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.


Introduction
Coherent optical modulations, i.e., techniques of light modulations propagating in optical fibers using modulations of the amplitude, frequency, and phase of electromagnetic radiation, as well as signal transmission in two polarizations, were widely researched in the 1980s. It is worth mentioning that a receiver for extract the information contained in the phase modulations is termed as a coherent receiver.
The interest in these studies was because fiber optic links using coherent modulations can support longer lengths than links where are used OOK (On-Off-Keying) modulations without the use of optical amplifiers. It is noteworthy that in addition to the advantages expansion of the depletion zone, as well as the change of the density of free charges (electrons and holes), and the consequent modification of the refractive index value (Soref and Bennett 1987;Nedeljkovic et al. 2011;Liow et al. 2014), as it is showed in Fig. 1.
Note that a reverse continuous voltage, together with the modulating electrical signal, is applied in the PN junction of the two arms of the MZM. It is noteworthy that the modulating digital signal, shown in Fig. 1, could be replaced by a radio frequency signal (RF). That occurs when, for example, the MZM operates in RF communications systems over optical fibers, i.e., on the so-called analog photonics links (APLs) (Gutierrez 2013).
As we can see by Fig. 1, each rib-waveguide consists of a PN junction located in its center, where an n-silicon is in contact with strongly doped silicon (n+) in one of its sides. Likewise, a p-silicon is in touch with a p+ -silicon, and the silicon layers, n+ , and p+ are in contact with a metal electrode.
To obtain a phase modulation, the modulating electrical signal is transformed into a bipolar NRZ signal, so that bits with a positive voltage are applied to one of the arms of the MZI, and bits with negative voltage go to the other arm of the MZI. These signals, together with the reverse voltage, supply the voltages in the two arms of the MZI, in order to obtain the phase variation between the optical signals that propagate in the two arms of the MZI. Hence, as the variation in the refractive index of silicon occurs due to the plasma dispersion effect, there is a change in the value of the effective index (neff) of the propagating mode, which provides a change in the value of the signal phase, at the output of the waveguide. Therefore, changing the value of the refractive index of silicon alters the values of the effective refractive index of the optical modes that propagate in the arms of the MZM, causing a change in the propagation speed of these propagating modes, which, consequently, causes a phase difference between the optical signals at the MZM output (Ogawa 2014(Ogawa ,2015Koyama and Iga 1988;Goi et al. 2012). Indeed, the increase in the value of the refractive index is caused by increasing the reverse voltage, as in this case, the free charges move away from the depletion zone. On the other hand, the decrease in the value of the refractive index is 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 × Vπ (where L is the length of the traveling wave electrode (TWE)), the greater the modulation efficiency.
Note in the upper part of Fig. 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 (Soref and Bennett 1987;Reed et al. 2010), 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 (Goi et al. 2012). 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 Fig. 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 (Liow et al. 2014). For most modulators mentioned above, the traveling wave electrode has lengths between 3 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 Fig. 2 [adapted from 10]. This subject is detailed in Sect. 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 Sect. 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 Sect. 3. We close this manuscript, where the conclusions are (Sect. 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 Fig. 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 ( g = intra + inter ), given by (Gan et al. (2012); Vakil and Engheta 2011): where is the graphene's chemical potential, E F is the Fermi level, n is the charge density in graphene, V F = 1 × 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 (n Vg ), which occurs due to the application of an external voltage (we will return to this subject later), is given by (1) σ intra (w, μ) = ie 2 μ g ω + iτ −1 πℏ 2 ; (2) In the deduction of Eqs. 1 and 2, it was considered μ >> K B T, being admitted that the ambient temperature is T = 300 K so that K B T ≈ 26 meV (K B is the Boltzmann's constant, T is the temperature in Kelvin). Note that for μ = 0, intra = 0 , and = inter = e 2 4ℏ = 0 (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, ℏw inter ∕ g = 2 → w inter = 2 g ∕ℏ , suffer this type of attenuation (Wirth-Lima et al. 2020).

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 (η 0 ≈ 377 Ω), the photon wave number in the vacuum (k0), as well as the conductivity of graphene, given by (Wang et al. (2012)): Using Eqs. 6 and 7 (µ g = 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 Fig. 4a (real part) and b (imaginary part).
We considered that h-BN provides the mobility of graphene µ m = 60,000 cm 2 /Vs, and we adopted the value 3.4 for its dielectric constant (Dean et al. 2010;Laturia et al. 2018;Wang et al. 2017).
In Fig. 4c and d, we can see the values for the graphene's dielectric constant, considering its chemical potential in the range 0.3 eV ≤ µ g ≤ 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 (Luo et al. 2013;Wang et al. 2012): where 0 = √ 0 ∕ 0 ≈ 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 (Lei et al. 2014). It is noteworthy that the edge of the graphene nanoribbon ("armchair" or "zigzag") does not change the coupling process mentioned above (Lei et al. 2014).
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 (Schedin et al. 2010).
Analysis of the occurrence of GSPPs modes in graphene nanoribbons with widths ranging from 25 to 100 nm made it possible to obtain a method to determine the dispersion ratios of the GSPPs, considering the energy of these modes (Christensen et al. 2012).
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 L p = 1/2 Imag (β) (Luo et al. 2013;Christensen et al. 2012). Fig. 4 a Real part of the graphene dielectric constant versus wavelength (1.36 µm ≤ λ ≤ 1,625 µm; µ g = 0.55 eV). b Imaginary part. c Real part of the graphene dielectric constant versus chemical potential (0.3 eV ≤ µ g ≤ 0.9 eV; λ = 1.55 µm). d Imaginary part 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 (Wang et al. 2017;Xue et al. 2011). The dielectric constant of h-BN has a value close to the SiO2 (3.9). We have adopted ε hBN = 3.4 (Laturia et al. 2018;Wang et al. 2017;Xue et al. 2011).
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 (Goniszewski et al. 2016;Nistor et al. 2012;Moser et al. 2008) 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 (Tsukagoshi et al. 2011;Vasko and Zozoulenko 2010).
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 (V bg ) 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 Fig 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 V bg = 0 eV, graphene would have minimal conductivity (σ min ), or maximum resistivity (ρ max ). 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 (V D ), 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 (Lafkioti 2010;Kong 2000). Hence, the residual charge density value in graphene (n0) is the range from ≈ 5 × 10 10 cm −2 up ≈ 30 × 10 10 cm −2 (p-type doping) for graphene on SiO 2 /Si, and is also related to the impurity density of charges in SiO 2 (Pince and Kocabas 2010). As we can notice, even without the application of gate voltage, graphene has a residual charge density (n 0 ), which is added to the charge density provided through the application of gate voltage.
For a back gate voltage, considering the application of V bg , the density of graphene charge carriers is given by: where n Vg is the charge density acquired due to the application of a gate voltage, C ox = d 0 ∕d 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 (Xia et al. 2009;Arya et al. 2016;Das Sarma et al. 2011) can be neglected for values of the dielectric constant of the insulator ε d ≈ 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, HfO 2 . However, the most used solution is the application of a top-gate voltage, according it is showed in Fig. 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 SiO 2 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 (Xu et al. 2015): where 0 d is the electrical permittivity of the dielectric, hd is the thickness of the dielectric (h-BN), and V D is the Dirac voltage (We considered V D = 4.5 V, h d = 10 nm, and ε d = 3.4 in our calculations). Note that from Eq. 11, considering V g = 0, we can see that graphene has an intrinsic chemical potential μ g ≈ − 0.161 eV (p-doped).
It is importante to state, that there are different values in the international scientific literature regarding V D . For example, a graphene film on SiO 2 /Si presented a low Dirac voltage (V D < 4.5 V, the average value of around 2.5 eV (Xu et al. 2015)). However, in an another Research, where a graphene layer over h-BN/SiO 2 /Si, was used, the Dirac voltage presented value V D ≈ 5 V. On the other hand, for graphene on SiO 2 /Si the Dirac value was V D ≈ 25 V (Kayyalha and Chen 2015).

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 E in = E 0 e jwt and and the signals that propagate in the two arms of the silicon MZMs (in the x-direction) are given by, E 1 = E 0 e j(wt−k c1 x+ 0) ; E 2 = E 0 e j(wt−k c2 x+ 0) , where the complex wavenumber kc = β − jα, β = 2π/λ is the phase constant (rad/m), λ is the wavelength of the propagating mode, α is the attenuation constant, and ϕ0 is the initial phase. As ϕ 0 does not influence what is showed below, we disregard this parameter. Therefore, after the coupling of the two signals mentioned above (on the output coupler), then the resulting signal is given by (E out ): Then, considering the input signal with cosine shape, the output signal is given by: As an example, we are showing the graphics referring to the behavior of the phases of the MZM output signal as a function of time, regardless of attenuation, (concerning Ein), in Fig. 6, considering β 1x = 45º, β 2x = 90º rad, β 2x = 135º, and β 2x = 225º, for λ 0 = 1.55052 μm (one of the ITU-T band C wavelengths, in the air).
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 β 1x = 45º just to show the theory involved in this subject. However, in silicon MZMs, usually, β 1x = 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: In Fig. 7, we are showing the schematic representation of the modulator we are presenting. As we can see this modulator consists of only one waveguide embedded in an h-BN insulating layer, supported by a silicon substrate. Indeed, the waveguide is a graphene nanoribbon with length Lw embedded in h-BN, the top layer having a thickness of 10 nm, and the bottom layer 20 nm. The gate voltage system consists of a metal electrode/ graphene/h-BN junction.
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 Fig. 6 Cos(ϕ) versus t for the signal in the air, ϕ = 45º, ϕ = 90º, and ϕ = 180º Fig. 7 Schematic representation of the modulator consisting of electrode/graphene/h-BN metal used in the BPSK and QPSK modulators we are presenting 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 (Wirth Lima et al. 2015).
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 L w = n2 , where n is an integer, i.e., the number of periods (in the spatial domain) between the input and output signals.
Notice, in Fig. 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 K c as a function of the graphene's chemical potential (0.5 eV ≤ μ g ≤ 0.8 eV; λ 0 = 1.55052 μm), in the left part of Fig. 8.
We also plotted the graph regarding the propagation length, i.e., L p = 1/2 Imag (kc) versus µg, for the same range of values mentioned above (right part of Fig. 8). Hence, we have adopted the value μ g = 0.65 eV and found L p ≈ 396 nm, which is sufficiently greater than the value of the length of the waveguide that we adopted (L w = 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 Fig. 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.
(15) Δ = L w . For our device to work as a BPSK modulator, we determined the DC gate voltage, number of atoms in the propagation direction, length of the waveguide, maximal propagation length, number of periods, graphene chemical potential due to the electrical modulator signal, and gate voltage needed for the electrical modulator signal: V b = 12.016; N a = 407; L w = 50.061 nm; L wm = 345.17 nm; Np = 17, μ g2π = 0.644 eV; V 2π = -0.313 V, μ gπ = 0.638 eV; V π = − 0.621 V, respectively (for TM modes).
It is worth mentioning that we determined the gate voltage values from Eq. 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 ) (Muralia et al. 2009), 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 (λ 0 = 5.99585 μm) for the signal to be modulated, the values for the chemical potential due to the DC gate voltage, that DC voltage, number of atoms in the propagation direction length of the waveguide, maximal propagation length, number of periods, graphene chemical potential due to the electrical modulator signal, and gate voltage needed for the electrical modulator signal are: µ gb = 0.2 eV; V b = − 2.936 V; N a = 1220; L w = 150.06 nm; L wm = 211.52 nm; N p = 9, μ g2π = 0.195 eV; V 2π = − 0.078 V, μ gπ = 0.19 eV; V π = − 0.153 V, respectively (for TM modes).
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.
It is noteworthy that the application of V π/4 occurs for bits "00", V π for bits "01", V 5π/4 for bits "11", and V 7π/4 for bits "10", as we can see by Fig. 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 (Eq. 14), for x = 0 and x = L w . We considered E 0 = 1 V (this value has been selected only to simplify the calculations), referring to the application of E in and the amplitude for V π (k cVπ = 2.196439621844192 × 10 9 + 1.444759823745566 × 10 6 i), to prove what is detailed above, as we can see in Fig. 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 ϕ a = k 0 L a (where L a is the propagation distance in the air), and in the waveguide is given by ϕ g = real(k cg )L g (where L g is the propagation distance in the waveguide), it is easy to see, that ϕ g = ϕ a real (k cg )L g /k 0 L a . 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 L w = 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.
The graphene-based modulator we are presenting has enormous advantages over conventional silicon modulators that operate through PN-junction rib-waveguide phase shifters. Note that in addition to the system being much simpler, it is cheaper, as only a waveguide and the gate voltage system are used. Furthermore, the signal modulation is much faster because, in the TWE, it is necessary, the movement of electrical charges via diffusion, in part of the operation, for the alteration of the refractive index through the modulator signal. On the other hand, in our device, we get the phase modulation by loading of the capacitive system graphene/h-BN/metallic electrode. Hence, the bit rate can be much higher than what silicon modulators can handle. Indeed, concerning the modulators that we are currently presenting, that transmission rate is limited by the speed of current electronic devices. It is worth mentioning that obtaining modulating signals with a bit rate greater than 100 Gbps is still a challenge. However, this bit rate evolved from 2.5 Gbps (Wirth Lima et al. 2015) in 2003 to more than 100 Gbps, between 2003 and 2010, and in 2010, bipolar SiGe transistors providing 500 GHz were already being researched, using BiCMOS technology (bipolar and MOS technology on the same chip) (Riishøj 1993;Möller 2010). As new technologies are emerging, we can say that the transmission rate in the Terabits range is possible.
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 (Miyazaki et al. 2010). 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.