1. Introduction

Semiconductor quantum dots (QDs) are, in many respects, an ideal light emitter for microcavity-coupled light sources. Small enough for strong quantum confinement effects but large enough to support a large transition dipole moment, QDs offer bright, tunable light sources for investigations of cavity effects on the spectral characteristics and dynamics of the spontaneous emission process [1]. QD-microcavity interactions are currently the subject of much research interest, owing in part to the potential applications that may stem from such systems [2]. Recent investigations have focused on single cavity-coupled QDs, both to maximize the enhancement and to study the detuning of the cavity and QD resonances [3,4]. In the case of direct gap QDs such as InAs or CdSe QDs, the fabrication methods are well established, the emission linewidths can be fractions of a meV at cryogenic temperatures, and they are sufficiently bright for single-particle spectroscopy [5–7]. While not offering these advantages to the same degree, indirect-gap QDs (especially silicon) coupled to microcavities may lead to a silicon-compatible light source [8] or silicon-based sensing devices [8] that feature benign, non-toxic chemistry and microelectronics compatibility.

In the strong QD-cavity coupling regime, the coupling rate exceeds both the cavity decay rate and the non-resonant QD decay rate, leading to the manifestation of cavity quantum electrodynamic (CQED) effects in the QD emission spectrum. One of the main technical motivations is the development of single photon sources for quantum information systems [9]; however, strong coupling imposes severe technical constraints on the cavity-QD system. In contrast, the weak coupling regime is characterized by a QD-cavity coupling rate smaller than the loss rates. Weak coupling is often referred to as the Purcell regime, in honor of E.M. Purcell’s seminal 1946 abstract in which the on-resonance rate enhancement was proposed [10]. This regime is much easier to attain experimentally, since it does not require cryogenic temperatures or high cavity quality (Q) factors. Coherent effects are not observed in weakly coupled systems; however, one obtains an enhancement or suppression of the spontaneous emission rate as a result of the modified optical density of states in the cavity. Cavity-induced spontaneous emission modifications are now under investigation by many groups, and could potentially lead to low threshold lasers or new fluorescent biosensing devices [11–13].

Curiously, few studies have investigated the Purcell regime in the more practical situation in which an ensemble of QDs is coupled to a microcavity. Notably, Gayral and Gérard recently showed that the QD photoluminescence (PL) Q-factors depend on the pump conditions and on the geometry of the cavity and measurement systems [14], although they considered only the theoretically more straightforward case in which the QD emission linewidth is negligible. There are no previous investigations of Purcell effects on the spontaneous emission dynamics in inhomogeneously broadened distributions of QDs with various sizes and finite linewidths, whose ensemble emission overlaps many cavity modes – despite the fact that these are among the simplest and most common systems to realize experimentally.

This paper is organized as follows. In the first section we will discuss the theory of QD-cavity coupling in three important cases: i) when the QD has a delta-function-like spectral width in comparison to the cavity (this is the “standard” case in which the largest Purcell factor will be observed); ii) when the cavity linewidth is a delta function compared to the emitter linewidth, which is achievable for silicon QDs at room temperature; and iii) the general case in which both cavity and QD have a Lorentzian profile. Using contour integrals, we will derive expressions that permit the decay rate and Purcell factor to be obtained in the general case. The subsequent section will present a model in which spontaneous emission rates are calculated for an arbitrary distribution of finite-linewidth QDs interfaced to a microcavity, where the ensemble emission spectrum may overlap several cavity modes. This arrangement is frequently reported in the literature, especially for microsphere-based fluorescent sensing applications [15]. Finally, we discuss the ensemble PL spectrum and decay dynamics when we have QDs with different sizes (and therefore different spontaneous emission rates) coupled to a microcavity.

2. Theory

The spontaneous emission rate, W_cav, for an emitter in a cavity is obtained from Fermi's Golden Rule:

Here, ρ(ω) and Λ(ω) represent the density of states for the cavity and the emitter, respectively. They are both typically given by Lorentzians; e.g., for the cavity we have:

2.1 Cavity wide, emitter narrow

This is the standard case solved in many quantum optics textbooks [16], so it will not be repeated in detail here. In short, when Λ(ω) is a delta function centered at ω₀, Eq. (1) becomes:

At the resonance ω_cav = ω₀; the maximum Purcell enhancement factor is obtained by setting Δω_cav = ω_cav/Q, ω₀ = 2πcn/λ, and ω_cav = ω₀₀, yielding:

Thus, the cavity enhances the spontaneous emission rate by a factor proportional to the Q/V ratio of the cavity.

2.2 Cavity narrow, emitter wide

This second case is identical to the first one, except that the cavity density of states is a delta function centered at ω_cav, yielding:

The Purcell enhancement will be identical to that in Eq. (7), except that Q is with respect to the emitter spectral width, not that of the cavity. In this case Q and V are decoupled, one being an emitter property and the other being a cavity property. For a given cavity, whether one sees an enhancement or a suppression of the spontaneous emission rate then depends on the cavity volume and the emitter linewidth.

2.3 The general situation: neither cavity nor emitter is a delta function

Finally, we consider the general case in which neither the cavity nor the emitter spectrum can be treated as a delta function. In this case, Eq. (1) has to be integrated over both Lorentzians and the matrix element. Essentially, we evaluate the following expression:

Here, A represents the constants ξ²πμ²/(εVћ), δ₀ and δ_cav are the half-widths-at-half-maximum of the cavity and QD spectra, respectively, ω₀ and ω_cav are the central frequencies of the QD and the cavity, and the Lorentzians are already normalized. As δ_cav or δ₀ approaches zero, they become delta functions and can be solved as above.

The first approach is to consider the case where both δ_cav << ω_cav and δ₀₀ << ω₀₀. This is generally the case in practice; mathematically it means that the integrand has decayed to zero by the time zero frequency is reached, so that there is little error in extending the lower limit of the integral to negative infinity. Then we can use contour integration and partial fractions to solve Eq. (9) as follows:

There are effectively four contour integrals in the above expression, each with two poles. The first and fourth integrals contain both poles in the upper and lower half planes, respectively, and therefore their contribution is zero. The remaining two integrals give the result:

Further simplification yields:

This expression equals the original one in Eq. (8) as long as δ₀ << ω₀ and δ_cav << ω_cav so that we can extend the integral to minus infinity, as done in Eq. (9). Written in terms of the full widths at half maximum (generally the experimentally reported parameter) and converting to the cavity and QD central frequency and FWHM we have:

This is the key equation. It is easy to show that dividing Eq. (12) by the free space rate and setting Δω₀ = 0 and ω_cav = ω₀ reduces to the special case of Eq. (6). In the more general case, Eq. (12) suffices for virtually all practical situations.

Equation (8) can also be solved exactly, but it is mathematically more difficult since contour integrals cannot be used in that case. Retaining the limits of integration and using decomposition by partial fractions yields, after some algebra:

In this expression it is clear that the left-hand term is the same result as that given in Eq. (11) and the remaining lines are small in comparison, particularly for δ₀ << ω₀₀ and δ_cav<< ω _cav. One limit that is obscured by this expression is that for the case of two identical Lorentzians ω_cav → ω₀ and δ_cav → δ₀, in which case the first and third terms on the right-hand side appear to contain divergences. In actual fact the divergences cancel and a simple limiting procedure leads to:

The first term agrees with that obtained from Eq. (11) (with ω_cav→ ω₀ and δ_cav →δ₀); the second term is an additional contribution that is usually negligibly small (e.g., for δ₀ = ω₀/700 the relative correction is on the order of 10⁻⁷).

2.4 Additional concerns

Several additional factors affect the spontaneous emission rate from QDs in microcavities. These include (i) the polarization degeneracy. In the case of planar cavities, polarization degeneracy increases the cavity emission rate by a factor of two. In cylindrical or spherical cavities, however, polarization degeneracy is lifted. (ii) The QDs may be randomly positioned inside the cavity. Since not all QDs are at the field maximum, this leads to a decrease in the ensemble emission rate by a factor of 3 in planar cavities in which the λ/2n layer is uniformly filled with emitters. In the case of cylindrical or spherical cavities with a thin layer of QDs near the field maximum, this factor can be ignored. (iii) Mode degeneracy: in spherical and cylindrical cavities, there are clockwise and counterclockwise senses of rotation. This degeneracy may be lifted if the cross section is not perfectly circular. Finally, (iv) the QDs may also couple into guided or leaky modes (planar and cylindrical cavities) or into free space (circular cavities), as illustrated in Fig. 1 . Therefore, in a spherical cavity there is always a competition between the free medium (W_FR) and cavity (W_cav) rates, which is quantified by the cavity coupling factor β_cpl = W_cav/(W_cav + W_FR). The global emission rate is W_G = W_cav + W_FR, and at the maximum cavity enhancement we have W_G = W_FR(F_p + 1). In the planar cavity case, Gayral and Gérard determined that the spontaneous emission rate into the guided modes of a planar cavity is ~0.8W_FR [17]. Thus, the actual rate measured experimentally is a result of the cavity and collection geometry: in the case of a sphere the W_G is the sum of the cavity and free space rates (at short timescales when collecting light from the “edge” of a sphere the cavity ringdown may also be observed as an initial fast decay); in a planar or cylindrical cavity the emission rates into guided modes appears as a non-radiative contribution when measuring only the cavity-coupled luminescence. Emission into other directions not collected in the experimental setup also appears as non-radiative additions to the experimentally observed emission dynamics.

 

Fig. 1 Illustration of the emissions into cavity, guided, and “free-medium” modes for (a) planar, (b) cylindrical, and (c) spherical cavities. The spherical cavity is, in many respects, the simplest one because of the lack of guided modes and overall fairly simple mode structure without too many radial modes. QDs emit into the different modes at rates given by Fermi’s golden rule for each geometry

Download Full Size | PDF

2.5 Emission rate distribution

In many planar [18], disk [19,20], and spherical microcavities [8,21], the fluorescence from a large number of QDs is coupled to the cavity modes. Since the QDs inevitably have a distribution of sizes, there will also be a distribution of spontaneous emission rates. If the cavity FSR is small, as is commonly the case, then some QDs will be on resonance and some not – in fact, the ensemble emission band may overlap many cavity modes. Therefore, we expect the ensemble lifetime distribution to be affected by the presence of the cavity in a more complicated manner than for single emitters.

When we have a distribution of decay rates, the ensemble decay takes the form of a stretched exponential I(t) = I₀exp(-(t/τ)^β), sometimes called the Kohlrausch decay function, after its 19th-century progenitor [22]. The stretching parameter, β, is between zero and unity, with lower values corresponding to a wider rate distribution. Mathematically, the rate distribution H(W) is the inverse Laplace transform of the measured I(t), which can be performed by solving the Bromwich integral or by using numerical approximations [23,24]. Here, the inverse transform was solved both by both methods, with similar results.

Our problem is to start with a distribution of QD sizes similar to that observed experimentally, determine the emission energy (rate) distribution H(E) (H(W)), and calculate the Purcell-induced rate enhancements or suppressions using Eq. (12) for the given cavity parameters, QD size distribution, and QD spectral width – for the entire distribution of QDs. This yields a cavity-modified rate distribution H(W) that can then be transformed to I(t) via:

Ultimately, this procedure will determine the effect of the cavity on the ensemble emission spectrum and dynamics in the general case when we have an arbitrary distribution of QDs coupled to an arbitrary cavity, as long as its mode structure is known.

3. Simulation of QD PL spectra in the presence of a cavity

In order to model the PL spectrum when numerous QDs are coupled to a microcavity, we proceed as follows. First, we define the QD properties and size distribution; the latter is usually obtained from transmission electron microscopy measurements. We assume a lognormal distribution, as expected for most volume-diffusion-driven nucleation and growth processes, consistent with previous observations of Si-QDs [25]. The QD linewidth can be estimated from single-particle measurements: in the case of Si QDs at room temperature, the spectral width of the luminescence has been proposed to be around 100 meV at room temperature, or ~2 meV at 35 K [26]. The Si-QDs are assumed randomly oriented in the cavity and with a bandgap given by

With D₁ = 4.8 eV²⋅nm² and ε_g being the bandgap of bulk Si [27]. The size dependent emission rate for the “free medium” (i.e., in a glass matrix) is calculated with W_FR = ξ²ω₀³μ²n³/(πεc³), with the dipole moments taken from the oscillator strengths given for silicon QDs in Ref. 28. There are certain assumptions inherent in these cited calculations; however, although these are important for calculating the Si-QD spectra and emission rates exactly, the exact values are not crucial for investigating cavity effects on QD ensembles generally. At this time, we consider all the QDs to be emissive; any non-radiative processes would simply represent a decay channel that is uncoupled to the microcavity. Once we have obtained the QD emission rates, they are mapped by linear interpolation onto a frequency space bounded by a bandpass filter with a selected frequency resolution (see Fig. 2 ). Application of a filter permits narrow regions of the emission spectrum to be simulated at high resolution, e.g., when the Q-factors are high. Without this filtering, for high Q-factors the simulation times can become excessive, owing to the large number of data points required for sufficient frequency resolution. In all subsequent calculations, every point in frequency space (i.e., the abcissa) will be considered to be a single QD (with the given linewidth) whose effect will be scaled by its probability of occurrence via the size distribution function.

 

Fig. 2 The PL spectrum calculated from I(λ) = P(λ)W(λ), for the Si-QD size distribution shown in the inset. The points in wavelength space are clearly visible, and are more widely separated at shorter wavelengths because of the non-linear relationship between QD radius and energy gap. The filtered spectrum is obtained by setting a bandpass filter and decreasing the data spacing – in this case the region between 790 and 850 nm is “expanded” to include 500 discrete points; as many as 10,000 can be used for a 12-hour simulation on a standard PC.

Download Full Size | PDF

Next, we select the cavity parameters including the free spectral range, the mode volume, and the mode linewidth. Since there is nothing in the simulation to prevent the entry of unphysical values, these should be obtained from analytical or numerical calculations or simulated separately. For this work, we used asymptotic approximations to calculate the mode volume and quality factor for the modes of spherical cavities [29,30]. Other expressions have been derived for the calculation of the mode volumes in planar cavities [31]. These approximations are typically within ~20% of the values calculated by numerical calculation of the complex solutions of the spherical Bessel and Hankel functions, as long as the microspheres are not too small [30], as we verified with finite difference time domain (FDTD) simulation of the electricfield distribution in 10-μm-diameter circular cross sections.

In Fig. 3 , we compare an example cavity spectrum with that of the QDs in order to illustrate how the calculation is performed. In this case, the QDs have a 3 meV spectral width and the cavity Q is ~700 (i.e., the cavity and QD linewidths are comparable, requiring the use of the general formula derived in Eq. (12) to calculate the emission rates). In terms of the overall spectrum, every defined point on the frequency axis represents a QD centered on that value. Its contribution to the PL spectrum and decay is subsequently weighted according to its probability, which is connected to the size distribution. The overall simulated spectrum will thus arise from the overlapping of the cavity spectra with an appropriate distribution of QDs.

 

Fig. 3 (a) A sample cavity spectrum (blue lines; FSR = 12.6 GHz corresponding to the TE modes of a ~5-μm-diameter sphere, Δf_cav = 0.5 GHz, Q ≈700) compared with that of single QDs (ΔE_QD = 3 meV; red curves), illustrating the relatives widths and overlap of the spectra. For clarity, only 40 QD Lorentzians are shown for 40 central wavelengths; the actual simulations used 500 to 2000 QD center frequencies uniformly distributed in wavelength space. Panels (b) and (c) show the variation of F_p and β_cpl as a function of wavelength (effectively, the range of QD central wavelengths) for the cavity-QD spectra shown in (a). In this case, F_p reaches a maximum near 0.2 on resonance, compared to an idealized Purcell factor of ~1. The coupling factor varies between 0.01 and 0.2, depending on the resonance overlap. The slight increase in the peak values with wavelength arises from the frequency dependence of the rate enhancement in Eq. (12). Note that the resonances in F_p and β are much wider than the intrinsic cavity resonances, due to the 3-meV QD spectral width.

Download Full Size | PDF

Calculating the “free medium” and cavity-modified ensemble PL spectrum and dynamics then proceeded as follows. The ensemble free space decay I(t) was calculated from the emission rates H(W), the latter obtained from the free space rates weighted by the size distribution P(R). The normalized free medium PL spectrum was then given by I(λ) = P(λ)W(λ), where P(λ) is the weighting function obtained by transforming P(R) into an energy (or wavelength) distribution. This assumes a linear relationship between the PL intensity and excitation rate which is observed at low to moderate excitation intensities [32], and that the excitation rate is the same for all QDs. In other words, since there is a variation in the decay rates with size, the PL spectrum is not simply a conversion of the size distribution to energy, but is weighted according to the QD emission rates. The cavity-modified rates are calculated as a function of emission frequency (wavelength) using Eq. (12), and similarly weighted by P(λ).

The Purcell factor and coupling factors are obtained at all frequencies via F_p = W_cav/W_FR and β_cpl = W_{ca v}/(W_cav + W_FR). Figure 3(b), 3(c) shows the oscillatory behavior of both of these properties for these QD and cavity parameters. On a cavity resonance, we see that F_p and β_cav are maximized as expected; furthermore, the largest Purcell factor from Eq. (12) for the cavity and QD parameters in Fig. 3 is considerably smaller than for the ideal case given by Eq. (6).

PL that is coupled into the modes can be collected preferentially, e.g., when obtaining PL scattered from the equator of a microsphere rather than from the “middle” of the sphere in a “vertical” viewing configuration. Figure 4 shows the simulated PL spectrum from an ensemble of Si-QDs in which the cavity:free-space collection ratio varies from 1:0.001 to 1:0.05. We observe that for the modes to appear in the PL spectrum, we need a high fraction of cavity-coupled PL to be collected; however, as discussed later, a similarly-appearing “wash out” effect occurs purely in the cavity component of the luminescence when the QD spectral width is much wide. Finally, we obtain the PL Q-factors by subtracting the background and fitting the PL spectrum with Lorentzian functions (Fig. 5 ). The lifetime distributions for emission into the cavity and into free space are obtained, transformed to a lifetime distribution, and the resulting cavity-modified ensemble PL decays are fit with the stretched exponential function and compared to the free space and pure cavity rates. This permits a determination of the effect of the cavity on the experimentally observed decay parameters in the realistic and experimentally common case where we have a distribution of QDs of finite linewidth coupled to a spherical microcavity.

 

Fig. 4 Simulated PL spectra for a microsphere with a diameter of 10 μm, for several different ratios of cavity:free-space collection efficiencies. The Si-QDs lognormal distribution parameters corresponded to a mean radius and deviation of 2.5 and 0.5 nm, respectively, and the linewidth was 5 meV.

Download Full Size | PDF

 

Fig. 5 (a) Simulated PL spectra for a microsphere with a d = 10 μm, and the corresponding background corrected emission spectrum. (b) Background-corrected spectra for a set of different microspheres with diameters ranging from 3 to 20 μm. The inset shows the “ideal” (cavity intrinsic) Q factors, in which there are only radiation losses. The observed PL Q-factors are much lower than these values (green fit) due to the 5 meV bandwidth of the QDs that limits the PL Q-factor to approximately 300, despite the much higher intrinsic cavity Q. The orange line in the inset shows the Q_PL calculated from the QD and cavity linewidths, and are quite close to those obtained by background subtraction and peak fitting.

Download Full Size | PDF

Calculating the PL spectra: summary

In the subsequent sections we will plot the simulated PL spectra for QDs coupled to different microcavities. The method established in the previous pages can be summarized inthe following list of steps:

4. Discussion

4.1 Effective Q factors

From Eq. (12), we can see that the Q-factors measured from PL spectra do not represent the true cavity Q. Previously, Gayral and Gérard found that, for a pillar-type microcavity with a narrow QD linewidth, the PL Q factors are decreased by a factor of $\sqrt{\left(F_p+1\right)}$ in a cavity in which there is a competition between mode emission and free-space emission, mainly because the PL peaks follow the coupling factor instead of the cavity modes. However, their work was based on QDs having the same emission rate and having a spectral shape approximated by a delta function. If instead we examine the general result in Eq. (12), we see that the PL factor is indeed decreased as suggested by the previous authors, but by an amount related to the QD linewidth. This can be seen from a straightforward analysis of Eq. (12), which can be rewritten as:

Here, A is 2ξ²μ²/(ħεV) and B is (ω₀Δω_cav + ω_cavΔω₀) / (Δω_cav + Δω₀), leaving the main part of the equation in the form of a standard Lorentzian. For a single emitter coupled to a cavity, therefore, the PL linewidth is given by Δω_cav + Δω₀, and the PL Q-factor is Q_PL = ω_cav/(Δω_cav + Δω₀). This value will be similar to the ensemble PL Q-factor, since the ensemble spectra are calculated from the product of the decay rates [Eq. (12)] and probability distribution. The latter will not have an appreciable effect on Q_PL unless there is a very rapid change in the probability distribution over the spectral bandwidth of a single dot. Actual measurements by data fitting can be slightly different, however, because of the underlying free-space emission requiring background subtraction, and because of the possibility for mode overlap when Δω_cav and/or Δω₀ are large. The latter effects are merely difficulties in obtaining accurate estimates of the PL Q-factors by background subtraction and fitting the spectra. Clearly, there is a difference between the intrinsic cavity Q-factor, given by Q = ω_cav/Δω_cav and Q_PL which naturally incorporates the spectral width of the emitter; the two are only equal when the emitter linewidth is a delta function.

In Fig. 5, we see the simulated PL spectra for a QD linewidth of 5 meV for several different spheres in which the radiation loss is the intrinsic cavity-Q-limiting mechanism (i.e., no other losses are limiting the cavity Q). Several phenomena are apparent: first, Q_PL is much lower than the intrinsic cavity Q for sphere diameters greater than ~3 μm. In the case of a 10-μm-diameter sphere, for example, the radiation-limited cavity Q is ~10⁷ but the theoretical and observed PL Q-factors are only ~100-300. The low Q observed in PL arises because of the fairly wide QD spectral width of 5 meV, which dominates the overall Q-factor for sphere sizes larger than ~3 μm, even for very high-intrinsic-Q spheres. In real microspheres, other loss mechanisms will become apparent at moderate diameters; however, the ultimate PL Q-factor will still be limited by the QD linewidth as long as the other loss mechanisms don’t lead to Q ≤ ~300. Thus, the Q-factors reported from PL measurements, as quite frequently done in the literature [12,19,20], can be a reasonable estimation under some circumstances but do not generally reflect the true cavity Q. The latter can be significantly larger than the PL Q-factors if the QD linewidth, ΔE_QD, is large – this should be a strong effect in the case of silicon QDs coupled to microcavities.

4.2 Lifetime distribution

There have previously been a few reports of enhanced spontaneous emission rates in Si-QDs coupled to a planar microcavity. In one case [18], the ensemble lifetime was decreased from 31.4 to 24.3 μs in the presence of a planar cavity; however, the stretching parameter β was not reported. Here, we show that cavity effects are manifested in the value of β as well as in τ ; the lifetime alone is an insufficient measure of the cavity effect on the ensemble decay. Effectively, the cavity may “stretch” the decay when some QDs are on resonance and some are not.

Modeling the rates for Si-QDs coupled to a microsphere is difficult, however. Essentially, because of the large mode volume the cavity-modified ensemble rates are smaller than the “infinite-medium” ones and the cavity is suppressing at all wavelengths, even on resonance (Fig. 6 ). When the cavity Q is large (e.g., for d ≈10 μm) the on-resonance rates do become enhanced but only if the QD linewidth is sufficiently narrow. Here, even for a 1 meV QD linewidth, the decay rates never become enhanced (even narrower QD emitters would be required. This does not imply that cavity modes cannot be observed in the luminescence spectra, since the collection efficiency for the modes can be very different, as discussed previously. Thus, the global lifetime will be similar to the free-medium one, since for long cavity lifetimes the coupling factor is low. In order to observe a rate enhancement we need a high Q/V cavity (e.g., d ~10 μm) combined with a narrow QD linewidth. In other words, the large mode volume of a spherical microcavity means that we need high Q and a correspondingly narrow QD spectral width in order to see on-resonance rate enhancements.

 

Fig. 6 Radiative rate as a function of Si-QD radius: green → “infinite” medium; red → 10-μm-diameter microsphere with ΔE_QD = 20 meV; blue → the same microsphere with ΔE_QD = 1 meV. The cavity rates are entirely suppressed compared to free space, and will only become enhanced at the resonance positions for still smaller values of ΔE_QD. The inset shows the corresponding ensemble lifetimes via Laplace transform of the data in the main panel. We see already hints of the general trend of β decreasing and τ increasing as the PL resonances become better defined (β appears somewhat higher for the red curve than the green, suggesting that the rate distribution could be more compressed). Although the general trends can be seen in diagrams like these, the exact values of β and τ obtained by fitting also depend sensitively on the sampling in time and frequency space and on the width of the filter.

Download Full Size | PDF

The difficulty in modeling a wide distribution of QDs coupled to a high-Q cavity is that one requires a wide frequency range combined with a high frequency resolution, which is computationally expensive. In order to simplify the following calculations to more easily demonstrate the effect of a cavity on an ensemble lifetime, we choose a cavity volume comparable to λ³, as would be typically the case of a photonic crystal cavity (PCC). While this makes the cavity parameters somewhat arbitrary, it nevertheless serves to illustrate the cavity effects on the decay parameters β and τ. Figure 7 demonstrates the case of a small modevolume as in a PCC; compared to Fig. 6 (for a sphere) the cavity-modified rate distribution is shifted upward and we can achieve rate enhancements close to the resonance for the same QD linewidths, while they are suppressed everywhere else. Despite the on-resonance rate enhancement, the overall ensemble enhancement should be small because only a small fraction of the QDs are near a cavity resonance (Fig. 7). In fact, on going from the free medium to a cavity, the lifetime first decreases because of the extra decay channel into the cavity. As the QD linewidth narrows, there is subsequently an increase in the ensemble cavity-coupled lifetime and a corresponding decrease in the stretching parameter β. Thus, as the cavity coupling becomes stronger, the ensemble lifetime becomes longer due to the off-resonance QDs, while the decay becomes increasingly stretched.

 

Fig. 7 Radiative rate for Si-QDs for a cavity with a small mode volume. In (a) the black curve shows the “infinite medium” rate, while the colors correspond to cavity-coupled emission with different values of ΔE_QD: red → 0.5 meV; orange → 1 meV; green → 5 meV; blue → 10 meV; navy → 20 meV. The corresponding cavity-only and global lifetimes are shown in (b) and (c), respectively, with the same color coding. In (c), the black line represents the free-medium (no-cavity) decay. The inset to (c) shows the corresponding values of τ (red) and β (blue) for the global decays on separate vertical axes. When ΔE_QD ≤ 0.5 meV, the trends of increasing β and decreasing τ with QD bandwidth can no longer be followed, since such narrow resonances cannot be well-sampled in frequency space. Here, the decrease in β with ΔE_QD is monotonic, suggesting that the difference observed between the red and green curves in Fig. 6 may be a data-fitting artifact.

Download Full Size | PDF

4.3 Effect of PL temperature and QD linewidth

Temperature plays an important role in the cavity spectra and dynamics through the QD emission linewidth; thus, most single-QD coupling experiments must be performed at cryogenic temperatures in order that Eq. (6) can approximately hold. However, most “real world” applications for QD-microcavities will be at room temperature and involve an ensemble of QDs (e.g., fluorescence biosensors, tunable quantum dot lasers, etc.). Equation (12) permits the effect of temperature (i.e., linewidth) on the QD-cavity spectra and emission dynamics to be quantified.

In addition, it is possible to estimate the QD spectral width by comparing the intrinsic cavity spectrum (e.g., measured evanescently) with the PL mode spectrum. Thus, insofar as we can consider an “average” linewidth, we avoid the necessity for laborious single-particle spectroscopy on Si-QDs. Previous work suggested that for single Si-QDs the room-temperature emission linewidth is around 100 meV, although, as stated in the original papers, it is experimentally difficult to ascertain whether the single-particle luminescence really comes from one particle or from a small cluster of particles [26,33], or whether QDs prepared by different methods may have considerably different luminescence spectral widths.

In Fig. 8 , we show the evolution of the emission spectra for a 20-μm diameter cavity coated with an ensemble of Si QDs whose linewidth increases from 1 to 100 meV (similar to the range reported experimentally between 35 and 300 K) [26]. The radiation-limited Q of this cavity is ~6 x 10¹³. As expected, the PL cavity modes become broader and increasingly “washed out” as the QD spectral width increases, until they virtually disappear when the single-QD Lorentzian linewidth, ΔE_QD, is near 20 meV. The PL Q values range from 1400 for a 1 meV QD linewidth and decrease to approximately 70 when ΔE_QD = 20 meV, consistent with Eq. (12). At this point, Δw₀ is becoming comparable to the free spectral range of the cavity, and it is no longer possible to observe modes in the PL spectrum.

 

Fig. 8 (a) Simulated TE-polarized PL for a 20-μm-diameter sphere with Si-QDs of different bandwidths ΔE_QD. The curves shown are, in order: blue (ΔE_QD = 1 meV), turquoise (ΔE_QD = 5 meV), green (ΔE_QD = 10 meV), orange (ΔE_QD = 20 meV), red (ΔE_QD = 100 meV). The mode structure becomes increasingly difficult to observe as the QD bandwidth increases. Panel (b) shows a comparison of simulated PL (red) with experimental PL data (blue) from a Si-QD-coated microsphere with a diameter of 20 μm (see PL image in the inset, in which the spectrometer slit can be observed on the right edge of the luminescent sphere) [34]. The simulation used ΔE_QD = 3 meV to provide a close fit to the experimental data. To maximize the visibility of the mode spectrum, we used a greater collection efficiency by a factor 10⁸ for mode PL compared with background PL, which would correspond to the energy trapped in a sphere with a Q factor near 10⁹, close to the maximum effectively possible for a silica sphere. Additional structure in the experimental spectra (especially some shoulders in the PL resonances) are due to the presence of the opposite polarization and/or higher-order radial modes [34].

Download Full Size | PDF

In Fig. 8(b) we plot the PL data for a 20-mm-diameter sphere coated with a layer of Si-QDs made in our lab for a different study [34]. The data match fairly closely to the simulation when ΔE_QD = 3 meV, reflecting the effective spectral width of the Si-QDs at room temperature. The experimental Q factors are in the range of 1500, while the simulated ones are ~1000, suggesting that the actual QD linewidth is slightly smaller than the 3 meV value used for the simulations. Interestingly, if the QD linewidth is 100 meV, as suggested previously, then the PL resonances should be washed out. This is not a collection efficiency issue: the mode spectrum is inherently destroyed by the wide QD spectral width, so that the width of the resonances becomes similar to or greater than the cavity free spectral range. Since mode spectra are frequently observed experimentally in the luminescence of Si-QDs coupled to microcavities [12,19–21] (as in Fig. 8) we conclude that the QD linewidth must be significantly smaller than 100 meV at room temperature. This suggests that either the single QDs observed previously [26] had unusually wide linewidths, or that what was thought to be a single QD was in fact a QD cluster with a separation distance smaller than the diffraction-limited resolution in fluorescence microscopy.

Such differences cannot be due to non-radiative effects, which were not modeled, since such processes decrease the QD-cavity coupling efficiency, resulting in lower PL Q-factors. Essentially, if ΔE_QD is really ~100 meV one would require some mechanism to increase the PL Q-factor significantly above that obtained from the above considerations, in order to produce the observed spectra. It may also be possible that the intrinsic cavity Q-factors are only on the order of the observed ones (Fig. 8), due to surface roughness, scattering, and other loss mechanisms. However, even if we assume a low cavity Q, we still require a QD spectral width sufficiently narrow to produce the observed spectra, which, once again, cannot be obtained with ΔE_QD much greater than 3 meV.

5. Conclusions

In this work, we derived an easily-applied but general expression that can be used to calculate the spontaneous emission rate for quantum dots coupled to an optical microcavity. It makes no assumptions concerning the spectral linewidths of the cavity or of the QD emitters. The PL Q-factors are given by ω_peak/(Δω_cav + Δω_QD) and can be controlled by the QD linewidth, which can be much wider than the cavity resonances. We extended the concept to investigate the effects that occur when ensembles of QDs are coupled to a single cavity, as is most often the case experimentally. In the case of QDs coupled to a microcavity, the ensemble luminescence decay – the most easily measured dynamical parameter experimentally – tends toward a stretched exponential function with decreasing values of β and increasing τ as the coupling efficiency increases. Although an increase in τ may be surprising for QDs coupled to a cavity, the explanation is simple: the increase occurs because, for a typical free spectral range, while some QDs are strongly enhanced, most are in fact suppressed. The magnitude of these effects also depends on the non-radiative rate, which contributes a cavity-independent decay channel. We next showed that the effective QD linewidth can be obtained via the microcavity PL Q factors. This method avoids the need for difficult single-particle spectroscopy studies (these are especially challenging in the case of indirect gap QDs such as Si). With this method, the effective linewidth of Si QDs prepared by physical vapor deposition must be around 3 meV, and it cannot be nearly as great as 100 meV without completely washing out the resonances in the PL spectra. Finally, we wish to note that at the time of submission of the current manuscript, a new paper in the March 12 edition of Physical Review Letters estimated a linewidth of 10 meV for Si-NCs using microcavity methods [35]. The work is generally consistent with our conclusions (3 meV linewidth), but without the new derivations leading to Eq. (12) and the subsequent application of the theory to ensembles of QDs. They also use an empirical formula for the PL Q factor. Ours, derived mathematically from the contour integration of the overlap integral, shows that their empirical formula is accurate.