Tunable terahertz graphene metamaterials

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Tunable terahertz graphene metamaterials Xiaoyong He Mathematics & Science College, Shanghai Normal University, No. 100 Guilin Road, Shanghai 200234, PR China

A R T I C L E I N F O

A B S T R A C T

Article history:

Using graphene metamaterial (MM) patterns, the tunable resonant properties of graphene–

Received 9 August 2014

SiO2/Si (GSiO2Si) structures deposited on flexible polymer substrates have been theoreti-

Accepted 22 October 2014

cally investigated in the terahertz regime. This study shows that the tuning mechanism

Available online 28 October 2014

of the GSiO2Si structure mainly depends on dipolar resonance, which is different from the conventional metallic MM structure based on the LC resonance. For graphene MM structures, the resonant transmission curves can be tuned over a wide range by controlling applied electric fields. The modulation depth of transmission is about 80%. As the Fermi level of the graphene layer increases, the resonant transmission become stronger, and the resonant dips significantly shift to higher frequency.  2014 Elsevier Ltd. All rights reserved.

1.

Introduction

Recently, terahertz (THz) science and technology have made rapid development in the aspects of sources [1–3] and detectors [4], but compared to the well-established neighboring infrared and microwave wave bands, the THz regime is still in need of fundamental advances. A great challenge in developing the THz technology further is that it is very difficult to find suitable materials to respond to the THz radiation strongly. Fortunately, this problem can be alleviated to a large extent with the help of metamaterials (MMs), that is, the composite materials designed to manifest exotic electromagnetic phenomena not observed in natural materials [5]. Their response to light is determined by their structures rather than by their composition. MMs provide a promising platform for the investigation of many phenomena, such as negative refractive index [6], super-focusing [7], and extraordinary transmission [8]. The most common unit cell structures for MMs include split-ring resonators (SRRs) [9,10], electric SRRs (eSRRs) [11], and H-shaped structures [12,13]. For actively controlled MMs, the responses can be tuned by using an external stimulus. Based on the MM structure, active THz modulators have been proposed over the past several years. For instance, E-mail address: [email protected] http://dx.doi.org/10.1016/j.carbon.2014.10.066 0008-6223/ 2014 Elsevier Ltd. All rights reserved.

an active MM switcher/modulator operated at THz frequencies has been suggested by Chen et al. [5], which can be designed to work at specific frequencies and exhibit good amplitude modulation of narrowband devices. By incorporating semiconductors in the critical regions of metallic SRRs, frequency-agile MM devices were produced, which operate as broadband THz modulators because of the causal relation between the amplitude modulation and the phase shift [14–16]. Recently, the investigation of active MM devices significantly benefits from the rapid development of graphene optoelectronics. With the merits of high carrier mobility and strong interaction with light by using the doping or regular structured patterning in broad-frequency regimes [17,18], graphene has attracted considerable attention for both fundamental physics and its enormous applications [19,20]. It can be regarded as an alternative to metallic materials and a good candidate for modulating MM and surface plasmon (SP) devices [21,22]. Graphene SPs exhibit strong confinement and relatively long propagation lengths [23–25]. Furthermore, graphene can also provide ultra-wideband tunability through electrostatic field, magnetic field, or chemical doping [26–28]. Presently, some research has been carried out to develop novel graphene MM devices [29–35]. For instance, on the basis

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of the graphene–dielectrics–metal layer structure, a tunable MM absorber can be achieved [30]. With the integrated graphene layer, the substantial gate-induced switcher has also been shown [32], manifesting the persistent photonic memory effects simultaneously. Valmorra et al. proposed the production of transmission modulators by using hybrid graphene/ MM structures; that is, the metallic eSRRs directly evaporated on the top of the large-area single graphene layer, and the modulation depth can reach about 12% [35]. As one of the key devices, a THz modulator is highly required to meet the need for short-range wireless THz communication or ultrafast interconnections. For current stateof-the-art semiconductor modulators [13,36–38], there still exist many problems, such as the small modulation depth and the need for cryogenic temperatures. Therefore, further improvements of the performance characteristics are required for the practical applications. By depositing graphene patterns on the SiO2/Si layers, we suggest the graphene–SiO2–Si (GSiO2Si) structure based on the flexible substrate to realize dynamically control of the propagation waves. Because the permittivity of the graphene layer can be varied via the applied electric fields or chemical doping, the transmission of the GSiO2Si structure can be conveniently modulated. Therefore, the tunable transmission properties of the proposed GSiO2Si structure were explored in the THz regimes, including the influences of the Fermi level of the graphene layer, the operation frequency, the thickness of the substrate, and the different kinds of MM unit cell patterns. The results show that the transmission of the proposed GSiO2Si structure can be modulated conveniently in a broadband range.

2.

Theoretic model and research method

Fig. 1(a) shows the sketch of the side view of the graphene MM structures, depositing on top of the SiO2/doped-Si layers. The thickness of the SiO2 layer is 30 nm, and the thickness of the doped Si layer is 1 lm. To reduce the influence of the substrate, the flexible polymer material was adopted, which is made from the polyimide layer. Fig. 1(b–d) shows the top views of the geometry for several kinds of unit cell structures, the eSRRs resonators (Fig. 1b), the H-shaped structure (Fig. 1c), and the cross-shaped structure (Fig. 1d). The period lengths along the x and y directions are px and py, respectively. The sub-wavelength MM structures are made of monolayer graphene. The normal incident waves transmit through the graphene MM structure along the z direction. Graphene can be considered as a two-dimensional material and is described by surface conductivity rg, which is related to the radiation frequency x, chemical potential lc (Fermi level, Ef), environmental temperature T, and relaxation time s. The conductivity of the monolayer graphene can be obtained from the Kubo formula [39]: 2

je ðx  js1 Þ rðx; lc ; s; TÞ ¼ rinter þ rintra ¼ 2 ph " Z 1 1 @f d ðeÞ @f d ðeÞ   de 2 @e @e ðx  js1 Þ 0 # Z 1 f d ðeÞ  f d ðeÞ de  2 2 ðx  js1 Þ  4ðe=hÞ 0

ð1Þ

Fig. 1 – (a) The side view of the GSiO2Si structure. The graphene unit cell structures are deposited on the SiO2/Si layers, the thickness of the SiO2 layer is 30 nm, and the doped Si layer is used to apply the gate voltage with the thickness of 1 lm. The substrate is the polyimide layer. (b–d) The top views of the geometry and dimension of the several kinds of MM unit cell structures, which have been connected by using the metallic wire. (b) The eSRR structure, lx = ly = 36 lm, g = 2 lm, D = 10 lm, T = 4 lm; (c) the H-shaped structure, w = 6 lm, h = 36 lm, and l = 48 lm; and (d) the cross structure, w = 6 lm and h = 36 lm. The periodic lengths along the x and y directions are both 60 lm. The green-shaded regions indicate the graphene layer, and the gray layer is SiO2. (A color version of this figure can be viewed online.)

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where fd(e) is the Fermi–Dirac distribution, j is the imaginary unit, e is the energy of the incident wave, h is the reduced Planck’s constant, and s is the scattering time. The first part of the above equation is the intra-band contribution, and the second part contributed to the inter-band contribution. Correspondingly, the dielectric constant of graphene layer can be expressed as follows: eg ¼ 1 þ j

rg xe0 D

ð2Þ

where D is the graphene layer thickness and e0 is the permittivity of free space. The Fermi level of the graphene layer depends upon the carrier concentration: Z 1 nd ¼ 2 ð3Þ f d ðeÞ  f d ðe þ 2Ef Þede p h v2F where vF  1 · 106 m/s is the Fermi velocity.

3.

Results and discussions

In order to compare the results with the conventionally metallic MM structure, first, we investigated the transmission of the graphene MMs based on the eSRR unit cell structure, which can provide a pure electric resonance with neither magnetic nor magnetoelectric responses. The geometric parameters of the eSRR unit cells are provided in the caption of Fig. 1(b). The simulation results were obtained from the well-established three-dimensional full-wave solver, CST Microwave Studio, which is based on the finite integration technique. The frequency domain solver was used. The transmission, reflection, and absorption curves can be obtained from the S-parameters, that is, T = |S21|2, R = |S11|2, and A = 1  T  R. As shown in the insets of Fig. 2, the graphene unit cell arrays were electrically connected with a thin metal wire to apply the bias voltage, just like the structures in Ref. [5]. The flexible substrate was made from the polyimide layer with a thickness of 2 lm. Fig. 2(a) shows the transmission curve of the graphene MMs; the polarization direction is along the y direction. Besides the broad resonant dips at high frequency, there is also a small resonant dip at low frequency when the Fermi level of the graphene layer is larger; for example, a resonant dip is observed at 0.76 THz if the Ef of the graphene membrane is 1.0 eV, but it will disappear when the Fermi level of the graphene layer is small. The reasons are as follows: the resonant dips of transmission at low frequency can be attributed to the LC resonance, which is similar to the phenomena occurring in the metallic SRR structure. Because the graphene layer thickness is very small, about 0.34 nm, and thus only on the condition that the Fermi level is very high, the graphene layer manifests remarkable ‘‘metallic’’ properties and LC resonance occurs. The results in Fig. 3(a–c), that is, the surface current density and electric fields, also confirm this. Because the LC resonance at low frequency is not significant, the modulation of the graphene MM structure mainly depends upon the dipolar resonance at high frequency, which results from the antenna-like coupling between the MM structure and the incident waves. As the Fermi level of the graphene layer increases, the resonant transmission of the MM structure becomes stronger, and the resonant dip shifts

Fig. 2 – The transmission of graphene MMs based on the eSRR unit cell structure. (a) The polarization of the incident wave is along the y direction. (b) The polarization is along the x direction. The Fermi levels of the graphene layer are 0.1, 0.2, 0.3, 0.5, 0.8, and 1.0 eV, respectively. (A color version of this figure can be viewed online.)

to higher frequency. It should be noted that the resonant dip at the Fermi level of 0.1 eV is attributed to the dipolar resonance and not the LC resonance, and it has been confirmed by the surface current density, which has not been shown. When the Fermi level of the graphene layer changes in the range of 0.1–1.0 eV, the resonant dips of the transmission curves can be tuned in the range of 0.75–1.90 THz, and the values of the transmission can be modulated in the range of 0.61–0.077. Correspondingly, the modulation depth of the frequency (fmod, fmod = Df/fmax) is 60.53%, and the modulation depth of the transmission (Tmod, Tmod = DT/Tmax) is 86.89%. The results for the case of the polarization direction of the incident light being along the x direction are presented in Fig. 2(b), manifesting significant dipolar resonance only. As the Fermi level of the graphene layer increases, the amplitude of the transmission decreases strongly, and the resonant dip significantly shifts to higher frequencies. When the Fermi level of the graphene layer changes in the range of 0.1– 1.0 eV, the resonant dip of the transmission modulates in the range of 0.55–1.30 THz, and the value of the transmission of the eSRR resonator structure is tuned in the range of 0.17– 0.72, respectively. Accordingly, the modulation depth of the frequency, fmod, is 57.69%, and the modulation depth of the transmission, Tmod, is 76.39%.

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Fig. 3 – (a–f) Show the surface current density, Ex, and Ey of the graphene MMs based on the eSRR unit cell structures at the resonant frequency of 0.76 THz (1.90 THz). The polarization direction of the incident light is along the y direction. (g–i) Show the results when the polarization direction of the incident light is along the x direction. The resonant frequency is 1.30 THz. The Fermi level of the graphene layer is 1.0 eV. (A color version of this figure can be viewed online.)

The electric current density can help in the understanding of the mechanisms of SP resonance. Fig. 3 shows the surface current density and the x (Ex) and y (Ey) components of the electric field of the graphene MMs based on the eSRR resonators. The Fermi level of the graphene layer is 1.0 eV. The results for the LC resonance at low frequency (0.76 THz) and those for the dipolar resonance at high frequency (1.90 THz) are shown in Fig. 3(a–f), respectively. The resonant frequencies are in accordance with the resonant dips in Fig. 2(a). Because the thickness of the graphene layer is very thin, and thus only on the condition that the Fermi level is larger, for example, 1.0 eV, the LC resonance of the suggested graphene GSiO2Si MM structure exists, as shown in Fig. 3(a). The incident THz fields drive the circulating surface currents in the inductive loops resulting in charge accumulation at the capacitive split gaps. The surface current in Fig. 3(a) is very similar to the results shown in Fig. 1(c) in Ref. [10] and Fig. 2(b) in Ref. [5], which confirms that the resonance at the

low frequency of 0.76 THz is LC resonance. For the proposed graphene MM structure, the tunable mechanism is mainly dependent on the dipolar resonance, which is closely related to the polarization of the incident light. Fig. 3(d–f) displays the results at the frequency of 1.90 THz. The surface current in Fig. 3(d) resembles the results in Fig. 1(d) in Ref. [10], confirming that the resonance at high frequency is the dipolar resonance. When the polarization of the incident light is along the x direction, the results of the dipolar resonances can be found in Fig. 3(g–i), and the corresponding resonant frequency is 1.30 THz. Additionally, Fig. 3 also shows that, near the region where the metal wire connects with the graphene unit cell, the electric field is very weak. The metal wire is also very thin, with its size in the range of several hundred nanometers. Based on the simulation calculation, when the diameter of the thickness of the metal wire is small, that is, <1 lm, the influences of the connecting wire on the transmission curves are very

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small and can be neglected. The diameter of the metal wire adopted in the calculation is 200 nm. For the suggested GSiO2Si structure, we mainly focus on the effects of the graphene patterns. Furthermore, some low-conductivity metal (Pb) or doped semiconductor (doped Si or InSb) can be used in the experimental fabrication to reduce the influence of the connecting wire. With the merits of having a simple structure to fabricate, the H-shaped unit cell structures were proposed to demonstrate a high refractive index and suppress the diamagnetic effect effectively. Fig. 4 shows the calculated propagation properties of the incident THz waves through graphene MMs based on the H-shaped unit cell structures at different Fermi levels in the THz regime. The structure parameters of the H-shaped MMs are presented in the caption of Fig. 1(c). The Fermi levels of the graphene layer are 0.1, 0.2, 0.3, 0.5, 0.8, and 1.0 eV, respectively. Fig. 4(a) depicts the transmission curve of the H-shaped unit cell structure when the polarization of the incident light is along the y direction. The transmission of the H-shaped unit cell structure shows obvious resonant dips. When the Fermi level of the graphene membrane changes in the range of 0.1–1.0 eV, the resonant dips of the transmission can be modulated in the frequency range of 0.33–0.72 THz, and the values of the transmission of the Hshaped unit cell structure can be tuned in the range of 0.15– 0.71. Accordingly, the modulation depth of the frequency is 54.17%, and the modulation depth of the transmission is 78.87%.

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Fig. 4(b) shows the transmission of the H-shaped unit cell structures; the polarization of the incident light is along the x direction. The reflection and absorption curves are given in Fig. 4(c) and 4(d), respectively. The interaction process of the incident light with the graphene MM structure can be understood as follows: the graphene MM structure captures the incident THz waves, and then the trapped energy is redistributed by means of absorption, transmission, and reflection. Due to the dipolar resonance, the transmission of the graphene MM structure shows a resonant dip. As the Fermi level of graphene layer increases, the resonant transmission becomes stronger. The reasons are as follows: according to Eq. (3), the carrier concentration of the graphene layer increases with the Fermi level. The graphene layer exhibits considerably more ‘‘metallic’’ properties at a high Fermi level, leading to the increase in the values of the reflection and absorption, as shown in Fig. 4(c) and 4(d). If the carrier concentration of the graphene layer is large enough, the effects of the reflection dominate and the absorption decreases. As the Fermi level increases, the combination of the reflection and absorption increases, resulting in a decrease in transmission. Furthermore, when the Fermi level of the graphene layer is low, the transmission dips mainly result from the absorption. The influences of reflection increase with the increase of the Fermi level of the graphene layer. When the carrier concentration is large enough, the reflection of the graphene layer contributes much more to the resonant dips of the transmission curves.

Fig. 4 – (a) The transmission of the graphene MMs based on the H-shaped unit cell structure. The polarization of the incident wave is along the y direction. (b–d) Show the transmission, reflection, and absorption of the H-shaped unit cell structure; the polarization is along x direction. The insets show the MM unit cell arrays connected with the metal wire. The Fermi levels of the graphene layer are 0.1, 0.2, 0.3, 0.5, 0.8, and 1.0 eV, respectively. (A color version of this figure can be viewed online.)

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For the MM unit cell element, the resonant dip of the transmission curve is expressed by xres = 1/(LC)1/2, where L and C are the effective inductance and capacitance, respectively. For most of the existing modulators, the tunable mechanisms are mainly based on the addition of a dielectric layer [38] or a change in the temperature of the substrate [40]. In these cases, the capacitance increases because the field expulsion effect usually dominates, resulting in an increase in capacitance and, in turn, a decrease in the resonant frequency. More details on this explanation are provided in Ref. [29]. However, for the suggested GSiO2Si structure, the resonant dip of the transmission curve manifests an obvious blue shift with an increase in the Fermi level of the graphene layer. The reasons are given as follows: because the dimension of the unit cell structure is smaller than the wavelength of the incident light, the graphene eSRR resonator can be regarded as a dipolar antenna. The relationship between the length of the dipolar antenna Lante and its resonant wavelength kres can be given by 2 Lante = kres. The resonant wavelength can be expressed as kres = a + b · neff, where a and b are the coefficients closely related to the device geometry and surrounding dielectric properties [41,42]. As the Fermi level increases, the effective index of the graphene SPs decreases, resulting in a decrease in the resonant wavelength, that is, an increase in resonant frequency, as shown in Fig. 4. When the Fermi level of the graphene layer changes in the range of 0.1–1.0 eV, the resonant dips of the transmission curves can be modulated in the range of 0.65–1.53 THz, and the values of the transmission resonant dips can be tuned in the range of 0.51–0.046. Correspondingly, the modulation depth of the frequency, fmod, is 57.52%, and the modulation

depth of the transmission, Tmod, is 90.20%, which are much larger than most of the presently published THz modulators. The resonant transmission spectrum of the graphene unit cell MM structure is broad, which indicates that the suggested graphene MM structures are very suitable for fabricating broadband modulators instead of filters or switchers. Furthermore, the full width at half maximum of the transmission (reflection or absorption) curve increases with the increase in Ef of the graphene layer, which can be explained as follows: As the Fermi level increases, the carrier concentration of the graphene layer increases, resulting in the enhancement of the graphene permittivity. Thus, the larger dielectric constant of the graphene layer at a higher Fermi level results in a larger reflection and absorption loss. Fig. 5(a–f) shows the surface current density and the electric fields for the y- and x-polarized MM structures based on the H-shaped unit cell, respectively. The Fermi level of the graphene layer is 1.0 eV. The simulation results are plotted at the resonant frequencies of 0.72 and 1.53 THz, which is in accordance with the dips shown in Fig. 4(a) and 4(b), respectively. The direction and size of the arrows in Fig. 5(a) and 5(d) indicate the direction and relative value of the surface current density, and the color map in Fig. 5(b–f) shows the relative local electric field amplitude. Because the circulating loop cannot form in the H-shaped unit cell structure, only dipolar resonance exists, which is also confirmed by the results shown in Fig. 5. Fig. 6(a) shows the transmission curves of the incident light through the graphene MM structure based on the cross-shaped unit cell at different Fermi levels. The length and width of the cross structure are 36 and 6 lm, respectively.

Fig. 5 – (a–c) Show the distribution of the surface current density; the x component of the electric field, Ex; and the y component, Ey, of the H-shaped graphene MM structure. The polarization direction of the incident light is along the y direction. The resonant frequency is 0.72 THz. (d–f) Show the surface current density, Ex, and Ey when the polarization direction of the incident light is along the x direction. The resonant frequency is 1.53 THz. The Fermi level of the graphene layer is 1.0 eV. (A color version of this figure can be viewed online.)

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Fig. 6 – (a) The transmission of graphene MMs based on the cross-shaped unit cell structures versus frequency in the THz regime at different Fermi levels of the graphene layer. The polarization of the incident light is along the y direction. The Fermi levels of the graphene layer are 0.1, 0.2, 0.3, 0.5, 0.8, and 1.0 eV, respectively. (b–d) Show the surface current density, Ex, and Ey of the cross-shaped graphene MM structure. The resonant frequency is 1.78 THz. The Fermi level of the graphene layer is 1.0 eV. (A color version of this figure can be viewed online.)

The polarization of the incident light is along the y direction. It can be found from Fig. 6(a) that the transmission curve of the graphene cross-shaped structure displays obvious resonant dips. As the Fermi level increases, the resonant dip of the transmission increases and the dip position shifts to higher frequency. When the Fermi level of the graphene layer changes in the range of 0.1–1.0 eV, the resonant dips of the transmission curves can be tuned in the range of 0.80– 1.78 THz, and the values of the transmission of the graphene stripe unit cell structure can be modulated in the range of 0.18–0.74. Correspondingly, the modulation depth of the frequency, fmod, is 55.06%, and the modulation depth of the transmission, Tmod, is 75.68%. Fig. 6(b–d) shows the results of the dipolar resonance of the graphene MM structures based on the cross-shaped unit cells, that is, the surface current density and electric fields. The polarization of the incident light is along the y direction. The Fermi level of the graphene layer is 1.0 eV and the frequency is 1.78 THz, which

corresponds with the resonant dip in Fig. 6(a). Additionally, because of the symmetrical structure of the cross shape, its transmission along the x direction exhibits similar properties, which have not been shown here. Thus, the cross-shaped structures can be used to fabricate polarization-insensitive devices. Table 1 shows the transmission properties of the several kinds of structure patterns mentioned above. The graphene MM structures can exhibit good frequency modulation (fmod) and amplitude tuning (Tmod) properties. The modulation depth of transmission is about 80%, which is considerably better than the properties of the existed THz modulators. For example, the modulation depth of the hybrid graphene/ MM structure dependent on the LC resonance is only about 12% [33]. For the hybrid graphene/MM structure, if the Fermi level of the graphene layer increases further, the carrier concentration increases, and the graphene membrane exhibits considerably more metallic properties. Consequently, like

Table 1 – The comparison of transmission properties of several kinds of different unit cell structures. Unit cell structure

Polarized direction

fmin  fmax (THz)

Df (THz)

Df /fmax (%)

Tmin  Tmax

DT

DT/Tmax (%)

eSRRs

x y

0.55–1.30 0.75–1.90

0.75 1.15

57.69 60.53

0.17–0.72 0.077–0.61

0.55 0.53

76.39 86.89

H-shaped

x y

0.65–1.53 0.33–0.72

0.88 0.39

57.52 54.17

0.046–0.51 0.15–0.71

0.46 0.56

90.20 78.87

Cross-shaped

x y

0.80–1.78 0.80–1.78

0.98 0.98

55.06 55.06

0.18–0.74 0.18–0.74

0.56 0.56

75.68 75.68

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the switching phenomena of the eSRR-based MMs fabricated on the nano-island superlattice substrates, the graphene layer will short out the SRRs gap and turn off the electric resonance [10], which hinders the increase in the modulation properties. By contrast, the tunable mechanism of the proposed GSiO2Si structure is mainly dependent on the dipolar resonance, and its modulation depth is much larger than those for the hybrid graphene/MM structures. In addition, the graphene MMs can exhibit good modulation properties along both the x and y directions; the cross-shaped structure especially exhibits polarization-insensitive properties due to its symmetrical structure. It is also worth noting that that the modulation depths for the several kinds of resonators are slightly different. The reasons are probably that the tunable mechanisms of the graphene MM structure are mainly dependent on the dipolar resonance, which is closely related to the dimension parameters of the MM structures. On the condition that the geometric parameters are roughly equal, the modulation depths of the different unit cell structures are almost similar. Because the thickness of the graphene layer is very small, the influence of the substrate layer on the properties is significant. In order to reduce the substrate effects, a thin flexible dielectric polymer layer (e.g., polyimide) is adopted. The refractive index of the lossless polyimide layer is about 1.8 [11]. Fig. 7 shows the effects of the substrate thickness on transmission based on the cross-shaped unit cell structure. The length and width of the cross structure are 48 and 12 lm, respectively. The polarization of the incident light is along the y direction. It can be demonstrated from Fig. 7 that, if the thickness of the substrate is not large, <10 lm, the structure exhibits good transmission resonance. As the substrate thickness increases, the resonant dips shift to low frequency, that is, red shift. For instance, the transmission resonant dip is about 1.74 THz without including the effects of the substrate. When the substrate thickness is taken into account, for example,

2 lm, the resonant dip is about 1.58 THz. The reasons are shown as follows: if the effect of the substrate layer is included, the dielectric constant of the polyimide substrate (3.24) is much larger than that of air, and the capacitance increases, resulting in the shift of the resonance frequency of the MM structures to low frequency. Additionally, the influence of the substrate thickness increases with the increase in the graphene Fermi level. When the thickness of the substrate changes from 0 to 10 lm, the resonant frequency is tuned in the range of 1.10–0.74 THz at the Fermi level of 1.0 eV, and the value changes from 0.82 to 0.67 THz at Fermi levels of 0.1 eV. Therefore, for the proposed transmission modulators, a thin substrate layer should be adopted to reduce the influence of the substrate and achieve better performance. A polyimide layer with a thickness of 2 lm is used in the simulation, which can be realized with the rapid development of experimental fabrication [12]. Finally, as the substrate thickness is very thin, the influence of the standing wave is not evident in this case.

4.

Conclusions

By depositing the planar arrays of graphene patterns on the SiO2/Si layers, the tunable transmission properties of the proposed GSiO2Si structure based on the flexible polymer substrate were shown in the THz regime. The influences of the operation frequency, the thickness of the substrate, the Fermi level of graphene layer, and the different kinds of graphene structure patterns, that is, eSRRs, H-shaped, and crossshaped resonators, have been given and discussed. Compared with the conventional metallic MM structure based on the LC resonance, the tunable mechanisms of the graphene MM structure are mainly dependent on the dipolar resonance. As the Fermi level of the graphene layer increases, the transmission resonant responses of the graphene MM structure become stronger, and the resonant dips shift to higher frequency, which is different from the red shift for most of the THz waveguide devices. By varying the applying bias voltage, significant modulation of the frequency and amplitude is achieved. The modulation depth of the transmission can reach about 80%. The results are very helpful in the design of novel plasmonic devices, such as broadband transmission modulators and sensors, which can be very beneficial in the application of optical communications and biomedical sensing.

R E F E R E N C E S

Fig. 7 – The influence of the substrate thickness on the transmission curves of the graphene MMs based on the cross-shaped unit cell structures. The substrate thicknesses are 0, 2, 5, and 10 lm, respectively. The polarization of the incident wave is along the y direction. (A color version of this figure can be viewed online.)

[1] Ferguson B, Zhang XC. Materials for terahertz science and technology. Nat Mater 2002;1(1):26–33. [2] Lu¨ JT, Cao JC. Coulomb scattering in the Monte Carlo simulation of terahertz quantum-cascade lasers. Appl Phys Lett 2006;89(21):211115. [3] Li H, Cao JC, Lu¨ JT, Han YJ. Monte Carlo simulation of extraction barrier width effects on terahertz quantum cascade lasers. Appl Phys Lett 2008;92(22):221105. [4] Guo XG, Tan ZY, Cao JC, Liu HC. Many-body effects on terahertz quantum well detectors. Appl Phys Lett 2009;94(20):201101.

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8 2 ( 2 0 1 5 ) 2 2 9 –2 3 7

[5] Chen HT, Padilla WJ, Zide JMO, Gossard AC, Taylor AJ, Averitt RD. Active terahertz metamaterial devices. Nature 2006;444(7119):597–600. [6] Zhang S, Park YS, Li J, Lu X, Zhang WL, Zhang X. Negative refractive index in chiral metamaterials. Phys Rev Lett 2009;102(2):023901. [7] He XY, Wang QJ, Yu SF. Investigation of multilayer subwavelength metallic–dielectric stratified structures. IEEE J Quantum Electron 2012;48(12):1554–9. [8] Li Z, Ma Y, Huang R, Singh R, Gu J, Tian Z, et al. Manipulating the plasmon-induced transparency in terahertz metamaterials. Opt Express 2011;19(9):8912–9. [9] Azad AK, Dai J, Zhang WL. Transmission properties of terahertz pulses through subwavelength double split-ring resonators. Opt Lett 2006;31(5):634–6. [10] Padilla WJ, Taylor AJ, Highstrete C, Lee M, Averitt RD. Dynamical electric and magnetic metamaterial response at terahertz frequencies. Phys Rev Lett 2006;96(10):107401. [11] Chen HT, Padilla WJ, Zide JMO, Bank SR, Gossard AC, Taylor AJ, et al. Ultrafast optical switching of terahertz metamaterials fabricated on ErAs/GaAs nano-island superlattices. Opt Lett 2007;32(12):1620–2. [12] Choi M, Lee SH, Kim Y, Kang SB, Shin J, Kwak MH, et al. A terahertz metamaterial with unnaturally high refractive index. Nature 2011;470(7334):369–73. [13] Benz A, Montano I, Klem JF, Brener I. Tunable metamaterials based on voltage controlled strong coupling. Appl Phys Lett 2013;103(26):263116. [14] Chen HT, O’Hara JF, Azad AK, Taylor AJ, Averitt RD, Shrekenhamer DB, et al. Experimental demonstration of frequency-agile terahertz meta-materials. Nat Photonics 2008;2(5):295–8. [15] Zhou Y, Yang J, Cheng X, Zhao N, Sun L, Sun H, et al. Electrostatic self-assembly of graphene–silver multilayer films and their transmittance and electronic conductivity. Carbon 2012;50(12):4343–50. [16] Gu J, Singh R, Liu X, Zhang X, Ma Y, Zhang S, et al. Active control of electromagnetically induced transparency analogue in terahertz metamaterials. Nat Commun 2012;3:1151. [17] Wright AR, Zhang C. Dynamic conductivity of graphene with electron-LO-phonon interaction. Phys Rev B 2010;81(16):165413. [18] Jablan M, Buljan H, Soljacˇic´ M. Plasmonics in graphene at infrared frequencies. Phys Rev B 2009;80(24):245435. [19] Yan H, Low T, Zhu W, Wu Y, Freitag M, Li X, et al. Damping pathways of mid-infrared plasmons in graphene nanostructures. Nat Photonics 2013;7(5):394–9. [20] He XY, Tao J, Meng B. Analysis of graphene TE surface plasmons in the terahertz regime. Nanotechnology 2013;24(34):345203. [21] Vakil A, Engheta N. Transformation optics using graphene. Science 2011;332(6035):1291–4. [22] He XY. Numerical analysis of the propagation properties of subwavelength semiconductor slit in the terahertz region. Opt Express 2009;17(17):15359–71. [23] He XY, Li R. Comparison of graphene-based transverse magnetic and electric surface plasmon modes. IEEE J Sel Top Quantum Electron 2014;20(1):4600106.

237

[24] Gao W, Shu J, Reichel K, Nicke DV, He X, Shi G, et al. Highcontrast terahertz wave modulation by gated graphene enhanced by extraordinary transmission through ring apertures. Nano Lett 2014;14(3):1242–8. [25] Gosciniak J, Tan DTH. Theoretical investigation of graphenebased photonic modulators. Sci Rep 2013;3:1897. [26] Koppens FHL, Chang DE, de Abajo FJG. Graphene plasmonics: a platform for strong light-matter interactions. Nano Lett 2011;11(8):3370–7. [27] Ju L, Geng B, Horng J, Girit C, Martin M, Hao Z, et al. Graphene plasmonics for tunable terahertz metamaterials. Nat Nanotechnol 2011;6(10):630–4. [28] He XY, Kim S. Graphene-supported tunable waveguide structure in the terahertz regime. J Opt Soc Am B 2012;30(9):2461–8. [29] Mousavi SH, Kholmanov I, Alici KB, Purtseladze D, Arju N, Tatar K, et al. Inductive tuning of Fano-resonant metasurfaces using plasmonic response of graphene in the mid-infrared. Nano Lett 2013;13(3):1111–7. [30] Andryieuski A, Lavrinenko AV. Graphene metamaterials based tunable terahertz absorber: effective surface conductivity approach. Opt Express 2013;21(7):9144–55. [31] Sensale-Rodriguez B, Yan R, Kelly M, Fang T, Tahy K, Hwang WS, et al. Broadband graphene THz modulators enabled by intraband transitions. Nat Commun 2012;3:780. [32] Lee SH, Choi M, Kim TT, Lee S, Liu M, Yin X, et al. Switching terahertz waves with gate-controlled active graphene metamaterials. Nat Mater 2012;11(11):936–41. [33] Zhu W, Rukhlenko ID, Premaratne M. Graphene metamaterial for optical reflection modulation. Appl Phys Lett 2013;102(24):241914. [34] Degl’Innocenti R, Jessop DS, Shah YD, Sibik J, Zeitler JA, Kidambi PR, et al. Low-bias terahertz amplitude modulator based on split-ring resonators and graphene. ACS Nano 2014;8(3):2548–54. [35] Valmorra F, Scalari G, Maissen C, Fu W, Scho¨nenberger C, Choi JW, et al. Low-bias active control of terahertz waves by coupling large-area CVD graphene to a terahertz metamaterial. Nano Lett 2013;13(7):3193–8. [36] Han J, Gu J, Lu X, He M, Xing Q, Zhang WL. Broadband resonant terahertz transmission in a composite metal– dielectric structure. Opt Express 2009;17(19):16527–34. [37] Tamagnone M, Fallahi A, Mosig JR, Perruisseau-Carrier J. Fundamental limits and near-optimal design of graphene modulators and non-reciprocal devices. Nat Photonics 2014;8(7):556–63. [38] Chiam SY, Singh R, Gu J, Han J, Zhang WL, Bettiol AA. Increased frequency shifts in high aspect ratio terahertz split ring resonators. Appl Phys Lett 2009;94(6):064102. [39] Gusynin VP, Sharapov SG. Magneto-optical conductivity in graphene. J Phys Condens Matter 2007;19(2):026222. [40] Singh R, Azad AK, Jia QX, Taylor AJ, Chen HT. Thermal tunability in terahertz metamaterials fabricated on strontium titanate single-crystal substrates. Opt Lett 2011;36(7):1230–2. [41] Novotny L. Effective wavelength scaling for optical antennas. Phys Rev Lett 2007;98(26):266802. [42] Yao Y, Kats MA, Genevet P, Yu N, Song Y, Kong J, et al. Broad electrical tuning of graphene-loaded plasmonic antennas. Nano Lett 2013;13(3):1257–64.