Figure 1B shows the simulated transmission spectrum of the proposed dual-band THz MM structure. It has two separate transmission dips, which correspond to two resonances, namely low-frequency resonant mode at 1.04 THz (f ₁ ) and high-frequency resonant mode at 2.95 THz (f ₂ ).

To elucidate the physical mechanisms of both resonant modes, the surface electric fields of the MM structure are collected at both frequencies of the resonant dips. Figures 2A,B depict the electric field distributions of the CRSRR at f ₁ and f ₂ , respectively. In the resonant mode of f ₁ , the enhanced electric field is located at the gap edge, behaving as a capacitance in an inductance-capacitance (LC) resonant circuit. Such resonance is related to the circular transient current induced by the incident THz field and it can be explained by the LC model [19, 20]. In the other resonance mode of f ₂ , the induced charges are concentrated at the gap on the horizontal rectangular ring and the two vertices on the left of the vertical rectangular ring, forming strong electric fields. It shows that the dipole is excited on the surface of the CRSRR, and this kind of resonance is caused by the dipoles [21]. As a result, the two resonant modes can be recognized as corresponding to LC resonance and dipole resonance, respectively.

To further explain the underlying physical mechanisms, the surface currents corresponding to the resonant modes are also studied. Figure 2C exhibits the surface current distribution of the CRSRR at f ₁ , the current directions are consistent in the whole structure, forming a current loop, which indicates that it is an LC resonance. Thus the equivalent circuit can be described as shown in Figure 2E. By neglecting the resistance generated by the inductance, the equivalent impedance of the circuit can be approximately expressed as follows: Z 1 ( ω ) = j ω L 1 + 2 × j ω L 1 ′ + 1 j ω C 1 (1)

Figure 2D exhibits the surface current distribution throughout the CRSRR at the high-frequency resonant mode f ₂ . Unlike the current loop in LC resonance, the direction of current on the left side of the structure is symmetrical to that on the right, which means the charges flow from any direction of point d to point c. Both surface currents parallel to the exciting electric field, which is similar to the electric dipole resonance [22]. The equivalent circuit based on this current distribution is depicted in Figure 2F, in which the impedance can be approximately expressed as follows [13]: Z 2 ( ω ) = 2 × 1 j ω C 2 ′ + j ω L 2 × ( j ω L 2 ′ × 2 + 1 j ω C 2 ) j ω L 2 + 2 × j ω L 2 ′ + 1 j ω C 2 (2) where, ω present angular frequency. The resonance occurs when the imaginary part of the impedance is zero. Hence, both resonant frequencies f ₁ and f ₂ of the CRSRR can be calculated according to the resonance formula as f = ω / 2 π , which are determined by the equivalent capacitance C related to the structure’s gap and the equivalent inductance L related to the structure of the metal unit itself [23, 24]. Therefore, the influence of the metamaterial’s geometrical dimensions on the equivalent capacitance and inductances are numerically simulated by CST. The effect of the CRSRR geometry on the resonance characteristics of the metamaterial is also discussed.

In order to obtain further insight into the resonances characteristic of the proposed MM, several geometrical parameters of the CRSRR structure are altered. By increasing the short edge length a of the rectangular ring from 16 to 24 µm without changing the other geometric parameters ( w = 3 μm, g = 2 µm), the resonant frequencies f ₁ and f ₂ have varied as shown in Figure 3A. The increase in the value of a makes the equivalent inductance L larger [25]. Considering the influence of L , both resonant frequencies in two modes exhibited the red-shift with the increase of a . Figure 3B shows the transmittance for different a , and the color bar shows the magnitude of transmission intensity. The positions of the resonant frequencies are indicated with the black trace, showing that the distance of the two resonances is decreased due to the a increment. The resonant frequencies and modulation depths under different a are displayed in Figure 3C. The frequency f ₁ shifts from 1.36 to 0.85 THz while f ₂ shifts from 3.29 to 2.50 THz when a raises from 16 to 24 µm. Meanwhile, the modulation depths of the dip in the low-frequency resonant mode increased to 97.9 % from 91.2 %, while it increased from 98.6 % to 99.2 % in the high-frequency resonant mode.

On the other side, the width of the proposed CRSRR structure has also been discussed. Figure 4A depicts the transmission spectra with different values of w . The width is inversely proportional to the equivalent inductance L , which meant that the inductance decreased with increasing width w [26]. Therefore, the resonant frequencies f ₁ and f ₂ were proportional to the ring width. As the value of w increases from 1 to 4 µm (at keeping a = 20 μm, g = 2 µm), the resonant frequencies increases, that is, the positions of the dips have blue-shifts, which agrees well with the results depicted in Figure 4B. Figure 4C shows the relations between resonance characteristics and the value of w . The frequency f ₁ moves from 0.95 to 1.09 THz while f ₂ moves from 2.77 to 3.03 THz with an increasing value of w . Meanwhile, the modulation depth of resonance dips also increased, increasing from 90.7 % to 96.5 % and from 95.7 % to 99.4 % in lower and higher frequency resonant modes, respectively.

To further investigate the variation trend of the resonant characteristics of the designed CRSRR, the gap distance g is set at values from 1 to 5 µm while keeping the other geometric parameters unchanged ( a = 20 μm, w = 3 µm). Figure 5A displays the transmission spectra with the variation of g . For the split-ring structure, the equivalent capacitance almost entirely comes from the charge accumulation at the gap. The larger the gap, the smaller the capacitance, and the resonant frequencies would be higher. Figure 5B shows the blue shifts in both frequencies as predicted. Unlike the effect of parameters a and w, the offsets induced by increasing g are minor. As shown in Figure 5C, the resonance frequency changes from 1.02 to 1.09 THz for f ₁ and from 2.89 to 3.04 THz for f ₂ with the variation of g . Also, the modulation depth of the transmission increases from 95.2 % to 96.7 % in the low-frequency resonant mode but reduces from 99.1 % to 98.9 % in the high-frequency resonant mode.

In summary, the metal layout and the gap distance affect the resonance characteristics of the proposed MM. Therefore, the THz MM sensor can be designed with desired resonant frequencies by simply designing the geometric parameters of the CRSRR structure. We proposed a dual-band sensor with initial resonant frequencies at 1.04 and 2.95 THz, in which the geometrical dimensions are of a = 20 μm, w = 3 µm and g = 2 µm. It can be used for detecting trace substance which has characteristic absorption peaks in the range of 0.2–3 THz.

The sensing principle of the THz MM is that, the changes in the surrounding permittivity caused by the analytes with different refractive indices will induce the variation of resonant frequencies in the relevant transmission spectrum [27]. Usually, refractive index sensitivity ( S ), figure of merit (FOM), and Q-factor are the three critical parameters for assessing the sensing performance of a sensor [28, 29]. Here, the RI sensitivity is defined as: S = | f a n a l y t e − f r e f | | n a n a l y t e − n r e f | = Δ f Δ n (3) where Δf is the frequency shift induced by the addition of the analyte layer and Δn is the change of RI. In order to compare the sensing characteristics in different frequency regions, the parameter FOM can be introduced as follows: F O M = S F W H M (4) where FWHM is the full width at a half-maximum of the resonant. In turn, the Q-factor represents the sharpness of the resonant, and the higher the Q-factor, the better the sensing performance. Here, the Q-factor can be calculated based on the formula below [29]: Q = f F W H M (5)

In practice, the influence of the analytes on the permittivity of the surrounding environment needs to be considered. In this case, the equivalent capacitance of the designed dual-band metamaterial sensor is approximately equal to the sum of the capacitance of the metamaterial ( C _res ) and the capacitance caused by in addition of the analyte layer ( C _analyte ) [30]. The C _analyte increases with the ambient permittivity, which in turn rises with the refractive index of the analyte increment. Therefore, the resonant frequency of the sensor would decrease due to the increasing permittivity, which can be described as follows: f a n a l y t e = 1 2 π L ( C r e s + C a n a l y t e ) (6)

Since the permittivity ɛ is proportional to the refractive index ( n ) as ɛ = n² [31], the sensing performance of the proposed THz MM sensor is investigated by changing n of the analyte layer from 1.0 to 1.3 with a changing step of 0.05. The analyte thickness should be in the range where the THz waves and MM can react strongly. Therefore, An analyte layer with a fixed thickness of t_a = 10 μm is placed on top of the MM sensor for numerical analysis.

Figure 6A shows the transmission spectra for analytes with different refractive indices. Both resonances are red-shifted with increasing RI. Moreover, the distance of the red-shift in the high-frequency resonant mode is more obvious than that in the low-frequency resonant mode. Figure 6B depicts the offset of the resonant frequency as a function of RI for both resonant modes. With the increase of RI, the positions of resonant dips shift in an approximately linear manner. In turn, the slopes of the fitting curves represented the RI sensitivity of the sensor at the current resonant frequency, which are 309 GHz/RIU for the f ₁ and 730 GHz/RIU for f ₂ , respectively.

To demonstrate the great sensing potential of the proposed dual-band THz MM sensor, the effect of tiny changes of RI on the resonant frequency is afterward numerically evaluated by increasing n of the analyte layer in a range [1.16, 1.2] with a changing step of 0.01. As shown in Figure 7, the spectral dips are red-shifted 13.3 GHz in resonant mode 1 and 21 GHz in resonant mode 2 when the value of n increased by 0.04, which means both resonances are still exhibiting visible frequency changes with the tiny increase of RI. It allows one to distinguish substances in the case of slight differences detected, that is, small changes in the dielectric constant caused by the presence of trace substances can be measured. Therefore, the proposed sensor based on the CRSRR structure meets the high-sensitivity requirements for sensing applications.

In order to obtain a quantitative description of the sensing performance, the RI sensitivity, FOM, and Q-factor of the designed sensor are calculated according to Eqs. 3–5. It is found that S(f ₁ ) = 309 GHz/RIU, FOM(f ₁ ) = 2.97 (RIU⁻¹), and Q-factor(f ₁ ) = 9.96 for the low-frequency resonant mode, while S(f ₂ ) = 730 GHz/RIU, FOM(f ₂ ) = 2.52 (RIU⁻¹), and Q-factor(f ₂ ) = 10.17 for the high-frequency resonant mode. Comparing these results with those most recently reported in the literature [13, 23, 24, 26, 27, 32, 33] (see Table 2), it could be concluded that the proposed dual-band THz MM sensor provides the highest sensitivity among the other analogous appliances. In addition, it enabled one to characterize identical samples at multi frequencies due to its two resonance modes, thereby increasing the selectivity and sensitivity of detection.