We present a theoretical model for metal-cavity submonolayer quantum-dot surface-emitting microlasers, which operate at room temperature under electrical injection. Size-dependent lasing characteristics are investigated experimentally and theoretically with device radius ranging from 5 μm to 0.5 μm. The quantum dot emission and cavity optical properties are used in a rate-equation model to study the laser light output power vs. current behavior. Our theory explains the observed size-dependent physics and provides a guide for future device size reduction.
© 2013 Optical Society of America
Metal-cavity microlasers have attracted extensive research interest in the past few years. Since the first demonstration of lasers in 1961 and the first vertical-cavity surface-emitting laser (VC-SEL) in 1979 , metals have been used by researchers for the miniaturization of semiconductor lasers. With the help of metals, micrometer- to nanometer-scale light sources are now promising candidates for low foot print high-speed optical communication system. Novel structures based on metal cavities, photonic crystal, or surface plasmons have been employed to confine the optical mode close to or even below diffraction limit. Recent work on metal cavity nanolasers have demonstrated operation not only under optical pumping [2,3], but also at room temperature under electrical injection [4, 5].
Low dimensional materials such as quantum dots (QDs) have been employed as the active material in lasers showing enormous advantages as compared to classical quantum-well lasers, such as high differential gain, low threshold current, high temperature stability, and wide modulation bandwidth due to their discrete density-of-states (DOS) . Submonolayer quantum dots (SML QDs) as an alternative to the Stranski-Krastanow grown quantum dots (S-K QDs), have been demonstrated with much higher modal gain, better uniformity, less inhomogeneous broadening, and sharper emission spectra . The smaller size and shape deviations of SML QDs allow a much higher saturated gain. Since SML QDs do not have wetting layers (WLs), the carrier population in WL bound states and the carrier scattering from WL states into QDs are avoided . Hence, both maximum gain and modulation bandwidth are expected to be larger.
One major challenge of achieving room-temperature electrical-injected microlasers is to have enough gain to balance the loss. Both the radiation and material loss increase rapidly as cavity size shrinks down. Heat accumulation from increasing series resistance is another limiting factor for electrical-injection microlasers. Thermal effects result in unstable threshold current and early output power roll-over. SML QDs have shown improved thermal stability in threshold current and differential efficiency in high-power high-speed VCSELs  and are also promising for micro-cavity lasers.
In this paper, we develop a theoretical model to investigate the size-dependent device performance of metal-cavity SML QD microlasers. Devices are demonstrated to lase at room temperature under electrical injection with device radius down to 2 μm for continuous wave (CW) and to 0.5 μm for pulsed mode operations. Using a quantum disk model for S-K grown QDs, we have successfully explained experimental results such as optical gain and linewidth enhancement factor . In this work, we extend the model for multi-stack SML QDs with strong vertical correlation, and consider them as effective quantum disks. Strain effects on the hetero-junctions are included in the Hamiltonian for calculating the electronic states in SML QDs. The QD material gain and the spontaneous emission rate are obtained with Fermi’s golden rule and both homogeneous and inhomogeneous broadening effects are considered. The characteristics of the laser optical cavity are solved using the Maxwell’s equations semi-analytically, with the effective index method and the transfer matrix method. We then use the rate-equation model to study the interaction between the injected carriers and the generated photons. The calculated QD gain and cavity properties are used as inputs and the light output power vs. current (L-I) behavior is predicted. Our theory agrees with experimental data for device radii from 5 μm down to 0.5 μm.
The coupling of spontaneous emission into the cavity modes becomes much more significant as the cavity size and the effective mode volume reduces . We derive a rigorous expression for the coupling factor (β factor) which accounts for both the emission properties of the QDs and the radiation environment modified by the cavity. The sub-threshold L-I behavior we observe is successfully explained by the increasing amount of carrier density-dependent spontaneous emission coupling into the cavity mode. The β factor at threshold increases drastically as we reduce the device size due to the more sparse mode distribution within the gain spectrum.
Figure 1(a) shows the schematic of the metal-cavity microlasers. The active region contains three groups of SML QDs. The device sidewall is passivated by silicon nitride (SiNx) for both electrical isolation and optical buffering to reduce metallic loss. The whole device is covered by silver to form the metal cavity. The top/bottom mirrors of the cylindrical 3λ/2nr microcavity are formed by 19/32 pairs of p-doped/n-doped AlGaAs/GaAs distributed Bragg reflectors (DBR), respectively.
2. Material gain of submonolayer quantum dots
The active region of the microlasers consists of three groups of SML QDs [8, 11], each being 8 nm thick and separated by 13 nm GaAs spacers. Each group of the SML QDs consists of ten stacks of 0.5-monolayer InAs QD layers, separated by 2.2-ML GaAs spacers. The nominal structure for each group of SML QDs is shown as Fig. 1(b) . Vertically-correlated SML QDs in each group are modeled as effective quantum disks. Extensive work  has been done to analyze the effect of size, shape, and piezoelectricity on QD optical properties. In our model, since the SML QD layers are thin, we can assume the effective quantum disks have no variance in the growth direction. Such cylindrical high-symmetry structure leads to a negligible shear strain, thus we consider the QD strain as biaxial, namely, εxx = εyy ≠ εzz, but εxy = εyz = εzx = 0.
Since the shear strain is assumed negligible, there are two effects that also become negligible: the piezoelectric effect [12,13] and valence-band mixing. For materials with zincblende crystal structure, the 1st-order piezoelectric polarization P1 is only dependent on the shear strain because the only non-zero elements in the strain tensor [e̿]3×6 are e14 = e25 = e36. The 2nd-order piezoelectric polarization P2 also vanishes when no shear strain exists [12, 13]. Due to the discrete energy levels of quantum dots, the band mixing effect is only caused by strain. That is, the off-diagonal terms in the coupled Hamiltonian [14–16] are only dependent on the shear strain (εxy, εyz, and εzx), where εxx − εyy is also negligible due to symmetry. Therefore, we can treat the conduction, heavy-hole, light-hole, and spin-orbit split-off bands as fully decoupled, and we can solve the single-band Hamiltonian for each band individually using the corresponding Γ-point effective mass. The biaxial strain effect is included through the extra strain terms from the Pikus-Bir Hamiltonian . The material parameters related to the band gap, strain and effective mass are taken from the experimental data summarized in . The eigen-problem formed by the Hamiltonian is solved numerically using the 3D finite-difference method. The Dirichlet boundary condition is used for an isolated QD when the dot density is low or the 2D fill factor is small. Periodic boundary conditions are used at high dot density and 2D fill factor to include the lateral coupling among QDs.
Figure 2(a) shows the isosurfaces of the 3D wavefunctions for the first two conduction band states (CB1 and CB2) and heavy-hole states (HH1 and HH2). The QDs are assumed to be isolated laterally. In this figure the effective diameter of the vertically-stacked SML QDs is assumed as 20 nm. We can see that the half maximum isosurface of the ground state and the 1st excited state wavefunctions are still well contained in the stacked SML QDs. Figure 2(b) shows the case when the lateral coupling is not negligible (2D fill factor being 62.8%) and a periodic boundary condition is used. In this case, the squared magnitude of the wavefunction at the mid-point between two unit cell centers is 19.8% of that at the unit cell center. When the 2D fill factor is 31.4%, the squared magnitude of the wavefunction at the mid-point between two unit cell centers becomes 0.72% of that at the unit cell center, which indicates the lateral coupling is negligible in this case.
The calculated ground state transition (C1 to HH1) energy as a function of the temperature and effective dot diameter is shown in Fig. 3(a). The temperature-dependent material band gap is used from the empirical Varshni equation [17, 18]. The measured ground state transition energy  from photoluminescence is shown for comparison. We conclude that the effective QD size is close to the size observed in .
Figure 3(b) shows the carrier-dependent quasi-Fermi level for the CB obtained from Eq. (1), together with the first 50 CB states, among which three are bound states (one ground state and two degenerate excited states). Similarly, the carrier-dependent quasi-Fermi level for the VB is shown in Fig. 3(c). The first 50 HH states shown are all bound states, while only the first LH state out of the 20 states shown is a bound state.Figure 4(a) and 4(b) show the carrier-dependent TE-polarized (E-field normal to growth direction) material gain and spontaneous emission rate, respectively, at T = 300 K.
3. Spontaneous emission coupling factor
Since the spontaneous emission is affected by the vacuum-field fluctuation and the interaction between the emitter and the optical modes, by modifying the radiation environment we can potentially control the spontaneous emission rate. It was discovered by E. Purcell  that spontaneous emission is not only an inherent property of the emitter, but also dependent on the density of optical modes. Such enhancement of the spontaneous emission rate is characterized by the Purcell factorEq. (4) is that the emitter is a two-level system. Therefore, the Purcell factor contains information about the optical properties (radiation environment) but lacks information on the electronic density-of-states of the emitter.
The spontaneous emission coupling factor (β factor) is defined as the ratio between the spontaneous emission coupled into the m-th mode Rsp,m and the spontaneous emission coupled into all modes Rsp [10, 14, 20]. Gérard et. al.  showed the relationship between the β factor and the Purcell factor as21]). Yamamoto et. al.  gave an empirical formula as 10]. We can see that the above empirical formula does not contain the detailed emission property of the emitter. Due to various linewidth broadening mechanisms, the quantum dot transition linewidth can be comparable to the cavity linewidth, and the detailed electronic density-of-states should be considered. Hence, we start with the discrete-mode spontaneous emission rate , as shown from Eq. (18) to Eq. (22). Omitting the inhomogeneous broadening integral ∫ dED() for simplicity (we can always add it back), we have Eq. (7) is proportional to the free-space spontaneous emission in Eq. (2). Therefore, we can write Eq. (8) we see that the discrete-mode spontaneous emission rate (s−1cm−3) reduces to an overlap integral between the free-space spontaneous emission spectrum (s−1cm−3eV−1) and the photon density spectrum (eV−1) for the m-th mode, expressed by a Lorentzian. The prefactor Dcav keeps the unit consistent and is dependent on the resonance wavelength for the m-th mode. By the comparison in Eq. (8), we see the prefactor is Dcav is closely related to the Purcell factor as Eq. (4). Therefore, by definition we can write the β factor for the m-th cavity mode as Eq. (9) and Eq. (10), we obtain the simplified form of β factor as Eq. (5). But now the β factor considers not only the radiation environment (photon density-of-states), but also the radiation property of the emitter (electronic density-of-states), which is embedded in the unitless parameter γ as
4. Size-dependent cavity properties and laser light output versus current
Solving the cavity properties of the microlaser usually requires 3D numerical methods such as finite-element method (FEM) and finite-difference time-domain method (FDTD). However, due to the metallic dispersion and the thin DBR layers, the computational cost for 3D methods is expensive. Alternatively, we can solve the 2D transverse waveguide modes and the effective indices neff from the Maxwell’s equationEq. (13) includes both the material dispersion and the modal dispersion due to size dependence. We use the neff for each layer in the 1D transfer matrix method, which calculates the longitudinal field distribution, as well as the reflection spectra from the top and bottom mirrors. The cavity resonance condition (round-trip phase matching condition) is obtained from the 1D Fabry-Pérot model , which has been shown to be in excellent agreement with the full-structure FDTD simulation and the experimental data.
The key size-dependent parameters to be obtained from the cavity model are the fundamental mode (HE11) lasing wavelength λr, quality factor Q, photon lifetime τp, mirror loss αm, and confinement factor ΓE. Size-dependent resonance wavelength, photon lifetime and the quality factor are calculated for the HE11 mode and shown in Fig. 5(a) and 5(b). The effective mode volume in terms of (λr/nr)3 is calculated based on the confinement factor obtained from the Fabry-Pérot model.Eq. (19). ηi is the intrinsic quantum efficiency, and vg is the group velocity. A is the surface recombination coefficient and C is the Auger coefficient. The A coefficient is dependent on the surface-to-volume ratio (Aa/Va) of the active region 26] as
The spontaneous emission coupling factor βsp,m(n) into the lasing mode in Eq. (14) is obtained from the previous section, and is dependent on the carrier density. The free-space spontaneous emission rspon increases with carrier density as shown in Fig. 4, thus the γ parameter in Eq. (12) is also carrier-dependent. Although the continuum-mode spontaneous emission life-time τsp,cont decreases with carrier density, the increase of the integral in Eq. (12) is faster if the free-space emission peak aligns with the cavity peak. Then, both the γ parameter in Eq. (12) and the βsp,m factor in Eq. (11) increase with the carrier density.
After the rate equations are solved for each given injection current, we can obtain the output light power as
Figure 6 shows a comparison between the theoretical and experimental light output power vs. current (L-I) curves for metal-cavity SML QD surface-emitting microlasers with different device diameters at T = 300 K. The L-I curves are measured under pulsed mode to eliminate laser self-heating for the ease of analyzing the size-dependent lasing characteristics. We can see that, as the device size reduces, the L-I curves exhibit more obviously a turn-on behavior below threshold. Such turn-on behavior was also observed for metal-cavity quantum-well microlasers in . This “upward-bending” L-I behavior below threshold can be explained by the increasing amount of spontaneous emission coupling into the cavity mode as the injection current increases.
In the rate-equation model, the surface recombination velocity vs is set to 6 × 105 cm/s and the Auger coefficient C is set to 1 × 10−29 cm6/s. However, we do not see significant change of the L-I curves when we vary vs in the 105 cm/s range or vary C in the 10−30 ∼ 10−29 cm6/s range. Due to the 1% pulsed mode operation, the thermal effect is negligible, and most carrier-dependent laser characteristics are pinned upon threshold. Table 1 summarizes the carrier density, material gain, threshold current density, and leakage current density at threshold. The threshold material gain gth increases as size reduces because of both larger radiation loss and larger material loss from metal. The threshold carrier density nth is dependent on gth as well as the lasing wavelength. Since the quantum dot emission bandwidth is narrow, the alignment between the cavity peak and gain peak is important. Furthermore, because nth is larger for smaller devices, the quasi-Fermi levels are closer to the quantum dot barrier, and the leakage current density at threshold Jl,th is also larger. As a result of larger nth and larger Jl,th, we can see that the threshold current density also increases when we reduce the device size. Figure 7(a) shows the Purcell factor calculated with Eq. (4). Fig. 7(b) shows the βsp factor at threshold extracted from the fitting of the L-I curves with the rate-equation model. We can see that the portion of spontaneous emission coupled to the lasing mode increases drastically when the cavity size reduces. The parameters used in our model are summarized in Table 2.
Metal-cavity submonolayer quantum-dot surface-emitting lasers are demonstrated under electrical injection at room temperature with device radius down to 0.5 μm. We have developed a comprehensive model for analyzing the size-dependent device performance. Our model yields the material gain and the spontaneous emission spectra of submonolayer quantum dots. We derive a rigorous expression for the spontaneous emission coupling factor. The laser cavity properties are solved with the effective index method and the transfer matrix method. As the future work, we can extend our cavity model for cases where multiple transverse modes are present. This can be done using the analytical vector mode-matching method, and the mode conversion and mismatch can be included. The resonance wavelength and photon lifetime for each individual mode can be obtained. This then can be followed by a multi-mode rate-equation model to produce the mode-resolved L-I curves. In this work, a single-mode model has shown good prediction of the cavity resonance for our application. With the information on the quantum-dot emission and the optical cavity, a single-mode rate-equation model is used to investigate the laser characteristics. Our theory shows excellent agreement with the experiments and directs our future work toward the miniaturization of metal-cavity lasers.
Appendix Derivations for the discrete mode spontaneous emission rate
Omitting the inhomogeneous broadening integral for now (we can add back anytime) and using the slow-varying approximation of cavity mode Em compared to crystal unit cells, the quantum-dot spontaneous emission coupled to the m-th cavity mode  is27] as Eq. (18) can be rewritten as
This work was supported by the DARPA NACHOS and EPHI programs in USA, and SFB 787 of DFG in Germany. The authors would like to thank Dr. Shu-Wei Chang, Dr. Chi-Yu Adrian Ni, and Professor Weng Cho Chew at University of Illinois at Urbana-Champaign for insightful discussions.
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