# Tuning of Graphene-Based Optical Devices Operating in the Near-Infrared

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}, WS

_{2}, MoSe

_{2}, ReS

_{2}, WSe

_{2}and others [44,45,46], and black phosphorus [47,48,49,50], are gaining popularity, but TMDs show a band gap of about 1 to 2.5 eV (which corresponds to frequencies from NIR to visible), and the band gap of phosphorus is highly variable depending on the number of layers: from 0.3 eV in the bulk to about 2 eV in an isolated monolayer [51,52,53]. Unlike the universal optical conductivity in graphene (single layer graphene absorbs ≈2.3% of vertical incident light in the FIR to UV range), semiconductor TMD can exhibit multiple absorption peaks from NIR to UV frequencies due to excitons and interband transitions (the absorption coefficients of two peaks in a TMD monolayer can reach ≈10% and ≈30%, respectively [54]), and the optical absorption of black phosphorus depends on the thickness, doping and polarization [55].

## 2. Graphene Optical Properties

_{G}, a quantity which can be obtained from either a microscopic model or from measurements [56].

#### 2.1. Graphene Conductivity

_{G}) is described by the high-frequency expression obtained from the Kubo model [57]

_{G}and the extinction coefficient k

_{G}can be retrieved by means of the following equations

_{c}> 0 eV) was considered, since the optical conductivity of graphene is symmetric for the positive and negative electrochemical potentials due to the symmetric band structure in graphene [8].

#### 2.1.1. Dependence of Graphene Complex Permittivity

_{G}= 0.34 nm, μ

_{c}= 0 ÷ 1 eV, Γ = 2e12 s

^{−1}and T = 300 K.

_{c}| < 0.2 eV—dependence tending to linear, 0.2 eV < |μ

_{c}| < 0.5 eV—transient process and |μ

_{c}| > 0.5 eV—dependence tending to linear (for the imaginary part, the boundaries of the zones are shifted at about ~0.3 eV and ~0.6 eV, respectively). This division is close to that presented in Reference [9]. The range |μ

_{c}| < 0.28 eV corresponds to the absorption region in which the absorption of the graphene remains high, and both the effective index and the absorption coefficient (α

_{G}= (4πf k

_{G})/c, which have a direct proportion with extinction coefficient k

_{G}) slightly deviate, ranging from 0.0005 to 1 dB/nm, respectively. Next, the step change region (<0.64 eV) is suitable for amplitude modulation since there is a significant ‘stepping up’ of the absorption index. The final region (>0.64 eV) corresponds to low absorption, and the effective index is linear (which is applicable for phase modulation).

_{c}| ≈ 0.4 eV or |μ

_{c}| = 0.515 eV [14]) is a transition from a lossy-dielectric response (amplitude modulator mode, with the central wavelength around 1552.5–1552.8 nm [9]) to a quasi-metallic one (transition to a phase-dependent mode [9]). The imaginary part of the dielectric constant decreases significantly for Fermi levels |μ

_{c}| > 0.4 eV, i.e., the absorption is significantly reduced. When a photon impinges on graphene, its energy may be absorbed to provoke an interband transition of a free carrier. Whether an electron or a hole is involved in this process depends on the sign of the Fermi level. According to the Pauli blocking principle, this transition can only occur if |μ

_{c}| < ħω/2, at λ = 1.55 μm, |μ

_{c}| < 0.4 eV. For an ideal, impurity-free graphene sheet (τ→∞) at zero temperature, this would reflect in an abrupt variation of absorption. At room temperature, a more gradual transmission from absorption to transparency can be expected [21]. It was found in Reference [9] that a large extinction ratio can be achieved in the operating region when |μ

_{c}| < 0.4 eV. In the real case, switching between modes occurs smoothly, which is associated with graphene defects created during the manufacturing process [11].

_{G}of the monolayer graphene is equal to about 0.34 nm while the thickness of multilayer graphene is equal to N times. Figure 2a shows the dependence of the permittivity on the chemical potential for the GE thicknesses from the considered works [5,8,9,11,13,14,18,19,20,21,24,25,26]. For example, values equal to 0.5 nm or 1 nm are used in numerical simulations to relax the mesh/grid (e.g., about 1 nm could correspond to a 3-layer graphene).

_{F}–Fermi velocity (=106 m/s)), which significantly affects the imaginary part of the dielectric constant (as seen in Figure 3). It is also worth noting that the ℏΓ is the relaxation energy that defines the quality of graphene (a shorter relaxation time corresponding to a lower quality of graphene, and τ→∞ corresponds to an ideally defect-free graphene sheet) [5,14,21]. It should be noted that the Kubo model from [13] distinguishes between the relaxation rates associated with interband and intraband transitions.

_{c}| < 0.4 eV from the change in the scattering rate. In this case, a strong dependence of absorption on Γ is observed, since in-band absorption becomes the dominant process here. In this regime, lower values for Γ, corresponding to high carrier mobility, give lower absorption [8]. The transient process has already been described in the section on the chemical potential, but, in addition to it, it is worth noting that the final values of the relaxation time have a significant effect since there is significant absorption due to intraband scattering interactions [21]. Imperfections in the graphene layer induce density inhomogeneities, varying the optical conductivity across the graphene sheet [61].

^{−1}to 1e14 s

^{−1}, with a minimum value of 4e9 s

^{−1}[18]. On the experimental side, a typical value is Γ = 8e13 s

^{−1}[9].

_{B}T ≈ 25 meV. The previous explanation also makes it clear that in the equilibrium state, photons at a wavelength of 1.55 μm will not be absorbed if the chemical potentials of the upper and lower graphene electrodes are equal to 0.425 eV. Therefore, when there is no applied voltage (${V}_{G}$ = 0), the graphene layers behave as transparent thin sheets [15].

_{3}N

_{4}or Si. The parameters of these materials have a higher thermal sensitivity, which can affect the stability of the tuning mechanism.

#### 2.1.2. Graphene Conductivity Formula

^{−1}and Γ = 2e12 s

^{−1}(since this parameter depends on the quality of graphene (Section 2.1.1 (c)) and it was interesting to look at the behavior of the permittivity curves when it changes). In this regard, the changes in the Kubo model curve when this parameter changes are one of the most important criteria when choosing a formula option. In this regard, it can be noted that the [13,25] and [17] options do not react to changes in the relaxation rate. The opposite case reflects the permittivity curve obtained by the formula from [15], demonstrating the highest sensitivity. The model from [18,19,20,26,58] is “smoothed” at Γ = 8e13 s

^{−1}(experimentally obtained result [9]), which makes it difficult to analyze the transient process. As can be seen from Figure 5a,d, the curves for the cases [5,21] show the stability of the real part of the dielectric constant to a change in the relaxation rate on the one hand, but the high sensitivity of the imaginary part on the other hand, which may be more effective, in comparison with other formulas according to the Kubo model, for modeling and analyzing graphene.

#### 2.2. Graphene Electrode Configurations

_{3}N

_{4}) that is indicated by W. The EGWC case also implies a non-use of PhC cavities (Electrolyte/ion-gel-Graphene-Waveguide).

_{I}), leads to a relatively weak light interaction.

_{s}is the graphene surface carrier density. The voltage needed to charge a GIW capacitor is a sum of two contributions: the first is the actual potential across the insulator; the second is due to the shift of the Fermi potential induced by the accumulated carriers on the graphene layer. Using Equation (6), it is possible to obtain

^{2}for ion-gel [19], ∼0.493 mF/m

^{2}for SiO

_{2}[15], ~27 mF/m

^{2}for polyethilenoxide (PEO) + LiClO

_{4}[7]), and ${V}_{Dirac}$ is the flat-band voltage corresponding to the charge-neutral Dirac point. ${V}_{Dirac}$ depends on the intrinsic surface carrier density on the graphene electrode due to band alignment, lattice imperfections, charged defects or other impurities in the fabricated graphene sheet. In the case of a GIG capacitor, Equation (7) should be modified to account for the shift of the Fermi potential in each of the two graphene layers. In this case, the last term of Equation (7) should be modified to read as $2\left|\mu \right|/q$ [21].

_{F}= ±ħω/2), both GE become transparent at the same time. The change in the sign of the excitation voltage only switches the roles of the capacitor plates as anode and cathode and gives a similar response to incident light [12].

_{F}= ±ħω/2), which occurs due to the accumulation of a positive charge. This leads to the absence of electrons, which are available for interband transitions, and graphene is “transparent”. The right boundary is due to the filling of electronic states at which interband transitions are not allowed [11,24].

## 3. Graphene-Based Optical Devices and Tuning

_{3}N

_{4}devices, quasi-TE-polarized modes exhibit stronger interactions with the graphene film than quasi-TM modes. The GIG-Si

_{3}N

_{4}has lower absorption than the GIW-Si device, but by embedding the capacitor in the Si

_{3}N

_{4}waveguide, the absorption level of the GIG-Si

_{3}N

_{4}-Emb can be increased again, almost to that of the GIG-Si, although the waveguide is large [66].

_{2}/Ar is normally used to remove polymer residuals for cleaning after transfer and removal of solvents/surfactants in LPE graphene) [70]. Additionally, one of the problems remains the cracking of single layer graphene at the edges of the steps of the waveguide [8].

_{2}O

_{3}[8,9,10,11,12,13,18,19,24,25,64], however, in addition to it, it is advisable to use Si

_{3}N

_{4}[21,24,26] and HfO

_{2}[20], as well as an electrolyte in the form of a mixture of LiClO

_{4}and polyethylene oxide (PEO) [72] or ion-gels [7,19,20]. In the Fermi levels |μ

_{c}| > 0.5 eV, the optimal choice of the insulating capacitor layer formed either by one GE or two GEs, is another issue since there is the problem of graphene absorption due to interband transitions.

_{c}| = 0.5 eV, which corresponds to the electric field on the capacitor insulator

_{3}N

_{4}, ${\epsilon}_{ox}$ = 7.5) is the best option among Al

_{2}O

_{3}and SiO

_{2}at ${E}_{0.5\mathrm{eV}}$ = 5e6 V/cm at a breakdown field of about 1e7 V/cm.

#### Wavelength Tuning

_{G}), the distance between graphene electrodes (D

_{G}) for the GIG structure, or the insulator thickness (t

_{I}) for others, and the applied voltage (V

_{G}) were analysed. Table 2 compares the analysed devices (all the devices are fabricated except for Reference [15]) showing that the wavelength shift ∆λ (last column) can vary from tens to thousands of picometers (last column in the table).

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Castro Neto, A.H.; Guinea, F.; Peres, N.M.R.; Novoselov, K.S.; Geim, A.K. The electronic properties of graphene. Rev. Mod. Phys.
**2009**, 81, 109–162. [Google Scholar] [CrossRef] [Green Version] - Novoselov, K.S.; Geim, A.K.; Morozov, S.V.; Jiang, D.; Katsnelson, M.I.; Grigorieva, I.V.; Dubonos, S.V.; Firsov, A.A. Two-dimensional gas of massless Dirac fermions in graphene. Nature
**2005**, 438, 197–200. [Google Scholar] [CrossRef] [PubMed] - Rinaldi, G. Nanoscience and Technology: A Collection of Reviews from Nature Journals. Assem. Autom.
**2010**, 30, 2. [Google Scholar] [CrossRef] - Wolf, E.L. Applications of Graphene; SpringerBriefs in Materials; Springer International Publishing: Cham, Switzerland, 2014; ISBN 978-3-319-03945-9. [Google Scholar]
- de Ceglia, D.; Vincenti, M.A.; Grande, M.; Bianco, G.V.; Bruno, G.; D’Orazio, A.; Scalora, M. Tuning infrared guided-mode resonances with graphene. J. Opt. Soc. Am. B
**2016**, 33, 426. [Google Scholar] [CrossRef] - Wang, F.; Zhang, Y.; Tian, C.; Girit, C.; Zettl, A.; Crommie, M.; Shen, Y.R. Gate-Variable Optical Transitions in Graphene. Science
**2008**, 320, 206. [Google Scholar] [CrossRef] - Abdollahi Shiramin, L.; Xie, W.; Snyder, B.; De Heyn, P.; Verheyen, P.; Roelkens, G.; Van Thourhout, D. High Extinction Ratio Hybrid Graphene-Silicon Photonic Crystal Switch. IEEE Photonics Technol. Lett.
**2018**, 30, 157–160. [Google Scholar] [CrossRef] - Mohsin, M.; Neumaier, D.; Schall, D.; Otto, M.; Matheisen, C.; Lena Giesecke, A.; Sagade, A.A.; Kurz, H. Experimental verification of electro-refractive phase modulation in graphene. Sci. Rep.
**2015**, 5, 10967. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shu, H.; Su, Z.; Huang, L.; Wu, Z.; Wang, X.; Zhang, Z.; Zhou, Z. Significantly High Modulation Efficiency of Compact Graphene Modulator Based on Silicon Waveguide. Sci. Rep.
**2018**, 8, 991. [Google Scholar] [CrossRef] - Cheng, Z.; Zhu, X.; Galili, M.; Frandsen, L.H.; Hu, H.; Xiao, S.; Dong, J.; Ding, Y.; Oxenløwe, L.K.; Zhang, X. Double-layer graphene on photonic crystal waveguide electro-absorption modulator with 12 GHz bandwidth. Nanophotonics
**2019**, 9, 2377–2385. [Google Scholar] [CrossRef] - Liu, M.; Yin, X.; Ulin-Avila, E.; Geng, B.; Zentgraf, T.; Ju, L.; Wang, F.; Zhang, X. A graphene-based broadband optical modulator. Nature
**2011**, 474, 64–67. [Google Scholar] [CrossRef] - Liu, M.; Yin, X.; Zhang, X. Double-Layer Graphene Optical Modulator. Nano Lett.
**2012**, 12, 1482–1485. [Google Scholar] [CrossRef] [PubMed] - Phatak, A.; Cheng, Z.; Qin, C.; Goda, K. Design of electro-optic modulators based on graphene-on-silicon slot waveguides. Opt. Lett.
**2016**, 41, 2501. [Google Scholar] [CrossRef] - Hu, Y.; Pantouvaki, M.; Van Campenhout, J.; Brems, S.; Asselberghs, I.; Huyghebaert, C.; Absil, P.; Van Thourhout, D. Broadband 10 Gb/s operation of graphene electro-absorption modulator on silicon: Broadband 10 Gb/s operation of graphene electro-absorption modulator on silicon. Laser Photonics Rev.
**2016**, 10, 307–316. [Google Scholar] [CrossRef] [Green Version] - Bahadori-Haghighi, S.; Ghayour, R.; Sheikhi, M.H. Double-layer graphene optical modulators based on Fano resonance in all-dielectric metasurfaces. J. Appl. Phys.
**2019**, 125, 073104. [Google Scholar] [CrossRef] - Koester, S.J.; Li, M. High-speed waveguide-coupled graphene-on-graphene optical modulators. Appl. Phys. Lett.
**2012**, 100, 171107. [Google Scholar] [CrossRef] - Luo, X.; Zhai, X.; Li, H.; Liu, J.; Wang, L. Tunable Nonreciprocal Graphene Waveguide With Kerr Nonlinear Material. IEEE Photonics Technol. Lett.
**2017**, 29, 1903–1906. [Google Scholar] [CrossRef] - Xu, C.; Jin, Y.; Yang, L.; Yang, J.; Jiang, X. Characteristics of electro-refractive modulating based on Graphene-Oxide-Silicon waveguide. Opt. Express
**2012**, 20, 22398. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Majumdar, A.; Kim, J.; Vuckovic, J.; Wang, F. Electrical Control of Silicon Photonic Crystal Cavity by Graphene. Nano Lett.
**2013**, 13, 515–518. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gan, X.; Shiue, R.-J.; Gao, Y.; Mak, K.F.; Yao, X.; Li, L.; Szep, A.; Walker, D.; Hone, J.; Heinz, T.F.; et al. High-Contrast Electrooptic Modulation of a Photonic Crystal Nanocavity by Electrical Gating of Graphene. Nano Lett.
**2013**, 13, 691–696. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sorianello, V.; Midrio, M.; Romagnoli, M. Design optimization of single and double layer Graphene phase modulators in SOI. Opt. Express
**2015**, 23, 6478. [Google Scholar] [CrossRef] - Phare, C.T.; Lee, Y.-H.D.; Cardenas, J.; Lipson, M. 30 GHz Zeno-based Graphene Electro-optic Modulator. In Proceedings of the CLEO: 2015, San Jose, CA, USA, 5–10 May 2015; OSA: San Jose, CA, USA, 2015; p. SW4I.4. [Google Scholar]
- Cai, M.; Wang, S.; Liu, Z.; Wang, Y.; Han, T.; Liu, H. Graphene Electro-Optical Switch Modulator by Adjusting Propagation Length Based on Hybrid Plasmonic Waveguide in Infrared Band. Sensors
**2020**, 20, 2864. [Google Scholar] [CrossRef] - Ding, Y.; Zhu, X.; Xiao, S.; Hu, H.; Frandsen, L.H.; Mortensen, N.A.; Yvind, K. Effective Electro-Optical Modulation with High Extinction Ratio by a Graphene–Silicon Microring Resonator. Nano Lett.
**2015**, 15, 4393–4400. [Google Scholar] [CrossRef] [Green Version] - Qiu, C.; Gao, W.; Vajtai, R.; Ajayan, P.M.; Kono, J.; Xu, Q. Efficient Modulation of 1.55 μm Radiation with Gated Graphene on a Silicon Microring Resonator. Nano Lett.
**2014**, 14, 6811–6815. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Yang, L.; Hu, T.; Shen, A.; Pei, C.; Li, Y.; Dai, T.; Yu, H.; Li, Y.; Jiang, X.; Yang, J. Proposal for a 2X2 Optical Switch Based on Graphene-Silicon-Waveguide Microring. IEEE Photonics Technol. Lett.
**2014**, 26, 235–238. [Google Scholar] [CrossRef] - Grande, M.; Bianco, G.V.; Laneve, D.; Capezzuto, P.; Petruzzelli, V.; Scalora, M.; Prudenzano, F.; Bruno, G.; D’Orazio, A. Gain and phase control in a graphene-loaded reconfigurable antenna. Appl. Phys. Lett.
**2019**, 115, 133103. [Google Scholar] [CrossRef] - Grande, M.; Bianco, G.V.; Perna, F.M.; Capriati, V.; Capezzuto, P.; Scalora, M.; Bruno, G.; D’Orazio, A. Reconfigurable and optically transparent microwave absorbers based on deep eutectic solvent-gated graphene. Sci. Rep.
**2019**, 9, 5463. [Google Scholar] [CrossRef] - Grande, M.; Bianco, G.V.; Capezzuto, P.; Petruzzelli, V.; Prudenzano, F.; Scalora, M.; Bruno, G.; D’Orazio, A. Amplitude and phase modulation in microwave ring resonators by doped CVD graphene. Nanotechnology
**2018**, 29, 325201. [Google Scholar] [CrossRef] [PubMed] - Sensale-Rodriguez, B.; Yan, R.; Kelly, M.M.; Fang, T.; Tahy, K.; Hwang, W.S.; Jena, D.; Liu, L.; Xing, H.G. Broadband graphene terahertz modulators enabled by intraband transitions. Nat. Commun.
**2012**, 3, 780. [Google Scholar] [CrossRef] - Sensale-Rodriguez, B.; Yan, R.; Rafique, S.; Zhu, M.; Li, W.; Liang, X.; Gundlach, D.; Protasenko, V.; Kelly, M.M.; Jena, D.; et al. Extraordinary Control of Terahertz Beam Reflectance in Graphene Electro-absorption Modulators. Nano Lett.
**2012**, 12, 4518–4522. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lee, C.-C.; Suzuki, S.; Xie, W.; Schibli, T.R. Broadband graphene electro-optic modulators with sub-wavelength thickness. Opt. Express
**2012**, 20, 5264. [Google Scholar] [CrossRef] [PubMed] - Ono, M.; Hata, M.; Tsunekawa, M.; Nozaki, K.; Sumikura, H.; Chiba, H.; Notomi, M. Ultrafast and energy-efficient all-optical switching with graphene-loaded deep-subwavelength plasmonic waveguides. Nat. Photonics
**2020**, 14, 37–43. [Google Scholar] [CrossRef] [Green Version] - Garmire, E. Nonlinear optics in daily life. Opt. Express
**2013**, 21, 30532. [Google Scholar] [CrossRef] [PubMed] - Hafez, H.A.; Kovalev, S.; Tielrooij, K.; Bonn, M.; Gensch, M.; Turchinovich, D. Terahertz Nonlinear Optics of Graphene: From Saturable Absorption to High-Harmonics Generation. Adv. Opt. Mater.
**2020**, 8, 1900771. [Google Scholar] [CrossRef] [Green Version] - Bowlan, P.; Martinez-Moreno, E.; Reimann, K.; Elsaesser, T.; Woerner, M. Ultrafast terahertz response of multilayer graphene in the nonperturbative regime. Phys. Rev. B
**2014**, 89, 041408. [Google Scholar] [CrossRef] - Hafez, H.A.; Kovalev, S.; Deinert, J.-C.; Mics, Z.; Green, B.; Awari, N.; Chen, M.; Germanskiy, S.; Lehnert, U.; Teichert, J.; et al. Extremely efficient terahertz high-harmonic generation in graphene by hot Dirac fermions. Nature
**2018**, 561, 507–511. [Google Scholar] [CrossRef] - Deinert, J.-C.; Alcaraz Iranzo, D.; Pérez, R.; Jia, X.; Hafez, H.A.; Ilyakov, I.; Awari, N.; Chen, M.; Bawatna, M.; Ponomaryov, A.N.; et al. Grating-Graphene Metamaterial as a Platform for Terahertz Nonlinear Photonics. ACS Nano
**2021**, 15, 1145–1154. [Google Scholar] [CrossRef] - Hasan, T.; Sun, Z.; Wang, F.; Bonaccorso, F.; Tan, P.H.; Rozhin, A.G.; Ferrari, A.C. Nanotube–Polymer Composites for Ultrafast Photonics. Adv. Mater.
**2009**, 21, 3874–3899. [Google Scholar] [CrossRef] - Sobon, G.; Sotor, J.; Pasternak, I.; Krzempek, K.; Strupinski, W.; Abramski, K.M. A tunable, linearly polarized Er-fiber laser mode-locked by graphene/PMMA composite. Laser Phys.
**2013**, 23, 125101. [Google Scholar] [CrossRef] - Jung, M.; Koo, J.; Park, J.; Song, Y.-W.; Jhon, Y.M.; Lee, K.; Lee, S.; Lee, J.H. Mode-locked pulse generation from an all-fiberized, Tm-Ho-codoped fiber laser incorporating a graphene oxide-deposited side-polished fiber. Opt. Express
**2013**, 21, 20062. [Google Scholar] [CrossRef] [PubMed] - Lin, Y.-H.; Yang, C.-Y.; Liou, J.-H.; Yu, C.-P.; Lin, G.-R. Using graphene nano-particle embedded in photonic crystal fiber for evanescent wave mode-locking of fiber laser. Opt. Express
**2013**, 21, 16763. [Google Scholar] [CrossRef] - Choi, S.Y.; Jeong, H.; Hong, B.H.; Rotermund, F.; Yeom, D.-I. All-fiber dissipative soliton laser with 10.2 nJ pulse energy using an evanescent field interaction with graphene saturable absorber. Laser Phys. Lett.
**2014**, 11, 015101. [Google Scholar] [CrossRef] - Morell, N.; Reserbat-Plantey, A.; Tsioutsios, I.; Schädler, K.G.; Dubin, F.; Koppens, F.H.L.; Bachtold, A. High Quality Factor Mechanical Resonators Based on WSe2 Monolayers. Nano Lett.
**2016**, 16, 5102–5108. [Google Scholar] [CrossRef] [Green Version] - Chhowalla, M.; Shin, H.S.; Eda, G.; Li, L.-J.; Loh, K.P.; Zhang, H. The chemistry of two-dimensional layered transition metal dichalcogenide nanosheets. Nat. Chem.
**2013**, 5, 263–275. [Google Scholar] [CrossRef] - Low, T.; Chaves, A.; Caldwell, J.D.; Kumar, A.; Fang, N.X.; Avouris, P.; Heinz, T.F.; Guinea, F.; Martin-Moreno, L.; Koppens, F. Polaritons in layered two-dimensional materials. Nat. Mater.
**2017**, 16, 182–194. [Google Scholar] [CrossRef] [Green Version] - Correas-Serrano, D.; Gomez-Diaz, J.S.; Melcon, A.A.; Alù, A. Black phosphorus plasmonics: Anisotropic elliptical propagation and nonlocality-induced canalization. J. Opt.
**2016**, 18, 104006. [Google Scholar] [CrossRef] [Green Version] - Xia, F.; Wang, H.; Jia, Y. Rediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronics. Nat. Commun.
**2014**, 5, 4458. [Google Scholar] [CrossRef] [Green Version] - Wang, H.; Yu, X. Few-Layered Black Phosphorus: From Fabrication and Customization to Biomedical Applications. Small
**2018**, 14, 1702830. [Google Scholar] [CrossRef] [PubMed] - Huang, Y.; Qiao, J.; He, K.; Bliznakov, S.; Sutter, E.; Chen, X.; Luo, D.; Meng, F.; Su, D.; Decker, J.; et al. Interaction of Black Phosphorus with Oxygen and Water. Chem. Mater.
**2016**, 28, 8330–8339. [Google Scholar] [CrossRef] [Green Version] - Li, L.; Yu, Y.; Ye, G.J.; Ge, Q.; Ou, X.; Wu, H.; Feng, D.; Chen, X.H.; Zhang, Y. Black phosphorus field-effect transistors. Nat. Nanotechnol.
**2014**, 9, 372–377. [Google Scholar] [CrossRef] [Green Version] - Das, S.; Zhang, W.; Demarteau, M.; Hoffmann, A.; Dubey, M.; Roelofs, A. Tunable Transport Gap in Phosphorene. Nano Lett.
**2014**, 14, 5733–5739, Correction in**2016**, 16, 2122–2122. [Google Scholar] [CrossRef] [PubMed] - Rudenko, A.N.; Yuan, S.; Katsnelson, M.I. Toward a realistic description of multilayer black phosphorus: From G W approximation to large-scale tight-binding simulations. Phys. Rev. B
**2015**, 92, 085419. [Google Scholar] [CrossRef] [Green Version] - Kozawa, D.; Kumar, R.; Carvalho, A.; Kumar Amara, K.; Zhao, W.; Wang, S.; Toh, M.; Ribeiro, R.M.; Castro Neto, A.H.; Matsuda, K.; et al. Photocarrier relaxation pathway in two-dimensional semiconducting transition metal dichalcogenides. Nat. Commun.
**2014**, 5, 4543. [Google Scholar] [CrossRef] [PubMed] - Low, T.; Rodin, A.S.; Carvalho, A.; Jiang, Y.; Wang, H.; Xia, F.; Castro Neto, A.H. Tunable optical properties of multilayer black phosphorus thin films. Phys. Rev. B
**2014**, 90, 075434. [Google Scholar] [CrossRef] [Green Version] - Depine, R.A. Electromagnetics of graphene. In Graphene Optics: Electromagnetic Solution of Canonical Problems; Morgan & Claypool Publishers: San Rafael, CA, USA, 2016; pp. 1–136. ISBN 978-1-68174-309-7. [Google Scholar]
- Hanson, G.W. Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene. J. Appl. Phys.
**2008**, 103, 064302. [Google Scholar] [CrossRef] [Green Version] - Polycarpou, A.C. Introduction to the Finite Element Method in Electromagnetics; Morgan & Claypool: San Rafael, CA, USA, 2006; pp. 1–126. ISBN 978-1-59829-047-9. [Google Scholar]
- Introduction—MEEP Documentation. Available online: https://meep.readthedocs.io/en/latest/Introduction/ (accessed on 19 April 2021).
- Wartak, M.S. Computational Photonics: An Introduction with MATLAB; Cambridge University Press: Cambridge, UK, 2013; ISBN 978-1-107-00552-5. [Google Scholar]
- Tan, Y.-W.; Zhang, Y.; Bolotin, K.; Zhao, Y.; Adam, S.; Hwang, E.H.; Das Sarma, S.; Stormer, H.L.; Kim, P. Measurement of Scattering Rate and Minimum Conductivity in Graphene. Phys. Rev. Lett.
**2007**, 99, 246803. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bao, Q.; Loh, K.P. Graphene Photonics, Plasmonics, and Broadband Optoelectronic Devices. ACS Nano
**2012**, 6, 3677–3694. [Google Scholar] [CrossRef] [PubMed] - Timurdogan, E.; Sorace-Agaskar, C.M.; Sun, J.; Shah Hosseini, E.; Biberman, A.; Watts, M.R. An ultralow power athermal silicon modulator. Nat. Commun.
**2014**, 5, 4008. [Google Scholar] [CrossRef] [Green Version] - Dalir, H.; Xia, Y.; Wang, Y.; Zhang, X. Athermal Broadband Graphene Optical Modulator with 35 GHz Speed. ACS Photonics
**2016**, 3, 1564–1568. [Google Scholar] [CrossRef] - Chen, X.; Wang, F.; Gu, Q.; Yang, J.; Yu, M.; Kwong, D.; Wong, C.W.; Yang, H.; Zhou, H.; Zhou, S. Multifunctional optoelectronic device based on graphene-coupled silicon photonic crystal cavities. Opt. Express
**2021**, 29, 11094. [Google Scholar] [CrossRef] - Abdollahi Shiramin, L.; Van Thourhout, D. Graphene Modulators and Switches Integrated on Silicon and Silicon Nitride Waveguide. IEEE J. Sel. Top. Quantum Electron.
**2017**, 23, 94–100. [Google Scholar] [CrossRef] - Sorianello, V.; Midrio, M.; Contestabile, G.; Asselberghs, I.; Van Campenhout, J.; Huyghebaert, C.; Goykhman, I.; Ott, A.K.; Ferrari, A.C.; Romagnoli, M. Graphene–silicon phase modulators with gigahertz bandwidth. Nat. Photonics
**2018**, 12, 40–44. [Google Scholar] [CrossRef] - Phare, C.T.; Daniel Lee, Y.-H.; Cardenas, J.; Lipson, M. Graphene electro-optic modulator with 30 GHz bandwidth. Nat. Photonics
**2015**, 9, 511–514. [Google Scholar] [CrossRef] - Ferrari, A.C.; Bonaccorso, F.; Fal’ko, V.; Novoselov, K.S.; Roche, S.; Bøggild, P.; Borini, S.; Koppens, F.H.L.; Palermo, V.; Pugno, N.; et al. Science and technology roadmap for graphene, related two-dimensional crystals, and hybrid systems. Nanoscale
**2015**, 7, 4598–4810. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Youngblood, N.; Li, M. Integration of 2D materials on a silicon photonics platform for optoelectronics applications. Nanophotonics
**2016**, 6, 1205–1218. [Google Scholar] [CrossRef] - Kim, K.S.; Zhao, Y.; Jang, H.; Lee, S.Y.; Kim, J.M.; Kim, K.S.; Ahn, J.-H.; Kim, P.; Choi, J.-Y.; Hong, B.H. Large-scale pattern growth of graphene films for stretchable transparent electrodes. Nature
**2009**, 457, 706–710. [Google Scholar] [CrossRef] [PubMed] - Li, H.-M.; Xu, K.; Bourdon, B.; Lu, H.; Lin, Y.-C.; Robinson, J.A.; Seabaugh, A.C.; Fullerton-Shirey, S.K. Electric Double Layer Dynamics in Poly(ethylene oxide) LiClO
_{4}on Graphene Transistors. J. Phys. Chem. C**2017**, 121, 16996–17004. [Google Scholar] [CrossRef]

**Figure 1.**Zoning of the Kubo model in the |μ

_{c}| range from 0 to 1 eV using the example of the results obtained from Equation (1) from Reference [5].

**Figure 2.**(

**a**) Real (blue curves) and imaginary (red curves) parts of the permittivity, (

**b**) refractive index and (

**c**) extinction coefficient as a function of chemical potential for different GE thicknesses from various publications ($\lambda =1.55$ µm, ${t}_{G}$ = 0.34 ÷ 1 nm, ${\mu}_{\mathrm{c}}$ = 0 ÷ 1 eV, Γ = 8e13 s

^{−1}, T = 300 K).

**Figure 3.**(

**a**) Real (blue curves) and imaginary (red curves) parts of the permittivity, (

**b**) refractive index and (

**c**) extinction coefficient as a function of the chemical potential for different values of relaxation rate from various publications ($\lambda =1.55$ µm, ${t}_{G}$ = 0.34 nm, ${\mu}_{\mathrm{c}}$ = 0 ÷ 1 eV, Γ = 5e11 ÷ 5e14 s

^{−1}, T = 300 K).

**Figure 4.**(

**a**) Real (blue curves) and imaginary (red curves) parts of the permittivity, (

**b**) refractive index and (

**c**) extinction coefficient as a function of the temperature from various publications ($\lambda =1.55$µ$\mathrm{m},$ ${t}_{G}$ = 0.34 nm, ${\mu}_{\mathrm{c}}$ = 0 ÷ 1 eV, Γ = 8e13 s

^{−1}, T = 0 ÷ 100 °C).

**Figure 5.**(

**a**) Real (blue curves) and imaginary (red curves) parts of the permittivity, (

**b**) refractive index and (

**c**) extinction coefficient as a function of the chemical potential from various publications, when $\lambda =1.55$ µm, ${t}_{G}$ = 0.34 nm, ${\mu}_{\mathrm{c}}$ = 0 ÷ 1 eV, Γ = 8e13 s

^{−1}and T = 300 K. (

**d**) Real and imaginary parts of the permittivity, (

**e**) refractive index and (

**f**) extinction coefficient as a function of the chemical potential from various publications, when $\lambda =1.55$ µm, ${t}_{G}$ = 0.34 nm, ${\mu}_{\mathrm{c}}$ = 0 ÷ 1 eV, Γ = 2e12 s

^{−1}, T = 300 K.

**Figure 6.**Sketches of the location of graphene electrodes: (

**a**) Graphene-Insulator-Waveguide (GIW), (

**b**) Graphene-Insulator-Graphene (GIG), (

**c**) Electrolyte/ion-gel-Graphene-Waveguide+PhC Cavity (EGWC) and (

**d**) Graphene-Insulator-Graphene inside waveguide (GEin). (G–Graphene electrode and E–electrode).

**Figure 7.**Examples of the device configurations: (

**a**,

**a1**) Mach–Zehnder graphene modulator (M-ZM), (

**b**,

**b1**) planar photonic crystal nanocavity-based modulator (CNM) and (

**c**,

**c1**)–micro-ring resonator (MRR).

Symbol | Description |
---|---|

λ and Δλ | wavelength and its shift |

σ (${\sigma}_{G}$), ${\sigma}^{\prime}$and ${\sigma}^{\u2033}$ | surface conductivity (graphene surface conductivity), intraband ${\sigma}_{intra}$ and interband ${\sigma}_{inter}$ conductivities |

ε (${\epsilon}_{G}$), ${\epsilon}_{G\_re}$ and ${\epsilon}_{G\_im}$ | complex permittivity (graphene complex permittivity), its real and imaginary parts |

n (${n}_{G}$) and Δn | refractive index (graphene refractive index) and its change |

n_{s} | graphene surface carrier density |

k (${k}_{G}$) | extinction coefficient (graphene extinction coefficient) |

e | electron charge |

$\omega $ | radian frequency |

$\u0127=h/2\pi $ | $\u0127=h/2\pi $, reduced Planck constant |

$\Gamma $ and$\tau $ | phenomenological scattering rate and $\tau =1/\Gamma $, relaxation time |

${f}_{d}\left(\xi \right)$ | ${f}_{d}\left(\xi \right)={\left({e}^{\left(\xi -{\mu}_{c}\right)/{k}_{B}T}+1\right)}^{-1}$, Fermi-Dirac distribution ($\xi $–energy) |

${k}_{B}$ | Boltzmann’s constant |

T | temperature |

${\mu}_{c}$ | chemical potential |

μ | charge carrier mobilities |

${v}_{F}$ | Fermi velocity |

${t}_{G}$ and ${t}_{I}$ | graphene and insulator thicknesses |

L | graphene length |

d | spacing between minima or free spectral range |

${D}_{G}$ | distance between graphene electrodes |

φ | phase shift |

${V}_{G}$ and ${V}_{Dirac}$ | applied voltage and flat-band voltage corresponding to the charge-neutral Dirac point |

${C}_{ox}$ | ${C}_{ox}=\epsilon {\epsilon}_{0}({W}_{m}l/{t}_{I}),$ oxide (insulator) capacitance per unit area |

GE | Graphene Electrode |

GIW | Graphene-Insulator-Waveguide |

GIG | Graphene-Insulator-Graphene |

EGWC | Electrolyte/ion-gel-Graphene-Waveguide+PhC Cavity |

GEin | Graphene-Insulator-Graphene inside waveguide |

M-ZM | Mach–Zehnder Graphene Modulator |

CNM | Crystal Nanocavity-based Modulator |

MRR | Micro-ring Resonator |

Ref. | t_{G}, nm | TMM | t_{I}, nm | Γ, s^{−1} | Insulator Material | V_{G} | λ, µm | Device Type | $\mathbf{\Delta}\mathit{\lambda},\mathbf{pm}$ |
---|---|---|---|---|---|---|---|---|---|

[7] * | - | EGWC | - | 3.8462e13 | PEO + LiClO_{4} | −1.2 ÷ 1.2 V | 1.56907 | CNM | 800 |

[8] * | 0.33 | GIG | 90 | 5e11 ÷ 1e14 | Al_{2}O_{3} | −40 ÷ 40 V | 1.53–1.57 | M-ZM | 140 |

[9] * | 0.7 | GIG | 10 | 5e12 ÷ 1e14, 8e13 (Γ_{exp}) | Al_{2}O_{3} | 0 ÷ 3 V 0 ÷ 1 V/1 ÷ 3 V | 1.552 | M-ZM | 173/385 |

[10] * | - | GIG | 5 | - | Al_{2}O_{3} | 0 ÷ 15 V | 1.549 | CNM | 66/165 |

[15] | 0.34 | GIG | 70 | 1e14 | SiO_{2} | −4.95 ÷ 4.95 V | 1.539–1.559 | WOM | 10 |

[19] * | 1 | EGWC | 5 | 2.2789e14 | Al_{2}O_{3} &ion-gel | −2 ÷ 0 V | 1.55 | CNM | 1000 |

[20] * | 0.34 | EGWC | 10 | 7.5963e13 | HfO_{2} &PEO + LiClO _{4} | −7 ÷ 6 V | 1.55 | CNM | 1000 |

[24] * | 1 | GIW | - | - | Al_{2}O_{3} | −12.5 ÷ 0 V | 1.548–1.557 | MRR | 2000 |

[25] * | 0.5 | GIW | 25 | 1e14 | Al_{2}O_{3} | −6 ÷ 6 V | 1.55 | MRR | 125 |

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**MDPI and ACS Style**

Vorobev, A.S.; Bianco, G.V.; Bruno, G.; D’Orazio, A.; O’Faolain, L.; Grande, M.
Tuning of Graphene-Based Optical Devices Operating in the Near-Infrared. *Appl. Sci.* **2021**, *11*, 8367.
https://doi.org/10.3390/app11188367

**AMA Style**

Vorobev AS, Bianco GV, Bruno G, D’Orazio A, O’Faolain L, Grande M.
Tuning of Graphene-Based Optical Devices Operating in the Near-Infrared. *Applied Sciences*. 2021; 11(18):8367.
https://doi.org/10.3390/app11188367

**Chicago/Turabian Style**

Vorobev, Artem S., Giuseppe Valerio Bianco, Giovanni Bruno, Antonella D’Orazio, Liam O’Faolain, and Marco Grande.
2021. "Tuning of Graphene-Based Optical Devices Operating in the Near-Infrared" *Applied Sciences* 11, no. 18: 8367.
https://doi.org/10.3390/app11188367