Ultra-fast tunable optoelectronic half adder/subtractor based on photonic crystal ring resonators covered by graphene nanoshells

This paper reports a new design of a tunable optoelectronic half adder/subtractor. Two photonic crystal (PhC) ring resonators are used to realize the proposed structure. Several silicon rods surrounded by silica rods covered with graphene nanoshells (GNSs) form every PhC ring resonator. Setting the chemical potential of GNS with an appropriate gate voltage, we can tune the PhC resonant mode as desired. The plane wave expansion technique is used to study the photonic band structure of the fundamental PC microstructure, and the finite-difference time-domain method is employed in the final design for solving Maxwell's equations to analyze the light propagation inside the structure. We systematically study the effects of physical parameters on the transmission reflection and absorption spectra. By optimizing the geometric dimensions, resonant absorption peaks can be excited at the same time for GNSs. Our numerical results also reveal the maximum time response is about 0.8 ps. The 200 µm2 area of the proposed half adder/subtractor makes it the building block of every photonic integrated circuit. Also, the design of various fast signal processing systems in optical communication networks is possible due to using tunable GNSs in PhC ring resonators. This study can introduce the use of two-dimensional materials in the design and implementation of logic circuits.


Introduction
Graphene is one of the most desirable two-dimensional (2D) materials from the ultraviolet to terahertz frequency ranges. It is due to graphene's key characteristics, such as fast optical response, broadband window, high carrier mobility, excellent band structure, and unique mechanical strength and flexibility (Sun et al. 2016;Tang et al. 2018;Ye et al. 2017). It also has electro-optic tunability, zero-band-gap (ZBG), and existing free electrons under zero doping conditions. Graphene has metallic properties in the terahertz frequency range (THz), which has made it a popular material among photonic designers. All mentioned features result in this 2D material becoming a suitable material for fabricating highly tunable optoelectronic and plasmonic devices. In the range from infrared to terahertz, graphene supports a strong plasmonic response, which significantly increases light and graphene interaction. From the ultraviolet (UV) to near-infrared (NIR) wavelength range, monolayer graphene can only absorb a very small fraction of light due to the lack of plasmonic response in this region. It is about ~ 2.3% for the visible to NIR range and ~ 9% for the UV range (Nair et al. 1308). In order to improve the absorption efficiency of graphene, researchers studied multiple atomic layers. Undoubtedly, the optimal performance of graphene as a 2D material, such as photon and electron conductivity, will decrease. Therefore, researchers have proposed various methods to enhance the interaction between single-layer graphene and incident light, some of which include providing structures with a wavelength-scale pattern for graphene sublayers. However, they complicate the manufacturing process and cause irreparable damage to single-layer graphene's electronic and photonic properties because the graphene band structure is susceptible to atomic-scale damage and environmental pollutants. Researchers have presented metamaterials based on periodic Al arrays to increase light absorption efficiency in graphene by plasmon excitation in the UV wavelength range (González-Campuzano et al. 2017). Unfortunately, a large amount of light energy applied to the structure is lost due to the high absorption of Al in the structure. A variety of structures based on the periodic dielectric grating array is used to trap light to increase total absorption. However, these efforts are mainly concentrated in the infrared range, and research is needed on other wavelength ranges as well (Guo et al. 2016). Therefore, a metamaterial absorber is required to increase the interaction of light with graphene in the ultraviolet wavelength range. Flexible tunability is also crucial to design and investigate optical communication circuits, and this demand is possible by graphene plasmons (GPs). According to the graphene sheet electron flexibility, the generation of certain Fermi energy band levels (E f ) can be obtained in the graphene layer by several methods such as adding substrate and biasing the gate voltage between the substrate and graphene layer. These methods permit us to modify the optical properties of GPs and the operating parameters of graphene-based optoelectronic devices, such as the central resonance wavelength and quality factor. Optoelectronic circuits are appropriate candidates to realize high-speed transmission in data processing systems. One possible solution to design and realize optoelectronic devices is the optical structures based on photonic crystals (PhCs). It is possible to confine and control the propagation of electromagnetic waves inside the desired PhCbased structures due to the available photonic band gap (PBG) in PhCs. So far a variety of all-optical devices such as filters (Butt et al. 2010;Mccrindle et al. 2014;Hosseinzadeh Sani et al. 2020a;Naghizade and Saghaei 2020a;Tavakoli et al. 2019;Naghizade and Mohammadi 2019;Foroughifar et al. 2021;Alden Mostaan and Saghaei 2021), modulators (Shrekenhamer et al. 2013;Watts et al. 2014), sensors (Wang et al. 2015;Yahiaoui et al. 2015;Hosseinzadeh Sani et al. 2020b;Farmani 2019), optical fibers (Ghanbari et al. 2500Saghaei et al. 2016aSaghaei et al. , 2015Saghaei et al. , 2016bAliee et al. 2020;Diouf et al. 2017;Saghaei 2017), fundamental logic gates Naghizade and Saghaei 2020b;Moniem 2016;Danaie and Kaatuzian 2012;Moradi et al. 2019), encoders (Naghizade and Saghaei 2020b;Mehdizadeh et al. 2017a;Naghizade and Khoshsima 2018;Haddadan et al. 2020), decoders (Alipour-Banaei et al. 2015;Parandin et al. 2018;Salimzadeh and Alipour-Banaei 2018), demultiplexers (Mehdizadeh and Soroosh 2016;Talebzadeh et al. 2017;Wen et al. 2012;Saghaei et al. 2011;Failed 2008;Naghizade and Mohammadi 2020;Naghizade and Sattari-Esfahlan 2017), PhC fibers (Saghaei 2018;Ebnali-Heidari et al. 2014;Raei et al. 2018;Saghaei and Van 2019;Saghaei and Ghanbari 2017), interferometers (Gu et al. 2007;, adders and subtractors (Moradi 2019;Moradi et al. 2018;Askarian et al. 2019;Naghizade and Saghaei (2021d); Parandin et al. 2017Parandin et al. , 2021Abdollahi and Parandin 2019;Parandin and Reza Malmir 2020;Saghaei 2021a, 2021b), analog to digital converters (Rostami and Rostami 2003;Tavousi et al. 2016;Fasihi 2014;Mehdizadeh et al. 2017b;Youssefi et al. 2012;Karkhanehchi et al. 2017;Failed 2021;Naghizade and Saghaei 2021c) have been designed and investigated using PhCs. Realizing the optical data processing systems can be obtained using computational operators designed based on PhC structures. Serajmohammadi et al. (Serajmohammadi et al. 2018) used waveguides and ring resonators (RRs) inside the 2D hexagonal lattice of the PhC structure to realize an optical half adder. The proposed structure works according to constructive and destructive interferences of optical waves. The 180-degree phase difference in inputs leads to destructive interference. For this purpose, different lengths of input waveguides in the XOR gate have been chosen. Their proposed structure has a delay time of 4 ps and a footprint of 1056 μm 2 . Neisy et al. (Neisy et al. 2018) proposed another optical half-adder consisting of two nonlinear resonant cavities (RCs). These RCs have different resonant modes due to the physical dimensions; therefore, their coupling operations depend on the power intensity. The proposed half adder has a very short time response of 3 ps. The main drawback of all studies is that half adders are not tunable. Although few proposed logic gates are tunable based on the applied intensity, this tunability is limited to a special range of optical wavelengths. An ultrafast all-optical half adder with a total footprint of 249.75 μm 2 was proposed by Hosseinzadeh Sani et al. . Their structure contained two power regulators, two nonlinear ring resonators, and several waveguides. They used nonlinear rods in resonators; thus, the effective refractive index of the resonator changes as the intensity of incoming light changes. The desired resonance wavelength was tuned by adjusting the linear and nonlinear rods' radii. Their simulation results revealed that the delay time of their designed half adder is about 3.6 ps. We designed a high-speed optical half-adder using PhC in an area of 158 µm 2 (Naghizade and Saghaei 2020c). One of our structure's advantages compared to similar studies was the non-use of high nonlinear dielectric rods. This resulted in no need to increase the input power to divert the incoming light emission to the desired output. The calculations demonstrated that the proposed half-adder has a steady-state time of 0.8 ps due to its small area. Simulations revealed that the minimum transmission of logic 1 and the maximum transmission of logic 0 are 4% and 75%, respectively. However, the maximum transmission of 75% for logic 1 is not an ideal value. Several materials such as graphene, MoS 2 , carbon nanotubes, GaA, and bisphenol A have already shown the tunable platforms for desired applications Gao et al. 2020;Sui et al. 2021;Yang et al. 2020;Song et al. 2020;Li et al. 2018;Chen et al. 2019;Balaei et al. 2021;Tabrizi et al. 2021).
This paper proposes a novel design of an optoelectronic half adder/subtractor to overcome the mentioned issues. It is achieved by using graphene nanoshell (GNS) material 380 Page 4 of 16 in two PhC ring resonators. The maximum delay time is about 3 ps in the proposed structure, and the total footprint is about 303 µm 2 . In all studies, the resonant wavelength of PhC depends on the rod radius or refractive index. In contrast, in this study, the resonant wavelength depends on the applied voltage to graphene. Furthermore, compactness, tunability, and low time response are the key advantages of our proposed half adder. This paper is organized as follows: In Sect. 2, we introduce the GNS material, PhC ring resonators, and its optical behavior. In Sect. 3, the optoelectronic design and its functionality will be discussed, and the results of the current study are presented. The paper is closed by the conclusion in Sect. 4.

Mathematical background
The graphene sheet's optoelectronic features make it a popular candidate for designing integrated optoelectronic devices. Some of these exciting features are electronic conductivity, high optical transmission efficiency, broad tunability, and mechanical flexibility. In practice, research groups proposed integrating graphene on insulator substrates such as Si and SiO2. In this work, a monolayer graphene shell thickness is supposed to be 0.5 nm, and the value of d is 10 nm. In other words, 20 fold of monolayer GNSs coated together around the SiO 2 rod. Charge carrier density ( n c ) of a GNS on SiO 2 substrate can be tuned by applying the gate voltage (V g ) between the substrate and GNS. The charge carrier density is computed as the following equation (Thi et al. 2018;Shi et al. 2016) where 0 and are the air permittivity and substrate's relative permittivity, respectively. e and h are the electron charge and the substrate height. In Eq. 1, the amount of 0 r ∕eh is equal to C, which is called the gate capacitance. The GNS chemical potential (µ c ) is calculated as follows (Ju et al. 2011;Farmani et al. 2017a): where, ℏ is the reduced Plank's constant and v f is the Fermi velocity. The resonance wavelength of the GNS, λ 0 , can be derived from the quasi-static analysis method as  where eff is the effective permittivity of the medium surrounding the GNS that is eff = SiO 2 + 0 ∕2 , and d is the thickness of GNS (Farmani et al. 2017b). The dimensionless constant = 3.1 is a fitting parameter. The conductivity of a GNS ( ) at T = 300 0 k for infrared to THz frequencies is approximated using Drude-like expression as follows (Naghizade and Saghaei 2020a;Casiraghi et al. 2007) (1) where τ and ω are the electron relaxation time and the angular frequency, respectively. Also, when the number of GNS layer (N) is more than one, the conductivity for N-layer GNS will be Nδ. Recently, published papers showed that the finite-difference time-domain (FDTD) method is a comprehensive technique to study the propagation of electromagnetic waves inside compact optical devices such as PhC-based structures. This method solves Maxwell's equation in a tiny volume of the intended PhC device (Johnson and Joannopoulos 2001;Gedney 2011). In this study, the plane wave expansion (PWE) method calculates the PBG region. The FDTD method is also used to solve Maxwell's equation and study the waveguide's light propagation. The following equation gives the optical absorption of graphene (Farmani et al. 2020) where c is the speed of light and in free space, V is the graphene volume, and E l is the local electric field. According to Eq. (5), the light absorption is proportional to the square of local electric field intensity. The UV absorption in graphene is significantly increased at λ = 273 nm due to the large increase in the electric field.

Optoelectronic half adder/subtractor
The fundamental 2D PhC used in this study has a square lattice of dielectric rods. The rod radius and refractive index are 118 nm and 3.46, respectively. The plane wave expansion (PWE) method has been employed to calculate the photonic band structure. Figure 1 shows the fundamental PhC has two PBG regions in TM mode where the first one is at 0.29 < a∕λ < 0.42. In this study, the lattice constant is assumed to be a = 590 nm; therefore, an optical beam with a wavelength of 1404 nm < λ < 2034 nm cannot propagate in any PhC direction. By removing several dielectric rods in the fundamental PhC structure, a waveguide can be created. Accordingly, the wavelengths in the PBG region can be propagated in this Fig. 1 The band structure diagram of fundamental PhC in TM mode waveguide. A typical half adder contains two input ports (labeled as X and Y shown in Fig. 2a) and two output ports, called sum (S) and carry (C). Table 1 represents the truth table of a typical half adder. As shown in the figure, when both inputs are inactive (logic 0 is used for OFF state), i.e., no signal is applied to them, or the amplitude of the input signals is less than a threshold value, then both output ports turn OFF. If only one input port is turned on (logic 1 is used for ON mode), S is activated, and C is turned off. When both inputs are active, S turns off, and C turns on. Figure 2a also shows the proposed half  subtractor consisting of two input ports (labeled as X and Y), and two output ports (labeled as D and B), where D is used for the difference bit, and B stands for the borrow bit. This logic circuit operates subtracting between the two inputs X and Y as X-Y, and the results of both half adder and half subtractor operators are summarized in Table 1.
To design the proposed operator, we created two input waveguides by removing several dielectric rods called W1 and W2. They were connected to form another waveguide labeled as W3. Three dielectric rods with the same radius and smaller than the radius of the rods that make up the structure have the function of light coupling from waveguides W1 and W2 to waveguide W3. Afterward, two output waveguides were made by removing some dielectric rods and called W4 and W5. Finally, two ring resonators were placed between W3 and the output waveguides of W4 and W5. We replaced the outer rods of the ring with defect rods made of silica covered by GNS. The refractive index of silica is assumed to be 1.4. In Fig. 2a, silicon and silica rods are shown in red and dark green, and GNS shown in dark red surrounds silica rods to tune the resonant mode of light behavior inside the structure depending on the applied gate voltage (Vg). The radius of silica rods is 108 nm, and the thickness of GNSs is about 10 nm. As seen in Fig. 2a, X and Y are used for inputs, and SUM & D and CARRY & B are used as output ports. Figure 2b shows a 3D perspective view of the proposed structure, where the SiO 2 plate shown in gray is the substrate with a thickness of h = 2 µm. The number of SiO 2 rods covered by GNS layers in every ring resonator is 24, and tuning the output wavelengths is obtained by applying the Vg to GNSs in every ring resonator (The dark gray region shown in the inset of Fig. 2b is the gate bias voltage). The FDTD method is a numerical method for solving Maxwell's equation in the intended environment and studying electromagnetic wave propagation in PhC-based structures.

Simulation and results
At first, by choosing the chemical potentials of GNSs with µc 1 = 0.5 eV and µc 2 = 0.5 eV corresponding toV 1 and V 2 , we aim to study the proposed structure's resonance wavelength at the output ports. Figure 3 shows the ring resonators' output spectra corresponding to SUM &D and CARRY & B ports where there is a strong resonant mode at 1550 nm.
In this study, an input optical source centered at 1550 nm with a pulse width of 3.5 ps is launched into the proposed half adder/subtractor. The gate voltages (Vg) are also applied to to control and tune the outputs as desired. For OFF and ON working states, the values of µc set to 0.3 eV and 0.5 eV, respectively. In other words, for µc = 0.5 eV, the resonance wavelength of ring resonators is 1550 nm. Every half adder/subtractor has four different states, which are discussed in the following subsections.

Optoelectronic half adder
Case #1 When both input ports X and Y are inactive, meaning that the signal strength applied to them is negligible, no light beam enters the structure. Therefore, there is no change in the status of the outputs, and the outputs remain in the OFF state.
Case #2 and Case #3 when one of the input ports (either X or Y) is ON, due to wavelength matching between the resonance wavelength of the ring resonators and the input optical beam at the appropriate chemical potential of µc = 0.5 eV (It is performed by biasing the V 1 ), the first ring resonator will couple the optical beams into W4. Therefore, S turns ON, and C remains (see Fig. 4a, b).
Case #4 Once both input ports are ON (active), we will set the chemical potential (µc) of the second ring resonator (R2) to 0.5 eV by biasing V 2 . Consequently, the first resonator does not couple the optical beam because the V 1 is off. Instead, the second resonator transmits light rays to W5. Thus, S turns off and C turns on (see Fig. 4c). Table 2 summarizes the simulation results. Comparison of Tables 1 with 2 reaffirms the designed PhC structure operates as an optoelectronic half adder. Figure 5 shows the output diagram of the proposed half adder. As shown in Fig. 5a, when one input port is active (ON state), the output intensities of S and C ports are about 85% and 1%, respectively. In this case, the delay time is about 0.8 ps. Figure 5b shows, when both input ports are active, the normalized intensities of S and C are about 3% and 170%, respectively. In this case, the delay time is about 0.8 ps. It can be seen that the margins of structure for logics 0 (M 0 ) and 1 (M 1 ) are equal to 3% and 85%, respectively. It means that the contrast ratio is 14.5 dB since this ratio is calculated by 10 × log(M 1 /M 0 ). The results show that the proposed structure has a shorter delay time, a greater contrast ratio, and a smaller footprint than previously proposed structures.

Optoelectronic half subtractor
Case #1 When both input ports X and Y are inactive (OFF state), no optical beam enters the structure. Therefore, the outputs turn OFF.
Case #2 When the input port X is active (ON state), by applying a gate voltage to the GNSs in the first resonator, Wavelength matching between the resonant mode and the input signal occurs at a suitable chemical potential, so the first resonator resonates and transmits light rays to W4. As a result, D turns ON, and B turns OFF (see Fig. 6a).
Case #3 When the input port Y is active (ON state), by applying gate voltages (V 1 and V 2 ) to the GNSs in both resonators, wavelength matching occurs between the resonant mode and an input signal for chemical potentials of µc 1 = 0.5 eV and µc 2 = 0.5 eV, the first and second ring resonators resonate and couple the optical beams into W4 and W5. Fig. 3 The ring resonators' output spectra corresponding to SUM &D and CARRY and B ports Therefore, both D and B output ports turn ON. It means the generated code is 11 that is -1 for signed numbers (see Fig. 6b).
Case #4 When both input ports are ON, we will set the first and second ring resonators' chemical potentials to 0.3 eV by biasing the V 1 and V 2 . As a result, both resonators cannot couple the optical beam into output ports. Therefore, D and B will be OFF. (See Fig. 6c). Table 2 summarizes the simulation results, and a comparison of Tables 1 and 2 confirms the use of the designed structure as an optoelectronic half subtractor. Figure 7 demonstrates the output diagram of the proposed half substractor. As shown in Fig. 7a, when the first  input port is active and the second one is inactive (i.e., X = 1 and Y = 0), the normalized intensities of output ports D and B are 45% and 5%, respectively. In this case, the delay time is about 0.8 ps. Our simulation results in Fig. 7b illustrate the normalized intensities of D and B are identical with a value of 45% when the first input port is inactive and the second one is active (X = 0 and Y = 1). In this case, the delay time is about 0.8 ps. When both input ports are active (X = 1 and Y = 1), D and B's normalized intensities are less than 3%, and the delay time remains unchanged. The half subtractor margins for logics 0 and 1 are equal to 5% and 45%, respectively. It means that the contrast ratio is 9.5 dB. The results demonstrate that the proposed structure has a shorter delay time, a greater contrast ratio, and a smaller footprint than previously reported structures (Tables 3, and 4).
Our numerical results demonstrated that the proposed structure has a short delay time, flexible tunability, and a small footprint. The main advantage of the proposed device is its multipurpose applications compared to previously reported structures so far. It is not sensitive to optical input power variation for switching mechanisms based on the optical beams phase's nonlinear Kerr phenomenon. In addition, our results confirm that by setting an appropriate

Conclusion
In summary, we presented a new PhC-based structure for realizing all-optical half adder and half subtractor in a single structure. Five waveguides and two PhC ring resonators formed the final structure. Every PhC ring resonator consisted of several silicon rods surrounded by silica rods covered by GNSs. Our results revealed that the proposed structure's total footprint and maximum delay time are about 379µm 2 and 0.8 ps, respectively. Compared to other recently published papers, the proposed structure's main advantage is its tunability by changing graphene's chemical potential. Graphene-based logic gates open a new window to develop and fabricate the multipurpose optoelectronic circuits for signal processing systems applications.
Funding This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Availability of data and material
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflict of interest
The authors declare no conflicts of interest.

Ethical standards
The authors have completely observed the ethical issues, including plagiarism, informed consent, misconduct, data fabrication and/or falsification, double publication and/or submission, redundancy.