The Influence of Defects on Proton Diffusion in

Perovskites АIIВIV1-xR

IIIxO3-δδδδ: Monte Carlo Study

V.I. Tsidilkovski1,a, M.Z. Uritsky1,b, A.N. Varaksin2,c, A.Ya. Fishman3,d

1Inst. of High-Temperature Electrochemistry, Russian Academy of Sciences 22 S.Kovalevskaya St., 620219 Ekaterinburg, Russia

2Inst. of Industrial Ecology, Russian Academy of Sciences 20-a, S.Kovalevskaya St., 620219 Ekaterinburg, Russia

3Inst. of Metallurgy, Russian Academy of Sciences 101, Amundsena St., 620016 Ekaterinburg, Russia

ae-mail: VTsidilkovski@ihte.uran.ru , be-mail: umz@ihte.uran.ru ,

ce-mail: varaksin@ecko.uran.ru, de-mail: fishman@uran.ru

Keywords: Proton, diffusion, conductivity, oxide, proton-conducting oxide, Monte Carlo

Abstract. The tracer diffusion coefficient D* and the mobility of protons in proton-conducting

oxides of the АIIВ

IV1-xR

IIIxO3-δ family have been calculated using the Monte Carlo method as

functions of temperature and concentration x of the acceptor impurity RIII. The effect of proton-

dopant interactions (proton trapping) and the effect of protonic sites blocking caused by proton-

proton and proton - oxygen vacancy interactions, are analyzed. It is shown that the proton

diffusivity depends significantly on the dopant content and is considerably reduced already at small

x. The D* value weakly depends on the concentration of oxygen vacancies at fixed, not too large

values of x. The conductivity dependence on the doping concentration σ(x) can have a maximum

due to the proton-defect and proton-proton interactions. The Haven ratio deviates slightly from

unity at the expected intensity of interparticle correlations. The calculated values of both the

diffusivity activation energy increase (with x), and the location of the σ(x) maxima agree with the

experimental data for a number of proton-conducting oxides.

Introduction

A number of perovskite-type oxides of the ABO3 family doped with lower valence elements have

high proton conductivities at elevated temperatures. Such high-temperature proton conductors

present great interest because of their potential use as electrolytes in fuel cells, hydrogen sensors

and other electrochemical devices [1, 2].

Protons are incorporated into these compounds when samples are exposed to the gas phase

containing water vapor. Upon dissociative dissolution of a water molecule, oxygen occupies

vacancies in the oxygen sublattice, while protons attach to oxygen ions and form (OH)− centers.

There exist equivalent potential minima, separated by small (less than 0.2 eV) potential barriers, for

proton at oxygen [3]. These multi-well states of (OH)− centers can provide unusual properties of

oxides at low temperatures [4]. The role of such minima in the proton migration processes at high

temperatures is determined primarily by the ratio of the characteristic time of the (OH)− centers

reorientation, τO, and the characteristic time of the proton transition between the minima related to

the adjacent oxygen ions, τOO. Generally, the mechanism of the proton transfer between oxygen ions

is rather complicated since this transition strongly depends on the lattice dynamics, especially on

the dynamics of the oxygen ions directly involved in the transfer [5 and Refs therein].

The role of the interaction of the protons with defects (impurities and vacancies) and of the

proton-proton correlations in the proton transport phenomena in proton-conducting oxides poses a

separate problem. Some experimental studies reveal the dependence of the mobility activation

energy on the dopant content, (see [5] and Refs therein). A maximum is observed in dopant

Defect and Diffusion Forum Vols. 258-260 (2006) pp 124-129Online available since 2006/Oct/15 at www.scientific.net© (2006) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/DDF.258-260.124

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,www.ttp.net. (ID: 129.187.254.47, UB der LMU Muenchen, Muenchen, Germany-04/07/14,12:59:13)

concentration dependences of the conductivity for some proton conductors (at x ~ 0.05-0.15), see,

e.g., [6]. Note that such maxima are quite common also for acceptor-doped oxygen conductors [7].

One of the obvious reasons for the dependence of the proton transport coefficients on the dopant

concentration is a possible trapping of H+ ions by R

III impurities with an effective negative charge.

The conclusion about the existence of such traps in proton-conducting oxides was made based on

the analysis of data obtained by various experimental techniques (neutron spectroscopy [8], muon

spin relaxation [9], etc.). Molecular statics and first-principles calculations also indicate the

presence of deeper wells for protons on oxygen ions near the RIII impurity [10, 11]. Experimental

and theoretical estimations generally give the value of the trap depth ∆E ~ 0.2-0.4 eV. In a recent study [11] ∆E was estimated to be ~ 0.2 eV in cubic In-doped BaZrO3, and the effect of traps on the

diffusion coefficient was calculated by the Monte Carlo method for the dopant concentration x =

0.05 and T = 600 K. The calculated diffusion coefficient proved to be almost an order of magnitude

smaller than in an undoped crystal.

In this paper we present our theoretical considerations of the effect of the random dopant

potential (different energy states of protons) and the effect of repulsive proton-proton and proton-

oxygen vacancy interactions on the proton transport in perovskite oxides. Using the Monte Carlo

algorithms [12], we calculate the tracer diffusion coefficient and the mobility of protons as

functions of temperature and the concentrations of dopants RIII and oxygen vacancies. The mobility

is obtained from the Monte-Carlo simulation of a steady-state proton current in a weak dc electric

field. Several realistic models of the potential relief for migration of protons with wells of different

depths near and far from the dopant are considered. The dependence of transport coefficients on

both the trapping energy ∆E of protons (∆E varies over reasonable limits) and the intensity of

repulsive interactions is studied.

Model and Approximations

The study deals with the hopping transfer of protons in cubic perovskites of the АIIВ

IV1-xR

IIIxO3-δ

family. Several sets of potential minimums near oxygen ions, which correspond to different

equilibrium orientations of (OH)− dipoles, were proposed for protons in these oxides (see, e.g., [3,

4, 10] and Refs therein). Further, in line with the conclusions drawn from the majority of

experimental and theoretical studies [5 and Refs therein], we assume the reorientation time for

(OH)− groups to be much smaller than the characteristic time for the proton transfer between

oxygen ions, τO << τOO. If τO << τOO, the probability of the proton transfer from the oxygen ion l to

the adjacent oxygen ion m may be written as W*li,mj = Pli •Wli,mj, where Pli is the average Gibbs

probability to find a proton on the site i on the oxygen ion l (if 1=∑i liP ) and Wli,mj is the

probability that a proton hops from the site (li) to the nearest site (mj) on the neighboring ion. It is

easily seen that the choice of a particular set of sites i is not important in the given situation if one

considers the prohibition of two protons being located on a single oxygen ion. Therefore, in what

follows we shall consider two variants of the so-called two-state model where one potential well is

available for a proton on an oxygen site. In this case, Pli transforms to Plm meaning the probability

of choosing the hopping direction l→m, while Wli,mj turns to Wlm meaning the probability of this

hop.

Figure 1 presents a schematic drawing of the potential barrier profiles in models 1 and 2. In both

models the potential well bottom for a proton near the trivalent RIII impurity is ∆E lower than in the

pure crystal. In model 2 the height of the potential barrier for the proton transport between sites near

trivalent impurities is equal to the potential barrier height, U0, in an undoped crystal, while in model

1 the barrier height for the analogous transfer equals U0+∆E.

Defect and Diffusion Forum Vols. 258-260 125

Fig. 1. Schematic drawing of the potential relief in models 1 and 2 (the left- and right-hand

drawings respectively)

We shall account for the repulsion between protons in terms of two simple approximations. The

main approximation (A), which is used in the study, assumes the condition of blocking, that is, the

prohibition of two protons being located on an oxygen ion. The interaction of protons and oxygen

vacancies is also described by the simplest means, namely the prohibition of a proton hopping to an

oxygen vacancy. In the approximation A the probability of hops has the form

}exp{)1(0 lmmmlm UnpAW ββω −−= .

Here A is a dimensionless numerical factor, β=(kBT)−1, ω0 is the characteristic frequency of the

promoting mode for protons, pm is the occupancy number of the site m with an oxygen ion (pm = 0

or 1), Ulm is the potential barrier for the l→m hop, and nm is the occupancy number of the site m

with a proton (nm = 0 or 1).

In order to analyze qualitatively possible consequences of a stronger repulsion between

positively charged defects, we shall dwell briefly on the approximation B, in which protons (or both

protons and oxygen vacancies) block not only the sites occupied by them, but also the nearest

neighboring oxygen ions (the repulsion is overestimated).

Note, that in this study, tracers mean labeled particles with the hopping probability and the

concentration similar to those of protons. We do not consider effects related to the isotopic

substitution of protons, which can be considerable for both the mobility and the concentration of

isotopes [13].

Simulation Technique

The Monte Carlo simulation was realized for cubic cells (30×30×30) of sites under periodic boundary conditions. The impurity concentration x changed from 0.02 to 0.5 in steps ∆x = 0.02. The temperature changed from 600 to 1200 in 25 K steps.

The diffusion simulation process started with random filling of the lattice with RIII impurities,

oxygen vacancies and protons. The filling procedure was realized taking into account the

aforementioned principles of sites blocking. Furthermore, the concentrations of protons cH, oxygen

vacancies cV, and the dopant x were related by the standard electro-neutrality condition

cH = x − 2cV. The proton migration was analyzed after the ensemble of protons relaxed to the

stationary state.

The tracer diffusion coefficient, D*, was determined by calculating the root-mean-square

displacement of an ensemble of labeled particles: D* = <X2>/(6Nt). Here <X

2> is the sum of

squared displacements of all particles as referred to their number, while Nt is the number of steps in

the Monte Carlo procedure.

The proton mobility µ was obtained by calculating the steady-state current in the system arising

upon application of a weak dc electric field E. For calculation of the current, the change of the

∆∆∆∆E

∆∆∆∆E

BIV R

III

O2- O

2- O

2- O

2-

BIV

RIII B

IV R

III

O2- O

2- O

2- O

2-

BIV R

III

∆∆∆∆E

U0

126 Diffusion in Solids and Liquids, DSL-2006 I

probability of a hop Wlm between neighboring sites l and m was written as a field-linear

approximation, WЕlm = Wlm × (1 ± qEa/2kT). Here Wlm is the probability of a hop l→m in the

absence of the field, q =eis the proton charge, a = (Rm−Rl) is the distance between the sites l and m, and E is the projection of the field to the vector (Rm−Rl). The current was calculated under

periodic boundary conditions, while the diffusion coefficient Dσ was defined via the mobility by the

standard relationship Dσ = µkT/q. The schemes used for calculation of the transport of interacting

particles by the Monte Carlo method are described in more detail elsewhere [12].

Results and Discussion

1. Figure 2 presents dependences of the diffusion coefficient of a labeled particle, D*(x), at

temperatures of 600 K and 1000 K for models 1 and 2 in the absence of oxygen vacancies (cV = 0

and cH = x) and with a weak proton-proton correlation (approximation A). One can see, that when

the impurity concentration is small (x < 0.1) and the trivalent impurities act as single traps, the

increase in x is followed by a sharp decrease in the coefficient D* (more drastic at lower

temperatures). As x increases, the diffusion coefficient decreases monotonically in model 1 and the

dependence D*(x) has a minimum in model 2. This behavior of D*(x) in model 2 is due to the fact

that in this model the potential barrier for proton hopping has the same height in undoped areas of

the crystal, free of RIII ions, and areas with R

III ions located in neighboring cells.

0 0,1 0,3 0,5

1

0,6

0,2

D*(x) / D*(0)

X

0 0,2 0,4 X

Ueff , eV

0,2

0,1

Fig. 2. Calculated coefficients D*(x) normalized

to D*(x→0) at T=600 K (open diamonds and

squares) and at T=1000 K (filled ones).

Diamonds and squares correspond to the results

calculated within different models: 1 and 2,

respectively. ∆E = 0.2 eV.

Fig. 3. The effective activation energy Ueff of

the coefficient D*(x) normalized to D*(x→0)

vs. dopant concentration x. The markers have

the same meaning as in Fig.2. ∆E = 0.2 eV.

2. Figure 3 presents the dependence of the effective activation energy Ueff(x) of the coefficient

D*(x) in models 1 and 2 (Ueff is referred to U0, the activation energy D* in the absence of defects).

It is seen that Ueff(x) behaves similarly in the two models up to the value x ~ 0.2. The growth of Ueff

with x correlates with observations made in some proton-conducting oxides, e.g., doped barium

cerate (see [5] and Refs therein). Note, that the Ueff(x) increase for barium cerate (~ 0.15 eV for x

~0.2) is close to the calculated Ueff(x) increase for the trapping energy ∆E ~ 0.2 eV. 3. Figure 4 shows the dependence D*(x) in model 1 at different depths of proton traps ∆E =

0.1, 0.2 and 0.4 eV. There is a visible difference between the coefficients D*, though ∆E values differ only a little.

Defect and Diffusion Forum Vols. 258-260 127

1000K

0,1eV

0,2eV

0,4eV

D*(x) / D*(0)

0

0,4

0,8

0 0,2 0,4

X

1200K

600K

σ(x) / σmax

1

0,01

0,0001

0 0,25 0,5 X

Fig.4. Dependence D*(x) (normalized to

D*(x→0)) calculated in model 1 for different

∆Е values, at T = 1000 K

Fig.5. Proton conductivity σ(x) normalized to

the maximum σ at T = 1200 K, as calculated in

model 1 in the approximations A and B (filled

and open diamonds respectively).

4. Figure 5 presents the dependences of the proton conductivity σ on the concentration of acceptor impurities in the absence of oxygen vacancies for weak (approximation A) and strong

(approximation B) proton-proton correlations. The conductivity is obtained by calculating the

steady-state current in the system in the presence of a weak dc electric field. The curves σ(x) are

given in the first model of the potential relief at ∆E = 0.2 eV. If proton-proton correlations are weak, σ(x) flattens out at x ~ 0.1-0.15. In the approximation B (a proton blocks its own oxygen ion

and 8 nearest oxygen ions), the curve σ(x) has a maximum at the same values of x. Flattening-out

and the position of the maximum shifts towards smaller x values as ∆E increases. Generally, the positions of the maxima correlate well with known experimental data for proton-conducting oxides,

see, e.g., [6]. It is interesting that positions of the σ(x) maxima for a number of acceptor-doped

perovskite oxygen conductors are located at approximately same x values [7].

It is worth mentioning that the presented dependencies are calculated for the maximum possible

concentration of protons, cH = x. This condition is not usually satisfied at high temperatures, and

experimental values of cH can be much smaller than x. In this case, in accordance with the defect

formation thermodynamics in these objects, cH changes with x approximately as √(x + const) at a constant temperature. This factor together with the weak dependence of the proton mobility on the

concentration of oxygen vacancies (see section 6 below) may give a maximum in the dependence

σ(x) even in the absence of strong proton-proton correlations.

5. The Haven ratio H = D*/Dσ can be found from the independent calculations of D* and the

mobility µ. In the approximation A, the H value slightly differs from unity up to the value x ~ 0.2

and decreases insignificantly as x increases further: H ~ 0.95 at x ~ 0.4-0.5. In the case of stronger

proton-proton correlations (the approximation B), the Haven ratio could largely deviate from unity:

H ~ 0.9 at x ~ 0.2. As x increases to 0.4-0.5, the H value decreases to ~ 0.7. We think that in the

approximation B the role of repulsion between protons is somewhat overestimated and,

consequently, these values can be used as the lower limit of the H value in the oxides at hand.

6. Finally, we shall briefly discuss the effect of oxygen vacancies on the proton diffusion

coefficient. In the models discussed above, the role of vacancies reduces to blocking of the oxygen

lattice sites for protons. In the approximation of weak correlations A, when vacancies are absent,

the number of blocked sites Nblock coincides with the dopant concentration (cH = x). If the vacancy

concentration cV is finite, then Nblock = x – cV, due to electro-neutrality condition (see above). It is

obvious that in this case D(cV) should increase slightly with growing cV, at fixed not too large

128 Diffusion in Solids and Liquids, DSL-2006 I

values of x. (This qualitative reasoning disregards the difference between blocking of sites by

mobile protons and frozen vacancies, since it has yet smaller effect.) According to the calculations,

at the temperatures ~ 1000 K the effect of vacancies is negligible, up to the value x ~ 0.4.

Conclusions

It is shown that the interaction between protons and defects can result in a considerable dependence

of the proton transport coefficients on the concentration x of the acceptor impurity in proton-

conducting oxides. If impurity traps have a characteristic depth, the proton diffusion coefficient is

much smaller than in an undoped crystal already at x ~ 0.05. The diffusion coefficient depends

slightly on the concentration of oxygen vacancies when the dopant concentration is not large. The

interaction of protons with traps and the prohibition for two protons to be located on a single

oxygen ion may yield a maximum in the x dependence of the conductivity. Stronger proton-proton

correlations can cause a drastic decrease in the diffusion coefficient with the increase of x, and

result in a considerable deviation of the Haven ratio from unity.

Acknowledgements

We would like to thank A.K. Demin for helpful discussions. The work was supported by the

Russian Foundation for Basic Research (grants No. 04-03-32390 and No. 04-03-32377).

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Defect and Diffusion Forum Vols. 258-260 129

Diffusion in Solids and Liquids, DSL-2006 I 10.4028/www.scientific.net/DDF.258-260 The Influence of Defects on Proton Diffusion in Perovskites АIIВIV11-xRIII

xO3-δ: Monte Carlo Study 10.4028/www.scientific.net/DDF.258-260.124

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