GaAs Quantum Dot in a Parabolic Microcavity Tuned to 87Rb D1

We develop a structure to efficiently extract photons emitted by a GaAs quantum dot tuned to rubidium. For this, we employ a broadband microcavity with a curved gold backside mirror that we fabricate by a combination of photoresist reflow, dry reactive ion etching in an inductively coupled plasma, and selective wet chemical etching. Precise reflow and etching control allows us to achieve a parabolic backside mirror with a short focal distance of 265 nm. The fabricated structures yield a predicted (measured) collection efficiency of 63% (12%), an improvement by more than 1 order of magnitude compared to unprocessed samples. We then integrate our quantum dot parabolic microcavities onto a piezoelectric substrate capable of inducing a large in-plane biaxial strain. With this approach, we tune the emission wavelength by 0.5 nm/kV, in a dynamic, reversible, and linear way, to the rubidium D1 line (795 nm).


Statistical evaluation and collection efficiency enhancement
To statistically estimate the collection efficiency enhancement we record micro-photoluminescence spectra and integrate the intensities of all peaks originating from a single quantum dot. We perform this integration for 36 quantum dots in parabolic microcavities, 27 quantum dots from reference quantum dots in planar structures and 12 as-grown quantum dots to obtain 3 sets of data. We then determine the average integrated intensity and corresponding error for each data set and also the maximum data point (see Figure S1(a)). The parabolic microcavity achieves up to 7789.7 kcnts/s on the CCD, which is 21.6 times more than the average as-grown quantum dot (360 ± 64) kcnts/s and still 8.5 times brighter than the brightest asgrown quantum dot with 910.7 kcnts/s. The average parabolic microcavity quantum dot with (2189 ± 289) kcnts/s achieves an enhancement of 6.2 compared to the average as-grown quantum dot. In comparison, the reference dots, with on average (577 ± 74) kcnts/s, show only a factor 1.6 enhancement.
For the estimation of the collection efficiency we excite a bright parabolic microcavity quantum dot with a red laser diode with 80 MHz repetition rate and record the microphotoluminescence spectrum. We identify the neutral exciton emission and integrate the emission with a 83 pm window (see Figure S1(b)), resulting in 105.672 kcnts/s. We characterize our setup with a tunable laser set to the emission wavelength of our quantum dot and use a pulse slicer to obtain a similar linewidth (inset in Figure S1(b)). We attenuate the laser in front of the spectrometer slit by 68.71 dB and obtain 75.988 kcnts/s integrated counts on the spectrometer for a measured power of 2.08 µW before the attenuation. From this, we derive a combined efficiency of spectrometer and CCD of η Spec = 6.8 %. In the remaining detection path we measure the transmission of each optical component and then derive the efficiency to be η Det = 15.8 %. The total setup efficiency is then the product η Setup = η Spec * η Det = 1.1 %. By correcting the integrated exciton counts for the setup we S3 obtain 9606 kcnts/s into the first lens, which corresponds to a collection efficiency of 12 % from the exciton emission.
This collection efficiency gives a lower bound as we excited the quantum dot above-band, which causes the population of several states and not only the neutral exciton state used for this estimation. For example neutral and charged exciton states can not exist simultaneously, which in turn means, that the states are not excited by every excitation pulse ( Integrated photoluminescence (cts/s) average maximum Figure S1: (a) Statistical evaluation: Bar plot of the integrated intensities of microphotoluminescence spectra from parabolic microcavites, planar mirror reference and as-grown quantum dots. The bars represent the average value and error and the stars the maximum value in the data set. (b) Collection efficiency measurement: Micro-photoluminescence spectrum of the exciton emission of the bright parabolic microcavity quantum dot with integration window marked in blue. Inset: Spectrum of the 80 MHz repetion rate attenuated laser (2.08 µW, ND 68.71) for setup calibration.

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The finite element method simulation allows us further to simulate an emitter that is not laterally in the center of the paraboloid. We select the paraboloid diameter d=1.45 µm, height h=500 nm and the dipole vertical position to match the focal length f=265 nm. Then we sweep the lateral position of the dipole towards the edge in 40 nm steps and record the collection efficiency, Purcell factor and far field emission pattern, for a dipole emission wavelength of λ=795 nm.  intensity is reduced by around 10 % and the maximum emission intensity is displaced from the center. In addition, the intensity profile is not gaussian anymore.
For our probabilistic approach of positioning the quantum dots we can now estimate the probability that we achieve a collection efficiency of more than 20 % into a lens with NA=0.8.
The quantum dot has to lie within a radius of 120 nm corresponding to an area of 0.045 µm 2 .
The paraboloid radius and area are 530 nm and 0.88 µm 2 , respectively, at the height of the quantum dot 265 nm above the apex. Therefore, the probability for a quantum dot to lie within 120 nm in the center is approximately 5 %.

Atomic force microscopy characterization of reflown photoresist and etched paraboloid
We employ atomic force microscope measurements (see Fig. S3) to characterize the height profile and roughness of the photoresist (Fig. S3(a-c)) and the etched structure ( Fig. S3(d-f)).
In Fig. S3(a,d) we show the height data with lighting from the sides. The etched parabola features a flat 0.5 µm wide apex which we attribute to incomplete removal of the photoresist during the dry etching and subsequent cleaning. In Fig. S3(b,e) we show the color-coded height data and a fit (dashed lines) to a paraboloid. In case of the etched paraboloid we consider for the fit only the top 0.5 µm of the paraboloid by clipping the rest, because everything below is beyond the sacrificial layer and not part of the final device. We observe that our paraboloids appear to be not fully round at the base, therefore, our fit model allows for two in-plane diameters (major-and minor axis of an ellipse). The resulting diameters for the fit of the etched paraboloid are 1.611 µm and 1.386 µm, which corresponds to a roundness of (84 ± 0.5)% as calculated from the ratio of the two diameters. In comparison, the roundness of the photoresist is (95 ± 0.5)%. To quantify the agreement of our model with the measured atomic force microscope data, we subtract the data from the fit and obtain a S6 height difference map (see Fig. S3(c,f)). We find that the deviations are at most 28 nm from the perfect shape for the etched paraboloids.

Simulation with atomic force microscopy data
To estimate how well a real life device could perform at best with our current fabrication precision, we import the atomic force microscopy height profile shown in Fig. S3 (d) and replace the previously used perfect parabola, while keeping the emitter centered. The results are shown in Fig. S4(a) where we plot the far field emission profile in the same range as in the main text. Compared to the perfect structure, the main intensity of the far field is now slightly off-centered which we attribute to the slight asymmetry of the real structure.

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Nevertheless, the collection efficiency plotted in Fig. S4(b) reaches up to 65 % for an NA of 0.8 and 63.5 % for 795 nm. We also notice that the overall shape of the curve appears more smooth and flat, which we attribute to the surface roughness smearing out the cavity effects.