Uniaxial transition dipole moments in semiconductor quantum rings caused by broken rotational symmetry

Semiconductor quantum rings are topological structures that support fascinating phenomena such as the Aharonov–Bohm effect and persistent current, which are of high relevance in the research of quantum information devices. The annular shape of quantum rings distinguishes them from other low-dimensional materials, and enables topologically induced properties such as geometry-dependent spin manipulation and emission. While optical transition dipole moments (TDMs) in zero to two-dimensional optical emitters have been well investigated, those in quantum rings remain obscure despite their utmost relevance to the quantum photonic applications of quantum rings. Here, we study the dimensionality and orientation of TDMs in CdSe quantum rings. In contrast to those in other two-dimensional optical emitters, we find that TDMs in CdSe quantum rings show a peculiar in-plane linear distribution. Our theoretical modeling reveals that this uniaxial TDM originates from broken rotational symmetry in the quantum ring geometries.

. The excitation source is a linearly polarized 633 nm HeNe laser, which is reflected by a non-polarizing dichroic beam-splitter and focused onto the sample with a NA = 1.4 oil-immersion objective. PL from the quantum rings is collected by the same objective, transmitted by the beam-splitter, and after being separated from the remaining excitation light using a long-pass filter, recorded by a charge coupled device (CCD) image array. In order to record the non-polarization selective angular intensity distributions of the emitters, a Bertrand lens following an intermediate tube lens 1 was chosen. Together with the selected resolution of the CCD array this results in a k-space resolution kx/k0 and ky/k0 of about 0.0667 per image pixel.
The intensity distributions I(kx/k0, ky/k0) in the back focal plane are calculated using the p-and s-polarized components of the electric fields radiated by a point dipole p on the air-glass interface depending on its in-plane orientation Φ and out off-plane orientation Θ, the polar emission angle and the distance from the center r: 2 ( , , Θ, Φ) = * + * . 1 Additional fixed parameters to describe the observed experimental configuration are the refractive indices n1 and n2 of the two half-spaces defining the interface as well as NA of the used microscope objective.
The emission pattern of a point-dipole oriented in the sample plane has a distinct form (Supplementary Figure 2). On the borders of the circle, the pattern has two half-moon shaped maxima on opposite sides. The line connecting the maxima is perpendicular to the real space orientation of the radiating dipole p. Therefore, the orientation of the emitting dipole can be determined from the orientation of the two maxima. In order to reduce photoluminescence blinking of the quantum rings for the recording of the emission patterns, a polystyrene thin film was spin-coated on top of the dispersed sample on a glass cover-slide. This was considered within the calculation by adjusting the refractive index of the upper half-space to n1 = 1.1, since the presence of the polystyrene thin film will increase the effective refractive index experienced by the QR emitter. The lower half-space refractive index n2 is defined by the glass cover slide, index matching oil and objective and given with n2 = 1.52.
To further demonstrate the in-plane uniaxial dipolar nature of the QR TDMs, additional cross-sections extracted from the emission patterns of the two representative QRs A and B in Fig

Supplementary Note 4 -Wavefunctions of thin CdSe quantum rings
We first look at two-dimensional infinite ring well potentials of a thin CdSe quantum well. The two outer and inner diameters of the ring are set to be equal to each other, OD1 = OD2 = 13 nm, ID1 = ID2 = 9.6 nm. In this case, there is a continuous rotational symmetry. The ground state has a s-type orbital, where the wavefunction is delocalized through the ring. The second lowest state is degenerate and has p-type orbitals, where there are two lobes along different axes.
Examples of these solutions to the Schrödinger equation, using the discrete variable representation, are shown below in Supplementary Fig. 6. Due to the continuous rotational symmetry, these solutions are not unique. A rotation of any amount of the p-type orbitals would also be a solution. A perfect ring of this type could be excited by linearly polarized light along any in-plane direction, as there are valid p-type solutions for any rotation.
Our theoretical modelling of the thin CdSe QRs using the empirical tight-binding method shows that the threedimensional, atomistic ring structures adds additional complexity to their electronic structures. The discrete nature of the atomic positions breaks the continuous rotational symmetry present in the infinite ring potential, even in the case of OD1 = OD2 and ID1 = ID2, and this effect is more apparent compared to that observed in thicker QRs (see Fig. 4 in main text and Supplementary Notes 5 below). The exact effect of this broken symmetry depends on the specific cut of the ring out of the bulk. Displacing the origin of the supercell slightly results in different edges. Supplementary Fig.   7 shows the conduction band minima and valence band maxima for three different displacements of the origin, where we used OD1 = OD2 =13 nm and ID1 = ID2 = 9.6 nm. For some of the displacements, the s-type orbital nature of the conduction band minimum is still present, but for others, the wavefunctions are highly localized. For CdSe quantum rings with OD1 ≠ OD2, both the oval nature of the quantum ring and the atomic structure break rotational symmetry. The oval shape of the ring sets natural locations of the lobes; the local atomic structure of those locations changes the relative energy of each of the lobes. This leads to a localization of the band edge states into individual lobes. Examples are shown in Supplementary Fig. 8, where we used OD1 = 13 nm, OD2 = 10 nm, ID1 = 9.6 nm, ID2 = 6.6 nm, and various displacements of the origin. Consequently, optical transition dipole moments between these band-edge states will be uniaxial and in-plane. Supplementary Figure 12: Wavefunctions of QRs with (right) and without (left) inverted aspect ratios of their inner ovals. Scale bars: 5 nm. Inverting the aspect ratio of their inner oval (right) creates much larger regions for the wavefunctions to localize, nearly creating two separate quantum dots. There is still some anisotropy, but it is not nearly as strong.
Supplementary Figure 13: Wavefunctions of QRs with the thicker end along the long (left) and short (right) axes.
Scale bars: 5 nm. Having the thicker end along the short axis direction allows the wavefunctions to be more spread out and destroys the uniaxial dipole moment, almost returning the circular symmetry of the conduction band edge present in the perfect ring structure (right column). The thicker part of the ring allows for more wavefunction localization while the elongation causes a preferred localization along the edges. These two effects, at least in some geometries, counteract to return wavefunctions distributed throughout the ring, even with a lack of circular symmetry (right column).