1. Introduction

Silicon is a promising candidate for photonic integrated circuits due to its low power consumption, low cost, and high integration density [1]. Additionally, silicon on an insulator provides a high index contrast platform [2,3] that enables the fabrication of extremely compact passive optical devices, such as low loss waveguides, [4,5] grating couplers, [6,7] mode converters, [8] multiplexers, [9] and more. However, silicon lacks active optical properties due to its indirect bandgap, which has impeded the development of silicon-based photonic integrated circuits. As an alternative, direct bandgap III-V semiconductors (e.g., GaAs) offer significant active optical properties such as light emission and absorption at telecom wavelengths. In particular, III-V nanophotonic devices can implement low threshold lasers, [10] amplifiers, [11] modulators, [12] and highly nonlinear devices [13]. The integration of III-V nanophotonic devices with silicon could pave the way towards ultra-compact low energy nanophotonic and optoelectronic devices. Researchers have increasingly explored the heterogeneous integration of III-V materials with silicon to utilize the advantages of both systems [2,14].

A commonly used method for this heterogeneous integration is achieved by placing a III-V nanophotonic device in close proximity to a silicon waveguide and relying on evanescent coupling between the two structures. Based on this technique, several nanophotonic designs (e.g., nanolasers) have been proposed and demonstrated [15–19]. However, this approach requires careful placement and alignment of the nanophotonic cavity patterned in the III-V material with the evanescent field of the waveguide to achieve efficient coupling. Such active alignment can significantly complicate the design and fabrication of integrated photonic devices [20]. An alternative approach is the use of hybrid photonic modes to couple silicon and III-V semiconductors without the need for precise alignment. In this approach, the mode of a silicon waveguide is carefully engineered to hybridize with a III-V semiconductor layer that serves as the active material [21]. Such hybrid modes have been used to engineer laser sources, [21,22] amplifiers, [23] and modulators [24]. However, their use in other nanophotonic devices, such as hybrid cavities, remains unexplored.

In this letter, we propose and analyze a hybrid mode approach for engineering a Si-GaAs nanophotonic cavity that enables low-power-threshold lasing and optical bistability at telecom wavelengths without the need for careful alignment. Unlike previous hybrid integration approaches, we utilize a patterned silicon cavity structure coupled to a wider, symmetric GaAs slab embedded with InAs quantum dots. This approach alleviates the need for careful alignment between the silicon photonic waveguide and III-V semiconductor material. We numerically design and characterize the structure using the 3D finite-difference time-domain (FDTD) method coupled to a Maxwell-Bloch equation model that accounts for the active material in the GaAs slab [25,26]. The proposed design provides a low lasing threshold of just 156 nW, which we attribute to the high spontaneous emission coupling ratio enabled by the low mode volume of the active media in the hybrid cavity [10]. Additionally, under continuous wave operation, we found this hybrid cavity has a low optical bistability threshold of 18.1 $\mu$W. This hybrid Si-GaAs cavity design can be optimized for different operating frequencies and integration with other active materials beyond quantum dots, such as Kerr nonlinear media [27]. This work could enable a simpler and more scalable approach for the heterogeneous integration of III-V materials with silicon nanophotonics.

2. Proposed hybrid Si-GaAs cavity design

Figure 1(a) illustrates the 3D structure of the hybrid device, which features a wide silicon waveguide with a 920 nm width that couples light and narrows down through a 3.25 $\mu$m linear taper. The narrower silicon waveguide with the cavity pattern couples to the GaAs slab, forming the hybrid nanocavity region. An additional linear taper enables efficient light transmission from the cavity through the wide silicon waveguide on the other side of the device. Figure 2(b) shows the cross-section of the hybrid structure without the cavity pattern, which consists of a silicon waveguide coupled to a GaAs slab containing InAs quantum dots. The waveguide is embedded in a SiO$_2$ cladding, and a spacer layer with thickness t separates it from the GaAs slab. We define the silicon waveguide’s height and width are $H_{Si}$ and ${W}_{Si}$, respectively, while the height of the GaAs slab is ${H}_{GaAs}$.

 

Fig. 1. (a) 3D schematic illustration of the Si-GaAs hybrid cavity, and (b) the corresponding 2D cross-section. (c) The transverse mode profile ($|E|$) of the hybrid mode (TE polarized) for different silicon waveguide widths ($W_{Si}$) of 200 nm, 580 nm, and 1200 nm at a fixed GaAs slab width of 2 $\mathrm{\mu}$m at an incident light wavelength of 1516.53 nm. The solid white lines represent the boundaries of the silicon waveguide and GaAs slab. (d) The silicon waveguide width controls the relative confinement of the hybrid mode in the silicon waveguide and GaAs slab region, where positions I, II, and III correspond to the different silicon waveguide widths marked in (c).

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We first analyze the hybrid mode structure of the silicon waveguide without a patterned cavity coupled to the GaAs slab. We simulate the mode structure of the waveguide by direct eigenmode expansion (ANSYS MODE solutions) for the case where $t$ = 60 nm, $H_{Si}$ = 220 nm [28] and $H_{GaAs}$ = 200 nm [13,29] Fig. 1(c) shows the transverse mode profile of the waveguide modes for different values of $W_{Si}$ (200 nm, 580, and 1200 nm). Consistent with previous work on hybrid modes, [2] as the width of the waveguide increases from 200 nm to 1200 nm, the mode continuously shifts from being predominantly confined in the GaAs slab to being confined in the silicon waveguide. Figure 1(d) shows the confinement factor in the silicon or GaAs region, defined as $\frac {\int _{Si~or~GaAs}\left |E\right |^2dA}{\int _{total}\left |E\right |^2dA}$, which shows the gradual shift between the two materials. The three vertical dashed lines labeled I, II, and III, correspond to the widths plotted in Fig. 1(c).

Without the GaAs layer, this device structure would lead to a highly localized mode within the silicon waveguide. But adding the GaAs layer causes the mode to hybridize. The key to designing a cavity mode for the hybrid Si-GaAs device is to carefully select the width $W_{Si}$. If this width is too large, most of the mode will be confined in the silicon, leading to strong optical confinement but poor overlap with the GaAs active material. In contrast, if $W_{Si}$ is smaller, the majority of the hybrid mode will be confined to the GaAs active material where it can experience large gain and nonlinearity. But in this latter case, the effective index contrast of the hybrid Si-GaAs photonic crystal will be low, which will result in poor optical confinement and low quality factors. To balance these conflicting requirements, we set the width of the waveguide to be 580 nm, as it enables a hybrid mode that is nearly equally confined in both the silicon and GaAs (label II in Fig. 1(c), (d)).

Next, to design the cavity, we first analyze a periodic photonic crystal in the silicon waveguide, specifically the mirror region of Fig. 2(a). We choose an ellipsoidal cavity for the mirror with a lattice period $a$ of 380 nm and radii of 140 nm and 130 nm for the major and minor ellipsoidal axes, respectively. We then calculate the photonic band structure by performing three-dimensional Finite Difference Time Domain simulations (ANSYS FDTD solution) with Bloch boundary conditions. Figure 2(b) shows the photonic band structure, which features a sufficiently wide photonic bandgap of 8.5 THz that spans within a telecom range of 192 THz to 200.5 THz (grey-shaded region).

 

Fig. 2. (a) Schematic diagram of the silicon waveguide cavity design, which features a total of 16 cavity pairs, including $N = 11$ ellipsoidal cavities on both sides and 5 centered pairs of tapered ellipsoids as defects. (b) The FDTD calculated band structure of the hybrid Si-GaAs device. The grey shaded area is the photonic bandgap (PBG), which corresponds to the difference between the calculated air bands (red curves) and dielectric bands of the silicon and GaAs (blue curves marked I and II, respectively). Here, $k_x$ is related to wavevector $k = k_x (2\pi /a)$, which is a function of the reciprocal lattice parameter $a$ specified by the Bloch boundary condition. The black line denotes the SiO$_2$ light line, and the horizontal black dashed line in PBG indicates the cavity resonance frequency at 197.68 THz. (c) The fundamental mode field (|E|) profile in the hybrid Si-GaAs cavity obtained using the 3D FDTD method. The white dashed lines denote the silicon and GaAs layers (220 nm and 200 nm thick, respectively), which are separated by a 60 nm spacer.

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In order to create a cavity mode, we start with a photonic crystal waveguide featuring 16 pairs of ellipses to which we introduce a defect. As shown in Fig. 2(a), the defect is composed of a taper where the major and minor axes of the five centered pairs of ellipses linearly increase from 140 nm to 150 nm and 130 nm to 140 nm, respectively, toward the center of the waveguide. This design results in a cavity resonance wavelength of 1516.53 nm, which is depicted as the black dashed line in the photonic bandgap plot in Fig. 2(b). Figure 2(c) plots the electric field profile $|E|$ of this hybrid Si-GaAs cavity design at the resonance wavelength, which reveals a fundamental cavity mode that extends in both the silicon waveguide and GaAs slab. The mode is highly localized in the cavity region and has a substantial overlap with both the silicon and GaAs, thus achieving a hybrid localized photonic crystal cavity mode. From the 3D-FDTD simulations, we calculate the mode volume to be $V_m \approx 2 (\lambda /n)3$ and the quality factor to be $Q=1.53\times {10}^4$. Based on this quality factor and mode volume, the theoretically predicted maximum Purcell factor is 794.17 (Supplement 1), which suggests the hybrid cavity design is a promising candidate for the enhancement of spontaneous emission [30]. This hybrid cavity mode should facilitate the interaction with the quantum dots embedded in the GaAs, which can be used to achieve nanolasing and optical bistability in silicon, as we discuss in the following sections.

3. Lasing

To numerically simulate lasing from the hybrid cavity, we incorporate four layers of InAs quantum dots at a planar density of 4$\times 10^{10}/cm^2$, with each layer separated by 50 nm intervals in the GaAs slab [31]. The quantum dots can be described as a gain material using a four-level atom model [25,26] (Supplement 1). To investigate the lasing capabilities of the hybrid Si-GaAs cavity, we use 3D-FDTD to calculate the cavity output power as a function of the input pump power (L-L curve) using a pump-probe technique. In our numerical simulation, we employ a 750 nm wavelength pump to excite the active material with a pulse width of 4.5 ps and a broadband probe of two orders of magnitude smaller than the pump amplitude centered at the hybrid cavity resonance wavelength of 1516.53 nm to observe the lasing. Figure 3 plots the resulting FDTD calculations of the hybrid cavity’s output lasing power (black circles), which we then fit using the cavity laser rate-equation model (Supplement 1) given by [32]

 

Fig. 3. Using 3D FDTD, we calculate the input power versus output power (log-log L-L curve) of the hybrid Si-GaAs cavity nanolaser (black circles) and fit the results using a rate equation model (Eq. (1)) as shown by the orange curve, which corresponds to a spontaneous emission coupling ratio of $\beta = 0.65$. The blue and yellow solid curves are also plotted using the rate equation with spontaneous emission coupling ratios of $\beta = 0.3$ and 1, respectively. The vertical dashed line is the lasing threshold obtained from the rate equation fit to the FDTD data.

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Here, $P_{in}$ is the input pump power, $\omega$ is the cavity resonance frequency, $\gamma = \omega /Q$ is the cavity decay rate, and $\beta$ is the spontaneous emission coupling ratio. We define the cavity photon number as $p = P_{out}/\hbar \omega \gamma \eta _{out}$, where $\eta _{out}$ is the output collection efficiency of the laser. Additionally, $\eta _{in}$ is the pumping efficiency, and $\zeta$ is the cavity photon number at transparency.

From Eq. (1), we calculate several log-log L-L curves (solid curves in Fig. 3) corresponding to different spontaneous emission coupling ratios (0.3, 0.65, and 1) and find the FDTD lasing data is best fit by a coupling ratio of 0.65 (orange curve). We then calculate the lasing threshold power from the condition when there is an average of one photon in the cavity ($p=1$). Using this value of p in Eq. (1), we obtain a threshold power of $P_{th} = 156$ nW, which is shown by the vertical dashed line in Fig. 3. We also observe an increase in the slope of the log-log curve near the lasing threshold in the FDTD data (black circles), which is a hallmark feature of lasing [33,34]. Our simulation demonstrates a lower lasing threshold compared to other devices using hybrid Si-III/V materials, [15–19] which have previously reported a lasing threshold power as low as 1 $\mu$W [19].

4. Bistability

Optical bistability, a nonlinear optical phenomenon, can be defined as an optical system possessing two different output states corresponding to the same input intensity. Such bistability has been demonstrated for a wide range of applications, including all-optical switches, [35] memories, [36] optical logic gates, [37] and optical transistors [38]. Our hybrid cavity can operate as a low-power bistable device by employing optical nonlinearity such as the saturable absorption of quantum dots. As saturable absorption does not require a gain material, we can simulate the bistability response of the quantum dots in the hybrid Si-GaAs cavity using a two-level one electron numerical model [39,40], which also enables faster device simulation.

To simulate the optical response of the InAs quantum dots in the GaAs region of the hybrid cavity, we incorporate the Maxwell-Bloch equations into numerical FDTD simulations using the method described by Shih-Hui Chang and Allen Taflove [25]. The Maxwell-Bloch equations describe the response of saturable absorbers to an incident electric field.

Here, $N_{11}$ and $N_{22}$ are the population densities of the two-level atoms for the ground and excited states, $\Omega = \mu E/\hbar$ is the optical Rabi frequency, and $\mu$ is the transition matrix elements of the two-level atom. We define the detuning frequency as $\Delta = \omega _0 - \omega$, while $\omega _0$ and $\omega$ are the resonant frequency and incident light frequency. We also describe the atomic decay rate as $\gamma ={\gamma _{non}+\gamma }_{rad}$, where $\gamma _{non}$ is the nonradiative decay rate and $\gamma _{rad}=\frac {\mu ^2\omega ^3}{3\pi \hbar \epsilon _0\left (1+\chi _D\right )c^3}$ is the radiative decay rate. Finally, $\beta =\gamma /2\ +1/T_2$, while $T_2$ is the dipole dephasing time.

For the simulation parameters, we set a layer of InAs quantum dots in the GaAs region of the hybrid cavity at a planar quantum dot density of $N =7\times 10^{10} /cm^2$ [41]. We set the decay rates of the quantum dots to their room temperature values of $\gamma _{rad}$ = 1 GHz [42] and $\gamma _{non}$ = 1 GHz, [43] and the dephasing time to $T_2 = 300$ fs [44]. In order to observe the bistability effect in the FDTD simulations, we specify a continuous wave (CW) input source at the cavity resonant wavelength of 1516.53 nm at one end of the hybrid cavity. Here, we define the source input power as a function of time such that the input power increases and decreases for the same period [47]. Finally, we calculate the output power at the other end of the hybrid cavity to observe the difference in the nonlinear behavior for increasing and decreasing source powers.

Figure 4 plots the simulated optical response of the saturable absorbers in the hybrid Si-GaAs cavity, which demonstrates hysteresis in the output power depending on whether the input power increases or decreases (blue and red curves, respectively). This behavior indicates bistability, in which the cavity exhibits two stable states of operation. The FDTD simulations also demonstrate good agreement with the theoretical bistability curve (black data points in Fig. 4), where the output power is a bistable function of the cavity input at a steady state [45,46] (Supplementary Note 3). The threshold power of the optical bistability corresponds to the transition of the output power between the two stable states in the hysteresis loop, which occurs at 18.1 $\mu$W (vertical purple dashed line). Our design provides a comparable threshold as observed in silicon photonic crystal cavities [36,48,49] and ring cavities [50–52]. The low bistability threshold of this hybrid cavity design is promising for various silicon photonic applications at telecom wavelengths requiring low-power optical nonlinearity.

 

Fig. 4. Output power vs. input power at the resonance wavelength 1516.53 nm of the hybrid Si-GaAs cavity, featuring quantum dots as saturable absorbers in the GaAs slab. The blue and red curves are the forward and reverse sweeps of the input power, respectively, while the black circles represent the theoretical bistability curve. The purple dashed line is the bistability threshold.

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5. Conclusion

In summary, we have shown that low power threshold lasing and optical bistability can be achieved using a new Si-GaAs hybrid cavity design. In this approach, the cavity is patterned in the silicon waveguide while the GaAs slab features quantum dots that are optically active at telecom wavelengths. This hybrid device does not require careful alignment and could be fabricated using the simple pick-and-place transfer printing method [53,54]. Moreover, the integration of quantum dots (QDs) within a hybrid cavity in III-V semiconductor materials holds great promise in quantum photonics. This technique could enable large-scale efficient integration of quantum emitters onto a silicon chip, offering exciting prospects for on-chip quantum applications such as quantum communication, computing, and information processing. Additionally, the device can operate at different frequencies by changing the cavity design parameters, with optimization enabling further improved performance. Our work represents a critical step toward the efficient heterogeneous integration of silicon and III-V materials for future low-power silicon nanophotonics.