Adiabatic response and quantum thermoelectrics for ac-driven quantum systems

We generalize the theory of thermoelectrics to include coherent electron systems under adiabatic ac driving, accounting for quantum pumping of charge and heat, as well as for the work exchanged between the electron system and driving potentials. We derive the relevant response coefficients in the adiabatic regime and show that they obey generalized Onsager reciprocity relations. We analyze the consequences of our generalized thermoelectric framework for quantum motors, generators, heat engines, and heat pumps, characterizing them in terms of efficiencies and figures of merit. We illustrate these concepts in a model for a quantum pump.

Figure 1

Sketch of the setup. A coherent quantum conductor is driven by time-periodic potentials and connected to two reservoirs biased by (a) a chemical-potential difference $\delta \mu$ or (b) a temperature gradient $\delta T$ or both. Charge ${\overline{\dot{N}}}_R$, heat ${\overline{\dot{Q}}}_R$, and power $\overline{\dot{W}}$ are exchanged between the reservoirs and the ac sources. The solid (dashed) arrow indicates (a) the motor (generator) mode of the device and (b) the heat-engine (heat-pump) mode.

Figure 2

Sketch of the device. A single-level quantum dot ($m=2$) is defined by two tunnel barriers ($m=1,3$). They are driven by periodic gate potentials $V_m\left(t\right)=V_j^{0}cos(\omega t+{\delta }_m)$, with $V_1^0=V_3^0=4, V_2^0=23, {\delta }_1=0, {\delta }_2=\pi /2$, and ${\delta }_3=\pi$. The tunneling amplitudes between barriers and the dot are $w=1$ and $w_L=w_R=0.7$ between barriers and reservoirs. The barriers and the dot are modeled by a discrete chain of $N=3$ sites with local energies ${\varepsilon }_1={\varepsilon }_3=3.3$ and ${\varepsilon }_2=-1$, respectively. The reservoirs have ${\mu }_L=\mu ,{\mu }_R=\mu -\delta \mu$, and temperature $T$.

Figure 3

Maximum efficiency ${\eta }^{\max }$ and transport coefficients at $T=0$ for the motor ($M$) or generator ($G$) modes. Inset: ${\eta }^{\max }$ for generator (motor) with $\mu =-0.7 (\mu =7.2)$.