Nonadiabatic rectification and current reversal in electron pumps

Pumping of electrons through nanoscale devices is one of the fascinating achievements in the field of nanoscience with a wide range of applications. To optimize the performance of pumps, operating them at high frequencies is mandatory. We consider the influence of fast periodic driving on the average charge transferred through a quantum dot. We show that it is possible to reverse the average current by sweeping the driving frequency only. In connection with this, we observe a rectification of the average current for high frequencies. Since both effects are very robust, as corroborated by analytical results for harmonic driving, they offer a new way of controlling electron pumps.

Figure 1

Schematic representation of the pumping device: The level $\varepsilon$ and the couplings ${\Gamma }_{\mathrm{L},\mathrm{R}}$ to left and right reservoir oscillate in time, inducing periodic charging and discharging. Time dependence of $\varepsilon$ (black line, left axis) and ${\Gamma }_{\mathrm{L},\mathrm{R}}$ (red and blue lines, right axis) according to Eqs. (1) for a phase delay of $\delta =3\pi /4$.

Figure 2

Pumped charge $Q$ per period according to Eq. (2) as a function of frequency $\Omega$ and phase shift $\delta$ obtained from an NEGF calculation. The parameters defining the driving and the reservoirs are as follows: ${\varepsilon }_0=0$, ${\varepsilon }_1=20\Gamma$, ${\eta }_{\mathrm{L},\mathrm{R}}=6$, ${\Gamma }_{\mathrm{L},\mathrm{R}}^0=\Gamma e^{-{\eta }_{\alpha }}/2$, ${\mu }_{\mathrm{L},\mathrm{R}}=0$, and $k_{\mathrm{B}}T=\Gamma /10$.

Figure 3

Pumped charge per period $Q$ for the same parameters as in Fig. 2. NEGF (black lines) and rate-equation (broken orange) results are indistinguishable. a) Current reversal for the phase shift $\delta =-3\pi /4$. b) Current rectification for the driving frequency $\Omega =\Gamma /10$. Parameters for which the time evolution is shown in Fig. 4 are marked by black points. Result for the harmonic model [Eq. (12) with ${\Gamma }_0={\Gamma }_1=\Gamma /20$] are shown as indigo/long-dashed line.

Figure 4

Time evolution within one period $\tau =2\pi /\Omega$ for three different parameters sets $(\Omega ,\delta )$. Note that the time is given in units of ${\Omega }^{-1}$, which implies that absolute times in column a) are by a factor of 10${}^3$ larger than in columns b) and c). Lower row: Currents through tunnel barriers $J_{\mathrm{L}}\left(t\right)$ and $J_{\mathrm{R}}\left(t\right)$ (red and blue lines). Upper row: Dot occupation $n\left(t\right)$ with its scale on the left side and transferred charge ${\int }^{t}d{t}^{\prime }(J_{\mathrm{L}}\left(t^{\prime }\right)-J_{\mathrm{R}}\left(t^{\prime }\right))/2$ with its scale on the right side. The gray area indicates the time span when the dot is charged, i.e., $\varepsilon \left(t\right)<0$. Red/blue arrows mark the times when left/right couplings are maximal.