Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots

The quality factor (Q), mode volume (Veff), and room-temperature lasing threshold of microdisk cavities with embedded quantum dots (QDs) are investigated. Finite element method simulations of standing wave modes within the microdisk reveal that Veff can be as small as 2(lambda/n)^3 while maintaining radiation-limited Qs in excess of 10^5. Microdisks of diameter D=2 microns are fabricated in an AlGaAs material containing a single layer of InAs QDs with peak emission at lambda = 1317 nm. For devices with Veff ~2 (lambda/n)^3, Qs as high as 1.2 x 10^5 are measured passively in the 1.4 micron band, using an optical fiber taper waveguide. Optical pumping yields laser emission in the 1.3 micron band, with room temperature, continuous-wave thresholds as low as 1 microWatt of absorbed pump power. Out-coupling of the laser emission is also shown to be significantly enhanced through the use of optical fiber tapers, with laser differential efficiency as high as xi~16% and out-coupling efficiency in excess of 28%.


Introduction
Optical microcavities with embedded quantum dots (QDs) have become a very active area of research, with applications to triggered single photon sources [1,2,3], strongly coupled lightmatter systems for quantum networking [4,5,6], and low threshold microcavity lasers [7,8,9]. For these applications some of the most important microcavity parameters are the quality factor (Q), mode volume (V eff ), and the efficiency of light collection from the microcavity (η o ). Q and V eff describe the decay rate (κ) and peak electric field strength within the cavity, respectively, which along with the oscillator strength and dephasing rate of the QD exciton determine if the coupled QD-photon system is in the regime of reversible energy exchange (strong coupling) or in a perturbative regime (weak coupling) characterized by a modification of the QD exciton radiative lifetime (the Purcell effect) [10]. The collection efficiency η 0 is of great importance for quantum networking [11] and linear optics quantum computing applications [12,13], where near-unity photon pulse collection values are required.
Microdisks supporting high-Q whispering-gallery resonances were first studied in the context of semiconductor microlasers in the early 1990s [14]. Since that time there has been extensive work on incorporating self-assembled InAs QD active regions within semiconductor microdisks for studying quantum interactions of light and matter [1,6,7,8,9,15]. With respect to lasers, the relatively small modal gain available from a single layer of QDs has typically resulted in device operation at reduced temperatures [7], or the use of multiple QD layers to achieve room temperature (RT) operation [8,9]. More recently, improvements in the cavity Q have resulted in RT operation in devices containing a single layer of QDs [16]. Furthermore, it has been shown that the collection efficiency of emitted power can be significantly increased by using optical fiber tapers to evanescently couple light from the microdisk [17]. As discussed above, these improvements in Q and η 0 are not only important for lasers, but for future experiments in cavity quantum electrodynamics (cQED).
In this article, we continue our study of taper-coupled microdisk-QD structures by considering device performance as the disks are scaled down in size. In Section 2, we use finite element simulations to examine the behavior of Q and V eff as a function of disk diameter. We relate these parameters to those used in cQED, and from this, determine that disks of 1.5 − 2 µm in diameter are optimal for use in future experiments with InAs QDs. Section 3 briefly outlines the methods used to fabricate and test devices consisting of a 2 µm diameter disk created in an Al-GaAs heterostructure with a single layer of self-assembled InAs QDs. In Section 4, we present experimental measurements of the fabricated devices. Through passive characterization, cavity Qs as high as 1.2×10 5 are demonstrated for devices with a predicted V eff ∼ 2.2(λ/n) 3 . In addition, photoluminescence measurements show that the devices operate as lasers with RT, continuous-wave thresholds of ∼1 µW of absorbed pump power. Finally, the optical fiber taper is used to increase the efficiency of out-coupling by nearly two orders of magnitude, so that an overall fiber-coupled laser differential efficiency of ξ ∼ 16% is achieved. We conclude by presenting some estimates of the number of QDs contributing to lasing and the spontaneous emission coupling factor (β) of the devices.

Simulations
To study Q and V eff of the microdisk cavities, finite-element eigenfrequency simulations [18,19] are performed using the Comsol FEMLAB commerical software. By assuming azimuthal symmetry of the disk structures, only a two-dimensional cross-section of the disk is simulated, albeit using a full-vectorial model. The cavity mode effective volume is calculated according to the formula [20]: where ε(r) is the dielectric constant, |E(r)| is the electric field strength, and V is a quantization volume encompassing the resonator and with a boundary in the radiation zone of the cavity mode under study. The resonance wavelength λ 0 and radiation limited quality factor Q rad are determined from the complex eigenvalue (wavenumber) of the resonant cavity mode, k, obtained by the finite-element solver, with λ 0 = 2π/Re(k) and Q rad = Re(k)/(2Im(k)). For this mode, λ ∼ 1265.41 nm, Q rad ∼ 10 7 , and V eff ∼ 2.8(λ/n) 3 . Figure 1(a) shows a scanning electron microscope (SEM) image of a fabricated microdisk. The devices are formed from a GaAs/AlGaAs waveguide layer that is 255 nm thick, and due to an emphasis of sidewall smoothness over verticality during fabrication [16], the etched sidewall angle is approximately 26 • from vertical. These parameters are included in the simulations as shown Figure 1(b). Here, we will focus on resonant modes in the 1200 nm wavelength band, corresponding to the low temperature (T=4 K) ground-state exciton transition of the QDs, relevant for future cQED experiments. We confine our attention to the more localized transverse electric (TE) polarized modes of the microdisk, and only consider the first order radial modes. In what follows we use the notation TE p,m to label whispering-gallery-modes (WGMs) with electric field polarization dominantly in the plane of the microdisk, radial order p, and azimuthal mode number m. The refractive index of the microdisk waveguide is taken as n = 3.36 in the simulations, corresponding to the average of the refractive indices of the GaAs and AlGaAs layers at λ = 1200 nm.
In addition, the modes that we study are standing wave modes that are superpositions of the standard clockwise (CW) and counterclockwise (CCW) traveling wave modes typically studied in microdisks. These standing wave modes form when surface scattering couples and splits the initially degenerate CW and CCW traveling wave modes [21,22,23]; this process is contingent upon the loss rates within the disk being small enough for coherent coupling between the traveling wave modes to occur. For the AlGaAs microdisk devices we have studied thus far [16,17], this has indeed been the case. The effective mode volume for a standing wave mode, as defined in Equation 1, is roughly half that of a traveling wave mode [23]. This is of particular relevance to cQED experiments involving such microdisks as the coherent coupling rate of light and matter scales as g ∼ 1/ √ V eff . A QD positioned at an anti-node of the standing wave will have an exciton-photon coupling rate which is √ 2 times larger than for the traveling wave mode.
Figures 1(b) and 2(a) show the results of the finite element simulations. We see that V eff for these standing wave modes can be as small as 2(λ/n) 3 while maintaining Q rad > 10 5 . In- deed, for microdisk average diameters D > 2 µm 1 , radiation losses are not expected to be the dominant loss mechanism as Q rad quickly exceeds 10 7 , and other sources of field decay such as material absorption or surface scattering are likely to dominate . To translate these results into the standard parameters studied in cQED, we calculate the cavity decay rate κ/2π = ω/(4πQ) (assuming Q = Q rad ) and the coherent coupling rate g between the cavity mode and a single QD exciton. In this calculation, a spontaneous emission lifetime τ sp = 1 ns is assumed for the QD exciton, and g = d · E/ is the vacuum coherent coupling rate between cavity mode and QD exciton, given by [10,20]: where c is the speed of light and n is the refractive index at the location of the QD. This formula assumes that the QD is optimally positioned within the cavity field, so that the calculated g is the maximum possible coupling rate. The resulting values for g and κ are displayed in Figure  2(b), and show that g/2π can exceed κ/2π by over an order of magnitude for a range of disk diameters. In addition, for all but the smallest-sized microdisks, κ/2π < 1 GHz. A decay rate of 1 GHz is chosen as a benchmark value as it corresponds to a linewidth of a few µeV at these wavelengths, on par with the narrowest self-assembled InAs QD exciton linewidths that have been measured at cryogenic temperatures [24]. Indeed, because dissipation in a strongly-coupled QD-photon system can either be due to cavity decay or quantum dot dephasing, in Figure 3 we examine the ratio of g to the maximum decay rate in the system assuming a fixed QD dephasing rate γ/2π=1 GHz 2 . This ratio is roughly representative of the number of coherent exchanges of energy (Rabi oscillations) that can take place between QD and photon. We see that it peaks at a value of about 18 for a disk diameter D ∼ 1.5 µm. For diameters smaller than this, loss is dominated by cavity decay due to radiation, while for larger diameters, the dominant loss mechanism is due to dephasing of the QD. For other types of atomic-like media besides the self-assembled InAs QDs considered here one need not assume a limit of γ/2π = 1 GHz, and we note that due to the exponential dependence of Q rad and approximately linear dependence of V eff on microdisk diameter, Q rad /V eff rapidly rises above 10 7 for microdisks of diameter only D = 2.5 µm. These values of Q rad and V eff are comparable to those found in recent high-Q photonic crystal microcavity designs [25,26,27,28,29]. In fact a similar scaling for high-Q planar photonic crystal microcavities, in which one may trade-off a linear increase in V eff for an exponential increase in Q, has recently been described by Englund, et al., in Ref. [30]. For now, however, we take the ratio g/max(γ, κ) with γ/2π = 1 GHz as our metric, and as such focus on 1.5-2 µm diameter microdisks. Fig. 3. Ratio of the calculated coupling rate g to the maximum decay rate in the system, max(γ,κ), as a function of the microdisk diameter at the center of the slab. A fixed QD decay rate γ/2π=1 GHz is assumed, and the cavity decay rate κ is taken to be solely due to radiation loss.

Growth, Fabrication, and Test Set-up
The samples used were grown by molecular beam epitaxy, and consist of a single layer of InAs QDs embedded in an In 0.15 Ga 0.85 As quantum well, which is in turn sandwiched between layers of Al 0.30 Ga 0.70 As and GaAs to form a 255 nm thick waveguide layer. This dot-in-awell (DWELL) structure is grown on top of a 1.5 µm thick Al 0.70 Ga 0. 30 As buffer layer that is later undercut to form the disk pedestal. Growth parameters were adjusted [31] to put the material's RT ground state emission peak at λ = 1317 nm. Fabrication of the microdisk cavities begins with the deposition of a 180 nm thick Si x N y etch mask layer through plasma-enhanced chemical vapor deposition. This is followed by electron-beam lithography to define a linear array of disks, a post-development reflow of the resist, and a SF 6 /C 4 F 8 inductively coupled plasma reaction ion etch (ICP-RIE) of the nitride mask. The DWELL material is then etched using Ar/Cl 2 ICP-RIE, and the array of disks is isolated onto a mesa stripe through standard photolithography and an ICP-RIE etch. Finally, the disks are undercut using a dilute solution of hydrofluoric acid (20:1 H 2 O:HF); an image of a fabricated device is shown in Figure 1. During the fabrication, two goals were given special consideration. As discussed within the context of silicon microdisks in Ref. [23], elimination of radial variations in the disk geometry is important for reducing scattering loss; this is accomplished through optimization of the electron-beam lithography, and in particular, use of the post-development resist reflow technique of Ref. [23]. In addition, a premium was placed on sidewall smoothness, even at the expense of sidewall verticality. This required an optimization of the ICP-RIE processes used to etch both the Si x N y mask and GaAs/AlGaAs waveguide layer. In particular, a low bias voltage, C 4 F 8 -rich plasma is used to etch the Si x N y , and a low Cl 2 percentage is used in the Ar/Cl 2 etch of the waveguide to eliminate any sidewall pitting due to excessive chemical etching.
The microdisks are studied in a photoluminescence (PL) measurement setup that provides normal incidence pumping and free-space collection from the samples. The pump laser is an 830 nm laser diode that is operated continuous-wave, and the pump beam is shaped into a Gaussian-like profile by sending it through a section of single mode optical fiber, after which it is then focused onto the sample with an ultra-long working distance objective lens (NA = 0.4). The free-space photoluminescence is first collected at normal incidence from the sample surface using the same objective lens for pump focusing, and is then coupled into a multi-mode fiber (MMF) using an objective lens with NA = 0.14. The luminescence collected by this MMF is wavelength resolved by a Hewlett Packard 70452B optical spectrum analyzer (OSA).
The PL setup has been modified [17] to allow for devices to be probed by optical fiber tapers. The fiber taper is formed by heating (with a hydrogen torch) and adiabatically stretching a single mode fiber until its minimum diameter is approximately 1 µm. It is mounted onto an acrylic mount that is attached to a motorized Z-axis stage (50 nm encoded resolution), so that the fiber taper can be precisely aligned to the microdisk, which is in turn mounted on a motorized XY stage. When doing passive measurements of cavity Q, the taper input is connected to a scanning tunable laser (5 MHz linewidth) with a tuning range between 1420-1480 nm, and the taper output is connected to a photodetector to monitor the transmitted power. Alternately, when collecting emission from the microdisk through the fiber taper, the taper input is left unconnected and the output is sent into the OSA.

Experimental Results
We begin our measurements by using the fiber taper to passively probe the Q of the microdisks. Based on the simulations presented in Section 2, we have focused on 2 µm diameter microdisks. Due to the small diameter of these microdisks, the finite-element-calculated free-spectral range of resonant modes is relatively large, with resonances occurring at 1265, 1346, and 1438 nm for the TE p=1 WGMs with azimuthal mode numbers m = 11,10, and 9, respectively. The simulations presented in Section 2 were done for the TE 1,11 mode in the λ = 1200 nm band due to the applicability of that wavelength region for future low temperature cQED experiments. However, for the current room-temperature measurements, the absorption due to the QD layer at those wavelengths is significant, so we probe the devices within the λ = 1400 nm band (∼100 nm red-detuned from the peak ground-state manifold QD emission). At these longer wavelengths the radiation-limited Q rad for a given disk diameter will be smaller than its value in the shorter λ = 1200 nm band. Table 1 summarizes the properties of the TE p=1 WGMs within the 1200-1400 wavelength band for a D = 2 µm microdisk with shape as shown in Fig. 1.   Figure 4(a) shows a wavelength scan of the transmitted signal when a fiber taper is positioned a few hundred nanometer away from the disk edge. The doublet resonance appearing at λ ∼ 1440 nm in the spectrum is the signature of the standing wave modes described earlier [21].
The measured linewidths correspond to Q factors of 1.2×10 5 , and in general Qs of 0.9-1.3×10 5 have been measured for these 2 µm diameter microdisks. The Qs of these modes are approaching the radiation-limited value of 3.7×10 5 , and are some of the highest measured values for near-IR wavelength-scale microcavities in AlGaAs [5,15,16,32]. The corresponding cavity decay rates are κ/2π ∼ 0.8 − 1.3 GHz, over an order of magnitude smaller than the predicted coupling rate g for an optimally placed QD. In addition, these Qs, if replicated within the QD emission band at λ = 1300 nm, are high enough to ensure that room-temperature lasing should be achievable from the single layer of QDs in these devices [31]. From calculations of the intrinsic radiation loss, the shorter 1300 nm wavelength modes should in fact have a significantly increased Q rad of 1.9×10 6 , although surface scattering may also slightly increase due to its approximate cubic dependence on wavelength [23].
The emission properties of the QD-containing microdisks are tested at room temperature by continuous-wave optical pumping through a high-NA objective lens at normal incidence and, initially, collecting the normal incidence emitted light through the same lens. A light-in versus light-out (L-L) curve for one of the D ∼ 2 µm microdisks with a resonant emission peak at λ ∼ 1345 nm is shown in Figure 4(b), and displays a lasing threshold kink at approximately 1.0 µW of absorbed pump power. The laser mode wavelength corresponds well with the TE p=1,m=10 mode from finite-element simulations (see Table 1). The absorbed pump power is estimated to be 11% of the incident pump power on the microdisk, and was determined assuming an absorption coefficient of 10 4 cm −1 for the GaAs layers and quantum well layer. This threshold level is approximately two orders of magnitude smaller than those in recent demonstrations of RT, continuous-wave microdisk QD lasers [8,9], although the active regions in those devices contain five stacked layers of QDs while the devices presented here contain only a single layer of QDs.
The low lasing threshold of the device presented in Figure 4(b) was consistently measured for the set of devices on this sample (approximately 20 devices). In Figure 5(a) we show another L-L curve, this time for a device that has a TE p=1,m=10 WGM emission peak at λ = 1330 nm and has a threshold absorbed pump power of 1.1 µW. As demonstrated in Ref. [17], the same fiber taper used to measure the cavity Q can efficiently out-couple light from the lasing mode. We do this by maintaining the free-space pumping used above while contacting a fiber taper to the side of the microdisk as shown in the inset of Figure 5(b). From the corresponding L-L curve (Fig. 5(b)) we see that the laser threshold under fiber taper loading has increased from 1.1 µW to 1.6 µW, but in addition the differential laser efficiency ξ is now 4% compared to 0.1% when employing free-space collection (Fig. 5(a)-(b)). Furthermore, because the microdisk modes are standing waves they radiate into both the forwards and backwards channels of the fiber. With collection from both the forward and backward channels the differentially efficiency was measured to be twice that of the single forward channel. Collecting from both channels and adjusting for all fiber losses in the system (roughly 50% due to fiber splices and taper loss), the total differential laser efficiency with fiber taper collection is 16%. Due to the difference in photon energy of the pump laser and microdisk emission, this laser differential efficiency corresponds to a conversion efficiency of 28% from pump photons to fiber-collected microdisk laser photons. 28% is thus a lower bound on the fiber-taper collection efficiency and/or quantum efficiency of the QD active region.
In addition to the improved laser differential efficiency of the TE p=1,m=10 laser mode when using the fiber taper to out-couple the laser light, we also see in the below-threshold spectrum of Figure 5(b) that two additional resonances appear at λ = 1310 nm and λ = 1306 nm. The long wavelength mode is identified as TM p=1,m=8 and the short wavelength mode as TE p=2,m=7 from finite-element simulations. These modes are not discernible in the free-space collected spectrum due to their low radiation-limited Q factors (800 and 5000 for the TE 2,7 and TM 1,8 , respectively), but show up in the taper coupled spectrum due to their alignment with the QD ground-state exciton emission peak and the heightened sensitivity of the taper coupling method. The single-mode lasing and limited number of WGM resonances (6 when including the degeneracy of the WGMs) in the emission spectrum in these D = 2 µm microdisks is a result of the large 80-100 nm free-spectral-range of modes in the 1300-1500 wavelength band. As a result, one would expect the spontaneous emission factor (β) of these microdisk lasers to be relatively high.
A log-log plot of the fiber taper coupled laser emission of Figure 5(b) is shown in Figure  6(a) along with a rate-equation model fit to the data. Of particular note is the well defined subthreshold linear slope of the log-log plot. In this case the sensitivity of the fiber taper collection allows for the sub-threshold slope to be accurately estimated at m=1.67, corresponding to a near quadratic dependence of spontaneous emission intensity on pump power ( Figure 6(b), inset) and indicating that there is likely significant non-radiative recombination. Assuming that radiative recombination occurs as a bi-particle process 3 , the larger than unity power law dependence of sub-threshold emission on pump power is indicative of single-particle non-radiative recombination processes such as surface recombination [34]. Given the close proximity of the WGM laser mode to the periphery of the microdisk and the above-band pumping, the presence of significant surface recombination is not surprising. Unfortunately, due to this large non-radiative component one can only provide a weak lower bound β ′ for the β-factor directly from the L-L curve. From Figure 6 we estimate β ≥ β ′ ∼ 3%.
A rate-equation model incorporating bi-particle spontaneous emission proportional to N 2 and surface recombination with a N 1.22 carrier dependence (the ratio of the power law dependences is set equal to the measured sub-threshold slope of m=1.67) is fit to the data and shown as a solid curve in Figure 6. In this model the measured fiber taper collection efficiency was used, along with the previously measured and estimated QD density, maximum gain, and quantum efficiency from stripe lasers [31]. An estimate for the actual radiative β-factor of 15.5% was used, corresponding closely with the partitioning of spontaneous emission amongst the 6 localized and high-Q WGM resonances within the QD ground-state manifold emission band 4 . The reference spontaneous emission lifetime of the ground-state QD exciton in bulk was taken as τ sp = 1 ns. The data was fit by varying only the effective surface recombination velocity. As seen in Figure 6, the fit is quite good over the entire sub-threshold and threshold regions of the laser data. The inferred surface recombination velocity from the fit is v s ∼ 75 cm/s, extremely slow for the AlGaAs material system [36] but perhaps indicative of the fast capture rate of carriers, and consequent localization, into QDs [37,38]. Due to the large perimeter-to-area ratio in these small D = 2 µm microdisks, even with this low velocity the model predicts that laser threshold pump power is dominated by surface recombination with an effective lifetime τ s ∼ 300 ps. Such a surface recombination lifetime has also been estimated by Baba, et al., in their recent work on QD-microdisk lasers [39].
The number of QDs contributing to lasing in these small microdisks can also be estimated. From the finite-element simulations the area of the standing wave WGM lasing mode in the plane of the QD layer is approximately 1 µm 2 , and the predicted QD density for this sample is 300 µm −2 , so that ∼ 300 QDs are spatially aligned with the cavity mode. Assuming a RT homogeneous linewidth on the order of a few meV [24], compared to a measured inhomogeneous Gaussian broadening of 35 meV, and considering the location of the lasing mode in the tail of the Gaussian distribution, we estimate < 10% of these dots are spectrally aligned with the cavity mode. By this estimate, on the order of 25 QDs contribute to lasing.

Conclusions
We have demonstrated fiber-coupled 2 µm diameter quantum-dot-containing microdisks that have a quality factor Q in excess of 10 5 for a predicted mode volume V eff as small as ∼ 2.2(λ/n) 3 . Such devices are predicted to be suitable for future experiments in single quantum dot, single photon experiments in cavity QED, where these Q and V eff values can enable strong coupling at GHz-scale speeds. An initial application of this work is in continuous wave, optically pumped microcavity lasers. Here, the high Q ensures that lasing can be achieved with the modest gain provided by a single layer of quantum dots, and combined with the ultra-small V eff , results in thresholds as low as 1.0 µW of absorbed power. In addition, the fiber taper coupling is shown to be an efficient method to collect the laser emission with a measured 28% lower bound on out-coupling efficiency.