Optimal performance of voltage-probe quantum heat engines

The thermoelectric performance at a given output power of a voltage-probe heat engine, exposed to an external magnetic field, is investigated in linear irreversible thermodynamics. For the model, asymmetric parameter, general figures of merit, and efficiency at a given output power are analytically derived. Results show a tradeoff between efficiency and output power, and we recognize optimum-efficiency values at a given output power are enhanced compared to a Büttiker-probe heat engine due to the presence of a characteristic parameter, namely, $d_m$. Moreover, similar to a Büttiker-probe heat engine, the universal bounds on the efficiency are obtained, and the efficiency at a given output power can exceed the Curzon-Ahlborn limit. These findings have practical implications for the optimization of realistic heat engines and refrigerators. By controlling the values of the asymmetric parameter, the figures of merit, and $d_m$, it may be possible to design more efficient and powerful thermoelectric devices.

Figure 1

Schematic drawing of a voltage-probe heat engine in the presence of an external magnetic $\mathrm{field}\left(\mathbf{B}\right)$.

Figure 2

The function $H_m$ versus $x_m$ for different $d_m$ values are color coded. Dashed black lines display $H$ [Eq. (52)].

Figure 3

Ratio ${\eta }_m/{\eta }_m\left({\mathcal{P}}_{\max }\right)$ as a function of $\varepsilon$ for different values of $y_m$ and $d_m$ in the region of $x_m>0$. Dashed black lines show $\mathcal{P}/{\mathcal{P}}_{\max }$. (a) Demonstrates the normalized efficiency of a Büttiker-probe heat engine as a function of $\varepsilon$ for different values of $y$. (b), (c), (d), (e), (f) Show the normalized efficiency of a voltage-probe heat engine for $d_m=0.1,0.5,1.0,3.0,5.0$, respectively, and different values of $y_m$ are color coded.

Figure 4

The normalized efficiency as a function of power gains $\mathrm{\Delta }\mathcal{P}$ for the favorable case ${\varepsilon }_{+}$. The rest parameters are set the same as the Fig. 3.

Figure 5

Efficiency bound ${\eta }_m^{\mathrm{bound}}/{\eta }_{c,m}$ versus the asymmetry parameter $x_m$ and the power gain $\mathrm{\Delta }\mathcal{P}$ for (a): favorable case ${\varepsilon }_{+}$ and (b): unfavorable case ${\varepsilon }_{-}$. The intersecting points of the red-circle lines represent the Curzun-Ahlborn limit (CA) efficiency in the linear response regime ${\eta }_{\text{CA},m}={\eta }_{c,m}/2$. For comparison, the bounds obtained by Benenti et al. [21] only from thermodynamic entropy production $\dot{\mathcal{S}}>0$ (black solid line) and Brandner et al. [34] on three-terminal setup with considering the role of current conservation (black dashed lines) are included.

Figure 6

(a) Shows a voltage-probe system consisting of a triple quantum dot with a single energy level, subjected to the external magnetic field $\left(\mathbf{B}\right)$ and connected to three electronic reservoirs, labeled $L, P$, and $R$, in different temperatures $(T_L>T_P>T_R)$ and chemical potentials ${\mu }_L, {\mu }_P$, and ${\mu }_R$. (b)–(d) Exhibit the contour plots of $d_L, d_P$, and $d_{LP}$, respectively, as a function of $\varphi$ and $\delta$ for the system illustrated in (a). The atomic site energies of the quantum dots connected to the reservoirs $L,P$, and $R$ are $E_L-\mu =1.0k_{B}T, E_P-\mu =1.0k_{B}T$, and $E_R-\mu =1.0k_{B}T$, respectively, where $\mu ={\mu }_R=0, T=T_R$, and $k_{B}T=1$. Coupling parameter: ${\gamma }_L={\gamma }_P={\gamma }_R=\gamma =0.5k_{B}T$. Hopping energy parameter between two atomic sites: $t=t^{\prime }{e}^{i\varphi /3}$, where $\varphi =2\pi \mathrm{\Phi }/{\mathrm{\Phi }}_0$ (${\mathrm{\Phi }}_0$ is quantum flux) and $t^{\prime }=1.0k_{B}T$.