Double quantum dot as a minimal thermoelectric generator

Based on numerical renormalization group calculations, we demonstrate that experimentally realized double quantum dots constitute a minimal thermoelectric generator. In the Kondo regime, one quantum dot acts as an $n$-type and the other one as a $p$-type thermoelectric device. Properly connected, a capacitively coupled double quantum dot provides a miniature power supply utilizing the thermal energy of the environment.

Figure 1

Schematic figure of the considered thermoelectric double quantum dot device. For suitable gate voltages of the electrodes 1,2 and 3,4, a negative current of (quasi-)electrons flows in the up quantum dot and a positive current (quasi-)holes in the down quantum dot. Hence, if the two dots are connected on the right-hand side, the up and down quantum dots provide on the left-hand side the negative and positive pole of a power supply energized by excess heat from the environment (the micrograph of the quantum dot at the center has been reproduced from Ref. [8]).

Figure 2

Conductance $G$ (upper panel), Seebeck coefficient $S$ (central panel), and thermal conductivity $K^e$ (lower panel) for the double quantum dot of Ref. [8] as a function of the applied gate voltages $V_u$ and $V_d$. The left (right) panels show the transport through the up (down) quantum dot as calculated by NRG; see text for the parameters. The Kondo resonance develops in proximity of the degeneracy line between (1,0) and (0,1) electrons in the (up, down) quantum dot, see charge stability diagram in the upper panels. Close to the maximum in the conductance, the Seebeck coefficient of the up (down) quantum dot is positive (negative), indicating charge transport of opposite sign through the two dots.

Figure 3

Left (right) panel: Spectral function $A_{u\left(d\right)}$ of the up (down) quantum dot for $V_u=-254.6$mV and $V_d=-14.8$mV, corresponding to a pronounced thermopower ($\times$) in Fig. 2 (solid lines). The main spectral weight of the Kondo resonance is below (above) the Fermi level at $\omega =0$. For comparison the spectral function for a reduced thermopower ($+$) are shown (dashed lines).