Electron localization and optical absorption of polygonal quantum rings

We investigate theoretically polygonal quantum rings and focus mostly on the triangular geometry where the corner effects are maximal. Such rings can be seen as short core-shell nanowires, a generation of semiconductor heterostructures with multiple applications. We show how the geometry of the sample determines the electronic energy spectrum, and also the localization of electrons, with effects on the optical absorption. In particular, we show that irrespective of the ring shape low-energy electrons are always attracted by corners and are localized in their vicinity. The absorption spectrum in the presence of a magnetic field shows only two peaks within the corner-localized state domain, each associated with different circular polarization. This picture may be changed by an external electric field which allows previously forbidden transitions, and thus enables the number of corners to be determined. We show that polygonal quantum rings allow absorption of waves from distant ranges of the electromagnetic spectrum within one sample.

Figure 1

$\mathcal{F}$ function of the 1D model whose zeros signal the allowed states. The insets show the corresponding 1D densities for each zero $\mathcal{F}$ with the vertical lines corresponding to the vertex positions. The number of vertices $N_v$ is indicated in each panel. In terms of the width $d$, the side length is fixed at $\ell =5d$ and the bend radii are $R=0.15d, 0.1d$, and $0.07d$ for the triangle, square, and hexagon, respectively.

Figure 2

Sample model: Polygonal constraints applied on a polar grid. For visibility we reduced the number of site points.

Figure 3

Energy levels for symmetric triangular (a), square (b), and hexagonal (c) samples, of external radii equal to 25 nm and side thicknesses 5 nm. The inset to (a) shows degeneracy of the two lowest energy levels of the triangular sample. Red points indicate purely corner-localized states.

Figure 4

Probability distribution for the 18 lowest states of a symmetric triangular ring. In (a) and (b) we show localization of the lowest states indicated in red in Fig. 3, and in (c), (d), (e), and (f) localization of the states above the energy gap in Fig. 3.

Figure 5

Triangular quantum rings with softened corners. (a) Round corners softened by circles of $r_{\mathrm{int}}=0.05R_{\mathrm{ext}}$ and $r_{\mathrm{ext}}=0.25R_{\mathrm{ext}}$ inscribed in the internal and external limiting polygon corners. (b) Corners softened by background ring radii reduced to $70\%$ of the distance from the center of the sample to the sharp internal and external corners.

Figure 6

Probability distribution for the states associated with the third (a) and (c) and fourth (b) and (d) energy levels of the soft-corner samples. (a) and (b) refer to the sample shown in Fig. 5, and (c) and (d) to the sample in Fig. 5.

Figure 7

Energy levels for a nonsymmetric triangular ring where the symmetry was broken by increasing the side thickness by $5\%$ (side parallel to the $y$ axis) and $10\%$ (subsequent side according to counterclockwise counting). Red points indicate corner-localized states.

Figure 8

Probability distribution for a few chosen energy levels of the nonsymmetric sample shown in Fig. 7. Results shown in (d), (e), and (f) have been graphically magnified by a factor of 2 ($\times 2$).

Figure 9

Probability distribution for a few chosen energy levels of a symmetric sample in the presence of an electric field applied along one of the sides ($E\approx 0.11$ mV/nm, $\phi =\pi /6$). Results shown in (d), (e), and (f) have been graphically magnified by a factor of 3 or 5 ($\times 3, \times 5$).

Figure 10

Probability distribution associated with the ground state (a) and with the third energy level (b) for a nonsymmetric sample in the presence of an electric field ($E\approx 0.67$ mV/nm, $\phi =\pi /6$).

Figure 11

The optically allowed transitions from the ground state in the presence of clockwise (green, solid) and counterclockwise (red, dashed) polarization superimposed on the density of states from Eq. (10) with $\mathrm{\Gamma }=0.056$ meV (gray, dotted), with the ground-state energy tuned to zero for triangular (a) and (b), square (c) and (d), and hexagonal (e) and (f) symmetric rings with sharp corners in a weak perpendicular magnetic field ($0.53$ T). (a), (c), and (e) correspond to transitions to corner-localized states and (b), (d), and (f) transitions to side-localized states. For visibility we use a logarithmic scale for the absorption functions.

Figure 12

As in Fig. 11 but in the presence of an electric field applied in the ring plane such that it is parallel to one or two ring sides [(a) and (b) $E\approx 0.0056$ mV/nm, $\phi =\pi /6$, (c) and (d) $E\approx 0.11$ mV/nm, $\phi =0$, (e) and (f) $E\approx 0.56$ mV/nm, $\phi =0$]. For visibility we use a logarithmic scale for the absorption functions.