Enhanced thermopower under a time-dependent gate voltage

We derive formal expressions of time-dependent energy and heat currents through a nanoscopic device using the Keldysh nonequilibrium Green function technique. Numerical results are reported for a metal-dot-metal junction where the dot level energy is abruptly changed by a step-shaped voltage pulse. Analytical linear responses are obtained for the time-dependent thermoelectric coefficients. We show that in the transient regime the Seebeck coefficient can be enhanced by an amount (as much as $40\%$) controlled by both the dot energy and the height of the voltage step.

Figure 1

Increase of the Seebeck coefficient in the transient regime of a metal-dot-metal junction: ${\mathrm{\varepsilon ̃}}_0=0.5$ with ${\mathrm{\gamma ̃}}_0=0.05$ (solid line), ${\mathrm{\gamma ̃}}_0=0.1$ (dashed line), and ${\mathrm{\gamma ̃}}_0=0.15$ (dotted line). We take symmetric barriers ${\Gamma }_R={\Gamma }_L$, $t_0=0$, ${\varepsilon }_F=0$, and $k_{B}T=0.1$. The unit of energy is $\Gamma$.

Figure 2

Left panel: Schematic representation of a metal-dot-metal junction with current directions for each lead. Right panel: Time dependence of the dot energy level. We define the chemical potentials and the temperatures of the left and right leads as ${\mu }_{L,R}={\varepsilon }_F\pm e\Delta V/2$ and $T_{L,R}=T\pm \Delta T/2$. The Fermi energy is ${\varepsilon }_F$ and $T=(T_R+T_L)/2$ is the average temperature.

Figure 3

Electric current (left panel) and heat current (right panel) through the left lead (solid lines) and the right lead (dashed lines) as a function of time $t$, for ${\Gamma }_R={\Gamma }_L$, $t_0=0$, ${\mathrm{\varepsilon ̃}}_0=0.5$, ${\mathrm{\gamma ̃}}_0=2.5$, $k_B{T}_L=1$, $k_B{T}_R=0$, and ${\mu }_{L,R}=\pm 0.5$. The insets show the dot occupation (left inset) and the dot heat (right inset). Dashed lines indicate the stationary limits at $t\to \infty$. The unit of energy is $\Gamma$.

Figure 4

Percentage of increase of the Seebeck coefficient maximum in the transient regime $S_{\max }^{\mathrm{tran}}$ as a function of ${\mathrm{\gamma ̃}}_0/{\mathrm{\varepsilon ̃}}_0$ for $k_{B}T=0.05$ (solid line), $k_{B}T=0.15$ (dashed line), and $k_{B}T=0.5$ (dotted line). We take ${\Gamma }_R={\Gamma }_L$ and ${\varepsilon }_F=0$. The unit of energy is $\Gamma$.