Enhanced photon-extraction efficiency from deterministic quantum-dot microlenses

The prospect of realizing building blocks for long-distance quantum communication is a major driving force for the development of advanced nanophotonic devices. Significant progress has been achieved in this field with respect to the fabrication of efficient quantum-dot-based single-photon sources. More recently, even spin-photon entanglement and quantum teleportation have been demonstrated in semiconductor systems. These results are considered as crucial steps towards the realization of a quantum repeater. The related work has almost exclusively been performed on self-assembled quantum dots (QDs) and random device technology. At this point it is clear that further progress in this field towards real applications will rely crucially on deterministic device technologies which will, for instance, enable the processing of bright quantum light sources with pre-defined emission energy. Here we report on enhanced photon-extraction efficiency from monolithically integrated microlenses which are coupled deterministically to single QDs. The microlenses with diameters down to 800 nm were aligned to single QDs by in-situ electron-beam lithography using a low-temperature cathodoluminescence setup. This deterministic device technology allowed us to obtain an enhancement of photon extraction efficiency for QDs integrated into microlenses as compared to QDs in unstructured surfaces. The excellent optical quality of the structures is demonstrated by cathodoluminescence and micro-photoluminescence spectroscopy. A Hong-Ou-Mandel experiment states the emission of single indistinguishable photons.

The prospect of realizing building blocks for long-distance quantum communication is a major driving force for the development of advanced nanophotonic devices [1]. Significant progress has been achieved in this field with respect to the fabrication of efficient quantum-dot-based single-photon sources [2][3][4][5][6][7]. More recently, even spin-photon entanglement [8,9] and quantum teleportation [10,11] have been demonstrated in semiconductor systems. These results are considered as crucial steps towards the realization of a quantum repeater. The related work has almost exclusively been performed on self-assembled quantum dots (QDs) and random device technology. At this point it is clear that further progress in this field towards real applications will rely crucially on deterministic device technologies which will, for instance, enable the processing of bright quantum light sources with pre-defined emission energy [12].
Here we report on enhanced photon-extraction efficiency from monolithically integrated microlenses which are coupled deterministically to single QDs. The microlenses with diameters down to 800 nm were aligned to single QDs by in-situ electron-beam lithography using a low-temperature cathodoluminescence setup. This deterministic device technology allowed us to obtain an enhancement of photon extraction efficiency for QDs integrated into microlenses as compared to QDs in unstructured surfaces. The excellent optical quality of the structures is demonstrated by cathodoluminescence and micro-photoluminescence spectroscopy. A Hong-Ou-Mandel experiment states the emission of single indistinguishable photons.
An important figure of merit of all non-classical photon sources is the photon-extraction efficiency (PEE), i.e. the probability of coupling a photon emitted by a QD into the external optics [13]. In the case of InGaAs QDs embedded in a GaAs matrix, this measure is limited by total internal reflection to about 1% for collecting optics with a numerical aperture (NA) of 0.8. While the PEE can be enhanced also by advanced waveguide structures or microcavities, the microlense-approach provides a simple and, therefore, very appealing concept to increase the photon flux of QD based quantum devices [14]. Moreover, microlenses provide broadband enhancement of the PEE without the need of complicated spectral tuning methods, and as such they are particular attractive for boosting the extraction of polarization-entangled photon pairs from the biexciton-exciton cascade of QDs 2 with diminishing fine-structure splitting (FSS) [15]. The microlense approach is very general and can also be applied in a straightforward way to enhance the PEE of materials such as wide-bandgap II/VI-semiconductors or InGaAs QDs on (111)-oriented GaAs with FSS close to zero [16][17][18] for which the growth of cavity structures is still challenging or has not been mastered yet at all. More than this, if combined with suitable cavity structures, significant further enhancement of PEE is expected for the microlens approach [14].
The application of solid immersion lenses (SILs) to enhance the PEE of photons emitted by a single QD in a semiconductor matrix was nicely discussed in Ref. [14]. Basically SILs circumvent restrictions imposed by total internal reflection at the semiconductor-air interface which lead to poor PEEs. Using commercially available mm-scale hemispheric SILs with a refractive index in between those of the semiconductor material and air the PEE is enhanced to 6 %. Significantly better performance and PEE = 17 % for a NA of the collecting optics of 0.8 can be achieved by integrating the lens directly into the semiconductor material.
However, this route is technologically more demanding since it is not trivial to find suitable processing parameters to "shape" the integrated SILs and, moreover, to align them to a single target emitter in an ensemble of self-assembled QDs.
In this work we apply a recently developed in-situ electron-beam lithography technique [19] to take advantage of the SIL concept to enhance the PEE. This lithography technique which is based on low-temperature cathodoluminescence spectroscopy allows us to realize deterministic and monolithically integrated QD microlenses which are spatially aligned precisely to single target QDs. While our integrated microlense approach can in principle be adopted to all semiconductor materials with embedded nanostructures, we have chosen the technically mature InGaAs/GaAs material system for demonstration purposes. We have also chosen a generic sample layout without additional optical elements such as a distributed Bragg reflector to focus solely on the geometrically enhanced PEE obtained by deterministically integrated microlenses.
Lens fabrication is sketched in Fig. 1(a-d). In the central processing step we invert the electron-beam sensitive resist polymethyl methacrylate (PMMA) covering the semiconductor material selectively at the positions of target QDs (see Supplementary Information for details on the sample growth). After development, the locally inverted resist acts as an etch mask which is transferred into the semiconductor material in the subsequent plasma etching step.
Here, the exposure dose profile determines the local thickness of the PMMA-etch mask which allows to tailor the microlens shape. A respective calibration curve obtained under an acceleration voltage of 15 kV is presented in Fig. 2 (black curve) and shows that the thickness of the remaining PMMA increases from 0 nm at 10.5 mC/cm 2 to 70 nm at 25.5 mC/cm 2 .
This thickness determines the corresponding etch depth which varies by almost 500 nm as a function of the dose as can be seen by the red trace in the same figure. Thus, in order to shape microlenses into the semiconductor material we use the calibration curve presented in Fig. 2 to calculate radial profiles of the exposure dose as indicated in Fig. 1 Supplementary Information). The results are presented in Fig. 5. The µPL spectrum of a single QD-microlens is depicted in Fig. 5(a) and shows discrete emission lines which are identified by photon cross-correlation measurements (not shown here) to stem from the recombination of negatively charged excitons X − , biexcitons XX, and neutral excitons X. Figure 5(b) shows the photon auto-correlation function g (2) (τ ) of the X − line. The value of g (2) (τ ) at zero delay yields single-photon emission with a very high suppression of multi-photon emission events associated with a measured value of 0.19. Taking the timing resolution of the setup into account we extract a deconvoluted value of g (2) (0) < 0.001. Furthermore, the radiative lifetime τ R = 779 ps is extracted from the fit.
A high degree of indistinguishability of photons emitted from the QD-microlenses is proven by analyzing the emission of the X − line via a fiber-coupled HOM setup. We performed measurements both for parallel and orthogonal configuration of the half-wave plate, not-switching or switching the polarization in one interferometer arm with respect to the other. For the parallel configuration, presented in Fig. 5(b), we observe an antibunching dip at zero delay clearly below 0.5 indicating indistinguishability of the emitted photons. The solid line shows a fit to the experimental data according to [20] and considering the timing resolution of the setup. We determine the coherence time τ C = 749 ps, g ⊥ (0) = 0.5 which confirms the perfect single photon emission of the QD.
In summary, we have demonstrated a versatile method to enhance the photon-extraction efficiency for semiconductor nanostructures by fabricating deterministic microlenses. The QD-microlens structures have proven that our approach has a high potential to boost the application of quantum light sources. Due to their broadband enhancement of photon extraction efficiency and the straightforward fabrication process they will be particularly appealing for the realization of polarization entangled photon pairs and sources of indistinguishable photons in quantum repeater networks.

Methods
The deterministic microlenses are fabricated in the following way. First, we spin-coat a 190 nm thick layer of PMMA on the sample before recording cathodoluminesence intensity maps in the CL-system at low temperature and a low dose of 10.5 mC/cm 2 . Out of a large number of embedded QDs we select target QDs which show reasonably spatially isolated emission spots, and which are therefore suitable to be deterministically integrated into microlenses. In the subsequent in-situ electron-beam lithography step we write lens-patterns into the resist as described above. Actually, we are writing concentric circles which are cen- The lens shape was deliberately chosen to be of Gaussian type: Three-dimensional electron beam lithography is quite challenging as he spatial dose profile must resemble the targeted lens profile. This is complicated by the omnipresent proximity effect in EBL whose local impact can be described by a Gaussian profile [25]. As the convolution of a Gaussian (lens) profile with another Gauss is still Gaussian, the targeted shape will be maintained in this way.

B. Determination of enhancement factor
To determine the enhancement factor we performed a statistical analysis in which we compare the CL emission intensity of a large number (25) of QDs in the unpatterned region with the QDs below our microlenses. These measurements were conducted as a function of excitation density. The reference intensity for each QD was the intensity of the singleexcitonic lines when they saturated. This procedure was chosen to rule out the effect of different excitation conditions for QDs in microlenses and in the unprocessed regions. The analysis yields an enhancement factor as high as 6.6 ± 2.7 for ideally shaped microlenses for NA = 0.8.

C. Hanbury-Brown and Twiss, and Hong-Ou-Mandel experiments
For details of the HBT setup please refer to [19].The Hong-Ou-Mandel experiment was carried out using an asymmetric Mach-Zehnder interferometer based on polarization maintaining fibers and a half wave plate (HWP) for switching the polarization in one arm of the interferometer [26,27]. Silicon avalanche photo diodes with a timing resolution of 350 ps acted as detectors. In the measurements also two side dips at ±15.25 ns down to 0.75 are observed. These dips originate from the optical delay in one interferometer arm and the symmetry of the side dips indicate a balanced second beam splitter (R 2 = T 2 = 50% ) [22].
All measurements were performed at a temperature of 15 K with an excitation power of P = 2.1 µ W corresponding to low power-power excitation well below saturation of the QD states.

II. NUMERICAL METHOD
The calculation of the photon-extraction efficiency and of the enhancement factor was performed in the framework of a finite-element method by using the commercially available software package JCMsuite by JCMwave (see http://www.jcmwave.com for details). Based on AFM results, a full 3D model structure of the lens was created and the light intensity distribution resulting from point sources placed in the semiconductor material below the microlens was computed. The angular integration of far field intensity for a given NA was performed in a post-processing step.