JOURNAL OF APPLIED PHYSICS 98, 103513 共2005兲
Predicting ionic conductivity of solid oxide fuel cell electrolyte
from first principles
Rojana Pornprasertsuka兲
Department of Materials Science and Engineering, Stanford University, Stanford, California 94305
Panchapakesan Ramanarayananb兲
Department of Mechanical Engineering, Stanford University, Stanford, California 94305
Charles B. Musgrave
Department of Chemical Engineering, Stanford University, Stanford, California 94305
Fritz B. Prinz
Department of Mechanical Engineering, Stanford University, Stanford, California 94305
共Received 1 April 2005; accepted 13 October 2005; published online 22 November 2005兲
First-principles quantum simulations complemented with kinetic Monte Carlo calculations were
performed to gain insight into the oxygen vacancy diffusion mechanism and to explain the effect of
dopant composition on ionic conductivity in yttria-stabilized zirconia 共YSZ兲. Density-functional
theory 共DFT兲 within the local-density approximation with gradient correction was used to calculate
a set of energy barriers that oxygen ions encounter during migration in YSZ by a vacancy
mechanism. Kinetic Monte Carlo simulations were then performed using Boltzmann probabilities
based on the calculated DFT barriers to determine the dopant concentration dependence of the
oxygen self-diffusion coefficient in 共Y2O3兲x共ZrO2兲共1−2x兲 with x increasing from 6% to 15%. The
results from the simulations suggest that the maximum conductivity occurs at 7 – 9 mol % Y2O3 at
600– 1500 K and that the effective activation energy increases at higher Y doping concentrations in
good agreement with previously reported literature data. The increase in the effective activation
energy for migration arises from the higher-energy barrier for oxygen vacancy diffusion across an
Y–Y common edge relative to diffusion across one with a Zr–Y common edge of two adjacent
tetrahedra. The binding energies between oxygen vacancies and dopants were extracted up to the
fourth nearest-neighbor interaction. Our results reveal that the binding energy is the strongest when
the vacancy is in the second nearest-neighbor position relative to the Y dopant atom. The
methodology was also applied to scandium-doped zirconia 共SDZ兲. Preliminary results from
quantum simulations of SDZ suggest that the effective activation energy for vacancy diffusion in
SDZ is lower than that of YSZ, in agreement with experimental observations. The agreement with
experimental studies on the two systems analyzed in this paper supports the use of this technique as
a predictive tool on electrolyte systems not yet characterized experimentally. © 2005 American
Institute of Physics. 关DOI: 10.1063/1.2135889兴
I. INTRODUCTION
One of the key objectives in developing solid oxide fuel
cells 共SOFC兲 is the improvement of ionic conductivity in
electrolyte materials. To aid the search for better ionic conductivity, we studied oxides with the CaF2 structure adopting
first-principles techniques. Yttria-stabilized zirconia 共YSZ兲,
the most common SOFC electrolyte, has been extensively
studied experimentally and is also the focus of the present
study. Our computational results are being compared with
previously reported experimental data.
To understand the underlying mechanism of oxygen-ion
diffusion in oxide materials, several theoretical approaches
have been reported previously 共see Refs. 1–3兲. Shimojo and
Okazaki1 performed molecular-dynamics 共MD兲 simulations
of oxygen diffusion in YSZ. Their results revealed that the
difference in conductivity at different dopant concentrations
a兲
Electronic mail: rohana@stanford.edu
Present address: Intel Corporation, Santa Clara, California 95052.
b兲
0021-8979/2005/98共10兲/103513/8/$22.50
is not caused by the preference of oxygen vacancies to remain clustered with dopant atoms but rather originates from
the strongly reduced migration probabilities of oxygen ions
when the dopant atoms are present in the common edge of
the tetrahedra.
Meyer et al.2 used Monte Carlo 共MC兲 simulations to
investigate the effect that vacancy-dopant interactions have
on the vacancy transport behavior. They employed MC simulations to study the anomalous conductivity of aliovalently
doped fluorite oxides. The analysis was based on three interaction potentials between dopants and vacancies. They concluded that the barrier model where the mobility of the vacancies is reduced in the neighborhood of the dopant ions
satisfactory agreed with the experimental results and the MD
simulations of Shimojo and Okazaki.1
Using potentials fit to the experimental data, Shimojo
and Okazaki1 and Meyer et al.2 were able to describe the
behavior of oxygen-ion diffusion in YSZ. Using densityfunctional theory 共DFT兲 with generalized gradient approxi-
98, 103513-1
© 2005 American Institute of Physics
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103513-2
J. Appl. Phys. 98, 103513 共2005兲
Pornprasertsuk et al.
mation 共GGA兲 to exchange, Eichler3 calculated the diffusion
barriers for oxygen-ion vacancies at several locations in the
supercell of tetragonal YSZ which provide a connected path
for vacancy migration to positions equivalent to the initial
position. The migration energy for oxygen diffusion was assumed to be the difference between the highest energy along
the path and the lowest energy. Recent research by Ramanarayanan et al.4 suggested that such an approach can serve
as an approximation only because the migration energy is
obtained from only a specified path.
Independent of the work recently published by Krishnamurthy et al.,5 we have been working 共along a similar approach used for SiGe alloys in Ref. 6兲 to develop an understanding of ionic conductivity in YSZ using GGA-based
DFT simulations to create a database of migration energy
barriers across different nearest-neighbor configurations 共tetrahedra兲 around the diffusing oxygen ion and oxygen vacancy. Then, kinetic Monte Carlo 共KMC兲 simulations based
on the DFT barriers are performed to determine the temperature dependence of oxygen-ion diffusion. In contrast to the
approach used by Krishnamurthy et al.,5 we account for a
broader range of diffusion barriers in the vicinity of a diffusing oxygen ion and an oxygen vacancy. We consider the
influence of all six cations in two adjacent tetrahedra containing diffusing oxygen ion and oxygen vacancy to the migration energy barrier. As a result, interactions between surrounding cations and the diffusing oxygen-ion–oxygenvacancy pair are directly included into the model.
Furthermore, the DFT calculations of different cation con··
YZr
figurations allow us to extract the binding energy of VO
⬘
and YZr
⬘ YZr
⬘ . The resulting binding energy enables us to understand more about the effect of defect interactions on the
diffusion mechanism.
We assume that the oxygen self-diffusion coefficient 共D兲
for vacancy diffusion consists of temperature-dependent and
-independent terms 关Eq. 共1兲兴:
··
兴exp共− ⌬HM /kT兲,
D = D0⬘ exp共− ⌬HA/kT兲 = fD0关VO
共1兲
··
兴 is the mole fraction of the oxygen vacancy and f
where 关VO
is the correlation factor. In general, ⌬HA will be composed of
two terms: the overall migration energy 共⌬HM 兲 and the vacancy formation energy 共⌬HV兲.7 Due to the high vacancy
formation energy in zirconia, the number of vacancies thermally generated is negligible. Most oxygen-ion vacancies
··
兲 are generated to satisfy the charge neutrality condition
共VO
resulting from the local charge imbalance of aliovalent dopants 共Y3+兲 substituting for host-cation sites 共Zr4+兲 共Eq. 共2兲兲:
2ZrO2
⬘ + VO·· + 3OOx .
Y2O3 ——→ 2YZr
共2兲
The migration energy calculated using DFT is used to
determine the probability that an oxygen ion jumps from its
lattice site onto an adjacent vacancy. Since different nearestneighbor configurations give rise to different migration energy barriers, the overall migration energy 共⌬HM 兲 cannot be
associated with the migration energy of any specific
configuration.4 Rather, ⌬H M needs to be obtained from statistically averaging a spectrum of migration pathways employing techniques such as the KMC.
In this study, D0 关containing information of lattice vibrational frequency 共␯0兲 and jump distance兴 was assumed to be
a constant at all dopant concentrations due to similar vibrational entropy contributions and only slight changes in the
jump distance at different dopant concentrations:
2.57– 2.58 Å from 6 to 15 mol % YSZ8 共assuming jump distance is half of lattice parameter兲. 共The correlation factor is
implicit in the KMC calculations.兲 After calculating the normalized diffusion coefficients 共D / D0兲 at different temperatures, the effective activation energy was extracted using an
Arrhenius plot. This effective activation energy is equivalent
to the overall migration energy 共⌬H M 兲 in Eq. 共1兲. The calculations were repeated by changing the dopant concentration.
Thus, the diffusion coefficients and the corresponding effective activation energies were extracted as a function of dopant concentration. The results are compared with the experimental results at different dopant concentrations and
different temperatures.
The present analysis provides useful insights into the
vacancy diffusion mechanism and elucidates the influence
that dopant type and concentration have on the ionic conductivity of solid oxide electrolyte materials. It also validates the
technique as a useful predictive tool.
II. QUANTUM SIMULATIONS: COMPUTATIONAL
DETAILS AND RESULTS
A. DFT calculations
Cubic ZrO2 has the CaF2 crystal structure: each unit cell
consists of four Zr4+ ions occupying fcc lattice sites and eight
O2− ions occupying the tetrahedral sites 共Fig. 1兲. Doping
with yttria 共Y2O3兲 replaces Zr by Y atoms and, for every two
Y atoms, one oxygen vacancy needs to be created to satisfy
charge neutrality. The calculations were performed using the
Vienna ab initio simulation package 共VASP兲 which employs
density-functional theory and expands the electronic structure using a plane-wave basis set.9–12 Electron-ion interactions are described using the projector-augmented wave
共PAW兲 method13,14 with plane waves up to the energy cutoff
at 400 eV 共29.4 Ry兲. We adopted the PW91 GGA exchangecorrelation functional proposed by Perdew and Wang.15 The
k-point sampling was restricted to a single gamma 共⌫兲 point:
共0,0,0兲. The supercell used in the calculations consisted of 30
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103513-3
J. Appl. Phys. 98, 103513 共2005兲
Pornprasertsuk et al.
FIG. 2. Positions of cations, diffusing oxygen ion, oxygen vacancy in the
two common tetrahedra, and the saddle-point plane. ①: cation positions in
the two adjacent tetrahedra, where the number label is used as a reference
for Table I; 씲: oxygen vacancy in the tetrahedral site 共first nearest Zr or Y
ions in positions 1, 2, 3, and 4兲; 쎲: diffusing oxygen ion, which is in
tetrahedral site 共first nearest Zr or Y ion in positions 3, 4, 5, and 6兲; - -:
saddle-point plane where the diffusing oxygen ion migrates to the same
plane as the common cation sites of the two adjacent tetrahedra 共three atoms
are allowed to relax in all directions within the plane兲.
Zr, 6 Y, and 69 O atoms corresponding to 8.3 mol % YSZ.
In principle, there is a range of pathways for an atom to
move to a neighboring vacant site. To determine the migration barrier of the oxygen vacancy, the saddle point was assumed to be located on the plane that has the following properties: 共i兲 it is perpendicular to the shortest path and 共ii兲 it
contains the other two cations 共Fig. 2兲. To calculate the energy at the saddle point, all three ions 共two cations and one
oxygen ion兲 were allowed to relax only within this plane
while the other atoms were allowed to relax in all directions.
The corresponding migration energy is the energy difference
between the saddle-point energy and the initial state energy
共Fig. 3兲.
Because the diffusion barrier depends on the local
atomic environment, migration energies of different configurations of neighboring atoms around the diffusing oxygen ion
and oxygen vacancy need to be calculated. The number of
different possible arrangements grows unmanageably large
unless approximations are made. We used the observation of
Shimojo and Okazaki1 to assume that the barrier depends
largely on the sites’ first nearest neighbors to the diffusion
center. All migration barrier calculations were performed
with a single dopant concentration of 8.3 mol % YSZ by
varying the location of dopant atoms within the supercell.
The results were used as barrier database for kinetic Monte
Carlo simulations, which will be discussed in the next section. 共Additionally, we also performed the same calculations
FIG. 3. Illustration of the migration energy barrier. The middle point is the
energy corresponding to the saddle point where three atoms 共diffusing oxygen ion and two cations that need to be diffused across兲 align in the same
plane.
with 14.3 mol % YSZ. Another migration energy database
for 6 – 15 mol % YSZ was then established by linearly interpolating between the 8.3 and 14.3 mol % YSZ data sets. The
results of the interpolation database turn out to be similar to
the 8.3 mol % YSZ database.兲
In this study, the migration energy was assumed to depend only on the arrangement of cations in the two adjacent
tetrahedra containing the diffusing atom and the vacancy 共atoms 1–6, Fig. 1兲. Consequently, the interactions of oxygenion vacancies and the other interactions beyond the components in the two tetrahedra were neglected. To further reduce
the computational complexity, we assumed that there were
no more than three Y ions in the six cation sites. Further··
more, the binding energies of VO
YZr
⬘ and YZr
⬘ YZr
⬘ up to fourth
nearest-neighbor interactions were extracted from the energy
differences between different defect arrangements modeled
by the various supercells considered. Because all supercells
had oxygen vacancies which were at least fifth nearest neigh·· ··
VO was not obbors to each other, the binding energy of VO
tained.
To investigate the dependence of the effective activation
energy on the type of dopants, migration energy calculations
were performed on scandium-doped zirconia 共SDZ兲 by employing the same approach as described above for YSZ.
B. Results and discussion
The lattice parameter of the 8.3 mol % YSZ obtained by
volume relaxation was 5.14 Å which is in good agreement
with the experimental value of 5.14– 5.16 Å.8,16 Diffusion
barriers of each combination of cations in the two adjacent
tetrahedra are summarized in Table I. The highest diffusion
barrier was found for oxygen-vacancy motion between two
adjacent tetrahedra containing Y–Y common edge, which is
in agreement with the results by Shimojo and Okazaki1 and
Krishnamurthy et al.5 The reason for the higher migration
energy across two tetrahedra containing Y–Y versus Zr–Zr
common edge could arise from two main factors: the smaller
space for the oxygen ion to move 共due to the bigger ionic
radius of Y3+ compared with Zr4+兲 and the binding energy
between YZr
⬘ and VO·· . Furthermore, the results 共Table I兲 suggest that the migration energy barriers are sensitive to the
surrounding atoms beyond the two common edge cations due
to an association effect between defects. With the same Y–Y
common edge, the activation changes from 1.23 to 1.40 eV
if an additional Y atoms is present at atom position 5 or 6
共Fig. 2兲. This result suggests that the barriers are sensitive to
the surrounding cations and the difference in the migration
energy barrier partly comes from the association between the
defects.
··
YZr
The binding energy of VO
⬘ and YZr
⬘ YZr
⬘ depends on the
identity of the other atoms surrounding these sites. We obtain
the average binding energy by using a least-squares fit to the
binding energies of the various cases where the identities of
the other atoms surrounding these sites are different. The
··
YZr
binding energies of VO
⬘ and YZr
⬘ YZr
⬘ at distance from first
to fourth nearest neighbors obtained from the least-squares
solution are listed in Table II. The plot of the binding energy
with respect to the distance between the defects 共in terms of
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103513-4
J. Appl. Phys. 98, 103513 共2005兲
Pornprasertsuk et al.
TABLE I. Summary of the DFT migration energy barriers in yttria-stabilized zirconia 共YSZ兲. The numbers in the first row indicate the positions of the cations
in the two tetrahedra containing the diffusing oxygen ion and the oxygen vacancy 共1, 2, 3, and 4 are the positions of the cations in the tratrahedron containing
the oxygen vacancy and 3, 4, 5, and 6 are the positions of cations in the tratrahedron containing the diffusion oxygen ion, as shown in Fig. 2兲.
1
2
3
4
5
6
Migration
energy
1
2
3
4
5
6
Migration
energy
Zr
Y
Zr
Zr
Zr
Zr
Zr
Y
Y
Y
Y
Y
Zr
Zr
Zr
Zr
Zr
Zr
Zr
Zr
Zr
Zr
Zr
Y
Zr
Zr
Zr
Zr
Y
Zr
Zr
Zr
Zr
Y
Y
Y
Y
Zr
Zr
Zr
Zr
Zr
Zr
Zr
Zr
Y
Zr
Zr
Zr
Zr
Y
Zr
Zr
Zr
Y
Zr
Zr
Zr
Y
Y
Y
Zr
Zr
Zr
Zr
Zr
Zr
Y
Zr
Zr
Zr
Zr
Y
Zr
Zr
Zr
Y
Zr
Zr
Y
Zr
Zr
Y
Y
Zr
Zr
Zr
Zr
Zr
Y
Zr
Zr
Zr
Zr
Y
Zr
Zr
Zr
Y
Zr
Zr
Y
Zr
Y
Zr
Zr
Zr
Zr
Zr
Zr
Zr
Y
Zr
Zr
Zr
Zr
Y
Zr
Zr
Zr
Y
Zr
Zr
Y
Zr
Y
0.67
0.20
0.20
1.19
1.19
0.80
0.80
0.23
0.88
0.88
0.39
0.62
0.88
0.88
0.62
0.39
1.23
0.71
0.71
0.71
0.71
Zr
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Zr
Zr
Zr
Zr
Zr
Zr
Zr
Zr
Zr
Zr
Zr
Y
Y
Y
Y
Zr
Zr
Zr
Zr
Zr
Zr
Y
Y
Y
Y
Y
Y
Zr
Zr
Zr
Zr
Zr
Y
Zr
Zr
Zr
Y
Y
Y
Zr
Zr
Zr
Y
Y
Y
Zr
Zr
Zr
Y
Y
Y
Zr
Zr
Zr
Y
Zr
Zr
Y
Zr
Zr
Y
Y
Zr
Y
Zr
Zr
Y
Y
Zr
Y
Y
Zr
Y
Y
Zr
Zr
Y
Zr
Zr
Y
Zr
Y
Zr
Y
Zr
Y
Zr
Y
Zr
Y
Y
Zr
Y
Y
Y
Zr
Zr
Zr
Y
Zr
Zr
Y
Zr
Y
Y
Zr
Zr
Y
Zr
Y
Y
Zr
Y
Y
Y
0.45
0.70
0.70
0.28
0.28
1.20
0.79
1.09
0.79
1.09
0.70
1.20
1.09
0.79
1.09
0.79
0.70
1.40
1.40
0.94
0.94
neighborhood position兲 is shown in Fig. 4. A contribution to
the error in the calculated binding energy may arise from
neglecting of other possible defect complexes such as
YZr
⬘ VO·· YZr
⬘ . The negative binding energy of VO·· YZr
⬘ suggests
··
that VO and YZr
⬘ are attracted to each other and, on the other
hand, the positive binding energy shows that YZr
⬘ and YZr
⬘
repel each other. The highest binding energies occur when
··
is in the second nearest-neighbor position relative to Y,
VO
which is in good agreement with the experimental
observations.17–19 Mott-Littleton defect calculations by
Zacate et al.20 using Born-like pairwise potentials also suggest that oversized dopants 共such as Y, Gd, and Sm兲 prefer a
cluster geometry in which the oxygen vacancy resides in the
second nearest-neighbor site.
··
YZr
The results for both VO
⬘ and YZr
⬘ YZr
⬘ show lower binding energies beyond the second nearest neighbors and approach zero at the fourth nearest-neighbor interaction. The
··
binding energies of VO
YZr
⬘ are 0.13– 0.35 eV which are at
least 25% of the migration energy barrier range of
0.2– 1.4 eV and, therefore, can influence the oxygen migration in YSZ. To obtain accurate results for the migration
energy barriers, surrounding cations around diffusing oxygen
and oxygen-vacancy pair up to the third nearest neighbors
should be taken into account. However, due to impractical
computational times for the DFT calculations, in this study,
we consider interactions only up to the second nearest··
YZr
neighbor VO
⬘ 共in the diffusing direction兲, which represents
the strongest association energy, in the migration energy barrier calculations.
For the SDZ calculations, the lattice parameter of
8.3 mol % SDZ obtained was 5.09 Å which is in good agree-
TABLE II. List of calculated binding energies of VO·· YZr
⬘ and YZr
⬘ YZr
⬘ from the
first nearest-neighbor 共1st NN兲 to the fourth nearest-neighbor 共4th NN兲 interactions. The binding energy is the least-squares solution of the combinations of total energies obtained from quantum calculations.
Associated defect
Binding energy
V – Y 1st NN
V – Y 2nd NN
V – Y 3rd NN
V – Y 4th NN
Y–Y 1st NN
Y–Y 2nd NN
Y–Y 3rd NN
Y–Y 4th NN
−0.2988
−0.3531
−0.1859
−0.1328
0.0335
0.1451
0.0048
−0.0973
FIG. 4. Plot of the binding energy from first nearest-neighbor to the fourth
nearest-neighbor YZr
⬘ YZr
⬘ and VO·· YZr
⬘ interaction extracted by a least-squares
fit of the total energies of the different arrangements of VO·· and YZr
⬘.
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103513-5
J. Appl. Phys. 98, 103513 共2005兲
Pornprasertsuk et al.
TABLE III. Comparison of DFT migration energies in scandia-doped zirconia 共SDZ兲 and yttria-stabilized
zirconia 共YSZ兲. The numbers in the first row indicate the positions of the cations in the two tetrahedra containing the diffusion oxygen ion and the oxygen vacancy 共1, 2, 3, and 4 are the positions of the cations in the
tetrahedron containing the oxygen vacancy and 3, 4, 5, and 6 are the positions of the cations in the tetrahedron
containing the diffusion oxygen ion, as shown in Fig. 2兲.
1
2
3
4
5
6
Migration energy
共SDZ兲
Migration energy
共YSZ兲
Zr
Zr
Zr
Zr
Zr
Zr
Zr
Sc/ Y
Sc/ Y
Zr
Zr
Sc/ Y
Zr
Zr
Zr
Zr
Zr
Zr
0.75
0.93
1.10
0.67
1.19
1.23
ment with the experimental value of 5.09 Å 共at 10 mol %
SDZ兲.21 The migration energy barriers of SDZ compared
with those of YSZ are shown in Table III. The calculations
suggest that without any dopants in the two adjacent tetrahedra, the migration energy of oxygen ions is slightly higher in
SDZ than that in YSZ because of the smaller lattice parameter of SDZ. However, the migration energy of the oxygen
ion in SDZ becomes lower than that in YSZ due to the presence of dopants in the two tetrahedra; the smaller radius of
Sc relative to Y provides for a larger space for the oxygen
ion—this effect wins over the smaller lattice parameter of
SDZ.
Experimental results also showed that the activation energy of 11 mol % SDZ is smaller than that in 11% YSZ
共0.7 eV in SDZ versus 0.9 eV in YSZ兲.22 Sc3+-substituted
Zr4+ in SDZ results in the same defect chemistry as Y3+ in
YSZ; this implies that both systems have the same oxygenvacancy concentrations. Furthermore, assuming that D0 of
both YSZ and SDZ are approximately the same, the diffusion
coefficient of SDZ is higher than YSZ at high dopant concentrations which was observed experimentally in
10– 15 mol % doped zirconia.21,22 The calculations of Zacate
··
ScZr
et al.20 show that VO
⬘ has the lowest binding energy compared with other trivalent dopants, which may be one of the
reasons for the lower migration energy and higher oxygen
self-diffusion coefficients in SDC compared with YSZ.
Although the migration energy barrier database gives
some insight into the dependence of the migration energy on
the ionic radius of the dopant and the lattice parameter, it is
only representative of atomistic processes at 0 K. We performed a KMC simulation using the database to study the
temperature dependence of the diffusivity and hence determine an effective activation energy for the entire diffusion
process.
III. KINETIC MONTE CARLO SIMULATIONS:
COMPUTATIONAL DETAILS AND RESULTS
A. Kinetic Monte Carlo simulations
The KMC technique attempts to capture the effects of
rare atomic processes that directly contribute to changes in
macroscopic properties. Therefore, the results of the simulations, after they are statistically averaged, are likely to reflect
macroscopic behavior. In this study, KMC was used to simulate an activated random-walk process in a randomly distributed landscape of vacancy and Y atoms. The KMC procedure
consists of a series of KMC moves. Each move consists of
the following steps.23 共i兲 Identify all possible events from the
current configuration. In fluorite oxides, there are six possible events for each oxygen vacancy 共six nearest-neighbor
oxygen sites兲, except in the case where any one of the first
nearest neighbors is also a vacancy. 共ii兲 Obtain the rates for
each of the events 共␯i兲. Rate is proportional to exp共
−⌬Em / kT兲, where the migration energy 共⌬Em兲 is obtained
from the ab initio calculations 共Table I兲. 共iii兲 Generate a
pseudorandom number ␥ between 0 and 1. 共iv兲 Advance the
time of each step 共⌬t兲 by −ln共␥兲 / 兺i␯i.24 共v兲 Choose one of
the events depending on the random number ␥, consistent
with the relative rate ␯i of all the events. 共vi兲 Reconfigure the
system according to the chosen event. 共vii兲 Update and
record the new position of the vacancy and time.
The diffusion coefficient of oxygen vacancy 共Dv兲 was
calculated as given by Eq. 共3兲:
Dv = limt→⬁ x2/6t,
共3兲
where t is the time calculated as the sum of all ⌬t of each
jump and x2 is the mean-squared displacement.7 Subsequently, the oxygen self-diffusion coefficient 共D兲 can be calculated as given by Eq. 共4兲.
··
兴Dv .
D = 关VO
共4兲
According to Eq. 共1兲, the overall migration energy or the
effective activation energy of the oxygen self-diffusion coefficient 共⌬HM 兲 can subsequently be obtained from the slope
of the Arrhenius plot of diffusion coefficients with respect to
the inverse temperature.
B. Simulation results and discussion
The periodic supercells used in the KMC simulations
were 5 ⫻ 5 ⫻ 5 unit cells of cubic ZrO2. Y atoms and oxygenion vacancies were introduced randomly according to the
concentration of Y2O3 dopants with the following restriction:
Y ions are made to be distributed such that there are no more
than three Y ions within the six nearest-neighbor positions of
the diffusing oxygen ion and the oxygen vacancy 共Figs. 1
and 2兲. 共The energy of the system which has more than three
Y ions will be so large that such configurations will be rare,
thus justifying this restriction.兲 The KMC simulation was run
until a total of 2 000 000 jumps were performed. Twelve different random initial distributions were used for each concentration between 6 and 15 mol % YSZ and at each tem-
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103513-6
Pornprasertsuk et al.
J. Appl. Phys. 98, 103513 共2005兲
FIG. 6. Summary of the effective activation energy with respect to yttria
doping concentration in YSZ. 共䉭兲 the experimental results obtained from
Ref. 26 at 950– 1000 ° C, 共䊐兲 KMC High T is the effective activation energy
at T ⬎ 1050 K, and 共⫻兲 KMC Low T is the effective activation energy at
T ⬍ 1050 K obtained from kinetic Monte Carlo simulations using 5 ⫻ 5 ⫻ 5
unit cell with 2 000 000 jumps averaged over 12 initial configurations.
FIG. 5. Arrhenius plot of normalized oxygen self-diffusion coefficients
共D / D0兲 obtained from kinetic Monte Carlo simulations with respect to temperature from 600 to 1500 K. 共a兲 8 mol % YSZ: the effective activation
energy at T ⬍ 1050 K is 0.68 eV and at T ⬎ 1050 K is 0.70 eV. 共b兲
15 mol % YSZ: the effective activation energy at T ⬍ 1050 K is 0.8 eV and
at T ⬎ 1050 K is 1.0 eV.
perature between 600 and 1500 K. The calculated values
using 10⫻ 10⫻ 10 unit cells with 3 000 000 jumps give
similar results.
The effective activation energy for the 8 mol % YSZ is
0.7 eV, which is slightly lower than the experimental value
of 0.83– 1.05 eV.25,26 This is probably caused by the exclusion of vacancy-vacancy interactions, the interactions with
extended neighbors beyond the two adjacent tetrahedra, and
inaccuracies in DFT calculations. However, at higher doping
concentrations, the activation energy obtained from KMC
splits into two regions at around 900– 1050 K 共Fig. 5兲. At
15 mol % YSZ, the activation energies at low-temperature
and high-temperature regions are 0.8 and 1.0 eV, respectively. This indicates that with decreasing temperatures, rate
controlling processes with lower activation energy become
statistically more prevalent, as expected.
The dependence of the effective activation energy with
respect to doping concentration is summarized in Fig. 6. The
behavior predicted by KMC at high temperatures 共T
艌 1050 K兲 qualitatively agrees with the experimental observations though 0.06– 0.25 eV is lower than the experimental
values.26 The calculations of Krishnamurthy et al.5 yield an
effective activation energy of 0.62 eV at concentration levels
as high as the 15 mol % YSZ and appear to differ from
observations26 by more than 0.5 eV. The higher effective
barrier our simulations predicted is likely due to the inclusion of pathways involving differing arrangements in all six
nearest-neighbor cations in the vicinity of the diffusing oxygen and vacancy pair. At finite temperatures, the additional
pathways we consider will have nonzero probabilities and
thus increase the effective barrier over models which preclude these paths.
The flat region at the beginning of the plot indicates
fewer interactions between oxygen vacancy and Y ions at
low doping concentrations. The activation energies at low
concentrations are close to the migration energy of the two
tetrahedra containing all Zr atoms. However, in the calculation, the flat region of the activation energy ranges from
6 to 8 mol % YSZ, while the experimental results show increasing activation energies above 6 mol % YSZ. This disparity may be the result of neglecting other interactions between oxygen vacancy and their outlying Y neighbors. At
high concentrations of YSZ, the KMC calculations indicate
an increase in the activation energy, similar to the experimental results.8,26
The logarithmic plot of normalized oxygen self-diffusion
coefficients at concentrations ranging from 6 to 15 mol %
YSZ 共5 ⫻ 5 ⫻ 5 unit cells, 2 000 000 jumps, and 12 configurations兲 at 1050– 1500 K is shown in Fig. 7共a兲. The highest
oxygen self-diffusion coefficients obtained from the KMC
simulations are between 7 and 8 mol % YSZ at 750– 1200 K
and slightly shift to 8 and 9 mol % at 1350– 1500 K. The
shift of the maximum conductivity has also been observed
experimentally8 关Fig. 7共b兲兴 and is also in good agreement
with the results of Krishnamurthy et al.5 Because of the general underestimation of the migration energy barriers by the
DFT calculations the magnitude of the KMC simulations appears to match the experimental values at higher temperature. The magnitude of the oxygen diffusivity at 1050 K is in
very good agreement with the experimental results at 1273 K
共Fig. 8兲.
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103513-7
J. Appl. Phys. 98, 103513 共2005兲
Pornprasertsuk et al.
Despite the decrease in oxygen-ion mobility, higher dopant
concentrations provide higher vacancy concentrations that
compensate for the loss in mobility up to 7 – 8 mol % YSZ as
suggested by the KMC simulations and the experimental
data.26 The KMC simulations were performed using the migration energy barrier database obtained by linearly interpolating the energy barriers at 8.3 mol % YSZ and 14.3 mol %
YSZ. The simulations show similar results as when using the
8.3 mol % YSZ migration energy database with a slight shift
in the peak of the ionic conductivity to around 8 – 10 mol %
YSZ at 600– 1500 K.
IV. SUMMARY
FIG. 7. 共a兲 Logarithmic plot of normalized oxygen self-diffusion coefficients vs Y2O3 doping concentration in YSZ obtained by kinetic Monte
Carlo 共KMC兲 simulations of 5 ⫻ 5 ⫻ 5 unit cell with 2 000 000 jumps averaged over 12 initial configurations at 1050– 1500 K. 共b兲 Logarithmic plot of
conductivity vs mol % Y2O3 in YSZ obtained experimentally from Ref. 8 at
1073– 1473 K.
The above results suggest that the decline in ionic conductivity at high doping concentration follows the decrease
in oxygen-ion mobility due to the higher diffusion barrier
across tetrahedral containing Y–Zr and Y–Y in the common
edge 共Table I兲 and more association between the defects.
Quantum simulations complemented with KMC simulations using DFT derived probabilities have been used to predict ionic conductivity in 6 – 15 mol % YSZ. This approach
provides results with qualitative agreement relative to the
experimental observations. The quantum simulations suggest
that the decrease in conductivity at high doping concentrations mostly arises from the higher migration energy required
to traverse across the two adjacent tetrahedra containing the
Y–Zr or Y–Y common edge during the diffusion process.
The calculated optimum ionic conductivity at 7 – 8 mol %
YSZ agrees well with the experimental observations. The
KMC results reveal that the increase in the overall migration
energy at higher yttria concentration is due to the higher
probability that oxygen vacancies encounter the Zr–Y and
··
YZr
Y–Y pairs. The binding energy of VO
⬘ and YZr
⬘ YZr
⬘ was
extracted to the fourth nearest-neighbor interactions. The
··
was the second
strongest binding energy was found when VO
nearest-neighbor to YZr
⬘ which is in a good agreement with
the experimental observations.17–19
We suggest that the present technique can be used to
predict ionic conductivity in different types of electrolytes to
find optimum dopant concentrations. Preliminary results
from SDZ calculations suggest that the migration energy of
SDZ will be lower than YSZ due to lower migration barrier
across Sc–Zr and Sc–Sc in agreement with the experimental
results.21,22 However, at very low concentrations of Sc, the
migration energy of SDZ could be slightly higher than YSZ
due to the smaller lattice parameter. Furthermore, by considering the effect of all cations in both adjacent tetrahedra, the
present technique could improve the ability to predict the
effect of the dopant concentration on the ionic conductivity.
This methodology can also be a predictive tool for more
complex structures such as ternary oxides.
ACKNOWLEDGMENTS
FIG. 8. 共䊐兲 Logarithmic plot of conductivity vs mol % Y2O3 in YSZ obtained experimentally from Ref. 26 at 1273 K. 共쎲兲 Logarithmic plot of
D / D0 vs Y2O3 doping concentration in YSZ obtained by kinetic Monte
Carlo 共KMC兲 simulation of 5 ⫻ 5 ⫻ 5 unit cell with 2 000 000 jumps averaged over 12 initial configurations at 1050 K.
We would like to thank Joseph Han for suggestions on
quantum simulations and administrating our supercomputer,
and Professor K. J. Cho for the help and suggestions on VASP
and kinetic Monte Carlo simulations. Computations were
performed on the supercomputer facilities at University of
California, San Diego, and BioX at Stanford University. This
research is supported by Office of Naval Research 共ONR兲
and The Global Climate Energy Project 共GCEP兲 at Stanford
University. Furthermore, we thank the members of the Rapid
Prototyping Laboratory at Stanford University and especially
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103513-8
the fuel cell team for their support and suggestions. This
work was supported by a scholarship from Thai government
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