Relaxors, spin, Stoner and cluster glasses

Abstract

It is argued that the main characteristic features of displacive relaxor ferroelectrics of the form A(B, B′)O₃ with isovalent B, B′ can be explained and understood in terms of a soft pseudo-spin analogue of conventional spin glasses as extended to itinerant magnet systems. The emphasis is on conceptual comprehension and on stimulating new perspectives with respect to previous and future studies. Some suggestions are made for further studies both on actual real systems and on test model systems to probe further. The case of heterovalent systems is also considered briefly.

Keywords:

• relaxors
• spin glassses
• cluster glasses
• localization
• non-ergodicity

Acknowledgements

The author thanks Prof. Roger Cowley for stimulating his interest in relaxors, Prof. Rasa Pirc for drawing his attention to the paper of Akbarzadeh et al. on BZT and Profs. Wolfgang Kleemann, Laurent Bellaiche and Peter Gehring for comments, criticisms and references. He also apologises to authors whose relevant work he has not cited. The aim has been to paint, broadbrush and concisely, a simple (probably oversimplified) but potentially stimulating picture of displacive relaxors as viewed from outside the mainstream of the topic.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. Note that in this paper the expression ‘field’ is reserved for the coefficients of terms of linear order in the Hamiltonian measuring the energy cost of displacements of ions away from their pure perovskite template. Some other authors effectively consider minimization with respect to part of the Hamiltonian and use the resulting polarization as providing effective fields on the rest of the Hamiltonian.

2. See also the second sentence of the previous footnote.

3. In fact there is evidence of a smaller order–disorder component as well as the main displacive one [24] but the discussion below allows straightforwardly for such a scenario.

4. For simplicity Equation (1) is restricted to scalar variables. Already in earlier numerical/simulational studies, such as [24] and [18], a more complete form is considered, but recall that the aim here is illustrative comparison for conceptualization, not numerical accuracy per se.

5. Field-theoretic studies of even nominally discrete/hard spin systems are often formulated in terms of a soft-spin formulation such as Equation (1), so as to allow for continuous variables, but finally the limit ($\kappa \to -\infty$, $\lambda \to \infty$, with $(\kappa /\lambda )\to$ const) is taken.

6. Retaining spin fluctuations but ignoring charge fluctuations and assuming uniformity of the resultant effective bare band structure.

7. For an explicit discussion see [24].

8. In fact purely antiferromagnetic interactions can be frustrated if of longer than nearest neighbour range.

9. The local fourth-order cross-terms and higher-order local terms are not written explicitly.

10. In their simulation of BZT(50/50) Akbarzadeh et al. [18] demonstrated explicitly that the strain terms are relatively less important in the relaxor phase of BZT.

11. These simplifications are not crucial to the conceptual argument.

12. If this is not the case then terms of higher order must be included, e.g. $\gamma u^6$

13. In fact, all the eigenfunctions are extended and the mobility edges, shown schematically in Figure 1, coincide with the band edges.

14. There is a similar transition in a conventional spin glass system as a function of the magnetic concentration between average periodic order (ferromagnetic) and (amorphous) spin glass order.

15. The initially proposed solution was not exact but further analysis has demonstrated both the solubility of the model and its solution.

16. Sometimes referred to as infinite-ranged, or as mean-field.

17. The $\kappa$ could be calculated using the first-principles techniques of [24] but the ionic radii are already highly suggestive of the similarities proposed above.

18. There will be second-order effects due to Nb displacements.

19. Actually, in first-principle calculations there are found deviations from these idealized values, but this is ignored here since the aim is illustrative of concepts rather than quantitative accuracy.

20. Note that the contributions from the bare positions of the Pb and O ions cancel, by symmetry.

21. This nomenclature relates to a mathematical method of solution [48, 53, 83].

22. For small field h the reduction in this mean-field model scales as $h^{2/3}$.

23. In this case the small h reduction in spin glass transition temperature scales as $h^2$.

24. For an earlier example of an experimental study of field dependence see [76].

25. Actually the randomness is correlated with the sublattice structure and is only of sign.