Modeling of Solid Solutions of LacSr1-cMnO3 uing Wien2k and Crystal codes
One of the extensively studied perovskite-type materials, LacSr1-cMnO3 (LSM), is of special interest due to numerous applications, particularly as the cathode for solid oxide fuel cells [733].
LSM was investigated both theoretically [734, 735] and experimentally [736] with a focus on the chemical-bonding nature, magnetic properties, metal–insulator transitions, structural transformations, and surface properties.
Numerous efforts were undertaken to study the phase transformations and phase stability in LSM in a wide range of solid solutions. These materials exhibit a complicated dependence of the properties on the concentration of the Sr dopant and oxygen nonstoichiometry. Nowadays, it is well recognized that the dielectric, piezoelectric, and other LSM properties are entirely related to the phase equilibrium
and the phase separation that occurs under different thermodynamic conditions.
In what follows we discuss the results of ab initio supercell studies of the relative stability of different LSM phases [737]. To perform the ab initio calculations, the program packages WIEN-2k (http://www.wien2k.at) and CRYSTAL-03 [21] were used.
The spin-polarized DFT FPLAPW and LCAO electronic and atomic-structure calculations [737] use the exchange–correlation PBE functional.
The basis set of augmented plane waves combined with local orbitals .APWClo/ is used in the WIEN-2k code for solving the Kohn–Sham equations. In this method the unit-cell volume is divided into two regions: (I) nonoverlapping atomic spheres centered at the atomic sites and (II) an interstitial region. In the two types of regions, different basis sets are used. Inside atomic sphere i of radius Ri , where electrons behave as they were in a free atom, a linear combination of radial functions times spherical harmonics is used. In the interstitial region between these atomic spheres, where the electrons are more or less “free,” a plane-wave expansion is used. On the sphere boundary, the wavefunctions of both regions are matched by a value. The APW C lo basis set has a significantly smaller size than the basis set in the LAPW method, and thus the computational time is drastically reduced. Nevertheless, these two schemes converge practically to identical results. The convergence of the method is controlled by a cutoff parameter RmtKmax, where Rmt is the smallest atomic sphere radius in the unit cell and Kmax is the magnitude of the largest k-vector in the reciprocal space. To improve the convergence of the calculations, it is necessary to increase this product. A reasonably large Rmt can significantly reduce the computational time. A value of Rmt D 1.7 a.u. was chosen, and a plane-wave cutoff RmtKmax = 9.
The calculations [737] are performed for the high-temperature cubic phase of LaMnO3 (LMO)-based crystals doped with Sr, substituting for La atoms in different fractions. This substitution results in a charge-compensating hole formation. The formation of other defects like oxygen or metal vacancies is neglected.
To model the LaMnO3 doped by Sr (LSM), a 2 2 2 supercell is used, which consists of eight primitive unit cells and thus contains 40 atoms. The WIEN-2k code generates the k-mesh in the irreducible wedge of the Brillouin zone (BZ) on a special-point grid that is used in a modified tetrahedron integration scheme (500 k-points were used). The accuracy in total energy calculations was 10 4 Ry. Different configurations of Sr atoms substituting for La atoms allow ordered LSM
solid solutions to be modeled. In particular, La0:875Sr0:125MnO3 is typically used in fuel cells and thus is the subject of detailed thermodynamic study. The calculations are carried out for the ferromagnetic spin alignment (all Mn spins in the supercells are oriented in parallel), which results in a metallic character of the resistivity [736]. However, the resistivity of this phase is larger by three orders of magnitude than that of a typical metal. This is confirmed by FP–LAPWband-structure calculations: a very small density of states (DOS) in the vicinity of the Fermi level is observed. The model used is in agreementwith the calculations [735] where the ferromagnetic state was revealed for layers of a cubic La0:7Sr0:3MnO3.
In [737], DFT–B3LYP–LCAO calculations for LacSr.1 c/MnO3 mixed crystals (c D 0, 0.125, 0.5, 1.0) were performed, see Fig. 10.8. La, Sr, and Mn core electrons were described by Hay–Wadt small-core (HWSC) pseudopotentials [467]. For the oxygen atoms, an all-electron 8-411.1d/G basis was taken from previous MnO calculations [612], performed with basis-set (BS) optimization. For La, Mn, and Sr ions, BSs 411.1d/G, 411.311d/G, and 311.1d/G were taken from La2CuO4 [738] calculations, the CRYSTAL web site [21], and SrTiO3 calculations [590], respectively.
To achieve a high numerical accuracy in the lattice and in the BZ summations, the cutoff threshold parameters of the CRYSTAL03 code for Coulomb and exchange integrals evaluation to 7, 7, 7, 7, and 14, respectively, were taken. The integration over the BZ has been carried out on the Monkhorst–Pack grid of shrinking factor 8 (its increase up to 16 gave only a small change in the total energy per unit cell). The self-consistent procedure was considered as converged when the total energy in the two successive steps differs by less than 10 6 a.u.
As the first step, B3LYP spin-polarized LCAO calculations for the cubic LaMnO3 and SrMnO3 (with one formula unit per primitive cell) are performed, using the maximal spin projection Sz D 2 for four d-electrons of the Mn3C ion. As we have seen in Chap. 9, such a spin projection ensures the lowest total energy compared with Sz D 0; 1. The optimized cubic lattice constants are a D 3.967 A° and a D
3.840 A° for LaMnO3 and SrMnO3, respectively. These values are in a reasonable agreement with the experimental lattice constants a D 3.947 A° and a D 3.846 A° , respectively. The two optimized cubic lattice constants for LaMnO3 and SrMnO3 were used for calculating the lattice constants of their solid solutions according to Vegard’s law (linear dependence of the lattice parameters on the composition). As follows from the WIEN-2k calculations, this is fulfilled quite well in this system.
To predict the relative stability of different phases, which might appear in the quasibinary phase diagram of LSM solid solutions in a wide range of dopant concentrations, the statistical thermodynamic approach combined with the ab initio calculations was used. Such an approach has been successfully applied to different systems (see, e.g., [739–741] and references therein).
The standard periodic ab initio approach could be used only for ground-state energy calculations and ordered structures and thus does not allow prediction of thermodynamic stability of these phases as the temperature grows. This forces the problem to be reformulated so as to permit the extraction of the necessary energy parameters from the calculations for the ordered phases, and to apply these parameters to the study of the disordered or partly ordered solid solutions, in order to get information on the thermodynamic behavior of LSMsolid solutions. From the experimental data [742], it follows that in these solid solutions Sr atoms substitute for La at all atomic fractions, 0 < c < 1. Therefore, it is possible to consider the LSM solid solution as formed by La and Sr atom arrays occupying the sites of a simple cubic lattice immersed in the external field of the remaining lattice of Mn and O ions. The thermodynamics of such solid solutions can be formulated in terms of the effective interatomic mixing potential, which describes the interaction of La and Sr atoms embedded into the field of the remaining lattice. The study is based on the calculation of the relative stabilities of different ordered LSM cubic phases.
Figure 10.8 illustrates four phases: three of them (a, b, c) correspond to c D0:5; the last one (i) with c D7=8 corresponds to 12.5% Sr-doped LaMnO3. The remaining phases in Table 10.18 are the same as those used in BacSr.1 c/O3 calculations [739].
In the concentration wave (CW) theory, [743] the distribution of atoms A in a binary A–B alloy is described by a single occupancy probability function n.r/. This is the probability to find the atom A (La) at the site r of the crystalline lattice. The configurational part of the free energy of solid-solution formation (per atom) includes the internal formation energy U, the function n.r/, a concentration of particles La(A), and the effective interatomic potentials between La atoms (A) and Sr atoms (B), for details see [737].
To find the internal formation energies, which are differences between total energies of superstructures and the reference-state energy, the energy of a heterogeneous mixture, cLaMnO3 C (1-c)SrMnO3, has been chosen for the reference state. This energy is calculated as the sum of weighted (according to the atomic fractions) total energies of the two pure limiting phases, LaMnO3 and SrMnO3. From ab initio calculations, the total energies Etot and equilibrium lattice constant for all superstructures are obtained, Table 10.18. The internal formation energies for ordered phases (Table 10.19) are calculated by the definition
All these energies calculated using two very different DFT methods (FPLAPW and LCAO) are negative, that is, the formation of these ordered phases is energetically favorable with respect to their decomposition at T D 0 K into a heterogeneous mixture of LaMnO3 and SrMnO3 phases. It is easy to see from Table 10.18 that at the stoichiometric composition c D 1/2, the ordered phases a, b, and c
(which have different local impurity arrangements in the supercell) are energetically more favorable than other phases. Also, these three phases differ slightly between themselves in the formation energies.
The absolute values for formation energies, given in Table 10.19, are larger in LCAO calculations than those from FPLAPW calculations, especially for the “i” configuration. In order to check this point, additional LCAO calculations were performed using two different hybrid exchange–correlation functionals (B3LYP and B3PW) and optimized lattice constants in all four configurations. However, the results are very close to those obtained by using Vegard’s law. Therefore, the only reason for the energy discrepancy is the use of different computational schemes. However, the use of two different methods allows more reliable information to be obtained. In this particular case, both methods give qualitatively similar results. Using these formation energies, it is possible to calculate the temperature evolution of the long-range order (LRO) parameters of the superlattices. The LRO parameters characterize the atomic ordering in sublattices of the ABO3-type perovskite. Their values were taken to be equal to unity (which corresponds to completely ordered phases at stoichiometric compositions). The concentration was taken to be equal to the stoichiometric compositions of the corresponding phases. Finally, the free energy of formation of the phase La0:875Sr0:125MnO3 was calculated (we refer the reader to [737] for details).
The calculations of LSM solid solution considered here demonstrate the efficiency of use of thermodynamic formalism based on combination of the ab initio electronic-structure calculations as developed in [737, 739–741].The main feature of this approach is the treatment of ordered superstructures presenting the La–Sr sublattice immersed in the field of the remaining lattice formed byMn and O atoms. The total energy calculations allow the formation energies of these superstructures for different compositions to be found, and their competition at T D0 K to be analyzed. These calculations for a series of ordered structures permit extraction of the key energy parameters—the Fourier transforms of the mixing potential, and thus the free energy for temperature-induced partly disordered structures to be determined. Using the 12.5% Sr-doped LaMnO3, a thermodynamic analysis was performed [737]. It was predicted, in particular, that disordering of this phase with respect to the decomposition into the heterogeneous mixture of LaMnO3 and SrMnO3 can occur only at temperatures above the melting point. This is in contrast to a similar study of isostructural BacSr.1 c/TiO3 solid solution where below a certain temperature Ba impurities in SrTiO3 tend to form BaTiO3 nanoclusters.
In this chapter we have seen that in modern quantum chemistry of solids the first-principles periodic calculations are successfully extended to formally aperiodic systems—defective crystals and solid solutions. In the next chapter we consider the application of periodic models in the calculations of crystalline surfaces and adsorption.
Reference: Quantum Chemistry of Solids: LCAO Treatment of Crystals and Nanostructures
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