Effects of Lanthanide Dopants on Oxygen Diffusion in Yttria-StabilizedZirconia

R. Krishnamurthy*,w and D. J. Srolovitz

Department of Mechanical and Aerospace Engineering and Princeton Institute for the Science and Technology ofMaterials, Princeton University, Princeton, New Jersey

K. N. Kudin and R. Car

Department of Chemistry and Princeton Institute for the Science and Technology of Materials, Princeton University,Princeton, New Jersey

The effects of lanthanide co-dopants on oxygen diffusion inyttria-stabilized zirconia (YSZ) are studied using a combinedfirst principles density functional theory (DFT)/kinetic MonteCarlo (kMC) modeling approach. DFT methods are used tocalculate barrier energies for oxygen migration in different localcation environments, which are then input into kMC simulationsto obtain long-time oxygen diffusivities and activation energies.Simulation results show a substantial increase in the maximumvalue of the oxygen diffusivity upon co-doping and in the dopantcontent at which this value is obtained for Lu-co-doped YSZ;while relatively little change is seen for Gd-co-doped YSZ. Ex-amination of the DFT barrier energies reveals a linear scaling ofbarrier heights with the size of cations at the diffusion transitionstate. Using this strong correlation, oxygen diffusivity is exam-ined in YSZ co-doped with several lanthanide elements. Theoxygen diffusivity decreases with dopant atomic number (anddecreasing dopant ion size) for co-dopants smaller than Y, andchanges relatively little when Y is replaced by co-dopants largerthan it. These results are broadly consistent with experiment,and are explained in terms of cation-dopant and vacancy con-centration-dependent correlation effects, with the aid of a simpleanalytical model.

I. Introduction

ZIRCONIA-BASED materials are used in several high-technologyapplications, including solid oxide fuel cells, oxygen sensors,

thermal barrier coatings, and refractory linings for high-tem-perature furnaces. While the monoclinic form of pure zirconia isthe stable phase at room temperature, the high temperature cu-bic phase can be stabilized at room temperature by doping withaliovalent oxides (e.g., Y2O3, CaO).1 Yttria-stabilized zirconia(YSZ) contains a large number of oxygen vacancies that are re-quired to neutralize the excess charge produced on the intro-duction of aliovalent dopants (Y, Ca etc.).2 These vacancies areresponsible for YSZ’s high oxygen ion conductivity. Dopingwith rare earth and other closely related trivalent elements suchas Sc are often considered favorably, owing to the enhancedoxygen diffusivity observed on doping with some of these ele-ments (e.g., Yb, Sc).3,4 Moreover, doping with these elementscan reduce the effective cation diffusivity in stabilized zirconia,5

thus enhancing the useful lifetimes of these materials in thermalbarrier coating and solid oxide fuel cell applications.6

The ionic conductivity of YSZ does not increase monotoni-cally with increasing vacancy concentration (or Y content);rather, it attains a maximum value at around 0.1 mole fractionyttria.7–9 As discussed below, several reasons have been offeredto explain this behavior.7,10 The earliest experimental study onthe effect doping zirconia with different aliovalent oxides (i.e.,Sc2O3, Y2O3, Yb2O3 etc.) has on its ionic conductivity was con-ducted by Strickler and Carlson.3 They found that doping withdifferent trivalent oxides produced essentially the same qualita-tive behavior in ionic conductivity versus dopant content, i.e., amaximum in conductivity at some fixed value of the dopantcontent. They also observed that the oxygen ion conductivity ofstabilized zirconia decreased upon doping with cations largerthan Zr. Subsequently, Storman and Cubican found from anexperimental study that the ionic conductivity of a 10 mole %(Y, Yb) decreased with increasing Yb substitution for Y; a resultthat is consistent with the simple ion size based argument pre-sented above (Y4Yb).4 Stafford et al.11 measured the ionicconductivity for stabilized zirconia doped with different trivalentions and found, in general, that the ionic conductivity increasedwith decreasing cation size. While they recognized that this trendcould be explained both in terms of an energetically more fa-vorable vacancy-oxygen exchange and a decreased tendencyfor vacancy–dopant binding with decreasing dopant size, theyfavored the latter explanation. Subsequent investigationsalso reflect the above relation between dopant ion size andconductivity.12,13

Theoretical explanations of the observed maximum in theoxygen diffusivity (or conductivity) versus dopant mole fractionfor fluorite-based oxides have focused either on vacancy–dopantassociation effects7 or on dopant-dependent vacancy–oxygenexchange energetics9,14–16 as possible causes for this behavior.Early X-ray17,18 and neutron scattering19 studies seem to indi-cate that dopant ions preferred to occupy sites adjacent to va-cancies, in agreement with a simple Coulomb-attraction picture,while more recent experimental studies,20,21 indicate that dopantions preferentially occupy next nearest neighbor sites to vacan-cies. Lattice statics calculations based on a Born–Mayer poten-tial also support the latter view.22 More recently, a com-prehensive molecular dynamics investigation of diffusion inYSZ by Shimojo et al. reveals that oxygen ions preferentiallymigrate across edges of cation tetrahedra whose corners are oc-cupied by Zr ions (i.e., a Zr–Zr edge), and almost never migrateacross Y–Y edges.14,15 Since the number of these dopant ion-containing edges (i.e., Y–Y, Y–Zr in YSZ) increases with in-creasing dopant content, oxygen ions find fewer paths that avoidY ions, thus leading to a decrease in the diffusivity. Monte Carloinvestigations by Meyer and Nicoloso using simple model po-tentials also indicate that among the three possibilities,i.e., vacancy–dopant attraction, vacancy–dopant repulsionand dopant-position-dependent vacancy–oxygen ion exchangeenergetics, only the third was able to successfully predict the

2143

JournalJ. Am. Ceram. Soc., 88 [8] 2143–2151 (2005)

DOI: 10.1111/j.1551-2916.2005.00353.x

G. S. Rohrer—contributing editor

Supported by NASA Glenn Research Center through agreement NRA-01-GRC-03.*Member, American Ceramic Society.wAuthor to whom correspondence should be addressed. e-mail: rkrishna@princeton.edu

Manuscript No. 11299. Received August 30, 2004; approved January 14, 2005.

observed trend in the oxygen diffusivity versus dopant content.16

In recent work, we combined first-principles density functionaltheory (DFT) calculations of migration across different cationedges in YSZ with kinetic Monte Carlo (kMC) simulations ofthe long time oxygen diffusivity and found that the competitionbetween increased oxygen vacancy concentration and activationenergy for oxygen diffusion explained the observed behavior.9

This approach was successful in predicting the experimentallyobserved variation in oxygen diffusivity with dopant contentand the magnitude of the oxygen diffusivity without using anyfitting parameters. These results clearly indicate that dopant-position-dependent oxygen ion-vacancy exchange energeticsplay the dominant role in determining the observed oxygen dif-fusivity versus dopant content behavior.

Theoretical and numerical simulation methods have also beenemployed to elucidate the effect of dopant ion size on oxygenion diffusivity in stabilized zirconia. Kilner, while reviewing lat-tice statics calculations of dopant ion–vacancy binding energies,observed that vacancy–dopant binding was minimal in fluoriteoxides doped with cations with sizes similar to that of the hostion (i.e., Zr for stabilized zirconia), and proposed that this wasresponsible for the high oxygen conductivity observed in simi-larly doped fluorite oxides.10 More detailed dopant–vacancy in-teraction studies performed with short-range potentials,electrostatic interactions and a shell model for the polarizabil-ity of the ions (only in Zacate et al.23), indicate that trivalentdopant ions larger than Zr (which includes all the rare earth el-ements and Sc) preferentially occupy next nearest neighbor po-sitions to vacancies in stabilized zirconia.23,24 A comparison ofthe vacancy–dopant binding energies for this next-nearest neigh-bor dopant–vacancy configuration for different dopants indi-cates that large dopant ions bind favorably to vacancies. Theincrease in oxygen ion conductivity with decreasing dopant ionsize was attributed to the dependence of the vacancy–dopantbinding energy on dopant ion size (assuming that this termdominates the oxygen diffusion activation energy).

Recently, first principles DFT calculations have been per-formed to address this issue. Stapper et al.25 used DFT calcu-lations of vacancy–dopant binding energy in a 96 atom unit celland found that Y ions prefer next nearest neighbor positions tovacancies as compared with nearest neighbor positions. Bog-icevic and Wolverton26 performed DFT and coupled frozenphonon calculations for several ordered compounds of zirconiaand trivalent dopant oxides, and concluded that a balance be-tween electronic and elastic effects determines the ordering pref-erences of dopants and vacancies. They found that a near exactbalance of elastic and electronic interactions occurs for Sc, andaccordingly, Sc exhibits the lowest vacancy–dopant binding en-ergy among all trivalent dopants. They argued that this is whySc-doped zirconia shows the largest oxygen ion conductivityamong trivalent ion doped zirconias. From the discussionabove, it is clear that dopant ion size dominates the vacancy–dopant binding energy. However, the relation between this bind-ing energy and the activation energy for oxygen diffusion re-mains unclear. On the other hand, models based on vacancy–dopant interactions alone were unable to reproduce the exper-imental trend in oxygen diffusivity versus dopant content influorite oxides. In contrast, models based on dopant-concentra-tion-dependent migration energies produced oxygen diffusivityversus dopant mole fraction trends that are in good agreementwith experiment. While this does not prove that vacancy–dopantinteractions are insignificant, it does suggest that the effect do-pant ions have on oxygen-vacancy exchange energies is the moreimportant one in determining oxygen diffusivity in stabilizedzirconia. Consequently, we investigate the important effect dop-ing with different trivalent dopants (mostly from the lanthanideseries) has on oxygen ion-vacancy exchange energies and henceon the oxygen diffusivity through a combined first principlesDFT/kMC approach.

We initially examine the effects of substituting the lanthanidesLu and Gd (Lu is smaller and Gd is larger than Y) for Y in YSZ.Following our previous work on YSZ, we determine the migra-

tion barriers for oxygen ion-vacancy exchange across differentcation edges from first principles DFT simulations, as discussedin Section II.9 We use these barriers in a kMCmodel to calculatethe long-time diffusivity of oxygen in YSZ, in which the yttriumhas been either partially or completely replaced by Lu or Gd (seeSection IV). As discussed in Section V, we find (from first prin-ciples calculations) that the migration energy across differentcation edges scales nearly linearly with the combined size of thecations that constitute the edges. Motivated by this observation,we map several lanthanide elements (and Sc) onto this relation-ship to obtain estimates for the energy barrier for oxygen ionmigration across edges whose corners are occupied by differentdopant ions and use these barriers in the kMC model to deter-mine the effect of dopant ion type (i.e., size) on oxygen ion dif-fusivity in different trivalent dopant oxide-stabilized zirconias.The results for the oxygen ion diffusivity are analyzed usingsimple correlation analysis in Section VI.

The first principles method requires no empirical input, andhence, does not suffer from some of the limitations inherent inusing empirical potentials to conduct molecular dynamics or at-omistic Monte Carlo simulations. On the other hand, first prin-ciples calculations are limited to relatively small simulation cells.As a result, the barriers for oxygen hopping may be dependenton the specific distribution of dopant and Zr ions in the simu-lation cell. Because of the relatively large computational expenseassociated with the first principles calculations, we assume thatthe migration barriers for oxygen hopping are affected only bythe two neighboring cations at the transition state. Thus, do-pant–vacancy association effects are not included in this model.This assumption can be relaxed by mapping a large number offirst principles calculations (for different local cation arrange-ments) using cluster expansion methods27/computational alche-my techniques28 and using these results in Monte Carlosimulations. The kMC method has the advantage over molec-ular dynamics simulations (which are restricted to very shorttimes) in properly sampling a large number of local environ-ments in a statistically meaningful manner. However, in contrastto the molecular dynamics approach, the present methodassumes that oxygen diffusion is well represented by oxygenvacancy hopping through the edges of cation tetrahedra.Nevertheless, our multiscale modeling approach represents aphysically realistic method for calculating experimentally rele-vant transport properties in complex oxides.

II. First-Principles Calculations

In this section, we describe our method for calculating the localactivation barriers for oxygen migration. These calculationswere performed using a DFT pseudopotential method in thelocal-density approximation (LDA) with appropriate general-ized gradient approximation (GGA) corrections.29 The specificdetails pertaining to these calculations, especially as applicableto Zr, Y and O have been published elsewhere.9 Ultrasoft pseu-dopotentials30 were also generated for Gd and Lu and total en-ergy minimization calculations were performed for cubic Gd2O3

and Lu2O3. Table I summarizes the lattice parameters obtainedfrom our GGA calculations (GGA-PBE), previous calculationsusing a different implementation of the GGA (GGA-PW91),33

and from experiment. The agreement between the lattice pa-rameters determined with both GGA implementations and theexperimental data is excellent. Other structural parameters, suchas internal coordinates and angles, not shown in Table I, alsoagree well with experiment and with the other theoreticalstudy.31

The methodology employed to calculate the migration barrieris similar to that used earlier.9 In essence, we use a simulationcell containing 2� 2� 2 conventional fluorite unit cells (i.e., 32Zr ions and 64 O ions), wherein six of the Zr ions were replacedby Y ions (and three oxygen ions were concurrently removed) tosimulate an yttria mole fraction ofB10%. This composition lieswell within the cubic phase field, and does not suffer from

2144 Journal of the American Ceramic Society—Krishnamurthy et al. Vol. 88, No. 8

significant distortions that were observed at lower yttriumconcentrations.9,25 Oxygen ion-vacancy exchange across all ofthe different possible cation edges were calculated at this totaldopant composition, x5 0.1. This restriction is tantamount tothe assumption that the activation barrier for oxygen migrationis unaffected by the yttria concentration and yttrium ion distri-bution. Clearly, this is not precisely the case. However, the goodagreement we observed between calculated results for oxygendiffusivity and experiment in our previous work on oxygen dif-fusion in YSZ suggests that this assumption is reasonable.9 Forcalculating migration barriers across cation edges containing Gdand Lu dopants (for e.g., Zr–Gd, Y–Lu, Gd–Gd), one or two Yions were replaced by the appropriate dopant ion. To determinethe migration barrier, we focussed on one oxygen ion-vacancyexchange across a particular cation edge. Y ions, other thanthose that constitute this cation edge, and oxygen vacancies,other than the one considered, are placed as far away from thehopping oxygen vacancy as possible to minimize their effect onthe migration energy. For the simulation cell considered here,this corresponds to placing these Y ions at fourth nearest neigh-bor positions (or farther away) except for the Zr–Zr edge wheresome Y ions were placed on third nearest neighbor positions.The saddle point in the energy landscape is approximated by thesplit-vacancy configuration corresponding to the oxygen ion oc-cupying a location midway between the equilibrium positions ofthe oxygen-vacancy pair. The energy barrier for oxygen migra-tion is calculated as the total energy difference between a fullyrelaxed equilibrium configuration and a saddle point configura-tion (relaxed subject to the specified constraint). This calculationis repeated for oxygen vacancy–ion exchange across all possiblecation edges in Lu or Gd co-doped YSZ (see Table II for thecorresponding migration energies).

III. kMC Simulations

The methodology used for the kMC simulations is similar tothat employed in our earlier study of O-ion diffusion in YSZ.9

Only a brief description is provided here. We model Lu or Gdco-doped YSZ using a cubic simulation cell of linear dimensionL5 50a, where a is the lattice parameter of the cation fcc unitcell (i.e., 0.5� 106 cation and 1.0� 106 anion sites). YSZ co-doped with a trivalent dopant lanthanide (we use the notationLn to represent a lanthanide ion) can be described by the che-mical formula, ðZrO2Þ1�xðY2O3Þð1�aÞx=2ðLn2O3Þax=2ðVOÞx=2,where VO represents the oxygen vacancy, x is the total concen-tration of dopant ions (i.e. Y1Ln), and a is the fraction of Yions that were replaced with Ln ions. Y and Ln ions of a numberappropriate for the chosen composition (i.e., chosen x and a),and oxygen vacancies of a number sufficient to ensure chargeneutrality are randomly distributed on the cation and anion sub-lattices, respectively. The rest of the sites are occupied by thehost ions (Zr and O). Oxygen ions exchange only with neigh-boring oxygen vacancies in /100S-directions across cation edg-

es that are oriented in the /110S-direction. The hopping ratesfor jumps across the six possible cation edges for each dopantLn (i.e. Zr–Zr, Zr–Y, Y–Y, Zr–Ln, Y–Ln, and Ln–Ln) are ob-tained using the Boltzmann relationship:

nAB ¼ n0 exp�EAB

kBT

� �(1)

where the migration energies, EAB, for hops across an AB cationedge are obtained from the first-principles calculations describedin the previous section. The frequency factor is fixed atn0 5 1013/s, as appropriate for many metal oxide ceramics.Note that the frequency factor n0 also includes a non-configu-rational entropy term. While the chosen value is commonly usedin solid state diffusion studies, there can be some variation inthis number from site-to-site and from system-to-system.34 Inthe present study, cations do not migrate. This is a reasonableassumption since cation diffusivities in this system are typicallymany orders of magnitude smaller than anion diffusivities. Pe-riodic boundary conditions are employed to minimize edgeeffects.

The standard N-fold way kMC simulation method is em-ployed here.35 Briefly, at any instant of time, an oxygen ion-vacancy exchange i is chosen from a list of all such possibleexchanges according to the condition:

Xi�1j¼1

n jAB

G< x1 �

Xij¼1

n jAB

G(2)

where njAB is the hopping rate for the jth oxygen vacancy-ionexchange, x1A[0, 1) is a random number and G is the sum of therates of all possible oxygen-vacancy exchanges,

G ¼Xnj¼1

n jAB (3)

and n is the total number of possible hops at that time. Vacancy–vacancy exchanges are forbidden and whenever two vacanciesoccupy adjacent sites, the rate for their exchange is set to zeroand the corresponding events are removed from the list of rates.The chosen hop i is executed and time is advanced stochastically,by

Dt ¼ � lnðx2ÞG

(4)

where x2A[0, 1) is another random number. Following an oxy-gen vacancy-ion interchange, the rates associated with the newvacancy position are calculated and added to the list of allowedinterchanges and those associated with the previous vacancyposition are removed.

This procedure is repeated until the target time is reached.Both time averaging and particle averaging are employed tocompute the mean square displacement of vacancies, /R2S.The mean square displacement /R2S was found to be a linearfunction of time (see Krishnamurthy et al.9). The vacancy dif-fusivityDv is extracted from this plot using the Einstein relation:

hR2i ¼ 6Dvt (5)

The oxygen self-diffusivity, DO is a function of the vacancyconcentration Cv and the vacancy diffusivity, Dv. Consideration

Table I. Comparison of our GGA Calculation (GGA–PBE)of the Cubic Lattice Parameters with Other Calculations

(GGA–PW91), and with Experiment

Parameter GGA–PBE GGA–PW91 31 Experiment 32

Cubic Gd2O3, a (A) 10.866 10.812 10.813Cubic Lu2O3, a (A) 10.339 10.358 10.391

Results for YSZ were given in Krishnamurthy et al.9 GGA, generalized gra-

dient approximation; YSZ, yttria-stabilized zirconia.

Table II. Activation Energies, EAB, for Oxygen Migration Across (AB)-Cation Edges

(AB) Zr–Zr Zr–Y Y–Y Zr–Gd Y–Gd Gd–Gd Zr–Lu Y–Lu Lu–Lu

EAB (eV) 0.473 1.314 2.017 1.648 2.164 2.333 0.783 1.766 1.569

August 2005 Effects of Lanthanide Dopants on Oxygen Diffusion 2145

of the jump balance shows:

DO ¼cv

1� cvDv (6)

IV. Effects of Gd and Lu Doping on the Oxygen Diffusivity

The variation of the oxygen diffusivity Do with the total dopantoxide fraction y (y5x/(2–x)) in Lu co-doped YSZ is shown forthree different values of a (the fraction of the total dopant con-tent that is Lu) at 1400 K in Fig. 1(a). The oxygen self-diffusivityexhibits a maximum at a total dopant fraction y between 0.08and 0.15, depending on a. This variation in the oxygen diffu-sivity with dopant content is consistent with experimental meas-urements of ionic conductivity and oxygen self-diffusivity inYSZ and similar fluorite-based mixed oxides.3,7,8,36 The origin ofthis behavior was explained using a dopant-concentration-dependent migration barrier model in Krishnamurthy et al.9

Figure 1(a) also shows that the maximum oxygen diffusivityincreases in value with increasing a (i.e., increasing Lu content).While experimental data for oxygen diffusion in Lu co-dopedYSZ are unavailable, this trend is consistent with similar in-creases in the maximum oxygen diffusivity in YSZ co-doped

with Yb and Sc (which, like Lu, are smaller ions than Y).4,12

This trend is also consistent with the ion size-based argumentspresented in the Introduction. The ion size of different lantha-nide ions relative to Y can be understood from Fig. 2(b), wherethese are shown as a function of the atomic number of the lan-thanide. The total dopant concentration where the maximum inthe oxygen diffusivity occurs, shifts to larger y with increasingLu substitution (i.e., increasing a). Although no experimentaldata on the changes produced in the oxygen diffusivity versusdopant fraction behavior upon co-doping YSZ with trivalentdopant oxides have been reported, a comparison of data forYSZ (corresponds to a5 0)8 and Yb–SZ (i.e., Yb–SZ is equiv-alent to a5 1 in Yb co-doped YSZ)3 indicates that the maxi-mum in the oxygen diffusivity shifts to larger y when smallerisovalent ions are substituted for Y. This is consistent with thecalculated trends discussed above. From Table II, it is clearthat migration across Lu ion containing edges is energeticallyfavored compared with migration across Y-containing edges.Accordingly, increasing substitution of Lu for Y should result inan increase in the oxygen diffusivity. This reasoning also ex-plains the observed shift in the maximum oxygen diffusivity tolarger y.

x

DO

(X 1

0−6 c

m2 /

s)

0.1 0.2 0.3 0.4

0.5

1

1.5

2

2.5

3

3.5

4

α = 0.5

α = 0

α =1

(a)

y

x

DO

(X 1

0−6 c

m2 /

s)

0.05 0.1 0.15 0.2 0.25

y0.05 0.1 0.15 0.2 0.25

0.1 0.2 0.3 0.4

0.5

1

1.5

2

2.5

3

α = 1

α = 0α = 0.5

(b)

Fig. 1. Oxygen diffusivityDO is shown as a function of the total dopantoxide mole fraction, y, and yttrium ion concentration, x, in (a) Lu co-doped yttria-stabilized zirconia (YSZ) and (b) Gd co-doped YSZ at 1400K for different values of a.

RA + RB − 2 RZr

EA

B -

EZ

rZr

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

(a)

(b)

Zi

Ri (

Å)

60 65 700.95

1

1.05

1.1

1.15

Nd

La

Ce

Sm Eu

Gd Tb

Dy Ho

Er Tm

YbLu

RY

Fig. 2. (a) First principles energy barrier for oxygen ion hopping acrossan AB cation edge versus the combined ion size RA1RB

37 and (b) Thesize of the eight-coordinated, trivalent, lanthanide ions Ri versus atomicnumber, Zi. The yttrium RY is indicated by a horizontal arrow in (b).

2146 Journal of the American Ceramic Society—Krishnamurthy et al. Vol. 88, No. 8

Oxygen diffusivity versus dopant oxide fraction for Gd co-doped YSZ is shown in Fig. 1(b) at 1400 K for three differenta’s. A comparison of the Gd- and Lu-doped YSZ results dem-onstrates that the oxygen diffusivity is less sensitive to changes inGd than Lu content. This can be understood by noting that thehopping frequency for oxygen migration is affected only mar-ginally when one set of very high migration barriers (hops acrossY–Zr and Y–Y) are replaced by another (hops across Gd–Zr,Gd–Y, and Gd–Gd). However, the small increase in oxygendiffusivity with increasing Gd substitution for Y cannot be ex-plained using this argument. In Section VI, we demonstrate thatthis is a correlation factor effect. Experimental measurements ofoxygen diffusivity versus dopant mole fraction in YSZ co-dopedwith larger trivalent dopants ions (e.g., Gd, Sm, La) are scarce.However, we note that oxygen diffusivity/ionic conductivity inLn–SZ, where Ln is a lanthanide ion larger than Y (i.e., Ln isGd, Sm, Nd, La etc.), is, in some cases smaller than in YSZ.3,11

This is in contrast to the simulations results shown in Fig. 1(b).Several reasons can be offered to explain this discrepancy

with experiment. First, our calculations assume that the co-doped YSZ stays cubic for all values of x and a and for all Lnco-dopants (larger or smaller than Y). Experimental observa-tions, on the other hand, indicate that for large trivalent dopants(e.g., La), the cubic phase is not stable at the temperatures ofinterest here.3 This may be because of the large elastic distor-tions caused by the size mismatch between these ions and thehost Zr ions. In the first principles simulations described previ-ously, ions around the Gd ion were displaced to a greater extentfrom their idealized cubic positions as compared with the cor-responding displacements around an Y ion in the base YSZ.While this suggests that elastic effects are more important forlarger ions, we did not perform an exhaustive study of elasticrelaxations as part of this research. Experimental observationsalso indicate that the minimum dopant concentration needed tostabilize the cubic phase increases with increasing dopant size.3

Therefore, the comparison of the present results with the exper-imental data is fine for Ln ions that are smaller than Y, but maybe inappropriate at small y for Ln ions that are larger than Y.Another possible effect that could explain the apparent discrep-ancy between Fig. 1(b) and the experimental data for large Lnions could be associated with the long-range elastic interactionsbetween the vacancy and the large Ln dopant, not included inthe present analysis.

V. Other Lanthanide Ions

Here we explore the effect that substituting other lanthanideions for Y has on the oxygen diffusivity through its effect on themigration barrier for oxygen ion-vacancy exchange. In order todetermine how doping modifies energy barriers for oxygen ion-vacancy exchange, we plot the energy barriers associated withexchange through different cation edges AB, as a function of theAB ‘‘bond’’ length (i.e., the sum of the A and B ion radii37). Thisis shown in Fig. 2(a) for all possible AB pairs in Lu- and Gd-doped YSZ using data obtained via first principles methods (seeSection II). These migration barriers scale almost linearly withthe ‘‘bond’’ length between the two cations nearest to the oxygenion in its diffusional transition state. While the data is well fit bythe linear relationship over the entire ‘‘bond’’ length range, it isespecially good for the high-energy migration barriers. Whilethis result may seem somewhat surprising, it is reasonable con-sidering that most of the lanthanide ions have stable trivalentoxidation states (i.e., similar Coulombic interactions), and theirion size varies gradually with atomic number (see Fig. 2(b)).37

Motivated by this good fit, we mapped the ionic size of severallanthanide elements (from Fig. 2(b)) onto the best fit straightline in Fig. 2(a) to extract estimates for the barriers for oxygenmigration across edges containing these elements (and Zr).Clearly, this is an approximation used to avoid the computa-tional expense associated with performing accurate first princi-ples calculations for each and every case. We also consider the

case of the (transition metal) Sc dopants in the same manner,since it is a widely used dopant (although not technically correct,we label Sc as one of the lanthanide dopants for notationalconvenience).

Using these migration energies for oxygen hopping acrossdifferent cation edges, we performed a series of kMC simula-tions of oxygen diffusion as a function of Ln-ion type and theLn dopant fraction a for yB0.1. Figure 3(a) shows the oxygendiffusivity at 1400 K as a function of ion size and Zi for trivalentLn dopants with ions smaller than Y in co-doped YSZ. Clearly,the oxygen diffusivity is largest for the smallest Ln ions (i.e., thepseudo-Ln, Sc) and decreases monotonously as we increase theLn ion size (from Sc to Y). This is not surprising in light ofthe observation that the oxygen-hopping barrier decreases with de-creasing cation size (‘‘bond’’ length). These results compare fa-vorably with experimental results for Sc and Yb co-doped YSZ;i.e., the oxygen diffusivity in co-doped YSZ decreases as we gofrom Sc to Yb to pure YSZ.4,12 The variation of the oxygendiffusivity with a is also consistent with experimental observa-tions.4,12 As discussed above, this oxygen diffusivity trend canalso be explained in terms of the increasing ease with which theoxygen ion can hop across AB edges containing smaller dopantions.

0.88 0.92 0.96 1

3

4

5

6

7LuSc ErYb Y

α = 1

α = 0.25

α = 0.5

α = 0.75

YSZ

Ri (Å)

Ri (Å)

DO

(X 1

0−6 c

m2 /

s)D

O (X

10−6

cm

2 /s)

1.04 1.08 1.12 1.16

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

GdTb LaSmY Nd

α=0.75

α=0.25

α=1

α=0.5

YSZ

(b)

(a)

Fig. 3. Oxygen diffusivity, DO, as a function of trivalent co-dopant ionsize for (a) dopant ions smaller than the Y ion, and (b) dopant ionslarger than the Y ion, and for different extents of co-doping (i.e., varyinga) at 1400 K.

August 2005 Effects of Lanthanide Dopants on Oxygen Diffusion 2147

Figure 3(b) shows the oxygen diffusivity at 1400 K as a func-tion of Ln ion size in Ln co-doped YSZ, where the dopant Lnions are larger than Y. In contrast to the results for small do-pants (see Fig. 3(a)), the oxygen diffusivity changes only mar-ginally when Y is replaced by larger trivalent Ln dopants.Moreover, in contrast to the monotonous decrease in conduc-tivity with increasing dopant ion size seen in Fig. 3(a), the ox-ygen diffusivity actually increases by a small amount when YSZis co-doped with lanthanides of ion size greater than Y. Thissurprising result is consistent, however, with the Gd results pre-sented above, in Fig. 1(b). Within the purview of our model, therelatively small change in diffusivity can be explained by the factthat the average hopping frequency for an oxygen ion is nearlyunaffected by replacing one set of high migration barriers withanother. In either case (particularly for low and moderate do-pant concentrations), the oxygen ion will avoid these high mi-gration barriers and predominantly hop across the lowmigration barrier Zr–Zr edges. However, the small increase indiffusivity with Ln dopant ion size (4RY) cannot be explainedthis way. Note that the oxygen diffusivity in YSZ actually rep-resents an averaged quantity computed over the very largenumber of hops that the oxygen ions make through the non-uniform energy landscape during the kMC simulations. Thehigher the barrier associated with a particular AB edge, thelower is the probability that the oxygen ion will hop through thistype of edge. Therefore, the average barrier height and the bar-rier height averaged over an oxygen ion trajectory can be quitedifferent. We return to this point below. While experimental re-sults for YSZ co-doped with dopants larger than Y are scarce,comparison of available ionic conductivity results for YSZ andLn–SZ (Ln4Y in ion size) show a small decrease in the con-ductivity with increasing Ln ion size (there are counter examplesas well),3,11 in contrast with the weak trend observed in Fig. 3(b).As we discussed above, either an incompletely stabilized cubicphase of YSZ or an enhanced importance of relatively long-range (compared with nearest neighbor interactions) dopant-va-cancy association effects for large dopants may explain this dis-crepancy.

The activation energy for diffusion, E, (based on 1200–2000K simulation data) is shown in Fig. 4 as a function of dopant ionsize for Ln co-doped YSZ. Clearly, substitution of trivalent co-dopant ions that are smaller than Y increases the activation en-ergy for oxygen diffusion and doping with ions that are larger

than Y leads to little change (there is a tendency for doping toslightly decrease the activation energy for oxygen diffusion), ascompared with undoped YSZ. While the activation energy doesdecrease upon large dopant ion substitution, it always remainsabove the barrier height associated with O ion diffusion across aZr–Zr edge. This is consistent with the expectation that for lowto moderate concentrations of the large dopants, the oxygenavoids Zr–Ln edges in preference for Zr–Zr edges. Hence, theactivation energy for diffusion differs little from the barrier forhopping across a Zr–Zr edge.

In contrast, when YSZ is doped with smaller sized cations (ascompared with Y), oxygen hopping across dopant-ion contain-ing edges is no longer prohibitive. Consequently, the activationenergy for diffusion includes more sampling of the Ln–Zr bondswith decreasing Ln ion size. Since such barriers are higher thanthat for hopping across Zr–Zr bonds, this leads to an increase inthe activation energy beyond that for the base YSZ. How canthe activation energy increase with decreasing dopant ion size atthe same time that the oxygen diffusivity itself also increaseswith decreasing dopant ion size? This must be associated withthe effect of dopants on the pre-exponential factor for diffusion(we return to this point below). It is interesting to note that theactivation energy for diffusion is reduced on changing the tri-valent co-dopant from Lu to the smaller Sc. This can be attrib-uted to the fact that the Sc ion size is very similar to that of thehost Zr ion, and hops across Sc–Sc cation edges are likewiseenergetically competitive with hops across Zr–Zr edges. On theother hand, hops across Lu–Lu are much less likely comparedwith hops across Zr–Zr edges. The absolute values for the ac-tivation energies for diffusion reported in Fig. 4 are smaller thanthose determined from ionic conductivity measurements by 0.2–0.5 eV,8,38 while they match reasonably well with those calcu-lated from tracer diffusivity measurements.39 This discrepancyhas been attributed to vacancy binding energy effects.24,40

The pre-exponential factor for oxygen diffusivity in co-dopedYSZ, D0

O, is shown as a function of the co-dopant ion size inFig. 5.D0

O is much larger than in the base YSZ when the dopantions are smaller than Y. In contrast, substituting large dopantions for Y in YSZ produces only very small changes in D0

O. Thisis consistent with the picture suggested above in which the fre-quency of oxygen ion hops across Ln–Ln and Ln–Zr increaseswith decreasing Ln ion size. The changes in D0

O with doping arelargely associated with the fact that this quantity is proportionalto the correlation factor. As the diffusion path becomes morerestricted, the correlation factor will decrease. Therefore, thefact that D0

O is independent of dopant size for large dopantssuggests that the oxygen diffusion path is nearly the same for allsuch dopants, as seen in Fig. 5. As the dopant size decreases, theoxygen diffusion path becomes less restricted and hence D0

O in-creases with decreasing dopant size.

VI. Correlation Effects

As discussed in the previous section, the oxygen diffusivity isstrongly affected by correlation effects. The oxygen diffusivityin Ln co-doped YSZ (given by ðZrO2Þ1�xðY2O3Þð1�aÞx=2ðLn2O3Þax=2ðVOÞx=2) can be written as

DO ¼cv

1� cv

� �

�ð1� xÞ2nZrZr þ 2xð1� xÞ ð1� aÞnZrY þ anZrLnð Þþð1� aÞ2nYY þ 2ð1� aÞanYLn þ a2nLnLn� �

x2

24

35

� a2

66f

(7)

where the term inside the square brackets represents the hoppingfrequency weighted by the composition of the different types ofcation edges and the last term is the product of the correlation

Ri (Å)

E (

eV)

0.9 1 1.10.47

0.475

0.48

0.485

0.49

0.495

0.5

0.505

0.51

0.515

0.52

0.525

0.53

0.535

EYSZ

EZrZr

LuSc Yb Er Nd LaSmGdTb

α = 1

α = 0.25α = 0.5

α = 0.75

Fig. 4. The activation energy for oxygen self-diffusion, E, as a functionof dopant ion size Ri, for different trivalent co-dopants substituted for Yin yttria-stabilized zirconia (YSZ). This data was obtained from fittingthe kinetic Monte Carlo results in the temperature range 1200–2000 K.The horizontal line labeled EYSZ is the activation energy for undopedYSZ and that labeled EZrZr is the barrier height for oxygen hoppingacross a Zr–Zr edge. The vertical arrow indicates the Y ion size.

2148 Journal of the American Ceramic Society—Krishnamurthy et al. Vol. 88, No. 8

factor f and the number of anion sites around an oxygen ion.The correlation factor f in co-doped YSZ itself has three con-tributions. The first contribution is a geometrical factor whichdepends only on the crystal structure and has been ignored inthis study.41 The second term is a consequence of the fact thateach oxygen ion hops across a varied energy landscape com-prising different numbers of energetically dissimilar migrationbarriers. A simple way to understand this term is to recognizethat hops over low energy barriers are likely to be followed by ahop returning the oxygen ion to its original position, while hopsacross high energy barriers are likely to be succeeded by hopsthat take the oxygen ion away from its original position. Clearly,hops of the former kind are negatively correlated (i.e., they willreduce the value of f ) and hops of the latter kind are positivelycorrelated. The relative energies of the different barriers and thenumbers in which they are present will determine the actualvalue of this contribution. The third factor is because of thepresence of a concentrated solution of vacancies. This factor canbe significant even when the interactions between vacancies areignored as the oxygen ions can possibly exchange with multiplevacancies, and the exchange that actually occurs will be inti-mately affected by the nature of the energy barriers (or the cat-ions occupying the corners of the tetrahedral edges) surroundingthem. Consequently, an initial random distribution of vacanciesis not likely to remain so after a large number of hops. Thus, thelocal concentration of vacancies can differ substantially from itsaverage value, cv, depending upon the arrangement of cations inthe vicinity of any location.

In a previous study, we considered correlation effects arisingfrom both a non-uniform energy landscape and from a concen-trated solution of vacancies in YSZ (i.e., YSZ not co-doped withother elements),9 using standard methods of correlation analysis(see, for example, Manning41and Le Claire42), as adapted to va-cancy self-diffusion. A simple approximation to the correlationfactor results when only the correlations because of cations thatare neighbors of a given vacancy–oxygen pair are considered.Hence, this approximation includes only the second term dis-cussed above. For this case, the correlation factor f is approx-imately:

f ¼ 1þ 2XE

rE ½nfEð1� cvÞ � nbE � (8)

where rE represents the number fraction of cation edge of typeE, n

fE is the probability that a jump over cation edge E in the 1

x-direction is succeeded by another jump in the same direction,and nbE is the probability that a jump over a cation edge E in the1x-direction is followed by a jump in the reverse direction.These quantities can be easily found using the approach outlinedin Krishnamurthy et al.9 in terms of the dopant fraction x andthe barrier energies nAB.

The oxygen diffusivity calculated using this approximation(i.e., using Eqs. (7) and (8)) is shown as a function of the totaldopant oxide fraction y for Lu co-doped YSZ in Fig. 6(a). Alsoshown in Fig. 6(a) are simulation results from Fig. 1(a). Theabsolute magnitude of the oxygen diffusivity determined usingthese expressions differs from the simulation results by 20%–50% and, in general, the match between the analytical and sim-ulation results is not quantitative. This is to be expected as theanalytical approximation ignores correlation effects because ofother vacancies (i.e., dilute vacancy concentration is assumed)and cations dopants other than those neighboring any particularvacancy–oxygen ion pair. Nevertheless, the model reproducesmost of the major trends observed in the simulation results. Forexample, the analysis clearly captures the increase in the max-imum value of the oxygen diffusivity as the smaller Lu ion is

y

DO

(10−6

cm

2 /s)

0.1 0.2 0.3

1

1.5

2

2.5

3

3.5

analysis

Monte Carlo

α = 0.5α = 0

α = 1

y

DO

(X

10−6

cm

2 /s)

0.1 0.2 0.3

1

1.5

2

2.5

3

3.5

4

Analysis,α = 0.5

MC, α=0.5

MC, α = 0

Analysis, α = 1

Analysis, α = 0

MC, α = 0.5

(a)

(b)

Fig. 6. The variation of the oxygen diffusivity with dopant oxide molefraction y, in ðZrO2Þ1�xðY2O3Þð1�aÞx=2ðLn2O3Þax=2ðVOÞx=2, where thetrivalent co-dopant Ln is (a) Lu and (b) Gd. These curves were obtainedfrom Eqs. (7) and (8) with nearest neighbor correlations (lines withoutdata points) and from the kinetic Monte Carlo simulations (lines withdata points), reproduced from Fig. 1, for comparison.

Ri (Å)0.9 1 1.1

0.5

1

1.5

2

2.5

3

3.5

4

YSZ

LuSc Er Nd LaSmGdTb

α =1

α =0.25

α =0.5

α =0.75

Yb

DO

(X 1

0−4 c

m2 /

s)0

Fig. 5. The pre-exponential factor in the Arrhenius expression for theoxygen diffusivity, D0

O, as a function of trivalent co-dopant ion size, inco-doped yttria-stabilized zirconia (YSZ). The results were obtainedfrom fits to the kinetic Monte Carlo data from 1200 to 2000 K.

August 2005 Effects of Lanthanide Dopants on Oxygen Diffusion 2149

increasingly substituted for Y in YSZ. Moreover, the analysisalso reproduces the shift in the maximum to higher dopant con-tents (i.e., higher y) with increasing Lu substitution observed inthe simulation results. Also, the total dopant content y at whichthe maximum value of the oxygen diffusivity occurs is repro-duced reasonably well.

Oxygen diffusivity versus total dopant oxide fraction y, cal-culated using the same analysis for Gd co-doped YSZ is shownin Fig. 6(b). Note that the trivalent Gd ion is larger than the Yion. Again, the match between the analytical model and thesimulation results is not quantitative. Yet, it captures the mainfeatures seen in the simulation results. For example, the maxi-mum value of the diffusivity changes very little (compared withthe Lu co-doped case, see Fig. 6(a)) and the total dopant oxidefraction y at which the maximum occurs is captured with rea-sonable accuracy. However, the analytical results do not capturethe small increase in diffusivity with enhanced Gd substitutionseen in the simulation results.

In both the small and large co-dopant ion cases (see Figs. 6(a)and (b)), the match between the analytical and simulation resultsis almost perfect at low dopant oxide fractions, i.e., yB0.03, anddeviates only at higher values of y, strongly suggesting that cor-relations because of a concentrated solution of vacancies andcorrelations because of cations other than the neighbors of theoxygen ion–vacancy pair are responsible for the discrepancy be-tween the two sets of results. While their inclusion is conceptu-ally simple, the resulting calculations are no longer analyticallytractable, as discussed for the undoped YSZ in Krishnamurthyet al.9 Moreover, Monte Carlo simulations, with large simula-tion cell sizes, are known to produce good estimates for thecorrelation factor.43,44 Consequently, we have not extended theanalytical solution to include these effects here. Briefly, we canexpect that these additional effects will result in a reduced valuefor the oxygen diffusivity, as they will further restrict oxygendiffusion paths in the YSZ. This is consistent with the results inFigs. 6(a) and (b).

These additional correlation effects act to enhance thestrength of the positive correlations of the hops across highmigration barrier dopant-ion containing edges when Y ionsare substituted by large dopant ions through their effect onthe local vacancy concentration, cv, in the vicinity of thedopant ion and away from it. Likewise, they reduce the strengthof the negative correlations of hops across Zr–Zr edges whenY ions are substituted by small dopant ions. This results inan increase in the oxygen diffusivity as compared with thebase YSZ.

VII. Conclusions

We have developed a first principles DFT/kMC approach tostudy the effects of co-doping YSZ with trivalent ions on theoxygen diffusivity. Oxygen ions reside inside tetrahedra, thecorners of which are occupied by cations. Following earlierwork on the base YSZ,9 migration barriers for oxygen ion-vacancy exchange across tetrahedral edges occupied by differentpairs of cations (i.e., Zr–Zr, Zr–Y, Ln–Y etc.) were calculatedusing DFT simulations. These calculations were performed fortwo co-dopants, Lu and Gd, which have ion sizes smaller andlarger than that of Y, respectively. The calculated migrationbarriers were input into kMC simulations to determine long-time oxygen diffusivity in co-doped YSZ.

The observed maximum in the oxygen diffusivity versus do-pant oxide fraction shifts to larger values and to higher dopantcontents with an enhanced level of Lu substitution for Y in Luco-doped YSZ. This behavior is in agreement with simple ion-size based arguments11 and with experimentally measured ionicconductivities in Yb co-doped YSZ (RY4RYb4RLu).

4,13 Themagnitude of the oxygen diffusivities and the dopant content atwhich a maximum value for the oxygen diffusivity is found alsoagree with experimental data for several different stabilized zir-conias.4,8,38 In Gd co-doped YSZ, the oxygen diffusivity changes

relatively little on substitution of Gd for Y, in agreement withthe limited experimental data available for Gd–SZ. However,these data show a decreased ionic conductivity in Gd–SZ ascompared with YSZ, in contrast with the simulation results pre-sented here. An incomplete stabilization of the cubic phase (weassumed that the material is always cubic) and/or vacancy–do-pant interactions beyond immediate cation neighbors of the va-cancy, which were neglected in the simulation model are possiblereasons for this discrepancy.

The migration barriers for oxygen ion hopping, as calculatedfrom first principles simulations, were found to scale linearlywith the sum of the sizes of the two cations that form the edgethrough which the oxygen ion is hopping. Using this correlation,estimates for the energies for oxygen ion hopping across allpossible cation edges in YSZ co-doped with several different Lnions were obtained. A series of kMC simulations were per-formed using these energies to study the effects of lanthanide co-dopants on oxygen diffusion in a 0.1 Ln–YSZ. Oxygen diffu-sivity was found to decrease monotonously with ion size for co-dopants Ln that are smaller than Y, and to increase marginallyon addition of co-dopants larger than Y. This suggests a criticalsize for the co-dopant ion beyond which the oxygen diffusivity isnot significantly affected by changing the co-dopant trivalention. The activation energy for oxygen diffusion decreases mo-notonously with increasing ion size, and is nearly equal to theenergy barrier for oxygen ion hopping across Zr–Zr edges, forlarge dopants. The pre-exponential factor for oxygen diffusivityis also large for small sized dopants and decreases as the size ofthe dopant is increased.

The observed values of the diffusivity and the increase in dif-fusivity on changing from YSZ to Ln–Sz agree well with exper-imental results for ionic conductivity in Yb and Sc co-dopedYSZ.4,12 The activation energies for oxygen diffusion are 0.2–0.5eV lower than those extracted from ionic conductivity measure-ments. However, vacancy–dopant interactions, which more for-mally constitute a part of the vacancy formation energy, are notsubtracted from the activation energies extracted from experi-ments. The calculated activation energies agree better with trac-er diffusivity measurements. The observed change in diffusivitywith dopant ion size is consistent with experiment for dopantions smaller than Y.3,4,12,13 While experimental results for co-doping YSZ with dopant ions larger than the Y ion are scarce,the existing information, in contrast, with the simulation results,shows a monotonous decrease in the diffusivity on co-dopingwith larger sized dopants.3,11 Reasons similar to those employedto explain the discrepancy in the calculated behavior and exper-iment for Gd co-doped YSZ can be used to explain these resultsas well.

References

1P. Aldebert and J. P. Traverse, ‘‘Structure and Ionic Mobility of Zirconia atHigh Temperature,’’ J. Am. Ceram. Soc., 68, 34–40 (1985).

2C. Wagner, ‘‘Uber den Mechanismus der Elektrischen Stromleitung imNernststift,’’ Naturwissenschaften, 31, 265–8 (1943).

3D. W. Stickler and W. G. Carlson, ‘‘Electrical Conductivity in the ZrO2-RichRegion of Several M2O3–ZrO2 Systems,’’ J. Am. Ceram. Soc., 48, 286–9 (1965).

4G. S. Corman and V. S. Stubican, ‘‘Phase Equilibria and Ionic Conductivity inthe System ZrO2–Yb2O3–Y2O3,’’ J. Am. Ceram. Soc., 68, 174–81 (1985).

5S. Weber, S. Scherrer, H. Scherrer, M. Kilo, M. A. Taylor, and G. Borchardt,‘‘SIMS Analysis of Multi-Diffusion Profiles of Lanthanides in Stabilized Zircon-ias,’’ Appl. Surf. Sci., 203-204, 656–9 (2003).

6D. Zhu and R. A. Miller, ‘‘Sintering and Creep Behavior of Plasma-SprayedZirconia- and Hafnia-Based Thermal Barrier Coatings,’’ Surf. Coat. Technol.,108–109, 114–20 (1998).

7R. E.W. Casselton, ‘‘Low Field DCConduction in Yttria-Stabilized Zirconia,’’Phys. Stat. Sol. (a), 2, 571–85 (1970).

8D. W. Stickler and W. G. Carlson, ‘‘Ionic Conductivity of Cubic Solid Solu-tions in the System CaO–Y2O3–ZrO2,’’ J. Am. Ceram. Soc., 47, 122–7 (1964).

9R. Krishnamurthy, Y.-G. Yoon, R. Car, and D. J. Srolovitz, ‘‘Oxygen Diffu-sion in Yttria Stabilized Zirconia: A New Simulation Model,’’ J. Am. Ceram. Soc.,87, 1281–9 (2004).

10J. A. Kilner, ‘‘Fast Ion Transport in Solids,’’ Solid State Ionics, 8, 201–7(1983).

11R. J. Stafford, S. J. Rothman, and J. L. Routbort, ‘‘Effect of Dopant Size onthe Ionic Conductivity of Cubic Stabilized ZrO2,’’ Solid State Ionics, 37, 67–72(1989).

2150 Journal of the American Ceramic Society—Krishnamurthy et al. Vol. 88, No. 8

12S. P. S. Badwal, F. T. Ciacchi, S. Rajendran, and J. Drennan, ‘‘An Investi-gation of Conductivity, Microstructure and Stability of Electrolyte Compositionsin the System 9% (Sc2O3– Y2O3)–ZrO2—(Al2O3),’’ Solid State Ionics, 109, 167–86(1998).

13H. Kaneko, F. Jin, H. Taimatsu, and H. Kusakabe, ‘‘Electrical Conductivity ofZirconia Stabilized with Scandia and Yttria,’’ J. Am. Ceram. Soc., 76, 793–5 (1993).

14F. Shimojo, T. Okabe, F. Tachibana, M. Kobayashi, and H. Okazaki, ‘‘Mo-lecular Dynamics Studies of Yttria Stabilized Zirconia. I. Structure and OxygenDiffusion,’’ J. Phys. Soc. Jpn., 61, 2848–57 (1992).

15F. Shimojo and H. Okazaki, ‘‘Molecular Dynamics Studies of Yttria Stabi-lized Zirconia. II. Microscopic Mechanism of Oxygen Diffusion,’’ J. Phys. Soc.Jpn., 61, 4106–18 (1992).

16M. Meyer and N. Nicoloso, ‘‘MC-Study of the Anomalous Conductivity ofFluorite-Type Solid Oxide Solutions,’’ Ber. Bunsenges. Phys. Chem., 101, 1393–8(1997).

17M. Morinaga, J. B. Cohen, and J. Jr. Faber, ‘‘X-ray Diffraction Study ofZr(Ca,Y)O2�x: I. The Average Structure,’’ Acta Crystallogr., A35, 789–95 (1979).

18M. Morinaga, J. B. Cohen, and J. Jr. Faber, ‘‘X-ray Diffraction Study ofZr(Ca,Y)O2�x: II. Local Ionic Arrangements,’’ Acta Crystallogr., A36, 520–30(1980).

19D. Steele and B. E. F. Fender, ‘‘The Structure of Cubic ZrO2: YO1.5 SolidSolutions by Neutron Scattering,’’ J. Phys. C: Solid State Phys., 7, 1–11 (1974).

20C. R. A. Catlow, A. V. Chadwick, G. N. Greaves, and L. M. Moroney, ‘‘EX-AFS Study of Yttria-Stabilized Zirconia,’’ J. Am. Ceram. Soc., 69, 272–7 (1986).

21B. W. Veal, A. G. McKale, A. P. Paulikas, S. J. Rothman, and L. J. Nowicki,‘‘EXAFS Study of Yttria Stabilized Zirconia,’’ Physica B, 150, 234–40 (1988).

22A. Dwivedi and A. N. Cormack, ‘‘A Computer Simulation Study of the DefectStructure of Calcia-stabilized Zirconia,’’ Philos. Mag. A, 61, 1–22 (1990).

23M. O. Zacate, L. Minervini, D. J. Bradfield, R. W. Grimes, and K. E. Sick-afus, ‘‘Defect Cluster Formation in M2O3 Doped Cubic ZrO2,’’ Solid State Ionics,128, 243–54 (2000).

24M. S. Khan, M. S. Islam, and D. R. Bates, ‘‘Cation Doping and Oxygen Dif-fusion in Zirconia: A Combined Atomistic Simulation and Molecular DynamicsStudy,’’ J. Mater. Chem., 8, 2297–307 (1998).

25G. Stapper, M. Bernasconi, N. Nicoloso, and M. Parrinello, ‘‘Ab Initio Studyof Structural and Electronic Properties of Yttria-Stabilized Zirconia,’’ Phys. Rev.B, 59, 797–810 (1999).

26A. Bogicevic and C. Wolverton, ‘‘Nature and Strength of Defect Interactionsin Cubic Stabilized Zirconia,’’ Phys. Rev. B, 67, 024106 (2003).

27L. G. Ferreira, S.-H. Wei, and A. Zunger, ‘‘First Principles Calculation ofAlloy Phase Diagrams: The Renormalized-Interaction Approach,’’ Phys. Rev. B,40, 3197–231 (1989).

28S. de Gironcoli, P. Giannozzi, and S. Baroni, ‘‘Structure and Thermodynamicsof SixGe1�xAlloys fromAb InitioMonte Carlo Simulations,’’ Phys. Rev. Lett., 66,2116–9 (1991).

29J. P. Perdue, K. Burke, and M. Ernzerhof, ‘‘Generalized Gradient Approx-imation made Simple,’’ Phys. Rev. Lett., 77, 3865–8 (1996).

30D. Vanderbilt, ‘‘Soft Self-Consistent Pseudopotentials in a GeneralizedEigenvalue Formalism,’’ Phys. Rev. B., 41, 7892–5 (1990).

31N. Hirosaki, S. Ogata, and C. Kocer, ‘‘Ab Initio Calculation of the CrystalStructure of the Lanthanide Ln2O3 Sesquioxides,’’ J. Alloys Compounds, 351, 31–4(2003).

32R. W. G. Wyckoff, pp. 1–22 in Crystal Structures, Vol. 2. Interscience, NewYork, 1967.

33J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D.J. Singh, and C. Fiolhas, ‘‘Atoms, Molecules, Solids and Surfaces: Applications ofthe Generalized Gradient Approximation for Exchange and Correlation,’’ Phys.Rev. B., 46, 6671–87 (1992).

34E. Kaxiras and J. Erlebacher, ‘‘Adatom Diffusion by Orchestrated Exchangeon Semiconductor Surfaces,’’ Phys. Rev. Lett., 72, 1714–7 (1994).

35C. C. Battaile and D. J. Srolovitz, ‘‘Kinetic Monte Carlo Simulation of Chem-ical Vapor Deposition,’’ Annu. Rev. Mater. Res., 32, 297–319 (2002).

36T. Suemoto and M. Ishigame, ‘‘Composition Dependence of the Ionic Diffu-sion Coefficients in Yttria-stabilized Zirconias,’’ Solid State Ionics, 21, 225–9 (1986).

37R. D. Shannon and C. T. Prewitt, ‘‘Effective Ionic Radii in Oxides and Flu-orides,’’ Acta Crystallogr., B25, 925–46 (1968).

38S. Ikeda, O. Sakurai, K. Uematsu, N. Mizutani, and M. Kato, ‘‘ElectricalConductivity of Yttria-stabilized Zirconia Single Crystals,’’ J. Mater. Sci., 20,4593–600 (1985).

39Y. Oishi and K. Ando, ‘‘Oxygen Self-Diffusion in Cubic ZrO2 Solid Solu-tions,’’ pp. 189–202 in Transport in Nonstoichiometric Compounds, Edited by G.Simkovich and V. Stubican. Plenum Press, New York, NY, 1985.

40M. Kilo, C. Fundenberger, C. Argirusis, M. A. Taylor, G. Borchart, M.Weller, and R. A. Jackson, ‘‘Experimental and Theoretical Investigation of Ox-ygen Diffusion in Stabilized Zirconia,’’ Radiat. Eff. Defects, 157, 1077–83 (2002).

41J. R. Manning, Diffusion Kinetics for Atoms in Crystals. Van Nostrand, Princ-eton, 1968.

42A. D. Le Claire, ‘‘Correlation Effects in diffusion in solid’’; pp. 261–330 inPhysical Chemistry: An Advanced Treatise, Vol. X, Edited by W. Jost. AcademicPress, New York, 1970.

43A. H. Bennett and B. J. Alder, ‘‘Persistence of Vacancy Motion in HardSphere Crystals,’’ J. Phys. Chem. Sol., 32, 2111–22 (1971).

44H. J. de Bruin and G. E. Murch, ‘‘Diffusion Correlation Effects in Non-Sto-ichiometric Solids,’’ Philos. Mag., 27, 1475–88 (1973). &

August 2005 Effects of Lanthanide Dopants on Oxygen Diffusion 2151