Strong correlations and thermal expansion in iron based superconductors

There is a nice preprint
Strong Correlations, Strong Coupling and s-wave Superconductivity in Hole-doped BaFe2As2 Single Crystals 
F. Hardy, A. E. Böhmer, L. de' Medici, M. Capone, G. Giovannetti, R. Eder, L. Wang, M. He, T. Wolf, P. Schweiss, R. Heid, A. Herbig, P. Adelmann, R. A. Fisher, C. Meingast

The figures below summarise some of the key physics. The top is the phase diagram.

The bottom shows the specific heat coefficient gamma as a function of alkali metal content (Cs to Rb to K, and then fractional K content (doping x).

Note that

a. The black curve shows values calculated from density functional theory (DFT) based calculations. The blue points are experimental data, which are as much as an order of magnitude larger, reflecting strong correlations.

b. As one goes K to Rb to Cs the correlations are enhanced, somehow reflecting the "negative pressure" associated with the increasing ion size.

c. The experimental trend is captured nicely by calculations using slave spins (SS) to treat the relevant multi-band Hubbard model with Hund's rule coupling and band structure from DFT.

The thermal expansion alpha is particularly interesting because it is dominated by electronic effects (unlike in most metals) and shows a coherent-incoherent crossover from a Fermi liquid (where alpha/T is constant) to a bad metal at a temperature T*.
As one goes K to Rb to Cs alpha/T is enhanced reflecting the increased correlations.

One reason I am particularly interested in the manifestation of strong correlations in the thermal expansion because this also occurs in organic charge transfer salts, as discussed at length in a recent paper I published with Jure Kokalj. But, we did struggle to obtain a detailed quantitative description of the experiments, partly because of the crystallographic complexity.
It would be nice to see if a DMFT + LDA treatment of the relevant model for these iron compounds could describe the data above.

I thank Christoph Meingast for bringing this work to my attention and helpful discussions about it.

What does the Hubbard model miss?

How is a Hubbard model related to Density Functional Theory?

Jure Kokalj and I recently wrote a paper where considered the effect of strong correlations on thermal expansion, all within the framework of a Hubbard model. This is mostly concerned with explaining anomalies in organic charge transfer salts at temperatures less than 100 K, i.e. much less than the Fermi energy.

One referee stated

“However conceptually this Hamiltonian can not capture the free energy of the relevant electrons loyally. Recall the total energy decomposition in density functional theory, the Hamiltonian corresponds only to the band energy part (which is a summation of occupied Khon-Sham states and different from the kinetic energy) plus interaction term. And the remaining Hartree part, exchange-correlated part and also ionic part, which depend on the lattice constants, are totally ignored. It is not known whether the contributions from such terms are trivial or monotonic especially when strong correlation is present. The neglect of such terms in the electronic model in use is not justified. In this sense, even though the parameters are taken from first principles estimations, it is not surprising that the results are not consistent with experimental data quantitatively and sometimes even qualitatively."

There are some subtle issues here that I would like to understand.
I am not sure I fully understand the referee's comments.
And, I am not sure I agree.

1. Do I understand that the referee is suggesting that the Hubbard model does not include the effects contained in the Hartree and exchange correlation term? Surely, this is not correct.

2. I agree that the Hubbard model will be missing all effects associated with core electrons and ionic terms. However, surely any effects associated with these will not vary significantly on energy and temperature scales of the order of 100 K?

I welcome any comments and insight.

Enhancement of thermal expansion by strong electronic correlations

Jure Kokalj and I just finished a paper
Enhancement of the thermal expansion of organic charge transfer salts by strong electronic correlations

Our main results concerning the electronic contribution to the thermal expansion alpha are as follows.

(i) At low temperatures strong correlations can increase the thermal expansion by as much as an order of magnitude.

(ii) A non-monotonic temperature dependence of alpha is possible.

(iii) Significant orientational dependence is possible, including the expansion having the opposite sign in different directions.

(iv) In the metallic phase the crossover from a Fermi liquid to a bad metal may be reflected in a maximum in the temperature dependence of alpha.

(v) In the Mott insulating phase a maximum in the temperature dependence of alpha can occur, at a temperature comparable to that at which a maximum also occurs in the specific heat and the magnetic susceptibility.

(vi) All of the above results are sensitive to the proximity to the Mott metal-insulator transition and the amount of frustration, reflected in the parameter values (U/t and t'/t) in the Hubbard model.

Although, we can describe many of the unusual qualitative features of experimental data for organic charge transfer salts, the overall magnitude of the thermal expansion coefficients that we calculate are
up to an order of magnitude smaller than observed. This disagreement may arise from uncertainties in how uniaxial stress changes the Hubbard model parameters, and uncertainty in the compressibilities
including not taking into account the effect of softening of the lattice associated with proximity to the Mott transition.

We welcome any comments.

Deconstructing iridates and many-body time scales

Iridates such as Sr2IrO4 have attracted considerable attention because they are 5d systems that exhibit a strong interplay between spin-orbit coupling and strong electronic correlations.
[See this earlier post].

Sr2IrO4 is a focus because it has been argued that it is a J=1/2 Mott insulator, just like La2CuO4, the parent compound for cuprate superconductors.
A current holy grail is to dope this material in the hope of producing high-Tc superconductivity. Many are trying. No one is succeeding.

There is actually a whole series of layered compounds, the Ruddlesden-Popper perovskites that differ, not just in their stoichiometry, but also their crystal structure, and consequently how the Iridium ions are coupled together. Sr_n+1Ir_nO_3n+1, where n is the number of SrIrO3 perovskite layers sandwiched between extra SrO layers.

Resonant-Inelastic-X-ray-Scattering (RIXS) experiments show that spin excitations in the Mott insulating phase of Sr2IrO4 appear to be well described by a spin-1/2 Heisenberg model with a small amount of spin anisotropy due to crystal field effects. However, for Sr3Ir2O7, RIXS suggests a large spin gap and spin anisotropy.

However, a different experiment suggests a small anisotropy.
In contrast, SrIO3 is a metal.

A major theoretical challenge is describe this whole family of materials and the disparate results, starting just from the crystal structures.
This has been done in an impressive paper,

Effective J=1/2 insulating state in Ruddlesden-Popper iridates: An LDA+DMFT study
Hongbin Zhang, Kristjan Haule, and David Vanderbilt

The calculations are based on GGA+DMFT, and represent another landmark achievement for the combination of Dynamical Mean-Field Theory with Density Functional Theory methods.

A key ingredient to understanding the apparent inconsistency between the results of different experimental probes is that in the many-body treatment the matrix describing the hybridisation of the three d-orbitals is frequency dependent. This is in contrast to the static matrix associated with crystal field theory.

The X-ray and thermodynamic experiments probe the system on different time scales. Thus they are respectively, more sensitive to the high- and low-frequency part of the hybridisation matrix.

The Figure below shows the calculated optical conductivity for the first three compounds in the RP series.

They also show how 0.2 per cent epitaxial lattice strain can have a big effect. [Aside: These are the kind of calculations I would like to see to address thermal expansion in organic charge transfer salts.]

I thank Kristjan Haule for explaining this work to me.

Pure plutonium is a strongly correlated metal

I often contrast elemental metals to strongly correlated electron materials such as cuprates, organic charge transfer salts, and heavy fermion compounds. However, this is not strictly correct. Some of the lanthanide and actinide elements are strongly correlated metals. This is most clearly demonstrated in the case of cerium and plutonium which undergo isostructural phase transitions involving large volume increases.

This is particularly nicely illustrated in the figure below, taken from a beautiful 2001 Nature news and views by Bob Albers, An expanding view of plutonium. Electronic structure methods based on Density Functional Theory (DFT) completely fail here.

A nice explanation of the figure was given by Savrasov, Kotliar, and Abrahams, in terms of strong electronic correlations that can be captured by Dynamical Mean-Field Theory (DMFT). In particular the expansion from the alpha to the delta phase is associated with a delocalised-localised transition of the f electrons. The lighter actinides have more delocalised f electrons leading to stronger chemical bonding and smaller volumes per atom in the crystal.

The strong correlations are reflected in other properties of plutonium.
This is highlighted in a nice review Plutonium condensed matter physics by Michael Boring and Jim Smith, which contrasts Pu to lighter elemental metals and heavy fermion compounds.

The figure below shows the temperature dependence of the thermal expansion of Pu and Iron. Note the magnitude of the slope is much greater for Pu.

The resistivity of plutonium is compared to potassium (K) and the heavy fermion compound UBe13. The resistivity for Pu is non-monotonic, saturating at a value comparable to the Mott limit, characteristic of a bad metal.

A challenge for the theory of organic superconductors

A straight-forward measurement for any superconductor is how the transition temperature varies with pressure (and thus volume). Calculating this variation is not easy. For example, in a simple BCS superconductor one would have to calculate how the phonon spectrum, electron phonon-coupling constant, and density of states vary with pressure. Presumably this can be done with some electronic structure method such as based on density functional theory (DFT).
But, how about in a strongly correlated electron system?  One would have to first find how the parameters in some Hubbard type model varied with pressure. Then one has the extremely tricky issue of calculating Tc as a function of the Hubbard model parameters.

It turns out that the volume dependence of Tc in organics is dramatically larger than in cuprates and elemental superconductors.

I was recently drawn to the paragraph below from this paper by a football team (11 authors!)
Comparative thermal-expansion study of β″-(ET)2SF5CH2CF2SO3 and 
κ-(ET)2Cu(NCS)2: Uniaxial pressure coefficients of Tc and upper critical fields

Note that simple models (e.g., Sommerfeld, Debye) predict that energy scales such as the Fermi energy and Debye temperature scale with the volume according to some exponent of order one. Hence, this exponent of 40 for the organics Tc is amazing!

I disagree somewhat with the conclusion of the last sentence of the paragraph: that this extreme sensitivity "underlies the role of the lattice degrees of freedom for the superconducting instability for this class of materials." I don't think this shows that electron-phonon coupling is involved directly in superconductivity. It could be just that Tc is quite sensitive to the Hubbard model parameters. Small variations in the latter do drive the system closer or further from the Mott transition.

Yet, the challenge remains to calculate this volume sensitivity from theory!
A very modest but useful first step would be to first see if a DFT-based calculation can reproduce the measured bulk compressibility.

Thermal expansion in heavy fermion compounds

Measuring the thermal expansion of a crystal sounds like a really boring measurement and not likely to yield anything of dramatic interest. This may be true for simple materials. However, for strongly correlated electron materials it reveals some interesting and poorly explained physics.

First, it is amazing that using fancy techniques based on capacitors one can measure changes in lattice constants of less than one part per million!

Second, there is some interesting thermodynamics that follows from the Maxwell relations. The isotropic thermal expansion is related to the variation in the entropy with pressure

Hence, in a Fermi liquid metal the thermal expansion versus temperature should have a linear temperature dependence at low temperatures.

Below, I show the temperature dependence of the thermal expansion (along two different crystal directions) for the heavy fermion compound CeRu2Si2, reported in a PRB article. The lower two curves are from the structural analogue LaRu2Si2 which does not have a contribution from 4f electrons.

The peak in the cerium compound is arguably associated with the formation of a coherent Fermi liquid below a coherence temperature of about 10 Kelvin. Below that temperature the thermal expansion is approximately linear in temperature.

How big is the effect? One way to quantify it is terms of the Gruneisen parameter [see Ashcroft and Mermin, page 493].

where T_i is the characteristic temperature scale of the entropy [here the Kondo or coherence temperature]. For heavy fermion compounds Gamma is two orders of magnitude larger than for elemental metals or the values of order unity, typically associated with phonons in simple crystals. This implies an extremely strong dependence of the characteristic temperature on volume. As far as I am aware there is no adequate theory of these large magnitudes.

There is a recent preprint which reports the temperature dependence of the thermal expansion in the iron pnictide KFe2As2 and relates it to a coherent-incoherent crossover like that discussed above.

Spin nematic fluctuations and elastic anomalies.

Previously I posted about some fascinating experimental results on anisotropic thermal expansion and elastic softening near superconducting and magnetic transitions in organic charge transfer salts.
Subsequently, I became aware that the new iron pnictide superconductors do exhibit somewhat similar phenomena. A combined theory-experimental PRL (10 co-authors!) describes shear acoustic mode spectroscopy in terms of nematic spin fluctuations.

They find in undoped BaFe2As2 that the shear modulus (C66) softens significantly as one approaches the magnetically ordered phase (which is a associated with a tetragonal-orthorhombic lattice distortion). For the optimally doped material there is a hardening of the lattice as one enters the superconducting phase.

The figure above explains the nematic order parameter and how it couples to shear lattice distortions. A key is the that the magnetic phase consists of Neel antiferromagnetic order on two separate sublattices . They are weakly coupled together and the nematic order parameter phi equals the dot product of m1 and m2, the antiferromagnetic order parameters on the two separate lattices. phi then couples directly to the shear strain.

The softening into the superconducting (SC) phase is explained by a coupling of the SC and AFM order parameters. This leads to a change in the static spin susceptibility upon entering the SC phase. This in turn effects the fluctuations in the nematic order parameter.

A couple of comments:

a. Experimental data is presented just for C66. It would be helpful to see it for longitudinal sound and for other transverse modes besides epsilon_s=epsilon_xy. These modes should not have significant coupling to superconductivity and magnetism if the nematic mode is where all the action is.

b. In other antiferromagnets lattice anomalies at magnetic transitions are explained in terms of the spin anisotropy (e.g. due to the Dzyaloshinsky-Moriya interaction) coupling to the different components of the stain tensor. This is reviewed here by Lines. Can that be ruled out in the pnictides?

A sound theory needed

The past few weeks I have been puzzling through the implications of some really nice experimental results on superconductivity in organic charge transfer salts. A group at Sherbrooke measured the speed of sound as a function of temperature for different polarisations. The Figure below, taken from their PRB, shows how anisotropic the elastic response is for the material κ-(BEDT-TTF)2Cu[N(CN)2]Br .

The sound is always propagating perpendicular to the layers, which lies in direction of the b axis of the crystal. C22 is the elastic constant for longitudinal sound. C44 has polarisation parallel to the c direction in the crystal which is the same direction as the t' (diagonal) hopping in the relevant Hubbard model [see picture below and this review]. C66 has polarisation in the a direction.

A few thoughts 

• A really helpful succinct summary of elasticity theory and sound velocity anomalies at Tc is found in Section II of this PRB, also from Sherbrooke [I believe there is an important typo in the expression for the sound velocity in terms of the strain tensor, just below equation (5) the j and k indices on the elastic tensor need to be interchanged.]
• These variations in the sound velocity near Tc by about 0.1% may seem small, but they are actually several orders of magnitude larger than in other unconventional superconductors. The authors point this out.
• There are fluctuations which extend to temperatures far above Tc. This is consistent with Nernst effect measurements, on the same materials, reported in Nature.
• The authors perform an elegant group theoretical argument that forces them to conclude that the anomaly in C66 and other experimental results (STM and thermal conductivity) require that the superconducting order parameter for this crystal must be mixed A_1g+B_1g 
• I am not that convinced by the STM and thermal conductivity experiments that claim to determine the locations of the nodes in the energy gap, because these experiments are surface sensitive.
• The proposed mixed symmetries are quite inconsistent with many microscopic calculations which predict the order parameter will have B_2g  symmetry, which competes with A_1g  near when t'~t and the lattice becomes that of the isotropic triangular lattice, as discussed here. This inconsistency is an important issue that needs to be resolved.
• The anisotropic response is somewhat reminiscent of the anisotropy in the thermal expansion near Tc (as shown below) found by Michael Lang's group [see this PRB].
• Similar anisotropies and variations in the sound velocity and thermal expansion are also seen near the Mott transition and the crossover from a Fermi liquid to a bad metal.