Highly Polarized Single Photons from Strain-Induced Quasi-1D Localized Excitons in WSe2

Single photon emission from localized excitons in two-dimensional (2D) materials has been extensively investigated because of its relevance for quantum information applications. Prerequisites are the availability of photons with high purity polarization and controllable polarization orientation that can be integrated with optical cavities. Here, deformation strain along edges of prepatterned square-shaped substrate protrusions is exploited to induce quasi-one-dimensional (1D) localized excitons in WSe2 monolayers as an elegant way to get photons that fulfill these requirements. At zero magnetic field, the emission is linearly polarized with 95% purity because exciton states are valley hybridized with equal shares of both valleys and predominant emission from excitons with a dipole moment along the elongated direction. In a strong field, one valley is favored and the linear polarization is converted to high-purity circular polarization. This deterministic control over polarization purity and orientation is a valuable asset in the context of integrated quantum photonics.

S3. Lorentz function fit to the PL spectrum S4. Second-order photon-correlation measurements S5. Excitons with weak but resolvable upper energy branch emission S6. Optical characterization of WSe 2 monolayer bubbles in a WSe 2 /hBN van der Waals stack S7. Localized exciton emission from WSe 2 clad on protruding wires S8. Model for excitons localized in a strain induced oval shaped potential S9. Model for quasi-1D excitons S10. Silicon-oxide pillars with different orientation S11. Jitter and B-dependence of the PL spectrum from excitons localized in a WSe 2 bubble S12. Circular polarization resolved PL spectra in the presence of a magnetic field S1. Scanning electron microscope and optical microscope image   Figure S2a shows the µ-PL spectrum of WSe 2 on a flat SiO 2 substrate. The peak at 1.75 eV is attributed to neutral exciton emission, while the peak at 1.68 eV corresponds to trion emission. Figure S2b compares the Raman spectrum of WSe 2 on flat SiO 2 (blue) with the spectrum of WSe 2 placed on a SiO x pillar (red). The E 2g + A 1g mode of WSe 2 on the SiO x pillar is red shifted by about 0.2 cm -1 . This indicates that 0.5 % of strain is induced by the pillar. 1, 2 Figure S2c compares µ-PL spectra recorded on WSe 2 placed on a flat SiO 2 substrate (blue) and on a SiO x pillar (red).

S2. Comparison of photoluminescence and Raman spectra on flat and strained WSe 2
Numerous sharp emission lines appear for the flake placed on top of the substrate protrusion. The sharp emission lines below 1.653 eV are attributed to localized dark excitons in view of the work reported in Ref [3]. Dark excitons in TMDCs possess a dipole moment that is oriented perpendicular to the plane of WSe 2 flakes, while for bright excitons the dipole moment is in the plane. Due to the orientation of the dipole, the detected emission intensity is normally weak.
However, due to the topography (pillars), part of the WSe 2 flake is oriented perpendicular to the substrate and the dipole moment of the dark excitons is parallel to the Si substrate plane. As a result, the collection efficiency of such localized dark excitons at the pillar edge can be as high as that for bright excitons. These dark excitons are not investigated any further in this manuscript. Figure S2d displays the µ-PL spectrum recorded on WSe 2 placed on a SiO x pillar for different temperatures. Emission from shallow trapped excitons appears below 4 K. The spectra were vertically offset for the sake of comparison. The monochromator grating has 600 grooves/mm. The excitation power is 0.1 µW. Figure S3 displays a Lorentz function fit of the peak marked with the blue dot in Figure 1b of the main text. The maximum is located at 1.704 eV and the peak has a full width at half maximum (FWHM) of 160 ± 14 µeV. Figure S3. Lorentz function fit to the µ-PL peak marked in Figure 1b of the main text.

S4. Second-order photon-correlation measurements
Second-order photon-correlation measurements were performed by using a Hanbury Brown and Twiss (HBT) setup. The experimental data were fitted with the second-order photon-correlation function (1) where t is the time delay between the coincidence counts, consists of pumping and radiative recombination rate, and a represents the value of . In some cases, bunching behaviour (2) is observed during second-order photon-correlation measurements. To include the bunching behavior visible in such cases, Eq. (1) is extended to where and consist of pumping and radiative recombination rates of the anti-bunching and ℎ bunching components, respectively, and b are constants. Figure S4 presents the g (2) (t) results from another emitter where bunching is superimposed to the anti-bunching. The data points are fitted with Eq. (2). At t = 0, . The extracted values from (2) (0) = (1 + ) = 0.27 ± 0.09 < 0.5 the fit are ns and ns. = 12.03 ± 2.58 ℎ = 9.06 ± 0.85 Figure S4. Second-order photon-correlation results for an emitter with bunching. The data points are fitted with Eq. (2).

S5. Excitons with weak but resolvable upper energy branch emission
The PL spectrum in Figure Figure S6a shows an optical microscope image of WSe 2 monolayer bubbles that form when rapidly assembling a WSe 2 /hBN stack. The PL spectrum from localized excitons in the WSe 2 bubble 1 is plotted in Figure S6b. Fine structure splitting (FSS) is observed. This splitting is attributed to an anisotropic dot confinement potential which is accompanied by anisotropic electron-hole exchange interaction. [4][5][6] The synchronized jitter in panel c of the lower and upper photoluminescence peaks indicates that both originate from the same quantum dot confinement potential ( Figure S6c). The emission for both is linearly polarized. Moreover, the emitted light is cross-polarized as illustrated in Figure S6d and e.

S8. Model for excitons localized in a strain induced oval shaped potential
The model of quasi-1D localized excitons in an elongated strain induced potential has been deduced from the established model of excitons subjected to an anisotropic confinement potential. [4][5][6] Therefore, the latter case will be treated here first.
For a WSe 2 monolayer, the valley and spin degrees of freedom are locked. Hence, optical emission from the K and K' valleys are right (σ + ) and left hand (σ -) circularly polarized, respectively. It is therefore convenient to use the notation and to represent excitonic states | ⟩ | ′⟩ that emit right and left hand circularly polarized light. For the sake of simplicity, we will assume an oval shaped confinement potential without D 2d symmetry, i.e. the x and y in-plane directions are not equivalent. [4][5][6] In such a potential, the excitonic states possess a dipole moment that is either oriented along the minor or major axis of the oval shape. The minor axis is referred to as the xdirection, whereas the major axis is aligned along the y-direction as shown schematically in panel a of Figure S8. The energy of the exciton with its dipole moment aligned along the x-direction is higher because of the tighter quantum confinement. The photoluminescence intensity from this higher energy exciton in x-direction is lower because of the smaller optical oscillator strength in the direction of the minor axis. [7][8][9][10][11][12][13] The eigenstates at zero magnetic field (B = 0) are a linear superposition of the localized and exciton states with equal weight and therefore can be | ⟩ | ′⟩ written as The index L refers to the exciton with lower energy, whereas U is used for the upper, higher lying exciton. These energies are given by Here, δ 0 is the zero-field splitting between radiative and dark excitons and δ 1 is the zero-field fine structure splitting caused by anisotropic electron-hole exchange interaction in the confinement potential without D 2d symmetry ( Figure S8b). In these expressions, is equal to , is the Bohr magneton, g the Landé or gyromagnetic 1 factor, and N L and N U are magnetic field dependent normalization constants. The energy difference between the high energy state and the low energy state is obtained from For strong magnetic fields, when , and , and vice versa for the = opposite sign of the magnetic field ( , and ). The expressions for the The key takeaways from this model are summarized in Figure S8b. The anisotropy in the straininduced confinement potential results in a fine structure splitting δ 1 in the absence of a magnetic field between the two possible exciton states whose emissions exhibit linear and cross polarization.
In the presence of a magnetic field, the fine structure splitting acquires a Zeeman term and varies according to .

S9. Model for quasi-1D excitons
We refer to quasi-1D excitons for the case where the ratio of the length of the major axis to the length of the minor axis of the anisotropic strain induced confinement potential approaches infinity.
This case of extreme anisotropy is illustrated in Figure S8c. While the expressions for the excitonic states and energies discussed in Section 8 still hold, optical transitions along the x-direction with its very tight confinement are strongly suppressed and only the low energy exciton state with | ⟩ the dipole oriented along the y-direction can be considered optically active because of the lack of optical oscillator strength in the direction of the minor axis. [7][8][9][10][11][12][13] The emission of the upper branch is therefore strongly suppressed. The emission properties of remain unaltered, i.e. linearly | ⟩ polarized in the absence of a magnetic field, while the polarization is converted from linear to circular for strong magnetic fields.

S10. Silicon-oxide pillars with different orientation
The linear polarization of the emission of localized excitons from WSe 2 clad onto the square shaped protrusion is aligned parallel to the edge where the emission appears. Hence the orientation of the linear polarization is fully controlled by geometry. This is illustrated clearly in Figure S9 on a sample where the orientation of the square shaped pillars is systematically altered. Pillars in each of the rows have the same orientation, however the pillars of adjacent rows are rotated by 10° with respect to each other. The angle of the pillars starts at 0° for the bottom row and ends at 90° for the top row. Figure S9a shows the SEM image of such a sample. Figure S9b displays a microscope image after placing a WSe 2 monolayer on top of this pre-patterned substrate with pillars. Typical PL emission spectra recorded at the edge of pillars with an angle of 10°, 50° and 90° are plotted in Figure S9c. The polarization direction of the emission is verified by plotting the integrated µphotoluminescence intensity as a function of the detection angle in a polar diagram ( Figure S9d).
As anticipated, the orientation follows the rotation angle of the pillar. The locations where the different data sets were recorded are marked by colored dots in Figure S9b.

S11. Jitter and B-dependence of the PL spectrum from excitons localized in a WSe 2 bubble
Figure S10a illustrates the photoluminescence (PL) spectrum recorded from Bubble 2 (See Figure   S6a) that has formed in the WSe 2 monolayer after it was placed on top of hBN. The photoluminescence exhibits two peaks whose energy positions jitter in a synchronized manner when recording the spectrum over an extended period of time ( Figure S10b). We therefore conclude that both emission features stem from excitons localized within the same confinement potential. With the increase of |B|, the two luminescence peaks split further in energy. The one with higher energy moves upward, while the one with lower energy moves downward ( Figure   S10c, d, and e). The energy difference between the upper branch and lower branch increases with the increase of |B| and can be fitted by the equation . The fitted zero field ∆( ) = 1 2 + ( ) 2 fine structure splitting, δ 1, is approximately 577 ± 13 µeV and the extracted g-factor equals 9.17 ± 0.04 ( Figure S10f). With the increase of B, the PL intensity of the upper branch quenches gradually because excitons have a higher probability to fill the lower energy branch as the energy splitting increases.  Figure S11 illustrates the magnetic field dependence of the circular polarization resolved PL spectra (left hand circular polarization σand right hand circular polarization σ + ) from the localized excitons emitting at 1.704 eV as discussed in the main text. The data sets shown are for 0 T, ±2 T, ±5 T, and ±10 T. Figure S11. Circular polarization resolved µ-PL spectra for quasi-1D localized excitons emitting at 1.704 eV recorded at different perpendicular magnetic field strengths. Blue lines correspond to left-hand circular polarization (σ -), whereas data for right-hand polarization (σ + ) are plotted in red. The magnetic field is equal to 0 T, ±2 T, ±5 T, or ±10 T.