Interactions of Fano resonances in the transmission of an Aharonov-Bohm ring with two embedded quantum dots in the presence of a magnetic field

Interactions of Fano resonances in the transmission for an Aharonov-Bohm ring with two embedded quantum dots are examined. As the interaction parameter between two quantum dots is modulated, two Fano dipoles (a resonance zero-pole pair) in the complex-energy plane form a quasiparticle, which behaves as a coupled object called a “Fano quadrupole.” In the strong overlapping regime of Fano resonance, the collision and merging of resonance zeros take place and these zeros move off from the real axis of energy in complex conjugate pairs. The periodic motion of both transmission zeros and resonance poles as a function of a magnetic field is discussed.

Figure 1

Geometry of the AB ring with coupled QDs. Three electrodes in the lower arm produce the depletion regions in the two-dimensional electron gas and play the role of the barriers for electrons with potentials $V_1$, $V_2$, and $V_3$. The interaction between two QDs is controlled by the middle electrode marked by a black color.

Figure 2

The total transmission of a nanoscale AB ring as a function of electron energy (the left column) and contour plots of the transmission in the complex-energy plane (the right column) with an increase of the interaction parameter $\xi$. The two distinct Fano resonances are shown in (a) and (d) for the weak overlapping regime $(\xi =2.5)$. Here, there are two poles [$E_b^R=(4.70-i0.11)\mathrm{meV}$ and $E_a^R=(5.30-i0.08)\mathrm{meV}$] and corresponding zeros [$E_b^0=4.81\mathrm{meV}$ and $E_a^0=5.21\mathrm{meV}$] that both zeros are placed on the real axis of energy. When $\xi ={\xi }_c=3.77$, the collision of Fano resonances and merging of transmission zeros appear in (b) and (e) with the same zero position as $E_b^0=E_a^0=5.12\mathrm{meV}$. In the strong overlapping regime of Fano resonances $(\xi =5.5)$, two transmission zeros $[E_{\pm }^0\approx (5.19\pm i0.11)\mathrm{meV}]$ move away in opposite directions from the real-energy axis in (c) and (f).

Figure 3

(Color online) The trajectories of resonance poles $({\tilde{E}}_a^R,{\tilde{E}}_b^R)$ and transmission zeros $(E_a^0,E_b^0)$ in the complex-energy plane with an increase of the coupling parameter $\xi (2<\xi <10)$. When $\xi$ increases, two zeros $E_a^0$ (red arrow) and $E_b^0$ (pink arrow) move toward each other and the collision and merging take place at ${\xi }_c=3.77$. After merging, two zeros move away from the real-energy axis in opposite directions as complex conjugate pairs (denoted by $E_{+}^0$ and $E_{-}^0$ with green arrows). The resonance pole ${\tilde{E}}_b^R$ associated with an even quasibound state moves to the higher energy (blue arrow), and ${\tilde{E}}_a^R$ arising from the odd quasibound state is nearly motionless $[{\tilde{E}}_a^R\approx (5.30-i0.08)\mathrm{meV}]$.

Figure 4

A schematic diagram for even and odd quasibound states with energies $E_b$ and $E_a$, connecting with the junction states of the ring by the appropriate matrix elements $V_{Lb}$, $V_{bL}$, $V_{aL}$, and $V_{La}$. The matrix element of the junctions through the upper arm is denoted by $V_r$.

Figure 5

(Color online) The trajectories of zeros and poles as a function of a magnetic flux for a fixed interaction parameter $\xi =5.5$. The two zeros for $\Phi =0$ at $E_{+}^0=(5.19+i0.11)\mathrm{meV}$ and $E_{-}^0=(5.19-i0.11)\mathrm{meV}$ start to move to the right (green arrow) and left (red arrow), respectively, and they reach to the real axis for $\Phi =\frac{1}{2}{\Phi }_0$ that is a full reflection of the transmission. These zeros continue to move on the other half of the trajectories and complete a stadiumlike orbit for $\frac{1}{2}{\Phi }_0<\Phi <\Phi$. The resonance poles at $E_b^R=(5.06-i0.12)\mathrm{meV}$ and $E_a^R=(5.30-i0.08)\mathrm{meV}$ for $\Phi =0$ move a short nearly straight line periodically.