# Seminar – Matthieu Verstraete: Ab initio thermoelectric properties

From the University of Liège. Work with Xu, Di Gennaro, Delhadj, Dewandre, Verstraete.

we are trying to scramble the post-petroleum energy scenarios. While nuclear seems dead, solar and wind rule, biomass sis not really ecological, the real key point is to make sure we use energy efficiently and recover it as much as possible. The projections on global energy consumption show that the projected growth in renewables will all go to fill in the energetic needs of fast-“developing” (my quotes) countries like China, India, Brazil, hence we will keep burning oil at the same rate.

Physical strategies to save energy are: Waste heat scavenging, exploiting phase transformations [Nature 502, 85 (2013)], magnetic induction, MEMS/NEMS, via sensors-smart clothes-WiFi-power plants, and most importantly thermoelectrics.

Thermoelectrics uses either the Peltier effect (for cooling), or its inverse the Seebeck effect (power generation), to produce thermodynamic cycles that transform heat flow in electric currents or viceversa. A crucial object to quantify the quality of thermoelectrics is the figure of merit:

$ZT=\frac{S^2 \sigma_{el}}{k_{latt}+k_{el}}$

One wants to make this number as big as possible. The denominator is the total thermal conductivity. The second term is due to electrons, while the first term is due to phonons (“phonon heat shunt”), which you can’t easily get rid of (this is part of the reason why thermoelectric is not so efficient). So one wants to minimize this phonon part and reduce the electronic one also, without reducing the electronic conductivity (at the numerator) so much. Finally, the coefficient $S$ is the so-called Seebeck coefficient whose ab-initio calculation this talk aims at.

In the 90’s there has been revived interest in thermoelectric, also in industrial, because of new niches, and also because of some new paradigms that allow to reduce thermal conductivity without reducing electrical conductivity too much. However, there is quite some controversy on the possibility that thermoelectrics will ever be efficient:

In particular the latter has argues that whatever people do doesn’t make much of a difference in the economics of beating other resources.

Nevertheless, theory since the 90s made quite some steps. Hicks & Dresselhaus [PRB 47 16631 (1993)] showed that in principle if you nano-structure normal materials, you can improve the figure of merit quite spectacularly, especially if you can move to 2D or 1D materials (5-10 Å layers). But Sofo and Mahan showed that the model was too simplistic [APPL 65 2690, (1994)], because layers should communicate if they are too thin (see also Sofo & Mahan, The Best Thermoelectric, PNAS). Another advance is “All-scale nanostructuring” [Biswas et al. Nature 489 414 (2012)], a technique designed to kill the phonons of different wave-length. This is done at different scales: from atomic doping, to nanoscale to mesoscale grains. So, by killing all phonon modes one can greatly improve ZT.

In this talk we learn about ab-initio calculations of the Seebeck coefficient, starting from the band structure of electrons in materials. Calculations are based on Boltzmann transport equation for the electrons, so arguments are semiclassical. Once one solves the equation one can obtain the currents (assuming diffusive scattering, no ballistic, no hopping, which works quite well thorough different scales, even in lower dimensional systems). From this setup he starts an ab-initio calculation of the Seebeck coefficients. One crucial approximation (that can eventually be waived, to some extent), that allows to solve the Boltzmann equations, is the constant relaxation time approximation (RTA)

$\frac{\partial f}{\partial t} = - \frac{f- f_0}{\tau}$

The RTA is sometimes very wrong, for example in simple metals. My understanding is that it corresponds to a “linear regime” approximation in nonequilibrium thermodynamics.

As an application, there are metals (Li, Au, Cu, Ag) where the Seebeck coefficient is anomalous, that is it has the wrong sign (it is positive: the majority of carriers pushed through the system are holes) (Robinson PR 161, Robinson and Dow PR 171, Jones Proc. Phys. Soc.). By their method they can explain the reason, based on some funny band structure:

• Xu and Verstraete, First Principles Explanation of the Positive Seebeck Coefficient of Lithium,  PRL 112, 196603 (2014)

They apply the theory to other metals, they get the right sign but not a good agreement in curves. The question now is: can one engineer $S$?

The reason why a positive as well as negative Seebeck coefficient are needed is that one needs materials with both properties in a thermoelectric device, to circulate currents. Hence, one wants these material to be chemically compatible, and the best would be to obtain both situations from one material only, by doping it appropriately.