Dark states in transport through triple quantum dots: The role of cotunneling

We analyze transport properties of coherent triple-quantum-dot systems in triangular geometry, weakly coupled to external leads. The real-time diagrammatic technique up to the second order is used to determine conductance and shot noise in the sequential tunneling and cotunneling regimes. In particular, we focus on the transport regime where the triple dot is trapped in a dark state and the system is in the interchannel Coulomb blockade, which leads to negative differential conductance. We show that although sequential transport is then exponentially suppressed, the blockade can be lifted by cotunneling processes. It is also shown that the shot noise in this transport regime is generally super-Poissonian, but the corresponding Fano factor is reduced when the cotunneling processes are included.

Figure 1

Schematic of triple quantum dot in a triangular geometry. The $i$th dot is coupled to the dot $j$ through the hopping $t_{\mathit{ij}}$. The first (second) dot is also coupled to the left (right) lead with the strength described by ${\Gamma }_L$ (${\Gamma }_R$).

Figure 2

The current $I$ (a) in units of $I_0=e\Gamma /\hslash$, the zero-frequency shot noise $S$ (b) in units of $S_0=e^2\Gamma /\hslash$ and Fano factor $F$ (c) as a function of the bias voltage $V$. The parameters are $k_{B}T=0.1$ meV ($T\approx 1.16$ K), $\Gamma =0.025$ meV, $t=-1$ meV, ${\varepsilon }_1={\varepsilon }_3=0$, ${\varepsilon }_2=-0.4$ meV, $U^{\prime }=10$ meV ($U=\infty$). The eigenenergies of the system are then $E_2=-2.146$ meV, $E_3=0.745$ meV, and $E_1=1.0$ meV. They are indicated in the figure (relative to the Fermi energy) by the vertical dotted lines for the transmitting states ($E_2$ and $E_3$) and by the vertical dashed line for the dark state ($E_1$). The electrochemical potentials of the leads are shifted asymmetrically according to ${\mu }_L=E_F+\mathit{eV}$ and ${\mu }_R=E_F$, where $E_F$ is the Fermi energy, $E_F=-2.2$ meV. For comparison, the dashed lines show the sequential tunneling results. The insets present the current and shot noise in the logarithmic scale, which clearly show the dominant role of cotunneling in the regimes where the system is trapped in a dark state. Here, $I_0\approx 6.08$ nA and $S_0\approx 9.74\times {10}^{-28}$ ${\mathrm{A}}^2$ Hz${}^{-1}$.

Figure 3

The current $I$ (a), zero-frequency shot noise $S$ (b), and Fano factor $F$ (c) as a function of the bias voltage $V$, calculated for ${\varepsilon }_2=0.4$ meV, while the other parameters as in Fig. 2. Now, the eigenenergies of the TQD system are $E_2=-1.870$ meV, $E_1=1.0$ meV (dark state), and $E_3=1.277$ meV.

Figure 4

Logarithm of the absolute value of the normalized current $I/I_0$ (a), (d), logarithm of the normalized shot noise $S/S_0$ (b), (e), and the Fano factor $F$ (c), (f) as a function of the bias voltage $V$ and level position of the second dot ${\varepsilon }_2$. Density plots (a)–(c) show the total (sequential plus cotunneling) results, while the plots (d)–(f) display results obtained using only sequential tunneling processes. The other parameters are the same as in Fig. 2. The Fano factor in linear response regime diverges, therefore this transport regime is marked by a white vertical stripe in (c) and (f). The dashed lines correspond to the cross sections shown in Figs. 2 and 3. To facilitate comparison between the sequential and total results the respective density plots are presented using the same color scale.

Figure 5

The current $I$ (a), zero-frequency shot noise $S$ (b), and Fano factor $F$ (c) as a function of the bias voltage $V$. The parameters are $k_{B}T=0.05$ meV ($T\approx 0.58$ K), $\Gamma =0.025$ meV, $t=1$ meV, ${\varepsilon }_1={\varepsilon }_3=0$, ${\varepsilon }_2=-0.4$ meV, $U^{\prime }=10$ meV ($U=\infty$). The corresponding eigenenergies of the TQD system are $E_2=-1.279$ meV, $E_1=-1.0$ meV (dark state), and $E_3=1.878$ meV. The electrochemical potentials of the leads are shifted asymmetrically: ${\mu }_L=E_F+\mathit{eV}$ and ${\mu }_R=E_F$, where $E_F$ is the Fermi energy, $E_F=-1.4$ meV. For comparison, the dashed lines show the sequential tunneling results. The insets present the current and shot noise in the logarithmic scale.

Figure 6

Current $I$ (a), zero-frequency shot noise $S$ (b), and Fano factor $F$ (c) as a function of the bias voltage $V$. The parameters are the same as in Fig. 5 except for ${\varepsilon }_2$, which now is ${\varepsilon }_2=0.4$ meV. The corresponding eigenenergies of the TQD system are $E_1=-1.0$ meV (dark state), $E_2=-0.746$ meV, and $E_3=2.145$ meV. The additional dotted (dotted-dashed) line in (c) shows the total Fano factor for $U^{\prime }=25(50)$ meV.

Figure 7

Logarithm of the absolute value of the normalized current $I/I_0$ (a), (d), logarithm of the normalized shot noise $S/S_0$ (b), (e), and the Fano factor $F$ (c), (f) as a function of the bias voltage $V$ and level position of the second dot ${\varepsilon }_2$. The density plots (a)–(c) show the total (sequential plus cotunneling) results, while the plots (d)–(f) display results obtained using only sequential tunneling processes. The other parameters are the same as in Fig. 5. The dashed lines correspond to the cross sections shown in Figs. 5 and 6.