Performance analysis of an interacting quantum dot thermoelectric setup

In the absence of phonon contribution, a weakly coupled single orbital noninteracting quantum dot thermoelectric setup is known to operate reversibly as a Carnot engine. This reversible operation, however, occurs only in the ideal case of vanishing coupling to the contacts, wherein the transmission function is delta shaped, and under open-circuit conditions, where no electrical power is extracted. In this paper, we delve into the thermoelectric performance of quantum dot systems by analyzing the power output and efficiency directly evaluated from the nonequilibrium electric and energy currents across them. In the case of interacting quantum dots, the nonequilibrium currents in the limit of weak coupling to the contacts are evaluated using the Pauli master equation approach. The following fundamental aspects of the thermoelectric operation of a quantum dot setup are discussed in detail: (a) With a finite coupling to the contacts, a thermoelectric setup always operates irreversibly under open-circuit conditions, with a zero efficiency. (b) Operation at a peak efficiency close to the Carnot value is possible under a finite power operation. In the noninteracting single orbital case, the peak efficiency approaches the Carnot value as the coupling to the contacts becomes smaller. In the interacting case, this trend depends nontrivially on the interaction parameter $U$. (c) The evaluated trends of the maximum efficiency derived from the nonequilibrium currents deviate considerably from the conventional figure of merit $zT$-based results. Finally, we also analyze the interacting quantum dot setup for thermoelectric operation at maximum power output.

Figure 1

Nanocaloritronics of a quantum thermoelectric transport setup. (a) A typical thermoelectric setup comprises the central quantum system described by the Hamiltonian ${\widehat{H}}_S$ sandwiched between two reservoirs labeled hot (cold), $\alpha =H\left(C\right)$. When this central system is subject to an electrochemical potential gradient and a temperature gradient, the resulting current $J$ drives an electrical power $P=-J{V}_{\mathrm{app}}$ via the electrical leads. Equal contact couplings ${\Gamma }_H={\Gamma }_C$ are assumed throughout. (b) A single orbital quantum dot is parametrized by its single-particle energy level $\epsilon$ and the Coulomb interaction parameter $U$. Transport is represented as transitions between states of the many-particle spectrum with electron numbers differing by $\pm 1$. Transport channels then comprise the energy difference $\epsilon ,\epsilon +U$ between those states with electron numbers differing by $\pm 1$. (c) Schematic depicting the thermoelectric effect under open-circuit conditions: The built-in or Seebeck voltage $V_S$ enforces zero current in the circuit. Thermoelectric operation occurs when the applied voltage $V_{\mathrm{app}}\in [0,V_S]$, where the condition $V_{\mathrm{app}}=V_S$ enforces open-circuit operation. The thermoelectric efficiency defined in the operating region $V_{\mathrm{app}}\in [0,V_S]$ is strongly affected by the energy difference $\epsilon -{\mu }_{\alpha }$, the applied temperature gradient $\Delta T=T_H-T_C$, and the magnitude $U$ of the Coulomb interaction.

Figure 2

Power and efficiency in the noninteracting $(U=0)$ limit for Carnot efficiency ${\eta }_C=0.23$ (black solid) and ${\eta }_C=0.33$ (green circles). The temperature at the cold contact is set to $T_C=100\text{K}$, and the equilibrium energy-level placement is set to $\epsilon -{\mu }_H=2k_B{T}_H$ at $V_{\mathrm{app}}=0$. The couplings to the reservoirs are taken as $\hslash {\gamma }_H=\hslash {\gamma }_C=0.01\text{meV}\approx {10}^{-3}{k}_{B}T$. (a) Plot of extracted power as a function of the applied bias $V_{\mathrm{app}}$. The span of the operating region $V_{\mathrm{app}}\in [0,V_S]$ broadens with an increase in the applied temperature gradient $\Delta T=T_H-T_C$. Results from the sequential tunneling approximation (dotted) and the exact calculation (bold) are identical. (b) Corresponding plots of efficiency in the operating region. Under the sequential tunneling approximation (dotted), the efficiency maximizes at the Carnot efficiency ${\eta }_C$ when the applied bias equals the built-in voltage $(V_{\mathrm{app}}=V_S)$. This corresponds to the reversible thermoelectric configuration (Refs. 8, 12, and 13) (see text). In the exact calculation (bold), however, the efficiency drops to zero under open-circuit conditions. The efficiency at maximum power lies in an intermediate operating point corresponding to the maximum power $P_{\max }$ shown in (a).

Figure 3

Power and efficiency at finite $U$ for ${\eta }_C=0.23$. (a) Power extracted in the operating region. The span of the operating region in the case of $U=k_B{T}_C$ (green circles) can be different from that of the noninteracting case (black solid). In general, the quantity $J_Q^H$ (see inset) is not identically zero when the electric current vanishes. (b) Variation of the efficiency in the operating region for different values of $U$: (i) $U=0$ (black solid), (ii) $U=k_B{T}_C$ (green solid), and (iii) $U=2.5k_B{T}_C$ (gray dashed). Note that with finite $U$ such that $U>\hslash \Gamma$, the efficiency is identically zero when the electric current vanishes under open-circuit conditions $(V_{\mathrm{app}}=V_S)$. The efficiency also reaches a maximum ${\eta }_{\max }$ at finite power operation.

Figure 4

Variation of the maximum efficiency with Coulomb interaction $U$. The maximum efficiency is equal to the Carnot efficiency for $U=0$ and asymptotically approaches it when $U\gg k_{B}T$. It reaches a minimum around $U\approx 2.7k_{B}T$. This variation is shown for (a) ${\eta }_C=0.23$ and (b) ${\eta }_C=0.5$. Also shown in each case is the comparison between the nonequilibrium calculation (bold) and that based on the figure of merit $zT$ (brown dotted). Note that the difference between them becomes more prominent for larger values of ${\eta }_C$ or larger temperature gradients $\Delta T$, thereby making the transport nonlinear and hence the concept of $zT$ less useful. The inset in (a) shows the variation of $1/zT$ with $U$ for the chosen level configuration $\epsilon -{\mu }_H=2k_B{T}_H$ at $V_{\mathrm{app}}=0$.

Figure 5

Variation of maximum efficiency with respect to ${\eta }_C$. (a) The maximum efficiency (gray dotted) approaches the Carnot efficiency and deviates more from the figure of merit $zT$-based calculation (brown dotted) as the Carnot efficiency increases. (b) Plot of the percentage deviation of maximum efficiency between the nonequilibrium evaluation and the $zT$-based evaluation. The maximum efficiencies at each value of ${\eta }_C$ here are taken from the respective global minimum $(U\approx 2.7k_{B}T)$ in their variation with respect to $U$ in Fig. 4.

Figure 6

Comparison between the efficiency at maximum power and various other limits for the interacting quantum dot setup. The efficiency at maximum power is evaluated at $U\approx 2.7k_{B}T$, which corresponds to the maximum deviation from ${\eta }_C$ in Fig. 4. The nonequilibrium evaluation for our setup using both symmetric electrostatic coupling (black diamonds) and fully asymmetric electrostatic coupling (brown circles) is shown. The nonequilibrium evaluation assuming no electrostatic coupling to the cold contact resembles the curves discussed in Ref. 16. The nonequilibrium evaluation of the efficiency at maximum power, in general, is dependent on the details of the setup and need not be strictly bound by limits discussed in Ref. 19.