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Tunable plasmon induced transparency in patterned graphene metamaterial with different carrier mobility
Superlattices and Microstructures 136 (2019) 106295
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Tunable plasmon induced transparency in patterned graphene metamaterial with different carrier mobility Chunzhen Fan *, Peiwen Ren, Wei Jia, Yuanlin Jia, Junqiao Wang School of Physics, Zhengzhou University, Zhengzhou, Henan, 450001, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Plasmon induced transparency Metamaterial Terahertz Graphene mobility
Dynamical manipulation of plasmon induced transparency (PIT) is realized in patterned graphene metamaterial through different carrier mobility, which is composed of two strips and a split ring resonator (SRR). Due to the near field coupling between these two bright modes, a prominent transparency window appears in the transmission spectrum. It can be well fitted with coupled Lorentz oscillator model, which furtherly verifies the accuracy and reliability of our results. On and off switch of the PIT transparency window can be regulated through the carrier mobility. Namely, the PIT transparency window becomes more evident with larger carrier mobility. The plasmon induced absorption (PIA) can also be synchronously realized and the absorption rate reaches up to a high value of 0.5. In addition, the group delay gets enhanced with larger mobility. And the delay time achieves 1.57 ps. In case of sensing application, the linear shift of the PIT transparency window can be obtained with different substrate. Thus, our work shadows great prospects in highly tunable optical switching, optical filters, integrated slow light and sensing devices through non-contact regulation.
1. Introduction Plasmonic metamaterials have emerged as a promising research topic in recent years due to its ability to realize unique optical effects with conventional materials [1]. They are artificial engineering materials with reasonably designed composition and arrangement of unit cell. These materials enable intriguing applications owing to their response to electromagnetic waves [2], sound waves [3] or thermal waves [4] and embody the excellent performance beyond the capability of natural materials [5]. Plasmon induced transparency (PIT) is one of these noticeable optical phenomena realizing with metamaterial. It is an analogue electromag netically induced transparency, arising from the destructive quantum interference with different excitation pathway [6,7], resulting in a sharp narrow transparency window in its transmission profile [8]. Accompanied effect is the varied dispersion behavior, namely, the group velocity of the propagation wave can be greatly reduced. Generally, PIT effect is caused to happen with two excitation mechanism. One is ascribed to the near field coupling between two bright modes, the other is attributed to the destructive interference between bright and dark modes [9]. The bright resonator can directly excite with the incident light acting as a dipole oscillator. Fu et al. investigated tunable PIT with two parallel graphene nanostrips [10], each strip serves as a bright mode. Jia et al. proposed a simple PIT structure composed of two coupled horizontal graphene strips attached to continuous vertical wire [11]. The dark mode is an indirect excitation, which has to be excited through destructive interference with the bright mode [12,13]. When the bright and
* Corresponding author. E-mail address: [email protected] (C. Fan). https://doi.org/10.1016/j.spmi.2019.106295 Received 10 September 2019; Accepted 11 October 2019 Available online 21 October 2019 0749-6036/© 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Schematic illustration of the patterned graphene metamaterial on the dielectric substrate. (a) Overall view, (b) top view of the unit cell. Parameters: l1 ¼ 10 μm, l2 ¼ 5.4 μm, l3 ¼ 1.7 μm, w1 ¼ 1.5 μm, w2 ¼ 1.0 μm, d ¼ 1.5 μm, Px ¼ Py ¼ 16 μm.
dark resonators are arranged in a periodic way in the THz region, the coupling between them can induce the narrow transparency spectra. Namely, it’s the bright and dark mode coupling. Our group has investigated the active control and large group delay in the plasmonic metamaterials, which consist of two identical graphene strips and a cross-shaped graphene resonator in the right middle [14]. Tunable PIT metamaterials consists of cut-wire pairs and U-shaped ring resonators were explored [15]. Ultra broadband PIT spectra can be achieved based on asymmetric split-ring resonator of Dirac semimetal metamaterials [16]. A classical PIT design including a periodic lattice with a cut wire and a pair of symmetry split ring resonators was also studied [17]. The PIT effect can remarkably produce a large dispersion and slow down photons velocities effectively [18], which has many wide applications in slow light devices [19], optical switcher [20] or optical sensors [21]. At the early stage of PIT development, its design usually composes of metallic unit cell, which has an inevitable intrinsic Ohm loss. Thus, the tunability of PIT transparency window can only be initiated through the varied geometrical parameters. It indicates that once the structures are fabricated, it is hard to tune its transmittance without re-fabrications. It hinders the performance of figure of merit (FOM) and the slow light behavior. To overcome such restrictions and reduce the radiations loss, graphene has emerged as a new fascinating materials. It has enhanced surface plasmon, lower propagation loss, and higher carrier mobility [22]. More importantly, it is capable of initiating a non-contact control of its transportation properties. Up to now, different types of graphene layers are investigated, like waveguide resonator [23], cut wire resonator [24], split ring resonator [25] and so on. However, much work has focused the tunable effect through the Fermi energy or the polarization angle in the non-contact way, little work is carried out on the tunable PIT with carrier mobility through chemical doping. In addition, the dual consideration of the PIT and PIA have also barely fully investigated. In this work, we numerically investigate the transmission spectra with graphene metamaterials, enabling the non-contact dynamic regulation. It consists of two graphene strips and a SRR. Both of them can directive couple with the incident light, and the near field coupling between these two bright modes results in a PIT transparency window. The physical mechanism of PIT phenomenon is elaborately explored in our manuscript with the Lorentz oscillator model through the analysis of the coupling parameters. Our theoretical results achieve good agreement with the numerical calculations. Furthermore, the on and off modulation of the PIT window is investigated in details with different carrier mobility. The position of the PIT window is kept unchanged. But the magnitude of transmittance is greatly got enhanced with a larger mobility. The corresponding absorption rate reaches up to 50%, which is a fascinating result compared to the absorption rate of current monolayer graphene in THz band. Then, we have also investigated the dispersion behavior around PIT transparency window with different mobility. Finally, the sensitivity of the structure can be dynamically regulated by changing the refractive index of the substrate, and the group delay of the graphene metamaterial achieve 1.57 ps. Our work provides a new strategy to get tunable PIT effect with graphene carrier mobility in THz region. 2. Structure design and methods The overall schematic representation of our patterned graphene structure on the dielectric substrate is shown in Fig. 1(a). The unit cell consists of two strips and a SRR, which is periodically arranged in the x and y direction. The periodicity is taken as Px ¼ Py ¼ 16μm. In our simulation, the incident THz wave is perpendicular along z direction and the electric field is polarized along y direction. The concrete parameters of the unit cell are shown elaborately in Fig. 1(b). The length of the cut wire is l1 and it takes as 10 μm: l2 is the length of SRR and it equals to 6 μm. l3 is the short length of the SRR and it sets as 1:7 μm. The width of the two resonator arew1 ¼ 1:5 μm and w2 ¼ 1:0 μm. d represents the distance between the strips and the SRR, which is fixed at 1.5μm. The transmission and 2
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Fig. 2. Contour diagram of the (a) real and (b) imaginary part of the graphene mobility in THz region.
Fig. 3. Transmission spectra in different patterned graphene metamaterial are investigated in THz region. (a) Only strips (black line), (b) only SRR (red line), and (c) combined structure (blue line). Parameters: l1 ¼ 10 μm, l2 ¼ 5.4 μm, w1 ¼ 1.5 μm, w2 ¼ 1.0 μm, d ¼ 1.5 μm, d1 ¼ d2 ¼ 1.7 μm, EF ¼ 1.5 eV and μ ¼ 3.0 μ0.
absorption spectra involved in this paper are obtained with the finite element method. For simplicity, we only investigate the unit cell in the structure, and set periodic boundaries in the XY direction of the work plane. Perfectly matched layer is taken above the proposed metamaterial to eliminate non-physical reflections at the boundary. The conductivity of the graphene includes the contribution from inter-band and intra-band parts, which can be got with the Kubo equation. Under room temperature, the graphene conductivity in the THz band is mainly related with the electron intra-band scat 2
tering and the inter-band transitions can be neglected. Thus, it can be expressed as [26,27], σðωÞ ¼ eπℏE2F
i
ωþiτ
1
, where e, ℏ; ω and EF
denotes the electron charge, the Planck constant, the angular frequency and the Fermi energy of graphene separately. τ represents the carrier relaxation lifetime and it can be obtained byτ ¼ μEF =eVF 2 . VF ¼ 1:0 � 106 m=s is the Fermi velocity [28]. μ is the carrier mobility in graphene. It is an important parameter reflecting the conductivity of carriers in semiconductors. And the magnitude of 3
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Fig. 4. z-component of the electric field distribution (Ez) at each resonant frequency 2.75 THz, 3.0 THz and 3.4 THz in the combined structure.
Fig. 5. The numerical simulation (left panel) and theoretical calculation (right panel) of the transmission spectra in our proposed graphene met amaterial with different carrier mobility.
carrier mobility is proportional to the conductivity of semiconductor material. To dynamic control the mobility of graphene, doping is regarded as one of the most feasible methods. Graphene doping generally falls into three categories, namely, the hetero atom doping, the chemical modification strategy and electrostatic field tuning [29]. For the hetero atom doping, the B and N atoms are the natural 4
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Table 1 The fitting parameters in Lorentz coupled oscillator model with different carrier mobility. μ
γ1
γ2
ω1
ω2
κ
g1
g2
0.1 μ0
0.380
0.360
2.75
3.41
0.210
0.01
0.79
0.5 μ0
0.155
0.143
2.75
3.41
0.106
0.01
0.76
1.0 μ0
0.098
0.087
2.75
3.41
0.070
0.01
0.76
1.5 μ0
0.077
0.066
2.75
3.41
0.055
0.01
0.76
2.0 μ0
0.066
0.056
2.75
3.41
0.047
0.01
0.76
2.5 μ0
0.063
0.042
2.75
3.41
0.036
0.02
0.75
3.0 μ0
0.060
0.040
2.75
3.41
0.035
0.02
0.74
candidates for doping in graphene. Dai group achieved N-doping in graphene nanoribbons through electro thermal reaction with NH3 [30,31]. Based on the organic molecules (nonaromatic or aromatic molecules), chemical modification can be employed to control graphene doping [32]. Colett et al. demonstrated that the charge-carrier density upon molecular deposition will be reduced. The carrier mobility of the graphene layer can also be regulated through the electrostatic field tuning with the construction of field effect transistor through the gate voltage connected with the Au electrode and the substrate [33]. Thus, with the above mentioned graphene doping method, predefined mount of carrier mobility of the uniform graphene layer can be obtained. Then, electron-beam lithography and isotropic plasma etching are usually employed for patterning graphene nanostructures [34]. The real and imaginary part of the graphene mobility in THz region is shown in Fig. 2. It can be clearly observed that, the real part of the conductivity decreases gradually with a larger incident frequency but the imaginary part of the conductivity is almost kept as constant, embodying the low loss property of graphene. 3. Results and discussion To figure out the optical response in the proposed graphene metamaterials, we numerically calculate the transmission spectra of isolated strips, SRR and the combined structure in THz domain in Fig. 3. Under the circumstance of only graphene strips, an obvious resonant dip at 2.75 THz exists owing to the direct interaction with the incident wavelength, acting as a dipole resonator. Similarly, the transmittance of SRR locates at 3.4 THz in the red line owning to the typical LC resonance. When the graphene strips and the SRR are integrated together within a unit cell, the weak hybridization between the two radiating modes induces a transparency window at 3.0 THz in the blue line. In this circumstance, the structure parameters are fixed as l1 ¼10 μm, l2 ¼5.4 μm, w1 ¼1.5 μm, w2 ¼1.0 μm, d ¼ 1.5 μm, d1 ¼ d2 ¼ 1.7 μm, EF ¼1.5 eV and μ ¼ 3.0 μ0. In order to better comprehend the physical mechanism behind the PIT phenomenon, z-component of the electric field distribution at the resonant frequency in the combined structure is illustrated in Fig. 4. The field distribution of the dip at 2.75 THz is mainly distributed around the edge of the two strips, which can be directly excited with the incident wave irradiation. In the case of the resonant frequency dip at 3.4 THz, the distribution of electric field localizes around the corners and edges in SRR in Fig. 4(c), behaving as a dipole resonator. The electric field transfers from the strips to the SRR in the combined structure confirming the near field coupling of strips and SRR. Thus, it leads to an outstanding transparency window in Fig. 4(b). By virtue of the transformation process of the electric field, the formation mechanism of PIT effect is revealed. One outstanding characteristic of the graphene is its tunable electric properties through carrier mobility, which can be initiated through chemical doping. The transmittance of the PIT resonant peak with carrier mobility varying from 0.1 μ0 to 3.0 μ0 is shown in the left panel of Fig. 5. It must be noted that on and off state of the transparency window can be achieved with increased graphene mobility. For a small mobility 0.1 μ0 , the PIT transparency window barely exists. When the mobility is getting larger, the PIT resonant peak becomes more clearly. It almost approaches 1.0 with 3.0 μ0 . But the position of the transparency window is kept unchanged, indicating its fascinating application in the optical modulator or switcher. With the coupled harmonic oscillator model, the fitting spectra of the transmission spectra is also presented in the right panel of Fig. 5. We have adapted the coupled Lorentz oscillator model to verify the validity of our calculation. Each individual resonator can be regarded as a bright mode under the excitation of the incident electromagnetic field E0eiωt . In the steady state condition, the vibration displacement of two harmonic oscillators can be represented as x1 ðtÞ ¼ b1 ðωÞeiωt andx2ðtÞ ¼ b2ðωÞeiωt . The coupled differential equations of the two resonators are described as following [35]: x€1 ðtÞ þ r1 x_1 ðtÞ þ ω1 2 x1 ðtÞ þ κx2 ðtÞ ¼ g1 ЕðtÞ
(1)
x€2 ðtÞ þ r2 x_2 ðtÞ þ ω2 2 x2 ðtÞ þ κx1 ðtÞ ¼ g2 ЕðtÞ
(2)
κ denotes the coupling coefficient between the two bright mode resonators, g1 and g2 indicate the metric measuring the coupling strength of the strips and SRR with the incident light. The corresponding damping factors and angular frequency of the two bright modes are expressed as γ 1 , γ2 and ω1 , ω2 . With the solution of equations (1) and (2), we can obtain the amplitudes in the forms as following:
5
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C. Fan et al.
Fig. 6. The tendency of each fitting parameter in Lorentz coupled oscillator model is presented.
b1 ðωÞ ¼
b2 ðωÞ ¼
ðω1 2
g1 ðir2 ω ω2 þ ω2 2 Þ g2 κ ω2 þ ir1 ωÞðω2 2 ω2 þ ir2 ωÞ
κ2
(3)
ir1 ω þ ω2 ω1 2 Þ þ g1 κ ω þ ir1 ωÞðω2 2 ω2 þ ir2 ωÞ
κ2
(4)
g2 ð ðω1
2
2
Therefore, the transmission of the graphene metamaterials structure can be calculated with the formula T ¼ 1 jb1 =E0 j2 [11]. The fitting transmission spectra with different graphene mobility based on the coupled Lorentz oscillators theory is shown in the right panel of Fig. 5. It can be clearly observed that the fitted transmission spectra are consistent with numerical calculation, which furtherly confirmed the validity of our results. The corresponding fitting parameters are listed concretely in Table 1 with different carrier mobility. The damping factor γ is an important parameter in the harmonic oscillator equations, which measures the rate of oscillation when the incident plane wave is removed. When the losses are present, the energy stored in the resonator decreases proportional to the average energy present at any time [36]. In our case, the damping factors γ 1 and γ 2 of two bright modes decreases with a larger carrier mobility, indicating that the autologous loss of graphene affects the damping coefficient of the two bright mode resonators. Another parameter κ denotes the coupling coefficient between these two resonator and it also becomes smaller with a larger carrier mobility. Thus, the PIT transparency windows is caused through the weak coupling between these two resonators. In what follows, the tendency of each fitting parameter in relation with the carrier mobility is shown in Fig. 6 for visualized comparison. It can be clearly found that the variation of the parameter g1 and g2 is small, which mainly relates to the incident wave interacting with each isolated resonator. With the increase of graphene carrier mobility, the weak coupling strength κ between two resonators has an obvious diminution, resulting in an enhancement strength of the transparent window. Moreover, γ 1 and γ2 have a significant decrease, indicating that the loss of graphene affect the damping coefficient between two bright resonators. In Fig. 7, the absorption spectra in our proposed graphene metamaterial with the carrier mobility increasing from 0.1 μ0 to 3.0 μ0 is 6
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Fig. 7. The absorption spectra in our proposed graphene metamaterial is shown with the carrier mobility increasing from 0.1 μ0 to 3.0 μ0.
7
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Fig. 8. The relationship of phase shift and group delay in our proposed graphene metamaterial with different mobility.
examined. The position of absorption peak keeps constant but the resonant intensity is changing. Furthermore, the absorption rate of the graphene metamaterial can reach as high as 50%, which is an incomparable advantage compared with the traditional graphene’s absorption rate of 2.3% in the THz domain. When the incident light wave propagates through our proposed plasmonic metamaterial, the impulse response is convolved with the input signal to yield the output signal. Group delay is a measure of device phase distortion. It denotes the transit time of a signal through a device versus frequency. In 1999, Kasapi et al. conducted the experimental verification of light pulse delay in the system for the first time [37]. Under the influence of the incident light, plasmon are generated on the graphene surface and propagate along the interface between graphene and the dielectric. In the transmission, the propagation constant is no longer linear with the angular frequency and the phase velocity of the electromagnetic wave also changes with the frequency, resulting in spectral dispersion. In order to observe the dispersion relation and phase diagram more clearly, we focus on the working frequency range from 2.5 THz to 4.0 THz, which is shown in Fig. 8. It is found that the phase delay is getting larger with increased carrier mobility. And the obtained group delay gets enhanced and it reaches a high value of 1.57 ps at 3.0 μ0, which creates a slow-light effect [38]. We have also plotted the variation of the resonant position in the phase diagram as a function of the carrier mobility in Fig. 9. It can be clearly found that two resonant dips are getting smaller with a lager carrier mobility, while the PIT resonant peak is getting enhanced with higher mobility. Finally, the influence on transparency window with different dielectric substrate is figured out in Fig. 10(a). The PIT transparency window is moving into the higher frequency region when refractive index of the substrate is taken from 1.3 to 1.6 with an interval of 0.1. Namely, an obvious red shift happens with a higher refractive index of the substrate n. The relationship between the resonant frequency and refractive index can be better quantified using the quality factor FOM ¼ jdf⁄ dnj=Δf [39], where jdf⁄ dnj represents the 8
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Fig. 9. The position at each resonant peak in the phase diagram with different carrier mobility are plotted.
Fig. 10. (a) The transmission with different refractive index of the substrate structure. (b) The value of FOM as a function of refractive index and its linear fit. Parameters: l1 l1 ¼ 10 μm, l2 ¼ 5.4 μm, w1 ¼ 1.5 μm, w1 ¼ 1.0 μm, d ¼ 1.5 μm, d1 ¼ d2 ¼ 1.7 μm, EF ¼ 1.5 eV and μ ¼ 3.0μ0.
ratio of the frequency of the transparency window to the substrate refractive index. Δf is the full linewidth at a half maximum of the resonance peak. It can reach up to a higher value of 3.64 with n ¼ 1.6 in Fig. 10(b). Linear fit of the FOM with the different n can also be obtained with the least squared method. 4. Conclusions In summary, tunable PIT effect is realized in patterned periodic graphene metamaterial in THz region with different carrier mobility, including a pair of strips and a SRR. The disclosed transparency window is ascribed to the near field coupling of two res onators. Resorting to the coupled Lorentz oscillator model, the fitting theoretical transmittance agrees well with the numerical simulation. Our work also achieves the dual modulation of PIT and PIA with different graphene carrier mobility. The dispersion 9
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behavior around the transparency window is also examined with the calculation of the group delay and phase shift, which strengthen its application in the slow light device. In addition, the sensing ability with our proposed structure by virtue of regulating refractive index of the substrate is also explored and the transparency window embodies a red shift with a larger n. Therefore, our work will deliver some potential applications in the tunable optical switchers, sensors and slow light equipment in the THz regime. Acknowledgments This work is supported by the key science and technology research project of Henan Province (1721023100107), the Natural Science Foundation of Henan Educational Committee (17A140002). References [1] M. Liu, Q. Yang, Q. Xu, X. Chen, Z. Tian, J. Gu, et al., Tailoring mode interference in plasmon-induced transparency metamaterials, J. Phys. D Appl. Phys. 51 (17) (2018) 174005. [2] B.F. Apostol, Scattering of the electromagnetic waves from a rough surface, J. 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Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Tunable plasmon induced transparency in patterned graphene metamaterial with different carrier mobility Chunzhen Fan *, Peiwen Ren, Wei Jia, Yuanlin Jia, Junqiao Wang School of Physics, Zhengzhou University, Zhengzhou, Henan, 450001, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Plasmon induced transparency Metamaterial Terahertz Graphene mobility
Dynamical manipulation of plasmon induced transparency (PIT) is realized in patterned graphene metamaterial through different carrier mobility, which is composed of two strips and a split ring resonator (SRR). Due to the near field coupling between these two bright modes, a prominent transparency window appears in the transmission spectrum. It can be well fitted with coupled Lorentz oscillator model, which furtherly verifies the accuracy and reliability of our results. On and off switch of the PIT transparency window can be regulated through the carrier mobility. Namely, the PIT transparency window becomes more evident with larger carrier mobility. The plasmon induced absorption (PIA) can also be synchronously realized and the absorption rate reaches up to a high value of 0.5. In addition, the group delay gets enhanced with larger mobility. And the delay time achieves 1.57 ps. In case of sensing application, the linear shift of the PIT transparency window can be obtained with different substrate. Thus, our work shadows great prospects in highly tunable optical switching, optical filters, integrated slow light and sensing devices through non-contact regulation.
1. Introduction Plasmonic metamaterials have emerged as a promising research topic in recent years due to its ability to realize unique optical effects with conventional materials [1]. They are artificial engineering materials with reasonably designed composition and arrangement of unit cell. These materials enable intriguing applications owing to their response to electromagnetic waves [2], sound waves [3] or thermal waves [4] and embody the excellent performance beyond the capability of natural materials [5]. Plasmon induced transparency (PIT) is one of these noticeable optical phenomena realizing with metamaterial. It is an analogue electromag netically induced transparency, arising from the destructive quantum interference with different excitation pathway [6,7], resulting in a sharp narrow transparency window in its transmission profile [8]. Accompanied effect is the varied dispersion behavior, namely, the group velocity of the propagation wave can be greatly reduced. Generally, PIT effect is caused to happen with two excitation mechanism. One is ascribed to the near field coupling between two bright modes, the other is attributed to the destructive interference between bright and dark modes [9]. The bright resonator can directly excite with the incident light acting as a dipole oscillator. Fu et al. investigated tunable PIT with two parallel graphene nanostrips [10], each strip serves as a bright mode. Jia et al. proposed a simple PIT structure composed of two coupled horizontal graphene strips attached to continuous vertical wire [11]. The dark mode is an indirect excitation, which has to be excited through destructive interference with the bright mode [12,13]. When the bright and
* Corresponding author. E-mail address: [email protected] (C. Fan). https://doi.org/10.1016/j.spmi.2019.106295 Received 10 September 2019; Accepted 11 October 2019 Available online 21 October 2019 0749-6036/© 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Schematic illustration of the patterned graphene metamaterial on the dielectric substrate. (a) Overall view, (b) top view of the unit cell. Parameters: l1 ¼ 10 μm, l2 ¼ 5.4 μm, l3 ¼ 1.7 μm, w1 ¼ 1.5 μm, w2 ¼ 1.0 μm, d ¼ 1.5 μm, Px ¼ Py ¼ 16 μm.
dark resonators are arranged in a periodic way in the THz region, the coupling between them can induce the narrow transparency spectra. Namely, it’s the bright and dark mode coupling. Our group has investigated the active control and large group delay in the plasmonic metamaterials, which consist of two identical graphene strips and a cross-shaped graphene resonator in the right middle [14]. Tunable PIT metamaterials consists of cut-wire pairs and U-shaped ring resonators were explored [15]. Ultra broadband PIT spectra can be achieved based on asymmetric split-ring resonator of Dirac semimetal metamaterials [16]. A classical PIT design including a periodic lattice with a cut wire and a pair of symmetry split ring resonators was also studied [17]. The PIT effect can remarkably produce a large dispersion and slow down photons velocities effectively [18], which has many wide applications in slow light devices [19], optical switcher [20] or optical sensors [21]. At the early stage of PIT development, its design usually composes of metallic unit cell, which has an inevitable intrinsic Ohm loss. Thus, the tunability of PIT transparency window can only be initiated through the varied geometrical parameters. It indicates that once the structures are fabricated, it is hard to tune its transmittance without re-fabrications. It hinders the performance of figure of merit (FOM) and the slow light behavior. To overcome such restrictions and reduce the radiations loss, graphene has emerged as a new fascinating materials. It has enhanced surface plasmon, lower propagation loss, and higher carrier mobility [22]. More importantly, it is capable of initiating a non-contact control of its transportation properties. Up to now, different types of graphene layers are investigated, like waveguide resonator [23], cut wire resonator [24], split ring resonator [25] and so on. However, much work has focused the tunable effect through the Fermi energy or the polarization angle in the non-contact way, little work is carried out on the tunable PIT with carrier mobility through chemical doping. In addition, the dual consideration of the PIT and PIA have also barely fully investigated. In this work, we numerically investigate the transmission spectra with graphene metamaterials, enabling the non-contact dynamic regulation. It consists of two graphene strips and a SRR. Both of them can directive couple with the incident light, and the near field coupling between these two bright modes results in a PIT transparency window. The physical mechanism of PIT phenomenon is elaborately explored in our manuscript with the Lorentz oscillator model through the analysis of the coupling parameters. Our theoretical results achieve good agreement with the numerical calculations. Furthermore, the on and off modulation of the PIT window is investigated in details with different carrier mobility. The position of the PIT window is kept unchanged. But the magnitude of transmittance is greatly got enhanced with a larger mobility. The corresponding absorption rate reaches up to 50%, which is a fascinating result compared to the absorption rate of current monolayer graphene in THz band. Then, we have also investigated the dispersion behavior around PIT transparency window with different mobility. Finally, the sensitivity of the structure can be dynamically regulated by changing the refractive index of the substrate, and the group delay of the graphene metamaterial achieve 1.57 ps. Our work provides a new strategy to get tunable PIT effect with graphene carrier mobility in THz region. 2. Structure design and methods The overall schematic representation of our patterned graphene structure on the dielectric substrate is shown in Fig. 1(a). The unit cell consists of two strips and a SRR, which is periodically arranged in the x and y direction. The periodicity is taken as Px ¼ Py ¼ 16μm. In our simulation, the incident THz wave is perpendicular along z direction and the electric field is polarized along y direction. The concrete parameters of the unit cell are shown elaborately in Fig. 1(b). The length of the cut wire is l1 and it takes as 10 μm: l2 is the length of SRR and it equals to 6 μm. l3 is the short length of the SRR and it sets as 1:7 μm. The width of the two resonator arew1 ¼ 1:5 μm and w2 ¼ 1:0 μm. d represents the distance between the strips and the SRR, which is fixed at 1.5μm. The transmission and 2
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Fig. 2. Contour diagram of the (a) real and (b) imaginary part of the graphene mobility in THz region.
Fig. 3. Transmission spectra in different patterned graphene metamaterial are investigated in THz region. (a) Only strips (black line), (b) only SRR (red line), and (c) combined structure (blue line). Parameters: l1 ¼ 10 μm, l2 ¼ 5.4 μm, w1 ¼ 1.5 μm, w2 ¼ 1.0 μm, d ¼ 1.5 μm, d1 ¼ d2 ¼ 1.7 μm, EF ¼ 1.5 eV and μ ¼ 3.0 μ0.
absorption spectra involved in this paper are obtained with the finite element method. For simplicity, we only investigate the unit cell in the structure, and set periodic boundaries in the XY direction of the work plane. Perfectly matched layer is taken above the proposed metamaterial to eliminate non-physical reflections at the boundary. The conductivity of the graphene includes the contribution from inter-band and intra-band parts, which can be got with the Kubo equation. Under room temperature, the graphene conductivity in the THz band is mainly related with the electron intra-band scat 2
tering and the inter-band transitions can be neglected. Thus, it can be expressed as [26,27], σðωÞ ¼ eπℏE2F
i
ωþiτ
1
, where e, ℏ; ω and EF
denotes the electron charge, the Planck constant, the angular frequency and the Fermi energy of graphene separately. τ represents the carrier relaxation lifetime and it can be obtained byτ ¼ μEF =eVF 2 . VF ¼ 1:0 � 106 m=s is the Fermi velocity [28]. μ is the carrier mobility in graphene. It is an important parameter reflecting the conductivity of carriers in semiconductors. And the magnitude of 3
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Fig. 4. z-component of the electric field distribution (Ez) at each resonant frequency 2.75 THz, 3.0 THz and 3.4 THz in the combined structure.
Fig. 5. The numerical simulation (left panel) and theoretical calculation (right panel) of the transmission spectra in our proposed graphene met amaterial with different carrier mobility.
carrier mobility is proportional to the conductivity of semiconductor material. To dynamic control the mobility of graphene, doping is regarded as one of the most feasible methods. Graphene doping generally falls into three categories, namely, the hetero atom doping, the chemical modification strategy and electrostatic field tuning [29]. For the hetero atom doping, the B and N atoms are the natural 4
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Table 1 The fitting parameters in Lorentz coupled oscillator model with different carrier mobility. μ
γ1
γ2
ω1
ω2
κ
g1
g2
0.1 μ0
0.380
0.360
2.75
3.41
0.210
0.01
0.79
0.5 μ0
0.155
0.143
2.75
3.41
0.106
0.01
0.76
1.0 μ0
0.098
0.087
2.75
3.41
0.070
0.01
0.76
1.5 μ0
0.077
0.066
2.75
3.41
0.055
0.01
0.76
2.0 μ0
0.066
0.056
2.75
3.41
0.047
0.01
0.76
2.5 μ0
0.063
0.042
2.75
3.41
0.036
0.02
0.75
3.0 μ0
0.060
0.040
2.75
3.41
0.035
0.02
0.74
candidates for doping in graphene. Dai group achieved N-doping in graphene nanoribbons through electro thermal reaction with NH3 [30,31]. Based on the organic molecules (nonaromatic or aromatic molecules), chemical modification can be employed to control graphene doping [32]. Colett et al. demonstrated that the charge-carrier density upon molecular deposition will be reduced. The carrier mobility of the graphene layer can also be regulated through the electrostatic field tuning with the construction of field effect transistor through the gate voltage connected with the Au electrode and the substrate [33]. Thus, with the above mentioned graphene doping method, predefined mount of carrier mobility of the uniform graphene layer can be obtained. Then, electron-beam lithography and isotropic plasma etching are usually employed for patterning graphene nanostructures [34]. The real and imaginary part of the graphene mobility in THz region is shown in Fig. 2. It can be clearly observed that, the real part of the conductivity decreases gradually with a larger incident frequency but the imaginary part of the conductivity is almost kept as constant, embodying the low loss property of graphene. 3. Results and discussion To figure out the optical response in the proposed graphene metamaterials, we numerically calculate the transmission spectra of isolated strips, SRR and the combined structure in THz domain in Fig. 3. Under the circumstance of only graphene strips, an obvious resonant dip at 2.75 THz exists owing to the direct interaction with the incident wavelength, acting as a dipole resonator. Similarly, the transmittance of SRR locates at 3.4 THz in the red line owning to the typical LC resonance. When the graphene strips and the SRR are integrated together within a unit cell, the weak hybridization between the two radiating modes induces a transparency window at 3.0 THz in the blue line. In this circumstance, the structure parameters are fixed as l1 ¼10 μm, l2 ¼5.4 μm, w1 ¼1.5 μm, w2 ¼1.0 μm, d ¼ 1.5 μm, d1 ¼ d2 ¼ 1.7 μm, EF ¼1.5 eV and μ ¼ 3.0 μ0. In order to better comprehend the physical mechanism behind the PIT phenomenon, z-component of the electric field distribution at the resonant frequency in the combined structure is illustrated in Fig. 4. The field distribution of the dip at 2.75 THz is mainly distributed around the edge of the two strips, which can be directly excited with the incident wave irradiation. In the case of the resonant frequency dip at 3.4 THz, the distribution of electric field localizes around the corners and edges in SRR in Fig. 4(c), behaving as a dipole resonator. The electric field transfers from the strips to the SRR in the combined structure confirming the near field coupling of strips and SRR. Thus, it leads to an outstanding transparency window in Fig. 4(b). By virtue of the transformation process of the electric field, the formation mechanism of PIT effect is revealed. One outstanding characteristic of the graphene is its tunable electric properties through carrier mobility, which can be initiated through chemical doping. The transmittance of the PIT resonant peak with carrier mobility varying from 0.1 μ0 to 3.0 μ0 is shown in the left panel of Fig. 5. It must be noted that on and off state of the transparency window can be achieved with increased graphene mobility. For a small mobility 0.1 μ0 , the PIT transparency window barely exists. When the mobility is getting larger, the PIT resonant peak becomes more clearly. It almost approaches 1.0 with 3.0 μ0 . But the position of the transparency window is kept unchanged, indicating its fascinating application in the optical modulator or switcher. With the coupled harmonic oscillator model, the fitting spectra of the transmission spectra is also presented in the right panel of Fig. 5. We have adapted the coupled Lorentz oscillator model to verify the validity of our calculation. Each individual resonator can be regarded as a bright mode under the excitation of the incident electromagnetic field E0eiωt . In the steady state condition, the vibration displacement of two harmonic oscillators can be represented as x1 ðtÞ ¼ b1 ðωÞeiωt andx2ðtÞ ¼ b2ðωÞeiωt . The coupled differential equations of the two resonators are described as following [35]: x€1 ðtÞ þ r1 x_1 ðtÞ þ ω1 2 x1 ðtÞ þ κx2 ðtÞ ¼ g1 ЕðtÞ
(1)
x€2 ðtÞ þ r2 x_2 ðtÞ þ ω2 2 x2 ðtÞ þ κx1 ðtÞ ¼ g2 ЕðtÞ
(2)
κ denotes the coupling coefficient between the two bright mode resonators, g1 and g2 indicate the metric measuring the coupling strength of the strips and SRR with the incident light. The corresponding damping factors and angular frequency of the two bright modes are expressed as γ 1 , γ2 and ω1 , ω2 . With the solution of equations (1) and (2), we can obtain the amplitudes in the forms as following:
5
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Fig. 6. The tendency of each fitting parameter in Lorentz coupled oscillator model is presented.
b1 ðωÞ ¼
b2 ðωÞ ¼
ðω1 2
g1 ðir2 ω ω2 þ ω2 2 Þ g2 κ ω2 þ ir1 ωÞðω2 2 ω2 þ ir2 ωÞ
κ2
(3)
ir1 ω þ ω2 ω1 2 Þ þ g1 κ ω þ ir1 ωÞðω2 2 ω2 þ ir2 ωÞ
κ2
(4)
g2 ð ðω1
2
2
Therefore, the transmission of the graphene metamaterials structure can be calculated with the formula T ¼ 1 jb1 =E0 j2 [11]. The fitting transmission spectra with different graphene mobility based on the coupled Lorentz oscillators theory is shown in the right panel of Fig. 5. It can be clearly observed that the fitted transmission spectra are consistent with numerical calculation, which furtherly confirmed the validity of our results. The corresponding fitting parameters are listed concretely in Table 1 with different carrier mobility. The damping factor γ is an important parameter in the harmonic oscillator equations, which measures the rate of oscillation when the incident plane wave is removed. When the losses are present, the energy stored in the resonator decreases proportional to the average energy present at any time [36]. In our case, the damping factors γ 1 and γ 2 of two bright modes decreases with a larger carrier mobility, indicating that the autologous loss of graphene affects the damping coefficient of the two bright mode resonators. Another parameter κ denotes the coupling coefficient between these two resonator and it also becomes smaller with a larger carrier mobility. Thus, the PIT transparency windows is caused through the weak coupling between these two resonators. In what follows, the tendency of each fitting parameter in relation with the carrier mobility is shown in Fig. 6 for visualized comparison. It can be clearly found that the variation of the parameter g1 and g2 is small, which mainly relates to the incident wave interacting with each isolated resonator. With the increase of graphene carrier mobility, the weak coupling strength κ between two resonators has an obvious diminution, resulting in an enhancement strength of the transparent window. Moreover, γ 1 and γ2 have a significant decrease, indicating that the loss of graphene affect the damping coefficient between two bright resonators. In Fig. 7, the absorption spectra in our proposed graphene metamaterial with the carrier mobility increasing from 0.1 μ0 to 3.0 μ0 is 6
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Fig. 7. The absorption spectra in our proposed graphene metamaterial is shown with the carrier mobility increasing from 0.1 μ0 to 3.0 μ0.
7
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Fig. 8. The relationship of phase shift and group delay in our proposed graphene metamaterial with different mobility.
examined. The position of absorption peak keeps constant but the resonant intensity is changing. Furthermore, the absorption rate of the graphene metamaterial can reach as high as 50%, which is an incomparable advantage compared with the traditional graphene’s absorption rate of 2.3% in the THz domain. When the incident light wave propagates through our proposed plasmonic metamaterial, the impulse response is convolved with the input signal to yield the output signal. Group delay is a measure of device phase distortion. It denotes the transit time of a signal through a device versus frequency. In 1999, Kasapi et al. conducted the experimental verification of light pulse delay in the system for the first time [37]. Under the influence of the incident light, plasmon are generated on the graphene surface and propagate along the interface between graphene and the dielectric. In the transmission, the propagation constant is no longer linear with the angular frequency and the phase velocity of the electromagnetic wave also changes with the frequency, resulting in spectral dispersion. In order to observe the dispersion relation and phase diagram more clearly, we focus on the working frequency range from 2.5 THz to 4.0 THz, which is shown in Fig. 8. It is found that the phase delay is getting larger with increased carrier mobility. And the obtained group delay gets enhanced and it reaches a high value of 1.57 ps at 3.0 μ0, which creates a slow-light effect [38]. We have also plotted the variation of the resonant position in the phase diagram as a function of the carrier mobility in Fig. 9. It can be clearly found that two resonant dips are getting smaller with a lager carrier mobility, while the PIT resonant peak is getting enhanced with higher mobility. Finally, the influence on transparency window with different dielectric substrate is figured out in Fig. 10(a). The PIT transparency window is moving into the higher frequency region when refractive index of the substrate is taken from 1.3 to 1.6 with an interval of 0.1. Namely, an obvious red shift happens with a higher refractive index of the substrate n. The relationship between the resonant frequency and refractive index can be better quantified using the quality factor FOM ¼ jdf⁄ dnj=Δf [39], where jdf⁄ dnj represents the 8
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Fig. 9. The position at each resonant peak in the phase diagram with different carrier mobility are plotted.
Fig. 10. (a) The transmission with different refractive index of the substrate structure. (b) The value of FOM as a function of refractive index and its linear fit. Parameters: l1 l1 ¼ 10 μm, l2 ¼ 5.4 μm, w1 ¼ 1.5 μm, w1 ¼ 1.0 μm, d ¼ 1.5 μm, d1 ¼ d2 ¼ 1.7 μm, EF ¼ 1.5 eV and μ ¼ 3.0μ0.
ratio of the frequency of the transparency window to the substrate refractive index. Δf is the full linewidth at a half maximum of the resonance peak. It can reach up to a higher value of 3.64 with n ¼ 1.6 in Fig. 10(b). Linear fit of the FOM with the different n can also be obtained with the least squared method. 4. Conclusions In summary, tunable PIT effect is realized in patterned periodic graphene metamaterial in THz region with different carrier mobility, including a pair of strips and a SRR. The disclosed transparency window is ascribed to the near field coupling of two res onators. Resorting to the coupled Lorentz oscillator model, the fitting theoretical transmittance agrees well with the numerical simulation. Our work also achieves the dual modulation of PIT and PIA with different graphene carrier mobility. The dispersion 9
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behavior around the transparency window is also examined with the calculation of the group delay and phase shift, which strengthen its application in the slow light device. In addition, the sensing ability with our proposed structure by virtue of regulating refractive index of the substrate is also explored and the transparency window embodies a red shift with a larger n. Therefore, our work will deliver some potential applications in the tunable optical switchers, sensors and slow light equipment in the THz regime. Acknowledgments This work is supported by the key science and technology research project of Henan Province (1721023100107), the Natural Science Foundation of Henan Educational Committee (17A140002). References [1] M. Liu, Q. Yang, Q. Xu, X. Chen, Z. Tian, J. Gu, et al., Tailoring mode interference in plasmon-induced transparency metamaterials, J. Phys. D Appl. Phys. 51 (17) (2018) 174005. [2] B.F. Apostol, Scattering of the electromagnetic waves from a rough surface, J. 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