Interstitial and vacancy mediated transport mechanisms in perovskites: A comparison of chemistry and potentials

Solid State Ionics 253 (2013) 18–26

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Interstitial and vacancy mediated transport mechanisms in perovskites: A comparison of chemistry and potentials Blas Pedro Uberuaga ⁎, Louis J. Vernon Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM, USA

a r t i c l e

i n f o

Article history: Received 25 April 2013 Received in revised form 24 July 2013 Accepted 19 August 2013 Available online 19 September 2013 Keywords: Perovskite Accelerated molecular dynamics Adaptive kinetic Monte Carlo Defect migration

a b s t r a c t Perovskites are important materials for fast-ion conduction applications and have been used extensively as model systems for irradiation studies, two situations where understanding defect mobility is critical for predicting performance. Using long-time scale simulation methods, we examine point defect mobility in perovskites as a function of the chemistry of the perovskite and the empirical potential used. We find that, while the basic mechanisms are the same regardless of these factors, the energies associated with the mechanisms vary significantly. We identify diffusion pathways for each type of interstitial, finding relatively complex behavior for A cation interstitials, which can diffuse one-dimensionally, and oxygen interstitials, which exhibit a twodimensional diffusion mechanism. We further find that several cation defects are immobile with a preference to transform into antisite complexes rather than migrate. These results provide new insight into the migration behavior of point defects in perovskites and complex oxides more generally. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Perovskites are technologically important ceramics for a wide variety of applications, from solid oxide fuel cells [1] to ferroelectricity [2]. For many applications, the mobility of defects is an important consideration. Because of this, many studies have examined the mobility of, in particular, oxygen in perovskites, because of the potential application as fast ion conductors [3]. In radiation damage environments, however, where energetic particles can create defects on all sublattices within the material, the mobility of cation defects is also of great importance. Given the importance of perovskites, many studies have been performed examining the mobility of defects within the perovskite structure. As we are describing modeling results, we will focus our overview of the literature on simulation studies, but first we touch on some experimental results. Experimentally, it is typically difficult to separate migration and formation energies from the overall activation energy for diffusion. However, this can be done more easily for non-stoichiometric compounds as the defects are not created thermally, but rather through modifying the chemistry of the material. This is often done extrinsically via doping [4], though, which adds an additional complication to understanding the mobility of defects as their interaction with the dopants must be accounted for [5]. That said, oxygen vacancy activation energies in hypostoichiometric doped perovskites are on the order of 0.75–0.85 eV [6]. Similar values have been found for Labased hypostoichiometric perovskites (see Ref. [7] for a summary), though values are slightly higher in LaAlO3-based materials than in ⁎ Corresponding author. Tel.: +1 5056679105. E-mail address: [email protected] (B.P. Uberuaga). 0167-2738/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ssi.2013.08.022

those based on LaScO3 [8]. Regarding cation transport, Miyoshi and Martin reported an activation energy for Mn diffusion of 0.6 eV in LaMnO3 + δ [9]. Turning to simulation studies, most have relied on empirical potentials (of the same type utilized here) and have focused on oxygen vacancy mobility, given the importance of oxygen transport for most technological applications and the predominance of oxygen vacancies in these materials, though doping can lead to hyperstoichiometry and oxygen interstitials as well [10]. A pioneering study by Kilner and Brook [5] determined oxygen vacancy migration energies to vary from 0.38 to 0.87 eV, depending on the chemistry of rare-earth aluminate perovskite, a dependence which they related to the size of the cations. Since then, a large number of perovskites, in both cubic and orthorhombic forms, have been examined with potentials, including (but not limited to) a number of LaBO3 perovskites with B = Mn, Cr, Fe, Co, and Ga [11–15], a number of BaBO3 perovskites with B = Sn, Zr, and Pr [16,17], SrTiO3 [18,19], NdCoO3 [20], and CaTiO3 [21]. DFT studies have examined, amongst other perovskites, MgSiO3 [22], SrTiO3 [18,23], and BaTiO3 [24]. The most studied defect, by far, is the oxygen vacancy, for reasons stated above. Overall, these studies agree that oxygen vacancy mobility is relatively high, with migration barriers that vary from 0.30 to 1.22 eV, depending on the chemistry and structure (cubic or orthorhombic) of the perovskite. This is true for studies using either potentials or DFT. As might be expected, anisotropy is observed in orthorhombic perovskites, as the lattice parameters are no longer equal in each direction [12]. Further, as oxygen vacancy concentrations are typically a consequence of doping or non-stoichiometry, studies have examined the migration barrier of oxygen vacancies near other types of vacancies [18] or substitutional dopants [21], finding that oxygen vacancy

B.P. Uberuaga, L.J. Vernon / Solid State Ionics 253 (2013) 18–26

mobility decreases in both cases. Finally, given that these are ionic materials, there are dependencies of the migration barrier on the charge state of the defect [24], though such effects are best examined with DFT. Studies of cation vacancies are much more limited. Islam examined cation vacancy migration mechanisms in LaMnO3, finding that both cation vacancies had very large barriers for migration (nearly 4 eV for the La vacancy and over 7 eV for the Mn vacancy) [12]. Similarly, Sr vacancy migration in SrTiO3 has a barrier approaching 3 eV as determined with DFT [18] and 3.9 eV as found via potentials [19]. Thomas et al. also report a migration energy for Ti vacancies in SrTiO3, which they found to be 11 eV with empirical potentials [19]. A more recent study using DFT + U methods found that the migration of La vacancies in LaMnO3 was 2.96 eV [25]. Finally, reports of migration mechanisms of interstitials in perovskites are rare, clearly because for most applications interstitial migration is irrelevant. The one study we are aware of is by Thomas et al. who report migration energies (but not mechanisms) for each of the six point defects in SrTiO3 [19]. They found that oxygen and Sr interstitials diffuse with a barrier of 0.3 eV while Ti interstitials have a much larger migration energy of 4.6 eV. One limitation of most of these studies is that the migration pathways and energies were determined via mapping the potential energy landscape along an assumed pathway between minimum energy sites of the relevant defect. By doing so, Islam determined that the migration pathway of oxygen vacancies is not linear in space, but curves around B cations [12]. While some studies have used methods such as nudged elastic band (NEB) [26] to more accurately determine the saddle point for a given path [18], the basic path itself was still assumed to be essentially linear between two sites. Here, motivated by the use of perovskites as model materials for irradiation studies [27–32], we examine the mobility of six point defects (three interstitials and three vacancies) in the perovskite structure using both temperature accelerated dynamics (TAD) [33] and adaptive kinetic Monte Carlo (AKMC) [34] employing empirical potentials of the Buckingham form to describe the interatomic interaction. In particular, we determine how the migration energies vary as a function of chemistry of the perovskite, changing the A cation in a II–IV perovskite as well as comparing migration in the II–IV perovskites to migration in a III–III perovskite. We also compare the migration mechanisms for the same perovskite but described with different potentials to determine what properties transfer between potentials and which do not. The use of TAD and AKMC allows one to study defect migration processes without any prior assumption about the relevant pathways. That is, both methods discover on-the-fly the relevant migration pathways and evolve the system from state to state depending on the discovered processes. While the details of the two methods differ in how they discover mechanisms, the end result is similar: a progression from state to state without any user bias. Thus, by using these methods, we can more accurately determine relevant pathways for migration of defects in complex materials such as perovskite. In contrast to most previous studies, we examine the mobility of all six possibly mobile point defects that might be present in an ABO3 perovskite: the A and B cation interstitials and vacancies, and the anion interstitial and vacancy. For most applications, interstitials are irrelevant as they are essentially non-existent at equilibrium. Similarly, cation vacancies have received relatively little attention as it is the oxygen transport, governed by oxygen vacancies, that is of utmost importance for most applications. That said, we are motivated by how these ceramics respond to irradiation. Indeed, materials such as SrTiO3 and BaTiO3 have been used as model ceramics for numerous irradiation studies [35–37]. In such cases, enough energy is supplied by the incident particles to create defects of all types. Thus, the evolution of these materials under irradiation is governed as much by the behavior of interstitials as vacancies, hence our examination of all types of point defects.

19

2. Methodology Perovskite, with the formula unit ABO3, is intimately related to the corundum structure [38]. Cubic perovskites have the space group Pm3m, though rotations of the oxygen octahedra, common in many perovskites, lead to an orthorhombic structure (space group Pnma) [39]. The structure of cubic perovskite consists of interpenetrating simple cubic lattices of A and B cations with a face-centered cubic lattice of oxygen. Alternating layers along the [100] direction are comprised of AO and BO2 planes. Typically, the A cation can have either a + 2 or + 3 formal valence state while the B cation can have a valence of either + 4 or + 3, respectively. In terms of Wyckoff notation, the cubic perovskite structure can be viewed as having A cations in the 1a positions (0, 0, 0), the B cations at the 1b positions (1/2, 1/2, 1/2), and the oxygen anions at the 3c positions (1/2, 1/2, 0; 1/2, 0, 1/2; 0, 1/2, 1/2). Not all of the perovskites discussed here are cubic at low temperatures. In particular, BaTiO3 becomes orthorhombic at low temperatures. LaAlO3 also exhibits a distortion away from ideal cubic. However, here, all materials were assumed cubic for simplicity and for ease of comparison of defect properties between the materials. We use long-time atomistic simulation methods to determine the migration mechanism and migration energy of each of the six point defects in each of the perovskite chemistries considered, the three interstitials and the three vacancies. We use a combination of both TAD and AKMC to simulate long enough trajectories to identify the migration mechanism for each defect. While the basic approach to identify migration pathways for each of these methods differs (TAD uses high temperature molecular dynamics and the AKMC simulations use saddle point searches), the basic information provided by each method is essentially the same: a topology of potential energy minima connected by saddle points. Using this data, we then identify the rate limiting step for diffusion and the associated energy barrier. To describe the interatomic interactions of the ions, we used the Buckingham form with long range electrostatics described by a Coulomb term. The potential parameters are taken from the literature. For SrTiO3, we examined two parameter sets, one by Grimes and coworkers [40–42] and another by Thomas et al. [19]. For the other perovskites, BaTiO3 and LaAlO3, we used parameter sets developed by Grimes and colleagues [41,43]. For convenience, the parameters are summarized in Table 1. In each case, we minimized the cell volume for the given potential. The resultant lattice constants are given in Table 2. One advantage of the potential sets derived by Grimes and coworkers is that they are transferable from one system to another. That is, the parameters are all fit such that the oxygen–oxygen interaction is the same for all materials. Thus, transferability is maximized at the expense of

Table 1 Potential parameters used in this study. The parameters originally appeared in Ref. [19] (Thomas potential), Refs. [40–42] (Grimes SrTiO3 potential), Ref. [41] (BaTiO3 potential), and Ref. [43] (LaAlO3 potential). The units of A are eV, of ρ angstroms, and of C eV*angstroms6. Perovskite Parameter

SrTiO3 (Thomas)

SrTiO3 (Grimes)

BaTiO3

LaAlO3

AA–O ρA–O CA–O AB–O ρB–O CB–O AO–O ρO–O CO–O qA qB qO

1769.51 0.319894 0 14,567.4 0.197584 0 6249.17 0.231472 0 1.84 2.36 −1.40

682.172 0.39450 0 2179.122 0.30384 8.986 9547.96 0.21916 32.0 2.0 4.0 −2.0

905.7 0.3976 0 2179.122 0.30384 8.986 9547.96 0.21916 32.0 2.0 4.0 −2.0

2088.79 0.3460 23.25 1725.20 0.28971 0 9547.96 0.21916 32.0 3.0 3.0 −2.0

20

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That is, this event describes two-dimensional migration. Each of the two anions that comprise the split structure has two oxygen neighbors within 2.5 Å, all of which are in the same (100) plane as the split interstitial. The migration event involves one of the two interstitial anions pushing one of those neighboring lattice anions out to form a new split within the same plane, but rotated. The migration energy for this event as given by the Grimes parameters is 0.43 eV. For three-dimensional diffusion, the O interstitial rotates to a new [100] oriented split interstitial that is centered on the original lattice site with a barrier of 1.01 eV, as illustrated in Fig. 1c–e. Fig. 2 illustrates the migration path for the Sr interstitial. As for the oxygen interstitial, the ground state structure for the Sr interstitial is again a [100] split interstitial with the two Sr ions on either side of an Sr lattice site. Also, the pathway for Sr interstitial migration is essentially identical to that of the O interstitial, though the lowest energy process corresponds to one-dimensional rather than two-dimensional diffusion. First, as shown in Fig. 2a–c, the interstitial can migrate onedimensionally along a [100] direction. The saddle point (Fig. 2b) has the Sr interstitial residing in a 3d (1/2, 0, 0) interstice. The barrier for this process is 0.52 eV. For three-dimensional diffusion, the Sr interstitial must execute a rotation which is illustrated in Fig. 2c–e. This rotation event leaves the Sr split structure centered on the same lattice site, but changes its orientation to a new [100] direction. The barrier for this rotation event is 0.84 eV, which is then the effective migration barrier for three-dimensional diffusion of this defect. The ground state of the Ti interstitial in SrTiO3 is shown in Fig. 3a. The structure of the Ti interstitial is more complicated than the others. It is still a split structure, lying along a roughly [100] direction (though slightly tilted) and sharing an Sr site. That is, the split Ti interstitial is mixed, with one of the two species being an Sr interstitial. As might be expected from such a structure, net diffusion is complicated. First, the Ti interstitial can diffuse with a barrier of 1.50 eV, via a process illustrated in Fig. 3. The Ti interstitial can oscillate between two Sr lattice sites via the process illustrated in Fig. 3a–c with a barrier of 1.14 eV. However, this alone cannot lead to net diffusion. To diffuse the Ti ion of the split interstitial (as opposed to the Sr ion component of the split interstitial structure) must rotate around the lattice site, as illustrated in Fig. 3c–e. This mechanism has a barrier of 1.50 eV and is the ratelimiting step. However, the more probable event is one that leads to the Sr interstitial breaking away from the Ti interstitial, which occupies the original Sr lattice site to create a TiSr antisite species. This event has a barrier of 1.26 eV, lower than the barrier for diffusion of the Ti interstitial described above. The Sr split interstitial then diffuses away, see Fig. 4b. This configuration is higher in energy than the initial structure in

Table 2 Lattice constants predicted for each of the systems studied in the cubic structure. Perovskite Lattice constant (angstroms)

SrTiO3 (Thomas)

SrTiO3 (Grimes)

BaTiO3

LaAlO3

a

3.905

3.898

3.993

3.823

accuracy. In contrast, the potential for SrTiO3 developed by Thomas et al. [19] was developed specifically to maximize the accuracy of the potential for SrTiO3. This potential would not be suitable for SrO, for example, as the charges on Sr and O are not of the same magnitude. Thus, an important difference between these two potentials is that the Grimes potential uses formal charges while the Thomas potential uses partial charges on the ions. One of our goals is to determine how these two approaches lead to different predictions for defect mobility within the material. 3. Results We first describe the defect migration mechanisms and energies, in each of the three perovskite chemistries mentioned above, as found by the TAD and AKMC simulations. We then compare and contrast the predicted behavior in the different perovskites. 3.1. SrTiO3 First, we describe the predicted pathways for point defect diffusion in SrTiO3 as predicted with the Grimes parameter set and then compare those to predictions made with the Thomas potential. The pathway for the oxygen interstitial is shown in Fig. 1. The ground state structure of the oxygen interstitial is a split structure, with two oxygen ions “sharing” one oxygen site in the perovskite structure, see Fig. 1a and c. Strictly speaking, the atoms do not sit on the same site, but rather lie on either side of a common oxygen lattice site. The axis of the split structure is [100]. Diffusion takes place by one of the two ions comprising the structure moving into the shared lattice site and pushing the second oxygen ion into a 12j interstice (1/8, 1/2, 1/8), which is the saddle point for the migration process (Fig. 1b). From the saddle, the interstitial oxygen ion pushes out another lattice oxygen ion to form a new split structure centered on another lattice site. As can be seen by comparing Fig. 1a and c, the diffusion process results in a rotation of the split interstitial structure. However, interestingly, this rotation is constrained such that only two possible orientations are visited by the oxygen interstitial via this event and the migration is constrained to a plane.

minimum

saddle

minimum

saddle

minimum

a

b

c

d

e

Fig. 1. Migration pathway for the oxygen interstitial in SrTiO3. Panels (a)–(c) represent a translation–exchange event that leads to the oxygen split structure reorienting to a new [100] direction but within the plane defined by the interaction of the interstitial with the lattice oxygen ions [the “bonds” in (a)]. This mechanism leads to two-dimensional diffusion. For three-dimensional diffusion, a rotation about the central vacancy of the split interstitial is required, as illustrated in panels (c)–(e). In this, and subsequent figures, only those ions that move more than 1 Å during the diffusion process are highlighted. The rest of the ions are in their positions in the original perovskite lattice. The large dark (green) spheres represent oxygen interstitials and the large light (green) square represents an oxygen vacancy. The rest of the ions, represented by small spheres, are ions in the perfect perovskite structure: oxygen is green, Sr is blue, and Ti is red. The defects are defined by comparing to the reference perovskite structure and identifying ions that do not have neighbors within 1 Å (the so-called reference lattice method of identifying defects [44,45]). The structure of the minima and the saddle points defining the migration mechanisms are indicated.

B.P. Uberuaga, L.J. Vernon / Solid State Ionics 253 (2013) 18–26

minimum

a

saddle

minimum

b

c

21

saddle

minimum

d

e

Fig. 2. Migration pathway of the Sr interstitial in SrTiO3. Panels (a)–(c) represent a simple translation of the interstitial along the axis of the split orientation, leading to one-dimensional diffusion of the interstitial. Panels (c)–(e) illustrate the rotation mechanism of the split interstitial to a new [100] orientation that is required for three-dimensional diffusion. Large dark (blue) spheres indicated Sr interstitials and the large light (blue) square indicates an Sr vacancy; the rest of the colors are as described in Fig. 1.

All of these migration energies, along with those for the other perovskites examined and discussed below, are summarized in Table 3.

Fig. 4a, but is entropically more favorable, and the energy of the configuration in Fig. 4b decreases as the Sr interstitial migrates further away from the antisite. This is reasonable considering that the Ti antisite, which has a net charge of +2, and the Sr interstitial, which also has a net charge of +2, would repel one another. It would thus seem that for the Ti interstitial to diffuse, most likely an Sr interstitial would have to diffuse to it, create a complex such as in Fig. 4a, and then diffuse away again, leading to a net translation of the TiSr antisite. Turning to vacancies, the migration path of the oxygen vacancy is given in Fig. 5. This is a very simple pathway, with the vacancy diffusing from a structure where it is localized on one oxygen lattice site (Fig. 5a) to an adjacent lattice site (Fig. 5c) via a split vacancy structure (Fig. 5b). At the saddle point (Fig. 5b), the structure is a split vacancy that spans two lattice sites with an oxygen ion sitting in the central interstice. The axis of the split structure is [110]. The barrier for this event is 1.22 eV. The angle formed by the interstitial with the two lattice sites that it spans is 174.2°, consistent with the results of Cherry et al. [13] that found the path to be curved between the two sites. The Sr vacancy is essentially immobile. It diffuses with a barrier of 5.03 eV. This process is a linear motion of the vacancy along a [100] direction. As for the Sr vacancy, the Ti vacancy is not mobile. Rather, it has a tendency to transform into a complex involving an Sr vacancy and an Sr antisite (SrTi), akin to the behavior of the Ti interstitial. The Sr complex is slightly higher in energy than the isolated Ti vacancy by about 0.28 eV with a barrier to go from the Ti vacancy to the Sr complex of about 1.90 eV. In this case, as for the Ti interstitial, both components of this defect complex have charges of the same sign (−2 for both), so we would expect them to repel each other. However, since the Sr vacancy mobility is so low, the two species are kinetically trapped and will not separate.

minimum

a

saddle

b

3.1.1. Comparison of potentials In order to understand how the predicted behavior of these point defects depends on the potential, we have determined the pathways of all six defects described above with the Thomas potential for SrTiO3. As mentioned, this potential was fit specifically to describe SrTiO3. The pathway for the oxygen interstitial predicted by the Thomas potential is the same as that illustrated in Fig. 1a–e, though the energetics differ. For 2D motion, the migration energy predicted by the Thomas potential is 0.32 eV while it is 0.91 eV for the rotation event that leads to 3D motion, values that are both about 0.1 eV smaller than those predicted by the Grimes potential. Similarly, the pathway for the Sr interstitial predicted by both potentials is very similar. For 1D migration of the Sr interstitial, the Thomas potential predicts a migration energy of 0.63 eV and for 3D migration, 0.72 eV. In this case, the barrier for 1D migration is higher than that predicted by the Grimes potential, but the 3D rotation mechanism is smaller, so there is not as large an anisotropy for Sr interstitial diffusion within the Thomas model as in the Grimes model. An important difference is in what the two potentials predict for the Ti interstitial. Whereas the Grimes potential predicted that while there is a diffusion mechanism for the Ti interstitial, it is more likely to break up into an antisite and an Sr interstitial, the Thomas potential predicts that the Ti interstitial can diffuse relatively easily with a barrier of 0.82 eV. No break-up events were observed with the Thomas potential, suggesting that if such a process can occur, it has a much higher barrier. The pathway involves an intermediate state of which there are several symmetry equivalent versions within the crystal, so the shortest

minimum

c

saddle

minimum

d

e

Fig. 3. Migration pathway of the Ti interstitial in SrTiO3. Panels (a)–(c) illustrate the shift in the center-of-mass of the split Sr–Ti mixed interstitial from one Sr lattice site to a neighboring site. For net migration, the split structure must also reorient, as illustrated in panels (c)–(e). The large dark (blue) sphere indicates the Sr interstitial, the large dark (red) sphere the Ti interstitial, and the large light (blue) square indicates the central Sr vacant lattice site. The rest of the colors are as described in Fig. 1.

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B.P. Uberuaga, L.J. Vernon / Solid State Ionics 253 (2013) 18–26

a

b

Fig. 4. Dissociation reaction for the Ti interstitial, which is slightly favored compared to the migration mechanism illustrated in Fig. 3. (a) The ground state structure of the split Sr–Ti mixed interstitial. (b) The dissociation product as the Sr component of the interstitial breaks away. The color scheme is the same as in Fig. 3 except in (b) the large, very dark (red) sphere indicates the TiSr antisite.

point, it is not clear what exactly leads to the more complicated behavior predicted by the Grimes potential.

pathway for net translation of the interstitial is not obvious. The structures of the two states are shown in Fig. 6. The oxygen vacancy pathway is the same for both potentials. The Thomas potential predicts a slightly lower migration energy than the Grimes potential (0.93 eV vs 1.22 eV) for the oxygen vacancy. This is also true for the Sr vacancy, with the Thomas potential giving 3.95 eV compared to the Grimes potential which gave 5.03 eV. As with the Grimes potential, the Thomas potential predicts that the Ti vacancy is immobile, converting to an Sr antisite and Sr vacancy complex. However, with the Thomas potential, the energetics of the antisite complex is much higher: the energy of the complex itself is 1.86 eV higher than the isolated Ti vacancy and the barrier to transform the Ti vacancy to the complex is 2.97 eV. These higher energies are consistent with the fact that the Thomas potential did not predict the Ti interstitial to form an antisite plus interstitial complex. Thus, it seems that the primary difference between the two SrTiO3 potentials relates not to the properties of interstitials and vacancies, but the propensity to form antisite defects. However, if we calculate the formation energy of a nearest neighbor antisite pair (TiSr + SrTi), the Grimes potential gives a value of 9.18 eV while this energy is 8.65 eV with the Thomas potential. Thus, it is not the overall energetics of the antisite pair that drives the behavior seen with the Grimes potential. At this

minimum

a

3.2. BaTiO3 To gain some understanding on how the chemistry of the perovskite influences the stability and migration of point defects within the material, we now consider BaTiO3. This is another II–IV perovskite, like SrTiO3. Ba has a larger ionic radius than Sr and thus BaTiO3 has a larger lattice constant than SrTiO3. This should influence the diffusion of point defects through the lattice, as should the different chemistry. For both the oxygen and the Ba interstitials, the behavior in BaTiO3 is essentially identical to that predicted by both potentials for SrTiO3, though the values change slightly. For the oxygen interstitial, the barrier for two-dimensional diffusion is 0.62 eV, higher than in SrTiO3, and for three-dimensional diffusion it is 1.05 eV, rather similar to what the Grimes potential predicts for SrTiO3. For the Ba interstitial, the disparity between one-dimensional and three-dimensional diffusion is not as great as for the Sr interstitial in SrTiO3: for 1D diffusion the barrier is 0.60 eV and for 3D it is only slightly higher, 0.65 eV. Interestingly, while the Grimes potential predicts that the Ti interstitial in SrTiO3 is more likely to transform to an antisite–vacancy complex,

saddle

b

minimum

c

Fig. 5. Migration pathway for the oxygen vacancy in SrTiO3. The color scheme is the same as in Fig. 1.

B.P. Uberuaga, L.J. Vernon / Solid State Ionics 253 (2013) 18–26 Table 3 Migration energies, in eV, as found using long-time simulation methods for the three interstitials (A, B, and oxygen) and the three vacancy (A, B, and oxygen) defects in each of the perovskites. The superscript on each value indicates the type of migration that energy describes, one-, two-, or three-dimensional migration. “n/a” indicates no migration mechanism was identified for that species. In those cases, the B cation vacancies, the observed mechanism was instead a transformation into a AB antisite plus A vacancy defect pair. Perovskite SrTiO3 (Thomas)

SrTiO3 (Grimes)

BaTiO3

LaAlO3

Interstitials Ai Bi Oi

0.631/0.723 0.823 0.322/0.913

0.521/0.843 1.503 0.432/1.013

0.601/0.653 1.893 0.622/1.053

1.561/1.603 0.583 0.312/1.853

Vacancies VA VB VO

3.95 n/a 0.933

5.03 n/a 1.223

6.68 n/a 1.123

5.03 n/a 0.70

Defect

it does not for BaTiO3. The pathway for Ti interstitial migration is illustrated in Fig. 7. The Ti interstitial causes significant displacement of nearby oxygen ions, displacements of greater than 1 Å. It diffuses with the interstitial moving from the ground state close to a 24k (19/32, 3/ 32, 0) interstice to a saddle point in which the interstitial is near a 48n (9/32, 1/32, 17/64) interstice. The barrier for this process is 1.89 eV. The migration pathway for the oxygen vacancy in BaTiO3 is the same as was found for this defect for both potentials in SrTiO3. The migration barrier is 1.12 eV, similar to the values in SrTiO3. The Ba vacancy diffuses via a process similar to that described above for the Sr vacancy in SrTiO3, though with a significantly higher barrier (6.68 eV), which essentially renders it immobile for all practical purposes. The Ti vacancy also exhibits similar behavior as in SrTiO3: the dominant mechanism is one in which the Ti vacancy transforms into a Ba vacancy plus Ba antisite (BaTi) complex. This complex is 1.27 eV higher in energy than the original Ti vacancy and the process occurs via an event that has a barrier of 2.65 eV.

3.3. LaAlO3 In contrast to SrTiO3 and BaTiO3, LaAlO3 is a III–III perovskite, where the formal charge on both cations is +3. Thus, not only does LaAlO3 represent a different chemistry as compared to the two titanates, but it is

a

23

also a perovskite with a different Madelung potential related to the types of cation–anion interactions that might be possible. The oxygen interstitial in LaAlO3 behaves very similarly to the oxygen interstitial in both SrTiO3 and BaTiO3. There is a two-dimensional migration mechanism with a barrier of 0.31 eV and a rotation mechanism, similar to that illustrated in Fig. 1c–e, that leads to three-dimensional diffusion with a barrier of 1.85 eV. Thus, the anisotropy between 2D and 3D diffusion is very large in this material. The La interstitial migration mechanism is also essentially the same in LaAlO3 as that of the Sr and Ba interstitials in the titanates. The migration energy for the one-dimensional diffusion mechanism is 1.56 eV while the rotation that leads to three-dimensional diffusion has a barrier of 1.60 eV. The Al interstitial behaves more like the Ti interstitial in SrTiO3 as predicted by the Thomas potential than by the Grimes potential and the Ti interstitial in BaTiO3. It induces a distortion in the local oxygen sublattice where one oxygen ion is displaced by more than 1 Å from its lattice site. The migration barrier, which leads to three-dimensional diffusion, is 0.58 eV. The mechanism is similar to that for the Ti interstitial in BaTiO3 as illustrated in Fig. 7, though only one oxygen ion is displaced significantly from its original lattice site. All of the vacancy species in LaAlO3 behave similarly as in the other perovskites. The oxygen vacancy exhibits three-dimensional migration with a barrier of 0.70 eV. The La vacancy, which can diffuse with a barrier of 5.03 eV, has a smaller barrier to transform into an antisite complex involving AlLa plus an Al vacancy. The barrier for the transformation is 4.81 eV. Finally, the Al vacancy does not exhibit any migration mechanisms, rather forming an LaAl/La vacancy complex with a barrier of 4.53 eV.

4. Discussion In perovskites used for fast-ion conduction applications, a number of factors ultimately govern the overall ionic conductivity. In developing models to predict which chemistries will ultimately provide the highest conductivity, researchers have identified the metal–oxygen bond energy [46], defect–dopant association [5,47,43], the Goldschmidt tolerance factor which measures the deviation away from the cubic structure [46,48,49], and the associated free volume of the crystal lattice [46,48] as factors that determine the mobility of oxygen within a given perovskite. As doping levels increase, conductivity reaches a maximum and begins to decrease, an observation that has been identified

b

Fig. 6. Migration pathway of the Ti interstitial in SrTiO3 as predicted by the Thomas potential. (a) The ground state structure of the Ti interstitial. (b) An intermediate state, of which there are several symmetrically equivalent versions as predicted by the potential, which the Ti interstitial passes through during migration to an equivalent site in the structure. The color scheme is the same as in Fig. 4.

24

B.P. Uberuaga, L.J. Vernon / Solid State Ionics 253 (2013) 18–26

minimum

saddle

minimum

b

c

a

Fig. 7. Migration pathway for the Ti interstitial in BaTiO3. The ground state structure in (a) exhibits a larger distortion of the oxygen sublattice than for the Ti interstitial in SrTiO3 (large dark (green) spheres indicate oxygen ions pulled off of their original lattice sites, which are indicated by the large light (green) squares). The color scheme is the same as in Fig. 4.

with the ordering of the relatively high concentration of oxygen vacancies and binding between oxygen vacancies and dopants [50,51]. Our results do not directly inform any of these factors. While we do see differences in oxygen vacancy mobility as a function of the perovskite chemistry, they are relatively small with variations due to the potential being almost as large as those due to chemistry. We find that oxygen vacancy migration is fastest in LaAlO3 and slowest in SrTiO3, the migration barriers differing by 0.52 eV. However, the variation in SrTiO3 from different potentials is 0.29 eV. Thus, it is hard to make definitive conclusions about the relative merits of the two potentials. This is in spite of the fact that the pair interactions as described by the two potentials are significantly different. The two potentials were fit with very different philosophies. The Grimes potential, to ensure maximum transferability, was fit as part of an overall procedure that included many different compounds at one time and the same anion– anion interaction was used for all materials. The potentials were fit primarily to bulk crystalline properties, including lattice and elastic constants. In contrast, the Thomas potential was optimized to fit SrTiO3 as accurately as possible. In this case, the fit included ionic charges as determined by DFT calculations. As illustrated in Fig. 8, the Grimes potential has more anion–anion repulsion and more cation–anion attraction than the Thomas potential. The optimal separation of the different species is also different, with the Grimes potential describing a much shorter Sr\O bond (for the Sr–O dimer) than the Thomas potential. Thomas, O-O Grimes, O-O Thomas, Sr-O Grimes, Sr-O Thomas, Ti-O Grimes, Ti-O

200

energy (eV)

150 100 50 0 -50 0

1

2

3

4

5

separation (angstroms) Fig. 8. Comparison of the interaction energy of anions with other species for the Thomas (solid lines) and the Grimes potential (solid lines with points) as a function of separation between the two species. The lines include both the Buckingham short range interaction and the Coulomb long range interaction.

Further, as mentioned above, the two potentials make very different assumptions about the charge state of the ions. In fact, the Thomas potential does not enforce charge neutrality for sub-stoichiometric units, such as SrO and TiO2. Even so, the two potentials predict very similar migration mechanisms for all of the defects considered (the only significant difference is the relative ease by which Ti interstitials convert to antisites with the Grimes potential) and even the predicted migration energies for the various defects are very similar. For O and Sr interstitials, migration energies agree to within about 0.1 eV, though there is greater disagreement for vacancy migration energies. In all of the perovskites considered, cation vacancies diffuse very slowly, with migration energies typically in excess of 4.0 eV. Further, B cation vacancies are immobile, with faster rates to transform into antisite complexes than to diffuse, similar to behavior observed in pyrochlore [52]. The most likely diffusion mechanism for B cations via a vacancy mechanism will be the dissociation of the A cation from the antisite and the subsequent encounter of another A cation with the antisite, moving it one lattice spacing. However, given the large diffusion barriers for A cations to diffuse, this is also very unlikely. The most significant differences between the various perovskites occur for interstitial migration. While A cation interstitial migration is relatively fast in the II–IV perovskites (faster than the oxygen vacancies) it is very slow in LaAlO3. The migration mechanisms are the same, so this is more of a consequence of the interaction of the interstitial with the surrounding cations. In contrast, if one considers the predictions of B cation mobility with the three Grimes potentials, it is fastest in LaAlO3. This suggests that cation interstitials with smaller charges diffuse more easily in II–IV perovskites and those with larger charges diffuse faster in III–III perovskites. Oxygen interstitial mobility is always predicted to be faster than oxygen vacancy migration, at least for twodimensional diffusion. The difference in the 2D vs 3D rates for oxygen interstitial diffusion is very large for LaAlO3 and is significantly smaller in the II–IV perovskites. These results also show that even in the relatively simple cubic perovskite structure, defect migration mechanisms can be somewhat complex. It is reasonable that A cation interstitials might exhibit one dimensional diffusion as they have a preferred orientation in the split structure and one can expect that there are two barriers, one describing translation and another rotation. What is surprising, however, is that the oxygen interstitial exhibits two-dimensional migration even though it also has a similar split interstitial structure. This illustrates the need to perform dynamical simulations that do not assume migration mechanisms in order to reveal such complex pathways. Overall, our results compare well with literature values for the species that have been examined in the past. As mentioned, oxygen

B.P. Uberuaga, L.J. Vernon / Solid State Ionics 253 (2013) 18–26

vacancy migration energy has been estimated to vary from 0.30 to 1.22 eV, and our values fall within this range. More specifically, Erhart and Albe [24] calculated with DFT the migration energies of all three vacancy species in BaTiO3. They found that the Ti vacancy has an extremely large barrier (almost 10 eV), consistent with the immobility that we find. For the charged versions of the Ba and oxygen vacancies, they found 6.00 eV and 0.89 eV, respectively. Our values, found with the potentials, are slightly higher (6.68 eV and 1.12 eV, respectively) but the trends are well reproduced. This is especially apparent when comparing the Sr vacancy migration in SrTiO3 with that of the Ba vacancy in BaTiO3. We find that the Sr vacancy is significantly more mobile (5.03 vs 6.68 eV), while Walsh et al. [18] found a barrier with DFT for the Sr vacancy that is significantly less than that for the Ba vacancy found by Erhart and Albe [24] (3.68 vs 6.00 eV). Similarly, the migration energy for La vacancies in LaMnO3 (2.96 eV found via DFT + U [25]) and Mg vacancies in MgSiO3 (3.47 eV with DFT [22]) are somewhat smaller than but similar to the values we find for the materials considered here. Clearly, the potentials agree qualitatively, and in some cases quantitatively, with the DFT results. In particular, the barriers as predicted by the Thomas potential for oxygen and Sr vacancy migration in SrTiO3 compare well with the DFT calculations of Walsh et al. [18]: 0.93 and 3.95 eV with the potential vs 0.53 and 3.68 eV with DFT. This suggests that the Thomas potential may provide somewhat better predictions of energy barriers than the Grimes potential. As we are motivated by radiation damage experiments in perovskites, it is important to discuss our results in that context. While overall we find that defect mobilities are rather similar in the various perovskites, important differences do appear. If we assume that at room temperature only processes with barriers less than 1 eV will be active (roughly the energy barrier at which processes take longer than an hour to occur at room temperature, assuming a prefactor of 5 × 1012/s), oxygen and A cation interstitials will be mobile in SrTiO3 and BaTiO3, though with both being somewhat more mobile in SrTiO3 than BaTiO3. The other defects in the II–IV perovskites are immobile (though the oxygen vacancy might be on the verge of mobility if the Thomas potential is correct). In contrast, in LaAlO3, the B cation interstitial and the two oxygen defects would be mobile at room temperature, though the oxygen interstitial would be constrained to two-dimensional diffusion. Thus, one might expect higher defect recombination rates in SrTiO3 than BaTiO3, consistent with the fact that BaTiO3 amorphizes at a smaller dose than SrTiO3 at room temperature [36]. At the same time, while the 2D mobility of oxygen interstitials in LaAlO3 is higher than in SrTiO3, the 3D mobility is much lower. However, the oxygen vacancy mobility is higher in LaAlO3. Thus, it is not obvious a priori in which material defect recombination would be greater. The fact that disordering processes (the formation of antisites) appear to be easier in SrTiO3 than in LaAlO3 (based on the relative energy of antisite complexes compared to vacancy species) suggests that SrTiO3 might be more radiation tolerant independent of kinetic considerations [53]. However, to be certain, higher level models, such as kinetic Monte Carlo or reaction–diffusion models, which account for all defect mobilities and defect reactions, including clustering which has not been considered here, would need to be developed to fully understand the competing processes between all of the defects and their ramification on radiation damage evolution. Finally, under equilibrium conditions, only a small subset of all possible defects will be present. For example, in SrTiO3, it is well established that the dominant disorder reactions are either Schottky or partial Schottky reactions [18], leading to the formation of, primarily, Sr and O vacancies. If the concentrations of these defects are high enough, and the temperature is low enough, these defects will often be bound into complexes. Such complexes are especially important under irradiation conditions, where defect concentrations are much higher and defect complexes can form directly within cascades [54]. Further, it is also well established that the mobility of ionic species are significantly influenced by this clustering [18,55], often to the point that the defect clusters have their own migration characteristics [44]. However, in the

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case of SrTiO3, Walsh et al. have found that the association or binding energy of Sr and O vacancies is very low, indicating that there will not be a significant thermodynamic tendency for such complexes to form. A complete examination of the possible migration mechanisms associated with defect clusters, while certainly of interest particularly for radiation damage studies, is beyond the scope of the current paper. 5. Conclusions Using modern atomistic simulation methods that simulate longtime dynamics, we have examined the migration mechanisms of interstitials and vacancies in perovskites with different chemistries and using different potentials. We find that the migration mechanisms for the various interstitial and vacancy species are very similar regardless of the potential used or the chemistry of the perovskite, though the energy barriers associated with those mechanisms are sensitive to both the potential used and the chemistry of the perovskite, as would be expected. We find that cation defects, particularly vacancies, are often immobile. Further, B cation defects have a tendency to transform to antisite complexes. Oxygen and A cation interstitials diffuse in a complex manner, exhibiting one- or two-dimensional diffusion mechanisms. These results show both the complexity of diffusion mechanisms even in relatively simple (cubic) oxides and the utility of long-time simulation methods to reveal those mechanisms. 6. Acknowledgments This work was supported as part of the Center for Materials at Irradiation and Mechanical Extremes, an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences under Award Number 2008LANL1026. Los Alamos National Laboratory, an affirmative action equal opportunity employer, is operated by Los Alamos National Security, LLC, for the National Nuclear Security Administration of the U.S. DOE under contract DE-AC52-06NA25396. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

S. Hull, Rep. Prog. Phys. 67 (2004) 1233–1314. S.R. Phillpot, S.B. Sinnott, A. Asthagiri, Annu. Rev. Mater. Res. 37 (2007) 239–270. B.C.H. Steele, Mat. Sci. Eng. B 13 (1992) 79–87. H.L. Tuller, P.K. Moon, Mater. Sci. Eng. B 1 (1988) 171–191. J.A. Kilner, R.J. Brook, Solid State Ionics 6 (1982) 237–252. T. Ishigaki, S. Yamauchi, K. Kishio, J. Mizusaki, K. Fueki, J. Solid State Chem. 73 (1988) 179–187. V.V. Kharton, F.M.B. Marques, A. Atkinson, Solid State Ionics 174 (2004) 135–149. D. Lybye, F.W. Poulsen, M. Mogensen, Solid State Ionics 128 (2000) 91–103. S. Miyoshi, M. Martin, Phys. Chem. Chem. Phys. 11 (2009) 3063–3070. A. Chroneos, R.V. Vovk, I.L. Goulatis, L.I. Goulatis, J. Alloys Compd. 494 (2010) 190–195. A. Jones, M.S. Islam, J. Phys. Chem. C 112 (2008) 4455–4462. M.S. Islam, J. Mater. Chem. 10 (2000) 1027–1038. M. Cherry, M.S. Islam, C.R.A. Catlow, J. Solid State Chem. 118 (1995) 125–132. R.A. De Souza, M. Martin, Monatsh. Chem. 140 (2009) 1011–1015. S.M. Woodley, J.D. Gale, P.D. Battle, C.R.A. Catlow, J. Chem. Phys. 119 (2003) 9737–9744. Y.Z. Wang, E. Bevillon, A. Chesnaud, G. Geneste, G. Dezanneau, J. Phys. Chem. C 113 (2009) 20486–20492. S.J. Stokes, M.S. Islam, J. Mater. Chem. 20 (2010) 6258–6264. A. Walsh, C.R.A. Catlow, A.G.H. Smith, A.A. Sokol, S.M. Woodley, Phys. Rev. B 83 (2011) 220301(R). B.S. Thomas, N.A. Marks, B.D. Begg, Nucl. Instr. Meth. B 254 (2007) 211–218. C. Tealdi, M.S. Islam, C.A.J. Fisher, L. Malavasi, G. Flor, Prog. Sol. State Chem. 35 (2007) 491–499. G.C. Mather, M.S. Islam, F.M. Figueiredo, Adv. Funct. Mater. 17 (2007) 905–912. B.B. Karki, G. Khanduja, Earth Planet. Sci. Lett. 260 (2007) 201–211. R.A. Everestov, E.A. Kotomin, Yu.F. Zhukovskii, Int. J. Quantum Chem. 106 (2006) 2173–2183. P. Erhart, K. Albe, J. Appl. Phys. 102 (2007) 084111. B. Puchala, Y.-L. Lee, D. Morgan, J. Electrochem. Soc. 160 (2013) F877–F882. G. Henkelman, B.P. Uberuaga, H. Jönsson, J. Chem. Phys. 113 (2000) 9901–9904. M.J. Zhuo, E.G. Fu, L. Yan, Y.Q. Wang, Y.Y. Zhang, R.M. Dickerson, B.P. Uberuaga, A. Misra, M. Nastasi, Q.X. Jia, Scripta Mater. 65 (2011) 807–810. M.J. Zhuo, B.P. Uberuaga, L. Yan, E.G. Fu, R.M. Dickerson, Y.Q. Wang, A. Misra, M. Nastasi, Q.X. Jia, J. Nucl. Mater. 429 (2012) 177–184.

26

B.P. Uberuaga, L.J. Vernon / Solid State Ionics 253 (2013) 18–26

[29] Z. Bi, B.P. Uberuaga, L.J. Vernon, E. Fu, Y.Q. Wang, N. Li, H. Wang, A. Misra, Q.X. Jia, J. Appl. Phys. 113 (2013) 023513. [30] Bi, Z.; in preparation. [31] J. Won, L. Vernon, A. Karakuscu, R. Dickerson, M. Cologna, R. Raj, Y. Q. Wang, S. J. Yoo, S. H. Lee, A. Misra, and B. P. Uberuaga, under review. [32] D. Yarotski, E.G. Fu, L. Yan, Q.X. Jia, Y.Q. Wang, A.J. Taylor, B.P. Uberuaga, Appl. Phys. Lett. 100 (2012) 251603. [33] M.R. Sorensen, A.F. Voter, J. Chem. Phys. 112 (2000) 9599–9606. [34] L. Vernon, S.D. Kenny, R. Smith, E. Sanville, Phys. Rev. B 83 (2011) 075412. [35] A. Meldrum, L.A. Boatner, R.C. Ewing, Nucl. Instr. Meth. B 141 (1998) 347–352. [36] A. Meldrum, L.A. Boatner, W.J. Weber, R.C. Ewing, J. Nucl. Mater. 300 (2002) 242–254. [37] M. Tang, P. Fuierer, P. Dickens, E. Fu, Phys. Status Solidi C 10 (2013) 216–220. [38] A. Navrotsky, Chem. Mater. 10 (1998) 2787–2793. [39] M.R. Levy, . , PhD thesis University of London, 2005. [40] M.A. McCoy, R.W. Grimes, W.E. Lee, Philos. Mag. A 75 (1997) 833–846. [41] R.W. Grimes, G. Busker, Nucl. Energy 35 (1996) 403–410. [42] G. Busker, A. Chroneos, R.W. Grimes, I.W. Chen, J. Am. Ceram. Soc. 82 (1999) 1553–1559. [43] L. Minervini, M.O. Zacate, R.W. Grimes, Solid State Ionics 116 (1999) 339–349. [44] B.P. Uberuaga, R. Smith, A.R. Cleave, F. Montalenti, G. Henkelman, R.W. Grimes, A.F. Voter, K.E. Sickafus, Phys. Rev. Lett. 92 (2004) 115505.

[45] B.P. Uberuaga, R. Smith, A.R. Cleave, G. Henkelman, R.W. Grimes, A.F. Voter, K.E. Sickafus, Phys. Rev. B 71 (2005) 104102. [46] R.L. Cook, A.F. Sammells, Solid State Ionics 45 (1991) 311–321. [47] J.A. Kilner, Solid State Ionics 129 (2000) 13–23. [48] H. Hayashi, H. Inaba, M. Matsuyama, N.G. Lan, M. Dokiya, H. Tagawa, Solid State Ionics 122 (1999) 1–15. [49] M. Mogensen, D. Lybye, N. Bonanos, P.V. Hendriksen, F.W. Poulsen, Solid State Ionics 174 (2004) 279–286. [50] D.A. Andersson, S.I. Simak, N.V. Skorodumova, I.A. Abrikosov, B. Johansson, Proc. Natl. Acad. Sci. U. S. A. 103 (2006) 3518–3521. [51] P.P. Dholabhai, S. Anwar, J.B. Adams, P. Crozier, R. Sharma, J. Solid State Chem. 184 (2011) 811–817. [52] Y.H. Li, B.P. Uberuaga, C. Jiang, S. Choudhury, J.A. Valdez, M.K. Patel, J. Won, Y.Q. Wang, M. Tang, D.J. Safarik, D.D. Byler, K.J. McClellan, I.O. Usov, T. Hartmann, G. Baldinozzi, K.E. Sickafus, Phys. Rev. Lett. 108 (2012) 195504. [53] K.E. Sickafus, R.W. Grimes, J.A. Valdez, A.R. Cleave, M. Tang, M. Ishimaru, S.M. Corish, C.R. Stanek, B.P. Uberuaga, Nat. Mater. 6 (2007) 217–223. [54] J. Won, L.J. Vernon, A. Karakuscu, R.M. Dickerson, M. Cologna, R. Raj, Y.-Q. Wang, S.J. Yoo, S.-H. Lee, A. Misra, B.P. Uberuaga, J. Mater. Chem. A 1 (2013) 9235–9245. [55] D.A. Andersson, B.P. Uberuaga, P.V. Nerikar, C. Unal, C.R. Stanek, Phys. Rev. B 84 (2011) 054105.