# Thermoelectric Efficiency of a Topological Nano-Junction

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## Abstract

**:**

## 1. Introduction

## 2. The Hybrid Nano-Junction

#### The Model

## 3. Non-Equilibrium Transport Through the Nano-Junction

#### Non-Equilibrium Green’s Function in the Floquet Basis

## 4. Results

#### 4.1. Electrical Current

#### 4.2. Thermoelectric Performance

## 5. Discussion

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Non-Equilibrium Green’s Function Method

#### Appendix A.1. Closed-Time Contour Formalism

**Figure A1.**Contour $\mathcal{C}={\gamma}^{+}\oplus {\gamma}^{-}\oplus {\gamma}^{M}$, considering ${t}_{0}=0$.

#### Appendix A.2. Non-Equilibrium Keldysh Formalism

**Figure A2.**Keldysh contour $\mathcal{C}={\gamma}^{+}\oplus {\gamma}^{-}$, considering ${t}_{0}\to -\infty $ and ${t}_{max}\to +\infty $.

## Appendix B. Thermoelectric Transport Coefficients and Onsager Relations

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**Figure 2.**Tunneling current through the nano-junction for zero (

**a**) and $H=0.3\Delta $ (

**b**) magnetic field. We set $\u03f5=-0.01\Delta ,T=0.1\Delta /{k}_{B}$ and the coupling constants ${\mathsf{\Gamma}}_{L}=0.7\Delta $ and ${\mathsf{\Gamma}}_{R}=0.05\Delta $ with ${j}_{0}=e\Delta /2\hslash $. Here $\Delta $ represents the BCS order parameter, chosen as the natural energy scale in the model.

**Figure 3.**(

**a**) A Cooper pair suffers an AR at the interface with the QD (red dashed line), and two AR’s at the TS interface and at the SC interface (dotted line) respectively, thus generating Andreev bound states; (

**b**) The Cooper pair splits into two electrons of opposite spin, in order to occupy the dot’s level, thus producing two backscattered holes at the TS interface. The AR of an electron (hole) is equivalent to the transfer of a single Cooper pair in (out) of the superconducting condensate.

**Figure 4.**Occupation number for different values of the magnetic field h, with effective Zeeman coupling $H=g{\mu}_{B}h$. The curves represent the dot occupation at $H=0$ (dotted line), $H=0.1\Delta $ (solid line) and $H=0.3\Delta $ (dashed line), respectively, with $\Delta $ the BCS order parameter, chosen as the natural energy scale in the model.

**Figure 5.**Thermoelectric figure of merit as a function of temperature at a fixed bias of 0.7 eV/$\Delta $, at different magnetic fields h, with $H=g{\mu}_{B}h$ the effective Zeeman coupling (

**a**). Notice the sharp response at $T\sim 0.5\Delta /{k}_{B}$ at an applied magnetic field $H=0.3\Delta $ (

**b**). Here $\Delta $ represents the BCS order parameter, chosen as the natural energy scale in the model.

**Figure 6.**Lorenz number for $H=0.3\Delta $. The minimum at $T\sim 0.5\Delta /{k}_{B}$ coincides with the sharp response in the thermoelectric figure of merit in Figure 5.

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Álamo, M.; Muñoz, E.
Thermoelectric Efficiency of a Topological Nano-Junction. *Entropy* **2018**, *20*, 366.
https://doi.org/10.3390/e20050366

**AMA Style**

Álamo M, Muñoz E.
Thermoelectric Efficiency of a Topological Nano-Junction. *Entropy*. 2018; 20(5):366.
https://doi.org/10.3390/e20050366

**Chicago/Turabian Style**

Álamo, Manuel, and Enrique Muñoz.
2018. "Thermoelectric Efficiency of a Topological Nano-Junction" *Entropy* 20, no. 5: 366.
https://doi.org/10.3390/e20050366