Two-particle dark state in the transport through a triple quantum dot

We study transport through a triple quantum dot in a triangular geometry with applied bias such that both singly- and doubly-charged states participate. We describe the formation of electronic dark states—coherent superpositions that block current flow—in the system, and focus on the formation of a two-electron dark state. We discuss the conditions under which such a state forms and describe the signatures that it leaves in transport properties such as the differential conductance and shot noise.

Figure 1

(Color online) The triple quantum dot (TQD) in triangular geometry is connected to two sources and one drain. In the infinite-bias limit, the rate at which electrons enter and leave the TQD is $\Gamma$, the same for all three leads. The QDs are coupled to each other by the coherent tunnel amplitudes $T_C$ as shown. Zero, one, or two electrons are allowed to be in the TQD at a time. The charging energy between two electrons in the same QD, e.g., (a) $U_{22}$ is, in general, larger than for two electrons in different QDs, e.g., (b) $U_{13}$.

Figure 2

(Color online) Stationary current $⟨I⟩$ through the TQD as a function of the charging-energy difference scaled with the tunnel amplitude ${\delta }_U/T_C$ for different ratios of $T_C/\Gamma$ and for two choices of charging energies. Results are shown for both GME and DME calculations (the DME calculations are independent of the ratio $T_C/\Gamma$) and both calculations show that at ${\delta }_U=0$, the current is zero. This is attributed to the formation of the two-electron dark state. In (a) (symmetric charging energies) the GME calculation shows an antiresonance of finite width about the DS, whereas in the DME calculation the DS appears as a point of discontinuity. For asymmetric charging energies, (b), both calculations show an antiresonance of finite width about the DS.

Figure 3

(Color online) Gap width in ${\delta }_U/T_C$ of the antiresonance at the DS as a function of tunnel rate per tunnel amplitude $\Gamma /T_C$ for the GME approach with the parameters of Fig. 2a. The size of the gap is given by the width of the antiresonance when the current has half the size of the maximum $⟨I_{\text{max}}⟩/2\left(e\Gamma \right)$.

Figure 4

(Color online) The Fano factor $F\left(0\right)$ as a function of the charging-energy difference with results for both GME and DME calculations. From the GME calculation for both (a) symmetric and (b) asymmetric charging energies, we observe that the shot noise is highly super-Poissonian $\left[F\left(0\right)>1\right]$ both near the dark state ${\delta }_U/T_C\sim 0$ and in the limit of ${\delta }_U/T_C⪢1$. In the DME approach, only the asymmetric case shows super-Poissonian behavior near the dark state. The Fano factor in (a) is Poissonian for ${\delta }_U/T_C<1$.

Figure 5

(Color online) Stability diagram for the TQD with parameters such that both one- and two-electron dark states can form; in particular, ${\delta }_U=0$. Here the current is plotted as a function of the source-drain bias voltage $V_{\text{Bias}}$ and gate voltage $V_{\text{Gate}}$. For positive bias, most electrons enter the system on the left side where two QDs couple to the environment, and leave to the right side where one QD is coupled to the environment. Only for this bias direction can the DSs occur. The DSs are visible as breaks in the borders of the Coulomb diamonds (marked as DS1 for the one-electron dark state and as DS2 for the two-electron dark state); the gap near the center is due to the single-particle dark-state, that to the right, the two-particle dark state. The parameters used are: $U_{11}=U_{22}=U_{12}=1\text{meV}$, $U_{33}=1.2\text{meV}$, $U_{13}=U_{23}=0.95\text{meV}$, tunnel amplitude $T_C=0.1\text{meV}$, tunnel rate $\Gamma =10\mu \text{eV}$, and temperature $T=150\text{mK}$.

Figure 6

(Color online) Differential conductance as a function of bias voltage $V_{\text{Bias}}$ and gate voltage $V_{\text{Gate}}$ with the same parameters as Fig. 5. For $V_{\text{Bias}}>0$, two lines of negative differential conductance (NC) are visible, which can be attributed to the one- and two-electron DS. When a DS is hit the current decreases to zero and the differential conductance becomes negative. As in the current, the diamond pattern is broken by the formation of DSs.

Figure 7

(Color online) As Fig. 6, but with finite charging-energy difference ${\delta }_U=0.5\text{meV}$ and symmetric charging energies. Despite the fact that the genuine two-electron dark state [Eq. (17)] can no longer form, lines of negative differential conductance are seen in positions similar to that in Fig. 6. A strong new line of NDC is also observed (marked NC2) which is related to the state of Eq. (24). More structure is visible here as compared with the ${\delta }_U=0$ plot since degeneracies have been lifted. The parameters used here were $U_{ii}=1\text{meV}$, $U_{ij}=0.5\text{meV}$ for $j\ne i$, $T_C=100\mu \text{eV}$, $\Gamma =10\mu \text{eV}$, and $T=100\text{mK}$.