Observation of renormalised quasi-particle excitations

A central concept of quantum-many body theory is that of coherent quasi-particles. Their key property is a well-defined relationship between energy and momentum (dispersion relation). Prior to the rise of ARPES (Angle-Resolved Photo-Emission Spectroscopy) over the past three decades, the existence of electronic quasi-particles was only inferred indirectly.

A very nice paper just appeared which shows a new way of measuring quasi-particle excitations in a
strongly correlated electron system. Furthermore, the experimental results are compared quantitatively to state-of-the-art theory, showing several subtle many-body effects.

Coherent band excitations in CePd3: A comparison of neutron scattering and ab initio theory 
Eugene A. Goremychkin, Hyowon Park, Raymond Osborn, Stephan Rosenkranz, John-Paul Castellan, Victor R. Fanelli, Andrew D. Christianson, Matthew B. Stone, Eric D. Bauer, Kenneth J. McClellan, Darrin D. Byler, Jon M. Lawrence

The mixed valence compound studied is of particular interest because with increasing temperature it exhibits a crossover from a Fermi liquid with coherent quasi-particle excitations to incoherent excitations, an example of a bad metal.

The figure below shows a colour intensity plot of the dynamical magnetic susceptibility

at a fixed energy omega, and a function of the wavevector Q. The top three panels are from the calculations of DFT+DMFT (Density Functional Theory + Dynamical Mean-Field Theory).

The bottom three panels are the corresponding results from inelastic neutron scattering.
A and B [D and E] are both at omega=35 meV and in two different momentum planes. C [F] is at omega=55 meV.
The crucial signal of coherence (i.e. dispersive quasi-particles) is that the shift of the maxima between the G and R points at 35 meV to the M and X points at 55 meV.

It should be stressed that these dispersing excitations are not due to single (charged) quasi-particles, but rather spin excitations which are particle-hole excitations.

The figure below shows how the dispersion [coherence] disappears as the temperature is increased from 6 K (top) to 300 K (bottom). The solid lines are theoretical curves.

The figure below shows that the irreducible vertex corrections associated with the particle-hole are crucial to the quantitative agreement of theory and experiment. The top (bottom) panel in the figure below shows the calculation at low (high) temperatures. The black (blue) curves are with (without) vertex corrections. The red curves are a rescaling of the blue curves by a numerical factor.

The correction has two effects: First, it smooths out some of the fine structure in the energy dependence of the spectra while broadly preserving both the Q variation and the overall energy scale; and second, it produces a strong enhancement of the intensity that is both energy and temperature dependent, for example, by a factor of ~6.5 at w = 60 meV at 100 K. This shows that the Q dependence of the scattering is predomi- nantly determined by the one-electron joint density of states, as expected for band transitions, whereas the overall intensity is amplified by the strong electron correlations. 

This landmark study is only possible due to recent parallel advances in theory, computation, and experiment. 
On the theory side, it is not just DMFT but also including particle-hole interactions in DMFT.
On computation, it is new DMFT algorithms and increasing computer speed. 
On the experimental side, it is pulsed neutron sources, and improvements in the sensitivity and spatial and energy resolution of neutron detectors.

Strongly correlated toplogical insulators

At the Journal Club for Condensed Matter Chandra Varma has a helpful commentary on
recent experimental papers reporting evidence that the mixed valence compound SmB6 is a topological insulator (see my earlier post and a recent Nature News article).

A few things I learnt from Varma's commentary.

The strongly correlated properties are not central to the topological properties. These are rather a property of the effective band structure which arises in a slave boson treatment or the Varma-Yafet variational wavefunction.

The Fu-Kane conditions for a topological insulator most likely hold in the mixed valence limit which is at the extreme of particle-hole asymmetry. In this limit the Kondo temperature is of the order of the hybridisation energy, in contrast to the Kondo limit when it is an order of magnitude smaller.

Some caution is in order because the existence of actual surface states [e.g. from ARPES] have not yet been definitively established.

Definitive signatures of the strongly correlated state might be seen in new low energy resonances that could arise from non-magnetic impurity substitution.

p.s. On his website Piers Coleman has a nice talk giving the background theory which preceded the experiments.

Topological insulators get more interesting

Topological insulators (TIs) are certainly a hot topic. However, there are two things that might make one nervous about all the excitement.

1. All the materials being studied as TIs [e.g. Bi2Se3] actually aren't TIs.
What!? A TI is by definition a bulk insulator with surface metallic states that are topologically protected. However, the actual materials turn out not to be bulk insulators. On a practical level this makes separating out bulk and surface contributions, particularly in transport measurements, tricky. But, also presents an ideological problem: one is not actually studying the phase of matter one wishes one was studying.

2. One could argue that TIs are "just a band structure effect", i.e., they do not involve any quantum many-body physics.

However, these objections are put to rest by a preprint
Discovery of the First True Three-Dimensional Topological Insulator: Samarium Hexaboride
Steven Wolgast, Cagliyan Kurdak, Kai Sun, J. W. Allen, Dae-Jeong Kim, Zachary Fisk

They report electrical transport measurements that show that SmB6 is a bulk insulator with surface metallic states.
This is of particular interest for several reasons

a. The material really is a true topological insulator.
b. The material is a Kondo insulator. [Although strictly the material is in the mixed valence rather than the local moment regime.] The insulating state emerges from strong electronic correlations.
c.  This resolves long standing puzzles about previous transport measurements on this material which did not show activated conductivity at low temperatures. This can now be explained as a sample dependent contribution from metallic surface states.
d. This material was predicted to be a topological Kondo insulator by Dzero, Sun, Coleman, and Galitski.

I also note a recent paper Actinide Topological Insulator Materials with Strong Interaction.

I thank Tony Wright for bringing the preprint to my attention.

Deconstructing solid oxide fuel cell materials

In Seattle, I had a really interesting and helpful discussion with Charlie Campbell about doped rare earth oxides.

Cerium oxides have attracted a lot of industrial attention because they have an amazing ability to reversibly release and uptake oxygen. [Just like hemoglobin in your blood!]. Hence, along with many others I thought this was a fundamental issue about pure cerium oxide. However, it turns out all the industrial materials (such as solid oxide fuel cells) are doped with transition metal ions. So the fundamental problem is the following: mixed alloys of ceria and zirconia (ZrO2) have this large uptake-release capacity; it is much larger than pure zirconia or pure ceria.

This paper [which my Indian colleagues made me aware of when I visited Bangalore earlier this year] examines the corresponding question for titania-ceria alloys. [A paper on zirconia-ceria is here.] They find that in the alloys there is a significant relaxation of the oxygen sublattice. In particular four of the metal-oxygen bonds become much longer, reflecting weak bonding of oxygen.

I wonder whether

-thinking about a Jahn-Teller distortion could be helpful here?

-there are high resolution crystal structure data that is amenable to the bond valence sum analysis similar to that performed here.

Where does the excess charge go?

Here is the current version of the slides for a talk, "Charge redistribution near oxygen vacancies in cerium oxides", that I am giving tomorrow in the Chemistry Department at University of Washington.

The main point of the talk is that the standard model of charge localisation (pictured above) is incorrect. A detailed discussion is contained in a review co-authored with Elvis Shoko and Michael Smith.

Are you local or non-local?

I wrote in a previous post about the importance of listening to referees. I recently got back a referee report for this review on oxygen vacancies in cerium oxides (written with Elvis Shoko and Michael Smith). The report ended:

Finally, I cannot avoid suggesting the authors to have a look at: “A Conversation on VB vs MO Theory: A Never-Ending Rivalry? Roald Hoffmann, Sason Shaik, Philippe C. Hiberty. Accounts of Chemical Research 2003 36 (10), 750-756”. Perhaps, they will hear some familiar tones.

I read and enjoyed the paper [inspiring the post Marriage Counseling for Chemists] and am now trying to make concrete the connection with our work.

By a bond valence sum analysis of the structure around oxygen vacancies we consider the charge distribution arising from the two electrons left behind by removing an oxygen atom. We find rather subtle charge distributions; the two electrons do not simply localise on the two Ce ions next to the vacancy [the standard picture which is either assumed or claimed ot be supported by density functional theory based calculations]. Instead the two electrons can delocalise over the next nearest neighbours, but do not delocalise into the whole crystal.

At first the connection with the VB vs. MO debate was not clear but on reflection there may be some profound ones such as:

• Valence Bond (VB) theory tends to localise electrons too much. Molecular Orbital (MO) theory tends delocalise electrons too much.
• Our empirical valence bond sum approach is a very local picture and somehow capturing the same physics/chemistry as VB theory.
• LDA is close to MO theory (it is a band theory, i.e., a non-local picture) and tends to delocalise electrons too much. Many of the LDA, and LDA+U calculations on cerium oxides artificially force electrons to localise on cerium ions.

Can we make any more connection than the above?

Examples of inhomogeneous mixed valence

In contrast, to homogeneous mixed-valence the inhomogeneous mixed-valence case involves a mixture of different integer valence ions which occupy inequivalent lattice sites in a static charge-ordered array. Examples of this are provided by Fe3O4, Eu3O4 and Eu3S4.

The following material and figure is copied from the chemexplore web site

Magnetite (Fe3O4) has the AM₂X₄ spinel structure, of the "inverse" type :

Magnetite has the empirical formula Fe₃O₄, or Fe²⁺(Fe³⁺O₂)₂, “ferrous ferrite”. Its formula as a spinel would be Fe³⁺_tetFe²⁺_octFe³⁺_octO₄ , where "tet" and "oct" stand for tetrahedral and octahedral coordinations by the oxide anions. In the above model, the blue spheres represent the tetrahedral iron(III) cations , and the red spheres are the octahedrally coordinated iron(II) and (III) cations. The oxide anions are shown as the green spheres. Because of the fortuitous inverse nature of the magnetite structure, ferrous and ferric cations are both in the similar octahedral coordination by oxides. In "normal" spinels, such as the mineral spinel itself (magnesium aluminate), the A cation is tetrahedral and the M cations are both octahedral:

However, this inverse-spinel charge ordering has recently been challenged in favour of the normal spinel charge structure where the Fe3+ ions exclusively occupy all the octahedral sites while the Fe2+ ions reside in the tetrahedral sites.

What happens in higher order oxides of cerium is not so simple, as discussed here.

Mixed valence: physicists vs. chemists II

In compounds which Varma characterized by homogeneous mixed-valence, each ion is assigned (at least by physicists) the same, non-integer, valence which is a result of a quantum mechanical superposition of two integral valences occuring on each ion.

Compounds exhibiting this type of mixed-valence include, CePd3, TmSe, SmB6 where the valences of the ions are 3.45, 2.72 and 3.7 for Ce, Tm, and Sm, respectively. In TmSe, the valence of 2.72 for the Tm ion is a result of valence fluctuations of this ion between the Tm2+ and Tm3+ states.

A distinctive experimental signature of homogeneous mixed valence is that the ground state is a spin singlet (and so has not net magnetic moment) even if one or both of the two oxidation states of the metal ion have a non-zero spin and magnetic moment. This is seen in the temperature dependence of the magnetic susceptibility. At high temperatures it has a Curie form characteristic of a magnetic moment. At low temperatures it saturates to a finite value. It also means there can be an absence of an Electron Spin Resonance (ESR) signal normally associated with metal ions with integer valence.

Other signatures were reviewed by Varma, including inter-ionic distances in the crystal structure that are intermediate between those normally associated with metal ions with integer valence. There can also be large Debye-Waller factors associated with large fluctuations in these distances.

Mixed valence: physicists vs. chemists I

The chemical concept of valence is of great utility. The valence of an atom, M, determines the number of neighbouring atoms with which M can form chemical bonds. For most metal ions, the valence, V, is equal to the oxidation number, O. Deviations from this equality occur when delocalization of electrons occurs. The simplest case is where V and O differ by one because one electron from each of the M atoms is completely delocalized in the conduction band. Mixed valency of transition metal and rare-earth ions in solids and compounds is a question of fundamental interest in materials physics, chemistry, and molecular biophysics.

In 1967, Robin and Day (chemists) published a classification scheme for mixed-valence that is still widely used today. Class 1 describes systems with two crystallographic sites that are clearly distinct and and the two sites have integral but unequal valence. There is a large energy associated with transfer of electrons between sites. At the other extreme is Class 3 for which there are two sites which are not distinguishable, and one assigns a non-integral valence to both sites. The classic case of this is the Creutz-Taube ion. The valence electrons are delocalised between the two sites. Class 2 is the intermediate case where the environments of the two sites are distinguishable but not very different. The energy associated with electron transfer is sufficiently small that it can be thermally activated and be associated with significant optical absorption in the visible range. On the time scale of the vibrations of the atoms the electrons may appear to be delocalised.

Classes 1 and 2 correspond to what Varma (a physicist) termed inhomogeneous mixed valence, although, perhaps, inhomogeneous integral valence may be more appropriate. Class 3 corresponds to homogeneous mixed valence.

more to come.....

A basic question about novel energy materials

A fundamental scientific question of technological importance concerning oxides of transition metals and rare earths is:

When an oxygen atom is removed from a bulk crystal of the oxide where do the two excess electrons go?

Elvis Shoko, Michael Smith, and I recently finished a review article which answers this question for the case of cerium oxide.

The approach we took was to consider high resolution crystal structures of

Ce11O20 and Ce7O12 and see how they could be viewed as ordered arrays of oxygen vacancies in an underlying CeO2 crystal. The charge distribution in the local environments of the O vacancies can then be deduced from the bond valence model.

An important finding we make is that the results are incompatible with the widely accepted standard picture of charge localization on two cerium ions next to the vacancy. Instead, we found that the charge distributes itself predominantly in the second coordination shell of cerium ions. Furthermore, one excess electron can be delocalised over more than one cerium ion.

Our conclusions concerning the charge distribution near oxygen vacancies are significant for several reasons.

First, they contradict many (but not all) atomistic simulations based on density functional theory.

Second, the actual charge distribution around the defect has important implications for the other questions we posed at the beginning of the review. For example,

1. the charge distribution has a significant effect on the relative stability of surface and subsurface vacancies.

2. the charge around oxygen surface and subsurface vacancies is not simply localised on Ce ions next to the vacancy this could change our understanding of the catalytic activity of these surfaces since it has been claimed or assumed that it is associated with Ce3+ ions at the surface.

3. the charge distribution around the vacancies has implications for the relative importance of electronic and ionic conduction, a subject we have discussed in this preprint.

Can strong electronic correlations save the planet?

Leone Spiccia (Monash University) gave a really interesting Chemistry seminar today concerning bioinspired manganese clusters for the photocatalytic oxidation of water. This chemical reaction is a key component of producing hydrogen from water for a hydrogen economy. The associated paper is here. The bioinspired material is centred around a cubic [Mn4O4] cluster. A similar (but not structurally identical) cluster is present in Photosystem II (PSII) and is the only known natural system that is able to oxidize water using visible light. These metal-oxide clusters are particularly interesting from the point of view of a strongly correlated electron system. In both the natural and artificial systems there appears to be still uncertainty about basic questions such as:

What is the oxidation and spin state of each of the four manganese ions at each of the stages of the photocatalytic cycle?

Are the electrons localised or delocalised over the manganese cluster?

Modelling of electron spin resonance experiments on the S2 oxidation state (this is just one of the five charge states the cluster takes during the cycle) in terms of Heisenberg model Hamiltonians has provided some constraints on the geometry, valence states, and magnetic interactions. Two papers I found informative are here and here. The second paper found a d electron configuration quite different to that found in similar synthetic systems.

A beautiful review has considered the interplay of electron magnetic exchange and electron transfer in protein metal complexes in terms of the same type of double exchange Hamiltonians that are relevant to colossal magnetoresistance materials.