JOURNAL OF CHEMICAL PHYSICS
VOLUME 119, NUMBER 18
8 NOVEMBER 2003
Oxygen ion migration in orthorhombic LaMnO3À ␦
Scott M. Woodleya)
Davy-Faraday Research Laboratory, The Royal Institution of Great Britain, London W1S 4BS,
United Kingdom
Julian D. Gale
Department of Chemistry, Imperial College of Science, Technology and Medicine, South Kensington SW7 2AY,
United Kingdom
Peter D. Battle
Inorganic Chemistry Laboratory, University of Oxford, Oxford OX1 3QR, United Kingdom
C. Richard A. Catlow
Davy-Faraday Research Laboratory, The Royal Institution of Great Britain, London W1S 4BS,
United Kingdom
共Received 19 May 2003; accepted 13 August 2003兲
Interatomic potentials that can model ligand field effects were used to investigate the properties of
vacancies in orthorhombic LaMnO3 . The minimum energy structures of LaMnO3⫺ ␦ 共where ␦⫽1/
192兲 were calculated for an oxygen vacancy on either the O1 or O2 site, respectively. It is predicted
that the ‘‘degenerate’’ activation energy 共and pathway兲 for oxygen diffusion in cubic LaMnO3⫺ ␦ is
lifted after a cubic–orthorhombic phase transition. Within the orthorhombic phase, one of the once
triply degenerate activation energies is lowered, indicating that there is a preferred migration
pathway, while one is increased, indicating an increased activation energy for unrestricted oxygen
migration 共and a much closer agreement to that observed in strontium doped systems兲. The lowest
energy pathways within the orthorhombic, as opposed to the cubic perovskite structure, are no
longer symmetric. The activation energies of migration indicate a preferential vacancy migration
between the O2 and O1(s) sites, where the migrating oxygen ion would simultaneously arc around
the central manganese ion with a bond length, Mn–O, which varied between 1.72 and 1.77 Å. This
type of pathway suggests that vacancies migrate along O2(m) – O1(s) – O2(m) – O1(s) chains.
© 2003 American Institute of Physics. 关DOI: 10.1063/1.1615759兴
INTRODUCTION
Vacancy migration is a key process in many ionic
materials.1–5 Of particular interest is the case of oxygen migration through the orthorhombic phase of LaMnO3⫺ ␦ , the
parent compound of the manganese perovskites that display
colossal magnetoresistance,6,7 where it is generally assumed
that the oxygen migrates as a doubly charged O2⫺ species.
There are also studies of the formation and migration of cation defects in lanthanum manganate8 and doped LaMnO3 is
used in solid oxide fuel cells.9
The structure of perovskites, ABO3 , can be viewed as a
network of corner sharing BO6 octahedra with the larger A
cations filling the holes between the octahedra 共see Fig. 1兲.
The cubic perovskite structure 共where the number of formula
units in the unit cell, Z, is one兲 is the high-pressure phase for
LaMnO3 . For this structure, it has been shown that O2⫺ can
migrate to oxygen vacancy sites along the octahedron
edges.10 More precisely, a neighboring O2⫺ ion moves along
a path which arcs around the central B cation to the vacant
oxygen site 共with the ‘‘vacancy’’ effectively propagating in
the opposite direction by ‘‘hopping’’ to where the migrating
O2⫺ started兲. The MnO6 octahedra have 12 edges. Thus,
there are potentially 12 nearest neighbor O2⫺ migration patha兲
Electronic mail: [email protected]
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9737
ways to investigate, although in the cubic phase these will all
be equivalent. Due to the limitations of interatomic potential
methods, previous investigations have assumed a cubic perovskite, as opposed to an orthorhombic structure, where the
octahedra are neither rotated nor distorted. This is a reasonable assumption, since at high temperatures, where ion migration is possible, the orthorhombic distortions are not as
large as those seen at room temperature. Modeling a cubic
perovskite has the advantage that there is only one pathway
to investigate, since by symmetry all the octahedral edges are
identical. Moreover, as there is a mirror plane which bisects
this pathway and the central B site, the computational task of
computing the minimum energy pathway, and hence the
saddle point, is reduced further.10
It is, however, clearly desirable to know how the activation energy and the migration path are affected by the distortion of the octahedron. The observed orthorhombic structure
of LaMnO3 has MnO6 octahedra that are both distorted and
rotated due to steric and electronic 共Jahn–Teller兲 effects.11
We have already shown that we can model Jahn–Teller distortions within LaMnO3 using a code primarily based on
interatomic potentials,12 modified by the addition of a ligand
field term. The formalism of the ligand field term that we use
to model the asymmetry of the Mn3⫹ ions is described in
Ref. 13. Due to the reduced symmetry, the orthorhombic
© 2003 American Institute of Physics
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9738
Woodley et al.
J. Chem. Phys., Vol. 119, No. 18, 8 November 2003
FIG. 1. The perovskite crystal structure for cubic LaMnO3 . Shaded polyhedra represent MnO6 corner sharing groups and the isolated spheres represent La cations.
phase has two, rather than one, distinct oxygen sites and
three 共labeled long, medium, short兲, rather than one, Mn–O
bond distances within the MnO6 octahedra. Each octahedron
has four O1 sites and two O2 sites, as shown schematically
in Fig. 2. The oxygen sites are the connecting corners of two
octahedra, with each O2 site forming two medium length
bonds with the manganese ions at the center of these octahedra, whereas each O1 site forms one short and one long
Mn–O bond. It is convenient to sublabel the O1 sites, O1(s)
and O1(l), when considering a particular MnO6 octahedron
关note that an O1(s) site for one octahedron is an O1(l) site
for the neighboring octahedron兴. Ignoring any structural relaxation due to the oxygen vacancy, it is now easy to see that
there are three different octahedron edges to investigate: the
paths from O2 to O1(l), O2 to O1(s), and O1(l) to O1(s).
Note that there is no O2–O2 octahedron edge. The migration
paths are expected to have different activation energies.
From a wider perspective, the O1 and Mn sites form layers
共or planes if there is no rotation of the MnO6 octahedra兲
which are connected by the O2 octahedra vertices. A large
difference in activation energies for the different paths would
imply either migration along chains, O2 – O1(l/s) – O2
– O(l/s) – ¯ or within layers – O1(s) – O1(l) – O1(s) – ¯ .
In this paper we calculate formation energies, E f , for the
creation of an oxygen vacancy on an O1 and an O2 site
within the orthorhombic structure of LaMnO3⫺ ␦ 共where ␦
changes from 0 to 1/192兲. We also investigate the structural
relaxation following vacancy formation and thus shall report
the bond distances and angles within the vicinity of the defect region. In removing an oxygen ion from the more symmetrical O2 site 共centered between two medium length
Mn–O bonds rather than one short and one long length
Mn–O bond兲, we anticipate the local ions will move such
that the local structure may no longer be symmetric. Thus,
either one of two, degenerate, relaxed structures will be
found. We next present the lowest energy pathways for oxygen migration and the resulting activation energy, E act . The
Mott–Littleton approach is ideal for modeling isolated
defects.14,15 However, we shall use the supercell method
rather than the Mott–Littleton in our study of defects within
the orthorhombic structure as the use of the potential functions needed to describe Mn3⫹ – O2⫺ interactions in the noncubic phase cannot yet be implemented in the latter approach. We are, however, able to calculate E f and E act for the
cubic phase 共where the environment about the Mn3⫹ is not
distorted兲 using both approaches which provides a useful
comparison.
NUMERICAL TECHNIQUES AND PARAMETERS
Lattice energy
FIG. 2. Schematic 3D picture of Mn cation positions 共shaded spheres兲, O
anion positions 共open spheres兲 and the short 共dotted line兲, medium 共dashed
line兲, and long 共solid line兲 Mn–O bond lengths. Anion numbers 5 to 12 and
1 to 4 represent O1 and O2 sites, respectively. Cations 1, 3, 4 with anions 5,
6, 9, 10 form a layer which is connected to a second layer 共cation 2 with
anions 7, 8, 11, 12兲 via two medium Mn–O bonds.
We apply standard lattice-energy minimisation techniques to model the stoichiometric bulk structure of LaMnO3
and nonstoichiometric LaMnO3⫺ ␦ 共where there is an oxygen
vacancy兲. In both cases we use a unit cell or supercell, respectively, and apply three-dimensional periodic boundary
conditions. The lattice energy is composed of four components: 共i兲 an Ewald summation16,17 to compute the Coulomb
term, where we use formal charges on the ions; 共ii兲 Born–
Mayer potentials to model the short-range ‘‘spherical’’ forces
between the cations and anions, La3⫹ – O2⫺ and
Mn3⫹ – O2⫺ ; 共iii兲 a Buckingham function to model shortrange interactions between the anions, O2 – O2⫺ ; 共iv兲 a
ligand field term for each transition metal ion within the
periodic cell, to provide the driving force for the Jahn–Teller
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J. Chem. Phys., Vol. 119, No. 18, 8 November 2003
Oxygen ion migration in orthorhombic LaMnO3⫺␦
distortions of the coordination sphere of Mn3⫹ . In addition,
we employ the shell model18,19 to describe ionic polarization
for the oxygen ions. The potential parameters are given in
Ref. 12.
For a regular octahedral environment about a transition
metal, the five d-orbitals split into two sets such that there
are three degenerate t 2g , nonbonding levels which are lower
in energy than the two degenerate e *
g , antibonding levels.
By distorting the octahedral environment, the degeneracy in
3⫹
is a high-spin d 4 ion such
the e *
g levels is removed. Mn
that only three electrons will populate the t 2g orbitals. The
distortion of the octahedral environment allows the electron
in the e *
g orbitals to lower its energy. The stabilization energy from our ligand field term,12
E JT⫽
⑀ td 共 O ts
兺
d ⫺1 兲 ,
tsd
共1兲
is due to the changes in these electronic energies. Here, ⑀ td
and O ts
d are the energy changes and occupations of the
d-orbitals for each transition metal, t, respectively, and s is
the spin of an electron. We obtain ⑀ td using an adaptation of
the angular overlap model 共AOM兲. This involves diagonalizing a 5⫻5 Hamiltonian matrix in order to find the orientation
of the d-orbitals, and the fitting of two potential parameters
per transition metal ion—ligand type pair in order to define
the distance dependence. Upon optimization of the lattice
energy with respect to either the cell parameters or ionic
coordinates, the degeneracy of the energy levels and the order of the energy levels may change. To prevent the energy
landscape becoming discontinuous, O ts
d can represent partial
occupancies via the implementation of a Fermi function. We
set the electronic or Fermi temperature to be that of room
temperature. Further details of our adaptation of the AOM
and its application to manganates is given in Ref. 12.
Our results will be obtained using the General Utility
Lattice Program 共GULP兲.20,21 The lattice energy for the bulk
structure of LaMnO3 will be minimized by relaxing both cell
dimensions and atomic coordinates at constant pressure using a Newton–Raphson procedure together with the BFGS
method22 for updating the Hessian. In the cases where this
procedure finds an unstable stationary point, we will relax
the imposed symmetry, apply a perturbation, and utilize the
rational function optimization 共RFO兲 method23 of minimization to ensure the Hessian is positive definite.
Supercells and relaxation
Supercells were generated from the relaxed cell parameters and ionic coordinates of orthorhombic LaMnO3 (Z
⫽4, where Z also represents the number of MnO6 octahedra
within each supercell兲.12 After removing one oxygen ion
from the supercell, we add a neutralizing uniform charge
background. Then, after fixing a lanthanum ion 共to stop
translations of the entire structure兲, these structures were optimized, but this time keeping the cell parameters fixed since
one isolated defect should not change the bulk lattice constants. To search the configuration space for the lowest local
energy minimum, E m , for each oxygen vacancy site, several
calculations were necessary whereby one of the nearest eight
9739
oxygen ions to the vacant site was displaced before allowing
the structure to relax. The energy required to form an oxygen
vacancy per supercell, E f , is defined as the difference between E m and the lattice energy of the perfect crystal.
The traditional problem, when modeling isolated defects
共␦→0兲 with a supercell approach, is that large supercells are
required in order to reduce the unwanted interaction between
a defect 共vacancy and local distortions兲 and its images. The
convergence of E f with respect to ␦ 共or the size of the supercell兲 can be improved by adding the term24
␣Q2
,
2 ⑀ rL
共2兲
where ␣ is the Madelung constant and ⑀ r the dielectric constant of the perfect crystal. We note that this term refers to
the Coulomb energy of a point charge Q 共charged defect兲,
immersed in a structureless dielectric, within a cubic unit cell
of length L with a neutralizing uniform charge background.24
Even when this term is included, the convergence of E f still
requires the use of fairly large supercells. For example, consider using one oxygen vacancy per unit cell to model an
isolated vacancy in LaMnO3 . During the relaxation of the
structure in this smallest supercell system, an oxygen ion
would migrate to the midpoint between ‘‘two’’ vacant oxygen sites. This is unfortunate as the migrating oxygen ion
relaxes to where the saddle point is expected to be located
共see below兲. However, as ␦→0 the contribution given by Eq.
共2兲 does dominate the energy difference between the lattice
with a periodic array of defects and that with an isolated
defect. We define
E D⫽
␣Q2
.
2L
共3兲
E D is very straightforward to compute using, for example,
GULP, as it is the lattice energy of the oxygen ion within an
empty supercell that has a charged uniform neutralizing
background. The supercell dimensions are chosen to be
equivalent, whereas the charged uniform neutralizing background is of opposite sign, to that which we will use in the
defect calculation. Estimating ⑀ r for our orthorhombic cell,
Eq. 共2兲 becomes
ED
1
3
兺 i ⑀ ii
,
共4兲
where ⑀ ii are the diagonal components of the diagonalized
static dielectric constant tensor.
In our final results we will use a 4⫻3⫻4 supercell (Z
⫽192) of the orthorhombic unit cell (Z⫽32). To test the
convergence and reliability of this supercell method we will
compare results produced using this and the Mott–Littleton
approach, but where the ligand-field effects are omitted.
Mapping the migration pathway
We assume that the migrating species is the O2⫺ . However, it is true that there are two possible scenarios as to how
the oxide ion might diffuse. First, it might migrate as O2⫺
共the fully ionic model as described here兲, or second, as O⫺ ,
followed by an electron hop as a second step. It is unlikely to
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9740
Woodley et al.
J. Chem. Phys., Vol. 119, No. 18, 8 November 2003
TABLE I. Predicted fractional coordinates for orthorhombic LaMnO3
共space-group Pnma with a⫽5.7483 Å, b⫽7.7199 Å, and c⫽5.5298 Å).
3⫹
La
Mn3⫹
O1 2⫺
O2 2⫺
x
y
z
0.5457
0.0000
0.6889
0.9782
0.2500
0.0000
0.5344
0.2500
0.0044
0.0000
0.7787
0.9333
hop as neutral oxygen since the first electron affinity of oxygen is exothermic. As far as we are aware, there is no experimental proof as to the exact mechanism. However, from
a quantum mechanical ‘‘point of view’’ the charge state on
hopping is hard to determine since it depends on how the
charge is partitioned. Although the so-called ionic model
transfers a formally charged oxide ion, because of the inclusion of polarization the Born effective charge for the oxygen
at the transition state will be less than the formal one. Hence
the model mimics the uncertainty of a quantum model. Assuming that the oxide ion is diffusing under the influence of
an external field 共which is the experimental reality of the
situation兲, it is again likely that the ionic species will diffuse
not the neutral one. Importantly the present study is to determine the contribution of ligand field effects to the activation
energy, and so it is the contrast between the same ionic
model with and without this term that is the most important
aspect.
In order to find the activation energy for ion migration, it
is necessary to locate the lowest saddle point on the energy
surface for the various possible pathways. The saddle point
of interest is the highest point along lowest energy pathway
for an oxygen ion to move between two vacant oxygen sites.
The activation energy for O2⫺ migration due to the presence
of vacancies is just the difference in lattice energy when the
moving O2⫺ is positioned on 共a兲 the saddle point and 共b兲
either 共or more precisely the lower when the oxygen sites are
not equivalent兲 of the two octahedron corners. Initially, we
only need to locate the approximate locations of the various
saddle points. Then we can invoke the method of RFO
optimization22 to find the nearest saddle point 共or stationary
point with one negative Hessian eigenvalue兲 for each pathway. With orthorhombic distortions present, guessing the approximate location of a saddle point is not easy. Thus, we
will first compute the energy surface as a function of the
migrating oxygen atom’s coordinates, which contains the approximate pathway for the oxygen ion migration. Each energy point is obtained after relaxing the orthogonal structure
about the fixed coordinates of the migrating ion.
As initially only approximate coordinates of the transition state are required then great precision for the migration
pathway is not necessary. Rather than just fixing the migrating ion within the supercell, the ions outside a radius cutoff
of 11 Å from the center of the defect region will also be fixed
to substantially reduce the number of variables. Moreover, to
reduce the size of calculations further, rather than initializing
all our defect calculations from bulk positions, we will estimate their relaxed positions as explained below. Consider the
local defect region: a fixed migrating oxygen ion and the
TABLE II. Predicted bond lengths 共Å兲 and angles 共°兲 within the MnO6 units
for orthorhombic LaMnO3 .
O
Mn–O
O1(s) – Mn–O
O2–Mn–O
O1(s)
O1(l)
O2
1.903
180.0
88.8 and 91.2
2.183
89.8 and 90.2
87.8 and 92.2
1.969
88.8 and 91.2
180.0
MnO4 v2 distorted octahedron, where v represents a vacancy.
We define s1 and s2 to be the vectors that connect the manganese cation to the migrating oxygen ion placed at the first
and second vacancy, respectively, after relaxing the defect
region. Thus, the vectors s1 and s2 point to the approximate
ends of the migration path. To compute the approximate
minimum energy path that connects two oxygen sites 共and
hence locate the approximate saddle point兲, we will initially
assume that this path lies within the plane that contains both
vectors s1 and s2 . The lattice energy will be calculated across
this plane such that 共i兲 the migrating oxygen ion is fixed at
the various grid points n/10s1 ⫹m/10s2 , where n and m are
integers 0 to 10. 共ii兲 Ions further than 11 Å from the central
manganese site or its images are fixed at their bulk relaxed
position. 共iii兲 Other ions are initially set to n/10r1i
⫹m/10r2i before being relaxed 共where r1i and r2i are the
relaxed coordinates of the ion i when calculating vectors s1
and s2 , respectively兲. The contours of the lattice energies,
E(n,m), of these relaxed structures will then be plotted. Part
共ii兲 reduces the size of the computational task from relaxing
192 octahedra to around 81 octahedra. Note that there will be
at least one layer of MnO6 octahedra fixed to their bulk
positions between the relaxed region and its images. If the
positions of the fixed ions in this outer region are compared
to the respective positions obtained during the calculation of
s1 and s2 we would find a maximum change of 0.07 Å.
Points near n⫽m⫽0 and n⫽m⫽10 will be omitted.
RESULTS AND DISCUSSION
Our predicted structural parameters, including the bond
length and angles within the distorted MnO6 units, for orthorhombic LaMnO3 are shown in Tables I and II. The formation of an oxygen vacancy, E f , and the changes in the lattice
energy for LaMnO3⫺ ␦ upon relaxation of the unrelaxed defect structure to that corresponding to the lowest energy state
are shown in Table III. Here ␦⫽1/192 such that one oxygen
ion has been removed from the supercell which initially contained 192⫻5 ions. For the unrelaxed defect structure, the
O2 site is the more stable position for the vacancy. However,
after relaxation the vacancy is predicted to be lower in energy on an O1 site. To compute E f for an isolated defect we
TABLE III. Predicted lattice energy differences between the unrelaxed and
relaxed orthorhombic LaMnO3⫺ ␦ (Z⫽192, ␦⫽1/192兲 structure, E u ⫺E r ,
and energy, E f , for the formation of the vacancy.
Vacancy site
E u – E r (eV)
E f (eV)
O1
O2
22.56
22.16
15.06
15.34
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J. Chem. Phys., Vol. 119, No. 18, 8 November 2003
Oxygen ion migration in orthorhombic LaMnO3⫺␦
TABLE IV. Predicted structural parameters 关bond lengths 共Å兲 and angles 共°兲
within the MnO5 units兴 for LaMnO3⫺ ␦ ( ␦ ⫽1/192), where the vacancy is on
共a兲 site O5គ and 共b兲 site O6គ . within the bulk, both sites would be equivalent
to O1 sites. The manganese and oxygen ion site labels are shown in Fig. 2.
共a兲
共b兲
O
O2
O6គ
O3គ
O10
O9
Mn1–O
O2–Mn1–O
O6គ – Mn1 – O
O3គ – Mn1 – O
O10–Mn1–O
1.766
2.236
88.9
88.9
152.6
90.2
1.797
152.6
84.5
84.5
170.2
2.016
90.2
170.2
91.9
91.9
1.873
103.1
91.4
103.6
98.2
O
O1
O6
O4
O10
O9
Mn4 –O
O1–Mn4 –O
O6–Mn4 –O
O4–Mn4 –O
O10–Mn4 –O
1.968
1.851
91.9
91.9
170.0
88.9
2.021
170.0
86.0
86.0
148.2
1.760
88.9
148.2
87.7
87.7
1.937
92.8
101.6
97.3
110.1
O
O2
O5គ
O3គ
O9
O10
Mn1–O
O2–Mn1–O
O5គ – Mn1 – O
O3គ – Mn1 – O
O9–Mn1–O
1.967
1.851
91.9
91.9
170.0
88.9
2.022
170.0
86.0
86.0
148.1
1.760
88.9
148.1
87.7
87.7
1.937
92.7
101.6
97.2
110.3
O
O1
O5
O4
O9
O10
Mn4 –O
O1–Mn4 –O
O5–Mn4 –O
O4–Mn4 –O
O9–Mn4 –O
1.766
2.234
88.9
1.797
152.7
84.6
2.017
90.2
170.2
91.9
1.873
103.1
91.5
103.6
98.2
88.9
152.7
90.2
84.6
170.2
91.9
still need to add the estimated correction given in Eq. 共4兲.
The static dielectric constant tensor for orthorhombic
LaMnO3 has the components ⑀ ii ⫽17.615, 15.069, 21.927,
⑀ i j ⫽0.0 otherwise. Thus, we estimate this additive correction
to the formation energies to be 0.20 eV, and the oxygen
vacancy formation energies to be 15.26 and 15.54 eV for the
O1 and O2 sites, respectively.
With the removal of an oxygen ion, there are two MnO5
units created. For example, in the case where the vacancy is
on an O1 site, then using the notation shown in Fig. 2,
Mn1O5 has the short Mn1 – O5គ bond missing and Mn4O5
has the long Mn4 – O5គ bond missing. Relaxation, as well as
causing a reduction in the distance between Mn1 and Mn4,
results in large distortions of the MnO5 units 共compare
Tables II and IV兲. Although equivalent in the bulk structure,
the degeneracies in the Mn–O bond lengths are removed.
Additionally, for Mn1O5 , the lengths of the other bonds
were reduced, whilst for Mn4O5 , the length of the long bond
共Mn4 –O9兲 and the short bonds are reduced, but one of the
medium length bonds increased. In the case where a vacancy
is created on an O2 site (O3គ in Fig. 2兲 such that there are two
fewer medium length bonds, we find two equivalent local
minima upon relaxation. Consider the chain of ions
– O2– Mn2 – O3គ – Mn1 – O2– in Fig. 2. The bond angles
O2– Mn(1 or 2) – Oi, where i⫽ 兵 5គ ,6គ ,9,10其 or 兵7,8,11,12其, in
the bulk structure are ⬃90°; that is, each set of four oxygen
ions are approximately coplanar as defined by the normal
parallel to the – O2– Mn2 – O3គ – Mn1 – O2– chain. Upon relaxing, the symmetry about the O2 site is lost as the oxygen
9741
TABLE V. Predicted structural parameters 共bond lengths 共Å兲 and angles 共°兲
within the MnO5 units兲 for orthorhombic LaMnO3⫺ ␦ ( ␦ ⫽1/192), where the
vacancy is on site O3គ . Within the bulk, this site would be equivalent to an
O2 site. Note that there are two sets with the same lattice energy. The
manganese and oxygen ion site labels are shown in Fig. 2.
共a兲
共b兲
O
O5គ
O6គ
O9
O10
O2
Mn1–O
O5គ – Mn1 – O
O6គ – Mn1 – O
O9–Mn1–O
O10–Mn1–O
1.723
1.995
91.1
91.1
139.5
85.7
1.807
139.5
92.0
92.0
174.0
2.406
85.7
174.0
87.1
87.1
1.873
115.2
101.8
103.6
84.1
O
O7
O8
O11
O12
O2
Mn2–O
O7–Mn2–O
O8–Mn2–O
O11–Mn2–O
O12–Mn2–O
2.123
1.797
89.0
89.0
163.8
90.1
2.048
163.8
86.8
86.8
163.3
1.807
90.1
162.3
89.2
89.2
1.897
92.7
100.3
103.3
97.4
O
O12
O11
O8
O7
O2
Mn2–O
O12–Mn2–O
O11–Mn2–O
O8–Mn2–O
O7–Mn2–O
1.728
1.995
91.1
91.1
139.5
85.7
1.807
139.5
92.0
92.0
174.0
2.405
85.7
174.0
87.1
87.1
1.873
115.2
101.8
103.6
84.1
O
O10
O9
O6គ
O5គ
O10
Mn1–O
O10–Mn1–O
O9–Mn1–O
O6គ – Mn1 – O
O5គ – Mn1 – O
2.123
1.797
89.0
2.048
163.8
86.9
1.807
90.1
162.3
89.2
1.897
92.7
100.3
103.4
97.4
89.0
163.8
90.1
86.9
163.3
89.2
ion at site O5គ moves more than the other seven anions at
sites Oi, towards the vacancy at O3គ , such that it forms a
bond angle O2– Mn1 – O5គ of 115.2° 共see Table V兲. An
equivalent relaxation can be found whereby an oxygen ion
共O12兲 about Mn2 is displaced towards the vacancy at O3គ .
Bond lengths and angles for the two MnO5 units for this
equivalent defect structure are given in the lower half of
Table V.
As discussed earlier, when considering the unrelaxed
structure of LaMnO3⫺ ␦ , there are three ideal migration paths
about the manganese cation to consider. After relaxing the
defect structure, the degeneracy of these paths breaks. We
compute the shortest approximate path for each ideal trajectory. In Fig. 2, these will be the paths connecting O3គ to O5គ
共O2 to O1(s)), O3គ to O6គ 共O2 to O1(l)), and O5គ to O6គ
(O1(s) to O1(l)), which will be referred to as paths A, B,
and C, respectively. The energy surfaces with respect to the
coordinates of the migrating oxygen ion are shown in Fig. 3.
Within the bulk structure, the O2 site contains an oxygen ion
that is corner shared by two equivalent MnO6 units. When
there is a vacancy on an O2 site, the surrounding ions relax
such that the two MnO5 units are no longer equivalent. For
example, from Fig. 2, if the vacancy is on an O3គ site the
oxygen ion at O5 can relax to one of two energy equivalent
sites. The MnO5 units will then have the parameters given in
the lower or upper part of Table V. Later we will refer to the
energy barrier, E b , between these two equivalent states. As
there are two possible paths for the route O3គ to O5គ 关O2 to
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9742
J. Chem. Phys., Vol. 119, No. 18, 8 November 2003
Woodley et al.
FIG. 3. The relaxed energy E i j across a plane containing vectors s1 and s2 that define the relaxed O1 or O2
anion sites relative to the central Mn1 cation averaged
position (i⫽0, j⫽0) for a migrating oxygen ion fixed
at points (i, j) in orthorhombic LaMnO3⫺ ␦ 共where
␦⫽1/192兲. In A, O3⬅共⫺10,0兲 and O5⬅共0,⫺10兲. In B,
O3⬅共0,10兲 and O6⬅共10,0兲. In C, O5⬅共0,10兲 and O6
⬅共⫺10,0兲. In D, O3⬅共10,0兲 and O5⬅共0,⫺10兲. The
solid circles represent the approximate coordinates for
the migrating anion when the relaxed crystal structure
has an energy that is at a saddle point in the energy
surface. The difference in the solid and dashed contours
is 0.2 eV for A – C and 0.1 eV for D. The dashed line
represents 0.8 eV.
O1(s)] we chose to show both energy surfaces in Fig. 3. We
will refer to the minimum energy path across this additional
surface as path D, which has one end of the path that corresponds to the point where O10 共rather than O5គ ) is the nearest
to the vacancy at O3គ site. The paths, A – D, arc around the
central manganese cation, rather than being direct straightline paths. It should be noted that the manganese cation is
not fixed at 共0,0兲 during relaxation 共in fact for path C, at the
saddle point Mn1 had moved 0.18 Å兲. A trajectory that bent
about the central manganese cation, as opposed to a direct
line path, was also predicted for the cubic phase.10 However,
the approximate position of the saddle points, marked on Fig.
3 with a filled circle, are not midway along the respective
lowest energy paths. In Fig. 4, we have shown the energy
barrier along the pathway shown on these energy surfaces.
Note that, these energy curves are plotted as a function of the
position of the migrating anion. Therefore, when the oxygen
ion is on the left-hand side of this figure, the vacancy is on
the right-hand side. It is clear that the approximate pathways
are not symmetrical and two of the paths, B and D, have an
additional feature of a minimum and therefore a double barrier. From these energy barriers we would predict that the
vacancy would prefer to migrate between the O2 and O1(s)
sites. Moreover, the migrating oxygen ion moves around the
central manganese ion with a varying bond length: between
short and medium. The energy barrier for the vacancy to
move to an O1(l) rather than an O1(s) site is higher. In fact,
the unexpected local minimum along path B is due to an
oxygen ion on an O1(s) site temporarily moving towards the
vacant O2 site, suggesting that as the migrating anion moves
FIG. 4. The energy barriers 共units of eV and arbitrary
origin兲 along paths A 共dashed line兲, B 共dot–dashed
line兲, C 共solid line兲, and D 共dotted line兲, as defined by
the dotted lines in Fig. 3.
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J. Chem. Phys., Vol. 119, No. 18, 8 November 2003
FIG. 5. The energy minimized perovskite crystal structure for cubic
LaMnO3 when the structure is modeled using the potential parameters of
Cherry et al. 共Ref. 10兲. Shaded polyhedra represent MnO6 corner sharing
groups and the isolated spheres represent La cations.
along path A, the oxygen ions on sites O1(l) may move in
order to lower the energy barrier. The smaller of the two
local maxima along path D is a consequence of the existence
of E b : from the additional local energy minimum, the migrating anion follows a path similar to that of path A. Refining the position of the migrating ion from the approximate
saddle points indicated within Fig. 3, we predict the activation energies, E act , of 0.30, 0.51, and 0.67 eV for
O2 – O1(s), O2 – O1(l), O1(s) – O1(l), respectively. The
observed Arrhenius energy of oxygen tracer diffusion in
Oxygen ion migration in orthorhombic LaMnO3⫺␦
9743
LaMnO3⫾ ␦ was 2.49⫾0.11 eV3 which includes the creation
of oxygen vacancies; the activation energy for a strontium
doped system is reported to be around 0.73 eV.25 When considering the calculated activation energies for migration, the
latter provides a better comparison since vacancies will already be present.
We should now consider the present results in the context of the earlier studies of Cherry et al.10 who used the
Mott–Littleton 共ML兲 method to model the oxygen migration
in the cubic perovskite structure for LaMnO3 which was calculated as 0.86 eV 共with, E f ⫽17.8 eV). However, using their
potential and the supercell 共SC兲 approach we found the energy required to create an oxygen vacancy to be 19.48 eV
关which includes the contribution from Eq. 共4兲 of 0.19 eV兴
and the activation energy for oxygen migration in
LaMnO3⫺ ␦ ( ␦ ⫽1/216) to be E act⫽0.50 eV. The difference
in the predicted activation and formation energies when using the same interatomic potential parameters was not the
result of applying different methods. In the ML method two
regions about the defect are defined using radial cutoffs, r 1
and r 2 . Within the inner region 共defined by r 1 ) the ions are
completely relaxed. Ideally, r 1 is increased until the defect
energy is converged. Typically the difference between two
defect energies, calculated using the same value of r 1 , may
converge faster than their absolute values. Thus, E act may be
computed using a smaller inner region than that used to compute E f . The ML method assumes that ions in region 2 are
harmonically relaxed around their ‘‘fully relaxed’’ bulk positions. At the time Cherry et al. published their results the
computation of phonon dispersion curves was rather an expensive procedure and enforcing cubic symmetry for perfect
lattice drastically reduced the computational effort. Using
their potential parameters 共without applying pressure兲 the cubic structure was not stable since all three acoustic phonon
modes became negative for k near the L point on the Bril-
FIG. 6. Predicted energy 共units of eV兲
required to create an oxygen vacancy
as a function of the radius r 1 共units of
Å兲 defining the inner defect region
used in the Mott–Littleton approach,
where the outer radius is fixed at r 2
⫽26.0 Å. The diamonds, crosses are
for the case where a cubic Z⫽1,
monoclinic Z⫽8 unit cell was used to
predicted bulk relaxed ionic coordinates, respectively.
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9744
Woodley et al.
J. Chem. Phys., Vol. 119, No. 18, 8 November 2003
louin zone boundary. The associated eigenvector corresponded to a MnO6 breathing mode. In order to find the true
local energy minimum, a 2⫻2⫻2 supercell (Z⫽8) was required 共thus folding back the phonon dispersion curve such
that the negative phonons at L are now at the ⌫ point兲. Upon
relaxation of the cubic structure within the Z⫽8 supercell,
the octahedra distorted slightly but rotated significantly:
Mn–O–Mn bond angles were 159° 共as opposed to 180°兲.
This distorted structure is shown in Fig. 5. Our supercell
approach for modeling defects included these rotations. In
the ML method, the relaxed Z⫽8 unit cell should be used for
supplying the ‘‘fully relaxed’’ bulk positions. Otherwise, in
the cubic LaMnO3 system, as r 1 increases more MnO6 octahedra are able to rotate. In Fig. 6 the variation with r 1 of
‘‘E f ’’ show that the ML method diverges or converges depending whether the cubic Z⫽1 or the fully relaxed Z⫽8
unit cell is used. From Fig. 6, E f ⫽17.8 eV implies r 1
⬇8.0 Å corresponding to an inner region of 148 ions. For the
largest value of r 1 shown, the inner region contains around a
thousand ions; a calculation which can now be routinely performed.
Using the Mott–Littleton method 共with the Z⫽8 unit
cell兲 for modelling a defect we found that E f ⫽19.50 eV and
E act⫽0.49 eV which agree remarkably well with the values
predicted using the supercell method of 19.48 and 0.50 eV,
respectively. The difference in the calculated activation energy, E act , for the orthorhombic and cubic phases suggest
that the ligand field distortions significantly lowers the energy barrier for O2⫺ migration. Moreover, the ligand field
effects influenced the direction for migration of the oxygen
vacancy. That is, for the O2 – O1(s) path E act was reduced
by 0.20 eV whereas in the other directions, O2 – O1(l) remained around 0.5 eV and for O1(s) – O1(l), E act increased
by 0.17 eV.
CONCLUSION
With ‘‘spherical’’ potential parameters reported in the
literature,10 we were able to show that the formation energy
for an isolated oxygen vacancy converged to the same value
when either the Mott–Littleton or the supercell approach was
utilized. Using new interatomic potentials, that can reproduce the distorted environment about the Mn3⫹ ion within
the orthorhombic phase of LaMnO3 , we predict the formation energy of an oxygen vacancy to be 15.26 and 15.54 eV
for the O1 and O2 sites, respectively. Furthermore, we were
able to model the oxygen ion lowest energy migration paths
through orthorhombic LaMnO3⫺ ␦ 共where ␦⫽1/192兲. We
found activation energies of 0.30, 0.51, and 0.67 eV for
O2 – O1(s), O2 – O1(l), O1(s) – O1(l), respectively. Thus,
the
lowest
energy
O2⫺
pathway
is
along
O2(m) – O1(s) – O2(m) – O1(s) chains and we therefore
predict a preferred direction of O2⫺ transport. With an increase in temperature the O2⫺ can switch between these
chains via the route O2(m) – O1(l) – O2(m), path B. Path C,
O1(s) – O1(l), which allows migration within the O1 layers,
has the highest energy barrier for O2⫺ migration out of the
pathways considered. Jahn–Teller distortions of the MnO6
octahedra split the once threefold degenerate energy barriers
or activation energies for O2⫺ migration and thus influences
the direction for migration of the oxygen vacancy. Without
the ligand field effect, we calculate that the activation energy
for oxygen migration is 0.50 eV, whereas the observed value
is expected to be similar to that measured for a strontium
doped system, 0.73 eV. Although the ligand field decreases
the activation barrier for oxygen migration along
– O1(s) – O2 – chains, if unrestricted oxygen migration is
modeled 共to include migration within the O1 layers兲 then the
calculated activation energy, 0.67 eV, is closer to that observed when Jahn–Teller distortions about the Mn3⫹ ions are
included.
ACKNOWLEDGMENTS
Financial support from E.P.S.R.C. is gratefully acknowledged, as well as useful discussions with colleagues Alexei
Sokol, Andrew Walker, the initial exploratory work of Neepa
Shah, and the helpful comments of Saiful Islam 共University
of Surrey兲 and John Kilner 共Imperial College London兲.
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