 2 0.75 0.25 *MnO La Sr O* , , and SrO . Here we accounted for precipitation of metals (La, Mn, Sr), Mn2O3 and La2O3 oxides, and LaMnO3 and SrMnO3 perovskites. These sections of the surface phase diagram indicate that the LSM crystal can be stable only within a small quadrangle region in the presented sections. At low, room temperature two of considered terminations - MnO2 and La0.75Sr0.25O - are stable. At the higher temperatures La0.75Sr0.25Oterminated surface gradually occupies a larger portion of the stability region. Already at SOFC operational temperatures (T1100 K) this termination becomes stable in the entire stability region. Thus, Sr dopant atoms in LSM cause a relative stabilization of the

Fig. 4. Sections of surface stability diagram for LSM (001) surface structures for O2 partial pressure *p*=0.2*p*0 and temperatures of (a) 300 K (RT), (b) 1100 K (SOFC operational temperature), and (c) 1500 K (sintering temperature) (Piskunov et al., 2008). The region, where LSM (*xb*= 1/8 ) is stable, is the shaded area between LaMnO3, La2O3, Mn2O3, and SrO precipitation lines. The numbers from 1 to 11 in the circles indicate precipitation lines for (1) La, (2) La2O3, (3) MnO, (4) Mn3O4, (5) Mn2O3, (6) MnO2, (7) Mn, (8) Sr, (9) SrO, (10) LaMnO3, (11) SrMnO3. (Some of the mentioned oxides are not considered in this Figure, but the numbering is designed to keep consistency of notations between figures.) Hollow arrows indicate the sides from respective precipitation lines where the precipitation occurs. Insets show magnified areas with the region of LSM stability (a shaded quadrangle). Reprinted with permission from Piskunov et al., 2008. Copyright 2008 American Physical Society.

**4.3 Stability of surface terminations for LSM: LCAO simulations** 

MnO2-terminated surfaces crosses the gap where LMO is stable, while PW91-GGA calculations described above and by Mastrikov et al., 2009, 2010 suggested that only the MnO2-terminated surface was stable. In calculations with hybrid B3LYP functional the MnO2-terminated surface seems to be stable, up to SOFC operational temperatures (1200 K) under ambient oxygen gas partial pressures (*pO2*=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 La2O3 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 <sup>0</sup> *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 SrTiO3, 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.

Fig. 3. Thermodynamic LaMnO3 [001] surface stability diagram as a function of O and Mn chemical potentials. It compares stabilities of both LaO- and MnO2-terminated [001] surfaces and accounts for precipitation conditions for (1) metal La, (2) La2O3, (3) MnO, (4) Mn3O4, (5) Mn2O3, (6) MnO2, and (7) metal Mn, the same set as at Figures 1 and 2. The stable region is shown as lightened area between precipitation lines 2,3, and 5. The right side shows a family of oxygen chemical potentials under different conditions. The label *m*  indicates the O2 gas partial pressure: 10*<sup>m</sup>* atm. Red line corresponds to oxygen partial pressure *p*=0.2*p*0 as in the ambient atmosphere.


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.

MnO2-terminated surfaces crosses the gap where LMO is stable, while PW91-GGA calculations described above and by Mastrikov et al., 2009, 2010 suggested that only the MnO2-terminated surface was stable. In calculations with hybrid B3LYP functional the MnO2-terminated surface seems to be stable, up to SOFC operational temperatures (1200 K) under ambient oxygen gas partial pressures (*pO2*=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

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

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 SrTiO3, such deviation drops down, from ~0.25 eV in LDA calculations (Johnston et al., 2004) to ~0.03

Fig. 3. Thermodynamic LaMnO3 [001] surface stability diagram as a function of O and Mn chemical potentials. It compares stabilities of both LaO- and MnO2-terminated [001] surfaces and accounts for precipitation conditions for (1) metal La, (2) La2O3, (3) MnO, (4) Mn3O4, (5) Mn2O3, (6) MnO2, and (7) metal Mn, the same set as at Figures 1 and 2. The stable region is shown as lightened area between precipitation lines 2,3, and 5. The right side shows a family of oxygen chemical potentials under different conditions. The label *m*  indicates the O2 gas partial pressure: 10*<sup>m</sup>* atm. Red line corresponds to oxygen partial

> (J/m2) orientation termination [001] LaO -1 -0.5 6.32 6.46 [001] MnO2 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

surface , *<sup>i</sup> La O* , *<sup>i</sup> La Mn*

produced with LCAO approach and hybrid B3LYP functional.

(54) in this case is noticeably smaller ( -0.40 eV) than

*i La* 

*i La* 

(*e*V/unit cell)

*O* 

eV in calculations (Heifets et al., 2007b) with the hybrid functional.

or La2O3 has to occur while LMO crystal is heated.

pressure *p*=0.2*p*0 as in the ambient atmosphere.

functional. The averaged correction <sup>0</sup>

### **4.3 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 *xb*= 1/8. Thus, Figure 4 shows the section of surface stability phase diagram under ambient oxygen gas partial pressure *pO2*=0.2 atm. and three various temperatures: a) 300 K room 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) surfaces: 2 1 , , *x x s s MnO La Sr O* in the last case concentrations of Sr atoms in the surface layer were varied: *xs*= 0.25, 0.5, 0.75 and 1 (which simulates a segregation effect). Only three terminations appear at the shown sections: 2 0.75 0.25 *MnO La Sr O* , , and SrO . Here we accounted for precipitation of metals (La, Mn, Sr), Mn2O3 and La2O3 oxides, and LaMnO3 and SrMnO3 perovskites. These sections of the surface phase diagram indicate that the LSM crystal can be stable only within a small quadrangle region in the presented sections. At low, room temperature two of considered terminations - MnO2 and La0.75Sr0.25O - are stable. At the higher temperatures La0.75Sr0.25Oterminated surface gradually occupies a larger portion of the stability region. Already at SOFC operational temperatures (T1100 K) this termination becomes stable in the entire stability region. Thus, Sr dopant atoms in LSM cause a relative stabilization of the

Fig. 4. Sections of surface stability diagram for LSM (001) surface structures for O2 partial pressure *p*=0.2*p*0 and temperatures of (a) 300 K (RT), (b) 1100 K (SOFC operational temperature), and (c) 1500 K (sintering temperature) (Piskunov et al., 2008). The region, where LSM (*xb*= 1/8 ) is stable, is the shaded area between LaMnO3, La2O3, Mn2O3, and SrO precipitation lines. The numbers from 1 to 11 in the circles indicate precipitation lines for (1) La, (2) La2O3, (3) MnO, (4) Mn3O4, (5) Mn2O3, (6) MnO2, (7) Mn, (8) Sr, (9) SrO, (10) LaMnO3, (11) SrMnO3. (Some of the mentioned oxides are not considered in this Figure, but the numbering is designed to keep consistency of notations between figures.) Hollow arrows indicate the sides from respective precipitation lines where the precipitation occurs. Insets show magnified areas with the region of LSM stability (a shaded quadrangle). Reprinted with permission from Piskunov et al., 2008. Copyright 2008 American Physical Society.

Thermodynamics of ABO3-Type Perovskite Surfaces 511

The adsorption energies for O atoms ( <sup>3</sup> : *surf Eads LaMnO Oads* and O2 molecules ( 3 2, : *surf Eads LaMnO O ads* ), as well as the surface and bulk formation energies

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 O2 molecule. In both cases the best adsorption site for both O atom and O2 molecule on MnO2-terminated surface is on top of surface Mn ion. Oxygen vacancies have smaller formation energy on MnO2-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.

0. 1.30 gaseous O2

I -1.1a) -0.42 1.36 tilted superoxide atop one

II -0.9a) -0.65 1.42 horizontal peroxide atop

III -0.9a) -0.69 1.41 horiz. peroxide atop Osurf

IV -2.8a) -1.19 1.50 tilted peroxide in .. V O

V -2.4a) -1.25 1.50 "vertical peroxide" in .. V O

V +0.9b) -1.25 1.50 TS of O- diffusion along

VIII -3.3b) -1.19 O ion in MnO2[001]

X -5.1b) -1.32 O ion in LaO[001] surface

IX -4.3b) -1.25 bulk O ion

Table 5. Bond lengths, Bader charges and "chemical assignment" of the different oxygen species. Experimental O-O bond lengths (NIST, 2010) for comparison: gaseous O2 1.21 Å, hydrogen superoxide radical HO2 1.33 Å, hydrogen peroxide H2O2 1.48Å. TS = transition state. Energies (compare Figure 4.6; for adsorbate coverage of 12.5 %): a) relative to gaseous O2 in triplet state over defect-free surface, b) relative to half a gaseous O2 over defect-free

Charge, e0

*bulk E LaMnO <sup>f</sup>* ) for oxygen vacancies are collected in Table

O-O distance,

<sup>Å</sup>"chemical assignment"

Mnsurf

one Mnsurf

TS of dissociation without .. VO , atop Osurf and bridging two Mnsurf

surface

adsorbed atop Mn

 adsorbed next to a surface vacancy

surface layer

layer

( ( : V ) 3 O

Label of configuration

surface.

*surf E LaMnO <sup>f</sup>* and ( : V ) 3 O


VII -0.5 a) -0.84 1.62

VI -1.1b) -0.62 O-

VI' -1.8 b) O-

<sup>1</sup> *x x s s La Sr O* - terminated surface with respect to the MnO2-terminated surface. However, as soon as Sr concentration *xs* at the 1 *x x s s 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 *pO2*=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 Piskunov et al, 2011), in order to check PW91-GGA prediction (discussed in previous subsection) that the MnO2-terminated surface is stabilized by adsorbed oxygen atoms.

Fig. 5. LCAO calculated cross-sections of surface stability diagram for LSM (001) surface structures along (a) SrO and (b) LaMnO3 precipitation lines for O2 partial pressure *p*=0.2*p*0 . Meaning of colors (terminations) and numbers (correspond to precipitation lines) are the same as in Figure 4.

### **4.4 Oxygen adsorption and vacancy formation in LMO**

As shown above, the MnO2-terminated (001) surface of LaMnO3 appears to be the most stable one. Therefore, we optimized the atomic structure of surface oxygen vacancies, as well as O atoms and O2 molecules adsorbed at different sites on this surface. For a comparison we also optimized the structure of oxygen vacancies in the LaMnO3 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. For example, the lowest energy for the adsorbed O2 molecule on MnO2-terminated surface is atop of Mn ion with the angle between O-O bond and Mn-O direction being ~50o .

<sup>1</sup> *x x s s La Sr O* - terminated surface with respect to the MnO2-terminated surface. However, as soon as Sr concentration *xs* at the 1 *x x s s La Sr O* -terminated surface becomes 0.5 or larger due

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 *pO2*=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 Piskunov et al, 2011), in order to check PW91-GGA prediction (discussed in previous subsection) that the MnO2-terminated surface is stabilized

Fig. 5. LCAO calculated cross-sections of surface stability diagram for LSM (001) surface structures along (a) SrO and (b) LaMnO3 precipitation lines for O2 partial pressure *p*=0.2*p*0 . Meaning of colors (terminations) and numbers (correspond to precipitation lines) are the

As shown above, the MnO2-terminated (001) surface of LaMnO3 appears to be the most stable one. Therefore, we optimized the atomic structure of surface oxygen vacancies, as well as O atoms and O2 molecules adsorbed at different sites on this surface. For a comparison we also optimized the structure of oxygen vacancies in the LaMnO3 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. For example, the lowest energy for the adsorbed O2 molecule on MnO2-terminated surface is atop of Mn ion with the

**4.4 Oxygen adsorption and vacancy formation in LMO** 

angle between O-O bond and Mn-O direction being ~50o .

to Sr segregation, such a surface becomes unstable.

(a) (b)

by adsorbed oxygen atoms.

same as in Figure 4.

The adsorption energies for O atoms ( <sup>3</sup> : *surf Eads LaMnO Oads* and O2 molecules ( 3 2, : *surf Eads LaMnO O ads* ), as well as the surface and bulk formation energies ( ( : V ) 3 O *surf E LaMnO <sup>f</sup>* and ( : V ) 3 O *bulk E LaMnO <sup>f</sup>* ) for oxygen vacancies are collected in 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 O2 molecule. In both cases the best adsorption site for both O atom and O2 molecule on MnO2-terminated surface is on top of surface Mn ion. Oxygen vacancies have smaller formation energy on MnO2-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.


Table 5. Bond lengths, Bader charges and "chemical assignment" of the different oxygen species. Experimental O-O bond lengths (NIST, 2010) for comparison: gaseous O2 1.21 Å, hydrogen superoxide radical HO2 1.33Å, hydrogen peroxide H2O2 1.48Å. TS = transition state. Energies (compare Figure 4.6; for adsorbate coverage of 12.5 %): a) relative to gaseous O2 in triplet state over defect-free surface, b) relative to half a gaseous O2 over defect-free surface.

Thermodynamics of ABO3-Type Perovskite Surfaces 513

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 O2 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 O2 molecule to the crystal surface. Similarly, the right hand panel refers to taking an O atom from a free O2 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.

comparison with Eqs. (60, 61).

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 O2 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 O2 molecules between oxygen gas and both crystal bulk and surfaces. For example, red arrow represents an adsorption of an O2 molecule atop surface Mn ion in the tilted position (configuration I) from oxygen gas under partial pressure *p*O2=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 O2 molecule. Similarly, the green arrow describes the adsorption of O atom atop Mn ion in MnO2 –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

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 *p*O2 = 1 atm. The formation of molecular adsorbates (superoxide I = red arrow, and peroxide II) is endergonic by ∆*rG* +2 eV per O2 since the entropy loss overcompensates the

is still slightly endergonic, by ∆*rG* +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 ∆*rG* -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.

(species VI, green arrow)

electronic energy gain. Even the formation of adsorbed atomic O-

Fig. 6. Spectrum of possible "one-particle" states, where "particles" are O atoms (right panel) and O2 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 MnO2[001]-terminated surface of LaMnO3, cf. Table 5. The axes on the left and right give the energy ∆*E*r relative to resting O2 molecule away from the surface (on the left) or an atom in such O2 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 *p*O2-dependence of the Gibbs energy of gaseous O2 (Table 1 and Eq.(50)), its energy scale refers to an O2 molecule on the left and to an O atom in an O2 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 *O*<sup>2</sup> on defect-free surface; green = formation of adsorbed O- atop Mn on defect-free surface; black = incorporation of oxygen into a surface oxygen vacancy.

**gaseous oxygen experimental** 

*O*  

**eV per O**

**3**

**H0**

 

**eV per O2**

Fig. 6. Spectrum of possible "one-particle" states, where "particles" are O atoms (right panel) and O2 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 MnO2[001]-terminated surface of LaMnO3, cf. Table 5. The axes on the left and right give the energy ∆*E*r relative to resting O2 molecule away from the surface (on the left) or an atom in such O2 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 *p*O2-dependence of the Gibbs energy of gaseous O2 (Table 1 and Eq.(50)), its energy scale refers to an O2 molecule on the left and to an O atom in an O2 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

**log pO2 =**

**-8**

**-6**

**-4**

**-2**

**0**

**I**

**IV**

 O 2 O Mn O

> O O Mn O

 O Mn O Mn

**III**<sup>O</sup>

**2**

**4**

**6**

500 1000 1500 2000

*T / K*

**-30 -20 -10**

**-2 -1**

**0 1**

**10**

**-4**

**-3**

**-2**

**-1**

**0**

**1**

**2**

formation of adsorbed O- atop Mn on defect-free surface; black = incorporation of oxygen

on defect-free surface; green =

**-4**

O2- ion in the bulk

**IX**

**VIII**

 Mn O Mn

 O Mn Mn **VI'**

 O O Mn O **VI**

 ½ O 2 O Mn O

**V**<sup>O</sup> Mn O Mn

**-3**

**-2**

 **G** 

*per* **O / eV**

**r**

**-1**

**0**

**1**

**2**

**3**

 O O Mn O

 **from DFT**

crystal and gas: red = formation of adsorbed superoxide *O*<sup>2</sup>

into a surface oxygen vacancy.

 

**-8**

**-6**

**-4**

 **G** 

*per* 

**O2 / eV**

**r** **-2**

**0**

 O O Mn O Mn

**VII**

**H0**

 **from DFT**

**II**

 O O Mn O Mn

> O Mn O Mn **V**

 O O O Mn O

**2**

**4**

**6**

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 O2 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 O2 molecule to the crystal surface. Similarly, the right hand panel refers to taking an O atom from a free O2 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 O2 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 O2 molecules between oxygen gas and both crystal bulk and surfaces. For example, red arrow represents an adsorption of an O2 molecule atop surface Mn ion in the tilted position (configuration I) from oxygen gas under partial pressure *p*O2=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 O2 molecule. Similarly, the green arrow describes the adsorption of O atom atop Mn ion in MnO2 –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 *p*O2 = 1 atm. The formation of molecular adsorbates (superoxide I = red arrow, and peroxide II) is endergonic by ∆*rG* +2 eV per O2 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 ∆*rG* +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 ∆*rG* -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.

Thermodynamics of ABO3-Type Perovskite Surfaces 515

Fister et al., 2008 and Herger 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.

EH is indebted to the Max Planck Institute for financial support through the honorary contract. EK, YM thank the EC FP7 NASA-OTM project (grant agreement N 228701) for partial financial support. S.P. is thankful for the financial support through the ESF project Nr. 2009/0216/1DP/1.1.1.2.0/09/APIA/VIAA/044. This study was supported by a grant of computer time at the EMS Laboratory at PNNL (Project No 42498). Authors are greatly indebted to R. Merkle, J. Fleig, R.A. Evarestov, D. Gryaznov, C. Noguera, and M. W. Finnis

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**6. Acknowledgements** 

for many stimulating discussions.

ISSN: 1550-235X

194, ISSN: 1432-2234

http://www.crystal.unito.it/

No.3, (July 2003), a.035418 , ISSN: 1550-235X

Washington, DC, USA, ISNB: 1563968312

**7. References** 

7690

235X
