Statistical analysis of the figure of merit of a two-level thermoelectric system: a random matrix approach

TL;DR

This paper uses random matrix theory to analytically and numerically analyze the statistical properties of thermoelectric transport in a two-level quantum dot system, revealing non-Gaussian distributions and optimal conditions for performance.

Contribution

It provides the first exact analytical expressions for the joint probability distributions of thermoelectric coefficients in a chaotic quantum dot system using random matrix theory.

Findings

Transport coefficient distributions are highly non-Gaussian for few modes.

Optimal thermoelectric performance occurs at half transparency.

Few conduction modes can enhance system performance.

Abstract

Using the tools of random matrix theory we develop a statistical analysis of the transport properties of thermoelectric low-dimensional systems made of two electron reservoirs set at different temperatures and chemical potentials, and connected through a low-density-of-states two-level quantum dot that acts as a conducting chaotic cavity. Our exact treatment of the chaotic behavior in such devices lies on the scattering matrix formalism and yields analytical expressions for the joint probability distribution functions of the Seebeck coefficient and the transmission profile, as well as the marginal distributions, at arbitrary Fermi energy. The scattering matrices belong to circular ensembles which we sample to numerically compute the transmission function, the Seebeck coefficient, and their relationship. The exact transport coefficients probability distributions are found to be highly non-Gaussian for small numbers of conduction modes, and the analytical and numerical results are in excellent agreement. The system performance is also studied, and we find that the optimum performance is obtained for half-transparent quantum dots; further, this optimum may be enhanced for systems with few conduction modes.