Optimized silicon nitride‑spaced graphene electro‑optic modulator with high efficiency and bandwidth

Vol.:(0123456789) Optical and Quantum Electronics (2025) 57:402 https://doi.org/10.1007/s11082-025-08310-0 Optimized silicon nitride‑spaced graphene electro‑optic modulator withhigh efficiency andbandwidth AshrafulIslamRaju1· PawanKumarDubey1· RasuoleLukose1· ChristianWenger1,2· AndreasMai1,3· MindaugasLukosius1 Received: 13 February 2025 / Accepted: 3 June 2025 / Published online: 28 June 2025 © The Author(s) 2025 Abstract Optical modulators with high modulation efficiency, large operational bandwidth, highspeed and low energy consumption is essential for the advancement of on-chip optical signal processing. To overcome the bandwidth-efficiency trade-off in graphene optical modulators, a buried silicon nitride waveguide-coupled double-layer graphene electro-absorption (EA) optical modulator has been proposed. In the proposed design, silicon nitride layer is also embedded between the two graphene layers as a dielectric spacer to enhance the graphene-light interaction. An extensive simulation has been performed to optimize the dielectric spacing layers between the two graphene for optimal device performance including the waveguide dimensions and optical modes profile. The simulated results show a high modulation efficiency of 1.1dB/V and a modulation depth of 0.16dB/µm, corresponding to a 15-dB extinction ratio for a 100µm device at 1550nm, with a 30nm spacer and 12V driving voltage. The proposed modulator achieves a 14GHz bandwidth and operates over a 1050nm broadband operation spectral range. The concurrent presence of high modulation bandwidth and efficiency renders these modulator designs highly viable for on-chip optical communication applications. Keywords Electro-optical modulator· Graphene modulator· Silicon nitride waveguide· Modulation efficiency 1 Introduction To accelerate the advancement of next-generation on-chip communication, there is a growing demand for electro-optical modulators that possess exceptional characteristics such as low-drive voltage, large optical bandwidth, ultra-high speed and compatibility of complementary metal-oxide semiconductor CMOS technologies. The state-of-the-art * Ashraful Islam Raju [email protected] 1 IHP-Leibniz Institute forHigh Performance Microelectronics, Im Technologiepark 25, 15236Frankfurt(Oder), Germany 2 BTUBrandenburg Technical University, Konrad-Wachsmann-Allee 13, 03046Cottbus, Germany 3 Technical University ofApplied Science, Hochschulring 1, 15745Wildau, Germany A.I.Raju et al. 402 Page 2 of 15 silicon-photonics technology-based modulators are currently a prime candidate for meeting these requirements (Chaisakul etal. 2019; Reed etal. 2010; Xiao etal. 2023), however they suffer from limited thermal ability, larger foot print, higher energy consumption and insertion loss due to weak plasma dispersion effect. To mitigate the drawbacks of silicon optical modulators, new materials and technology have already been explored. Graphene, a zero-bandgap two-dimensional (2D) material is one of the promising materials that could revolutionize silicon photonics due to its noticeable electrical, mechanical, thermal and optical properties (Bao and Loh 2012; Bonaccorso etal. 2010; Falkovsky 2008; Hendry etal. 2010; Lukosius etal. 2024). Graphene demonstrated an excellent optical property with an ultra-high carrier mobility, wide operational bandwidth, and the ability to tune conductivity electrically. An atomically thinned single layers graphene absorbs 2.3% of the perpendicularly incident light which further can be enhanced by integrating graphene on hybrid waveguide structures (Liu etal. 2011). The absorption of graphene can be tuned by actively tuning the Fermi level through applying voltages. These nailing optical properties of graphene have made it a major contender in various optoelectronics applications including high-speed photodetectors (Schall etal. 2014; Xia etal. 2009), optical polarizers (Bao etal. 2011) and electro-optical modulators (Li etal. 2014; Liu etal. 2012; Romagnoli etal. 2018). In the past few years, different kinds of graphene-based waveguide-coupled optical modulators with single-layer graphene (SLG) and double-layer graphene (DLG) have been studied and demonstrated, based on graphene-oxide-silicon and graphene-oxide-graphene configuration respectively. To date, graphene-based electro-optic modulators with DLG architecture have already shown promising characteristics such as broadband optical bandwidth (Liu etal. 2011), high-speed operation (Giambra etal. 2019; Koester and Li 2012), considerable modulation efficiency (Liu etal. 2012), and larger temperature stability (Dalir etal. 2016). For instance, a DLG based EA optical modulator was proposed in 2012 with a modulation depth of 0.10 dB/µm at modulating voltage 5V, however the modulation bandwidth was limited to 1GHz (Li etal. 2014). In 2016, a graphene-based broadband modulator with a modulation depth of 2dB and high bandwidth of 35 GHz demonstrated (Dalir etal. 2016). However, a large 30 V modulating voltage needed to operate the device to achieve such high bandwidth. Recently, a DLG EA optical modulator was demonstrated with 39 GHz modulation bandwidth and 2.2 dB/V modulation efficiency at moderate drive voltage (Agarwal etal. 2021). However, a 2D–3D hetero-integration of dielectric materials between the two graphene layers are required to achieve such performance. Nevertheless, it should be emphasized that most of the previously reported graphene-based EA optical modulators are based on the silicon-on-insulator (SOI) integrated waveguide. However, the propagation loss of SOI waveguide is relatively large (about 2dB/cm to 3dB/cm) due to the surface roughness of silicon stripe. In comparison to SOI waveguides, silicon nitride (Si3N4) waveguide has much lower propagation loss of 0.1 dB/cm. The utilizing of Si3N4 waveguides also offers several advantages such as lower refractive index, broader wavelength transparency windows, reduced thermal-optic effects, and increased fabrication tolerance (Shao etal. 2016). Therefore, in the last few years, Si3N4 waveguide -onSiO2 platform has gained attention for realizing photonic integrated circuits (PICs) (Lee etal. 2021; Lukose etal. 2023; Phare etal. 2015). Despite having above mentioned properties of Si3N4 waveguide -onSiO2 platform, only few graphene-based Si3N4 waveguide-coupled EA optical modulators have been proposed and demonstrated (Lee etal. 2021; Phare etal. 2015). In these reports, graphene EA optical modulators on the Si3N4 platform employ mainly aluminum oxide (Al2O3) or a 2D–3D material based heterostructures as spacer layer between two graphene layers. On one hand, Al2O3, may be used in device concepts where its higher dielectric constant Optimized silicon nitride‑spaced graphene electro‑optic… Page 3 of 15 402 provides benefits, but the increased optical losses must be considered, whereas the complexity and scalability of 2D-3D heterostructures should also be addressed. On the other hand, Si3N4 is as an alternative dielectric spacer which has a much lower absorption coefficient in the near-infrared range, making it a better choice for minimizing optical losses and enhancing the efficiency of graphene-based modulators. It is also a material with higher dielectric strength compared to Al2O3, enabling it to withstand stronger electric field without breakdown. Si3N4 also offers superior insulating properties, resulting in lower leakage currents compared to Al2O3. Additionally, it preserves the high mobility and low doping of intrinsic graphene, enables to operate beyond the transparency regime ensuring superior dynamic performance. Furthermore, Si3N4 exhibits strong thermal stability, making it capable of withstanding high-temperature processing and CMOS compatible. To the best of our knowledge, no reports have yet demonstrated graphene-based modulators integrating Si3N4 as a waveguide as well as a spacer layer between the graphene, highlighting for further exploration and development in this area. In this paper, an ultra-high-speed and highly efficient buried waveguide-coupled DLG EA optical modulator on Si3N4 platform using a fabrication wise simpler approach has been designed and simulated where the waveguide material and the dielectric spacer between the two graphene layers are Si3N4.This approach not only enhance the graphenelight interaction but also amplifies the capacitance of the EA optical modulators while maintaining their resilience against high voltages. The impact of graphene quality and the waveguide configuration on the modulation efficiency and bandwidth are investigated by finite element method. Subsequently, the dielectric spacer layer thickness between the two graphene layers and graphene size are carefully designed based on the optimized waveguide structure to enhance the modulator’s performance. According to the simulation results, the proposed modulator demonstrated a modulation efficiency of 1.1 dB/V and a modulation depth of 0.16 dB/µm equivalent to an extinction ratio up to 15dB. The proposed modulator achieves a high modulation bandwidth of 14 GHz and operates over a 1050 nm broadband operation spectral range. This research showcases the potential of the proposed design in achieving high-performance electro-optic graphene modulators integrated within Si3N4 waveguides platform. 2 Device structure andmodelling 2.1 Device concept Figure1 illustrates the schematic structure of the proposed double-layer graphene EA optical modulator. The proposed graphene EA optical modulator comprises a dual-layer graphene capacitor integrated with a Si3N4 waveguide buried on a 2.1 µm-thick buried oxide (BOX: SiO2) layer. The capacitor consists of two graphene sheets (Bottom: GRAP1 and Top: GRAP2) separated by a 20 nm Si3N4 dielectric spacer. The capacitor is enclosed by another 20 nm Si3N4 following by a 1.5 µm silica (SiO2) on the top it as a top cladding (TOX) layer. The Si3N4 dielectric spacer between the two graphene layers plays a crucial role in determining the overall performance (trade-off between modulation efficiency and bandwidth) of the modulator. Therefore, to optimize the modulator’s performance, we conducted a design exploration and optimization for different Si3N4 dielectric spacer thicknesses ( dSL ), specifically 10 nm, 20 nm, and 30 nm. Efficient metallic contact can be achieved with two Palladium/Gold (Pd/Au) pads positioned on each side of the graphene A.I.Raju et al. 402 Page 4 of 15 to electrically tune the graphene electro-optical properties. Both materials offer exceptional thermal and electrical conductivity, coupled with low contact resistance and strong adhesion properties, rendering it ideal choices for contacting graphene. we designed the gap ( g G−C) between the waveguide and each metal electrode/pad about 1.5 µm in order to avoid any optical interference between them. The operational principle of our proposed modulator is very straightforward: applying a voltage to the graphene electrodes induces simultaneous electron doping in one electrode and electron depletion in the other. This process leads to a shift in the Fermi energy (chemical potential) and consequently brings about changes in the refractive index and absorption of the waveguide material within the capacitor region. 2.2 Modelling methodology The simulations in this work are conducted using the finite difference eigenmode (FDE) solver, while the properties of graphene are described using a surface conductivity model (Hanson 2008). The complex surface conductivity of a monolayer graphene can be obtained through the application of the Kubo formula (Hanson 2008). where ω is the radian frequency of incident light, μc is the chemical potential that can be electro-statically controlled, Γ is the scattering rate which is inversely proportional to scattering time 𝜏(𝜏=1∕Γ) , T is the temperature, E is the energy, ℏ is the reduced Plank’s constant, e is the charge of an electron and fd(E) is the Fermi–Dirac distribution function: where KB the Boltzmann’s constant. The first term in Eq. (1) represents the conductivity contribution arising from intra-band electron-photon scattering process, while the second term is due to inter-band scattering. The permittivity of the graphene can be calculated as a function of the complex conductivity of graphene using a volumetric method (Shao etal. 2016). (1) 𝜎( 𝜔,𝜇c,Γ,T ) =ie 2 (𝜔+i2Γ) 𝜋ℏ 2 [∞ ∫ 0 E (𝛿f d (E) 𝛿 E( − 𝛿f d (−E) 𝛿 E) dE ) − ∞ ∫ 0(𝛿f d (−E)−𝛿f d (E) ( 𝜔 +i2Γ) dE )], (2) fd(E)= ( e E−𝜇c KBT+1 )−1 Fig. 1 Proposed device schematic hWG : waveguide height WWG : waveguide width, dSL : spacer layer thickness between two graphene, gG−C : waveguide to electrode distance and LG : length of the graphene Optimized silicon nitride‑spaced graphene electro‑optic… Page 5 of 15 402 where 𝜀0 is the permittivity of the air medium and hG is the thickness of the single layer graphene respectively. Figure2 shows the simulated conductivity and permittivity of a single layer graphene as a function of graphene chemical potential. The following calculations are based on the incident light with λ=1550 nm , T=300 K, hG=0.34 nm and for three different scattering time τ , e.g. 13 fs, 65 fs and 100 fs that corresponds to graphene mobility of 293, 1466 and 2200 cm2V−1 s−1 at μc=0.4eV respectively (Romagnoli etal. 2018). The scattering time directly impacts the material mobility, which, in turn, plays a crucial role in determining the material’s quality and, consequently, the modulator performance. Figure2a illustrates the graphene optical surface conductivity normalized by 𝜎0=60𝜇S , the universal graphene conductivity and Fig.2b illustrate the dielectric permittivity for different τ concerning the Fermi level of a graphene mono-layer, determined using Eq. (1) and Eq. (3) respectively. As the chemical potential ( 𝜇c ) increases, the real part of the permittivity, which primarily governs the material’s refractive properties, gradually rises until it reaches a maximum point when the Pauli blocking condition is met ( 𝜇 c ≈ℏ𝜔∕2 ) . Afterwards, it experiences a sharp decrease as the 𝜇c increases, it is noteworthy that the sign of the real part of the permittivity constant can transition from positive to negative. This shift implies that the properties of graphene can be altered from dielectric-like to metallic-like characteristics (Giambra etal. 2019). The imaginary part of the permittivity is mainly associated with material absorption and exhibits distinct characteristics depending on the chemical potential. For 𝜇c below the Pauli blocking threshold ( 𝜇 c < �𝜔∕2 ) , a constant region is observed, where inter-band electron-photon scattering dominates. As the 𝜇c approaches the Pauli blocking energy threshold, both inter-band and intra-band processes become significant, leading to a notable change in the imaginary part of the permittivity. Finally, at 𝜇c above the Pauli blocking energy ( 𝜇 c > �𝜔∕2 ) , the imaginary part of the permittivity remains nearly constant and is primarily influenced by the intra-band scattering process. Additionally, the variation in complex permittivity demonstrates elevated values for τ =65 fs and τ =100 fs, owing to their associated higher mobility. In contrast, the change is relatively lower for τ =13 fs due to its correspondingly lower mobility. These findings suggest that higher mobility corresponds to a somewhat superior material quality in comparison (3) 𝜀|| =1+i𝜎∕(𝜔𝜀0hG) Fig. 2 Graphene properties: a complex conductivity, b complex permittivity as a function of chemical potential simulated at 𝛌=1550𝐧𝐦 , 𝐓=300𝐊 A.I.Raju et al. 402 Page 6 of 15 to lower mobility. The changes of graphene’s permittivity influence the complex effective mode index ( ηeff ) in the waveguides and consequently the modulators performance. 3 Results andanalysis 3.1 Graphene‑Si3N4 waveguide design andoptimization The effective mode index is a critical parameter in the design of the graphene EA optical modulator. The interaction between the optical mode distribution and the graphene layer plays an essential role in designing the graphene EA optical modulator. Optical absorption serves as a measure of this interaction, making it an essential factor to investigate when considering different waveguide configurations. The absorption can be calculated by: Here 4.343 ×10−6 is the constant for converting from m−1 to dB/µm, 𝜅0= 2𝜋 ∕𝜆 is the wavenumber and imag( η eff) is the imaginary part of the refractive index. Figure3 shows the absorption for various dielectric layer thickness dSL(10 nm, 20 nm and 30 nm) at τ =65 fs (mobility: 1466 cm2V−1 s −1) and explores how optical absorption varies with waveguide dimensions at the neutrality point ( 𝜇c=0) . It is observed that the transverse electric (TE) mode exhibits higher interaction with the graphene layer compared to the transverse magnetic (TM) mode. This discrepancy arises from the strong longitudinal electric field component present at the top interface of the Si3N4 waveguide, which contributes significantly to the interaction in the TE mode. However, the variation of dSL has a very low impact in designing the optimum waveguide dimensions. Figure3a illustrates the variation of absorption as a function of waveguide width (WWG ) , while maintaining a constant waveguide height (hWG ) of 300 nm. As the waveguide width increases, the absorption also rises, reaching its maximum value at a width of 1.2 µm. At this point, the largest absorption values for the TE and TM modes are 0.16 dB/µm and 0.023 dB/µm, respectively. Subsequently, the absorption remains almost constant as the waveguide width continues (4) 𝛼 (dB∕𝜇m)=2𝜅 0 ×imag ( η eff) ×4.343 ×10 −6 Fig. 3 a TE and TM mode absorption as a function of waveguide width, b TE and TM mode absorption as function of waveguide height simulated at 𝛌=1.55𝛍𝐦 , 𝐓=300𝐊 Optimized silicon nitride‑spaced graphene electro‑optic… Page 7 of 15 402 to increase. This behavior is attributed to the fact that the mode distribution is primarily concentrated in the waveguide and enlarges with the increment in the waveguide width up to 1.2 µm. In contrast, Fig.3b demonstrates the absorption variation as a function of waveguide height, while maintaining a constant waveguide width of 1.2 µm. As the waveguide height increases, the absorption also rises, peaking at a waveguide thickness of 250 nm before gradually decreasing. This behavior arises from the interplay between the waveguide thickness and its interaction with graphene and the TE mode. At lower waveguide thicknesses, around 200–300 nm, the interaction with graphene and the TE mode is more pronounced, leading to the observed peak in absorption. This insight underscores the significance of optimizing the waveguide height for achieving optimal performance. 3.2 Optimization of Si3N4 spacer forenhanced modulation efficiency Following the optimization process, a TE Mode waveguide with a height of 300 nm and a width of 1.2 µm is chosen for subsequent simulations in order to achieve a single mode condition as well as maximum absorption. Figure4a shows optical mode profile of the proposed modulator for a waveguide width and height 1.2 µm and 300 nm respectively at dielectric spacer thickness of 10 nm. The relationship between the TE effective refractive index and chemical potential, 𝜇c is investigated under various dielectric spacer thickness dSL , and it shown in Fig.4b at a scattering time of τ =65 fs. The real part of the refractive index, primarily responsible for phase changes, exhibits slight variations near the Pauli blocking threshold ( 0.3 eV<μc<0.5 eV ) and then experiences a sudden shift as chemical potential 𝜇c increases from 0.5 eV to 1eV. The changes in the real part of the refractive index within this region for dSL of 10 nm, 20 nm, and 30 nm are 0.01, 0.008, and 0.007, respectively, indicating that a phase modulator can be effectively designed within this range (Sorianello etal. 2018, 2015). In contrast, the imaginary refractive index, mainly responsible for absorption, undergoes abrupt changes from 0.3 eV<μc<0.5 eV due to the inter and intra band domination as explained in the previous section. The changes of imaginary refractive index for different dielectric thickness dSL of 10 nm, 20 nm, and 30 nm are 0.45 ×10−3, 0.44 ×10−3and0.43 ×10−3 respectively which can be used to calculate the absorption using Eq. (4) and can be Fig. 4 a Opticalmode profile of proposed modulator (WWG = 1.2 µm and hWG = 300 nm), b Real refractive index (Re (𝛈𝐞𝐟𝐟 ) ) and Imaginary refractive index (Imag (𝛈𝐞𝐟𝐟 ) ) of fundamental TE mode as a function of 𝝁c under different 𝐝𝐒𝐋 at 𝛕=65𝐟𝐬 A.I.Raju et al. 402 Page 8 of 15 effectively designed an EA optical modulator in this region. Both the real and imaginary mode index increase with the reduction of dielectric spacer thickness dSL which is due to the enhancement of the reciprocal action between the graphene and the optical modes. Based on these findings, we choose this specific range to design our EA optical modulator, which promises significant potential for achieving efficient and controllable modulation capabilities. To explore the effects of chemical potential 𝜇c on graphene, a gate voltage is applied to the graphene capacitor, leading to a shift in its chemical potential. This alteration in the Fermi level subsequently influences the absorption properties of graphene. The investigation of the relationship between the gate voltage and the chemical potential of graphene is carried out using (Ye etal. 2014). Here 𝜈F =1.1 ×10 6 m∕ s is the Fermi velocity, Cox is the capacitance per unit length of the device and |Vg −VDirac| is the applied voltage where VDirac is the voltage corresponding to the charge-neutral Dirac point ( VDirac =0.7 V). In Fig.5a, the relationship between the gate voltage and the chemical potential is depicted for different dSL of 10 nm, 20 nm, and 30 nm. The thicker dielectric spacers (dSL =30 nm) require higher voltages to drive the Fermi level of graphene compared to thinner dielectric layers (dSL =10 nm) . For instance, to reach the 𝜇c=0.4eV , the corresponding gate voltages required are 4V, 6V, and 10 V for dSL of 10 nm, 20 nm, and 30 nm, respectively. Based on the analysis above, we investigated of the optical absorption concerning the voltage applied to the graphene capacitor. Figure5b illustrates the changes in absorption as a function of gate voltage for different dSL , including 10 nm, 20 nm, and 30 nm. The curves are also plotted for three distinct values of τ (13 fs, 65 fs and 100 fs). While varying τ has a minor impact on the maximum absorption at lower voltages, it significantly affects absorption at higher voltages, leading to notable consequences for the modulator’s insertion loss (IL) and extinction ratio (ER). On the other hand, the voltage required to switch the graphene absorption from high to low strongly depends on dSL . A thicker dSL layer necessitates higher voltages for switching, thereby influencing (5) | 𝜇 c| =ℏ𝜈 F( 𝜋C ox| V g −V Dirac|)1∕2 Fig. 5 a Chemical potential at different voltages for 𝐝𝐒𝐋 of 10 nm, 20 and 30 nm at 𝛕=100𝐟𝐬 , b TE mode absorption as function of applied voltage for 𝐝𝐒𝐋 of 10 nm, 20 and 30 nm at three distinct 𝛕(13𝐟𝐬, 65𝐟𝐬,𝐚𝐧𝐝100𝐟𝐬) Optimized silicon nitride‑spaced graphene electro‑optic… Page 9 of 15 402 the modulator’s efficiency and speed significantly. From Fig.5b, we can now calculate essential parameters such as modulator’s efficiency, IL, and ER. Modulation efficiency is calculated by using: here 𝛼90% and 𝛼10% represents the maximum and maximum absorption at corresponding minimum (V 𝛼 10%) and maximum (V 𝛼 90%) voltages and LG is the graphene length. In Fig.6a, the modulation efficiency of devices with different graphene lengths is depicted for different dSL (10 nm, 20 nm, and 30 nm) at three distinct values of τ (13 fs, 65 fs and 100 fs). It is shown that modulation efficiency is strongly influenced by factors of dSL, LG and τ . In particular, the efficiency increases with larger LG and decreases with decreasing dSL . The quality of graphene also plays a substantial role in enhancing modulation efficiency; higher values of τ are associated with achieving greater modulation efficiency and vice versa. At a driving voltage of 12 V and a moderate τ (65 fs), the modulation efficiency for a 100 µm-long device is approximately 2.8 dB/V, 1.5 dB/V, and 1.10 dB/V for dSL of 10 nm, 20 nm, and 30 nm, respectively. With the same driving voltage and device geometry, but with a smaller τ of 13 fs, the modulation efficiency is 2.2 dB/V, 1.2 dB/V, and 0.85 dB/V for dSL of 10 nm, 20 nm, and 30 nm, respectively. (6) (𝛼 90% −𝛼 10% )×L G ∕(V 𝛼 90%−V 𝛼 10%) Fig. 6 a Modulation efficiency, b insertion loss, c extinction ratio for different 𝐝𝐒𝐋 (10 nm, 20 nm and 30 nm) at three distinct 𝛕(13𝐟𝐬, 65𝐟𝐬,𝐚𝐧𝐝100𝐟𝐬) of the proposed modulator at 12 V driving voltage d) FoM for different 𝐝𝐒𝐋 (10 nm, 20 nm and 30 nm) at three distinct 𝛕(13𝐟𝐬, 65𝐟𝐬,𝐚𝐧𝐝100𝐟𝐬) at various voltages