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Abstract
The gate-controllable electronical property of graphene provides a possibility of active tuning property for THz modulator. However, the common modulation technology which only depends on voltage cannot solve the problem of power consumption limitation in communication applications. Here, we demonstrated a Brewster angle-controlled graphene-based THz modulator, which could achieve a relatively high modulation depth with low voltage. First, we explored the complex relationships among the Brewster angles, reflection coefficients and the conductivities of graphene. Then, we further investigated the optimal incident angle selection based on the unusual reflection effect which occurs at Brewster angle. Finally, an improved scheme by dynamically adjusting the incident angle was proposed in this paper. It would make it possible that the modulator could achieve a modulation depth of more than 90% with a Fermi level as low as 0.2eV at any specific frequency in the range of 0.4THz-2.2THz. This research will help to realize a THz modulator with high-performance and ultra-low-power in quantities of applications, such as sensing and communication.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Terahertz technology offers a variety of applications, including sensing [1,2], noninvasive imaging [3,4], and wireless communications [5,6]. However, most of the practical applications depend on effective wave modulation, which leads to surging demand for THz modulators. For example, the key to a high data transmission rate in wireless communications is to develop an efficient amplitude and/or phase modulator for encoding information in the carrier wave [7]. Modulators can be classified by the wave properties they are designed to manipulate, such as amplitude [8,9], phase [10,11] and polarization state [12,13] of electromagnetic (EM) waves. Among the above classification, amplitude modulators have been studied most thoroughly.
Over the past decade, terahertz response modulation technology based on traditional metamaterial or metasurface has attracted great attention [14,15]. Nevertheless, these methods which can only work at a fixed frequency after fabrication are difficult for active tuning. This is far from enough for communication applications where a large on/off ratio is required at serval selected frequency by changing the response of THz magnitude. In this context, two-dimensional(2D) materials have been extensively investigated for active control of terahertz radiation [16,17]. The gate-controllable electronic property of graphene makes it a promising material for efficient THz modulation, which provides the possibility for operating frequency and modulation depth selectivity of the terahertz modulator [18,19]. Recently, it has been proven that the graphene in a field effect transistor (FET) configuration could provide broadband THz amplitude modulation and is suitable for communication applications [20].
When it comes to the application of graphene-based modulator, it’s worth noting that the unusual reflection of electromagnetic radiation at the interface which occurs under Brewster angle incident has been widely studied [21,22]. Give the condition that the Brewster angle may be altered by adjusting the surface conductivity of the medium [23,24], the modulation mechanism of reflection wave can be used in the design of amplitude modulator [25]. However, the complex relationship among the incident angle, operating frequency and modulation depth has not been fully explored for Brewster angle modulators. As far as present, without selecting proper incident angle, a large Femi level change of graphene is generally required to achieve an ideal modulation depth at a specific operating frequency. However, the change of Fermi level is usually realized by applying a gate voltage, as previous study suggested [26]. In other words, for THz modulator, the larger the change of Fermi level required for modulation, the greater the applied gate voltage. As a result, the large gate voltage greatly increases the power consumption of the modulator, which limits the further application of the THz modulator in wireless communication.
In this paper, we demonstrate a cost-effective THz amplitude modulator based on tunable Brewster reflection, which can operate at a selected frequency with high modulation depth. The investigation starts with the impacts of graphene’s Fermi level and carrier mobility on the Brewster angle of the modulator. On this basis, we calculated the variations of modulation depth with Fermi level and incident angle at several frequency points within THz band. The detailed analysis of the calculation results indicates that for a specific operating frequency, the optimal modulation efficiency of THz amplitude modulator could be achieved at low Fermi level by adequately tuning the incident angle of THz waves at a specific operating frequency. Therefore, the power consumption of the Brewster angle-controlled graphene-based THz modulator we proposed in this paper could be reduced for the THz communication.
2. Model and simulation of modulator based on the Brewster angle
Here, as a proof of concept, the THz modulator proposed in this paper is shown in Fig. 1. On top of the modulator, the single-layer graphene is chosen to ensure the controllable electronical property and is transferred to the dielectric gate. Below the dielectric gate, there is a back gate. The bottom layer is a quartz substrate. The reflection of dielectric gate to terahertz is negligible, and the back gate needs to have a large impedance, which will help reduce the insertion loss of the modulator. Based on previous experience, Al2O3 can be used as dielectric gate and TiOx can be used as back gate. As shown in Fig. 1, the two metal electrodes are respectively connected with the back gate and graphene for the gate voltage. P-polarized terahertz wave is incident of θ. By adjusting the gate voltage to change the Fermi level, the conductivity could be altered and finally the Brewster angle of incident terahertz wave can be modulated. For this modulator, its dielectric gate thickness $t_{d}$ and back gate thickness $t_{b}$ are designed to be 100 nm and 50 nm, respectively. In addition, the thickness of quartz ($t_{s}$) for the modulator substrate of is about 0.5 mm, which is comparable to the wavelength dimension.
The terahertz response of the graphene modulator was calculated by using COMSOL Multiphysics software and finite element method. Finite element method is one of the most widely used numerical calculation methods in engineering analysis [27]. Due to that the size of the dielectric gate and the back gate belonging to nanometer scale which is much smaller than the terahertz wavelength, the influence of the gate can be ignored during simulation. The simplified model of the Brewster angle modulator based on graphene is shown in Fig. 2(a), which describes the reflection and refraction of incident terahertz waves at the interface. When p-polarized terahertz wave is incident from air (optically thinner medium) to graphene-substrate interface (optically denser medium), its reflection coefficient will become 0 supposing that the incident angle satisfies some conditions. In this case, the p-polarized terahertz incident wave will be totally refracted. The above incident angle which makes the p-polarized wave completely unable to be reflected is called Brewster angle.
According to Fresnel equation, Maxwell equation and their boundary conditions, the reflection coefficient of terahertz wave in this physical system can be described as:
The band structure of graphene and the photon transition process in graphene are shown in the inset of Fig. 2(a). Red arrows and green arrows indicate intraband transitions and interband transitions, respectively. Intraband transition is dominant under terahertz irradiation. Thus, the conductivity of graphene can be expressed by the simplified Drude model [28].
where EF and μ are the Fermi level and the carrier mobility. Fermi velocity and Planck constant are denoted by vF, ℏ, whereas e and ω correspond to electron charge and angular frequency. As can be seen from Eq. (2), the gate voltage can be used to dynamically modulate the conductivity of graphene by changing the Fermi level. In our work, we used 10000cm2V−1S-1and 106 m/s as the ideal value of carrier mobility μ and Femi velocity vF, which is consistent with previous study [29].Monolayer graphene, which is a thin film, can be modeled as a two-dimensional existence according to transitional boundary conditions. The transitional boundary condition should be used on interior boundaries to model a geometrically thin sheet of a medium in the COMSOL simulation. In this study, the thickness of monolayer graphene is on the order of nm. It does really meet the requirement that the medium should be geometrically thin. Therefore, it is feasible and reasonable to treat graphene with transitional boundary conditions in our study. Given the condition that its thickness tg is 0.6 nm, its dielectric constant can be described by the following equation:
Based on the above mathematical relationship, the reflection coefficient r is simultaneously depended on the incident angle θi and graphene conductivity σg supposing that the substrate material is determined. This shows that in addition to adjusting the incident angle to reach Brewster's angle, the complete refraction of p-polarized terahertz waves can also be realized by changing the electrical conductivity of graphene σg. Therefore, it is indeed feasible to realize the Brewster angle modulation of the whole system and minimize the reflection coefficient of p-polarized terahertz wave by adjusting the conductivity of graphene.
The schematic diagram of simulation model is shown in Fig. 2(b). Due to that the size of the dielectric gate and the back gate belong to nanometer scale which are much smaller than the terahertz wavelength, the influence of the gate can be ignored during simulation. The left and right boundaries are set as Periodic Boundary Conditions (PMC), and the upper and lower boundaries are defined as Perfect Matching Layers (PML) to truncate the domain to achieve complete absorption. Terahertz waves are incident from the port 1. For more detailed simulation steps, see S1 in Supplement 1.
3. Results and discussion
3.1 Electrical tunability Brewster angle of the modulator
Eq. (2,3) shows that the conductivity of graphene is determined jointly by Fermi level EF and carrier mobility μ. Thus, in order to investigate the relationship between the graphene conductivity and the Brewster angle of the terahertz modulator at a specific frequency, the modulation effects of Fermi level and carrier mobility should be taken into account in turn.
On the one hand, the modulator's reflection coefficients at various Fermi levels are calculated and simulated given the condition that the operating frequency is 0.8THz and the incident angle varies from 0° to 89°. The result shown in Fig. 3(a) indicated that the Brewster angle and reflection coefficient of the modulator both significantly increase with the enhancement of Fermi level. On the other hand, Fig. 3(b) also clearly suggested that the carrier mobility μ could affect the modulation performance of the modulator by change the conductivity σg. Comparing the results presented in Fig. 3(a) and Fig. 3(b), we came to realize that the influence of the carrier mobility μ on Brewster angle and reflection coefficient is much smaller than that of Fermi level EF. In view of this case, the influence of carrier mobility would be ignored in the subsequent study. Its value will be assigned to 10000cm2V-1S−1 as mentioned above and only the modulation effect of Fermi level would be considered.
Fig. 3. The modulation of Brewster angle with (a) different Fermi levels and (b) different carrier mobility @0.8THz;(c) The modulation of Brewster angle with different Fermi levels at different frequencies
In order to further explore the relationship between Brewster angle and Fermi level within the broad frequency band, the Brewster angles of this graphene modulator at different Fermi levels in the range of 0.2THz-2.2THz are calculated and simulated as shown in Fig. 3(c). The results demonstrated that at a specific frequency point, the Brewster angle of the terahertz modulator would change with the Fermi level which could be adjusted by tuning gate voltage. In other word, this modulator could achieve electrical tunability Brewster angle. In addition, the results also indicated that the Brewster angle would alter with the frequency given the condition of the same Fermi level. Within the low frequency range (0.2THz-1.0THz), the Brewster angle decreases monotonically with the increase of frequency at the same Fermi level. However, within the high frequency range (1.4THz- 2.2THz), the Brewster angle shows a complex change with the increase of frequency at the same Fermi level. It sometimes would even increase with the increase of frequency, which is completely opposite to the law presented in the low-frequency range. The above analysis indicated that the behavior of Brewster angle modulation in the high frequency terahertz range becomes more complex and further research is needed. We suspect that this may be due to the size limitation of the device, which makes it cannot match well with the shorter wavelengths at higher frequency.
3.2 Optimal modulation depth of the modulator
Modulation depth is one of the most important criteria to evaluate the modulation performance of the modulator. The modulation depth of the device can be described using the reflection coefficient of terahertz modulator. It can be seen from Eq. (1) and (2) that the reflection coefficient of the THz modulator proposed in this paper is determined by the incident angle, the Fermi level, and the frequency simultaneously. Due to that the reflection coefficient of this modulator is very sensitive to the selected parameters as presented in Fig. (3), the slight deviation of incident angle and conductivity would greatly affect its modulation depth. Therefore, the modulation effect of incident angle and Fermi level and their influence on the modulation performance of graphene modulator were further analyzed in this section given the condition of a specific frequency. Supposing that the incident angle of terahertz wave is θ, the modulation depth of the graphene-based modulator with different Femi level can be described as:
The results we have obtained from Fig. 3(c) also clearly suggested that in the high frequency range, the Brewster angle shows a more complex change with the increase of Fermi level at the same frequency point. Therefore, we have focused on the high frequency range and studied the effects of incident angle and Fermi level on modulation depth at 1.6THz, 1.8THz, 2.0THz and 2.2THz. The effects of incident angle and Fermi level on modulation depth at 0.4-1.4THz also have been studied and shown in S2 in Supplement 1. The results as shown in Fig. 4 suggested that for these different frequency points, provided that the appropriate incident angle is selected, the Thz modulator alwasys could achieve a modulation depth greater than 90% with a relatively small changes in Fermi level (ΔEF = 0.2 eV). In addition, the modulation depth at 0.2 eV, which varies under different incident angles, are shown in the inset of Fig. 4. The studies we have performed further proved the significance of the incident angle selection to achieve considerable modulation depth. If the incident angle is not selected properly, even a small deviation would cause a sharp decrease in modulation depth and seriously affect the performance of the modulator. Taking 1.6THz as an example, when the incident angle changes from 63° to 66°, the modulation depth at 0.2 eV decreases from 97% to 81%.
Fig. 4. Modulation depth varies with the incident angle and Fermi level at (a) 1.6THz; (b) 1.8THz; (c) 2.0THz; (d) 2.2THz
To investigate the effect of incident angle on modulation depth more comprehensively, the modulation depth in the low frequency range was also calculated and simulated on the basis of the above simulation results in the high frequency range. With a 0.2THz interval in the range of 0.4THz-2.2THz, we calculated the incident angle corresponding to the maximum modulation depth given the condition of low-power consumption (ΔEF = 0.2 eV). As shown in Fig. 5(a), the results indicated that at different frequency points, the optimal incident angle corresponding to the maximum modulation depth at low power consumption are not the same. It’s worth noting that the optimal angle is 63° when the frequency is 0.4 THz, 0.6 THz, and 1.2 THz whereas the optimal angle is 64° when the frequency is 1.0THz, 1.6THz, and 2.2THz. Therefore, 63 ° and 64 ° were selected as the incident angles to further calculate the modulation depth at different frequencies under low power consumption. The results presented in Fig. 5(b) suggested that only when the modulator is always at the optimal incident angle, can it ensure more than 90% depth modulation at low power in the frequency range of 0.4THz-2.2THz. The studies we have performed indicated that the modulation depth of the THz modulator based on graphene is highly susceptible to the incident angle. For any specific frequency, the modulator can achieve a large modulation depth at low-power consumption supposing that the appropriate incident angle is selected. For the modulation depth when the Fermi level change is higher than 0.2 eV, even if the incident angle is not properly selected, the modulation depth over 79% (ΔEF = 0.4 eV) and over 84% (ΔEF = 0.6 eV) can be achieved under a lager Fermi level change (i.e., greater power consumption). Therefore, in order to exhibit and highlight our research findings, we only showed modulation depth with the Fermi level change of 0.2 eV (i.e., at a low power consumption). See S3 in Supplement 1 for the modulation depth varies under the Fermi level change over 0.2 eV.
Fig. 5. (a) The optimal incident angle corresponding to frequencies; (b) Incident angle- dependent modulation depths with low-power consumption (ΔEF = 0.2 eV)
4. Conclusion
In this work, we explored the modulation scheme of the p-polarized terahertz modulator based on the Brewster angle by emphasizing the significance of incident angle selection. It is found that by dynamically tuning the incident angle for different frequencies, the modulation depth of the modulator could be higher than 90% even if the Femi level change is as low as 0.2 eV. Therefore, our work could make it possible that the modulator can achieve a large modulation depth at low-power consumption supposing at any specific frequency in the range of 0.4THz-2.2THz. The scheme emphasized in this paper can achieve electrical tunability Brewster angle and optimal efficiency of graphene-based THz modulator by selecting appropriate incident angle, which could potentially provide a new possibility for terahertz modulators with a large on/off ratio and offer a new option for applying terahertz technology to high-speed real-time 6 G applications.
Funding
National Natural Science Foundation of China (62005014); National Key Research and Development Program of China (2021YFA0718901); Fundamental Research Funds for the Central Universities (FRF-TP-20-015A1).
Acknowledgments
This work was supported by the National Key R & D Program of China (No. 2021YFA0718901) and the National Natural Science Foundation of China (No. 62005014). This work was also supported by the Fundamental Research Funds for the Central Universities (FRF-TP-20-015A1) at the University of Science and Technology Beijing.
Disclosures
The authors have no conflicts to disclose.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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