bounds derived for relaxation-free systems, however, a general proof of this remains elusive.'
author:
- 'Robert S. Whitney'
date: 'March 16, 2015'
title: |
    Finding the quantum thermoelectric with maximal efficiency\
    and minimal entropy production at given power output
---

Introduction
============

Thermoelectric effects in nanostructures [@Pekola-reviews; @Casati-review; @Sothmann-Sanchez-Jordan-review; @Haupt-review] and molecules [@Paulsson-Datta2003; @Reddy2007] are of great current interest. They might enable efficient electricity generation and refrigeration [@books; @DiSalvo-review; @Shakouri-reviews], and could also lead to new types of sub-Kelvin refrigeration, cooling electrons in solid-state samples to lower temperatures than with conventional cryostats [@Pekola-reviews], or cooling fermionic atomic gases [@Grenier2012; @Brantut-Grenier-et-al2013; @Grenier2014]. However, they are also extremely interesting at the level of fundamental physics, since they allow one to construct the simplest possible quantum machine that converts heat flows into useful work (electrical power in this case) or vice versa. This makes them an ideal case study for [*quantum thermodynamics*]{}, i.e. the thermodynamics of quantum systems [@QuantumThermodyn-book].

![\[Fig:thermocouple\] (a) The simplest heat-engine is a thermocouple circuit made of two thermoelectrics (filled and open circles). The filled and open circles are quantum systems with opposite thermoelectric responses, an example could be that in (b). For a heat-engine, we assume $T_L > T_R$, 