1. Introduction

There has been rapid progress in the development of integrated, triggered, single-photon sources for quantum communication and quantum information processing applications; sources in which a single photon is produced upon a deterministic “triggering”. In particular, single-photon sources based on self-assembled InAs QDs embedded in GaAs/AlGaAs micropillar cavities can now be electrically triggered [1], and in-principle can be integrated into in-plane waveguides [2]. Other InAs QD-based single-photon sources that incorporate photonic crystal microcavities [3] could be easily coupled directly into in-plane photonic “circuits” monolithically integrated in the GaAs host wafer [4, 5]. Triggered single-photon sources based on coupling quantum-dot exciton emission directly to waveguide modes, rather than cavity modes have also been predicted [6, 7], developed [8, 9], and demonstrated experimentally using InAs QDs [10]. Fully-integrated, non-classical photonic circuits in III–V wafers are on the horizon.

Silicon-on-insulator (SOI) is an alternative platform for developing optical circuitry. Motivated by the potential advantages of facile integration with sophisticated electronics, silicon photonics has grown remarkably through both industry and research. On-chip detectors [11–13], sources [14, 15], and passive waveguides, resonators, and microcavities have all been demonstrated using complimentary metal-oxide-semiconductor (CMOS)-compatible optical stepper foundry resources. To our knowledge there have not yet been any integrated, triggered single-photon sources reported in SOI. Indeed, the lack of epitaxial QDs with strong oscillator strengths in silicon poses a considerable challenge in this regard. A hybrid approach is almost surely required to achieve an integrable, triggered single-photon source in SOI.

One solution might be via the epitaxial growth of III–V heterostructure nanowires on silicon (111) surfaces [16,17]. InP-based nanowires on silicon have been reported with emission at 1.5 μm [18]. An alternative – and potentially simpler – quantum emitter at 1.5 μm would be PbSe or PbS colloidal QDs with diameters of ∼ 5 nm. There are many reports of silicon microcavity-coupled exciton emission from such QDs [19–27]. Progress towards deterministic localization of a single QD to the microcavity mode volume has also been made: we employed a site-selective process for integrating few PbSe QDs on the surface of silicon-host “L3” microcavities to within the main antinode of the fundamental mode [22]. In this approach, cavity-coupled exciton emission from these QDs stands above the background luminescence by at least a factor of 10 at room temperature for cavities with Q factors of several thousand.

Here we report the successful coupling of the cavity-enhanced PbSe exciton emission into and out of sub-μm sized channel waveguides in SOI “circuits” fabricated using optical stepper lithography through ePIXfab [28] at IMEC [29]. The coupled light intensity saturates strongly as the power of a continuous wave (CW), non-resonant excitation laser increases. To quantify this saturation behaviour, the cavity-coupled exciton emission from a stand-alone cavity containing PbSe QDs selectively located within the dominant cavity antinode was studied. The saturation threshold corresponds to the intensity of a few-μW CW HeNe laser focused to a spot size of diameter 2 μm on the cavity center. By carefully modelling the electromagnetic environment (including excitation source scattering, local radiative density of states at PbSe QD locations, and depolarization factors), and exciton dynamics within the QDs, this saturation is shown to be due to ground state depletion as excitons are trapped in non-radiative states.

2. Experiment

2.1. Integrated source

Integrated single-photon source schemes based on luminescent QDs as emitters often invoke a microcavity to efficiently collect the QD excitonic emission. A cavity design for which large Q and small mode volumes, and thus potentially efficient photon collection, have been realized in the SOI platform is the “L3” configuration, shown in Fig. 1, along with an intensity profile of its fundamental in-gap cavity mode. Such cavities can be coupled to photonic crystal waveguides that are impedance matched to low-loss, sub-μm scale silicon channel waveguides [30].

 

Fig. 1 (A) Schematic and scanning electron micrograph of an “L3” microcavity. (B) Fundamental in-gap cavity mode electric field intensity at the silicon-air interface, with etched holes outlined. Axes originate at the L3 slab centroid, and ẑ is perpendicular to x̂ and ŷ.

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The integrated source geometry used here is shown in Fig. 2(A) and Fig. 2(B), and likewise utilizes QDs (PbSe, colloidal) inside an L3 cavity coupled to a photonic crystal waveguide. The photonic crystal waveguide is in-turn coupled to a channel waveguide, and for the sake of easy and efficient free-space collection of cavity-waveguide-coupled photoluminescence, the channel waveguide is coupled to a large multi-mode tapered waveguide, terminated with a diffraction grating-coupler. This design is an optimization of an earlier realization in which we measured efficient cavity-waveguide coupling [30]. Fabrication was through ePIXfab and IMEC.

 

Fig. 2 (A) Close-up scanning electron micrograph of the photonic crystal cavity, photonic crystal waveguide, and channel waveguide region. (B) An optical image of the entire cavity, waveguides, and grating structure, with excitation spot (centered on the cavity) and collection spot (centered on the grating coupler) indicated. (C) Example ŷ-polarization-filtered PL spectrum for the excitation/collection geometry indicated in (B), and a plot of the cavity-enhanced, waveguide-coupled PL versus pump power.

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As-received SOI photonic chips were spin-coated with 2 μm of AZ P4110 photoresist to ensure the cavity-frequency emission would emanate from the grating coupler at a convenient collection angle. Photoresist over the photonic crystals (including photonic crystal cavities and waveguides) was removed by exposure to ultraviolet light and subsequent chemical development, while leaving resist over the tapered waveguides and grating couplers intact. Photonic crystals (again including photonic crystal cavities and waveguides) and channel waveguides were undercut by dipping the sample in a buffered oxide etch solution for 20 minutes. The etch was arrested by rinsing the sample in deionized water. The sample was then dried under nitrogen flow, dipped in a hexane solution of PbSe QDs (synthesized according to [31,32]) in a nitrogen environment, and finally placed under vacuum.

Excitation laser light of 1064 nm wavelength was focused onto the cavity surface to a spot size of approximately 20 μm by 30 μm (Fig. 2(B)), and emission was collected from one of the couplers with a 15 X reflecting microscope objective. Collected photoluminescence from the grating only (Fig. 2(B)) was passed through a small aperture and diverted to a Bruker RFS100 Fourier transform interferometer. Cavity-coupled photoluminescence excites only TE-polarized (ẑ · E = 0) channel waveguide modes that couple out of the grating couplers with ŷ-polarization. Figure 2(C) shows an example ŷ-polarized spectrum. The sharp cavity-enhanced photoluminescence peak sits atop an approximately 100 meV wide feature that matches the grating coupler spectrum and is due to PbSe QDs resting directly on the waveguides adjacent to the microcavity.

2.2. Stand-alone cavity

To facilitate a quantitative analysis of the saturation behaviour, data was obtained from a stand-alone L3 photonic crystal cavity in which PbSe QDs were grafted selectively into the cavity surface center, as depicted in Fig. 3(A), and described previously [22]. Since the QDs are confined to within 200 nm of the dominant antinode of the cavity mode, variations in the laser excitation intensity and the efficiency of coupling to the cavity are reduced.

 

Fig. 3 Experimental setup and resulting data modeled in this article. (A) Schematic of excitation/collection geometry: excitation (at 633 nm) and collection performed with a common 100 X microscope objective. Red-filled circle indicates 1/e excitation spot intensity. Shaded square indicates span of grafted PbSe QDs. (B) Example PL spectrum with cavity-coupled emission indicated, and cavity-coupled PL versus pump power.

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The center of the stand-alone cavities was excited at normal incidence with continuous wave 633 nm HeNe laser light polarized in the ŷ-direction, using a 100 X microscope objective that also collected all of the light scattered normal to the sample surface. The 1/e² intensity diameter was 2.0 μm on the cavity surface, and the excitation power was varied by attenuation with various neutral density filters. Measurements were performed with the sample in vacuum.

3. Model

Extensive modeling of microcavity-coupled exciton emission at low temperature in epitaxial InAs-based QD systems has revealed several features distinctive of quantum electrodynamics in the solid state environment. In particular, quantitative explanations of experimentally-measured emission spectra as a function of cavity-exciton detuning, and excitation power, require explicit treatment of acoustic phonon-scattering, rather than treating it phenomenologically, and do not require consideration of non-radiative recombination processes (e.g., see Refs. [33–40]). In contrast, room temperature experiments reported here involve solid-state formulations of colloidal PbSe QDs with dephasing rates of tens of meV [41, 42], in excess of the cavity detuning, and quantum yields much less than unity [32, 42–44]. We find it necessary to explicitly include non-radiative decay processes to explain the saturation behaviour, and a phenomenological treatment of phonon interactions is sufficient (e.g., through a pure dephasing rate to broaden the zero phonon line).

3.1. Master equation model

A simplified Hilbert space shown schematically in Fig. 4 was used to model the observed saturation behaviour. A single exciton state, |X〉, represents the low-energy, “brightest” component of the ground state manifold of excitonic states that is split by many factors in PbSe QDs. This is the state responsible for exciting the cavity mode. Inclusion of more than 2 cavity mode Fock states did not change the calculated cavity photon population. A single higher-lying state, |P〉, resonantly absorbs energy from the HeNe excitation source and rapidly transfers it to the |X〉 state. In order to fit the observed saturation behaviour, and to be consistent with the small quantum yield of exciton emission from monolayers of PbSe QDs on silicon surfaces, a non-radiative decay channel via state |Y〉 was also included.

 

Fig. 4 Minimal Hilbert space necessary to accommodate observed saturation behaviour: four states for the QD subspace and two for the cavity subspace. Significant decay paths indicated by solid blue arrows, of which squiggly lines are radiative and the remainder non-radiative. Laser field of Rabi coupling frequency Ω “pumps” the |P〉 state. The cavity is “fed” by coupling to the |X〉 ↔ |G〉 transition with electric-dipole coupling strength g.

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The model system Hamiltonian consists of the bare Hamiltonian, H₀, electric dipole coupling of the excitonic transition to the cavity field, H_cav, and a continuous wave laser field resonantly coupled to the higher energy electronic transition, H_pump:

We calculated the steady state density matrix ρ with the Lindblad-form master equation, which allows explicit inclusion of the population decay (γ_jk) and pure dephasing (γ′_j) rates:

To solve for the steady-state behaviour, we first transformed these equations to a picture with no explicit time dependence in the rotating wave approximations for ω_cav ≈ ω_X – ω_G = ω_XG and ω_pump ≈ ω_P – ω_G. The collection of steady state density matrix element equations dρ/dt = 0 was then cast into superoperator form, 𝒧ρ^v = 0, in which the steady state density matrix elements are components of the eigenvector of the zero eigenvalue of the superoperator 𝒧, and calculated via the inverse power method. From the steady state density matrix we calculated the cavity population 〈a^†a〉, which is proportional to the observed cavity intensity emission.

3.2. Depolarization influence, depolarization model, and dielectric environment

Depolarization effects typically do not play an important role in III–V based quantum electrodynamic modeling of InAs QDs as there is negligible dielectric contrast between the InAs QDs and the surrounding host material, and absorption and emission properties of these QDs are measured in environments of dielectric contrast similar to those in which they are utilized. However, comparable depolarization effects must be included in similar modeling of PbSe QDs, as absorption and emission properties are typically measured in dilute solutions whereas utilization is in vacuum, on the surface of a silicon photonic crystal in our experiments, where depolarization is substantially different.

Accurate accounting of depolarization effects due to our specific dielectric environment – the L3 cavity and many quantum dots – requires computational methods, as no wholly suitable closed-form expression exists. However, despite the necessity and capabilities of computational methods, partial use of closed-form expressions is still advantageous, particularly expressions that are consistent with both the numerical calculations performed and, where available, experimental data. Towards this end, electric field depolarization at the quantum dot-environment interface was accounted for by the Lorentz local field factor [45, 46], |(2 +ε_QD/ε_med)/3|, for a QD of permittivity ε_QD in medium of permittivity ε_med. The particular computational method we used was finite-difference time-domain (FDTD) electrodynamic simulations, with FDTD Solutions software from Lumerical Solutions, Inc. [47].

We accounted for the influence of depolarization by first using experimental data obtained from dilute solutions of PbSe QDs, and the Lorentz local field factor, in order to extract an “intrinsic”, free-space dipole transition moment for the emissive exciton transition, |μ_XG,0|. We then considered this intrinsic dipole to be located within a model dielectric environment, and FDTD simulations were used to evaluate the electric field at the dipole, which included all depolarization factors with computational exactitude. Details below and in following subsections.

The model dielectric environment of the QDs in our experiments is shown in Fig. 5. The total permittivity, ε(r,ω) = ε_L3(r,ω) +ε_QDs(r,ω), consists of the silicon-host L3 photonic crystal cavity ε_L3(r,ω) (w/ backing silicon) and a close-packed hexagonal array of 45 QDs on the cavity surface ε_QDs(r,ω) with pitch varied from 6 nm to 8 nm. The QD arrangement is based on electron microscope images of QDs on silicon surfaces that exhibit short-range hexagonal order, packed with a pitch within the range adopted in our model. The intrinsic dipole was then considered to be located at the center of the centroidal QD, at position r_QD. The L3 cavity slab has a thickness of 198 ± 4 nm, pitch of 420 ± 4 nm, air hole radius of 122 ± 10 nm, and the two holes on the x̂-axis are shifted away from the cavity centroid by 10 ± 4 nm. The distance between the L3 cavity slab and backing silicon is 1193 ± 10 nm.

 

Fig. 5 Model dielectric environment ε(r,ω) = ε_L3(r,ω) +ε_QDs(r,ω). Nanocrystal array ε_QDs(r,ω) on left, centered on the L3 cavity surface. The computational volume for FDTD calculations (see text) is restricted to the 3 μm cube centered about the centroidal QD. The intrinsic “test” dipole is located at the center of centroidal QD, position r_QD. The device silicon slab is surrounded by vacuum above and below, with backing silicon 1.2 μm below.

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Justification of the Lorentz local field factor is provided by its successful use in (1) relating PbSe QD threshold absorption coefficients and oscillator strengths to radiative decay rates, in an analysis by Moreels et al. [48], and (2) in modeling short-wavelength, continuum PbSe QDs absorption characteristics, particularly accounting for absorption coefficient differences of PbSe and PbSe/CdSe core/shell QDs in terms of dielectric depolarization effects alone, in an analysis by De Geyter et al. [49]. The Lorentz local field factor was also found adequate in local-field modeling of emission lifetimes of CdSe/ZnS colloidal QDs on various dielectric substrates [45], and it has been suggested to be appropriate in various theoretical formulations [46, 50, 51]. In addition, we have verified that this approximation is consistent with results obtained using full FDTD simulations and, finally, because the Ohmic loss due to a small, non-resonant imaginary part of the QD dielectric response at the exciton wavelength is negligible compared to that due to non-radiative decay to the |Y〉 state.

3.3. “Simple” model parameters

Before calculation of the intrinsic dipole moment and other depolarization-sensitive quantities, let us establish the many “simple” model parameters that are known already and unaffected by depolarization specific to our dielectric environment.

The cavity mode frequency ω_cav was taken directly from the PL spectra, and for data fitting we set the exciton transition energy h̄ω_XG = h̄(ω_X – ω_G) = h̄ω_cav. The pumped transition energy h̄(ω_P – ω_G) was set to the HeNe excitation photon energy, h̄ω_pump = hc/633.0nm, where h is Planck’s constant and c is speed of light in vacuum.

The measured cavity Q of 3 × 10³, along with ω_cav, sets γ_cav = 7 × 10¹⁰ Hz. QD inclusion had negligible effect on the decay rate. The population decay time ${\gamma }_{PX}^{-1}$ from |P〉 to |X〉 was taken to be 5 picoseconds, based on a variety of measured and calculated values [52–56]. Other significant decay parameters are described in the “fit parameter” subsection, below.

A pure dephasing rate of γ′_X = 4.8 × 10¹³ Hz was taken from an analysis of PbSe thick film PL spectra [42]. The |Y〉 state pure dephasing was taken to be γ′_Y = γ′_X, but the exact value is neither known or consequential for our analysis. The ground state pure dephasing γ′_G was assumed negligible. For completeness (see “Laser-QD coupling Ω” subsection), a value of γ′_P = 10¹⁴ Hz was used in the simulations reported here, although our modeling results are independent of γ′_P for a large range of γ′_P, owing to its large value.

3.4. Intrinsic, free-space dipole moment |μ_XG,0| of the radiative exciton transition |X〉 ↔ |G〉

We intuitively define the intrinsic, free-space dipole moment as the |X〉 ↔ |G〉 dipole transition moment when depolarization and non-free-space photonic density of states are “factored out”. Quantitatively this was achieved by defining |μ_XG,0| in relation to a reliably-measured spontaneous emission rate in an environment for which effects on radiative decay are accountable.

The environment we chose was a dilute solution of QDs in solvent, for which the radiative lifetime was taken to be ${\gamma }_{\mathit{XG},\text{QD}+\text{sol}}^{-1}=(3\pm 1)\mu \text{s}$, consistent with many photoluminescence lifetime and quantum efficiency reports of approximately (1–2)μs and several to many tens of percent, respectively. Relating this to |μ_XG,0| was achieved by firstly relating |μ_XG,0| to the free-space spontaneous emission rate γ_XG,0 via Fermi’s Golden Rule, then relating γ_XG,QD+sol to γ_XG,0 with aid of the Lorentz field factor. In this approach, we obtained:

The resulting values are $\left|{\mathit{\mu }}_{\mathit{XG},0}\right|=7_{-2}^{+3}$ Debye and ${\gamma }_{\mathit{XG},0}=5_{-2}^{+6}\times {10}^6$ Hz, where super- and sub-scripts denote uncertainty deviations from in-line numerical value. The corresponding in-bulk PbSe spontaneous emission rate, for n_bulk ≈ 5 and ε_QD,nr(ω_XG) comparable to bulk PbSe permittivity at ω_XG [58], is γ_XG,bulk ≈ n_bulk ${\gamma }_{\mathit{XG},0}=2_{-1}^{+3}\times {10}^7$ Hz, which is 10 to 200 times slower than the typically 0.5 to 2 GHz spontaneous emission rates of epitaxial InAs QDs in bulk semiconductor hosts [59–61]. The “intrinsic” dipole transition moments of the PbSe excitons are on order 3 to 7 times smaller than those of typical InAs QD excitons [62–65].

3.5. Spontaneous emission rate γ_XG = γ_XG,ε(r)

The spontaneous emission rate of an exciton associated with a QD located within a hexagonally-packed array of QDs on the L3 cavity surface (our ε(r) environment), into all electromagnetic modes except the cavity mode, γ_XG,ε(r), was calculated as follows.

In three separate simulations, a point dipole source of orientation x̂, ŷ, or ẑ was placed at r_QD in the ε(r) environment, and the radiated power, P_rad,ε(r), was extracted. The ratio of P_rad,ε(r) to the power radiated by the same dipole source in free space, P_rad,0, is equal to the ratio of the corresponding radiative decay rates for point dipole emitters, γ_XG,ε(r) and γ_XG,0 [66]:

This relationship, along with γ_XG,0 from the previous subsection, provided us with a means to calculate γ_XG,ε(r). In doing so, we also (1) averaged γ_XG,ε(r) over all three dipole orientations, and (2) excluded the cavity mode contribution to γ_XG,ε(r), which is already accounted for in g, by considering only the power radiated at several cavity mode linewidths above the cavity mode frequency (i.e. at ω_cav +δω). The result is essentially independent of δω, and equals

Thus the spontaneous emission rate of excitons in this complex dielectric environment turns out to be very similar to isolated QDs in solution (γ_XG,QD+sol). A decomposition of influences on the spontaneous emission rate is as follows: taking the QDs out of solution and into air increases the dielectric contrast, resulting in a smaller spontaneous emission rate by a factor of 4 to 7 (depending on the originating solvent). In contrast, despite the presence of the photonic band gap, the L3 cavity – even excluding the cavity mode – increases an otherwise free-space spontaneous emission rate by a factor of ∼ 4, when averaged over three orthogonal dipole orientations. The spontaneous emission rate enhancement due to the L3 slab, excluding the cavity mode and QD array, is presented in more detail in Fig. 6. Such behaviour at the surface of a uniform, hexagonal, silicon-host photonic crystal has been previously reported [67].

 

Fig. 6 Spontaneous emission rate of a point dipole source of frequency ω_cav + δω, for positions along the ẑ-axis of the L3 cavity, excluding the cavity mode and QD array, for electric dipole orientations along axes x̂, ŷ, or ẑ (see text and orientation definitions in Fig. 1). All values normalized to the free-space spontaneous emission rate γ_XG,0.

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3.6. Cavity-QD coupling g

Given the intrinsic dipole transition moment, calculation of the cavity-QD coupling $g={\mathit{\mu }}_{\mathit{XG},0}\cdot {\mathbf{E}}_{\text{cav}}^{\text{vac}}({\mathbf{r}}_{\text{QD}})/\overline{h}$ was reduced to calculation of the cavity mode vacuum electric field at the QD position, ${\mathbf{E}}_{\text{cav}}^{\text{vac}}({\mathbf{r}}_{\text{QD}})$. This too was calculated using FDTD Solutions, here by exciting the cavity mode in the full ε(r) dielectric environment, letting all electric fields except the cavity mode field E_cav(r) decay entirely, then evaluating:

Integration was over the entire computational volume, and the value is consistent with the value from a corrected formula for non-Hermitian modes [68]. QD inclusion has an insignificant effect on the cavity mode volume. The cavity-QD coupling strength is $g=6_{-2}^{+4}\times {10}^9\hspace{0.17em}\text{Hz}$. The scattering rate into the cavity mode is $R_{\text{cav}}=g^2/{\gamma }_X^{\prime }=8_{-4}^{+13}\times {10}^5\hspace{0.17em}\text{Hz}$.

3.7. Laser-QD coupling Ω

Direct calculation of Ω = μ_GP,0 · E_pump,ε(r)(r_QD)/h̄ is impossible, as we know of no way to directly determine the laser-pumped dipole transition moment μ_GP,0 from experimental data. However, the saturation behaviour of interest depends not on Ω alone, but instead on the absorption rate R of the QD, related to Ω and the electric field intensity inside the QD by

Thus, as can be seen through this closed-form expression and as confirmed for our simulations, neither a specific value of Ω or γ′_p was necessary for simulating the saturation behaviour. Instead, knowledge of the absorption rate R, particularly for a calculable laser field amplitude at the intrinsic dipole location, is adequate.

The in-solution absorption rate per QD per incident electric field intensity at our pump wavelength of 633 nm was obtained from Moreels et al. [69] by way of their reported absorbance at 400 nm, A_400nm = 1.246 cm⁻¹, of a known concentration C_Q–PbSe = 0.32 μM of PbSe QDs, and an absorption spectrum showing that the absorbance at 633 nm relative to the absorbance at 400 nm is A_633nm/A_400nm ≈ 0.2. In terms of the laser field amplitude inside the QD, |E_pump(r_QD)|, this absorption rate per QD is

To finalize application of these equations to our system, we needed to relate the laser field amplitude inside the QD for our full ε(r) dielectric environment, |E_pump,ε(r)(r_QD)|, to the measured power, P₀, of our HeNe excitation source. This was accomplished by (1) FDTD simulations from which we extracted |E_pump,ε(r)(r_QD)|/|E_pump,0|, where |E_pump,0| is the incident laser field amplitude, and (2) relating |E_pump,0| to P₀ via the Gaussian beam profile and our measured minimum 1/e² beam radius W₀ = 1.0 μm, ${\left|{\mathbf{E}}_{\text{pump},0}\right|}^2/2{\eta }_0=2P_0/\pi W_0^2$. Combining, we obtained:

 

Fig. 7 Intensity profiles of HeNe excitation field, as modulated by the L3 cavity ε_L3(r). Gaussian laser field was injected along the ẑ axis towards increasingly negative z, indicated by black arrow. Air-silicon interfaces lined in black. (left) Profile several nanometers above the slab surface, the plane containing the PbSe QDs. (right) Profile in the x = 0 plane.

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3.8. Fit parameter, ${\tau }_{\text{non}-\text{rad}}={\gamma }_{\mathit{YG}}^{-1}+{\gamma }_{\mathit{XY}}^{-1}$

In previous subsections, we established the pump rate and rates for the two radiative decay paths, i.e. |X〉 → |G〉 and through the cavity. Of all the model parameters, the two not addressed in previous subsections are both associated with the non-radiative decay path, i.e. γ_XY and γ_YG. Neither of these are known for our particular QDs beyond the modeling efforts here. However, the limit of γ_XY ≫ γ_XG + R_cav enables a unique determination of the non-radiative decay time ${\tau }_{\text{non}-\text{rad}}={\gamma }_{\mathit{YG}}^{-1}+{\gamma }_{\mathit{XY}}^{-1}$ for a particular dielectric environment, and we chose this to be our sole fit parameter to characterize the non-radiative decay path (results next section). Drawing from ${\gamma }_{\mathit{XG}}=4_{-2}^{+5}\times {10}^5\hspace{0.17em}\text{Hz}$ and $R_{\text{cav}}=8_{-4}^{+13}\times {10}^5\hspace{0.17em}\text{Hz}$ as presented in previous subsections, the limit in which we extract τ_non–rad equates to ${\gamma }_{\mathit{XY}}\gg {1.2}_{-0.6}^{+1.8}\times {10}^6\hspace{0.17em}\text{Hz}$.

This enabling limit is motivated by (1) measurements indicating low quantum yields of exciton emission from monolayers of PbSe QDs on unpatterned SOI wafers, and (2) FDTD simulations indicating R_cav + γ_XG calculated for our textured dielectric environment is not significantly larger than the spontaneous emission rate of QDs in a monolayer on unpatterned SOI. These measurements consistent of published [42] thick-film integrated PL versus temperature in which the room-temperature PL is observed to be a factor of 10 lower than the low-temperature PL, thus establishing a maximum 10% quantum efficiency at room temperature, and unpublished PL lifetime measurements establishing non-radiative relaxation in monolayers of PbSe QDs on unpatterned SOI wafers occuring several times faster than in thick films.

4. Modeling results

Attempts to exclude the non-radiative decay path were unsuccessful in accommodating the observed cavity-coupled saturation behaviour, as seen in Fig. 8, in which the best (minimum χ²) 3-state fits are inadequate. The best 3-state fits are presented for the smallest and largest saturation powers consistent with the model parameters. The smallest 3-state fit saturation power is still 5 times larger than the observed saturation power. The fourth state |Y〉 can, however, accommodate the lower observed saturation power if it possesses a sufficiently long lifetime, i.e. if it serves as a non-radiative “population-trapping” state, as seen in Fig. 8, for which the 4-state fit is adequate. Consequently, minimally four QD states are necessary to model our system.

 

Fig. 8 Best (minimum χ²) fits to cavity-enhanced photoluminescence for only three electronic levels (left), i.e. without a non-radiative state, and for four electronic levels (right), i.e. including a non-radiative “trap” state.

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Over our model parameter space, including all estimated uncertainties, we find the τ_non–rad values required for a 4-state fit are ${\tau }_{\text{non}-\text{rad}}\approx 3_{-2}^{+3}\mu \text{s}$. Further, because ${\tau }_{\text{non}-\text{rad}}={\gamma }_{\mathit{YG}}^{-1}+{\gamma }_{\mathit{XY}}^{-1}$ and ${\gamma }_{\mathit{XY}}\gg {1.2}_{-0.6}^{+1.8}\times {10}^6\hspace{0.17em}\text{Hz}$, as established in the previous section, our fit parameter approximately coincides with the trap state lifetime, ${\tau }_{\text{trap}}\equiv {\gamma }_{\mathit{YG}}^{-1}$. Thus the trap state lifetime consistent with our model parameters is ${\tau }_{\text{trap}}\approx 3_{-2}^{+3}\mu \text{s}$.

The span of τ_trap is dominated physically by (1) uncertainty in the QD packing density, (2) uncertainty in the L3 slab thickness, and (3) stated uncertainty in the solvent permittivity. The first two of these three have a direct and significant bearing on the pump field inside the QD, so for the purpose of graphically representing the influence of these factors on the trap state lifetime, it is convenient to define a plot parameter directly related to the pump field inside the QD. We defined an effective depolarization parameter, DPF, that is equal to the pump field inside the QD (for our full dielectric environment), normalized to the pump field in the QD for an otherwise isolated QD in free space, for a fixed pump power:

Figure 9 contains the plot of τ_trap versus DPF. Variation of τ_trap with DPF is primarily due to uncertainty specific to our dielectric environment, e.g. relating to the photonic crystal cavity or QD array, whereas variation of τ_trap for a fixed value of DPF is attributable primarily to the assumed uncertainty in the solvent permittivity that entered into our model through the estimated absorption rate R, and radiative lifetime ${\gamma }_{\mathit{XG},\text{QD}+\text{sol}}^{-1}=(3\pm 1)\mu \text{s}$, of the QDs in solution.

 

Fig. 9 Trap state lifetime ${\tau }_{\text{trap}}={\gamma }_{\mathit{YG}}^{-1}$ required to fit the data. Parameterization of τ_trap is in terms of the “effective depolarization”, DPF, which is defined in-text (see Eq. (18)) and is equal to the laser field inside the QD in the full model dielectric environment relative to the laser field inside the same QD in vacuum. Variation of τ_trap with DPF is dominated by uncertainties in parameters specific to our dielectric environment (photonic crystal cavity, QD array), whereas variation of τ_trap for a particular DPF is dominated by parameter uncertainty not specific to our dielectric environment (e.g. solvent permittivity from solvent-based QD properties). Points are sampled from the model parameter space.

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The fact that the effective depolarization parameter is near unity reflects the fact that, despite the multitude of significant depolarization mechanisms (e.g. QD array, SOI platform, L3 cavity), the laser field inside the QD is comparable to the laser field inside the same QD isolated in vacuum for the same pump intensity. This is a coincidence of this particular dielectric environment, and the contributing factors can be decomposed as follows: the SOI platform alone results in a decrease of the pump field in the QD by roughly a factor of 5 (relative to a free-space pump field), the L3 cavity texture increases the pump field by a factor of roughly 3 (relative to a bare SOI substrate), and the surrounding QD array increases the pump field inside the QD by another factor of roughly 1.4; compounding these results in a value of roughly 0.8 for the central set of parameters. Note that non-inclusion of any of these factors would result in a pump field amplitude outside our estimated uncertainties.

5. Conclusions

Realization of an integrated, silicon-host, triggered single-photon source requires substantial progress on several fronts. Our work here, exploring the integration of colloidal PbSe QDs with finished photonic structures, is no exception. We have demonstrated room-temperature coupling of microcavity-coupled QD photoluminescence into a channel silicon waveguide. Elimination of unwanted background photoluminescence is an outstanding challenge, but this should be dramatically reduced by using our already established site-selective binding technique to restrict QDs primarily to the main anti-node of the microcavity mode.

Extensive quantitative modeling of the cavity-coupled emission saturation highlights the importance of dealing with large depolarization factors, or local field corrections, in this system, as compared to the more commonly studied InAs epitaxial QD systems. The conclusion drawn from the modeling is that non-radiative trap states retain excited excitons, and prevent them from returning to the ground state with a time constant on the order of ∼ 3 μs. This occurs when the rate γ_XY at which excitons decay to this non-radiative state far exceeds their total radiative decay rate to both the cavity and radiation modes, i.e. γ_XY ≫ 1 × 10⁶ Hz. These conclusions are consistent with low quantum yields observed for exciton emission from PbSe QDs on silicon surfaces, in vacuum.