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Quantum Thermodynamics of Nanoscale Thermoelectrics and Electronic Devices

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Thermodynamics in the Quantum Regime

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 195))

Abstract

This chapter is intended as a short introduction to electron flow in nanostructures. Its aim is to provide a brief overview of this topic for people who are interested in the thermodynamics of quantum systems, but know little about nanostructures. We particularly emphasize devices that work in the steady-state, such as simple thermoelectrics, but also mention cyclically driven heat engines. We do not aim to be either complete or rigorous, but use a few pages to outline some of the main ideas in the topic.

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Notes

  1. 1.

    The negative sign in P is because we take the electric current I to be positive when it flows from the reservoir at bias V to the reservoir at zero bias. Then I has the same sign as V when it flows “downhill” (from a region of higher bias to one of lower bias) turning electrical power into Joule heating. This means that if the device is to generate power, then I must have the opposite sign to V, so it is pushing electrical current “uphill”.

  2. 2.

    In Landauer-Büttiker scattering theory, the mesoscopic electronic device is viewed as a scatterer onto which electronic wavefunctions, incoming from the leads, impinge and are transmitted or reflected. This well-known powerful approach is particularly useful for systems with weak Coulomb interaction and underlies the reasoning of various sections of this chapter. Details can be found in various text books, see for example [18,19,20].

  3. 3.

    We know from the textbook problem of a quantum particle hitting a barrier, that the transmission probability will be a smooth function of energy (going from zero at low energies to one at high energies), because the solution of the wave equation allows for tunnelling through the barrier.

  4. 4.

    The Seebeck coefficient S is often called the thermopower, even though it does not have either the meaning or the units of power.

  5. 5.

    In some special cases, one can construct a more exotic linear response theory when \(k_\mathrm{B}\Delta T\) or eV are small compared to an energy scale which is not temperature.

  6. 6.

    There is ambiguity in the literature about the sign of S. One can choose either sign, as long as one is consistent.

  7. 7.

    This simple relation is valid only if the direct capacitive coupling between the upper dots (which tunnel-couple to the hot and cold reservoirs) is so strong that they cannot be occupied simultaneously. Note however, that the effects described here continue to exist also in the presence of a smaller capacitive coupling, \(U_\text {HC}\sim U_\text {MH},U_\text {MC}\).

  8. 8.

    The thermoelectric response measured in a series of terminals has been used to probe how energy relaxation takes place along the edge, involving an elegant demonstration of chirality [25, 26].

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Acknowledgements

We acknowledge the support of the COST Action MP1209 “Thermodynamics in the quantum regime” (2013-2017), which enabled us to meet regularly to learn about and discuss much of the physics presented in this chapter. RW acknowledges the financial support of the French National Research Agency’s “Investissement d’avenir” program (ANR-15-IDEX-02) via the Université Grenoble Alpes QuEnG project. RS is supported by the Spanish Ministerio de Economía y Competitividad via the Ramón y Cajal program RYC-2016-20778. JS acknowledges support from the Knut and Alice Wallenberg foundation and from the Swedish VR.

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  1. Laboratoire de Physique et Modélisation des Milieux Condensés (UMR 5493), Université Grenoble Alpes and CNRS, Maison des Magistères, BP 166, 38042, Grenoble, France

    Robert S. Whitney

  2. Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Rafael Sánchez

  3. Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology, S-412 96, Göteborg, Sweden

    Janine Splettstoesser

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  1. School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

    Felix Binder

  2. School of Mathematical Sciences, University of Nottingham, Nottingham, UK

    Luis A. Correa

  3. ICFO: Institut de Ciencies Fotoniques, Barcelona Institute of Science and Technology, Castelldefels, Spain

    Christian Gogolin

  4. CEMPS, Physics and Astronomy, University of Exeter, Exeter, UK

    Janet Anders

  5. School of Mathematical Sciences, University of Nottingham, Nottingham, UK

    Gerardo Adesso

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Whitney, R.S., Sánchez, R., Splettstoesser, J. (2018). Quantum Thermodynamics of Nanoscale Thermoelectrics and Electronic Devices. In: Binder, F., Correa, L., Gogolin, C., Anders, J., Adesso, G. (eds) Thermodynamics in the Quantum Regime. Fundamental Theories of Physics, vol 195. Springer, Cham. https://doi.org/10.1007/978-3-319-99046-0_7

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