w.elsevier.com/locate/ssi

Solid State Ionics 177

Thermodynamic stability and disordering in LacSr1�cMnO3 solid solutions

D. Fuks a, L. Bakaleinikov b,c, E.A. Kotomin d,e, J. Felsteiner f, A. Gordon b,

R.A. Evarestov d,g, D. Gryaznov d,e,*, J. Maier d

a Department of Materials Engineering, Ben Gurion University, P.O.B. 653, 84105 Beer Sheva, Israelb Department of Mathematics and Physics, Haifa University, 36006 Tivon, Israel

c Ioffe Physico-Technical Institute, Polytechnicheskaya str. 26, St. Petersburg 194021, Russiad Max-Planck-Institut fur Festkorperforschung, Heisenbergstr.1, D-70569 Stuttgart, Germany

e Institute for Solid State Physics, Kengaraga str. 8, Riga LV-1063, Latviaf Department of Physics, Technion, 32000 Haifa, Israel

g Department of Quantum Chemistry, St. Petersburg University, St. Peterhof 198504, St. Petersburg, Russia

Received 17 August 2005; received in revised form 17 October 2005; accepted 17 October 2005

Abstract

Two different types of ab initio electronic and atomic structure calculations for ordered solid solutions are combined with solid solution

thermodynamics in a study of the Sr-doped LaMnO3. Unlike Ba in the isostructural (Sr,Ba)TiO3, Sr aggregation in the LaMnO3 matrix potentially

leading to decomposition into heterogeneous mixture of LaMnO3 and SrMnO3 phases is energetically unfavorable. We demonstrate that for a

particular solid solution with 12.5% Sr the order–disorder transition occurs only at very high temperatures otherwise Sr is supposed to be

periodically distributed in the LaMnO3 matrix.

D 2005 Elsevier B.V. All rights reserved.

PACS: 64.75.+g; 64.60.-i

Keywords: ab initio calculations; LacSr(1�c)MnO3; Thermodynamic stability; Effective mixing interatomic potential; Concentration wave theory; Internal formation

energy

1. Introduction

During the last decade transition metal oxides have

attracted a great attention, because of a wide range of unusual

and sometimes unexpected properties. These materials dem-

onstrate in particular mixed conductivity, ferroelectricity,

ferromagnetism and even high-temperature superconductivity.

Recently, the effect of the colossal magnetoresistance was

discovered in some manganese oxides. One of the extensively

studied perovskite-type materials, LacSr(1�c)MnO3 (LSM), is

of special interest due to numerous applications [1,2],

particularly as cathode for solid oxide fuel cells [3]. LSM

was investigated recently both theoretically [4,5] and exper-

imentally [6] with a focus on the chemical bonding nature,

0167-2738/$ - see front matter D 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.ssi.2005.10.014

* Corresponding author. Max-Planck-Institut fur Festkoerperforschung, Hei-

senbergstr.1, D-70569 Stuttgart, Germany. Tel.: +49 711 689 1771; fax: +49

711 689 1722.

E-mail address: d.gryaznov@fkf.mpg.de (D. Gryaznov).

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 (e.g., [7]).

These materials exhibit a complicated dependence of the

properties on the concentration of the Sr dopant and oxygen

nonstoichiometry. For instance, it was shown [8] that the

properties of the ferromagnetic insulating state at low

temperatures hardly depend on the dopant element fraction.

There is a very strong concentration dependence of the

specific heat and the magnetization. While the specific heat

anomaly strongly increases upon doping due to ferromagnetic

order, the jumps of both entropy and magnetization at the

charge order transitions decrease drastically.

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 this paper, we

report the results of ab initio studies of the relative stability of

(2006) 217 – 222

ww

Table 1

Total energies per 2�2�2 cell, Etot in Eq. (7), stoichiometric LacSr(1�c)MnO3

compositions cst and equilibrium lattice parameters, aeq for different super

structures structures (a– i) plotted in Fig. 1 and Ref. [24]

Structure cst WIEN-2k CRYSTAL-03

Etot, Ry aeq, A E tot, Ry aeq, A

a 1/2 � 115574.4034 3.881 � 5771.8383 3.903

b 1/2 � 115574.4248 3.881 � 5771.8688 3.903

c 1/2 � 115574.4178 3.879 � 5771.8812 3.903

d 1/4 � 94302.8877 3.862

e 3/4 � 136845.8880 3.901

f 1/4 � 94302.8725 3.865

g 3/4 � 136845.8763 3.901

h 1/8 � 83667.1051 3.855 (3.828)*

i 7/8 � 147481.5822 3.914 � 5776.8921 3.951

LaMnO3 – � 158117.2795 3.921 3.947** � 5778.3816 3.967

SrMnO3 – � 73031.3269 3.848 (3.806)* � 5764.9568 3.840

* Experimental data from Ref. [18].

** Experimental data from Ref. [4].

D. Fuks et al. / Solid State Ionics 177 (2006) 217–222218

different LSM phases. We analyze the structural disordering of

La0.875Sr0.125MnO3 as the temperature grows. For this purpose

we calculate the temperature dependence of the long-range

order parameters.

2. Computational details

To perform the ab initio calculations we used the program

packages WIEN-2k [9] and CRYSTAL-03 [10]. The first

package performs the electronic and atomic structure calcula-

tions based on the spin polarized Density Functional Theory

(DFT). We have used here the exchange-correlation functional

in a generalized gradient approximation (GGA) [11]. The basis

set of Augmented Plane Waves combined with local orbitals

(APW+lo) 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) non-overlapping 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 wave functions of both regions are matched by

a value. The APW+lo basis set has a significantly smaller size

than the basis set in 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 cut-off 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. In our calculation

we choose a muffin-tin MT radius Rmt=1.7 a.u. and a plane-

wave cut-off RmtKmax=9.

We performed the calculations for the high-temperature

cubic phase of LaMnO3 (LMO)-based crystals doped with Sr,

substituting for La atoms in different fractions. This substitu-

tion results in a charge compensating hole formation. Accord-

ing to experimental studies [12,13], formation of other defects

like oxygen or metal vacancies is neglected. The primitive unit

cell of pure LaMnO3 consists of five atoms. To model the

LaMnO3 doped by Sr (LSM), we used a 2�2�2 supercell,

which consists of eight primitive unit cells and thus contains

5�8=40 atoms. WIEN-2k code generates the k-mesh in the

irreducible wedge of the Brillouin Zone (BZ) on a special point

grid which is used in a modified tetrahedron integration scheme

[14]. In our case 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 us to model ordered

LSM solid solutions. In particular, La0.875Sr0.125MnO3 is

typically used in fuel cells and thus is the subject of our

detailed thermodynamic study. The calculations are carried out

for the ferromagnetic spin alignment (all Mn spins in the

supercells are oriented in parallel), what results in a metallic

character of the resistivity [6]. However, the resistivity of this

phase is by three orders of magnitude larger than that of typical

metal. This is confirmed by our band structure calculations: we

observe very small density of states (DOS) in the vicinity of the

Fermi level. Our model is in agreement with the calculations

[5] where the ferromagnetic state was revealed for layers of a

cubic La0.7Sr0.3MnO3.

We performed also calculations for LSM mixed crystals

(c =0, 0.125, 0.5, 1.0) using the DFT-LCAO formalism as

implemented into the CRYSTAL-03 computer code [10]. In

this formalism the crystalline orbitals are presented in a form of

Linear Combination of Atomic Orbitals (LCAO). The atomic

orbitals themselves are expanded into a set of localized atom

centered Gaussian-type orbitals (GTO). We applied the hybrid

B3LYP functional [15] which uses in the exchange part the

mixture of the exact Fock (20%) and Becke (80%) 3-parameter

exchange, whereas in the correlation part Lee–Yang–Parr [16]

non-local GGA functional is employed.

La, Sr and Mn core electrons were described by Hay–Wadt

small core (HWSC) pseudopotentials [17]. For the oxygen

atoms an all-electron 8–411(1d)G basis was taken from

previous MnO calculations [18], 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 [19]

calculations, the CRYSTAL web site [20] and SrTiO3 calcula-

tions [21], respectively.

To achieve a high numerical accuracy in the lattice and in

the BZ summations, we set the cut-off threshold parameters of

CRYSTAL-03 code [10] for Coulomb and exchange integrals

evaluation (ITOL1 to ITOL5) to 7, 7, 7, 7 and 14, respectively.

The integration over the BZ has been carried out on the Pack–

Monkhorst grid [22] 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, we performed B3LYP spin-polarized

LCAO calculations for the cubic LaMnO3 and SrMnO3 (with

-

Fig. 1. Four structures used to calculate the values V(0), V1, V2 and V3

necessary for a study of the disordering in La0.875Sr0.125MnO3.

D. Fuks et al. / Solid State Ionics 177 (2006) 217–222 219

one formula unit per primitive cell), using the maximal spin

projection Sz=2 for four d-electrons of the Mn3+ ion. Such a

spin projection ensures the lowest total energy compared with

Sz=0, 1 [23].

We optimized the cubic lattice constants to be a0=3.967 A

and a0=3.840 A for LaMnO3 and SrMnO3, respectively

(Table 1). These values are in a reasonable agreement with

the experimental lattice constants. Since the structure optimi-

zation in CRYSTAL code is quite time-consuming, 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.

3. Thermodynamic analysis

To predict the relative stability of different phases, which

might appear in the quasi-binary phase diagram of

LacSr(1�c)MnO3 solid solutions in a wide range of dopant

concentrations, the statistical thermodynamic approach com-

bined with the ab initio calculations was used. We have

successfully applied such an approach recently to different

systems (see, e.g., [24–26] 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 forced us to

reformulate the problem as to permitting us to extract 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 LSM solid

solution. From the experimental data [27] 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 solution 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. This may be successfully done in the framework of

Concentration Wave (CW) theory [28]. Our study is based on

the calculation of the relative stabilities of different ordered

LSM cubic phases. The phases under consideration are

isostructural to those reported in our recent study of BacSr(1�c)-

TiO3 solid solutions [24], we use hereafter the same notations

and expressions for the internal formation energies. These

notations and phase compositions are also presented in Table 1,

whereas Fig. 1 illustrates four phases: three of them (a, b, c)

correspond to c =0.5, the last one (i) with c =7/8 corresponds to

12.5% Sr-doped LaMnO3.

In the CW theory the distribution of atoms A in a binary A–

B alloy is described by a single occupancy probability function

n rYð Þ. This is the probability to find the atom A (La) at the site rY

of the crystalline lattice. The configurational part of free energy

of solid solution formation (per atom) is given by

DF ¼ DU � TDS ¼ 1

2N

XrY; rYV

rY rYV

VV rY; rYVð Þn rYð Þn rYVð Þ

þ kTXrY

n rYð ÞIln n rYð Þ þ 1� n rYð Þð ÞIln 1� n rYð Þð Þ½ �:

ð1Þ

Here VV rY; rYð Þ is the effective mixing interatomic potential

which is connected with the effective interatomic potentials

between La atoms (A), VAA rY; rYVð Þ, between Sr atoms

(B),VBB rY; rYVð Þ, and between La and Sr atoms, VAB rY; rYVð Þ, bythe relation

VV rY; rYVð Þ ¼ VAA rY; rYVð Þ þ VBB rY; rYVð Þ � 2VAB rY; rYVð Þ: ð2Þ

The summation in Eq. (1) is performed over the positions of

sites of the Ising lattice (a simple cubic lattice in our case) with

atoms La and Sr distributed on it. The function n rYð Þdetermining the distribution of solute atoms in the ordered

phase may be expanded in the Fourier series. It is presented as

a superposition of CWs,

n rYð Þ ¼ cA þ 1

2

Xjs;S

Q kYjs

� �eikY

js rYþ Q4 k

Yjs

� �e�ik

Yjs rY

h i; ð3Þ

where cA is a concentration of particles A, eikYjs rYis a CW, k

Yjs is a

nonzero wave vector defined in the first BZ for the Ising lattice

of the disordered binary alloy, the index { js} numerates the

wave vectors in the BZ, that belong to the star s, and Q kYjs

� �is

the CW amplitude. As shown in Ref. [28], all Q kYjs

� �are linear

functions of the long-range order (LRO) parameters of the

superlattices, gs, that may be formed on the basis of the Ising

lattice of the disordered solid solution:

Q kYjs

� �¼ gscs jsð Þ: ð4Þ

Fig. 2. Formation energies for nine LacSr(1�c)MnO3 ordered phases (per atom

on the La/Sr sublattice) as calculated using the WIEN-2k code. The notations

are taken from Ref. [24], four phases are given in Fig. 1, and phase

compositions are also presented in Table 1.

D. Fuks et al. / Solid State Ionics 177 (2006) 217–222220

The first part of Eq. (1), the formation energy, DU, for the

ordered superstructure reads [28]

DU ¼ 1

2VV 0ð ÞIc2 þ 1

2

Xc2s jsð Þg2s VV k

Yjs

� �: ð5Þ

Here VV kYjs

� �is the Fourier transform of the interatomic mixing

potential, VV 0ð Þ is Fourier transform for kYjs ¼ 0, c is the

concentration of La atoms in the superstructure. The cs( js) arecoefficients that determine the symmetry of the probabilities

n rYð Þ describing the distribution of atoms A in a binary A–B

solid solution. The superstructure vectors kYjs define the

positions of the additional X-ray reflections that appear when

the binary system changes from a disordered state on the Ising

lattice to an ordered or partly ordered state. These vectors are

chosen according to the Lifshitz criterion [29,30].

All possible ordered structures on the simple cubic lattice

that satisfy the Lifshitz criterion are displayed in Fig. 1 of Ref.

[24], which deals with the isostructural BacSr(1�c)TiO3 solid

solution. Occupancy probabilities, n rYð Þ, stoichiometric compo-

sitions cst, the coefficients cs( js) and the relations for the

formation energies DU, for the ordered phases in LacSr(1�c)

MnO3 solid solutions are given in Table 1 of [24].

To find the internal formation energies, which are differences

between total energies of superstructures and the reference state

energy, we have chosen for the reference state the energy of a

heterogeneous mixture, cLaMnO3+(1�c)SrMnO3. This ener-

gy is calculated as the sum of weighted (according to the atomic

fractions) total energies of the two pure limiting phases,

LaMnO3 and SrMnO3. From our ab initio calculations we

obtained the total energies Etot and equilibrium lattice constant

for all superstructures (Table 1). The internal formation energies

DU for ordered phases (Table 2) are calculated by the definition

DU ¼ Etot � cELaMnO3

tot þ 1� cð ÞESrMnO3

tot

� �: ð7Þ

All these energies calculated using two very different methods

are negative, i.e. the formation of these ordered phases is

energetically favorable with respect to their decomposition at

T=0 K into heterogeneous mixture of LaMnO3 and SrMnO3

phases. The formation energies (per atom on the La/Sr

sublattice) for all phases calculated using the WIEN-2k code

are presented also in Fig. 2. It is easy to see that, for example, at

the stoichiometric composition cst =1/2 the ordered phases a, b,

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.

Table 2

Formation energies of different superstructures, Eq. (7), in electronvolts per

atom on La/Sr sublattice, as calculated by means of the WIEN-2k and

CRYSTAL-03 codes

Configuration � DU, eV

WIEN-2k CRYSTAL-03

a 0.170 0.288

b 0.207 0.339

c 0.195 0.360

i 0.080 0.320

The internal formation energies given by Eq. (5) for the ordered

superstructures shown in Fig. 1 read as

DUa ¼1

2VV 0ð Þc2a þ

1

8V1V1g

21; ð8aÞ

DUb ¼1

2VV 0ð Þc2b þ

1

8V2V2g

22; ð8bÞ

DUc ¼1

2VV 0ð Þc2c þ

1

8V3V3g

23; ð8cÞ

DUi ¼1

2VV 0ð Þc2i þ 3

128VV 1g

21 þ 3

128VV 2g

22 þ 1

128VV 3g

23:

ð8dÞ

The values for DUa, DUb, DUc and DUi are collected in

Table 2, whereas the Fourier transforms VV 0ð Þ, VV 1, VV 2 and VV 3

and the k-vector stars corresponding to these structures are

discussed in Ref. [24]. The CRYSTAL-03 absolute values are

larger than those from WIEN-2k calculations, especially for the

‘‘i’’ configuration. In order to check this point, we performed

additional CRYSTAL-03 calculations using two different

hybrid exchange-correlation functionals (B3LYP and B3PW)

and optimized the lattice constants in all four configurations.

However, the results are very close to those obtained by using

the Vegard’s law. The only reason of the energy discrepancy

obtained by WIEN-2k and CRYSTAL-03 codes is due to the

use of different approximations employed in these codes: MT

spheres and plane waves in WIEN-2k versus localized orbitals

and pseudopotentials in CRYSTAL-03. The use of two

different methods allows us to obtain more reliable informa-

tion. In this particular case, both methods give qualitatively

similar results.

The LRO parameters here characterize the atomic ordering

in sublattices of the ABO3-type perovskite. Putting these values

equal to unity (which corresponds to completely ordered

phases at stoichiometric compositions) and choosing the

concentration equal to the stoichiometric compositions of the

Table 3

The Fourier transforms of the mixing potential (in electronvolts per atom on La/

Sr sublattice) derived using Eqs. (8a)– (8d) from data in Table 2

Mixing potential WIEN-2k CRYSTAL-03

V(0) 9.6I10� 3 � 0.548

V1 � 1.372 � 1.756

V2 � 1.664 � 2.168

V3 � 1.569 � 2.332

Fig. 3. Free energy for La0.875Sr0.125MnO3 phase (in electronvolts) minimized

in the space of three LRO parameters (solid line) and that for the disordered

phase (dashed) as a function of temperature.

Fig. 4. Temperature dependence of LRO parameters for La0.875Sr0.125MnO3.

The vertical dotted line corresponds to the temperature of the order–disorder

phase transformation.

D. Fuks et al. / Solid State Ionics 177 (2006) 217–222 221

corresponding phases, ca=cb=cc=0.5, and ci =7/8, one can

solve Eqs. (8a)–(8d) and get the values of VV 0ð Þ, VV 1, and VV 3

presented in Table 3.

Let us study now the temperature evolution of the LRO

parameters for a particular La0.875Sr0.125MnO3 phase. The

occupancy probability for this superstructure at stoichiometric

composition is [24]

n rYð Þ ¼ 7=8 � 1=8 Ig1 I eipx þ eipy þ eipz� �

� 1=8 Ig2

I eip xþyð Þ þ eip xþzð Þ þ eip yþzð Þh i

� 1=8 Ig3 I eip xþyþzð Þ

ð9Þ

Using this expression, one can calculate the entropy term in the

configurational part of the free energy of phase formation,

Eq. (1)

� TDS ¼ kT I1

8IX4i¼1

kiIniIln nið Þ þ kiI 1� nið ÞIln 1� nið Þf g;

ð10Þwhere ki=1, for i=1, 4 and ki =3, for i=2, 3, and

n1 ¼ 78� 3

8g1 � 3

8g2 � 1

8g3; n2¼ 7

8� 1

8g1 þ 1

8g2 þ 1

8g3

n3 ¼ 78þ 1

8g1 þ 1

8g2 � 1

8g3; n4¼ 7

8þ 3

8g1 � 3

8g2 þ 1

8g3:

Finally, the free energy of formation of this phase at

stoichiometric composition can be presented as

DF ¼ 1

2VV 0ð ÞI 7

8

� 2

þ 3

128V1V1g

21 þ

3

128V2V2g

22

þ 1

128V3V3g

23 � TDS g1; g2; g3ð Þ ð11Þ

where the term TDS is defined by Eq. (10). We believe that

contributions into the free energy of formation of vibrational

entropies for ordered and disordered phases under study are

similar and largely cancel each other.

4. Results

The three equilibrium LRO parameters g1, g2, g3 at each

temperature T are determined by the absolute minimum of the

functional, Eq. (11), in three-dimensional space of these

parameters, i.e. in 3D cube [0,1]� [0,1]� [0,1]. As temperature

increases, the position of the minimum on the free energy

surface moves in this space in such a way that the LRO

parameters, corresponding to this minimum, decrease becom-

ing less than unity.

The analysis of Eq. (11) topology in the (g1, g2, g3) space

based on the WIEN-2k calculations shows that there is only

one minimum in the temperature range between 0 and 2600 K.

Above this temperature another minimum arises. This new

local minimum is situated at the (0, 0, 0) point and thus

corresponds to completely disordered phase. Up to T=2890 K

the free energy at (0, 0, 0) is larger than that for another

minimum, i.e. the disordered state is unstable as compared with

partly ordered state characterized by 0<g1, g2, g3<1. But at

T >2890 K the disordered state becomes stable, and thus the

phase La0.875Sr0.125MnO3 is expected to be totally disordered.

In Fig. 3 the free energy minimized in the (g1, g2, g3) space isplotted as a function of temperature (solid line). The dotted line

gives the free energy for the completely disordered phase. As

one can see, they intersect at T=2890 K. Similar CRYSTAL-

03 calculations suggest T=3200 K for the appearance of the

second minimum, and T=3817 K for the order-disorder

transition. Obviously, these temperatures are even higher than

those obtained by WIEN-2k.

In Fig. 4 the temperature dependence of the LRO

parameters characterizing the disordering process is plotted.

The states with g1, g2, g3m0 to the left of the vertical dotted

D. Fuks et al. / Solid State Ionics 177 (2006) 217–222222

line in Fig. 4 correspond to the ordered or partly ordered

phase. At these temperatures the partly ordered state is

preferable. The temperature of the order–disorder phase

transformation is far above the melting temperature (1500 K

[31]). This means that the La0.875Sr0.125MnO3 phase remains

ordered or partly ordered till the melting point. However, the

decomposition of this phase into the two-phase mixture of the

disordered state and another phase with different stoichio-

metric composition may occur at temperatures lower than that

of order–disorder transformation in La0.875Sr0.125MnO3. Such

a comprehensive analysis of the thermochemical stability

demands an additional study of the temperature and concen-

tration dependencies of the free energies of all competing

phases, in principle, also of the nonstoichiometric compounds.

The results of this challenging investigation will be presented

elsewhere. It is interesting to note in conclusion that the

ordered structures (a) and (c) in Fig. 1 indeed were detected

experimentally [32] using high-resolution transmission elec-

tron microscopy (HRTEM) in thin La(1�c)SrcCoO3 film

grown on LaAlO3 (001) substrate by metal-organic chemi-

cal-vapor deposition (MOCVD) technique. It was concluded

there that these structures result directly from the La lattice

substitution for Sr. Another structure denoted as n-

La0.33Sr0.67CoO3 was also observed in [32], and its atomic

structure model was determined. It was pointed out that this

structure is a minor phase. This phase is not defined by the

Lifshitz stars of k-vectors and is the result not only of the La

lattice site substitution by Sr but also of the fluctuations in the

local chemical composition. This or analogous phases do not

satisfy the Lifshitz criterion, and thus are not considered in

our study.

5. Summary

In this paper we investigated the LacSr(1�c)MnO3 solid

solutions using the thermodynamic formalism based on

combination of the ab initio electronic and atomic structure

calculations as developed by Fuks et al. [24–26]. The main

feature of our approach is the treatment of ordered super-

structures presenting the La–Sr sublattice immersed in the

field of the rest lattice formed by Mn and O atoms. The total

energy calculations allow us to find the formation energies of

these superstructures for different compositions and to analyze

their competition at T=0 K. These calculations for a series of

ordered structures permit us to extract the key energy

parameters – the Fourier transforms of the mixing potential

– and thus to determine the free energy for temperature

induced partly disordered structures. Using the 12.5% Sr-

doped LaMnO3, La0.875Sr0.125MnO3, and combining two

different ab initio methods, we performed a thermodynamic

analysis. We predicted, in particular, that disordering of this

phase with respect to the decomposition into the heteroge-

neous mixture of LaMnO3 and SrMnO3 can occur only at

temperatures above the melting point. This is in contrast to

our study of isostructural BacSr1� cTiO3 solid solution where

below certain temperature Ba impurities in SrTiO3 tend to

form BaTiO3 nanoclusters.

Acknowledgements

This study was supported by the German–Israeli Founda-

tion (GIF) (grant #G-703-41.10/2001) and A. von Humboldt

Award to R.E. Authors are greatly indebted to J. Fleig and R.

Merkle for stimulating discussions.

References

[1] R. von Helmolt, J. Wecker, B. Holzapfel, L. Shultz, K. Samwer, Phys.

Rev. Lett. 71 (1993) 2331.

[2] S. Jin, T.H. Tiefel, M. McCormack, R.A. Fastnacht, R. Ramesh, L.H.

Chen, Science 264 (1994) 413.

[3] J. Fleig, K.D. Kreuer, J. Maier, Handbook of Advanced Ceramics,

Elsevier, Singapore, 2003, p. 57.

[4] G. Banach, W.M. Temmerman, Phys. Rev. B 69 (2004) 054427.

[5] H. Zenia, G.A. Gehring, G. Banach, W.M. Temmerman, Phys. Rev.

B 71 (2005) 024416.

[6] J. Geck, P. Wochner, D. Bruns, B. Buchner, U. Gebhardt, S. Kiele, P.

Reutler, A. Revcolevschi, Phys. Rev. B 69 (2004) 104413.

[7] A.N. Grundy, B. Hallstedt, L.J. Gauckler, Comput. Coupling Phase

Diagrams Thermochem. 28 (2004) 191.

[8] R. Klingeler, J. Geck, R. Gross, L. Pinsard-Gaudart, A. Revkolevshi, S.

Uhlenbruck, B. Buchner, Phys. Rev. B 65 (2002) 174404/1.

[9] P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, J. Luitz, Computer Code

WIEN-2k, Inst. for Materials Chemistry, TU Vienna, 2002.

[10] V.R. Saunders, R. Dovesi, C. Roetti, R. Orlando, C.M. Zicovich-Wilson,

N.M. Harrison, K. Doll, B. Civalleri, I.J. Bush, Ph. D’Arco, M. Llunell,

CRYSTAL-2003 User’s Manual, 2003.

[11] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865.

[12] I.G. Krogh Andersen, E. Krogh Andersen, P. Norby, E. Skou, J. Solid

State Chem. 113 (1994) 320.

[13] J. Nowotny, M. Rekas, J. Am. Ceram. Soc. 81 (1998) 67.

[14] P.E. Blochl, O. Jepsen, O.K. Andersen, Phys. Rev. B 49 (1994) 16223.

[15] A.D. Becke, J. Chem. Phys. 98 (1993) 5648.

[16] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785.

[17] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82, 270, 284, 299 (1984).

[18] M.D. Towler, N. Allan, N.M. Harrison, V.R. Saunders, W.C. Mackrodt, E.

Apra, Phys. Rev. B 50 (1994) 5041.

[19] J.K. Perry, J. Tahir-Kheli, W.A. Goddard III, Phys. Rev. B 63 (2001)

144510.

[20] http://www.chimifm.unito.it/teorica/crystal/basis_sets/mendel.html.

[21] S. Piskunov, E. Heifets, R.I. Eglitis, G. Borstel, Comput. Mater. Sci. 29

(2004) 165.

[22] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188.

[23] R.A. Evarestov, E.A. Kotomin, Yu Mastrikov, D. Gryaznov, E. Heifets, J.

Maier, Phys. Rev. B 72 (in press).

[24] D. Fuks, S. Dorfman, S. Piskunov, E.A. Kotomin, Phys. Rev. B 71 (2005)

014111.

[25] D. Fuks, S. Dorfman, E.A. Kotomin, Yu Zhukovskii, A.M. Stoneham,

Phys. Rev. Lett. 85 (2000) 4333.

[26] D. Fuks, S. Dorfman, J. Felsteiner, L. Bakaleinikov, A. Gordon, E.A.

Kotomin, Solid State Ionics 173 (2004) 107.

[27] L. Rormark, S. Stolen, K. Wik, T. Grande, J. Solid State Chem. 163

(2002) 186.

[28] A.G. Khachaturyan, Theory of Structural Transformations in Solids,

Wiley, New York, 1983.

[29] E.M. Lifshitz, J. Phys. 7, 61, 1942; 7, 251, 1942.

[30] L.D. Landau, E.M. Lifshitz, Statistical Physics, 2nd Revised and Engl.

Ed., Pergamon Press, Oxford, 1978, p. 433 (Translated from the Russian

by J.B. Sykes and M. J. Kearsley).

[31] J.C. Cohn, J.J. Neumeier, C.P. Popoviciu, et al., Phys. Rev. B 56 (1997)

R8495;

A. Cady, D. Haskel, J.C. Lang, et al., Rev. Sci. Instrum. 76 (2005)

063702.

[32] Z.L. Wang, J. Zhang, Phys. Rev. B 54 (1996) 1153.