1. Introduction

Nanophotonics is a sought-after approach to high-data-rate transmission and processing applications [1–7]. Integrated photonics potentially offers lower power consumption and higher operation speed than their electronic counterparts [8–11]. However, to achieve full-scale integrated-photonics processing, optical components (e.g., switches and modulators) must have a small footprint and a large optical bandwidth (BW). Satisfying these conditions simultaneously requires strong light–matter interaction at the nanoscale. Two main approaches toward integrated nanophotonics are hybrid electronic–photonic [9,10] and all-optical systems [8,12–15].

Hybrid electronic–photonic devices have potentially low cost and broad BW [10]. Silicon-based hybrid devices have the extra advantage of high compatibility with CMOS technology. However, the hybrid electronic–photonic circuits have two key disadvantages. First, metallic interconnects as well as electrical-to-optical conversion cause high power consumption at high bit rates [4]. Second, owing to the capacitive nature of modulation in these circuits, the modulation speed and, consequently, the whole circuit’s operation speed are limited to a few GHz [10]. Unlike hybrid electronic–photonic circuits, all-optical circuits rely on nonlinear optical effects for light control and modulation [16–18]. By unloading metallic interconnects and capacitive modulation, all-optical processing can potentially acquire much faster modulation speed and lower power consumption. However, the realization of all-optical processing is contingent upon materials with sufficiently large optical nonlinearity.

Recently, we have shown that, in the long-wavelength regime, $\sim$10-nm-wide graphene nanoribbons (GNRs) have a remarkably large optical nonlinearity in the telecom frequency range (0.75–0.95 eV or, equivalently, 1.3–1.65 $\mu$m) [19]. The third-order Kerr susceptibility of $\sim$10-nm-wide GNRs can be as large as $10^{-10} \textrm {m}^2\textrm {V}^{-2}$, which translates to a nonlinear refractive index of $10^{-8} \textrm {m}^2 \textrm {W}^{-1}$, significantly larger than the nonlinear refractive indices of silicon ($\sim 10^{-18} \textrm {m}^2 \textrm {W}^{-1}$) and graphene ($\sim 10^{-11} \textrm {m}^2 \textrm {W}^{-1}$) in the same frequency ranges [8]. The GNRs’ large Kerr nonlinearity results in their strongly field-dependent susceptibility as well as refractive index and, consequently, makes them a suitable core material for self-amplitude-modulation (SAM) in ultrafast pulse generation applications [9,20–30]. The all-optical SAM potentially provides much stronger pulse-shaping capability than any active-electronic modulation [24]. For such applications, the SAM device must have high absorption at low optical intensity and low absorption at high optical intensity. This behavior is known as saturable absorption and is associated with strong Kerr nonlinear effects. However, in the long-wavelength regime, unlike in the plasmonic regime [31–33], the poor optical confinement significantly diminishes the nonlinear optical effect. One way to increase the optical concentration in the vicinity of GNRs is by embedding GNRs in on-chip semiconductor waveguides.

In this paper, we design, numerically model, and simulate a dielectric waveguide with embedded GNRs for all-optical SAM applications. Owing to the strongly field-dependent susceptibility of GNRs, the proposed design has a broad BW and a very compact footprint. First, we characterize the GNR optical nonlinearity via the density matrix theoretical formalism that carefully accounts for material-specific properties and scattering mechanisms [19,34,35]. Then, we design a vdW heterostructure comprising undoped GNRs and hexagonal boron nitride (hBN) sheets. We embed the GNR-hBN vdW heterostructure (GBNH) in a rib silicon waveguide and optimize it for the maximum optical concentration near the GBNH. Given the advances in the seed-initiated parallel-GNR synthesis [36–38] and multi-layer graphene/hBN vdW heterostructures fabrication [39–41], the GBNH employed here are well within current experimental capabilities. Encasing in hBN enables the stacking of GNRs with high density without changing the individual GNR’s electronic or optical properties. hBN also protects undoped GNRs from unintentional doping [42]. We show that implanting different-width GNRs in the GBNH broadens the saturable absorption BW. Also, by increasing the number of GNR layers in the GBNH, the modulation strength rises up to 0.03 dB$\mu$m$^{-1}$ over the telecom frequency range, which removes the need for dynamical tuning. For narrowband applications, an appropriately engineered GBNH yields even larger modulation depth, as large as 0.7 dB$\mu$m$^{-1}$. The combined advantages of small footprint, low power consumption, large modulation depth, and broad BW underscore the capability of the GBNH-based dielectric waveguide as a saturable absorber for ultrafast pulse generation applications [9].

2. Characterization of the GNR optical nonlinearity

In the telecom frequency range, the GNR optical response is dominated by intersubband optical transitions, because the photons have enough energy to mediate the intersubband electronic transitions. Moreover, due to relatively weak screening in GNRs as quasi-one-dimensional systems, electron scattering has a critical effect on their optical response. Given these, calculating the optical nonlinearity of GNRs requires a full-quantum mechanical approach. We use the self-consistent-field approximation within the Markovian master-equation formalism (SCF-MMEF) [19,34,35] to calculate undoped GNRs’ optical nonlinearity. (While graphene and GNRs are notoriously sensitive to surface transfer doping, hBN appears to have a stabilizing effect on carrier density [42]. Therefore, we assume that undoped GNRs will indeed remain nearly undoped within GBNH). Here, we focus on armchair graphene nanoribbons (aGNRs), because they demonstrate stronger optical nonlinearity than the zigzag graphene nanoribbons [19]. For the details of the calculations, see Methods.

Based on the number of carbon pairs, so-called dimers, oriented along the ribbon in a unit cell, aGNRs are categorized into three families: $3N$, $3N + 1$, and $3N + 2$, with $N$ being an integer. ($3N + 2$)-aGNRs are semi-metallic, unlike the other two family that are semiconducting. Figure 1 shows the nonlinear refractive indices ($n_2 + ik_2$) of 8- to 12-nm-wide aGNRs of different families at room temperature. The nonlinear refractive indices are very similar and comparable across the three families. The aGNR nonlinear refractive indices are as large as $10^{-8} \textrm {m}^2 \textrm {W}^{-1}$, considerably larger than the nonlinear refractive indices of silicon ($\sim 10^{-18} \textrm {m}^2 \textrm {W}^{-1}$) and graphene ($\sim 10^{-11} \textrm {m}^2 \textrm {W}^{-1}$) in the same frequency range [8]. $n_2$ and $k_2$ peak at the resonant frequencies corresponding to intersubband electronic transitions. These intersubband resonances are critically dependent on the aGNR width. Since dynamical tuning of the aGNR width is impractical, we incorporate different-width aGNRs in a vdW heterostructure to broaden the ensemble optical nonlinearity.

 

Fig. 1. The real part ($n_2$) and imaginary part ($k_2$) of the nonlinear refractive indices of undoped aGNRs. The width of aGNRs is in the range of 8–12 nm. The numbers denote the number of dimers in the aGNR’s unit cell. The aGNRs are assumed to be sandwiched between two hBN films. The nonlinear refractive index of aGNRs can be as large as $10^{-8} \textrm {m}^2 \textrm {W}^{-1}$.

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3. Design and optimization

Here, we design a vdW heterostructure comprising hBN and aGNR layers, so-called GNR-hBN vdW heterostructure (GBNH) [Fig. 2(a)]. The average width of the aGNRs ($W$) and the average distance between two adjacent aGNRs are chosen to be $10$ nm. We note that these aGNR widths are well within modern fabrication capabilities of GNRs. The GBNH is made up of a combined 100 layers. Given the equal thickness of aGNRs and hBN monolayer ($\sim 0.7$ nm), the GBNH is almost 35 nm thick. The waveguide’s characteristics, namely propagation modes, have a very low sensitivity to the exact distribution of aGNRs in the GBNH. Because the GBNH cross-section area is a tiny fraction of the effective mode area, the waveguide’s characteristic is governed by the effective, average optical behavior of the GBNH (see Methods for details). There are two key parameters determining the GBNH effective optical behavior: (a) $\rho$, the percentage of aGNR layers in the GBNH and (b) $\delta W$, the maximum difference of an aGNR’s width from the average width ($W$). For a given $\delta W$, the width of the aGNRs in the GBNH is in the range of $W-\delta W$ to $W+\delta W$. For instance, a GBNH with $\delta W = 0.2$ nm comprises only three aGNRs (80-, 81-, and 82-aGNRs). The seed-initiated parallel-GNRs synthesis yields accurate edge shapes yet does not acquire a narrowly limited growth rate [36,37]. Therefore, having a nonzero $\delta W$ is the very realistic approach to capture the width variation. The aGNRs’ widths have a normal-like distribution [36], which, for simplicity, we approximate with a uniform distribution in the $[W-\delta W, W+\delta W]$ interval. Given the advances in the seed-initiated parallel-GNR synthesis [36–38] and the multi-layer graphene/hBN vdW heterostructures fabrication [39–41], the fabrication of GBNH is well within current experimental capabilities.

 

Fig. 2. (a) Schematic of a GBNH-embedded rib silicon waveguide. The zoomed-in heterostructure consists of graphene nanoribbons (gray) encased in hBN (pink). (b) The cross-section view of the waveguide. (c) $E_{\textrm {GBNH}}/E_{\textrm {max}}$ as a function of the silicon strip thickness, $T_s$. $E_{\textrm {GBNH}}$ denotes the electric field at the center of the GBNH and $E_{\textrm {max}}$ is the maximal electric field across the device. The closer the ratio is to unity, the more concentrated the optical field is near the GBNH. $T_s \approx 45$ nm yields optimal optical confinement. The insets show the electric-field profile across the waveguide for narrow ($T_s = 20$ nm), optimal ($T_s = 45$ nm), and thick ($T_s = 100$ nm) silicon strips at 950 meV.

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Next, we embed the GBNH in a rib silicon waveguide. The structure of the modulator is schematically illustrated in Fig. 2(a). Figure 2(b) shows the cross-section view of the device with air cladding ($\varepsilon _r = 1.0$). The GBNH is 400-nm wide and 35-nm thick, sitting on a 300-nm-thick SiO$_2$ layer ($\varepsilon _r = 3.9$). The silicon strip width is 600 nm and is connected to a 50-nm-thick silicon layer ($\varepsilon _r = 11.7$). We have chosen typical widths used previously for similar structures [10] and do not optimize for width here. The thickness of the silicon strip ($T_s$) is of critical importance for optical confinement. The design objective is maximizing the optical concentration in the vicinity of the GBNH. A figure of merit to quantify the optical concentration is the ratio between the electric field at the center of the GBNH ($E_{\textrm {GBNH}}$) and the maximum electric field in the waveguide ($E_{\textrm {max}}$). The closer the $E_{\textrm {GBNH}}/E_{\textrm {max}}$ is to unity, the more concentrated the optical field is near the GBNH. Figure 2(c) shows $E_{\textrm {GBNH}}/E_{\textrm {max}}$ as a function of $T_s$ for the outermost frequencies of the telecom frequency range. The optimal value of $T_s$ is almost 45 nm. For the details of the calculations, see Methods.

4. Results and discussion

We begin by studying the effect of $\delta W$ on the waveguide characteristics, namely the TM propagation modes and their corresponding propagation constants ($\beta _r + i\beta _i$). In the GBNH-based waveguides, the phase constant ($\beta _r$) weakly changes with the optical intensity, which results in poor phase modulation. Therefore, we only focus on amplitude modulation. Figure 3(a) shows the attenuation constant ($\beta _i$) as a function of optical intensity. The solid curves show $\beta _i$ averaged over the telecom frequency range and the shaded areas represent $\beta _i$ calculated for the frequencies in the range of 0.75–0.95 eV. The attenuation constant and consequently, absorption, decrease with increasing optical density – a signature of saturable absorption. Moreover, for a fixed percentage of aGNR layers (i.e., fixed $\rho$), larger $\delta W$ results in considerably smaller variation of $\beta _i$ as a function of frequency. To better understand the latter effect of $\delta W$, we calculate the modulation depth, $\alpha (I) \equiv 20\log [\beta _i(0)-\beta _i(I)]$, as a function of frequency [Fig. 3(b)]. The optical intensity is chosen to be $7\times 10^{7}$ Wm$^{-2}$, which is at least 20 times smaller than the minimum saturation optical intensity. For low $\delta W$, the modulation depth vastly varies with frequency, with high values of 0.2 dB$\mu$m$^{-1}$. On the other hand, increasing $\delta W$ flattens the modulation depth and broadens the modulation depth over the whole telecom frequency range. It should be emphasized that the broadband response is self-sustaining and does not require any dynamical tuning.

 

Fig. 3. (a) The average attenuation constant as a function of normalized optical intensity. The shaded area represents the range of attenuation constants calculated for the frequencies $0.75-0.95$ eV. (Within the shaded area, the curves associated with a single frequency are nonmonotonic.) (b) The modulation depth as a function of frequency for three values of $\delta W$. Increasing $\delta W$ flattens the modulation depth and broadens the BW. (c) The saturation optical intensity as a function of frequency and $\delta W$. For large $\delta W$, the saturation optical intensity varies less with frequency. (d) The modulation depth as a function of frequency for three values of $\rho$. Increasing $\rho$ enhances the modulation strength. For $\rho = 20\%$, a self-sustaining broadband modulation depth of at least $\sim$ 0.03 dB$\mu$m$^{-1}$ is achieved over the frequency range.

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To further investigate the role of $\delta W$ on the saturable absorption’s BW broadening, we calculate the waveguide’s saturation optical intensity as a function of frequency and $\delta W$ [Fig. 3(c)]. For small $\delta W$, the saturation optical intensity drastically changes with frequency. With increasing $\delta W$, the variation range of $I_{\textrm {sat}}$ becomes smaller. For instance, a GBNH with $\delta W = 0.2$ nm and a GBNH with $\delta W = 2$ nm have comparable average saturation intensities ($\sim 10^9$ Wm$^{-2}$), but, in the latter, the variation range of $I_{\textrm {sat}}$ is 5 times smaller.

Unlike $\delta W$, $\rho$ (the percentage of GNR layers in the GBNH) negligibly change the waveguide’s BW. Figure 3(d) shows the modulation depth as a function frequency for different values of $\rho$. In Fig. 3(d), the modulation depths are calculated for fixed $\delta W$ and optical intensity. Increasing $\rho$ merely results in an upshift of $\alpha$ and leaves the BW intact. For $\delta W$= 2 nm and $\rho$= 20%, the modulation depth is at least 0.03 dB $\mu$m$^{-1}$ (the minimal value across the range). This means that, to achieve a 3 dB contrast, the device length should be 100 $\mu$m and the required optical density would be $\sim$10 kW cm$^{-2}$, which is worth comparing with the operational optical intensity of graphene-based all-optical modulators ($\sim$ 10$^{6}$ kWcm$^{-2}$). For narrowband applications, the footprint of the modulator can be even more compact. At a given frequency, the GBNH can be engineered to achieve the modulation depth as large as 0.7 dB$\mu$m$^{-1}$. This modulation depth translates to the device length of $\sim$5 $\mu$m, which is much shorter than $\sim 140 \mu$m, the length of corresponding graphene-based all-optical modulators [8].

5. Methods

To calculate the linear susceptibility $\chi ^{(1)}$, third-order Kerr susceptibility $\chi ^{(3)}$, linear refractive index ($n_0+ik_0$) and nonlinear refractive index ($n_2+ik_2$) in response to a TM-polarized illumination, we base our analysis on the self-consistent field approximation within the Markovian master-equation formalism, SCF-MMEF [19,34,35]. (The electronic system strongly couples only with the electric field component along the ribbon [10], which is why we ignore TE modes.) This method accurately captures the the intersubband optical transitions as well as the electron scattering via concurrent competing mechanisms. The aGNRs are assumed to be sandwiched between two hBN films. For the SCF-MMEF, we calculate the band structure of hydrogen-passivated aGNRs via the third-nearest-neighbor tight-binding method. In order to capture the effect of bond shortening near the edges of aGNRs, we accordingly modify the edge-bond lengths. Our calculations yield an excellent agreement with the results of the ab initio calculations [43]. In the SCF-MMEF, we include electron scattering via intrinsic phonons, ionized impurities, surface-optical (SO) phonons, and line-edge roughness (LER). Although the seed-initiated synthesis of aGNRs yield precise edge shape, we assumed an exponential correlation function for the aGNR LER, with the conservative values of 1 nm for the rms roughness and 3 nm for the correlation length. The relevant parameters for the electron scattering via acoustic phonons, longitudinal-optical phonons, ionized impurities, LER, and SO phonons are provided in Ref. [34]. Knowing the linear and third-order Kerr susceptibility, the aGNR linear and nonlinear refractive indices are given by [31,44]:

To characterize the GBNH, first, we need to calculate the aGNR field-dependent optical susceptibility, $\chi ^{}_{\textrm {G}}(E)$, with $E$ denoting the electric field. Using the well-known saturation model [16,45], $\chi ^{}_{\textrm {G}}(E) = \chi ^{(1)}/[1+(E^2/E_{\textrm {sat}}^2)]$, with $E_{\textrm {sat}}$ denoting the saturation electric field. The saturation electric field can be readily obtained from $E_{\textrm {sat}}^2 = {\big (} \frac {1}{2}\varepsilon _0\varepsilon _r v_p{\big )}^{-1} I_{\textrm {sat}}$, with $I_{\textrm {sat}} = -k_0/(2k_2)$ and $v_p$ and $\varepsilon _r$ being the phase velocity and effective relative permittivity, respectively [8,16,24]. By investigating the expressions for $k_0$ and $k_2$, it can be seen that, unlike $I_{\textrm {sat}}$ which is proportional to $\varepsilon _r v_p$, $E_{\textrm {sat}}$ is solely dependent on the GNR’s optical response, i.e., $\chi ^{(1)}$ and $\chi ^{(3)}$. The saturation model is perfectly accurate in the $E^2 \ll E_{\textrm {sat}}^2$ (or equivalently, $I \ll I_{\textrm {sat}}$) limit, in which two-photon absorption (TPA) and non-saturable absorption are completely negligible.

Knowing the aGNR field-dependent optical susceptibility, we calculate the effective susceptibility of the GBNH:

Finally, obtaining the GBNH effective susceptibility from Eq. (2) and using the bulk susceptibility for other materials, we employ the finite-element method to calculate the TM propagation modes, and their corresponding propagation constant and electric field profile across the device. The calculated effective mode area ($A_{\textrm {eff}}$) is $\sim$0.3 $\mu$m$^2$. The cross-section area of the GBNH is 0.014 $\mu$m$^2$, less than 5% of $A_{\textrm {eff}}$. The fact that the GBNH cross-section area is a tiny fraction of $A_{\textrm {eff}}$ implies the extremely low sensitivity of our SAM device characteristics to the exact distribution of aGNRs in the GBNH. In other words, the propagating modes are governed by the average optical behavior of the GBNH. We expect that the effects stemming from different GNRs seeing slightly different electric fields (because of spatial nonuniformity of a mode) to be quite small.

6. Conclusion

In the long-wavelength limit, armchair graphene nanoribbons have a strong optical nonlinearity. However, the naturally poor optical confinement in the long wavelength limits the use of the GNR optical nonlinearity. Here, we designed and numerically modeled a rib silicon waveguide with embedded graphene nanoribbons for all-optical SAM applications. At the core of the waveguide, we embed a van der Waals heterostructure comprising armchair graphene nanoribbons and hexagonal boron nitride. By implanting graphene nanoribbons with different widths in the van der Waals heterostructure, the modulator acquires a self-sustaining broadband modulation strength over the telecom frequency range, without a need for dynamical tuning. The compact footprint along with the self-sustaining broad bandwidth suggest the proposed device as a suitable saturable absorber for the nonlinear nanophotonic applications, particularly ultrafast pulse generation.