Tunable electromagnetically induced transparency based on graphene metamaterials

Binggang Xiao; Binggang Xiao; Shengjun Tong; Alexander Fyffe; Zhimin Shi

doi:10.1364/OE.382485

1. Introduction

Electromagnetically induced transparency (EIT) is a coherent nonlinear optical effect that produces a narrow spectral window of transmission near an absorption line [1,2]. Traditional EIT implementations are enslaved to rigorous experimental conditions, such as stable gas lasers and low temperature environments [3]. In order to overcome the strict experimental conditions, EIT-like phenomenon based on metamaterial structures have been extensively studied due to the high flexibility in tailoring the effective optical properties of metamaterials [4,5].

The photonic EIT-analogy is often interpreted using the coupling between bright and dark modes. It is generally believed that a bright mode can be easily excited in a medium, but the dark mode requires near field coupling [6]. Thus far, it has been shown that there are two possible mechanisms that can be used to achieve EIT behavior [5,7]. The first mechanism is to couple two bright modes, which involves the weak coupling of two transmissive resonances at adjacent frequencies [8]. The second mechanism is to couple a bright mode with a dark mode, creating a transparency window at a desired frequency by coupling a highly transmissive resonance and a highly reflective or absorptive resonance at the same frequency [9,10].

Metamaterials that exhibit an EIT-like effect are widely attractive for many applications including sensing [11,12], slow light control [13,14], modulation [15], optical buffering [16], and filtering [17]. Graphene has recently attracted a tremendous amount of research interest due to its many superior electronic and optical properties [18–21]. Specifically, its conductivity can be easily tuned by external voltage gating and consequently the adjusting of its Fermi level [22,23]. Such flexibility has opened new opportunities to incorporate graphene as a tunable material in the design of many photonic and electronic devices [24–26]. In the past few years, many sensors based on graphene have been proposed and studied. For example, in 2016, Huang et al. proposed a gate-controlled on-chip graphene metasurface consisting of a monolayer graphene sheet and silicon photonic crystal-like substrate with a refractive-index sensitivity of 1267nm / RIU and FOM of 8.89 [27]. In 2017, Wenger et al. proposed a graphene on a subwavelength dielectric grating with an improved performance without any dynamic tunability [28].

In this paper, we present a novel graphene-based metamaterial structure that can exhibit tunable EIT behavior by applying an external voltage. The metamaterial is composed of two graphene nano-disks and a graphene nano-strip, which rest on a homogeneous isotropic substrate. An ion gel layer with a high capacitance is spin-coated on the graphene pattern, and a gold electrode is manufactured on the ion gel layer. The graphene Fermi level can be adjusted by changing the voltage between the metal electrode and the underlying doped silicon substrate. We present a field characterization to show that the dipole resonance of the disks can couple effectively with the quadrupole resonance of the strip to create an EIT-like response. In this study, we show how the geometric parameters of the metastructure can affect the EIT response, and we also show that the EIT-like response of the metamaterial can be dynamically tuned by adjusting the Fermi level of the graphene. Lastly, the performance of the proposed metamaterial device for sensing and group delay applications is analyzed. Compared with previous works, our metamaterial structure shows better performance with flexible post-fabrication adjustability.

2. Model and simulation parameters

Our metamaterial, shown schematically in Fig. 1(a), is composed of two mono-layer graphene nano-disks of equal radius separated by a mono-layer graphene nano-strip arranged in a rectangular lattice with period of the unit cell P_x and P_y in the x and y direction, respectively. Here the ion gel strip is described by a nondispersive permittivity ɛ = 1.82, which is overlaid on the graphene pattern structures, with a thickness of 100 nm and a width of 100 nm. The metastructure sits on top of a 200-nm-thick layer of CaF₂ with a relative permittivity of 2.05. The substrate beneath the CaF₂ is doped silicon with a relative permittivity of 11.7. The geometry of the metamaterial is defined by the nano-disk radius R, the nano-strip length L and width W, and the coupling distance between the nano-disks and the nano-strip S. The thickness of the mono-layer graphene is set at its theoretical value t = 0.34 nm. The conductivity of graphene can be expressed as the sum of the intraband transition and the interband transition as follows [29]:

(1)$$\sigma = \frac{{i{e^2}{E_f}}}{{\pi {\hbar^2} ({\omega + i{\tau^{ - 1}}} )}} + \frac{{i{e^2}}}{{4\pi \hbar }}\ln \left[ {\frac{{2|{{E_f}} |- \hbar ({\omega + i{\tau^{ - 1}}} )}}{{2|{{E_f}} |+ \hbar ({\omega + i{\tau^{ - 1}}} )}}} \right]$$

Where e is electronic charge, $\hbar \equiv h/2\pi$ is the reduced Planck’s constant, E_f is the chemical potential energy, $\tau = \mu {E_f}/ev_f^2$ is the relaxation time, ${v_f} = c/300$ is the Fermi velocity, c is the speed of light in free space, $\mu = 10000\;c{m^2}{V^{ - 1}}{s^{ - 1}}$ is the measured DC mobility [30].

Fig. 1. (a) Top view of the unit cell of our proposed metamaterial structure, which is characterized by the follow parameters: The period of the unit cell, Px and Py in the x and y directions, respectively, radius R of the nano-disks, width W and length L of the nano-strip, and coupling distance S between the nano-disks and the nano-strip. (b) Schematic diagram of the proposed graphene metamaterial device with light incident from the top. The graphene pattern structures integrate with the CaF₂/Si layers, the ion gel strip is overlaid on the graphene, and a gold electrode is manufactured on the ion gel layer.

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In order to study the EIT-like response of the proposed graphene metamaterial, electromagnetic simulations are performed using commercial frequency domain finite element method software CST Microwave Studio. The schematic diagram of the simulation process is shown in Fig. 1(b). Gold gate contacts are fabricated on the ion-gel layers, so we can tune graphene’s Fermi energy directly by controlling the bias voltages between the gold contact and the silicon substrate. The excitation source is chosen to be a plane wave which is normally incident onto the metamaterial propagating in the –z-direction, and the electric field is polarized in the x-direction.

In order to produce a significant EIT response at any desired working frequency, the graphene nano-disk’s dipole resonance frequency and the strip’s quadrupole resonance frequency are simultaneously chosen to be close to the desired center frequency. The radius R of the graphene nano-disk and the physical length L of the nano-strip can be estimated by the following analytic formulas [22,31–34]:

(2)$$R \approx \frac{{\alpha c{L_1}{E_f}}}{{\pi \hbar {\omega _0}^2({{\varepsilon_1} + {\varepsilon_2}} )}}$$

(3)$$L \approx \frac{{\eta {e^2}{E_f}}}{{{\hbar ^2}{\omega _0}^2{\varepsilon _0}({{\varepsilon_1} + {\varepsilon_2}} )}}$$

where α ≈ 1/137 is the fine-structure constant, L₁ is independent of frequency and disk size but depend on the symmetry of the plasmon under consideration. and for a dipole nano-disk this value is taken as 12.5, ω₀ is the desired center frequency of the metastructure, ɛ₁ and ɛ₂ are the dielectric permittivities of the materials above and below the graphene, respectively, and η = 1.34 is a dimensionless correction factor whose value is determined from the simulation results. These analytic guidelines make our design versatile and capable of operating at any center frequency within a wide working range. As an example, we here wish to design an EIT-like metamaterial with a center frequency of 27.45 THz. According to Eqs. (2) and (3), the radius of the nano-disk and the length of the nano-strip can be calculated to be 100 nm and 400 nm, respectively, which are very close to our final optimized values. Unless otherwise noted, the values of the geometric parameters are chosen as follows: the nano-disk radius R = 100 nm, the nano-strip length L = 400 nm and width W = 60 nm, and the coupling distance S = 70 nm. The unit cell period is set at P_x = P_y= 640 nm. Note that when the nano-strip width and the nano-disk diameter are larger than 20 nm, the quantum confinement effects associated to the type of edges can be ignored [35].

To understand the metastructure’s EIT-like response, three simulations with a fixed graphene Fermi energy (0.7 eV) are performed with (i) only the graphene nano-disks in the unit cell, (ii) only the graphene nano-strip in the unit cell, and (iii) the full structure with both the nano-disks and nano-strip. As one sees in Fig. 2(a), the transmission spectrum of the graphene nano-disks (green dashed line) exhibits a strong resonance at 27.66 THz, while the transmission spectrum of the graphene nano-strip (blue dashed line) displays a weak resonance at 28.67 THz. When both the nano-disks and the nano-strip are present, the two resonances interplay with each other, which leads to a characteristic EIT-like response centered at 27.45 THz in its transmission as shown in the red solid line in Fig. 2(a). The spatial profile of the electric field distribution at several critical frequencies marked as points b to g in Fig. 2(a) are shown in Figs. 2(b) – 2(g), respectively. As one sees from the figure, the resonance of the nano-strip at 11.76 THz (point b) corresponds to the dipole mode, while the resonance at 28.67 THz (point c) corresponds to the quadrupole mode. Figure 2(d) shows the dipole mode of the nano-disks at 27.66 THz (point d). When both nano-disks and nano-strip are present, the quadrupole mode of the nano-strip couples strongly with the dipole mode of the nano-disk, resulting in a characteristic EIT-like, split-resonance lineshape. Figure 2(e) shows the strong field excitation of an anti-symmetric mode at 26.96 THz (point e), where the field within the nano-disks is out of phase with respect to the field within the nano-strip. On the other resonance dip at 27.95 THz (point g), the field within the nano-disks and the nano-strip is in phase, indicating a symmetric mode resonance. The symmetric and anti-symmetric modes destructively interfere at 27.45 THz, which results in a transparency window with a maximum transmission coefficient of 83.2% and with no strong field excitation within the metastructure as shown in Fig. 2(f).

Fig. 2. (a) Transmission spectra of graphene metamaterial composed of only the nano-disks (green dotted line), only the nano-strip (blue dotted line), and both nano-disks and nano-strip (red solid line). (b)-(d) Field distributions E_z of dipole strip, quadrupole strip, dipole disks. (e)-(g) Field distributions E_z of Transmission peak and resonance dips.

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The analogy between the atomic EIT system and our metamaterial system is shown in Fig. 3, where state |1 > is the ground state, state |2 > is the metastable state, and state |3 > is the excited state. The arrows in the figure denote allowed transitions. There are two paths to move the atom from the ground state to the excited state with a strong control field applied: direct excitation |1>-|3 > and indirect excitation |1>-|3>-|2>-|3 > . Each transition produces a phase shift of π/2, so there is a phase difference of π between the indirect excitation and direct excitation, leading to the occurrence of destructive interference, eliminating the transition and forming a transparency window [1]. The resonance modes of these metamaterials are equivalent to the role of atomic energy levels. The quadrupole mode (dark mode) of the nano-strip is similar to the metastable state of the atomic system; the dipole mode (bright mode) of the nano-disks is similar to the excited state. States |2 > and |3 > are related to the coupling between the two modes. The destructive interference between the two paths and the relation between the incident field and the excited modes can be described by the following simplified Lorentz oscillator model [27]:

(4)$$\begin{array}{l} {E_1}({\omega - {\omega_{01}} + i{r_1}} )+ k{E_2} ={-} g{E_0}\\ k{E_1} = {E_2}({\omega - {\omega_{02}} + i{r_2}} )\end{array}$$

where ω₀₁, ω₀₂; E₁, E₂; r₁ and r₂ are the resonance frequency, amplitude and damping factor of the bright mode and the dark mode, respectively. k is the coupling coefficient of states |2 > and |3>, and g is a geometric parameter indicating the coupling strength between the bright mode and the incident probe field E₀.

Fig. 3. The analogy between the proposed structure and atomic EIT system with three levels.

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To understand the dependence of the EIT behavior on the geometric parameters of the metastructure, a parametric study is performed by scanning the values of the coupling distance S and the disk radius R. As one can see in Figs. 4(a) and 4(b), as the coupling distance changes from 50 nm to 90 nm, the anti-symmetric resonance shows a significant blue shift while the symmetric resonance does not change significantly. Furthermore, as the coupling distance increases, both the width of the transparency window and the overall transmittance decrease. This is primarily due to a weakening of the coupling between the two structures as S increases. Meanwhile, as the disk radius varies, the dipole resonance frequency of the nano-disk varies, which also lead to change in the shape of the EIT-like resonance.

Fig. 4. (a) Transmission spectrum of the graphene metamaterial when the coupling distance S is 50 nm, 60 nm, 70 nm, 80 nm, and 90 nm. (b) Transmission spectrum of the graphene metamaterial when the radius R of the graphene disks is 90 nm, 95 nm, 100 nm, 105 nm, and 110 nm. (c) Transmission spectrum of the graphene metamaterial when the width W of the graphene strip is 40 nm, 50 nm, 60 nm, 70 nm, and 80 nm. (d) Transmission spectrum of the graphene metamaterial when the length L of the graphene strip is 360 nm, 380 nm, 400 nm, 420 nm, and 440 nm.

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To study the effect of the geometric parameters of the nano-strip on the EIT resonance, a parametric study is performed by varying the nano-strip width, W, and length, L. As shown in Figs. 4(c) and 4(d), one can see that when W = 60 nm, L = 400 nm, the minimum values of the two transmission dips are approximately equal. As W or L gradually deviates, the quadrupole mode resonance frequency deviates from the dipole mode of the disks. The energy of the disks coupled to the strip decreases, and the transparency window gradually disappears.

3. Performance analysis for applications

Since the EIT-like response of our metamaterial is sensitive to the change of refractive index of the ambient medium, it can be used for biochemical molecular sensing applications [12,36]. We simulate the sensitivity of our graphene metamaterial in terms of its resonance response to the change in the refractive index n of the top surrounding medium. As shown in Fig. 5(a), when the refractive index of the medium above the graphene metamaterial changes from 1 to 1.3, the EIT transparency window undergoes a significant red shift from 11.02 um to 11.93 um. As shown in Fig. 5(b), the wavelength change of the peak of the transparency window linearly depends on the refractive index of the incident medium. The sensing performance can be quantified using the sensitivity S and a figure of merit (FOM) as the quality factor as follows [37,38]:

(5)$$S = \frac{{\Delta \lambda }}{{\Delta n}}$$

(6)$$FOM = \frac{S}{{FWHM}}$$

where Δλ is peak wavelength shift of the transparency window and FWHM is the full width at half maximum of the transparency window. As shown in Fig. 5(b), the sensitivity of the EIT metastructure is calculated to be 3016.7 nm/RIU with the FOM reaching more than 12.0 for the refractive index of the surrounding media to vary from 1.0 to 1.3. This result exceeds the performance of similarly reported values for graphene-based metamaterial sensors [27–28,39], as shown in Table 1.

Fig. 5. (a) Transmission spectrum of the graphene metamaterial when the refractive index n is 1, 1.1, 1.2, and 1.3. (b) Wavelength change of the transparency window peak versus the refractive index (red solid line). FOM change of the transparency window versus the refractive index (blue solid line).

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Table 1. Comparison of performance reported in various graphene sensors

View Table

Unlike the EIT effect achieved in traditional metal metamaterials, the EIT response in our graphene-based metamaterial can be modified significantly by an external electric field. It has been shown that applying a voltage to a graphene sheet can effectively change its Fermi energy, which modifies the carrier density in graphene and consequently modifies the surface conductivity of a graphene sheet [22,23]. Similar to our previous analysis, the geometric parameters are chosen such that the nano-disk radius R = 100 nm, the nano-strip length L = 400 nm, width W = 60 nm, and the coupling distance between the nano-disk and the nano-strip S = 70 nm. Figure 6(a) shows that when the Fermi Level of the graphene is changed from 0.5 eV to 0.8 eV, a significant blue shift appears in the transparency window. In Fig. 6(b), one can see that the frequency shift of the transparency window peak is linearly proportional to the squareroot of the Fermi Energy, E_f^1/2, and the tunable range can be as large as almost 6 THz. Therefore, the working frequency of our device can be tuned post-fabrication using external voltage control.

Fig. 6. (a) Transmission spectra of the metastructure at various Fermi levels of the graphene. (b) The frequency shift of the transparency window as a function of E_f^1/2.

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Additionally, it is known that EIT-like effects can significantly modify the group velocity of electromagnetic waves in the transparency window frequency range [40,41]. The performance of the slow light effect can be characterized by group delay τ_d and group index n_g, which are defined as [3,42]:

(7)$${\tau _d} ={-} \frac{{d\varphi (\omega )}}{{d\omega }}$$

(8)$${n_g} = \frac{c}{{{v_g}}} = \frac{c}{t}{\tau _d}$$

where $\varphi (\omega )$ is the phase response of the transmission spectrum, ω is the angular frequency of the field, c is the speed of light in vacuum, v_g is the group velocity and t is the thickness of the structure. In Figs. 7(a) and 7(b), we show that light within the transparency window can experience significant positive group delay of up to 0.15 ps (E_f = 0.7 eV), which corresponds to a group index of 200. Furthermore, the slow light effect can be adjusted by changing the Fermi level of the graphene (i.e. applying an external voltage). As shown in Figs. 7(a) and 7(b), as the Fermi level varies from 0.5 eV to 0.8 eV, the group delay and group index gradually increase. Thus, there is a great potential of the use of such atomically-thin device for tunable optical retardation applications.

Fig. 7. (a) Group delay of the metastructure as a function of frequency at various Fermi levels. (b) Group index of the metastructure as a function of frequency at various Fermi levels.

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4. Conclusion

In this paper, we have proposed and demonstrated a graphene metamaterial design that can exhibit tunable EIT-like response at mid-infrared frequencies. We have shown that the EIT-like response is produced by the strong coupling between the dipole resonance of the nano-disks and the quadrupole resonance of a nano-strip, which creates a transparency window between a symmetric resonance mode and an asymmetric resonance mode. Such an EIT-like response of the proposed graphene metastructure can be tuned over a wide frequency range by adjusting the Fermi Energy of graphene with an applied bias voltage. We have shown that this metamaterial structure can be used as an effective refractive index sensor with a sensitivity of 3016.7 nm / RIU and a FOM above 12.0. We have also demonstrated that our metamaterial can also be used as a tunable slow-light device with a group index of 200. Our design provides a new tunable photonic platform that can facilitate the development of biochemical molecule testing, tunable optical retarders, etc.

Funding

Natural Science Foundation of Zhejiang Province (LY16F010010); Zhejiang Province Public Welfare Technology Application Research Project (2015C34006).

Disclosures

The authors declare no conflicts of interest.

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	Our performance	Ref [27]	Ref [28]	Ref [39]
Sensitivity (nm/RIU)	3016.7	1267	2500.	587.8
*FOM*	12.0	8.89	10.7	9.9

Tunable electromagnetically induced transparency based on graphene metamaterials

Abstract

1. Introduction

2. Model and simulation parameters

3. Performance analysis for applications

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (7)

Tables (1)

Equations (8)

Optics Express