Separation of heat and charge currents for boosted thermoelectric conversion

In a multiterminal device the (electronic) heat and charge currents can follow different paths. In this paper we introduce and analyze a class of multiterminal devices where this property is pushed to its extreme limits, with charge and heat currents flowing in different reservoirs. After introducing the main characteristics of this heat-charge current separation regime, we show how to realize it in a multiterminal device with normal and superconducting leads. We demonstrate that this regime allows us to control independently heat and charge flows and to greatly enhance thermoelectric performances at low temperatures. We analyze in detail a three-terminal setup involving a superconducting lead, a normal lead, and a voltage probe. For a generic scattering region we show that in the regime of heat-charge current separation both the power factor and the figure of merit $ZT$ are highly increased with respect to a standard two-terminal system. These results are confirmed for the specific case of a system consisting of three coupled quantum dots.

Figure 1

The heat-charge current separation scheme. A generic scattering region is connected to three reservoirs labeled by the letters S (superconducting lead), P (voltage probe), and N (normal metal lead). In the main text we assume the superconducting reservoir (S) to be the reference. In any case, as pointed by the arrows, only charge flows inside lead S whereas only heat flows inside lead P.

Figure 2

Plot of the ratio $\mathrm{\Lambda }/{\mathrm{\Lambda }}_0$ and thermopower $\mathcal{S}$ for fixed $K=10\left(3k_{B}T\right)/{\pi }^{2}h$ as a function of ${\rho }_2\left(0\right)$. Here we show that using only one parameter $\left[{\rho }_2\left(0\right)\right]$ we cannot control separately the thermopower and the ratio $\mathrm{\Lambda }=K/\left(GT\right)$. The other parameters are: ${\rho }_1\left(0\right)=0.8,g^{\prime }\left(0\right)=0.001{\left(k_{B}T\right)}^{-1},{\rho }_1^{\prime }\left(0\right)=0.03{\left(k_{B}T\right)}^{-1}$ and ${\rho }_2^{\prime }\left(0\right)=0.3{\left(k_{B}T\right)}^{-1}$.

Figure 3

Thermopower $\mathcal{S}$ from Eq. (13), with $K=10\left(3k_{B}T\right)/{\pi }^{2}h$ as a function of ${\rho }_2^{\prime }\left(0\right)$ [in units of ${\left(k_{B}T\right)}^{-1}$]. Using the additional degrees of freedom provided by the derivatives of the parameters [here we use ${\rho }_2^{\prime }\left(0\right)$], we gain the control of the thermopower $\mathcal{S}$ without affecting the ratio $\mathrm{\Lambda }$. The other parameters are: ${\rho }_1\left(0\right)=0.8,g^{\prime }\left(0\right)=0.001{\left(k_{B}T\right)}^{-1}$, and ${\rho }_1^{\prime }\left(0\right)=0.1{\left(k_{B}T\right)}^{-1}$.

Figure 4

(left) Probability histogram of the power factor $Q=G{\mathcal{S}}^2$ [in units of $k_B^2/h$] for the SPN system (black curve) and for the corresponding normal two-terminal system (red curve). The maximum value of $Q$ for the SPN setup is about $0.5 k_B^2/h$, while it is about $0.2 k_B^2/h$ for the two-terminal case. (right) Probability histogram of the figure of merit $ZT$ for the SPN system (black curve) and for the corresponding normal two-terminal case (red curve). The maximum of the SPN setup is just above 0.1, while it is about 0.05 for the two-terminal system.

Figure 5

Correlation between the values of $Q$ and $ZT$ for the same data as for Fig. 4: Each point corresponds to a given random realization. Red (black) points are relative to the two-terminal (SPN) setup. The green dashed curve represents the bound of Eq. (15) that holds for both the SPN and the two-terminal system. The green solid curve, instead, represents the bound of Eq. (16) that is the stronger bound given by the unitarity on the two-terminal system.

Figure 6

A three coupled QDs (“tridot”) system in the SPN setup. ${\gamma }_N$ and ${\gamma }_P$ label the coupling to the reservoirs N and P, respectively. The three QDs are coupled to a superconducting lead (S), with a fixed chemical potential.

Figure 7

Correlation between the power factor $Q=G{\mathcal{S}}^2$ and the figure of merit $ZT$ relative to “tridot” systems for the SPN setup (black points) and the two-terminal setup (red points). The green curve corresponds to the bound of Eq. (15), given by the unitarity of the scattering matrix, and sets a maximum value for $Q$ as a function of $ZT$. In panel (a) we show the correlation for the GOE and in panel (b) for the GUE. Both plots show that for the SPN setup both $Q$ and $ZT$ are one order of magnitude larger with respect to the corresponding values for the two-terminal system. Moreover it is possible to see that the increase of the performance is not due to the breaking of the TRS. The plot refers to ${10}^5$ Hamiltonian realizations, taking ${\gamma }_N={\gamma }_P=\gamma ={10}^3{k}_{B}T$.