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.

Quantum of thermal conductance

Here are a couple of things I find surprising about the electronic transport properties of materials.

1. One cannot simply have materials, particularly metals, that have any value imaginable for a transport coefficient. For example, one cannot make the conductance or the thermopower as large as one wishes by designing some fantastic material.

2. Quantum mechanics determines what these fundamental limits are. Furthermore, the limiting values of transport coefficients are often set in terms of fundamental constants [Planck's constant, Boltzmann's constant, charge on an electron].

The fact that this is profound is indicated by the fact that this was not appreciated until about 25 years ago. A nice clean example is the case of a quantum point contact with N channels. The conductance must be N times the quantum of conductance, 2e^2/h. This result was proposed by Rolf Landauer in 1957 but many people did not believe it until the first experimental confirmation in 1988.

The thermal conductance through a point contact should also be quantised. The quantum of thermal conductance is

Asides:
1. note that the Wiedemann-Franz ratio is satisfied.
2. this sets the scale for the thermal conductivity of a bad metal.

A paper in Science this week reports the experimental observation of this quantisation.

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.

Strange metal. Strange insulator. Strange material.

Jaime Merino and I just finished a paper Effective Hamiltonian for the electronic properties of the quasi-one-dimensional material Li0.9Mo6O17

This is a really strange and interesting material. It has featured in earlier posts. We discuss how the observed properties of both the "metallic" phase and the "insulating" phase are quite unusual and don't seem to fit into any "standard" picture [Fermi liquid, Luttinger liquid, quantum critical, ...]. We then propose the simplest possible lattice model Hamiltonian that might capture its properties. This is worthy of further study.

We thank Nigel Hussey for getting us interested in this fascinating material. He has a forthcoming PRL about the unconventional (possibly triplet) superconductivity.

Comments welcome.

Deconstructing the Nernst effect in electron doped cuprates

The graph below shows the temperature dependence of the Nernst signal measured in the normal metallic state of a family of electron doped cuprates Pr_{2-x}Ce_xCuO_4. It is taken from a 2007 PRB by Li and Greene.

A few noteworthy features
-the signal is proportional to B and so not due to superconducting fluctuations
-the signal is proportional to temperature at low temperatures but has a non-monotonic temperature dependence
-the magnitude of the linear temperature dependence is an order of magnitude smaller than predicted by the simple quasi-particle theory of Behnia.

The authors consider how the data can be explained by a two-band with both electrons and holes, but point out such a model is inconsistent with the single hole Fermi surface seen in ARPES.

It would be interesting to re-consider this data in light of the recent experiments on these materials which showed a linear temperature dependence of resistivity (and thus a quasi-particle scattering rate) with a magnitude proportional to Tc [as in overdoped hole doped cuprates].

Can strongly correlated electrons save the planet II?

Several earlier posts discussed the thermoelectric effect in strongly correlated electron materials. The Seebeck coefficient S is a quantitative measure of the effect. At low temperatures it can be orders of magnitude larger than in elemental metals. 

The figure above illustrates how thermoelectric couples can be used to either perform refrigeration or generate electrical power from waste heat. It is taken from a nice Perspective in Science Smaller is Cooler by Brian Sales which reviews state of the art materials in 2002.

The thermoelectric figure of merit, ZT is a dimensionless ratio which is a good measure of how useful a material will be in thermoelectric applications.

sigma is the conductivity and kappa the thermal conductivity.

Currently used materials such as Bi2Te3 [also a topological insulator!] have values of ZT ~1. If materials can be found with ZT~4 then thermoelectric refrigerators will be competitive with traditional compressor refrigerators, which are less reliable and environmentally dirtier.

So how good are strongly correlated electron materials?

There is a nice review by Paschen on heavy fermions.

It is important to note that the thermal conductivity is the sum of electronic and phonon contributions. If one neglects the latter (for the moment) and uses the Wiedemann-Franz ratio then ZT ~ S^2 where S is in units of k_B/e. This is indeed its magnitude near the coherence temperature in strongly correlated electron materials. Hence, ZT ~ 1 (but not larger) seems possible.

BUT, this argument neglects the thermal conductivity due to phonons which is much larger that the electronic contribution in this temperature regime. So one needs to find a way to reduce this. This leads to the idea of a Phonon Glass Electron Crystal.

Candidate strongly correlated materials may be skutterudites which exhibit heavy fermion behaviour (e.g. SmPt4Ge12).

Thermopower reveals destruction of quasi-particles

There is a nice preprint, Thermoelectric Power of the YbT2Zn20 (T = Fe, Ru, Os, Ir, Rh, and Co) Heavy Fermions by E. D. Mun, S. Jia, S. L. Bud’ko, and P. C. Canfield

The figure below shows the temperature dependence of the thermoelectric power for the title compounds. Note that it non-monotonic, being linear at low temperatures, reaching a maximum magnitude of order k_B/e ~ 80 microV/K at a temperature T_min.

 The next figure shows that T_min (left scale) is correlated with the single ion Kondo temperature (horizontal scale) and the temperature at which the resistivity is a maximum (right scale).

 The magnitude of the linear temperature dependence at low temperatures is simply related to that for the specific heat (see also this earlier post) as shown in the Figure below.

Aside: the inset on the lower right considers the Kadowaki-Woods ratio but does not make use of recent work concerning its universality.

All of the above features seem to be characteristic of broad classes of strongly correlated electron metals, as emphasized by Jaime Merino and I, in a PRB published in 2000. The temperature scales in the middle figure are characteristic of that at which there is a crossover from a Fermi liquid at low temperature to a bad metal at higher temperatures.

The Nernst effect in strongly correlated electron materials

The Nernst effect is a thermal conduction analogue of the Hall effect for electrical conductivity, i.e., it measures the transverse electrical current induced by a longitudinal thermal current in the presence of a magnetic field perpendicular to both currents.
It was considered an obscure (and very small) effect in elemental metals. However, the past 15 years it has become a powerful probe of strongly correlated metals, initially because of its sensitivity to superconducting fluctuations, as discussed by Ong.

A nice helpful review is The Nernst Effect and the Boundaries of the Fermi Liquid Picture by Kamran Behnia.

He argues that the magnitude of the Nernst signal at low temperatures for a wide range of materials is proportional to the ratio of the charge carrier mobility to the Fermi energy. This is supported by the Figure below. Note the logarithmic scales.

A few notes.

1. The simple Fermi liquid expression [equation (8)] gives the Nernst signal as proportional to the energy derivative of the scattering time. For a Fermi liquid form of the scattering rate, the energy dependence is quadratic, and the signal will vanish. Essentially Behnia's replacement of the the energy derivative by the ratio of the scattering time and the Fermi energy means he is assuming that the scattering has a marginal Fermi liquid form. This is worth considering in more detail.

2. As the temperature increases there should be a crossover from a Fermi liquid with coherent quasi-particles with well-defined wavevectors to a "bad metal" with incoherent excitations. How this is manifested in the Nernst signal is an outstanding question.

3. A PRB by Kontani has given a general expression for the Nernst coefficient in a Fermi liquid including vertex corrections. A detailed analysis (within the framework of FLEX) claims that vertex corrections are important and due to antiferromagnetic fluctuations a large Nernst signal is possible in the pseudogap phase. [See Section 5.2 of this review].

Optimal doping corresponds to maximum entropy

What is so unique about the optimal doping at which the superconducting transition temperature is a maximum in the cuprates?

There is an interesting paper Unified electronic phase diagram for hold-doped high-Tc cuprates by Honma and Hor. It builds on their earlier work which argued the existence of a universal planar hole scale (P_pl), which can be characterised by the thermopower at T=290 K, denoted S^290. P_pl is independent of the nature of the dopant, the number of CuO2 plane layers per unit cell, the structure, and the sample quality. The figure below shows S^290 versus P_pl for a wide range of cuprates.

Note that the thermopower changes sign at a doping of P_pl ~ 0.25 which is comparable to that at which  Tc is a maximum [except for Sr doped La214]. 

What is the significance of this sign change of the thermopower?

For a simple Fermi liquid it would correspond to a change in the sign of the charge carriers, i.e., from electrons to holes.

Recently Peterson and Shastry interpreted this sign change in terms of the Kelvin formula for the thermopower, which gives the thermopower as -1/e times the derivative of the entropy with respect to the particle number. This can be related to the temperature dependence of the chemical potential via the Maxwell relation,

Here s=specific entropy, c_h=hole density, mu_h = chemical potential.

It is rather surprising that a transport property can be expressed in terms of a thermodynamic property.

Thus the change in sign of the thermopower means that the entropy is a maximum as a function of hole doping.

Indeed, a maximum in the entropy near optimal doping is was found for the t-J model via 

Finite temperature Lanczos calculations on small lattices of up to 20 sites and summarised in a 2000 review by Jaklic and Prelovsek.

The paper also considers a Kelvin type relation for the thermopower, but does not mention Kelvin, and shows how the maximum in the entropy vs. doping is associated with a change in sign of the thermopower.

[Peterson and Shastry do not mention this earlier work.]

A recent preprint by Garg, Shastry, Dave, and Philips, argue that the sign change reflects an underlying quantum critical point at optimal doping. However, I wonder about the extent that one can see a quantum critical effect on lattices as small as 20 sites. 

This is related to a 2009 PRB by Mark Jarrell and collaborators who calculated the temperature and doping dependence of the entropy using the cluster dynamical approximation. They found entropy was a maximum at optimal doping [~0.15-0.2]. They also don't mention the earlier work by Jaklic and Prelovsek.

A very strange metal

The linear chain compound Li0.9Mo6O17 exhibits a subtle competition between superconductivity, a "bad" metal, and a strange "insulating" phase. Recently large deviations from the Weidemann-Franz law were reported by Nigel Hussey's group.

The graph below shows the temperature dependence of the electrical resistance for current parallel to the chain direction. It has a "metallic" temperature dependence above about 30 K, and an "insulating" temperature dependence between the superconducting transition temperature around 1 K and 30 K. This is rather unusual and puzzling since one normally sees a direct transition from a metallic phase to a superconducting phase. Although there are other cases such as reported in this PRB [see Fig. 2 inset] for an organic charge transfer salt where a superconducting state occurs close to a charge ordered insulator [see also the Table in this PRL].

The data is taken from a Europhys. Lett. by Chen et al. which also reports a rather strange angular dependent magnetoresistance.

Deconstructing the chemical potential of the cuprate superconductors

I have been reading through the nice review Finite temperature properties of doped antiferromagnets by Jaklic and Prelovsek from 2000. They summarise their studies of the t-J model by the Finite temperature Lanczos method.

At first sight the graph below of the temperature and doping dependence of the chemical potential does not look particularly interesting [at least to me]. However, they highlight its significance.

Here are a few points.

• In a simple Fermi liquid the chemical potential has a positive, quadratic and weak temperature dependence. This is only seen for doping c_h=x=0.3
• For a wide doping range [0.05 < c_h < 0.3] the temperature dependence is approximately linear. The slope changes sign for approximately optimal doping (c_h ~ 0.15).
• The weak temperature dependence for c_h ~ 0.15 means that optimal doping corresponds to maximum entropy!  [This can be deduced via the Maxwell relation below. Don't you love thermodynamics!]

• This relation is also related [approximately] to the thermopower via a relationship [equation 8.6], which is essentially a restatement of the Kelvin formula [discussed  by Peterson and Shastry].
• The latter means the thermopower should change sign around optimal doping, as is indeed observed [more on that later].
• The large entropy near optimal doping emerges from the interplay of the localised spins [from the remnants of the Mott insulator] and frustration of the antiferromagnetic spin interactions via doping.

I would be interested to see a similar calculation for the Hubbard model on the anisotropic triangular lattice at half filling to see how the chemical potential varies as a function of U/t as the Mott insulator is approached from within the metallic phase.

Seeking new thermoelectric materials

Thermoelectric materials are of significant technological interest and present some fundamental scientific questions.
An extremely useful concept is the dimensionless thermoelectric figure of merit. The optimum material with have a high electrical conductivity and thermoelectric power (Seebeck coefficient), but also a low thermal conductivity. This has led to the notion of a Phonon Glass Electron Crystal (PGEC): a material which has the low thermal conductivity characteristic of a glass and the high electrical conductivity characteristic of a crystal. How might one achieve this?
In simple kinetic theory [with well-define acoustic phonon quasi-particles] the thermal conductivity is proportional to the phonon velocity and the phonon mean-free path. In glasses there is so much structural disorder the concept of phonon quasi-particles and a mean-free path is ill defined. In the quasi-particle picture one could reduce the thermal conductivity either by decreasing the phonon mean-free path or by decreasing the phonon speed, or both. The former can happen via a large anharmonicity, which is what is responsible for phonon-phonon scattering. The latter can happen in a soft material or by strongly coupling the acoustic phonons to low frequency optical phonons.

There is a really nice News and Views article Thermoelectrics: Half-full Glasses by Cronin Vingin in Nature Materials that puts in context neutron scattering experiments, on two different classes of materials. [I thank Elvis Shoko for bringing this article to my attention and for helpful discussions]. The first class of materials are clathrates and the second skutterudites. Examples, are methane hydrate and BaFe4Sb12, respectively.

[Aside: previously I posted about superconductivity in skutterudites]. The picture below shows a clathrate structure.

Although, chemically distinct a common structural feature is that both have large cavities within which an atom can "rattle" around in. This means that associated with these motions there are low frequency "optical" phonons which are very anharmonic. These modes then couple to acoustic phonons via coupling to motions of the cage.

The figure is taken from a recent PRB, the introduction to which I found gave a particularly helpful overview.

Deconstructing thermal transport in the pseudogap state

A key question concerning the cuprate superconductors is what is the relationship between the pseudogap state and superconductivity? I previously posted about some beautiful STM data that shows no change in the excitation spectrum of underdoped cuprates when the temperature increases above the transition temperature.

Similar physics is seen in ARPES data and in the thermal conductivity data shown below, taken from this PRL by Doiron-Leyraud et al.

The bottom panel shows that the zero temperature value of kappa(T)/T changes little as a function of doping. In a d-wave superconductor this quantity has a universal non-zero value that is determined by the size of the gap. The upper panel shows how the transition temperature varies with doping, vanishing below p=0.05.

It is striking that the thermal conductivity suggests that the excitation spectrum is unchanged in the non-superconducting samples.

This fact that the thermal conductivity is a powerful probe of the pseudogap state led Michael Smith and I to examine how different models for the pseudogap state (particularly arcs vs. pockets) lead to qualitatively different predictions for the thermal conductivity. Our results appeared in Phys. Rev. B this week.

Deconfined spinons take the heat

There is a nice paper in Science this week which provides experimental evidence for gapless spin excitations (possibly deconfined spinons) in an organic material that may have a (Mott insulating) spin liquid ground state.

[I thank Andrew Bardin and Ben Powell for bringing it to my attention].

The Figure below shows the temperature dependence of the thermal conductivity kappa(T). The fact that as T->0, kappa/T has a non-zero intercept is a clear signature of gapless excitations. The magnitude of the intercept is comparable to its value in the metallic phase of other organic charge transfer salts, and an order of magnitude larger than what one gets in the d-wave superconducting state due to nodal quasi-particles. [see for example this PRL].

[This is a good example of a Figure because it compares results for several materials so you can see just how clearly different the dmit-131 material is].

The authors also measure the thermal conductance tensor in a magnetic field. Within error the thermal Hall angle was zero. This was motivated by this recent PRL by Katsura, Nagaosa, and Lee, predicted a sizeable thermal Hall effect in quantum spin liquids with deconfined spinons. [Patrick Lee wrote a guest post about this a while back].