Dynamically tunable Fano resonance with high Q factor based on asymmetric Dirac semimetal split-ring structure

We propose an asymmetric split-ring (ASR) structure based on Dirac semimetal, which has Fano resonance with a high quality (Q) factor in the terahertz (THz) band, the Q factor can reach a maximum value of 20.19. Amplitude modulation can be achieved by increasing the degree of asymmetry Δθ of the asymmetric split ring. As a result, in this study, an amplitude modulation of 27.19% has been achieved by increasing the asymmetry from 10° to 40°. Furthermore, our full-wave electromagnetic simulations show that the frequency sensitivity values of Fano and quadrupole resonance are as high as 0.6 THz/refractive index unit (RIU) and 0.933 THz/RIU, respectively. In addition, the sensing range can be adjusted by changing the Fermi levels of Dirac semimetal. Our study can guide the practical application of ultrasensitive THz sensors.


Introduction
The Fano resonance is caused by the coherent coupling and interference between continuous and discrete states [1,2]. Unlike the Lorentz resonance, the Fano resonance has a sharp and asymmetric line shape. Fano resonance, because of its narrow linewidth and high Q factor, can be used in biological and chemical sensors, waveguide modulators, optical switch, and slow light field [3][4][5][6][7][8][9][10], which resulted in gathering people's attention related to research [11][12][13][14][15]. To date, various metamaterial structures, ranging from microwave to optical frequencies, based on Fano resonance have been proposed. For instance, Singh R et al obtained Fano resonance with a Q factor of up to 50 in the terahertz (THz) band by changing the asymmetry of the aluminum split ring [16]. Wen et al proposed a plasma nanometer sensor that uses Fano resonance based on the metal-insulator-metal waveguide structure [17]. Liu et al studied the optical properties of double-crossed nanowires, which exhibited Fano resonance in the near infrared region [18]. However, its applications have the following restraints: high metal loss, low modulation range, and complex permittivity functions. To avoid the constraints, twodimensional (2D) materials like graphene are introduced in the field of sensing. Xia et al proposed a both graphene pattern and complicated bias gate free method to demonstrate the excitation and modulation of the graphene surface plasmons (GSPs) in a graphene monolayer [19], and the highest sensitivity of 1.77 μm/RIU and 2.3 μm/RIU are obtained, respectively. They also discuss more possibilities of anisotropic 2D materials beyond anisotropic response by using vertically stacked nanostructures (NSs) to achieve polarizationindependent optical absorption in a pure anisotropic region [20]. The sensitivity of 2.43 μm/RIU and 2.08 μm/ RIU has been obtained, respectively.
Three-dimensional (3D) Dirac semimetals are also a 3D analog of graphene, which is a new quantum matter discovered recently. 3D Dirac semimetals as the 3D analog of graphene have all advantages of graphene as that of photosensitive devices. 3D Dirac semimetals exhibit ultrahigh mobility that reaches 9×10 6 cm 2 V −1 s −1 at 5 K [21], which is higher compared to graphene (2×10 5 cm 2 V −1 s −1 at 5 K) [22], because of their crystalline symmetry protection against gap formation [23][24][25][26][27]. Most importantly, Dirac semimetals can also dynamically control their surface conductivity by altering the Fermi energy. Chen  Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. material into two parallel strips to construct plasmon-induced transparency (PIT) metamaterials (MMs) and obtained novel PIT optical responses by using the weak hybridization between the two strips [28]. Therefore, it will be interesting to incorporate Dirac semimetals in the design of MMs to achieve dynamically tunable Fano resonance for sensing applications.
In this paper, we propose a two-gap THz asymmetric split-ring (ASR) structure based on Dirac semimetal films (DSFs). Fano resonance with a high Q factor transpires from a gentle symmetry-breaking structure. The Q factor of Fano resonance can be as high as 20.19, and the Q factor of quadrupole resonance can be as high as 22.19. Amplitude modulation can be achieved by increasing the degree of asymmetry Δθ of the asymmetric split ring. In this study, an amplitude modulation of 27.19% has been achieved by increasing the asymmetry from 10°t o 40°. Moreover, the sensitivity values of Fano and quadrupole resonance of our designed resonator can reach 0.6 THz/RIU and 0.933 THz/RIU, respectively. Mainly, the sensing range of Fano and quadrupole resonance of the designed structure can be controlled by altering the Fermi energy. Our study has implications in various potential applications, such as, biological and chemical sensors, switches, and other THz devices. According to the random phase theory, the complex surface conductivity of a Dirac semimetal can be obtained by the Kubo formula. Considering the contribution of intraband and interband processes, the conductivity can be expressed as [27]:

Structural and method
Ec is the cutoff energy, and g is the degeneracy factor. Correspondingly, the permittivity of 3D Dirac semimetals can be obtained by using the two-band model [27]: e e s we where ε=1 for g=40 (for AlCuFe quasicrystals [29]) and σ 0 is the permittivity of vacuum. Fano resonator is modeled and studied by CST microwave studio frequency domain method. In the calculation, the unit cell boundary conditions in the x-y plane and Floquet ports in the z direction have been adopted. Plane waves are incident perpendicular to the metamaterial surface.

Results and discussion
The transmission spectrum measured by SSR (red) and ASR (blue) structures is shown in figures 2(a) and (b); the electric polarization is perpendicular to and parallel to the two gaps, and the Fermi level of the DSFs is set to 90 meV. In figure 2(a), the SSR shows only a single wide dipole resonance at 5.5 THz, which is directly excited by free space light and is called bright dipole mode, as the dipole resonances is strongly coupled with free space. It has high radiation properties and a wide bandwidth. The ASR shows two types of resonances, one is similar to the SSR and recognized as the dipole mode, and the other exhibits a very sharp asymmetric Fano resonance at 4.92 THz; the latter occurs in a subradiative dark mode because of the weak coupling between the structure and the free space; the radiation loss is completely suppressed, leading to the asymmetric resonance. At this time, the Q factor reaches 20.19. As shown in figure 2(b), the SSR has only one dipole resonance at 9.04 THz, and the ASR has two types of resonances at 8.85 THz (dipole) and 9.63 THz (asymmetric). The Q factor of the asymmetric resonance at 9.63 THz is 22.2. Most electromagnetically induced transparent resonances has broad frequency band and with low Q factor. For example, the [30] presented a dynamically tunable electromagnetically induced reflection (EIR) based on the complementary graphene metamaterials composed of the wire-slot and split-ring resonators slot (SRRs-slot) array structures for the terahertz region. But the value of Q factor is less than 8. Compared with our resonator, the Q factor we get is almost three times as high.
To understand the resonance of the split ring resonator, we simulated the electric field distribution at resonant frequency corresponding to figures 2(a) and (b) labelled A, B, C, D, E, F, respectively. As show in figures 3(a) and (d), two parallel current loops are generated at the dipole resonance of SSR, which results in broad resonance. When split ring resonator is asymmetric, as show in figures 3(c) and (e), the electric field distribution at the frequencies of 5.63 THz and 8.85 THz is similar to that of dipole resonance in SSR, so it is also dipole resonance. In figure 3(b), due to the increase of the upper arc and the decrease of the lower arc, the energy of the upper arc increases and the energy of the lower arc decreases. As the bright mode, the upper arc affects the dark mode of the lower arc. We can observe the anti-parallel current loop, which leads to the Fano resonance. Similarly, as show in figure 3(f), we observe the anti-parallel current distribution of four nodes, which leads to quadrupole resonance.
Next, we introduce the degree of asymmetry (Δθ=θ 1 − θ 2 ) by increasing the central angle (θ 1 ) corresponding to the upper arc and decreasing the central angle (θ 2 ) corresponding to the lower arc. The Fermi level of DSFs has been set at 90 meV, and the effect of different degrees of asymmetry on Fano resonance has been discussed. As shown in figure 4, the resonance intensity and bandwidth increase with the increase of Δθ, and the corresponding Q factor decreases gradually. An amplitude modulation of 27.19% has been achieved when Δθ has been increased from 10°to 40°.
To verify the asymmetric resonance of the structure and understand its inherent mechanism, we introduce the coupled Lorentz oscillator model and analyze it in combination with an electric field [31,32]: where x 1 and x 2 , γ 1 and γ 2 , and m 1 and m 2 represent the amplitudes, damping rates and effective mass of the bright and dark modes, respectively. ω 0 and ω 0 + δ are the resonant frequencies of the bright and dark modes, respectively, and κ and δ are the coupling coefficients of the two modes and the shifts of the resonant frequency,

( ) /
where P is the polarization intensity and ε 0 is the vacuum permittivity. As the Q factor is large, and the incident field interaction is weak, we assume that g 2 is zero. The EIT transmission in the THz region can be expressed as [33]:  Table 1 shows the change of coupling coefficient with the increase of Δθ. It can be observed that κ is proportional to Δθ, which determines the modulation intensity of Fano resonance. In addition, the damping factors γ 1 , γ 2 correspond to the decay rate in the atomic physics, which is inversely proportional to the spectral  linewidth. We observe that γ 1 and γ 2 are inversely proportional to Δθ, which indicates that the coupling of bright mode and dark mode leads to the excitation of dark mode and the appearance of Fano resonance, and the resonance spectrum will become wider with the increase of Δθ. For Fano resonance with different degrees of asymmetry, the electric field is shown in figure 6. When Δθ=10°, the dipole mode is displayed at 4.92 THz; therefore, there is more or less no resonance. When Δθ increases to 20°, the opposite type of charge accumulates in the gap; the upper and lower rings generate a reverse parallel current, showing a magnetic dipole mode; and the Fano resonance appears. If Δθ increases further, the charge density of the gap increases further, and the corresponding Fano resonance peak increases.
In addition, we also observed the results by changing the polarization angle. Figure 7(a) is the result of our simulation. Since our structure is asymmetric, the polarization angle increases from 0°to 45°and the resonance result changes obviously. Figure 7(b) is the result obtained by theoretical fitting, which is basically consistent with the simulation curve in figure 7(a), indicating that the resonator we designed is of practical significance.

Sensing performance
To study the sensitivity of the structure, we assume that the DSFs layer is covered with an additional layer of the object to be tested. First, the effect of the thickness of the analyte on the Fano and quadrupole resonance is analyzed. We set the Fermi level and refractive index at 90 meV and 1.43 (Poly tetra fluoroethylene (PTFE)), respectively, and the thickness of the object to increases from 60 to 240 nm. As shown in figure 8(a), when the electric polarization is perpendicular to the two gaps, the thickness of the object increases, the Fano resonance   shows a monotonous red-shift because of the increase in the effective dielectric constant of the top layer. The frequency shift is 0.13 THz. Similarly, when the electric polarization is perpendicular to the two gaps, as shown in figure 8(b), the quadrupole resonance also shows a red shift with increasing thickness, and the frequency shift is 0.21 THz. Next, we study the resonance transmission spectrum of different analytes with different refractive indexes, setting the thickness to 60 nm and the Fermi level to 90 meV, respectively. The results are shown in figures 9(a) and (b), as the refractive index of the analyte increases, both the frequencies of the Fano resonance and the quadrupole resonance show red shift. The following refractive indexes represent important technical indicators for the THz sensing technology; Chemicals-PTFE (n=1.43), Hexogen (RDX (n=1.66)), and TNT (n=1.76); biological materials-air-dried Herring DNA (n≈1.65) and Ovalbumin (n≈1.15) [34].
To quantitatively describe the sensitivity of the designed structure, we calculate and plot theresonant frequency shift as a function of the refractive index of the analyte for a fixed thickness of 60 nm. As shown in figure 10(a), the frequency shift of the Fano and quadrupole resonance increases linearly with the increase in the refractive index of the analyte. The slope of the linear shift of quadrupole resonance is greater than the Fano resonance, which signifies that the quadrupole resonance has high frequency sensitivity. Defining the frequency sensitivity as df/dn, the frequency sensitivity values of the Fano resonance and the quadrupole resonance are 0.2 THz/RIU and 0.38 THz/RIU, respectively. Then, we calculate the influence of the thickness of different analytes on the sensitivity. Figure 10(b) shows the quadratic fitting curve. It can be seen that as the thickness of the analyte increases, the frequency sensitivity becomes larger. The frequency sensitivity values of the Fano resonance and the quadrupole resonance reach 0.6 THz/RIU and 0.933 THz/RIU, respectively, when the thickness of the analyte is 240 nm. In addition, the rate of sensitivity tend to saturate when the thickness is>240 nm.
Finally, to analyze the tunable property of the asymmetric Dirac semimetal split ring structure, we simulated the amplitude transmission spectra of DSFs at different Fermi levels. The simulated results are shown in     figures 11(a) and (b); as the Fermi level increases from 90 to 100 meV, the frequencies of the Fano and quadrupole resonance show blue-shift. The frequency peaks of the Fano and quadrupole resonance can be adjusted within the range of 4.92-5.33 THz and 9.63-10.72 THz, respectively. The corresponding modulation depth of the frequency is 7.69% for Fano resonance and 10.17% for quadrupole resonance.

Conclusion
We have prove that the ASR structure based on Dirac semimetals offers Fano resonance with a high Q factor in the THz band, and the Q factor can reach 20.19. An amplitude modulation of 27.19% has been achieved by increasing Δθ of the asymmetric split ring from 10°to 40°. Meanwhile, our full-wave electromagnetic simulations show that the frequency sensitivity values of Fano and quadrupole resonance are as high as 0.6 THz/ RIU and 0.933 THz/RIU. In addition, the sensing range can be adjusted by altering the Fermi levels of Dirac semimetals. The results contribute to the detection of biological and chemical molecules in the THz range, and the design of ultrasensitive sensors.