II. Graphene Nanophotonic Modulators for Near-Infrared Applications
2.4. Experimental Investigation of the Solid-Electrolyte-Gated Graphene-Covered Metal-
2.4.3. Experiment Results – (2) Modulation Characteristics
Figure 2.4.5. Modulation characteristics of the solid-electrolyte-gated graphene-covered MISIM waveguide. (a)-(b) Modulated FFIL of the 10-μm-long MISIM waveguide at the modulation frequency of (a) 100 kHz and (b) 1 MHz. (c) Relation of the electro-optic S21 parameter to the modulation frequency. The square symbol is estimated S21 value for each frequency and the red curve is the transfer curve of the first-order low-pass filter.
To confirm the 3-dB bandwidth, the FFIL modulation of the combinations with the MISIM waveguide of length 10 μm in chips 2 and 3 was measured by a square wave voltage signal applied between the electrodes. The applied signal oscillated between 1 V and 3 V. The modulated FFILs are shown in Figure 2.3.6. To determine the time constant of each modulation, the time response is curve- fitted calculated with an expression –10log10[ΔPexp(–(t – t0)/τ)+P0] to the rising edge of each FFIL curve. In addition, for the evaluating falling edge of each FFIL curve, the time response is fitted with a curve calculated with an expression –10log10{ΔP[1–exp(–(t–t0)/τ)]+P0}. In the case of the combination in chip 2, the time constants from the rising and falling edges are 0.144 s and 0.114 s, respectively. In the case of the combination in chip 3, those are 0.0895 s and 0.0937 s, respectively.
Since the 3-dB bandwidth is given by (2πτ)-1, it is estimated to be 1.11 or 1.40 MHz for the combination in chip 2 and 1.78 or 1.70 MHz for the combination in chip 3. These values are similar to that obtained from Fig. 5(c).
On one hand, the 3-dB bandwidth is small since the polarization response of the solid electrolyte is inherently slow [68] and the solid-electrolyte and graphene electrodes are 1 mm distant from each other. However, on the other hand, the modulation is much faster than other operations driven by solid-electrolyte gating [69]. One possible reason for this is that the solid electrolyte could be affected by the high humidity during the measurement [70],[71].
Figure 2.4.6. Modulation by a square wave voltage signal of the frequency of 10 kHz. (a) FFIL response of 10-μm-long MISIM waveguide in chip 2. (b)-(c) Rising edge and falling edge of the FFIL response in chip 2. (d) FFIL response of 10-
2.4.4. Theoretical Investigation of the MISIM waveguide Integrated with a Graphene Capacitor
Up to this point, we have confirmed that the MISIM waveguide covered with graphene is really useful for a modulator with a large modulation depth. This may lay the cornerstone of developing the MISIM waveguide covered with a graphene capacitor, which can have a larger modulation depth and, possibly, a much larger 3-dB bandwidth. A Si photonic modulator using the MISIM waveguide covered with a graphene capacitor is schematically shown in Figure2.4.7(a), and the cross-section of the MISIM waveguide is shown in Figure 2.4.7(b). The graphene capacitor consists of two graphene layers and an aluminum oxide (Al2O3) layer of thickness tox (tox was set at 10 nm). The width of the gap between the electrodes (wg) can be made small since the MISIM waveguide mode is strongly confined in the insulator layers between the Si strip and the Cu blocks. In other words, the electrodes do not affect the MISIM waveguide mode if wg is larger than about 300 nm. By narrowing the gap, it is possible to decrease the resistance of the graphene layers and the capacitance of the capacitor. This is a clear advantage of the MISIM waveguide. In the case of a conventional Si strip waveguide covered with a graphene capacitor, the gap between two electrodes for such a graphene capacitor should be wider than a few micrometers.
Figure 2.4.7. Structure of the MISIM waveguide integrated with the graphene capacitor. (a) Schematic of the device. (b) Cross-sectional structure of the device.
To analyze the modulator, first, the modulation depth of the MISIM waveguide need to be calculated at a wavelength of 1550 nm. For the calculation, there is an assumption that the electrodes are infinitely separated and the graphene layers are infinitely wide. The calculated relation of the propagation loss of the MISIM waveguide to the graphene chemical potential c is shown in Figure 2.3.8(a) (assuming that the chemical potential of the upper graphene layer is c while that of the lower graphene is –c). As c increases from 0 eV to 0.6 eV, the propagation loss decreases from 0.649 dB/m to 0.237 dB/m, and the modulation depth (i.e., the propagation loss decreases due to the c increases from 0 eV to 0.6 eV) is 0.412 dB/min this case. Next step is determining the width of the region where the two graphene layers overlap with the Al2O3 layer in between. The modulation depth is shown as a function of the overlap region width wo in Figure 2.3.8(b). It almost does not change for wo > 300 nm, and wo was set at 300 nm. Finally, the electrode gap width wg was set at 800 nm. The theoretical modulation depth of 0.412 dB/m may be undoubtedly achieved from a realized device since the realized MISIM waveguide covered with single-layer graphene really has the modulation depth similar to the calculated value.
For an extinction ratio of 4.6 dB, which is that of the Ge electroabsorption modulator in [40], the MISIM waveguide should be 11.2 m long. Then, the total length of the modulator including the two couplers is 12.4 m, and the insertion loss of the modulator is 3.65 dB. The 3-dB bandwidth of the modulator can be estimated in the following way. Under the assumption that the 3-dB bandwidth is limited by the RC time constant, we determined the device resistance R and capacitance C of the modulator. When the sheet resistance of graphene and the contact resistance between the electrode and graphene are denoted by rs and rc, respectively, the device resistance is given by R = [2rc+rs (wg+wo)]/lg, where lg is the MISIM waveguide length. For rs = 125 /sq [9] and rc = 100 ∙μm [13], R = 30.2 .
When the quantum capacitance of graphene and the capacitance of the graphene overlap region are denoted by Cq and Co, respectively, the device capacitance is given by C = (1/Co+2/Cq)–1. Cq is approximately given by 2e2μclgwg/(πħ2vF2) [53], and Co is given by ε0εrlgwo/tox, where 0 and r are the vacuum permittivity and the RF dielectric constant of Al2O3, respectively. With Cq evaluated for c = 0.4 eV, C = 28.5 fF. In consequence, the 3-dB bandwidth, which is given by (2πRC)–1, is 185 GHz.
(Actually, it highly depends on rs and rc, which depend on fabrication processes. For example, if they are five times larger than used for the estimation, it decreases to 37 GHz. Finally, we confirmed that the extinction ratio changes by less than 0.1 dB from 4.6 dB in the wavelength range between 1400 nm and 1695 nm (Figure 2.3.8(c)). Therefore, the optical bandwidth of the modulator reaches almost 300 nm.
Figure 2.4.8. Characteristics of the MISIM waveguide covered with the graphene capacitor. (a) Relation of the propagation loss to the chemical potential of graphene. (b) Modulation depth as a function of the overlap region width,
III. Graphene Nanophotonic Modulators for Mid-Infrared Applications
3.1. Introduction to Mid-Infrared Photonics
Mid-infrared (mid-IR) photonics is the field of generating, manipulating, and detecting photons in the spectral range between 2 and 20 μm [72]. Recently, it has attracted considerable attention due to many advantages of the mid-IR. For example, many molecules possess their own unique absorption spectra in the mid-IR, which are called fingerprints. The fingerprints are essential for identifying molecules. In addition, there are two atmospheric transmission windows in the mid-IR: 3-5 μm and 8-13 μm spectral ranges. Mid-IR photonics has been used for military applications such as infrared countermeasures operating in the 3-5 μm spectral range [73]. Its promising applications include chemical or biological sensing [74],[75], astronomical instrumentation [76] and free space communications [77],[78].
Although generation and detection of mid-IR photons have been actively studied [79],[80]
integrated-optical manipulation of such photons seems to be still in its infancy. Mid-IR integrated photonics naturally becomes of great importance to develop compact and rugged devices. Following this direction, a variety of mid-IR waveguides have been investigated [81]-[85]. Among them, waveguides based on group IV materials are very attractive since silicon (Si) and germanium (Ge) with large refractive indices are transparent in the mid-IR and Si photonics technology for telecom wavelengths is quite mature [86]. Passive Si or Ge waveguide devices such as splitters, interferometers, resonators, and (de)multiplexers have been reported [87]-[90]. However, just a few Si or Ge waveguide modulators exist now. One is a thermo-optic Mach-Zehnder modulator which operates at a wavelength (λ) of 3.8 μm [91]. Its 3-dB bandwidth and power consumption are 23.8 kHz and 47 mW, respectively.
Another modulator is an electro-optic Mach-Zehnder modulator based on a Si-on-lithium-niobate substrate [92]. It operates at λ = 3.39 μm and it may have a bandwidth larger than 10 GHz. Both the modulators are more than 1,000 times longer than the wavelengths. In addition, there is an all-optical absorption modulator based on a Ge-on-Si substrate, which operates at λ ≤ 3.2 μm and has a bandwidth of a few tens of MHz [92]. Electrorefraction modulators based on the free carrier plasma dispersion effect may be developed. However, such modulators operating at λ = 4 μm may also require waveguides longer than 250λ [93]. Moreover, they require laborious fabrication processes to form pn or pin junctions. These facts indicate that ultracompact modulators with a large bandwidth and low power consumption can be realized only when a whole new way of modulation is introduced.
3.2. Theoretical Investigation of a Mid-Infrared Graphene Plasmonic Modulator based on a Coupling of a Hybrid Plasmonic Waveguide Mode to a Graphene Plasmon
Graphene plasmons (GPs) are collective electron oscillations mainly existing in the mid-IR to terahertz range [94],[95] The electromagnetic wave associated with a GP, called a GP polariton (GPP), is extremely confined around graphene, and its characteristics are dynamically controllable by using electrostatic gating. Diverse GP waveguides based on a graphene nanoribbon [96] uniform graphene controlled by a nonuniform gating electric field [97] or uniform graphene on a patterned substrate [98]
have been used to demonstrate extremely compact functional devices working in the mid-IR. Such GP devices include electroabsorption modulators, switches, Bragg filters, resonators, etc., [99]-[102]. The fact that their dimensions are quite smaller than 0.1λ is very attractive and highlighted, but coupling of GP devices to mid-IR sources, detectors, or photonic waveguides is usually overlooked. Grating couplers have been used to excite a GPP from a free-space beam [103],[104], and a tapering structure of silicon carbide can couple a surface phonon polariton to a GPP [105]. However, the conversion efficiency of the latter is just 25 %. This means that there is practical difficulty in integrating GP devices with photonic devices. In this regard, it is necessary to exploit GPs in a different way. In this chapter, a Ge waveguide modulator based on grating-assisted coupling of a hybrid plasmonic waveguide mode to a GP-based waveguide mode is theoretically investigated around λ = 8 μm [106]. The design of the modulator is explained, and it is demonstrated that intensity modulation larger than 25 dB is achieved at a subwavelength scale with an insertion loss smaller than 2 dB. The investigated modulator may open the door to ultracompact high-speed mid-IR waveguide modulators consuming small energy.
3.2.1. Structure of the Modulator and Analysis Method
The structure of the investigated modulator is schematically shown in Figure 3.2.1. It consists of an input photonic waveguide, a hybrid plasmonic waveguide containing graphene on a grating in its slot region, and an output photonic waveguide. Both the photonic waveguide and the hybrid plasmonic waveguide share in common the Ge core layer and the low-index lower cladding or the substrate. For the hybrid plasmonic waveguide to be formed, the metal layer and the thin low-index insulator layers with graphene in between are placed on the Ge core. As well known, the hybrid plasmonic waveguide supports the so-called hybrid plasmonic waveguide mode (HPWM) which is strongly confined in the insulator layers, namely the slot region (see Figure 3.2.2(c)). In addition to the HPWM, there is another waveguide mode named the GP-based waveguide mode (GPWM) which is intrinsically a GPP and substantially bound to the graphene (see Figure 3.2.2(d)). The magnetic field of the GPWM is discontinuous across and almost antisymmetric about the graphene, but that of the HPWM is almost continuous across the graphene. The fundamental transverse magnetic (TM) mode of the input photonic waveguide is incident on the hybrid plasmonic waveguide and excites very efficiently the HPWM (see Figure 3.2.2(b)). While the HPWM is propagating, it is coupled to the GPWM propagating in the same direction via the grating engraved on the insulator layer below the graphene. The grating is essential for the co-directional coupling between the HPWM and the GPWM at the coupling wavelength λc due to the enormous phase mismatch between them. After the grating, the HPWM is incident on the output photonic waveguide and excites the fundamental TM mode there. Because of the coupling, there is a dip or rejection band centered at λc in the transmission (i.e., the ratio of the output fundamental TM mode power to the input fundamental TM mode power) spectrum of the output photonic waveguide.
As the graphene chemical potential μc is adjusted, the dip shifts such that intensity modulation of the fundamental TM mode can be achieved at a fixed wavelength. For the intensity modulation, the metal layer functions as an electrode to apply the electric modulation signal which uniformly tunes μc in the graphene. The essence of the modulator operation is the grating-assisted coupling between the disparate waveguide modes. If a waveguide has two modes with very different propagation constants but small propagation losses, the power of one mode with a smaller propagation loss can be reduced to zero by grating-assisted coupling between such modes. However, when a waveguide has a mode with small propagation constant and loss and another mode with huge propagation constant and loss, the power of the low-loss mode almost does not change without an intense overlap between the two modes even though phase matching between them is achieved via a grating (Supplementary Information S1 in ref.
[106]). The hybrid plasmonic waveguide is a unique structure which not only enables the required intense overlap between the HPWM and the GPWM in its slot region but also is easily and efficiently
Figure 3.2.1. Structure and operation principle of the mid-infrared waveguide modulator. The modulator consists of the input photonic waveguide, the hybrid plasmonic waveguide, and the output photonic waveguide. Their common substrate and core are calcium fluoride (CaF2) and germanium (Ge), respectively. The hybrid plasmonic waveguide is made by placing the aluminum (Al) layer and thin low-index insulators 1 and 2, which are hafnium oxide (HfO2) and zinc sulfide (ZnS), respectively, on the Ge core. The graphene layer is between insulators 1 and 2, and the grating is the periodic groove array engraved on insulator 2. The hybrid plasmonic waveguide supports the hybrid plasmonic waveguide mode (HPWM) and the graphene-plasmon-based waveguide mode (GPWM), which have propagation constants H and G, respectively. The photonic waveguide mode (PWM), which has a propagation constant P, of the input photonic waveguide efficiently excites the HPWM, and the HPWM is coupled to the GPWM via the grating at the coupling wavelength. This co-directional coupling results in a rejection band at the coupling wavelength in the transmission spectrum of the output photonic waveguide. The transmission is high out of the rejection band since the HPWM can travel in the hybrid plasmonic waveguide without large loss and then excite efficiently the PWM. By adjusting the driving voltage Vd between the Al and graphene layers, the coupling wavelength is tuned. In consequence, intensity modulation is achieved at a fixed wavelength.
The realization of the investigated modulator demands materials which are transparent in the spectral range between 7 and 9 μm and, especially regarding the insulators, processable in ways compatible with graphene. Calcium fluoride (CaF2) is chosen as the substrate material. It has been widely used for mid-IR photonics and it has a small refractive index (1.3499 at λ = 8 μm) [107]. Chen et al. reported the Si-on-CaF2 substrate which is made by transferring and bonding a detached Si membrane on a CaF2 substrate [108]. This technology could be employed to make the Ge-on-CaF2
substrate necessary for the modulator. Zinc sulfide (ZnS) is chosen as the lower insulator since it is one of limited low-index dielectrics transparent in the 7-9 μm spectral range [109]. It is possible to deposit ZnS on a Ge substrate by using thermal evaporation [110], and ZnS is patternable by reactive ion etching [111]. In addition, graphene was well transferred onto ZnS films to realize photodetectors [112].
Hafnium oxide (HfO2) is chosen as the upper insulator since it can be formed on graphene by using atomic layer deposition without serious deterioration of graphene properties [113],[114]. It has been used as a top gate oxide of graphene-based transistors [114]. Finally, aluminum (Al) is chosen as the metal. In the mid-IR, Al is as good for plasmonics as gold or silver [115]. The thicknesses of the HfO2, ZnS, and Ge layers are denoted by tH, tZ, and tG, respectively. For convenience of simulation, it is assumed that the Al layer and the CaF2 substrate are semi-infinite. The wavelength-dependent refractive indices of the materials [116]-[120] are used for the analysis of the modulator. Graphene is treated as a boundary with a surface current density σgEt, where σg is the graphene conductivity and Et is the electric field tangential to the graphene.48 σg is calculated by using the analytic expression in Reference 49, which depends on λ, μc, and the graphene scattering rate Γ. Γ is given by (evF2)/(2μμc) , where e, vF, and μ represent the electron charge, the Fermi velocity, and the graphene carrier mobility, respectively.50 In this work, μc and μ are normally assumed to be 0.6 eV and 10,000 cm2/V/s, respectively.
To design and analyze the modulator, waveguide analysis and wave propagation analysis are required. The waveguide analysis was carried out by using the transfer matrix method taking into account boundaries with surface current densities. The wave propagation analysis was mainly carried out by using the finite difference time domain (FDTD) method (FDTD Solutions, Lumerical Inc.).
However, it takes a quite long time to analyze wave propagation through the grating using the FDTD method. Thus, as reference during the design of the modulator, the wave propagation in the hybrid plasmonic waveguide was approximately but quickly analyzed by using the mode-matching method which is applied to the forward-going and backward-going guided modes of the hybrid plasmonic waveguide. When the FDTD method was used, the meshes used in the slot region had dimensions of 1 nm by 1 nm. The meshes used outside the slot region were gradually enlarged such that the largest meshes had dimensions of 10 nm and 8 nm in the horizontal and vertical directions, respectively. For the analysis based on the FDTD method, the fundamental TM mode was launched as a source in the
output photonic waveguide to extract the output fundamental TM mode power. Since the hybrid plasmonic waveguide contains metal and graphene, the powers of the HPWM and the GPWM at a position in the hybrid plasmonic waveguide were calculated by the following steps: First, the electromagnetic fields at the position were monitored. Second, the unconjugated orthogonality was applied to the fields to determine the mode expansion coefficients of the HPWM and the GPWM for the fields. Last, the powers were calculated using the coefficients.