MODELING OF DIFFUSION MIGRATION OF ANIONS IN
IONIC CRYSTALS
Bannikov V.V, Leont’ev L.I., Mitrofanov V.Ya.,
Institute of Metallurgy, Ural Department of Russian Academy of Science,
Russia, Ekaterinburg, Amundsen St.,101
The modeling of formation of oxygen diffusion potential barrier in NiO1- crystal was
performed taking into account the re-distribution of excess electrons and lattice
relaxation. It is shown that the competition of short-range energy and of electrostatic
energy plays the important role in forming of diffusion potential barrier.
INTRODUCTION
The interest to the diffusion phenomena in oxides is caused by the following reasons.
1. The vacancy in anion sub-lattice and anion (oxygen in 3d-oxides) migration
determines essentially the behavior of some magnetic characteristics (see for e.g. [1]).
2. The change of charge states of 3d-ions near vacancies in anion sub-lattice takes
place in the 3d-oxides and the related systems were the vacancy diffusion is the
dominating mechanism. The diffusion process in this case is accompanied by the
reaction of the type 3dn3dn1 in the cation subsystem. In a crystal excess electrons
(holes) are usually localized by Coulomb interactions nearby the vacancies with
formation of the complexes containing 3d-ions of mixed valence (MV). The
redistribution of the excess charges in the process of diffusion atom migration may
essentially manifest itself on the potential barrier form and the diffusion coefficient
value. The presence of mixed valence states at cations near vacancy can result also in
charge transfer effects (double exchange interaction). Double exchange interactions
are sensitive both to the metal –anion distance and geometry of superexchange bond
[2] and often can change strongly during anion migration [3].
3. Use of approximation of pair interactions is the traditional approach to calculation
of structural, magnetic, diffusion and other properties of ideal and defective ionic
dielectrics, containing 3d-ions. Adequacy of model of pair interactions is completely
determined by reliability of interionic interaction potentials. For effective modeling
structural and diffusion characteristics of ionic dielectrics it is desirable, that pair
potentials have been set by analytical expressions. The vacancy model of anion
diffusion in cubic crystal lattice is chosen as object for the analysis.
The primary goal of the present work was modeling of the potential barrier
form for the diffusing oxygen atom in 3d-oxides with use of analytical expressions for
pair potentials of interaction, the analysis of influence of distortion of an ideal crystal
lattice nearby the vacancy and redistribution of excess charges on a potential barrier
form. As a simple model system the crystal NiO1- containing structural vacancies in
oxygen sub-lattice has been chosen. It was supposed that the mechanism of
compensation of an excess positive charge takes place due to presence of Ni+ ions in
cation sub-lattice together with Ni2+ ions in some vicinity of crystal defect-pair of
vacancies - migrating anion. Concentration of vacancies is considered to be small, so
interaction between MV centers can be neglected.
APPROXIMATION OF PAIR POTENTIALS. DESCRIPTION OF THE
MODEL.
The energy of impurity crystal can be written down as
E  E in  E int  E out ,
(1)
3 - 51
where E int is the energy corresponding to interactions of ions inside area I which
directly surrounds the impurity (vacancy), E out is the energy of interaction of atoms in
the area II of which coincides with all crystal except for area I, E int is the energy of
interaction of atoms in areas I and II. We shall consider, that area I can be chosen
enough large, such, that atoms in the area II occupy the positions corresponding to
structure of an ideal crystal.
The explicit expressions for pair potentials describing interactions in pairs
2+
2Ni -O and O2--O2- are taken from the paper [4]. It is supposed, that potentials Vij
describing pair interaction (pair energy) of i and j ions which are taking place on
distance r, can be written in the following form
Vij  Zi Z j / r  Uij ,

(2)

U ij  g ij rij  l i  l j  U ijpol  f ij (r ),
(3)
where Zi , Z j - charges of ions. The first term in (2) corresponds to energy of
interaction of pair the ions considered as point charges. The second term U ij
g ij (exchange and
describes short-range quantum-mechanical contributions
pol
dispersion), the electrostatic correction U ij distinguishing classical Coulomb
interaction from interaction of point charges, and effects of interosculation (or
shielding) electronic densities of interacting ions f ij [4,5]


g ij rij  li  l j  Cij  exp(Dij  rij  li  l j ) -
U polij 
f ij  
Xi X j
r

Yi X j
rij  li

X i Yj
rij  l j

λ ij
rij  li  l j
Yi Yj
rij  li  l j

Zi Z j
r
6
,
(4)
1
1
 K i li2  K jl 2j , (5)
2
2
A ij
exp(Bij r ).
(6)
r
Here X i , Yi are charges of a core and shell of i ion ( Z i  X i  Yi ), l i - shift of a
shell of i ion in relation to its core, K i is an elastic constant in the shell models,
r  rij - distance between cores of i and j ions. It is necessary to note, that in work [5]
for the first time it was offered to consider the Born - Mayer contribution to the
potential g ij as function of distance between shells, instead of cores of ions.
Parameters of the calculated short-range potential f ij responsible for the account of
interosculation of electronic densities of ions, and the parameters of short-range
potentials found with use of the empirical data, are received in work [4] and resulted
2
2
in Table 1. The constant in the Van-der-Vaals interaction potential of O  O
matters  OO =4.513 a.u., and elastic constants - K O =3.99, K Ni =36.7 a.u. [4].
In the present research the impurity center was modeled as a cubic cluster of
the size 5x5x5 (in terms of equilibrium distance cation-anion a0), containing 624 ions
and oxygen vacancy in an initial position (0,0,0). The consideration of the low3 - 52
symmetry structural distortions caused by a relaxation of a lattice near to vacancy, and
also by displacement of the ionic environment nearest to migrating oxygen was taken
into account only in limits of MV cluster.
Table 1. Values of parameters of pair potential interaction in atomic units [4].
A
B
C
D

2— 2O O
44.8624
1.05927
21.7
1.320
4.513
2+
2Ni -O
20.3381
1.30241
62.8
1.630
0
Modeling of the potential barrier form for migrating oxygen ion was carried
out as follows. It was set certain value of displacement of oxygen ion from an initial
position (1,1,0), further energy of a crystal was minimized depending on set of lowsymmetry displacements of ions of MV cluster, and also depending on re-distribution
of excess electrons on nickel ions.
DISCUSSION OF THE RECEIVED RESULTS
Within the framework of the present model various ways of oxygen migration
have been analyzed, and it has been shown, that the minimal energy of activation
corresponds to a diffusion way in XY, XZ,YZ planes (Fig.1). Furthermore, in has
been shown that the minimal value of activation energy corresponds to migration of
an ion in a direction [1,1,0], i.e. along a diagonal of a side of a cube.
Fig.1 The geometry of oxygen ion diffusion in NiO lattice
The present calculations have shown that energy of allocated MV cluster essentially
depends on the account of effects of a relaxation of cluster ions, and also from effects
of re-distribution excess electrons near to vacancy. The low symmetry displacement
of oxygen and nickel ions takes place near to vacancy (characteristic displacement in
oxygen sub-lattice makes size about 0.05a0), and the re-distribution of excess
electrons takes place while the one of oxygen ions is migrating, as shown on Fig. 2
(2D analogue of NiO lattice). In particular at the initial oxygen point (Fig.2, a) takes
place threefold orientation degeneracy of an electronic subsystem state that can be
removed due to the double exchange caused by correlated hopping of electronic pair.
Calculation of dependence of system energy from the value of displacement of
a migrating ion for a preferable (diagonal) way of diffusion has been performed. The
result of potential barrier calculations is present on Fig. 3. The evaluated height of a
potential barrier (the minimal energy of activation of diffusion) is equal
approximately to 2.7 eV (The experimentally estimated value is about 2.5 eV [6,7]). It
is necessary to note some prominent features of the received results. First, the present
calculations show the importance of taking into account the effects of lattice
relaxation in the MV cluster. For example the case a on Fig.2, corresponds to small
displacements of oxygen ions nearest to vacancy (xy0.045a0), the displacements
of nearest nickel ions is about.0.001-0.002 of a0.
Fig.2 The scheme of system behavior on the different stages of oxygen ion migration.
Second, it is proved that important role in forming of potential barrier is played with
re-distribution of excess electrons on various ions of nickel during diffusion. The
scheme of re-distribution is shown on Fig.2. During diffusion the flowing of excess
charges from old vacancy to new in some vicinity of saddle point actually takes place.
We shall note, that the flowing takes place at stages c, d, e of Fig. 2 - on which the
3 - 53
most essential distortions of MV cluster take place (displacement of the ions of nickel
nearest to migrating oxygen). The neglect of effects of excess electrons re-distribution
and of lattice relaxation at calculations of potential barrier form for migrating oxygen
ion results in anomaly high values of activation energy (~10-11 eV) and the
asymmetric form of a potential barrier (Fig. 4). Third, the calculations show, that in
the range of values x/a0 (0.05 - 0.3, 0.75-0.95) sharp growth of energy of system
takes place, whereas in the vicinity of saddle point (x/a0 (0.35 - 0.7)), the barrier is
considerably more flat. (Fig.3).
3,0
E, eV
2,5
2,0
1,5
1,0
0,5
x/a0
0,0
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
Fig.3 The calculated potential barrier for migrating oxygen ion in NiO system.
3 - 54
14
E, eV
12
10
8
6
4
2
0
0.0
0.2
0.4
0.6
0.8
 /a
x
1.0
0
Fig.4 Asymmetric potential barrier, corresponding of rigid electron localization on
the nickel ions in positions (0,0,1) and (0,0,-1)
The explanation of features of formation of the potential barrier shown on Fig.
3, can be given from the comparative analysis behavior of two components of total
crystal energy – the short-range and long-range (electrostatic) energies. The
comparative dependence of these components on displacement  of migrating oxygen
is shown on Fig. 5. Comparison of them allows to make a conclusion on a
competition of these two components at formation a diffusion barrier. It is clear, that
both components vary in enough wide limits (~10-15 eV), that practically in all a
range of values x they approximately compensate each other (the difference between
them just gives the potential barrier presented on Fig. 3). The competition between
these components leads to smoothing of -dependence of total energy in the range
30
Short-range
Electrostatic
E, eV
25
20
15
-25
-30
-35
0.2
3 - 55
0.4
0.6
0.8
 /a
x
0
x/a0(0.35 - 0.7). The estimated contribution of short-range energy to activation
energy for oxygen ion diffusion is about 10 eV, and the contribution of electrostatic
energy is about -7.5 eV.
Fig.5 The comparison of -dependence of short-range energy and electrostatic energy
CONCLUSION
The performed research exhibits that the potential barrier form for migrating
oxygen ion in NiO1- is determined in equal degree by properties of electronic and
lattice subsystems. The important role in forming of potential barrier belongs to
competition of two contributions – short-range and electrostatic energies.
To consider the role of electron subsystem more correctly one should to take
into account the influence of low symmetry fields of vacancies, which essentially
modify electronic structure of Ni+ ions with two-fold degenerated ground state in
cubic crystal field [4]. The Jahn-Teller distortions of Ni+ oxygen environment must
also be taken into account. The double exchange plays the important role in
discussion of magnetic properties of the crystals containing MV complexes [2], in this
case its contribution to activation energy also may be significant.
In this way the considered vacancy model of diffusion of oxygen ions in
oxides with NaCl structure containing MV complexes of 3d-ions can adequately
describe the oxygen diffusion in these compounds. This model can be easily
generalized for the description of cation migration in similar systems [8].
The authors wish to thank Prof. A.Ya. Fishman for his interest and stimulating
discussions.
The work was realized at the support of RFFI (project 02-03-32877 and Leading
Scientific School project 468.2003.3)/
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