Quantum Chemistry of Solids / 24-Surface Modeling in LCAO Calculations of Metal Oxides

11.3 Slab Models of SrTiO3, SrZrO3 and LaMnO3 Surfaces 509

We discuss here in more detail the results of a hybrid HF-DFT LCAO comparative study of cubic SrZrO3 and SrTiO3 (001) surface properties in the single-slab model [825]. As known from [824], the consideration of systems with 7–8 atomic layers is su cient to reproduce the essential surface properties of cubic perovskites. Three di erent slab models have been used in [825]. The first (I) and the second (II) ones consist of 7 crystalline planes (either SrOor MO2-terminated, respectively) being symmetrical with respect to the central mirror plane but nonstoichiometric (see Fig. 11.4). The central layer is composed of MO2 (M = Ti, Zr) units in the model I and SrO units in the model II. Both models I and II have been applied for studying the surface properties of titanates by ab-initio calculations [832]. The asymmetric model III is stoichiometric and includes 4 SrO and 4 MO2 atomic planes. Accordingly, it is terminated by a SrO plane on one side and by a MO2 plane on the other side and there is no central atomic layer. The model III has been included in the simulation to investigate the influence of the stoichiometry-violation in the symmetrical models I and II on the calculated surface properties. For all slabs a 1 × 1 surface unit cell has been taken. For the 2D translations in slabs the experimental bulk lattice constants of SrZrO3 (4.154 ˚A) and SrTiO3 (3.900 ˚A) were used that does not di er significantly from DFT B3PW LCAO theoretical values (4.165 ˚A and 3.910 ˚A respectively).

In Table 11.20 are presented the results obtained for charge distribution and atomic relaxations for models I and II (Mulliken atomic charges and vertical atomic displacements are given). The displacements in the normal to the surface direction are given with respect to atomic positions in the unrelaxed bulk structure. The negative displacements correspond to the relaxation inwards, towards the bulk, while the positive ones relax outwards, i.e. to the vacuum side. It should be noted that the positions of atoms in the central plane exactly fit the corresponding bulk sites and can not relax due to the imposed mirror symmetry. There are no such restrictions for the model III. However, the absolute atomic displacements cannot be calculated unambiguously in this case due to uncertainty in the zero-level choice when the relaxed and unrelaxed structures are superimposed. This problem may be overcome by fixing the positions of the two middle layers at the bulk geometry. The model III does not satisfy this condition, so the corresponding absolute atomic displacements have not been calculated in model III.

The obtained displacement d values for the SrTiO3 practically coincide with the shifts from calculations [831] where almost the same LCAO procedure has been used. Calculated vertical shifts for the upper Sr atoms are noticeably greater than the shifts of other atoms in both models I and II, though they are negative in the first and positive in the second case. Also, for SrZrO3 they are larger than for SrTiO3. Displacements of other atoms are very similar on SrZrO3 and SrTirO3 surfaces, except the inward shift of the top oxygen atom on the ZrO2 surface that is much greater than the shift of the corresponding oxygen atom on TiO2 surface. The X-ray di raction data [828] for SrTiO3 (001) surface are presented in the last column of Table 11.20. Note that the experimental values correspond to 300 K whereas, the theoretical values correspond formally to 0 K. As well as in other ab-initio investigations, the satisfactory agreement between the theory and experiment was found only for the displacements of the top Sr atoms in the model I. The comparison of calculated and measured displacements for the deeper atoms in SrTiO3 slabs is di cult as the experimental data reveals large errors. Furthermore, in these slab simulations, as in most other

510 11 Surface Modeling in LCAO Calculations of Metal Oxides

Table 11.20. Mulliken charges1 q (|e|) and vertical atomic displacements d (˚A) for the SrOand TiO2-terminated slabs models (I and II, respectively)

1Bulk charges: SrZrO3: q(Sr) = 1.88, q(Zr) = 2.12, q(O) = –1.33; SrTiO3: q(Sr) = 1.87, q(Ti) = 2.53, q(O) = –1.47.

2 Experimental data [828] for SrTiO3 at 300 K.

3 Results of DFT PW calculations [829] using the optimized lattice constant a = 3.86 ˚A.

studies, the cubic bulk unit cell is used to generate the surface unit cell and the corresponding symmetry (in the surface plane) is kept fixed; whereas the unit cell must be doubled and symmetry should be lowered to tetrahedral for reproducing of the ferroelectric distortions. Most certainly, the surface phase transition can also take place in the case of zirconate due to the existence of the lower-symmetrical orthorhombic modification at room temperature. Table 11.20 also presents the results of a previous LDA PW simulation [829] for SrTiO3. LCAO displacements satisfactory correlate with the data of PW calculations.

The Mulliken atomic charges are also given in Table 11.20. These quantities can be used for analysis of the electron redistribution in the surface layers, which may be important for adsorption of another species on the surface. Table 11.20 clearly shows that the Sr atomic charge is close to the bulk one and exhibits insensitivity to the kind of M atom, type of termination and number of layers. On the contrary, calculated Ti and Zr charges give evidence of considerable covalency of M–O bonds in both crystals. It can also be seen that deviations of the surface oxygen charges from their bulk values are relatively large and exhibit the opposite sign for the two types of surface termination. The calculated Mulliken charges are very similar to the charges obtained in [833] via the Bader density decomposition.

When using models I and II the surface energy Es can be determined as a sum of two parts, Es = Eunrel + Erel(T ). The first term Eunrel is equal to one half of the energy for crystal cleavage into SrOand MO2-terminated slabs. It can be written as

11.3 Slab Models of SrTiO3, SrZrO3 and LaMnO3 Surfaces 511

where Eslabunrel(SrO) and Eslabunrel(MO2) are the energies for unrelaxed SrOand MO2- terminated slabs, Ebulk is a bulk energy per primitive cell, and S is the area of the

2D cell. The second term is the relaxation energy for SrOor MO2-terminated slabs and is computed as a di erence between the relaxed and unrelaxed slab energies:

where T = SrO or MO2.

The surface energy of model III (stoichiometric slab) equals the di erence between the energy of the slab and the quadruple energy of bulk primitive cell divided by 2S. The results for surface energy are summarized in Table 11.21.

Table 11.21. The calculated surface energies Es (J/m2) and bandgaps BG (eV)

1Experimental BG in bulk crystals: 5.9 eV for SrZrO3 [835], and 3.3 eV for SrTiO3 [834]. 2The results of DFT PW calculations [829] are given for comparison in parentheses.

For both crystals Es for SrO termination is slightly less than that for MO2 termination. It is important that the surface energies averaged over models I and II are very close to Es for the model III (this is valid both for relaxed and unrelaxed systems). The obtained values show that the surface energy of SrZrO3 is smaller than the surface energy of SrTiO3 crystal. The calculated value for the surface energy of SrTiO3 (1.24–1.25 J/m2, see Table 11.21) agrees satisfactorily with the value calculated in [829] using the DFT PW framework (1.36 J/m2).

In Table 11.21 the calculated bandgaps (BG) for slabs are compared. It is clearly seen that in all cases the BG for SrZrO3 systems is wider. This agrees with the larger ionicity of SrZrO3 in comparison with SrTiO3. One can notice that in the case of SrTiO3 the BG for the model I is markedly greater than that for models II and III. The substantial reduction of BG for the TiO2 surface relative to the bulk value is due to an extended shoulder in the valence-band electronic DOS (see below). The same picture has been found in DFT PW calculations [829], in spite of the fact that the LDA gaps are half of others (see Table 11.21).

In all figures below the zero energy is taken at the Fermi level of the bulk crystal (SrZrO3 or SrTiO3) and all curves were shifted so that the centers of O 1s bands

512 11 Surface Modeling in LCAO Calculations of Metal Oxides

coincide. Also, the DOS values have been reduced to one oxygen atom to ensure the same scale for bulk crystal and di erent slabs.

In Fig. 11.21 the total oxygen DOS for the surface model III and bulk crystal are compared. The bulk and slab VB have similar shapes and widths except the small areas at the top and bottom that reflect the surface contributions.

Fig. 11.21. Oxygen-projected VB DOS for bulk crystal and for surface model III (a – SrZrO3; b – SrTiO3)

These contributions are resolved in Fig. 11.22 where the DOS projected to the surface oxygens is plotted. Figures 11.22a and b demonstrate that the oxygen DOS distribution for SrO and TiO2 terminations are qualitatively di erent, whereas the distinctions between the model types (I and III or II and III) are less significant.

Fig. 11.22. Projected VB DOS for oxygen atom in the first surface layer for all three models (a – SrZrO3; b – SrTiO3). For the model III the first oxygen projection corresponds to the SrO-side and the second to the MO2-side

It is clearly seen in Fig. 11.22 that the new O-subbands appear in the slabs at the top of the bulk VB. The corresponding peaks for the oxygen on an SrO-terminated SrZrO3 surface (in models I and III) are significantly higher than the peaks for oxygen on ZrO2-terminated surfaces. Also, the charge of the O atom in the first layer in SrOterminated slab of a SrZrO3 crystal is noticeably lower than its bulk value (1.50 vs.

11.3 Slab Models of SrTiO3, SrZrO3 and LaMnO3 Surfaces 513

1.33 in Table 11.20). This can be explained by the larger ionicity of the M–O bond in the zirconate than in the titanate.

These features allow one to expect that a (001) SrO-terminated surface can accumulate the excess negative charge in the case of SrZrO3 crystal and it should be more basic in nature than a SrO-terminated surface of SrTiO3 crystal. In the case of SrTiO3 slabs the corresponding oxygen contribution to SrO surface states is lower, whereas the new surface states originated from the TiO2 termination form a low but extended shoulder that is alike in models II and III. The indicated shoulder reduces the bulk BG noticeably (see Table 11.21). This means that electronic shells on the TiO2-terminated SrTiO3 surface are more polarizable than electronic shells on the other surface types regarded. The similarities of the results obtained for the surface energy and DOS from the surface models I and II on the one hand, and from the model III on the other hand confirm the weak interaction between the opposite slab faces if its thickness is su ciently large, resulting in the mutual independence of two types of termination (SrO and MO2) in the surface models of perovskites considered.

One can conclude that disagreement between the theory and experiment for the vertical displacements of the SrTiO3 surface atoms does not depend on the actual (LCAO hybrid HF-DFT or PW DFT) ab-initio method used. The dependence on the slab model and thickness may also be excluded from the possible reasons. Most probably, this discrepancy can be explained by the possible surface phase transition or surface reconstruction under the experimental conditions. The ab-initio calculations using the extended tetragonal surface unit cell can resolve the indicated contradiction. The values obtained show that relaxation of the top surface atoms is more prominent in the case of SrZrO3 than in the case of SrTiO3, and the corresponding surface energy for zirconate is lower by about 0.1 J/m2. This fact may be attributed to the larger ionicity of SrZrO3 bulk crystal in comparison with SrTiO3 bulk crystal. The results under consideration give evidence that the SrO surface of a SrZrO3 crystal is more basic than the SrO surface of a SrTiO3 crystal. Based on this conclusion one can suppose, for example, a stronger interaction of oxygen atoms on the SrO surface with the H atoms of the adsorbed water molecules (with possible water ionization) occurs in the case of zirconate than in the case of titanate.

11.3.2 F Center on the SrTiO3 (001) Surface

In Sect. 10.3.1 we considered the calculations of the oxygen vacancy in a bulk SrTiO3 crystal (bulk F centers). The atomic and electronic structure of surface F centers is practically unknown.

Only a few theoretical papers deal with the analysis of the F centers on the TiO2- terminated SrTiO3(001) surface: one of them is based on semiempirical INDO calculations [846] and another one on GGA and LDA+U slab calculations [847]. In both papers, the structure of the F centers was studied for very high defect concentrations as the small 2D supercells were used. It has been demonstrated above in the bulk F center calculations that the results for low defect concentrations might be significantly di erent from those corresponding to the infinite dilution limit. Moreover, it is very likely that in the calculations [846] both lattice relaxation and vacancy-formation energies are not converged to the infinite dilution limit and no surface defect-migration energies were presented.

514 11 Surface Modeling in LCAO Calculations of Metal Oxides

In [848] the surface F centers on the TiO2-terminated unreconstructed (001)SrTiO3 surface are studied in more detail by the DFT method. A stoichiometric slab containing six atomic TiO2 and SrO planes (6 bulk unit cells , i.e. 30 atoms) has been chosen as a material model. The surface F center was studied for 2D supercells 2 × 2(120 atoms) and 3 × 3(270 atoms). The formation energy for relaxed (unrelaxed) surface oxygen vacancies was found to be 6.22 eV (8.86 eV) and 5.94 eV (8.81 eV) for 120atom and 270-atom supercells, respectively. Again, one must be aware of the fact that the present values might not be completely converged to the infinite-dilution limit. This surface vacancy-formation energy could be compared to the 7.73 eV and 7.17

axis. The conclusion could be drawn that the defect-formation energy on the TiO2- terminated surface is considerably smaller than in the bulk; it is roughly reduced by 1.5 eV or 20–25%. This is similar to what has been obtained for other oxides and it is due to the reduced coordination at the surface, [850]. The following relaxation of the atoms nearest to the vacancy and in the Ti–O–Ti surface chain is reported in [848]. The Ti and O nearest neighbors of the vacancy are displaced by 7%a0 and 4%a0, respectively (a0 is the parameter of bulk cubic lattice). The former displacements exceed by a factor of two those in the bulk and result from the half-coordination sphere left when the surface is formed. Unlike the bulk, where atoms move towards and outwards from the O vacancy, on the surface the direction of atomic displacements is more complicated.

A well-pronounced strong anisotropy in the atomic displacements along the Ti − VO − Ti axis was clearly observed. Atomic displacements in the smaller supercell show a nonmonotonic decay with the distance from the vacancy that is caused by the interference e ects on the border of the nearest supercells and demonstrates that this supercell is not big enough to avoid defect–defect interactions on the surface. The calculated activation energy for defect migration for the two supercells of 120 and 270 atoms is 0.19 and 0.11 eV, respectively. This demonstrates that (i) an increase of the distance between defects reduces the migration energy (due to the reduced repulsion energy between periodically distributed defects), and (ii) compared to the bulk migration, the defect-migration energy on the perovskite surface is largely reduced.

This is in line with the calculations of the F-center migration on the MgO surface where the activation energy obtained was also considerably smaller than in the bulk [850]. In addition, high vacancy mobility makes the surface reconstruction easier to see in the experiments. The di erence electronic density map for the surface F center on the 2 ×2 ×3 slab shows that the electronic density around it is more delocalized than that corresponding to the bulk F center. The calculated Bader e ective charges of the two nearest Ti atoms give an electronic density increase of 0.4 e per atom, whereas the rest of the missing O charge is spread in the vicinity of the vacancy. Finally, the defect ionization energy of the surface F center is almost half that in the bulk (0.25 eV vs.

0.49eV). Its dispersion is still not negligible (0.14 eV), being comparable with that

√√ √

for the bulk 2 2 ×2 2 ×2 2 supercell (0.15 eV). The single bulk F center in SrTiO3 is a small-radius defect. The surface F center is predicted to be more delocalized than that in the bulk. This is in agreement with previous findings for ionic oxides such as MgO, [850] or Al2O3 [849] and results from the reduced Madelung potential and atomic coordination at the surface.

11.3 Slab Models of SrTiO3, SrZrO3 and LaMnO3 Surfaces 515

The oxygen vacancies on the SrTiO3 (001) surface essentially change its properties and they have to be taken into account when adsorption modeling. Unfortunately, the surface-defect calculations in the slab model are complicated due to the delocalized nature of the surface oxygen vacancy electron states. Such a study requires both su ciently thick slabs (in modeling the surface) and large 2D supercells (in modeling the single vacancy on the surface).

While the theoretical study of the atomic and electronic structure of titanate surfaces is active, for manganites such a study began only in 2004, as is seen in the next section.

11.3.3 Slab Models of LaMnO3 Surfaces

Of primary interest for fuel-cell applications are the LaMnO3 (LMO) surface properties, e.g. the optimal positions for oxygen adsorption, its surface transport properties, as well as the charge-transfer behavior. In fuel-cell applications, the operational temperature is so high (T > 800 K) that the LMO unit cell is cubic and thus Jahn–Teller (JT) lattice deformation around Mn ions and related magnetic and orbital orderings no longer take place.

The first ab-initio calculation of cubic LaMnO3 surface properties [851] has been made by the HF LCAO method for (110) LMO surface in the single-slab model. The extension of these calculations to the (001) LMO surface was made in [852]. The bulk cubic unit cell atoms are distributed over atomic planes (normal to the surface direction) in the following way: for the (110) surface – O2–LaMnO–O2–LaMnO · · · ; for the (001) surface – LaO–MnO2–LaO–MnO2 · · · (see Fig. 11.4). In both cases the surfaces are polar.

In LMO (110) surface calculations [851] the O2-terminated slab consisting of seven planes was taken, i.e. four O2 planes and three LaMnO planes. Such a symmetrical slab is nonstoichiometric, i.e. it does not consist of an integer number of formula units. To restore the stoichiometry of the 7-plane slab, one oxygen atom has been removed from both O2-planes terminating the slab, i.e. the slab of three bulk primitive unit cells with periodically repeated surface oxygen vacancies was used. Such an approach is justified since it is well known that the polar surfaces are stabilized by surface defects and surface-atom relaxation. The surface energy Es (per surface unit cell) for such a slab equals:

where EOslab and Ebulk are the total energies for the O-terminated slab and bulk unit cells, respectively. On the other hand, the defectless surfaces could be modeled using nonstoichiometric slabs: O2-terminated or LaMnO-terminated. In this case the surface energy Es could be calculated as the average:

where EOslab2 and ELMOslab are the total energies of the slabs with O2 and LaMnO terminations, respectively.

The comparison of energies for the stoichiometric and nonstoichiometric 7-plane slabs permits conclusions to be drawn on the role of surface oxygen vacancies in

516 11 Surface Modeling in LCAO Calculations of Metal Oxides

surface stabilization. The O-terminated slab consists of three Mn-containing planes, each Mn atom is supposed to have oxidation state +3, i.e. 4 valence electrons are localized on Mn3+ ions. The UHF LCAO calculations have been performed for both the FM (the number of α − β electrons is 3 × 4 = 12) and the AFM (α − β = 4 electrons) states. Since the SCF calculations of these slabs are extremely slow and timeconsuming, to a first approximation, the relaxed surface geometry optimized by means of the classical shell model [853] based on atom–atom potentials has been used. Since one of the two surface O atoms (per 2D unit cell) is removed, the remaining atoms reveal displacements not only perpendicularly to the surface, but also inplane.

In slab calculations [852] the displacements in the first two top planes were taken into account, which are considerably larger than those in deeper planes. Similar to the bulk calculations, the FM stoichiometric (110) slab for the O termination turns out to be energetically more favorable than AFM, by 0.9 eV per Mn. The calculated O- terminated surface energy is 3.5 eV for the unrelaxed slab and 0.7 eV for the relaxed one, i.e. the relaxation energy (per surface unit cell) is 2.8 eV.

Table 11.22. The e ective atomic charges Q(|e|) in four unrelaxed top layers of the LMO (110) surface, both stoichiometric O-terminated (Q1), and nonstoichiometric O2- and LaMnO-terminated (Q2 and Q3), as well as the relevant deviations of plane charges, ∆Q(|e|) from those in the bulk (Q(La)=2.56; Q(Mn)=2.09; Q(O)=–1.55)

Plane Atom Q1 ∆Q1 Q2 ∆Q2 Q3 ∆Q3

IO –1.16 0.39 –0.77 1.56 –1.86(O) –1.40

1.73(La)

1.83(Mn)

Table 11.22 shows the e ective atomic charges Q of slab atoms and the deviation of the plane charges ∆Q (per unit cell) from those calculated with the bulk atomic charges. For example, in the plane II for the O-terminated stoichiometric surface the e ective charge of the La deviates from that in the bulk by 2.45 e – 2.56 e= –0.11 e. The e ective charges of Mn and O deviate from those in the bulk by 0.1 e and – 0.05 e, respectively. That is, the LaMnO plane’s charge deviates from that in the bulk by –0.06e. This value characterizes the charge redistribution in near-surface planes compared to the bulk. For stoichiometric surfaces the sum of ∆Q over all planes is zero. Thus, the e ective charge of a surface O atom is considerably reduced with respect to that in the bulk. The charges of both metal atoms in the second plane are slightly more positive, which is almost compensated by the charge of a more negative

11.3 Slab Models of SrTiO3, SrZrO3 and LaMnO3 Surfaces 517

O atom. Charges of the two O atoms in the third plane are close to those in the bulk. Surprisingly, in the central, fourth plane, the Mn charge turns out to be considerably (by 0.22 e) less positive than in the bulk. This is enhanced by the same trend for the O atoms, which results in a considerable (–0.35 e) e ective charge of the central plane with respect to the bulk.

The calculation for the surface energy for the 7-plane nonstoichiometric slabs, obtained using (11.13), gives Es=6.8 eV, i.e. about a factor of two larger than the surface energy of the stoichiometric slab. This demonstrates that oxygen vacancies strongly stabilize the polar (110) surface. The charge distribution for the nonstoichiometric O−2 and LaMnO-terminated surfaces is summarized in Table 11.22. As one can see, the e ective charge of Mn in the central plane of the O2-terminated surface with respect to the bulk value is –0.22 e, i.e. the same in absolute value but with the opposite sign to that on the O-terminated surface. The two surface O atoms share nearly the same charge as a single O atom possesses on the stoichiometric surface. Other planes are only slightly perturbed. Since the O2- and LaMnO-terminated surfaces complement each other, their total charges (with respect to those in the bulk) are expected to be equal in magnitude but of the opposite signs, which indeed takes place.

Calculations for the asymmetric, 8-plane stoichiometric LaO · · · MnO2 slab show that the surface energy is larger by 1 eV than that for the stoichiometric 7-plane slab with surface vacancies. This means that the surface vacancies serve as a better stabilizing factor than the charge redistribution near the surface compensating for the macroscopic dipole moment. From the relevant charge redistribution one can see that both surfaces of this 8-plane slab, the O2- and LaMnO-terminated ones, are charged strongly positively with respect to the bulk charges, whereas the internal planes are charged mostly slightly negatively.

The LMO (001) surface was modeled in [852] by similar symmetric 7-plane slabs with two kinds of terminations (LaO · · · LaO and MnO2 · · · MnO2) and an 8-plane LaO · · · MnO2 slab. The former is nonstoichiometric, the latter is stoichiometric (four bulk unit cells per surface unit cell). Unlike the (110) O-terminated surface, it is not easy to make the 7-plane (001) slab stoichiometric through introduction of surface vacancies. If we count the formal ionic charges, La3+, Mn3+, O2−, these two slabs have the total charges of 1 e (LaO) and –1 e (MnO2). In the SCF calculations, slabs are assumed to be neutral by definition, which results in the electronic-density redistribution between atoms in di erent planes.

The calculated defectless surface energy for the (001) using (11.13) equals 2.04 eV. This is smaller by a factor of 3 than that for a similar nonstoichiometric (110) surface.

Analysis of the charge redistribution for the LaOand MnO2-terminated (001) surfaces shows that the largest charge perturbation with respect to the bulk charges is observed for the top and bottom planes, similarly to the (110) case. The deviations of the e ective atomic charges starting the second plane are quite small, of the order of 0.1–0.2 e. These two surface terminations are complementary, which is why the total charges of the LaOand MnO2-terminated slabs are equal in absolute values and have opposite signs of –1 e and 1 e, respectively. As to the use of the asymmetric, stoichiometric 8-plane slab LaO · · · MnO2, its main problem was believed to arise due to the macroscopic polarization caused by a dipole moment perpendicular to the surface as a consequence of the alternating oppositely charged (1 e, –1 e) planes.

518 11 Surface Modeling in LCAO Calculations of Metal Oxides

However, as was demonstrated for the SrTiO3 (110) polar surface [854], the charge redistribution near the surface is able to cancel the macroscopic surface polarization. For this, the absolute value of the e ective charge of the top plane should be smaller than that in the slab center. As was found in [852], the e ective charge of the fourth MnO2-plane of the 8-plane slab is –0.86 e, whereas that of the top LaO-plane 0.37 e, i.e. the condition is fulfilled and the (001) plane is stable. The same is true for the 8-plane (110) slab. For the 8-plane (001) slab only the top and bottom planes of a slab are considerably perturbed as compared to the bulk charges, charges in all the other planes are only slightly modified. The e ective charges of the LaO and MnO2 (top and bottom) planes are close in magnitude and opposite in sign. The relevant surface energy calculated using (11.2) turns out to be as small as 1.02 eV/cell for the FM state. This means that the (001) surface is stable and energetically favorable. The (0 0 1) surface energy in the AFM state is formally even lower, 0.83 eV. However, the relevant total energies for the bulk unit cell (per formula unit) and the slab in the AFM state are higher (by 0.22 and 0.56 eV, respectively) than the FM states.

The LCAO single-slab calculations [852] of the electronic structure of the polar LnMnO3 (001) and (110) surfaces clearly demonstrate that the stoichiometric slabs have considerably lower energies than the nonstoichiometric ones. It should be stressed that the structural oxygen vacancies are energetically required and hence are essential elements of the (110) polar surface structure. Their formation makes the (110) slabs stoichiometric and energetically more favorable than the stoichiometric slabs stabilized by the near-surface electronic density redistribution necessary to compensate the macroscopic dipole moment perpendicular to the asymmetric LaO · · · MnO2 surfaces.

The comparative DFT B3PW LCAO (single-slab) and GGA PW (periodic-slab) calculations of LaMnO3 (001) and (110) surfaces in slab models has been made in [855], we refer the reader to this publication for details of the calculations. It is important that the B3PW LCAO and GGA PW calculations for the LaMnO3 surfaces show reasonable agreement for atomic displacements, e ective charges (in the PW calculations, Bader analysis was used), and surface energies. The e ective charges of surface atoms markedly depend on the surface relaxation and less on the particular (FM or AAF) magnetic configuration. The HF-based conclusion (see discussion above) that the polar (001) surface is energetically more favorable than the (110) one was confirmed. This conclusion is important for the modeling of surface adsorption and LaMnO3 reactivity. The surface relaxation energy is typically of the order of 1–1.5 eV (per square unit a2, where a is the cubic lattice parameter) i.e. much larger than the tiny di erence between various magnetic structures. Moreover, the calculated surface energy for the slab built from orthorhombic unit cells is close to and even slightly larger than that for the cubic unit cells. These two facts justify the use in surface and adsorption modeling of slabs built from the cubic cells. This is a very important observation since the detailed adsorption and migration modeling, e.g., for surface O atoms at moderate coverages that is relevant for fuel-cell applications, is very time consuming even for the smallest slab thicknesses.

The discussion of LMO surface modeling concludes Part II (Applications) of this book. We tried to demonstrate, via numerous examples, the e ciency of modern quantum-chemical approaches to calculate the properties of crystalline solids – bulk and defective crystals, surfaces and adsorption.