Helical thermoelectrics and refrigeration

The thermoelectric properties of a three-terminal quantum spin Hall (QSH) sample are examined. The inherent helicity of the QSH sample helps to generate a large charge power efficiently. Along with charge the system can be designed to work as a highly efficient spin heat engine too. The advantage of a helical over a chiral sample is that, while a multiterminal quantum Hall sample can only work as a quantum heat engine due to broken time reversal (TR) symmetry, a multiterminal QSH system can work effectively as both a charge or spin heat engine and as a charge or spin refrigerator as the TR symmetry is preserved.

Figure 1

3T QSH thermoelectric system. The blue dashed line represents the spin-up edge mode and the maroon solid line represents the spin-down edge mode. The voltage bias $\mathrm{\Delta }V$ is applied between terminals 1 and 2. The thermal gradient is applied at terminal 3, which acts as a voltage probe too.

Figure 2

Two types of energy-dependent transmission: (a) the QPC type, described by a saddle point potential, and (b) the resonant tunneling type, due to the presence of an antidot.

Figure 3

(a) Spin-up and (b) spin-down conductances (in units of $\frac{e^2}{h}$) are shown for QPCs at constriction $X$ and for resonant tunneling at constriction $Y$. (c) Spin-up and (d) spin-down Seebeck coefficients (in units of $\frac{k_B}{e}$) ($S^{\uparrow }$ and $S^{\downarrow }$) are shown for QPCs at constriction $X$ and resonant tunneling at constriction $Y$. Parameters are $\hslash {\omega }_0=0.1k_B\theta , \mathrm{\Gamma }=2k_B\theta$, and $\theta =0.1$ K.

Figure 4

(a) Maximum power for charge currents ${\mathcal{P}}_{\mathrm{ch}}^{\max }$ in units of $\frac{{\left(k_B\mathrm{\Delta }\theta \right)}^2}{h}$ and (b) efficiency at that power in units of ${\eta }_c$ for both in configuration 2. (c) Maximum power for spin currents ${\mathcal{P}}_{\mathrm{sp}}^{\max }$ in units of $\frac{{\left(k_B\mathrm{\Delta }\theta \right)}^2}{h}$ and (d) efficiency at that power in units of ${\eta }_c$ for both in configuration 1. Parameters are $\hslash {\omega }_0=0.1k_B\theta$ and $\theta =0.1$ K.

Figure 5

(a) Maximum cooling power for charge currents $J^Q\left({\eta }_{\mathrm{ch}}^{r,\max }\right)$ in units of $\frac{\left(k_B^2\theta \mathrm{\Delta }\theta \right)}{h}$ and (b) maximum efficiency ${\eta }_{\mathrm{ch}}^{r,\max }$ in units of ${\eta }_c^r$ for configuration 2. (c) Maximum cooling power for spin currents $J^Q\left({\eta }_{\mathrm{sp}}^{r,\max }\right)$ in units of $\frac{\left(k_B^2\theta \mathrm{\Delta }\theta \right)}{h}$ and (d) maximum efficiency ${\eta }_{\mathrm{sp}}^{r,\max }$ in units of ${\eta }_c^r$ for configuration 1. Parameters are $\hslash {\omega }_0=0.1k_B\theta$ and $\theta =0.1\mathrm{K}$.