Charge pumping in strongly coupled molecular quantum dots

The interaction between electrons and the vibrational degrees of freedom of a molecular quantum dot can lead to an exponential suppression of the conductance, an effect which is commonly termed Franck-Condon blockade. Here, we investigate this effect in a quantum dot driven by time-periodic gate voltages and tunneling amplitudes using nonequilibrium Green's functions and a Floquet expansion. Building on previous results showing that driving can lift the Franck-Condon blockade, we investigate driving protocols which can be used to pump charge across the quantum dot. In particular, we show that due to the strongly coupled nature of the system, the pump current at resonance is an exponential function of the drive strength.

Figure 1

Schematics of the Hamiltonian from Eq. (1). As a result of electron-vibron coupling, the vibrational spectrum has different ground-state energies depending on the electron number $n=〈d^{†}d〉$, with the unoccupied sector ($n=0$) having an energy increase of ${\lambda }^2/\mathrm{\Omega }$. The energy of the dot electron $\epsilon \left(t\right)$ and the couplings to the leads $V_{\text{L,}}\text{R}\left(t\right)$ are subject to periodic driving.

Figure 2

Keldysh integration contour $C$, running from $\tau =-\infty$ past the larger of the times $\tau$ and ${\tau }^{\prime }$ in the lower half plane before returning to $-\infty$ in the upper half plane.

Figure 3

Frequency-resolved dc current integrand ${〈I_{\alpha }\left(\omega \right)〉}_{00}$ through a biased polaron dot (${\mu }_{\text{L}}=-{\mu }_{\text{R}}=\mathrm{\Omega }/4$), evaluated near the zeroth-order resonance. We use Eq. (15) with Floquet matrix dimension $N_{\text{Fl}}=5$ for different values of the dot drive amplitude $A$ and a coupling drive amplitude of $\mathrm{\Delta }=0.6$, as well as a bare electronic tunneling rate $\mathrm{\Gamma }=0.002\mathrm{\Omega }$. Increasing $A$ causes substantial widening of the peak, signifying strong current increase.

Figure 4

Exponential FCB ($\lambda =4\mathrm{\Omega }$) lifting by driving the dot energy in the presence of a bias window ${\mu }_{\text{L}}=-{\mu }_{\text{R}}=\mathrm{\Omega }/4$. The Floquet result (blue) with dimension $N_{\text{Fl}}=5$ is compared with the result (red) from Ref. [33], corresponding to $N_{\text{Fl}}=1$. Inset: Increasing the coupling drive for fixed dot drive leads to a small-scale quadratic rise in tunneling current $I_{\mathrm{\Delta }}$.

Figure 5

Frequency-resolved dc current integrand ${〈I_{\alpha }\left(\omega \right)〉}_{00}$, through an unbiased polaron dot, evaluated near the peak locations (peak order $k$), using Eq. (15) with Floquet matrix dimension $N_{\text{Fl}}=21$. The coupling drive amplitude and bare tunneling rate are given by $\mathrm{\Delta }=0.6$ and $\mathrm{\Gamma }=0.002\mathrm{\Omega }$, respectively. The side resonances appearing outside the bias window are comparable in size to the central one at $k=0$, but different sides lead to partial cancellation upon integration. Inset: Modification of the shape of the central resonance for different values of phase shift.

Figure 6

The dc current ${〈I〉}^{\text{dc}}$ as a function of the dot drive amplitude $A$ for different values of phase difference ${\varphi }_{\text{L}}=-{\varphi }_{\text{R}}=\varphi /2$, and fixed coupling drive amplitude $\mathrm{\Delta }=0.6$. Dots show the simulation of Eq. (16), and lines show the numerical fit proportional to $\frac{A}{\mathrm{\Omega }}{e}^{A/\mathrm{\Omega }}sin\varphi$. The low-amplitude regime behaves in a way similar to purely electronic pumping, with exponential lifting of FCB ($\lambda =4\mathrm{\Omega }$) evident for larger amplitude. Inset: Sine-shaped phase dependence of ${〈I〉}^{\text{dc}}$ for a fixed value of $A$.