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The normal metal-quantum dots-superconductor hybrid system is a good platform to study the mechanism of thermoelectric conversion. In terms of non-equilibrium Keldysh Green's function formalism and linear response theory, the charge and spin thermoelectric transport characteristics of a normal-double quantum dot-superconductor hybrid system with spin-orbit coupling are studied in this paper. We deeply discuss the relationship between thermoelectric coefficients and the system parameters, and find both charge and spin thermoelectric coefficients exhibit distinct symmetry characteristics in the parameter space composed of temperature and energy. An increase in temperature leads to a decrease in the conductance within the energy gap, which is attributed to a reduction in Andreev transport. However, outside the energy gap, the conductance gradually increases, and the thermal conductance is gradually enhanced. This is because more quasiparticles outside the energy gap participate in thermoelectric transport, and a large charge thermopower is generated in the region far from the energy gap. It is found that the thermoelectric figure of merit is greater than 1, indicating a strong violation of the Wiedemann-Franz law. With the increase of temperature, the large spin thermopower as well as spin thermoelectric figure of merit can be obtained outside the energy gap. The charge (spin) thermopower and the thermoelectric figure of merit show the rich evolutionary characteristics as functions of the energy level and the Zeeman energy. With the disappearance of the charge thermopower, the spin thermopower still has a finite value, which leads to the emergence of a pure spin Seebeck effect. This is helpful for designing a pure spin current thermoelectric generator. Due to a competitive mechanism between the spin-orbit coupling effect and the Zeeman field, thermoelectric coefficients are decreased with increasing the strength of spin-orbit interaction, but one still can obtain the spin thermoelectric quantities which meet the practical needs by regulating the strength of spin-orbit coupling and the Zeeman energy. The evolution pattern of the thermoelectric coefficients in the energy space indicates that the enhancement of thermoelectric conversion efficiency can be achieved by modulating the energy levels of double quantum dots. In addition, this hybrid system can function as a heat engine to achieve the conversion of heat to work. Although its power and efficiency do not evolve synchronously, in some parameter regions, people can still obtain the thermodynamic performance that meets practical needs. The research results of this paper hold theoretical and practical significance for understanding the thermoelectric transport and thermodynamic performance of hybrid thermoelectric systems.
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Keywords:
- Hybrid quantum dot system /
- Spin-orbit coupling /
- Thermoelectric transport /
- Power and efficiency
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图 1 混合型双量子点结构模型. N表示与量子点1连接的金属电极, S表示与量子点2连接的超导电极. $t_{c}$为量子点之间的耦合强度, θ为自旋-轨道耦合场${\alpha}$与z轴方向的外磁场B 之间的夹角
Figure 1. The model of hybrid double quantum dots, where N is a normal-metal electrode that is attached to the quantum dot 1, and S represents the superconducting electrode that is connected with the quantum dot 2. $t_{c}$ is the interdot coupling strength, and θ denotes the included angle between the spin-orbit coupling field ${\alpha}$ and the external field B along the z axis.
图 2 (左列)电荷热电系数: (a)电导$G_{c}$、(b)热导$\kappa_{e}$、(c)热功率$S_{c}$和 (d)品质因子$Z_{c}T$作为能级$\varepsilon_{d}$与温度$k_{B}T$的函数. 不同温度条件下, 电导(a')$G_{c}$、(b')热导$\kappa_{e}$、(c')热功率$S_{c}$和 (d')品质因子$Z_{c}T$的截面图被呈现在右列. 其他参数选为$\alpha=0.2\Delta$, $\Delta_{z}=\Delta$以及$\theta=\pi/2$
Figure 2. Charge thermoelectric coefficients: (a) conductance $G_{c}$, (b) heat conductance $\kappa_{e}$, (c) thermopower $S_{c}$ and (d) figure of merit $Z_{c}T$ as a function of the energy level$\varepsilon_{d}$ and temperature $k_{B}T$(left column). For different temperatures, the cross sections of (a') conductance $G_{c}$, (b') heat conductance $\kappa_{e}$, (c') thermopower $S_{c}$ and (d') figure of merit $Z_{c}T$ are shown in the right column. The other parameters are $\alpha=0.2\Delta$, $\Delta_{z}=\Delta$, and $\theta=\pi/2$.
图 3 (左列) 自旋热电系数: (a)热功率$S_{s}$和 (b)品质因子$Z_{s}T$作为能级$\varepsilon_{d}$与温度$k_{B}T$的函数. 不同温度条件下, (a')热功率$S_{s}$和 (b') 品质因子$Z_{s}T$的截面图被呈现在右列. 其他参数选为$\alpha=0.2\Delta$, $\Delta_{z}=\Delta$以及$\theta=\pi/2$
Figure 3. Spin thermoelectric coefficients: (a) thermopower $S_{s}$ and (b) figure of merit $Z_{s}T$ as a function of the energy level$\varepsilon_{d}$ and temperature $k_{B}T$(left column). For different temperatures, the cross sections of (a') thermopower $S_{c}$ and (b') figure of merit $Z_{s}T$ are shown in the right column. The other parameters are $\alpha=0.2\Delta$, $\Delta_{z}=1\Delta$, and $\theta=\pi/2$.
图 4 (a)电荷热功率$S_{c}$和(b)自旋热功率$S_{s}$作为能级$\varepsilon_{d}$与塞曼能$\Delta_{z}$的函数. (c)不同塞曼能条件下, $S_{c}$(实线)和$S_{s}$(虚线)作为$\varepsilon_{d}$的函数. (d)电荷品质因子$Z_{c}T$和(e)自旋品质因子$Z_{s}T$作为能级$\varepsilon_{d}$与塞曼能$\Delta_{z}$的函数. (f)不同塞曼能条件下, $Z_{c}T$(实线)和$Z_{s}T$(虚线)作为$\varepsilon_{d}$的函数. 其他参数选为$\alpha=0.2\Delta$, $k_{B}T=0.3\Delta$以及$\theta=\pi/2$
Figure 4. (a) Charge thermopower $S_{c}$ and (b) spin thermopower $S_{s}$ as a function of the energy level$\varepsilon_{d}$ and the Zeeman energy $\Delta_{z}$. (c) For different Zeeman energies, $S_{c}$ and $S_{s}$ as a function of the energy level$\varepsilon_{d}$. (d) Charge figure of merit $Z_{c}T$ and (e) spin figure of merit $Z_{s}T$ as a function of the energy level$\varepsilon_{d}$ and the Zeeman energy $\Delta_{z}$. (f) For different Zeeman energies, $Z_{c}T$ and $Z_{s}T$) as a function of the energy level$\varepsilon_{d}$. The other parameters are $\alpha=0.2\Delta$, $k_{B}T=0.3\Delta$, and $\theta=\pi/2$.
图 5 (左列) 自旋热电系数: (a)热功率$S_{s}$和 (b)品质因子$Z_{s}T$作为能级$\varepsilon_{d}$与自旋-轨道耦合强度α的函数. 不同α条件下, (a')热功率$S_{s}$和 (b') 品质因子$Z_{s}T$的截面图被呈现在右列. 其他参数选为$\Delta_{z}=0.5\Delta$, $k_{B}T=0.3\Delta$以及$\theta=\pi/2$
Figure 5. Spin thermoelectric coefficients: (a) thermopower $S_{s}$ and (b) figure of merit $Z_{s}T$ as a function of the energy level$\varepsilon_{d}$ and spin-orbit coupling strength α(left column). For different temperatures, the cross sections of (a') thermopower $S_{c}$ and (b') figure of merit $Z_{s}T$ are shown in the right column. The other parameters are $\Delta_{z}=0.5\Delta$, $k_{B}T=0.3\Delta$, and $\theta=\pi/2$.
图 6 电荷热电系数: (a)热功率$S_{c}$和 (b)品质因子$Z_{c}T$作为量子点能级$\varepsilon_{1}$与$\varepsilon_{2}$的函数. 自旋热电系数: (c)热功率$S_{s}$和 (d) 品质因子$Z_{s}T$作为量子点能级$\varepsilon_{1}$与$\varepsilon_{2}$的函数. 其他参数为$\alpha=0.2\Delta$, $\Delta_{z}=\Delta$, $k_{B}T=0.3\Delta$以及$\theta=\pi/2$
Figure 6. Charge thermoelectric coefficients: (a) thermopower $S_{c}$ and (c) figure of merit $Z_{c}T$ as a function of the quantum dot's levels $\varepsilon_{1}$ and $\varepsilon_{2}$. Spin thermoelectric coefficients: (b) thermopower $S_{s}$ and (d) figure of merit $Z_{s}T$ as a function of the quantum dot's levels $\varepsilon_{1}$ and $\varepsilon_{2}$. The other parameters are $\alpha=0.2\Delta$, $k_{B}T=0.3\Delta$, and $\theta=\pi/2$.
图 7 (a)最大功率$P_{max}$ (以$P_{0}=(k_{B}\Delta T)^{2}/h$为单位) 和 (b) 最大功率时的效率$\eta_{maxP}$ (以 卡诺效率$\eta_{c}$为单位) 作为量子点能级$\varepsilon_{1}$与$\varepsilon_{2}$的函数. 其他参数为$\alpha=0.2\Delta$, $\Delta_{z}=\Delta$, $k_{B}T=0.3\Delta$以及$\theta=\pi/2$
Figure 7. (a) Maximum power $P_{max}$ (in units of $P_{0}=(k_{B}\Delta T)^{2}/h$) and (b) efficiency at maximum power $\eta_{maxP}$ (in units of Carnot efficiency $\eta_{c}$) as a function of the quantum dot's levels $\varepsilon_{1}$ and $\varepsilon_{2}$. The other parameters are $\alpha=0.2\Delta$, $\Delta_{z}=\Delta$, $k_{B}T=0.3\Delta$ and $\theta=\pi/2$.
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