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Abstract
This paper describes the design and numerical simulation of a two-dimensional comb-shaped graphene structure. Based on the principle of using the simplest possible design for the intended function of the bandpass filter, the proposed structure includes one graphene nanoribbon (GNR) and several lateral GNRs vertically placed near the main GNR. The transmission characteristics of the bandpass filter can be tuned by adjusting geometric parameters, or, by adjusting the chemical potential of graphene. These control approaches are, together, more convenient for tunability than a conventional metallic structure. It can be observed that by increasing the gate voltage, the length or width of the periodic part, one can move the peak of the transmission spectrum towards smaller wavelengths. We believe that the presented structure will be useful for optical integrated components and other compact optical devices in the future.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Surface plasmon’s polarition(SPP) has attracted interest of many researchers [1]. SPP’s edge modes at metal-insulator-metal(MIM) interface have been widely utilized as plamonic filters [2], sensors [3], and sources [4]in the frequencies of optical range. However, their development was hindered by their low transmission efficiencies and the inconvenience that cannot be tuned freely once the geometry of the structure is fixed.
On the other hand, the field of graphene plasmonics has burgeoned rapidly in recent years [5,6]. Because of graphene’s unique electronic structure, consisting of a monolayer of carbon atoms arranged in a honeycomb lattice, it is able to easily overcome the low transmission efficiencies problems of MIMs and also have capability for dynamic tunability of the device just by applying a variable bias [7]. Thus, a great deal of attention has been devoted to graphene-based waveguides. Numerous devices based on graphene plasmons polarizations have been numerically simulated or experimentally demonstrated, including modulators [8], optical switches [9], filters [10,11], phase shifters [12], etc.
Graphene-based plasmons are easily induced over a wide range from the near infrared to the terahertz range; this is similar the wavelength range of MIM-based plasmonic devices [13]. The peak of the transmission spectra of a structure can be changed by applying an external voltage which actively modifies the chemical doping of graphene [14]. SP modes supported by graphene ribbon constructions have been studied and classified in terms of their edge modes and waveguide modes; their resonant frequency is strongly dependent on their original size and also their material properties [14–16]. The edge mode causes a strong field confinement near the graphene rims and hence enhances the electromagnetic coupling between objects [14].
In most studies related to GNR filter or waveguide published, the transmission spectra are sharp with narrow passbands [10,11,17]. In some applications, however, a wideband filtering may be needed [18,19], calls for a different structure with characteristic of a bandpass filter with a broad pass band.
Hence, in this study, a bandpass filter with broad pass-band characteristic is proposed based on monolayer GNRs that are placed on top of a silicon/silica slab. With a comb-like geometry, the structure consists of a set of GNRs, in which several GNRs are placed perpendicularly next to the input-output GNR. By simply changing the bias voltage applied to the graphene, the transmission characteristics of a broadband bandpass filter are altered and therefore room temperature tunable filtering can be achieved. Also, more details are studied of the geometrical parameters of GNRs that are pertinent to the transmission spectra. We believed that this GNR-based midinfrared bandpass filter has good potential for developing ultra-compact optoelectronic devices such as low-cost spectrometers and hyperspectral imaging sensors.
2. Theory, design, and methods
At mid-infrared frequencies, the inter-band transition is suppressed. The complex surface conductivity of graphene can be calculated from the simplified Kubo formula [20]:
Among the factors that determine the conductivity of graphene , the chemical potential $\mu _c$ is the most commonly used, for it can be easily tuned by the gating voltage. When a gating voltage $V_{bias}$ is applied, it will change the Fermi level $E_F$ and the intra-band losses of graphene. The changed Fermi level $E_F$ shifts away from the Dirac point and is highly doped. This changes the value of the chemical potential $\mu _c$ with surface conductivity $\sigma _g$. The relationship between the applied voltage ($V_{bias}$) and the chemical potential ($\mu _c$) is [21]:
where $t, \varepsilon _{SiO_2}$ is the thickness and permittivity of the thin $SiO_2$ layer, $v_f$ is the Fermi velocity.The waveguide effective refractive index $n_{\textrm {eff}}$ can be obtained by solving the dispersion equation [22]
where $\eta _0 (\approx 377 \Omega )$ is the impedance of air.In this study, the graphene is treated as an ultra-thin anisotropic material with a thickness of $\Delta =0.34 \textrm {mm}$. The inplane permittivity of graphene is [23,24]
where $\varepsilon _0$ is the permittivity of vacuum, and the out-of-plane permittivity is a constant $\varepsilon _{\perp }=2.5$.To verify the theory and research these graphene structures’ performance, the FDTD method with a PML absorbing boundary condition was used to investigate these structures’ transmission properties. Two power monitors are set at $Port_{in}$ and $Port_{out}$ as shown in Fig. 1 to detect the input power $P_{in}$ and the transmitted power $P_{out}$. The transmissivity could then be calculated as $T = P_{out}/P_{in}$. When the SPP wave is excited, the GSP edge mode is coupled into the lateral ribbons. The GSP wave that satisfied certain resonance conditions could be coupled to the exit port.
Fig. 1. a. A 3D-schematic drawing of the single tooth GNR structure supported on its substrate. b. Top view of the proposed single-tooth GNR structure
First, for simplicity, a single-tooth GNR structure was proposed and studied. A graphical representation of the single tooth GNR-based bandpass filter structure is shown in Fig. 1. Table 1 gives the structural parameters of the single-tooth GNR structure. In order to eliminate the excitation of waveguide modes in graphene ribbon, we choose the width of GNR of 15 nm to construct the plasmonic bandpass filter, in which only edge modes can be excited. Moreover, the doped silicon is used as substrate which one can easily apply gate voltage on it. Figure 2 shows the biasing sketch, as well as the relationship between the gating voltage and the chemical potential.
Fig. 2. a. The side view of the proposed filter with a monolayer graphene ribbon deposited on the $SiO_2/Si$ substrate. A gating voltage $V_{bias}$ is applied between a layer of metal $Au$ and the $Si$ substrate to produce the desired chemical potential of graphene. b. Relationship curve between the gating voltage and the chemical potential with the thickness of t = 50 nm.
Table 1. Geometry Parameters of the Proposed Single-tooth GNRs Structure
A graphene nanoribbon is distributed along the y-coordinate axis, with a single short coplanar nanoribbon placed vertically. These nanoribbons are located on top of a substrate. When a bias voltage is applied, the chemical potential of the graphene can be modified. SPPs will be strongly excited along the edge of the graphene layer when a broadband mid-infrared Gaussian wave is incident upon the input port of the structure.
For a single tooth comb type structure, a physics-based theory about the relationships between transmission and the incident wavelengths from the source can be modeled as [25]:
It can be concluded from (5) that if the phase satisfies $\phi (\lambda ) = (2m+1)\pi \quad (m=0,1,2\ldots )$ the two terms inside the absolute value sign on the right hand side of the equation cancel one other and so that the transmittance $T$ will be minimized. Hence, the wavelength $\lambda _m$ of the trough transmission is determined to be:
It was noted that the wavelength $\lambda _m$ is positively correlated with the tooth depth $d$ and that it depends on tooth width $w$ due to relationship between $n_{\textrm {eff}}$ and $w$.A single tooth structure (as shown in Fig. 1) was simulated using the commercial software Lumerical FDTD solutions. In the simulation, a dipole source was placed 2 nm upon graphene ribbon to excite the fundamental edge mode in the GNRs. When it comes to experiment, a probe tip can be placed on the top of the graphene ribbon to excite the SPPs. Meanwhile, a scattering-type scanning near-field optical microscopy (s-SNOM) can be used to measure the SPP propagation and field distribution. To ensure simulation accuracy, the minimum FDTD mesh scale was set to 0.1 nm for z direction, and 1 nm for both x and y directions. Moreover, boundary conditions of perfectly matched layers (PML) were applied in x, y and z directions. Figure 3 shows the y-z axis profile of the structure. The influences of a device’s geometric parameters on the transmission spectrum was studied, as shown in Fig. 4.
Fig. 3. The $E_z$ field distributions of the SPPs of GNR in the y-z axis section at a wavelength of 5000 nm. It can be observed that the edge mode was excited and hence confined the energy of SPP along the edge of the GNR
Fig. 4. a. Transmission characteristics versus lateral GNR length $d$ varied from 30 nm to 40 nm b. Transmission characteristics versus width $w$ changed from 12.5 nm to 17.5 nm.
The transmission results in Fig. 4a shows obviously that when teeth length was set to 30 nm, 35 nm and 40 nm, red shift occurs in the transmission dips. Also, as depicted in Fig. 4b, when the teeth widths changed from 12.5 nm to 17.5 nm where in step of 2.5 nm, the transmission dip corresponds to lower wavelength. The reasons for those results can be explained by the equations introduced above.
Electrical tunability provides graphene an extraordinary property. Electrical turning can be realized by modulating the chemical potential of the GNR. We studied the transmission spectrum of the filter with $\mu _c$ set to 0.3 eV, 0.4 eV and 0.5 eV. The tooth widths are fixed at 15 nm and the tooth lengths is 35 nm. Transmittance results, depicted in Fig. 5 show that there is a rapid blue shift when $\mu _c$ is increasing.
Fig. 5. Transmission characteristics of a single-tooth filter for different chemical potential values
Naturally, there is a fundamental interest in extending knowledge about the characteristics of single tooth structures to multiple-tooth structures (as shown in Fig. 6) to observe the differences between them. Table 2 gives the summarized values fo the proposed multiple-teeth GNR structure. Acting as a filter, multiple-tooth structures based on metal-insulator-metal (MIM) have been studied before, withband-pass or band-stop characteristics. It can be predicted that similar structures based on graphene have the same characteristics. The difference is that because the graphene nanoribbon multi-tooth filter can work in the mid-infrared up into the THz band, and because of the strong mode constraint of graphene, it can compress operating wavelengths of the devices from tens of microns to within 500 nm. In theory, if the distance between the teeth of the multi-tooth filter is sufficiently far (beyond the height of the graphene surface plasmon evanescent wave), a multi-tooth filter is a simple extension of a single-tooth filter. Therefore, the filter multi-tooth will use the transmission spectrum of the single-tooth filter as the center frequency to form a broadband band-stop filter. Considering the size of the device, however, space between individual teeth is less than distance of surface plasmon evanescent wave, so there is coupling between them. This will give these filters band-pass characteristics, where the simulation results are shown in the next section.
Fig. 6. a. A 3D-schematic drawing of the multiple-tooth GNR structure on its substrate. b. Top view sketch of the proposed multiple-tooth bandpass filter structure.
Table 2. Summarized Values of the Proposed Multiple-Teeth GNRs Structure
3. Results and discussion
Figure 7 shows the transmission characteristics of the proposed structure in Fig. 6. As illustrated in the transmittance figure, the peak of the structure are located at about $\lambda$ = 7000 nm with a maximum transmittance of 0.86 for a 200 nm long GNR. Those frequencies with a standing wave resonance pattern can be efficiently transmitted to the right-side waveguide. Table 3 gives the filter performance comparison between the proposed structure with the reported GNR-based filters. It can be seen from Table 3 that the wider fractional transmission band of 37.4% can be achieved by the proposed filter.
Fig. 7. Transmission characteristics of the multiple-tooth structure filter with a width $w$ of 15 nm, length $d$ of 30 nm and a chemical potential of 0.3 eV
Table 3. Comparison between the proposed filter and some reported filters
The $E_z$ field distribution in the x-y axis of the transmittance peak corresponding to the wavelength of 7000 nm is depicted in Fig. 8a. It should be noticed that only TM SPP could be supported within the desired frequency region of the proposed structure. In such a case, z-component of electromagnetic field would best describe the propagation of TM SPP waves. Furthermore, the $E_z$ field distribution for transmission dipped corresponding to the wavelength of 7000 nm is depicted in Fig. 8b, which shows that there is no excitation and the field in the resonator is very weak, as the waves cannot pass through the filter.
Fig. 8. The contour profiles of the field $E_z$ of the bandpass filter with wavelength of a. 7000 nm and b. 8200 nm.
The transmission spectra depicted in Fig. 9a corresponding to different widths of the graphene nanoribbons. In the simulation, only one parameter was changed, while the others were kept constant. The simulation results reveal that with increase of width $w$ from 12.5 to 17.5 nm in step of 2.5 nm, the transmission peaks show a blue-shift trend and the passband wavelength narrows the effective index $n_\textrm {eff}$ decreases with the addition of the ribbon width $w$, leading to a decreasing resonant frequency. The length of the GNRs affect the transmission spectra and the result is demonstrated in Fig. 9b, for lengths from 30 to 50 nm in 10 nm steps. It is shown that the transmission peaks exhibited redshift with the increasing ribbon length $d$, due to the increase of the resonant wavelength.
Fig. 9. Transmission spectrum dependence on different parameters:a. widths of GNR, $w$; b. length of lateral GNRs, $d$; c. period, $\Lambda$; and d. chemical potential of GNR.
We also studied the effect of the period $\Lambda$ of the teeth on the transmission spectrum when other parameters are consistent. Results for values of $\Lambda$ of 25 nm, 30 nm, 35 nm are shown in Fig. 9c, the change in period has a significant impact on the bandwidth of the filter, and causes a change in the center frequency of the bandpass. This change is caused by the change of the coupling condition between adjacent teeth. This feature is significantly different from MIM-based multi-tooth filters; the mode energy of the MIM waveguide is concentrated in the gap between the metal sections, in the dielectric layer: a relatively small tooth distance is required to eliminate the influence on the transmission spectrum of the filter of mutual coupling between the teeth. However, the mode energy of graphene nanoribbons is mainly concentrated at the edge of the nanoribbon. If the distance between each tooth is larger than the height of a surface plasmon, the influence of the period of the tooth on the filters’ performance is greatly reduced, which is expressed as an increase in the filter passband.
As adverted before, one of the most attractive properties of graphene that makes it a promising material for applications such as optical integrated on-chip circuits is its capability for dramatic optical property alterations simply by turning gate voltage. By using a gate voltage turning (an electrostatic gate voltage or chemical doping) to the GNRs of the structure, surface charge carrier densities are varied dynamically. Alterations in the surface charge carrier densities modify the chemical potential and the surface conductivity of the GNRs. Due to these modifications, the optical response of the GNRs and thus optical response of the whole filter structure is altered, bringing on peak variations for the device. In Fig. 9d, the transmission properties are demonstrated. As shown Fig. 9d, increasing the $\mu _c$ shifts the center peak wavelength downward, while at the same time its peak amplitude slightly increases.
4. Conclusion
In this paper, a set of comb-shaped graphene nanoribbons structure has been proposed as a compact bandpass filter with a broad passband based on the edge mode. After simulation using the 3D FDTD method, the results show that the transmission spectra can be affected by the width and length of the graphene nanoribbons. Furthermore, the structure also shows sensitivity to the chemical potential of graphene. We shall find potential applications in highly integrated optical circuits and other relative field.
Funding
National Natural Science Foundation of China (51607050, 61871171); Fund of Science and Technology on Electronic Information Control Laboratory (6142105180615).
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