Tuning Fano resonances by magnetic forces for electron transport through a quantum wire side coupled to a quantum ring

We consider electron transport in a quantum wire with a side-coupled quantum ring in a two-dimensional model that accounts for a finite width of the channels. We use the finite difference technique to solve the scattering problem as well as to determine the ring-localized states of the energy continuum. The backscattering probability exhibits Fano peaks for magnetic fields for which a ring-localized states appear at the Fermi level. We find that the width of the Fano resonances changes at high magnetic field. The width is increased (decreased) for resonant states with current circulation that produce the magnetic dipole moment that is parallel (antiparallel) to the external magnetic field. We indicate that the opposite behavior of Fano resonances due to localized states with clockwise and counterclockwise currents results from the magnetic forces which change the strength of their coupling to the channel and modify the lifetime of localized states.

Figure 1

(Color online) (a) The system studied in the scattering problem. The electron is assumed to come from the channel below the contact to the ring and the vertical channel has an infinite length. (b) The model system used for determination of the ring-localized states. $L$ is the finite length of the channel, which varies in the calculation. The confinement potential is assumed 0 within the blue (gray) area and 200 meV in the outside.

Figure 2

(Color online) (a) Energy spectrum for the channel of a finite length $\left(L\right)$ for $B=0$. (b) The peaks indicate the energies of the localized states extracted of the energy spectrum according to Eq. (12). [(c)–(e)] Charge densities of the five lowest energy eigenstates for $L=352\text{nm}$. Plot (c) corresponds to the ground state, (d) to the first excited state, etc.

Figure 3

(Color online) (a) Positions of localized states in the energy continuum as calculated by formula (12). The darker the shade of red the larger value of $N\left(E\right)$. The thin vertical lines indicate energies of 2.4, 2.65, and 3.4 meV that are considered in detail below. The green dashed line indicates the continuum threshold (ground-state energy of the electron within the channel for $k=0$). (b) Energy spectrum of the closed circular quantum ring that is not connected to the channel.

Figure 4

(Color online) (a) Black line shows the electron backscattering probability for energy $E=2.65\text{meV}$ and the red one—the value of the resonance detection counter $N\left(E\right)$ [Eq. (12)]. The inset shows a zoom of high-$B$ part of the figure. (b) The value of $\chi$ plotted in green indicates the direction of the current circulation inside the ring: $\chi =1\left(-1\right)$ corresponds to counterclockwise (clockwise) direction. The blue line shows the fraction of the probability density $r$ that is contained within the ring, i.e., for $x>32.5\text{nm}$ of the computational box of Fig. 1a.

Figure 5

(Color online) The contours show the charge density for two peaks of backscattering probability of Fig. 4 obtained for $E=2.65\text{meV}$ (a) at $B=0.5\text{T}$ and (b) at $B=0.751\text{T}$. The arrows show the probability current distribution.

Figure 6

(Color online) Backscattering probability (black line, left axis), the resonance counter [Eq. (12), red line, right axis], and the direction of the current circulation within the ring in the scattering eigenstates [green line, $\chi =+1\left(-1\right)$ corresponds to counterclockwise (clockwise) orientation]. Plots (a) and (b) were prepared for $E=2.4\text{meV}$ and 3.4 meV, respectively.

Figure 7

(Color online) The backscattering probability averaged according to the linear response formula (11) for the Fermi energy assumed at 2.65 meV for 0 K (black line), 100 mK (blue line), and 300 mK (red line).

Figure 8

(Color online) Same as Fig. 4a but for the channel width reduced to 32 nm. The inner and outer radii of the side-coupled ring 76 nm and 108 nm, respectively. The geometry is displayed in the inset. The thin vertical lines indicate energies of 5.05, 5.16, and 5.26 meV that are considered in Fig. 9.

Figure 9

(Color online) Backscattering probability (black line, left axis) and the direction of the current circulation within the ring in the scattering eigenstates [green line, $\chi =+1\left(-1\right)$ corresponds to counterclockwise (clockwise) orientation]. Plots (a)–(c) were prepared for $E=5.05$, 5.16, and 5.26 meV, for the structure parameters of Fig. 8.