Orbital (dis)order: A tale of two temperatures

Not long ago, we published a pair of papers on orbital order and disorder that have collectively been in the pipeline for seven years, consuming the effort and generous patience of a shamefully large number of DPhils, MChems, and collaborators. The two materials in question are Y2Mo2O7—a notoriously unusual spin glass—and LaMnO3—the parent of the LCMO family of colossal magnetoresistance manganites. While I would like to think we have now learnt a great deal more about the science of orbital disorder in these two systems, the process has also taught me a number of other important lessons: the limitations of PDF and RMC analyses, the traps of (my) scientific naïvety, and the potential pitfalls of pushing for high-profile publications. In this extended post, I wanted to try to cover both angles—scientific and developmental alike.

But let’s start with orbital disorder: what does it mean and why is it important? This is not an easy story, so the aim is to proceed gently (at the expense of sounding ridiculously patronising!)…

The phenomenon turns up most frequently in transition-metal compounds with specific electronic configurations. The key is going to be the degeneracy of these configurations—in other words, the number of equivalent ways of distributing the valence electrons amongst the five valence transition-metal d-orbitals. In the case of Sc3+, which we consider to have no valence electrons (d0), the configuration is singly-degenerate because there is only one way of putting zero electrons in five orbitals. Zn2+ (d10) is another trivial case: each orbital must contain its full complement of two electrons and so again there is no choice. But intermediate d-electron counts will often result in some non-trivial degeneracy. For these cases there are multiple energetically-equivalent ways of occupying the valence d-orbitals, which gives rise to an electronic (hence orbital) degree of freedom.

Probably the simplest example is that of Cu2+ in an octahedral coordination environment, such as occurs in KCuF3. An octahedral crystal field splits the five d-orbitals into a lower-energy t2g set (3 orbitals) and a higher-energy eg set (2 orbitals). The d9 configuration of Cu2+ means that almost all of these orbitals are doubly occupied; there is a single “hole”, which can be placed in either one of the two eg-type orbitals (or indeed any linear combination thereof). Hence there are two orthogonal but energetically-equivalent ways of distributing the valence electrons—we might think of these as corresponding to the hole occupying either a dz2 or a dx2–y2-type orbital. The electronic configuration is doubly degenerate. Formally, its symmetry is characterised by the irreducible representation Eg, where the label “E” reflects the twofold degeneracy.

At the heart of the Jahn-Teller theorem is the idea that the degeneracy of an electronic configuration can be quenched by coupling to a structural distortion of the same symmetry. Taking the particular case of Cu2+, what this means is that any distortion of Eg symmetry will break the electronic degeneracy and hence modify the energy of the complex. The set of possible distortions can be thought of as occupying a two-dimensional space, represented in this diagram (cf. a triply-degenerate configuration, which would couple to a three-dimensional space of possible distortions). Symmetry can’t tell us which particular distortion lowers the energy most: whereas in the case of Cu2+ the most favourable distortion is nearly always a pure tetragonal elongation (Q3 in the diagram, or its symmetry-related distortions along x and y), for high-spin Mn3+ (d4, also Egsymmetry) the ground-state is usually Q2-like. Whichever particular distortion a system chooses, as soon as the distortion takes hold then the electronic degeneracy is lost. This is what is meant by quenching. The choice of orbital occupation is now linked entirely to the distortion, leaving a single option in every case. So, for example, if a Cu2+ complex elongates along z then it is the dz2 orbital that is doubly occupied and the hole sits in the dx2–y2 orbital.

Despite this quenching there is a new configurational degeneracy that arises from the underlying symmetry of the distortion space. For example, an elongation along z—coupled to the corresponding double occupation of the dz2-type orbital—is energetically equivalent to the same type of distortion along y and occupation of dy2, or along x and occupation of dx2. So if we can establish a fixed coordinate system—such as that provided by the unit cell axes of a crystal—then we can assign to each transition-metal centre an orbital/distortion state that behaves as a discrete degree of freedom. Such states are usually well-described by the n-state Potts model, where the value of n depends on the particular distortion involved: n = 3 for Q3 distortions (cf. dx2, dy2, dz2) and 6 for Q2 distortions (cf. dy2–z2, dz2–y2, dx2–z2, dz2–x2, dx2–y2, dy2–x2).

In a solid containing a macroscopic number (N, say) of transition-metal centres, there are in principle nN possible orbital/distortion-state configurations and a bulk configurational entropy of Rln(n). Yet because the state of any one centre will usually influence those of its neighbours, these individual degrees of freedom are often strongly coupled. If this coupling is sufficiently strong then the entropy term can vanish altogether. In such a case, the resulting ensemble of states must exhibit long-range periodicity and the system is said to show orbital order . This is the case in our examplar KCuF3 and in most Mn3+-containing ceramics. By contrast, orbital disorder occurs whenever there is no long-range periodicity amongst the orbital/distortion states; in other words, some configurational entropy is retained.

Why on earth does any of this matter? The key is that d-orbital occupation mediates both the (super)exchange pathways responsible for cooperative magnetism and also the overlap responsible for electronic conductivity. So in phenomena such as CMR—where the application of a magnetic field results in a transition from insulating to metallic behaviour—the presence or absence of patterns in the occupation of orbital states is a crucial microscopic ingredient.

When writing my first-ever EPSRC grant, it struck me that one thing missing from the field was an experiment-driven microscopic picture of what an orbital-disorder state actually looked like. This felt like the sort of thing that a combination of total scattering and reverse Monte Carlo (RMC) should be particularly good at sorting out, not because total scattering is at all directly sensitive to orbital occupation, but because it should pick out both the local orbital-occupation-driven distortions and also the correlations between these distortions. Once we know the distortions, we can work back to the orbital occupations. While leaders in the field such as Thomas Proffen and Simon Billinge had already used total scattering to look at local distortions in key orbital disorder phases such as high-temperature LaMnO3, the emphasis had been on the persistence of the distortions themselves rather than their longer-range correlations. In fact Thomas and I had talked offline about the idea of using LaMnO3 as a case study with which to compare-and-contrast the small-box and big-box modelling approaches for interpreting total scattering data. We always expected to end up at the same basic description, with two slightly different but complementary viewpoints.

So when Ed Beake joined my fledgling research group in 2010 and asked for a computational MChem project, it felt reasonable to suggest he carry out some RMC refinements of LaMnO3 total scattering data collected across its thermally-driven orbital order–disorder transition. Thomas had kindly given us access to his previously-published NPDF data so we could be sure that any comparisons against the earlier PDF studies were entirely meaningful. Ed did a great job of all the refinements, but to our surprise his RMC configurations for the orbital disorder phase showed a fundamentally different type of local distortion to the one we expected.

The Q2 distortion in LaMnO3 means that the six Mn–O bonds around each Mn centre fall into three pairs: two ‘short’, two ‘medium’, and two ‘long’ bonds. In the room-temperature orbital order phase, the Mn–O bonds of each pair are opposite one another. This arrangement is known from the crystal structure, and is consistent with all the symmetry analysis given above. Ed also saw this same arrangement within his RMC configurations for this ordered phase. So far, so good. But at higher temperatures, where the long-range orbital order vanishes, Ed’s configurations consistently showed a different arrangement of the same set of bonds, with the two long bonds adjacent to one another. Within the group we referred to these states as trans and cis, drawing on conventional coordination chemistry nomenclature. Not only was this cis configuration unexpected, it didn’t really make sense from a symmetry perspective.

Over the three years or so that followed we did every possible check we could think of. Callum Young, a DPhil student in the group, picked up on Ed’s work. He made a new sample of LaMnO3, collected a new high quality set of neutron total scattering data, and supplemented this with new X-ray total scattering data. He fitted PDFs and reciprocal space data and Bragg profiles. Each time our RMC analysis came back with the same result. Moreover, things seemed to get even more exciting: not only did the cis distortion recur amongst our RMC refinements as we improved the number quality of the data sets, but we seemed to find evidence of a domain structure. Dave Keen invested a huge amount of time in checking the effects of data correction (no sensible changes made any difference to our RMC results). Over time we found ways of rationalising the cis distortion: it had been seen in one or two other Mn3+-containing compounds; it also breaks the degeneracy of the Eg Mn3+ configuration via an indirect mechanism; the application of symmetry arguments is never clear in disordered states anyway; the configurational entropy is much larger than for the transdistortion; and there were a host of unusual observations about LaMnO3 buried in the literature that it might well have explained.

In truth, I wanted to believe the result. The idea that a system might switch between two different types of Jahn-Teller distortions was immensely intellectually appealing, and reminded me strongly of the phenomenology of spin-transition compounds (a topic close to home). All we would need would be for one distortion type to be enthalpically favoured (this would be the Q2distortion in our LaMnO3 model, cf. the low-spin state in spin crossover) and the other entropically favoured (the cis distortion, cf. the high-spin state). I was also heavily influenced by the really beautiful work of my friends Greg Halder and Karena Chapman at Argonne/APS who had shown how hydrostatic pressure could manipulate Jahn-Teller distortions in a Cu2+ system. If a pΔV term can switch distortion types, then why not TΔS? The trans/cis geometries provided a very natural ΔS argument: there are four times as many cis distortions than trans. We wrote the work up for Science, and I started presenting our results (prematurely!) at conferences—including in an embarrassingly poor showing at the ICNS meeting in Edinburgh that still makes me cringe.

The reviews were pretty merciless. LaMnO3 was such a well-studied system. Any shift in conceptual understanding was going to be met with resistance—something I found intensely irritating at the time, but now an aspect of scientific method I have come to value thoroughly. How much of what we were seeing was really dynamical in origin? Could there be hidden inconsistencies between our X-ray and neutron experiments? Had we tested all possible models? What of the known uniqueness problems of RMC? How meaningful were our error bars? How could we really be sure? The paper trickled down the usual list of very-high-then-not-quite-so-very-high-then-slightly-lower-again-impact journals, gnawing away at our collective enthusiasm and patience in the process. To my shame, and in part out of a sense of duty to my students, my focus was on getting the paper published rather than really listening to the message of the referees. What is perhaps worse is that, as the months went on, I learned how to argue better and even got to the point with Nature Communications that with a little more effort the manuscript may well have been accepted (though obviously I do not pretend to know for sure).

By this stage Peter Thygesen had picked up where Callum had left off, and we decided to focus on checking our results without resort to RMC. In part this decision was motivated by suggestions from the referees, and in part because we were starting to understand better the limitations of RMC through other projects underway in the group. Peter’s approach was to develop small-box models that we could test directly against the X-ray and neutron PDFs. Because this type of modelling was much simpler than RMC we could compare a large number of different local models of orbital order / distortion types, while also keeping the number of refinement parameters constant, or roughly so. We were learning how to use symmetry to our advantage, and in doing so the differences we’d noticed using RMC started to disappear. Our various local models—despite the enormously meaningful physical differences (e.g. is local inversion symmetry broken or not?)—really gave very similar fits to data. The parameters that described static distortions correlated strongly with those describing thermal motion. It wasn’t just that RMC had a uniqueness problem, it was that (for this particular system) PDF had a uniqueness problem. It became extremely difficult to see how we might argue that any one model was favoured exclusively by the experimental data. In the absence of this sensitivity, Bayes would advise us to take into account model likelihood. And, however intellectually attractive it might have been, the cis distortion just wasn’t very likely. This was essentially the argument being made by the referees. It was time to listen.

So we took a break from LaMnO3 and focussed instead on Y2Mo2O7, an ostensibly unrelated compound that Joe Paddison had been working on with Chris Wiebe and his group. The key scientific question regarding Y2Mo2O7concerns its magnetism: it behaves like a canonical spin glass, but unlike all other known spin glasses there is no obvious mechanism for generating the requisite disorder in magnetic interactions.

The natural explanation has always been off-centering of the Mo4+ cations, which occupy the vertices of a pyrochlore lattice. Crystallographically there is a clear elongation of the Mo anisotropic displacement tensor that persists even at very low temperatures, where thermal effects are essentially non-existant. EXAFS measurements also showed substantial disorder in Mo–O bond lengths. Yet an earlier neutron PDF study carried out in Simon Billinge’s group ruled out any splitting of the Mo–Mo peak in the PDF, which was taken as evidence against Mo displacements.

Are Mo displacements at all expected from symmetry arguments? The electronic configuration of Mo4+ (d2) at its site in the pyrochlore lattice (D3d point symmetry) is certainly degenerate, and again one might expect an Eg-type distortion to occur. Yet in Mo4+ systems there is the additional complication of Peierls-type distortions, where orbital overlap between neighbouring Mo sites drives spin dimerisation and loss of inversion symmetry. In this case, it is the partially-occupied eg orbitals that point directly along the Mo–Mo vectors, such that even for complete dimerisation the system would remain magnetic (unpaired a1gspins).

So we made a new sample of Y2Mo2O7 (with Mike Hayward‘s help), collected some new high-quality neutron and X-ray PDF data, and then Peter used his small-box approach to test the sensitivity of these data to different local distortion models. Lo and behold, a model containing Mo off centering was now noticeably better at fitting the PDF data. Moreover, this model did an excellent job of explaining the spin glass state. Each Mo dimerises with one—and only one—of its six neighbours. These “orbital dimers” uniformly decorate the pyrochlore lattice but because there is a macroscopic degeneracy of possible decorations, the system remains disordered to low temperatures. The disorder in orbital dimer orientations leads to disorder in Mo–Mo separations, which in turn leads to disorder in the strength of magnetic interactions.

Using what we had learned from studying Y2Mo2O7, we were able to return to the problem of LaMnO3 and provide some tentative statistical basis in support of the so-called anisotropic 3-state Potts model for its orbital disorder state. But what emerged very clearly in these two studies is that the sensitivity of PDF techniques to local structure is heavily system dependent. The big difference between the two problems is the temperature scale involved. In the case of LaMnO3, orbital disorder emerges at such high temperatures (750 K) that disentangling thermal and static contributions to the PDF is all but impossible. This situation of Y2Mo2O7 is entirely different, because off-centering persists to the very lowest measurable temperatures (< 10 K). This difference is something I’d missed in my EPSRC grant proposal, and something I’d been unwilling to accept when faced with the various rounds of negative referee reports. The fact is that PDF analysis is sometimes less effective than I might like, and I personally don’t think it hurts our community for me to admit this. Part of the motivation for going into all the detail that I’ve given here is to make clear the fallibility of the approaches I’ve taken in the past, and also to share the ways in which we’ve tried to learn from our mistakes.

As a final aside, it’s worth mentioning that all the hours spent thinking about the relationship between local distortion symmetry and configurational degeneracy in the RMC-based cis and trans distortion models for LaMnO3 were far from wasted. As part of the process I had developed some simple models for “square ices” that found their way into the review I wrote with Dave Keen for Nature. Likewise these same ideas laid the ground work for Ali Overy’s geometric enumeration of “procrystalline” states that we later published in Nature Communications. And, who knows, maybe one of these days we’ll discover an orbital-transition phase after all!

Orbital Dimer Model for Spin-Glass State in Y2Mo2O7
P M M Thygesen, J A M Paddison, R Zhang, K A Beyer, K W Chapman, H Y Playford, M G Tucker, D A Keen, M A Hayward, and A L Goodwin
Physical Review Letters 118, 067201 (2017)

Local structure study of the orbital order/disorder transition in LaMnO3
P M M Thygesen, C A Young, E O R Beake, F Denis Romero, L D Connor, T E Proffen, A E Phillips, M G Tucker, M A Hayward, D A Keen, and A L Goodwin
Physical Review B 95, 174107 (2017)