A valuable new book on thermoelectricity

Kamran Behnia has published a book Fundamentals of Thermoelectricity

Such a monograph is overdue. I think the topic is particularly important and interesting for several reasons. (This is illustrated by the fact that I have written almost 40 blog posts on the topic).

• The thermoelectric power is a transport property that presents a number of rich and outstanding puzzles.
• The sign, magnitude, spatial anisotropy, and temperature dependence of the thermopower can put significant constraints on theories because the thermopower is so sensitive to particle-hole asymmetry. In comparison, often it may not be too hard to cook up a theory can get a resistivity that agrees with experiment. However, the thermopower is another story.
• Thermoelectric materials are technologically important. Furthermore, if someone can find a material with a "Figure of merit" that is just twice that of the best current materials we could throw out all our refrigerators with moving parts!

The book has a nice preface. Here are a few choice quotes.

To many readers of this book, it should be a surprise to learn that a consistent and unified theory for phonon drag is still missing.... 

Three chapters devoted to a survey of experimental facts aim to revive a number of forgotten puzzles...

But the embarrassment [discussed below] has vanished thanks for our forgetfulness and not to our cleverness....

Even more enigmatic than the positive Seebeck coefficient of noble metals at room temperature is their thermoelectric response at very low temperatures.... 

Before beginning to write this book, I did not know that there is an three-orders -of-magnitude gap between theory and experiment regarding the thermoelectric response of Bogoliubov quasi-particles of a superconductor....

Why such facts have gradually faded from the collective memory of the condensed matter physics community is another question that deserves to be raised but is not addressed by this book.

Section 6.5 "Origin of the Positive Seebeck Coefficient of Noble Metals"
begins with the following quote from Robinson in 1967.

For more than thirty years the absolute thermoelectric power of pure samples of monovalent metals has remained a nagging embarrassment to the theory of the ordinary electronic transport properties of solids. All familiar simple theory has promised us that in these materials the sign of the electron-diffusion contribution to the thermopower should be that of the charge carriers as determined by the Hall effect, i.e. negative; but instead it turns out to be positive for Cu, Ag, Au and—even more perversely—for Li alone of the solid alkalies. At least two generations of experimentalists have remained completely unshaken in testifying to these results as obstinate facts of life.

A great value of the book is that it brings together a diverse set of experimental data from a wide range of materials.

I have a few minor quibbles.

I could find no mention of:

a. the Kelvin formula and the associated nice treatment of it by Michael Peterson and Sriram Shastry.

b. Dynamical Mean-Field Theory (DMFT) and how it nicely describes the thermopower as there is a crossover with increasing temperature from a Fermi liquid to a bad metal.

c. experimental techniques. What are the challenges, problems and obstacles to accurate and reliable measurements?

The caption of Figure 8.5 claims that for an organic charge transfer salt kappa-(BEDT-TTF)2Cu(NCS)2 "The expectations of a tight-binding model is in good agreement with the experimental data". The text says this is a "rare achievement in the case of correlated metals".
However, this "agreement" requires an arbitrary and unjustified rescaling of all the band energies by a factor of about five! This data and the theoretical challenge it presents is discussed in detail here.

The book is written by an experimentalist. I learnt from the back cover that there is also a new book, Modern Theory of Thermoelectricity by Zlatic and Monnier. I am looking forward to reading that.

Kamran Behnia has done a great service to the community by writing the book. Thank you!

Kelvin formula for thermopower in bad metals

Jure Kokalj and I just finished a paper,
Enhancement of the thermoelectric power by electronic correlations in bad metals: a study of the Kelvin formula 

In many strongly correlated electron metals the thermoelectric power has a non-monotonic temperature dependence and values that are orders of magnitude larger than for elemental metals. Kelvin proposed a particularly simple expression for the thermopower in terms of the temperature dependence of the chemical potential. We consider a Hubbard model on an anisotropic triangular lattice at half filling, a minimal effective Hamiltonian for several classes of organic charge transfer salts. The finite temperature Lanczos method is used to calculate the temperature dependence of the thermopower using the Kelvin formula. We find that electronic correlations significantly enhance the magnitude of the thermopower and lead to a non-monotonic temperature dependence. The latter reflects a crossover with increasing temperature from a Fermi liquid to a bad metal. Although, the Kelvin formula gives a semi-quantitative description of some experimental results it cannot describe the directional dependence of the sign of the thermopower in some materials.

Scale of the Nernst effect in a bad metal

A science fiction fantasy is that we should be able to make "materials by design" that have any physical property (density, thermal conductivity, hardness, thermoelectric figure of merit, heat capacity...)  that we desire. However, it seems that there are certain physical constraints that determine the overall scale of many physical properties.

I find it helpful to have a feel for typical orders of magnitude. What is particularly interesting is that sometimes these magnitudes are related to fundamental constants [electronic charge (e), Boltzmann's constant (k_B), Planck's constant (hbar)] and basic length scales such as the lattice constant a of a crystal.

Here are three scales I have emphasised before

Resistivity ~ hbar a / e^2 ~ 100 microohm-cm  which is associated with the Mott-Ioffe-Regel limit.

Thermoelectric power,  S ~ k_B/e ~ 86 microvolt/K

Mobility, mu ~ e a^2/ hbar ~ 1 cm^2 V/sec

One can find these scales by dimensional analysis or by doing things like looking a formulas from transport theory and (assuming a bad metal) setting the mean-free path comparable to the lattice constant. One can debate whether one uses hbar or h, but for little purpose.

How about the Nernst signal, nu?

nu ~ k_B a^2 / hbar ~ 0.01 microV/KT

A few minor notes.

1. One can get this scale from the above expressions for S and mu if one uses the observation that in some strongly correlated materials
nu ~ S * Hall mobility.

2. One Volt/Tesla = m^2/sec  [One can see this easily from F = q(E + vxB)].

3. Given that the Nernst effect involves charge transport I find it surprising that the electronic charge does not appear.

The figure below, taken from a nice review by Behnia, shows that this is the right scale for bad metals such as cuprates, and heavy fermions above the coherence temperature.

One also sees this scale in recent DMFT calculations for a doped Hubbard model (see Figure 2d in this PRL ) and recent measurements (see Figure 4) on organic charge transfer salts.

Large thermal conductivity of correlated semiconductors

Previously I posted about the challenge of understanding the colossal thermoelectric effect in FeSb2 and the puzzles of the classic Kondo insulator FeSi.

I talked about the former yesterday at the cake meeting [take 7 minutes to convince everyone they should read a particular paper]. I noticed for the first time just how large the thermal conductivity is, actually comparable to diamond at low temperatures.

The red curve is FeSb2 and the black curve FeAs2, which is less correlated.

This is of interest for at least two reasons.

1. The large thermal conductivity is bad for thermoelectric applications as it will significantly reduce the thermoelectric figure of merit.

2. It needs to be explained theoretically, including the large difference between FeSb2 and FeAs2. [Presumably the phonons are similar in the two compounds].

3?. Does this make reliable thermoelectric measurements harder or easier?

For comparison below I show the temperature dependence of the  thermal conductivity of diamond and copper, taken from here. [n.b. the vertical scale is different by a factor of 100 compared to the above graph].

I welcome comments.

The challenge of colossal thermoelectric power in FeSb2

There is an interesting paper
Highly dispersive electron relaxation and colossal thermoelectricity in the correlated semiconductor FeSb2
Peijie Sun, Wenhu Xu, Jan M. Tomczak, Gabriel Kotliar, Martin Søndergaard, Bo B. Iversen, and Frank Steglich.

The main results that are a struggle to explain are in the figure below.
The top panel shows the temperature dependence of the thermopower [Seebeck coefficient] of FeSb2 [red] and the isoelectronic FeAs2.
First, notice the vertical scale is tens of mV/K. In an elemental metal the thermopower is less than a microV/K. In a strongly correlated metal it can be tens of microV/K. [see for example this earlier post].
Why is it so large? Why is the Sb compound so much larger than the As compound?
In a simple model of a band semiconductor S ~ k_B/e * gap/k_B T. But here the Sb compound has the smaller gap.
Also, why is there a maximum in the temperature dependence, S(T) going to zero with decreasing temperature.

In an attempt to elucidate these subtle issues the authors have also measured the Nernst effect and the magnetoresistance. The Nernst signal is also colossal, being of the other of mV/KT for FeSb2, which is two orders of magnitude large than that for FeAs2.

The authors also consider a simple analytical model of a semiconductor with an energy dependent scattering rate to see what properties that can explain: some but not all. A strongly energy dependent scattering rate is also needed; this can occur in the case of Kondo physics, for example.
They also find some interesting relations between Seebeck, Nernst, Hall mobility, magnetoresistance, and the thermal mobility.

It is helpful to read the paper in conduction with an experimental review and this earlier theory paper,
Thermopower of correlated semiconductors: Application to FeAs2 and FeSb2
Jan M. Tomczak, K. Haule, T. Miyake, A. Georges, and G. Kotliar

To further complicate all of the above it seems that the results can be quite sample dependent, and the results varying significantly, even by orders of magnitude between different groups. One clue is that it seems that FeSb2 is very close to a metal-insulator transition, seen in some samples but not others…

Much remains to be done...

Universal? properties of thermoelectric power in bad metals

There is a nice preprint Universal thermopower of bad metals
Veljko Zlatic, G.R. Boyd, Jim Freericks

It contains calculations of the temperature and doping dependence of the thermoelectric power for the Falicov-Kimball model within the approximation of Dynamical-Mean Theory [DMFT].

This spinless fermion model is even "simpler" than the Hubbard model. Yet it captures some of the same physics, particularly the Mott metal-insulator transition. It also has the advantage that DMFT has an exact analytical solution. One does not need an "impurity solver", such as for the Hubbard model. There is an extensive Rev. Mod. Phys. on this, by Freericks and Zlatic.

Below I discuss one significant disadvantage of the model.

The figure below shows the calculated temperature dependence of the thermopower for several different dopings. The solid lines are the result from the Kubo formula [essentially exact] and the dashed line is the approximate Kelvin formula [the derivative of the chemical potential with respect to temperature].

Note that both the magnitude [of order k_B/e=80 microVolt/K] and non-monotonic temperature dependence are similar to what one sees in many strongly correlated electron materials. [Compare for example this post about heavy fermion compounds.]

Furthermore, it is striking that the Kelvin formula gives semi-quantitative results that are reliable.

However, when it comes to detailed comparison with experiment on actual materials, it is important to keep in mind a significant shortcoming of the Falicov-Kimball model. It does not seem to have a low-energy coherence scale associated with the formation of Fermi liquid quasi-particles. In many strongly correlated electron materials this energy scale is much less than the bare energy scale t, of the intersite hopping. In the Figure above one can see that the temperature dependence of the thermopower occurs on a scale of order some significant fraction of the hopping t. For example, in organic charge transfer salts this is of order 400 K, and in the cuprates t is of order 4000 K. In these materials the thermopower varies on a scale that is one order of magnitude smaller.

I thank Nandan Pakhira for bringing the preprint to my attention.

Seminar on bad metals at Rutgers

On Tuesday I am giving a Condensed Matter Seminar at Rutgers.

Here is the current version of the slides for my talk.

The main results in the talk are in a recent PRL, written with Jure Kokalj.

The organic charge transfer salts and the relevant Hubbard model are discussed extensively in a review, written with Ben Powell.

Emergence of dynamical particle-hole asymmetry

Largely due to the work of Sriram Shastry I have recently become aware that particle-hole asymmetry in strongly correlated electron systems is an important issue (and challenge).
This was flagged in an earlier post.

There are a number of experimental anomalies that suggest the asymmetry is much larger than that associated with band structure effects. These include:

-highly asymmetric ARPES line shapes in the cuprates
-the slope of the I-V characteristics for some STM spectra
-a thermoelectric power that is large and changes sign with temperature in some cuprates

Theoretically it has been a puzzle that theoretical calculations for doped Mott insulators often give self energies that have a large particle-hole asymmetry. See for example Figure 3 in this PRL, Figure 13 of this PRB, and the figure below. It is very different from the perfect particle-hole symmetry implicit in Fermi liquid theory and marginal Fermi liquid theory. Also the quadratic frequency dependence only appears over a narrow frequency range, leading to kinks in the quasi-particle dispersions.

There is a new preprint
Extremely Correlated Fermi Liquid study of the U=infinity Anderson Impurity Model
by Sriram Shastry, Edward Perepelitsky, and Alex Hewson

The frequency dependence of the self energy for a range of impurity occupations n is shown below.

The authors show how this asymmetry emerges naturally in terms of Shastry's theory of an Extremely Correlated Fermi liquid that has two Fermi liquid type "self energies", elucidated in this PRB and particularly in this talk. In particular, there is an emergent low-energy scale Delta associated with the asymmetry.

I thank Sriram and Edward for helpful discussions about their work.

Relating non-Fermi liquid transport properties to thermodynamics

On tuesday I had nice discussion with Raghu Mahajan, Maissam Barkeshli, and Sean Hartnoll about their recent preprint Non-Fermi liquids and the Wiedemann-Franz law.

Aside: I generally find that discussing a paper with the authors before/after I have read it greatly increases my understanding. Here are a few things that became clearer to me.

In this paper "almost conserved quantities" means quantities for which the relaxation time is very long. Thus in a Fermi liquid the quasi-particles have very long lifetimes and so one can think of the quasi-particle number for every wave-vector near the Fermi surface as being "almost conserved". This means there are many conserved quantities.

However, they consider a system in which there is a Drude peak in the frequency dependent conductivity but fermionic quasi-particles are poorly defined due to large scattering. Optimally doped cuprates might be an example of a real material with this property. I thought that one dimensional models that exhibit this are Luttinger liquids. They have a Drude peak due to a collective bosonic mode but no fermionic quasi-particles. However, they are close to integrability which corresponds to having an infinite number of conserved quantities.

Note, this is different from most of the bad metals I discuss on this blog: they have no Drude peak and no quasi-particles. Although Aristomenis Donos and Sean recently considered a model (based on the holographic correspondence) that does have this property.

A Drude peak but no quasi-particles means there is one dominant relaxation timescale, that for momentum relaxation. This is what they mean by only one almost conserved quantity. This is a bit like hydrodynamics.

Central to the paper is a "memory matrix formalism" for transport properties. Some justification (and an intuitive understanding) for that is given in this paper. Central to that is the real part of static correlation functions [thermodynamic quantities] between the total momentum P and the electrical current J and heat current Q.

A Wiedemann-Franz type ratio can given in terms of these thermodynamic functions. The actual Lorenz ratio is much less than one.

This is because of a cancellation of the two terms in 

where the first term obeys the modified ratio

This is the central result of the paper. The ratio of two transport quantities is determined by the ratio of two thermodynamic quantities.

It will be nice to see extensions of this approach to give the thermopower (alpha/sigma=Seebeck coefficient) and the Hall coefficient. Both these quantities are fairly independent of scattering time in a Fermi liquid.

I think that in the absence of thermal conductivity due to phonons (unrealistic) the thermoelectric figure of merit could be larger than one.

In some sense this work is similar in spirit to that of Shastry on the Hall coefficient and thermopower. He considered the high frequency limits of these quantities for strongly correlated electron models and showed they could be related to equal time expectation values of operators (thermodynamic quantities).
He also considered Kelvin's formula for the thermopower.

I have one minor quibble. They say that CeCoIn5 violates Wiedemann-Franz (WF) at low temperatures. However, in a PRL Michael Smith and I showed that the relevant experimental paper in Science involves a spurious extrapolation to low temperatures. At sufficiently low temperatures we claim WF will hold. I think this alternative point of view should be stated in the paper.
It does seem awfully hard to find violations of Wiedemann-Franz.

Nernst effect as a probe of quasi-particle coherence

There is an interesting PRL

Nernst Effect: Evidence of Local Kondo Scattering in Heavy Fermions

by Peijie Sun and Frank Steglich 

They measure the temperature dependence of the Nernst coefficient for two different heavy fermion compounds and compare them to an isostructural compound without 4f electrons.

The temperature dependence is correlated with that of the thermoelectric power. 
An important question is Nernst signal is related to the coherence temperature associated with the formation of quasi-particles associated with the Fermi liquid.
But there is a complexity associated with this identification.

They argue that Nernst effect is largely a reflection of the single-ion Kondo effect and is measuring the strong energy dependence of the scattering rate.

They suggest the low temperature minimum and the high temperature maximum should be respectively identified with the Kondo temperature for the doublet ground state of the crystal effective field
and the Kondo temperature of the Hund's rule J=5/2 state of Ce3+.

Long live Fermi liquid theory!

There is a very nice preprint
Hidden Fermi Liquid, Scattering Rate Saturation and Nernst Effect: a DMFT Perspective
by Wenhu Xu, Kristjan Haule, and Gabriel Kotliar

I think it is original and important. I wish I had written it!

They consider the metallic phase of a two-dimensional Hubbard model at (close to optimal) hole doping 0.15 away from the Mott insulator, within Dynamical Mean Field Theory (DMFT). 

The surprising result (to me) is that one can talk about quasi-particles (i.e. poles in the one electron Green's function) up to much high temperatures than one might expect (specifically, far beyond the temperature T_FL, below which the scattering rate has a quadratic temperature dependence).

One just has to allow the quasi-particle weight Z to be temperature dependent, as shown in the Figure below.

This leads to a temperature dependent band structure.

Furthermore, most of the transport properties calculated within DMFT are quantitatively described by a quasi-particle approximation and Sommerfeld expansion, even into the bad metal region. The graph below shows the temperature dependence of the thermopower. Note the change of sign.

A few comments:

1. The authors suggest there may be a connection to Nigel Hussey's phenomenology of the cuprates, particularly with regard to saturation of the scattering rate at high temperatures.

These ideas are developed more in a recent PRB by Jure Kokalj, Nigel and I.

[But as the authors point out DMFT cannot capture the anisotropy observed in the cuprates].

2. I am not sure about calling this a "Hidden Fermi liquid" since that terminology is associated with a specific idea of Phil Anderson which I discussed here. It is not clear to me that these "Fermi liquids" are the same thing. In particular, Anderson's seems much more exotic.

3. The emergence of the different temperature scales and "strange metal" behaviour confirms my prejudice (and Anderson's) that the AdS/CFT approach is not relevant.

4. Minor quibble: It would be helpful to have units on the axes in Figure 2. I can see that the thermopower S is in units of k_B/e but have no idea about the Nernst signal. This would help in comparing the magnitude to experimental values for the cuprates.

I welcome more comments on this work.

Some ultra-cold atom experiments I would like to see

I have been having some stimulating interactions with my Australian cold atom colleagues, including Matt Davis, Chris Vale, Andy Martin, and Kris Helmerson.

As I see it ultracold atomic gases and solid-state materials have complementary strengths and weaknesses for investigating emergent quantum many-body phenomena. Solid state materials are much easier to bring to spatially uniform thermal equilibrium, achieve temperatures much less than characteristic temperatures (such as the Fermi temperature), and perform high precision thermometry. On the other hand it is hard to drive solid state systems far from equilibrium, to investigate non-equilibrium phenomena such as turbulent charge flows, and the time scales for relaxation to equilibrium are often too fast to be observed.

In contrast, ultra-cold atomic gases make it is much easier to access non-equilibrium states, and image them and their time evolution. The two platforms are also complementary in the access they provide to tune-ability, control and design. Solid state systems can be tuned considerably by temperature, pressure, magnetic field, electric field, and chemical substitution. However, sometimes it is hard to know how these variations produce changes in the underlying microscopic interactions between the constituent particles. In contrast, some of the underlying interatomic interactions in ultracold atom systems can be readily tuned from weak to strong in a precise and the known manner. However, a major challenge remains to expand the repertoire of possible tune able interactions, particularly to include some of the more common interactions found in solid state systems (e.g., the coupling of orbital motion of fermions to a magnetic field and the Heisenberg antiferromagnetic spin interaction in Mott insulators).

Here a few experiments that I would particularly like to see done and may be "relatively straight-forward", i.e, feasible in the next few years. Of particular interest would be observing these phenomena in fermionic atom systems in which one can tune the strength of the interactions, observe the BEC-BCS crossover, and universal behaviour associated with scattering close to unitarity.

Probing Thermoelectric transport with cold atoms
and
Quantum oscillations in ultracold Fermi gases: Realizations with rotating gases or artificial gauge fields
Charles Grenier, Corinna Kollath, Antoine Georges

The "Higgs boson"!
Visibility of the amplitude (Higgs) mode in condensed matter
Daniel Podolsky, Assa Auerbach, and Daniel P. Arovas

For bosonic systems there is a recent experimental paper from Immanuel Bloch's group
The ‘Higgs’ amplitude mode at the two-dimensional superfluid/Mott insulator transition

This then connects to
Conductivity of hard core bosons: A paradigm of a bad metal
by Lindner and Auerbach
An earlier post discussed this paper, suggesting calculation of the thermopower.

Observation of an d-wave pseudogaps. For the s-wave case see
Observation of a pairing pseudogap in a two dimensional Fermi gas.

Postdoc available in condensed matter theory at UQ

I have just advertised for a new postdoc to work with me on the theory of strongly correlated electron materials. You can see more details via the official advertisement. The flavour of some of the research can be seen from my blog posts under the "thermoelectric" label on this blog.

Transition from a band insulator to a bad metal

Many previous posts have considered how in a metallic phase close to a Mott insulator one can observe a crossover from a Fermi liquid to a bad metal with increasing temperature.

One observes something quite different in FeSi (iron silicide) which has been a subject of debate for several decades. Different paper titles include the following words: Kondo insulator, ferromagnetic semiconductor, unconventional charge gap, strong electron-phonon coupling, Anderson-Mott localization, singlet semiconductor, covalent insulator, correlated band insulator, ferromagnetic metal, ....

At low temperatures FeSi is a semiconductor with a gap of about 50 meV (500 K). Both the spin susceptibility and the resistivity are gapped. However,  around 200 K there is a crossover to a bad metal.
The spin susceptibility has a maximum versus temperature around 400 K and above that can be fitted to a Curie-Weiss form, suggesting the presence of local moments.
The thermopower has a maximum around 50 K with a colossal value of 700 microVolts/Kelvin, making the material attractive for thermoelectric applications. The thermopower changes sign at about 150 K and 200 K.
With increasing temperature the optical conductivity shows redistribution of spectral weight on the electron Volt (eV) scale, an important signature of strong electronic correlations.

There is a really nice paper which provides a compelling theoretical description and explanation of what is going on.
Signatures of electronic correlations in iron silicide
Jan Tomczak, Kristjan Haule, and Gabi Kotliar

The authors perform electronic structure calculations combining Density Functional Theory (DFT) [at the level of Generalised Gradient Approximation (GGA)] with DMFT [Dynamical Mean-Field Theory].
They reproduce the main features of the experimental data.

Here is some of the key physics.
FeSi is a band insulator at low temperatures.
With increasing temperature there is a crossover to incoherence, i.e. the Bloch wavevector is no longer a good quantum number.
Fe is in a mixed valence state with a mean valence (no. of d electrons) of 6.2 and a variance of 0.9.
There is a preponderance of S=1 states, contrary to earlier suggestions that FeSi is a singlet insulator.
The incoherence arises because of fluctuations in the local moment, which is to a large extent non-local.
The results are controlled by the Hund's coupling J rather than the Hubbard U, something also seen recently in other systems with orbital degeneracy [see this two-faced post or discussion of strontium ruthenate or a recent review].

Caveats about thermopower interpretation

The thermoelectric power of a metal is rather complex. Even Ashcroft and Mermin suggested that it was difficult to interpret and relate to the theoretical calculations.
Earlier posts have considered some of the subtleties, particularly in strongly correlated electron systems.

To me a couple of recent experimental papers present beautiful data but are not cautious enough in their interpretation. They need to rule out alternative explanations [see below] before I will be convinced of the explanations that they propose.

Fermi-surface reconstruction by stripe order in cuprate superconductors
F. Laliberté, J. Chang, N. Doiron-Leyraud, E. Hassinger, R. Daou, M. Rondeau, B.J. Ramshaw, R. Liang,  D.A. Bonn, W.N. Hardy, S. Pyon, T. Takayama, H. Takagi, I. Sheikin, L. Malone, C. Proust, K. Behnia, and Louis Taillefer

They observe a sign change in the thermopower and associate this with a Fermi surface reconstruction, stripe formation, and a quantum phase transition.

Spin fluctuation driven enhancement of thermopower near the quantum critical point in Ba(Fe(1-x)Co(x))2As2

S. Arsenijević, H. Hodovanets, R. Gaál, L. Forró, S. L. Bud'ko, P. C. Canfield

In both papers, the fact that S/T for a specific doping has a logarithmic temperature dependence over about a decade in temperature is equated with quantum criticality.

Why am I not convinced?

1. A recent preprint shows the cuprate data can be explained using the semi-classical Boltzmann equation and a Functional Renormalisation Group treatment of the Hubbard model (see Figure 10). The sign change is associated with a the van Hove singularity and a Lifshitz transition.

2. Dynamical mean-field theory (DMFT) shows how the thermopower can change sign as a function of temperature due to strong correlations and the associated low coherence temperatures. (See e.g. Figure 4 in this preprint which I discussed in an earlier post).

I suspect that a DMFT treatment of the relevant multi-band Hubbard model with Hund's rule coupling [a la Park, Haule, Kotliar] will be able to explain the thermopower data for the pnictides.

How good metals turn bad

There is a really nice preprint
How bad metals turn good: spectroscopic signatures of resilient quasiparticles
by Xiaoyu Deng, Jernej Mravlje, Rok Zitko, Michel Ferrero, Gabriel Kotliar, and Antoine Georges

They study a doped Hubbard model using Dynamical Mean-Field Theory (DMFT). Although the ground state is a Fermi liquid this is only a good description at very low temperatures. In particular, the quadratic temperature dependence (characteristic of a Fermi liquid) only occurs below a temperature of about 0.05 delta D [where delta=doping and D=band width]. But, well-defined quasi-particles still exist all the way up to the "bad metal" region at which the mean-free path is comparable to the lattice constant.

I found this resilience of quasi-particles somewhat surprising [I am not quite sure why] and interesting. Furthermore, in this intermediate temperature regime there is large entropy and significant local magnetic moments, and Kelvin's formula gives a good description of the temperature dependence of the thermoelectric power.

The strong correlations lead to a significant particle-hole asymmetry (in the self energy and single-particle density of states) via the subtle interplay of the lower Hubbard band with the quasi-particle peak in the density of states.

To me this all illustrates how much (but obviously not all) of the rich physics seen in strongly correlated materials can be captured by DMFT.

Note added. I am told that the axis label on the left figure is in error and it should be 1/Z not Z, i.e. Z actually increases with temperature.

Cake meeting talk on thermopower

This week I talk about Kelvin formula for the thermopower. Here are the slides. Not sure that they will be much use for people not at the meeting. At least, they give a flavour of some of the experimental results of particular interest to me. Most have been featured in earlier blog posts on the subject.

I made one small discovery in my preparation: a specific example where the Kelvin formula is exact. If one takes the temperature dependence of the chemical potential for non-interacting fermions with a general density of states [equation 2.77 in Ashcroft and Mermin] then the thermopower from the Kelvin formula is exactly that given by the Mott-Heikes formula obtained from solving the Boltzmann equation [equation 13.62 in Ashcroft and Mermin] for the case where the scattering time and velocity have no energy dependence.

A limit to my understanding

One of the many things I find hard to understand about quantum many-body theory is the fact that the order in which you take the limits of functions really does matter. I fear I was brainwashed by too many calculus classes which always considered well defined analytical functions!

Two significant and profound examples concern the theory of superconductivity and the the thermoelectric effect.

Example one: electromagnetic response of a superconductor

Consider the frequency and wavevector dependence of the current-current correlation function, denoted Lambda(q,omega) below, where q is the wavevector and omega the frequency. The q to 0 limit can also be viewed as taking the thermodynamic limit.

The following equations are taken from an important 1993 paper Insulator, metal, or superconductor: the criteria by Doug Scalapino, Steve White, and Shoucheng Zhang

The first equation defines the superfluid density D_s, and the last one the Drude weight D. The two quantities are equal in a BCS superconductor but are not equal in a non-superconducting metal or in unconventional superconductors such as cuprates or organic charge transfer salts.

The second equation is required by gauge invariance. Comparing to the first equation we see that one gets a different answer depending on the order in which q_x (parallel to the current) and q_y goes to zero.

Example two: thermoelectric response

A very elegant 2010 paper Kelvin formula for thermopower, Michael Peterson and Sriram Shastry considered the significance of taking limits in different orders. They started with the exact Kubo expression for the frequency and wavevector dependent thermoelectric response S(q,omega).

The correct value for the Seebeck coefficient is obtained by first taking the q to 0 (i.e. thermodynamic limit of infinite system size) and then taking the static (omega to 0) limit.  However, if one reverses the order of these two limits  S(q,omega) reduces to Kelvin's (1854) formula which gives the Seebeck coefficient (n.b. a transport property) in terms of purely thermodynamical variables, i.e. S is proportional to the derivative of the chemical potential with respect to temperature. This formula is only approximate, but is very useful particularly for getting magnitudes and trends.

The slide below from a 2012 talk by Shastry nicely summarises the above.

Can someone give other examples?

Signatures of "band-like" transport in organic electronic materials

I used to regularly write posts about charge transport in organic electronic materials. Some of these generated lively discussion in the comments section.

This morning I read an interesting paper Band-Like Electron Transport in Organic Transistors and Implication of the Molecular Structure for Performance Optimization
by Nikolas Minder, Shimpei Ono, Zhihua Chen, Antonio Facchetti, Alberto Morpurgo

They correctly distinguish "band-like" transport where a charge carrier is delocalised over just a few molecules from true band transport where it is delocalised over a large number of molecules [or unit cells in a crystalline semiconductor such as silicon].

They claim that a signature of band-like transport is the common observation of a mobility that decreases with increasing temperature and a Hall effect signal. I agree with the former but am confused about the latter. I thought for incoherent polaron transport one could still have a Hall effect, as discussed in a classic paper by Friedman and Holstein. 

The authors overlook the fact that a signature of band-like transport is that the mobility should be larger than e a^2/hbar ~ 1 cm^2/Vsec. Ignorance of this old and important result seems to be common in the field.

Previously, I pointed out that comparing the relative magnitudes of the energy gaps respectively associated with mobility, optical conductivity, and thermopower is a nice way to distinguish coherent from incoherent transport.

Characteristics of optimal doping in cuprates

In the cuprate superconductors there is a value of the doping at which the superconducting transition temperature is a maximum (optimal doping). Coincidentally (?) this also seems to the doping at which the metallic phase is most non-Fermi liquid like. Some theories (especially due to Varma) try and connect these two phenomena via a quantum critical point below the superconducting dome. An earlier post discusses how the entropy is maximal and the thermopower changes sign near optimal doping.

A cluster DMFT (Dynamical Mean-Field Theory) calculation by Kristian Haule reproduces the correlation between high-Tc and anomalous metallic properties. The figure below shows the Matsubara frequency dependence of the imaginary part of the self energy (at wave vector (0,pi) = anti-nodal region) (top) and the anomalous self energy (related to the superconducting pairing) for different dopings.

In a simple Fermi liquid the slope of the upper curve at low frequencies is related to the quasi-particle weight.
Haule concludes the quasi-particles are most incoherent and the scattering rate the largest around optimal doping.