Photon-assisted electron transport through a three-terminal quantum dot system with nonresonant tunneling channels

We have studied the electron transport through a quantum dot coupled to three leads in the presence of external microwave fields supplied to different parts of the considered mesoscopic system. Additionally, we introduced a possible nonresonant tunneling channel between leads. The quantum dot charge and currents were determined in terms of the appropriate evolution operator matrix elements and under the wide-band limit the analytical formulas for time-averaged currents and differential conductance were obtained. We have also examined the response of the considered system on the rectangular-pulse modulation imposed on different quantum dot-lead barriers as well as the time dependence of currents flowing in response to suddenly removed (or included) connection of a quantum dot with one of the leads.

Figure 1

Schematic picture of the QD coupled with three leads. It can serve as a possible STM experimental setup when the $L$ lead corresponds to the STM tip.

Figure 2

The averaged current $⟨j_L\left(t\right)⟩$ against the QD energy level ${\epsilon }_d$ in the system of a QD coupled with three (solid line) or two (broken line) leads. The microwave field is applied to the QD with the amplitude ${\Delta }_d=6$ and frequency $\omega =5$. The lead chemical potentials ${\mu }_L$, ${\mu }_{R_1}$, and ${\mu }_{R_2}$ are equal 0.2, 0.0, and 0.0, respectively, and the coupling between $L$ and $R_1,R_2$ leads is absent, $V_{LR}=0$. All energies are in $\Gamma$ units and the current in $e\Gamma ∕\hslash$ units.

Figure 3

The averaged current $⟨j_L\left(t\right)⟩$ against ${\epsilon }_d$ in the system of a QD coupled with three (thick lines) or two (thin lines) leads. The upper (lower) panel corresponds to $x=0(x=0.28),x$ being the measure of the coupling between leads, $x=\pi V_{LR}∕D$. In a QD–three leads system ${\mu }_L=0.1$, ${\mu }_{R_2}=-0.1$, and ${\mu }_{R_1}=-0.1$, 4, and 10 (solid, broken, and dotted curves, respectively). In a QD coupled with two leads ${\mu }_L=0.1,{\mu }_R=-0.1,4$, and 10 (solid, broken, and dotted curves, respectively). $\omega =5,{\Delta }_d=8,{\Delta }_{R_1}={\Delta }_{R_2}={\Delta }_L=0$. The values of solid curves have been multiplied by a factor of 10 for the illustrating purposes.

Figure 4

The averaged current $⟨j_L\left(t\right)⟩$ against ${\epsilon }_d$ in the system of a QD coupled with three (thick lines) or two (thin lines) leads. The solid, broken, and dotted curves correspond to $x=0,0.28$, and 0.7, respectively. ${\Delta }_d=8,{\Delta }_{R_1}={\Delta }_{R_2}={\Delta }_L=0,{\mu }_L=-{\mu }_{R_1}=-{\mu }_{R_2}=0.1,\omega =5$.

Figure 5

The averaged current $⟨j_L\left(t\right)⟩$ against ${\epsilon }_d$ for the QD-2l systems: ${\mu }_L=-{\mu }_R=0.1$—thin solid lines and ${\mu }_L=0.1,{\mu }_R=4$—thick solid lines and the QD-3l system: ${\mu }_L=-{\mu }_{R_1}=0.1,{\mu }_{R_2}=4$—broken lines. The upper (lower) part corresponds to $x=0(x=0.28)$ and $\omega =5,{\Delta }_d=1,{\Delta }_{R_1}={\Delta }_{R_2}={\Delta }_L=0$. The dotted lines correspond to the sum of the currents flowing in two QD-2l systems.

Figure 6

The averaged currents $⟨j_L\left(t\right)⟩$, $⟨j_{R_1}\left(t\right)⟩$, and $⟨j_{R_2}\left(t\right)⟩$ (solid, broken, and dotted lines, respectively) against ${\epsilon }_d$ in the system of a QD coupled with three leads. The upper (lower) panel corresponds to $x=0(x=0.28)$. ${\Delta }_d=8$ and other parameters are as in Fig. 4. The values of $⟨j_{R_2}\left(t\right)⟩$ for $x=0$ (upper panel) have been multiplied by a factor of 20 for illustrating purposes.

Figure 7

The averaged currents against ${\epsilon }_d$ and $\omega$. The upper (middle) panel shows $⟨j_L\left(t\right)⟩\left[⟨j_{R_2}\left(t\right)⟩\right]$ for $x=0$ and the lower panel gives $⟨j_L\left(t\right)⟩$ for $x=0.28$. ${\Delta }_d=8,{\Delta }_{R_1}={\Delta }_{R_2}={\Delta }_L=0,{\mu }_L=-{\mu }_{R_1}=-{\mu }_{R_2}=0.2$.

Figure 8

The averaged current $⟨j_L\left(t\right)⟩$ against ${\epsilon }_d$ for different values of the overdot coupling strength between the left and right leads. The thick solid, solid, dotted, and broken lines correspond to $x=0$, 0.14, 0.28, and 0.7, respectively. $A=(e∕\hslash )2x^2(2{\mu }_L-{\mu }_{R_1}-{\mu }_{R_2})∕{(1+2x^2)}^2$ and ${\Delta }_{R_1}=3,{\Delta }_L={\Delta }_{R_2}={\Delta }_d=0,{\mu }_L=-{\mu }_{R_1}=0.2,{\mu }_{R_2}=0,\omega =5$.

Figure 9

The averaged current $⟨j_L\left(t\right)⟩$ against ${\epsilon }_d$ for the case of the microwave field applied only to the QD with ${\Delta }_d=3$ (thin line) or ${\Delta }_d=6$ (thick line)—upper panel. The lower panel corresponds to the case of the microwave field applied to the QD and $R_2$ lead with ${\Delta }_d=2{\Delta }_{R_2}$ and ${\Delta }_{R_2}=1.5$ or 3 (thin or thick lines, respectively). The broken lines show the results when the microwave field applied to the QD and $R_2$ lead are out of phase (with the phase difference of $\pi$), ${\mu }_L=-{\mu }_{R_1}=0.2,{\mu }_{R_2}=0,\omega =5$.

Figure 10

The averaged current derivatives $d⟨j_L\left(t\right)⟩∕d{\epsilon }_d$ against ${\epsilon }_d$ and $\omega$ for $x=0$. We compare the results obtained for the QD coupled with two leads—the upper (middle) panel—for ${\mu }_L=5,{\mu }_R=-8,{\Delta }_L=8,{\Delta }_d=4,{\Delta }_R=0({\mu }_L=5,{\mu }_R=0,{\Delta }_L=8,{\Delta }_d=4,{\Delta }_R=2)$ with the results obtained for the QD coupled with three leads—the lower panel—${\Delta }_L=8,{\Delta }_d=4,{\Delta }_{R_1}=2,{\Delta }_{R_2}=0,{\mu }_L=5,{\mu }_{R_1}=0,{\mu }_{R_2}=-8,\omega =5$.

Figure 11

The time-dependent current flowing in the system of a QD coupled with the three leads: $L$, $R_1$, and $R_2$. The $L$ lead is coupled with the QD only—the left panels, and with the QD and two other leads, $V_{L{R}_1}=V_{L{R}_2}=4$—the right panels. The couplings between the QD and $R_1,R_2$ leads are changed periodically. The upper, middle, and lower panels correspond to ${\mu }_L=3,0$, and $-3$, respectively. $-{\mu }_{R_1}={\mu }_{R_2}=3,{\epsilon }_d=0$. The thin, thick, and broken curves correspond to $j_L$, $j_{R_1}$, and $j_{R_2}$ currents, respectively. The time is in $\hslash ∕\Gamma$ units and currents are in $e\Gamma ∕\hslash$ units.

Figure 12

(Color online) The time-dependent currents flowing from the $L$, $R_1$, and $R_2$ leads in the system shown in Fig. 1. The thin solid, thick solid, and broken curves correspond to different couplings of the $L$ lead to the other elements of the system: $(V_{L{R}_1}=4,V_{Ld}=V_{L{R}_2}=0)$, $(V_{L{R}_1}=4,V_{Ld}=4,V_{L{R}_2}=0)$, and $(V_{L{R}_1}=V_{Ld}=V_{L{R}_2}=4)$, respectively.