Theory and experiment of submonolayer quantum-dot metal-cavity surface-emitting microlasers

Pengfei Qiao; Chien-Yao Lu; Dieter Bimberg; Shun Lien Chuang

doi:10.1364/OE.21.030336

1. Introduction

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 [1], 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) [6]. 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 [7]. 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 [8]. 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 [8] 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 [9]. 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 [10]. 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 (SiN_x) 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/2n_r microcavity are formed by 19/32 pairs of p-doped/n-doped AlGaAs/GaAs distributed Bragg reflectors (DBR), respectively.

Fig. 1 (a) Schematic of a submonolayer (SML) quantum-dot (QD) metal-cavity surface-emitting laser. The active region contains 3 groups of SML QDs. (b) Schematic of each group of SML QDs, consisting of 10 stacks of 0.5 ML InAs QDs separated by 2.2 ML thick GaAs spacers. [8] (c) A scanning electron micrograph of a 0.5-μm-radius microlaser before SiN_x passivation and metal coating.

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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) [11]. Vertically-correlated SML QDs in each group are modeled as effective quantum disks. Extensive work [12] 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, ε_xx = ε_yy ≠ ε_zz, but ε_xy = ε_yz = ε_zx = 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 (ε_xy, ε_yz, and ε_zx), where ε_xx − ε_yy 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 [16]. The material parameters related to the band gap, strain and effective mass are taken from the experimental data summarized in [17]. 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.

Fig. 2 (a) Wavefunctions of the first two conduction band states (CB1 and CB2) and the first two heavy-hole states (HH1 and HH2). The wavefunctions are shown at the isosurface of ${| Ψ |}^{2} = 0.5 {| Ψ |}_{max}^{2}$ . The blue disks are the 10-fold vertically-correlated submonolayer quantum dots, assuming no lateral coupling. (b) Wavefunction of the conduction band ground state (CB1), considering lateral coupling. The wavefunction is shown at the isosurface of ${| Ψ |}^{2} = 0.16 {| Ψ |}_{max}^{2}$ . In this example, the quantum-dot 2D fill factor is 62.8% and the 2D dot density is 2 × 10¹¹ cm⁻².

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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 [8] from photoluminescence is shown for comparison. We conclude that the effective QD size is close to the size observed in [11].

Fig. 3 (a) Ground state transition energy as a function of the temperature for different effective dot sizes. The measured photoluminescence peak is shown as the star. [8] (b) The quasi-Fermi level F_c for conduction band (blue circles) as a function of injected carrier density. Horizontal lines show 50 conduction band states with bound states circled. (c) The quasi-Fermi level F_v for valence band (red circles) as a function of injected carrier density. Horizontal lines show 50 heavy-hole and 20 light-hole states.

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To account for inhomogeneous broadening [6,14] due to QD size variations, the electron and hole carrier densities (n and p) are related to the quasi-Fermi levels (F_c and F_v) as

\begin{array}{l} n = 2 \frac{N_{dot}^{2 D}}{L_{z}} \sum_{i} \int d E [\frac{1}{\sqrt{2 π} σ_{c}} e^{- {(E - E_{c}^{i})}^{2} / 2 σ_{c}^{2}}] f_{c} (E, F_{c}), \\ p = 2 \frac{N_{dot}^{2 D}}{L_{z}} \sum_{j} \int d E [\frac{1}{\sqrt{2 π} σ_{v}} e^{- {(E - E_{v}^{j})}^{2} / 2 σ_{v}^{2}}] f_{v} (E, F_{v}) \end{array}

respectively. The subscripts c and v indicate conduction and valence bands, respectively. The square brackets are the linewidth broadening functions, with linewidths being σ_c and σ_v.

N_{dot}^{2 D}

is the 2D dot density, f_c and f_v are the Fermi occupation probabilities, and

E_{c}^{i}

and

E_{v}^{j}

are the energies for the i-th conduction band and j-th valence band, respectively. L_z is the thickness of each QD group.

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.

Once the quasi-Fermi levels are obtained, we can calculate the carrier-dependent material gain (cm⁻¹) and the free-space spontaneous emission rate (cm⁻³s⁻¹eV⁻¹) as [6, 9, 14]

\begin{array}{l} g (\bar{h} ω) & = \frac{2 N_{dot}^{2 D}}{L_{z}} C_{0} \sum_{i, j} \int d E {| M_{env}^{i j} |}^{2} {| \hat{e} \cdot p_{c v} |}^{2} D (E, E_{c v}^{i j}) L (E, \bar{h} ω) (f_{c, i} - f_{v, j}), \\ r_{spon} (\bar{h} ω) & = \frac{2 N_{dot}^{2 D}}{L_{z}} B_{0} C_{0} \sum_{i, j} \int d E {| M_{env}^{i j} |}^{2} {| \hat{e} \cdot p_{c v} |}^{2} D (E, E_{c v}^{i j}) L (E, \bar{h} ω) f_{c, i} (1 - f_{v, j}) \end{array}

where

M_{env}^{i j}

is the overlap integral between the envelop functions of i-th CB and j-th VB states, and

E_{c v}^{i j}

is the transition energy between the two states. ê · p_cv is the bulk momentum matrix element. The linewidth functions due to inhomogeneous broadening D() and homogeneous broadening L() are expressed as

\begin{array}{l} D (E, E_{c v}^{i j}) = \frac{1}{\sqrt{2 π (σ_{c}^{2} + σ_{v}^{2})}} exp [- {(E - E_{c v}^{i j})}^{2} / 2 (σ_{c}^{2} + σ_{v}^{2})], \\ L (E, \bar{h} ω) = \frac{Γ_{c v}}{π} \frac{1}{Γ_{c v}^{2} + {(E - \bar{h} ω)}^{2}}, B_{0} = \frac{n_{a}^{2} ω^{2}}{π^{2} \bar{h} c^{2}}, C_{0} = \frac{π e^{2}}{n_{a} c ε_{0} m_{0}^{2} ω} \end{array}

where the linewidth Γ_cv accounts for carrier scattering processes, and n_a is the refractive index for the active region. 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.

Fig. 4 (a) TE-polarized material gain and (b) TE-polarized spontaneous emission rate calculated for the submonolayer quantum dots at T = 300 K, with carrier densities from n = 6 × 10¹⁷ cm⁻³ to n = 5.4 × 10¹⁸ cm⁻³.

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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 [19] 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 factor

F_{p} = \frac{3 Q}{4 π^{2} V_{eff}} {(\frac{λ}{n_{r}})}^{3}

where Q is the quality factor of the optical cavity, V_eff is the effective mode volume, and n_r is the refractive index. The assumption to arrive at the Purcell factor given in Eq. (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 R_sp,_m and the spontaneous emission coupled into all modes R_sp [10, 14, 20]. Gérard et. al. [21] showed the relationship between the β factor and the Purcell factor as

β = \frac{F_{p} / 3}{g F_{p} / 3 + 1}

where g is the mode degeneracy (g=2 for circular pillars and disks [21]). Yamamoto et. al. [22] gave an empirical formula as

β = \frac{λ^{4}}{4 π^{2} V Δ λ ε^{3 / 2}}

which is equivalent to the expression derived by Baba et. al. [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 [23], as shown from Eq. (18) to Eq. (22). Omitting the inhomogeneous broadening integral ∫ dED() for simplicity (we can always add it back), we have

\begin{matrix} R_{sp, m} = \int d (\bar{h} ω) {[\frac{2 N_{dot}^{2 D}}{L_{z}} \frac{c}{V_{eff} n_{a}} C_{0} \sum_{i j} {| M_{env}^{i j} |}^{2} {| \hat{e} \cdot p_{c v} |}^{2} f_{c, i} (1 - f_{v, j}) \frac{Γ_{c v} / π}{Γ_{c v}^{2} + {(E_{c v}^{i j} - \bar{h} ω)}^{2}}] \\ \times \frac{Γ_{m} / π}{Γ_{m}^{2} + {(\bar{h} ω_{m} - \bar{h} ω)}^{2}}} \end{matrix}

where Γ_m is the half-width half-maximum (HWHM) of the optical density-of-states, determined by the cavity quality factor (h̄ω_m/2Γ_m = Q). We notice that the term in the square bracket in Eq. (7) is proportional to the free-space spontaneous emission in Eq. (2). Therefore, we can write

\begin{array}{l} R_{sp, m} & = \int d (\bar{h} ω) \frac{c / (V_{eff} n_{a})}{B_{0}} r_{spon} (\bar{h} ω) \frac{Γ_{m} / π}{Γ_{m}^{2} + {(\bar{h} ω - \bar{h} ω_{m})}^{2}} \\ \approx D_{cav} (\bar{h} ω_{m}) \int d (\bar{h} ω) r_{spon} (\bar{h} ω) \frac{Γ_{m} / π}{Γ_{m}^{2} + {(\bar{h} ω - \bar{h} ω_{m})}^{2}} \end{array}

From 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 D_cav 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 D_cav is closely related to the Purcell factor as

\begin{array}{l} D_{cav} (\bar{h} ω_{m}) & = \frac{c / (V_{eff} n_{a})}{B_{0}} = \frac{π^{2} \bar{h} c^{3}}{V_{eff} n_{a}^{3} ω_{m}^{2}} = \frac{\bar{h} ω_{m}}{8 π V_{eff}} {(\frac{λ_{m}}{n_{a}})}^{3} = \frac{2 Γ_{m} Q}{8 π V_{eff}} {(\frac{λ_{m}}{n_{a}})}^{3} \\ = π Γ_{m} [\frac{Q}{4 π^{2} V_{eff}} {(\frac{λ_{m}}{n_{a}})}^{3}] = π Γ_{m} \frac{F_{p}}{3} \end{array}

where the square bracket is exactly the same as the Purcell factor in Eq. (4). Therefore, by definition we can write the β factor for the m-th cavity mode as

\begin{array}{l} β_{sp, m} = \frac{R_{sp, m}}{R_{sp}} = \frac{R_{sp, m}}{g R_{sp, m} + R_{sp, cont}} \\ = \frac{D_{cav} (\bar{h} ω_{m}) \int d (\bar{h} ω) r_{spon} (\bar{h} ω) \frac{Γ_{m} / π}{Γ_{m}^{2} + {(\bar{h} ω - \bar{h} ω_{m})}^{2}}}{g D_{c a v} (\bar{h} ω_{m}) \int d (\bar{h} ω) r_{spon} (\bar{h} ω) \frac{Γ_{m} / π}{Γ_{m}^{2} + {(\bar{h} ω - \bar{h} ω_{m})}^{2}} + R_{spon, cont}} \end{array}

where R_sp,cont accounts for the coupling into the continuum modes of the cavity, and g accounts for the degeneracy of the m-th mode. If the cavity supports other discrete modes, those can be lumped into R_sp,cont as well. Combining Eq. (9) and Eq. (10), we obtain the simplified form of β factor as

β_{sp, m} = \frac{γ F_{p} / 3}{γ g F_{p} / 3 + 1}

We can see the expression is similar to the empirical formula in 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

\begin{array}{l} γ = \frac{1}{R_{sp, cont}} \int d (\bar{h} ω) r_{spon} (\bar{h} ω) \frac{Γ_{m}^{2}}{Γ_{m}^{2} + {(\bar{h} ω - \bar{h} ω_{m})}^{2}} \\ = τ_{sp, cont} V_{a} \int d (\bar{h} ω) r_{spon} (\bar{h} ω) \frac{Γ_{m}^{2}}{Γ_{m}^{2} + {(\bar{h} ω - \bar{h} ω_{m})}^{2}} \end{array}

To evaluate the spontaneous emission coupled into the continuum modes is nontrivial. Not only that the continuum electric field mode has to be used in the optical matrix element, but also the continuum photon density-of-states ρ_cont(h̄ω) is needed. Furthermore, the amount of coupling is depending on injected carrier density. In this case, we can replace R_sp,cont by a phenomenological lifetime τ_sp,cont for an active region volume of V_a.

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 n_eff from the Maxwell’s equation

[\nabla_{t}^{2} + n^{2} (ρ) k_{0}^{2}] [\begin{matrix} E_{z} \\ H_{z} \end{matrix}] = k_{z}^{2} [\begin{matrix} E_{z} \\ H_{z} \end{matrix}] = n_{eff}^{2} k_{0}^{2} [\begin{matrix} E_{z} \\ H_{z} \end{matrix}]

where the

k_{0} = ω \sqrt{ε_{0} μ_{0}}

is the free-space wavenumber, and k_z is the wavenumber in the propagation direction. n(ρ) is the transverse profile for the refractive index. The effective index n_eff solved from Eq. (13) includes both the material dispersion and the modal dispersion due to size dependence. We use the n_eff 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 [24], 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 λ_r, quality factor Q, photon lifetime τ_p, mirror loss α_m, and confinement factor Γ_E. 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 (λ_r/n_r)³ is calculated based on the confinement factor obtained from the Fabry-Pérot model.

Fig. 5 (a) Calculated lasing wavelength and the photon lifetime as a function of the cavity diameter for the fundamental HE₁₁ transverse mode. (b) Calculated cavity quality factor and effective mode volume as a function of the cavity diameter for the HE₁₁ transverse mode.

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To study the light output power vs. current behavior of the metal-cavity microlasers, we use the rate equations [14, 20, 25] of the carrier density n and the photon density S

\begin{array}{l} \frac{d n}{d t} = η_{i} \frac{I - I_{l} (n)}{q V_{a}} - (A n + C n^{3}) - R_{sp} (n) - v_{g} g (n) S, \\ \frac{d S}{d t} = Γ_{E} v_{g} g (n) S - \frac{S}{τ_{p}} + Γ_{E} β_{sp, m} (n) R_{sp} (n) \end{array}

where the material gain g(n) and spontaneous emission rate R_sp(n) are obtained from our QD gain model which accounts for the inhomogeneous broadening effect. The photon lifetime τ_p, and confinement factor Γ_E are obtained from our cavity transfer matrix model with effective indices. The rigorous definition for the energy confinement factor Γ_E is given in Eq. (19). η_i is the intrinsic quantum efficiency, and v_g 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 (A_a/V_a) of the active region

A = \frac{A_{a}}{V_{a}} v_{s} = \frac{4}{D} v_{s} (cylindrical)

where v_s is the surface recombination velocity and D is the cavity diameter. Before the laser threshold is reached, as more carriers are injected, the quasi-Fermi level gets closer to the quantum dot barrier and more carriers leak out of the quantum dots without radiative recombination. Therefore such carrier leakage has important effect on the threshold carrier density and threshold current. We consider such leakage current [26] as

I_{l} (n) = I_{l 0} \cdot exp (\frac{E_{g, barrier} - [F_{c} (n) - F_{v} (n)]}{k T})

where E_g,_barrier is the band gap of the quantum dot barrier.

The spontaneous emission coupling factor β_sp,_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 r_spon 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 τ_sp,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 β_sp,_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

P = β_{c 1} \bar{h} ω \frac{V_{a}}{Γ_{E}} v_{g} α_{m} S + β_{c 2} \bar{h} ω R_{sp} V_{a}

The first term is the contribution from stimulated emission, which depends on the photon density S and the photon escape rate v_gα_m, and V_a/Γ_E is the effective mode volume. The second term is the contribution from spontaneous emission. β_c₁ and β_c₂ are the coupling efficiencies to the detector for the stimulated emission and spontaneous emission, respectively, which account for the loss of light power from the experimental setup.

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 [25]. 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.

Fig. 6 Theoretical and measured light output power vs. current (L-I) curves for submono-layer quantum-dot metal-cavity surface-emitting microlasers with different device diameters at T = 300 K. The turn-on behavior below lasing threshold is explained by the increasing β_sp factor with carrier injection, i.e., increasing amount of spontaneous emission coupled into the cavity mode.

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In the rate-equation model, the surface recombination velocity v_s 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 v_s 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 g_th increases as size reduces because of both larger radiation loss and larger material loss from metal. The threshold carrier density n_th is dependent on g_th 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 n_th is larger for smaller devices, the quasi-Fermi levels are closer to the quantum dot barrier, and the leakage current density at threshold J_l,_th is also larger. As a result of larger n_th and larger J_l_,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 β_sp 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.

Table 1. Size-dependent laser characteristics extracted from the rate-equation model.

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Fig. 7 (a) Calculated Purcell factor for the fundamental mode in the metal-cavity micro-lasers with different cavity diameters. (b) Extracted spontaneous emission coupling factor β_sp from the fitting of device light output power vs. current (L-I) curves with the rate-equation model.

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Table 2. Parameters used in our theoretical model.

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5. Conclusion

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 E_m compared to crystal unit cells, the quantum-dot spontaneous emission coupled to the m-th cavity mode [23] is

\begin{array}{l} R_{sp, m} = \frac{2 N_{dot}^{2 D}}{L_{z}} \frac{2 π}{\bar{h}} \sum_{i, j} {| 〈 ψ_{i}^{c} | μ \cdot \frac{E_{m}}{2} | ψ_{v}^{j} 〉 |}^{2} \frac{Γ_{c v} + Γ_{m}}{π} \frac{f_{c, i} (1 - f_{v, j})}{{(E_{c v}^{i j} - \bar{h} ω_{m})}^{2} + {(Γ_{c v} + Γ_{m})}^{2}} \\ \approx \frac{2 N_{dot}^{2 D}}{L_{z}} \frac{2 π}{\bar{h}} [\int_{V_{a}} \frac{d^{3} r}{V_{a}} \frac{{| E_{m} (r) |}^{2}}{4}] \sum_{i, j} {| 〈 ψ_{c}^{i} | μ \cdot \hat{e} | ψ_{v}^{j} 〉 |}^{2} \frac{Γ_{c v} + Γ_{m}}{π} \frac{f_{c, i} (1 - f_{v, j})}{{(E_{c v}^{i j} - \bar{h} ω_{m})}^{2} + {(Γ_{c v} + Γ_{m})}^{2}} \end{array}

The energy confinement factor is rigorously defined [27] as

\frac{V_{a}}{V_{eff}} = Γ_{E} = \frac{\int_{V_{a}} d^{3} r ε_{0} n_{a}^{2} {| E_{m} (r) |}^{2} / 2}{\int_{V} d^{3} r ε_{0} n^{2} (r) {| E_{m} (r) |}^{2} / 2} = \frac{\int_{V_{a}} d^{3} r ε_{0} n_{a}^{2} {| E_{m} (r) |}^{2} / 2}{\bar{h} ω_{m}}

where the refractive indices n_a (active region) and n have to be replaced by

n^{2} \to \frac{1}{2} [{\frac{\partial [ω^{'} n^{2} (ω^{'})]}{\partial ω^{'}} |}_{ω^{'} = ω} + n^{2} (ω)]

The momentum matrix in Eq. (18) can be rewritten as

{| 〈 ψ_{c}^{i} | μ \cdot \hat{e} | ψ_{v}^{j} 〉 |}^{2} = {| M_{env}^{i j} |}^{2} {| μ_{c v} \cdot \hat{e} |}^{2} = {| M_{env}^{i j} |}^{2} \frac{e^{2}}{m_{0}^{2} ω_{i j}^{2}} {| \hat{e} \cdot p_{c v}^{i j} |}^{2}

The discrete mode spontaneous emission rate becomes

R_{sp, m} = \frac{2 N_{dot}^{2 D}}{L_{z}} \frac{2 π}{\bar{h}} (\frac{Γ_{E} \bar{h} ω_{m}}{2 ε_{0} n_{a}^{2} V_{a}}) \sum_{i, j} \frac{e^{2}}{m_{0}^{2} ω_{i j}^{2}} {| M_{env}^{i j} |}^{2} {| \hat{e} \cdot p_{c v} |}^{2} \frac{Γ_{c v} + Γ_{m}}{π} \frac{f_{c, i} (1 - f_{v, j})}{{(E_{c v}^{i j} - \bar{h} ω_{m})}^{2} + {(Γ_{c v} + Γ_{m})}^{2}}

Considering ω_m ≈ ω_ij due to the narrow linewidth for the discrete mode, and splitting the Lorentzian function as the convolution of two Lorentzian functions

\begin{array}{l} R_{sp, m} = \frac{2 N_{dot}^{2 D}}{L_{z}} \frac{2 π}{\bar{h}} \sum_{i, j} {\frac{\bar{h} e^{2}}{2 ε_{0} n_{a}^{2} V_{eff} m_{0}^{2} ω_{m}} {| M_{env}^{i j} |}^{2} {| \hat{e} \cdot p_{c v} |}^{2} f_{c, i} (1 - f_{v, j}) \\ \int d (\bar{h} ω) \frac{Γ_{c v} / π}{{(E_{c v}^{i j} - \bar{h} ω)}^{2} + Γ_{c v}^{2}} \frac{Γ_{m} / π}{{(\bar{h} ω - \bar{h} ω_{m})}^{2} + Γ_{m}^{2}}} \\ = \int d (\bar{h} ω) [\frac{2 N_{dot}^{2 D}}{L_{z}} \frac{c C_{0}}{V_{eff} n_{a}} \sum_{i, j} {| M_{env}^{i j} |}^{2} {| \hat{e} \cdot p_{c v} |}^{2} f_{c, i} (1 - f_{v, j}) \frac{Γ_{c v} / π}{{(E_{c v}^{i j} - \bar{h} ω)}^{2} + Γ_{c v}^{2}}] \frac{Γ_{m} / π}{{(\bar{h} ω - \bar{h} ω_{m})}^{2} + Γ_{m}^{2}} \end{array}

Acknowledgments

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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Device diameter (μm)	Threshold carrier density n_th (×10¹⁸ cm⁻³)	Threshold material gain g_th (cm⁻¹)	Leakage current density at threshold J_l_,th (kA/cm²)	Threshold current density J_th (kA/cm²)

10	1.96	160.72	1.488	5.730
5	1.97	163.01	6.305	15.83
4	1.98	165.96	14.21	24.89
3	2.29	175.02	25.96	39.82
2	2.77	216.83	128.7	150.2
1	3.58	685.68	411.4	452.1

Name and symbol	Value	Name and symbol	Value

Surface recombination velocity v_s	6 × 10⁵ cm/s	Injection efficiency η_i	0.4∼0.7
Auger coefficient C	1×10⁻²⁹ cm⁶/s	Quantum dot diameter	20 μm
Coupling efficiency β_c₁	0.05∼0.15	Quantum dot 2D density	1×10¹¹ cm⁻²
Coupling efficiency β_c₂	0.005∼0.05	Series leakage current parameter I_l₀	100∼300 mA
Homogeneous broadening linewdith Γ_cv	15 meV	Inhomogeneous broadening linewidth for conduction band σ_c	15 meV
Inhomogeneous broadening linewdith for valence band σ_v	2 meV

Device diameter (μm)	Threshold carrier density n_th (×10¹⁸ cm⁻³)	Threshold material gain g_th (cm⁻¹)	Leakage current density at threshold J_l_,th (kA/cm²)	Threshold current density J_th (kA/cm²)

10	1.96	160.72	1.488	5.730
5	1.97	163.01	6.305	15.83
4	1.98	165.96	14.21	24.89
3	2.29	175.02	25.96	39.82
2	2.77	216.83	128.7	150.2
1	3.58	685.68	411.4	452.1

Name and symbol	Value	Name and symbol	Value

Surface recombination velocity v_s	6 × 10⁵ cm/s	Injection efficiency η_i	0.4∼0.7
Auger coefficient C	1×10⁻²⁹ cm⁶/s	Quantum dot diameter	20 μm
Coupling efficiency β_c₁	0.05∼0.15	Quantum dot 2D density	1×10¹¹ cm⁻²
Coupling efficiency β_c₂	0.005∼0.05	Series leakage current parameter I_l₀	100∼300 mA
Homogeneous broadening linewdith Γ_cv	15 meV	Inhomogeneous broadening linewidth for conduction band σ_c	15 meV
Inhomogeneous broadening linewdith for valence band σ_v	2 meV

Theory and experiment of submonolayer quantum-dot metal-cavity surface-emitting microlasers

Abstract

1. Introduction

2. Material gain of submonolayer quantum dots

3. Spontaneous emission coupling factor

4. Size-dependent cavity properties and laser light output versus current

5. Conclusion

Appendix Derivations for the discrete mode spontaneous emission rate

Acknowledgments

References and links

Cited By

Figures (7)

Tables (2)

Equations (23)

Optics Express