Graphene-based BPSK and QPSK modulators working at a very high bit rate (up Tbps range)

We are presenting graphene-based Binary Phase Shift Keying (BPSK) and Quadrature Phase Shift Keying (QPSK) modulators, which can operate in the range from the TeraHertz up to the infrared. It is noteworthy that these devices have huge advantages over the silicon Mach-Zehnder optical modulators (MZMs) with lateral PN-junction rib-waveguide phase shifters. Among the countless advantages, we can mention, for example, that these modulators consist of only one waveguide and have a much simpler application system of the modulator signal (gate voltage) than in silicon-based MZMs. Other huge advantages are greater efficiency, and yet, they are cheaper and have shorter lengths (and consequently, greater integration in photonic integrated circuit (PIC)). The first step to present these modulators was to detail the graphene theory that is involved in this device. After this step, we show the project, numerical simulations, and analyses related to our graphene-based BPSK and QPSK modulators. We believe that these modulators will contribute to the generation of new devices made up of 2D materials, which should revolutionize this area of science.

provided with the decrease in the reverse voltage, since in this case, more load carriers go to the depletion zone.
The voltage between the two arms of the MZI that provides a π-phase shift (Vπ) determines the efficiency of the modulation, i.e., the lower the value L x Vπ (where L is the length of the traveling wave electrode (TWE)), the greater the modulation efficiency.
Note in the upper part of Figure 1, the schematic representation of the simplified equivalent circuit of PN junctions for this push-pull operation for zero-chirp modulation, through a TWE. Generally, the lengths of the two arms of the MZM are of the same length (symmetrical MZM), to avoid the occurrence of greater insertion loss in the longer arm.
A disadvantage of this type of modulation is that the change in the refractive index occurs in a non-linear manner [4,11], causing signal distortion, due to several factors. Therefore, in this type of modulation, the linearity of the phase shift and the minimization of the carrier-induced optical loss are of great importance for obtaining a good performance of the MZM [9]. Hence, the MZM has to be biased at the quadrature point (π/2), which can be achieved with a bias voltage Vbias = Vπ/2 i.e., with a phase shifter of π/2. Note that, in this case, in one of the arms, the resultant voltage value due to the modulating signal can be null (-Vπ/2 + Vπ/2; bit 1), while in the other arm can be -Vπ/2 -Vπ/2 = Vπ (bit 0), as we can see by Figure 1. Moreover, the traveling wave electrode must be designed to match the group velocity of electrical signal to the group velocity of the light. The velocity matching allows high-speed modulation required in the range from 40 to 100 Gb/s [6]. For most modulators mentioned above, the traveling wave electrode has lengths between 3 mm to 4 mm.
It is noteworthy that a single MZM shown above can be used for BPSK modulations.
However, to obtain QPSK modulation, two nested MZMs are used, as it is shown in Figure 2 [adapted from 10]. This subject is detailed in section 3.
We are presenting BPSK and QPSK modulators consisting of graphene-based waveguides, which can replace PN-junction rib-waveguides, with huge advantages. That modulators consist of only one waveguide and one modulator signal application system, which is much simpler than in the MZMs mentioned above. In addition to being more efficient and cheaper, they have shorter lengths, and consequently, greater integration in photonic integrated circuits (PICs). This is possible because graphene has free electrons in plasmonic undulations so that to change its dielectric constant (and consequently its refractive index), the external voltage can be applied directly to graphene (gate voltage).
Then, the change in the propagation constant of graphene occurs more quickly. It is worth mentioning that to increase the efficiency concerning to the contact area with graphene, we replaced silica with h-BN (Hexagonal Boron Nitride).
This work is constituted as follows: In section 2, we detail the part of physics encompassing graphene, which is necessary for detailing the graphene-based modulators, which we are presenting. We detail the BPSK and QPSK modulators, showing the design, involved theory, analysis via numerical simulations, and advantages of these modulators, in section 3. We close this manuscript, where the conclusions are (section 4).

Graphene
In order to be able to detail the BPSK and QPSK modulators consisting of graphene, we will show, in this section, the necessary theory regarding this 2D material.
The planar monolayer honeycomb lattice of graphene, consisting of two primitive lattice vectors, is shown in Figure 3.

(Figure 3)
It is worth noting the fact that for the most accurate possible determination of the lengths of graphene nanoribbons used in the devices we are presenting, it is necessary to take into account the distance between the carbon atoms in their atomic structure. As the lateral ends of the graphene are of the zigzag type, the lengths of the nanowires are a multiple of 0.123 nm.

Physical parameters for the graphene used in our nanophotonic modulator
In this section, we show the Equations that determine the operation of graphene and the respective control of the modulator we are presenting.
Optical conductivity of graphene After some mathematical manipulations, we can arrive at the mathematical expressions that determine the intraband and interband conductivity of graphene (σ = σ + σ ), given by [12,16]: is the graphene's chemical potential, EF is the Fermi level, n is the charge density in graphene, VF = 1 x 10 6 m/s is the Fermi velocity, ℏ is the reduced Planck's constant, e is the electron charge, and w is the angular frequency.
It is noteworthy that τ is the electron relaxation time related to graphene, given by μm being the charge mobility in graphene.
The charge density in graphene (nVg), which occurs due to the application of an external voltage (we will return to this subject later), is given by Note that for μ = 0, σ = 0, and σ = σ = ℏ = (universal conductivity). In the deduction of Equations 1 and 2, it was considered μ >> KBT, being admitted that the ambient temperature is T =300 K so that KBT ≈ 26 meV (KB is the Boltzmann's constant, T is the temperature in Kelvin). Note that for μ = 0, σ = 0 and σ = σ = ℏ = (universal conductivity).
To avoid losses due to the change in conductivity caused by interband transitions (firstorder process), it is necessary to block these interactions by increasing the value related to the graphene's chemical potential. In this way, only surface plasmons polaritons in graphene (GSPPs) with a frequency above the limit, that is, ℏ ⁄ = 2 → = 2 ℏ ⁄ , suffer this type of attenuation [17].

Electrical permittivity of graphene
Considering the graphene's thickness t = 0.34 nm, its effective relative permittivity is given by: Furthermore, there is another Equation related to the effective relative permittivity, which takes into account the air impedance (η0 ≈ 377 Ω), the photon wave number in the vacuum (k0), as well as the conductivity of graphene, given by [15]: Using Equations 6 and 7 (µg = 0.55 eV), we plotted the values of the graphene's dielectric constant, considering the graphene embedded in a Hexagonal Boron Nitride (h-BN), and wavelength range 1.36 µm ≤ λ ≤ 1.625 µm (E, S, C, and L bands of ITU-T).
Then, we found that the results were identical, according to what is showed in Figures 4a (real part) and 4b (imaginary part).
In Figures 4c and 4d, we can see the values for the graphene's dielectric constant, considering its chemical potential in the range 0.3 eV ≤ µg ≤ 0.9 eV and wavelength λ = 1.55 µm.
Wavenumber related to the wavevector in the direction of propagation (propagation constant) in a single layer of graphene The complex wave number, in the direction of propagation, referring to a graphene nanoribbon embedded in a single dielectric to graphene, for GSPP TM modes is given by [14,15]: where = ⁄ ≈ 377 Ω is the intrinsic impedance of the free space.
On the other hand, for GSPP TE modes, the complex wave number in the direction of propagation for these referred GSPP TE modes is given by: Graphene nanoribbon working as a waveguide GSPPs in a layer of graphene, acting as a waveguide, can be obtained through direct coupling between photons emitted by an optical emitter and surface plasmons (SPs) located on that layer. In a Research, it has been demonstrated that the optical emitter located on the same plane as the graphene layer (with polarization in the direction of the width of the graphene strip) provides coupling between photons emitted by the emitter and GSPPs, ten times greater than the emitter positioned over the graphene layer [22]. It is noteworthy that the edge of the graphene nanoribbon ("armchair" or "zigzag") does not change the coupling process mentioned above [22].
GSPPs can occur in structures wider than 10 nm since, for dimensions below this value, plasmons occur into several resonances due to the characteristics of the carbon structures, as well as the quantum nature of their optical excitations [23].
Analysis of the occurrence of GSPPs modes in graphene nanoribbons with widths ranging from 25 nm to 100 nm made it possible to obtain a method to determine the dispersion ratios of the GSPPs, considering the energy of these modes [24].
The propagation length of the GSPPs modes (the distance that a GSPP mode travels, until it drops to 1/e from its initial intensity) is given by Lp = 1/2 Imag (β) [14,24].
We considered the graphene layer embedded between two layers of hexagonal boron nitride (h-BN). The reason for choosing h-BN is that the roughness of the h-BN layer is much less than that of the silica surface. Therefore, a graphene layer located between two layers of h-BN has much higher charge mobility and homogeneity than embedded in SiO2. Additionally, as there are few charge traps on the graphene/h-BN interface, the electronic properties of the device have much better values than on the graphene SiO2 interface [20,25]. The dielectric constant of h-BN has a value close to the SiO2 (3.9). We have adopted εhBN = 3.4 [19,20,25].
Gate voltage applied to a graphene waveguide The two most used methods for obtaining control of the physical/optical parameters of graphene is through chemical doping [26,27,28] and via gate voltage.
An efficient way to controlling of the chemical potential of graphene is to use a gate voltage. For example, it is possible to localize the Fermi level for graphene in the valence band, applying a positive voltage to the graphene surface. On the other hand, the application of a negative voltage raises the Fermi level of graphene for the conduction band. It occurs because of the effect of the electric field so that the metallic electrode/h-BN/graphene region operates similarly to a capacitor [29,30].
There are three ways to apply a gate voltage to graphene: back voltage, top voltage, and back/top voltage. Figure 5a shows an example of a scheme for obtaining a back gate voltage (Vbg) between silicon and a graphene nanoribbon. As we can see, the graphene nanoribbon was deposited on a substrate (in this case, silicon oxide) with a well-defined thickness, supported on highly doped silicon.
Taking into account that highly doped silicon works similarly to a metallic contact, silicon dioxide is insulating, and graphene is a zero-gap semimetal, we can say that an electric field effect similar to that inside a capacitor of parallel plates occurs. Hence, it is possible to control the charge density in graphene through the application of gate voltage.
We can use back gate voltage in graphene-based nanophotonic devices, such as, for example, a transistor, as we can see in Figure 5a (adapted from ref. [31]).
( Figure 5) Therefore, the position of the Fermi level, whose value is practically the same as the chemical potential of graphene, can be controlled, in this case, through back gate voltage.
Note that while applying a positive voltage to silicon (type p, highly doped) and a negative voltage to graphene causes electron transfer to graphene, and holes to silicon, applying a negative voltage to silicon, and a positive to graphene, occurs the transfer of holes to graphene and electrons to silicon.
It is worth noting that considering graphene as ideal, for Vbg = 0 eV, graphene would have minimal conductivity (σmin), or maximum resistivity (ρmax). However, in reality, considering real graphene, minimum conductivity does not occur in this condition because of the intrinsic doping of charges in graphene from the environment, and the device's manufacturing process, among other factors.
The value of the Dirac voltage (VD), is definided as the value of the gate voltage where the minimum conductivity (or maximum resistivity) of graphene occurs. The influence in the value of Dirac voltage depends on graphene manufacturing and cleaning process, as well as the interface itself, in which there are energy levels that behave as acceptors/charge donors [32,33]. Hence, the residual charge density value in graphene (n0) is the range from ≈ 5 x 10 10 cm -2 up ≈ 30 x 10 10 cm -2 (p-type doping) for graphene on SiO2/Si, and is also related to the impurity density of charges in SiO2 [34]. As we can notice, even without the application of gate voltage, graphene has a residual charge density (n0), which is added to the charge density provided through the application of gate voltage.
For a back gate voltage, considering the application of Vbg, the density of graphene charge carriers is given by: where nVg is the charge density acquired due to the application of a gate voltage, = ⁄ is the capacitance per unit area for the graphene/insulator/Si structure (dielectric capacitance), and d and εd are the thickness, as well as the dielectric constant of the insulator layer, respectively.
It is noteworthy that the quantum capacitance of graphene [35,36,37] can be neglected for values of the dielectric constant of the insulator εd ≈ 4, gate voltage bigger than a few millivolts, thickness of the insulator bigger than a few Angstroms. It is worth mentioning that, in the back voltage configuration, the thickness of the insulator is generally greater than 200 nm.
As in the back voltage configuration, the electric field has a low value it is necessary to apply large potential differences (PD) to obtain moderate increments of charge density.
On the other hand, the application of high PD values (PD > 100 V) can cause damage to the insulator layer. Therefore, taking into account the above, we can affirm that the back gate voltage process is limited. To overcome this problem it is necessary to decrease the thickness of the insulator layer, but obtaining thinner layers of this dielectric requires more advanced techniques. Another solution is to use a dielectric with a high dielectric constant value, such as, for example, HfO2. However, the most used solution is the application of a top-gate voltage, according it is showed in Figure 5b.
In this top-gate structure, a graphene nanoribbon is deposited on silicon, and the insulator is deposited above the graphene, which is in contact with the electrode ("gate electrode"). An option to further increase the efficiency of the top-gate voltage is to replace SiO2 with h-BN, the thickness of the h-BN layer being thin enough to increase the generation of charges in graphene (usually between 10 at 20 nm).
We used that gate voltage configuration in our devices because besides being more efficient, the intrinsic charge of graphene (+) is added to the charge generated by the gate voltage (positive too), which contributes to a lower applied voltage.
The Fermi level of graphene due to the applied voltage is given by [38]: where is the electrical permittivity of the dielectric, hd is the thickness of the

Graphene-based BPSK and QPSK modulators
Before we start detailing the modulators we are presenting, let's make an analytical comparison between the silicon-based modulators based on a traveling wave electrode and our modulators based on waveguides made of graphene.
The signal considered at the entrance of the MZM is given by  Figure 6, considering β1x = 45º, β2x = 90º rad, β2x = 135º, and β2x = 225º, for λ0 = 1.55052 μm (one of the ITU-T band C wavelengths, in the air).
( Figure 6) Note that in these cases, the MZM output signals are out of phase related to the input signal by ϕ = 45º, 90º, and 180º (destroyed), respectively. Also, note that although we are only interested in phase modulation, the output signal also has amplitude modulation.
We considered β1x = 45º just to show the theory involved in this subject. However, in silicon MZMs, usually, β1x = 90º since, in this way, the MZM output signals have more symmetrical lags concerning the input signal.
It is worth mentioning that, in general, the reverse voltage values for the modulator signals applied in silicon-based MZMs are determined experimentally.
Without loss of generality, we considered a monochromatic cosine signal continuous wave (CW) laser to simplify the technical details of the modulator we are presenting. Therefore, the output signal, considering only a waveguide with length Lw, is given by: We can obtain the phase shift between the input and output signals by varying the complex wavenumber using a gate voltage. Thus, we obtained the phase shifts, as well as the optical losses suffered by the optical signal that propagates inside the waveguide, through the complex wavenumber referring to the GSPP mode in the direction of its propagation. In this graphene-based modulator, non-linear effects were not considered, since the amplitude of the inserted optical signal is chosen, so that these effects are avoided [40].
Taking into account that we will obtain the phase shift in the spatial domain, the phase difference Δϕ between the waveguide output signal (which suffers delay) and the inserted signal is given by: Therefore, we can get the same phase as the inserted signal, considering where n is an integer, i.e., the number of periods (in the spatial domain) between the input and output signals.
Notice, in Figure 7, that for the device we are presenting to operate as a phase modulator, firstly, we should insert a continuous voltage (DC) into the graphene, which works as a waveguide. This voltage (gate voltage) determines the chemical potential necessary for obtaining the real part of the complex wavenumber for GSPP TM modes.
As we can see, it is shown the real and imaginary parts of Kc as a function of the graphene's chemical potential (0.5 eV ≤ μg ≤ 0.8 eV; λ0 = 1.55052 μm), in the left part of We also plotted the graph regarding the propagation length, i.e., Lp =1/2 Imag (kc) versus µg, for the same range of values mentioned above (right part of Figure 8). Hence, we have adopted the value μg = 0.65 eV and found Lp ≈ 396 nm, which is sufficiently greater than the value of the length of the waveguide that we adopted (Lw = 50.061 nm), as we will show further.
We used the parameters mentioned above for the initial operation of the modulator and we applied the changing of the chemical potential, using the electrical modulator signal, to provide the desired phases of the output signal, as shown in Figure 8. It is noteworthy that we also found, numerically, the values referring to the maximal propagation length (Lpm) that GSSPs modes can propagate, with attenuation within the acceptable limit.
Therefore, we can be sure that the output signal suffers no attenuation, i.e., it will have sufficient amplitude.
For our device to work as a BPSK modulator, we determined the DC gate voltage, It is worth mentioning that we determined the gate voltage values from Equation 11.
It is relevant to state that Np is applied to all periods in the temporal domain related to a bit of that modulated signal, taking into account the duration time of one bit of the modulator signal.
Concerning the resistance of graphene, as a function of the applied voltage, a graphene nanoribbon can withstand voltages up to the value that causes its breakdown current density, that is, (10 8 A/cm 2 ) [41], therefore much higher than the values we used.
Another great advantage of graphene is the possibility of using the signal to be modulated in the THz frequency range. For example, considering the frequency 50 THz However, the main advantage regarding the modulator we are presenting is the fact that it can also operate as a QPSK modulator. That is because changing the complex wavenumber of graphene via gate voltage makes it possible, in an efficient way, the 45º, 135º, 225º, and 315º phases (referring to dibits 00, 01, 11, and 10, respectively) without the need for obtaining the phase difference between two BPSK signals, i.e., without the need for a thermo-optic (TO) phase shifter.
Using the same length as the waveguide used in the BPSK modulator, we determined the parameters for this QPSK modulator (for λ0 = 1. It is noteworthy that the application of Vπ/4 occurs for bits "00" Vπ for bits "01"and V5π/4 for bits "11", and V7π/4 for bits "10", as we can see by Figure 7. Note, also, that in both cases, occurs very small attenuations. To prove what is detailed above, we plotted two periods related to the amplitude versus time (Equation 14), for x = 0 and x = Lw. We considered E0 = 1 V (this value has been selected only to simplify the calculations), referring to the application of Ein and the amplitude for Vπ (kcVπ = 2.196439621844192 x 10 9 + 1.444759823745566 x 10 6 i), to prove what is detailed above, as we can see in Figure 9.
( Figure 9) Complementing this subject, as the phase difference in the air (as a function of the propagation distance, considering the spectral domain), is given by ϕa = k0La (where La is the propagation distance in the air), and in the waveguide is given by ϕg = real(kcg)Lg (where Lg is the propagation distance in the waveguide), it is easy to see, that ϕg = ϕareal (kcg)Lg/ k0La. So, for example, considering that the wavelength in the waveguide is half the value of the wavelength in the air, to obtain a phase difference Δϕ = π, between the wave propagating in the air and the wave propagating in the waveguide, the waveguide length must be Lw = 1.16289 µm, considering only a period of the wave propagating in the air.
In the data reported above, we opted for the strict control of the gate voltage to obtain the necessary values for each phase change. It is noteworthy that our QPSK modulator can also operates in the TeraHertz range. In experiments on graphene field-effect transistor, the source and drain electrodes were manufactured by electron-beam lithography and thermal evaporation of the metals, with Au (50 nm thick) and Ti (5 nm thick), with resistance less than a few kΩ. For the gate voltage electrode, the aluminum electrode was 30 nm thick and was deposited directly on graphene. However, it was found that between graphene and the electrode a natural insulator appeared after that device was exposed to the air for several hours [44]. This natural oxide, with a dielectric constant between 5 and 9, as well as the thickness between 5 and 9 nm, was used as a insulating layer between the electrode and graphene. We can use this aluminum electrode to apply the gate voltage over the h-BN, but in our case, this natural oxide must be avoided to improve the efficiency of the gate voltage system using h-BN as the insulation layer.

Conclusions
We are presenting a graphene-based BPSK and QPSK modulators, which can operate from the TeraHertz range, to the infrared and has enormous advantages over the silicon modulators with lateral PN-junction rib-waveguide phase shifters. For example, these graphene modulators have only a much simpler waveguide and gate voltage application system than those used in silicon-based MZMs. Besides, they are more efficient, cheaper, and have shorter lengths (and consequently, greater integration in photonic integrated circuit (PIC)).
In this manuscript, we first detailed the issues relating to the theory involving silicon MZMs and graphene, which is need to detailing of the modulator we are presenting. After this step, we show the project, numerical simulations, and analyzes concerning this device.
We believe that this modulator is part of the new generation of devices made up of 2D materials, which should revolutionize this area of science.

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