Supporting Info for "Deterministic Integration of Quantum Dots into on-chip Multimode Interference Beamsplitters Using in Situ Electron Beam Lithography" Peter Schnauber, Johannes Schall, Samir Bounouar, Theresa Höhne, Suk-In Park, Geun-Hwan Ryu, Tobias Heindel, Sven Burger, Jin-Dong Song, Sven Rodt, and Stephan Reitzenstein, Institut für Festkörperphysik, Technische Universität Berlin, Hardenbergstraÿe 36, 10623 Berlin, Germany Zuse Institute Berlin, Takustraÿe 7, 14195 Berlin, Germany Center for Opto-Electronic Material and Devices Research, Korea Institute for Science and Technology, Hwarangno 14-gil 5, Seongbuk-gu, Seoul 02-791, Republic of Korea E-mail: stephan.reitzenstein@physik.tu-berlin.de Multimode interference (MMI) coupler geometry The dimensions of the ideal (simulated) MMI are shown in Fig. S 1 a and it's eld distribution in Fig. S 1 b. This eld distribution is identical to that in the main text, but shown with the correct aspect ratio of the gure here. The actual MMI coupler containing QD1 deviates from the simulated MMI in the following parameters: the access and exit taper width is 1
(1.1 ± 0.1) µm, the exit waveguide separation is (3.05 ± 0.01) µm, the length of the MMI is (69.3 ± 0.3) µm. Figure S 1: a) Geometry of the simulated MMI. b) Simulation of the eld distribution inside the device shown in a) (same as in the main text). Multi-mode interference coupler transmission measurement Analogous to the propagation and bend loss, the MMI transmission is determined using U-shaped waveguide structures. The propagation-loss-corrected emission intensity of the control port is compared to the combined emission intensity of the two signal ports, see Fig. S 2. In this case the QDs were integrated into the waveguides by chance, as the matching of waveguide dimensions between waveguides in mapped regions and non-mapped regions had not been fully established at the time of this measurement. MMI transmission vs. MMI length As the MMI length is the parameter that deviates the most from the simulated parameters, we calculate the MMI transmission with respect to the MMI length to estimate the eect on the device transmission. Fig. S 3 shows the simulation results. As expected, the transmission is barely reduced by the slight mismatch in MMI length. 2
QD QD QD 100µm Control Signal Figure S 2: Microscope image of U-shaped waveguide structure to measure the MMI transmission. QD light is coupled into the MMIs as well as the control waveguide section. Figure S 3: Simulation of MMI transmission vs. MMI length. A 500 nm mismatch reduces the transmission by 0.6 percentage points only. 3
Fit function for g (2) (τ) QD1 shows two decay times in time resolved measurements, which are t 1 = 1.00 ns accounting for recombination processes and t 2 = 2.89 ns accounting for recapture processes. The model to t the pulsed on-chip Hanbury-Brown and Twiss measurement therefore includes a two-sided bi-exponential decay. The bi-exponential decay is described by the side peak amplitudes P 1 and P 2 (free parameters) as well as the central peak amplitudes p 1 and p 2 (free parameters), with the times t 1 and t 2 being xed. The model is: g (2) (τ) = i=4 i 0; i= 4 ( p 1 exp τ ) t 1 ( + p 2 exp τ t 2 ( ) τ 12.5 ns i P 1 exp + t 1 ) + i=4 i 0; i= 4 ( ) τ 12.5 ns i P 2 exp t 2 As stated in the main text, there appears to be some uncorrelated background in our measurement. In order to obtain a conservatively estimated g (2) (0) value, we do not account for this background in our model, which is the reason for the slight deviations between t and experimental data in between coincidence peaks. Yield for manufacturing multi-emitter quantum photonic circuits Moving forward to the manufacturing of multi-emitter quantum photonic circuits, only deterministic device approaches are feasible. Non-deterministic approaches suer from a very low device yield, that renders the manufacturing of multi-qubit circuits impossible. The insitu EBL can ensure device yields >49 % with six quantum emitters even for low quantum dot densities and moderate ne-tuning ranges. In the following, we present an exemplary calculation of the manufacturing yield of an N-node quantum circuit using semiconductor quantum dots and spectral ne-tuning. To determine the corresponding device yield we assume: 4
A low QD density of 10 7 /cm 2 (and 10 8 /cm 2, 10 9 /cm 2 in the table below), which is favorable to limit the number of unwanted spectator quantum dots in the waveguides. A CL preselection map size of 50x50 μm 2, which is readily available with our technique. Considering the QD density of 10 7 /cm 2, then inside this 50x50 μm 2 map there are about 250 QDs. We could easily go to sizes of 100x100 μm 2 with proper cryostat drift control. The in-situ EBL write eld size used in this work is 300x300 μm 2. A spectral tuning range of 1 mev, which has been demonstrated by other groups for electro-optical 1 and strain tuning. 2 An inhomogenous broadening of the QD ensemble of 52.9 mev with the ensemble distribution being described by a simple Gaussian normal distribution with central energy of µ = 1.3375 ev and a one sigma distribution width of σ = 22.5 mev. We consider two (three, four, ve and six) QDs that will be integrated into two (three, four, ve and six) simple 450 nm wide single mode waveguides that cover much less area than the CL map. Thus, we can neglect the area inside the map that is covered by the WGs and is not available for selecting QDs. Using these assumptions, we calculate the probability to nd a rst QD of energy E QD1 in a ±5 mev range around the center of the QD ensemble: P QD1 = µ+5 mev µ 5 mev NormalDistr(µ, σ, E)dE = 18 % (1) This means, that on average there are 18 % 250 QDs = 45 QDs inside the 50x50 μm 2 map with energies close to µ. Thus, nding a good starting QD with energy close to µ is easily done. Now we calculate the probability to nd a 2nd QD within a ±0.5 mev tuning range around the rst QD: P QD2 = E(QD1)+0.5 mev E(QD1) 0.5 mev NormalDistr(E(QD1), σ, E)dE = 1.8 % (2) 5
As a result, having 250 QDs inside the mapping eld, the probability to nd exactly k QDs inside the energy tuning range is ( ) 250 P k = (P QD2 ) k (1 P QD2 ) 250 k. (3) k The probability to nd at least N QDs inside this energy range is P N = 250 k=n 1 ( ) 250 (P QD2 ) k (1 P QD2 ) 250 k. (4) k Note that the '-1' in the sum subscript originates from the fact that the energy of the rst QD can be basically arbitrarily chosen. With a deterministic approach, all N QDs (within the selected spectral range) can then be integrated into a nanophotonic device with N emitter nodes. In contrast, with a random device processing approach, one must consider the probability to coincidentally integrate an energetically suitable QD at the center of the randomly positioned nanostructure. For simple WGs as used in this work, the QD should roughly be within a ±50 nm window around the center of the WG to ensure good emitter-mode coupling. Assuming a 10 μm long WG 'light catching' section, the expectation value F for the number of QDs which are coincidentally inside the 'light catching' region of the device is: F = 100 nm 10 µm QD-Density (5) The probability for each of those QDs to lie inside the energy tuning range is P QD2, which has been calculated earlier. The probability, that at least one of the QDs inside the 'light catching' region lies within the energy tuning range is: P 1 = F ( ) F (P QD2 ) k (1 P QD2 ) F k k k=1 (6) 6
Where F is rounded to the next integer for simplicity. Now, the nal probability that a device with N 'light catching' nodes is randomly aligned in a way that it incorporates in all N nodes at least one QD within the energy tuning range, is: P N,rdm = P N 1 1 (7) Where the '-1' originates from the fact that the energy of the rst emitter can be arbitrarily chosen. In Table S 1, we compare the yield of our deterministic device processing approach with a random EBL approach by calculating and presenting P N and P N,rdm for certain QD densities and device node numbers N. It is obvious that the deterministic approach is superior and leads to yields 49 % for all considered cases. In contrast, with suitably small QD density and/or a number of N > 2 QDs to be integrated in the photonic circuit the yield of the standard process falls below 1 %. This clearly shows that it is not suitable for scaling up to N > 2 systems. Table S 1: Expected yield per write eld for successfully fabricating a nanophotonic device with N quantum dot nodes that are all within the resonance tuning range. A 50 50 μm 2 deterministic pre-selection eld and a 100 nm 10 μm spatial positioning tolerance is assumed. Values below 10 4 are rounded to 0 and above 0.9999 rounded to 1. Random yield P N,rdm Deterministic yield P N QD Density (cm ) 10 N 10 8 10 9 10 7 10 8 10 9 2 0.02 0.17 0.99 3 <10 3 0.03 0.94 4 0 0.005 0.83 1 5 0 <10 3 0.66 6 <10 3 0.49 7
References (1) Bennett, A. J.; Patel, R. B.; Skiba-Szymanska, J.; Nicoll, C. A.; Farrer, I.; Ritchie, D. A.; Shields, A. J. Applied Physics Letters 2010, 97, 031104. (2) Trotta, R.; Zallo, E.; Magerl, E.; Schmidt, O. G.; Rastelli, A. Phys. Rev. B 2013, 88, 155312. 8