Plasma wave resonances in Graphene channels under controlled gate for high frequency applications

Several devices based on 2D materials have become interesting for high-frequency applications especially sensors, amplifiers and modulators of Terahertz frequencies. Moreover, High Electron Mobility Transistors (HEMTs) have emerged as important competitors owing to the high quality of resonances associated with plasma-wave oscillations in the channel. In this study, the plasma wave resonances in a Graphene channel of High Electron Mobility Transistors (HEMTs) were studied. The calculations were based on our small-signal model and therefore can determine the resonances and voltage gain of the Monolayer and Bilayer graphene channels. The influence of the dielectric substrates between the gate and the channel, impurity depth positions and channel materials (InGaAs, Bilayer and Monolayer graphene) on the dynamic behavior of the Graphene transistor was investigated. This analysis can extract the high performance conditions of HEMTs Graphene amplifiers.


Introduction
In recent years, several studies have shown that two-dimensional (2D) materials are the best candidates for terahertz devices [1][2][3][4][5], particularly for optical communications devices [6], nanodetectors coupled with THz antennas [7] and phototransistors based on black phosphorus materials [8]. The superior properties of 2D materials lie in the oscillations of plasma waves under conditions of high carrier mobility [2][3][4][5]. One of the 2D material and most important then the 2D electron gasis because of its high electronic mobility, unique structure of massless Dirac fermions, weak electron-phonon interactions and unique ballistic transport [9,10]. This makes graphene a good candidate for saturable absorption in different optical devices [11,12], infrared devices, electro-optic devices, and field effect transistors (FETs) for radio frequency applications [13].
Furthermore, High Electron Mobility Transistors (HEMT) have important electrical properties for several terahertz applications, such as detectors, emitters and amplifiers [14][15][16]. The study of HEMTs based on InGaAs material shows the existence of THz plasma-wave oscillations in the channel which leads to the appearance of terahertz resonances in the small-signal admittance spectrum as refered [17][18][19][20]. High resonances in the smallsignal response appeared at frequencies up to 7 THz when a short gate covered the total nano-channel length of the transistor [21,22]. For the THz antenna and detection, the analytical [17,23] and experimental [24,25] studies demonstrate the possibility of amplifying some terahertz frequencies (no more than 2 THz) by the InGaAs HEMTs when the resonant plasma frequency coincides with the frequency of the incoming monochromatic THz radiation. The analysis of HEMTs based on InGaAs material has been described in [14,17] and focus their behavior on terahertz frequencies. Several high-frequency HEMTs applications, sensing and amplification require the performance of resonances based on strong plasma mode oscillations in the transistor channel. Therefore, we seek to improve the quality of the resonances by using Graphene as a channel to provide multiple choices for HEMTs applications in the high-frequency domain.
In this study, analytical calculations are proposed to describe the behavior of HEMTs with graphene channels. The calculations were based on the small-signal model developed for InGaAs HEMTs in [17]. The model was used to determine the response and voltage gain spectra in THz frequencies for different scenarios of Graphene materials (Graphene Monolayer (GML) and Graphene Bilayer (GBL)) and dielectric substrates (H 2 O, Ethanol (160K), HfO 2 and SiO 2 ). The effects of different graphene parameters (relaxation rate, effective mass and dielectric permittivity) and the impurity position in the dielectric substrate were investigated.

Analytical model
We consider the HEMTs as a channel of bulk Graphene entirely covered by the gate where the channel is separated from the gate by a dielectric substrate. The structure is schematically shown in figure 1 where the dielectric of distance d between the channel and gate is SiO 2 . The transistor channel is characterized by a length L, thickness δ = 15 nm, gate-to-channel distance d and electron concentration n 0 .
As a reference, the source potential is zero ΔV s = 0 where the drain-source and gate-source voltages are ΔV d and ΔV g , respectively.
The carrier oscillations behavior in the channel is controlled by the gate voltage ΔV g which leads to the control of the voltage amplification in the following calculations. A numerical description of the transistor was obtained using a one-dimensional analytical equations when we neglect the dependence of any physical phenomenon on the thickness of Graphene (one-dimensional Graphene structure).
The admittance elements Y dd and Y dg were determined using the hydrodynamic model coupled with a pseudo-two-dimensional (P2D) Poisson equation developed in [23,26,27]. The admittances spectra are obtained through the relation of drain current-to-ΔV d and ΔV g voltages: According to [28], the expressions of the admittance elements are given by : is the free-carrier 3D plasma frequency, is a parameter describing the transverse geometry of the device, β 2 = w wn w w w n - where ν is the relaxation rate, m * is the effective mass, d is the distance between the channel-gate and ò c,d is the dielectric constants in the channel and in the dielectric between the channel and the gate, respectively.
As that studied in [17], the voltage gain spectrum is determined when the transistor is loaded with an external charge Y C introduced between the source and drain contacts. The spectral voltage gain was studied for a real Graphene device characterized in [29] for Terahertz range as shown in figure 2.  In this case, the drain current is given by the equivalent electronic transistor-load circuit as: The voltage gain was obtained using equations (1) and (2), respectively: In this article, the analysis of the Graphene HEMTs response is based on Graphene parameters such as relaxation rate ν, effective mass m * and permittivity constants ò c,d .

Admittances and voltage gain of HEMTs graphene
In this section, we analyze the Terahertz behavior of HEMTs Graphene channels with the dielectric substrate SiO 2 . The results of figure 3 are obtained for the graphene-SiO 2 where the effective mass m * = 0.021 m 0 [30], the relaxation rate ν = 0.5 ×10 12 s −1 [31], the permittivity constants of the dielectric SiO 2 and the graphene are ò c = 4 and ò d = 2.3823 [32], respectively. Figure 3 shows a series of resonance peaks appearing at frequencies up to 20 THz. The resonances in the admittances spectra Y dd and Y dg are associated with the excitation of plasma modes in channel of HEMTs. The admittance Y dd presents all resonances corresponding to all excitation modes when only the odd resonance appears in the Y dg spectrum. Therefore, resonances originating from the oscillations modes in the channel can be controlled by the gate.
The voltage gain spectrum calculated using equation (3) and reported in figure 4 for different real charge Y C The voltage gain exhibits resonances at frequencies up to 20 THz. Graphene HEMTs can be used as a voltage amplifier in a high range of terahertz frequencies where the amplification is assisted by plasma resonances. For the high admittance Y C , the gain spectrum exhibits resonances of odd modes corresponding to Y dg behavior; therefore, the potential is fixed at the channel extremities and symmetrical boundary conditions are achieved. In the absence of charge load and approaching open-circuit conditions (Y C ≈ 0), a shift of the resonance frequencies is observed, and the case of an asymmetrically in the channel appears.

Effect of dielectric substrates on HEMTs graphene response
We will focus in this section the behavior of graphene HEMTs on different dielectric substrates: H 2 O, Ethanol (160K), HfO 2 and SiO 2 . The change in dielectric substrates introduces a variation in two physical terms: relaxation rate and permittivity according to tables 1 and 2, respectively [31,32].
Scattering time for different dielectric graphene extracted from [31].
In order to analyze the effect of the dielectric substrates on the admittances Y dd and Y dg , we kept all the Graphene parameters constants (relaxation rate ν = 10 13 s −1 , effective mass m * = 0.192 × 10 −31 Kg, dielectric constants in the channel and the length L) and we modified separately the dielectric constant values of H 2 O, Ethanol (160K), HfO 2 and SiO 2 (table 1). Figure 5 illustrates the modification introduced by different dielectric substrates on admittances Y dd and Y dg . As shown in figure 3, we observe a series of resonance peaks caused by plasma wave oscillations along the channel where the resonant frequency depends on the free-carrier 3D plasma frequency ω 3D . It can also be noticed that the amplitude of the resonance peaks decreases and can disappear when the dielectric between the channel and the gate is changed (see the solid lines in figures 5(a) and (b)). Therefore, high-quality resonances were obtained in the case of SiO 2 between the channel and gate. The appearance of this quality of resonances is related to the decrease in the SiO 2 dielectric constant ò c and therefore the increase in bowth: the gate control and the free-carrier 3D plasma frequency are given by the analytical terms k y and ω 3D , respectively.
However, the resonances disappeared when the Graphene channel was coupled with dielectric substrates H 2 O, Ethanol (160K), HfO 2 . Therefore, all the dielectrics H 2 O, Ethanol (160K), HfO 2 present the same behavior effect on the terahertz Graphene HEMTs response (absence of plasma oscillations modes).
For a real study, the results are extracted for the relaxation rate and the dielectric constants of HfO 2 and SiO 2 according to tables 1 and 2. The choice of HfO 2 had the same effect as that shown by H 2 O and Ethanol (160K) on the admittance spectra in figure 5.
The response of HEMTs Graphene-HfO 2 and Graphene-SiO 2 is reported in figure 6. We observed a high amplitude of resonances associated with the high oscillations plasma modes in the Graphene channel with SiO 2 substrate. Therefore, good gate control is obtained by the effective substrate SiO 2 (see figure 6(b)). The dielectric SiO 2 presents a low value for both the permittivity ò c and the relaxation rate ν, compared to HfO 2 (see tables 2 and 3), which can produce a high free-carrier 3D plasma frequency (resonances  appearing at high frequency) and high carrier mobility (significantly increasing the quality of the resonance), respectively.

Impurity effect on the graphene HEMTs response
In this section, we consider that the impurity located at a distance Z in SiO 2 substrate is the highest in the Graphene region as shown in figure 7. The scattering time is given in table 3 for monolayer n-type Graphene with a surface electronic density n = 10 12 cm −2 according to the [33].
In other hand, the effective mass of monolayer graphene is calculated by : Where v F is the Fermi velocity istimated as 1.1 × 10 6 m/s [34], and E F = 0.12 eV is the Fermi energy. The effective mass of monolayer Graphene depends on the Fermi energy which significantly depends on the impurity   High-quality resonances were obtained when the impurity was far from the SiO 2 -graphene interface. Therefore, a low amplitude of resonances appears for a decrease in the position Z by conserving the oscillations plasma modes (the odd and all resonances). A significant variation in the scattering time τ was observed when the impurities were located at several positions. The scattering time decreases with increasing of the position Z (τ decreases when the impurities are far from the interface SiO 2 -Graphene). The impurity position Z effect relies on the influence of the relaxation rate which directly depends on the variation in the scattering time. where the electron mobility of GBL is μ e = 2.5 × 10 5 cm 2 V −1 s −1 and the mobility of GML is 1.43 × 10 6 cm 2 V −1 s −1 [34]. Therefore, the exact value of the relaxation rate of the Graphene bilayer considered in this section is ν = 0.17 ×10 12 s −1 [35].

Response of graphene Bilayer channels HEMTs
The responses of the Graphene Bilayer (GBL) and Graphene Monolayer (GML) channel HEMTs are shown in figure 9.
We observe a slight displacement of resonances towards the low terahertz frequencies for the GBL channels associated with the decreases in the 3D plasma frequency modes which depends on the effective mass. A significant decrease in the amplitude of the resonances corresponds to a decrease in the relaxation rate of the GBL compared to that of the GML (see figure 9). In |Y dg | spectrum, the control of the gate decreases at a high terahertz frequency, therefore the frequency positions and the amplitude of resonances decrease. Figure 10 illustrates the voltage gains of the HEMTs with InGaAs, GML and GBL channels. The results in figure 10 can be used to analyze the material effect on HEMTs amplification through the physical parameters m * , ν, ò d and ò r of InGaAs, GML and GBL materials.  The results in figure 10 show the amplifier process realized by different materials considered as channels of the HEMTs. An ideal amplifier can be obtained by a Graphene Bilayer for low terahertz frequencies (less than 2 THz) and by a Graphene Monolayer for high Terahertz frequencies (greater than 5 THz). The importance of InGaAs material for terahertz applications is negligible compared to Graphene with mono/Bi layers.

Comparison of InGaAs and graphene channels HEMTs response
In this section, we describe the study of the Terahertz response obtained using the parameters of the InGaAs and graphene materials. We do not consider the terahertz behavior of the HEMTs on an InGaAs material which was studied in [17].
The parameters of graphene and InGaAs material are illustrated in table 4. Figure 11 shows the admittance and voltage gain of the InGaAs and Graphene channel HEMTs. The voltage gain was calculated in the absence of any real charge Y C connected between the source and drain of the HEMT terminals.
According to the response discussion in section 3, we note that a high amplitude of resonances is obtained by the Graphene-SiO 2 owing to the high electron mobility in the channel. Indeed, as discussed in several sections the frequency range of resonances is controlled by the 3D free-carrier plasma mode which is related to the dielectric constant in the channel and in the dielectric between the channel and the gate.
An inevitable consequence of the Graphene parameters is that the Graphene channel presents a series of resonances with an amplitude 10 times greater than the resonances of the InGaAs channel. Moreover, a series of resonances appeared at high terahertz frequencies (up to 20 THz) for the Graphene material when an InGaAs  channel was generated resonances in the frequency range limited to 6 THz. For the voltage gain comparaison, the high-frequency resonances obtained by the Graphene channel HEMTs lead to improved amplifier quality to tens of terahertz (see figure 11(c)). These results show the Terahertz frequency domain of amplification of several material-channel HEMTs.

Conclusion
Analytical calculations have been proposed to determine the terahertz response of Graphene-channels HEMTs. This study characterizes the HEMTs for the graphene Monolayer, graphene Bilayer and graphene with different dielectric substrates : H 2 O, Ethanol (160K), HfO 2 and SiO 2 . The calculation considers the small-signal model of HEMTs developed in [17] where the results were extracted for the Graphene parameters as the effective mass, relaxation rate and permittivitty constants.
The results can lead to a real comparaison between the Terahertz efficiency of the InGaAs and the Graphene channels HEMTs. The influence of the Graphene parameters, Graphene type (Monolayer and Bilayer) and various materials (InGaAs, GBL and GML) have been shown.
The admittances Y dd and Y gd spectra show high resonances quality at the frequency range up to 17 THz. Similar to the InGaAs channels HEMTs, the Y dd of Graphene HEMTs exhibits all the oscillations modes in the channel where the gate-drain admittance Y gd presents only the odd resonances due to the control of the gate.  According to the results of the voltage gain spectra, a high amplification can be obtained by the Graphene channels HEMTs at frequencies from 1 THz to 17 THz. The InGaAs and Graphene channels responses show that resonances with high amplitudes in the high terahertz frequency range are generated by the Graphene channel compared with InGaAs. Indeed, high-quality resonances in high terahertz frequencies are also obtained by the SiO 2 dielectric substrate which is associated with an increase in the plasma frequency and relies on the long scattering time.
Analysis of the impurity position in the dielectric substrate shows that good resonance peaks appear when the impurities are far from the Graphene-dielectric interface.
A significant increase in the resonances can be obtained by Monolayer Graphene (GML) compared to Bilayer Graphene (GBL) and InGaAs materials. Therefore, amplification is guaranteed by the GML channel HEMTs and allowed in the frequency range f < 20 THz corresponding to their electronic parameters.
These results can be useful for adjusting the material in HEMTs devices for nanometric terahertz amplifiers based on plasmonic oscillations modes in the channel of HEMTs.

Acknowledgments
This work is partly supported by the Algerian ministry of higher education and research.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

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