1. Model for B Site Ordering in PMN Eric Cockayne Benjamin P. Burton Material Measurement Laboratory, NIST, Gaithersburg

2. Observed Cation Ordering in Various ABB'O 3 perovskites (Burton & Cockayne, PRB 60, R12542 (1999))

3. Randall & Bhalla (Jpn. J Appl. Phys. 29, 327 1990): short-range ordering correlated with relaxor behavior

4. Akbar and Davies (J. Phys. Chem. Sol. 61, 159 (2000)): 1:1 Ordering in PMN: ``random site” model: One sublattice is Nb, the other a random arrangement of Mg 2/3 Nb 1/3 Coulomb model (Bellaiche and Vanderbilt PRL 81, 1318 (1998)): predicts 1:2 [111] ordering for AB 1/3 B' 2/3 O 3 with heterovalent B cations Holds true for BMN (Dias and Luiz- Moreira, J. Appl. Phys. 94, 3414 (2003)).

5. Prosandeev et. al., PRB 70, 134110 (2004) 1:1 [111] A Example of ``1:1-type” structure” Ab-Initio-Calculation-Based Proposed Ground State of PMN

6. Burton & Cockayne, PRB (1999) Calculated energies of Pb- and Ba- complex perovskites

7. Various explanations for observed 1:1 ordering in PMN: 1. Pb 4+ substitution on B sublattice (Bellaiche & Vanderbilt) 2. Theories with multibody terms Pb bonding to underbonded oxygen (Burton and Cockayne) empirical multibody terms (A. Y. Gufan, Phys. Solid State 47, 1134 (2005)) charge transfer to A site depending on surrounding B environment (Wu and Krakauer, PRB 63, 184113 (2001). This talk: simple quantitative ”multibody” model: Coulomb interactions + effect of local electric fields at A sites Coulomb effects do not explain cation ordering in PMN. But Why Not?

8. Starting point: existing effective Hamiltonian approach to local polarizations in Pb-based perovskites (Cockayne, Bellaiche and Burton, AIP Conf.l Proc. 2001)) H eff = H onsite (  i ) + H ij (  i,  j ) + H(  i,E loc (  )) + H(  i,e) → H eff = H onsite (  i ) + H(  i,E loc (  )) + H Coulomb (  ) H onsite (  i ) → quartic; isotropic

9. “Centered” A site Ion; e.g. Ba Ionic energetics vs. strength of local field “Off-center” ion, eg. Pb  U ~ a E 2  U ~ a 1 |E| + a 2 E 2 Sitewise addition for electric fields in a configuration: U = U Coulomb + C 1  i |E i | + C 2  i |E i | 2 (C 1 = 0 for centered ions) This is physical origin of multibody terms in model C 1 and C 2 always negative: Similutaneously want low Coulomb energy (ordered) and strong local fields (disorder)!

10. Calculations of local field strengths and Plausibility argument for stability of 1:1 [111] A

11. Configuration exploration Brute force: 6x6x2 simulation cell; compute all 1:1-type structures 36 sites on each Mg 2/3 Nb 1/3 sublattice. Explore all arrangements of Mn 24 Nb 12 on these sites (exploit symmetries to reduce burden) Find ground state for various linear and quadratic coupling constants Map phase diagram; identify small-unit-cell candidates for PMN ground state. Ab initio electronic structure VASP used (DFT) code Projector augmented wave pseudopotentials optimized to PBE version of GGA Run using PBEsol version of GGA 400 eV plane wave cutoff; Common k-point grid equivalent to 6x6x6 Monkhorst-Pack on primitive perovskite cell Methods

12. 1:2 1:1 A 1:1 B 1:1 C Results: qualitative phase diagram Magnitude of C 2 Magnitude of C 1 1:2 1:1 A 1:1 B 1:1 C

13. Ab initio results Structure Atoms/cell Symmetry U/(5 atoms) (eV) 1:2 [111] 15 rhomb -36.616 1:1 [111] A 30 mono -36.603 1:1 [111] B 60 ortho -36.598 1:1 [111] C 60 mono -36.640 1:2 [110] 15 ortho -36.587 1:1 [001] 15 tetrag -36.520 1:1 [111] C has lowest energy of any known configuration! Caveat: only symmetry-preserving relaxation has been allowed further relaxation (e.g. oxygen octahedral tilting) could lower energy further.

14. Conclusions Simple quantitative model for cation ordering in complex pervoskites Competition between Coulomb effects, favoring highly ordered structures, and relaxation due to strong local fields favoring disorder Whether an ion tends to be central (Ba) or off-centered (Pb) affects the relative stability of the different configurations Several ``random-site type” configurations are identified as candidates for the ground state of PMN and related materials One of these structures has the lowest energy in DFT calculations