Seebeck Power Generation and Peltier Cooling in a Normal Metal-Quantum Dot-Superconductor Nanodevice

Abstract

We theoretically investigate the Seebeck and Peltier effect across an interacting quantum dot (QD) coupled between a normal metal and a Bardeen–Cooper–Schrieffer superconductor within the Coulomb blockade regime. Our results demonstrate that the thermoelectric conversion efficiency at optimal power output (optimized with respect to QD energy level and external serial load) in NQDS nanodevice can reach up to $58\%\eta _\text{C}$, where $\eta _\text{C}$ is Carnot efficiency, with output power $P_{\text {max}}\approx 35\,\text{fW}$ for temperature below the superconducting transition temperature. Further, the Peltier cooling effect is observed for a wide range of parameter regimes, which can be optimized by varying the background thermal energy, QD level energy, QD-reservoir tunneling strength, and bias voltage. The results presented in this study are within the scope of existing experimental capabilities for designing miniature hybrid devices that operate at cryogenic temperatures.

Appendix

In this section, we present the calculation for the Green’s functions and occupancy of the QD. In Nambu representation, we define the single particle retarded Green’s function of the QD as a $2\times 2$ matrices [45]

$$\begin{aligned} \mathbf{{G}}^{r}_{d}(\omega )={\left\langle \left\langle { \begin{pmatrix} d_{\uparrow }\\ d_{\downarrow }^{\dagger }\\ \end{pmatrix} \begin{pmatrix} d_{\uparrow }^{\dagger }&d_{\downarrow } \end{pmatrix} }\right\rangle \right\rangle }= \begin{pmatrix} \langle \langle {d_{\uparrow }|d_{\uparrow }^{\dagger }}\rangle \rangle &{} \langle \langle {d_{\uparrow }|d_{\downarrow }}\rangle \rangle \\ \langle \langle {d_{\downarrow }^{\dagger }|d_{\uparrow }^{\dagger }}\rangle \rangle &{} \langle \langle {d_{\downarrow }^{\dagger }|d_{\downarrow }}\rangle \rangle \\ \end{pmatrix}= \begin{pmatrix} G^r_{d,11}(\omega ) &{} G^r_{d,12}(\omega ) \\ G^r_{d,21}(\omega ) &{} G^r_{d,22}(\omega ) \\ \end{pmatrix} \end{aligned}$$

where the diagonal components of $\mathbf{{G}}^{r}_{d}(\omega )$ represents the single particle retarded Green’s function of electron with spin $\sigma =\uparrow$ and hole with spin $\sigma =\downarrow$ , respectively. The off-diagonal component represents the superconducting paring correlation on the QD. The Fourier transform of the single particle retarded Green’s function for QD

$$\begin{aligned} \begin{aligned} G_{d,11}^r(\omega )=\langle \langle {d_{\uparrow }|d^\dagger _{\uparrow }}\rangle \rangle \nonumber =-\frac{i}{2\pi }\lim _{\delta \rightarrow 0^{+}}\int {\theta (t)\langle \{d_{\uparrow }(t),d^\dagger _{\uparrow }(0)\}\rangle e^{i(\omega +i\delta )t}}dt \end{aligned} \end{aligned}$$

where $\theta (t)$ is Heaviside function, must satisfy the following EOM [38]

$$\begin{aligned} \omega \langle \langle {d_{\uparrow }|d^\dagger _{\uparrow }}\rangle \rangle =\langle \{d_{\sigma },d^\dagger _{\uparrow }\}\rangle +\langle \langle [d_{\uparrow },H]|d^\dagger _{\uparrow }\rangle \rangle . \end{aligned}$$

By evaluating different commutator and anti-commutator brackets we drive the following coupled equations for the single particle Green’s functions

$$\begin{aligned} \left\{ \omega -\epsilon _d-\sum _k\frac{|\mathcal {V}_{k,N}|^2}{\omega -\epsilon _{k,N}}-\sum _k|\mathcal {V}_{k,S}|^2\left( \frac{|u_k|^2}{\omega -E_k}+\frac{|v_k|^2}{\omega +E_k}\right) \right\} \langle \langle {d_{\uparrow }|d_{\uparrow }^{\dagger }}\rangle \rangle = \\ 1+\left\{ {\sum _k|\mathcal {V}_{k,S}|^2u^*_kv_k\left( \frac{1}{\omega -E_k}-\frac{1}{\omega +E_k}\right) }\right\} \langle \langle {d_{\downarrow }^{\dagger }|d_{\uparrow }^{\dagger }}\rangle \rangle +U\langle \langle {d_{\uparrow }n_{\downarrow }|d_{\uparrow }^{\dagger }}\rangle \rangle \end{aligned}$$

$$\begin{aligned} \left\{ \omega +\epsilon _d-\sum _k\frac{|\mathcal {V}_{k,N}|^2}{\omega +\epsilon _{k,N}}-\sum _k|\mathcal {V}_{k,S}|^2\left( \frac{|u_k|^2}{\omega +E_k}+\frac{|v_k|^2}{\omega -E_k}\right) \right\} \langle \langle {d_{\downarrow }^{\dagger }|d_{\uparrow }^{\dagger }}\rangle \rangle = \\ \left\{ {\sum _k|\mathcal {V}_{k,S}|^2u_kv^*_k\left( \frac{1}{\omega -E_k}-\frac{1}{\omega +E_k}\right) }\right\} \langle \langle {d_{\uparrow }|d_{\uparrow }^{\dagger }}\rangle \rangle -U\langle \langle {d_{\downarrow }^{\dagger }n_{\uparrow }|d_{\uparrow }^{\dagger }}\rangle \rangle \end{aligned}$$

$$\begin{aligned} \left\{ \frac{\omega -\epsilon _d-U}{\langle {n_{\downarrow }}\rangle }\right\} \langle \langle {d_{\uparrow }n_{\downarrow }|d_{\uparrow }^{\dagger }}\rangle \rangle = 1+\left\{ {\sum _k|\mathcal {V}_{k,S}|^2u^*_kv_k\left( \frac{1}{\omega -E_k}-\frac{1}{\omega +E_k}\right) }\right\} \langle \langle {d_{\downarrow }^{\dagger }|d_{\uparrow }^{\dagger }}\rangle \rangle \\ +\left\{ \sum _k\frac{|\mathcal {V}_{k,N}|^2}{\omega -\epsilon _{k,N}}+\sum _k|\mathcal {V}_{k,S}|^2\left( \frac{|u_k|^2}{\omega -E_k}+\frac{|v_k|^2}{\omega +E_k}\right) \right\} \langle \langle {d_{\uparrow }|d_{\uparrow }^{\dagger }}\rangle \rangle \end{aligned}$$

$$\begin{aligned} \left\{ \frac{\omega +\epsilon _d+U}{\langle {n_{\uparrow }}\rangle }\right\} \langle \langle {d_{\downarrow }^{\dagger }n_{\uparrow }|d_{\uparrow }^{\dagger }}\rangle \rangle = \left\{ {\sum _k|\mathcal {V}_{k,S}|^2u_kv^*_k\left( \frac{1}{\omega -E_k}-\frac{1}{\omega +E_k}\right) }\right\} \langle \langle {d_{\uparrow }|d_{\uparrow }^{\dagger }}\rangle \rangle \\ +\left\{ \sum _k\frac{|\mathcal {V}_{k,N}|^2}{\omega +\epsilon _{k,N}}+\sum _k|\mathcal {V}_{k,S}|^2\left( \frac{|u_k|^2}{\omega -E_k}+\frac{|v_k|^2}{\omega +E_k}\right) \right\} \langle \langle {d_{\downarrow }^{\dagger }|d_{\uparrow }^{\dagger }}\rangle \rangle \end{aligned}$$

The terms with summations over k appearing in above equations can be simplified by replacing $\sum _k \rightarrow \int {\rho (\epsilon )d\epsilon }$ and then solving these expressions using the complex contour integration in the wide-band limit. Finally after solving above coupled equations, we arrive at the expression for the retarded Green’s function of electron with spin $\sigma =\uparrow$ and off-diagonal superconducting pairing correlation on the QD, i.e.,

$$\begin{aligned} G_{d,11}^{r}(\omega ) =\frac{\alpha _1(\omega )}{\omega -\epsilon _d+\left( \frac{i\Gamma _N}{2}+\beta (\omega )\right) \alpha _1(\omega )-\frac{\alpha _1(\omega )\,\alpha _2(\omega )\left( \frac{\Delta }{|\omega |}\beta (\omega )\right) ^2}{\omega +\epsilon _d+\left( \frac{i\Gamma _N}{2}+\beta (\omega )\right) \alpha _2(\omega )}} \end{aligned}$$

$$\begin{aligned} G_{d,21}^{r}(\omega ) =\frac{\alpha _2(\omega )\left( \frac{\Delta }{|\omega |}\beta (\omega )\right) }{\omega +\epsilon _d+\left( \frac{i\Gamma _N}{2}+\beta (\omega )\right) \alpha _2(\omega )}\times G_{d,11}^{r}(\omega ) \end{aligned}$$

where

$${\alpha _1(\omega )=1+\frac{U\langle {n_{\downarrow }}\rangle }{\omega -\epsilon _d-U}},\quad {\alpha _2(\omega )=1+\frac{U\langle {n_{\uparrow }}\rangle }{\omega +\epsilon _d+U}}$$

and

$$\begin{aligned} \beta (\omega )=\frac{\Gamma _S}{2}\rho _S(\omega )=\frac{\Gamma _S}{2}\frac{\omega }{\sqrt{\Delta ^2-\omega ^2}}\theta (\Delta -|\omega |)+\frac{i\Gamma _S}{2}\frac{|\omega |}{\sqrt{\omega ^2-\Delta ^2}}\theta (|\omega |-\Delta ) \end{aligned}$$

where $\rho _S(\omega )$ is the modified BCS density of states, with the real part accounting for the Andreev bound states within the superconducting gap. The other matrix elements is given by $G_{d,22}^{r}(\omega )=-G_{d,11}^{r}(-\omega )^{*}$ and $G_{d,12}^{r}(\omega )=G_{d,21}^{r}(-\omega )^{*}$. These retarded Green’s functions allow us to calculate the advanced and lesser/greater Green’s functions and eventually the thermoelectric transport properties.

The average occupancy on the quantum dot ($\langle {n_{\uparrow }}\rangle$=$\langle {n_{\downarrow }}\rangle$ for non-magnetic system) is calculated using the self-consistent integral equation of the form

$$\begin{aligned} \langle {n_{\sigma }}\rangle =\frac{-i}{2\pi }\int ^{\infty }_{-\infty }G^{<}_{d,11}(\omega ) d\omega \end{aligned}$$

where the lesser Green’s function $G^{<}_{d}$ obeys the Keldysh equation [38, 39]

$$\begin{aligned} \mathbf{{G}}^{<}_{d\sigma }(\omega )=\mathbf{{G}}^{r}_{d\sigma }(\omega )\mathbf{{\Sigma }}^{<}(\omega )\mathbf{{G}}^{a}_{d\sigma }(\omega ). \end{aligned}$$

The advanced Green’s function matrix is $\mathbf{{G}}^{a}_{d\sigma }(\omega )=\left[ \mathbf{{G}}^{r}_{d\sigma }(\omega )\right] ^{\dagger }$ and the lesser self-energy matrix is obtained using Ng ansatz [38, 46], i.e.,

$$\begin{aligned} \mathbf{{\Sigma }}^{<}(\omega )= -\sum _{\alpha \in N,S}\left[ \mathbf{{\Sigma }}^{r}_{\alpha }-\mathbf{{\Sigma }}^{a}_{\alpha }\right] f_{\alpha }(\omega -\mu _{\alpha }) \end{aligned}$$

This ansatz satisfies the continuity equation in steady state, allowing us to derive the lesser Green’s function to examine the transport properties.

Now using retarded and advanced self-energy, we get

$$\begin{aligned} \mathbf{{\Sigma }}^{<}(\omega )= \begin{pmatrix} \;\Sigma ^{<}_{11}(\omega ) &{}\quad \Sigma ^{<}_{12}(\omega )\;\\ \;\Sigma ^{<}_{21}(\omega ) &{}\quad \Sigma ^{<}_{22}(\omega )\; \end{pmatrix} \end{aligned}$$

with

$$\begin{aligned} \Sigma ^{<}_{11}(\omega )&= -i\Gamma _{N}f_N(\omega -\mu _N)-\frac{i\Gamma _S|\omega |}{\sqrt{\omega ^2-\Delta ^2}} \theta (|\omega |-\Delta )f_S(\omega -\mu _S)\\ \Sigma ^{<}_{12}(\omega )&= \Sigma ^{<}_{21}(\omega )=\frac{i\Gamma _{S}\Delta }{\sqrt{\omega ^2-\Delta ^2}}\theta (|\omega |-\Delta )f_S(\omega -\mu _S)\\ \Sigma ^{<}_{22}(\omega )&= -i\Gamma _{N}f_N(\omega +\mu _N)-\frac{i\Gamma _S|\omega |}{\sqrt{\omega ^2-\Delta ^2}} \theta (|\omega |-\Delta )f_S(\omega -\mu _S) \end{aligned}$$

Now, multiplying matrices in the expression of $\mathbf{{G}}^{<}_{d\sigma }(\omega )$, we get the lesser Green’s function for electrons on the QD as

$$\begin{aligned} \begin{aligned} G^{<}_{d,11}(\omega )&= i\Gamma _{N}f_N(\omega -\mu _N)|G_{d,11}^{r}(\omega )|^2+ i\Gamma _{N}f_N(\omega +\mu _N)|G_{d,12}^{r}(\omega )|^2+\\&\frac{i\Gamma _S|\omega |}{\sqrt{\omega ^2-\Delta _2}}\;\theta (|\omega |-\Delta )f_S(\omega -\mu _S)\times \\&\left[ |G_{d,11}^{r}(\omega )|^2+|G_{d,12}^{r}(\omega )|^2 -\frac{2\Delta }{|\omega |} Re\left( G_{d,11}^{r}(\omega ).G_{d,12}^{a}(\omega )\right) \right] \end{aligned} \end{aligned}$$

where $f_{\alpha \in N,S}(\omega \mp \mu _{\alpha })=\left[ {exp((\omega \mp \mu _{\alpha })/k_BT_{\alpha })+1}\right] ^{-1}$ is the Fermi-Dirac distribution function of reservoirs with temperature $T_{\alpha }$ and chemical potential $\pm \mu _{\alpha }$ (measured from Fermi level $\mu _f=0$).