Tuning terahertz transitions in a double-gated quantum ring

We theoretically investigate the optical functionality of a semiconducting quantum ring manipulated by two electrostatic lateral gates used to induce a double quantum well along the ring. The well parameters and corresponding interlevel spacings, which lie in the THz range, are highly sensitive to the gate voltages. Our analysis shows that selection rules for interlevel dipole transitions, caused by linearly polarized excitations, depend on the polarization vector angle with respect to the gates. In striking difference from the conventional symmetric double well potential, the ring geometry permits polarization-dependent transitions between the ground and second excited states, allowing the use of this structure in a three-level lasing scheme.

Figure 1

A diagram of the system's geometry and parameters. $R$ is the ring radius, $d_1$ and $d_2$ are the distances of the charged wires from the ring center, with charge densities ${\lambda }_1$ and ${\lambda }_2$ respectively. The angular position along the ring is given by $\phi$, measured from the horizontal axis as shown. The dotted green line is the projection of the linearly polarized radiation onto the plane of the ring, where $\theta$ is the angle between this projection and the horizontal axis.

Figure 2

Potential and lowest three energy levels with the corresponding wave functions sketched on top for $\beta =2$ and $d/R=100$: (a) is for $\gamma =1$ and (b) is for $\gamma =1.003$. The potentials and energy levels have been shifted by a constant to bring the well minima to zero.

Figure 3

Lowest lying eigenenergies plotted as a function of $\beta$ for reasonable values of asymmetry parameter: (a) is for $\gamma =1$, (b) is for $\gamma =1.009$, and a large value (c) is for $\gamma =1.2$.

Figure 4

Energy separation between ground state and first excited state as a function of $\beta$. The solid curve is for $\gamma =1$, subsequently the dot-dashed curve is for $\gamma =1.003$, the small-dashed curve is for $\gamma =1.006$, and $\gamma =1.009$ is the large-dashed curve. Dotted lines represent approximate results obtained by perturbation theory and the WKB method, not visible for $\gamma =1.009$ within the plotted range and scale of $\beta$.

Figure 5

Square of dimensionless transition dipole matrix element between initial ($i$) and final ($f$) single-electron states plotted as a function of $\beta$ for different values of $\gamma$. The upper branch is $|P_{01}{|}^2/{\left(eR\right)}^2$ at $\theta =\pi /2$ and the lower branch is $|P_{02}{|}^2/{\left(eR\right)}^2$ at $\theta =0$. The full lines denote the results for $\gamma =1$, dashed lines are for $\gamma =1.003$, and dot-dashed lines are for $\gamma =1.006$. The inset is a schematic showing the optical selection rules between the eight lowest energy eigenstates. The full blue arrows are the allowed transitions for when $\gamma =1$, the red dot-dashed arrows are forbidden transitions which become allowed when $\gamma \ne 1$.

Figure 6

Square of dimensionless transition dipole matrix element between first and second excited states, for the asymmetric system and $\theta =\pi /2$, plotted as a function of $\beta$. The solid line is $\gamma =1.007$, dashed line is $\gamma =1.005$, dot-dashed is $\gamma =1.003$, and dotted is $\gamma =1.001$.