Graphene, a single-layer, two-dimensional crystalline material composed of carbon atoms arranged in a honeycomb lattice, has attracted considerable attention in the past 20 years due to its unique electronic and optical properties [1–3]. Graphene has a wide broadband optical response from visible to mid-infrared wavelength with 2.3% absorption [4, 5]. Graphene exhibits a remarkably low resistivity with mobility of up to 20,000 cm² V⁻¹ s⁻¹ originating from the zero effective mass for the carriers [6]. Thus, graphene-based applications have rapidly emerged, such as detectors, sensors, modulators, etc [7–10]. In 2011, Liu et al. [2] prepared the world’s first graphene electro-optical modulator by covering the upper surface of a silicon waveguide with a single layer of graphene. Although the device had a 3-dB modulation bandwidth of 1.2 GHz, a modulation depth of 0.1 dB/μm, and an extinction ratio of about 4 dB, it was a very meaningful attempt to fabricate a modulator using a two-dimensional material. In 2012, they improved the optical modulator’s performance by using double-layer graphene instead of a monolayer. The optical absorption increased, while the modulation depth was not greatly improved, and the modulator eventually achieved a modulation depth of 0.16 dB/µm and an extinction ratio of 6.5 dB [3]. After that, various configurations such as ring resonators, Mach-Zehnder, and hybrid plasmonic waveguide have been proposed to improve the modulation bandwidth and extinction ratio [11–13]. The modulation bandwidth achieved highs of ~190 GHz and 35 GHz in simulation and experiment studies [3, 14]. However, the gapless energy band structure of graphene prevents effective modulation of the ON and OFF states, thus limiting its application in the field of on-chip optoelectronic devices.

Due to its distinct electrical structure, black phosphorus (BP), a new component of two-dimensional materials, has drawn considerable interest in the research of nano optoelectronic devices. In 2014, Reich and Cheng [15, 16] successfully peeled BP into two to three atomic layers independently using the same method as graphene. BP has tunable optical properties that are highly sensitive to thickness, doping, and optical polarization [17–20] due to its orthorhombic crystal structure made up of folded layers [21] and small effective masses of electrons, and holes along the armchair crystal direction [4]. In addition, BP possesses a direct band gap structure with optical band gaps of 1.73 eV, 1.15 eV, 0.83 eV, and 0.3 eV for monolayer, bilayer, trilayer, and bulk BP, respectively [22].

The optical properties of BP can be modulated by either an electrical signal (electro-optical modulation) or an optical signal (all-optical modulation). The currently reported studies on electro-optical modulation of BP are mainly based on the change of absorption, which can be achieved by tuning the Fermi energy levels, and the applied electric field can modulate the light absorption in BP [21]. Deng et al. [23] investigated the bandgap of thin-film BP with a vertical electric field by analyzing the carrier transport properties in a double-gate BP transistor. The bandgap of BP was experimentally extracted from the minimum conductance of the transistor in different vertical electric fields and at different temperatures. The results show that the bandgap of a 10 nm thick BP sheet decreases from 0.3 eV to 0.05 eV at a vertical displacement field of 1.1 V/nm. Such voltages can be easily realized in high-quality integrated circuits. Liu et al. [24], using low-temperature scanning tunneling microscopy, found that up to 35.5% bandgap modulation was achieved by applying an electric field of 0.1 V/nm to fewer layers of BP. The results show that effective bandgap tuning can be achieved at a small bias voltage, which makes BP a promising material for broadband optoelectronics. In the field of nano-photonics, BP is therefore anticipated to be a novel two-dimensional material distinct from graphene and TMDs [25].

According to the reports, BP can be employed in field-effect tube photocatalysis, photodetectors, sensors, light modulators, and other devices [26–29]. However, relatively few articles reported on the application of multilayer BP to optical modulators. These modulator schemes suffer from the following disadvantages: They are hard to fabricate due to a complex structure that usually involves etching and materials transfer several times, have a large planar size, have a high cost, and have complementary metal-oxide semiconductor (CMOS) incompatibility. Therefore, based on the shortcomings of a large footprint, fabrication difficulty, and CMOS incompatibility, the ridge-type waveguide is considered a structural tradeoff between high performance, small device size and ease of manufacturing for on-chip optical modulators.

In this work, we have designed an optical modulator based on graphene and BP heterojunction. The geometrical parameters of the modulator waveguide are optimized using the finite-difference time-domain (FDTD) method and the distributions of the components are observed in TE and TM modes with different BP orientations. The simulation results show that the extinction ratio of our proposed structure is 0.166 dB, and a modulator with a working length of 3 μm can realize a modulation depth of 0.498 dB. The device has a 3-dB bandwidth of up to 2.65 GHz with 27.23 fJ/bit energy consumption. The extinction ratio and bandwidth of the proposed modulator increased by 66% and 120.83%, respectively, compared to the monolayer graphene-based ridge-type waveguide modulator [2]. Energy consumption was reduced by 97.28% compared to the double-layer graphene-based modulator [3].

The structure of the proposed modulator is shown in Fig. 1. Silicon dioxide (SiO₂) is used as the substrate, followed by an embedded Si waveguide, which is then covered with triple-layer BP and monolayer graphene, where the thickness of the SiO₂ layer and Si layer are 2 μm and 220 nm, respectively. In numerical simulation with the FDTD method, the thickness of a single graphene layer and triple-layer BP are equivalently set to 5 nm and 3 nm.

Figure 1. Scheme for the proposed modulator.

Graphene possesses unique optical properties, and its complex conductivity can be dynamically tuned by applying a driving voltage. The conductivity of graphene consists of two components that are related to intra-band and inter-band carrier excitation. Here, the conductivity of graphene can be calculated using the Kubo formula as follows [30]:

where ω is the angular frequency, e is the electron charge, κ_B is the Boltzmann constant, µ_C is the chemical potential, ħ is the reduced Planck constant, Τ is the temperature and τ is the inter-band relaxation time. After obtaining the conductivity of graphene, the dielectric constant of graphene can be obtained by the Drude formula:

where Ԑ₀ is the vacuum dielectric constant, and ԁ is the thickness of graphene, which is set to 5 nm here. Figure 2(a) represents the variation of the graphene dielectric constant at 1,550 nm for different chemical potentials. It can be clearly seen in Fig. 2 that the dielectric constant of graphene decreases sharply at ~0.4 eV.

Figure 2. Dielectric constants of graphene and black phosphorus (BP). (a) Real and imaginary parts of the dielectric constant of graphene at 1,550 nm. (b) The dielectric constant of triple-layer BP.

When µc < 0.51 eV, the real and imaginary parts of the graphene dielectric constant are positive, and graphene shows dielectric material properties; When µc > 0.51 eV, the real part of the graphene dielectric constant becomes negative, the imaginary part is close to zero, and graphene shows metallic material properties. Therefore, we can further control the optical parameters of graphene by controlling the chemical potential. The graphene chemical potential versus the voltage is given by [31–33]

where ħ is the reduced Planck constant, v_F is the Fermi velocity, which is about 1.1 × 10⁶ m/s, n₀ is the carrier concentration, ε₀ is the vacuum permittivity, ε_r is the relative permittivity of the medium, and V_g is the externally applied voltage. V_d is the equivalent voltage of the initial Fermi energy level shift due to chemical doping or impurities, etc. For pure undoped graphene, V_d = 0. d is the thickness of the dielectric layer and e is the electronic charge. In this work, we added a BP layer below the graphene layer to improve the performance of the modulator. Since BP exhibits strong anisotropy in the armchair (x-axis) and zig-zag (y-axis) directions, we consider the different modulation properties when there is incident light with different modes along different BP orientations in the simulations. The photonic properties of single-layer BP can be described by the Drude model. Previous studies have shown that triple-layer BP, which has a ~0.8 eV band gap, works best at 1,550 nm in the wavelength band commonly used for communication, but there are few reports on the application of triple-layer BP in optical modulators. In our simulation, we calculated the dielectric constant of triple-layer BP by the first-principles method [34, 35] as shown in Fig. 2(b).

To achieve a strong field localization effect, we first optimized the geometric parameters of the waveguide. The impact of the waveguide’s height (H) and width (W) on its effective refractive index in TE and TM mode when the graphene layer and BP layer are excluded is illustrated in Fig. 3. Figure 3(a) shows that the effective refractive index increases with waveguide height. In proportion to the height increase, the localization effect of the waveguide is enhanced, thereby it could enhance the interaction between the light and 2D materials. Figure 3(b) shows that the effective refractive index increases with waveguide width. As the width increases, the effective refractive index of the TE and TM modes increases due to the fact of that wider waveguides are better able to localize the energy, which allows the energy to be more confined within the silicon waveguide, while the footprint will linearly increase simultaneously. The height of the silicon waveguide affects the effective refractive index similarly to the width. In consideration of a small footprint, CMOS compatibility, and low cost, the width of the silicon waveguide and height are set to 400 nm and 220 nm, respectively.

Figure 3. The impact of the waveguide’s height (H) and width (W) on its effective refractive index in TE and TM mode. (a) Effect of waveguide height on effective refractive index. (b) Effect of waveguide width on effective refractive index.

In this work, we added a BP layer below the graphene layer to improve the performance of the modulator. Therefore, we used the following optimized parameters in the process of simulation setup: The width and height of the silicon waveguide are set to 400 nm and 220 nm, and the equivalent thickness of graphene and triple-layer BP are set to 5 nm and 3 nm, respectively. In order to obtain more accurate results, the graphene layer and black phosphorus layer are finely divided into grids. When the thickness of the material is along the x-direction, the grid size steps dx, dy, and dz are 0.5 nm, 1 nm, and 1 nm, respectively. While the thickness of the material is along the z-direction, the step sizes dx, dy, and dz of the grid size are 1 nm, 1 nm, and 0.5 nm, respectively [36].

In what follows, the loss, the effective refractive index, and the distribution of the field in different modes, and a comparison of the results with those obtained when only the graphene layer is presented are discussed in detail. Figure 4 shows the effects of the chemical potential on the real and imaginary parts of the effective refractive index in the TE and TM modes. After adding graphene and BP layers to the silicon waveguide while keeping the chemical potential of the BP layer unchanged, we adjusted the chemical potential of graphene. It can be seen that the imaginary part of the effective refractive index gradually decreases, while the real part is almost unchanged. Furthermore, the trends are similar in both modes, but the magnitude of the changes differs.

Figure 4. The relationship between the real and imaginary parts of the effective refractive index and the change of chemical potential in TE and TM modes.

Figure 5(a) illustrates the loss of this modulator with various modes and transmission directions by tuning the chemical potentials at 1,550 nm. The results reveal a common trend in the behavior of TE and TM modes regarding transmission along the armchair and zig-zag directions. Moreover, a higher chemical potential correlates with reduced loss. The absorption efficiency of the TE mode is notably high when light propagates in the armchair direction, whereas the TM mode exhibits high absorption efficiency when light propagates in the zig-zag direction. Therefore, it becomes evident that absorption efficiency is contingent not only on the photoelectric field’s distribution but also on the direction of light propagation. Figure 5(b) illustrates the transmission of the modulator as a function of bias voltage at a wavelength of 1,550 nm. As the bias voltage is augmented, the loss progressively diminishes, concurrently with a gradual enhancement in transmittance, and both parameters ultimately stabilize.

Figure 5. The losses and transmission of the modulators. (a) Modulator losses in different modes and transmission directions for different chemical potentials at an operating wavelength of 1,550 nm. (b) The transmission of the modulator as a function of bias voltage at a wavelength of 1,550 nm.

Figure 6 shows the electric field distribution of the waveguide at W = 400 nm and H = 220 nm. Since BP exhibits strong anisotropy in the armchair and zig-zag directions, the electric field distributions will determine the strength of the interaction of BP and incident light. By comparing the TE and TM modes, it can be found that in the TE mode, the Ex electric field mainly concentrates in the two interfaces where the Si waveguide contacts with SiO₂. This can significantly enhance the effect of BP interaction with the photoelectric field, which in turn enhances the absorption efficiency of BP. Referring to Fig. 6, it becomes apparent that BP occupies the location associated with the highest intensity, corresponding to the Ex component of the TE mode. Consequently, the most suitable option for this modulator is the transmission of the TE mode along the armchair direction.

Figure 6. Mode field and distribution of each component for TE and TM modes of waveguide at W = 400 nm and H = 220 nm.

Figure 5(a) shows that the loss begins to steeply decline at 0.3 eV and levels off at less than 0.5 eV. Therefore, 0.5 eV is considered to be the ON state, and the corresponding loss is 0.141 dB/µm. Meanwhile, 0.3 eV is considered to be the OFF state, and the corresponding loss is 0.307 dB/µm. Modulation depth is one of the most critical factors and parameters affecting the optical modulator. With the calculation of the modulator’s extinction ratio of 0.166 dB and a 3 µm working length, the modulator can achieve a modulation depth of 0.498 dB. The flat plate capacitance can easily be obtained from the flat plate capacitance formula. However, considering that the contact resistance of metal-BP varies greatly in different literature reports, this paper only adopts a rough estimation method and considers that the contact resistance is about R = 1~10 kΩ, f_3dB ≈ 0.265 to 2.65 GHz. We can calculate the energy consumption by E_bit = C(ΔV)² / 4, where C is the capacitance and R is the amount of resistance, respectively. ΔV is the difference of applied voltage. The modulation bandwidth for this structure is 2.65 GHz with 27.23 fJ/bit energy consumption. With the current waveguide design, the modulation footprint is about 10 µm².

Finally, we compare our investigated results with the results of recent studies in Table 1. Our proposed modulator has less energy loss and a high modulation depth compared to these studies. At the same time, we have used a ridge-type waveguide structure, which is simpler in the fabrication process. Thus, compared to other modulators, our proposed modulator has some advantages considering all the performance characteristics of the modulator in terms of size and fabrication cost.

TABLE 1. Comparison between the performance characteristics of our proposed modulator with recent studies.

References	Modulation Depth (dB/µm)	f3dB (GHz)	Ebit (fJ per bit)	Device Footprint (µm2)
[2]	0.1	1.2	-	25
[3]	0.16	1	1000	-
[23]	0.1	35	1400	18
[33]	0.28	46.4	630	10
[36]	0.58	21.2	52.7	-
This Study	0.166	2.65	27.23	10

In this paper, we investigated an optical modulator consisting of triple-layer BP covered with monolayer graphene on a silicon waveguide. The width and height of the silicon waveguide are set to 400 and 220 nm, respectively, by geometry parameter sweep. The extinction ratio and 3 µm work length modulation depth of this optimized modulator could achieve 0.166 dB and 0.498 dB. Considering the effect of the resistor value on the 3dB bandwidth, the modulator has a 3dB bandwidth of up to 2.65 GHz with good electrode contact (1 kΩ), an energy consumption of 27.23 fJ/bit, and a footprint of 10 μm². The extinction ratio and bandwidth of the proposed modulator increased by 66% and 120.83%, respectively, compared to the monolayer graphene-based ridge-type waveguide modulator [2]. Energy consumption was reduced by 97.28% compared to the double-layer graphene-based modulator [3]. Our work may offer new insights into designing high-performance on-chip optical modulators.

Natural Science Foundation of Sichuan Province (Grant no. 2022NSFSC1800).

The authors declare no conflicts of interest.

Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.