Nonlinear Dynamics of Silicon-Based Epitaxial Quantum Dot Lasers under Optical Injection

Abstract

For silicon-based epitaxial quantum dot lasers (QDLs), the mismatches of the lattice constants and the thermal expansion coefficients lead to the generation of threaded dislocations (TDs), which act as the non-radiative recombination centers through the Shockley–Read–Hall (SRH) recombination. Based on a three-level model including the SRH recombination, the nonlinear properties of the silicon-based epitaxial QDLs under optical injection have been investigated theoretically. The simulated results show that, through adjusting the injection parameters including injection strength and frequency detuning, the silicon-based epitaxial QDLs can display rich nonlinear dynamical states such as period one (P1), period two (P2), multi-period (MP), chaos (C), and injection locking (IL). Relatively speaking, for a negative frequency detuning, the evolution of the dynamical state with the injection strength is more abundant, and an evolution path P1-P2-MP-C-MP-IL has been observed. Via mapping the dynamical state in the parameter space of injection strength and frequency detuning under different SRH recombination lifetime, the effects of SRH recombination lifetime on the nonlinear dynamical state of silicon-based epitaxial QDLs have been analyzed.

1. Introduction

Semiconductor lasers exhibit rich dynamical behaviors after being subjected to external perturbations such as optical injection [1], optical feedback [2], and optoelectronic feedback [3], which can be applied in various scenarios, including chaotic secure communications [4], photonic microwave amplifiers [5], radio-over-fiber uplink transmission [6], reservoir computing [7], and random bit generation [8,9].

In recent years, with the development of fabrication technology, quantum dot lasers (QDLs) have shown many performance advantages over traditional semiconductor lasers, such as a lower threshold current density [10], improved stability against optical feedback [11,12], minimal impact from the environmental temperature [13], lower relative intensity noise (RIN) [14], and a small linewidth enhancement factor [15,16]. These above advantages arise from the unique self-assembly growth mode of quantum dots (QDs) structures. Three-dimensional quantum confinement of QDs leads to a complete discretization of the energy levels, with the atom-like degeneracy into the form of a delta-function-like state, which gives quantum dot lasers (QDLs) a different form of emission from other semiconductor lasers [17]. Similar to traditional semiconductor lasers, QDLs can also emit from the ground state (GS). Furthermore, it is possible to observe QDLs emitting from both the GS and the excited state (ES). In QDLs, the GS is excited when a slight bias current is applied, followed by the ES once the current surpasses the second threshold. As the current rises, the ES lasing becomes dominant while the GS lasing weakens. It has also been shown that the lasing state of a QDL can be directly switched from ES to GS under certain conditions when there is external optical feedback as an external perturbation [18].

The rapid increase in internet traffic has made the development of low-cost, high-speed, and low-power optical communication technology in data centers become a hot topic. As a result, scientists have turned their focus towards innovating the next generation of information and communication technology by looking into silicon-based photonic integrated circuits (PICs) [19]. For silicon-based PICs, both optically active and passive components are monolithically integrated on a single chip, offering a high bandwidth density, high energy efficiency, and low latency. Unfortunately, silicon, which is an indirect band gap material, is not suitable for the generation of lasers. In contrast, direct bandgap III-V compound semiconductors have robust optical properties that can be tailored for lasers operating at various wavelengths with high efficiency, a large modulation bandwidth, and sufficient optical power output [20]. Therefore, epitaxial growth of III-V QDLs directly on silicon substrates is considered as one of the most promising methods for achieving on-chip light sources in silicon-based PICs [21,22,23,24,25]. For QDLs directly grown on silicon, the lattice mismatch between silicon and QDs semiconductor materials and thermal expansion coefficients lead to the generation of threaded dislocations (TDs), which act as non-radiative recombination centers through the Shockley–Read–Hall (SRH) recombination. In silicon-based epitaxial QDLs, TDs densities are typically 10⁶~10⁸ cm⁻². Despite significant efforts to improve the quality of silicon-based epitaxial QDLs, the achievable TDs densities are still around 10⁵~10⁶ cm⁻² [26,27]. Previous investigations demonstrated that, for silicon-based epitaxial QDLs under optical feedback, the static and dynamic characteristics are influenced by the SRH recombination. For a shorter SRH recombination lifetime, a smaller linewidth enhancement factor but a larger damping factor are obtained. Furthermore, any reduction in the SRH recombination lifetime will shrink the chaotic region and shift the first Hopf bifurcation point to stronger feedback [28]. As a result, for inspecting the dynamic characteristics of silicon-based epitaxial QDLs under optical feedback, the SRH recombination should be taken into account. Since optical feedback can be regarded as a special light injection from a certain perspective, it can be predicted that, for analyzing the dynamic characteristics of silicon-based epitaxial QDLs under optical injection, the SRH recombination should also be considered. However, we have noticed that relevant reports on the nonlinear dynamical characteristics of silicon-based epitaxial QDLs under optical injection are almost based on the model excluding the SRH recombination.

In this paper, based on a three-level model including SRH recombination lifetime, we investigate the dynamical behaviors of silicon-based epitaxial QDLs under optical injection. By changing the injection strength and frequency detuning, different nonlinear dynamical states are observed, and the route for silicon-based epitaxial QDLs entering into chaos is revealed. In addition, the impact of the SRH recombination lifetime on the nonlinear dynamical state of silicon-based epitaxial QDLs under optical injection has also been analyzed.

2. Theoretical Model

Figure 1 shows a schematic diagram of the carrier dynamics of silicon-based epitaxial QDLs, considering charged electrons and holes as neutral excitons. The emission time of the spontaneous emission of carriers in the ground state (GS), excited state (ES) and reservoir state (RS) as shown in Figure 1 is ${\tau }_{GS}^{spon}$, ${\tau }_{ES}^{spon}$, ${\tau }_{RS}^{spon}$. It should be emphasized that the rate of SRH recombination is characterized by the SRH recombination lifetime τ_SRH. Carriers are pumped directly into the RS through the Auger recombination process; some carriers are trapped from RS to the ES with a captured time ${\tau }_{ES}^{RS}$, and some carriers relaxed from ES to GS with a relaxation time ${\tau }_{GS}^{ES}$. In addition, through thermal excitations, some carriers in GS escape into ES with an escape time ${\tau }_{ES}^{GS}$, and from ES escape into RS with an escape time ${\tau }_{RS}^{ES}$; stimulated radiation occurs only in the GS. After introducing optical injection, the rate equation for silicon-based epitaxial QDLs including the SRH recombination lifetime can be described as follows [29]:

$${\displaystyle \frac{d{N}_{RS}}{dt}}={\displaystyle \frac{I}{q}}+{\displaystyle \frac{N_{ES}}{{\tau }_{RS}^{ES}}}-{\displaystyle \frac{N_{RS}}{{\tau }_{ES}^{RS}}}(1-{\rho }_{ES})-{\displaystyle \frac{N_{RS}}{{\tau }_{RS}^{spon}}}-{\displaystyle \frac{N_{RS}}{{\tau }_{SRH}}}$$

(1)

$${\displaystyle \frac{d{N}_{ES}}{dt}}=\left({\displaystyle \frac{N_{RS}}{{\tau }_{ES}^{RS}}}+{\displaystyle \frac{N_{GS}}{{\tau }_{ES}^{GS}}}\right)(1-{\rho }_{ES})-{\displaystyle \frac{N_{ES}}{{\tau }_{GS}^{ES}}}(1-{\rho }_{GS})-{\displaystyle \frac{N_{ES}}{{\tau }_{RS}^{ES}}}-{\displaystyle \frac{N_{ES}}{{\tau }_{ES}^{spon}}}-{\displaystyle \frac{N_{ES}}{{\tau }_{SRH}}}$$

(2)

$${\displaystyle \frac{d{N}_{GS}}{dt}}={\displaystyle \frac{N_{ES}}{{\tau }_{GS}^{ES}}}(1-{\rho }_{GS})-{\displaystyle \frac{N_{GS}}{{\tau }_{ES}^{GS}}}(1-{\rho }_{ES})-{\Gamma }_p{v}_g{g}_{GS}{S}_{GS}-{\displaystyle \frac{N_{GS}}{{\tau }_{GS}^{spon}}}-{\displaystyle \frac{N_{GS}}{{\tau }_{SRH}}}$$

(3)

$${\displaystyle \frac{d\mathrm{\varphi }}{dt}}={\displaystyle \frac{1}{2}}{\Gamma }_p{v}_g(g_{GS}{\alpha }_H^{GS}+g_{ES}{\alpha }_H^{ES}+g_{RS}{\alpha }_H^{RS})-\Delta {\omega }_{inj}-k_c\sqrt{k_{inj}{S}_{inj}/S_{GS}}\sin \mathrm{\varphi }$$

(4)

$${\displaystyle \frac{d{S}_{GS}}{dt}}=\left({\Gamma }_p{v}_g{g}_{GS}-{\displaystyle \frac{1}{{\tau }_p}}\right)S_{GS}+{\beta }_{sp}{\displaystyle \frac{N_{GS}}{{\tau }_{GS}^{spon}}}+2k_c\sqrt{k_{inj}{S}_{inj}{S}_{GS}}\cos \mathrm{\varphi }$$

(5)

where N, S, ϕ are the carrier number, photon number, and phase, respectively. I is the bias current; q is the electron charge; Γ_p is the optical confinement factor; v_g is the group velocity, defined as v_g = c/n_r, where c is the speed of light and n_r is the refractive index. ${\alpha }_H^{GS,ES,RS}$ is the GS, ES, and RS contribution to the α_H factor, respectively. β_sp is the spontaneous emission factor; Δω_inj is the frequency detuning; k_c is the coupling coefficient of master and slave lasers, defined as k_c = v_g(1 − R)/(2L(R)^1/2) where L is the laser cavity length; R is the facet reflectivity; and τ_p is the photon lifetime. S_inj is the photon number of injection light, and k_inj is the injection strength. The material gain coefficient g_GS,ES,RS can be described by the following:

$$g_{GS}={\displaystyle \frac{a_{GS}}{1+\epsilon S_{GS}}}{\displaystyle \frac{N_B}{V_B}}\left(2{\rho }_{GS}-1\right)$$

(6)

$$g_{ES}=a_{ES}{\displaystyle \frac{N_B}{V_B}}\left(2{\rho }_{ES}-1\right)$$

(7)

$$g_{RS}=a_{RS}{\displaystyle \frac{D_{RS}}{V_{RS}}}\left(2{\rho }_{RS}-1\right)$$

(8)

where a_{GS, ES, RS} are the differential gain of different states, ε is the gain compression factor, N_B is the total dot number of QDs, V_B is the volume of the active region, D_RS is the total number of states in the RS, V_RS is the volume of the RS, ρ_GS = N_GS/(2N_B), ρ_ES = N_ES/(4N_B) and ρ_RS = N_RS/D_RS represent the occupancy probabilities of carriers in GS, ES, and RS, respectively. For simplicity, the noises have been neglected. These rate equations can be numerically solved using the fourth-order Runge–Kutta method through MATLAB R2022a software.

The relationship between the SRH recombination lifetime and the threading dislocation (TD) density in silicon-based epitaxial QDLs can be written as follows [30]:

$${\displaystyle \frac{1}{{\tau }_{SRH}}}={\displaystyle \frac{1}{{\tau }_{SRH}^0}}+{\displaystyle \frac{D{\pi }^3\sigma }{4}}$$

(9)

where ${\tau }_{SRH}^0$ = 1877 ns is the lifetime of dislocation-free GaAs-based QDLs, and D = 221 cm² s⁻¹ is the average diffusion coefficient. σ is the TD density value, and its value is within the range of (10⁵~10⁸) cm⁻² [31], which is at least two orders of magnitude higher than in GaAs-based QDLs [26]. Therefore, the SRH recombination in silicon-based epitaxial QDLs is stronger than that in GaAs-based QDLs. The relationship between TD density and the SRH recombination lifetime is shown in Figure 2. As shown in this diagram, for the current attainable TD density of approximately (10⁵~10⁶) cm⁻², the corresponding SRH recombination lifetime is about 5~0.5 ns. In this case, the SRH recombination lifetime is equivalent to the spontaneous emission lifetime, and, therefore, the SRH recombination cannot be ignored. The other parameters utilized during the simulation are as follows: ${\tau }_{ES}^{RS}$ = 6.3 ps, ${\tau }_{GS}^{ES}$ = 2.9 ps, ${\tau }_{RS}^{ES}$ = 2.7 ns, ${\tau }_{ES}^{GS}$ = 10.4 ps, ${\tau }_{GS}^{spon}$ = 1.2 ns, ${\tau }_{ES}^{spon}$ = 0.5 ns, ${\tau }_{RS}^{spon}$ = 0.5 ns, S_inj = 3 × 10⁵, Γ_p = 0.06, n_r = 3.5, ${\alpha }_H^{GS}$ = 0.5, ${\alpha }_H^{ES}$ = 0.122, ${\alpha }_H^{RS}$ = 0.037, β_sp = 1.0 × 10⁻⁴, k_c = 10 × 10¹⁰ s⁻¹, τ_p = 4.1 ps, τ_SRH = 0.5~5 ns, a_GS = 5.0 × 10⁻¹⁵ cm², a_ES = 10.0 × 10⁻¹⁵ cm², a_RS = 2.5 × 10⁻¹⁵ cm², ε = 2.0 × 10⁻¹⁶ cm³, N_B = 1.0 × 10⁷, V_B = 5.0 × 10⁻¹¹ cm³, D_RS = 4.8 × 10⁶, V_RS = 1.0 × 10⁻¹¹ cm³ [32].

3. Results and Discussion

In this work, we consider the case that the SRH recombination lifetime (τ_SRH) of a silicon-based epitaxial quantum dot laser (QDL) is set at 0.5 ns, 1 ns, 5 ns, respectively. For the convenience of comparison, the case of ignoring SRH recombination is also considered. It should be pointed out that, when ignoring the SRH recombination, the last term in Equations (1)–(3) is absence, and the rate equations are transferred into those reported in Ref. [33]. Figure 3 illustrates the function of the GS photon number associated with the bias current under different values of τ_SRH for a silicon-based epitaxial QDL at free-running. From this diagram, it can be seen that, with the increase in τ_SRH from 0.5 ns to 5 ns, the threshold current decreases from 50 mA to 20 mA. When τ_SRH takes 5 ns, the variation trend of the output power is very close to that obtained when ignoring the SRH recombination. In other words, we can ignore the effect of the SRH recombination when τ_SRH is greater than the spontaneous emission lifetime. In the following, the bias current is set at 150 mA.

The above results show that the SRH recombination lifetime affects the output photon number under a given current. Next, we will analyze the influence of the SRH recombination lifetime on the dynamical state of silicon-based epitaxial QDL under optical injection. Figure 4 presents the time series, power spectra, and phase portraits of the silicon-based epitaxial QDL under optical injection with Δω_inj = −6.3 GHz for different injection strengths, where red and blue correspond to ignoring SRH recombination and τ_SRH = 0.5 ns, respectively. We firstly analyze the case that the effect of SRH recombination is ignored. For k_inj = 0.1 (Figure 4(a1–a3)), the time series of the silicon-based epitaxial QDL exhibits periodic oscillations, and the fundamental frequency can be obtained in the power spectrum, which is about 14.5 GHz. At this time, the trajectory of the phase diagram is characterized by a clear limit ring. Therefore, it can be determined that the silicon-based epitaxial QDL operates at a period-one (P1) state. For k_inj = 0.27 (Figure 4(b1–b3)), a periodic waveform with two peak intensities can be observed in the time series, and a subharmonic frequency of about 7.6 GHz appears in the power spectrum, and the corresponding phase diagram possesses two rings. As a result, the silicon-based epitaxial QDL exhibits a period-two (P2) state. For k_inj = 0.5 (Figure 4(c1–c3)), similar to the analysis method of Figure 4(b1–b3), the laser also operates at a P2 state. For k_inj = 1.1 (Figure 4(d1–d3)), multiple peaks with varying intensities emerge in the time series, new frequency components manifest in the power spectrum, and the phase diagram exhibits overlapping alternation of multiple loops. Hence, the dynamics of silicon-based epitaxial QDL can be characterized as a multi-period (MP) state. When k_inj = 1.3 (Figure 4(e1–e3)), the peak intensity of the time series shows irregular fluctuation; at this point, the power spectrum broadens, and the phase diagrams display strange attractors. With these characteristics, we can determine that the dynamical state of silicon-based epitaxial QDL can be classified as the chaos (C) state. For k_inj = 1.4 (Figure 4(f1–f3)), the dynamical state can be determined as MP, similar to Figure 4(d1–d3). For k_inj = 2 (Figure 4(g1–g3)), the time series output is stable and there are no obvious peaks in the power spectrum, and the corresponding phase portrait is a stable point. Under this case, the laser frequency of the silicon-based epitaxial QDL is exactly equal to the frequency of the injected light. As a result, the silicon-based epitaxial QDL operates in the injection locking (IL) state. In the same way, as the above analyses, the dynamical states can be distinguished under different k_inj when τ_SRH is equal to 0.5 ns. By comparing the dynamical states obtained under τ_SRH = 0.5 ns with those obtained under ignoring SRH recombination, it can be seen that for k_inj taken as 0.5, 1, 1.3, 1.4, the obtained dynamical states are different. Therefore, we could conclude that different τ_SRH may cause different dynamical states in the silicon-based epitaxial QDLs under optical injection even if other parameters are the same. It should be pointed out that the determination of the dynamical state is referred to in Ref. [34].

Figure 5 gives the bifurcation diagram of the power extreme peak series of the silicon-based epitaxial QDL under optical injection with Δω_inj = −6.3 GHz for τ_SRH taken as different values. For ignoring SRH recombination (as shown in Figure 5a), there exist two branches when k_inj is varied within the range of (0.02~0.25), and, therefore, it can be judged that the silicon-based epitaxial QDL is working at P1 oscillation. For k_inj located in the region of (0.25~1), there exist four branches, and it can be determined that the laser operates at P2 oscillation. For k_inj located in the range of (1~1.1), the silicon-based epitaxial QDL exhibits an MP state. For k_inj varied within (1.1~1.4), the silicon-based epitaxial QDL enters a C state due to coherent collapse. When k_inj is within (1.4~1.75), the silicon-based epitaxial QDL enters the MP state again. Under a stronger injection strength, the laser will be locked. The dynamical evolution route of P1-P2-MP-C-MP-IL is observed. Similarly, the dynamical evolution paths for τ_SRH taken with other values can be obtained. As shown in Figure 5b–d, the dynamical evolution paths are P1-P2-MP-C-MP-IL, P1-P2-C-P2-MP-C-MP-IL, and P1-P2-C-P2-MP-C-MP-C-MP-IL, respectively.

Figure 6 displays the bifurcation diagram of the power extreme of peak series for the silicon-based epitaxial QDL under optical injection with Δω_inj = −8 GHz for different τ_SRH. Via the same analysis as those in Figure 5, the dynamical evolution paths of Figure 6a–d are P1-P2-MP-C-MP-C-MP-IL, P1-P2-MP-C-MP-C-MP-IL, P1-P2-MP-P1-MP-IL, and P1-P2-MP-P2-MP-P2-MP-P1-MP-IL, respectively. Combining Figure 5 with Figure 6, it can be concluded that the laser has different evolution paths for different frequency detuning. However, some common characteristics can be observed. When τ_SRH is much larger than the spontaneous emission lifetime, the bifurcation diagram is similar to that obtained when ignoring the SRH recombination. When τ_SRH is less than or comparable to the spontaneous emission lifetime, the bifurcation diagram is different from that obtained when ignoring SRH recombination. Under this case, the SRH recombination cannot be not ignored.

The above result is based on the conditions of Δω_inj = −6.3 GHz and Δω_inj = −8 GHz. In order to systematically analyze the characteristics of silicon-based epitaxial QDL under optical injection, the distribution of the dynamical states of the laser in the parameter space of frequency detuning and injection strength are given in Figure 7, where the red dashed line indicates a frequency detuning of −6.3 GHz and the white dashed line indicates −8 GHz. As shown in this diagram, in the positive frequency detuning region, the C state is almost invisible, and the laser is directly transited from the P1 state to the IL state with the increase in k_inj, and meanwhile, the dynamical states are much richer in the negative frequency detuning region. For τ_SRH = 5 ns (Figure 7b), the distribution of the dynamical state is almost the same as that obtained when ignoring the SRH recombination (Figure 7a). Therefore, the SRH recombination can be neglected when τ_SRH is much larger than the spontaneous emission lifetime. When τ_SRH and the spontaneous emission lifetime are comparable (Figure 7c,d), with the decrease in τ_SRH, the region for IL state shrinks, and meanwhile, the regions for C state and P2 state expand.

4. Conclusions

In summary, we have theoretically studied the nonlinear dynamical characteristics of a silicon-based epitaxial quantum dot laser (QDL) under optical injection after taking the influence of Shockley–Read–Hall (SRH) recombination into account. The results indicate that, for the silicon-based epitaxial QDL operating at free-running, its threshold increases with the decrease in the SRH recombination lifetime (τ_SRH). After introducing optical injection with different parameters, the silicon-based epitaxial QDL can exhibit a series of nonlinear dynamical states including period-one (P1), period-two (P2), multi-period (MP), chaos (C), and injection locking (IL). Through comparing the nonlinear dynamical state distributions of a silicon-based epitaxial QDL in the parameter space of injection strength and frequency detuning under different τ_SRH values, the influence of SRH recombination on the state of the laser is revealed. The results demonstrate that ignoring the SRH recombination may lead to an incorrect estimation of the laser state when τ_SRH is comparable to the spontaneous emission lifetime.