Charge Transfer Model of Atomic Ordering in Complex Perovskite Alloys

Transcription

1 Charge Transfer Model of Atomic Ordering in Complex Perovskite Alloys Zhigang Wu and Henry Krakauer Department of Physics, College of William and Mary Williamsburg, VA Abstract. We introduce an electrostatic model including charge transfer, which is shown to account for the observed B-site ordering in Pb-based perovskite alloys. The model allows charge transfer between A sites and is a generalization of Bellaiche and Vanderbilt s purely electrostatic model. The large covalency of Pb 2+ compared to Ba 2+ results in an environment dependent effective A-site charge. This model successfully reproduces the long range order of both Pb-based and Ba-based complex perovskite alloys. It yields smaller structural energy differences for Pb-based compared to Babased A(B 1/3 B 2/3 )O 3 systems, consistent with first-principles calculations and the fact that Ba-based disorder at higher temperatures than the Pb-based alloys. INTRODUCTION In this paper we focus on the differences in B-site atomic ordering between Ba and Pb based perovskite alloys.for example, at 1640 C, increasing the tetravalent composition x in (1 x) Ba(MgNb)O 3 + x BaZrO 3 (BMN-BZ) alloy, the following sequence of B-site ordering is observed: [111] 1:2 order for x < 5%; then [111] 1:1 order for 5% < x < 25%; and finally disorder for larger x [1]. Other Ba-based perovskites, e.g., (1 x) Ba(Mg 1/3 Ta 2/3 )O 3 + x BaZrO 3 (BMT- BZ) [1], (1 x) Ba(Mg 1/3 Nb 2/3 )O 3 + x BaZrO 3 (BMN-BZ) [3], display a similar sequence of B-site order. On the other hand, for Pb-based systems, e.g., (1 x) Pb(Mg 1/3 Ta 2/3 )O 3 + x PbZrO 3 (PMT-PZ), [111] 1:2 order is not observed at x=0; instead, annealing between 1325 C and 1350 C results in [111] 1:1 order all the way down to x=0 [4,10]. Other Pb-based perovskites, e.g., Pb(Mg 1/3 Nb 2/3 )O 3 (PMN) [5 7], display similar B-site ordering. ELECTROSTATIC MODEL WITHOUT CHARGE TRANSFER Bellaiche and Vanderbilt (BV) presented a very simple but remarkably successful ionic model to explain the B-site ordering of perovskite alloys [8].They argued that

2 the electrostatic energy between point-charge ions in the ideal cubic structure is the dominant factor leading to the observed B-site ordering.in their model the total (electrostatic) energy in A(BB )O 3 perovskite compounds is: E = e2 2 (lτ) (l τ ) Q lτ Q l τ, (1) ɛ R lτ R l τ where R lτ is the position of the atom on site τ of cell l (τ={a,b,o 1,O 2,O 3 }), and ɛ is the electronic dielectric constant.for alloys with Pb or Ba occupied A- sites, Q l,a = q A =+2andQ l,o = q O = 2.BV showed that up to a constant, the configurationally averaged total energy depends only on the B-site charges, because the A-sites and O-sites charges have fixed values.the B-site charges Q l,b are conveniently written as Q l,b = q B + q l,b =4+ q l,b, (2) and the configurationally averaged electrostatic energy depends only on the B- sublattice up to a constant: E B = e2 q l,b q l,b, (3) 2ɛa l l l l where a is the cubic lattice constant, and R l = la. Bellaiche and Vanderbilt performed Metropolis Monte Carlo simulations [9] with a=7.7 a.u., ɛ=10, and supercells to study the thermodynamic behavior as a function of temperature and the tetravalent composition x.they used a B-site-only notation to describe the different classes of alloys.for example, IV x IV 1 x denotes a homovalent binary alloy having tetravalent B-atoms, e.g., Pb(ZrTi)O 3 ;andii (1 x)/3 IV x V 2(1 x)/3 indicates a heterovalent ternary, such as (1 x) Ba(MgNb)O 3 + x BaZrO 3.The predicted behaviors are described as [8]: (1) For III (1 x)/2 IV x V (1 x)/2 and II (1 x)/2 IV x VI (1 x)/2 heterovalent ternaries, a rocksalt-type [111] 1:1 order becomes disordered with progressively increasing x, consistent with experimental observations.(2) For II (1 x)/3 IV x V 2(1 x)/3 heterovalent ternaries, the model predicts a richer phase diagram, with [111] 1:2 order for x less than about 5%, [111] 1:1 order for x greater than 5%, followed by disorder for large x.this is in good agreement with observations for Ba based systems but disagrees for Pb based systems, where [111] 1:1 order persists down to x =0.They proposed a possible explanation [8], postulating a small amount of Pb 4+ on the B-sublattice to explain the stabilization of the 1:1 phase.as far as we know, there is no evidence of the presence of Pb 4+ on the B-sublattice. ELECTROSTATIC MODEL WITH CHARGE TRANSFER First-principles calculations [14] show that typical structural energy differences can be as large as 60 kj/(mol-abo 3 ) for Ba based alloys, an order of magnitude

3 larger than in the same Pb based alloys.such large energy differences in the Ba alloys can be accounted for by BV s electrostatic model [14].The reduction of energy differences in Pb based alloys is attributed to the large covalency of Pb compared to ionic Ba, which permits local Pb-related relaxation to compensate for the large purely electrostatic energy differences.an initial estimate of the energy scale of Pb atomic relaxation might be taken from the ferroelectric well-depths of PbTiO 3, which are about 10 kj/(mol-abo 3 ) without strain [15].However, in the heterovalent alloys, the covalency of Pb also permits an effective charge transfer between A-sites, which could greatly increase the local relaxation energy in Pb alloys.since the A-sites in II (1 x)/3 IV x V 2(1 x)/3 ternaries can have a non-neutral nearest neighbor shell of B-sites, it seems reasonable that charge transfer could be an important mechanism, allowing the Pb based alloys to compensate the purely electrostatic energy differences arising from the B-site configuration. The difference in covalency between Ba and Pb suggests a nearest-neighbor configuration-dependent effective Q l,a : Q l,a = q A + q l,a =2+ q l,a, (4) where q A = λ( 1 q i,b ), (5) 8 i=n.n.b sites where the parameter λ controls the magnitude of the charge transfer onto the A- sites, and the total energy now depends on the A- and B-sublattices E AB = e2 2ɛ (lτ) (l τ ) Q lτ Q l τ, (6) R lτ R l τ where τ ={A,B}.The value of the additional parameter λ can be chosen depending on the species occupying the A-site.For Ba-occupied A-sites, no charge transfer is possible, λ=0, and our model reduces to Bellaiche and Vanderbilt s model.for Pb-occupied A-sites, the value of λ can be fit to first-principles calculations. The form of our charge transfer electrostatic model is closely patterned after similar models that were introduced to model the configuration dependence of Coulomb energies for pseudobinary alloys [16 18], where calculated mixing enthalpies were found to be commensurate with that of lattice mismatch energies [16].The trend of stability of (AC) n (BC) n superlattices predicted by these models is also consistent with first-principles total-energy calculations for lattice-matched semiconductor superlattices [18]. RESULTS AND DISCUSSION The parameters in our model are chosen as follows. ɛ is chosen to reproduce the energy differences of selected Ba-based alloys (we used ɛ = 10, as in BV).The

4 value of λ is then fit to selected Pb-based alloy energy differences.fig.1 shows selected structural energy differences (x = 0) as a function of λ.to reproduce the observed [111] 1:1 order for Pb based alloys, Fig.1 indicates that λ Pb should be in the range 0.6 <λ Pb < 0.8, although this is only indicative of the expected values. To confirm this, we perform Metropolis Monte Carlo simulations [9] as a function of temperature and concentration x. We performed simulations for λ=0.7 and 0.3, setting a=7.7 a.u. and ɛ=10, using a supercell with periodic boundary conditions.in order to obtain good statistics, we use a very large number of moves (up to 10 7 ) at each temperature. We start the calculations at high temperature (e.g., 4,000K) with a random configuration.the temperature is then slowly decreased until no trial moves are accepted by the Monte-Carlo program.following BV we compute the Fourier transform of the charge-charge correlation function η(k) to determine the B-site ordering from the Monte-Carlo outputs for a thermodynamic state at a specific temperature [8], η(k) =α ll q l q l+l exp( ik l ), (7) where α is a normalization factor, the sum runs over the B-sublattice, and k is the wavevector in the Brillouin zone of the unit cubic cell.the charge-charge correlation function η(k) is directly proportional to the ensemble average of the square of the Fourier transform of the charge distribution, e.g., a large value of η at k =2π( 1 2, 1 2, 1 2 )ork =2π( 1 3, 1 3, 1 3 ) corresponds to a strong [111] 1:1 or [111] 1:2 order respectively. Fig.2 shows the predicted behaviors of η(k) vs.tetravalent composition x at final temperatures of T =1000K and 1500K for II (1 x)/3 IV x V 2(1 x)/3 heterovalent ternaries, at λ=0.3 and 0.7 respectively. For λ=0.3, which permits only a small charge structure1 structure2 structure3 structure4 1:1 order 1:2 order λ FIGURE 1. Energy as a function of λ for six structures.

5 transfer between A-sites, Bellaiche and Vanderbilt s prediction is found (also for λ=0.0, not shown), i.e., the B-site ordering sequence of Ba-based perovskite alloys is reproduced.for λ=0.7, Fig. 2 shows there is no [111] 1:2 ordered phase.instead, there is a continuous transformation from [111] 1:1 order at x=0 to a disordered state at large x.increasing the final temperature to T=1500K does not change this ordering sequence, but only decreases the value of x at which disorder appears, as would be expected.moreover, the [111] 1:1 structure is similar to the randomsite model [1], in that it has alternating planes of mixed (II-V)-planes and pure (V)-planes, as was also found by BV.These results are in good agreement with experimental observations of the ordering sequence of Pb(Mg 1/3 Nb 2/3 )O 3 (PMN) [5 7], (1 x) Pb(Mg 1/3 Ta 2/3 )O 3 + x PbZrO 3 (PMT-PZ) [4] and Pb(Mg 1/3 Ta 2/3 )O 3 (PMT) [10]. Our model is also compatible with BV s prediction of III (1 x)/2 IV x V (1 x)/2 and II (1 x)/2 IV x VI (1 x)/2 ordering.for these structure, the A-sites will have on average a neutral nearest neighbor shell of B-sites, leading to zero charge transfer on average, i.e., for any value of λ, Q A =0, and our model reduces to Bellaiche and Vanderbilt s model.results of Metropolis Monte Carlo simulations for III-IV-V alloys confirm this, and our model results in both Ba-based and Pb-based perovskite alloys having λ = 0.3, Τ = 1000Κ λ = 0.7, Τ = 1000Κ 1:1 1:2 x x λ = 0.3, Τ = 1500Κ λ = 0.7, Τ = 1500Κ x x FIGURE 2. Monte Carlo simulations of the long-range order parameter η(k) vs. tetravalent atomic composition x for the charge transfer model for II (1 x)/3 IV x V 2(1 x)/3 ternaries, with supercells, a=7.7 a.u., and ɛ=10.

6 thesamelrosequenceforiii (1 x)/2 IV x V (1 x)/2 ternaries and II (1 x)/2 IV x VI (1 x)/2 ternaries. We now examine more carefully the structural energy differences of Ba-based and Pb-based perovskites.the smaller structural energy differences of the Pb-based perovskites has been to used to explain [14] the fact that Pb-based A(BB )O 3 perovskites disorder at lower temperatures than Ba-based perovskites, especially in the A(B 1/3 B 2/3 )O 3 systems [2,11,7,10,12,14].Burton and Cockayne interpreted this phenomenon as a consequence of enhanced Pb-O hybridization between the Pb 6s and O 2p states of underbonded oxygens in B 2+ -O-B 2+ or B 2+ -O-B 2+ environments [14].They calculated three 15-atom perovskite structures ([111] 1:2, [110] 1:2 and [001] 1:2 ) in each of the eight possible stoichiometries of A(B 1/3 B 2/3 )O 3 (A=Pb 2+,Ba 2+ ;B=Mg 2+,Zn 2+ ;B =Nb 5+,Ta 5+ ).As shown in the left panel of Fig. 3, E Pb 0.1 E Ba.Also shown in the left panel, are the BV model predictions, which agree well with the Ba based alloys.these first-principles calculations are for fully relaxed structures, including lattice parameters (strain and volume) and internal atomic coordinates.we also calculated total energies of these three structures using our charge transfer model.the results as a function of λ are shown in the right panel of Fig.3.The structural energy differences become comparable to the Pb-based perovskites for λ Comparing both sides of Fig.3, there is remarkable agreement between our simple model and the ab initio calculations. For III-V A(B 1/2 B 1/2 )O 3 systems (not shown), the structural energy differences of the Pb-based systems are more similar to their Ba-based counterparts [14], and we believe this may largely be due to strain relaxation of the Pb-based III-V alloys. As mentioned, our model results in zero charge transfer for the III-V alloys.our [001]1:2 [110]1:2 [111]1: λ FIGURE 3. Total energies (per mol=abo 3 ), relative to E [111]1:1, for the perovskite based supercells with compositions A(B 1/3 B 2/3 )O 3. Burton and Cockayne s [14] results on the left, ours on the right.

7 TABLE 1. Comparison of PZN ordering energies, LAPW (Ref. 13) vs. Model (λ =0.7). Energy in kj/mol-abo 3. Structure LAPW Model [111] 1:1 (RS2) [111] 1: [111] 1:1 (SC) [111] 1:1 (RS1) [001] 1: model thus qualitatively explains the contrast between the large structural energy differences of A(B 1/3 B 2/3 )O 3 perovskites compared to A(B 1/2 B 1/2 )O 3 perovskites. We can also compare some structural total energy differences using our model with those from first-principles LAPW calculations for PZN [13].Table I presents results for five structures: two 15-atom cells, which are the same as those of Burton and Cockayne but with no strain, [111] 1:2 and [001] 1:2, a 30-atom structure that simulates the space-charge (SC) model [1], [111] 1:1 (SC), and two 30-atom structures that simulate the random-site (RS) model [1], [111] 1:1 (RS1) and [111] 1:1 (RS2). As indicated in Table 1, the charge transfer model predicts the same energy sequence as the LAPW calculations.except for the [111] 1:1 (SC) structure, which our model predicts as the lowest energy state among the five structures, while LAPW predicts [111] 1:1 (RS2) instead. SUMMARY We have demonstrated that the charge transfer model is able to explain the observed B-site compositional ordering in Pb-based ternary perovskite alloys.the model also reproduces trends of structural energies seen in first-principles calculations.the success of this model suggests that, for Pb 2+ and possibly other covalent ions, the net charge on the A-site can depend on the environment.with the inclusion of charge transfer in these cases, we conclude that the electrostatic interaction is the leading mechanism responsible for the B-site ordering in perovskite alloys, extending Bellaiche and Vanderbilt s finding. ACKNOWLEDGMENT This work is supported by the Office of Naval Research grant N We are grateful to Suhuai Wei for useful discussions on electrostatic models incorporating charge transfer.

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