Nanostructuration is one of the most promising strategies in the quest for next-generation thermoelectric devices. This led us to study the thermoelectric conversion and its fluctuations using a scattering approach suitable for coherent quantum transport. Studying the thermopower of a chaotic quantum dot, we have found that a new family of universal distributions occurs when this is not the bulk but the edges of the dot spectrum which are probed at the Fermi energy. We have also studied disordered nanowires and calculated their thermopowers as well as their figures of merit and their maximal output powers to determine the optimal working conditions for the low temperature thermoelectric conversion.
The application of a temperature gradient ΔT across a conductor causes charge carriers to diffuse from hot to cold regions. This gives rise to a voltage ΔV= SΔT, S being the thermopower or Seebeck coefficient. The Seebeck effect enables the conversion of heat into electricity while the Peltier effect enables refrigeration. Until now, most thermoelectrics have been based on doped bulk semiconductors. Nanostructuration provides opportunities to reduce the phonon thermal conductivity and allows tailoring of the electronic band structure, a necessary condition to achieve large thermopowers. Quantum wells, quantum wires and quantum dots have already led to improved thermoelectric conversion in comparison to their bulk counterparts. That’s why for a few years, thermoelectricity has gradually gained attention from the mesoscopic community, which faces with the problem of energy management at the nanoscale a new challenge.
In this context, we have studied the thermopower of a chaotic quantum dot using a random matrix approach. In the late nineties, large mesoscopic fluctuations of the thermopower around zero averages had been measured by Molenkamp et al. in a chaotic dot, upon varying the shape of the cavity. A non-gaussian distribution was reported in the quantum limit, when the number N of opened channels per lead is small. One year earlier, a non-gaussian distribution of the thermopower had been derived by Van Langen, Silvestrov and Beenakker, in the single-channel case (N=1). Those calculations were deduced from a seminal work of Brouwer et al. concerning the distribution of the time-delay matrix and were in particular obtained in the so-called wide-band limit. While this approximation is valid when the Fermi energy EF lies in the bulk of a wide conduction band, it fails as soon as the edges of the conduction band are approached. To go beyond this approximation, we have considered a 1D lattice embedding a chaotic scatterer of M sites. This model is exactly solvable numerically and also analytically tractable in some limits. We have studied the thermopower distribution as EF approaches the edges of the conduction band and hence the spectrum edges of the scatterer. Our results showed that this distribution becomes different near the edges than the universal one obtained in the bulk. Moreover, the thermopower distributions near the edges turned out to be also universal after an energy rescaling similar to the one used by Tracy and Widom for the energy level spacing at the spectrum edges of large Gaussian matrices. We thus obtained a family of asymptotic thermopower distributions between the one derived in the bulk and a new one valid at the band edges that we also derived analytically. Similar conclusions were drawn for the distribution of the delay-times, making our results also relevant for other waves than electrons .
A natural extension of this work was to consider the crossover from the 0D chaotic regime towards the 1D localized regime . Following an earlier work of Beenakker and coworkers, we have calculated the low temperature limit of the thermopower of disordered nanowires, as one varies the Fermi energy EF. We have shown that the thermopower is drastically enhanced when EF approaches the edges of the spectrum of the nanowire, while the conductance drops down. An analytical formula describing perfectly this behavior was derived from a previous work of Derrida and Gardner giving the localization length of the Anderson model near the spectrum edges. We also characterized the distribution of the thermopower as a lorentzian distribution, with a width proportional to the density of states evaluated at EF. In addition, much attention was paid to the analysis of the maximal power that can be extracted from the device and of the figure of merit governing the efficiency of the thermoelectric conversion. While the former is optimized in the bulk for short nanowires, the latter is maximized near the spectrum edges for long nanowires. From a practical point of view, our results allow the determination of the doping level which optimizes the thermoelectric conversion in semiconductor nanowires.
 A. Abbout, G. Fleury and J.-L. Pichard, Delay time and thermopower distributions at the spectrum edge of a chaotic scatterer, Phys. Rev. B 87 (2013) 115147
 R. Bosisio, G. Fleury and J.-L. Pichard, Gate-modulated thermoelectric conversion in disordered nanowires: I. Low temperature coherent regime, arXiv:1310.4923 (2013)