Dynamical responses of quantum dots by pulsed fields

Tunneling currents of a quantum dot induced by pulsed fields are investigated theoretically using time-dependent tunneling rates. Taking into account nonadiabatic effects, we present simple analytic expressions for tunneling currents, which can be used to study the evolution of the occupation probabilities in the quantum dot as well as extract system parameters governing tunneling measurements.

Figure 1

A systematic drawing of a tunneling geometry is shown. Tunneling barriers are model with capacitance $C_S$ and $C_D$, respectively, and a square pulsed field $V\left(t\right)$ is applied to the drain with respect to the source lead.

Figure 2

For ${\gamma }_D={\gamma }_S=0.5\gamma$ and $V_0=2070\hslash \gamma ∕e$, we plot the tunneling rates available to each level in (a). For electron-phonon relaxation rates of $w=0$ (thin), $5\gamma$ (thick line), occupation probabilities and tunneling currents are shown as a function of time in (b) and (c), respectively. Here, a pulse width ${\tau }_p$ are assumed to be $1∕\gamma$.

Figure 3

For a symmetric geometry $C_D=C_S$ and ${\gamma }_D={\gamma }_S$, the currents $i_{\mathrm{dc}}$ (defined in the text) are plotted as a function of pulse duration for several pulse amplitudes. The dotted lines represent results in the presence of the energy relaxation of $w=5\gamma$, however, in the most cases they are not resolved from those of $w=0$ except for $V_0=2080$. Here, currents and tunneling rates are measured in units of $e\gamma$ and $\gamma$, respectively.

Figure 4

We compare calculated results (solid lines) of Fig. 3 with the current expressions of Eqs. (21, 22). For pulse amplitudes of $V_0=2080$, $1960\hslash \gamma ∕e$ in (a) and (b), respectively, approximated currents are represented by dashed and dotted lines. In the case of (a), the difference between solid and dashed lines are not resolved.