Tunable slow light effect based on dual plasmon induced transparency in terahertz planar patterned graphene structure

Journal Pre-proofs Tunable slow light effect based on dual plasmon induced transparency in terahertz planar patterned graphene structure Mingzhuo Zhao, Hui Xu, Cuixiu Xiong, Baihui Zhang, Chao Liu, Wenke Xie, Hongjian Li PII: DOI: Reference:

S2211-3797(19)33052-9 https://doi.org/10.1016/j.rinp.2019.102796 RINP 102796

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Results in Physics

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14 October 2019 5 November 2019 5 November 2019

Please cite this article as: Zhao, M., Xu, H., Xiong, C., Zhang, B., Liu, C., Xie, W., Li, H., Tunable slow light effect based on dual plasmon induced transparency in terahertz planar patterned graphene structure, Results in Physics (2019), doi: https://doi.org/10.1016/j.rinp.2019.102796

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Tunable slow light effect based on dual plasmon induced transparency in terahertz planar patterned graphene structure Mingzhuo Zhao 1, 2, Hui Xu 3, *, Cuixiu Xiong 1 , Baihui Zhang 1, Chao Liu 1, Wenke Xie 1, *, and Hongjian Li 1, * 1

School of Physics and Electronics, Central South University, Changsha 410083, China School of Physics and Electronics, Hunan University of Science and Technology, Xiangtan 411201, China 3 School of Mathematics and Statistics, Hunan University of Technology and Business, Changsha 410205, China E-mail: [email protected] (Hui Xu), [email protected] (Wenke Xie), [email protected] (Hongjian Li) 2

Abstract We have studied a simple novel graphene ribbon structure. A very excellent and prominent dual graphene plasmon induced transparency phenomenon could be achieved by the destructive interference resulted from the excited plasmonic modes in terahertz band. Using the simple relationship between graphene and applied voltage, a good tunable effect of this structure can be achieved. The transmission of this proposed structure is theoretically investigated by using the equivalent resonator coupled mode method. The theoretical data from our proposed method are in good agreement with the numerical simulation results. Moreover, utilizing the high dispersion property, we have also researched the slow light effect for this proposed system. The results of theoretical research have indicated that the group refractive index of our proposed structure can maintain an excellent numerical value. This investigation can play a significant role in the tunable graphene-based slow light devices. Keywords: Plasmon-induced transparency; Graphene planar metamaterial; Slow light device.

devices based on graphene surface plasmon, such as slow light, have been analyzed and designed [15-17]. According to the surface plasmon polaritons stimulated from monolayer graphene, researchers can also design different structures to achieve some special phenomena, such as Fano effect and plasmon induced transparency (PIT) effect. Fano effect is an asymmetric linear spectral phenomenon obtained by destructive interference in a quantum or optical system [18]. In optical system, it generally requires two different optical modes, and the two modes are treated as bright mode and dark mode according to the degree of excitation. The bright mode as a continuous state can be fully excited by the exciting light, while the dark mode as a discrete state cannot be excited by the exciting light. The destructive interference of these two modes or states can lead to the Fano effect. Moreover, PIT effect can be considered as a special case of the Fano effect. According to the theoretical analysis of the two effects, people have investigated different structures to detect these two phenomena[19, 20] and have designed various optical devices and applications [21-27]. In this study, we have designed a terahertz planar patterned graphene-based structure. The optical properties of this designed structure are numerically simulated by the finite-difference time-domain (FDTD) method, and a very prominent dual graphene plasmon induced transparency phenomenon is achieved. This structure is composed of a patterned graphene ribbon. Three optical modes can be obtained by the response of graphene at different positions to the exciting light source. Thus, a very prominent dual PIT

1. Introduction With the rise of two-dimensional (2D) material graphene, its various applications have attracted more and more attention of the researchers. So far, the researchers have studied and discussed many different devices with these various applications of graphene, such as sensors [1], modulators [2], photodetectors [3, 4] and so on [5, 6]. After years of research, scientists have gradually grasped most of the properties of graphene and then scientists have found that the monolayer graphene exhibits a metal-like property in some wave bands [7, 8], that is to say, the complex conductivity of monolayer graphene exhibits a very similar model to that of metal. Moreover, the permittivity of monolayer graphene all depends on its complex conductivity [9]. If the imaginary part of the graphene conductivity is above 0, the real part of the equivalent permittivity would be under 0, which means that the monolayer graphene exhibits the characteristics liking a metal film. This particular property can make the monolayer graphene support the propagation of transversemagnetic (TM) electromagnetic modes, and thus it can stimulate the excitation of surface plasmon polaritons. In addition, the imaginary part of the equivalent permittivity decided on the real part of graphene conductivity can represent the propagation loss for the excited graphene plasmonic wave. In recent years, the surface plasmon polaritons has been successfully stimulated by researchers [10-14]. Based on these properties and foundations, various

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effect can be obtained after a destructive interference generated by these three modes. By altering the Fermi energy, we can achieve a good tuning function. Through the theoretical analysis of coupled mode theory (CMT), we find that the theoretical fitting results can match the simulated numerical data very well. Moreover, we have further analyzed the group refractive index of this system by the theoretical transmission coefficient, realizing a theoretical analysis of the slow light effect. From the obtained results, we have discovered that this design keeps an excellent slow light performance since the maximum value of the group refractive index can reach to 642. This achievement may provide a good research basis for the fabrication of micronano slow light device. Since all the graphene in our structure are connected together to form a patterned graphene ribbon, this structure has a distinct advantage compared with those discontinuous patterned structures, that is to say, it is easier and more comfortable to inject a voltage to regulate the Fermi energy. So the tuning function of this device can be realized without changing the physical size and structure parameters. This excellent tuning function may provide a theoretical basis for reconfigurable micro-nano tuning devices. The simple construction of this design can also make the structure easier to implement in experiments. Having all these advantageous features, our proposed design may be employed in the tuning or slow-light devices and our investigations also can supply some theoretical basis for the realization of these components.

The schematic illustration of this dual PIT planar graphene ribbon structure is shown in Fig. 1(a). It can be seen that this structure is composed of many periodic patterned graphene ribbons, and four graphene micro-chips with a same width and different length are placed on the both sides of each unit cell of main graphene band, as shown in Fig. 1(b). A planar monolayer graphene is placed in middle of the dielectric silicon (dielectric constant is 11.9, palik[28]). The thickness of the upper and lower silicon layer is 50 nm and 0.25 m, as shown in Fig. 1(c). We put the control electrodes on top of the upper silicon layer to form a parallel plate capacitor, regulating the electron concentration, thus it can tune and affect the Fermi energy of graphene. The plane excitation source is incident along the negative z-axis direction. In the caption of Fig. 1, we have given the detailed structural parameters. Due to the arrangement of carbon atoms in graphene-based micro-nano structure, it is necessary to take into account the edge effects. Moreover, the optical responses of nanostructured graphene plasmonics described by classical local electromagnetic theory and first-principles calculations are in good agreement when the size of graphene nanoribbons are greater than 10 nm[29, 30]. Thus, in our work, we can use classical descriptions to investigate the optical response and reasonably neglect any quantum finitesize and edge effects (Armchair and Zigzag configurations). Moreover, we perform the FDTD simulations (FDTD Solutions) with a perfect matched layer boundary condition at z direction and periodic boundary conditions at x and y direction, respectively.

2. Structural design and theoretical model

Figure 1. (a) The schematic illustration of this outstanding dual PIT graphene ribbon structure. (b) A unit cell of this structure, the detailed structural parameters are as follows: L=3m, wg=0.6m, w=1m, lg =1m, lg1 =0.7m, lg2 =0.4m, lg3 =0.9m, lg4 =0.7m. (c) The voltage-modulated Fermi energy diagram. It shows that the graphene is embedded in the dielectric silicon. The distance between the monolayer graphene and gold electrode is 50nm. The specific thickness of lower silicon layer is 0.25m. (d) The three theoretical coupled resonators of coupled mode theory in our design. 2

In the introduction, it is mentioned that the monolayer graphene exhibits a metal-like property and its conductivity can well describe its photoelectric properties. However, due to the special structural properties of two-dimensional monolayer graphene, its complex conductivity isn’t a volume conductivity and is generally surface conductivity of graphene. According to the theoretical and experimental studies, the relevant surface conductivity of monolayer graphene in this system can be completely given by the Kubo formula derived from Random Phase Approximation (RPA)[31-33]:  f ( ) f d ( ) e 2 (  i 1 ) 1  s ( , c , , T )  [ ( d  )d  ]   i 2 (  i 1 ) 2 0  f d ( )  f d ( )  d 0  (  i 1 ) 2  4( ) 2

excitation light is set at terahertz region, respectively. In these cases, the condition is satisfied as EF ≫ (ћω, kBT), thus the second term can be negligible. Then, the two terms can be simplified into a metal-like Drude model, namely: ie2 EF  g   intra  2 , (3)  (  i 1 ) here, carrier relaxation time =EF / (evF2)[35]. The carrier mobility  can climb to a very high value about 4 m2/(V·s) at a room temperature according to data of experimental articles [33, 36], thus we can set it to be 1 m2/(V·s) as a fixed value in our system. According to the basic phenomenon that the graphene plasma polaritons are located at the surface of graphene, we can reasonably set the condition that the propagation constants of graphene ribbons in this structure can be replaced with the propagation constants of complete monolayer graphene, namely:

  intra   inter

,(1) The physical parameters in this formula are described as follows: Fermi-Dirac distribution f d ( )  1/ (1  e(  c )/( kBT ) ) , carrier energy , chemical potential c (Fermi energy EF), angular frequency , temperature T, electron charge e, Fermi velocity vF of graphene (≈106 m/s), carrier relaxation time , reduced Planck constant , and Boltzmann constant kB. The above formula can be divided into the sum of two terms: intra+inter. Here, the first term and second term respectively result from the contributions of intraband electron-photon scattering and the contributions of direct interband electron transitions. After a further calculation, the two terms can be written separately[34]:

  k0  Si  (

2 Si 2 ) ,  g0

(4)

Where,  is propagation constant of the graphene plasma wave, k0 is the wave numbers in a vacuum, Si is the relative dielectric constant of silicon in this system, g is the conductivity of graphene, and 0 is the intrinsic impedance in a vacuum, respectively. According to this formula and the definition of effective refractive index: neff =/k0, we can further receive the numerical value of the effective refractive index. In the Fig. 2(a-b), we have plotted the relationship between the real and imaginary part of the effective refractive index and the Fermi energy. The value of Qi can be obtained by dividing the real and imaginary parts of the effective refractive index, namely, Qi=Real (neff)/Imag (neff). From these curves, we can make a brief summary that both the real and imaginary parts of the effective refractive index would decrease as the increase of Fermi energy, but their growth rates are different. The real part increases rapidly, while the imaginary part increases smoothly.

 intra  2ie 2 k BT  ln[2 cosh( c )]  2 1 2 k  (   i  )  BT  2 1 , (2) G ( )  inter  ie (  i )  d 2 1 2 2 2   0 4 k BT (  i ) / (2k BT )   

here, G( )  sinh  / [cosh( EF / kBT )  cosh  ] , and    / kBT . In our article, the Fermi energy is set as 0.7eV to 1.2eV, the ambient temperature is set as 300K, and the frequency of

Figure 2. (a) The evolution between frequency and the real part of effective refractive index as different Fermi energy. (b) The evolution between frequency and the imaginary part of the effective refractive index as different Fermi energy.

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mode theory (CMT) method to fit the simulation data [38-41]. Moreover, according to the property of the dual graphene PIT phenomenon, we can assume three equivalent coupled resonators to be equivalent to the three modes of the dual PIT effect [40]. Then, as required by the coupled mode theory, we can choose a1, a2 and a3 to represent the complex amplitudes of three equivalent coupled resonators, respectively. The A1, A2 and A3 are the wave traveling toward or backward the three hypothetical equivalent coupled modes as shown in Fig. 1(d). A denotes the incoming or outgoing waves of each equivalent resonators (where the superscript in denotes an incoming wave, the out denotes an outgoing wave, the subscript + denotes that a positive propagating direction, and - denotes that a negative propagating direction, respectively). in(n=1, 2, 3) represent the internal loss of the nth mode. on(n=1, 2, 3) represent the external loss of the nth mode. μnm represent the coupling loss between the nth mode and mth mode (m=1, 2, 3, mn), respectively. Thus, those three equivalent coupled modes can be described as follows: i 12 i 13   a1   1     2 i  23    a2    i 21  i  i 32  3   a3  31  , (5)   o1/12 0 0   A1in  A1in       o1/22 0    A2in  A2in   0  0 0  o1/32   A3in  A3in   here,  n  (i  in   in   on ) (ωn is the resonant angular frequencies for nth equivalent coupled mode). Besides, on=ωn/(2Qon) and in=ωn/(2Qin) (Here, Qon and Qin are respectively the external and internal loss quality factor for nth equivalent coupled mode,). Furthermore, 1/Qtn = 1/Qon + 1/Qin, and Qtn is the total quality factor for nth equivalent coupled mode (Qtn=f/Δf. Δf and f are respectively the full width of half maximum and resonant frequency for nth equivalent coupled mode,). As mentioned earlier, Qi=Real (neff)/Imag (neff). Then the numerical value of each Qin can be obtained by the resonant frequency. By analyzing the transmission spectra obtained by the numerical simulation and above definitions of those quality factors, we can gain the numerical values of the three equivalent coupled modes and can describe the relationship between them and Fermi energy in Fig. 4, respectively.

3. Results and analysis Using the FDTD method [37], we can simulate the structural optical performance and achieve the numerical simulation transmission, as shown in red solid line of Fig. (3). It can be seen from the curves that this proposed structure realizes a prominent dual PIT effect. In addition, the structural parameters in a unit cell are as follows: L=3m, wg=0.6m, w=1m, lg =1m, lg1 =0.7m, lg2 =0.4m, lg3 =0.9m, lg4 =0.7m. These values are fixed in this system. In Fig. 3, we can see that the resonant frequency of our proposed system occurs a blue shift while the Fermi energy increases.

Figure 3. The transmittance results respectively obtain from numerical simulation method and theoretical calculation method. The red solid line describes the numerical simulation data and the blue dot line describes the theoretical results. Here, the Fermi energy EF =0.7eV, 0.8eV, 0.9eV, 1.0eV, 1.1eV in this dual graphene PIT system

For a more detailed theoretical analysis of this acquired dual graphene PIT effect, we have developed the coupled

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Figure 4. (a) The relationship between the resonant frequencies and the Fermi energy. (b) The relationship between the numerical values of quality factor of the three equivalent coupled modes and the Fermi energy.

Moreover, according to the energy conservation between the three equivalent coupled resonators, we can achieve the following relationships: in1 Anin  A(out , A(inn 1)   Anout ein (n  2,3) , n 1)  e

appropriate and can describe the theoretical properties of the system. Moreover, we have known that the Fermi energy of monolayer graphene has a good tuning function, namely, the Fermi energy can be adjusted by chemical doping or electrostatic pressure (electric doping). Therefore, we can also theoretically analyze this tuning relationship in our paper. The voltage regulation relationship of graphene Fermi energy is as follows [42, 43]:  0 SiVg 1/ 2 E F  vF ( ) , (9) tSi e Here, tSi is the thickness between the two electrodes, 0 is the permittivity of free space, and Vg is the applied bias voltage, respectively. Based on this formula, we have plotted their evolutionary relationships in Fig. 5 (a). From this formula, we can know that the Fermi energy of monolayer graphene can be tuned smoothly by changing the bias voltage, and the dual PIT phenomenon of this structure can be successfully tuned. Our designed structure also has a very good advantage, that is, the graphene ribbons in this structure are complete and uninterrupted. Therefore, compared with other discontinuous patterned graphene-based structures, this structure can easily control the applied bias voltage of graphene in experiment and this structural design make it easier to realize in actual manufacture. Based on the coupled mode theory, we have calculated the theoretical transmittance of this structure with different Fermi energy, as shown in Fig. 5 (b). It can be seen that the resonant frequency has a significant blue shift with the increase of Fermi energy, which can be attributed to the increased resonant energy as the increased Fermi energy. With an increased Fermi energy, a higher energy must be needed in order to achieve the resonance condition. Therefore, the corresponding resonant frequency will move towards the direction of an increased value. Thus, the blue shift phenomenon mentioned above can happen. Also, it can be seen from the figure that the resonant frequency almost presents a linear relationship, and the tuning property is very reliable and convenient as the previous analysis.

(6)

A  A   a , A  A   a (n  1, 2,3) , (7) Here, n is the phase shift of the excited plasmonic wave between the (n-1)th and nth equivalent coupled modes. But the value of them can be as zero due to the same wavefront. Thus, with Eqs. (5-7) and A3in  0 , we can get a result of the transmission coefficient: 1/2 1/2 1/2 Aout  ei (1 2 ) t1   o 2 ei2 t2   o3 t3 t  3in  t0  o1 , (8) t4 A1 Here, t0  ei (1 2 ) ,  1/ 2 1/ 2 i1 t1  ( 22 33   23 32 ) o1  ( 12 33   13 32 ) o 2 e   ( 12 23   13 22 ) o1/32 ei (1 2 ) ,  t  (     ) 1/ 2  (     ) 1/ 2 ei1 21 33 23 31 o1 11 33 13 31 o2  2   ( 11 23   13 21 ) o1/32 ei (1 2 ) ,  1/ 2 1/ 2 i1 t3  ( 21 32   22 31 ) o1  ( 11 32   12 31 ) o 2 e   ( 11 22   12 21 ) o1/32 ei (1 2 ) ,  t4   11 23 32   11 22 33   12 21 33   12 23 31    13 21 32   13 22 31 .  out n

in n

1/ 2 on n

out n

in n

1/ 2 on n

and，   11   1 ,  12  i 12  ( o1 o 2 )1/ 2 ei1 ,  13  i 13  ( o1 o3 )1/ 2 ei (1 2 ) ,   1/ 2 i 1/ 2 i   21  i 21  ( o1 o 2 ) e 1 ,  22   2 ,  23  i 23  ( o 2 o3 ) e 2 ,    i 31  ( o1 o3 )1/ 2 ei (1 2 ) ,  32  i 32  ( o 2 o3 )1/ 2 ei2 ,  33   3 .   31

Thus, we can obtain the theoretical transmittance of this dual PIT system (T=|t|2). Then we can get the fitting transmission spectra of this dual PIT phenomenon by using the above quality factors, as shown in the blue dot line in Fig. 3. From the matching degree of the two curves, we can make a conclusion that this coupled mode theory is very

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Figure 5. (a) The evolutionary relationships between the Fermi energy and the applied bias voltage. (b) The theoretical transmittance spectra versus Fermi energy and frequency.

coupled mode theory: =arg (t). With the help of this complex phase, we have calculated the numerical values of the group refractive index [44-46]: dk c d ng  c  , (10) d  lSi d  Here, c is the velocity of light in a vacuum and lSi=0.2m is the thickness of this graphene-based structure.

In addition, the good dispersion property is a very important property for monolayer graphene. Although this property can sometimes be annoying, it has a unique advantage in the slow light applications. High dispersion means a high equivalent refractive index and it can lead to a very high group refractive index. Moreover, high group refractive index means an outstanding slow light effect. In this system, we can obtain the theoretical phase change according to the transmission coefficient obtained by the

Figure 6. (a-f) The group refractive index (blue lines) and phase shift (red lines) versus frequency as the Fermi energy EF=0.7eV, 0.8eV, 0.9eV, 1.0eV, 1.1eV, 1.2eV respectively.

From Fig. 6, we can successfully understand that the evolutionary relationship of the phase shift and group refractive index versus frequencies at different Fermi energy in this system. Furthermore, we can know that the phase shift and group refractive index at the induced transparent window

have great changes. This phenomenon can be explained as high dispersion. In the vicinity of transparent window, destructive interference of the light and the three equivalent coupled modes can lead to a high dispersion for the excited plasma wave and also cause a significant phase change, 6

leading to a great change in group refractive index. As can be obtained from Fig. 6, the maximum numerical value of the group refractive index with different Fermi energy shows a same trend as the increase of Fermi energy, and its maximum numerical value can be up to 642 at Fermi energy EF=1.2eV. Such a high value is very advantageous in the same type of slow light device and this research can provide a basic foundation for the fabrication of micro-nano slow light device.

No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication. I would like to declare on behalf of my co-authors that the work described was original research that has not been published previously, and not under consideration for

4. Conclusions In conclusion, we have designed a novel single-layer planar graphene terahertz structure. Through the FDTD numerical simulation method, we can have successfully obtained a very outstanding dual graphene PIT effect. Moreover, this simple structure makes it easier to adjust the Fermi energy of graphene for achieving a good PIT tuning function. By the analysis and derivation of the equivalent resonator coupled mode theory, a theoretical fitting transmittance of this system can be achieved. It is found that the theoretical fitting results from the CMT method match the numerical simulation data well, so it is proved that the equivalent resonator coupled mode theory method can be accurate and reliable. By means of the group refractive index, we have known that this system can achieve a very superior slow light capability. The maximum group refractive index can reach to a very high value about 642, and its performance is far greater than that of other slow light devices. The simple design, tunable dual graphene PIT effect and high slow light performance of this structure can provide a great advantage for our device. With these advantages, our proposed investigation can be available on the tunable slow light devices and this research can also offer some theoretical basic foundation for the fabrication of these micro-nano devices.

publication elsewhere, in whole or in part. All the authors listed have approved the manuscript that is enclosed.

Acknowledgments: This work is supported by the Natural Science Foundation of Hunan Province (Grant No. 2019JJ50147) and the National Natural Science Foundation of China (Grant No. 61275174).

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Author statement

This research was planned by Mingzhuo Zhao and Hongjian Li. Mingzhuo Zhao developed the analytic theory. Numberical simulation was performed by Mingzhuo Zhao, Hui Xu, Cuixiu Xiong, Baihui Zhang, Chao Liu. The authors Mingzhuo Zhao, Hui Xu, Cuixiu Xiong, Baihui Zhang, Chao Liu, Wenke Xie and Hongjian Li discussed the results. Mingzhuo Zhao wrote the original manuscript. Mingzhuo Zhao, Hui Xu, Wenke Xie and Hongjian Li revised the manuscript.

Conflict of interest 7

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1. We have proposed a novel type of monolayer graphene structure. 2. A very obvious dual plasmon induced transparency effect can be successfully achieved by the destructive interference in the terahertz region. 3. Compared with the devices based on patterned or discrete graphene structure, our design keeps the graphene monolayer in a continuous form. 4. It has the benefit of preserving the high mobility of graphene and also simplifies the fabrication processes. 5. Our proposed graphene-based metamaterial structure has an excellent slow light performance since the maximum vaule of the group refractive index can reach to 642.

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